Phase Transformations: Examples from Titanium and Zirconium Alloys

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Phase TransformationsExamples from Titanium and Zirconium Alloys

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PERGAMON MATERIALS SERIES

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Department of Materials Science and Metallurgy, University of Cambridge,Cambridge, UK

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Phase TransformationsExamples from Titanium andZirconium Alloys

S. Banerjee and P. MukhopadhyayBhabha Atomic Research Centre, Mumbai, India

Amsterdam • Boston • Heidelberg • London • New York • OxfordParis • San Diego • San Francisco • Singapore • Sydney • Tokyo

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This book is dedicated to the memory of Robert W. Cahn, who sadlydied in April 2007.

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Contents

Foreword xviiPreface xixAcknowledgements xxi

CHAPTER 1Phases and Crystal Structures 3

Symbols and Abbreviations 31.1 Introduction 41.2 Polymorphism 41.3 Phase Diagrams of Elemental Titanium and Zirconium 7

1.3.1 Introductory remarks 71.3.2 Titanium 91.3.3 Zirconium 101.3.4 Epilogue 111.3.5 Phase stability and electronic structure 131.3.6 Some features of transition metals 18

1.4 Effect of Alloying 211.4.1 Introductory remarks 211.4.2 Alloy classification 211.4.3 Titanium alloys 211.4.4 Zirconium alloys 231.4.5 Stability of titanium and zirconium alloys 24

1.5 Binary Phase Diagrams 261.5.1 Introductory remarks 261.5.2 Ti–X systems 271.5.3 Zr–X systems 291.5.4 Representative examples of Ti–X and Zr–X

phase diagrams 291.6 Non-Equilibrium Phases 43

1.6.1 Introductory remarks 431.6.2 Martensite phase 44

1.6.2.1 Crystallography 441.6.2.2 Transformation temperatures 471.6.2.3 Morphology and substructure 48

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1.6.3 Omega Phase 491.6.3.1 Athermal and isothermal � 491.6.3.2 Crystallography 501.6.3.3 Morphology 511.6.3.4 Diffraction effects 51

1.6.4 Phase separation in �-phase 521.7 Intermetallic Phases 53

1.7.1 Introductory remarks 531.7.2 Intermetallic phase structures: atomic layer stacking 551.7.3 Derivation of intermetallic phase structures from

simple structures 611.7.4 Intermetallic phases with TCP structures in Ti–X and

Zr–X systems 621.7.5 Phase stability in zirconia-based systems 62

1.7.5.1 ZrO2 polymorphs 621.7.5.2 Stabilization of high temperature polymorphs 631.7.5.3 ZrO2–MgO system 651.7.5.4 ZrO2–CaO system 661.7.5.5 ZrO2–Y2O3 system 67

References 67Appendix 73

CHAPTER 2Classification of Phase Transformations 89

Symbols and Abbreviations 892.1 Introduction 892.2 Basic Definitions 902.3 Classification Schemes 92

2.3.1 Classification based on thermodynamics 932.3.2 Classifications based on mechanisms 1012.3.3 Classification based on kinetics 105

2.4 Syncretist Classification 1052.5 Mixed Mode Transformations 115

2.5.1 Clustering and ordering 1152.5.2 First-order and second-order ordering 1162.5.3 Displacive and diffusional transformations 1202.5.4 Kinetic coupling of diffusional and displacive

transformations 120References 122

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CHAPTER 3Solidification, Vitrification, Crystallization and Formation ofQuasicrystalline and Nanocrystalline Structures 127

List of Symbols 1273.1 Introduction 1283.2 Solidification 128

3.2.1 Thermodynamics of solidification 1283.2.2 Morphological stability of the liquid/solid interface 1353.2.3 Post-solidification transformations 1403.2.4 Macrosegregation and microsegregation in castings 1413.2.5 Microstructure of weldments of Ti- and Zr-based

alloys 1453.3 Rapidly Solidified Crystalline Products 150

3.3.1 Extension of solid solubility 1523.3.2 Dispersoid formation in rapidly solidified Ti

alloys 1533.3.3 Transformations in the solid state 153

3.4 Amorphous Metallic Alloys 1573.4.1 Glass formation 1573.4.2 Thermodynamic considerations 1593.4.3 Kinetic considerations 1653.4.4 Microstructures of partially crystalline alloys 1713.4.5 Diffusion 1763.4.6 Structural relaxation 1803.4.7 Glass transition 182

3.5 Crystallization 1843.5.1 Modes of crystallization 1853.5.2 Crystallization in metal–metal glasses 1873.5.3 Kinetics of crystallization 1923.5.4 Crystallization kinetics in Zr76�Fe1−xNix�24 glasses 200

3.6 Bulk Metallic Glasses 2053.7 Solid State Amorphization 212

3.7.1 Thermodynamics and kinetics 2153.7.2 Amorphous phase formation by composition-induced

destabilization of crystalline phases 2203.7.3 Glass formation in diffusion couples 2203.7.4 Amorphization by hydrogen charging 2253.7.5 Glass formation in mechanically driven systems 2263.7.6 Radiation-induced amorphization 229

3.8 Phase Stability in Thin Film Multilayers 237

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3.9 Quasicrystalline Structures and Related Rational Approximants 2413.9.1 Icosahedral phases in Ti- and Zr-based systems 248

References 252

CHAPTER 4Martensitic Transformations 259

Symbols and Abbreviations 2594.1 Introduction 2604.2 General Features of Martensitic Transformations 261

4.2.1 Thermodynamics 2614.2.2 Crystallography 2664.2.3 Kinetics 2774.2.4 Summary 280

4.3 BCC to Orthohexagonal Martensitic Transformation inAlloys Based on Ti and Zr 2814.3.1 Phase diagrams and Ms temperatures 2824.3.2 Lattice correspondence 2894.3.3 Crystallographic analysis 294

4.3.3.1 Morphology and substructure 3044.3.3.2 Transition in morphology and substructure 320

4.3.4 Stress-assisted and strain-induced martensitic transformation 3244.4 Strengthening due to Martensitic Transformation 326

4.4.1 Microscopic interactions 3294.4.1.1 Lath boundaries 3294.4.1.2 Twin boundaries and plate boundaries 331

4.4.2 Macroscopic flow behaviour 3354.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 339

4.5.1 Transformation sequences 3404.5.2 Crystallography of the B2 → R transformation 3424.5.3 Crystallography of the B2 → B19 transformation 3424.5.4 Crystallography of the B2 → B19′ transformation 3454.5.5 Self-accommodating morphology of Ni–Ti martensite

plates 3474.5.6 Shape memory effect 3524.5.7 Reversion stress in a shape memory alloy 3564.5.8 Thermal arrest memory effect 360

4.6 Tetragonal � Monoclinic Transformation in Zirconia 3624.6.1 Transformation characteristics 3624.6.2 Orientation relation and lattice correspondence 3634.6.3 Habit plane 366

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4.7 Transformation Toughening of Partially Stabilized Zirconia (PSZ) 3694.7.1 Crystallography of tetragonal → monoclinic transformation

in small particles 372References 373

CHAPTER 5Ordering in Intermetallics 379

List of Symbols 3795.1 Introduction 3805.2 Theoretical Treatments 383

5.2.1 Alloy phase stability 3845.2.2 Order–disorder transformations 386

5.2.2.1 Historical developments 3875.2.2.2 Static concentration wave model 3895.2.2.3 Cluster variation method 392

5.2.3 The ground states of the Lenz and Ising model 3975.2.4 Special point ordering 401

5.2.4.1 BCC special points 4045.2.4.2 HCP special points 4065.2.4.3 FCC special points 407

5.2.5 Concomitant clustering and ordering 4075.2.6 A case study: Ti–Al system 412

5.3 Transformations in Ti3Al-based alloys 4165.3.1 � → D019 ordering 4165.3.2 Phase transformations in �2-Ti3Al-based systems 4175.3.3 Structural relationships 4215.3.4 Group/subgroup relations between BCC (Im3m),

HCP (P63/mmc) and ordered orthorhombic (Cmcm)phases 424

5.3.5 Transformation sequences 4285.3.5.1 Transformation sequence in the alloy Ti–25 at.%

Al–12.5 at.% Nb 4305.3.5.2 Transformation sequence in the alloys Ti–25 at.%

Al–25 at.% Nb, Ti–28 at.% Al–22 at.% Nb andTi–24 at.% Al–15 at.% Nb 431

5.3.6 Phase reactions in Ti–Al–Nb system 4325.4 Formation of Zr3Al 436

5.4.1 Metastable Zr3Al (D019) phase 4375.4.2 Formation of the equilibrium Zr3Al (L12) phase 4395.4.3 �+Zr2Al → Zr3Al peritectoid reaction 441

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5.5 Phase Transformation in �-TiAl-Based Systems 4435.5.1 Structural relationship between �2- and �-phases 4435.5.2 Phase reactions 446

5.5.2.1 Ti-34-37 at.% Al; � → � → �2 4465.5.2.2 Ti-38-40 at.% Al; � → �2 → �2 +� 4475.5.2.3 Ti-41-48 at.% Al; � → �+� → �2 +� 4485.5.2.4 Ti-49-50 at.% Al; � → � 4505.5.2.5 Ti49-50 at.% Al; � → � 450

5.5.3 Transformation mechanisms 4515.5.3.1 Formation of the �2 +� lamellar microstructure 4515.5.3.2 Mechanism of the � → � massive transformation 4535.5.3.3 Discontinuous coarsening of the lamellar

�2 +� microstructure 4565.6 Site Occupancies in Ordered Ternary Alloys 458

5.6.1 Ordering tie lines 4585.6.2 Kinetic modelling of B2 ordering in a ternary

system 4605.6.3 Influence of binary interaction parameters 4625.6.4 B2 ordering in the Nb–Ti–Al system 464

References 465

CHAPTER 6Transformations Related To Omega Structures 473

List of Symbols 4736.1 Introduction 4746.2 Occurrence of the �-Phase 475

6.2.1 Thermally induced formation of the �-phase 4756.2.2 Formation of equilibrium �-phase under high

pressures 4796.2.3 Combined effect of alloying elements and pressure in

inducing �-transition 4816.3 Crystallography 484

6.3.1 The structure of the �-phase 4846.3.2 The �–� lattice correspondence 4856.3.3 The �–� lattice correspondence 488

6.4 Kinetics of the � → � Transformation 4906.4.1 Athermal � → � transition 4916.4.2 Thermally activated precipitation of the �-phase 4926.4.3 Pressure-induced � → � transformation 494

6.5 Diffuse Scattering 495

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6.6 Mechanisms of �-Transformations 4996.6.1 Lattice collapse mechanism for the � → � transformation 4996.6.2 Formation of the plate-shaped � induced by shock pressure

in �-alloys 5046.6.3 Calculated total energy as a function of displacement 5066.6.4 Incommensurate �-structures 5096.6.5 Stability of �-phase and d-band occupancy 516

6.7 Ordered �-Structures 5186.7.1 Structural descriptions 5186.7.2 Transformation sequences in Zr base alloys 5226.7.3 Transformation sequences in Ti base alloys 5306.7.4 Ordered �-structures in other systems 5336.7.5 Symmetry tree 534

6.8 Influence of �-Phase on Mechanical Properties 5366.8.1 Hardening and embrittlement due to �-phase 5366.8.2 Dynamic strain ageing due to �-precipitation 539

References 550

CHAPTER 7Diffusional Transformations 557

List of Symbols 5577.1 Introduction 5587.2 Diffusion 560

7.2.1 Diffusion mechanisms 5607.2.2 Flux equations: Fick’s laws 5627.2.3 Self- and tracer-diffusion coefficients in �-Zr and �-Ti 5647.2.4 Self- and tracer-diffusion coefficients in �-Zr and �-Ti 5667.2.5 Interdiffusion 5707.2.6 Phase formation in chemical diffusion 578

7.2.6.1 Phase nucleation 5807.2.6.2 Phase growth 581

7.2.7 Diffusion bonding 5847.3 Phase Separation 587

7.3.1 Phase separation mechanisms 5897.3.2 Analysis of a phase diagram showing a miscibility gap 5977.3.3 Microstructural evolution during phase separation in

the �-phase 6037.3.4 Monotectoid reaction – a consequence of

�-phase immiscibility 606

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7.3.5 Precipitation of �-phase in supersaturated �′-phaseduring tempering of martensite 609

7.3.6 Decomposition of orthorhombic �′′-martensite during tempering 6167.3.7 Phase separation in �-phase as precursor to precipitation

of �- and �-phases 6187.4 Massive Transformations 623

7.4.1 Thermodynamics of massive transformations 6237.4.2 Massive transformations in Ti alloys 626

7.5 Precipitation of �-Phase in �-Matrix 6327.5.1 Morphology 6337.5.2 Orientation relation 6427.5.3 Invariant line strain condition 6437.5.4 Interfacial structure and growth mechanisms 6487.5.5 Morphological evolution in mesoscale 655

7.6 Precipitation of Intermetallic Phases 6577.6.1 Precipitation of intermetallic compounds from dilute

solid solutions 6577.6.2 Precipitation in ordered intermetallics: transformation

of �2-phase to O-phase 6627.7 Eutectoid Decomposition 670

7.7.1 Active eutectoid systems 6757.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system 676

7.8 Microstructural Evolution During Thermo-MechanicalProcessing of Ti- and Zr-based Alloys 6837.8.1 Identification of hot deformation mechanisms through

processing maps 6847.8.2 Development of microstructure during hot working of

Ti alloys 6877.8.2.1 �-alloys 6877.8.2.2 �+� alloys 6877.8.2.3 �-alloys 6897.8.2.4 Ti-aluminides 690

7.8.3 Hot working of Zr alloys 6917.8.3.1 � and near-�-Zr alloys 6927.8.3.2 �+� alloys 6967.8.3.3 �-alloys 701

7.8.4 Development of texture during cold working of Zr alloys 7017.8.5 Evolution of microstructure during fabrication of

Zr–2.5 wt% Nb alloy tubes 706References 710

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CHAPTER 8Interstitial Ordering 719

List of Symbols 7198.1 Introduction 7208.2 Hydrogen In Metals 721

8.2.1 Ti–H and Zr–H phase diagrams 7228.2.2 Terminal solid solubility 725

8.3 Crystallography and Mechanism of Hydride Formation 7288.3.1 Formation of �-hydride in the �- and �-phases 7288.3.2 Lattice correspondence of �-, �- and �-phases 7298.3.3 Crystallography of � → � transformation 7308.3.4 Crystallography of � → � transformation 7358.3.5 Mechanism of the formation of �-hydrides 7378.3.6 Hydride precipitation in the �/� interface 7378.3.7 Formation of �-hydride 739

8.4 Hydrogen-Related Degradation Processes 7418.4.1 Uniform hydride precipitation 7428.4.2 Hydrogen Migration 7438.4.3 Stress reorientation of hydride precipitates 7458.4.4 Delayed hydride cracking 7468.4.5 Formation of hydride blisters 747

8.5 Thermochemical Processing of Ti Alloys by TemporaryAlloying With Hydrogen 753

8.6 Hydrogen Storage In Intermetallic Phases 7548.6.1 Laves phase compounds 7548.6.2 Thermodynamics 7568.6.3 Ti- and Zr-based hydrogen storage materials 756

8.6.3.1 Ti-based hydrogen storage materials 7588.6.3.2 Zr-based hydrogen storage materials 759

8.6.4 Applications 7618.7 Oxygen Ordering In �-Alloys 764

8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O 7648.7.2 Oxidation kinetics and mechanism 769

8.8 Phase transformations in Ti-rich end of the Ti–N system 772References 780

CHAPTER 9Epilogue 785References 800

Index 801

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Foreword

The present volume looks at phase transformations essentially from a physicalmetallurgist’s view point, in consonance with the background and the researchexperience of the authors, and has some distinguishing features. Some, though notall, of these are enumerated in the following.

Almost all types of phase transformations and reactions that are commonlyencountered in inorganic materials, such as alloys, intermetallics and ceramics,have been covered and the underlying thermodynamic, kinetic and crystallographicaspects elucidated.

It has generally been customary in metallurgical literature to draw examplesfrom iron-based alloys for describing the characteristic features of different typesof transformations, in view of the wide variety of transformations occurring inthese alloys. The authors of this monograph have cited examples of all the phasetransformations and reactions discussed from titanium- and zirconium-based sys-tems and have successfully demonstrated that these alloys, intermetallics andceramics exhibit an even wider range of phase changes as compared to ferroussystems and that the simpler crystallography involved renders them more suitablefor developing a basic understanding of the transformations.

Phase transformations are brought about due to changes in external constraintswhich include thermodynamic variables such as temperature and pressure. Tillrecently, the emphasis in metallurgical literature has been on the delineationof temperature-induced transformations. In this book, transformations driven bypressure changes, radiation and deformation and those occurring in nanoscalemultilayers have also been brought to the fore, while accepting the pre-eminentposition occupied by the temperature-induced ones.

Order–disorder transformations, many of which constitute very good examplesof continuous transformations, have been dealt with in a comprehensive manner.It has been demonstrated that first principles calculations of phase stability canyield meaningful results, consistent with experimental observations.

Displacive transformations, both shear dominated (martensite, shock pressureinduced omega) and shuffle dominated (omega), have been covered in a cogentmanner.

Some crystallographic bcc to hcp transformations, which occur by diffusionalas well as by displacive modes, have been identified, compared and contrasted, interms of the experimentally observable features which characterize them.

The authors, who have a lifetime of experience in investigating phase trans-formations in zirconium and titanium alloys, have handled an ambitious project

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by trying to bring diverse topics under the same cover. And they have certainlynot failed in their endeavor. One could always point out that non-metallic sys-tems have not been adequately represented in their treatment. However, in quitea few instances, they have compared phase transformations occurring in alloys,intermetallics and ceramics and have demonstrated that the underlying principlespertaining to all these systems are basically the same.

The multidisciplinary and interdisciplinary interest in the area of phase changeshave engendered a variety of approaches with regard to the study of phase transfor-mations, each exhibiting some distinctive features. Physicists are interested primar-ily in the motivation or, in other words, the why of a transformation. They concernthemselves mainly with higher order, continuous phase transitions occurring insimple, composition-invariant systems. Chemists, metallurgists and ceramists, bycontrast, focus a major part of their attention on phase transformations (and phasereactions) involving alterations in crystal structure, chemical composition and stateof order. Of great concern to metallurgists are the mechanisms, or the how, ofsuch transformations. Phase changes of interest to geologists are similar to thoseencountered in metallic and ceramic systems but generally take place over muchmore extended temporal and spatial scales under extreme conditions of tempera-ture and pressure. The present volume will be useful to students, research workersand professionals belonging to all these disciplines.

In my judgment, the authors of this volume have done a commendable job whileaddressing phase transformations and phase reactions, drawing apposite examplesfrom titanium-and zirconium-based systems, and have been able to produce amonograph which was not there but which should very much have been there.I congratulate them on this count.

C.N.R. Rao, F.R.S. Linus Pauling Research Professor,Jawaharlal Nehru Centre for Advanced Scientific Research,

Bangalore, India

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Preface

Studies on phase transformations in metallic materials form a major part of physicalmetallurgy. The terms phase transitions and phase transformations are often usedin an interchangeable manner in metallurgical literature although it is realized thatthe former generally refers to transitions between two phases having the samechemical composition while the latter spans a wider range of phenomena, includingphase reactions leading to compositional changes. Having made this distinction,we would like to mention at the outset that the present volume deals with phasetransformations.

We started our respective research careers almost four decades back by look-ing into some phase transformations and phase reactions occurring in zirconiumalloys. As we gathered more and more experience, we realized that these alloys,together with titanium alloys, exhibit nearly all types of phase transformationsencountered in inorganic materials and that in this respect these are more versatilethan even ferrous alloys. Moreover, the crystallographic features associated withthe phase changes are often simpler in these systems, making them more suitablefor providing a basic understanding of the relevant phenomena.

In earlier days, some of the important issues in the area of phase transformationsin alloys, intermetallics and ceramics pertained to the following:

(1) crystallographic aspects of martensitic transformations, including the role ofthe lattice-invariant shear, in determining martensite morphology and substruc-ture and the strengthening contribution of the latter;

(2) distinguishing features of diffusional and displacive transformations and mech-anisms of hybrid transformations;

(3) analysis and synthesis of phase diagrams and the prediction of the sequenceof phase transformations on the basis of phase diagram analyses;

(4) spinodal decomposition leading to a homogenous phase separation processand the evolution of microstructure in systems exhibiting instability in respectof concentration and/or displacement waves of short and/or long wavelengths;

(5) driving force, kinetics and mechanisms relevant to displacive phase trans-formations and the role of strain fluctuations and their localization in thenucleation of such transformations;

(6) formation of amorphous structure in metallic materials, stability of the amor-phous phase and the modes of crystallization on appropriate processing;

(7) the effect of factors such as pressure, deformation and radiation on phasetransformations.

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A number of research groups all over the world, including our group, respondedto the challenges thrown by these issues. The background was well set as theinformation and knowledge accumulated on the basis of metallography observa-tions (mainly at light microscopy levels), X-ray diffraction results and studies onkinetics had already provided a fair understanding of the mechanisms of differ-ent types of phase transformations. Theoretical developments such as the phe-nomenological theory of martensite crystallography, the thermodynamical theoryof spinodal decomposition and the theory of the growth kinetics of precipitateshad had noteworthy success in making quantitative predictions regarding manyan aspect of solid state phase transformations. That was also the time whentransmission electron microscopy emerged as a powerful technique for makingobservations, morphological as well as crystallographic, at a much higher resolu-tion than hitherto available, enabling physical metallurgists to resolve a numberof mechanism-related problems which had been raised on the basis of theoreticaland experimental investigations carried out earlier.

We are happy to state that each of the issues listed above has been addressed, insome manner or the other, by our colleagues and by us over the years to enhanceour understanding and appreciation of these.

If one scans today’s literature on phase transformations, one will find that mostof these issues, though better comprehended than before, continue to be in the lime-light. However, the experimental tools now available have enormously improvedour ability to study phenomena at much higher levels of spatial as well as temporalresolution. This superior experimental capability, supplemented by tremendouslyenhanced computing power, is providing a much better understanding of phasetransformation phenomena. We do hope that the readers of this volume will get aflavour of these advancements.

The book is divided into nine chapters. The first of these provides some sortof an introduction to the various types of phase changes covered later on. Thesecond chapter delineates different schemes of classification of phase transforma-tions in a general manner. The following six chapters deal with specific types oftransformations. An attempt has been made to elucidate the basic principles per-taining to the relevant transformations, in general terms, at the beginning of eachof these chapters because we have felt that this would be pedagogically advan-tageous for developing a clear understanding of the subject. However, we havetaken care to ensure that all the illustrative examples are drawn from titanium-and zirconium-based systems. The final chapter is in the nature of an epilogue.

Srikumar BanerjeePradip Mukhopadhyay

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Acknowledgements

This book reflects the totality of the experience gained by us during our researchcareer which, in the formative years, was under the guidance of R. Krishnan inMetallurgy Division, Bhabha Atomic Research Centre (BARC). Our research hasbeen almost entirely supported by this institute (BARC), where a sustained activityon the physical metallurgy of zirconium has remained in focus for nearly fourdecades. It is here that we have had the benefit of interacting with M.K. Asundi,V.S. Arunachalam, P. Rodriguez, B.D. Sharma, R. Chidambaram, C.V. Sundaramand C.K. Gupta over the years. Interactions with other major centres of physicalmetallurgy research in the country have also been of considerable help. In thisconnection, we would like to especially acknowledge the fruitful discussionson many aspects of phase transformations research with P.R. Dhar of IndianInstitute of Technology (IIT), Kharagpur; S. Ranganathan and K. Chattopadhyayof Indian Institute of Science (IISc), Bangalore; T.R. Anantharaman, P. Rama Rao,P. Ramachandrarao and S. Lele of Banaras Hindu University (BHU), Varanasi;D. Banerjee and K. Muraleedharan of Defence Metallurgical Research Laboratory(DMRL), Hyderabad; and V.S. Raghunathan of Indira Gandhi Centre for AtomicResearch (IGCAR), Kalpakkam.

One of us (P. Mukhopadhyay) was introduced to ordering reactions intitanium aluminides by P.R. Swann at Imperial College, London, while the other(S. Banerjee) has had productive collaborations with R.W. Cahn and B. Cantor atUniversity of Sussex, Brighton; M. Wilkens and K. Urban at Institut für Physik,Max-Planck Institut für Metallforschung, Stuttgart; and H.L. Fraser, R. Banerjeeand J.C. Williams at the Ohio State University, Columbus. We also have hadseveral occasions to imbibe pertinent ideas from H.I. Aaronson of Carngie-MellonUniversity, Pittsburgh; J.W. Cahn and L.A. Bendersky of National Institute ofStandards and Technologies (NIST), Washington, D.C.; J.W. Christian of Univer-sity of Oxford and V.K. Vasudevan of University of Cincinatti.

We must acknowledge our indebtedness to the authors of many of the publica-tions which have been instrumental in nurturing our understanding of the topicscovered in this book.

We have been extremely fortunate in having a continuous stream of brightcolleagues in the course of our professional career. They have perhaps given usmuch more in terms of ideas and concepts than whatever advice and guidancewe have been able to offer. We take this opportunity to list the names of someof those colleagues in the approximate sequence of our coming in contact with

xxi

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xxii Acknowledgements

them: S.J. Vijayakar, G.E. Prasad, L. Kumar, V. Seetharaman, E.S.K. Menon,M. Sundararaman, V. Raman, R. Kishore, U.D. Kulkarni, J.K. Chakravartty,G.K. Dey, K. Madangopal, D. Srivastava, R. Tewari, A.K. Arya, R.V. Ramanujam,J.B. Singh. Needless to say, this list is, by no means, complete.

During the preparation of the manuscript of this book we received substantialhelp from many of our colleagues, notably G.K. Dey, D. Srivastava, A.K. Arya,R. Tewari, A. Laik, G.B. Kale, K. Bhanumurthy, R.N. Singh, S. Ramanathanand J.K. Chakravartty. We have received sustained secretarial assistance fromM. Ayyappan and P. Khattar. P.B. Khedkar and A. Agashe have been mainlyresponsible for preparing the illustrations.

We are grateful to Elsevier Publishers for their patience and readiness to help.Above all, we are greatly indebted to Robert Cahn, whose constant encouragementand occasional reprimands have contributed considerably to the completion of thiswork. He passed away at a time when this volume was in the proof-setting stage.His death has indeed created a void in the physical metallurgy community thatwill take a long time to be filled. To us it has been an irreparable loss, professionaland personal.

We dedicate this book to the memory of our parents and of Prof. Robert W.Cahn.

Srikumar BanerjeePradip Mukhopadhyay

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Chapter 1

Phases and Crystal Structures

1.1 Introduction 41.2 Polymorphism 41.3 Phase Diagrams of Elemental Titanium and Zirconium 7

1.3.1 Introductory remarks 71.3.2 Titanium 91.3.3 Zirconium 101.3.4 Epilogue 111.3.5 Phase stability and electronic structure 131.3.6 Some features of transition metals 18

1.4 Effect of Alloying 211.4.1 Introductory remarks 211.4.2 Alloy classification 211.4.3 Titanium alloys 211.4.4 Zirconium alloys 231.4.5 Stability of titanium and zirconium alloys 24

1.5 Binary Phase Diagrams 261.5.1 Introductory remarks 261.5.2 Ti–X systems 271.5.3 Zr–X systems 291.5.4 Representative examples of Ti–X and Zr–X phase diagrams 29

1.6 Non-Equilibrium Phases 431.6.1 Introductory remarks 431.6.2 Martensite phase 441.6.3 Omega phase 491.6.4 Phase separation in �-phase 52

1.7 Intermetallic Phases 531.7.1 Introductory remarks 531.7.2 Intermetallic phase structures: atomic layer stacking 551.7.3 Derivation of intermetallic phase structures from

simple structures 611.7.4 Intermetallic phases with TCP structures in Ti–X

and Zr–X systems 621.7.5 Phase stability in zirconia-based systems 62

References 67Appendix 73






Chapter 1

Phases and Crystal Structures

Symbols and AbbreviationsA � Elastic anisotropy ratio (A = C44/C

′)Cij � Elastic stiffness modulus (elastic constant)C ′ � Elastic shear stiffness modulus; shear constant;

(C ′ = �C11 −C12�/2)Cp � Specific heat at constant pressure

e/a � Electron to atom ratioG: Gibbs free energy (G = H −TS)H � EnthalpyP � PressureS � EntropyT � TemperatureV � VolumeVa � Atomic volume�p � Piston velocity�s � Shock velocity�ij � Thermodynamic interaction parameter between elements i and jbcc: Body centred cubicfcc: Face centred cubichcp: Hexagonal close packed

�-phase: hcp phase in Ti- and Zr-based alloys�-phase: bcc phase in Ti- and Zr-based alloys

�′ � hcp martensite�′′ � Orthorhombic martensite�m � Generic martensite (�′ or �′′)Ms � Temperature at which martensite starts forming during quenchingMf � Temperature at which martensite formation is completed during

quenching�s � Temperature at which the �m → � reversion starts on

up-quenchings � Temperature at which athermal phase starts forming during

quenchingTo � Temperature at which the free energies of the parent (�) and

product (�m) phases are equal.AIP: Ab initio pseudopotential

3



4 Phase Transformations: Titanium and Zirconium Alloys

ASA: Atomic sphere approximationASW: Augmented spherical wave

FPLAPW: Full potential linear augmented plane waveLAPW: Linear augmented plane wave

LCGTO: Linear combination of gaussian type orbitalsLDA: Local density approximation

LMTO: Linear muffin tin orbitalMC: Monte carloMD: Molecular dynamicsMT: Muffin tin

NFE: Nearly free electronQMC: Quantum monte carloQSD: Quantum structural diagram

TB: Tight binding

1.1 INTRODUCTION

Titanium (Ti), zirconium (Zr) and hafnium (Hf) are transition metals belongingto Group 4 (nomenclature as per the recommendations of IUPAC 1988) of theperiodic table of elements. The interest in the metals Ti and Zr and in alloys basedon them has gained momentum from the late 1940s in view of their suitabilityfor being used as structural materials in certain rapidly developing industries; par-ticularly, the aerospace and chemical industries in the case of Ti alloys and thenuclear power industry in the case of Zr alloys. Some important characteristics ofthese metals are listed in Table 1.1. It can be seen from this table that the elec-tronic ground state configurations of these metals are Ar�3d24s2 and Kr�4d25s2,respectively. The similarity in the dispositions of the outer electrons, i.e. the fourelectrons (two s electrons and two d electrons) outside the inert gas shells (Mshell for Ti and N shell for Zr) is, to a large extent, responsible for the similaritiesin some of the chemical and physical properties of these two metals and as acorollary, in many aspects of their chemical and physical metallurgy, includingalloying behaviour.

1.2 POLYMORPHISM

Apart from existing in solid, liquid and gaseous states, many elements exhibita special feature: they adopt different crystal structures in the solid state underdifferent conditions of temperature or pressure or external field. The transition from



Phases and Crystal Structures 5

Table 1.1. Some characteristics of elemental Ti and Zr.

Property Element

Ti Zr

Atomic number (Z) 22 40Number of naturally occurring isotopes 5 5Atomic weight 47.90 91.22Electronic ground state configuration Ar�3d24s2 Kr�4d25s2

Density at 298 K �kg/m3� 4510 6510Melting temperature (K) 1941 2128Boiling temperature (K) 3533 4650Enthalpy of fusion (�Hf ) kJ/mol 16.7 18.8Electronegativity 1.5 1.4Metal radius (nm) 0.147 0.160

References: Froes et al. 1996, Kubaschewski et al. 1993, McAuliffe and Bricklebank 1994, Soloveichik 1994.

one modification (allotrope) to another is termed a polymorphous transformationor a phase transformation (transition).

A phase transition is associated with changes in structural parameters and/orin the ordering of electron spins (Steurer 1996). It will be discussed in a laterchapter that two basically different types of phase transitions may be encountered:first-order transitions and second-order (or higher order) transitions. A transitionof the former type is associated with discontinuous changes in the first derivativesof the Gibbs free energy, G = H − TS, while a transition of the latter type ischaracterized by discontinuous changes in the second (or higher order) derivativesof the Gibbs free energy and there are no jumpwise changes in the first deriva-tives. In either type of transition, the crystal structure undergoes a discontinuouschange at the transition point (e.g. transition temperature or transition pressure).It is not necessary to have a symmetry relationship between the parent and theproduct phases in a first-order transition. However, in a second-order transitiona group/subgroup relationship can always be found in relation to the symmetrygroups associated with the crystal structures of the two phases.

Elemental Ti and Zr (and Hf) exhibit temperature induced as well as pressureinduced polymorphism. The pertinent phases, transition temperatures and transitionpressures are listed in Table 1.2 and Table 1.3. It can be seen from Table 1.2 that forboth Ti and Zr, the high temperature phase, termed the �-phase, has the relatively“open” bcc structure while the low temperature phase, termed the �-phase, hasthe close packed hcp structure. The hcp structure of the �-phase is, however,slightly compressed in the sense that the value of the axial ratio is smaller thanthe ideal value of 1.63. It has been pointed out (McQuillan 1963, Collings 1984)that the more open bcc structure has a higher vibrational entropy as compared to




Table 1.2. Allotropic forms of elemental Ti and Zr at atmospheric pressure (Massalski et al. 1992)(Variable: temperature).

Element Phase Temperatureregime (K)

Enthalpy oftransformation(kJ/mol)

Crystal structure

Ti Alpha(�) Up to 1155 4 17�4 2� hexagonal close packedBeta(�) 1155–1943 body centred cubic

Zr Alpha(�) Up to 1139 (1136) 4 103�3 9� hexagonal close packedBeta(�) 1139–2128 body centred cubic

Note: The figures in parentheses are from Kubaschewski et al. 1993.

Table 1.3. Allotropic forms of elemental Ti and Zr at room temperature (Steurer 1996) (Variable:pressure)

Element Phase Pressure regime (GPa) Crystal structure

Ti Alpha(�) Up to 2 hexagonal close packedOmega() > 2 hexagonal

Zr Alpha(�) Up to 2 hexagonal close packedOmega() 2–30 hexagonalOmega prime (′) > 30 body centred cubic

the close packed hcp structure and as a consequence of this, the free energy of acompeting bcc lattice will decrease more rapidly than that of the hcp lattice withincreasing temperature; a temperature will ultimately be reached at which the freeenergy of the former will be less than that of the latter so that the bcc form will bemore stable. The -phase can be obtained from the �-phase by the application ofsufficient pressure in elemental Ti and Zr. Some crystallographic data pertainingto all these phases are presented in Table 1.4. The structure of the -phase hasbeen determined to be either hexagonal, belonging to the space group P6/mmm(Silcock 1958), or trigonal, belonging to the space group P3m1 (Bagariatskii et al.1959), depending on the solute concentration. The equivalent positions in the unitcell of the structure are 000; 2/3 1/3 (1/2−z); 1/3 2/3 (1/2+z). For the ideal structure with hexagonal (P6/mmm) symmetry, z = 0 while 0 < z < 1/6 definesa non-ideal structure with trigonal (P3m1) symmetry. There are three atoms inthe unit cell. The axial ratio is close to �3/8�1/2. The symmetry of the structureis high and as in the case of the simple hexagonal lattice, there are 24 point groupoperations (Ho et al. 1982). The packing density (� 0 57) associated with thehexagonal (hP3) structure of the -phase is lower than that for the bcc (� 0 68)and the hcp (� 0 74) structures. The occurrence of such an open structure in metals




Table 1.4. Crystal structures and lattice parameters of allotropic forms of elemental Ti and Zr(Massalski et al. 1992, Steurer 1996).

Element Ph Crystal structure Lattice paramaters(nm)

Axialratio

P SN PS SG

Ti �-Ti Mg A3 hP2 P63/mmc a = 0 29506 1 5873Va�nm�3 = c = 0 4683517 65×10−3 �-Ti W A2 cI2 Im3m a = 0 33065 1 0

-Ti -Ti − hP3 P6/mmm a = 0 4625 0 6082c = 0 2813

Zr �-Zr Mg A3 hP2 P63/mmc a = 0 32316 1 5929Va�nm�3 = c = 0 5147523 28×10−3 �-Zr W A2 cI2 Im3m a = 0 36090 1 0

-Zr -Ti − hP3 P6/mmm a = 0 5036 0 617c = 0 3109

′ W A2 cI2 Im3m − −Ph – Phase; P – Prototype structure; SN – Strukturbericht notation; PS – Pearson symbol; SG – Space group.Notes:1. The lattice parameter values of �- and - phase correspond to a temperature of 298 K.2. The quantity Va refers to the atomic volume under ambient conditions.

with metallic d-bonding is somewhat unusual. Normally, the transition metals haveclose packed (fcc, hcp) or fairly close packed (bcc) structures. Open structuresare common among the p-electron systems or the actinide elements (Duthie andPettifor 1977, Skriver 1985). The stability of this phase has been attributed tothe covalent bonding contribution from s-d electron transfer (Steurer 1996). Inthe case of Zr (and Hf), it has been found that on the application of substantiallyhigher pressures (Table 1.3) the -phase transforms to the ′-phase, which hasthe bcc structure. Although a similar transformation has not been observed in thecase of Ti, even at a pressure as high as 87 GPa, theoretical considerations indicatethat this metal too would undergo such a transformation at still higher pressures(Ahuja et al. 1993, Steurer 1996). This issue is addressed in greater detail in alater chapter of this volume.

1.3 PHASE DIAGRAMS OF ELEMENTAL TITANIUMAND ZIRCONIUM

1.3.1 Introductory remarksFrom the point of view of the phase rule, a pure element represents a singlecomponent system which may exhibit different phases. The phase rule imposes




the condition f +p = c+ 2, where f is the number of degrees of freedom in thepressure–temperature–composition space, p is the number of phases and c thenumber of components. For an element under temperature and pressure conditionsof interest, f = 3−p. This implies that a single phase is represented by an area inthe pressure–temperature plane (p = 1� f = 2), a two-phase mixture is representedby a curve (p = 2� f = 1), which may be termed a phase boundary or phase line,and a three-phase mixture by a point (p = 3� f = 0), generally known as a triplepoint. A single component phase diagram is essentially a plot of areas representingphases, which are demarcated by phase boundaries, in the pressure–temperature orthe P–T plane. A typical phase diagram of an element will generally show a vapourphase, a liquid phase and one or more solid phases. The phase boundaries haveto abide by a few thermodynamic rules. The entropy change (�S) and the volumechange (�V) across a phase boundary are related to the slope of the boundary bythe Clausius–Clapeyron equation:

dP

dT= �S

�V(1.1)

This slope can be positive or negative: �S must be positive for increasingtemperature by the second law of thermodynamics, but �V can be either positiveor negative.

The second derivative, d2PdT2 , gives a measure of the curvature and can be

expressed as (Partington 1957):

d2P

dT 2= − 1

�V

[d�V

dP

(dPdT

)2

+2d�V

dTdPdT

− �Cp

T

](1.2)

For relatively incompressible solids like the transition metals, the terms on theright-hand side are small with the result that the phase boundaries have very smallcurvature and look like straight lines over the experimentally available pressureranges (Young 1991).

Experimental work on pressure-induced phase transformations in transition met-als has been somewhat limited because of their low compressibility; phase changesmay occur only at very high pressures which are difficult to achieve. Shock waveexperiments are at present the most effective means of studying the high-pressurephase diagrams of these metals (Young 1991). A shock wave is a disturbance prop-agating at a supersonic speed in the medium. One can imagine the shock to bearising from a piston which moves into the medium at a constant velocity �p. Theboundary between the compressed and the uncompressed material will move aheadof the piston with a certain velocity �s, which is termed the shock velocity. Thebasic objective of shock wave experiments is to measure the velocities �p and �s




and to determine from them the thermodynamic state of the host material. For mostmaterials, �p and �s bear a linear relationship. But at a phase boundary this relation-ship may break down and the �s versus �p plot may show a discontinuity (McQueenet al. 1970). The reason for this is that a steady shock wave needs a sound speedthat increases with compression and that this requirement is violated by a first-order phase transition, with the result that the shock wave breaks up into a low-pressure wave (representing the untransformed material) and a high-pressure wave(representing the transformed material). The detectors register the arrival of onlythe first (i.e. low-pressure wave) and the two-wave region appears as a flat segmentof constant �s on the �s versus �p plot; a third segment appears on the plot whenthe shock velocity in the transformed material exceeds that corresponding to theuntransformedmaterial (Young1991).Theappearanceofdiscontinuities in the�s–�p

plane is thus a good indication of the occurrence of a first-order phase transition.

1.3.2 TitaniumAs stated earlier in this chapter, elemental Ti exists as the hcp �-phase at roomtemperature under atmospheric pressure. On raising the pressure, while keepingthe temperature constant, Ti transforms to the hexagonal -phase at around 2 GPapressure. The �– phase boundary has been reported to have a negative slope(Zilbershteyn et al. 1975, Vohra et al. 1982). This transition is associated with alarge hysteresis and the equilibrium phase boundary has not been determined accu-rately (Young 1991). Further compression at room temperature to pressures upto87 GPa has not shown any phase other than the -phase until recently (Xia et al.1990a,b). As indicated earlier, this point will be covered in a subsequent chapter.

Under atmospheric pressure, the �-phase transforms to the denser �-phase (bcc)at 1155 K. The �–� phase boundary has been determined by high temperature,static pressure measurements (Bundy 1963, Jayaraman et al. 1963). The triplepoint at which the �-, �- and -phases meet occurs at about 9.0 GPa and 940 K(Young 1991). The �– phase boundary has been experimentally determined uptoa pressure of 15 GPa (Bundy 1963). No phase other than the �-, �- and –phaseshas been found in Ti. Shock wave experiments conducted on elemental Ti haveshown a discontinuity in the �s–�p curve; it has been suggested that this maycorrespond to the �– or –� transition (McQueen et al. 1970, Kutsar et al. 1982,Kiselev and Falkov 1982). The experimentally determined pressure–temperaturephase diagram of Ti is shown in Figure 1.1 (Young 1991).

Linear muffin tin orbital (LMTO) calculations which take into considerationthe hcp, bcc, and fcc structures have predicted the stability of the -phase forpressures up to 30 GPa (Gyanchandani et al. 1990). The disposition of the �–boundary (Figure 1.1) is not inconsistent with the theoretical prediction that at 0 Kthe -phase is the equilibrium phase in the case of Ti.




Ti2.5

2.0

1.5

1.0

0.5

00 6 12 18

P (GPa)

bcc(β)

hcp(α)

hex(ω)

Liquid

T (

× 1

03 K

)

Figure 1.1. Experimentally determined temperature–pressure phase diagram for Ti.

1.3.3 ZirconiumAs in the case of Ti, elemental Zr exists as the hcp �-phase at room temperatureand pressure, while on pressurization at this temperature it gets converted to thehexagonal -phase at a pressure of about 2 GPa. In this case also, the �– phaseline exhibits a negative slope (Jayaraman et al. 1963, Zilbershteyn et al. 1975,Guillermet 1987). A precise determination of the equilibrium transition pressurehas, however, not been possible due to the occurrence of hysteresis (Young 1991).Static pressure experiments at room temperature have established that the -phasetransforms to a bcc phase (′) at a pressure of 30 GPa (Xia et al. 1990a,b). Thisbcc phase has been found to be the same as the �-phase.

It has been mentioned earlier that under atmospheric pressure, �–Zr transformsto �–Zr at 1139 K. The �–� phase boundary for elemental Zr has been studied byhigh-temperature, static pressure experiments (Jayaraman et al. 1963, Zilbershteynet al. 1973). The –� boundary has been determined upto a pressure of 7.5 GPa(Jayaraman et al. 1963). The �–�– triple point has been found to occur at975 K and 6.7 GPa. As mentioned earlier, the �-phase appears to be identicalto the ′-phase that occurs at room temperature under high pressures and thisimplies that the –� phase boundary has to turn backward towards the T = 0 Kaxis at high pressures (Young 1991). Shock wave experiments conducted on Zrare reported to show a discontinuity in the �s versus �p curve as in the caseof Ti and this has been interpreted as being suggestive of the occurrence of a




0 2 4 6 8 100

1

2

3Zr

P (GPa)

T (

× 1

03 K

)

bcc(β)

hcp(α) hex

(ω)

Liquid

Figure 1.2. Experimentally determined temperature–pressure phase diagram for Zr.

phase transition (McQueen et al. 1970, Kutsar et al. 1984). The experimentallydetermined pressure–temperature phase diagram of Zr is shown in Figure 1.2.

In the case of Zr, LMTO calculations predict that the �– and –′ transi-tions should occur at pressures of 5 GPa and 11 GPa, respectively (Gyanchandaniet al. 1990).

1.3.4 EpilogueThe occurrence of the -phase at high pressures in elemental Ti and Zr and atroom pressures in alloy systems such as Ti–V and Zr–Nb and the similarity ofthe � and the structures have been interpreted as being indicative of the factthat the phase diagrams of Ti and Zr exhibit the phenomenon of s-d electrontransfer (Sikka et al. 1982). Effecting an increase in the number of d-electrons,either by the application of pressure or by alloying with elements relatively richerin d-electrons, drives the structure towards the bcc structure characteristic of thenext group of elements to the right (i.e. V or Nb). The specific form of the structure, which may be regarded as a hexagonal distortion of the bcc structure,may be related to the details of the Fermi surfaces (Myron et al. 1975, Simmonsand Varma 1980).

The crystal structures of Ti, Zr and Hf under pressure have recently been studiedby Ahuja et al. (1993) by means of first principles, total energy calculations basedthe local density approximation. These calculations correspond to zero temperature




but many of the results obtained by them, especially for Zr, are in good agreementwith experimental observations made at room temperature. The observed crystalstructure sequence: hcp �hP2� → �hP3� → bcc �cI2� with increasing pressurehas been validated for Zr and Hf and it has been predicted that the same sequenceshould apply to Ti. The equilibrium volumes obtained for Ti, Zr and Hf are0.0160, 0.0222 and 0 0201 nm3, respectively, which compare reasonably well withthe experimental values of 0.0176, 0.0233 and 0 0223 nm3 for these metals. Thecalculated c/a values corresponding to the minimum total energy are also in goodagreement with the experimental values. Some of the disagreement between thetheoretical predictions and the room temperature experimental observations couldbe ascribed to thermal effects. For example, the calculations indicate that at thetheoretical equilibrium volume, the hP3 structure is slightly more stable than thehP2 structure; but room temperature observations show that the reverse is true—aresult that matches with the calculations at the experimental volume. An importantpoint is that the calculations do show that the energy difference between the �- andthe -phases is small for both Ti and Zr, which is consistent with the fact thatthe pressure induced � → transition can be brought about in these metals atmoderately high pressures.

The calculations of Ahuja et al. (1993) indicate that the charge density for the-phase has a substantial non-spherical component, reflecting covalent bonding.This is quite different from the chemical bonding prevailing in the fcc, hcp andbcc structures where the charge density is predominantly spherical around theatomic positions and flat in the intervening regions. The chemical bonding forthese structures is metallic. However, despite the difference in the nature of thechemical bonds for the various structures, band filling arguments can be used, atleast to some extent, to explain the crystallographic sequence encountered in thesetetravalent metals.

At zero temperature and sufficiently high pressures, all the three metals – Ti,Zr and Hf – are predicted to assume the bcc structure. Again, at zero pressureand high temperaturess, these elements are known to transform from the hcpto the bcc structure. There is thus the possibility that the two bcc regions in apressure–temperature phase diagram will be in contact. A schematic phase diagram,pertinent to these metals, has been constructed by Ahuja et al. (1993) and is shownin Figure 1.3. These authors have also examined the issue of the stability of thebcc phase. They have shown that the tetragonal shear constant, C ′ = �C11 −C12�/2,has a negative value at zero pressure for the bcc structure. This corresponds to amechanically unstable situation. However, the sign of C ′ changes with increasingpressure. For the high pressure bcc phase, the calculated C ′ values are all positive,in agreement with the observed high pressure bcc phase in Zr and Hf. This can beexplained as an effect of s–d electron transfer; for example, the d-band occupation




Pressure

Tem

pera

ture

hcpbcc

bcc I II

L

ω

Figure 1.3. Schematic temperature–pressure phase diagram for the metals Ti, Zr and Hf.The bcc phase is mechanically unstable in the region I and mechanically stable in the region II at lowtemperatures.

of Zr increases under pressure, making it behave more like the element to its right,i.e. Nb, which has a bcc crystal structure.

Even though the bcc structure, according to calculations, is mechanically unsta-ble at zero pressure, the high temperature �-phase of all the three metals is knownto posses this structure. This can be explained in terms of the high entropy associ-ated with the bcc structure. The �-phase of these elements shows some anomalousproperties including its well known anomalously fast diffusion behaviour. Thisbehaviour might be related to the intrinsic mechanical instability associated withthe value of the C ′ parameter. Another possible explanation suggested for theanomalous diffusion behaviour invokes -embryos acting as activated complexconfigurations in the atom–vacancy exchange process (Sanchez and de Fotnaine1975). The fact that the -phase is calculated to have a lower total energy than the�-phase at the equilibrium volume for all the three metals lends support to suchan interpretation.

The mechanical instability of the bcc phase becomes less severe with increasingpressure in the sense that the value of C ′ becomes less negative with decreasingvolume. Therefore, as the pressure increases, a progressively lower temperature isneeded to restore the stability of the bcc structure (Ahuja et al. 1993).

1.3.5 Phase stability and electronic structureThe stability of phases, the dependence of this stability on parameters like tem-perature and pressure and the selection of phases that are actually observed andrecorded in phase diagrams are determined by the result of the competition amongseveral possible phases (and, therefore, structures) that could be stable in a given




system. This competition is based on the respective values of the Gibbs free energycorresponding to the various pertinent phases and their variation with temperature,pressure, composition and parameters such as magnetic, electric or stress fields,dose rates of particle and photon irradiation, etc. A number of factors contribute tothe enthalpy, H , and the entropy, S. A very important contribution to the entropyarises from the statistical mixing of atoms. There may be additional contributionsfrom vibrational effects, clustering of atoms, distribution of magnetic moments,long range configurational effects, etc. The statistical mixing of atoms contributesto the enthalpy as well. These contributions are related to the interaction energies:those corresponding to nearest neighbour atoms, next nearest neighbour atoms andfurther distant atoms in a given structure. These interaction energies may arisefrom various origins – electronic, magnetic, elastic and vibrational. A formidableproblem in the context of the assessment of phase stability is that the relativestability among the competing crystal structures is usually dictated by very smallenergy differences between large values of the cohesive energy. Apart from this, acorrect prediction implies the prediction of the lowest free energy structure amongthe chosen structural alternatives. This, in turn, stipulates a prior algorithm togenerate all probable structures. Even when all these difficulties are overcome, itis needed to incorporate the roles of variables like temperature and pressure inrealistic terms. These are, indeed, difficult tasks.

The success of a theory of phase stability is largely determined by its abilityto make predictions that are consistent with experimental observations. Thereis a need to be able to calculate phase stability from “first principles” if thebasic microscopic parameters that dictate the free energy of a phase are to beproperly understood. It should also be possible to make use of such calculationsfor predicting phase diagrams in systems where the experimental determination ofsuch diagrams is difficult. The understanding and prediction of phase stability inrespect of disordered and ordered alloys in terms of electronic structure calculationsconstitute an area of considerable importance in materials science and significantprogress has been made with regard to the “first principles” approach to the bandtheory of such materials (Massalski 1996).

The computation of an alloy phase diagram from first principles implies itsdelineation from a knowledge of the electronic structure of the alloy. In a truly abinitio calculation, one begins with a periodic array of nuclei of charge Ze togetherwith Z electrons per nucleus, and then solves the Schrodinger equation for thetotal energy of the system. When Z is small (e.g. for H, He and Li), it is possibleto handle this problem by Quantum Monte Carlo (QMC) methods (Ceperley andAlder 1986) which are exact in principle. However, the QMC method is not yetpractical for heavier atoms, and the development of the density functional theoryand its computational version, the local density approximation (LDA), has been of




great value. Here the full many-body wave function is approximated as a productof one-electron functions, and the exchange–correlation energy is expressed as afunction of the local electron density, n�r�, given by n�r� = ��k�r��

2��k�r� beingthe one-electron wave function for the occupied state k (Young 1991).

In the density functional theory, the total energy of a system of nuclei andelectrons is considered to be a unique functional of n�r� and is a minimum at thetrue ground state. The total energy, Et , is expressed as Et = E1 +E2 +E3 +E4 +E5

where the terms on the right-hand side represent the kinetic (E1), electron–nucleus(E2), electron–electron (E3), exchange–correlation (E4) and nucleus–nucleus (E5)energies.

The different approaches used to solve the one-electron Schrodinger equation,with the imposition of the lattice periodicity (Bloch condition) as a boundarycondition, have engendered a variety of band-structure methods; some of these are(Young 1991): ab initio pseudopotential (AIP); linear muffin tin orbital (LMTO);augmented spherical wave (ASW); linearized augmented plane wave (LAPW);full-potential LAPW (FPLAPW) and linear combination of Gaussian-type orbitals(LCGTO).

The LMTO method, which has been extensively used, is based on some addi-tional approximations. While the muffin tin (MT) potential implies that the atomicpotential V�r� is spherically symmetric within a sphere inscribed in the primitiveunit cell and is constant in the interstitial region, the LMTO method brings in afurther simplification by way of the atomic-sphere approximation (ASA), wherebythe spherical potential is extended to the full atomic volumes, reducing the netinterstitial volume to zero. The Bloch condition is implemented by effecting thecancellation of all neighbour wave functions within the atomic sphere (Skriver1984). The ‘L’ in LMTO implies the approximation that the basis functions aremade energy-independent; this permits the eigenfunctions to be obtained in a sin-gle diagonalization operation, speeding up the calculation enormously and thuscontributing to the efficacy of the method, a major limitation of which is therestriction to high-symmetry crystal structures imposed by the ASA (Young 1991).The LMTO method has been used to predict the stability of different phases withregard to the pressure–temperature phase diagrams of many transition metals,including Ti and Zr.

Total energy calculations based on the LDA, which use only atomic numbers asinputs, have been very successful in the estimation of 0 K ground state propertiesof the elements and of ordered compounds. In fact, the implementation of theLDA by many an investigator, combined with the development of efficient linearmethods for studying the electronic structure of solids, has led to fully ab initiocalculations of the total energy at 0 K of pure solids, relatively simple compoundsand disordered alloys (Sanchez 1992). By making it possible to assess a wide range




of physical properties quite close to the corresponding experimentally obtainedvalues, these quantum mechanical total energy computations have provided veryfavourable evidence in support of the LDA method, which can be applied, togetherwith appropriate statistical models, to address the difficult problem of alloy stabilityat non-zero temperatures.

Even though the LDA method has been quite successful, it has some non-triviallimitations including the underestimation of band gap energies and the inabilityto predict narrow band Mott-transition phenomena (Young 1991). A more generalmethod for calculating the equilibrium state of matter at finite temperatures is thequantum molecular dynamics method (Car and Parrinello 1989). In this approach,the LDA wave function is solved for a small number of nuclei in an arbitraryconfiguration and the Hellman–Feynman theorem is used for finding the netforce on each nucleus; the nuclei are then moved in accordance with classical(Newtonian) dynamics and the LDA calculation is undertaken again for the newconfiguration of the nuclei. This approach has been found to be useful for arrivingat band structures and bonding details in respect of solids and liquids at finitetemperatures (Young 1991).

In the context of statistical models, it is appropriate to make a mention hereof the Monte Carlo (MC) (Binder 1986) and molecular dynamics (MD) (Hoover1986) methods. Like QMC, these methods are exact in principle. Although it ispossible to undertake direct calculation of free energy by MC, the technique is notyet very suitable for the determination of phase stability and accurate delineation ofphase boundaries. As of now, the MD method also suffers from similar limitations.It is true that isobaric–isothermal ensemble versions of MC and MD have beensuccessfully employed to predict the most stable crystal structures of certain solids(Parrinello and Rahman 1981), but these methods have found their most importantuse in providing a standard for comparing and refining approximate statisticalmechanics models (Young 1991).

Some of the aspects briefly outlined in the preceding paragraphs have beencovered in greater detail in a subsequent chapter.

It is to be noted that a major shortcoming of many of the ab initio phasediagram calculations concerns the inadequate treatment of local volume and elasticrelaxations and the neglect of vibrational modes. Even in crystalline solids, atomsare in perpetual motion; they move from one lattice site to another by diffusionat non-zero temperatures and also vibrate about their equilibrium positions. In amulticomponent system like an alloy, a given lattice site is occupied by atomsof different species at different times. If a large atom replaces a small one, theenvironment of the lattice site responds by expanding. Likewise, when a smallatom replaces a large atom, the neighbouring atoms relax towards the lattice sitein question. It should be possible to address the accompanying strain fluctuations




within the same type of first principles framework that is pertinent to fluctuationsin concentration. However, the treatment of local relaxations of this kind presents avery difficult problem and not many attempts appear to have been made to includethis effect in first principles calculations of phase stability and phase diagrams(Sanchez 1992, Gyorffy et al. 1992).

Apart from the direct quantum mechanical route, many semi-empirical schemespertaining to phase stability have also been pursued, often with a good deal of suc-cess. Many of these schemes involve the construction of certain phenomenologicalscales on which various aspects of bonding and structural characteristics are mea-sured (Raju et al. 1995). These scales include parameters like the electronegativityfactor, the size factor, the coordination factor, the electron concentration (e/a)factor, the promotion energy factor, etc. that are used to systematize a variety ofstructural features. The resulting structure maps are essentially graphical represen-tations of the relative structural stability of alloy phases. They are two-dimensionaldiagrams, constructed by using suitable alloy theory coordinates for sorting outdifferent crystal structures that are compatible with a chosen alloy stoichiometry.The efficacy of these structure maps depends crucially on the appropriate choiceof coordinates. What are needed are those “bond indicators” which are transparentin their physical content, are transferable in their applicability and have a bear-ing on the alloy formation situation in terms of a validated model (Raju et al.1995). In the classical approach, the emphasis has been on the construction ofphysically simple and transferable coordinates that may systematize the observedtrends in relation to the occurrence of alloy phases. The major limitations of theclassical formalism lie in the linear dependencies among many of the differentphenomenological scales and the absence of a microscopic model that connectsone or more of these directly to a real space alloy physics (Raju et al. 1995).Quantum mechanical considerations have been invoked in order to tide over thesedeficiencies with the result that the classical coordinates have been replaced bywhat are known as quantum structural parameters and classical structure diagramsby quantum structural diagrams (QSD). There have been numerous applicationsof QSD to various classes of solids including intermetallics, quasicrystals, high Tc

superconductors and permanent magnetic materials (Phillips 1991). Even thoughnot all of these have served to elucidate the issue of structural stability of con-densed phases, these have been very useful in ordering the vast available data baseinto certain systematics. There are, indeed, quite a few examples of QSD whichhave really enhanced the understanding of the physicochemical factors governingphase stability.

Most of the existing models pertaining to phase stability, ranging from thoseoffering detailed density maps and electronic parameters of alloys to the semi-empirical ones, suffer from a major difficulty in the context of the construction




of phase diagrams in that a theoretical treatment of the temperature dependenceof energy is not straightforward and tractable (Massalski 1996). The calculationsused for predicting enthalpy at 0 K (first principles calculations) or at some unde-fined temperature (semi-empirical models) are seldom able to furnish adequateinformation regarding the thermal behaviour of such enthalpies or the thermalentropy contributions to the free energy. The prediction of entropies, particu-larly for relevant metastable phases in phase diagrams, has to be realized forthe utilization of the full potential of the theoretical methods of phase stabilitycalculations.

1.3.6 Some features of transition metalsElements belonging to the family of transition metals, of which Ti and Zr aremembers, are generally characterized by certain interesting features. Some of thesewill be briefly covered in this section.

Elements of Groups 3–10 in the periodic table constitute the transition metalswhich have in common that their d-orbitals (3d, 4d and 5d) are partially occupied.These orbitals are only slightly screened by the outer s-electrons, resulting insignificantly different chemical properties of these elements going from left to rightin the periodic table; the atomic volumes rapidly decrease with increasing numberof electrons in the bonding d-orbitals, because of cohesion, and then increase asthe anti-bonding d-orbitals get filled (Steurer 1996).

Transition metals are characterized by a fairly tightly bound (and partially filled)d-band that overlaps and hybridizes with a broader nearly-free-electron (NFE)sp-band. The d-band (with a large density of states near the Fermi level) is welldescribed within the tight-binding (TB) approximation by a linear combinationof atomic d-orbitals and the difference in behaviour between the valence sp andd electrons arises from the d-shell lying inside the outer valence s-shell, therebyresulting in a small overlap between the d-orbitals in the bulk (Pettifor 1996).

In general, the transition metals exhibit high densities, cohesive energiesand bulk moduli, with some exceptions. These characteristics arise from strongd- electron bonding. Plots of molar volume, cohesive energy and bulk modulusagainst the number of d-electrons yield roughly symmetrical curves with extremevalues approximately at the middle of the series (Young 1991). An exception tothis trend occurs with the 3d magnetic elements. The values of these parametersfor the transition elements are shown in Table 1.5. The general behaviour alludedto the above can be rationalized in terms of the Friedel model of transition metald-bands (Harrison 1980). Cohesive energy versus group number plots for 3d, 4dand 5d transition metals are shown in Figure 1.4.

The sequence of the observed room temperature (and pressure) crystal structuresin the case of 3d, 4d and 5d transition metals is presented in Table 1.6. This




Table 1.5. Values of molar volume, cohesive energy and bulk modulus for transition metals(Young 1991).

Z Element Molar volume (m3/M mol) Cohesive energy (kJ/mol) Bulk modulus (GPa)

21 Sc 15 00 376 0 54 622 Ti 10 64 467 0 106 023 V 8 32 511 0 155 024 Cr 7 23 395 0 160 025 Mn 7 35 282 0 90 426 Fe 7 09 413 0 163 027 Co 6 67 427 0 186 028 Ni 6 59 428 0 179 039 Y 19 88 424 0 41 040 Zr 14 02 607 0 94 941 Nb 10 83 718 0 169 042 Mo 9 38 656 0 261 043 Tc 8 63 688 0 −44 Ru 8 17 650 0 303 045 Rh 8 28 552 0 282 046 Pd 8 56 376 0 189 071 Lu 17 78 428 0 47 472 Hf 13 44 619 0 108 073 Ta 10 85 781 0 191 074 W 9 47 848 0 308 075 Re 8 86 774 0 360 076 Os 8 42 788 0 −77 Ir 8 52 668 0 358 078 Pt 9 09 564 0 277 0

observed sequence (hcp → bcc → hcp → fcc) indicates that close packed structuresare preferred at either end of the series, while the more open bcc structure ispreferred in the middle. Pettifor (1977) has carried out a TB orbital calculationand shown that the structure sequence across the series is the result of the fillingof the d-band and that the s-p electron number is nearly constant. While thismodel correctly predicts the structure sequence hcp → bcc → hcp → fcc, it doesnot predict the structures of all the elements correctly. In the tight binding model,to a first-order approximation, the cohesive energy turns out to be independentof structure; the relative structural stability arises from small differences in bandstructure contribution to the total electronic energy, an adequate description ofwhich calls for the inclusion of higher order moments for describing the densityof states curve (Raju et al. 1996). A fully self consistent LMTO calculation leadsto a still better agreement between theory and experiment (Skriver 1984).




43 5 6 7 8 9 10

3d

4d

5d

250

350

450

550

650

750

850

950

Group number

Coh

esiv

e en

ergy

(kJ

/mol

)

Figure 1.4. Cohesive energy versus group number plots for 3d, 4d and 5d transition metals.

Table 1.6. Crystal structures of d-transition metals at room temperature and pressure.

HCP BCC HCP FCC

3d series Sc Ti V Cr Fe Mn Co Ni4d series Y Zr Nb Mo Tc Ru Rh Pd5d series Lu Hf Ta W Re Os Ir Pt

Note: The actual structure of Mn is complex though it is listed under bcc in this table.

A systematic theoretical study with regard to the phase transitions that can beexpected to occur in unalloyed transition metals at ultra-high pressures has not yetbeen attempted. However, it is, in general, expected that the early transition metalswill assume the structures of their right-hand side neighbours as the s–d electrontransfer will lead to the filling of the d-band under pressure; for the later membersof the series, pressure is expected to have the effect of emptying the d-band, thusreversing the earlier trend (Young 1991).

Obviously, transition metal phase transitions can also be driven by alloying,whereby the number of electrons populating the d-band can be altered. Fairlygeneral theoretical arguments suggest that alloys of transition metals with roughlyhalf-filled d-bands exhibit ordering tendencies, while those with nearly empty ornearly full d-bands show clustering tendencies in the disordered state and thus tendto phase separate at low temperatures; this prediction appears to be borne out bya considerable body of experimental data, even though there are many exceptionsto this rule (Gyorffy et al. 1992).




1.4 EFFECT OF ALLOYING

1.4.1 Introductory remarksIn alloys based on Ti or Zr, a very important effect of an alloying element pertainsto the manner in which its addition affects the allotropic �-phase to �-phasetransformation temperature. Some elements stabilize the �-phase by raising thistemperature while some others lower it, thereby stabilizing the �-phase. Elementswhich, on being dissolved in Ti or Zr, cause the transformation temperatureto increase or bring about little change in it are known as �-stabilizers. Theseelements are generally non-transition metals or interstitial elements (like C, N andO). Elements which, on alloying with Ti or Zr, bring down the transformationtemperature are termed �-stabilizers. These elements are generally the transitionmetals and the noble metals with unfilled or just filled d-electron bands. Amongthe interstitial elements, H is a �-stabilizer. Unlike in pure Ti or Zr, in alloysthe single phase � and the single phase � regions are separated by a two-phase�+� region in the temperature versus composition phase diagram. The width ofthis region increases with increasing solute content. The single equilibrium �- to�-phase transformation temperature associated with elemental Ti or Zr is replacedby two equilibrium temperatures in the case of an alloy: the �-transus temperature,below which the alloy contains only the �-phase, and the �-transus temperature,above which the alloy contains only the �-phase. At temperatures between thesetwo temperatures, both the �- and the �-phases are present.

1.4.2 Alloy classificationThe allotropic transformation exhibited by Ti and Zr forms the basis of the classi-fication of commercial alloys based on these metals. Such classification is effectedon the basis of the phases present in these alloys at ambient temperature (andpressure). The relative proportions of the constituent phases are determined by thenature (�-stabilizing or �-stabilizing) and the amounts of the alloying elements.In the case of alloys, the �- and �-phases contain various amounts of the differentalloying species in solid solution.

1.4.3 Titanium alloysTechnical alloys of Ti, which are generally multicomponent alloys containing�-stabilizing as well as �-stabilizing elements, are broadly classified as � alloys,�+� alloys and � alloys. Within the second category, there are the subclasses“near �” and “near �” alloys, referring to alloys whose compositions place themnear the �/��+�� or the ��+��/� phase boundaries, respectively.




Unalloyed Ti and its alloys with one or more �-stabilizing elements consist fullyor predominantly of the �-phase at room temperature and are known as � alloys.The �-phase continues to be the primary phase constituent of most of these alloys attemperatures well beyond about 1040 K (Froes et al. 1996). These alloys generallyexhibit good strength, toughness, creep resistance and weldability, together withthe absence of a ductile-to-brittle transition (Collings 1984). However, they arenot amenable to strengthening by heat treatment.

The compositions of �+� alloys are such that at room temperature they containa mixture of the �- and �-phases. These alloys have one or more of �- as wellas �-stabilizing elements as alloying additions. In general, �+� alloys possessgood fabricability. They are very strong at room temperature and moderately soat high temperatures (Collings 1984). The relative volume fractions of the �- and�-phases in these alloys can be varied by heat treatment, which provides a handlefor adjusting their properties.

In �-alloys, the �-phase is stabilized by the addition of adequate amounts of�-stabilizing elements and can be retained at room temperature. These alloysgenerally contain significant amounts of one or more of the transition metals V,Nb, Ta (Group 5) and Mo (Group 6). These “�-isomorphous” alloying elements donot form intermetallic compounds through eutectoid decomposition of the �-phaseand are generally preferred to eutectoid forming �-stabilizing elements such asCr, Cu, Ni; however, elements of the latter category are sometimes added to �(and �+�) alloys for improving their hardenability and response to heat treatment(Froes et al. 1996). The strength of � alloys is generally greater than that of�+� and �-alloys. Moreover, they exhibit excellent formability (Wood 1972).But they have relatively high densities, are prone to ductile–brittle transition atlow temperatures and generally possess inferior creep resistance as compared to� and �+� alloys (Collings 1984, Froes et al. 1996).

The archetypical �-stabilizing and �-stabilizing alloying additions to Ti areAl and Mo, respectively. It is useful to be able to describe a multicomponentTi-based alloy in terms of its “equivalent” Al and Mo contents. The two pertinentexpressions often quoted in this context (Collings 1994) are:

Al�eq = Al�+ Zr�/3+ Sn�/3+10 O�

Mo�eq = Mo�+ Ta�/5+ Nb�/3 6+ W�/2 5++ V�/1 25+1 25 Cr�

+1 25 Ni�+1 7 Mn�+1 7 Co�+2 5 Fe�

where [X] indicates the concentration of the element X in weight per cent in thealloy. It can be seen that while Al and O are strong �-stabilizers, Sn and Zr arerelatively weak ones. It can also be seen that the efficacy of the transition elements




with regard to the stabilization of the �-phase progressively increases in the order:Ta, Nb, W, V, Mo, Cr and Ni, Mn and Co, and Fe, the last being the strongest�-stabilizer.

It may be mentioned here that Ti can form extensive substitutional solid solutionswith most of the elements with atomic size factor within about 20% and this facthas opened up many alloying possibilities for exploitation.

Some examples of important commercial Ti base alloys are: Ti-5Al-2.5Sn(� alloys); Ti-8Al-1Mo-1V, Ti-6Al-2Sn-4Zr-2Mo (near � alloys); Ti-6Al-4V,Ti-6Al-2Sn-6V, Ti-3Al-2.5V (�+ � alloys); Ti-6Al-2Sn-4Zr-6Mo, Ti-5Al-2Sn-2Zr-4Cr-4Mo, Ti-3Al-10V-2Fe (near � alloys); Ti-13V-11Cr-3Al, Ti-15V-3Cr-3Al-3Sn, Ti-4Mo-8V-6Cr-4Zr-3Al, Ti-11.5Mo-6Zr-4.5Sn (� alloys).

1.4.4 Zirconium alloysUnlike Ti, Zr is not quite amenable to alloying. One of the reasons for this couldbe the relatively large size of the Zr atom. Most of the elements have very limitedsolubilities in �-Zr, with a few exceptions such as Ti, Hf, Sc and O. By comparison�-Zr is a much better solvent, but it is generally quite difficult to retain the �-phaseat room temperature in a metastable state by quenching (Froes et al. 1996). Theoccurrence of non-equilibrium phases in �-quenched Ti- and Zr-based alloys hasbeen dealt with in a later section.

According to the exhaustive compilation made by Douglass (1971), the retentionof the �-phase during quenching has been found to be feasible in the binary Zr–Mo,Zr–Cr, Zr–Nb, Zr–U, Zr–V and Zr–Re systems. The minimum concentrations ofalloying additions for complete retention of the �-phase in the first four systemsare 5 wt%, 7.2 wt%, 15 wt% and 20 wt% respectively. Retention of cent per cent�-phase is not possible in the systems Zr–V and Zr–Re; alloys containing themaximum amounts of V or Re in solution at quenching temperatures as highas 1573 K have been found to contain the -phase in addition to the �-phase(Petrova 1962). The retention of quite large volume fractions of a metastable,Zr-rich �1-phase has been observed in relatively solute-lean alloys (Zr-2.5 wt%Nb and Zr-5 wt% Ta) belonging to the monotectoid Zr–Nb (Banerjee et al. 1976,Menon et al. 1978) and Zr–Ta (Mukhopadhyay et al. 1978, Menon et al. 1979)systems.

The most common Zr alloys of commercial importance are the zircaloys, namelyzircaloy 2: Zr-1.5Sn-0.1Cr-0.1Fe-0.1Ni, Cr + Fe + Ni not to exceed 0.38 wt%;zircaloy 4: Zr-1.5Sn-0.15Cr- 0.15Fe, Cr + Fe not to exceed 0.3 wt% and theZr-2.5% Nb, Zr-1% Nb and Zr-2.5Nb-0.5Cu alloys. These alloys contain onlysmall amounts of �-stabilizing elements and are all basically �-alloys, with the�-phase as the predominant constituent phase.




1.4.5 Stability of titanium and zirconium alloysThe aspect of lattice stability or, in other words, of structural phase stability isan important issue with regard to pure metals like Ti and Zr and alloys based onthese. It has been stated in an earlier section that the crystal structures of the threelong periods of transition metals follow the sequence hcp → bcc → hcp → fcc asthe group number increases from 3 to 10 (3d: Sc to Ni; 4d: Y to Pd; 5d: Lu to Pt).It appears that there is a correlation between the crystal structure and the groupnumber in the case of the elemental transition metals and between the crystalstructure and the average group number or the electron to atom (e/a) ratio in thecase of alloys. The occurrence of correlations like this testifies to the fact that theelectronic structure is a key factor in determining phase stability. The e/a ratiois a parameter which relates to many properties of binary transition metal alloys,particularly Ti–X alloys, where X represents a transition metal (Collings 1984).A qualitatively similar situation is obtained with Zr–X alloys also. However, ageneral and comprehensive theoretical explanation rationalizing the correlationbetween phase stability and electron concentration (which is the same as or isclosely related to the e/a ratio) in the case of transition metal systems is still toevolve (Faulkner 1982).

The issue of the stability of equilibrium phases in Ti (and Zr) alloys can alsobe addressed by adopting a thermodynamic approach (Kaufman and Bernstein1970, Kaufman and Nesor 1973). In this approach, the energywise competitionbetween the relevant phases is duly considered while assessing phase stability inunalloyed metals as well as in alloys. This quantitative thermodynamic approachhas been used for the computation of phase diagrams pertaining to binary as wellas multicomponent systems.

It has been mentioned earlier in the context of Ti–X and Zr–X alloys that�-phase stabilizers are generally non-transition or simple metals, while �-phasestabilizers are generally transition metals and noble metals. Collings (1984) hasput forward a qualitative explanation, based on electron screening considerations,with regard to the phase stabilizing action of �-stabilizer and �-stabilizer solutes.This is outlined in the following paragraphs.

When a simple metal X is dissolved in Ti (or Zr), most of the electrons belongingto X atoms occupy states in the lower part of the band and only very few appear atthe Fermi level. The d-electrons belonging to the host (solvent) tend to avoid thesolute atoms and this leads to a dilution of the Ti (or Zr) sublattice. A consequenceof this is to emphasise any pre-existing Ti–Ti (or Zr–Zr) bond directionality andthereby to preserve the hcp structure characteristic of Ti (or Zr). As more and moreX atoms are added, the field of Ti- (or Zr)-like �-stability is ultimately terminated,generally by the appearance of an intermetallic phase of the stoichiometry Ti3X(or Zr3X), which is also based on or is closely related to the hcp structure.




Coming next to the case of �-phase stabilization, one may first recall thatthe crystal structures of transition metals change from hcp to bcc as the e/aratio increases from 4 to 6. Collings (1984) has pointed out that it is possibleto rationalize this stabilization of the bcc structure within the framework of anelectron screening model which stipulates that a high concentration of conductionelectrons, by enhancing the screening of ion cores, may cause a symmetrical(i.e. cubic) crystal structure to be favoured. Thus an increase in the electrondensity (as in elements belonging to Groups 5 and 6) will tend to symmetrize thescreening, thereby enhancing the stability of the bcc structure. The fact that the sixd-transition metals belonging to Groups 3 and 4 undergo the hcp → bcc structuraltransformation at high temperatures indicates that symmetrization can also beaccomplished through lattice vibrations (Collings 1984). Given this background,one can see that the addition of transition metals belonging to Groups 5–10 to Tior Zr increases the electron density and as a consequence, stabilizes the bcc or�-phase. Thus, such elements are �-stabilizers. Ageev and Petrova (1970) havepointed out in the context of Ti alloys that the �-stabilization brought about bytransition metal solutes is more effective the farther they are from Ti in the periodictable and that for the retention of the �-phase during quenching from the �-phasefield, the nature and the concentration of the �-stabilizer has to be such that thevalue of the e/a ratio is at least 4.2.

In the context of the stability of bcc transition metals, it has been shown (Fisherand Dever 1970, Fisher 1975) that the magnitude of the elastic shear modulus C ′,defined as �C11 −C12�/2, can be used for comparing the stabilities of these metalsand their alloys. A cubic monocrystal is characterized by three fundamental stiff-ness moduli, C11�C44 and C12. The shear stiffness modulus, C ′, though made upof two fundamental moduli, is obtainable directly by experiment. The ultrasonicwaves needed for the measurement of these moduli are (Collings 1984): a longitu-dinal wave in a <100> direction for C11; a transverse wave in a <100> direction,polarized along <100> or a transverse wave in a <100> direction, polarizedalong <100> for C44; and the other transverse wave in a <100> direction,polarized along < 110> for C′. Since C44 is governed by the transverse <100>wave, <100> polarized, and C ′ by the same wave, <110> polarised, C ′ = C44 inan isotropic cubic material. Collings and Gegel (1973) have studied the variationof the parameter C ′ with the e/a ratio and have demonstrated that alloying Group4 elements with elements occurring to the right of them in the periodic tableenhances the stability of the bcc structure and that this effect is maximized at aboute/a= 6 (for the elements Cr, Mo and W). They have also found that C ′ almost van-ishes at e/a = 4 1 and that this value corresponds to the compositional thresholdfor martensitic transformation. In an anisotropic cubic material, the extent of thedeparture from isotropy is indicated by the value of the so called Zener anisotropy




ratio, A = C44/C′. While in simple bcc metals like Na, the values of A are quite

large, these can be quite low for bcc transition metals; for example, for the Group 6metals Cr, Mo and W, the values of A are 0.71, 0.72 and 1.01, respectively (Fisher1975). Fisher (1975) has also pointed out that while the C44 shears are resistedprimarily by nearest neighbour repulsion, the C ′ shear depends mainly on the nextnearest neighbour forces. The large values of C ′ for bcc transition elements arethought to be a consequence of the cohesive contributions of the d-electrons. Theparameter C ′ appears to be interpretable as a bcc stability parameter. Thus, forthe highly stable bcc transition metals of Group 6, C ′ is about 1 5 × 1011 N/m2

but its values decrease rapidly with decreasing e/a ratio, approaching zero at roomtemperature for alloys which exhibit -phase instabilities or under a martensitictransformation at ordinary temperatures (Collings and Gegel 1973).

When Ti (or Zr) is alloyed with transition metals of higher group numbers, theincreasing stability of the �-phase is reflected in a continuous lowering of the�/��+�� transus temperature. It is mentioned later in this chapter that in the caseof �-stabilized binary Ti alloys, two types of phase diagrams are encountered:�-isomorphous and �-eutectoid. Collings (1984) has pointed out that a generaltrend is that as the group number of the solute increases, there is a tendency forthe phase diagram to change from the former to the latter type.

1.5 BINARY PHASE DIAGRAMS

1.5.1 Introductory remarksBinary Ti–X and Zr–X (X being any element other than Ti and Zr, respectively)phase diagrams exhibit multifarious forms and reflect various kinds of phasereactions. The equilibrium phases are the �- and �-phases and numerous inter-metallic phases. These are the phases that are shown in the equilibrium phasediagrams. However, many non-equilibrium phases such as the martensite phase(hcp and orthorhombic), the -phase and a large number of metastable intermetal-lic phases also occur in binary Ti and Zr base alloys. Some of these will be coveredin detail in the succeeding chapters.

There have been many attempts to categorize Ti and Zr alloy phase diagrams,taking cognizance of the fact that basically there are two types of systems, namely�-stabilized and �-stabilized systems. As mentioned earlier, in the former caseX is usually a non-transition or simple metal, while in the latter X is usually atransition or a noble metal. It has been suggested in the context of Ti–X systemsthat the regular solution thermodynamic interaction parameter, �ij , is positive for�-stabilized alloys, indicating a clustering tendency, and negative for �-stabilizedalloys, indicating an ordering tendency (Collings and Gegel 1975).




For a given element X, the differences in the nature of the binary Ti–X andZr–X equilibrium phase diagrams generally arise from the relative inefficiency of�-Zr and �-Zr with regard to taking X in solid solution as compared to �-Ti and�-Ti, particularly when X is a substitutional element.

1.5.2 Ti–X systemsMargolin and Nielsen (1960) have suggested that �-stabilized Ti–X systems canbe basically subdivided into three classes: (a) �–�-isomorphous systems whereX is completely soluble in the �- as well as �-phases (e.g. Ti–Zr, Ti–Hf);(b) �-isomorphous systems where X is completely soluble in the �-phase andhas limited solubility in the �-phase (e.g. Ti–V, Ti–Mo) and (c) �-eutectoidsystems where X has a limited solubility in the �-phase which decomposes eutec-toidally into the �-phase and an appropriate intermetallic phase, TimXn, on cool-ing. Depending on the kinetics of �-phase decomposition, this class is furthersubdivisible into “active” (rapid, e.g. Ti–Cu, Ti–Ni) and “sluggish” (e.g. Ti–Cr,Ti–Mn) eutectoid systems. They have also suggested that �-stabilized Ti–X sys-tems can be subdivided into two categories, depending on the degree of �-phasestabilization: (a) systems exhibiting a “limited” degree of �-stability, where the�-phase is related to the �- and an appropriate intermetallic phase by a peritec-toid reaction (e.g. Ti–B, Ti–Al); and (b) systems characterized by a “complete�-phase stability” where the �-phase can coexist with the liquid phase (e.g. Ti–N,Ti–O).

An exhaustive classification scheme for binary Ti–X phase diagrams has sub-sequently been suggested by Molchanova (1965) who has classified the availableequilibrium phase diagrams into three broad groups, each of which contains a fewsubgroups. This classification, as reported by Collings (1984), is shown below:

Group I: Systems where X shows continuous solid solubility in the �-phaseSubgroup I (a): Complete solubility in the �-phase (X: Zr, Hf)Subgroup I (b): Partial solubility in the �-phase (X: V, Nb, Ta, Mo)Subgroup I (c): Partial solubility in the �-phase and eutectoid decomposition

of the �-phase (X: Cr, U)Group II: Eutectic systemsSubgroup II (a): Partial solid solubility in the �- and �-phases; eutectoid decom-

position of the �-phase (X: H, Cu, Ag, Au, Be, Si, Sn, Bi, Mn, Fe, Co, Ni,Pd, Pt)

Subgroup II (b): Partial solid solubility in the �- and �-phases; peritectoid �–�transformation (X: B, Sc, Ga, La, Ce, Nd, Gd, Ge)

Subgroup II (c): Extremely limited solid solubility in the �- and �-phases(X: Y, Th)




Group III: Peritectic systemsSubgroup III (a): Simple peritectic (X: N, O)Subgroup III (b): Partial solid solubility in the �- and �-phases (X: Re)Subgroup III (c): Partial solid solubility in the �- and �-phases; eutectoid

decomposition of the �-phase (X: Pb, W)Subgroup III (d): Partial solid solubility in the �- and �-phases; peritectoid �–�

transformation (X: Al, C).

In a simpler classification, Molchanova (1965) has suggested that binary Ti–Xequilibrium phase diagrams can be divided into four categories: �-isomorphous(including �–� isomorphous), comprising subgroups I (a), I (b) and III (b);�-eutectoid, comprising subgroups I (c), II (a) and III (c); simple peritectic, com-prising subgroup III (a); and �-peritectoid, comprising subgroups II (b) and III (d).This classification scheme is shown in Figure 1.5 in which the legends �, � and �stand for the�-phase, the�-phase and the pertinent intermetallic phase, respectively.

LL + β L + α

L + β

L + γL + β

L + γ

β + γ

α + γα + γ α + γ

β + γ

L +

β

Ti

Tem

pera

ture

Binary Ti alloys

β-stabilized

β-eutectoid β-peritectoidβ-isomorphous

α-stabilized

Simpleperitectic

SolutesN,O

SolutesB,Sc,Ga,La

Ca,Gd,Nd,GeAl,C

SolutesV,Zr,Nb,Mo,

Hf,Ta,Re

SolutesCr,Mn,Fe,Co,Ni,Cu

Pd,Ag,W,Pt,AuH,Be,Si,Sn,Pb,Bi,U

L LL

Ti Ti Ti

Solute content

ββ β β

α +

β

α +

β

β + α

β + αα

α

α

α

Figure 1.5. A classification scheme for binary Ti–X equilibrium phase diagrams. The legends ��and � stand, respectively, for the �-phase, the �-phase and the pertinent intermetallic phase.




It is to be noted that quite a few Ti–X systems, designated earlier as�-isomorphous systems, are not so in reality (Massalski et al. 1992). Below atransus delineating the upper boundary of a region referred to as a “miscibilitygap”, a homogeneous, single-phase �-solid solution decomposes into a thermo-dynamically stable aggregate of two bcc phases, one Ti-rich (�1) and the othersolute-rich ��2� � � → �1 +�2. The former participates in a monotectoid reaction:�1 → �+�2, the monotectoid temperature and composition varying from systemto system. Examples of Ti–X systems where such a monotectoid reaction occursinclude Ti–V, Ti–Mo, Ti–Nb and Ti–W.

1.5.3 Zr–X systemsIt has been pointed out earlier that inspite of the similarity in the electronic andcrystal structures of Ti and Zr (both of these transition metals belong to Group 4of the periodic table of elements), the alloying behaviour of these elements exhibitnoteworthy differences, largely due to the size factor. While one encounters the�–� isomorphous, �-eutectoid and �-stabilized types of equilibrium diagrams inZr–X systems, �-isomorphous type phase diagrams do not occur in these alloys.Alloying elements, X, which give rise to �-isomorphous equilibrium phase dia-grams with Ti, yield either �-eutectoid (e.g. X: V, Mo, Re) or �-monotectoid (e.g.X: Nb, Ta) types of equilibrium diagrams with Zr.

For a Pauling valence of 4, the second Brillouin zone is the one most nearlyfilled for both the �- and �-phases in the case of Zr. This zone for �-Zr is boundedby the

{1012

}and

{1120

}planes and has a volume of 3.6 electrons per atom;

the excess electrons, 0.4 per atom, overlap into the third zone on the{1012

}side

of the second zone (Luke et al. 1965). The second Brillouin zone for the �-phaseis bounded by �200� and �211� planes and has a volume of eight electrons peratom. The inscribed Fermi sphere accommodates 4.19 electrons per atom and doesnot touch the zone boundaries. The larger volume of the �-phase second zone incomparison with the �-phase zone implies that the �-structure can accommodatemore electrons and thus the solubility of some transition elements is greater in the�- phase than in the �-phase.

1.5.4 Representative examples of Ti–X and Zr–X phase diagramsIn this section representative examples of a few types of Ti–X and Zr–X binaryequilibrium phase diagrams will be introduced: Ti–Zr, Ti–Mo, Ti–V, Ti–Cr, Ti–Al,Zr–Nb, Zr–Fe, Zr–Sn, Zr–Al, Ti–N,Zr–H and Zr–O. The phase diagrams presentedhere are based on those appearing in Massalski et al. (1992). Subsequent updateshave been published in respect of some of these binary systems. These updateshave been referred to at appropriate places.




The Ti–Zr system is an example of an �–� isomorphous system, while Ti–Moand Ti–V constitute important examples of so called �-isomorphous systems andform the basis of several commercial � and �+� alloys. The Ti–Cr system is a typ-ical �-eutectoid system, while the �-stabilizer related Ti–Al system is pertinent toseveral technical � and �+� alloys. The Zr–Nb system, which relates to the impor-tant family of commercial Zr–Nb alloys, is a �-monotectoid system. The Zr–Fephase diagram exemplifies a �-eutectoid system. The Zr–Sn and Zr–Al systemsexhibit �-phase stabilization. The former is very relevant with regard to importanttechnical Zr alloys such as zircaloys, while the latter is germane to the Zr3Al inter-metallic phase which has been considered as a potential nuclear reactor structuralmaterial. In all these cases, X is a substitutional solute. In the Ti–N, Zr–H and Zr–Osystems, all of which are of technological importance, X is an interstitial solute.

The Ti–Zr system (Figure 1.6) appears to be a truly isomorphous system, thoughperhaps not as close to an “ideal solution” situation as the Zr–Hf system. Theequilibrium phases occurring in the Ti–Zr system are the liquid (L), ��Ti�Zr�,��Ti�Zr�, �-Ti, �-Ti, �-Zr and �-Zr. Apart from these, the metastable �′ (marten-site) and -phases are also encountered. The special points of the Ti–Zr systemare listed in Table 1.7 (Murray 1987, Massalski et al. 1992.)

2128 K

L

1813 K

1943 K

1155 K

~878 K

1136 K

100 20 30 40 50 60 70 80 90 100

Weight per cent Zr

2270

2070

1870

1670

1470

1270

1070

870

670

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Ti Atom per cent Zr Zr

(β-Ti, β-Zr)

(α-Ti, α-Zr)

Figure 1.6. Equilibrium phase diagram for the Ti–Zr system.




Table 1.7. Special points of the Ti–Zr system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Zr)

L� ��Ti� �Zr� Congruent 1813±15 38±2L� �Ti Melting 1943 0L� �Zr Melting 2128 100��Ti� �Zr�� Ti��Zr� Congruent 878±10 52±2�Ti� �Ti Allotropic 1155 0�Zr� �Zr Allotropic 1136 100

~1123 K

~968 K

1155 K

~12

1943 K

L

2896 K

100 20 30 40 50 60 70 80 90 100

Weight per cent Mo

2870

2670

2470

2270

2070

1870

1670

1470

1270

1070

870

670

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Ti Atom per cent Mo Mo

(β-Ti, Mo)

(α-Ti)

Figure 1.7. Equilibrium phase diagram for the Ti–Mo system.

In the Ti–Mo system (Figure 1.7), the equilibrium solid phases that are encoun-tered are: the bcc (�-Ti, Mo) solid solution, in which Ti and Mo are com-pletely miscible above the allotropic transformation temperature of Ti (1155 K),the hcp �-Ti (Mo) solid solution in which the solubility of Mo is restricted(maximum of about 0.4 at.%), �-Ti, �-Ti and Mo. This system exhibits a mis-cibility gap in (�-Ti, Mo) and a monotectoid reaction: ��-Ti) � (�-Ti) + (Mo)(Terauchi et al. 1978), the monotectoid temperature being about 968 K. Themetastable martensite (hcp �′ and orthorhombic �′′) and -phases also occur in




Table 1.8. Special points of the Ti–Mo system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Mo)

L� ��Ti� Melting 1943 0L�Mo Melting 2896 100��Ti�Mo�� Ti�+ �Mo� Critical ∼ 1123 ∼ 33��Ti�� Ti�+ �Mo� Monotectoid ∼ 968 (12) (0.4) �∼ 60��Ti� �Ti Allotropic 1155 0

the Ti–Mo system. The special points of the Ti–Mo system are shown in Table 1.8(Murray 1987, Massalski et al. 1992).

The equilibrium phase diagram of the Ti–V system (Figure 1.8) also showsa miscibility gap in the bcc (�-Ti, V) phase and a monotectoid reaction occur-ring at 948 K: (�-Ti) � (�-Ti) + (V) (Nakano et al. 1980). Above 1155 K, Tiand V are completely miscible in the (�-Ti, V) solid solution. The solubilityof V in the hcp (�-Ti) phase is restricted, with a maximum of 2.7 at.% V.The metastable phases, martensite (�′ or �′′, depending on the V content) and , are

1155 K

948 K

1123 K

1943 K

1878 K

2183 K

L

100 20 30 40 50 60 70 80 90 100

Weight per cent V

2170

1970

1770

1570

1370

1170

970

770

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Ti Atom per cent V V

(β-Ti, V)

(α-Ti)

Figure 1.8. Equilibrium phase diagram for the Ti–V system.




Table 1.9. Special points of the Ti–V system.

Phase reaction Type of reaction Temperature (K) Composition (at.% V)

L� ��Ti�V� Congruent 1878 32L� �Ti Melting 1943 0L� V Melting 2183 100��Ti�V�� Ti�+ �V� Critical 1123 ∼ 50��Ti�� Ti�+ �V� Monotectoid 948 (18) (2.7) (∼ 80)�Ti� �Ti Allotropic 1155 0

also encountered in this system. The special points pertinent to the Ti–V system arelisted in Table 1.9 (Murray 1987, Massalski et al. 1992). Subsequently, an updatehas been published by Okamoto (1993a) with regard to the Ti–V phase diagram.

Figure 1.9 shows the Ti–Cr equilibrium phase diagram. The equilibrium con-densed phases encountered are the liquid (L), the bcc (�-Ti, Cr) solid solution,the hcp (�-Ti) solid solution, the topologically close packed intermetallic phases�-TiCr2, �-TiCr2 and �-TiCr2, and, of course, �-Ti, �-Ti and Cr. In a narrow tem-perature range below the congruent melting temperature, Ti and Cr are completely

100 20 30 40 50 60 70 80 90 100

Weight per cent Cr

2270

2070

1870

1670

1470

1270

1070

870

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Ti Atom per cent Cr Cr

L

1683 K

1943 K

1155 K

940 K

1643 K

~1543 K

~1493 K

~1073 K

(β -Ti, Cr)

(α -Ti)

γ-TiCr2

β -TiCr2

α -TiCr2

2136 K

Figure 1.9. Equilibrium phase diagram for the Ti–Cr system.




Table 1.10. Special points of the Ti–Cr system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Cr)

L� ��Ti�Cr� Congruent 1683±5 44L� �Ti Melting 1943 0L� Cr Melting 2136±20 100��Ti�� TiCr2 Congruent 1643±10 ∼ 66��Ti�� Ti�+�TiCr2 Eutectoid 940±10 (12 5±0 5) (0.6) (∼ 63)��Ti�+�TiCr2 � �TiCr2 Peritectoid ∼ 1493 (39) (∼ 63)(∼ 65)�TiCr2 � �TiCr2 Unknown ∼ 1543 ∼ 65 to 66�TiCr2 � �TiCr2 +Cr Eutectoid ∼ 1073 (∼ 65) (∼ 66) (96)�Ti� �Ti Allotropic 1155 0

miscible in the (�-Ti, Cr) phase. The maximum solubility of Cr in the (�-Ti) phaseis 0.6 at.%. The martensitic �′ and the -phase also form in this system. Thespecial points germane to the Ti-Cr system are presented in Table 1.10 (Murray1987, Massalski et al. 1992).

In the Ti–Al equilibrium phase diagram, (Figure 1.10), the solid phases thatappear are: the bcc (�-Ti) and the hcp (�-Ti) solid solutions, the ordered inter-metallic phases, Ti3Al (also referred to as �2), TiAl (also referred to as �), TiAl,

1970

1770

1570

1370

1170

970

770

Tem

pera

ture

(K

)

1943 K

1155 K

~1558 K

~1398 K

Ti3Al

TiAl

(Al)

938 K933 K

TiAl3

TiAl2

α-TiAl3

L

δ

100 20 30 40 50 60 70 80 90 100

Weight per cent Al

0 10 20 30 40 50 60 70 80 90 100

Ti Atom per cent Al Al

(β-Ti)

(α-Ti)

Figure 1.10. Equilibrium phase diagram for the Ti–Al system.




Table 1.11. Special points of the Ti–Al system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Al)

L� ��Ti�Al� Congruent ∼ 1983 11L + ��Ti�� TiAl Peritectic ∼ 1753 (53) (47.5) (51)L +TiAl� � Peritectic ∼ 1653 (73.5) (69.5) (71.5)L +�� TiAl3 Peritectic ∼ 1623 (80) (72.5) (75)L +TiAl3 � �Al� Peritectic 938 (99.9) (75) (99.3)L� �Ti Melting 1943 0L� Al Melting 933 100��Ti�+TiAl� ��Ti� Peritectoid ∼ 1558 (43) (49) (45)��Ti�� Ti3Al Congruent ∼ 1453 ∼ 32��Ti�� Ti3Al +TiAl Eutectoid ∼ 1398 (40) (39) (48)TiAl +�� TiAl2 Peritectoid 1513 (65) (70) (67)�� TiAl2 +TiAl3 Eutectoid ∼ 1423 (71.5) (68) (75)TiAl3 � �TiAl3 Unknown ∼ 873 75�Ti� �Ti Allotropic 1155 0

� and TiAl3, and the (Al) solid solution. The addition of Al to Ti stabilizes the(�-Ti) phase relative to the (�-Ti) phase. The maximum solubilities of Al in (�-Ti)and (�-Ti) are about 48 and 45 at.%, respectively while that of Ti in Al is around0.7 at.%. The phase boundaries for the TiAl2 and � phases are yet to be ascer-tained. The metastable martensitic �′ phase also forms in the Ti–Al system. Thespecial points of this system are indicated in Table 1.11 (Murray 1987, Massalskiet al. 1992). Two updates (Okamoto 1993b, 1994) pertaining to the Ti–Al phasediagram have appeared later.

The equilibrium phases encountered in the Zr–Nb system are: the liquid (L),bcc (�-Zr, Nb), (�-Zr) and (Nb) solid solutions and the hcp (�-Ti) solid solution.The bcc (�-Zr, Nb) solid solution exhibits a miscibility gap and a monotectoidreaction: (�-Zr) ←−−→ (�-Zr) + (Nb) occurs. The phase diagram (Abriata andBolcich 1982, Massalski et al. 1992) is shown in Figure 1.11 and the special pointspertinent to the system are listed in Table 1.12. The metastable martensite (�′) and-phases form in this system. The Zr–Nb phase diagram has subsequently beenupdated (Okamoto 1992).

The equilibrium Zr–Fe phase diagram (Arias and Abriata 1988, Massalski et al.1992) is shown in Figure 1.12. The equilibrium phases are: the liquid (L); the bccterminal solid solution, (�-Zr), in which the maximum solubility of Fe is about6.5 at.%; the hcp terminal solid solution, (�-Zr), in which Fe has a maximumsolubility of 0.03 at.%; the four intermetallic phases, Zr3Fe, Zr2Fe, ZrFe2 andZrFe3; the high temperature bcc terminal solid solution, (�-Fe), in which Zr hasa maximum solubility of about 4.5 at.%; the fcc terminal solid solution, (�-Fe)




91.0

2128 K

1136 K

0.6

18.5

893 ± 10 K

60.61261 K

21.7

2013 K

2742 K

L

100 20 30 40 50 60 70 80 90 100

Weight per cent Nb

2770

2570

2370

2170

1970

1770

1570

1370

1170

970

770

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Zr Atom per cent Nb Nb

(β-Zr, β-Nb)

(α-Zr)

Figure 1.11. Equilibrium phase diagram for the Zr–Nb system.

Table 1.12. Special points of the Zr–Nb system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Nb)

L� ��Zr� Nb� Congruent 2013 21.7L� �Zr Melting 2128 0L� Nb Melting 2742 100��Zr� Nb�� Zr�+ �Nb� Critical 1261 60.6��Zr�� Zr�+ �Nb) Monotectoid 893±10 (18.8) (0.6) (91.1)�Zr� �Zr Allotropic 1136 0

which shows a maximum solubility of around 0.7 at.% Zr; and the low temperaturebcc terminal solid solution, (�-Fe), in which the maximum solubility of Zr is onlyabout 0.05 at.%. Table 1.13 shows the special points relevant to the Zr–Fe system.Amorphous Zr–Fe alloys have been produced over a wide range of compositionsby rapid solidification processing. The metastable -phase also forms in thissystem. An update of the Zr–Fe equilibrium diagram has appeared later (Okamoto1993c).

The assessed Zr–Sn phase diagram (Abriata et al. 1982, Massalski et al. 1992)is shown in Figure 1.13. In this diagram, there appears to be uncertainty regarding




1009080706050403020100

Weight per cent Fe

100 20 30 40 50 60 70 80 90 100

Zr Atom per cent Fe Fe

270

470

670

870

1070

1270

1470

1670

1870

2070

2270

Tem

pera

ture

(K

)

(α-Zr)

4.00.03

~573 K

1003 K

~6.51201 K

1158 K~24.0

1048 K

1946 K66.7

1755 K

1610 K

1811 K

1667 K90.2 ~99.3

1630 K(γ-Fe)

1198 K 1185 K~99.91043 K

Magnetic trans

ZrFe3

(α-Fe)

Magnetic trans~548 K

1247 K

Lδ Fe

ZrFe2

(β-Zr)

Zr3Fe

Zr2Fe

2128 K

~95.5

Magn

trans

Figure 1.12. Equilibrium phase diagram for the Zr–Fe system.

Table 1.13. Special points of the Zr–Fe system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Fe)

L� �Zr Melting 2128 0L� ��Zr�+Zr2Fe Eutectic 1201 �∼ 24� �∼ 6 5� �31�L� ZrFe2 Congruent 1946 66.7L� ZrFe3 + ��Fe� Eutectic 1610 (90.2) (75) (∼ 99 3)L� �Fe Melting 1811 100L +ZrFe2 � Zr2Fe Peritectic 1247 (∼ 25) (66) (33.3)L +ZrFe2 � ZrFe3 Peritectic 1755 (86.7) (∼ 72 5) (75)��Fe�� L + ��Fe� Catatectic 1630 (∼ 95 5) (90.8) (∼ 99 3)�Zr� �Zr Allotropic 1136 0��Zr�� Zr�+Zr3Fe Eutectoid 1003 (4) (0.03) (24)��Zr�+Zr2Fe� Zr3Fe Peritectoid 1158 (∼ 6) (31) (∼ 25)Zr2Fe� Zr3Fe +ZrFe2 Eutectoid 1048 (33.3) (26.8) (66)ZrFe3 + ��Fe�� Fe� Peritectoid 1198 (75) (?) (∼ 99 95)�Fe� �Fe Allotropic 1667 100�Fe� �Fe Allotropic 1185 100




2128 K

2261 K

1865 K

1600 K

1415 K

1255 K

11.8

17.0 19.1

~505 K505 K

Zr 4

Sn

Zr 5

Sn 3

ZrS

n

L

100 20 30 40 50 60 70 80 90 100

Weight per cent Sn

2470

2270

2070

1870

1670

1470

11361270

1070

870

670

470

Tem

pera

ture

(K

)

0 10 20 30 40 50 60 70 80 90 100

Zr Atom per cent Sn Sn

(β-Zr)

(β-Sn)

(α-Zr)

Figure 1.13. Equilibrium phase diagram for the Zr–Sn system.

Table 1.14. Special points of the Zr–Sn system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Sn)

L� �Zr Melting 2128 0L� ��Zr�+Zr5Sn3 Eutectic 1865 (19.1) (17) (40)L� Zr5Sn3 Congruent 2261 40L� �Sn Melting 505 100L +Zr5Sn3 � ZrSn2 Peritectic 1415 (79) (40) (66.6)��Zr�+Zr5Sn3 � Zr4Sn Peritectoid 1600 (11.8) (40) (20)��Zr�+Zr4Sn� �Zr Peritectoid 1255 (4.9) (20) (7.3)�Zr� �Zr Allotropic 1136 0�Sn� �Sn Allotropic 286 100

most of the liquidus and the entire region between about 30 and 50 at.% Sn.The special points pertaining to the Zr–Sn system are listed in Table 1.14. Themetastable martensitic �′ phase forms in this system.

The equilibrium phases encountered in the phase diagram of the Zr–Al system(Massalski et al. 1992) shown in Figure 1.14 are: the liquid (L); the bcc (�-Zr) andthe hcp (�-Zr) solid solutions, the ten intermetallic phases, Zr3Al, Zr2Al, Zr5Al3,




26

22.5

10080706050403020100

0 10 20 30 40 50 60 70 80 90 100

Zr Atom per cent Al Al

934 K(Al)

870

1070

1270

1470

1670

1870

2070

2270Te

mpe

ratu

re (

K)

Weight per cent Al

L

1853 K1918 K

1863

K

1463

K

1758 K

1803 K1668 K

29.5

37

3949

1753 K

1548 K1623 K

1523 K

1261 K12.5

11.5

1213 K1136

(α-Zr)

(β-Zr)

2128

ZrA

l 3

Zr 3

Al

Zr 2

Al

Zr 3

Al 2

Zr 4

Al 3

Zr 2

Al 3Z

rAl

ZrA

l 2

59

73.5

Zr 5

Al 3

Zr 5

Al 4

Figure 1.14. Equilibrium phase diagram for the Zr–Al system.

Zr3Al2, Zr4Al3, Zr5Al4, ZrAl, Zr2Al3, ZrAl2 and ZrAl3, and the fcc (Al) solidsolution in which the maximum solubility of Zr is about 0.07 at.%. The addition ofAl stabilizes (�-Zr) relative to (�-Zr) and the maximum solubilities of Al in thesetwo phases are about 11.5 and 26 at.%, respectively. The special points of the Zr–Al system are shown in Table 1.15. Subsequently, three updates in respect of theZr–Al phase diagram have appeared (Murray et al. 1992, Okamoto 1993d, 2002).

The equilibrium condensed phases that occur in the binary Ti–N system are: theliquid (L), the terminal bcc solid solution (�-Ti), the terminal hcp solid solution(�-Ti), and the three stable nitride phases, Ti2N, TiN and �′. Both the terminal solidsolutions have wide ranges of composition. The dissolved N (�-stabilizer) extendsthe stability regime of the �-Ti phase to a temperature (2623 K) much abovethe melting point of elemental �-Ti. Two of the nitride phases, Ti2N and �′, arestable over narrow composition ranges while the third, TiN, exhibits stability overan extensive composition range. Figure 1.15 shows the Ti–N equilibrium phasediagram; the special points of this system are listed in Table 1.16 (Massalski et al.1992). An update pertaining to the Ti–N phase diagram has appeared subsequently(Okamoto 1993e).

Figure 1.16 (Zuzek et al. 1990, Massalski et al. 1992) shows the solid phasesencountered in the Zr–H phase diagram. These are the bcc terminal solid solution




Table 1.15. Special points of the Zr–Al system.

Phase reaction Type of reaction Temperature (K) Composition (at.% Al)

L� �Zr Melting 2128 0L� ��Zr�+Zr5Al3 Eutectic 1623 (29.5) (26) (37.5)L +Zr3Al2 � Zr5Al3 Peritectic 1668 (∼ 37) (40) (37.5)L +Zr5Al4 � Zr3Al2 Peritectic 1753 (∼ 39) (44.4) (40)L� Zr5Al4 Congruent 1803 44.4L� Zr5Al4 +Zr2Al3 Eutectic 1758 (49) (44.4) (60)L +ZrAl2 � Zr2Al3 Peritectic 1868 (∼ 59) (66.7) (60)L� ZrAl2 Congruent 1918 66.7L� ZrAl2 +ZrAl3 Eutectic 1763 (73.5) (66.7) (75)L� ZrAl3 Congruent 1853 75L +ZrAl3 � �Al� Peritectic 934 (99.97) (75) (99.93)L� Al Melting 933 100�Zr� �Zr Allotropic 1136 0��Zr�+Zr5Al3 � Zr2Al Peritectoid 1523 (22.5) (37.5) (33.3)��Zr�+Zr2Al� Zr3Al Peritectoid 1261 (12.5) (33.3) (25)��Zr�+Zr3Al� �Zr Peritectoid 1213 (9.2) (25) (11.5)Zr5Al3 � Zr2Al +Zr3Al2 Eutectoid ∼ 1273 (37.5) (33.3) (40)Zr3Al2 +Zr5Al4 � Zr4Al3 Peritectoid ∼ 1303 (40) (44.4) (42.9)Zr5Al4 � Zr4Al3 +ZrAl Eutectoid ∼ 1273 (44.4) (42.9) (50)Zr5Al4 +Zr2Al3 � ZrAl Peritectoid 1548 (44.4) (60) (50)

770

1270

1770

2270

2770

3270

37700 2 4 6 8 10 15 20 25

Weight per cent N

Tem

pera

ture

(K

)

0 5 10 15 20 25 30 35 40 45 50 55

Ti Atom per cent N

1943 K (β-Ti)

(α-Ti )

1155 K

1323 K 33.3 1373 K

30 3323

341073 K

3937.5 δ′Ti2N

2623 K

3563 K47.4

2293 K

15.2

12.56.24.0

L

20.528

TiN

Figure 1.15. Equilibrium phase diagram for the Ti–N system.




Table 1.16. Special points of the Ti–N system.

Phase reaction Type of reaction Temperature (K) Composition (at.% N)

L ←→ �Ti Melting 1943 0L ←→ TiNa Congruent ∼ 3563 47.4L + ��Ti� ←→ ��Ti� Peritectic 2293±25 (4.0) (12.5) (6.2)L +TiN ←→ ��Ti� Peritectic 2623±25 (15.2) (2.8) (20.5)��Ti�+TiN +Ti2N Eutectoid 1323±60 (23) (30) (33)

or PeritectoidTiN ←→ Ti2Nb Congruent ∼ 1373 33.3Ti2N +�′ +TiN Peritectoid 1073±100 (34) (37.5) (39)

(Probably)�Ti ←→ �Ti Allotropic 1155 0

a Observed under pressure >∼1 MPa.b Occurrence if �� Ti�+TiN +Ti2N equilibrium is eutectoid.

0 10 20 30 40 50 60 70 80

Zr Atom per cent H

270

470

670

870

1070

1270

Tem

pera

ture

(K

)

Weight per cent H

(β-Zr)

823 K

5.93 ~37.5 56.7

(α-Zr)

1136 K

δ

ε

Figure 1.16. Equilibrium phase diagram for the Zr–H system.

(�-Zr), which decomposes eutectoidally at 823 K at a H concentration of 37.5 at.%,the hcp terminal solid solution (�-Zr) which exhibits a maximum H solubility of5.9 at.% at 823 K and the hydride phases � (fcc) and � (fct).

The Zr–O phase diagram (Abriata et al. 1986, Massalski et al. 1992) is shown inFigure 1.17. The equilibrium condensed phases are the liquid (L), the bcc terminalsolid solution (�-Zr), the hcp terminal solid solution (�-Zr) and the oxide phases,�-ZrO2−x (cubic, cF12), �-ZrO2−x (tetragonal, tP6) and �-ZrO2−x (monoclinic,mP12). The special points of the Zr–O system are shown in Table 1.17.




0 10 20 30Weight per cent O

0 10 20 30 40 50 60 70

Zr Atom per cent O

470

870

1270

1670

2070

2470

2870

3270T

empe

ratu

re (

K)

(β-Zr)

1136 K

(α-Zr)

(α′-Zr)

(α1″-Zr)(α2″-Zr)

(α3″-Zr)

(α4″-Zr)

P = 1 atmL

10

2128 K

10.5 19.525

2403 K 2338 K35 40 62

2983 K L + G

~2650 K

γ-Z

rO2-

x

β-Z

rO2-

xα-

ZrO

2-x

28.6

29.1

29.8

31.2

66.7

66.7

66.563.6~1798 K

~1478 K

~1243 K

~773 K

2243 K

Figure 1.17. Equilibrium phase diagram for the Zr–O system.

Table 1.17. Special points of the Zr–O system.

Phase reaction Type of reaction Temperature (K) Composition (at.% O)

L ←→ �Zr Melting 2128 0

L ←→ ��Zr� Congruent 2403±10 25±1

L ←→ �ZrO2−x Congruent 2983±15 66.6

L ←→��Zr�+�ZrO2−x

Eutectic 2338±5 �40±2��35±1��62±1�

L + ��Zr� ←→ ��Zr� Peritectic 2243±10 �10±0 5��19 5±2��10 5±0 5�

L +�ZrO2−x +G 2983 �∼ 66 6� �∼ 66 6� �∼ 100�

�ZrO2−x ←→ ��Zr�+�ZrO2−x

Eutectoid ∼ 1798 �63 6±0 4� �31 2±0 5��66 5±0 1�

�ZrO2−x ←→ ��Zr�+�ZrO2−x

∼ 1478 �∼ 66 5� �29 8±0 5� �∼ 66 5�

�ZrO2−x +�ZrO2−x +G ∼ 2650 �∼ 66 6� �∼ 66 6� �∼ 100�

�ZrO2−x +�ZrO2−x +G ∼ 1478 �∼ 66 6� �∼ 66 6� �∼ 100�

�ZrO2−x ←→ �ZrO2−x Congruent ∼ 2650 66.6

�ZrO2−x ←→ �ZrO2−x Congruent ∼ 1478 66.6

�Zr ←→ �Zr Allotropic 1136 0




1.6 NON-EQUILIBRIUM PHASES

1.6.1 Introductory remarksPhases such as the �-, �- and intermetallic phases mentioned earlier are equilibriumphases and the corresponding phase fields are delineated in equilibrium phasediagrams of the type described in the previous section. However, non-equilibriumor metastable phases, as distinct from equilibrium phases, are quite important inrespect of many alloy systems, including those based on Ti and Zr. Equilibriumphase diagrams are usually developed by deducing the initial states of alloyswhich have been quenched from different temperatures to room temperature. Butthe quenching process may lead to the formation of non-equilibrium phases. Twoimportant examples of such non-equilibrium phases in Ti–X and Zr–X systems arethe martensite and the athermal -phases. Both these phases are formed throughathermal displacive transformations.

It will be seen in a later chapter that one way of classifying phase changes is todivide them into two broad classes: reconstructive and displacive (Roy 1973,Christian 1979, Banerjee 1994). Transformations of the former kind involve break-ing of the bonds of atoms with their neighbours and re-establishment of bonds toform a new configuration in place of the pre-existing one. Such a process requiresatomic diffusion comprising random atomic jumps and disturbs atomic coordina-tion. Atomic movements in displacive transformations, on the other hand, can bebrought about by a homogeneous distortion, by shuffling of lattice planes, by staticdisplacement waves or by a combination of these. Cooperative movements of alarge number of atoms in a diffusionless process accomplish the structural changein displacive transformations. Unlike the diffusional atomic jumps which are ther-mally activated, the displacive movements do not require thermal activation andcannot, therefore, be suppressed by quenching. A structural transition involving peri-odic displacements of atoms from their original positions can be described in termsof a displacement wave and the introduction of a displacement wave in the par-ent lattice requires coordinated atom movements in an athermal process; the ather-mal martensitic and -transformations can, respectively, be described in terms oflong wavelength and short wavelength displacement waves (Banerjee et al. 1997).

In the present chapter brief accounts of the martensite and the -phases and ofphase separation in the �-phase will be provided with reference to Ti–X and Zr–Xalloys. A detailed coverage in respect of the same will be found in three of thesubsequent chapters.

The martensitic transformation, which is diffusionless and involves coopera-tive atom movements, proceeds by the propagation of a shear front at a speedthat approaches the speed of sound in the material, leading to the formationof the metastable martensite phase. This transformation occurs in many alloy




systems, including Ti–X and Zr–X systems, in which the major component exhibitsallotropy. The -phase, which is an equilibrium phase in Group 4 metals (Ti, Zr,Hf) at high pressures, forms in several alloys based on these metals and also inmany other bcc alloys at ambient pressure as a metastable phase.

On rapidly quenching Ti–X and Zr–X alloys, X being an �-stabilizing element,from the �-phase field, the martensite phase, �m, which has the hcp structure, isobtained. The situation is somewhat different when X is a �-stabilizing element,such as a transition metal. During the process of rapid cooling from the �-phasefield, when a composition-dependent temperature (known as the martensitie start orMs temperature) is crossed, the bcc �-phase commences to transform spontaneouslyby the martensitic mode to the martensite phase �m whose structure may be hcp(�′) or orthorhombic (�′′), depending on the alloy composition. However, in thecase of these alloys, another athermal process, namely, that associated with theformation of the athermal -phase, competes with the martensitic process. At anytemperature compatible with the formation of both �m and -phases, there is anarrow range of composition (or electron to atom ratio), just beyond the martensiteformation regime, over which the athermal -phase forms from the parent �-phase.If a s temperature, akin to the Ms temperature, is conceived as being associatedwith the start of athermal -phase formation, then one may visualize that thes locus lies above the Ms locus in the narrow composition range referred toabove, if temperature is plotted against composition. In the composition regimeof martensite formation, which lies to the left of this narrow range, the Ms locuslies above the s locus. Even though the -phase appears athermally on rapidquenching from the �-phase field only over a narrow range of electron to atomratio, this phase occurs over a broader composition range as a precipitation productof �-phase decomposition. The typical structures exhibited by rapidly �-quenchedbinary Ti–X or Zr–X alloys, X being a �-stabilizing element, are indicated in theschematic shown in Figure 1.18. Beyond the �+ region (where these two phasescoexist), the �-phase is retained in a metastable (susceptible to decomposition onageing) or stable manner on quenching.

It may be noted that similar values of the electron to atom ratio (∼ 4 15)characterize the limit of the stability of the bcc �-phase with respect to either ofthe two athermal transformations (Collings 1984).

1.6.2 Martensite phase1.6.2.1 CrystallographyThe phenomenological crystallographic theories of the martensitic transformationare based on the concept that the interface between the martensite and the parentphases is macroscopically invariant. The central theme of these theories is that thetotal macroscopic shear consists of three components: (a) the lattice shear or the




M s

ωs

Tem

pera

ture

Concentration of X

I III IVII

Figure 1.18. Schematic showing the Ms and s loci for a binary Ti–X or Zr–X system, X being a�-stabilizing element, on rapid cooling from the �-phase field. Region I corresponds to martensite(�m) formation; in regions II and III, the �-phase co-exists with the athermal - and the aged-phases, respectively; in region IV only the �-phase occurs in a metastable or stable state.

Bain strain which brings about the necessary change in the lattice (e.g. bcc to hcp);(b) a lattice invariant inhomogeneous shear which provides an undistorted plane;and (c) a rigid body rotation to ensure that the undistorted habit plane is unrotatedas well. The inhomogeneous shear accompanying the martensitic transformation isinstrumental in generating the martensite substructure which, in most cases, is toofine to be resolved under the light microscope. Transmission electron microscopy(TEM) techniques have been extensively used for resolving this substructure andfor obtaining information regarding the orientation relationship, the habit planeand the nature of the inhomogeneous strain for individual martensite crystals.

A unique feature of the � → �′ martensitic transformation in Ti and Zris that the necessary lattice strains approximately satisfy the invariant planestrain condition. Because of this, the magnitude of the lattice invariant shear iscomparatively small and it is relatively simple to characterize the substructure ofthe martensite in Ti–X and Zr–X alloys.

There are a number of choices for relating the lattices of the parent (�) andproduct (�′) phases. The correct choice of lattice correspondence is generallymade by selecting the one which involves the minimum distortion and rotation ofthe lattice vectors. In the case of the transformation in Zr, it has been suggested(Burgers 1934) that the �011�� plane forms the basal plane �0001��′ , while theclose packed 111�� and 111�� directions lying on that plane correspond to theclose packed �1120��′ directions. This accounts for four of the six �1120��′directions; the remaining two are derived from the 100�� and 100�� directions.




Having chosen this lattice correspondence, the next step would be to determine themagnitudes of the strains which would deform the distorted hexagonal structureinto a regular one, having lattice parameters consistent with those of �-Zr.

If ao� a and c refer to the lattice parameters of the �- and �-phases, respectively,the magnitudes of two of the principal lattice distortions, �1 (along 100��) and �2

(along 011��) are given by �1 = a/ao and �2 = �3/2�12 a/ao. The distortion �3 along

011�� is �1/2�1/2��a/ao� where � = c/a. On substituting the values for the latticeparameters at the transformation temperature, the magnitudes of the principal strainsfor pure Zr are seen to be as follows: 2% expansion along 011��, 10% expansionalong 011�� and 10% contraction along 100��. The situation is analogous in the caseof Ti and the corresponding principal strains are 1% expansion, 11% expansion and11% contraction, respectively, along the aforementioned directions.

A pair of planes remains undistorted under the action of a homogeneous latticestrain if and only if one of the principal strains is zero and the other two are ofopposite signs (Wayman 1964). A special feature of the martensitic bcc to hcptransformation in Ti and Zr is that the principal strain along the 011�� directionis very small and the other two principal strains are of opposite signs. If theprincipal strain along the 011�� direction, �3, were zero, the lattice shear wouldhave left a plane undistorted. Since �3 is very small in the case of Ti and Zr, itis not unreasonable to treat the transformation with the approximation that �3 iszero (Kelly and Groves 1970).

It has been reported (Bagaryatskii et al. 1959, Flower et al. 1982) that the nor-mally observed hcp (�′) structure of the martensite is distorted to an orthorhombicstructure (�′′) in many Ti–X systems, X being a transition metal, when the marten-site is supersaturated beyond a certain limit. The orthorhombic distortion increaseswith increasing solute content. It has been noticed that the deformation inducedmartensite, mentioned later in this section, almost invariably has an orthorhom-bic structure (Williams 1973). This is not surprising when one considers the factthat this type of martensite can occur only in alloys which are so enriched in�-stabilizing solutes that they are not transformed on �-quenching.

It has been demonstrated (Otte 1970) that the � → �′ transformationinvolves the activation of the shear systems �112��111�� ≡ �2112��′2113��′ and�101��111�� ≡ �1011��′ [2113��′ . The habit plane associated with this transfor-mation has been found to be very close to �334�� (Williams 1973, Shibata andOno 1977), although in some �-stabilizing solute enriched Ti–X alloys �344��habit has also been reported (Liu 1956, Gaunt and Christian 1959, Hammond andKelly 1970). The orientation relationship between the �- and �′-phases has beenobserved to be: �011�� 0002��′ ; < 111 >�< 1120 >�′ (approximately), whichis consistent with the approximate orientation relation deduced by Burgers withregard to the bcc → hcp transformation in elemental Zr (Burgers 1934).




In the case of the orthorhombic �′′ martensite, the orientation relationship hasbeen reported to be: 100�′′� and 010��′′ inclined by about 2o from �001�� and< 110 >�, respectively; 001�′′� < 110 >� (Hatt and Rivlin 1968).

1.6.2.2 Transformation temperaturesThe martensitic transformation is characterized phenomenologically by the assign-ment of several temperatures. The most common among these are: Ms, the temper-ature at which martensite starts forming during quenching; Mf , the temperature atwhich the transformation is completed; �s, the temperature at which the �m → �reverse transformation starts during up-quenching (in Ti and Zr alloys); and To,the temperature at which the free energies of the parent and martensite (� and �m)phases are equal. If Ms and �s are very close to each other, it is indicated that thedriving force for the transformation is small and also that To can be taken to bethe mean of these two temperatures (Collings 1984).

A thermodynamic analysis (Kaufman 1959) of the �→� transformation in sev-eral Ti and Zr base alloys has shown that the To temperatures are about 50 K higherthan the experimentally observed Ms temperatures. This implies that the supercool-ing (To–Ms), necessary to initiate the martensitic transformation in these systemsis relatively low. The change in free energy accompanying the transformation atthe Ms temperature is significantly lower as compared to that in ferrous systems.

At the Ms temperature, the chemical driving force necessary to start a martensiticreaction depends on the shear modulus of the alloy at the transformation tempera-ture, the magnitude of the homogeneous shear associated with the transformationand the magnitude of the inhomogeneous shear. The strain energy associated withmartensite formation is determined by the homogeneous lattice strains and theshear modulus while the surface energy corresponds to the energy of the interfacebetween the parent and the product lattices.

If the austenite–martensite reaction in ferrous systems is compared with the� → �′ transformation in Ti and Zr base alloys, it is found that although there isnot much difference in the homogeneous strain values in the two cases, the shearmodulus of ferrous alloys at the transformation temperature is much higher. Again,the energy associated with the parent–martensite interface can also be expectedto be much smaller in the case of Ti- and Zr-based alloys because only a smallamount of inhomogeneous shear is necessary to make the total strain an invariantplane strain in the case of the �→�′ transformation. These considerations indicatethat the “back stress”, which arises from the strain and surface energies opposingmartensite formation, is much smaller in Ti- and Zr-based alloys as compared toferrous alloys. This explains why a small driving force is adequate for initiatingmartensite formation in the former. The chemical driving force which balances




the “back stress” at the Ms temperature can be assisted by external stress, leadingto stress-assisted or stress-induced or deformation-induced martensite formation.

The Ms temperature is composition dependent. Again, the measured Ms tem-perature for a given alloy composition may exhibit a dependence on the rate ofcooling (Jepson et al. 1970). In �-stabilized Ti–X and Zr–X alloys, Ms increaseswith increasing solute content and may lie a little below the ��+��/� transus; in�-stabilized alloys, Ms decreases with increasing concentration of X and alwayslies in the (�+�) field (Collings 1984). In dilute alloys of the latter type, the Ms

temperature is relatively high and water quenching may not be sufficiently rapidfor completely suppressing thermally activated atom movements, leading to somesegregation of the solute atoms prior to the transformation and consequently tothe retention of some �-phase. If the solute content in the alloy increases, theMs temperature decreases and the diffusional contribution is inhibited, with theresult that a full transformation to the martensite phase comes about. It may bementioned here that the quench rates necessary to achieve the structural trans-formation while preserving compositional homogeneity depend strongly on thenature of the alloying element, X, or more specifically, on its diffusion kinetics inthe �-phase. A quench rate that is adequate when X is an early transition metalsuch as V, Nb or Mo, may not be so when X is a late transition metal like Fe,Co or Ni. This is so because metals belonging to the latter category diffuse muchfaster in the �-phase; for instance, in the context of diffusion in �-Ti at 1273 K,it may be noted that the diffusion coefficients of Co and Mo are in the ratio200:1 (Collings 1984). As the solute concentration increases further, a stage isreached where the Mf temperature drops below the temperature of the quenchingbath; in this situation, the retention of some untransformed �-phase again becomesfeasible.

1.6.2.3 Morphology and substructureIf dilute Ti–X and Zr–X alloys are quenched from the �-phase field, maintainingan adequately fast cooling rate, one generally obtains a hcp martensite phase(�′) which is known as lath or packet or massive martensite and consists ofrelatively large, irregular packets or “colonies” which are populated by near-parallel arrays of much finer platelets or laths. No retention of the �-phase occursin a lath martensite. As the solute content increases, the average packet size andthe average lath size decrease. Beyond a certain level of solute concentration,which depends on the nature of the solute, a transition occurs in the martensitemorphology, resulting in the formation of plate or acicular martensite. In contrast tothe arrangement of near-parallel units in the lath morphology, the martensite unitsform in various intersecting directions in the plate or acicular structure. Anotherimportant difference between the lath and the plate morphologies is that the sizedistribution in the latter case is much broader than in the former. This is essentially




due to the fact that in the plate morphology the martensite units continuouslypartition the parent �-grain and as a result of this the space available for the growthof the plates belonging to the subsequent generations gets more and more limited.The transition from lath to plate morphology is not abrupt and the two may coexistover some range of composition. When the solute concentration is sufficientlyhigh, the martensitic transformation may be incomplete and some �-phase, whichis usually trapped between the platelets of the acicular martensite, may be retained.Not far removed from the “� plus acicular martensite” quenched structure is theWidmanstatten arrangement consisting of groups of �-phase needles lying withtheir long axes parallel to the �110� planes of the retained �-phase.

The term “substructure” of a martensite generally refers to the structure withinthe martensite unit as revealed under the transmission electron microscope (TEM).This substructure arises from (a) the lattice invariant component of the transfor-mation strain which may be slip, twin or a combination of both; and (b) thepost-transformation strain resulting from the accommodation effect. There hasbeen considerable interest in characterizing the internal structure of martensiteplates for determining the nature of the inhomogeneous shear participating inthe transformation process, as envisaged in the phenomenological theory of themartensitic transformation. For this it is necessary to be able to identify and sepa-rate the inhomogeneities introduced by matrix constraints from those produced bythe lattice invariant component of the transformation strain. Such a separation isnot straightforward. In a twinned martensite plate, a set of transformation twins isexpected to appear periodically at almost equal intervals within the plate; the ratioof the thicknesses of the twinned and the matrix portions should be consistent withthe value predicted by the theory and the specific variant of the twin plane shouldbe consistent with the observed habit plane. When a set of twins in a martensiteplate satisfies all these conditions, the twins are taken to be transformation twins.In a dislocated martensite crystal, it is more difficult to separate the transformationinduced dislocations from those introduced by post-transformation stresses. A ruleof the thumb appears to be that only those dislocations which are arranged in regu-lar arrays and are observed very frequently may be taken to have been produced bythe inhomogeneous shear. Generally, a transition from the dislocated to the twinnedsubstructure is found to occur with increasing concentration of alloying elements.

1.6.3 Omega phase1.6.3.1 Athermal and isothermal �It has been mentioned earlier that under ambient pressure, the -phase can occur ina metastable manner in alloys in which the �-phase is stabilized with respect to themartensitic � → �m transformation. The composition range over which this phasemay be encountered is a characteristic of the alloy system under consideration. Ithas also been indicated that this phase can be obtained either by rapidly quenching




from the �-phase field (athermal ) or as a product of thermally activated �-phasedecomposition (isothermal or aged ).

The athermal � → transformation is displacive, diffusionless and of thefirst-order and the -phase so obtained has a composition very close to thatof the �-phase. The thermally activated transformation, on the other hand, isaccompanied by solute rejection by diffusional processes from the to the �-phaseand is thus partially replacive in nature. The athermal � → transformationcannot be suppressed even by extremely rapid quenching and is completely andcontinuously reversible with negligible hysteresis. The special characteristics ofthis transformation also include the appearance of an extensive diffuse intensitydistribution in diffraction patterns, with the maximum intensity located close tothe positions of ideal -reflections, as a precursor to the transformation event andthe stability of the dual phase �+ structure with extremely fine (∼1–4 nm) particles distributed in the �-matrix along �111�� directions (Banerjee et al. 1997).The number density of the particles is extremely large and this fact lends supportto the contention that the transformation does not involve long range diffusion.

The volume fraction of the isothermal -phase forming in the �-matrix is afunction of the reaction time. This dependence of the volume fraction on time arisesessentially due to the diffusion controlled partitioning of the solute between solutelean and solute rich � regions. The solute lean regions are eventually transformedto the -phase. The composition of the isothermal -phase corresponds to themaximum solubility of the solute in the -phase. Thus after prolonged ageingat temperatures lower than about 770 K, a metastable + � state is attained,characterized at a given temperature by a fixed volume fraction and compositionof each of the and � terminal points (Hickman 1969). After sufficiently longageing periods at 720–770 K, �-phase precipitation can be expected.

An early model of isothermal -phase development visualized an initial struc-tural transformation of the lattice into and � (as in the case of the athermal-phase in its pertinent composition regime), followed by an exchange of soluteand solvent atoms across the /� interface (Courtney and Wulff 1969). It hassubsequently been suggested that initially a composition fluctuation occurs andthis is followed by a structural � → transformation within a solute lean zone,triggered by a longitudinal phonon with a 2/3 �111� wave vector; it is the insta-bility of the bcc lattice with respect to this disturbance that is responsible for theathermal transition (de Fontaine et al. 1971).

1.6.3.2 CrystallographyThe crystal structure of the -phase has been described in an earlier section. Theorientation relationship between the � and -phases has been determined by a largenumber of investigators and has been unanimously accepted as: �111�� 0001�;




�110�� 1210�. It has been found that this orientation relationship is validboth for the athermal -phase and the isothermal -phase (Williams 1978). Thisrelationship implies that there are four possible crystallographic variants of the-structure, depending on which one of the �111�� planes is parallel to the (0001)plane. Again, for the same variant of the -structure, there are three �110� direc-tions so that in all 12 variants of the -structure are possible. But since the basalplane of this structure has six-fold symmetry, the three variants for a given �111��plane will appear identical and, therefore, the contribution from only four variantswill be seen in selected area diffraction (SAD) patterns. The lattice parameters, a

and c of the structure and a� of the � (bcc) structure, are related as follows:

a = √2a�� c = �

√3/2�a�

1.6.3.3 MorphologyPrecipitates of the athermal -phase that evolve during rapid �-quenching are veryfine (<5 nm) and it is difficult to assign any well-developed geometrical shape tothese particles which have a tendency to be aligned along �111�� directions. Theshapes of isothermal -phase precipitates are more readily discernible and gener-ally two types of morphologies, ellipsoidal and cubic, are encountered, dependingon the linear lattice misfit, �V −V��/3V�, where V represents the unit cell volumedivided by the number of atoms in the unit cell (Hickman 1969). If this misfitis small (<0 5%), as is often the case when the solute is a 4d-transition metalsuch as Nb or Mo, the precipitate morphology is dominated by surface energyconsiderations, leading to an ellipsoidal shape (Hickman 1969). If the misfit islarge (>1%), as is generally the case when the solute is a 3d-transition metal likeV, Cr, Mn or Fe, the minimization of elastic strains in the cubic matrix dictates acubic morphology (Hickman 1969, Blackburn 1970).

1.6.3.4 Diffraction effectsPronounced diffuse scattering has been observed in electron, X-ray and neutrondiffraction patterns prior to the formation of the -phase in all -forming systems.These diffuse intensity patterns are closely associated with the non-diffuse (sharp)reflections corresponding to the crystalline -phase. In view of this close associa-tion, the diffuse intensity distribution has been attributed to non-ideal -structures.It has been mentioned earlier that the ideal hexagonal -structure is obtained whenthe parameter z has the value zero and the non-ideal trigonal -structure resultsif 0 < z < 1/6. Selected area electron diffraction patterns obtained from the trulyathermal -phase are characterized by sharp spots and straight lines of intensitywhile broad reflections and either straight or curved diffuse lines of intensity




(diffuse streaking) originate from the “diffuse” -phase (Collings 1984). A modelproposed by Sass and co-workers (Dawson and Sass 1970, McCabe and Sass 1971,Balcerzak and Sass 1972) envisages an ensemble of -particles, 1–1.5 nm in diam-eter and 1.5–2.5 nm apart, arranged in rows along �111�� directions. According tothis model, clusters of such rows contribute to the sharp spots and straight linesof intensity, while the broad reflections and diffuse streaking arise from eitherindividual rows of particles or isolated particles.

It has been demonstrated that a transition from diffuse to sharp -reflectionsoccurs in �-quenched specimens in response to either decreasing solute content(Sass 1972) or decreasing temperature (de Fontaine et al. 1971). In both cases,curvilinear lines of diffuse intensity become straight and well defined. It has beenpointed out (Williams 1973) that since the diffuse streaking tends to coincide withthe positions of the -reflections when they are present, compositionwise there isno sharp line of demarcation separating the regions of athermal and diffuse .

A soft phonon mechanistic model of the -phase reaction (de Fontaine et al.1971) has been able to provide a rationalization, in terms of lattice dynamics,for the temperature and composition dependences of the athermal and diffuse-phases. After examining electron diffraction patterns belonging to several zonesand considering the symmetry of the reciprocal lattices, de Fontaine et al. (1971)have constructed a three-dimensional model of the diffuse intensity which isdistributed on quasi-spherical surfaces centred around the octahedral sites of thereciprocal of the bcc �-lattice. These spheres of intensity touch all the �111� facesof the octahedra surrounding them. When this intensity distribution in the recipro-cal space is sectioned to reveal the diffuse intensity pattern in a plane correspondingto any zone, the pertinent shifts of the diffuse intensity maxima from the positionsof ideal -reflections and asymmetry in intensity distribution are manifested.

The lattice dynamical model for phase stability, with special reference to thequenched -phase, has been further developed by Cook (1975). It does appear thatthe 2

3�111� soft mode, interacting with a lattice of composition and temperaturedependent relative stability, is responsible not only for athermal but also fordiffuse , which represents in varying degrees, dynamical fluctuations betweenthe �- and -phases (Collings 1984).

1.6.4 Phase separation in �-phaseBelow a transus representing the upper boundary of a region in the equilibriumphase diagram known as a miscibility gap, a previously homogeneous single phase� solid solution decomposes into a thermodynamically stable aggregate of twobcc phases, one solute lean and the other solute rich, designated respectively asthe �1- and �2-phases. Two ideal examples of systems where such a � → �1 +�2

decomposition occurs are the Zr–Nb and Zr–Ta systems, both of which exhibit




the monotectoid reaction �1 → �+�2. The equilibrium phase diagrams for thesesystems have many points of similarity with those of Zr–X and Ti–X (X being atransition metal) eutectoid systems; in the case of the latter, Ti is replaced by Zr andthe intermetallic phase TimXn replaced by �2. A bimodal free energy (G) versussolute concentration (X) curve is associated with either of the Zr–Nb and Zr–Tasystems, representing the occurrence, in equilibrium, of two � solid solutions.This situation is somewhat different from that representing the coexistence of say,the �-and �-phases in equilibrium. While the latter situation is described in termsof two independent free energy parabolas, equilibrium phase separation, with theabsence of structural change, has to be described by a continuous curve with twominima, separated by an intervening maximum, for which the second derivativeof free energy with respect to solute concentration is negative.

Although above the monotectoid temperature, both �1 and �2 are equilibriumphases, the former ceases to be an equilibrium phase below this temperature. How-ever, the �1-phase has been found to occur in a metastable manner at temperaturesclose to but lower than the monotectoid temperature in the Zr–Nb (Banerjee et al.1976, Menon et al. 1978) as well as the Zr–Ta (Mukhopadhyay et al. 1978, Menonet al. 1979) systems.

Phase separation in the �-phase has also been reported in some Ti–X systems(X: Cr, V, Mo, Nb). In situations where the temperature (Williams et al. 1971) orthe solute concentration (Williams 1973) is too high to be conducive for -phaseprecipitation, a solute lean bcc phase, designated as �′ separates from the �-phase.The � → �′ +� phase separation reaction can be considered to be a clusteringreaction characteristic of alloy systems which exhibit positive heats of mixing(Chandrasekaran et al. 1972) or similar manifestations of a tendency for the alloy-ing constituents to unmix. It is interesting to note that �-stabilizing elements suchas Al, Sn and O, when added to Ti–V and Ti–Mo alloys in sufficient quantities,appear to increase the stability of the bcc lattice in that -phase formation issuppressed in favour of �-phase separation (Williams 1971). Thus these solutes,which are certainly not �-stabilizers in the conventional sense, can be regarded asstabilizers of the bcc lattice against the instability. Ageing in the �′ +� phasefield, which lies just outside the + � phase field, would eventually result in�′-nucleated �-phase precipitation.

1.7 INTERMETALLIC PHASES

1.7.1 Introductory remarksA large number of intermetallic phases are encountered in binary Ti–X and Zr–Xsystems and these exhibit a variety of crystal structures. Many of these structures




are derived from three simple crystal structures, namely, face centered cubic(fcc, Al), body centered cubic (bcc, A2) and hexagonal close packed (hcp, A3)structures, which are commonly associated with pure metals and disordered solidsolutions.

It may be noted that the formation of intermetallic phases does not appear tooccur in these binary systems when X is an alkali or alkaline earth metal (with theexception of Be) or a transition metal belonging to Group 3 or Group 4. Likewise,no intermetallic phases form when X is a rare earth (RE) element. Ti–RE andZr–RE phase diagrams are normally characterized by the absence of intermetallicphases, limited mutual solubilities in the solid state, and quite often, a miscibilitygap in the liquid state. In general, in the intermetallic phase ZrmXn, X is a transitionmetal from Group 5 (only V) to Group 10 or a simple (non-transition) metal fromGroup 11 to Group 16. In a similar manner in the intermetallic phase, TimXn, X isa transition metal from Group 6 (only Cr) to Group 10 or a non-transition metalfrom Group 11 to Group 16. In both TimXn and ZrmXn families, X is more often anon-transition metal than a transition metal. However, this does not imply that theincidence of X being a transition metal is infrequent: close to a hundred binaryintermetallic phases of this type have been reported.

Almost all the important binary intermetallic phases that have been observed inTi–X and Zr–X alloys, together with hydrides, borides, carbides, nitrides, oxides,phosphides and sulphides have been listed in Tables A1.1 and A1.2, respectively.The composition range, space group, Pearson symbol and strukturbericht designa-tion associated with each of these phases have been incorporated in these tables.The nomenclatures of crystal structures, in terms of their strukturbericht designa-tions and the corresponding Pearson symbols, are listed in Table A1.3 for readyreference.

A survey of Table A1.1 and Table A1.2 shows that more than 80% of TimXn

and ZrmXn type intermetallic phases are almost equally distributed among threecrystal classes: cubic, tetragonal and hexagonal. The orthorhombic system comesnext (∼15%) while less than 5% belong to the rhombohedral and monoclinicsystems. The unit cells of a majority (∼62%) of these phases are of the primitivetype, followed by body centred (∼21%), face centred (∼10%) and base centred(∼7%) cells. The occurrence of body centred unit cells is most common in tetrag-onal phases, of face centred cells in cubic phases and of base centred cells inorthorhombic phases. The more frequently encountered structures in binary Ti–Xintermetallics are the B2 (cP2, CsCl type), C11b (tI16, MoSi2 type), L12 (cP4,AuCu3 type), A15 (cP8, Cr3Si type), L1o (tP4, AuCu type) and D019 (hP8, Ni3Sntype) structures. In the case of Zr–X intermetallics, these are the B2, C11b, C15(cF24, Cu2Mg type), D88 (hP16, Mn5Si3 type), C16 (tI12, Al2Cu type), C14 (hP12,MgZn2 type) and Bf (oC8, CrB type) structures.




Generally speaking, there are many intermetallics that can be put to a vari-ety of uses. For example, there has been a considerable interest in developingstrong alloys based on intermetallic phases for structural applications. However,such intermetallics are normally brittle and for this reason, their processing andapplication are difficult. But it has to be pointed out here that as a class of mate-rials intermetallics, in which atomic bonding is at least partly metallic, tend to beless brittle than ceramics, where atomic bonding is mainly covalent or ionic innature. Broadly speaking, alloys based on intermetallic phases are hard to deformplastically as compared to pure metals or disordered alloys because of their strongeratomic bonding and the resulting ordered distribution of atoms which gives riseto relatively complex crystal structures. The brittleness of intermetallics generallyappears to decrease with increasing crystal symmetry and decreasing unit cell size.In view of this, intermetallics with relatively high crystal symmetry (e.g. cubic,such as B2, D03, L12, or nearly cubic, such as L1o, D022, where a slight tetragonaldistortion is present) are thought to have good potential for structural applications(Sauthoff 1996). In the context of Ti–X and Zr–X systems, the intermetallics thathave been considered for structural applications include Ti3Al�D019�, TiAl�L1o�,TiAl3�D022� and Zr3Al�L12�. Another potential application area pertains to hydro-gen storage: certain Ti and Zr bearing Laves phase intermetallics show promisewith regard to applications as hydrogen storage materials (Sauthoff 1996).

1.7.2 Intermetallic phase structures: atomic layer stackingThe structures of many intermetallic phases can be considered to be formedby the sequential stacking of certain polygonal nets of atoms. These structuralcharacteristics can be readily described by using specific codes and symbols, whichcan be very useful for a compact presentation and comparison of the structuralfeatures of different materials. Various notations have been devised for describingthe stacking patterns (Pearson 1972, Ferro and Saccone 1996). Without goinginto details of these, only the Schlafli notation, PN , will be introduced here. Inthis notation, PN describes the characteristics of each node in the network in thefollowing manner: the superscript N is the number of P-gon polygons surroundingthe node. Thus P = 3 corresponds to a triangle, P = 4 to a rectangle or a square,P = 5 to a pentagon, P = 6 to a hexagon and so on. Some of the very commonlyoccurring nets are 36 (triangular, T net), 44 (square, S net), 63 (hexagonal, H net)and 3636 (kagome net or K net). These four types of nets are shown schematicallyin Figure 1.19(a). If a network has nodes which are not equivalent in terms ofthe polygons surrounding them, the net can be described by listing successivelythe different corners. For example, if a net is described as 32434+3342 (2:1), theimplication is that in this net there are two types of nodes, 32434 and 3342, and thatthey occur with a relative frequency of 2:1 (Figure 1.19(b)). A node of the first




T-net S-net

K-netN-net

(a)

32434 + 3342(2:1)

(b)

Figure 1.19. (a) Schematic representation of a 36 (triangular) T net, a 44 (square) S net, a 63

(hexagonal) H net and a 3636 (kagome) K net of points. (b) The net shown is more complex andcontains two types of nodes. This net can be described by the notation 32434 + 3342 (2:1). Theimplication is that these two types of nodes occur with a relative frequency of 2:1. A node of thefirst type (32434) is surrounded, in the given order, by two triangles, one square, one triangle andone square, while a node of the second type (3342) is surrounded by three triangles and two squares.

type is surrounded, in the given order, by two triangles, one square, one triangleand one square while a node of the second type is surrounded by three trianglesand two squares.

A close packed layer of atoms forms a 36 net composed of equilateral triangles.However, not all 36 nets of atoms correspond to close packed layers. To cite anexample, the triangles of 36 nets of bcc �110� layers are not equilateral but haveangles of 55o, 55o and 70o approximately. In the case of close packing (i.e. a 36

net comprising equilateral triangles), the nodes of one net lie over the centres of




the triangles of the nets immediately above and below. Such a situation is notobtained in respect of the stacking of the triangles of the 36 nets of bcc �110�layers (Pearson 1972).

The morphologically triangular, 36 (close packed), hexagonal, 63, and kagome,3636, nets, together with those made up of squares, are of frequent structuraloccurrence. A 36 net can be subdivided into a 63 and a larger 36 net (the ratio ofnumber of sites being 2:1) or into a kagome net and a larger 36 net (the ratio ofnumber of sites being 3:1) (Pearson 1972). A primary classification of structuresin terms of the stacking of (nearly) planar layers of atoms is quite instructive. Forexample, numerous structures can be formed by the stacking of T, H or K (triangle–hexagon) layer nets of atoms one over the other sequentially. It is a characteristicof each such layer that it can be positioned about one of three equivalent sites, A,B and C, and this leads to the possibility of varying the stacking sequence and/orthe succession of the net types. Moreover, planes of atoms can be constituted ofa combination of different layer networks (e.g. hexagonal plus triangular), eachof which is occupied by a different chemical species. It is possible to derive avery large number of structure types by permuting the stacking and net sequences.Geometrically close packed (GCP) structures are obtained when the permutationinvolves only the stacking sequences of the equilateral triangular net. The numberof possible structure types may be further increased by chemically ordering thecomponent atoms on the triangular nets.

Other structures may be generated by stacking together layer networks ofatoms comprising only squares or squares along with triangles, pentagons and/orhexagons. The squares, pentagons or hexagons of one net may or may not becentred by atoms of nets above and below (Pearson 1972).

It is interesting to see how several frequently encountered structures in respectof Ti–X and Zr–X intermetallics can be described in terms of the stacking ofdifferent types of layer networks of atoms. Before that a brief introduction totopologically close packed (TCP) structures will be provided in view of the factthat quite a few of these intermetallic phases have such structures.

Octahedral and tetrahedral voids (Figure 1.20) are the two most common types ofinterstitial voids present in the simple spherically close packed (i.e. GCP) metallicstructures (fcc and hcp). The former are larger and are surrounded by six atomswhich form the corners of a triangular antiprism (octahedron). The latter, which aresmaller, are enclosed by four atoms which are tetrahedrally disposed. The primitiveunit cell of the hcp structure contains two atoms with coordinates (000) and ( 2

313

12 ).

There are thus two atoms associated with each lattice point. If the axial ratiohas the ideal value (c/a = �8/3�1/2), then the largest interstices (octahedral) havecoordinates ( 1

323

14 ) and ( 1

323

34 ). There are two such interstices per unit cell. The next

largest interstices (tetrahedral) occur at (00 38 ), (00 5

8 ), ( 23

13

18 ) and ( 2

313

78 ), there being




a 1

a 2

a 1

a 2

Atoms

Octahedral voids

Tetrahedral voids

120°

Figure 1.20. This figure shows the locations of octahedral and tetrahedral voids in the hcp structure.

four such interstices per unit cell. The region around a tetrahedral void representsthe densest packing of equal sized spheres; all topologically close packed structuresare characterized by exclusively tetrahedral voids which may be geometricallyimperfect because of the differences in the sizes of the component atoms (Sinha1972). The coordination polyhedron of an atom is defined by the lines joining thecentres of atoms in the shell of close neighbours around it. The coordination istwelve-fold in the cases of the fcc and hcp structures and the polyhedra formedby the twelve neighbours assume the shapes of a cubo-octahedron (fcc) and atwinned cubo-octahedron (hcp), respectively. In the case of the TCP structuresyet another type of twelve-fold coordination polyhedron, in which all the facesare triangular, becomes important. This polyhedron is the icosahedron which hastwenty faces in the shape of equilateral triangles and thirty edges which correspondto nearest neighbour distances. Each of the other two twelve-fold coordinationpolyhedra mentioned earlier has 24 edges. In the case of the icosahedron, thedistance between the central atom and any atom on the polyhedron surface isaround 10% smaller than that between the atoms on the surface. The atoms onthe icosahedron surface are more close packed than in the fcc or hcp structures;however, because of the five-fold symmetry axis associated with the icosahedron,it is not possible to have a lattice like arrangement made up solely of icosahedra(Sinha 1972).

The condition that only tetrahedral interstices may be present in a TCP structurebrings in the requirement that besides a number of atoms having an icosahedralenvironment, certain others with higher coordination polyhedra around them mustalso be present; thus TCP structures are characterized by some or all of CN 12,CN 14, CN 15 and CN 16 polyhedra (Kasper 1956). All these Kasper polyhedra




have exclusively triangular faces and tetrahedral arrangement. The fractions ofsites with different coordination numbers change from structure to structure. Byway of illustration, let three TCP structures, be considered: A15 (cP8, Cr3Si type),C14 (hP12, MgZn2 type) and C15 (cF24, Cu2Mg type); in the first, 25% of thesites correspond to CN 12 and 75% to CN 14 while in either of the other (Lavesphase) structures, 67% of the sites correspond to CN 12 and 33% to CN 16(Sinha 1972). It has been pointed out (Frank and Kasper 1958, 1959) that thereis a significant consequence of there being, in a crystal structure, coordinationpolyhedra of CN 12, CN 14, CN 15 and CN 16 and exclusively tetrahedral voidsand this is that the resulting structure is generally a layered structure. In fact, mostof the TCP structures can be regarded as layered structures. The main atomiclayers, referred to as primary layers, are tesselated and contain arrays made up oftriangles, pentagons and hexagons. The triangular meshes in the primary layerscorrespond to the nearest neighbour atoms. Besides the primary layers, generallythere are secondary layers in which the coordination does not correspond to nearestneighbours (Sinha 1972). The layer stacking has, of course, to be effected in sucha manner that only tetrahedral interstices are present.

It has been mentioned in the previous section that certain structures are quitefrequently encountered in respect of Ti–X and Zr–X intermetallic phases. A briefaccount will now be presented to illustrate how these structures can be viewedin terms of the layer stacking sequence representation. The two atomic speciesconsidered will be designated as X and Z.

First, the case of superstructures based on close packed layers stacked in closepacking will be taken up. If X and Z are each on a 44 subnet in each closepacked layer, then a family of polytypic structures with XZ stoichiometry and arectangular arrangement of the components in close packed layers is obtained. Anexample of such a structure is the L1o (tP4, AuCu type) structure. This structurecan also be described in terms of stacking alternate 44 layers of X and Z atomsin succession in the [001] direction. One can next consider the case of XZ3

stoichiometry in each close packed layer, with X atoms on a 36 subnet and Z atomson a 3636 subnet. A family of polytypic structures with a triangular arrangementof X atoms is obtained. The L12 (cP4, AuCu3 type) and D019 (hP8, Ni3Sn type)structures belong to this family. In the former, the close packed layers, which lienormal to the [111] direction, are stacked in the sequence ABCABC� � � so that alllayers are surrounded cubically. In the latter, the close packed layers are arrangedin hexagonal ABAB� � � , stacking. The D019 structure is a hexagonally stackedprototype of the L12 structure. It is thus a superstructure of the hcp (hP2) structurein the same way as the L12 structure is of the fcc (cF4) structure (Pearson 1972).

Some superstructures based on bcc packing may now be considered. If oneconsiders the XZ stoichiometry, with X atoms on 44 nets and Z atoms also on 44




nets, one arrives at the B2 (cP2, CsCl type) structure. In this structure one speciesoccupies the cube corners and the other the body centre, so that alternate layers ofthe two species occur along <100> directions. The X and Z atoms together formtriangular nets parallel to �110� planes with X occupying one of the rectangular 44

subnets resulting from the geometry of the triangles of the 36 net, and Z occupyingthe other. Next, let the case of the XZ2 stoichiometry be taken up. If one considers36 close packed layers in bcc [110] stacking with Z atoms on a 63 subnet and Xatoms on a larger 36 subnet, one arrives at a family of polytypic structures XZ2

with close packed layers stacked in bcc sequence. The C11b (tI6, MoSi2 type)structure belongs to this family.

The C16 (tI12, CuAl2 type) structure can be visualized as being made up ofsquare-triangle, 32434 nets of Z atoms at z = 0 and z = 1

2 which are orientedantisymmetrically with respect to each other; the squares in these Z layers whichlie over the cell corners and basal face centre, are centred by a 1

2 44 net of X atoms

at z = 14 and z = 3

4 (Pearson 1972).Even though many structures contain atoms in triangular prismatic coordination,

there are some in which this is the main feature of the atomic arrangement. Inone class of such structures, 36 and 63 nets of atoms, occupying the same stackingsequence, are stacked alternately in the “paired-layer” sequence (e.g. AaAa). TheC32 (hP3, AlB2 type) structure can be regarded as the prototype of this family ofstructures and one of the simpler structures obtained from this structure is the Bf

(oC8, CrB type) structure, which is made up of independent layers of triangularprisms of X atoms parallel to the (010) plane with the prism axes oriented inthe [100] direction. The prisms are centred by atoms which form zigzag chainsrunning in the [001] direction (Pearson 1972).

An example of structures generated by the stacking of pentagon-triangle nets ofatoms is the D88 (hP16, Mn5Si3 type) structure.

In the tetrahedrally close packed A15 (cP8, Cr3Si type) structure of XZ3 stoi-chiometry, the X atoms form a bcc array and lines of Z atoms run throughout thestructure parallel to the edges of the body centred cell formed by the X atoms.This structure is of the Frank–Kasper type and can be visualized as being formedby the alternate stacking of primary triangle-hexagon 3262 +3636 (2:1) layers andsecondary 44 layers, with the result that each X atom is surrounded icosahedrallyby 12 Z atoms and each Z atom is surrounded by four X atoms and 10 Z atomsin a CN 14 polyhedron with triangular faces.

The Laves phase structures have XZ2 stoichiometry and belong to a familyof polytypic structures in which three closely spaced 36 nets of atoms are fol-lowed by a 3636 kagome net parallel to the (001) plane when the structures aredescribed in terms of a hexagonal cell. The former types of nets are stacked onthe same sites as the latter. Alternatively, the Laves phases can be visualised




as having Frank–Kasper structures in which pentagon–triangle primary layers ofatoms are stacked alternately, parallel to the (110) planes of the hexagonal cell,with secondary 36 triangular layers whose atoms centre the pentagons of the mainlayers (Pearson 1972). Thus, the C14 (hP12, MgZn2 type) structure is generatedby stacking together pairs of primary pentagon–triangle 3535 + 353 (2:3) layersand secondary 36 layers parallel to the (110) plane. The cubic C15 (cF24, MgCu2

type) structure can also be regarded as being built up by stacking consecutivelythree triangular (36) layers and a kagome (3636) layer of atoms which lie in planesnormal to the [111] direction in respect of the cubic cell. Each X atom is sur-rounded by a CN 16 polyhedron of 12 Z and four X atoms while each Z atom isicosahedrally enclosed by six X and six Z atoms.

1.7.3 Derivation of intermetallic phase structures from simple structuresThe structures of many intermetallics can be regarded as being derived from threesimple structures, namely, fcc (A1), bcc (A2) and hcp (A3) structures, which arecommonly associated with pure metals and disordered metallic solid solutions.The most common structures exhibited by binary intermetallic phases are listed inTable A1.4 (Ferro and Saccone 1996).

Typical intermetallic phase structures derived from the fcc structure include L12

(cP4, AuCu3 type), C15b (cF24, AuBe5 type), L′12 (cP5, Fe3AlC type), D022 (tI8,TiAl3 type), L1o (tP4, AuCu type), D1a (tI10, Ni4Mo type), L11, (hR32, CuPt type)and Pt2Mo type (oI6) structures (Pitsch and Inden 1991, Sauthoff 1996). Generallythese structures are cubic, tetragonal (often with an axial ratio close to unity),rhombohedral or orthorhombic. Examples of intermetallic phases with some ofthese structures in Ti–X and Zr–X systems are: TiCo3, TiIr3, �′-TiNi3, �-TiPt3,TiRh3, TiZn3, Zr3Al, ZrHg3, Zr3In, ZrIr3, ZrPt3, ZrRh3�L12�; ZrNi5�C15b�; TiAl3,TiGa3, �-ZrIn3�D022�; TiAl, �′′-TiCu3, TiGa, TiHg, Ti3In2, �-TiRh, ZrHg (L1o);and TiAu4, �-TiCu4, TiPt8�D1a�.

Common intermetallic phase crystal structures derived from the bcc structureincldue B2 (cP2, CsCl type), B32 (cF16, NaTl type), D03 (cF16, BiF3 type), andL21 (cF16, Cu2AlMn type) structures. (Pitsch and Inden 1991, Sauthoff 1996).Generally these structure are cubic. Examples of intermetallic phases with theB2 structure in Ti–X and Zr–X systems are: �-TiAu, TiBe, TiCo, TiFe, �-TiIr,TiNi, TiOs, �-TiPd, �-TiPt, �-TiRh, TiRu, TiTc, TiZn, S-ZrCo, ZrCu, ZrIr, ZrOs,�-ZrPt, �-ZrRh, ZrRu and ZrZn. Intermetallics with other bcc-based structures arerare in these systems.

Prominent among the intermetallic phase crystal structures derived from thehcp structure are: Bh (hP2, WC type), D019 (hP8, Ni3Sn type), B19 (oP4, AuCdtype), C49 (oC12, ZrSi2 type) and D0a (oP8, �-Cu3Ti type) structures. Generally,these structures are hexagonal or orthorhombic. Examples of intermetallic phases




with these structures in Ti–X and Zr–X systems include Zr3Se2�Bh�; Ti3Al, Ti3Ga,Ti3In, Ti4Pd, Ti4Sb, Ti3Sn, Zr3Co, ZrNi3�D019�; �-TiAu, �-TiPd, �-TiPt (B19);ZrGe2, ZrSi2 (C49); and �-TiCu3, ZrAu3�D0a�.

1.7.4 Intermetallic phases with TCP structures in Ti–Xand Zr–X systems

It has been mentioned earlier that quite a few of the intermetallic phases occurringin Ti–X and Zr–X systems have topologically closed packed (TCP) structures.These phases are mostly A15 (cP8, Cr3Si type) phases or Laves phases (C14,C15 or C36 structures). Examples of such phases are: Ti3Au, Ti3Hg, Ti3Ir, Ti4Pd(stoichiometric), Ti3Pt, Ti3Sb, Zr3Au, Zr3Hg, Zr4Sn, Zr4Tl (A15); �-TiCr2, TiFe2,TiMn2, TiZn2ZrAl2, �-ZrCr2, ZrMn2, ZrRe2, ZrRu2, ZrTc2 (C14); TiBe2, TiCo2,�-TiCr2, �-ZrCr2, ZrFe2, ZrIr2, ZrMo2, ZrV2, ZrW2, ZrZn2 (C15); TiCo2, �-TiCr2

and �-ZrCr2 (C36, hP24, MgNi2 type). The phase Zr4Al3 (hP7) is also a TCPphase the structure of which can be described either in terms of pentagon–triangleprimary and 44 secondary nets parallel to the (110) plane, or with hexagon–triangleprimary and 36 secondary nets parallel to the (001) plane (Pearson 1972).

Some of the Laves phases mentioned above can absorb very significant quanti-ties of hydrogen and, for this reason, are considered for applications as hydrogenstorage materials. Reference must be made in this context to the phases ZrV2,ZrCr2, ZrMn2 and TiCr2 which exhibit high sorption capacities with hydrogen tometal ratio (H/M) values of 1.8, 1.3, 1.2 and 1.2, respectively (Sauthoff 1996).

1.7.5 Phase stability in zirconia-based systemsZirconia (ZrO2)-based systems are among the most extensively investigated ceram-ics in so far as phase transformation studies are concerned. Not only do theyexhibit interesting phase transformations, but also the properties of these ceram-ics can be engineered by suitably controlling the stability of different competingphases and by inducing phase transformations in a desired manner. In view of this,zirconia-based systems are pedagogically very appropriate systems for illustratinghow phase transformations can be effectively utilized for controlling microstruc-ture and, in turn, properties – mechanical, thermal, electrical and optical – ofceramics. Crystal structures and stability of different phases in zirconia ceramicsare briefly described in this section.

1.7.5.1 ZrO2 polymorphsPure ZrO2 exhibits three polymorphic forms under ambient pressure; these belong,respectively, to monoclinic, tetragonal and cubic crystal systems (Garvie 1970). Thecrystal structures and lattices parameters of these polymorphs are given in Table 1.18




Table 1.18. Crystal structures and lattice parameters of zirconia polymorphs.

Phase Crystal structure Lattice parameters (nm)

Pearson symbol Space group

m-ZrO2 mP12 P21/c a = 0.5156b = 0.5191c = 0.5304� = 98 9o

t-ZrO2 tP6 P42/nmc a = 0.5094c = 0.5177

c/a = 1.016c-ZrO2 cF12 Fm3m a = 0.5124

(Stevens 1986, Massalski et al. 1992). The occurrence of an orthorhombic form ofZrO2 under high pressures has also been reported (Lenz and Heuer 1982).

The monoclinic phase (generally designated as m-ZrO2) is stable upto about1443 K where it transforms to the tetragonal phase (t-ZrO2) which is stable upto2643 K; at still higher temperatures, the cubic phase (c-ZrO2) is encountered whichis stable upto the melting temperature of 2953 K (Stevens 1986). Among thesethree phases, the monoclinic phase, has the lowest density.

In m-ZrO2, the Zr4+ ion has seven-fold coordination with O ions, with a rangeof Zr–O bond lengths and bond angles. The OII coordination is close to tetrahedralwith only one angle (134 3o) differing significantly from the angle of the tetrahe-dron (109 5o) while the OI coordination is triangular. The Zr ions are located inlayers parallel to (100) planes, separated by OI and OII ions on either side. Theaverage Zr–OI and Zr–OII distances are 0.207 and 0.221 nm, respectively (Stevens1986). Figure 1.21 shows a schematic of the idealized ZrO7 polyhedron.

Each Zr4+ ion in t-ZrO2 is surrounded by eight O ions. There is some distortion inthis eight-fold coordination due to the fact that while four of the O ions are at a dis-tance of 0.2065 nm, in the form of a flattened tetrahedron, the other four are at a dis-tance of 0.2455 nm in an elongated tetrahedron rotated through 90o (Stevens 1986).

The high temperature cubic phase, c-ZrO2, has the fcc fluorite (CaF2) typestructure, in which each Zr4+ ion is coordinated by eight equidistant O ions whichare arranged in two equal tetrahedra. A layer of ZrO8 groups in c-ZrO2 is shownin Figure 1.22.

1.7.5.2 Stabilization of high temperature polymorphsAn important concept which is often utilized in zirconia ceramics is to “alloy” pureZrO2 with another suitable oxide to fully or partially stabilize high temperaturepolymorphs of ZrO2 to lower temperatures.




a a

Zr

O

Figure 1.21. Schematic showing the idealised ZrO7 polyhedron pertinent to m-ZrO2.

Zr

O

II

II

I

II

IIII

Figure 1.22. This figure shows a layer of ZrO8 groups in c-ZrO2.

The tetragonal to monoclinic transformation, which is martensitic in nature, isaccompanied by a large (3–5%) volume expansion which is sufficient to exceedelastic and fracture limits even in relatively small grains of pure ZrO2 and can onlybe accommodated by cracking. A consequence of this is that the fabrication of largecomponents of pure ZrO2 is not possible due to spontaneous failure on cooling.




The addition of cubic stabilizing oxides in appropriate amounts can permit thecubic polymorph to be stable over a wide range of temperatures: even from roomtemperature to its melting temperature. The oxides that are commonly used toform solid solutions with ZrO2 include MgO (magnesia), CaO (calcia) and REoxides such as Y2O3 (yttria) and CeO2 (ceria). These oxides exhibit extensive solidsolubility in ZrO2 and are able to form fluorite type phases which are stable overwide ranges of composition and temperature. If the amount of stabilizing oxideadded to ZrO2 is insufficient for complete stabilization of the cubic phase, then apartially stabilized zirconia (PSZ) is obtained rather than a fully stabilized form.The PSZ usually comprises a mixture of two or more phases. Both the cubic solidsolution and the tetragonal solid solution are present and the latter may transformto the monoclinic solid solution on cooling.

It may be mentioned here that the volume expansion associated with the tetrag-onal to monoclinic transformation may be used to advantage for improving tough-ness and strength. This aspect will be discussed in a later chapter. A relativelytough, partially stabilized zirconia ceramic, consisting of a dispersion of metastabletetragonal ZrO2 inclusions within large grains of stabilized cubic ZrO2, can bederived by inducing a stress induced tetragonal to monoclinic transformation.

An appraisal of the phase equilibria of zirconia with other oxide systems is veryimportant with regard to the application of zirconia as an engineering ceramic.However, many difficulties are encountered while determining the equilibriumphase diagrams of even the simplest binary zirconia systems. First, the reactionsin these systems at relatively low temperatures (<1680K) are somewhat sluggishowing to low diffusivity. Since diffusional reactions such as precipitation, eutectoiddecomposition and ordering proceed very slowly, equilibrium is often difficult toachieve. Secondly, coherency strain between different phases may influence theirrelative stabilities in two phase regions and may be responsible for the retentionof metastable phases. Finally, at high temperatures, the presence of impurities ornon-stoichiometry may influence the equilibria.

The salient features of binary phase diagrams of three important zirconia sys-tems, namely, ZrO2–MgO, ZrO2–CaO and ZrO2–Y2O3, are summarized in thefollowing sections.

1.7.5.3 ZrO2–MgO systemThe polymorphic transformations of pure ZrO2, the eutectoid decomposition ofthe fluorite type solid solution into a tetragonal solid solution and MgO (theeutectoid temperature and composition being ∼1673 K and and ∼13 mole % MgO,respectively), and a very limited (<1 mole %) solubility of MgO in tetragonalZrO2 at 1573K are some important data (Grain 1967) for the construction of thephase diagram which still remains somewhat tentative. The phase boundaries of




the fluorite type solid solution field at high temperatures, the solubility of MgO inmonoclinic ZrO2 and the occurrence of a eutectoid reaction in the tetragonal solidsolution field still appear to be uncertain (Stubican 1988).

In this system, the occurrence of a fluorite related ordered compound, Mg2Zr5O12

(28.57 mole % MgO) has been established but not the regime of its stability.The formation of another metastable ordered compound, MgZr6O13 (14.28 mole% MgO) has also been reported (Rossell and Hannink 1984).

1.7.5.4 ZrO2–CaO systemThe ZrO2-rich side of the ZrO2–CaO system in in some ways similar to that of theZrO2–MgO system in that the terminal cubic as well as tetragonal solid solutionphases decompose by eutectoid reactions. However, the solubility limits of CaOin both the tetragonal and monoclinic phases are considerably higher than thoseof MgO. Two ordered phases CaZrO9 (�1) and Ca6Zr19O44 (�2) have been foundto occur in this system and these are stable in the temperature ranges 1393 to1502 ± 15 K and 1398 to 1528 ± 15 K, respectively. The CaZrO3 phase exhibitsa polymorphic transformation at 2023 ± 30 K from a high temperature cubic toa lower temperature orthorhombic form. The presence of the two polymorphhsof the CaZrO3 phase and the ordered �1 and �2 phases introduces several phasereactions in the c-ZrO2 +CaZrO3 phase field. The ZrO2 rich side of the ZrO2–CaOphase diagram (Stubican 1988) is shown in Figure 1.23.

3270

2770

2270

1770

1270

7700 10 30 40 5020

Mol % CaO

Tem

pera

ture

(K

)

MT + M

TT + C

1413 K

M + φ(o)

φ2φ1φ2 + φ(o)1628 K

C + φ(o)

2023 K

C + φ(c)CC + L

L

L + φ

ZrO2 CaZrO3

Figure 1.23. The ZrO2-rich side of the ZrO2–CaO phase diagram. M, T and C stand, respectively,for the monoclinic, tetragonal and cubic (fluorite type) solid solutions and L for liquid; the symbols�, �(c), �(o), �1 and �2 represent the phases CaZrO3, CaZrO3 (cubic), CaZrO3 (orthorhombic),CaZrO9 and Ca6Zr19O44, respectively.




0 2 4 6 8 10 12 14 16

Mol % Y2O3

2670

2070

1470

870

C + TC

T

M + T

M

M + C

M + ZYO

Tem

pera

ture

(K

)

Figure 1.24. This figure shows the solid state phase relations in the ZrO2 rich end of the ZrO2–Y2O3

phase diagram. M, T and C stand, in that order, for the monoclinic, tetragonal and cubic solidsolutions, while ZYO stands for the phase Zr3Y4O12.

1.7.5.5 ZrO2–Y2O3 systemThe low Y2O3 region of the ZrO2–Y2O3 system is very important for technologicalapplications. For this reason the subsolidus equilibria in this portion of the phasediagram have been examined extensively. Figure 1.24 depicts the phase relationsin the ZrO2-rich side of this system (Stubican 1988) subject to the possible limi-tations imposed by the fact that below ∼1500 K it is extremely difficult to attainequilibrium, even with reactive powders. The ordered compound, Zr3Y4O12, isstable up to 1655±5 K. At still higher temperatures, it disorders to a fluorite-typesolid solution.

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APPENDIX

Table A1.1. Crystal structures of important binary intermetallic phases in Ti–X systems (Massalskiet al. 1992).

Phase Composition(at.% X)

Space group Pearson symbol Strukturberichtdesignation

Ti2Ag 33.3 14/mmm tI6 C11b

TiAg 48–50 P4/nmm tP4 B11Ti3Al 22–39 P63/mmc hP8 D019

TiAl 48–69.5 P4/mmm tP2 L1o

Ti3Al∗5 58–63 14/mbm tP32 � � �TiAl2 65–68 141/amd tI24 � � �TiAl3 75 14/mmm tI8 D022

Ti3Au 25 Pm3n cP8 A15� TiAu (HT) 38–58 Pm3m cP2 B2� TiAu (MT) 49–50 Pmma oP4 B19� TiAu (LT) 50 P4/nmm tP4 B11TiAu2 66.7 14/mmm tI6 C11b

TiAu4 79–82 14/m tI10 D1a

TiB 49–50 Pnma oP8 B27Ti3B4 56.1 Immm oI14 D7b

TiB2 65.6–66.7 P6/mmm hP3 C32TiBe2 66.7 Fd3m cF24 C15TiBe3 75 R3m hR12 � � �� Ti2Be17(HT) 89.5 P63/mmc hP38� Ti2Be17(LT) 89.5 R3m hR19 � � �TiBe12 92.3 14/mmm tI26 D2b

TiBe ∼50 Pm3m cP2 B2TiC ∼32–48.8 Fm3m cF8 B1Ti2C ∼32–36 Fd3m cF48 � � �Ti2Cd 33.3 P4/mmm tI6 C11b

TiCd 50 P4/nmm tP4 B11Ti2Co 32.9–33.3 Fd3m cF96 E93

TiCo 49–55 Pm3m cP2 B2TiCo2 66.5–67 Fd3m cF24 C15TiCo2 68.75–72 P63/mmc hP24 C36TiCo3 75.5–80.7 Pm3m cP4 L12

� TiCr2(LT) 63–65 Fd3m cF24 C15� TiCr2(MT) 64–66 P63/mmc hP12 C14� TiCr2(HT) 64–66 P63/mmc hP24 C36

(Continued)




Table A1.1. (Continued)



Ti3Cu 25 P4/mmm tP4 L6o

Ti2Cu 33.3 I4/mmm tI6 C11b

TiCu 48–52 P4/nmm tP4 B11Ti3Cu4 57.1 I4/mmm tI14 � � �Ti2Cu3 60 P4/nmm tP10 � � �TiCu2 66.7 Amm2 oC12 � � �TiCu4 78–80.9 Pnma oP20� TiCu4 ∼78–∼80.9 I4/m tI10 D1aTiCu∗

3 � � � Pmmn oP8 D0a

Ti–Cu �′′ phase∗ � � � P4/mmm tP2 L1o

TiFe 48–50.2 Pm3m cP2 B2TiFe2 64.5–72.4 P63/mmc hP12 C14Ti3Ga 25 P63/mmc hP8 D019

Ti2Ga 33.3 P63/mmc hP6 B82

Ti5Ga∗3 37.5 14/mcm tI32 D8m

Ti5Ga∗4 44.4 P64/mcm hP18 � � �

TiGa∗ 50 P4/mmm tP2 L1o

Ti3Ga∗5 62.5 P4/mbm tP32 � � �

TiGa3 75 14/mmm tI8 D022

Ti5Ge3 37.5 P63/mcm hP16 D88

TiGe2 66.7 Fddd oF24 C54� hydride 1.05–2.0 Fm3m cF12 C1� hydride 1.72–2.0 I4/mmm tI2 L′2b

� hydride∗ 0.01–0.03 P42/n tP6 � � �

Ti3Hg 25 Pm3n cP8 A15TiHg 50 P4/mmm tP2 L1o

Ti3In >21 P63/mmc hP8 D019

Ti3In2 � � � P4/mmm tP2 L1o

Ti3Ir 25–27 Pm3n cP8 A15� TiIr 35–57.5 Pm3m cP2 B2TiIr3 73–77 Pm3m cP4 L12

TiMn2 60–70 P63/mmc hP12 C14TiMn4 81.5 R3m hR53 � � �Ti2N ∼33 P42/mnm tP6 C4TiN 28 to > 50 Fm3m cF8 B1�′ nitride ∼38 I41/amd tI12 Cc

Ti2Ni 33.3 Fd3m cF96 � � �

TiNi 49.5–57 Pm3m cP2 B2TiNi3 75 P63/mmc hP16 D024

�′ TiNi∗3 ∼83–88 Pm3m cP4 L12

Ti3O ∼20–∼30 P31c hP ∼ 16 � � �

Ti2O ∼25–33.4 P3m1 hP3 � � �

� TiO 34.9–55.5 Fm3m cF8 B1







Ti3O2 ∼40 P6/mmm hP ∼ 5 � � �

Ti2O3 59.8–60.2 R3c hR30 D51

Rutile ∼66.7 P42/mnm tP6 C4Anatase∗ � � � I41/amd tI12 C5Brookite∗ � � � Pbca oP24 C21TiOs 38–51 Pm3m cP2 B2Ti3P 25 P42/n tP32 � � �Ti5P3 ∼36–∼ 39 P63/mcm hP16 D88

TiP 48–50 P63/mmc hP8 Bi

TiP2 66.7 I4/mcm tI12 C16Ti4Pb ∼20 P63/mmc hP8 D019

Ti4Pd 20 Pm3n cP8 A15Ti2Pd 33.3 I4/mmm tI6 C11b

� TiPd (HT) 47–53 Pm3n cP2 B2� TiPd (LT) 47–53 Pmma oP4 B19Ti2Pd3 60 Cmcm oC20 � � �Ti3Pd5 62.5 P4/mmm tP8 ∼C11bTiPd2 65–67 I4/mmm tI6 C11b

TiPd3 75 P63/mmc hP16 D024

Ti–Pd � phase 75–84 P4/mmm cP4 L12

TiPo 50 P63/mmc hP4 B81

Ti3Pt 22–29 Pm3n cP8 A15� TiPt (HT) 46–54 Pm3m cP2 B2� TiPt (LT) 46–54 Pmma oP4 B19Ti3Pt5 62.5 Ibam 0I32 � � �TiPt3 <75 P63/mmc hP16 D024

Ti–Pt � phase 75–81 Pm3m cP4 L12

TiPt8 89–98 I4/m tI18 D1a

Ti5Re24 82.8 I 43m cI58 A12Ti2Rh 33.3 I4/mmm tI6 C11b

� TiRh (HT) ∼38–58 Pm3m cP2 B2� TiRh (LT) ∼38–58 Pm3m tP2 L1o

Ti3Rh5 62.5 Pbam oP16 � � �

TiRh3 73–78 Pm3m cP4 L12

TiRu 45–52 Pm3m cP2 B2TiS ∼49.7 P63/mmc hP4 B81

TiS2 64.4–66.7 P3m1 hP3 C6TiS3 ∼75 P21/m mP8 � � �Ti4Sb >20.1–<25 P63/mmc hP8 D019

Ti3Sb 25 Pm3n cP8 A15TiSb 50 P63/mmc hP4 B81

(Continued)







TiSb2 66.2–67.1 I4/mcm tI12 C16Ti8Se9 ∼52.9 R3m hP12 � � �Ti3Se4 55–57.6 C2/m mC14 � � �Ti3Si 25 P42/n tP32 � � �Ti5Si3 35.5–39.5 P63/mcm hP16 D88

TiSi 50 Pmm2 oP8 � � �Pnma oP8 B27

TiSi2 66.7 Fddd oF24 C54Ti3Sn 23–25 P63/mmc hP8 D019

Ti2Sn 32.7–35.9 P63/mmc hP6 B82

Ti5Sn3 37.5 P63/mcm hP16 D88

� Ti6Sn5 (HT) 45.5 P63/mmc hP22 � � �� Ti6Sn5 (LT) 45.5 Immm oI44 � � �

TiTc ∼50 Pm3m cP2 B2Ti–Tc � phase ∼85 I 43m cI58 A12Ti5Te4 44.4 I4/m tI18 � � �TiTe >40 ∼P63/mmc hP16 ∼B81

Ti3Te4 55–59.2 C2/m mC14 � � �

TiTe2 ∼60–66.7 P3m1 hP3 C6TiU2 66.7 P6/mmm hP3 C32TiZn15 93.7 Cmcm oC68 � � �

TiZn3 75 Pm3m cP4 L12

TiZn2 66.7 P63/mmc hP12 C14TiZn 50 Pm3m cP2 B2Ti2Zn 33.3 I4/mmm tI6 C11b

∗Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.

Table A1.2. Crystal structures of important binary intermetallic phases in Zr–X systems (Massalskiet al. 1992).



Zr2Ag 33.3 14/mmm tI6 C11b

ZrAg 50 P4/nmm tP4 B11Zr3Al 25 Pm3m cP4 L12

Zr2Al 33.3 P63/mmc hP6 B82

Zr5Al3 37.5 I4/mcm tI32 D8m

Zr3Al2 40 P42/mmm tP20 � � �







Zr4Al3 42.9 P6 hP7 � � �Zr5Al4 44.4 P63/mcm hP18 � � �ZrAl 50 Cmcm oC8 Bf

Zr2Al3 60 Fdd2 oF40 � � �ZrAl2 66.7 P63/mmc hP12 C14ZrAl3 75 I4/mmm tI16 D023

Zr3Au 25 Pm3n cP8 A15Zr2Au 33.3 I4/mmm tI6 C11b

ZrAu2 66.7 I4/mmm tI6 C11b

ZrAu3 75 Pmmn oP8 D0a

ZrAu4 80 Pnma oP20 � � �ZrB2 66.7–68 P6/mmm hP3 C32ZrB12 92.4 Fm3m cF52 D2f

ZrBe∗ 50 Cmcm oC8 Bf

ZrBe2 66.7 P6/mmm hP3 C32ZrBe5 83.3 P6/mmm hP6 D2d

Zr2Be17 89.5 R3m hR19 � � �

ZrBe13 92.9 Fm3c cF112 D23

ZrC 33–50 Fm3m cF8 B1Zr2Cd 33.3 I4/mmm tI6 C11b

ZrCd3 75 P4/mmm tP4 L6o

Zr3Co 25 Cmcm oC16 E1a

P63/mmc hP8 D019

Zr–Co � phase 33.3 I4/mcm tI12 C16Zr–Co phase ∼50 Pm3m cP2 B2Zr–Co � phase >65 to ∼73 Fd3m cF24 C15Zr–Co � phase 79.3 Fm3m cF116 D8a

� ZrCr2(HT) 64–69 P63/mmc hP12 C14� ZrCr2(MT) 64–69 P63/mmc hP24 C36� ZrCr2(LT) 64–69 Fd3m cF24 C15Zr2Cu 33.3 I4/mmm tI6 C11b

ZrCu 50 Pm3m cP2 B2Zr3Fe 24–26.8 Cmcm oC16 E1a

Zr2Fe 31–33.3 I4/mcm tI12 C16ZrFe2 66–72.9 Fd3m cF24 C15ZrFe3 75 Fm3m cF116 D8a

Zr2Ga 33.3 I4/mcm tI12 C16Zr5Ga3 37.5 P63/mcm hP16 D88

Zr3Ga2 40 P4/mbm tP10 D5a

Zr5Ga4 44.4 P63/mcm hP18 � � �ZrGa 50 I41/amd tI16 Bg

(Continued)







Zr2Ga3 60 Fdd2 oF40 � � �Zr3Ga5 62.5 Cmcm oC32 � � �ZrGa2 66.7 Cmmm oC12 � � �ZrGa3 75 I4/mmm tI16 D023

Zr3Ge 25 P42/n tP32 � � �Zr5Ge3 37.5 P63/mcm hP16 D88

Zr5Ge4 44.4 P41212 tP36 � � �ZrGe 50 Pmma oP8 B27ZrGe2 66.7 Cmcm oC12 C49� hydride 56.7–66.4 Fm3m cF12 C1� hydride 63.6 I4/mmm tI6 L′2b

� hydride∗ ∼1.0 P42/n tP6 � � �

Zr3Hg 25 Pm3n cP8 A15ZrHg 50 P4/mmm tP2 L1o

ZrHg3 75 Pm3m cP4 L12

Zr3In 25 Pm3m cP4 L12

Zr2In 33.3 P4/mmm tP2 L1o

ZrIn 50 Fm3m cF4 A1ZrIn2 66.7 I41/amd t124 � � �� ZrIn3(HT) 75 I4/mmm tI8 D022

� ZrIn3 (LT) 75 I4/mmm tI16 D023

Zr3Ir 25 I 42m tI32 � � �Zr2Ir 33.3 I4/mcm tI12 C16Zr5Ir3 37.5 P63/mcm hP16 D88

ZrIr 48–53 Pm3m cP2 B2ZrIr2 66.7 Fd3m cF24 C15ZrIr3 70–81 Pm3m cP4 L12

ZrMn2 60–79.2 P63/mmc hP12 C14ZrMo2 60–67 Fd3m cF24 C15ZrN >40 Fm3m cF8 B1Zr2Ni 33.3 I4/mcm tI12 C16ZrNi 50 Cmcm oC8 Bf

ZrNi3 74–75.5 P63/mmc hP8 D019

ZrNi5 81.6–85.2 F 43m cF24 C15b

� ZrO2−x(HT) 61–66.7 Fm3m cF12 C1� ZrO2−x (MT) 66.5–66.7 P42/nmc tP6 � � �� ZrO2−x (LT) 66.7 P21/c mP12 C43ZrOs 50 Pm3m cP2 B2ZrOs2 >61–∼70 P63/mmc hP12 C14Zr3P 25 P42/n tP32 � � �







� ZrP (HT) 50 Fm3m cF8 B1� ZrP (LT) 50 P63/mmc hP8 Bi

ZrP2 66.7 Pnma oP12 C23Zr5Pb3 37.5 P63/mcm hP16 D88

Zr2Pd 33.3 I4/mmm tI6 C11b

ZrPd 50 Fm3m cF4 A1ZrPd2 66.7 I4/mmm tI6 C11b

ZrPd3 75 P63/mmc hP16 D024

ZrPo 50 P63/mmc hP4 B81

Zr5Pt3 37.5 P63/mcm hP16 D88

� ZrPt (HT) 50 Pm3m cP2 B2� ZrPt (LT) 50 Cmcm oC8 Bf

ZrPt3 75 Pm3m cP4 L12

P63/mmc hP16 D024

Zr3Pu 26 P6/mmm hP3 C32ZrPu4 70–90 P4/ncc tP80 � � �ZrRe2 66.7 P63/mmc hP12 C14Zr5Re24 ∼82.8 I 43m cI58 A12Zr2Rh 33.3 I4/mcm tI12 C16� ZrRh (HT) 50–62 Im3m cI2 A2� ZrRh (LT) > 50 Pm3m cP2 B2Zr3Rh5 62.5 Cmcm oC32 � � �

ZrRh3 72–82 Pm3m cP4 L12

ZrRu 48–52 Pm3m cP2 B2ZrRu2 66–68 P63/mmc hP12 C14Zr2S 33.3 Pnnm oP36 � � �

Zr3S2 40 P6m2 hP2 Bh

ZrS 50 Fm3m cF8 B1P4/nmm tP4 B11

ZrS2 66.7 P3m1 hP3 C6ZrS3 75 P21/m mP8 � � �

Zr3Sb 25 I 4 tI32 D0e

Zr5Sb3 36 P63/mcm hP16 D88

ZrSb2 66.7 Pnnm oP24 � � �Zr2Se 33.3 Pnnm oP36 � � �

Zr3Se2 40 P6m2 hP2 Bh

Zr2Se3 60 P63mc hP8 � � �

ZrSe2 64.9–66 P3m1 hP3 C6ZrSe3 75 P21/m mP8 � � �Zr3Si ∼25 P42/n tP32 � � �

I 4 tI38 D0e

(Continued)







Zr2Si 33.3 I4/mcm tI12 C16Zr5Si3 37.5 P63/mcm hP16 D88

Zr3Si2 40 P4/mbm tP10 D5a

� ZrSi (HT) 50 Cmcm oC8 Bf

� ZrSi (LT) 50 Pnma oP8 B27ZrSi2 66.7 Cmcm oC12 C49Zr4Sn ∼20 Pm3n cP8 A15Zr5Sn3 33–∼40 P63/mcm hP16 D88

ZrSn2 66.7 Fddd oF24 C54ZrTc2 66.7 P63/mmc hP12 C14ZrTc6 85.7 I 43m cI58 A12Zr3Te 25 R3m hR12 � � �Zr5Te4 44.4 I4/m tI18 � � �ZrTe 50 P63/mmc hP4 B81

ZrTe2 55–66.7 P3m1 hP3 C6ZrTe3 75 P21/m mP8 � � �

Zr4Tl 20 Pm3n cP8 A15U–Zr � phase 22–37 P6/mmm hP3 C32ZrV2 ∼66.7 Fd3m cF24 C15ZrW2 ∼66.7 Fd3m cF24 C15Zr2Zn 33.3 I4/mmm tI16 D023

Zr3Zn2 39.5 P42nm tP20 � � �

ZrZn 50 Pm3m cP2 B2ZrZn2 66.7 Fd3m cF24 C15

∗Metastable phase; HT: High temperature phase; MT: Medium temperature phase; LT: Low temperature phase.

Table A1.3. Nomenclature of crystal structures: strukturbericht designations and correspondingPearson symbols (Massalski et al. 1992).

Strukturberichtdesignation

Prototype phase Space group Pearson symbol

Aa � Pa I4/mmm tI2Ab � U P42/mnm tP30Ac � Np Pnma oP8Ad � Np P4212 tP4Af HgSn6−10 P6/mmm hP1Ag � B P42/nnm tP50Ah � Po Pm3m cP1Ai � Po R3m hR1







Ak � Se P21/c mP64Al � Se P21/c mP32A1 Cu Fm3m cF4A2 W Im3m cI2A3 Mg P63/mmc hP2A4 C (diamond) Fd3m cF8A5 � Sn I41/amd tI4A6 In I4/mmm tI2A7 � As R3m hR2A8 � Se P3121 hP3A9 C (graphite) P63/mmc hP4A10 � Hg R3m hR1A11 � Ga Cmca oC8A12 � Mn I 43m cI58A13 � Mn P4132 cP20A14 I2 Cmca oC8A15 Cr3Si Pm3n cP8A16 � S Fddd oF128A17 P (black) Cmca oC8A20 � U Cmcm oC4Ba CoU 1213 cI16Bb AgZn P3 hP9Bc CaSi Cmmc oC8Bd � NiSi Pbnm oP8Be CdSb Pbca oP16Bf CrB Cmcm oC8Bg MoB I41/amd tI16Bh WC P6m2 hP2Bi TiAs P63/mmc hP8Bk BN P63/mmc hP4Bl AsS P21/c mP32Bm TiB Pnma oP8B1 NaCl Fm3m cF8B2 CsCl Pm3m cP2B3 ZnS (sphalerite) F 43m cF8B4 ZnS (wurtzite) P63mc hP4B81 NiAs P63/mmc hP4B82 Ni2In P63/mmc hP6B9 HgS P3121 hP6B10 PbO P4/nmm tP4B11 � CuTi P4/nmm tP4

(Continued)







B13 NiS R3m hR6B16 GeS Pnma oP8B17 PtS P42/mmc tP4B18 CuS P63/mmc hP12B19 AuCd Pmma oP4B20 FeSi P213 cP8B26 CuO C2/c mC8B27 FeB Pnma oP8B29 SnS Pmcn oP8B31 MnP Pnma oP8B32 NaTl Fd3m cF16B34 PdS P42/m tP16B35 CoSn P6/mmm hP6B37 SeTl I4/mcm tI16Ca Mg2Ni P6222 hP18Cb CuMg2 Fddd oF48Cc ThSi2 I41/amd tI12Ce PdSn2 Aba2 oC24Cg ThC2 C2/c mC12Ch Cu2Te P6/mmm hP6Ck LiZn2 P63/mmc hP3C1 CaF2 Fm3m cF12C1b AgAsMg F 43m cF12C2 FeS2(pyrite) Pa3 cP12C3 Ag2O Pn3m cP6C4 TiO2 (rutile) P42/mnm tP6C6 CdI2 P3m1 hP3C7 MoS2 P63/mmc hP6C8 SiO2 (high quartz) P6222 hP9C9 SiO2 (� crystobalite) Fd3m cF24C10 SiO2 (� tridymite) P63/mmc hP12C11a CaC2 I4/mmm tI6C11b MoSi2 I4/mmm tI6C12 CaSi2 R3m hR6C14 MgZn2 P63/mmc hP12C15 Cu2Mg Fd3m cF24C15b AuBe5 F 43m cF24C16 Al2Cu I4/mcm tI12C18 FeS2(marcasite) Pnnm oP6C19 �Sm R3m hR3C21 TiO2 (brookite) Pbca oP24C22 Fe2P P62m hP9







C23 Co2Si Pnma oP12C28 HgCl2 Pmnb oP12C32 AlB2 P6/mmm hP3C33 Bi2Te3 R3m hR5C34 AuTe2 (calaverite) C2/m mC6C35 CaCl2 Pnnm oP6C36 MgNi2 P63/mmc hP24C37 Co2Si Pbnm oP12C38 Cu2Sb P4/nmm tP6C40 CrSi2 P6222 hP9C42 SiS2 Ibam oI12C43 ZrO2 P21/c mP12C44 GeS2 Fdd2 oF72C46 AuTe2 (krennerite) Pma2 oP24C49 ZrSi2 Cmcm oC12C54 TiSi2 Fddd oF24D0a � Cu3Ti Pmmn oP8D0c SiU3 I4/mcm tI16D0′

c Ir3Si I4/mcm tI16D0d AsMn3 Pmmn oP16D0e Ni3P I 4 tI32D02 CoAs3 Im3 cI32D03 BiF3 Fm3m cF16D09 ReO3 Pm3m cP4D011 Fe3C Pnma oP16D017 BaS3 P421m oP16D018 Na3As P63/mmc hP8D019 Ni3Sn P63/mmc hP8D020 Al3Ni Pnma oP16D021 Cu3P P63cm hP24D022 TiAl3 I4/mmm tI8D023 ZrAl3 I4/mmm tI16D024 TiNi3 P63/mmc hP16D1a MoNi4 I4/m tI10D1b Al4U Imma oI20D1c PdSn4 Aba2 oC20D1d Pb4Pt P4/nbm tP10D1e B4Th P4/mbm tP20D1f Mn4B Fddd oF40D1g B4C R3m hR15D13 Al4Ba I4/mmm tI10

(Continued)







D2b Mn12Th I4/mmm tI26D2c MnU6 I4/mcm tI28D2d CaCu5 P6/mmm hP6D2e BaHg11 Pm3m cP36D2f UB12 Fm3m cF52D2g Fe8N I4/mmm tI18D2h Al6Mn Cmcm oC28D21 CaB6 Pm3m cP7D23 NaZn13 Fm3c cF112D5a Si2U3 P4/mbm tP10D5b Pt2Sn3 P63/mmc hP10D5c Pu2C3 I 43d cI40D5e Ni3S2 R32 hR5D5f As2S3 P21/c mP20D51 � Al2O3 R3c hR10D52 La2O3 P3m1 hP5D53 Mn2O3 Ia3 cI80D54 Sb2O3 (senarmonite) Fd3m cF80D58 Sb2S3 Pnma oP20D59 Pt2Zn3 P42/nmc tP40D510 Cr3C2 Pnma oP20D511 Sb2O3 (valentinite) Pccn oP20D513 Al3Ni2 P3m1 hP5D7a � Ni3Sn4 C2/m mC14D7b Ta3B4 Immm oI14D71 Al4C3 R3m hR7D72 Co3S4 Fd3m cF56D73 Th3P4 I 43d cI28D8a Mn23Th6 Fm3m cF116D8b ! CrFe P42/mnm tP30D8c Mg2Zn11 Pm3 cP39D8d Al9Co2 P21/c mP22D8e Mg32�Al�Zn�49 Im3 cI162D8f Ge7Ir3 Im3m cI40D8g Ga2Mg5 Ibam oI28D8h W2B5 P63/mmc hP14D8i Mo2B5 R3m hR7D8k Th7S12 P63/m hP20D8l Cr5B3 I4/mcm tI32D8m W5Si3 I4/mcm tI32D81 Fe3Zn10 Im3m cI52D82 Cu5Zn8 I 43m cI52







D83 Al4Cu9 P43m cP52D84 Cr23C6 Fm3m cF116D85 Fe7W6 R3m hR13D86 Cu15Si4 I 43d cI76D88 Mn5Si3 P63/mcm hP16D89 Co9S8 Fm3m cF68D810 Al8Cr5 R3m hR26D811 Al5Co2 P63/mmc hP28D101 Cr7C3 Pnma oP40D102 Fe3Th7 P63mc hP20E01 PbFCl P4/nmm tP6E07 FeAsS P21/c mP24E1a Al2CuMg Cmcm oC16E1b AgAuTe4 P2/c mP12E11 CuFeS2 I 42d tI16E21 CaTiO3 Pm3m cP5E3 Al2CdS4 I 4 tI14E9a Al7Cu2Fe P4/mnc tP40E9b Al8FeMg3Si6 P62m hP18E9c Al9Mn3Si P63/mmc hP26E9d AlLi3N2 Ia3 cI96E9e CuFe2S3 Pnma oP24E9e Fe3W3C Fd3m cF112E94 Al4SiC4 P63mc hP18F5a FeKS2 C2/c mC16F01 NiSbS P213 cP12F51 CrNaS2 R3m hR4F56 CuSbS2 Pnma oP16H11 Al2MgO4 Fd3m cF56H24 Cu3VS4 P43m cP8H26 Cu2FeSnS4 I 42m tI6L′11 Fe4N P43m cP5L′12 AlFe3C Pm3m cP5L′2b ThH2 I4/mmm tI6L′3 Fe2N P63/mmc hP3L1a CuPt3 Fm3c cF32L1o AuCu P4/mmm tP2L11 CuPt R3m hR32L12 AuCu3 Pm3m cP4L2a � CuTi P4/mmm tP2L21 AlCu2Mn Fm3m cF16L22 Sb2Tl7 Im3m cI54L6o CuTi3 P4/mmm tP4




Table A1.4. Binary intermetallic phases: most commonly exhibited structures.

Structure type Number of binary phasesexhibiting the structure

Rank Strukturberichtdesignation

Pearsonsymbol

Prototypephase

1 A1 cF4 Cu 5202 A3 hP2 Mg 3623 B1 cF8 NaCl 3184 A2 cI2 W 3095 B2 cP2 CsCl 3076 L12 cP4 AuCu3 2667 C15 cF24 MgCu2 2438 D88 hP16 Mn5Si3 1779 C14 hP12 MgZn2 148

10 C32 hP3 AlB2 12211 Bf oC8 CrB 12012 D73 cI28 Th3P4 11713 D2d hP6 CaCu5 10614 D011 oP16 Fe3C 10115 B81 hP4 NiAs 10116 C23 oP12 Co2Si 9517 C1 cF12 CaF2 8718 L1o tP2 AuCu 8219 A15 cP8 Cr3Si 8220 C38 tP6 Cu2Sb 7421 B27 oP8 FeB 7322 � � � hP38 Ni17Th2 6223 C42 oI12 CeCu2 6124 B82 hP6 Ni2In 5425 C2 cP12 FeS2 5026 D8a cF116 Mn23Th6 4927 � � � hR36 Be3Nb 49



Chapter 2

Classification of Phase Transformations

2.1 Introduction 892.2 Basic Definitions 902.3 Classification Schemes 92

2.3.1 Classification based on thermodynamics 932.3.2 Classifications based on mechanisms 1012.3.3 Classification based on kinetics 105

2.4 Syncretist Classification 1052.5 Mixed Mode Transformations 115

2.5.1 Clustering and ordering 1152.5.2 First-order and second-order ordering 1162.5.3 Displacive and diffusional transformations 1202.5.4 Kinetic coupling of diffusional and displacive transformations 120

References 122






Chapter 2

Classification of Phase Transformations

Symbols and AbbreviationsG: Gibbs free energy ( G = H-TS)T: Temperature

Tc: Equilibrium transition temperatureS: EntropyP: PressureH: Enthalpy

Cp: Specific heat at constant pressure�: Generalized order parameter

Tm: Melting temperature�-phase: hcp phase in Ti and Zr based alloys�-phase: bcc phase in Ti and Zr based alloys

�∗: Order parameter corresponding to maximum in free energy�e: Equilibrium order parameter for a first order transition��

A: Chemical potential of component, A in the phase, �Cij: Elastic stiffness modulus (elastic constant)Ms: Temperature at which martensite starts forming during quenchingV: Specific volume

Vi, Gid: Velocity and dissipated free energy associated with interface processVd, Gdd: Velocity and dissipated free energy associated with diffusion process

2.1 INTRODUCTION

The study of phase transformations is of interest to metallurgists, geologists,chemists, physicists and indeed to all scientists concerned with the states of aggre-gation of atoms. Due to the multidisciplinary interest in this subject, a wide varietyof nomenclature, sometimes even misleading, has been introduced in the litera-ture for the characterization of different types of phase transformations. It is notuncommon that different sets of terminologies are used in different disciplinesfor describing essentially similar phase transformations which, in a generalizedmanner, can be defined as a change in the macrostate of an assembly of interac-ting atoms or molecules as a result of some variation in the external constraints.The diversity of scientific interest and the complexity of the possible interac-tions between individual atoms of the assembly naturally lead to many different

89




approaches for the study of phase transformations. Physicists primarily focus theirattention on higher-order continuous phase transitions in single-component systemssuch as magnetic, superconducting and superfluid transitions. In contrast, met-allurgists and chemists are mainly concerned with phase transformations (whichinclude phase reactions) involving changes in crystal structure, chemical com-position and order parameter (both long and shortrange). Phase transformationsencountered by geologists, though quite similar to those observed in metallic andceramic systems, usually occur over much more extended time and length scalesunder extreme conditions of pressure and temperature.

Phase transformations also occur in organic materials such as polymers, bio-logical systems and liquid crystals. Many of the relevant concepts developed forinorganic systems have parallels in organic systems. However, no attempt will bemade in this chapter to compare and contrast phase transformations in organic andinorganic systems as the nature of atomic interactions responsible for the transfor-mations is quite different in these two classes of materials. Alloys, intermetallicsand ceramics form a group of materials in which phase transformations can bediscussed on a common conceptual basis and, therefore, a single classificationscheme can be used for appropriately grouping different types of transformationsin these systems. As mentioned earlier, Ti- and Zr-based systems, which includealloys, intermetallics and ceramics, exhibit nearly all possible types of phasetransformations and, therefore, serve as excellent examples for studies on phasetransformations in inorganic materials in general.

Phase transformations can be classified on the basis of different criteria, namely,thermodynamic, kinetic and mechanistic (Christian 1965, Roy 1973, Rao and Rao1978). A comparison of the characteristic features of different types of trans-formations is presented in this chapter with a view to providing a coarse-brushpicture of these in a generalized manner. The chapters which follow will describethese transformations more elaborately, taking illustrative examples from Ti- andZr-based systems.

2.2 BASIC DEFINITIONS

In order to resolve some of the confusion and controversy which are of a semanticnature a summary of some basic definitions is presented here.

A phase is a portion of a system bounded by surfaces and with a distinctive andreproducible structure and composition. Within a single phase, minor fluctuationsin structure and/or in composition can occur. One phase can be distinguished froma second phase if at the contacting surface there is a sharp (within one or two atomlayers) first-order change in composition and/or structure and hence properties.



Classification of Phase Transformations 91

The two terms phase transformation and phase transition are often used inter-changeably. Sometimes a distinction between them is implied but rarely specified.In a paper entitled “A synchretist classification of phase transitions", Rustum Roy(1973) has addressed the controversy which exists in the literature in this regard.Generally the word phase transition is restricted to transitions between two phaseswhich have identical chemical compositions, while the term phase transformationcovers a wider spectrum of phenomena which include phase reactions leadingto compositional changes. In the metallurgical literature, phase transformationsinclude precipitation of a second phase, �, of a different crystal structure andchemical composition from the parent �′ phase (�′ → �+�), �′ and � having thesame crystal structure but different chemical compositions, eutectoid decomposi-tion (� → �+�) and many such processes which, in the chemistry literature, aregrouped as phase reactions (Rao and Rao 1978).

A more subtle point concerns the meaning of identical chemical composition.The equilibrium point defect concentration may be different in two polymorphs.Though in a strict sense they cannot be considered as identical in chemicalcomposition, transformations between such polymorphs are usually classified ascomposition-invariant transformations.

In considering equilibria between two phases, the requirement of reversibilitymust be taken into account. Several relationships pertaining to equilibria betweentwo phases can be explained using the free energy versus temperature plot of asingle component system (Figure 2.1). The liquid (L) to crystal (A) transition,

T L/A

Liquid

Crystal B

Crystal A

TA/B

Metastable A

EquilibriumGlass

Tg

Supercooled liquid

Temperature

Monotropic

Enantiotropic

Fre

e en

ergy

A′

T2T1

Figure 2.1. Free energy versus temperature plots showing phase transformations in a single-component system. The differences between monotropic and enantiotropic transitions and betweenstable equilibrium and metastable equilibrium transitions are highlighted.




L� A, at the melting point, TL/A, and the crystal A to crystal B transition, A� B,at the transition temperature, TA/B, are stable equilibrium transitions. The transitionbetween a metastable phase A and another metastable phase A′, A (metastable)� A′ (metastable), can also be an equilibrium transition and can be grouped alongwith the former two cases as being enantiotropic, i.e. reversible and governed byclassical thermodynamics.

In contrast, when we consider transitions which can be represented by verticallines in this diagram, such as A′ (metastable) → B (stable) at T1 and Glass →A (stable) at T2, the reversibility criterion is not met. These irreversible trans-formations, defined as monotropic transitions, proceed only in one direction andit is not possible to establish an equilibrium between the parent and the productphases.

Polymorphic transformations are generally defined as those which involve astructural transition without a change in the chemical composition. Sometimesthese transformations are also referred to as congruent processes. There are,however, several examples, such as the transformation of crystalline oxygen tocrystalline ozone and transformations of position isomers, which satisfy the afore-mentioned definition of a polymorphic transition, but cannot even be consideredas phase transitions. This is because ozone and oxygen, in the phase rule sense, aretwo different substances (or components) which survived even the solid → liquid→ vapour transitions while preserving their individuality. Similarly each positionisomer is an individual component and, therefore, isomeric transitions cannot beconsidered as phase transitions. In view of this, the definition of polymorphic trans-formations needs to be restricted to transformations involving phases with differentcrystal structures which are part of a single component system. In multicomponentmetallic alloys and intermetallics, chemical composition-invariant crystallizationis a good example in which the parent phase transforms to the product withoutallowing any partitioning of the constituent elements (or components) betweenthe two phases. In this sense, the system behaves as if it is a single-componentsystem.

2.3 CLASSIFICATION SCHEMES

There are several ways in which phase transformations can be classified, based onthermodynamic, kinetic and mechanistic criteria. A single classification schememay not be adequate to include all types of transformations encountered in allvarieties of materials. In this chapter, an attempt is made to evolve a classi-fication scheme which is applicable to phase transformations in metals, alloys,intermetallics and ceramics.




2.3.1 Classification based on thermodynamicsEhrenfest (1933) proposed a classification based on the successive differentiationof a thermodynamic potential, usually the Gibbs free energy function, with respectto an external variable such as temperature or pressure. The order of a transfor-mation is then given by the lowest derivative which shows a discontinuity at thetransition point. In the generalized sense, for an nth order transition

(nG

Tn

)P

�= 0�

(n−1G

Tn−1

)= 0 at T = Tc (2.1)

where G represents Gibbs free energy and Tc is the equilibrium transition tem-perature. It is to be noted that the Ehrenfest classification can be used only forequilibrium transitions of a single component system. Substituting n = 1 and 2 inEq. (2.1), at T = Tc we get for first-order transitions

G = 0�

(G

T

)P

= −S = H

Tc

(2.2)

and for second-order transitions

(G

T

)P

= −S = −(

H

Tc

)= 0�

(2G

T 2

)P

= 1Tc

(H

T

)P

= Cp

Tc

�= 0

(2.3)

A comparison between a first- and a second-order transition can be made inschematic plots of different thermodynamic quantities as functions of temperature(Figure 2.2). First-order transitions are characterized by discontinuous changes inentropy, enthalpy and specific volume. The change in enthalpy corresponds to theevolution of a latent heat of transformation, and the specific heat at the transitiontemperature, as a consequence, is effectively infinite.

In contrast, second-order transitions are characterized by the absence of alatent heat of transformation (as H , S and V do not undergo a discontinuouschange at Tc) and a high specific heat at the transition temperature. There areexperimental results which show that in some instances of second-order transitionsthe specific heat at Tc exhibits infinity rather than a finite discontinuity. A truesecond-order transition is, therefore, defined as one showing a finite discontinuityin the second derivative of the Gibbs function while a so-called lambda pointtransition exhibits an infinity. Though the Ehrenfest classification examines thepresence of a discontinuity in the nth derivative of the Gibbs function for decidingthe order of a transition, in the modern literature transitions with n ≥ 2 are grouped




G

H

CP

T

T

T

CP

TTM Tc

Tc

Tc

G

TTM

S

L

H

TTM

S

L

q

Temperature

First order Second order

Figure 2.2. Changes in the thermodynamic quantities, free energy, enthalpy and spe-cific heat, at the transition temperatures corresponding to first-order and second-orderphase changes.

together as “higher-order transitions” which are characterized by a continuousfirst derivative, �G/T P = 0, followed by either a discontinuity or infinity for“higher” derivatives.

In a multidimensional plot of free energy against temperature, pressure, etc. eachphase can be represented by a well-defined surface, as illustrated in Figure 2.3.The equilibrium transformation conditions between two phases are then definedby the intersection of two such surfaces. Moreover the free energy surface fora given phase may be extrapolated into conditions where that phase is not inthermodynamic equilibrium, and the difference in free energy, which is representedby the separation of the free energy surfaces corresponding to the two phases, canbe regarded as the driving force for a first-order transformation from one phaseto the other. This concept of a metastable phase is not readily applicable to asecond-order transformation where it is more appropriate to consider that there isa single continuous free energy surface.

Most of the phase transformations encountered in metallic systems are of thefirst-order type. Ferromagnetic ordering and some chemical ordering processesare examples of higher-order transitions in metallic systems. These transitionscan be represented in “mean field” descriptions of cooperative phenomena wherethe respective order parameters continuously decrease to zero as the temperature




Temperature

Fre

e en

ergy

Pressure

β

βα

α

Figure 2.3. Free energy surfaces for two phases, � and �, as functions of pressure (P) and temper-ature (T ). The projection of the line of intersection of the two surfaces on the P–T plane representsthe �–� phase boundary in the P–T phase diagram.

is raised to the transition temperature (Curie temperature or the critical orderingtemperature), as shown in Figure 2.4. Any transition which can be described interms of a continuous change in one or more order parameters can be treated interms of a generalized Landau equation (Landau and Lifshitz 1969) which states

(a) (b)

Tc

η

T

Second order

Tc

η

T

First order

Figure 2.4. Order parameter (�) versus temperature (T ) plots for (a) second-order and (b) first-ordertransitions.




that close to the critical temperature the free energy difference, G, between finiteand zero values of the order parameter, �, may be expanded as a power series:

G = A�2 +B�3 +C�4 +· · · (2.4)

the coefficients A, B, C, etc. being functions of pressure and temperature.The fundamental differences between the first- and higher-order transitions can

be explained on the basis of the corresponding Landau plots. For higher-ordertransitions, the free energy must be an even function of � which means thatB = 0 (and similarly the coefficients of the odd-powered terms of � are zero).Figure 2.5(a) shows the G versus � plots for higher-order transitions at differenttemperatures, both above and below Tc. When T > Tc, the system exhibits a singlestable equilibrium at � = 0 which corresponds to a positive value of A. As thetemperature approaches Tc, the curvature (2G/�2 at � = 0 gradually decreasesand as the temperature is lowered below Tc, the curvature as well as the value ofA become negative. This essentially means that the system becomes unstable atT = Tc and any infinitesimal fluctuation in the order parameter leads to a lowering

(a)(b)

Second order

Fre

e en

ergy

–ve +ve

Tc = Ti

0

T > Tc

T ≈ Tc

Tc > T

η

First order

Free

ene

rgy

Tc > T > Ti

Tc >> Ti

Ti > T

T > Tc

T ≈ Tc

0η

η = ηc

Figure 2.5. Free energy as a function of order parameter (�) for (a) second-order and (b) first-order transitions. In the case of second-order transitions, the parent phase becomes unstable,(2G/�2) < 0 at � = 0 at the transition temperature, Tc, which is the same as the instabilitytemperature, Ti. For some first-order transitions an instability temperature, Ti (which is much lowerthan the equilibrium transition temperature, Tc), can be identified where the parent phase becomesunstable at � = 0.




of the free energy. The negative curvature of the G versus � plot also implies thatwith an increase in the order parameter, the free energy progressively decreases,finally reaching the stable equilibrium positions defined by the minima present inthe plots corresponding to T < Tc. The two minima corresponding to the positiveand negative � values represent two equivalent states associated with antiphasedomains of the same ordered structure.

Landau plots for a first-order transition are shown in Figure 2.5(b). In this case,the value of B in the Landau expansion (Eq. (2.4)) is not zero. At the transformationtemperature, the G versus � curve shows two minima, one at � = 0 and the otherat � = �c, the two minima being separated by a free energy barrier. At � = 0 thesystem is not unstable as the curvature remains positive at this point at T = Tc.A continuous increase in the order parameter, therefore, will initially raise thefree energy which will drop only after the peak of the free energy hill is crossed.Since the system as a whole is not unstable either at Tc or at temperatures closeto but below Tc, a gradual transition of the system in a homogeneous manner tothe free energy minimum at or near � = �c is not possible. A phase transitionunder such a situation can initiate only if localized portions of the system areactivated to cross the free energy barrier to reach a point beyond �∗ where �can grow further spontaneously. The formation of such localized product phaseregions (where � has nearly reached the �c value) is known as nucleation. Theproduct nuclei remain separated from the parent phase by sharp interfaces and thephase transition proceeds through the growth of these nuclei.

The presence of two free energy minima separated by a free energy hill near theequilibrium in the case of a first-order transformation brings out its characteristicfeatures, namely, the coexistence of the parent and the product phases and thediscrete nature (involving nucleation and growth) of the transformation. In contrast,all higher-order transitions, by definition, are homogeneous in the sense that theparent and the product phases cannot be distinguished at any stage of the transitionand there is no question of having an interface between the two phases. It isinteresting to note that a discussion on Landau’s theory, which is strictly concernedonly with the equilibrium state, has led us to consider continuous vis-à-vis discretetransformations.

Continuous or homogeneous transitions are those in which the parent phaseas a whole gradually evolves into the product phase without creating a localizedsharp change in the thermodynamic properties and the structure in any part ofthe system. Such a process can occur only when the system becomes unstablewith respect to an infinitesimal fluctuation which leads to the transition and thefree energy of the system continuously decreases with the amplification of sucha fluctuation. All higher-order transitions, by definition, satisfy the condition ofhomogeneous/continuous transformation at equilibrium. In contrast, all first-order




transitions at equilibrium are discrete transitions which necessarily involve nucle-ation and growth. When we consider Landau plots for first-order transitions attemperatures far below Tc (Figure 2.5(b)) we notice that at temperatures below theinstability temperature, Ti, the curvature of the G versus � plot becomes negativeat � = 0 and the free energy continuously drops with increase in �, finally reachingthe value corresponding to �c. Therefore below Ti a first-order transition can alsoproceed in a continuous mode. It must be emphasized here that the occurrence ofan instability temperature is not universal for all first-order transitions. Only inrather a limited number of cases can first-order transitions be described in terms ofa Landau representation. Spinodal clustering, spinodal ordering and displacementordering processes are some examples in which continuous first-order transitionsare encountered in conditions far from equilibrium.

Equilibrium phase diagrams showing a miscibility gap correspond to solidsolutions which exhibit a clustering tendency. The boundary of the equilibriumtwo-phase field, �1 + �2, in the phase diagram (Figure 2.6) is determined byequating the chemical potentials � of the two components, A and B, in the twophases, �1 and �2, in equilibrium at a given temperature, T1:

��1A�B = �

�2A�B (2.5)

α2α1

X 2X 1

A

B

C

D

α

Atom fraction (X)

Tem

pera

ture

EquilibriumsolvusCoherentsolvus

Coherentspinodal

T1

Figure 2.6. Schematic phase diagram for a clustering system which remains a homogeneous solidsolution in region A. A phase separation process occurs by a discrete nucleation and growth mech-anism in regions B and C. Spinodal decomposition occurs in region D.




The concentrations, X1 and X2, at which the tie line at T1 intersects the miscibilitygap correspond to those of �1 and �2. A homogeneous solid solution � decomposesinto an incoherent mixture of �1 and �2 as it is brought from the region A to theregion B. Such a decomposition reaction, involving the creation of sharp interfacesbetween the two phases, is undoubtedly of the first order. A coherent mixture ofthe two phases can exist in the region C defined by the coherent miscibility gap.The coherent spinodal region D, fully residing within the region C, defines theconcentration–temperature field where the phase separation process can initiate byintroducing long wavelength (relative to interatomic distances) concentration fluc-tuations and a continuous amplification of such fluctuations. Though the processis continuous, the transformation is of the first order except at the point wherethe coherent spinodal touches the coherent solvus, where it may be considered tobe of second order. The driving force for the phase separation arising from thenegative curvature of the free energy (G)–concentration (X) plot ((2G/X2)< 0in the spinodal region) is opposed by the gradient energy and the coherency strainenergy. All these factors and a correction factor for thermal fluctuations determinethe wave number for which the amplification rate is the highest.

Conceptually a chemical ordering process in which the ordered superlattice canbe created only by replacement of atoms in the lattice of the disordered phasecan be described in a manner similar to the spinodal clustering process. In thecase of continuous ordering, concentration modulations with wavelengths of theorder of the interatomic spacing need to be introduced. For an ordering system,the effective gradient energy is negative, and many of the predictions are oppositeto those of spinodal decomposition. The amplification factor is negative beyond acritical wavelength and the maximum amplification corresponds to a wavelengthequal to a small lattice vector.

As shown in Figure 2.7(a) and (b), continuous ordering can be envisaged forboth first-order and second-order reactions. Any change in the lattice dimensionsdue to ordering introduces a third-order term in the Landau equation (Eq. (2.4))which makes the transformation first order. Continuous ordering in the first-ordercase requires finite supercooling below the coherent phase boundary. De Fontaine(1975) has also distinguished spinodal ordering from continuous ordering. Inthe former, the early stages of ordering are characterized by the ordering wavevectors which maximize the amplification factor but the amplification of thesedoes not evolve the equilibrium ordered structure. In true continuous ordering, theequilibrium ordered structure continuously evolves from a low-amplitude quasi-homogeneous concentration wave. De Fontaine (1975) has also examined theLandau–Lifshitz symmetry rules for determining whether a specific ordering wavevector qualifies to be a candidate for a second-order transformation.




XX

Tem

pera

ture A

B

αo

Second order First order

α dα d

A

D

C

B

Equilibriumphase

boundaryCoherentphaseboundary

Coherentinstabilityboundary, T i

(a) (b)

Figure 2.7. Schematic phase diagram for an ordering transition of (a) second order and (b) first orderin which continuous ordering occurs under supercooling below the instability temperature, Ti. RegionA represents the phase field where the disordered solid solution, �d, is stable while Region Bcorresponds to the stability domain of two-phase mixture, �o + �d. Region C corresponds to stabilitydomain of ordered �o. In the case of a second-order process the �d → �o transition occurs ina continuous manner below the equilibrium phase boundary. In contrast, continuous ordering ispossible in Region D only below the coherent instability boundary, Ti, in some first-order transitions(where the symmetry elements of the ordered structure form a subset of those of the parent disorderedstructure).

A product phase can evolve from the parent phase through a continuous dis-placement of the parent lattice. Two types of displacement, namely, a homoge-neous lattice deformation and a relative displacement of atoms within unit cells(often called shuffles), can take place, either singly or in combination. The lat-tice deformation produces a change in volume and external shape and it seemsvery improbable that this could be accomplished continuously or homogeneouslyunless the principal lattice strains are very small (within the limits of linear elasticstrain). Most martensitic transformations in metals and alloys involve much largervalues of lattice strains and are, therefore, not candidates for continuous transfor-mations. However, the possibility of a continuous displacive transformation hasto be considered if the lattice deformation is very small.

Transformations in some ferroelectric crystals of low symmetry are believed tobe of the second order and occur by the progressive development of an instabilityin a dynamic plane displacement which becomes, at Tc, a static wave extendingthrough the crystal. The approaching soft-mode instability is indicated as a pre-transition effect above the transition temperature by a reduction in an appropriatelattice stiffness. Low and reducing values of the shear constant, 1

2 (C11 −C12), are




found in the parent phase as Ms is approached from higher temperatures in manynoble metal bcc alloys and in fcc In–Tl; but in other martensitic transformations,especially those in steels, there are no anomalies of this kind and, therefore, thesetransformations are clearly of the first order.

The athermal bcc → � transformation, which is envisaged as a pure shuffletransition, is another example of a continuous displacive transformation which isa first-order transformation from the consideration of symmetry rules. Though itis possible to accomplish this transformation by a continuous amplification of adisplacement wave, the observed fine particle and dual phase (� + �) morphologysuggests that �-particles form in a quasiperiodic manner through heterophasefluctuations which have the form of ellipsoidal wave packets of displacement wave(Cook 1974).

2.3.2 Classifications based on mechanismsBuerger (1951) has introduced a classification based on mechanisms, namely,reconstructive and displacive transformations. In the metallurgical literature, how-ever, a mechanistic classification groups transformations into (a) nucleation andgrowth and (b) martensitic types. The introduction of the term nucleation andgrowth in this context has created considerable confusion, as all first-order trans-formations including martensitic transformations require the nucleation and thegrowth steps. In the current literature, usage of such confusing nomenclature isavoided and a mechanistic classification designates the two classes as (a) diffu-sional and (b) displacive transformations. The former corresponds to the recon-structive transformation in which atom movements from the parent to the productlattice sites occur by random diffusional jumps. This implies that near neighbourbonds are broken at the transformation front and the product structure is recon-structed by placing the incoming atoms at appropriate positions which results inthe growth of the product lattice. In contrast, atom movements in a displacivetransformation can be accomplished by a homogeneous distortion, shuffling oflattice planes, static displacement waves or a combination of these. All thesedisplacive modes involve cooperative movements of large numbers of atoms ina diffusionless process. Displacive (which includes martensitic) transformationsinitiate by the formation of nuclei of the product phase, and the growth of thesenuclei occurs by the movement of a shear front at a speed that approaches thespeed of sound in the material under consideration.

In order to differentiate the mechanisms of atom movements across the trans-formation front, Christian (1965, 1979) has compared the movements involved indiffusional and displacive transformations with civilian and military movements,respectively. In the latter case, if the atoms are labelled in the parent lattice, thecoordination between the neighbours can be shown to be essentially retained in




A

B

C

D

E

F

G H I J A G H I JB

C

D

E

FShear

(b)

(a)

Shear

RM

N

Q

O

S

P

N′

M′

O′

Q′

P′

Figure 2.8. Schematic illustration of lattice correspondence in a two-dimensional lattice. Thoughthe parent and product lattices in both (a) and (b) are identical, the lattice correspondences in the twocases can be distinguished if the dots representing atoms occupying the lattice sites can be labelled.It is through the establishment of the lattice correspondence that the nature of the homogeneousshear and shuffle, if required, can be identified.

the product lattice, though the bond angles undergo changes. This point is illus-trated in Figure 2.8(a) which shows how a set of atoms (labelled A, B, C, etc.)decorating the parent lattice changes to the product lattice. The existence of alattice correspondence implies that a vector in the parent lattice, defined by thesequence of atoms ABCD� � � , becomes a vector in the product lattice with theatoms arranged in the same sequence, although the spacing between them getsaltered to match the product lattice dimensions. Such a transformation can beviewed as a homogeneous deformation of the parent lattice (a simple shear in the




case of the two-dimensional illustration in Figure 2.8(a), shear directions beingshown by arrows). The importance of lattice correspondence in determining theshear can be illustrated by Figure 2.8(b) which shows identical arrays of spotsrepresenting the parent and the product structures but with a different lattice cor-respondence. The atomic rows MNO and MQR are shown in both the parent andthe product structures. It is evident from this drawing that the product structureis derived by a combination of a homogeneous deformation and a shuffle. Theshear, as indicated by arrows, transforms the rectangle MOPQ in the parent toa parallelogram M′O′P′Q′ in the product and the atom N is shifted to its newposition by a shuffle.

The best experimental evidence for the inheritance of the atomic coordina-tion through a displacive transformation is provided by the observation that thechemical order present in the parent structure is fully retained in the productstructure. A similar correspondence also exists for crystallographic planes. A rela-tionship of this kind in which straight lines transform to straight lines and planesto planes is described mathematically as an affine transformation. Physically itmay be considered as a homogeneous deformation of one lattice into the other.The correspondence associates each vector, plane and unit cell of the parent witha corresponding vector, plane and unit cell of the product. In general, the corre-sponding lattice vectors and the spacings of corresponding lattice planes are notequal in the two structures, and the angular relation between any pair of latticevectors in the parent structure is not preserved in the product. It is to be notedthat the lattice correspondence does not by itself imply any orientation relationbetween the phases, since the transformation may involve a rigid body rotation ofthe product structure with respect to the parent structure.

In diffusional transformations, such lattice correspondences are not present.Even in those cases of diffusional transformations in which the chemical compo-sitions of the parent and the daughter phases are identical and a strict orientationrelationship exists between them, random jumps of atoms from the parent to theproduct lattice positions do not permit the lattice correspondence to be preserved.Such composition-invariant diffusional transformations proceed by atomic jumpsacross the advancing transformation fronts which separate the parent and the prod-uct phases. In order to understand the basic difference between displacive anddiffusional transformations let us again consider labelling the atoms as A, B, C,D, etc. in the parent lattice and the same set of atoms as A′, B′, C′, D′ etc. in theproduct lattice in Figure 2.9(a) and (b), respectively. The transformation front hasbeen shown to advance by a single atomic layer. This schematic drawing showsthat the sequence in which these atoms were placed before the transformation isnot the same as that in the product lattice. This can happen if each atom breaksthe bonds with its neighbours in the parent lattice and shifts to a new position




A

B B′

CC′

DD′

EE′

F F′

Transformationfront

A

A′

B′

B

CC′

D

D′E

E′

FF′

(a)

(b)

Figure 2.9. Schematic diagram of atom movements across the transformation front in a (a) displacivetransformation; (b) diffusional transformation.

which corresponds to a lattice point in the product structure. In this manner, thetransformation boundary proceeds towards the parent phase, converting the parentto the product phase. The jumps of atoms A, B, C, etc. are random and are notcorrelated with those of their neighbours, unlike the case of displacive transfor-mations. Since diffusional transformations involve the breaking of bonds betweenneighbouring atoms and the reconstruction of bonds to form the product phasestructure, they are also known as reconstructive transformations.




2.3.3 Classification based on kineticsPhase transformations are also grouped in terms of the kinetics of the process.The most important distinguishing kinetic criterion is the requirement of thermalactivation. First-order transformations necessarily occur by the nucleation of theproduct followed by its growth by the propagation of the interface between theparent and the product phases. The movement of the interface can be eitherthermally activated or athermal. The atom transfer process across the interface isthermally activated in the case of the former while it does not require the assistanceof thermal fluctuations in the latter.

The kinetic classification, originally introduced by Le Chatelier (Roy 1973),divides phase transformations into two main groups: (a) rapid or nonquenchableand (b) sluggish or quenchable. Transformations belonging to the former classare so fast that the parent phase, which is stable at high temperatures (or highpressures), cannot be retained by a rapid quench to ambient conditions. In contrast,sluggish transitions are slow to the extent that the high-temperature (or high-pressure) phase can be retained metastably on quenching. The basic idea behindthis classification scheme also centres around the requirement of thermal activation.This classification, however, suffers from the limitation that the experimentalability to rapidly change temperature and pressure is continuously improvingand transformations which are grouped as “non-quenchable” today may become“quenchable” tomorrow.

A truly displacive transformation occurs through the passage of a glissile inter-face which is essentially a displacement (or shear) front, the movement of whichis not assisted by thermal activation. Such transformations are non-quenchableirrespective of the quenching rate employed.

2.4 SYNCRETIST CLASSIFICATION

The fundamental parameters on the basis of which a phase transformation isclassified are thermodynamic, mechanistic and kinetic. A syncretist classifica-tion scheme has been introduced by Roy (1973) by taking all these aspects intoaccount. Figure 2.10 shows a three-dimensional matrix with the x-, y- and z-axesrepresenting the mechanistic (structural), thermodynamic and kinetic parameters,respectively. Along the x-axis the two major classes, namely, diffusional (recon-structive) and displacive transformations, are separated by a “mixed” class oftransformations which have attributes of both displacive and diffusional transfor-mations. Examples of each of these are available in Ti- and Zr-based systems.Martensitic transformations of the bcc (�) phase of pure Ti, Zr and of alloysbased on these metals have been discussed in detail in Chapter 4 which also deals




Thermodyn

amic

Kin

etic

Mechanistic (Structural)

Rap

id

Ath

erm

al

Inte

rmed

iate

Slo

w

The

rmal

ly a

ctiv

ated

Displacive MixedΔH and ΔV = 0

order

Second

order

ΔH and ΔV ≠ 0

First

Mixed

pretransit

ion

effect

with

small Δ

H , ΔV

x

y

z

Diffusional

Figure 2.10. Syncretist classification scheme of phase transformations based on mechanistic (struc-tural), thermodynamic and kinetic criteria.

with martensites in NiTi-based intermetallics and ZrO2-based ceramics. A host ofdiffusional transformations such as precipitation, amorphous-to-crystalline phasetransformations, massive transformation, eutectoid phase reactions have also beenencountered in these systems and they are discussed in Chapters 4 and 7.

Displacive transformations can be further divided into different subgroups,depending on whether the transformation is dominated by lattice strains (marten-sitic transformation) or by shuffles (e.g. omega transition and ferroelectric tran-sitions). The � → � transition which is frequently observed in several Ti- andZr-based systems is unique with respect to lattice registry in three dimensions,pretransition effects and transformation product morphology. A detailed accountof the �-transformation is presented in Chapter 7.

The classifications based on thermodynamic criteria, represented along they-axis of Figure 2.10, divide phase transformations into first-order, higher-orderand “mixed” transformations. The distinctions between these classes of transfor-mations are illustrated in Figure 2.11 which depicts the variation of thermodynamicquantities such as specific volume (V ), enthalpy (H) and entropy (S) at and nearthe transition temperature. In a first-order transition, there is a step change in thesequantities at this temperature and there is no need for V , H and S of one phase toshow a tendency to gradually approach the value corresponding to the other phaseas the transition temperature is approached. A second-order transition is charac-terized by a gradual change in V , H and S and the absence of a step change in




T T T

Specificvolume

Enthalpy

Entropy

Temperature

(a) First order (b) Mixed (c) Higher order

Figure 2.11. Changes in specific volume (V ), enthalpy (H) and entropy (S) at and near the transitiontemperature in (a) second-order and (b) and (c) first-order transitions; (b) shows a “mixed” character,exhibiting pretransition effects as well as step changes at the transition temperature.

these parameters at the transition temperature. The mixed situation is encounteredin a large number of transitions such as ferroelectric and ferromagnetic transitions.In such mixed transformations, though a pretransition second-order-like effect isobserved, there is a finite discontinuous jump in the value of these thermodynamicquantities at the transition temperature.

Considering kinetics as the third variable, phase transformations can be groupedinto thermal and athermal classes. All true displacive transformations are athermalwhich cannot be suppressed by quenching. In contrast, reconstructive or diffusionaltransformations are invariably thermally activated and, therefore, such transfor-mations are, in principle, suppressible on quenching. The required quenching ratefor suppressing a diffusional transformation, however, varies depending on theincubation period involved.

Having discussed different schemes of classification of phase transformationsin alloys, intermetallics and ceramics, we will now examine how a given phasetransformation can be classified on the basis of thermodynamic, kinetic and mech-anistic criteria. A classification tree (Figure 2.12) can be constructed by addressingappropriate questions at different levels.

The first question to raise is whether the transformation is homogeneous (orcontinuous) or discrete. Higher-order transitions are continuous while first-ordertransitions are discrete at the equilibrium transition temperature, Tc. Some first-order transitions exhibit an instability temperature, Ti (Ti << Tc), below whichthe transformation initiates in a continuous manner. Homogeneous transforma-tions can further be subdivided depending on the nature of fluctuations (waves),the amplification of which represents the progress of the transformation, and the




Phase transformations

Higher order First orderHomogeneous

continuousNecessarily Below instabilitytemperature

in selected cases

DiscreteNucleation

Athermal Thermally activated

(Kinetic)

Stable particles of new phase

Growth(Mechanistic)

Diffusionalreconstructive Displacive

Long range Short range

Thermally activatedatom transport

Cooperative movementsof atoms, athermal

Lattice straindominated

Shuffledominated

Interfacediffusion

controlled

Bulk diffusioncontrolled

Interfacediffusioncontrolled

Solute partitioning Compositioninvariant

Polymorphiccrystallization

Massivetransformation

Chemical ordering

Recrystallization

Cellularprecipitation

Eutectoiddecomposition

Homogeneousand

heterogeneousprecipitation

Displaciveomega

transformations

Martensitictransformations

Composition invariant

fcc → bcc, bctin Fe alloys

bcc → hcp inTi, Zr alloys

bcc → ωin

Ti, Zr alloys

(Thermodynamic)

Figure 2.12. Classification tree for phase transformations.




Table 2.1. Lattice wave descriptions of homogeneous transformations.

Nature of wave Long wavelength Short wavelength

Replacive Spinodal clustering Spinodal orderingDisplacive Martensitic OmegaElectric Ferroelectric AntiferroelectricMagnetic Ferromagnetic Antiferromagnetic

wavelength associated with these waves. Table 2.1 lists different types of homo-geneous transformations which can be distinguished on the basis of the nature ofthe wave and the wavelength.

Discrete transformations, by definition, occur by the nucleation and growthprocess in which a small particle of the product phase (or a phase structurally andchemically close to the product phase) appears in the parent matrix. The two phasesare separated by a sharp interface which gradually moves towards the parent phase,converting it to the product phase. Discrete transformations are subdivided intodiffusional (or reconstructive) and displacive, based on the mechanism of atomtransfer across the advancing transformation front. When the atom jump across theinterface occurs by random diffusional jumps, the transformation is designated as adiffusional transformation. A subset of diffusional transformations is the replacivetransformation which is accomplished by replacement of atoms in the lattice ofthe parent phase without destroying the parent lattice. Good examples of suchtransformations are those chemical ordering reactions which lead to the formationof a superlattice, without involving lattice reconstruction but rearrangement ofatoms in the parent lattice.

Phase transformations in which the chemical composition of the parent phase isnot inherited by the product naturally involve partitioning of different componentsthrough long-range diffusion. Identification of such transformations as belongingto the category of diffusional transformations is rather trivial. Those cases whereboth the parent and the product phases have an identical composition, for exam-ple, the polymorphic transformation of a single-component system, the massivetransformation of a multicomponent alloy or the martensitic transformation, canbe grouped into the two categories, namely, displacive and diffusional, dependingon the mechanism of atom transport across the transformation front.

The mechanisms of atomic movements are difficult to observe through directexperiments. Therefore, the growth mechanism is inferred from a variety of crys-tallographic, morphological and kinetic observations such as presence of latticecorrespondence, orientation relation, shape of the product phase, habit plane,macroscopic shape deformation, inheritance of atomic order and thermally acti-vated or athermal nature of the growth process. These experimental observables




are good indicators for deciphering the mechanisms of atomic movements and in agreat majority of cases the growth mechanisms established from these observationshave received universal acceptability. There are, however, some cases for whichunanimity regarding the mechanism of atomic movements has not been arrived at.There is also a continued debate as to whether the operation of a shear mechanismin a transformation can be inferred from the existence of lattice correspondenceand from the fact that the habit plane obeys the invariant plane strain condition.Some of these issues are discussed in later chapters of this book.

The nature of the nucleation step in discrete transformations needs to be exam-ined for recognizing the class of a given transformation. For example, a martensitictransformation can have either an athermal or a thermally activated nucleationstep. In the former case, the overall transformation assumes an athermal char-acter, while in the latter the volume fraction of the martensite grows with timesigmoidally at a constant temperature between the start (Ms) and finish (Mf )temperatures associated with martensite formation (Figure 2.13). Athermal nucle-ation is also encountered in diffusional transformations such as crystallization ofmetallic glasses. In this case, the nuclei of crystalline phases present in the as-quenched amorphous matrix are activated for growth during the crystallizationprocess.

Diffusional transformations can be further subdivided into different cate-gories based on the diffusion lengths of the different atomic species requiredto accomplish the transformation. In composition-invariant transformations, suchas polymorphic transformations of single-component systems, polymorphic crys-tallization, massive transformation and partitionless solidification, the random

Temperature

Mar

tens

ite (

%)

Ms

Temperature

Mar

tens

ite (

%)

Mf

~100%Athermal Athermal (Burst)

(a) (b)

Time

Mar

tens

ite (

%)

T1

Ms > T1 > Mf

Isothermal

(c)

Figure 2.13. (a) and (b) Volume fraction transformed as a function of temperature in the caseof athermal nucleation of martensite. (c) Increase in volume fraction transformed with time at a con-stant temperature, T (Ms > T > Mf ), resulting from thermally activated nucleation of martensite.




diffusive atom movements occur only across the advancing transformation front.The distances involved in the diffusion process are in the range of the nearestneighbour atomic distances. These transformations are, therefore, designated asshort-range diffusional transformations.

The partitioning of atomic species occurs between the product phases mainlyby interfacial diffusion in cases such as eutectoid decomposition, eutectic crystal-lization and cellular precipitation, resulting in a two-phase lamellar product. Theinterlamellar spacing can vary from a few nanometres to a few micrometres. Thesetransformations can be grouped into the category of intermediate-range diffusionaltransformations. Sometimes classification is made on the basis of the nature ofdiffusion – whether bulk or interfacial – which dominates in the transformationprocess. If one draws a comparison between the eutectic and the eutectoid decom-positions, one can see that in the former case diffusion is mainly in the liquidphase ahead of the transformation front whereas in the latter case, partitioning ofdifferent components occurs primarily at the transformation front.

As indicated earlier, displacive transformations are those which can be accom-plished by introducing a lattice deformation in the parent lattice. In this classof transformations, a perfect lattice correspondence is maintained and chemi-cal order, if present in the parent phase, is retained in the product phase. Avery wide range of transformations are grouped in this class, which covers soft-mode ferroelectric and ferroelastic transitions in materials such as SrTiO3 andBaTiO3, the omega transition in Ti and Zr alloys, shear transformations in �(bcc) phases in noble metal alloys and martensitic transformations in intermetallicssuch as nickel aluminides and nickel titanides, cubic-to-tetragonal or cubic-to-orthorhombic transitions involving small lattice strains and classical systems ofiron-based alloys. Since the characteristics of all of them are not the same, theyhave been further subdivided into different groups using different criteria forclassification. In recent literature, Cohen et al. (1979), Delaye et al. (1982) andChristian (1990) have proposed these classification schemes which are summarizedin Table 2.2.

It has been mentioned earlier that a displacive transformation involves a homo-geneous strain of the parent lattice which is accompanied by atomic shuffles(relative displacements of atoms within unit cells) in some cases. The relativecontributions of the lattice strain and the shuffle can be used as important criteriafor grouping martensites into two classes, namely, the shuffle dominated and thelattice strain dominated transformations.

There are a number of examples of displacive transformations which exhibitpretransition softening of elastic moduli. Amongst these, Ni–Al alloys, contain-ing 30–50% Al, constitute the most widely studied systems; they show weakto moderate first-order character. The fact that the high temperature �2-phase




Table 2.2. Classification of displacive transformations.

Criteria Classification

Magnitude of shuffle and ofhomogeneous lattice strain(Cohen et al. 1979)

Shuffle dominated FerroelectricFerromagneticOmega

Lattice strain dominated Martensitic: high strainenergy controlsmorphology and kinetics

Quasimartensitic: low strainenergy; can occurcontinuously

Presence of precursormechanical instability(Delaye et al. 1982)

No mechanical instability asMs is approachedStrongly first order

Allotropic changes in pureelements

Transformations in primarysolid solutions

Moderate indications ofmechanical instabilities

Moderate first order

�-phases of noble metalalloys and Ni-based shapememory alloys

Marked mechanical instability Cubic → tetragonalWeakly first order, second order Cubic → orthorhombic with

small lattice strains

Structural basis(Christian 1990)

Between close packed layerstructure

fcc, hcp, 9R, 18R, 4H, etc.;including monolithic andorthorhombic distortions

Between fcc, bcc and derivedstructure

Between bcc, hcp and derivedstructure

Between cubic and tetragonal andslight distortions

(CsCl structure) prepares itself for the transformation as the temperature is low-ered towards the martensitic start (Ms) temperature is well reflected in X-rayand electron scattering as well as in acoustic measurements. The entire �4��0�transverse acoustic phonon branch (corresponding to the shear modulus C ′ in thelimit � → 0) is unusually low and the energy decreases considerably (though notto zero) at certain wave vectors as the temperature approaches Ms. This partialsoftening and the evolution of diffuse scattering due to elastic strain along the��0� directions indicate the existence of localized fine-scale displacement pat-terns. In the Ni62�5Al37�5 alloy, the presence of localized displacements, which areremarkably similar to that required for the formation of the 7R martensite (the




stacking sequence of close packed planes being ABAC), has been observed in high-resolution electron microscopy images just prior to the transformation (Tanner etal. 1990). As the parent phase approaches the Ms temperature, the density of suchmicroregions deformed by lattice strains increases. The nucleation event in suchcases can be viewed as a collapse or growth of microdomains associated withnon-uniform lattice strains into a macrodomain of a size larger than the criticalsize of a nucleus and of homogeneous and nearly appropriate lattice strain.

On the question of precursors in martensitic transformations, there are conflict-ing observations reported in different systems. Martensitic transformations can,therefore, be divided into different subclasses based on the nature and extent ofprecursor mechanical instability. Strongly first-order martensitic transformationsdo not show any softening of elastic constants as the system approaches the Ms

temperature. In contrast, there are systems where moderate or marked indicationsof softening of elastic constants are present in the vicinity of the Ms temperature.Generally these “mixed” transformations, exhibiting partial mode softening, areassociated with small lattice strains. Displacive-type structural transitions accom-panying ferroelectric transitions in BaTiO3 can be cited as examples. These areassociated with the displacement of a whole sublattice of ions of one type rela-tive to another sublattice. The perovskite structure with a generalized compositionABO3 consists of a three-dimensionally linked network of BO6 octahedra, withA ions forming AO12 cuboctahedra to fill the spaces between BO6 octahedra.In view of these topological and geometrical constraints, there are only threestructural degrees of freedom: (a) displacement of cations A and B from thecentres of their cation coordination polyhedra, AO12 and BO6, respectively; (b)distortions of the anionic polyhedra, coordinating A and B atoms; and (c) tilt-ing of the BO6 octahedra about one, two or three axes. The first of these isthe most important for the occurrence of ferroelectricity, since a separation ofthe centres of positive and negative charges corresponds to an electric dipolemoment. Structural phase changes in BaTiO3 with temperature are shown inFigure 2.14.

In cubic paraelectric BaTiO3, both Ba and Ti have zero displacements, withperfectly regular polyhedra of coordinating oxygen ions. The tetragonal, mon-oclinic (orthorhombic) and rhombohedral forms, which are all ferroelectric, areassociated with displacements of the ionic species and distortions of the polyhe-dra. In tetragonal BaTiO3, both AO12 and BO6 are elongated along the c-axis, asc/a = 1�0098. Displacements of 9.68 pm for the Ba2+ ion and 11.50 pm for theTi4+ ion along the tetragonal axis are responsible for creating a dipole momentwith the polarization vector along the same direction. Smaller displacements ofthe oxygen ions contribute to distortions, with the four oxygen ions in the BO6

octahedron being perpendicular to the tetragonal axis displaced by 3.63 pm, in




Cubic

a2

a2

a1

a1

a1

a1

a3

a3

a2a3

130°C

Tetragonal

c0°C

At0°C

a 1 = a 2 = 3.992 Å

At 130°C; a 1 = a 2 = a 3 = 4.009 Å

a 1 = a 2 = 4.013 Åc = 3.976 Åβ = 98° 51′

c = 4.035 Å

Monoclinic

β

α

At–90°C

Rhombohedral

α

α

At–90°C

a 1 = a 2 = a 3 = 3.998 Å

α = 89° 52.5′

At–90°C

b ′ = 5.681 Åa ′ = 5.667 Å

c ′ = 3.989 Å

a ′c

Orthorhombic –90°C

c ′

b ′

Figure 2.14. Sequence of phase transformations which occur in BaTiO3 at 130, 0 and −90C. Unitcell dimensions and the orientation of the polarization vector are also indicated.

the opposite direction to the Ti4+ displacement. In the rhombohedral structure,displacements and distortions are correlated. In this case, displacements are paral-lel to the threefold axis which passes through two opposite triangular faces of theoctahedron. The orthorhombic structure, by virtue of its lower symmetry, presentsa wider range of polyhedral distortions.

As is illustrated in Figure 2.14 the structure of BaTiO3 does not remain stableover the whole temperature range below the first ferroelectric Curie point and ittransforms successively into lower symmetry variants, namely, cubic → tetragonal→ monoclinic (orthorhombic) → rhombohedral. The vector directions of polar-ization are also indicated within the unit cells of these structures (Eric Cross 1993).

Christian (1990) has grouped martensitic transformations in subcategories basedon the structures of the parent and the product phases, as listed in Table 2.2.




2.5 MIXED MODE TRANSFORMATIONS

The classification scheme discussed so far makes an attempt to assign a giventransformation to a specific category based on thermodynamic, kinetic and mecha-nistic criteria. It must be emphasized that there exist several phase transformationsin real systems which do not fall exclusively in a single category. These transfor-mations, which exhibit characteristics of different classes of transformations, areoften called “mixed mode” or “hybrid” transformations (Banerjee 1994). Some ofthese cases are briefly discussed here for the purpose of illustration.

From thermodynamic considerations one can cite cases which show pretransi-tion effects similar to those of second-order transitions and at the same time a sharpdiscontinuity of thermodynamic functions at the transition temperature. Thoughthese have been designated as “mixed” in Figure 2.11 the sharp discontinuity at thetransition temperature makes them first-order transitions as per the thermodynamicdefinition. Pretransition effects in these often arise due to quasistatic structuralfluctuations (modulations in chemical composition or displacement) or modula-tions in the polarization of electric or magnetic vectors associated with latticepoints. The interplay between more than one homogeneous phase transformationscan be illustrated by (a) concomitant ordering and clustering processes and (b)sequential operation of spinodal clustering and magnetic ordering.

2.5.1 Clustering and orderingThe formation of an ordered intermetallic phase from a supersaturated dilute solidsolution often requires concomitant ordering and clustering. The conditions foreither simultaneous or sequential operation of clustering and ordering processeshave been identified (Kulkarni et al. 1985, Khachaturyan et al. 1988, Soffa andLaughlin 1989) in terms of instabilities associated with the clustering wave vector(k close to �000�) and the ordering wave vector (k terminating at one of the specialpoints of the reciprocal space). Let us consider an fcc solid solution which experi-ences the influences of �100� ordering instability and <000> clustering instability.The following situations can arise and the transformation sequence is accordinglyselected: (a) The disordered solid solution is initially unstable with respect to�100� ordering but metastable against clustering. Ordering of the solid solution toan optimum level can introduce a tendency towards phase separation in the opti-mally ordered structure. (b) The disordered solid solution simultaneously exhibits�000� clustering and �100� ordering tendencies. Both the processes can proceedsimultaneously, their relative kinetics determining the rates of progress of the twoprocesses. (c) The peak instability temperature for �000� clustering is higher thanthat for �100� ordering. In this situation, clustering occurs first, creating solute-richregions within which the ordering process sets in once the condition of ordering




A Composition (%B)

Free

ene

rgy

A Composition (%B)

Tem

pera

ture

αpαfParamagnetic

Tricriticalpoint

Tc (X ) T1

αp

αf

Ferromagnetic

RQP

(a) (b)

Figure 2.15. (a) A phase diagram showing a two-phase region introduced by a ferromagneticordering. (b) Free energy–concentration diagram at T1 showing the introduction of a spinodalclustering region in the ferromagnetic phase.

instability is fulfilled. Such coupled clustering–ordering processes are discussed inChapter 7 in connection with phase transition sequences in Zr–Al and Cu–Ti alloys.

Higher-order transitions like magnetic ordering can also induce a clusteringtendency in a solid solution, resulting in the appearance of multicritical points inthe phase diagram. Allen and Cahn (1982) have discussed these issues in greatdetail. Let us consider a binary fcc system of components A and B, where Ais ferromagnetic and the Curie temperature, Tc(X), of the A–B solid solutionchanges with composition in the manner shown in Figure 2.15. In the absence ofthe magnetic contribution, the system behaves like an ideal solution while withthe introduction of the magnetic contribution, the free energy of the �-phase isreduced from that corresponding to paramagnetic �p to that of ferromagnetic �f .At temperatures below the tricritical point, a two-phase region emerges between�f and �p and two spinodes can also be identified in the free energy–compositionplot (Figure 2.15(b)). At a temperature T1 an alloy at the point Q experiencesa clustering instability and initially decomposes spinodally to two ferromagneticphases, the one enriched in the non-magnetic component eventually undergoinga ferromagnetic → paramagnetic disordering. At the points P and R, the alloysare metastable with respect to spinodal clustering and, therefore, the paramagneticphase, �p, in the former and the ferromagnetic phase, �f , in the latter can formonly by nucleation and growth.

2.5.2 First-order and second-order orderingThere are some instances where a system exhibits simultaneously a second-orderand a first-order chemical ordering tendency. The ordering process in Ni4Mo can




be cited in this context (Banerjee 1994, Arya et al. 2001). The Ni4Mo alloy,quenched from the high-temperature disordered (fcc) phase field, exhibits (Spruielland Stansbury 1965, Ruedl et al. 1968) a short-range ordered (SRO) state char-acterized by diffraction intensity at

{1 1

2 0}

fccpositions and a complete extinction

of intensity at �210�fcc positions in the reciprocal space. The{1 1

2 0}

reflectionsdo not coincide with the superlattice reflections of the equilibrium Ni4Mo (D1a)structure (Figure 2.16). While the SRO state consists of heterospace fluctuations inthe form of concentration wave packets of size 2–5 nm, with wave vectors

⟨1 1

2 0⟩,

the equilibrium D1a structure is associated with wave vectors, 15 �420�. The two

competing superlattice structures, as shown in Figure 2.16(c) and (d), respectively,can be described in terms of �420� planes of all Ni (N) and all Mo (M) atoms in thestacking sequences of MMNNMMNN and MNNNNMNNNN, respectively. Whilethe

⟨1 1

2 0⟩

ordering fulfils all the symmetry criteria for a second-order transition,the 1

5 �420� ordering is necessarily a first-order transition. In order to examinethe relative strengths of the two ordering tendencies, namely, the first order 1

5�420� and the second order⟨1 1

2 0⟩, the free energy (F ) surfaces, as functions of

the respective order parameters, �⟨1 1

2 0⟩

and � 15 �420�, have been calculated using

first principles thermodynamic calculations (Arya et al. 2001). The instability ofthe system with respect to fluctuations corresponding to the two order parameterscan be determined by examining the curvature (2F/�2) of the F versus � plotsat � = 0 along the two directions, �

⟨1 1

2 0⟩

and � 15 �420�. With decreasing tempera-

tures, T1, T2, T3 and T4, the following four situations arise, the corresponding freeenergy surfaces being depicted in Figure 2.17(a)–(d).

(1) Tc(D1a) < Tc (1 12 0) < T1: positive curvatures for both �1 1

2 0� and 15 �420� order-

ing, implying stability of the disordered state, � = 0.(2) Tc(D1a) < T2 < Tc (1 1

2 0): negative curvature for⟨1 1

2 0⟩

and positive curvaturefor 1

5 �420� ordering, implying instability of the system for⟨1 1

2 0⟩

ordering andno tendency towards D1a ordering.

(3) Ti(D1a) < T3 < Tc (D1a), < Tc (1 12 0): negative curvature for

⟨1 1

2 0⟩

and positivecurvature for 1

5 �420� ordering at � = 0 but a dip in the free energy with respectto the latter (D1a ordering) near � 1

5 �420� = 0�8. This implies that the systemexperiences simultaneously tendencies towards

⟨1 1

2 0⟩

ordering (second order)and D1a ordering (first order).

(4) T4 < Ti (D1a) < Tc (D1a) < Tc (1 12 0): negative curvatures along both

⟨1 1

2 0⟩

and15 �420� ordering implying that the system is unstable with respect to thedevelopment of both

⟨1 1

2 0⟩

and 15 �420� ordering. In this situation, homoge-

neous ordering is feasible for both⟨1 1

2 0⟩

and D1a ordering. A mixed stateconsisting of concentration waves with wave vectors ranging from

⟨1 1

2 0⟩

to15 �420� is encountered on the path of the ordering process. This mixed state




(a) (b)

(420)

M

M

N

1234

N4 M

N1 2 3 4 5p: 0

M

N4 M

(d)

(c)

MMNN

-Tile

N2 M2(420)

-Unit Cell

N

M Period

p : 0 1 2 3 4

N2 M2

Figure 2.16. Electron diffraction patterns corresponding to (a) the “short-range ordered” structurecharacterized by

{1 1

2 0}

reflections and complete extinction of �210� reflections and (b) the D1a

ordered structure in the Ni–25 at.% Mo alloy. Real lattice descriptions of the fcc-based superstructuresin terms of stacking of �420� planes in the [001] projections and static concentration waves areshown in (c) for

⟨1 1

2 0⟩

ordering and in (d) for D1a-Ni4Mo ordering with wave vector 15 �420�. The

sequences of Ni (N) and Mo (M) layers of (420) planes and subunit cell clusters are also shown.




η /ηmax

T 1 > T 2 > T 3 > T 4

T 1

T 1

T 2

T 2

T 3

T 3

T 4T 4

ηcηc

8

6

4

2

0

F or

d (K

)

–2

–4

6

4

2

0

–2

–4

–60.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 2.17. The ordering free energy of the Ni–25 at.% Mo alloy, exhibiting the⟨1 1

2 0⟩

and the15 �420� ordering tendencies, plotted as a function of order parameters for the corresponding orderingwave vectors at four different temperatures, T1, T2, T3 and T4, respectively, pertinent to the situationsdescribed in the text.

is characterized by diffraction patterns (Figure 2.18) which show a spread ofdiffracted intensity linking

{1 1

2 0}

and 15 �420� positions and by the presence of

subunit cell clusters (or motifs) representing⟨1 1

2 0⟩

and 15 �420� ordered struc-

tures (as illustrated in Figure 2.16) in lattice resolution images (Figure 2.18).

(a) (b)

Figure 2.18. Microstructure and diffraction pattern corresponding to mixed⟨1 1

2 0⟩

and 15 �420�

ordering: (a) diffraction pattern showing intensity distribution linking{1 1

2 0}

and 15 �420� spots;

(b) high-resolution electron micrograph showing motifs of D1a and N2M2 structures (as schematicallyillustrated in Fig. 2.16 (c) and (d)).




2.5.3 Displacive and diffusional transformationsPhase transformations are classified as displacive and diffusional on the basis ofthe nature of atom movements across the advancing transformation front. One canenvisage transformations which occur by a coupling between a displacive and adiffusional mode of atomic movements. The formation of ordered �-structuresfrom the disordered parent bcc �-phase can be cited as an appropriate example ofa mixed diffusive/displacive transformation in which the bcc lattice is transformedinto the hexagonal �-structure by a periodic displacement of lattice planes while thedecoration of the �-lattice by different atomic species occurs through diffusionalatom movements. These two processes can as well be designated as displaciveand replacive ordering, respectively, and the overall process can be viewed asa superimposition of a displacement wave and a concentration wave on the bcclattice (Banerjee et al. 1997). This mode of transformation (discussed in Chapter 6)is encountered in several bcc Ti and Zr alloys, leading to the formation of a widevariety of ordered �-structures.

Displacive and diffusional atom movements can also be coupled through kineticconsiderations in several cases, one of the best examples being the formationof �-hydride precipitates in either the �- or the �-phase matrix alloys – a topicdiscussed in detail in Chapter 8.

The formation of the �-hydride phase from either the �- or the �-phase involvesa shear transformation of the parent lattice accompanied by partitioning of hydro-gen atoms (discussed in Chapter 8). The latter process being exceptionally rapid,the displacive lattice shear and hydrogen partitioning can occur nearly concur-rently. Hydride formation in Zr- and Ti-based alloys can be compared with bainiteformation in steels.

2.5.4 Kinetic coupling of diffusional and displacive transformationsOlson et al. (1989) have analysed the kinetics of a transformation process in whichthe product phase forms with a partial redistribution of the interstitial element dur-ing non-equilibrium nucleation and growth. The rate at which the advancing trans-formation front moves depends both on its intrinsic mobility and on the ease withwhich the interstitial element diffuses ahead of the moving interface. The intrinsicmobility is related to the process of structural change across the moving interface.

The growth, involving partial supersaturation in which local equilibrium is notestablished at the interface, seems to be an unstable process. This is because aperturbation in composition towards equilibrium would lead to a reduction in freeenergy which would drive the system to attain a local equilibrium. Stability of thenon-equilibrium growth mechanism can, however, be brought about by anotherprocess, such as structural transformation across the interface, occurring in series.A displacive structural transition involving movement of a glissile interface and




α

γGdd

G id

Free

ene

rgy

X α Xl

Xl

Xm

Xα

Carbon concentrationX

Xγ

Distance →

Car

bon

→

(a)(b)

Figure 2.19. (a) Free energy–concentration plots for ferrite (�) and austenite (�) showing the freeenergy component dissipated in structural change at the interface, Gid, and the free energy componentdissipated in diffusive movements of interstitial atoms, Gdd, ahead of the transformation front; (b)shows concentration distribution in the �- and �-phases.

diffusive movements of interstitial atoms ahead of the interface are, therefore,modelled as coupled processes resulting in the bainitic transformation in steels.Both these processes dissipate the net free energy, G (as indicated in Figure 2.19)which is made up of Gid and Gdd amounts dissipated in the interface process andthe diffusion process, respectively. The interfacial velocities can be calculated forthe two processes and expressed as

Vi = ��Gid and Vd = ��Gdd (2.6)

where � and � are response functions relating the velocity to the appropri-ate dissipation. For the process to be kinetically coupled, the interface velocity,V = Vi = Vd. The interface velocity is calculated on the basis of thermally acti-vated motion of dislocations which constitute the ferrite/ austenite glissile interfaceand is found to be comparable with the diffusion field velocities computed fordifferent levels of interstitial supersaturation.

It may be noted that the two types of thermally activated events which arecoupled in this treatment operate on widely differing size scales. The mannerin which the unit processes of the displacive and the diffusional aspects of thetransformation couple at the microscopic level can be understood in terms ofthe discontinuous nature of the thermally activated interfacial motion. During the“waiting time” prior to a thermally activated event, the solute partitioning in thevicinity of the interface can cause a steady increase in the local interfacial drivingforce till it reaches a threshold value where the interface is driven to its nextposition of temporary halt. The movement between these positions occurs througha diffusionless “free glide” motion.




This coupled transformation model has been applied to bainitic transformationsin steels. The model predicts an increasing interfacial velocity and supersaturationduring growth with decreasing transformation temperature, while the nucleationvelocity passes through a maximum, giving C-curve kinetics.

In this context, it should be mentioned that there is a different view point onthe mechanism of the bainitic transformation in which atom transport across theinterface is considered to occur through diffusional random jumps. The displaciveprocess, involving coordinated atomic jumps from a parent lattice site to a pre-destined site in the product lattice, has not been accepted as a requirement for abainitic transformation in the diffusionist view which has been summarized in anexcellent manner by Reynolds et al. (1991).

REFERENCES

Allen, S.M. and Cahn, J.W. (1982) Bull. Alloy Phase Diagrams, 3, 287.Arya, A., Banerjee, S., Das, G.P., Dasgupta, I., Saha-Dasgupta, T. and Mookerjee, A.

(2001) Acta Mater., 49, 3575.Banerjee, S. (1994) Solid → Solid Phase Transformations (eds W.C. Johnson, J.M. Howe,

D.E. Laughlin and W.A. Soffa) TMS, Warrendale, PA, p. 861.Banerjee, S., Tewari, R. and Mukhopadhyay, P. (1997) Prog. Mater., Sci., 42 (1–4), 109.Buerger, M.J. (1951) Phase Transformations in Solids, Wiley, New York, p. 183.Christian, J.W. (1965) Physical properties of martensite and bainite, Special Report 93,

Iron and Steel Institute, London, p. 1.Christian, J.W. (1979) Phase Transformations, Vol. 1, Institute of Metallurgists, London,

p. 1.Christian, J.W. (1990) Mater. Sci. Eng., A127, 215.Cohen, M., Olson, G.B. and Clapp, P.C. (1979) Proceedings of International Conference

on Martensite, MIT Press, Cambridge, MA, p. 1.Cook, H.E. (1974) Acta Metall., 22, 239.de Fontaine, D. (1975) Acta Metall., 23, 553.Delaye, L., Chandrasekaran, M., Andrade, M. and Van Humbeck, J. (1982) Solid–

Solid Phase Transformations (eds H.I. Aaronson, D.E. Laughlin, R.F. Sekerka andC.A. Wayman) TMS, Warrendale, PA, p. 1429.

Ehrenfest, P. (1933) Proc. Acad. Sci. Amst., 36, 153.Eric Cross, L. (1993) Ferroelectric Ceramics., (eds Nava Setter, Enrico L. Colla and

Birkhaeuser Basel), Switzerland.Khachaturyan, A.G., Lindsey, T.F. and Morris, J.W. (1988) Metall. Trans., 19A, 249.Kulkarni, U.D., Banerjee, S. and Krishnan, R. (1985) Mater. Sci. Forum, 3, 111.Landau, L.D. and Lifshitz, E.M. (1969) Statistical Physics, Pergamon Press, Oxford.Olson, G.B., Bhadeshia, H.K.D.H. and Cohen, M. (1989) Acta Metall., 37, 381.Rao, C.N.R. and Rao, K.G. (1978) Phase Transitions in Solids, McGraw Hill, New York.




Reynolds, W.T., Jr, Aaronson, H.I. and Spanos, G. (1991) Mater. Trans. JIM, 32, 737.Roy, R. (1973) Phase Transitions (ed. L.E. Cross) Pergamon Press, Oxford, p. 13.Ruedl, E., Delavignette, P. and Amelinckx, S. (1968) Phys. Status Solidi, 28, 305.Soffa, W.A. and Laughlin, D.E. (1989) Acta Metall., 37, 3019.Spruiell, J.E. and Stansbury, E.E. (1965) J. Phys. Chem. Solids, 26, 811.Tanner, L.E., Schryvers, D. and Shapiro, S.M. (1990) Mater. Sci. Engi., A127, 205.






Chapter 3

Solidification, Vitrification, Crystallization andFormation of Quasicrystalline and NanocrystallineStructures

3.1 Introduction 1283.2 Solidification 128

3.2.1 Thermodynamics of solidification 1283.2.2 Morphological stability of the liquid/solid interface 1353.2.3 Post-solidification transformations 1403.2.4 Macrosegregation and microsegregation in castings 1413.2.5 Microstructure of weldments of Ti- and Zr-based alloys 145

3.3 Rapidly Solidified Crystalline Products 1503.3.1 Extension of solid solubility 1523.3.2 Dispersoid formation in rapidly solidified Ti alloys 1533.3.3 Transformations in the solid state 153

3.4 Amorphous Metallic Alloys 1573.4.1 Glass formation 1573.4.2 Thermodynamic considerations 1593.4.3 Kinetic considerations 1653.4.4 Microstructures of partially crystalline alloys 1713.4.5 Diffusion 1763.4.6 Structural relaxation 1803.4.7 Glass transition 182

3.5 Crystallization 1843.5.1 Modes of crystallization 1853.5.2 Crystallization in metal–metal glasses 1873.5.3 Kinetics of crystallization 1923.5.4 Crystallization kinetics in Zr76�Fe1−xNix�24 glasses 200

3.6 Bulk Metallic Glasses 2053.7 Solid State Amorphization 212

3.7.1 Thermodynamics and kinetics 2153.7.2 Amorphous phase formation by composition-induced

destabilization of crystalline phases 2203.7.3 Glass formation in diffusion couples 2203.7.4 Amorphization by hydrogen charging 225

125



3.7.5 Glass formation in mechanically driven systems 2263.7.6 Radiation-induced amorphization 229

3.8 Phase Stability in Thin Film Multilayers 2373.9 Quasicrystalline Structures and Related Rational Approximants 241

3.9.1 Icosahedral phases in Ti-and Zr-based systems 248References 252



Chapter 3

Solidification, Vitrification, Crystallization andFormation of Quasicrystalline and NanocrystallineStructures

List of SymbolsGl/s: Free energy of the liquid/solid phase�l/s: Chemical potential of the liquid/solid phasecl/s: Composition of the liquid/solid phase�l/s: Activity coefficient for the liquid/solid phaseTl/s: Liquidus/solidus temperatureKl/s: Thermal conductivity in the liquid/solid phaseko: Partition coefficient representing the ratio of the slopes of the

liquidus and solidus lines�: Wavelength of a perturbation�: Amplitude of a perturbation�: Rate of growth/decay of a perturbationGc: Composition gradient of the liquid phaseDl: Diffusion coefficient of the solutetf : Local solidification time

Ms��: Martensitic start temperature for the phaseT i

m: Melting point of the ith componentHi

f : Heat of fusion of the ith componentI: Rate of nucleation of a crystalline phaseZ: Frequency of the atomic jumps across the interfaceA∗: Surface area of the critical nucleusN ∗: Number density of critical nucleiW ∗: Work done to form a critical nucleusGv: Volume free energy change for the formation of critical nucleus

�: Induction timef : Fraction of the transformed volumeQ: Activation energy of a given processTp: Temperature at which transformation rate reaches the peak valueu: Growth rateD: Interdiffusion constant

Hv: Enthalpy of vacancy formation

127




kB: Boltzmann constant�±: Frequency of ordering �+� or disordering �−� jumpsZ �: Number of nearest neighbor �-sites around an -site

3.1 INTRODUCTION

In this chapter, transformations involving liquid, amorphous, nanocrystalline andquasicrystalline phases are discussed. Many of these transformations occur underconditions far removed from equilibrium. With the continuous development ofnon-equilibrium processing techniques, an increasing number of novel transfor-mation products, some of them possessing exotic properties, have been discoveredin the recent past. Ti and Zr alloys have had their due share in this excitingscientific development of production of metastable microstructures by techniquessuch as liquid and vapour state processing, mechanical attrition, interdiffusion andradiation processing. Phase transformations associated with the liquid, amorphous,quasicrystalline and nanocrystalline states are discussed in this chapter by citingexamples taken from Ti- and Zr-based alloys.

3.2 SOLIDIFICATION

Solidification and melting are strong first-order transformations which are oftremendous importance in various technological applications such as ingot casting,foundry casting, single crystal growth and welding. An understanding of themechanism of solidification is very important for predicting how different para-meters such as temperature distribution and cooling rate influence microstructure,alloy partitioning and mechanical properties of cast and fusion welded materials.The objective of this section is to introduce some of the concepts which willbe needed for the discussions in later sections dealing with rapid solidification,amorphization and devitrification.

3.2.1 Thermodynamics of solidificationAs in the case of any other transformation, solidification cannot proceed at equi-librium. Depending on the extent of departure from the equilibrium condition,a hierarchy of different conditions has been identified by Boettinger and Biloni(1996), the solidification rate increasing as the system is driven away from equi-librium. These conditions are listed as follows in the order of increasing departurefrom equilibrium.



Solidification, Vitrification and Crystallization 129

(1) Full diffusional equilibrium is established globally. Under this condition, thereare no gradients of chemical potential and temperature in the system. Thecompositions of the liquid and the solid phases attain the equilibrium values.The solidification process cannot continue after this condition is attained.

(2) Chemical equilibrium is established locally at the liquid/solid interface. Thecompositions of the liquid and solid phases at the interface are given by theequilibrium phase diagram, any correction due to the interface curvature beingtaken into account.

(3) Local interfacial equilibrium is established between the liquid and a metastablesolid phase. Such a situation arises when the equilibrium phase cannot nucleateor grow fast enough to compete with the metastable phase. The compositionsof the solid and liquid phases at the interface are given by the pertinentmetastable phase diagram.

(4) Local equilibrium condition is not established at the liquid/solid interface.Here the temperature and compositions at the interface are not given by eitherequilibrium or metastable phase diagrams.

The condition of local equilibrium, whether stable or metastable, is that thechemical potentials of the components in the liquid and the solid phases are equalacross the liquid/solid interface. Condition (4) relates to a situation where therapidly moving interface does not permit the chemical potentials of the compo-nents to equalize across the interface. The rapid growth rates which result underlarge supercooling can trap the solute into the freezing solid at levels exceedingthe equilibrium value for the corresponding liquid composition prevailing at theinterface. If one considers only the chemical potential of the solute, it increasesupon being incorporated in the freezing solid by a process called solute trapping.However, to make this process thermodynamically possible, a decrease in thechemical potential of the solvent, leading to a net decrease in the free energy,becomes essential (Baker and Cahn 1971).

Let us consider the thermodynamics of solidification in a binary system compris-ing the components A and B, which is represented by the free energy–compositionplots, Gl and Gs (Figure 3.1), corresponding to the liquid and solid phases at atemperature which is between the solidus and the liquidus temperatures. In orderto determine the composition range of solids which can form from a liquid ofcomposition co, a tangent is drawn to the Gl curve at co. This tangent intersectsthe Gs curve at two points, cs1 and cs2. The Gs curve between cs1 and cs2 remainsbelow the tangent, indicating that it is thermodynamically possible to form a solidin this composition range from the liquid of composition, co. At temperaturesabove the liquidus, tangents to any point on the Gl curve do not intersect the Gs

curve and at the liquidus temperature, the tangent to the Gl curve touches the Gs




A

Gs

Gl

Tl

cs1 cs(eq) co c l(eq)cs2

BAtom fraction of B

P M

Q N

c (To)

*cn*cs∗Aμs (cn)

∗Bμs (cn)

∗Aμs (cs)

∗Bμs (cs)

Aμ l (co)

Aμ l (co)

Figure 3.1. Free energy–concentration plots,Gl andGs, of the liquid and the solid phases, respectively,at a temperature between liquidus (Tl) and solidus (Ts). From thermodynamic considerations, a liquidof composition, co, can form a solid of any composition between cs1 and cs2.

curve only at one point which gives the equilibrium solid composition, cs�eq�. Ifwe define the chemical potentials of the components A and B in the liquid and thesolid phases at the interface as �A

l , �Bl , �A

s and �Bs , respectively, the free energy

change during solidification, G, is given by

G = [��A

s −�Al ��1− c∗

s �− ��Bs −�B

l �c∗s

](3.1)

where c∗s is the composition of the solid phase being separated. Considering that

the liquid phase is homogeneous in composition, �Al and �B

l correspond to theliquid composition, co, and are given by the intercepts made by the tangent to Gl

at co on the free energy axes corresponding to pure A and pure B, respectively.The free energy change associated with the formation of the solid of compositionc∗

s is represented by the drops MN and PQ shown in Figure 3.1 for two differentvalues of cs (c∗

n and c∗s ). It is to be noted that for cs = c∗

s , solidification involvesa lowering of the chemical potentials, �A and �B, of both the components. Incontrast, for cs = c∗

n, �A = �Al −�A

s < 0 but �B = �Bl −�B

s > 0.Based on the free energy–composition plots for the liquid and the solid phases

at a given temperature, the domains of all possible solid compositions, whichare allowed to form from thermodynamic considerations, can be presented in anisothermal plot of the solute content of the liquid versus that of the solid at theinterface (Figure 3.2). At a given temperature, the liquidus composition gives the




X

Y Slope 1

ΔG = 0

ΔG > 0

Slope 1

Slope kE

Equilibrium

P cs1

cs2

cl(eq)O

BΔμ > 0

AΔμ > 0

AΔμ = 0

AΔμ < 0

BΔμ = 0

Δμ < 0B

Liquid composition at interface,atom fraction of B (c l)

Sol

id c

ompo

sitio

n at

inte

rfac

e,at

om fr

actio

n of

B (

c s)

c (To)

cs(eq)

Figure 3.2. The domain OXYEP of all possible solid compositions that can form from various liquidcompositions. This can be divided into three distinct regions, the shaded region OEP where chemicalpotentials of both A and B decrease on solidification, the region above the line OE where solutetrapping occurs and the region below the line PE where solvent trapping occurs (after Baker andCahn 1971).

maximum solute content of the liquid at the interface from which solidification canoccur, and in case solidification occurs without any solute partitioning, the solidinherits the composition of the liquid. The condition of partitionless solidificationis met at a point where Gl and Gs curves intersect (Figure 3.1(b)) (i.e. where theintegral molar free energies of the two phases are equal). The locus of these pointsdefines the To line.

The domain of all possible solid compositions that can form from various liquidcompositions at a given temperature can be defined by the curve OXYEP. Themaximum limit of solute concentration in the solid is given by the point Y , whichis fixed by the point of intersection of the c�To� line and the line OY (slope = 1)representing identical compositions of the liquid and the solid phases. The point Ecorresponds to the maximum limit of solute concentration in the liquid, given bythe liquidus composition, cl�eq�, and the equilibrium solidus composition, cs�eq�.At the boundary of the OXYEP domain, the condition of the change in the integralmolar free energy for solidification being zero (G= 0) is satisfied. As the liquidcomposition is changed to lower the solute content, a tangent intersects the Gs

curve at two points. This situation is illustrated in Figure 3.1 by the tangent at co

which intersects Gs at cs1 and cs2, and solids of any composition between thesetwo limits can solidify from the liquid of co composition. At the composition,




c�To�, where Gl and Gs intersect, the range of solid compositions spans from zero(point P) to c�To� (point Y ).

The domain OXYEP can be divided into three regions. In the shaded regionOEP, the chemical potentials of both A and B decrease on solidification, while thechemical potential of only one of the components decreases outside this region.The overall free energy change favours solidification in the entire OXYEP domain,but outside the region OEP, one of the components enters the solid phase withan increase in the chemical potential. Such a process is termed solute trapping (inthe region OXYE) or solvent trapping (below the line PE, where �A > 0 but�B < 0�.

For a quantitative description of the solute trapping concept, let us consider asimple case of a dilute binary solution for both liquid and solid. Then the chemicalpotentials are given by Henry’s law for the minor component:

�Bs = Bs +RT ln �scs (3.2)

�Bl = Bl +RT ln �lcl (3.3)

where Bs�l and �s�l are related constants which depend on the temperature andreference state. Under equilibrium conditions, where cs = cs�eq� and cl = cl�eq�,

�Bs −�B

l = Bs −Bl +RT ln�scs�eq��lcl�eq�

= 0 (3.4)

The change in chemical potential of the minor component across the solidifica-tion front can, therefore, be expressed as

�B = RT lncscl�eq�cs�eq�cl

(3.5)

In terms of the distribution coefficient, k, at the interface, k = cs/cl,

�B = RT ln�k/k�eq�� (3.6)

The straight line OE with a slope k�eq� (in Figure 3.2) represents the equilibriumcondition in which the minor component experiences no change in chemicalpotential. When k > k�eq�, in the region above the line OE, �B > 0, whichcorresponds to solute trapping.

For the major component, Raoult’s law holds, and the change in the chemicalpotential of the component A on solidification can be written as




�A = RT ln�1− cs��1− cl�eq��1− cs�eq��1− cl�

(3.7)

The straight line PE, of slope �1−cs�eq��/�1−cl�eq��= 1, through the equilib-rium composition point �cs�eq�, cl�eq�� in Figure 3.2 represents the equilibriumcondition. Solvent trapping occurs below the line PE where �A > 0.

The G = 0 curve is given by the condition

�1− cs��A + cs�

B = 0 (3.8)

and the curve passes through the points O, X, Y , E and P.Thermodynamics places general restrictions on the composition limits of the

solidifying phases, but it does not specify the solid composition under a givensupercooling (T ) and solidification rate (V , the velocity of the solid/liquidinterface). Boettinger (1982) has shown the allowable composition ranges of thesolid superimposed on phase diagrams of binary systems (Figure 3.3(a) and (b)).The shaded regions in these diagrams indicate thermodynamically allowed solidcompositions that may be formed from a liquid of composition co at varioustemperatures. The To curve gives the highest temperature at which partitionlesssolidification �co = cs� can occur. Figure 3.3(b) shows the case where the To curveplunges and partitionless solidification is not permitted for a liquid of compositionco. These curves can be used to determine the limit of the extension of solubilityobtainable by rapid quenching of the liquid phase. If the To curve plunges to a

To

co

To

coL

α

a

L

(a) (b)

Atom fraction of B Atom fraction of B

Tem

pera

ture

Tem

pera

ture

Figure 3.3. Thermodynamically allowed solid compositions which can form from a liquid of com-position co are shown by the shaded regions in two schematic phase diagrams. The To curve givesthe highest temperature at which partitionless solidification (co − cs) can occur. While in (a), parti-tionless solidification of a liquid of composition co is possible, in (b) where the To curve plunges,partitionless solidification is not permitted for the same liquid composition (after Boettinger 1982).




very low temperature, as in Figure 3.3(b), the -phase with solute contents beyondthe To curve cannot be formed from the melt. In fact, for phase diagrams with aretrograde solidus, the To curve plunges sharply resulting in a limited extension ofsolubility. Eutectic systems with plunging To curves are good candidates for easyformation of metallic glasses. This point is elaborated in Sections 3.4.1 and 3.4.2.In contrast, alloys with To curves which are only slightly depressed below theliquidus curves (as shown in Figure 3.3(a)) make good candidates for extensionof solubility and are unlikely ones for glass formation.

In the analysis of a solidification problem, the condition of local equilibriumis often invoked. This assumption is valid whenever the deviation from equilib-rium, expressed as T , cs − cs�eq�, or cl − cl�eq�, is small compared to the totaltemperature and composition ranges pertinent to the solidification process. Thisis indeed so for most solidification problems that involve rather low velocities ofthe liquid/solid interface.

Let us now consider the steady state plane front condition. We impose avelocity V on the liquid/solid interface and assume that, after an initial transient,the temperature and composition profiles and the position of the interface movewith the velocity V . Under this condition, the composition of the solid, cs, must beequal to the overall composition, co, of the alloy. Because cs = co, the process hassome aspects of “partitionless” solidification, but it includes the situation in whicha liquid layer of a different composition (cl �= co) remains ahead of the advancingsolidification front (Figure 3.4). Under steady state conditions, this layer remainsunchanged with time and may be hard to detect. It will have a thickness of about

ko < 1co

Initialtransient (I)

Initial transient (II)Final transient (III)

Δco

co ko co/koco

Tl (co)

Ts (co)

Tm

(a)

Tem

pera

ture

Composition

Solid Liquid

Δco

co ko

co ko

co

(b)

Com

posi

tion

Distance

→

→

Figure 3.4. (a) A schematic phase diagram showing the liquidus and solidus temperatures corre-sponding to an alloy of composition co. (b) The composition profile of the liquid and the solid in thevicinity of the solidification front under steady state solidification. ko is the equilibrium partitioningratio.




Dl/V where Dl is the diffusivity in the liquid. The thickness of the layer turnsout to be less than 1�m when V exceeds 1 cm/s. The possibility of establishingsteady state conditions can be examined by redrawing in Figure 3.1 the domainsof thermodynamically allowable liquid interface compositions at differenttemperature – composition regions, defined, respectively, as Regions I, II and III.

In Region I, the point B1, the maximum solute concentration in the solid,remains below co, and therefore, it is not possible to establish the steady statefrom thermodynamic considerations. This is essentially because G > 0 for allpossible values of cl. In Region II, co lies between BII and EII, implying that steadystate solidification is possible, provided the composition of the liquid remainswithin the band marked in the figure. The solid formed remains metastable withrespect to partial remelting. There is, however, a diffusional instability for steadystate solidification in Region II. A downward fluctuation in the solid composition(cs < co) will lead to a shift of the liquid composition to the right (cl increasing).Since the solid that is forming is below the average composition co, excess soluteis rejected to the liquid, making the liquid further enriched in the solute. Thismay result either in the break-up of the plane front interface or in a reduction inthe interface temperature. Steady state solidification, though thermodynamicallypossible, is diffusionally unstable. In Region III, steady state growth is not onlythermodynamically possible but also stable. If solidification starts at the pointEIII, cs > co and the liquid will be depleted of excess solute and the system willspontaneously leave the point EIII and settle on the horizontal line cs = co forpartitionless solidification.

3.2.2 Morphological stability of the liquid/solid interfaceIn the previous section, it has been assumed that the solid–liquid interface ismicroscopically planar. Under this condition, the composition profile induced inthe solid varies only in the direction of growth. A planar interface may becomeunstable to small changes in shape even if the heat flow remains unidirectional.The stability of the liquid–solid interface during solidification is considered here,first for pure metals and then for alloys. In pure metals, solidification can bedescribed in terms of the latent heat being conducted away from the liquid–solidinterface, i.e.

Ks

(dTdx

)s

= Kl

(dTdx

)l

+ vLv (3.9)

where Ks�l are the thermal conductivities of and �dT/dx�s�l are the temperaturegradients in the solid and the liquid, respectively, Lv is the latent heat of fusion perunit volume and V is the velocity of the liquid–solid interface. The morphological




stability of this interface is governed by the sign of the temperature gradient in theliquid ahead of the transformation front. When the solid grows into a superheatedliquid, i.e. for a positive gradient, �dT/dx�l > 0, the interface remains stable.This can be understood from the following argument. If a small protrusion ofsolid develops at the plane solidification front due to a local increase in V , thetemperature gradient in the liquid ahead of the protrusion will increase, whilethat in the solid will decrease. Consequently, more heat will be conducted intothe protruding solid and less away, resulting in a decrease in the growth rate inthis localized region compared to that in the planar region. As a consequence,the protrusion will disappear. The process in which a perturbation of the planarmorphology dies down when the latent heat is extracted through the solid isschematically illustrated in Figure 3.5(a).

Solid LiquidLiquidSolid

Heat flow

Tm

TmHeat flow

T

x

Solid Liquid

Heatflow

(a) (b)

Solid Liquid

Heatflow

Isotherms

vT

x

v

Figure 3.5. Temperature distribution during solidification (a) for extraction of heat through the solidand (b) for heat flow into the liquid.




During the solidification process, in which the solid grows into a supercooledliquid, the gradient Gl �= dT/dx� ahead of the interface is negative. If a protrusionforms on such an interface, the negative temperature gradient becomes morenegative and, therefore, heat is removed more effectively from the tip of theprotrusion than from the surrounding flat regions. A perturbation created on theinterface, therefore, tends to grow with time, indicating an inherent instability ofthe solidification front, as shown in Figure 3.5(b).

The instability of the solid–liquid interface is responsible for developing arms oncrystals nucleated in a supercooled liquid. These arms grow along crystallographicdirections of easy heat transfer and during their growth create secondary andtertiary arms, resulting in a dendritic structure. Dendrites in pure metals are knownas thermal dendrites to distinguish them from those forming in alloys primarily dueto the constitutional supercooling phenomenon, which is described in the followingby considering the case of one-dimensional movement of the solidification frontin a binary alloy of composition, co, as shown in the corresponding phase diagramFigure 3.4(a). As in this case the solidification process leads to a partitioning ofthe solute preferentially towards the liquid phase (partition coefficient, the ratioof the slopes of the liquidus and solidus lines, ko < 1), the liquid ahead of thesolidification front becomes solute enriched. After the initial transient, a steadystate is established when the liquid in contact with the solidification front attains acomposition, co/ko, and the solid/liquid interface reaches the solidus temperature,Ts�co�. Under this condition, a local equilibrium is established at the solidificationfront. Tiller et al. (1953) have expressed the composition of the liquid ahead of thesolidification front in terms of the solute diffusion coefficient, Dl, in the liquid,the distance, z, from the interface and the velocity, v, of the interface:

cl = co

[1+ 1−ko

ko

exp(−vz

Dl

)](3.10)

The profile of the solute concentration ahead of the solidification front is shownin Figure 3.6(a) while the liquidus temperature of the solute-enriched region infront of the solidification front is depicted in Figure 3.6(b). The correspondingliquidus temperature for the composition in front of the interface is given by

Tl�z� = Tm +mlco

[1+ 1−ko

ko

exp(−vz

Dl

)](3.11)

where ml is the slope of the liquidus line. Figure 3.6(b) also shows three pos-sible profiles of the actual temperature. These are labelled as cases (a), (b) and(c). For the case (a), the actual temperature remains above Tl�z�, which meansconstitutional supercooling does not occur ahead of the solid–liquid interface.




(a)

Solid Liquid

Solute

Slope Gc

Interface

Distance, z

Tem

pera

ture

(b)

ac

bTl (z)

Distance, z

Tem

pera

ture

Temperaturegradient, Gs

Temperature gradientin liquid, Gl

(c)

Figure 3.6. (a) Solute concentration, (b) temperature profile ahead of the solidification front for asystem with k0 < 1; z is measured from the solid/liquid interface and (c) early stages of the devel-opment of morphological instability of liquid–solid interface as revealed in dendrites of the rapidlysolidified Zr54�5Cu20Al10Ni8Ti7�5 alloy frozen in the amorphous matrix (magnification = 10�000X).

Case (b) represents the situation where the actual temperature profile, T�z�, istangent to the Tl�z� line at z = 0, while for case (c), the T�z� line remains belowTl�z� for some distance ahead of the solid–liquid interface where the condition ofconstitutional supercooling prevails. The case (b) essentially depicts the limitingcondition between the presence and the absence of constitutional supercoolingahead of the solidification front, and this condition can be obtained by equatingthe slopes of the Tl�z� and T�z� lines at z = 0 which yields

Gl

V≥ mlco

Dl

ko −1ko

(3.12)

for constitutional supercooling and the resulting instability in the liquid–solid inter-face to occur. This criterion for constitutional supercooling, which was obtainedby Tiller et al. (1953), serves as a model to understand the major cause of themorphological instability of the solid–liquid interface, but it does not yield anyinformation about the size scale of the modulation developing on the solid–liquidinterface.

The analysis of morphological stability of the moving solid–liquid interface hasbeen originally reported by Mullins and Sekerka (1963) and many assumptions




of the original theory have subsequently been relaxed, as summarized by Sekerka(1986). An outline of the analysis of the morphological stability of the solid–liquidinterface can be presented as follows.

A perturbation of amplitude, �, and wavelength, �, is introduced on a flat solid–liquid interface growing in the z-direction. For a two-dimensional �z� x� analysis,the perturbed surface can be represented as

z = � exp��t+2�ix/�� (3.13)

where � is the rate of growth (or decay) of the perturbation. The value of �is determined by solving the steady state heat flow and diffusion equations withappropriate boundary conditions for small values of � (linear theory). The planarinterface is stable if the real part of � is negative for all values of � and � = 0will define the condition of the stability/instability transition which is given by thefollowing equation:

G−mlGc�c + 4�2Tm�

�2= 0 (3.14)

where � is the surface energy and G, the conductivity weighted temperaturegradient, is given by

G = KsGs +KlGl

Kl +Ks

(3.15)

Kl and Ks are the conductivities of liquid and solid, respectively. Gc, thecomposition gradient in the liquid, can be obtained from Eq. (3.10) as

Gc = vco�ko −1�koDl

(3.16)

The parameter, �c, can usually be set to unity. However, �c may deviatesignificantly from unity under rapid solidification conditions. In general, �c isgiven by

�c = 1+ 2ko

1−2ko −[1+ ( 4�Dl

V�

)2]1/2 (3.17)

The stability of the solid–liquid interface is determined by the sign of the lefthand side of equation (3.14). If it is positive, the interface is stable with respectto the perturbation introduced. The first term G has a stabilizing influence for




a positive temperature gradient. For a single component material, this is the onlyterm which is present. Therefore, in such a case, morphological instability canset in only during the growth of the solidification front into a supercooled liquid(negative temperature gradient). The second term in Eq. (3.14) represents the effectof solute diffusion in the liquid and being negative has always a destabilizinginfluence. The third term, which involves capillarity, has a stabilizing influence forall wavelengths, the minimization of the total surface energy being the motivatingfactor. This factor becomes more prominent at short wavelengths and, therefore,acts as a balancing force against any reduction in the wavelength of the modulationof the solid–liquid interface. The undulation at the solidification interface in caseof a Zr–Al alloy is shown in Figure 3.6(c).

3.2.3 Post-solidification transformationsMicrostructures of as-solidified alloys based on Ti and Zr are influenced bythe solute migration resulting from the solidification process. The first phase tosolidify in these alloys in the � (bcc) phase which undergoes subsequent solid statephase transformations depending on the local chemical composition (Figure 3.7).This point is explained in Section 3.2.5 by using a hypothetical binary phasediagram for an alloy with a �-stabilizing element. Enrichment of the liquid with� stabilizing solutes causes a local depression of the Ms temperature. During

Figure 3.7. Bright-field microstructure showing martensitic structure cutting across the cellularboundaries. The martensite has formed from the � phase which was the first phase to solidifyfrom the liquid.




Table 3.1. Sequences of transformations during solidification processingin Ti- and Zr-based alloys.

Alloy system Sequence of phase transformations

Zr (with Si and L → directlyO impurities)

Zr – 1 at.% Nb L → �′ → ′

Zr – 8.5 at.% Nb L → � → ��Zr – 27 at.% Al L → � → �+�′ → Zr2Al�B82�+��Ti – 50 at.% Al L → � → 2 → 2 +�Zr3Al–Nb L → � → �+�′ → �+Zr5Al3 → Zr2Al

post-solidification cooling, some regions encounter a martensitic (� → ′) or aWidmanstatten (�→ ) transformation, depending on the prevailing cooling rate.Regions which are enriched in the �-stabilizing solute beyond a certain level eitherretain the � phase fully or transform into a �+� structure. The martensitic andthe retained � (with or without � dispersion) structure is superimposed on thedendritic or cellular structure in the final microstructure, with the local chemicalcomposition determining the nature of the post-solidification transformation. Thesolute enrichment process in the interdendritic regions can also induce other phasereactions such as the formation of an ordered intermetallic phase, �→ Zr2Al �B82�,or a peritectoid reaction, �+ Zr5Al3 → Zr2Al, or � → 2 → 2 +� (as in TiAl-based alloys). Transformation sequences during solidification processing havebeen determined in some limited studies on Ti- and Zr-based alloys. Table 3.1summarizes the results in a few representative cases.

3.2.4 Macrosegregation and microsegregation in castingsThe solidification theory discussed in the preceding section is utilized in assessingthe extent of macro- and microsegregation in ingots and other castings. Macroseg-regation causes non-uniformity in alloy composition that occurs over large dis-tances, while microsegregation is over distances comparable to the dendrite armspacing. The extent of microsegregation is expressed in terms of the segregationratio (= local maximum solute concentration/local minimum solute concentration)or the volume fraction of the non-equilibrium secondary phase which forms as aresult of segregation.

As the solidification front grows into the liquid metal pool, the dendrite armsisolate the liquid into microscopic pools in the mushy zone (Figure 3.8). The regionbetween two adjacent dendrite arms can be taken as a characteristic volume sincethe dendrite spacing, d, in an alloy under a given cooling condition remains quiteuniform. By applying the equilibrium partition ratio, ko, the interface composition




Solid Solid + liquid(mushy zone) Liquid

λSolid

Liquid

xR xr

λ = 0 (Dendrite spine)

λ = λ i (Solid–liquid interface)

λ = d /2 (Midpoint between twodendrites)

Figure 3.8. Schematic diagram showing dendritic areas of the solid phase growing intothe mushy zone.

of the solid and the liquid phases can be estimated to be koco and koc′l where co is

the average composition and c′l is the liquidus composition at the non-equilibrium

solidus, as shown in Figure 3.9. The redistribution of the solute within the soliddendrite by diffusion can be computed from the material balance equation

�c∗l − c∗

s �dfs =∫ fs

0Ds

�2cs

�x2dfsdt+ �1−fs�dc

∗l (3.18)

where c∗l and c∗

s are the compositions at the interface of the liquid and the solid,respectively, fs is the weight fraction solid within the volume element and Ds isthe diffusion coefficient of the solute in the base metal. The solute redistributiondue to diffusion can be computed by substituting values for diffusion coefficient,dendrite arm spacing, solidification time and by employing numerical techniquesdescribed in literature (Brody and Flemings 1966).

An estimate of the extent of microsegregation can be made by evaluating theparameter

a = 4Dstfd2

(3.19)

where tf is the local solidification time which is defined as the difference in timebetween the passing of the liquidus and solidus isotherms for a given point inthe ingot. For a � 1, diffusion in the solid is negligible and microsegregation ismaximum, whereas for a � 1 microsegregation is negligible. Since

d = Ctnf �0�3 < n < 0�5�� (3.20)

microsegregation does not change with the cooling rate when n = 0�5.




c R x T

Solid + liquidSolid Liquid

Distance

Tem

pera

ture

T l

T ′s

c R x TDistance

Liqu

id c

ompo

sitio

n

c o

c ′I

c o

Tem

pera

ture

Compositionc ok o

T l

T ′sc ′I

Figure 3.9. Temperature–composition diagram showing interface composition of the solid (koco)and the liquid phases (koc

′l ) where co is the average composition and c′

l is the liquidus compositionat the non-equilibrium solidus. The liquid composition profile ahead of the solidification front isalso shown.

Brody and David (1970) have compared the microsegregation parameters ofseveral binary Ti alloys. It is noted that the extent of microsegregation increaseswith an increase in the freezing range and with a steep slope of the liquidus. Ternaryand more complex alloys are expected to show more extensive microsegregationwhenever the addition of alloying elements results in a lowering of the freezingrange.

Macrosegregation arises out of the bulk movement of solute rich liquid (forko<1) in the mushy zone. Mechanical stress, either applied or thermally induced,may cause opening up of tears in the nearly solid regions. The tears may befilled then by the liquid from the neighbourhood which will have a much higher




concentration of the solute. This mechanism leads to the formation of films ofhigh solute content that run roughly parallel to isotherms. The flow of solute-enriched liquid to feed the solidification shrinkage is another mechanism ofmacrosegregation.

Consumable electrode arc melting is the most extensively used technique formaking Ti and Zr alloy ingots. The nature of microsegregation in such ingots canbe visualized by considering the movement of the liquid pool during the growth ofan ingot. Figure 3.10 shows a schematic of the consumable electrode arc meltingprocess in which a liquid pool and a mushy zone traverse upwards as the electrodeis progressively consumed. At the start of the cast, the operation will not be ata steady state which is attained after the liquid zone moves to a certain extent.The steady state growth, during which the size of the liquid pool and mushyzone remains constant, may be disrupted if the heat input to the ingot suddenlychanges. This can happen due to a physical discontinuity in the electrode or to ashorting of the arc. As a result of a decrease in the heat flux, the local cooling rateincreases and a semicontinuous band of positive segregation develops parallel tothe isotherms. Such defects in the ingot are called “freckles”. During the growthof the ingot, the shape and size of the liquid pool and mushy zone change. Initiallythey remain thin and nearly horizontal. At steady state, the liquid pool and mushyzone are larger and deeper at the centre than at the sides. Liquid in the mushy zoneat the centre will flow not only axially but also radially to feed metal to the sidesfor compensating solidification shrinkage. The centre would generally experiencea negative segregation while the outer regions a positive segregation (Brody andDavid 1970).

Electrode

Liquid pool

Liquid pool + solid zone(mushy)

Solid region

Starter pad

Cooled metal mould

↓

↓↓

Figure 3.10. Schematic drawing showing consumable electrode arc melting; liquid, mushy and solidzones at steady state and the directions of liquid movement for feeding solidification shrinkage (bothradial and axial) are indicated.




The following mechanisms of macrosegregation in ingots produced by consum-able arc electrode melting have been identified:

(1) Flow of solute-enriched liquid into hot tears.(2) Interruption of steady state feeding of liquid metal for compensating solidifi-

cation shrinkage.(3) Radial segregation resulting from transient conditions at start-up or near

completion of a cast or at other times when the operating conditions aresignificantly changed.

3.2.5 Microstructure of weldments of Ti- and Zr-based alloysWelding is observed to severely alter the microstructure and consequently themechanical properties of Ti- and Zr-based alloys. Usually properties such as fatiguelife, fracture toughness and tensile ductility are adversely affected in the fusionzone and heat-affected zones of weldments. As the name suggests, the fusionzone refers to the region which undergoes melting and subsequent solidification.For an or an +� alloy, a solid state transformation, leading to a conversioninto the full � phase, occurs during heating prior to melting. During the coolingdown stage, the first solidifying phase is � and as the temperature falls below the�/( +�) transus, the -phase starts forming. It is because of these four (two in thesolid state and the other two involving melting and solidification) transformationsteps that the microstructure of the fusion zone undergoes substantial alteration.

The zone known as the heat-affected zone is the one which goes through onlysolid state transformations, such as from or +� to � followed by � to ′

or + �. Since this region does not undergo the melting–solidification cycle,microstructural changes are introduced only through solid state transformationsincluding recrystallization and grain growth.

Microstructural modifications which occur in the fusion and heat-affected zonesof weldments due to phase transformations are primarily governed by the localalloy composition and the heating/cooling rate imposed. Since the diffusivity inthe liquid phase is quite high, it is reasonable to assume that the liquid poolcreated during welding corresponds to the average alloy composition unless afiller material of a different composition is used. There is, however, a possibilityof picking up interstitial elements such as O, N and H in the molten pool as theseelements have high solubilities in the base metals. In order to avoid the picking upof interstitial contaminants, welding of Ti and Zr alloys is performed invariablyin a protective environment (either inert gas cover or vacuum).

The solidified microstructure, as revealed in an optical microscopy examinationof the fusion zone, shows either a dendritic or a cellular structure, depending on thegrowth rate and the local thermal gradient in the liquid. For revealing the solidified




microstructure, a special etching technique known as “dendrite decoration” is usedin which the oxide formed on the sample surface during etching is retained.

Solidification in the fusion zone begins by the epitaxial growth of partiallymelted grains at the interface of the molten pool. The grains grow into the liquidmetal pool. Concurrently grains are nucleated at the upper molten pool surface andinterfaces of these grains in contact with the liquid pool tend to develop a morpho-logical instability leading, generally, to a cellular structure. This variation of thestructure from the cellular at the upper surface to a fully dendritic in the lower sur-face of the fusion zone arises due to a substantially lower value of G/R (where R isthe cooling rate) at the lower surface compared to that at the upper. As the transi-tion from cellular to dendritic growth occurs as a function of position in the fusionzone, the observed growth direction shows an increased dependence on the shapeof the receding weld pool. Near the lower surface, a single variant of the den-dritic structure extends nearly from the heat-affected zone to the weld centre line,suggesting that the growth direction deviates to a significant extent from the maxi-mum thermal gradient. Closer to the upper weld surface, the solidification processadheres more strictly to the maximum thermal gradient; changes in growth direc-tion occur both by secondary nucleation of more favourably oriented grains andby deviation in the growth direction from the crystallographically favoured one.

The 100� directions are those which correspond to the maximum growthrate. In the vicinity of the lower weld surface, the growth of dendrites occurspredominantly along 100� directions, often at the expense of deviating fromthe maximum thermal gradient. If the deviation is too large, dendritic growthcontinues along more favourably oriented secondary arms or by nucleation ofmore favourably oriented grains. Near the upper weld surface, where the valueof G/R is much higher, the growth is more stable and a cellular solidificationstructure is generally produced. The cellular structure often becomes curved sothat the growth direction adheres to the maximum thermal gradient.

The next point which needs to be addressed is the segregation of alloying ele-ments. In order to evaluate the segregation tendency of a given alloying element,the parameter 1/k= −�ml/Dl��1−ko� can be used, where ml is the liquidus slope,Dl is the diffusivity in the liquid and ko is the partitioning ratio. This parameter, ascaling factor in instability equations for the movement of the solidification front,provides a measure of the extent to which a given alloying element promotes thebreakdown of a planar liquid/solid interface. Large values of ko correspond to alloy-ing elements which promote interface instability, while small values favour planarinterface movement. Table 3.2 lists the ko parameter for different alloying elementsin binary alloys of Ti and the relative concentrations of these elements which cancause an interface breakdown at a given value of G/R normalized to the elementwith largest ko (Gould and Williams 1980). Experiments have shown that out of




Table 3.2. Values of the parameters K and critical requiredconcentrations at a given G/R for the various alloyingadditions.

Alloying addition K = −mlDl�1−k�

C∗

C∗Fe( �C− s

wt%− cm2

)Fe 1�23×106 1Cu 9�80×105 1.26V 8�79×104 14.0Al 7�02×104 17.5Sn 2�35×104 52.3

the alloying elements compared in Table 3.2, Cu has a much stronger tendency tosegregate as compared to V, Al and Sn, consistent with the predictions based onthe ko parameter. However, Fe shows an anomalously low segregation tendency.

We have, so far, considered the solidification morphology, though the finalmicrostructure of the fusion zone is developed by a superimposition of solidstate transformations on the compositionally graded solidification product. Theheat-affected zone also undergoes compositional changes due to redistribution ofalloying elements between the and the � phases. During the final cooling stage,a variety of phase transformations occur within the �-phase, depending on thelocal alloy composition and the cooling rate. The possible phase transformationsin the �-phase can be indicated in a schematic phase diagram (Figure 3.11). Let us

Tem

pera

ture

Atom Fraction of c

4

α

β

Ms(α′)

Ms(ω)3

2

1

Tic1 c2 c3 c4

Figure 3.11. Schematic phase diagram indicating possible phase transformations in the �-phase.




consider an alloy (either Ti–X or Zr–X) with the average concentration of X beingc1. During welding, the fusion zone in the liquid state will attain a homogeneousconcentration x1, provided there is no pick-up of alloying elements from the fillermaterial or environment. As discussed earlier, the solidified �-phase will exhibit arange of compositions within the fusion zone, depending on the extent of segrega-tion occurring during solidification. The heat affected zone can be subdivided intoseveral regions, A, B, C, etc. on the basis of the temperature each region has seenduring welding. Region A (represented by point 1 in Figure 3.11) corresponds togoing into the full �-phase field. Depending on the cooling rate imposed, this regionwill either transform into the martensitic ′ phase or will exhibit a Widmanstatten +� structure. Region B (represented by point 2) enters into the +� phasefield and the �-phase, during cooling, transforms either martensitically to ′ or bya diffusional process to the two-phase +� structure. The final microstructureof region B, therefore, consists of a primary and a transformed � (either ′

or ( +�)) phase mixture. The volume fractions of the primary and the trans-formed � phases will depend on the temperature at which the –� equilibrationresults in partitioning of alloying elements, �-stabilizing elements preferentiallymigrating to the �-phase (and similarly -stabilizing elements to the -phase).This causes further stabilization of the �-phase. With increased �-phase stabilitycorresponding to a lower equilibration temperature (as in the case of point 3 inFigure 3.11), the �-phase transforms into the �+� structure on cooling (region C).Finally, in region D, very small volume fraction of the �-phase present in theintervening space of primary is retained on cooling (point 4 in Figure 3.11). Inthis manner, the local composition of the alloy, as dictated by the solidificationprocess or by the –� equilibration process in the fusion and the heat-affectedzones, determines the microstructure. As the peak temperature reached during thewelding process decreases from the melting temperature at the centre line of theweld to the ambient, the regions A, B, C and D will appear successively withincreasing distance from the centre line. The thickness of each zone, however, willdepend on the energy input during welding, the thermal gradient and the durationof the thermal exposure at elevated temperatures.

The foregoing discussion on microstructure development in the heat-affectedzone assumes attainment of equilibrium which rarely takes place during the shortduration of thermal excursion in welding. As a consequence, an incomplete parti-tioning of the alloying elements in the and the � phases occurs and correspond-ingly subsequent phase transformations take place.

The development of microstructure in the fusion and the heat-affected zones ofa weldment is strongly influenced by the thermal history of the various regions, thesizes of which are primarily determined by welding parameters such as power den-sity, traverse speed and the thermal properties of the material. Figure 3.12 shows




Figure 3.12. Welding microstructures produced in Ti–6Al–4 V by laser welding.

the microstructure produced in Ti–6Al–4 V by laser beam welding (LW). Thenotable differences between gas-tungsten arc welding (GTAW) and LW processeshave been found to be as following: (a) a deeper penetration can be achieved byLW compared to GTAW at comparable values of power input and traverse rate;(b) Both the heat-affected and the fusion zones associated with GTAW welds showWidmanstäatten -plates, indicating a diffusional � → transformation whilethese zones produced in LW consist mainly of the martensitic ′-phase.

The deeper penetration in laser welding (and also in electron beam welding) isthe consequence of the mechanism of energy transfer, commonly known as the keyhole effect. As the high power density beam falls on the workpiece, vaporizationoccurs at the surface of the workpiece creating a hole. The hole acts as a radiationtrap, or black body, enhancing the coupling of the beam energy into the workpiece. Vapour is ejected from the key hole at a very high speed, dragging with itmolten material and stabilizing the molten walls of the key hole. Such a processleads to the formation of a fusion zone with a high depth-to-width ratio. As theenergy input is much smaller than that required in the case of GTAW to achievethe same level of penetration, the thickness of the heat-affected zone also remainsrestricted in laser welding.

A larger pool of molten metal and a larger heat-affected zone are responsiblefor a relatively slower cooling rate in these zones in GTAW welds as com-pared to laser welds. This is reflected in the microstructure produced in thesezones: � →Widmanstatten transformation in GTAW welds as against marten-sitic � → ′ transformation in laser welds.




3.3 RAPIDLY SOLIDIFIED CRYSTALLINE PRODUCTS

Rapid solidification processing (RSP) techniques, which induce extension of solidsolubility (thereby enlarging the alloying range), refinement of microstructure,suppression of segregation during solidification and dispersion of precipitatingphases in a finer spatial scale have been applied to Ti and Zr alloys only to a limitedextent. Because of the high reactivity of these alloys, RSP techniques involvingcontainment by crucibles, nozzles, etc. cannot be used. Special techniques, viz.,plasma rotating electrode process, inert gas atomization, melt extraction frommolten pool or pendant drop, laser or electron beam surface melting, cold heartharc melt spinning and hammer and anvil splat quenching, are adopted for rapidsolidification of these alloys. These techniques are illustrated in Figure 3.13 andtheir important features are described.

The plasma rotating electrode technique (Figure 3.13(a)) consists of local melt-ing of a rotating consumable electrode of the alloy to be processed, by strikingan arc with a tungsten electrode. The molten metal is thrown out of the pool,by the action of the centrifugal force, in the form of atomized particles whichundergo solidification in the flight path from the centre to the periphery of thechamber maintained in an inert gas environment. Powder particles produced insuch a process are typically 50–200 �m in diameter and show a dendritic struc-ture. Cooling rates achieved in this process remain in the range of 103–105 K/s.This method is used for the production of rapidly solidified Ti alloy powders incommercial scales. The process can be modified by using a high-intensity laserbeam heat source in place of the arc. The laser heat source allows the introductionof a high-velocity inert gas jet to supplement centrifugal atomization and increasesthe rate of heat extraction.

The inert gas atomization technique involves cold hearth melting and bottomejection through a refractory metal-skull nozzle, Figure 3.13(b). Typically, powdersin the size range of 200–400 �m of Ti alloys can be produced using this techniquein kilogram scale in a single run.

Melt extraction is performed by extracting a solidified filament from a moltenpool, either contained in a crucible or created by local heating of the melt stock,Figure 3.13(c). A cold rotating wheel is used for extracting the heat and ejectingthe solidified filament. This method is usually adopted in the laboratory, but somederivatives of this have the potential for scaling up.

Rapid solidification can be effected by scanning an intense laser or electron beamon the surface of finished alloy components, Figure 3.13(d). This requires a suitablecontrol of the energy density of the incident beam and of the interaction timebetween the beam and the material so that a thin layer is melted and subsequentlyrapidly solidifies, heat from the molten layer being extracted by the bulk of the




D

C

(a)

Melt stockHeat source

Wiper

Solidfilament

(c)

2

1

3

4

5

(e)

(b)

AB

B

A

(d)

B A

3

1 2

4

Figure 3.13. RSP techniques (a) plasma rotating electrode, (b) inert gas atomization technique,(c) melt extraction/pendant drop, (d) laser/electron beam surface melting and (e) hammer and anvilsplat quenching.

component. Very intimate melt–substrate contact is achieved which results in highcooling rates during and after solidification.

Cold hearth arc melt spinning involves the ejection of a stream of liquid alloyfrom a cold hearth arc melted pool on to a rapidly spinning metallic wheel. Theribbon produced is thin (about 15–50 �m thick) and a cooling rate of about 105 K/scan be achieved.

Hammer and anvil splat quenching is carried out by squeezing a liquid dropletbetween impacting piston and anvil heat extracting surfaces, Figure 3.13(e). Thesolidification rate can be as high as 107 K/s but the process can produce onlymilligram scale quantity in a single run.




The rapid solidification techniques outlined here are all suitable for the process-ing of Ti and Zr alloys. RSP techniques have been extensively used for modifyingthe microstructure of crystalline Ti alloys, mainly with the aim of controllingthe microstructure of Ti alloy powders which have several industrial applications.Powder metallurgy applications for Zr alloy products are, however, quite limited,and therefore, not many investigations pertaining to the effect of RSP on themicrostructure of crystalline Zr alloys have been reported.

3.3.1 Extension of solid solubilityOne of the objectives of rapid solidification processing is to extend the limit ofsolid solubility. The extent to which such a metastable extension of solubility iseffected depends on the rate of quenching during solidification, the increment inthe chemical potential of the solute element above the equilibrium value and thediffusivities of the solute in the liquid and the solid phases. Ti and its alloys usuallysolidify as a bcc �-Ti phase which may either be retained at room temperature ormay undergo solid state phase transformations, depending on the relative stabilityof the �-phase. Table 3.3 shows the solubility limits (in atom percent) of somealloying elements under equilibrium and rapid solidification conditions (Roweet al. 1987).

The metastable solubility extension shown in Table 3.3 refers to the solubilitylimits at the solidification temperature. With lowering of temperature, most ofthese supersaturated solid solutions reject solutes which get precipitated in theform of second phase dispersions which can be very effective in improving hightemperature strength and creep resistance of rapidly solidified Ti alloys.

Table 3.3. Solubility limits (in at.% of some alloying elements under equilibrium and rapid solidi-fication conditions.

Element Matrix Maximum equilibriumsolubility in the�-phase

Extended solubilityin �-phase in RSPalloys

RSP technique

B Binary 0.5 6.0 SplatC Binary 3.1 10.0 SplatSi ∼10 at.% Zr 5.0 6.0 SplatNd Binary ∼0.3 >1.0 EBSQY ∼8 Al ∼0.3 >2.0 LaserGd Binary ∼0.3 >0.5 EBSQEr ∼5 Al ∼0.3 >1.5 Splat

EBSQ, electron beam splat quenching.




3.3.2 Dispersoid formation in rapidly solidified Ti alloysAttempts to disperse hard refractory compounds in Ti alloys have not been quitesuccessful in the ingot metallurgy of these alloys. This is due to the fact thatstable compounds segregate to a very significant extent during solidification at theconventional ingot cooling rates. Rapid solidification processing has, therefore,been employed in several Ti alloys with the aim of producing a finely dispersedstructure. Rapidly solidified Ti alloys with additions of (a) Er or (b) Ce andS have shown very fine (50–100 nm) dispersions of Er2O3, CeS and Ce4O4S3.These dispersed phases have shown excellent resistance against coarsening attemperatures as high as 1223 K.

Compounds of Ti with the metalloid elements C, B, Si and Ge have high meltingpoints and good chemical stability. These compounds are, however, less stable inthe Ti alloy matrix than the rare earth oxides and sulphides. Rapidly solidified Tialloys containing C show a distribution of fine spherical TiC precipitates whilethose containing B produce a fine dispersion of needle-shaped TiB precipitates. Tialloys with Si and Ge, in the rapidly solidified condition, contain fine dispersionsof Ti5(Si,Ge)3. Both carbides and silicides coarsen rapidly at 973 K while TiBremains stable upto about 1073 K.

Rapid solidification processing is also applied for alloys in which eutectoiddecomposition takes place. The cooling rate required for solidification withoutany significant segregation is lower in the case of eutectoid-forming alloys thanin rare earth or metalloid-containing alloys. A higher volume fraction of thesecond phase makes the eutectoid-forming alloys (containing one or more of thealloying elements Cr, Mn, Fe, Ni, W and Cu) suitable candidates for high strengthalloys which can retain their strength up to intermediate temperatures. The uppertemperature limits of the stability of intermetallic compounds present in eutectoidalloys restrict their high temperature applications.

3.3.3 Transformations in the solid stateThe influence of the rapid solidification treatment on the subsequent phase trans-formations in the solid state has not been studied extensively. Inokuti and Cantor(1979) have reported refinement of the martensitic structure in rapidly solidifiedFe based alloys. The refinement is attributed to the small size of austenite grainsforming from the liquid phase which, in turn, limits the size of the martensiteplates.

Banerjee and Cantor (1979) have reported the microstructure produced in unal-loyed Zr and Zr–Nb alloys by rapid quenching from the liquid state. The possibilityof the formation of the -phase directly from the liquid phase, skipping the inter-mediate equilibrium �-phase, has been examined. The thermodynamic feasibility




α (Zr) β (Zr) L (Zr)

β (Zr)

α (Zr)

L (Zr)

Tα L (1700 K)

T β L (2125 K)Tα β (1135 K)

0

–100

–200

–300

–400

–500

G (

kJ/m

ol)

1000 2000 3000 4000Temperature (K)

↓

↓

↓

Figure 3.14. Schematic free energy versus temperature plots for the liquid, the - and the �-phasesfor pure Zr. The temperatures T�/l and T /l correspond to those at which the liquid/� and the liquid/ equilibrium are established.

of such a process is illustrated schematically in Figure 3.14. The evidence ofa direct L → transformation has been recorded in unalloyed Zr samples con-taminated with Si and O. The cellular structure of the -phase with Si-enrichedintercellular region (Figure 3.15(a)) observed in rapidly solidified samples pointsto the fact that the -phase cells originated directly from the liquid. The extentof supercooling required for making such a direct L → transformation possi-ble can be reduced by alloying elements which enhance the relative stability ofthe -phase in comparison with the �-phase. Since both Si and O are strong -stabilizers, contamination of these elements in Zr is expected to raise the Tl/

temperature and thereby reduce the extent of supercooling required for the direct solidification.

In alloys containing a �-stabilizing element, two types of displacive trans-formations, namely the � → ′ martensitic and the � → � transformations, areencountered. Which of these processes is selected by a given alloy is determinedby the alloy composition. As has been explained in Chapter 1, the � → ′ trans-formation operates in compositions where Ms�

′� is higher than Ms�� and viceversa. The transition from the ′ structure to the �+� structure is noticed in Zr–Nb




(a) (b)

(c)

Figure 3.15. Microstructures developed in rapidly solidified Zr alloys undergoing post-solidificationphase transformation: (a) cellular morphology of the -phase which formed directly from the liquid.Intercell boundary regions are richer in Si and O. (b) Propagation of ′-martensite laths acrossthe boundaries of �-cells which formed as a solidification product. (c) �+� microstructure inZr-5.5 wt% Nb alloy.

samples quenched from the �-phase field (solid state quenched) when the Nb levelexceeds ∼7 at.%, the composition at which Ms�

′� line intersects the Ms�� line(Figure 1.18). Banerjee and Cantor (1979) have shown that in rapidly solidifiedZr–Nb samples, this transition is shifted to a lower (∼5.5%) level of Nb. Thispreference for � → � transformation over the � → ′ martensitic transformationunder rapid solidification is attributed to the retention of a higher concentrationof vacancies which are known to stabilize the �-like defects prior to the trans-formation (Kuan and Sass 1976). Figure 3.15(b) and (c) shows the martensitic ′

structure and the �+� structure, respectively, in rapidly solidified Zr–Nb alloys.It may be noted that the ′ laths in (b) are cutting across the intercell boundaries,implying that the concentration profiles across the cell boundaries are such thatMs�

′� remains above the quenching temperature all along the growth path ofmartensite laths.




Figure 3.16. Dark-field micrograph of Zr-27 at.% Al alloy showing cellular structure leading to Alenrichment at cell boundaries where higher density of Zr2Al particles are seen.

In a study on a rapidly solidified Zr-27 at.% Al alloy, Banerjee and Cahn (1983)investigated the sequence of transformation events which led to the formationof the ordered Zr2Al phase (B82 structure) in this alloy. The analysis of themorphological features (Figure 3.16) of the transformation products led to theconclusion that the following sequence of transformation events occurred:

(1) Formation of a supersaturated �(bcc)-phase mainly by a partitionless solidi-fication process. The limited alloy partitioning in some localized areas led tothe formation of a cellular structure where the cell boundaries were decoratedby a higher number density of Zr2Al particles as shown in Figure 3.16.

(2) Spinodal decomposition of the supersaturated �-phase (Zr–Al) during continu-ous cooling subsequent to solidification. This process resulted in the formationof a compositionally modulated structure with modulations along the elasti-cally soft 100� directions which, in turn, produced Al-rich cuboids of about20 nm size.

(3) A combined chemical and displacement ordering within the Al-rich cuboidsresulting in the formation of the Zr2Al structure – details of which are discussedin Chapter 6.

(4) � → � transformation in the Al-depleted regions in the intervening spacebetween the cuboids.




3.4 AMORPHOUS METALLIC ALLOYS

Amorphous metallic alloys or metallic glasses have emerged as a new class ofengineering materials after vitrification of metallic alloys by using the techniqueof ultrarapid quenching of molten alloys has become possible. These materials,which do not have any long-range crystalline order but retain metallic bonding,exhibit several interesting properties emanating from their unique structure whichis isotropic and homogeneous in the microscopic scale. Extremely high hardnessand tensile strength, exceptionally good corrosion resistance and very low mag-netic losses in some soft magnetic materials are some of the attractive propertiesassociated with amorphous metallic alloys. There are three technologically impor-tant classes of amorphous alloys, namely (a) the metal–metalloid alloys, such asFe–B, Fe-Ni-P-B and Pd–Si, (b) the rare earth–transition metal alloys, such asLa–Ni and Gd–Fe, and (c) the alloys made up of a combination of early and latetransition metals such as Ti–Cu, Zr–Cu, Zr–Ni and Nb–Ni. With the availabil-ity of metallic glasses, several important issues concerning the stability of thesemetastable phases and the kinetics of their thermal decomposition have attractedthe attention of the phase transformation research community. It will be shown inthis chapter that the formation and the decomposition of Ti- and Zr-based metal-lic glasses offer some unique opportunities for studying several aspects of phasestabilities and transformations in metallic glasses. These include glass-formingabilities (GFAs), diffusion mechanisms, modes and kinetics of crystallization andformation of bulk metallic glasses. A major advantage of studying these systemsis that a number of binary metal–metal alloys based on Ti and Zr are amenable toeasy glass formation.

3.4.1 Glass formationThe three-dimensional lattice arrangement of atoms in a crystalline solid isdestroyed as it melts. In the liquid state, the long-range order both translationaland orientational of the crystalline solid is not retained as the atoms vibrate aboutpositions which are rapidly and constantly interdiffusing. Melting being a stronglyfirst-order transition, thermodynamic quantities such as specific volume, enthalpyand entropy undergo a discontinuous change at the melting temperature as a crys-talline solid transforms into a liquid. At temperatures above the melting point, aliquid is in a state of internal equilibrium and its structure and properties are inde-pendent of its thermal history. A low viscosity, which is essentially the inability toresist a shear stress, characterizes the liquid state. On cooling, a liquid transformsinto a crystalline solid under the equilibrium cooling condition. Vitrification ofa liquid is possible only when the liquid is cooled at a rate sufficiently rapidto escape a significant degree of crystallization so that the “disordered” atomic




configuration of the liquid state is frozen in. Glass formation is easy in a num-ber of non-metallic systems such as silicates and organic polymers. The natureof bonding in these systems places severe limits on the rate at which crystallineorder can be established during cooling. Thus the melt solidifies into a glass evenat low cooling rates (often less than 10−2 K/s). Metallic melts, in contrast, havenon-directional bonding which allows a very rapid rearrangement of atoms intothe crystalline state. Hence very high cooling rates need to be imposed for formingmetallic glasses by avoiding crystal formation. Cooling rates exceeding 105 K/s arenecessary for the formation of metallic glasses in several binary and ternary alloysystems based on Ti and Zr. In recent years, several Zr-based alloys with a numberof components have been found to be amenable to vitrification at substantiallyslower cooling rates (Inoue 1998). This has opened up the possibility of obtainingmetallic glasses in the bulk form. The formation and properties of bulk metallicglasses are discussed in Section 3.6.

The process of vitrification of a liquid under non-equilibrium cooling can becompared with the equilibrium crystallization process in plots of viscosity, �,and thermodynamic quantities such as specific volume, V , and specific heat, Cp,against temperature (Figure 3.17). While the liquid to crystal transformation isaccompanied by a step change in these properties, a progressive change in viscosityand enthalpy precedes the vitrification event as the liquid is undercooled below theequilibrium melting temperature, Tm. It is evident from these plots that vitrificationis possible only if the equilibrium crystallization process is avoided. The coolingrate, therefore, needs to be sufficiently high so that insufficient time for nucleationand/or growth does not permit the formation of the crystalline phase to a detectablelevel. Although the driving force for nucleation continuously increases with theextent of undercooling, the rapid increase in viscosity is responsible for decreasingatomic mobility and thereby effecting the kinetic suppression of crystallization.Eventually, the atomic configuration of the liquid becomes homogeneously frozenat the glass transition temperature, Tg. This structural freezing to the amorphousstate is, by convention, considered to occur when the viscosity reaches a value of1013 poise. Since the atomic configuration of the amorphous state does not corre-spond to a unique equilibrium structure, Tg and the glass structure are both coolingrate dependent, variations in the latter resulting in glasses with different states ofstructural relaxation. Figure 3.17(a) shows the glass transition temperatures, Tg1

and Tg2, for two glasses, G1 and G2, forming under different rates of quenching.The GFAs of different metallic systems have been assessed in terms of both

relative thermodynamic stabilities of the amorphous and the equilibrium andmetastable crystalline phases which compete to form during cooling and kineticfactors which determine the critical cooling rate necessary for avoiding thecrystallization process. The thermodynamic and kinetic criteria for glass formation




9.0

8.5

V c

m3

mol

–1

G2

Temperature

↓↓

Tg1

500

G1

Tg2

1000

X Tm

L

(c)

log

τ

4

–10

–5

0

515

10

5

0

2

log

η

1/T →

L

Crystal (X)

Tg1

Tg2

G2

G1

q2 = 1 Ks–1

q1 = 105 K s–1

1/Tm

glass, G

(a) (b)

80

100

40

20

0

60

Cp /g

tw K

CP

Temperature K300100 200

Tm

500400 600 800700

→

→

→

C

xp

C

lp

C

gp

→

Figure 3.17. (a) Viscosity (�) as a function of reciprocal temperature showing the liquid to glasstransitions at Tg1 and Tg2 for two different cooling rates, q = 1 and 105 K/s. The liquid to crystal,(X), transformation is also shown in the same figure. (b) Specific volume, V , and (c) specific heat,Cp, as a function of temperature showing step changes at the liquid to glass transition. CX

p , Cgp and

CLp refer to crystalline, glassy and liquid phases, respectively.

have been discussed in the following section with special reference to Ti- andZr-based systems.

3.4.2 Thermodynamic considerationsPure metals are extremely difficult to vitrify under the conditions of rapid solid-ification which typically attain a cooling rate of about 106 K/s. Thin sections of




splat quenched foils of Ni with dissolved gaseous impurities to a level of about2% have been reported to vitrify. This, however, requires a cooling rate as highas 109–1010 K/s, which is estimated to be attainable in thin sections (<100 nm) ofsplats. The difficulty in the vitrification of pure metals can be ascribed to the poorstability of the amorphous phase with respect to the equilibrium crystalline struc-ture and the ease of spontaneous crystallization. The latter is expected to requirevery little structural adjustment and consequently a very low thermal activation.

From the assessment of GFAs of a number of alloys, it has been observedthat GFA is enhanced substantially in several systems with increasing additionsof solute elements in certain composition ranges. The origin of enhanced GFAcan be traced to factors which can be grouped into thermodynamic and kineticfactors. This section deals with the relative thermodynamic stability of the liquid/amorphous phase with respect to the competing crystalline phases.

The alloy systems in which glass formation occurs most readily are those whichexhibit either one or more deep eutectics or a steep decrease in the liquidustemperature, Tl, with increasing levels of solute content. Phase diagrams of anumber of Ti- and Zr-based alloys (e.g. Ti–Cu, Ti–Be, Zr–Cu, Zr–Ni, Zr–Ni–Cu)show such characteristics. Let us take the example of the Zr–Ni binary system forexamining the relative stabilities of different competing phases. Figure 3.18 showsthe free energy–concentration (G–c) plots for the ordered intermetallic phases andthe supercooled liquid (amorphous) phase. Such free energy–concentration plotshave been constructed on the basis of an analysis of the phase diagram (Katgerman1983) and by using the computed values of formation energies of intermetallicphases (Miedema 1980). Sharply bent U-shaped G–c plots (shown as vertical linesin Figure 3.18) for the intermetallic phases reflect the stability of these phaseswithin narrow composition limits.

The G–c plots for the liquid and the � phases intersect at a point where theintegral molar free energies of these phases are equal (Figure 3.19(a)). This marksthe composition limit up to which partitionless solidification is thermodynamicallypossible. The locus of these intersection points at different temperatures definesthe To�c� line, the shaded area below, which defines the region of stability of the�-phase with respect to the liquid phase under the condition of partitionless solid-ification (Figure 3.19(b)). Since rapid solidification at a sufficiently high coolingrate does not permit partitioning of the alloying elements between the parent liq-uid and the crystalline product, the composition of the liquid is inherited by thesupersaturated crystalline phase. The To�c� curve can, therefore, be considered asthe polymorphic melting curve of a multicomponent system.

The addition of solute atoms to a pure metal, specially in the case where thesolute and the solvent atoms differ in size and in chemical character, makesthe diffusive rearrangement of atoms a necessary step during solidification into




20

16

12

8

4

0

4

8

12

16

20

24

28

32

36

40

Zr Ni

ΔG k

J/gm

– a

tom

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

650 K

A

Zr 2

Ni

ZrN

iZ

r 9N

i 11

Zr 7

Ni 1

0

Zr 8

Ni 2

1Z

r 2N

i 7Z

rNi 5

c (at.% Ni) →

Figure 3.18. Free energy (G) versus composition (c) plot at 650 K corresponding to the amorphousphase and ordered intermetallic phases in the Zr–Ni system.

a crystalline phase. As a consequence, glass formation becomes easier. The glasstransition temperature, Tg, generally increases with solute concentration, as shownin Figure 3.19(b). Hypothetical G–c plots consistent with the phase diagrams ofTi–Be and Ti–Fe systems (Figure 3.20) show that a metastable equilibrium betweenthe liquid (amorphous) phase and the �-phase can be established at a sufficientlylarge supercooling provided, the formation of equilibrium phases is suppressed.The point of intersection of the To�c� and Tg�c� lines indicates the minimumsolute concentration limit for an alloy to vitrify. An alloy having a solute contentexceeding this limit cannot solidify into the crystalline solid through a partitionlesssolidification as it cools down to Tg, where the viscosity of the liquid rapidlyrises to about 1013 poise. A thermodynamic basis of glass formation can thus beexplained by considering the competition between the partitionless polymorphicsolidification of the liquid to supersaturated � on one hand and the liquid to glasstransition on the other. It must be emphasized here that the avoidance of theformation of the equilibrium crystalline phases, namely the equilibrium � or the




(a)

(b)

L

A c (at fraction B) →

T1

T

→

To

Te

Tg

T2

T3

L1S1

S2

C1

C2

ce

L2

β + γ

L + γ

β

L

L1

S1c1

γ

L

S2L2

c2

G

β

γ

β

c → c →

Figure 3.19. (a) Schematic G–c plots and (b) corresponding phase diagram representing the equi-librium between the liquid, � and intermetallic � phases. The integral molar free energies of theliquid and the � phases are equal at c1 and c2, corresponding to T1 and T2. The locus of such pointsdefines the To line.

equilibrium intermetallic phases, cannot be explained on a thermodynamic basis.The suppression of nucleation and growth of equilibrium crystalline phases areessentially controlled by kinetic factors which are discussed in Section 3.4.3.

Metallic glasses are generally grouped into the following major categories,depending on the constituent elements:

(1a) Late transition metals (TL) alloyed with metalloids (M): TL includes GroupVIIB, Group VIII and Group IB noble metals while M includes Si, B andSb. Typical examples of this category are Fe–B, Ni–B, Fe-Ni-Co-B, Pt–Sbglasses. In several binary systems, deep eutectics are encountered at com-positions in the range of 13–25% M. Many of the amorphous alloys in thiscategory are of technological importance because of their soft ferromagneticproperties which arise due to the presence of a combination of Fe, Co and Ni.




(a)

TiBe2

L (T1)

L (T2)

L (T3)

80

m-TiBe

G

c (at.%Be)40Ti 20 60

ce

β

α

↓

(b)

G

Ti 20

c (at.%Fe)40 60 80

TiFe

ce

L (T1)

L (T2)

L (T3)

β

α

↓

→

→

Figure 3.20. Hypothetical G–c plots consistent with the phase diagrams of Ti–Be and Ti–Fe systems.

(1b) Early transition metals (TE) alloyed with metalloids (M): Early transitionmetals in this class of amorphous alloys include Ti, Zr and Nb which formeutectics at compositions in the range of 15–30% of metalloids such asSi and B.

(2) Lanthanide metals (RE) alloyed with late transition (TL) and Group IB metals:Many of these systems, exhibiting deep eutectics, are easy glass-formingalloys. Examples of systems of this type include La78Ni22�Gd–Fe32−50 andGd–Co40−50.

(3) Early transition metals (TE) alloyed with late transition metals (TL): Theaddition of a TL (or a Group IB) metal generally leads to a sharp decreasein the liquidus temperature, Tl, which finally drops down to a low eutectictemperature. The Zr–Ni system, which has been discussed earlier, representsthis category of potential glass formers which includes Ti–Cu, Zr–Cu, Zr–Fe,Zr–Co, Zr–Ni–Fe and several other Ti- and Zr-based systems.

In all the categories of easily glass-forming alloys, a high GFA is almostinvariably associated with either the presence of one or more deep eutectics ora sharp drop in the liquidus temperature. Both these aspects of phase diagramsstrongly indicate an enhancement of the stability of the liquid phase with increas-ing additions of alloying elements in the relevant composition range. The freeenergy difference between the supercooled liquid and the competing equilibriumcrystalline phase is thereby substantially reduced.




The extent of lowering of Tl is usually expressed in terms of a reduced glasstemperature, T r

g , defined as the ratio, Tg/Tl. The lower is the Tl and the higher isthe Tg, the higher is the GFA. The highest known values of T r

g for metallic glassesare in the range of 0.66–0.69.

Free energy, G�c�, plots at different temperatures for the competing crystallineand amorphous phases are indicative of the relative stabilities of these phases.Though the free energy of the supercooled liquid (glass) is brought down substan-tially by alloying additions in potentially glass-forming alloys, the G�c� curve forthe liquid/amorphous state still remains at a level higher than that correspondingto the equilibrium crystalline phases. It is, therefore, essential that kinetic factorsplay a major role in the vitrification process by suppressing the nucleation andlimiting the growth of crystals. It is also clear from G�c� plots (e.g. Figure 3.19(a))that the stability of the liquid phase is not necessarily maximum at the eutecticcompositions. In fact it will be shown later using kinetic arguments that GFA isthe highest in some systems not at the eutectic composition but at the compositioncorresponding to one of the intermetallic phases.

Various semiempirical methods have been proposed for relating GFA withthe depression of Tl below some mean value. Marcus and Turnbull (1976) haveproposed a normalized parameter T/T o

l , where T �= T ol − Tl� is expressed

as the deviation of Tl from the ideal solution liquidus temperature, Tol , which is

given by

T ol = HA

f TAm

HAf −R ln�1− c�TA

m

(3.21)

where HAf and TA

m are the heat of fusion and melting point, respectively, of thesolvent metal and c is the mole fraction of the solute. Large positive values ofthis parameter indicate a large deviation from the ideal solution behaviour whichcorresponds to an ordering tendency and a high GFA. It has been pointed outby Zielinski et al. (1978) that a combination of two factors, namely the unlikeatom bond strength, expressed by the magnitude of the excess negative enthalpyof mixing, Hm, and the depression of the melting point, (T o

l −Tl), can be veryeffective in categorizing different alloy systems into readily glass-forming (RGF)and non-glass-forming alloys. This is to be expected since a deep depressionin the liquidus temperature is invariably associated with a strong interaction onmixing of the alloy components. Free energy plots, G�c�, for the Zr–Ni system(Figure 3.18) clearly illustrate this point. The G�c� plot for the liquid phase showssharp drops with increasing addition of Ni (up to about 50%) in Zr in the vicinityof compositions where a series of intermetallic phases, which are all chemicallyordered structures, are present. The atomic radius ratio, rA/rB, where A represents




the smaller size atom in an A–B alloy, is another parameter which is seen to havea strong correlation with GFA. A survey of a number of alloys shows that thealloys which readily form an amorphous phase correspond to an rA/rB ratio ofless than 0.85.

3.4.3 Kinetic considerationsThe essential requirement for the formation of a glassy state in an alloy duringcooling from the liquid state is that the equilibrium crystalline phases are notallowed to form to a detectable level prior to the rapid rise in the viscosity finallyleading to vitrification. A quantitative estimation of the cooling rate required forachieving the condition of avoidance of crystal formation can, therefore, be madeby considering the kinetics of the nucleation and growth of crystalline phaseswhich compete with glass formation. For a glass to form, one of the followingconditions has to be satisfied: (a) complete suppression of nucleation of crystallinephases; (b) limited nucleation of crystals but no significant growth of these,resulting in a distribution of quenched-in nuclei of crystals in an amorphous matrix;(c) nucleation of a very few isolated crystalline nuclei which grow to relativelylarge sizes but the number density of these crystals embedded in the amorphousmatrix is too low to be detected by X-ray diffraction (XRD).

Condition (a) is rarely satisfied in metallic glasses including Ti- and Zr-basedones. Even in a process such as laser glazing, where the cooling rate of the thin layer(<10�m) of the surface is very high, it is difficult to suppress the nucleation ofcrystals completely. A certain number density of quenched-in nuclei is invariablypresent, and therefore, glass formation is essentially controlled by the fact thatthese nuclei do not get an opportunity to grow. Recent high-resolution electronmicroscopy work (Savalia et al. 1996) has clearly demonstrated the presence ofsuch quenched-in nuclei in several Zr-based amorphous alloys. In some instances, alimited number of large-size crystals have been detected in metallic glass samples,suggesting that condition (c) is operative in such cases.

An estimation of the critical cooling rate required for the avoidance of crystalnucleation can be made for both homogeneous nucleation (either steady state ortransient) and heterogeneous nucleation.

Theories of nucleation originally developed with regard to the formation ofliquid drops in a supersaturated vapour have been extended later to condensedphases. The rate of nucleation, I , of a crystalline phase from the liquid/amorphousmatrix phase is given by

I = ZA∗N ∗ (3.22)

where the factor Z is related to the frequency of atoms jumping across the inter-face between the liquid/amorphous matrix and the critical nuclei and is expressed




in terms of the diffusion constant, D, the jump frequency � , and the jump dis-tance, �, as

Z = N 2/3� = 6N 2/3�D/�2� (3.23)

and A∗ and N ∗ are, respectively, the surface area and the number density of acritical nucleus which contains N atoms. Volmer and Weber (1926) proposed thatthe concentration of the critical nucleus, N ∗, is given by

N ∗/N = exp�W ∗/kT� (3.24)

where W ∗ is the work done to form the critical nucleus and is related to the volumefree energy change, Gv, associated with the formation of the critical nucleus, thesurface energy required for creating the surface of the nucleus and the elastic strainenergy resulting from the volume changes that occur during the transformation(Figure 3.21). W ∗ can be expressed in terms of the specific surface energy, �, thevolume free energy change and the strain energy density, E, as

−W ∗ = A�−GvV +EV (3.25)

where A and V represent the surface area and volume of the critical nucleus.

O

Interfacialenergy

w *

r *

Volume freeenergy

r

Wor

k do

ne fo

r nu

clea

tion

(w)

w

Aγ = K 1r 2

ΔGv = K 2r

3

Figure 3.21. Volume free energy change (Gv), interfacial energy and the net work done (W ) forcreating a nucleus of radius r.




The diffusion coefficient, D, can be expressed as D = Do exp�−Q/kT� forsolids, where Q is the activation energy for diffusion and D= kT/3�� for liquids(Stokes–Einstein equation, where � is the diameter of the diffusing molecule andcan be taken as equal to the jump distance).

Combining Eqs. (3.22)–(3.25), the nucleation rate, I , can be written as

I = 6A∗N 5/3�Do/�2� exp�−Q+W ∗�/kT (3.26)

for solids and

I = 6A∗N 5/3�D/�2� exp�W ∗/kT� (3.27)

for liquids. The preexponential factors in Eqs. (3.26) and (3.27) are usually in therange of 1035–1040/cm3 s. For a spherical nucleus of radius r, the work done, W ,for creating a nucleus can be expressed as

W = 4�r2�− 43�r3�Gv −E� (3.28)

The variation of W with r for a homogeneous nucleus is schematically shownin Figure 3.21 and W ∗ and r∗ for the critical nucleus can be obtained by findingout the maximum of the W versus r plot. Differentiation of Eq. (3.28) yields

r∗ = 2�Gv −E

(3.29)

and

W ∗ = 16��3

3�Gv −E�2(3.30)

In the case of the nucleation of a solid phase in a liquid matrix, the strain energycontribution can be neglected and Eq. (3.30) can be reduced to

W ∗ = 16��3

3�Gv�2

(3.31)

Combining Eqs. (3.27) and (3.31), the nucleation rate of crystals in a liquidmatrix can be expressed in terms of the driving force, Gv, as

I = 1035

�exp

(16��3

3�Gv�2kT

)(3.32)




where � is the viscosity of the liquid. Some time is required for establishing thesteady state rate of nucleation. The nucleation frequency rises sigmoidally withtime from zero to the steady state value, following the relation

I = Io exp�−�/t� (3.33)

where Io is the steady state rate of nucleation and � is the induction time. Kaschiev(1969) derived an equation for the induction time � by assuming that transportwas controlled by jumps across the interface, which on modification leads to thefollowing form for comparison with experiments,

� = 48��2�/�V�Gv�2 (3.34)

Nucleation occurring before the attainment of the steady state, that is, within theinduction time �, is known as transient nucleation. Substitution of appropriatevalues in Eq. (3.34) yields an induction time of several microseconds. Since glassformation in metallic systems requires a quenching rate of about 106 K/s, nucle-ation events occurring within microseconds also play an important role in vitrifyinga molten alloy. The steady state nucleation rate and the rate of transient nucleationare both dependent on the degree of undercooling, which strongly influences thevalues of Gv. The volume free energy change, Gv, can be expressed in one ofthe following three forms:

Gv = HfT

Tm

(3.35)

Gv = HfT

Tm

[�1− �Tm + �1+ �T

T +Tm

](3.36)

Gv = HfT

Tm

− CpT2

2T

(1− T

6T

)(3.37)

where = T ln�T/Tm�.Dubey and Ramachandrarao (1984) have pointed out that Eq. (3.37) is the most

appropriate for the estimation of Gv as a function of undercooling as comparedto Eqs. (3.35) (Jones and Chadwick 1971) and (3.36) (Thompson and Spaepan1979). Savalia et al. (1996) have plotted the estimated Gv against temperature inplots A, B and C in Figure 3.22 as obtained from Eqs. (3.35), (3.36) and (3.37),respectively. It is to be noted that as per Eq. (3.37), the estimated Gv increaseswith the extent of undercooling, reaches a maximum and then decreases withfurther increase in undercooling. The reduced value of Gv at high undercooling,




A

B

C

9 × 103

5 × 103

2 × 103

3 × 102 7 × 102 103

Temperature (K)

ΔGC (

J/m

ol)

Figure 3.22. Estimated GC (J/mol) as a function of temperature (K). Curve A corresponds toEq. (3.35) (Jones and Chadwick 1971), curve B corresponds to Eq. (3.36) (Thompson and Spaepan1979) and curve C to Eq. (3.37) (Dubey and Ramachandrarao 1984).

corresponding to Eq. (3.37), as compared to those obtained from Eqs. (3.35) and(3.36), leads to a substantial reduction (2–3 orders of magnitude) in the nucleationfrequency.

During rapid solidification, an undercooled liquid can vitrify only if both homo-geneous and heterogeneous nucleation – either transient or steady state – areessentially bypassed. How the number density of nuclei increases with time underthese four conditions is schematically shown in Figure 3.23. Transient nucleationtimes can be calculated by using the formulations proposed by Kelton et al. (1983,1986) and Ghosh et al. (1991). Savalia et al. (1996) have calculated the transient

Time

a b

d

Num

ber

of n

ucle

i

c

Transientnucleation

Figure 3.23. Schematic diagram showing the variation of number density of nuclei with time duringrapid solidification. Curves a and b are for transient homogeneous and heterogeneous nucleation,respectively. Curves c and d are for steady state homegeneous and heterogeneous nucleation, respec-tively.




15.0

10.0

5.0

0.0

–5.0

–10.00.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Reduced temperature (T /T m)

Log

(t n)

Figure 3.24. Transient nucleation time as a function of reduced temperature (T/Tm) for the ternaryZr76Fe16Ni8 alloy.

nucleation time, tn, as a function of the reduced temperature (T/Tm) using theapproach of Kelton et al. (1983, 1986) and the experimental data on enthalpychanges occurring during solidification and crystallization, Hf �=13 kJ/mol� andHc �= 5�98 kJ/mol�, respectively, for the ternary Zr76Fe16Ni8 alloy (Figure 3.24).The estimated value of the transient nucleation time being comparable to the incu-bation period, the role of transient nucleation cannot be neglected. In fact, in thisspecific case, the estimated value of the transient nucleation time has been foundto be larger than the estimates made earlier (Ghosh et al. 1991) due to lowervalues of Gv. It may be emphasized here that a longer transient nucleation timeimplies a reduced contribution of steady state nucleation to the overall crystalnucleation process in a situation of rapid solidification which corresponds to a veryshort solidification time. Transient nucleation time, therefore, is of considerablesignificance under rapid solidification and a longer transient nucleation time willimply a better GFA.

The rate of steady state homogeneous nucleation is given by Eq. (3.32), indi-cating the influence of the viscosity, �, of the liquid which can be expressed interms of the Vogel–Fulcher–Tamman equation (Busch et al. 1998):

��T� = � exp(

�

T −T ′

)(3.38)

where � and � are constants and T ′ is the temperature where the excess configura-tional entropy of the free volume is zero. By substitution of appropriate values of




20

15

10

5

0.0

–5

–10

–15

–20

–25

–30

–35

–400.2 0.3 0.4 0.5 0.6 0.7 0.8

Reduced temperature (T /T m)

Log

I ss

Figure 3.25. Steady state nucleation frequency as a function of the reduced temperature (T/Tm) forZr76Fe16Ni8 alloy.

these quantities (� = 5�46 × 1011 Ns/m2; � = 18221�6 K; T = 248�4 K) obtainedon the basis of the analysis of Battezzatti and Greer (1989), the viscosity of themolten alloy Zr76Fe16Ni8 has been calculated as a function of temperature (Savaliaet al. 1996). Using Eqs. (3.32) and (3.38), the steady state nucleation frequencyhas been obtained for Zr76Fe16Ni8 as a function of the reduced temperature, T/Tm

(Figure 3.25). A comparison of calculated values of the steady state nucleationfrequency for different alloy compositions reveals their relative GFAs.

In systems where homogeneous nucleation frequencies are very low, hetero-geneous nucleation dominates. Glass formation in such systems is possible atsomewhat slower cooling rates, provided heterogeneous nucleation sites are elim-inated. The Pd40Ni40P20 alloy is one such system which was known about threedecades back as amenable to vitrification at a cooling rate of 103 K/s in the absenceof heterogeneous nucleation. With the discovery of bulk metallic glasses in mul-ticomponent Zr alloys in the recent past, it is now demonstrated that avoidance ofhomogeneous and heterogeneous nucleation of crystals and thereby formation ofglass are possible in several alloy systems at much slower cooling rates (typically100 K/s).

3.4.4 Microstructures of partially crystalline alloysRapid solidification of an alloy at a cooling rate close to the critical rate necessaryfor avoiding nucleation of crystals develops a condition under which the crystal




formation and the vitrification processes are closely competitive. As has beenmentioned earlier, the total avoidance of crystal nucleation is difficult to achievein most of the potential glass forming alloys, and therefore, a large majority ofpredominantly glassy alloys do contain a distribution of crystalline particles. Inthis section, the morphologies of crystalline particles embedded in the amorphousmatrix have been examined with a view to gaining an understanding of the nucle-ation and growth processes of such crystals which are preserved in the frozen-incondition in the early state of their growth.

Studies on partially crystalline Zr alloys have been reported in Zr–Ni, Zr–Feand Zr–Fe–Ni systems by Dey and Banerjee (1985, 1986, 1998) and in the Zr–Besystem by Tanner and Ray (1979). Partially crystalline structures can be groupedinto the following two categories, essentially based on the size of the crystallineparticles: (a) the structure containing a negligible volume fraction of extremelyfine crystalline particles or quenched-in nuclei which did not get an opportunity togrow to any significant extent and (b) partially crystalline structures consisting ofa distribution of crystalline particles of a micrometre size range in an amorphousmatrix, maintaining a low volume fraction of the crystalline phase; such a structurecan form when nucleation of a few crystals and their limited growth are possiblewithin the time available prior to vitrification but the nucleation frequency is solow that the crystalline particles remain few and far between.

The presence of quenched-in nuclei in the amorphous matrix is not detectablein XRD and electron diffraction patterns which show the broad maxima corre-sponding to the first, second and third peaks in the radial distribution function.High-resolution electron microscopy of alloys, which had been characterized tobe fully amorphous by diffraction experiments, has revealed the presence of smalldomains of crystalline particles. Figure 3.26 shows one such example in whichlattice fringes associated with quenched-in nuclei are seen in the amorphous Zr–Nimatrix. The measured lattice spacing of about 0.25 nm matches closely with d110

of the � (bcc)-Zr phase. This is not unexpected as the crystalline phase most likelyto form in this alloy is the �-phase which can even inherit the composition of theliquid phase.

The frequency of composition invariant homogeneous nucleation in Zr76Fe16Ni8

has been estimated in Section 3.4.3. An undercooling to the extent of T/TM = 0�6,as per this estimation, produces a homogeneous nucleation frequency of about1010 m−3/s. Therefore, a limited number of nuclei are created even under a rapidcooling rate of 105 K/s. These quenched-in nuclei remain very small in size asthey do not get an opportunity to grow prior to the vitrification of the matrix.The presence of such quenched-in nuclei has a strong influence on the kinetics ofcrystallization as will be discussed in Section 3.5.3.




Figure 3.26. Small crystalline particles distributed within the amorphous matrix as revealed byhigh-resolution electron microscopy (HREM).

Partially crystalline structures in Zr3�FexNi1−x� alloys produced under differentcooling rates show a distribution of several crystalline phases in the amorphousmatrix, as revealed by XRD. Since the cooling rate during melt spinning drops asone traverses from the side which comes in contact with the copper wheel (heatextracting surface) to the air side, the phases present in the melt-spun ribbon in thevicinity of one side are sometimes different as compared to those on the other side.The results of phase analyses of alloys of different compositions are summarizedin Table 3.4. TEM investigations on these partially crystalline structures showthat while single isolated crystals are present in some cases, crystal aggregatesof special morphologies are often encountered. An aggregate morphology whichis frequently observed consists of a central core crystal surrounded by peripheralcrystals (Figure 3.27(a)). The central core exhibits both spherical and dendritic

Table 3.4. Phase analysis of rapidly solidified Zr–Fe–Ni alloys.

Alloy composition Nature of phases present

Zr76Fe24 ��Zr3Fe

Zr76Fe20Ni4 ��Zr3Fe



Zr76Fe8Ni16 ��Zr3Fe, Off-stoichiometric Zr2Ni

Zr76Fe4Ni20 ��Zr3Fe, Off-stoichiometric Zr2Ni

Zr76Fe8Ni24 Off-stoichiometric Zr2Ni




(a)

0.2 μm 1.0 μm

(b)

0.2 μm

(c)

Figure 3.27. Morphology of crystals in amorphous matrix in partially crystalline alloys. (a) Anaggregate morphology consisting of a central core crystal surrounded by peripheral crystals, (b) het-erogeneous nucleation of peripheral crystals around the core crystals is occasionally suppressed,under the rapid cooling condition of piston–anvil splat quenching, as is revealed from the presenceof �-crystals embedded in the amorphous matrix and (c) “sunflower” morphology.

morphologies. Electron diffraction and energy dispersive spectroscopy (EDS) haveidentified the core crystals to be of the �-phase supersaturated with Fe and Ni, bothof which are strong �-stabilizers. The peripheral crystals are of the Zr3�Fe�Ni�phase in alloys with higher levels in Fe (c > 0�5) and of the Zr2Ni phase in alloyswith higher Ni contents (c < 0�5).




The defect-free dendritic morphology of the core crystals of the supersaturated�-phase suggests that they form from the liquid phase and not from the amorphousphase. The composition of the core crystals indicates that alloy partitioning occursonly to a limited extent during solidification. The enrichment of �-stabilizingelements in the core crystals is responsible for the retention of this phase even atroom temperature. The peripheral crystals are nucleated heterogeneously on thesurface of the core crystals in either of the following two situations. (a) Nucleationof peripheral crystals from the liquid phase can occur heterogeneously on thesurface provided by core crystals. As the � core crystals grow with a limited solutepartitioning, the composition of the liquid ahead of the solidification front graduallychanges. Finally, a stage is attained in which the formation of �-phase becomes lesspreferred than the formation of intermetallic phases such as Zr3�Fe�Ni� or Zr2Nidue to either thermodynamic or kinetic considerations. (b) Nucleation of peripheralcrystals can occur after the vitrification of the matrix. As the � core crystalsgrow in continuously cooling liquid phase, the viscosity of the surrounding liquidrises sharply, finally leading to vitrification. The peripheral crystals subsequentlynucleate and grow in the amorphous matrix.

Depending on the prevailing cooling rate at the interface of the core crystals,fresh nucleation of intermetallic phases can take place. Under the rapid coolingcondition of piston–anvil splat quenching, heterogeneous nucleation of peripheralcrystals around the core crystals is occasionally suppressed as is revealed fromthe presence of �-crystals embedded in the amorphous matrix (Figure 3.27(b)).In contrast, peripheral crystals of Zr3�Fe�Ni� and Zr2Ni are often seen around� core crystals in melt-spun alloys. The interfaces between the core crystals andthe surrounding amorphous matrix act as heterogeneous nucleation sites on whichZr3�Fe�Ni� or Zr2Ni crystals nucleate (Figure 3.27(a)). As mentioned earlier, thecrystallization products in Zr3�FexNi1−x� alloys are Zr3�Fe�Ni� and Zr2Ni foralloys with x > 0�5 and x < 0�5, respectively. In the former case, a partitionlesscrystallization occurs as is reflected in faceted crystal/amorphous interfaces. Crys-tals of Zr3�Fe�Ni� phase, which has a Re3B type structure, tend to form a specialorientation relationship with the core �-crystals. Different crystallographic vari-ants, which are twin related, are often found to form adjacent to each other, sharingthe twin interface between them. The resultant morphology of the crystal aggre-gate, as illustrated in Figure 3.27(c), is described as the “sunflower” morphology.Electron diffraction patterns from individual “petals” and dark field imaging haverevealed that the opposite petals have the same orientation and that each “petal”is twin related with the two adjacent petals. The growth of crystal aggregates with“sunflower” morphology is encountered quite frequently in Zr3�FexNi1−x� alloyswith x > 0�5 in which polymorphic crystallization can occur. The growth of suchthree dimensionally symmetric aggregates can occur only if the matrix is fully




isotropic as in the case of crystallization of an amorphous phase. From the consid-eration of the symmetry relation between the parent and the product phases, theamorphous to crystalline phase transformation is analogous to crystal formationfrom a liquid or vapour phase. This explains why the morphology of the crystalaggregates in the amorphous matrix often bears a similarity with the crystallineproducts from liquid or vapour phase.

3.4.5 DiffusionThe thermal stability of metallic glasses is a subject of vital concern as one desiresto produce glasses which will retain their amorphous nature at as high a temper-ature as possible. The current limit is about 1300 K, for some W-based glasses.Sometimes the amorphous phase is used as an intermediate which can be trans-formed into crystalline phases of desired grain structures. Such an approach is oftenadopted for the production of nanocrystalline structures using the amorphous phaseas a precursor state. Phenomena such as diffusion, structural relaxation and crystal-lization need to be discussed for assessing the thermal stability of metallic glasses.

A clear understanding of the changes occurring in metallic glasses during heattreatments is dependent on the elucidation of diffusion parameters and mecha-nisms. There have been very few direct measurements of diffusion in amorphousalloys. The experimental difficulties are considerable since the diffusion distanceis very small for the accessible temperatures where crystallization can be avoided.When diffusion distances are in the range of 100 nm, techniques which permitcomposition analysis at a very high depth resolution need to be employed formeasuring the concentration-depth profiles. These include Auger Electron Spec-troscopy (AES) with sputter etching, Rutherford Backscattering Spectroscopy(RBS) and Secondary Ion Mass Spectroscopy (SIMS). Some indirect methods forthe measurement of the diffusivities are also employed.

In one of the early diffusion experiments, Gupta et al. (1975) determined the dif-fusivity of Ag in Pd81Si19. The surface of the alloy was sputter-deposited with 110Ag

radioactive isotope. After diffusion annealing, the surface was sputter-etched. TheAg concentration was determined, from the radioactivity of the material removed,as a function of etching depth. One of the problems with such experiments is thatdifferent points on the surface are sputtered at different rates. Birac and Lesueur(1976) used a neutron beam, and the (n, ) reaction of 6Li nuclei, for measuringits diffusion in Pd80Si20.

Cahn et al. (1980) succeeded for the first time in measuring the self-diffusionof B in Fe40Ni40B20. A layer of the same composition, containing 10B and 11B inthe ratio, 96:4, was sputtered onto the metallic glass, which contained the naturalisotopes in the ratio 20:80. After diffusion annealing, the surface was sputteredand 10B/11B ratio was measured by using SIMS.




Several indirect methods for the measurement of diffusivity are available. Forexample, it is possible to use the rate of primary crystallization. The diffusivitiesof B and C in Fe-B-C alloys have been deduced from measurements of the crystalgrowth rate (Koster and Herold 1980, Koster 1983). They obtained a diffusioncoefficient, D, of 2 × 1019 m2/s and an activation energy of 180 kJ/mol. Thesevalues led them to surmise that B diffuses as a substitutional atom rather thanas an interstitial atom. This method has been applied to Fe40Ni40P14B6 by Tiwariet al. (1981).

There have been some attempts, by Taub and Spaepan (1979), to evaluate thediffusion coefficient from viscosity data. In the Pd–Si system, the diffusivity ofgold in the as-quenched glass has been reported to be some orders of magnitudelarger than the value of D� as the glass is made to relax by annealing at atemperature below but close to Tg. These results strongly suggest that frozen-in‘defects’ in metallic glasses, which are present in the as-quenched state, play animportant role in the mechanism of diffusion.

Let us focus our attention on Ti- and Zr-based metallic glasses which areprimarily grouped under the class of metal–metal amorphous alloys with early andlate transition metals as constituents. As has been indicated in Section 3.4.2, a fairlylarge number of binary alloys of this type are easy glass formers. Interpretationsof diffusion data obtained in such systems are expected to be straightforward ascomplications due to multicomponent interactions will be absent.

Measured values of diffusion coefficients, D, of different diffusing species inbinary metal–metal amorphous alloys show an Arrhenius type dependence ontemperature (Figure 3.28) when the data are replotted as log D against reciprocalof normalized temperature, Tg/T . This observation is indicative of the fact thatfor a given diffusing species and a given amorphous alloy, a single thermallyactivated diffusion mechanism remains operative over the entire temperature rangestudied. The influence of the atomic size of the diffusing species on the diffusionconstant can be seen from the diffusivity data for Cu, Al, Au and Sb in a givenmetallic glass, Zr61Ni39. At any given temperature, it has been observed thatDCu >DAl >DAu >DSb which is consistent with the fact that rCu < rAl < rAu < rSb

where r is the respective atomic radius. The D values for Cu were found to behigher than the corresponding values for Al by about an order of magnitude in thetemperature range of 556–621 K (Sharma and Mukhopadhyay 1990). The valuesof the activation energy, Q, for diffusion evaluated on the basis of the observedArrhenius type temperature dependence of D, were found to be 1.33 ± 0.17 eVfor Cu and 1.68±0.13 eV for Al. The corresponding values of the preexponentialfactor, Do, were 10−7�57±1�46 and 105�35±1�75 m2/s, respectively. This comparison alsoreveals that the activation energy of diffusion in a given amorphous alloy increaseswith increasing atomic size of the diffusing species. Such a rule is expected to be




1.41.21.0

T g / T

10–18

10–22

10–26

Diff

usio

n C

oeffi

cien

t, D

(m

2 s–1

)

Figure 3.28. Diffusion coefficients, D, of different diffusing species in binary metal-metal amor-phous alloys as a function of normalized temperature Tg/T showing an Arrhenius type dependenceon temperature.

valid only in cases where the diffusing species have more or less similar chemicalinteractions with the amorphous matrix.

There have been a number of investigations to find out whether the diffu-sivities of a given species in an alloy in crystalline and amorphous states aresignificantly different. Contradictory results have been reported from these inves-tigations. Valenta et al. (1981) have shown significantly slower diffusion of P andFe in Fe40Ni40P14B6 when the amorphous alloy is crystallized. Contrary to this,Akhtar et al. (1982a,b) have shown a much faster diffusion of Pt in Ni33Zr67 aftercrystallization. Such contradictory results stem from the fact that there is a widevariety of crystallization mechanisms which result in a variety of crystal structures,phase distributions and grain sizes in the crystallized products. Diffusion dataobtained from the homogeneous amorphous phase cannot be compared with thatobtained in the crystallized product of the same material. Such a comparison issomewhat meaningful only in cases where the amorphous phase crystallizes into asingle phase crystalline state with a large grain size (as in the case of polymorphiccrystallization described in Section 3.5.1).

Structural changes within the amorphous phase induced by heat treatmentscausing relaxation, plastic deformation and irradiation are expected to bring aboutchanges in diffusivity in metallic glasses. Cantor and Cahn (1983) have reviewedthe experimental data to arrive at the conclusion that diffusivity is very sensitiveto relaxation in amorphous alloys which are very rapidly cooled through Tg.




Diffusion data reported in Zr61Ni39 in the temperature range of 551–621 K showthat a relaxation heat treatment does not affect diffusivity significantly (Sharmaand Mukhopadhyay 1990). Autorelaxation of the glass during the cooling downfrom Tg appears to have reduced the influence of the subsequent relaxation heattreatment.

Akhtar et al. (1982a,b) have reported the diffusion coefficient of Au in amor-phous Zr67Ni33 at four different temperatures using as-quenched, relaxed and plas-tically deformed specimens. At each temperature, the diffusivity in the deformedcondition is found to be higher than that in the as-quenched condition, the latterin turn being higher than that in the relaxed condition.

Cahn et al. (1982) have observed that Au diffusion coefficients in amorphousZr64Ni36 decrease after the amorphous alloy is irradiated with fast neutrons.They have argued that the irradiation-enhanced chemical short-range order causesa reduction in atomic volume with a corresponding reduction in the diffusioncoefficient.

Diffusion of small size H atom in Ti- and Zr-based metallic glasses assumesa great significance because of the possibility of H storage in some of thesematerials. The activation energy of H diffusion in Zr67Pd33 has been reported tobe 0.25 eV between 270 and 365 K, increasing to about 0.7 eV in the temperaturerange of 430–490 K. Similar characteristics have been observed in amorphousTi–Cu alloys. Diffusion of H in hydrogenated Ti–Cu and Zr–Pd amorphous alloysis between one and two orders of magnitude faster than in the correspondingcrystalline hydrides.

Cantor and Cahn (1983) considered various experimental and theoreticalinformation when available regarding diffusivities in amorphous alloys forarriving at possible atomistic mechanisms of diffusion in these systems. Since anArrhenius type relation has been found to be valid for the temperature dependenceof diffusivity in amorphous alloys, it is attractive to consider whether atomicmodels of diffusion in crystalline materials can also be applied to amorphousalloys. This approach can be further justified in view of the fact that the localarrangements of atoms in and the densities of amorphous and crystalline alloysare somewhat similar.

The atom-vacancy exchange process is known to be the most important mech-anism for both self- and impurity diffusion in crystalline alloys. In the absence ofa reference lattice, a vacancy in an amorphous alloy can be defined as an emptyspace in the amorphous structure of atomic or near atomic dimensions. Severalinvestigations have been made with a view to examining the stability of vacantsites of atomic dimensions in an amorphous alloy. From modelling work, it hasbeen shown that if an atom is removed from the dense random packed structureof an amorphous alloy, atomic vibrations quickly redistribute the excess space




over a large volume. This tendency of smearing the excess volume created by theremoval of an atom does not allow the presence of a near atomic size vacant spacein the amorphous structure. The structure and size of soft sphere dense randompacked models of amorphous alloy structures show that most interstitial sites aresurrounded by distributed tetrahedral and octahedral groups of atoms with moretetrahedral and fewer octahedral sites than in an equivalent close-packed crystal.The size distribution in Figure 3.29 shows that a small fraction of octahedralinterstices have sizes in the range of 0.6–0.7 of the atomic diameter, and theselarge interstices can be considered as vacancies in the amorphous structure.

3.4.6 Structural relaxationThe structure and properties of a glass depend on the quenching rate. When a glassis annealed, its structure will first relax to that of a glass formed at lower coolingrates and ultimately tend towards that of an “ideal” glass. Egami (1983) usedenergy dispersive XRD methods and showed that two types of change occurredduring relaxation. One is related to the topological short-range order. Duringthe process of atomic movement, the tetrahedra per se are not affected by theirrelative configurational change. A higher degree of topological short-range orderis established by a highly collective phenomenon involving the cooperation ofa number of atoms. The second change is related to the chemical short-rangeordering during which site interchange of atoms takes place.

0.3

0.2

0.1

0.2 0.4 0.6

Vacant site radius/atomic radius

Fra

ctio

n of

site

s

Radius of vacantsites in amorphousstructure

Tetrahedral octahedral – fcc interstitial radii

Figure 3.29. Size distribution of vacant sites in amorphous structure.




Further studies have indicated that relaxation mechanisms can be divided intoreversible and irreversible types. Reversible changes appear to be associated withchanges in the chemical short-range order. Kurusmovic and Scott (1980) haveshown that the Young’s modulus of the Fe40Ni40B20 glass may be cycled reversiblybetween the values which are characteristic of different annealing temperatures.The analogy with short-range order in crystalline alloys is particularly striking.The irreversible change appears to be associated with a change in the topologicalshort-range order.

During structural relaxation, the alloy becomes denser. Significant changesoccur in many properties such as the electrical resistivity, magnetic anisotropy,Curie temperature, elastic modulus and mechanical properties. Some changes arebeneficial, while others are detrimental. The changes in magnetic and mechanicalproperties are discussed below.

Luborsky (1983) have reported remarkable improvements in the magnetic prop-erties of Fe–Ni-based glasses after stress-relief annealing for 2 h at 100 K belowtheir glass transition temperature. These changes include an increase in rema-nence, a decrease in the saturation field and a change in the Curie temperature.However, if the treatment leads to crystallization, then the magnetic propertiesdeteriorate. Hence, it is important to ensure that the compositions of metallic glassferromagnets are chosen so as to have good thermal stability.

The Curie temperature can be determined either by magnetic measurements orby locating the appropriate thermal anomaly in a differential scanning calorimeter(DSC) run. Egami (1983) found that cycling Fe27Ni53P14B6 repeatedly between 523and 573 K caused the Curie temperature to cycle between 368 and 375 K. Thesereversible changes are due to a change in the position of Fe and Ni atoms andthe consequent alteration in the chemical short-range order. Further exploration ofthe use of low-temperature annealing in bringing about such beneficial changes inmagnetic properties appears to be desirable.

One of the attractive properties of metallic glasses is their large ductility in bend-ing and compression. In many cases, low-temperature annealing treatments leadto the loss of this ductility. There have been several investigations of this temperembrittlement phenomenon, and it appears to be dependent on the composition.

In the case of Fe40Ni40P14B6, a heat treatment at 373 K for 2 h leads to embrit-tlement. AES has been used for demonstrating that segregation of P occurs duringstructural relaxation which leads to embrittlement (Walter and Bertram 1978,Walter 1981). In general, metal–metal glasses and Cu60Zr40 appear to be immuneto this type of embrittlement.

An interesting suggestion with regard to the embrittlement tendency has comefrom the group of Davies (1978). Alloys which eventually crystallize as an fccphase do not exhibit the embrittling tendency, whereas alloys which crystallize




into a bcc or hcp phase become brittle on relaxation. This implies that the structuralgroupings of the crystalline phases responsible for embrittlement can be partiallyobserved in the amorphous phase. This might also explain why embrittlementoccurs within a certain composition range in a given system.

3.4.7 Glass transitionThe nature of the glass transition is still a matter of controversy. Experimentalevidence and theoretical models suggest the glass transition to be a first-order phasetransition, based on the free volume approach, a higher order phase transition or nophase transition at all, e.g. kinetic freezing. However, there is a general agreementthat the maximum undercooling of a liquid is limited to the isentropic temperaturein order to avoid the paradoxical situation described by Kauzman (1948) wherethe configurational entropy of the supercooled liquid becomes smaller than theconfigurational entropy of the ordered equilibrium phase. Consequently, as longas crystallization can be prevented, the undercooled liquid will freeze to a glassclose to the ideal glass transition temperature, T o

g , where the entropy differencebetween the liquid and the equilibrium crystalline phase would vanish. In reality,the glass transition sets in at a temperature somewhat above T o

g . Depending onthe deviation of the glass transition temperature, Tg, from the ideal glass transitiontemperature, Tg, the glass attains different relaxation states. Only at an infinitelyslow cooling rate, if the liquid is vitrified (by avoiding crystallization), the liquidto glass transition occurs at T o

g and the resulting glass attains a fully relaxed state.Such a transformation can be considered as a second-order transition at T o

g . Underrealistic cooling rates, a glass having excess entropy and consequently not in thefully relaxed state forms at Tg. Depending on the extent of relaxation of the productglass, the glass transition temperature measured from experiments varies with theimposed cooling rate, a higher cooling rate yielding a higher value of Tg, as shownin Figure 3.17 in which T 1

g and T 2g are the glass transition temperatures for the

cooling rates, G1 and G2 (G1 >G2), respectively.The glass transition event is also encountered during heating experiments. Con-

tinuous heating experiments in a DSC often show an endothermic event priorto the large exothermic event of crystallization. A thermogram obtained duringheating of the Zr-35 at.% Ni glass at a heating rate of 5 K/min clearly revealsthe endothermic event attributable to the glass to liquid transition preceding thecrystallization event (Figure 3.30).

Taking into account experimental values of the specific heat of a stable andhighly undercooled liquid and measured values of the enthalpy of crystallizationof the amorphous alloy, the undercooled liquid at Tg is found to exhibit onlya very small excess entropy in comparison with the stable crystalline phase.Specific heat, thermal expansion and Mossbauer spectroscopy data on several fully




150 250 350 400Temperature, °C

Hea

t flo

w e

xoth

erm

al50

000

mW

↓

Figure 3.30. DSC thermogram of Zr-35 at.% Ni glass obtained at a heating rate of 5 K/min showingthe endothermic nature of the glass to liquid transition.

relaxed amorphous alloys reveal that the glass transition can be approached underinternal equilibrium conditions (Tg approaching T o

g and becoming independent ofthe heating rate).

Usually amorphous alloys form in composition ranges where the heat of mixingbetween the components is negative (strong ordering tendency). However, there areinstances where a positive deviation from the ideal solution behaviour is noticed innarrow composition ranges where the amorphous phase tends to separate into twophases, both having amorphous structure. Such a system is expected to exhibit twoglass transition temperatures. Experimental observations of two glass transitionevents and of a phase separated microstructure of the amorphous alloy have ledTanner and Ray (1980) to infer the presence of a two-phase amorphous structurein the Zr36Ti24Be40 alloy. Using an atom probe microscope, Grunse et al. (1985)have shown the presence of spatially extended concentration waves in as-quenchedTi50Be40Zr10.

Observations on structural relaxation and localized fluctuations in structure andcomposition of metallic glasses have prompted a model of glass transition basedon the heterogeneous glass in terms of density and concentration. In this model,it is visualized that a glass consists of liquid-like regions of large free volumeor high local free energy and solid-like regions with small free volume or low




T 1 > T g

τ *Fr

eque

ncy

(a)

Relaxation time (τ) ↓

T 2 < T g

Freq

uenc

y

(b)

Relaxation time (τ) ↓

Figure 3.31. Schematic diagram showing relaxation spectra at (a) temperatures above Tg and(b) temperatures below Tg.

free energy. Each region undergoes a transition at a frequency much smallerthan the Debye frequency (∼1013 s−1) between local energy minima correspondingto different configurational states, the relaxation time, �i, being proportional toexp�−�i/kBT ) where �i is the energy barrier between these states. The relaxationspectra at temperatures below and above Tg are schematically shown in Figure 3.31.At T1 > Tg, the whole spectrum lies to the left of the time of measurement �∗

(for example, 30 s), so that the whole system undergoes frequent configurationaltransformations and is liquid-like. With lowering of temperature below Tg, (T2 <Tg), the whole spectrum shifts to longer times such that relaxation times for mostof the regions are greater than �∗, and the isolated liquid-like regions are small involume fraction and are embedded in the rigid solid matrix. At such temperatures,localized, short-range structural relaxation can occur, leading to glass transition inthese small domains.

In the close vicinity ofTg, the peak of the distribution of the relaxation time is loca-ted near �∗, leading to a long-range, cooperative structural relaxation which causesa rapid decrease in the relaxation time and an accompanying rise in the viscosity.

3.5 CRYSTALLIZATION

Several experimental techniques have been used to monitor the crystallization ofmetallic glasses. Among these, DSC and TEM have proved to be particularlyuseful. In the case of DSC, crystallization gives rise to distinct exothermic peaks.The heat of crystallization can be measured and is found to be of the order of 40%of the heat of fusion of the alloy, the remaining enthalpy having been extractedfrom the liquid during quenching. Using electron microscopy, the morphology




and structure of crystals can be established and the mechanism and kinetics ofcrystallization can be followed.

Heating of an amorphous alloy leads to several other changes apart from relax-ation. These include glass–liquid transition, phase separation and crystallization.Not many studies have been made on the glass–liquid transition.

Although phase separation into two amorphous phases is a well-documentedevent in the case of oxide glasses, metallic glasses appear to undergo phaseseparation in only rare instances. Evidence for phase separation has been reportedin Pd74Au8Si18 by Chou and Turnbull (1975) and in B40Ti24Zr36 by Tanner and Ray(1979). DSC plots indicate the occurrence of two glass transition temperatures.Banerjee (1979) reported a spinodal decomposed amorphous microstructure in theZr-24% Fe alloy. Piller and Haasen (1982) have used the sensitive atom probefield ion microscope in order to demonstrate that Fe40Ni40B20 decomposes intotwo amorphous regions, one having composition corresponding to (Fe,Ni)3B andthe other being a B-deficient region.

3.5.1 Modes of crystallizationFrom symmetry rules, it can be shown that the amorphous to crystalline phasetransition is necessarily a first-order transition. This is consistent with the observednucleation and growth processes encountered by different investigators study-ing crystallization. As there is no periodic arrangements of atoms in the parentamorphous structure, there is no possibility of achieving a lattice correspondencebetween the parent and the product structures. Therefore, the occurrence of crys-tallization via “military” atom movements is ruled out. Hence, the transition isexpected to occur essentially by diffusional atom movements. Depending on thediffusion distances involved in the crystallization process, one can broadly classifythe mechanisms of crystallization into three broad categories, namely (a) polymor-phic crystallization, (b) eutectic crystallization and (c) crystallization involvinglong-range diffusion – primary followed by eutectic or primary followed by poly-morphic crystallization. These three modes of crystallization are schematicallyillustrated in Figure 3.32.

(a) Polymorphic crystallization (A → ):When the compositions of the parent amorphous phase and of the product

crystalline phase are the same, the crystallization process involves diffusionalatomic jumps across the advancing transformation front. This situation isanalogous to that in massive transformation or in “partitionless solidification”processes and is illustrated in Figure 3.32 for the alloy composition c1. Kosterand Herold (1980) have termed this type of crystallization as “polymorphiccrystallisation”.




α

Fre

e en

ergy

(G

)

E

A

G

A G

β

G

α

A Bc1 c4

c, atom fraction of Bc6

c2 c5c3 c7

E

α

A

A′

α

Distance →

A

E

Ac3AA

c1αc

→

Distance → Distance →Distance →

αEA

c5

c2

c4

α

A (c1)→ α (c1) A′ (c5) → α (c6) + β (c 7)A → α (c4) + A′ (c5) A (c3) → α (c6) + β (c7)

Polymorphic(c 1)

Primary + Eutectic(c 2)

Eutectic(c 3)

α

Figure 3.32. Schematic representation of different modes of crystallization: polymorphic, primaryfollowed by eutectic and eutectic occurring in amorphous alloys of compositions given by c1, c2

and c3, respectively. Free energy changes which motivate the crystallization process are indicated byarrows in the free energy – concentration (G–c) plots corresponding to the amorphous (A) and thetwo crystalline phases, and �. In polymorphic crystallization, the amorphous alloy of compositionc1 transforms into the -phase of the same composition. In primary crystallization, phase ofcomposition c4 forms first from the amorphous alloy of composition c2; the composition of thelatter gradually changes to c5 which finally decomposes into an eutectic mixture (E) of and �. Ineutectic crystallization, the amorphous phase of composition c3 directly transforms into the eutecticmixture (E) of and �. Concentration profiles across a crystalline particle is shown below each ofthe schematic micrographs.

(b) Eutectic crystallization (A → +�):The partitioning of alloying elements into two crystalline phases (illustrated

in Figure 3.32 for the composition c3), which are forming simultaneously fromthe parent amorphous phase in a cellular transformation, requires diffusion at ornear to the transformation front. Eutectic/eutectoid decomposition and cellularprecipitation are the analogous phase transformations in crystalline systems.




Eutectic crystallization, which has been encountered in several systems suchas Fe–B, Fe–Ni–B, Mo–Ni, and Zr–Fe, occurs at a relatively low rate ascompared to polymorphic crystallization. Decomposition of an amorphousphase into a mixture of a crystalline and a second amorphous phase, in acellular transformation mode, appears to be possible in a system in whichthe amorphous phase has a tendency towards phase separation (or unmixing).However, in the case of metallic glasses, such a transformation, which isanalogous to the monotectoid reaction, has not been encountered.

(c) Primary crystallization followed by eutectic or polymorphic crystallisation:During the formation of a crystal having a composition which is different

from that of the parent amorphous phase, a long-range diffusion field is estab-lished ahead of the transformation front. Primary crystallization, as designatedby Koster and Herold (1980), involves such a process, the kinetics of whichare generally controlled by the mechanism of long-range atom transport inthe amorphous matrix. The composition of the matrix amorphous phase mayfinally transform via one of the many possible phase reactions. In the caseof Fe–B, the amorphous matrix gradually attains the Fe75B25 composition andthen transforms into the Fe3B phase via a polymorphic crystallization process.By analogy with various liquid-to-crystal phase reactions, one can visualizemany possible reactions, such as the following:(a) Primary crystallization followed by a peritectic/peritectoid reaction

(A→ (primary) + A′ → �): There exists a possibility of this trans-formation sequence which has not been reported in any metallic glasssystem.

(b) Primary crystallization followed by an eutectic reaction(A → (primary)) + A′ → (primary) + � +�� (eutectic): A and A′

are amorphous phases having different compositions, and and � aredifferent crystalline phases as shown in Figure 3.32 for composition c2.The appearance of more than one exothermic peak in DSC thermogramsappears to originate from such successive phase reactions (Figure 3.33).

3.5.2 Crystallization in metal–metal glassesMetal–metal glasses have not received the same degree of attention as metal–metalloid glasses, although this situation appears to be changing with increasinginterest in these glasses. Metal–metal glasses present several features of interest.Unlike metal–metalloid glasses, which are generally restricted to compositionsof around 20 at.%, metal–metal glasses can be formed over a wider range ofcompositions. Both metal–metalloid and metal–metal glasses can be preparedaround compositions corresponding to deep eutectics. In addition, metal–metal




T g T g

T x1 T x2

T x

Glasstranstion

Crystallization

PrimaryEutectic

orpolymorplic

12

TemperatureTemperaturePolymorplic or

eutectic crystallization Primary followed byeutectic or

polymorplic crystallization

Rat

e of

hea

t flo

w (

H )

·

Figure 3.33. Differential scanning calorimetry thermograms showing, dH /dt, the rate of heat flowversus temperature, T , at a constant rate of heating (dT /dt = constant) for (a) polymorphic andeutectic crystallization and (b) primary followed by eutectic or polymorphic crystallization. Theendothermic event of glass transition at Tg and the exothermic events of crystallization at Tx areindicated. While for polymorphic and eutectic crystallization a single exothermic event is observed,as shown in (a), two distinct thermal events are noticed in primary crystallization followed by eithereutectic or polymorphic crystallization as shown in (b).

glasses can form at compositions which correspond to stoichiometric compoundshaving high melting points. There are also indications of structural differences.Polyhedral packing, characterized as Kasper polyhedra, appears to be a dominantstructural feature in metal–metal glasses. Systems which have been investigatedin detail include Cu–Zr, Ni–Zr, Fe–Zr, Ni–Nb, Ti–Be–Zr and Mg–Zn. In orderto illustrate the type of investigations made on these alloys, the Ni–Zr and Fe–Zrsystems are used as examples in the discussion which follows.

The equilibrium diagram for the Ni–Zr system contains four well-defined eutec-tics at 8.8, 36.3, 63.5 and 75.9 at.% Zr. Glasses are formed at compositions whichcorrespond to these eutectics as well as at compositions which correspond to theequilibrium intermetallic phases. The results of Dong et al. (1981) and Dey et al.(1986) with regard to four alloys are described below.

The Ni36�5Zr63�5 alloy corresponds to an eutectic between the Zr2Ni and ZrNiphases. Crystallization of this amorphous alloy involves two steps: primary crys-tallization of Zr2Ni crystals, followed by the formation of ZrNi crystals. Theactivation energy for the nucleation of primary crystals has been reported to be500 kJ/mol, and the diffusion controlled growth of these crystals of average diam-eter, d, has been shown to be characterized by the relationship, d t1/2 where tis the annealing time.




The Ni33�3Zr66�7 alloy exhibits polymorphic crystallization, which results in theformation of Zr2Ni crystals. The activation energy associated with the crystalliza-tion process has been determined to be about 320 kJ/mol from both DSC experi-ments performed in the continuous heating mode and measurement of crystal sizeas functions of time and temperature. Dey et al. (1986) have observed the presenceof very closely spaced planar faults within the Zr2Ni crystals (Figure 3.34).

The Zr72Ni28 alloy crystallizes into an off-stoichiometric Zr2Ni phase which hasa C16 (tetragonal) structure. The crystals exhibit a dendritic morphology but acloser examination reveals a spherulitic morphology. The activation energy forgrowth has been measured to be 180 kJ/mol.

DSC experiments carried out on Zr76Ni24 have revealed two exothermic peaks.The first has been attributed to the formation of the hcp -phase, identified by TEMstudies. Using peak shift observations obtained from DSC runs made at differentheating rates, the activation energy associated with the primary crystallizationevent has been determined to be 310 kJ/mol. This value agrees closely with thatobserved in Ti50Be40Zr10, where primary crystallization into has been reported.As the -phase is expected to be solute lean, the rate-controlling process isidentified as being the diffusion of the solute element in the amorphous matrix.The second step in the crystallization process could be either an eutectic reaction,leading to simultaneous formation of and Zr2Ni, or a polymorphic reaction,leading to the formation of Zr2Ni. The activation energy associated with thesecond step has been found to be 180 kJ/mol (Table 3.5). Such a low valueof the activation energy is consistent with the occurrence of a eutectic or a

Figure 3.34. Bright field TEM micrograph showing the presence of very closely spaced planarfaults within Zr2Ni crystal.

ElsevierUK

Code:

PTA

Ch03-I042145

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1Color

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190P

haseT

ransformations:

Titanium

andZ

irconiumA

lloys

Table 3.5. Crystallization sequence (mode), glass transition temperature, Tg, crystallization temperature, Tx, peak crystallization temperature,Tp, activation energies for (i) overall crystallization, E; (ii) nucleation, En, and (iii) growth, Eg, isothermal annealing temperature, T andAvarami exponent for (i) single-step process (n) and (ii) two-step processes (n1 and n2).

Composition Sequence(mode)

Bulk orsurface

Tg

80 K/minT1 (K)20 K/min

Tp (K)20 K/min

E/En/Eg

(kJ/min)T (K) n/n1/n2

Zr76Fe24 (1) Bulk 650.0 653.5 656.0 272.0 (E)/545.0(En)/168.0(Eg)

626.0 3.10 (n)

Zr76Fe24Ni4 (2) Bulk 641.0 653.0 655.0 286.0 (E) 631.0 2.71 (n)

Zr76Fe16Ni7 639.0 650.7 652.0 278.0 (E) 634.0 2.65 (n)

Zr76Fe12Ni12 (3) Bulk 627.0 646.0 648.0 275.0 (E) 633.0 2.55 (n)

Zr76Fe8Ni16 (4) Surface 624.0 634.0 647.0 274.0 (E) 631.0 2.21 (n1)/2.73 (n2)

Zr76Fe4Ni20 (5) Bulk 624.0 634.0 647.0 236.0 (E) 634.0 1.98 (n1)/4.00 (n2)

Zr76Ni24 (7) Bulk 650.0 652.0 654.0 271.0 (E)410.0 (En)

631.0 3.20 (n)

(1) A → Zr3Fe (polymorphic); (2) A → Zr3 (Fe,Ni)+A′ (primary) → + Zr2Ni (eutectic); (3) A → Zr3 (Fe,Ni) (polymorphic); (4) A → Fe rich (Fe,Ni)3, Zr(Ll2)+A′ (primary);(5) A → Zr3 (Fe,Ni); (6) A′ (primary), A′ → Zr2Ni+ (eutectic); (7) A → Zr2Ni+ (eutectic).




polymorphic crystallization process, both of which involve only short-range atomtransport at the transformation front for the distribution of the solute into the twoproduct phases. However, Dong et al. (1981) have reported single step polymorphiccrystallization. Such differences can arise due to variations in the initial conditionsof the amorphous phase (e.g. the extent of relaxation which the glass has undergoneduring the melt-spinning operation).

Koster and Herold (1980) have reported that polymorphic crystallisation occursin Fe40Zr60, Fe30Zr70 and Fe24Zr76 glasses. The activation energy associated withthe growth of crystals has been found to be about 170 kJ/mol. The resulting crys-tals in the aforementioned alloys are FeZr2, FeZr2 and FeZr3 respectively. Dey andBanerjee (1986) have carried out DSC experiments employing both isothermalholding and continuous heating runs and have found that the activation energy asso-ciated with the overall crystallization process in the Fe24Zr76 glass is 270 kJ/mol.TEM studies have revealed the presence of Zr3Fe crystals (orthorhombic Re3Bstructure), with planar boundaries separating the crystalline and amorphous phases.This is possible because of the absence of any long-range atom transport duringthe polymorphic crystallization process.

Crystal aggregates of a fascinating shape have been observed in both partiallycrystalline as-melt-spun tapes and crystallized samples of fully amorphous Zr76Fe24

glass. These crystal aggregates, which presumably had formed at a very early stagein the crystallization process, consisted of six petals originating from a centralspherical crystal, thus giving rise to a “sunflower”-like appearance (Figure 3.35).

Figure 3.35. Bright field TEM micrograph showing crystal aggregate having a “sunflower” mor-phology.




Crystallographic analyses of each of these petals and of the core have permitted theorientation relationship to be determined. The petals consisted of an ordered phasecrystals of Zr3Fe which can be viewed as being an ordered structure based on -Zr, while the core had a bcc structure. The formation of such an agglomerate isenergetically favourable because the unique orientation relationship of the adjacentcrystals permits the formation of low energy interfaces between them.

3.5.3 Kinetics of crystallizationIt has been mentioned earlier that the amorphous to crystalline transition (devit-rification) occurs in different modes such as polymorphic, eutectic and primaryfollowed by eutectic. The kinetics of the process is, therefore, governed by themode which operates in a given system. Devitrification, being a strongly first-ordertransition, occurs by the nucleation of crystals in the amorphous matrix followedby the growth of these nuclei by the movement of the crystal/amorphous interface,resulting in a progressive consumption of the matrix amorphous phase. The overallkinetics of devitrification is determined by the number density of quenched-innuclei, the rate of thermally activated fresh nucleation and the rate of growth ofthe crystalline phase.

If the nucleation rate is so high at the early stages of the transformation thatall the quenched-in nuclei are consumed before appreciable growth occurs andthermally activated fresh nucleation is limited, the number density of crystallineparticles will remain more or less constant and their sizes will remain uniformduring the growth process. On the other hand, when fresh nucleation continuesalong with growth, a distribution of particle sizes will result.

As mentioned earlier, the composition of a growing crystal or the averagecomposition of a two-phase nodule, which are the products of polymorphic andeutectic crystallization respectively, remains the same as that of the amorphousmatrix during the growth process. Therefore, in these cases, there is no long-range concentration field ahead of the growing crystals. In contrast, if the growingcrystalline phase has a composition different from that of the amorphous matrix(as in the case of primary crystallization), a long-range diffusion field is createdahead of the crystal/amorphous interface.

It is possible to identify the mode of crystallization from the analysis of thekinetics of the overall process (consisting of nucleation and growth) using theJohnson, Mehl and Avrami (J–M–A) formulation (Burke 1965) which relatesthe actual fraction, f , of the transformed volume with t, the duration of thetransformation:

f = 1− exp�−Ktn� (3.39)




where n, known as the Avrami exponent, assumes different values for differentmechanisms and geometries of the growing crystals, as discussed later in thissection. The temperature dependence of K is given by the Arrhenius equation:

K = Ko exp(

− Q

RT

)(3.40)

where Q is the activation energy of the process. Data on the fraction transformed ata given temperature for different durations of the transformation can be analysed toobtain the value of n for an isothermal crystallization process in a glass. It may benoted that it is not possible to determine unequivocally the nucleation and growthbehaviour of a crystallization process from data on the time dependence of thetransformed volume only, as is often attempted. It is essential to have additionalknowledge about the process, best gained from direct microscopic observationson the growing crystals as a function of time. Table 3.6 shows the values of theAvrami exponents, n, for different types of nucleation and growth transformations.

DSC is often used for studying the crystallization kinetics mainly because ofthe precise control of temperature and heating rates and the high sensitivity ofrecording events of heat release associated with this technique. Both continuousheating and isothermal holding experiments are carried out for gaining infor-mation regarding the crystallization kinetics. The methodology of this analysishas been discussed later in this section for illustrating typical cases representingpolymorphic, eutectic and primary crystallization.

In the context of devitrification of metallic glasses, it is to be noted that duringcontinuous heating experiments, crystallization occurs at temperatures only slightlyhigher than the glass transition temperature, Tg. Since structural relaxation of the

Table 3.6. Values of Avrami exponent, n, for different types of nucleation andgrowth transformation.

Geometry Nucleation rate n

Interface controlledPlate Rapid, depleting 1Cylinder Rapid, depleting 2Sphere Rapid, depleting 3Sphere Constant 4

Long-range diffusion controlledSphere Rapid, depleting 3/2Sphere Constant 5/2Cylinder Rapid, depleting 1Plate Rapid, depleting 1/2




glass occurs during heating it up to Tg, causing substantial reductions in atomictransport rates, the nucleation and growth rates of crystals depend not only on thetemperature but also on the thermal history. In continuous heating calorimetricstudies, when the temperature is increased linearly with time, the thermal effectsof glass transition and devitrification may overlap, making the analysis of theresults very difficult. It is important, therefore, to select such systems for studies ondevitrification kinetics for which the kinetic crystallization temperature is severaldegrees celsius above Tg. In isothermal kinetics studies, the occurrence of a suitablylong incubation period prior to detectable transformation makes it convenient torecord the thermal evolution data on a stable baseline. Isothermal DSC results canbe analysed to obtain the fraction transformed, f�t�, as a function of time. The zerotime is defined by the instant when the isothermal holding temperature is reached.The total heat evolution due to crystallization is reflected in the exothermic peakobserved during the isothermal holding at a given temperature. The measurementsof the area under the heat evolution curve up to different time periods give the f�t�values which, when presented in a (J–M–A) plot of ln�− ln�1 − f�) versus ln t,gives a straight line fit, the slope of the straight line giving the J–M–A exponent,n. This method of analysis of results of isothermal kinetics experiments in DSChas been illustrated later with examples of polymorphic crystallization in binaryZr76Fe24 and primary crystallization in Zr67Ni33 glasses.

Even though the interpretation of isothermal kinetics data is straightforward inprinciple, a number of problems are encountered in practice. It is often difficultto maintain the baseline flat over the entire length of the transformation time.The variation in the specific heat (Cp) of the amorphous and crystalline phasescontributes to the baseline shift, which can be computed by taking into account theCp value of the mixture of the amorphous and the crystalline phases prevailing atdifferent stages of the transformation. The temperature range over which isother-mal experiments can be conducted is limited: if the temperature is too high, thetransformation may start even before the isothermal holding is reached and willcertainly overlap instrumental transients; if it is too low, the rate of reaction is sosluggish that the fraction transformed cannot be determined accurately. For thesereasons, and also because of its greater speed and convenience, non-isothermalDSC (such as continuous heating experiments) is often practiced while conductingstudies on kinetics of crystallization. This technique is also useful in cases wherethe crystallization process exhibits more than one exothermic peak, suggesting theoccurrence of more than one crystallization event, such as primary followed byeutectic crystallization. In a continuous heating experiment, the differential powerrequired for maintaining the temperature of a sample and a reference materialis measured as a function of temperature, the heating rate being kept constant.




Essentially such an experiment yields the rate of enthalpy change of the sampleundergoing a first-order phase transition as a function of temperature.

An outline of the reaction kinetics of a phase transformation under a constantheating rate condition, as worked out by Kissinger (1957), is presented here.

The kinetics of a solid state reaction can be described by the equation

df

dt= A�1−f�n exp�−Q/RT� (3.41)

where df /dt is the rate of change in the fraction transformed, n is the order of thereaction and Q is the activation energy. As the temperature is raised at a constantrate,�= dT/dt�, the transformation rate, df /dt, rises to a maximum value at T = Tp

and then falls due to a continuous reduction in the untransformed volume, (1 − f ),which becomes zero at the completion of the process. By differentiation of Eq. 3.41

d

dt

(dfdt

)=[Q

RT 2−An�1−f�n−1 exp�−Q/RT�

]dfdt

AtT = Tp�ddt

(dfdt

)= 0

(3.42)

Therefore, the maximum transformation rate and Tp, the temperature at whichthis rate reaches the peak value, are related by

Q

RT 2p

= An�1−f�n−1 exp�−Q/RTp� (3.43)

This can be reduced to (

T 2p

)= A′ exp�−Q/RTp� (3.44)

A′ is a temperature-independent factor, provided the fraction transformed cor-responding to the peak transformation rate remains the same at all temperatures.Henderson (1979) has shown that the peak in df /dt occurs at f = 0�63 for J–M–Akinetics and linear heating. Equation (3.44) suggests that a plot of ln��/T 2

p � versus1/Tp (Kissinger plot) can be expected to yield a linear fit, the activation energy ofthe overall transformation process being given by the negative slope of the plot.

Experimental determination of the activation energy of the overall crystallizationprocess can, therefore, be made by recording DSC thermograms at different heat-ing rates (), which give Tp values for different . The usefulness of the Kissingermethod for studying kinetics of crystallization is illustrated in Section 3.6.1 by con-sidering examples of polymorphic, eutectic and primary + eutectic crystallizationmodes in Zr–Fe–Ni glasses.




The activation energy, Q, determined from the Kissinger peak shift methodunder the continuous heating condition refers to the activation energy of the overallprocess which includes both nucleation and growth. Ranganathan and Heimendahl(1981) have proposed a methodology for separately determining the activationenergies associated with the nucleation and growth processes, using experimentaldata from isothermal kinetics studies.

For considering the kinetics of the nucleation and growth processes separately,let us take the case of a constant nucleation rate, I , which can be expressed as

I = Io exp(

− Qn

RT

)(3.45)

where Qn is the activation energy for nucleation, which is a sum of W ∗, the energyrequired to form a critical nucleus and Qd, the activation energy of diffusion. Qn

can be determined from the slope of a plot of ln I against 1/T . TEM examinationsof samples aged for different durations at a few selected temperatures can providedata on the nucleation density as a function of time. The activation energy fornucleation of crystals can be determined using such data analysed on the basis ofEq. 3.45.

The measurements of the size of crystals (for polymorphic and primary crystal-lization) or of nodules (for eutectic crystallization) can provide data which can beused for obtaining the activation energy for growth. The radius, r (or a represen-tative linear dimension), of crystals or nodules growing with time, t, following alinear growth law

r = A1t (3.46)

is pertinent to polymorphic and eutectic crystallization, while the growth isparabolic

r = A2�Dt�1/2 (3.47)

for primary crystallization which is usually bulk diffusion controlled. The growthrate, u�= dr/dt�, can be written as

u = uo exp(

− Qg

RT

)(3.48)

where Qg is the activation energy for growth, which can be determined from theslope of the plot of lnu against 1/T .




An illustrative example of such a kinetic analysis of nucleation and growthprocesses separately from the experimental data obtained from isothermally treatedsamples is shown in Figure 3.36.

Isothermal holding of metallic glass samples in a DSC close to the crystallizationtemperature, and monitoring the rate of heat evolution as a function of time, givesdata on the fraction transformed as a function of time which can be analysed usingthe J–M–Avrami formulation (Eq. 3.39) for obtaining the Avrami exponent. Themethod of this analysis for polymorphic crystallization in Zr-33 at.% Ni glass isshown in Figure 3.37.

Let us now consider the relationship between the activation energies for theoverall process (Q) and for the nucleation (Qn) and the growth (Qg) steps for thefollowing situations:

Case (i): Nucleation rate, I = 0 and linear growth, u = constant.This situation arises when a fixed number (N ) of quenched-in nuclei operate

and no fresh thermally activated nucleation occurs. A linear growth rate is acharacteristic feature of polymorphic and eutectic crystallization in which atommovements are essentially confined to the vicinity of the transformation front. Ifthe growing particles (for polymorphic) and nodules (for eutectic) are assumed tobe spherical, the fraction transformed, f , can be expressed as

f = 1− exp(

−43�Nu3t3

)(3.49)

From Eqs. 3.39, 3.40, 3.48 and 3.49, we get the Avrami exponent, n = 3 andQ = 3Qg.

In the case of two dimensionally growing particles, i.e. discs with fixed thick-ness, the same analysis could be applied with n= 2 and Q= 2Qg. For the case ofone-dimensional growth (needles) n = 1 and Q = Qg.

Case (ii): Constant nucleation rate, I > 0 and linear growth, u = constant.In this case, the fraction transformed can be written for spherical particles as

f = 1− exp(−�

3u3It4

)(3.50)

Substituting the values of I and u from Eqs. 3.45 and 3.48, we obtain

f = 1− exp[−�

3Iou

3o exp

(−Qn +3Qg

RT

)t4

](3.51)

Therefore Q = Qn +3Qg and the Avrami exponent, n = 4.



198 Phase Transformations: Titanium and Zirconium AlloysM

ax c

ryst

al s

ize

(μm

)

605 K615 K

595 K

Time1/2 (min1/2)24201612840

0.8

0.6

0.4

0.2

(d)(c)

In I S

S In

G

170150

1/T (10–3 K–1)

135–5.0

160

–2.0

–4.0

–3.0

–1.0

0.0

ISS

G

1.0

G

ISS

Zr76Fe24

Zr76Ni24

(a)

623 K

623 K 623 K

623 K

623 K

623 K

Zr76Fe24

Zr76Ni24

0 100 200 300

Time (min)

40

30

20

10

0

50

Num

ber

of n

ucle

i (μm

)–3

(b)

610 K600 K

623 K613 K

603 K

590 K

580 K

593 K

Zr76Fe24

Zr76Ni24

200 30010000.0

1.0

2.0

3.0

4.0

5.0

Time (min)

Max

dia

met

er (

μm)

Figure 3.36. Kinetic analysis of nucleation and growth processes from the experimental dataobtained from isothermally treated samples at different temperatures: (a) number of nuclei as afunction of time, (b) maximum diameter as a function of time, (c) crystal size as a function of timein case of primary crystallization and (d) plots of nucleation rates and growth rates against 1/T forZr76Fe24 and Zr76Ni24. Activation energies of the nucleation and growth processes are determinedfrom these plots.




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Log t

0.0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.6

–0.7

–0.8

–0.9

–1.0

–1.1

–1.2

–1.3

–1.4

–1.5

677 K 675 K673 K

672 K679 K

Log

(– lo

g(1

– x )

)

675 K 673 K672 K

677 K679 K

x

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Time (min)987654321

(a) (b)

Figure 3.37. (a) Fraction transferred as a function of time for polymorphic crystallization in Zr-33at.% Ni. (b) Plots drawn to estimate the Avrami exponent using the fraction transformed data.

This case also applies to the primary recrystallization process of cold workedmetals and alloys.

Case (iii): Nucleation rate, I = 0 and parabolic growth, r = A2�Dt�1/2

When N number of spherical quenched-in nuclei are operating, the fractiontransformed at time, t, is given by

f = 1− exp[−4

3NA3

2D3/2o exp

(−1�5Qd

RT

)](3.52)

Following a similar procedure as the one for cases (i) and (ii), we get Q= 3/2Qd

and n = 3/2.For a two dimensionally growing particle, n = 2/2 = 1 and Q = Qd

Case (iv): Constant nucleation rate I > 0 and parabolic growth, r = A2�Dt�1/2

For three-dimensional growth of spherical particles in duration t,

f = ∫tt=0 I

43�A3

2D3/2t3/2 dt = 8

15�IA3

2D3/2t5/2 (3.53)




3.5.4 Crystallization kinetics in Zr76�Fe1−xNix�24 glassesDetailed kinetics studies on the crystallization process in binary Zr–Fe, Zr–Ni and ternary Zr–Fe–Ni glasses have revealed the characteristics of differentmodes of crystallization. A number of investigations (Buschow 1981, Dey andBanerjee 1985a,b, Dey et al. 1986, Ghosh et al. 1991) have established that whilethe Zr76Fe24 glass crystallizes polymorphically to the Zr3Fe phase (body cen-tred orthorhombic, Re3B type structure), the Zr76Ni24 glass decomposes into amixture of the hcp and the Zr2Ni (body centred tetragonal) phases on crys-tallization. As Ni substitutes for Fe in ternary alloys, Zr76�Fe1−xNix�24, (x =0�4�8�12�16�20�24), the crystallization products are expected to change as theequilibrium Zr3Fe and Zr2Ni phases have limited solubilities in respect of Ni andFe, respectively. The partitioning of Ni and Fe atoms among the growing crys-talline phases during crystallization is also expected to have a significant influenceon the kinetics and mechanisms of the crystallization process. The work of Deyet al. (1998) has demonstrated how ternary additions influence the crystallizationbehaviour of Zr76�Fe1−xNix�24 alloys. In view of the fact that the common modes,namely polymorphic, eutectic, primary followed by eutectic and surface crystal-lization, are all encountered in this system, the results of this work are summarizedhere for highlighting the different kinetic features which characterize these modesof crystallization.

The number and the nature of the thermal events accompanying crystallisationin these glasses are shown in DSC thermograms (Figure 3.38(a)), all of whichcorrespond to a constant heating rate of 10 K/min. The glass transition tempera-ture, Tg, the crystallization start temperature, Tx, and the peak transformation ratetemperature, Tp, are listed along with the sequence and the mode of crystallizationin Table 3.6. The progress of crystallization, as detected during isothermal holdingin a DSC, in all these alloys is shown in Figure 3.38(b). Some special featuresof these exotherms need special mention: (a) the presence of a well-defined andsufficiently long incubation period prior to crystallization in glasses of the follow-ing compositions: Zr76Fe24, Zr76Fe20Ni4, Zr76Fe16Ni8, Zr76Fe12Ni12 and Zr76Ni24;(b) very short incubation periods for the glasses Zr76Fe8Ni16 and Zr76Fe4Ni20;and (c) alloys grouped in (a) show symmetric exotherms while those grouped in(b) exhibit very broad asymmetric exotherms.

Phase analysis by XRD and electron microscopy and diffraction on specimenscrystallized at temperatures in the range of 573–673 K for different durations haveprovided information regarding the identity of the crystalline phases formed andthe mode of crystallization as well as the nucleation density and growth rate ofcrystalline particles as functions of temperature and time. These are complementaryto kinetics data obtained from the bulk samples. The occurrence of surface crys-tallization could be detected from metallographic examinations of cross-sections




(a)

720700680660640620600

Temperature (K)

Rat

e of

hea

t flo

w

2

45

6

7

1

3

(b)

00 05 10 15 2520 30 35 40 5045

Rat

e of

hea

t flo

w

5

620 K6

3

4

2

632 K

631 K

631 K

1

Time (min)

7

633 K630 K

632 K

Figure 3.38. (a) DSC thermogram showing the number and the nature of the thermal events accom-panying crystallization in Zr76�Fe1−xNix�24 glasses at compositions (i) Zr76Fe24, (ii) Zr76Fe20Ni4,(iii) Zr76Fe16Ni8, (iv) Zr76Fe12Ni12 (v) Zr76Ni24, (vi) Zr76Fe8Ni16 and (vii) Zr76Fe4Ni20; (b) DSCthermogram showing the progress of crystallization during isothermal holding Zr76�Fe1−xNix�24

glasses.

of partially crystallized specimens and by carrying out XRD of specimens beforeand after the removal of surface layers of about 10�m thickness. Taking intoaccount the results obtained from microscopy and DSC experiments, the followinginferences could be drawn.

DSC thermograms for the alloys Zr76Fe8Ni16 and Zr76Fe4Ni20, which do notshow long incubation periods, are associated with the surface crystallization pro-cess which precedes bulk crystallization. The product phase resulting from surfacecrystallization has been identified to be an ordered cubic phase with the L12 struc-ture (lattice parameter: 0.420 nm) and an approximate composition of Zr3(Fe,Ni).The exotherms for these two glasses show a limited overlap of two peaks – the first




one being asymmetric, characteristic of the primary crystallization of Zr3(Fe,Ni)occurring on the surface and the second symmetric peak corresponding to thepredominant eutectic crystallization occurring in the bulk. The J–M–A analysisof the kinetic data which requires a deconvolution of the two peaks yields theAvrami exponents, n1 and n2, for the two processes.

The Zr76Fe24 glass crystallizes polymorphically to yield the equilibrium Zr3Fephase. The observed Avrami exponent of 3 corresponds to an interface-controlledgrowth of a spherical transformed product (Zr3Fe crystals in the present case). Asmentioned earlier, polymorphic crystallization is a composition invariant process inwhich the kinetics are controlled by the mechanism of short-range atom transportacross the crystal/amorphous matrix interfaces. The activation energy obtainedfrom the J–M–A analysis of the isothermal kinetics corresponds to the overallkinetics including both nucleation and growth; activation energies for these stepscan be determined separately by measurement of the number density and the sizeof largest crystals formed in specimens annealed at different temperatures fordifferent durations. TEM examinations of samples which have undergone differentextents of crystallization provide such informations (plotted in Figure 3.36 forboth the Zr76Fe24 and the Zr76Ni24 glasses).

The Zr76Ni24 glass crystallizes in an eutectic mode to produce a mixture of the -Zr and the Zr2Ni phases. Nodules of this two-phase mixture nucleate and growto consume more and more of the amorphous matrix. No long-range diffusion fieldis created around these growing nodules. As a consequence, these nodules growuntil they come in contact with adjacent nodules. No amorphous matrix is retainedat this stage. The observed Avrami exponent, n= 3�2, is again consistent with aninterface-controlled growth of crystalline aggregates in three dimensions. Thoughthere is a partitioning of solutes at the interface between the amorphous matrixand the two product phases, the absence of a long-range diffusion field ahead ofthe growing particle makes the process interface controlled. Figure 3.36(a) and(b) shows that the rates of nucleation and growth remain constant (linear growth)with time for both polymorphic (in the case of Zr76Fe24) and eutectic (in the caseof Zr76Ni24) crystallization. For such cases, the temperature dependence of thegrowth rate, dr/dt, can be expressed as

dr/dt = �� exp�−Qg/RT��1− exp�−G/RT�� (3.54)

where r is the radius of the growing crystal at time t, G is the change in thechemical free energy per mole accompanying crystallization, Qg is the activationenergy for growth, is the characteristic frequency and � is the interface width.Since crystallization experiments are carried out at temperatures where glasses aresupercooled to a great extent, G<< RT and the growth rate approximates to




drdt

= � exp�−Qg/RT� (3.55)

The slopes of plots of ln�dr/dt� versus 1/T (Figure 3.36(d)) yield the activa-tion energy for the growth process. The activation energy values obtained frommeasurements of isothermal growth rates of crystalline particles are found to bedifferent from those corresponding to the overall kinetics, as determined from thefraction transformed as a function of time (isothermal, J–M–A analysis) and fromthe Kissinger peak shift analysis of data obtained from constant heating rate exper-iments. As explained earlier, this difference arises due to the fact that the overallcrystallization process includes both the thermally activated formation of freshnuclei and the growth of the quenched-in nuclei as well as of the continuouslyforming nuclei.

In some cases, crystallization occurs in two distinct steps as reflected in twowell-separated peaks in the DSC thermograms recorded in continuous heatingexperiments (Figure 3.39). Quenching of samples from different temperatures,selected between the first and the second peak temperatures, arrest the crystal-lization reaction at different stages, and microstructural investigations on thesesamples reveal that the first peak is associated with a primary crystallization eventwhich is often followed by either a polymorphic or a eutectic crystallization step.

The two-stage crystallization occurs only when the growing crystals duringthe primary event do not inherit the composition of the parent amorphous phase;in such a situation, solute rejection and the creation of a long-range diffusionfield ahead of the growing crystal/amorphous interface take place. The dendritic

710 720 730 740 750 760 770

Heating rate10 K/minZr60 Ni40

T C1 T C2

Hea

t flo

w

↓

↓ ↓

Temperature↓

Figure 3.39. DSC thermogram obtained in a continuous heating experiment showing two-stepcrystallisation.




morphology of the primary crystals is one of the manifestations of the soluterejection process.

In the case of crystallization of Zr76Fe4Ni20 glass, the primary crystallizationproduct is the Zr3(Fe,Ni) phase. The growth of these crystals in the Zr76Fe4Ni20

glass matrix will necessitate a rejection of Ni atoms from the growing crystalsto the matrix. How the radius, r , of such crystals changes with time, t, can beexpressed in the following form:

r = √Dt (3.56)

where is a dimensionless constant depending on the crystal shape and D is theinterdiffusion constant. The value of D, determined from measurements on thegrowth rate of Zr3(Fe,Ni) crystals in a Zr76Fe4Ni20 matrix, is 3�5×10−18 m2 s at atemperature of 595 K. This value is close to that reported for diffusion of Ni inZr-rich Zr–Ni glasses (Sharma et al. 1989), indicating that long-range diffusion ofsolute elements is the rate-limiting step of diffusional growth of crystals formingduring the primary crystallization event. The Avrami exponent determined for thefirst step (n1 = 1�98) and for the second step (n2 = 4�00) is also consistent withthe two modes – primary for the first and eutectic for the second crystallizationevents, the products of eutectic crystallization being Zr2Ni and -Zr.

The crystallization behaviour of the ternary Zr76(Fe1−xNix)24 system also bringsout the fact that the Zr3(Fe,Ni) phase can accept Ni atoms to a considerableextent. As a consequence, polymorphic crystallization is possible in the range ofamorphous alloys corresponding to x < 12. On the other hand, the Zr2Ni phase,having a very limited solubility for Fe, rejects Fe atoms into the amorphous matrixduring its growth.

The mode of crystallization in amorphous Zr76Fe20Ni4 depends on the crystal-lization temperature. At T < 623 K, the Zr3Fe phase forms by primary crystalliza-tion while at higher temperatures, 623 K < T< 673 K, the Zr3(Fe,Ni) phase formsby polymorphic crystallization. The temperature dependence of the solubility ofNi in the Zr3(Fe,Ni) phase appears to be responsible for this behaviour.

It is to be noted that the activation energy for polymorphic growth in a largenumber of Zr-based metal–metal glasses is of the same order as that associated withthe diffusion of metal atoms in these amorphous alloys (Dey and Banerjee 1999).This is unlike the case of polymorphic crystallization in metal–metalloid glasseswhere the activation energy for growth is much larger than that for diffusion.This implies that the atom transport mechanism at the crystallization front is quitedifferent for metal diffusion and metalloid diffusion in such amorphous alloys.In the former case, a group of atoms get involved in the elementary activationprocess, while metalloid diffusion occurs primarily by a single, atom-free volumeexchange process.




An analogy can be drawn with regard to polymorphic crystallization, parti-tionless solidification and massive transformation. The common features of thesetransformations are (a) a rapid movement of the transformation front, under theinfluence of a large driving force created by a high degree of undercooling and(b) the absence of composition difference across the front. Perepezko and Mas-salski (1972) have suggested that in a massive transformation the breakdown ofthe parent structure occurs within a diffuse region, a part of which is expected tobe amorphous. The thickness of the transformation front, as estimated from themeasured growth rate for polymorphic crystallization in a typical Zr-based glass,has been found to be about 2 nm, which compares favourably with that associatedwith massive transformation in alloys.

3.6 BULK METALLIC GLASSES

The formation of metallic glasses in several Ti- and Zr-based binary and ternaryalloys, as discussed in previous sections, requires a minimum critical cooling rateabove 105 K/s. In recent years, a number of multicomponent alloys have beenidentified, which can be amorphized at a cooling rate range as low as 10−1–102 K/s. The sample thickness which can be amorphized in these alloys canbe increased to several centimetres. These alloys are generally known as bulkamorphous alloys which are characterized by a higher value (in the range of0.60–0.75) of the reduced glass transition temperature, Tg/Tm. Furthermore, thesemulticomponent amorphous alloys have a much wider supercooled region beforecrystallization, i.e. the difference Tx −Tg is as large as 50–100 K. Table 3.7 listsa few Zr-based bulk metallic glasses and the corresponding values of Tg/Tm andTx −Tg. It has been pointed out by Inoue (1998) that bulk metallic glasses aregenerally associated with (a) multicomponent (more than three components) alloycompositions, (b) large differences in the atomic sizes of constituent atoms (above12%) and (c) negative heats of mixing. These empirical criteria for easy glassformation can be rationalized in terms of thermodynamic (relative stabilities ofamorphous and crystalline phases) and kinetic considerations.

As in the case of binary and ternary amorphous alloy compositions, the increasedtendency of glass formation results from a low value of the free energy, GL−X,of the transformation in respect of the liquid to crystalline phase. This correspondsto a low heat of fusion, Hf , and a high value of the entropy of fusion Sf . Theconfigurational entropy of the system being dependent on the number of possiblemicroscopic states, a multiplicity of components leads to an increase in Sf . Ahigh value of the reduced glass transition temperature, on the other hand, relatesto a low Hf .




Table 3.7. Table showing Tg� Tx and Tx −Tg for some Zr-based glasses.

S. No. Alloy Method of preparation Tg (K) Tx (K) Tx =Tx −Tg(K)

1. (I) Zr65Cu7�5Ni10Al7�5 Mechanical alloying ofGFC mixture

650 746 96

MA of elementalpowder mixture

638 694 56

2. (I) Zr57Ti5Cu20Ni8Al10 MA of GFCmixture

672 794 122

MA of elementalpowder mixture

647 711 64

3. (II) Zr64Ni36(60 h) milling time MA – 760 –

4. (II) Zr60Ni25Al15(60 h) MA 725 797 72

5. (II) Zr63Ni10Cu17�5Al17�5C2 (30 h) MA 665 747 82

6. (III) (Zr0�65Al0�07Ni0�10

Cu0�175)100−xWx for 2% W– 658 720 62

7. (III) (Zr0�65Al0�07Ni0�10

Cu0�175)100−xWx for 4% W– 637 706 59

8. (IV) Zr34Ni55Al11 835 853 18

9. (IV) Zr10Ni65Al25 (melt spinning) can

10. (IV) Zr30Ni55Al11Y4 be cast into bulk 797 826 29

11. (IV) Zr24Ni55Al11Y10 rod (1 mm diameter) fully 756 798 42

12. (IV) Zr20Ni55Al11Y14 comprising amorphous phase 736 781 45

13. (IV) Zr17Ni55Al11Y17 717 771 54

14. (IV) Zr12Ni55Al11Y22 696 733 37

15. (V) Zr65Al10Ni10Cu15 656 756 105 K

16. (V) Zr65Al10Ni10Cu15Be12�5 689 818 142 K

17. (VI) Zr54Cu46 (heating rate5 K/min)

642.6 689.5

The ease of glass formation essentially depends on the avoidance of crystalnucleation and growth during the solidification process. The homogeneous nucle-ation and growth rates of crystalline phases are discussed in Section 3.5. Anexamination of Eqs. 3.32 and 3.35 reveals that a significant reduction in the nucle-ation and growth rates can be achieved by increasing the viscosity, �, of the liquidand the heat of fusion, Hf , and by lowering the liquid/crystal interfacial energy,�, and the entropy of fusion, Sf .




Furthermore, the reason for the high GFA of a particular multicomponent alloycan be investigated from the point of view of interatomic distances. The importanceof atomic size differences on the GFA is revealed from the observation thatthe multicomponent bulk metallic glass forming alloys consist of elements withsignificant differences in atomic sizes (above 12%). A combination of negativeheat of mixing and large difference in atomic sizes is expected to promote ahigh random packing density of the constituent atoms in the supercooled liquid.Corresponding to such a structure, an increased difficulty in atomic rearrangementscan cause a significant reduction in the atomic diffusivity. The formation of ahigher degree of dense random packed structure has also been confirmed from theXRD profiles obtained on Zr–Al–Ni amorphous alloys. The reported changes inthe atomic distances and coordination number corresponding to different atomicpairs occurring during the amorphous to crystalline transformation show thatthese changes are insignificant for Ni–Zr and Zr–Zr pairs, while the coordinationnumber for Zr–Al pairs undergoes a drastic change on crystallization. This isindicative of the requirement of a large-scale reorganization of Zr–Al pairs duringthe crystallization event. It is because of this reason that Al addition in Zr–Cu-basedglasses has a pronounced influence on Tx (=Tx −Tg) as shown in Figure 3.40. Thereasons for the attainment of the high GFA for some ternary and multicomponentalloys have been summarized schematically in Figure 3.41.

The relation between the GFA and Tx can be demonstrated clearly in the caseof the ternary Zr–Al–Ni system. Figure 3.42 shows the contours of equal valuesof Tx in the ternary diagram. The Tx value is maximum (77 K) for the alloyZr60Al15Ni25; Tx values of above 50 K are obtained in a wider composition range,as indicated in Figure 3.42. Having examined a large number of alloy compositions,Inoue (1998) has identified the generic alloy composition, Zr65Al7�5TM27�5, whichis associated with the largest Tx values (about 1.5 times of that for other bulkglass forming systems). Based on this, a number of multicomponent alloys havebeen formulated which are good candidates for making metallic glasses in the bulkform. The alloy Zr65Al7�5Cu2�5 Co10Ni15 is an example of this set of bulk glassforming alloys.

DSC experiments on the crystallization of three glass compositions in the familyof Zr65Al7�5TM27�5 have revealed that crystallization occurs in a single exothermicevent (Figure 3.43), with H ranging from 3.13 to 3.74 kJ/mol. Prior to thecrystallization event, all these alloys exhibit an endothermic reaction occurringover a wide temperature range below Tx. It has also been shown that the amorphousphase in these alloys transforms gradually to the supercooled liquid state over awide composition range, followed by crystallization at about 750 K.

Inoue (1998) has examined a number of five-component alloys Zr65−x Al10

Cu15 Ni10 Mx where M is Ti, Hf, V, Nb, Cr and Mo, for finding out suitable




600

650

700

750

800

100

50

0 5 10 15 20

at. % Al

ΔT x

(= T

x –

T g)/

K

Zr65AIx Cu35–x

T x,T

g/K

T x

T g

Figure 3.40. Plots showing effect of Al addition in Zr–Cu-based glasses on Tx, Tg and Tx

(= Tx −Tg) (after Inoue, 1998).

compositions having a strong tendency for the formation of the glassy phase inbulk. For the convenience of experimentation, they have scanned arc-melted ingotsof a cross-section of approximately 100 mm2 for determining the area fractionof the ingot which remains in the amorphous phase in the as-cast condition, theamorphous area fraction being a good measure of the GFA. The results of theseexperiments have shown that the glass forming tendency in the arc-melted ingotsis not always related to Tx, which reflects the thermal stability of the supercooledliquid. This is evident from the easy GFA in alloys Zr-Al-Cu-Ni-M (M = Ti, Nbor Pd) in which Tx is rather small, ranging between 47 and 65 K. This apparentanomaly can be explained by considering the fact that arc-melted ingots alwayscontain a number of crystalline nuclei in the region which remains in contact withthe watercooled copper mould. Nucleation of crystals is, therefore, not a limitingfactor for crystal formation under this condition. It is the difficulty of the crystalgrowth process which is responsible for the vitrification of the molten mass. It is,therefore, inferred that the addition of Ti, Nb or Pd to Zr-Al-Cu-Ni alloys is usefulin suppressing the growth of crystallites.




The constituent elements have large negative heats of mixing and large atomicsize ratios (above 10%)

Formation of amorphous phase with a higher degree of dense random packedstructure

Increase of solid/liquidinterfacial energy

Difficulty of atomic rearrangement ofthe constituent elements on a long-

range scale

Supression of nucleation of crystallinephases Supression of crystal growth

Figure 3.41. Schematic diagram summarizing the reasons for the attainment of the high glassforming ability for some ternary and multicomponent alloys.

Al 10 20 30 40 50 60 70 80 90 ZrZr (at.%)

9080

7060

5040

3020

10

Ni

1020

3040

5060

7080

90

Ni (at.%

)Al (a

t.%)

70605030

20

ΔT x

ΔT x ( = T x – T g) (K)

Figure 3.42. Zr–Al–Ni ternary diagram in which the contours of equal values of Tx are shown(after Inoue, 1998).

Attempts of substituting Zr by Y to a large extent have been quite successful.Alloys with compositions in the range Zr60−xYxAl15Ni25 (x = 15–30) have beenfound to be easy glass formers. The crystallization process of these amorphousalloys is quite special as they exhibit two glass transitions and two supercooled




500 600 700 800Temperature (K)

0.67 K/sZr65Al7.5Ni5Cu22.5

Zr65Al7.5Ni10Cu17.5

Zr65Al7.5Ni15Cu12.5

T g T x

↓↓↓

Rat

e of

hea

t flo

w (

H)

·

Figure 3.43. DSC thermograms showing the crystallization of three glass compositions in the familyof Zr65Al7�5TM27�5 (after Inoue, 1998).

liquid regions before the completion of crystallization. Figure 3.44 shows the DSCplots in which the first stage glass transition Tg1, the first stage exothermic crys-tallization Tx1 and the second stage glass transition Tg2, are indicated. The distinctsplitting of the glass transition and crystallization reactions into two stages appearsto originate from the tendency of phase separation of Zr-rich and Y-rich regions.The separation of the two amorphous phases is also revealed from the modulatedcontrast exhibited by the as-quenched Zr-Y-Al-Ni glasses. On annealing at tem-peratures between Tx1 and Tx2, a microstructure containing a fine dispersion ofY-rich crystalline particles in the Zr-rich glassy matrix is produced. Homogeneousprecipitation of the nanoscale Y-rich crystalline particles has a significant influ-ence on the viscosity and atomic diffusivity of the remaining amorphous phasewhich becomes enriched in Zr and depleted in Y. The increased viscosity (to anextent of about 30%) enhances the thermal stability of the remaining amorphousphase as is reflected in the wide supercooled region (Tx ∼ 100 K).

The crystallization products resulting from metallic glasses based on Zr arequite similar to those appearing in the melt-spun binary alloys described earlier.The crystalline phases resulting from the amorphous Zr–Al–Cu alloys are Zr2Al(orthorhombic), ZrAl (bct), Zr2Al (B82) and Zr2(Al,Cu). However, the crystal-lization kinetics for alloys with large Tx have been found to be different fromthose corresponding to binary amorphous alloys with low Tx. This point can beillustrated using the example of the crystallization behaviour of the Zr65Al7�5Cu27�5




400 500 600 700 800 900

Temperature (K)

0.67 K/s

Zr45Y15Al15Ni25

Zr33Y21Al15Ni25

Zr33Y27Al15Ni25

Zr35Y30Al15Ni25

Tg1

Tg1

↓562Tg2

↓775

Tg2

↓763

Tg2

↓772

Tg2

↓775

↓562

Tx1↑562

Tx2↑813

Tx1↑562

Tg1

↓562

Tg1

↓562

Tx1↑562

Tx1↑562

Tx2↑821

Tx2↑812

Tx2↑805

Rat

e of

hea

t flo

w (

H )

→

⋅

Figure 3.44. DSC thermograms of Zr60−xYxAl15Ni25 (x = 15–30) amorphous alloys showing thefirst stage glass transition, Tg1, the first stage exothermic crystallization, Tx1, and the second stageglass transition, Tg2 (after Inoue, 1998).

alloy which has a Tx of 90 K. J–M–A analysis of the crystallization kinetics ofZr65Al7�5Cu27�5 reveals that the Avrami exponent (n value) is not constant over theentire temperature range covering the amorphous solid and the supercooled liquid.The value of n increases from 3 to 3.7 with increasing annealing temperature,Ta (Figure 3.45). Inspite of the fact that the crystalline phases forming due toannealing at temperatures ranging from 663 K (<Tg) to 723 K are the same, i.e.Zr2Cu and ZrAl, the variation in the n value has been attributed to a change inthe nucleation process. While at temperatures below Tg, the crystallization pro-ceeds by the growth of a constant number of nuclei in the amorphous matrix, thenumber of nuclei in the supercooled liquid grow at a constant rate. The growthmechanism of crystals remains the same interface-controlled process in both theamorphous solid and the supercooled liquid matrices. Based on the kinetics data,




4.0

3.5

3.0

2.5660 680 700 720 740

Ta(K)

Ava

ram

i exp

onen

t (n

)

Zr65Al7.5Cu27.5

T g

↓

Figure 3.45. Avrami exponent (n) as a function of annealing temperature (Ta) obtained during thecrystallization of Zr65Al7�5Cu27�5 alloy (after Inoue, 1998).

the activation energy for nucleation has been estimated to be 400 kJ/mol in the for-mer and 230 kJ/mol in the latter matrix. The influence of the matrix is also noticedon the morphology of the crystalline products. While Zr2Cu crystals formed in theamorphous solid have been found to be nearly spherical with a smooth interface, adendritic morphology is associated with the crystals forming from the supercooledliquid (Inoue 1988).

The existence of the supercooled liquid over a wide temperature range (Tx ∼90 K) is attributed to the retardation of the growth of the Zr2(Cu,Al) phase, whichrequires redistribution of Al around Zr in a highly dense random packed structure.Such a dense random packing arises due to significantly different atomic sizesand strong attractive bonding between the three constituent elements. The highdegree of dense random packing is also responsible for a high value of the liquid–solid interface energy, which causes a retardation in the precipitation of crystallinephases.

3.7 SOLID STATE AMORPHIZATION

The transition from crystal to glass can be compared with the melting process sincethe glass can be considered thermodynamically as a frozen, highly undercooledliquid. In spite of a longstanding interest in the phenomenon of melting, a generally




accepted theory of the transition between a crystalline and a liquid phase has notbeen developed yet even for a single component system. All melting theories arebased on some form of instability or catastrophe which are either vibrational, elas-tic, isochoric, defective or entropic in nature. These theories of melting envisage ahomogeneous melting process though it is well established that melting is a first-order transition which occurs by the creation of a well-defined interface betweenthe liquid and the crystalline phases. Heterogeneous nucleation on external andinternal surfaces plays an important role in the melting process. A considerabledegree of superheating of crystals is possible if effective heterogeneous nucleationsites are eliminated.

While considering melting and freezing of a pure metal, Fecht et al. (1989)have indicated the limits of supercooling of liquids and of superheating of crystals.In analogy to the Kauzman temperature, T o

g , where the excess entropy of theundercooled liquid vanishes, one can identify a temperature where the entropyof the superheated crystal would become equal to that of the liquid. It has beenshown that this isentropic limit of stability of the solid phase for pure Al islocated at 1.38 Tm. The isoentropic limits of stability of superheated crystals andof supercooled liquids for pure Al are shown in Figure 3.46 (a) and (b). It maybe noted here that before reaching the isentropic limits during supercooling orsuperheating, a succession of catastrophe barriers are encountered. Those includea thermodynamic barrier (equilibrium melting and crystallization) and a kineticbarrier (glass formation and shear instability for melting). Experimental data onspecific heat and thermal expansion of a system under supercooled and superheatedconditions are required for the estimation of these barriers. In the absence of suchdata, a realistic estimation of these limits is difficult to make. However, the general

20

15

10

5

0

T m

T I

Liquid

Temperature (K)

(a)

Crystal

Undercooledliquid

Ato

mic

spe

cific

hea

t

T m

T g

T I

Temperature (K)

(b)

Free

ene

rgy

chan

ge

Figure 3.46. Isoentropic limits of stability of (a) supercooled liquids and (b) superheated crystalsfor pure Al.




observation that the superheating of crystals above their melting points is muchmore restricted than supercooling of liquid is due to the pronounced asymmetrybetween the two processes as indicated in Figure 3.46(b).

Let us now consider the case of the melting of a binary alloy in which thereexists a possibility of redistribution of solutes between the crystalline and the liquidphases. In this case, the kinetics of melting for crystalline solid solutions dependon the imposed heating rate. While equilibrium melting occurs under low heatingrates, superheating above the equilibrium solidus temperature can be achieved athigh heating rates. In case the extent of superheating is high enough to exceed theTo temperature (where the integral molar free energies of the crystalline and theliquid phases are equal), the crystalline phase can undergo a partitionless meltingprocess.

The crystal to amorphous transition, which is necessarily a non-equilibriumprocess, can be induced under certain constraints which prevent the formation ofequilibrium phases. This point can be explained with the help of a polymorphousphase diagram which becomes relevant under the circumstance where the par-titioning of the solute is not possible. Figure 3.47 shows the equilibrium phasediagram (drawn by broken lines) superimposed on a polymorphous phase diagramwhich indicates the stability domains of equilibrium and of metastable abovethe To line. One must use this diagram in conjunction with a time scale. This timescale should be long enough to permit ergodic sampling of a set of microstates

Liquid

T

*

C

*Glass

Metastableα above T o

Metastableα below T o

Tem

pera

ture

Equilibriumα

T o

T is

T go

Concentration↓

Figure 3.47. Equilibrium phase diagram (drawn by dashed lines) superimposed on a polymorphousphase diagram which indicates the stability domains of equilibrium and of metastable above theTo line.




belonging to the metastable phases but short enough so that partitioning of thecomponents in different phases is prevented.

The situations under which solid state amorphization is possible can be groupedinto the following two broad categories:

(1) composition induced destabilization and vitrification of crystalline alloys underisothermal conditions, and

(2) amorphization under externally driven conditions.

While glass formation in diffusion couples and hydrogen-induced vitrificationbelong to the former category, radiation-induced and mechanically driven amor-phization represent the latter.

3.7.1 Thermodynamics and kineticsThe stability/instability criteria of a solid with respect to melting include the effectsof both shear strain and hydrostatic pressure. The conditions of stability of a solidrequire the elastic moduli (both shear and bulk moduli) to be positive,

��

�!�T�Ci =� > 0 (3.57)

1V

�P

�V�T�Ci =B > 0 (3.58)

where �, ! and � are shear stress, strain and modulus, respectively, and B is thebulk modulus. The elastic constants of a superheated crystal, � (= �C11 −C12�/2,isothermal shear modulus), and B (= �C11 + 2C12�/3, isothermal bulk modulus),vanish at the critical temperatures, T� and TB, respectively, which are abovethe melting temperature, Tm. Experimental data on elastic moduli as functionof temperature (Tallon and Wolfenden 1979), when extrapolated to temperaturesabove Tm, show that the instability sets in at T ∼ 1�6Tm (Figure 3.48).

The increase in thermal disorder as the temperature is raised leads to an ultimateshear instability of a metastable superheated crystalline solid. Other types ofdisorder, namely defects and chemical disorder, can also contribute in bringingabout the shear instability. Thorpe (1983) has theoretically studied the mechanicalstability of a model crystal, consisting of balls of equal mass M connected withnearest neighbour balls by springs with force constant K. Beginning with theperfect crystal, the springs are gradually removed from random locations. Usingcomputer simulation, the shear and bulk moduli are calculated as functions of thefraction of springs removed. The results of this theoretical work have shown thatboth � and B vanish where a critical fraction of springs are removed (Figure 3.49).




200 600 1000 1400 1800

2

4

6

8

Temperature

T mB

Instabilityμ

Ela

stic

mod

uli

1010

Pa

Figure 3.48. Variation of elastic moduli as a function of temperature.

B

μ

Ela

stic

mod

uli (

arb.

uni

t)

Fraction of removed springs

Figure 3.49. Shear and bulk modulus as function of fraction of springs removed.

Egami and Waseda (1984) have considered a binary solid solution containingatoms of two different sizes. By carrying out a simple analysis of local strain effectsusing an elastic continuum approach, they have shown that such a solution becomestopologically unstable when the concentration of the smaller atoms (A atoms)




reaches a critical concentration, c∗A, which depends on the ratio of atomic sizes

r = RA/RB, where RA and RB are the atomic radii of the two types of atoms.Considering the instability arising out of a critical level of strain disorder, theconcentration level at which the solid solution becomes topologically unstable hasbeen found to be

c∗A = 2�R3

B/�R3B −R3

A�+higher order terms (3.59)

Let us now examine the free energy – concentration diagram of a system at atemperature below Tg (for the alloy compositions under consideration). Figure 3.50shows such a diagram which depicts the condition of metastable equilibriumbetween the crystalline -phase and the amorphous phase (a). The presence of theequilibrium intermetallic phase is denoted here with a dotted line. If the formationof the intermetallic phase is kinetically prevented, nucleation of the amorphousphase becomes thermodynamically possible only when the concentration of Batoms in the -phase, c B exceeds the limit c1

B. The maximum free energy changeassociated with the nucleation of the amorphous phase from an -phase havinga composition given by c B is shown by the vertical line Gm. Such a nucleationprocess, which is facilitated at heterogeneities like high-energy grain boundaries,involves the partitioning of B atoms preferentially towards the nucleating amor-phous phase. If, however, the -phase is enriched to a composition where c B > co,a massive (i.e. partitionless) → a transformation becomes possible.

A BC B

1c B2c B

αc B

Free

ene

rgy

c o

aα

ΔG m

Figure 3.50. Free energy–composition diagram showing metastable equilibrium between the crys-talline -phase and the amorphous phase.




As discussed earlier in Section 3.4.1, the phase diagrams of typical glass formingalloys are characterized by steeply plunging To-lines, as seen for Zr–Ni and Zr–Cualloys. For such alloys, a generic non-equilibrium phase diagram can be developed,neglecting the kinetically excluded intermetallic compounds, for illustrating thepossible thermodynamic states of a metastable system constrained to be a singlephase. For alloys with large negative slopes for the To lines, the To line mustcross the ideal glass transition line T ∗

g at a certain composition, c∗. Under thiscondition, a triple point (c∗, T ∗) is defined in respect of the supersaturated crystal,undercooled liquid and ideal glass (Figure 3.51). The entropic instability line, T s

i ,against melting (shown by broken line) should pass through the triple point sincethe entropy difference between the crystal and the liquid vanishes at this point. Thispolymorphous phase diagram (Figure 3.51) essentially depicts that the followingtwo conditions are satisfied: G = H − TS = 0 (condition for polymorphicmelting) and S = 0 (Kauzman condition) at the triple point. This also shows thatthe composition-induced disorder reduces the polymorphic melting temperatureof the crystalline solid solution to the ideal glass transition temperature. The slopeof the To line, dTo/dc = −��G/�c�/S at the triple point approaches infinity.Fecht et al. (1989) have predicted the triple point for -Zr supersaturated withNi to be 638 K and 11.5 at.% Ni. The To line extends below the triple point as a

Composition

Glass

Crystal

Liquid

Tem

pera

ture

T o

T *

C *

T g*

Tsi

Figure 3.51. Schematic diagram indicating a triple point (c∗, T ∗) as defined in respect of thesupersaturated crystal, undercooled liquid and ideal glass.




straight line with infinite slope as long as non-ergodicity prevails and the Kauzmanargument holds. Below the triple point, the transition between the crystal andthe glass is isentropic and, therefore, truly continuous in volume as long as themetastable constraints are maintained.

The presence of non-equilibrium lattice defects such as vacancies and anti-sitedefects play a major role in providing the constraint under which the meltingtemperature of the crystal can be considerably reduced. These defects, which havevery low mobility at relatively low processing temperatures, remain frozen in thelattice. If one considers the vacancy as a second component in the system, onecan draw a phase diagram like the one shown in Figure 3.52.

The free energy difference, G = H −TS, between a liquid and a singlecrystal for pure metals can be realistically estimated to be (Fecht et al. 1989)

G = 7SfT�T�Tm +6T�� (3.60)

and for glass forming alloys by

G = 2SfT�T�Tm +T� (3.61)

where T is the undercooling below the melting point, Tm, and Sf the entropyof fusion. The increase in free energy of the crystalline phase can be expressed as

Gv = cv�Hv −TSv�+kBT ��cv ln cv + �1− cv� ln�1− cv�� (3.62)

0.02 0.02 0.04 0.06 0.100.08

Vacancy concentration (c v)

0

0.2

0.4

0.6

0.8

1.0

T/ T

m

ΔH = 0

ΔS = 0

ΔG = 0

T∗Crystal

Glass

Liquid

c °v

Figure 3.52. Phase diagram illustrating the role of non-equilibrium lattice defects such as vacanciesand anti-site defects in providing the constraint under which the melting temperature of the crystalcan be considerably reduced.




Combining these equations, the decrease in melting temperature of the defectivecrystal can be expressed as a function of defect concentration.

3.7.2 Amorphous phase formation by composition-induceddestabilization of crystalline phases

There are a number of experimental results to demonstrate that a crystallinematerial can be transformed into an amorphous one by progressively introducingalloying elements. It has often been noticed that a crystalline phase is destabilizedwhen loaded with some specific alloying elements to a level exceeding a certainthreshold. Introduction of alloying elements can be effected by several means,for example, (a) by isothermal annealing of diffusion couples, (b) by mechanicalalloying, (c) by introducing hydrogen by diffusion and (d) by ion implantation.All these treatments, which are essentially isothermal but are implemented underchemically non-equilibrium conditions, can lead to the formation of amorphousphases. It is the excess chemical energy associated with the initial configurationwhich permits the glassy state to be adopted and retained as a metastable product.Some experimental results pertaining to the aforementioned treatments will nowbe described, and glass formation will be rationalized in terms of thermodynamicsand kinetics of the pertinent process.

3.7.3 Glass formation in diffusion couplesThe early observations on glass formation in diffusion couples were reported ininitially crystalline multilayers of Au–La (Schwartz and Johnson 1983) and insamples of Si coated with a thin film of Rh (Herd et al. 1983). The formation of anamorphous layer in the reaction zone of binary diffusion couples has been observedin a number of systems in which one of the components is Zr or Ti. These are thewell-known glass forming systems such as Zr–Cu, Zr–Ni, Zr–Co, Zr–Fe and Ti–Ni.A variety of techniques have been employed for detecting the amorphous phase.Serial sectioning of diffusion couples near the reaction zone provides samples fora plan view examination by XRD and TEM, while cross-sectional TEM revealsthe presence and distribution of different layers forming at the reaction zone.The composition profile is determined by using electron proble microanalysisand Rutherford backscattering. The presence of the amorphous phase can also bedetected by the observation of a crystallization event during heating in a DSC.

Let us examine some of the reported experimental results in order to understandthe thermodynamics and kinetics of the formation of amorphous layers in the reac-tion zones of diffusion couples. Cross-sectional TEM studies on Zr–Ni diffusioncouples by Newcomb and Tu (1986) have shown the presence of a well-definedplanar interlayer of an amorphous phase in diffusion couples reacted at 573 K fordurations of 1.5 and 4 h. The formation of the ordered intermetallic ZrNi phase has




(b)(a)

Amorphousinterlayer

Kirkendalvoids

Ni NiZrZr

550 K

HEX

BCC

Amorphous

FCC

= Equilibrium compounds

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90Ni ZrcZr

–60000

–50000

–40000

–30000

–20000

–10000

10000

20000

ΔG (

kJ/m

ol)

Figure 3.53. (a) Schematic diagram showing the distribution of different phases and of Kirkendalpores in a Zr–Ni diffusion couple reacted at 573 K for 12 h, (b) free energy concentration diagramin Zr–Ni system.

been observed when the reaction time is extended to 12 h. A schematic illustration(Figure 3.53(a)) shows the distribution of different phases and of Kirkendal poresin a Zr–Ni diffusion couple reacted at 573 K for 12 h (Newcomb and Tu 1986).Free energy–concentration plots for the different competing phases in this system(Figure 3.53(b)) can be used for illustrating two possible mechanisms for theformation of the amorphous layer. The normal downhill interdiffusion process isexpected to produce a hcp solid solution of Ni in Zr and an fcc solid solution of Zrin Ni. The amorphous phase can either nucleate heterogeneously at the interface,the local metastable equilibrium being dictated by tangent construction on the freeenergy curve for the terminal solid solution or by solute enrichment of the solidsolution to an extent where it reaches beyond the co limit (when the system isforced below the To temperature corresponding to the composition, co) leading topolymorphic vitrification.

In order to find out whether a solid solution phase in the Ni–Zr system canbecome unstable with respect to the Gibbs criterion, it is necessary to determinethe minimum level of solute content which allows a massive or polymorphicvitrification. For the solution of Ni in hcp Zr, the To�cN i) line reaches 0 K atcNi = 0�22. The criterion based on the local strain, as proposed by Egami (1983),gives the composition limits for instability as cNi = 0�26 and cZr = 0�13 for thehcp and fcc solid solutions, respectively. From these estimations of instabilitylimits, it appears possible that the destabilization of a solid solution phase canindeed occur, leading to a polymorphic vitrification process. It should, however,




be noted that the nucleation of an amorphous phase by heterogeneous nucleationbecomes thermodynamically feasible at much lower levels of the solute contentin terminal solid solutions. The importance of the heterogeneous nucleation of anamorphous phase in diffusion couples can also be realized from the observationthat an amorphous phase does not form in a couple made by depositing Ni on asingle crystal of Zr.

The question of polymorphic vitrification vis-a-vis nucleation of an amorphousphase of a composition given by the local metastable equilibrium condition hasbeen addressed by Bhanumurthy et al. (1988, 1989) while analysing the resultspertaining to Zr–Cu diffusion couples. These experiments have shown that aninterface reaction between Zr and Cu at a temperature of 873 K results in theformation of an ordered intermetallic phase, Zr2Cu, whereas at a temperature closeto 600 K, an amorphous layer forms in the reaction zone. At such low temperatures,the formation of the ordered intermetallic phase is fully suppressed.

An examination of hypothetical free energy – concentration plots for the , the� and the liquid phases (Figure 3.54) reveals that the metastable solubility of Cuin the -Zr lattice is considerably extended when intermetallic phase formationis suppressed. The equilibrium conditions between the different competing phasesare shown by the common tangents AB (between hcp and Zr2Cu), CD (between and the amorphous a-phase), EF (between and the bcc �-phase) and finally JK(between � and L). The points of intersection in the free energy – concentration

0.50.40.30.20.1

H

G

aG

α

I JF M

G

β

T~600 K

G

Zr2Cu

T

KD

P

B

NACE

JK: G

β, G

a

EF: G

α, G

β

CD: G

α, G

a

ET: G

α at E

AB: G

α, G

Zr2Cu

Atomic fraction (Cu)

Fre

e en

ergy

(G

)

Figure 3.54. Free energy – concentration plots for , �, Zr2Cu and the liquid phases.




curves for different phases mark the limits at which polymorphic transformationsbetween them become thermodynamically possible. For the -phase to becomeamenable to polymorphic vitrification, the Cu enrichment of this -phase shouldgo beyond the point H. Before attaining such a high Cu concentration, the super-saturated -phase becomes amenable to a composition invariant massive to �transition, which has not been observed in the reaction zone of Zr–Cu diffusioncouples. Microanalysis of the different phases present in the reaction zone hasshown that the maximum concentration of Cu in the -phase, which lies in contactwith the amorphous layer, is about 4 at.%. Based on these observations, it hasbeen inferred that the formation of the amorphous phase in the reaction zone ofZr–Cu diffusion couples occurs by its nucleation from the -phase, enriched toa Cu content of about 4 at.% and its subsequent growth by the coalescence ofindependently nucleated amorphous regions. Once the -phase is enriched in Cu toa level shown by the point E, the maximum free energy change for the nucleationof the amorphous phase is given by the drop NP where P is given by a tangentdrawn on the Ga plot which is parallel to ET. A continuous amorphous layereventually develops which grows in thickness by consuming the adjacent -phaselayer till the supply of Cu atoms gets restricted by the formation of Kirkendalpores on the Cu-rich side. The Cu level required for the destabilization of the -Zrlattice, on the basis of the Egami criterion, has been estimated to be about 28 at.%.This lends further support to the contention that the amorphous layer forms in theZr–Cu system not by the destabilization of the -lattice but by the nucleation ofa Cu-rich amorphous phase which can establish a metastable equilibrium with an -Zr–Cu alloy containing about 4 at.% Cu.

The kinetics of the one-dimensional growth of the amorphous interlayer in thereaction zone of a binary diffusion couple can be described by a set of coupleddifferential equations (Johnson 1986):

�c

�t= D

�2c

�x2(3.63)

D�c

�x�x1

= �1− c1�dx1

dt(3.64)

−D�c

�x�x2

= c2

dx2

dt(3.65)

dx2

dt=K2�c2 − co

2� (3.66)

dx1

dt=K1�c1 − co

1� (3.67)




1 2GLASS

X

dX1

dtdX2

dt

X1 X2

1.0

0

c1

μ1

μ2

D~

K1

K2

c

01

c

02

Figure 3.55. Schematic diagram showing the concentration profile of metal 1 and chemical potentialprofile of metal 1 and metal 2.

where D is the interdiffusion constant in the amorphous phase, c(x) is the concen-tration profile of metal 1 in the amorphous phase, co

1 and co2 are the concentrations

of metals 1 and 2, respectively, in the amorphous phase which are in equilibriumwith pure metal 1 and pure metal 2 (as shown in Figure 3.55); x1 and x2 give thepositions of the interfaces separating the amorphous layer and the metals 1 and 2,K1 and K2 are the kinetic response parameters at these interfaces and c1 and c2

are abbreviated forms of c�x1� and c�x2�.Introducing some simplifying assumptions such as the interdiffusion constant D

being independent of composition and the response parameters K1 and K2 beinglinear and being related as

K1/K2 = �co1/�1− co

2�� (3.68)

The equations have been solved numerically and for long times (t → ),

x2 = −D/K2 +√

2aDt (3.69)




where a is a constant of order unity. For short times (t → 0) the solution is in thefollowing form:

x2 = constant�K2t+negligible higher order terms (3.70)

These results predict a linear growth law at short times (for a thin amorphousinterlayer) and a shifted t1/2 law in the limit of long times. This means that whenthe amorphous layer is thinner than the characteristic length, l�= D/K�, the growthis interface controlled while a diffusion controlled growth mechanism operateswhen the amorphous interlayer is much thicker than l.

Analysis of experimental data on the growth of amorphous interlayers in diffu-sion couples of Ni–Zr, Co–Zr and Ni–Hf has yielded D, the interdiffusion constant,as a function of temperature at T < Tg. Johnson (1986) has shown that D valuesmatch very closely with the diffusion constant for impurity diffusion of Ni inthe amorphous Ni67Zr33 alloy. This observation points to the fact that the inter-diffusion process is strongly dominated by the migration of the smaller atoms ofthe late transition metals (Ni in the case of Zr–Ni) and practically no migrationof Zr atoms. The formation of Kirkendal voids along the interface separating Nifrom the amorphous layer is a direct evidence that Ni is the moving species. Thevoid formation is responsible for reducing interfacial contact and ultimately forcutting off the supply of Ni atoms. At this stage, the growth of the amorphouslayer terminates.

Experiments have shown that the growth of amorphous interlayers can leadto a layer thickness of 100–200 nm without any accompanying formation ofcrystalline intermetallic compounds. Since this thickness is much larger than thesize of critical crystalline nuclei, the avoidance of crystalline nucleus formationduring interdiffusion annealing for a time scale of about 104 s appears improbable.The fact that only the late transition metal atoms are mobile with virtually nomigration of Zr atoms can perhaps explain why nucleation of ordered intermetalliccompounds does not occur during the growth of the amorphous interlayer.

3.7.4 Amorphization by hydrogen chargingYeh et al. (1983) have reported that a bulk polycrystalline partially ordered fccsolid solution of Zr0�75Rh0�25 composition transforms to an amorphous phase whenhydrided by exposure to hydrogen gas at temperatures between 425 and 500 K.This reaction can be expressed as

Zr0�75Rh0�25�cryst�+H2�gas�Zr0�75Rh0�25H1�14�amorphous�T < 500 K (3.71)




At higher temperatures (T > 500 K) the equilibrium product forms as per thefollowing equation:

Zr0�75Rh0�25�cryst�+H2�gas�ZrH2�cryst�+Rh�cryst�T > 500 K (3.72)

TEM observations have revealed that at T < 500 K, amorphization proceedsby the nucleation of glassy zones along the grain boundaries of the crystallinestarting material, followed by the expansion of these amorphous regions into thegrain interiors. The boundary between the crystalline and the amorphous regionsis sharp, suggesting a strong first-order transition. The entire process proceedsmuch as melting would proceed in a polycrystalline sample. X-ray studies haveindicated that the fcc Zr0�75Rh0�25 solid solution dissolves some hydrogen prior totransforming to the glassy phase. This suggests that with hydrogen entry, the freeenergy of the alloy is raised above that of the glassy phase. As a consequence, thesuperheated crystalline phase transforms into an amorphous phase.

The temperature threshold, 500 K, below which amorphization occurs, is dictatedby the relative values of the diffusion constants of H and the metal atoms. Above500 K the equilibrium product consists of two crystalline phases, namely fcc Rhand fcc ZrH2. For the formation of such a product, metal atom redistribution mustoccur by thermally activated diffusion over a length scale at least of the order of therespective critical nuclei sizes of the two crystalline phases. This is apparently notpossible below 500 K. The temperature dependence of the chemical rate constantsof Eqs. 3.71 and 3.72 is expected to be considerably different as these processes arecontrolled by hydrogen and metal atom diffusion, respectively. It is this differencein temperature dependence that enforces a kinetic constraint on the separation ofthe two crystalline phases and instead allows the hydrogen-charged Zr75Rh25 alloyto undergo amorphization in a manner similar to melting.

3.7.5 Glass formation in mechanically driven systemsHigh energy ball milling can lead to glass formation from elemental powdermixtures as well as by amorphization of intermetallic compound powders. Solidstate amorphization by high energy milling has been demonstrated in a number ofTi- and Zr-based and other alloy systems such as Ni–Ti, Cu–Ti, Al–Ge–Nb, Sn–Nb, Ni–Zr, Cu–Zr, Co–Zr and Fe–Zr. The process of ball milling is illustrated inFigure 3.56. Powder particles are severely deformed, fractured and mutually coldwelded during collisions of the balls. The repeated fracturing and cold welding ofpowder particles result in the formation of a layered structure in which the layerthickness keeps decreasing with milling time. A part of the mechanical energyaccumulates within these powder particles in the form of excess lattice defectswhich facilitate interdiffusion between the layers. The continuous reduction in thediffusion distance and the enhancement in the diffusivity with increasing milling




Powderparticle

B

Hard ball

A

Figure 3.56. Schematic diagram illustrating the process of ball milling.

time tend to bring about chemical homogeneity of the powder particles by enrichingeach layer with the other species being milled together. The sequence of the eventsthat occur during milling can be followed by taking out samples from the ballmill at several intervals and by analysing these powder samples in respect of theirchemical composition and structure. Let us describe one such experiment in whichelemental powders of Zr and Al were milled in an attritor under an Ar atmosphere.

Elemental powders of Zr and Al of 99.5 purity, when milled in an attritor using5 mm diameter balls of zirconia as the milling media and keeping the ball topowder weight ratio at 10:1, showed a progressive structural change as revealedin XRD patterns (Figure 3.57(a) and (b)). Diffraction peaks associated with theindividual elemental species remained distinct upto 5 h of milling at a constantmilling speed of 550 rpm. All particles and the balls appeared very shiny in theinitial stages. With increasing milling time, the particles lost their lustre, the111 and 200 peaks of fcc Al gradually shrunk and the three adjacent low-anglepeaks of hcp -Zr, corresponding to 1010, 0002 and 1011, became broader. Afterabout 15 h of milling, XRD showed only -Zr peaks which shifted towards thehigh angle side, implying a decrease in the lattice parameters resulting from theenrichment of the -Zr phase with Al. After 20 h of milling, all Bragg peaksexcept one broad peak close to the {1010} peak disappeared. Powders milledfor 25 h showed an extra reflection corresponding to a lattice spacing of 5.4 nm,which matches closely to a superlattice reflection of a metastable D019 (Zr3Al)phase. On further milling, the powders transformed into an amorphous phase.The sequence of structural evolution could be described as -Zr + Al −→ -Zr(Al) solid solution+Al −→ nanocrystalline solid solution+ localized amorphousphase −→ Zr3Al (D019) + -Zr (Al) solid solution + amorphous phase −→ bulkamorphous phase.




Zr (101)

3822 26 30 34

Inte

nsity

(A

.U.)

d

c

b

a

Zr (100)

Zr (002)

42 46 50

a – 5 hb – 10 hc – 15 hd – 20 h

Zr (102)

Al (200)

Al (111)

(a)

2 – θ

(b)

2 – θ

3616 20 2824 32

Inte

nsity

(A

.U.)

4440 48

a – 25 hb – 30 hc – 45 h

Figure 3.57. XRD patterns showing a progressive structural change for different times when ele-mental powders of Zr and Al of 99.5 purity were milled in an attritor using 5 mm diameter balls ofzirconia with a ball to powder weight ratio of 10:1.

The mechanism of solid state amorphization during mechanical alloying hasbeen studied on the basis of experimental observations made on several alloysystems. One of the probable mechanisms, based on local melting followed byrapid solidification, has not found acceptance as evidence of melting could not beseen in experiments. The example of ball milling of elemental Zr and Al powdershas demonstrated that the amorphisation process is preceded by the enrichmentof the -Zr phase to a level of approximately 15 at.% Al. The solute concen-tration progressively changes during milling. The various stages encountered inthe course of amorphization can be explained in terms of schematic free energyversus concentration plots for the , the metastable D019, and the amorphousphases (Figure 3.58). With increasing degrees of Al enrichment, the free energyof the interface region gradually moves along the path 1-2 (Figure 3.58). Oncethe concentration crosses the point 2, it becomes thermodynamically feasible tonucleate the Zr3Al phase which has the metastable D019 structure. Although theequilibrium Zr3Al phase has the L12 structure, it has been shown (Mukhopadhyayet al. 1979) that the metastable D019 structure is kinetically favoured during theearly stages of precipitation from the -phase. This is not unexpected as the hcp




Fre

e en

ergy

Amorphous

Zr3 Al (DO19)

α

432

1

Zr 25Aluminium content (at.%) →

2′

3′

Figure 3.58. Schematic free energy – concentration plots in Zr–Al system for the , the metastableD019 and the amorphous phases illustrating the various stages encountered in the course of amor-phization.

structure and the D019 structure (which is an ordered derivative of the former)follow a one-to-one lattice correspondence and exhibit perfect lattice registry.

With further Al enrichment, as the concentration crosses the point 3, nucleationof the amorphous phase becomes possible. It is to be emphasized that the changein composition occurs gradually from the interface to the core of the particles,with the result that the amorphous phase starts appearing at interfaces while thecore remains crystalline. As the Al concentration in the powder particles crossespoint 4, each particle can turn amorphous by a polymorphic process. The observedsequence of solid state amorphization in the case of ball milling of elemental Zrand Al powders suggests the occurrence of amorphization by a lattice instabilitymechanism which is brought about by solute enrichment of the -phase beyond acertain limit (point 4 in Figure 3.58).

3.7.6 Radiation-induced amorphizationIt was discovered in the early 1980s that some intermetallic compounds undergoa crystalline to amorphous (C −→ A) transition under irradiation by energeticparticles. It was also recognized that the C −→ A transition results from displace-ment damage and not from ionization damage. Displacement damage occurs dueto the momentum transfer from the projectile particles (incident electron, ion orneutron) to atoms occupying lattice sites in the target material. For example, ifone considers an elastic collision between an incident particle of mass m andan atom of atomic weight M , the struck atom can receive the maximum energy,




Emax = 4Mm/En�M +m�2, where En is the energy of the incident particle. Whilean 1 MeV neutron can transfer several hundred keV energy to a target atom, alighter particle such as an electron of the same energy (1 MeV) is capable oftransferring only some tens of eV energy to a target atom. The threshold energy,Ed, for displacing an atom from its lattice site is in the range of 20–80 eV (� 25 eVfor Cu) and so an electron of 1 MeV energy can displace only one or two atomsfrom lattice sites. In contrast, a 1 MeV neutron can impart an energy of as muchas 200 000 eV to a Cu atom. Such an atom, called a primary “knock-on”, causesfurther damage by displacing secondary, tertiary, etc. “knock on”s. The damagestructure produced by a single energetic electron will, therefore, consist of a sin-gle (or two) vacancy – interstitial pair, the separation between the vacancy andthe interstitial being dictated by the length of the replacement collision sequence.Figure 3.59(a) and (b) shows schematically the defect production process due to anincident energetic electron. In contrast, a high-energy neutron (or an acceleratedion) produces a row of primary “knock-on”s, each of which triggers a series ofdisplacements in its path, as shown in Figure 3.59(c). The cascade of displace-ments finally terminates where the energy transferred to the target atoms fallsbelow the threshold energy for atomic displacement. At these termination points,the displaced atoms deposit their energies by thermal excitation of neighbouringatoms. Such a thermal excitation has a very short life span (10−12 s) and is known

I

V

e

V

I

e

V

I

(a) (b)

(c)

Ion

Figure 3.59. Schematic diagrams showing the defect production process due to an incident energeticelectron ((a) and (b)). In contrast, a high-energy neutron (or an accelerated ion) produces a row ofprimary “knock-on”s, each of which triggers a series of displacements in its path (c).




as a thermal spike. The region over which structural change occurs due to a cas-cade of displacements is known as a “displacement cascade”, the size of whichis determined by the mass and the energy of the projectile particle and the massand the threshold displacement energy of the target atoms. Since a large numberof atoms are expelled from the core of the cascade, this depleted region containsa high density of vacancies while the periphery of the cascade gets enriched ininterstitials. Often interstitials are produced at the end of a replacement collisionsequence chain which propagate along close packed directions from the core tothe periphery of the cascade.

The brief description of the radiation damage processes, pertinent to irradi-ation by electrons and relatively heavy particles (such as neutrons and ions),given here provides a background for gaining an understanding of why and howradiation damage induces the crystal to amorphous transformation primarily inintermetallics.

It has been observed that some compounds undergo amorphization while othersremain crystalline under similar irradiation conditions. The observed difference inthe susceptibility to amorphization under irradiation has led to the identificationof several empirical criteria which promote amorphization.

(1) Directional bonding such as ionic and covalent bonding is a requirement asevidenced from the fact that pure metals (with the exception of Ga) and disorderedsolid solutions cannot be amorphized under irradiation (Cahn and Johnson 1986).The melting point of Ga at ambient pressure is anomalously low (Tm = 302 K, theheat of fusion being 0.6 kcal/mol). This corresponds to a very small difference inthe free energies of the crystalline and amorphous states of Ga at low temperatures.The small requirement of enthalpy for the crystal to amorphous transition can bemet by the energy stored in the form of point defects which are produced underirradiation. Semimetals such as Si, Ge and Bi are also amenable to amorphizationunder irradiation – an observation which is consistent with this criterion.

(2) Intermetallic compounds which exhibit narrow solubility ranges in the phasediagram tend to amorphize under irradiation, while those with wide solubilityranges remain crystalline. Though this criterion is not universal, one can arguethat a narrow solubility range of composition corresponds to a high energy beingassociated with the anti-site defects created during irradiation. A large energystorage through these defects may eventually lead to amorphization.

(3) The presence of deep eutectics and of a number of line compounds in thesame region of a phase diagram is indicative of a relatively high stability of theliquid phase and of an inclination towards chemical ordering in the system. Thesethermodynamic features are usually associated with a tendency for amorphizationnot only under irradiation but also during rapid solidification (see Section 3.4).Binary alloys with constituents from the early and the late transition metals such




as Zr–Ni, Zr–Fe, Zr–Cu, Ti–Ni and, Ti–Cu exhibit phase diagrams which satisfythis criterion and are known to be amenable to amorphisation under irradiation.

Since the amorphous state is metastable with respect to the unirradiated crys-talline state, the occurrence of a crystal to amorphous (C → A) transformationunder irradiation is possible only if the increase in free energy, Girr, due toirradiation is greater than the difference between the free energies, GC→A, of thecrystalline and amorphous phases, i.e.

Girr > GC→A (3.73)

Some of the energy input from the incident radiation must, therefore, be storedpermanently (or for a time comparable to the amorphization time) in the material.Since most of the irradiation energy is dissipated as heat, a question arises asto the mechanism by which enough energy could be accumulated in the latticefor fulfilling the above criterion. There are two important mechanisms of energystorage within an irradiated material: first, by accumulation of point defects, bothvacancies and interstitials, much in excess of their equilibrium concentrations, andsecond, by the creation of anti-site defects produced by random displacements andby replacive collision sequences.

Let us now examine the relative merits of these two conceivable energy storagemechanisms. One can estimate the critical vacancy concentration required to raisethe enthalpy of a metallic crystal at T = 0 K by an amount equal to the heat offusion, Hf, at the melting temperature, Tm. Doyama and Koehler (1976) haveshown that the critical vacancy concentration, cv, required for raising the freeenergy of a crystal at T = 0 K to the free energy of an amorphous phase is given by

cv = Hv/Hf � 0�008 (3.74)

where Hv is the enthalpy of formation of a vacancy.The question that now arises is whether the steady state vacancy concentration

in a crystal can be raised to a level as high as 8×10−3 under irradiation. The steadystate vacancy concentration under irradiation depends on the rate of productionof vacancy – interstitial pairs, the rate at which they recombine and the rate ofdisappearance of vacancies at sinks such as surfaces, interfaces and dislocationloops. It has been found that even at low temperatures and at high displacementrates (∼103 displacements per atom), the steady state vacancy concentration doesnot exceed about 10−3. This is primarily because of the high probability of recom-bination of vacancies and interstitials as their concentrations increase with thedisplacement rate. Even when the enthalpy of formation of interstitials (whosesteady state concentration remains at one or two orders of magnitude lower than




that of vacancies) is added to that of vacancies, the free energy of the crystallinephase of a pure metal cannot be raised above the free energy of its amorphousphase by introducing point defects to a realistic maximum concentration.

The second mechanism of storing energy in the crystal lattice exposed to irra-diation is by the creation of anti-site defects. Obviously such a mechanism canoperate only in chemically ordered structures. Let us consider the example of theB2 (CsCl type) ordered structure. Here the lattice can be viewed as comprisingtwo interpenetrating simple cubic sublattices, denoted as and �. In the fullyordered condition, the -sites are occupied by only A atoms and the �-sites areoccupied by only B atoms. Each -site is surrounded by eight nearest neighbour�-sites and vice versa. In such a structure, vacancies in - and �- (denoted by and �) sites can be distinguished and their exchanges with atoms lead to a changein the order parameter, S.

Under irradiation, a high steady state vacancy concentration promotes atom–vacancy exchange process, as shown below:

A +v�� A� +v (3.75)

B� +v � B +v� (3.76)

A +B�� A� +B (3.77)

Equation 3.77 shows the exchange of A and B atoms from the to �-sitesand vice versa, leading to disordering in the forward reaction and to ordering inthe backward reaction. The overall reaction, however, involves two steps corre-sponding to two successive atom–vacancy site exchanges as shown in Eqs. 3.75and 3.76.

The energy barriers for the reactions of A and B atoms with vacancies onthe and � sublattices in a fully ordered (order parameter, S = 1) and a fullydisordered (S = 0) alloy are shown in Figure 3.60. Four different types of atom–vacancy interchange processes are indicated. In a completely disordered alloy,such exchanges do not lead to any change in the energy of the system and thereforein the activation barrier, Em, since such jumps are symmetric. In the partiallyor fully ordered alloy sublattice, changes of vacancies contribute to an alterationin the degree of order and lead to a decrease or increase in the energy of thesystem by an amount U . By linear interpolation, one obtains for the contributionat the saddle point, one half of this energy. The jump frequency can, therefore, bewritten as

�± = �o exp[−(Em ∓ U

2

)/kB

](3.78)




Em

S = 0

S = 1

Aβ + Bα = Aα + Bβ

u

Figure 3.60. Schematic diagram showing the energy barriers for the reactions of A and B atomswith vacancies on the and � sublattices in a fully ordered (S = 1) and disordered (S = 0) alloy.

where kB denotes Boltzmann’s constant. The lower value of the activation energyholds for a vacancy jump in which an atom changes from a wrong to the rightsublattice, i.e. for an ordering jump ("+). A higher value of activation energyrefers to a disordering jump ("−).

The overall rate of change of order in an irradiation environment can be writtenas (Banerjee and Urban 1984) a sum of three terms:

dS/dt = �dS/dt�c + �dS/dt�r + �dS/dt�t (3.79)

where (dS/dt)c and (dS/dt)r refer to the disordering rate due to the replacementcollision sequence and the random defect annihilation process, respectively, while(dS/dt)t is the rate of change of order parameter arising from the thermally activatedexchange of A and B atoms from the and � sites and vice versa:

�dS/dt�t = K+c�Ac

B −K−c

Ac

�B (3.80)

where c�A, c � , etc. represent concentration of A in �-site and B in -site, respec-tively, and the rate coefficients K+ (ordering) and K− (disordering) correspond tothe reverse and forward reactions shown in Eq. 3.77 and are expressed as

K± = ��±�Z� c v Z �c

�v

Z� cv +Z �c�v

(3.81)

where Z� is the number of nearest neighbour -site around a �-site.




(a) (b)

dS/d

t →

0 0.1 0.2 0.3–0.3

–0.2

–0.1

0

0.1

0.2

0.3

1150

1140

1130

T = 1120 K

Order parameter (S ) →

dS/d

t →

0 1.00.5–0.010

–0.005

0

0.005

0.010

T = 580 K

570

560

553540

Order parameter (S ) →

Figure 3.61. dS/dt versus S plots showing the influence of irradiation in creating a high concen-tration of anti-site defects for (a) thermal disordering and (b) irradiation disordering of a B2 alloy.

The influence of irradiation in creating a high concentration of anti-site defects(in other words, chemical disordering) can be illustrated by comparing the dS/dtversus S plots (Figure 3.61) for (a) thermal disordering and (b) irradiation disorder-ing of a B2 alloy. While the disordering temperature, Tc, is 1140 K (above whichdS/dt is negative for all values of S) under thermal disordering condition, thedisordering temperature under irradiation T ∗

c is 553 K as shown in Figure 3.61(b).For details on kinetics of order–disorder transformation in alloys under irradiation,readers may refer to Banerjee and Urban (1984).

The creation of anti-site defects (in other words, chemical disordering) plays avery important role in irradiation-induced amorphization, and the contribution ofpoint defects is relatively less important. However, a quantitative estimation ofthe contributions of these components requires experiments using different typesof radiations that have different replacement to displacement ratios and modellingof the irradiation-induced microstructural evolution using chemical rate equationsas well as a molecular dynamics approach.

Extensive experimental results on irradiation-induced amorphization of theLaves phase Zr�Cr�Fe�2 precipitates in a zircaloy-2 matrix under electron, ion andneutron irradiation are available. Some of the important results are summarizedhere with a view to making comparisons of the efficacy of different types ofradiation with regard to bringing about amorphization:

(1) Amorphisation occurs with all the three types of radiation when the irradia-tion (electrons, ions, neutrons) temperature is below a critical temperature, Tc. Theradiation doses required for amorphization under 1.5 MeV electron, 127 MeV Arion and neutron irradiation are shown in Figure 3.62.




Ion NeutronElectron60

40

20

0200 300 400 500 600

Temperature (K)

Dos

e (d

pa)

Figure 3.62. Irradiation dose (dpa) as a function of temperature (K) quantifying the radiation dosesrequired for amorphization under 1.5 MeV electron, 127 MeV Ar ion and neutron irradiation.

1.00

0.50

0.000.00 0.50 1.00

Fraction of irradiationtime

Electron

Crystallinefraction

Neutron

Figure 3.63. Crystalline fraction as a function of irradiation time showing the sharp drop in thedegree of crystallinity with electron irradiation.

(2) Under electron irradiation, amorphization occurs homogeneously withinthe entire volume of the precipitates. The degree of crystallinity drops sharply,as shown in Figure 3.63. Since electron irradiation is carried out on thin foils(10–20 nm thickness), interstitials migrate to the surface swiftly, allowing vacancysupersaturation to build up in the centre of the foil so that there is a substantialaccumulation of defect energy within the foil. Chemical disordering also con-tributes towards amorphization.




(3) Under neutron irradiation, the amorphization of Zr�Cr�Fe�2 precipitatesinitiates from the precipitate–matrix interface, suggesting that cascade (or ballistic)mixing in the thin layer close to this interface is responsible for amorphization.The ballistic mixing of the two phases at the interface can bring about a significantdeparture from stoichiometry which causes a large increase in the free energy ofthis thin layer. With the increase in radiation dose, the amorphous layer graduallypropagates towards the core of the precipitates, as shown in Figure 3.63.

3.8 PHASE STABILITY IN THIN FILM MULTILAYERS

When thin films are deposited on suitable substrate surfaces, they can exhibitcrystal structures which are metastable with respect to those associated with thesame materials in bulk form. Such metastable structures have been documentedin literature for many systems including metal/metal and metal/semiconductorsystems. Experimental observations on multilayers include instances where eitherone or both layers can exist in the metastable state. With the increase in introductionof multilayered nanostructures in a variety of applications, it is pertinent to examinethe reasons for the shift in the relative stabilities of the relevant phases withvariation in the thicknesses and in the thickness ratio of the constituents. Onceagain it is seen that the knowledge gained from recent researches on Ti- andZr-based multilayer systems has provided an insight into this important issue.To illustrate this point, experimental observations on Ti/Al multilayers reportedby Fraser and coworkers (Ahuja and Fraser 1994a,b, Banerjee et al. 1996) aresummarized here.

An examination of multilayered structures comprising alternate Al and Ti layers,each of several nm thickness, has revealed that in systems in which individuallayers are of equal thickness and the unit bilayer thickness �≥ 20 nm, both metalsassume their stable structures, namely fcc for Al and hcp for Ti. However, for �<xnm, both metals have the hcp structure and for intermediate thickness ranges (x <y<�) both show the fcc structure. Thus, with increase in �, the Al layer transformsonce from hcp to fcc, while the Ti layer transforms twice: from hcp to fcc andthen back to hcp. This behaviour of Ti is unexpected in two ways: first becauseit transforms from its stable structure (hcp) to a metastable structure (fcc) uponincrease in thickness, and second, the transition from fcc/fcc to hcp/fcc multilayersoccurs upon increase in thickness with a decrease in the misfit at the interfaces.

The observations showing structural changes in multilayers have been rational-ized by Banerjee et al. (1999) in the manner outlined here. A bilayer “unit system”comprising two layers and two interfaces (Figure 3.64) is defined and the specificfree energy of this “unit system” is expressed as




A

B

A

B

A

Figure 3.64. A schematic diagram showing a bilayer “unit system” comprising two layers and twointerfaces.

g = 2�+ �GAfA +GBfB�� (3.82)

where g is normalized by the area of the interface, � is the change in inter-facial energy, Gi�= G (metastable) − G (stable)) and fi are, respectively, theallotropic free energy change per unit volume of the reference phase for and thevolume fraction of the metal i in the reference bilayer. Equation (3.81) describesa thermodynamic potential surface that varies as a function of two independentvariables, f and �. The specific free energy of this biphase system, as describedin Figure 3.65(a), can be represented by a surface in the f −�−1 space, and theequilibrium structure will be the one with the lowest specific energy of forma-tion, g. The transformation from one biphase configuration to another (as e.g.Tifcc +Alfcc → Tihcp +Alfcc) occurs when the g surface for a given combinationof biphase (e.g. Tifcc +Alfcc) intersects the g surface for a different biphase (e.g.Tihcp +Alfcc). In this manner, the stability regimes of different biphase configura-tions can be depicted in biphase diagrams. It may be noted here that the interfacialenergy term, �, includes a chemical term arising out of dissimilar metals bond-ing at interfaces as well as a structural term due to the disregistry of the twocontiguous lattices at the interfaces. The strain energy and the bulk chemical freeenergy terms are included in Gi which is volume dependent.

Let us consider the case of the Al/Ti multilayers studied by Ahuja and Fraser(1994a,b) who reported a transition of Ti from the bulk, stable hcp form to ametastable fcc form below a critical value of ��= �∗

fcc/fcc� and both Al and Tibecoming hcp below another value of ��= �∗

hcp/hcp�. These two transformationsof the biphase (Al/Ti) system can be depicted conveniently by a constant volumefraction cut of the g�� f� surface, i.e. a plot of g versus �, as shown inFigure 3.65(b). For an arbitrarily large value of �, the lowest free energy of




hcp/hcpfcc/fcc

hcp/fcc

hcp Ti + fcc Alfcc Ti + fcc Al

(i)

(ii)

(iii)

2Δγ hcp/hcp

2Δγ fcc/fcc

0.0

Δg

hcp Ti+

fcc Al

ΔGAl/AlΔGTi / Ti

Bilayer thickness, λ

fcc/hcp

Volume fraction of Ti2 fTi

0.1

1/λ,

λ = b

ilaye

r th

ickn

ess

(nm

)

0.00.0

0.05

0.25

0.15

0.2

0.3

(1)

0.2 0.4

fcc/fcc

(2)

1

(3)

hcp/hcp

0.6 0.8

hcp/hcpfcc/hcp

fcc/fcc

(a) (b)

(c)

a

Ti

AlBA

BA

Ti 1 nm

λ∗hcp/hcp λ∗

fcc/fcc

Figure 3.65. (a) The specific free energy of a biphase system represented by a surface in the f −�−1

space; (b) g versus � plot showing a transition of Ti from the bulk, stable hcp form to a metastablefcc form below a critical value of ��= �∗

fcc/fcc� and both Al and Ti becoming hcp below anothervalue of ��= �∗

hcp/hcp�; and (c) HRTEM image showing the Al/Ti multilayered sample, with thethickness of the Ti and Al layers being 5.0 and 2�0 nm, respectively. The beam direction is parallelto <1120> (after R. Banerjee et al.).

the system is achieved if both metals assume their stable structures, i.e. GTi =GAl = 0 (line (i) in Figure 3.65(b)). Below � = �∗

fcc/fcc, the lowest free energyof the system corresponds to a bilayer configuration in which Ti has adopted ametastable (fcc) structure (line (ii) in Figure 3.65(b)). Below a still lower valueof ��= �∗

hcp/hcp�, the energy of the system is minimized by the transition of both




metals to the hcp structure (line (iii) in Figure 3.65(b)). The stability diagramshown in Figure 3.65(b) provides a physical basis for the formation of metastablephases in nanolayered materials. In the hcp Ti+hcp Al regime, Ti remains in itsstable state, making GTi = 0. For this regime, Eq. 3.82 is then reduced to

g = 2�hcp/hcp +GAlfAl� (3.83)

This means the intercept and the slope of the line (iii) are 2�hcp/hcp and GAlfAl,respectively. Based on a similar argument, the intercept and the slope of the line(ii) are 2�fcc/fcc and GTifTi. The line (i) corresponds to the combination of hcpTi and fcc Al, both in their respective stable states, implying that �, GAl andGTi are all equal to zero. It is, therefore, evident that the metastable states suchas fcc Ti and hcp Al are stabilized in nanolayered structures due to the negativevalues of � which more than compensate for the increase in G resulting fromthe formation of the metastable phases. Figure 3.65(b), which is consistent withexperimental observations on the hierarchy of biphase stability, originates fromthe following inequalities:

GAl < GTi and �hcpTi/hcpAl < �fccTi/fccAl < 0 (3.84)

Figure 3.65(b) is constructed for a fixed value of the volume fraction, fTi.Biphase diagrams for cases where both the volume fraction and the bilayer thick-ness are variable can be constructed in the fi − �−1 space. This is illustratedfor Ti/Al multilayers in Figure 3.65(a). In the absence of coherency, the bound-aries in this type of biphase diagrams are straight lines. Non-linear dependenceof composite moduli on volume fraction will introduce curvature in the varia-tion of �1/��, with f in the biphase diagram. The biphase diagram for the Al/Timultilayers has been constructed on the basis of the slopes of the lines (1), (2)and (3) respectively, which are given by GTi/2�fcc/fcc, GAl/2�hcp/hcp and(GAl +GTi�/2��hcp/hcp −�fcc/fcc).

Experimental data points are superimposed on the biphase diagram of the Al/Timultilayered system shown here. The symbols shown in the inset represent fccAl/fcc Ti, hcp Al/hcp Ti and fcc Al/hcp Ti multilayers observed for differentvalues of fTi and �. These experimental data points are consistent with the biphasediagram. Application of the biphase diagram concept in predicting the structure ofseveral multilayer systems such as Co/Cr, Zr/Nb and Ti/Nb has been successful(Thompson et al. 2003). A HRTEM image (Figure 3.65(c)) shows the Al/Timultilayered sample with the thickness of the Ti and Al layers being 5.0 and2.0 nm, respectively (Banerjee et al. 1996).




3.9 QUASICRYSTALLINE STRUCTURES AND RELATEDRATIONAL APPROXIMANTS

Quasicrystalline structures can be defined as those with long-range aperiodic orderand crystallographically forbidden rotational symmetries (e.g. 5-, 8-, 10- and 12-fold rotation axes). The observation that certain intermetallic compounds exhibitsharp diffraction peaks displaying the “non-crystallographic” icosahedral rotationalpoint group has generated a great deal of excitement. It is well known that thetranslational periodicity of atoms allows only certain rotational operations about anaxis which bring the arrangement back into registry with the unrotated assembly.For three-dimensional periodic crystals, the allowed rotation operations are two-,three-, four- and sixfold, about appropriately chosen axes. Taken together withother operations such as translations, reflections and inversions, these point groupoperations define all of the 230 space groups.

It was generally believed that only periodic arrangements of atoms can producesharp diffraction peaks. The discovery of a quasicrystalline structure in a rapidlyquenched Al–Mn alloy, schematic diffraction patterns from which are shown inFigure 3.66(a), has laid this myth (Shechtman et al. 1984) to rest. It is now realizedthat the occurrence of sharp Bragg diffraction peaks does not require the presenceof long-range periodic translational order, but rather of long-range positional order,which may or many not be specified by a periodic function in three dimension. Thispoint can be explained in one dimension by considering the Fibonacci sequence:

1�1�2�3�5�8�13�21�34# # #

where every term of the series is generated by the addition of the two immediatelypreceding terms. Let us consider two translation vectors S (short) and L (long)along a given direction and generate a series using an algorithm in which S isreplaced by L and L is replaced by LS in every successive series. It may be notedthat the repeat period grows as per the Fibonacci sequence of numbers (as shownin Figure 3.67). The ratio of the number of L to the number of S segments changesin recursive steps and finally converges to the “golden mean” (� = �1+√

5�/2 �1�618) as the repeat period grows to infinity. This illustrates how an array of twosegments, S and L, can be created in such a way that translational symmetry isabsent even when the one-dimensional sequence is extended upto infinity.

Penrose (1974) has shown that by using two specially shaped tiles, as designatedby a kite and a dart (Figure 3.68), it is possible to cover a plane with fivefoldsymmetry. There exist “matching rules” for the construction of the Penrose lattice.The ratio of the sides of these tiles (kite and dart) is given by �, as shown inFig. 3.68. Some important properties of the Penrose pattern are (a) orientationalorder, (b) quasiperiodic translational order and (c) self-similarity.

ElsevierUK

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1Color

Recto

242P

haseT

ransformations:

Titanium

andZ

irconiumA

lloys

79.2°

58.29°

37.37°

63.43

31.72

(b)

(2)

20.91°

010(1)

(6)

20.91°

(5)

(4)

10.81°

0τ21

13τ + 1τ

(3)

1τ20

10.81° 1 τ 0

13.28°

τ2τ41

13.28°

(a)

Figure 3.66 Schematic diffractive pattern from a quasicrystalline phase showing (a) fivefold, threefold and twofold symmetries andthe observed angles between the corresponding zone axes and (b) stereogram showing matching of observed symmetry elementswith those of an icosahedron.




••••••PERIOD: 1 A/B = 1/0 = 00

A A A A A(a)

(b)

(c)

(d)

(e)

• • • • • • •PERIOD: 2 A/B = 1/1 = 1

A AA B A B B

•• •••••PERIOD: 3 A/B = 2/1 = 2

AA A AB B

• • • • • • • •PERIOD: 5 A/B = 3/2 = 1.5

A AAA B B B

•••••••••PERIOD: 8 A/B = 5/3 = 1.66

A A A A AB B B

Figure 3.67. Fibonacci sequence of numbers.

T1

36°216°

H136°

T

72°

H

τ τ

(a) (b)

H72°

1T

1144°

72°

τ τ

72°

T

H

Figure 3.68. Schematic diagrams showing two specially shaped tiles (a) kite and (b) dart. The ratio,�, of the sides of these tiles (kite and dart) is also shown in the figure.

Mackay (1982) has extended Penrose tiling to three dimensions (3D) and hasdemonstrated that by making use of a pair of acute and obtuse rhombohedral tiles(as shown in Figure 3.69), filling of space with fivefold symmetry is possible.The interesting feature of 2D and 3D Penrose lattices is that the Fourier transformof their structures gives rise to sharp diffraction peaks displaying icosahedralsymmetry for the latter.

Quasilattices can be constructed with any arbitrary orientational symmetry andarbitrary quasiperiodicity. The existence of octagonal (Wang et al. 1987), decago-nal (Bendersky 1985, Chattopadhyay et al. 1985) and duodecagonal (Ishimasa et al.1985) quasicrystalline structures in various alloy systems has been established byexperiments.




(a)(b)

(c)

(d)(e)

Figure 3.69. Penrose tiling in three dimension (3D) demonstrating filling of space with fivefoldsymmetry by using a pair of acute and obtuse rhombohedral tiles.

The diffraction patterns shown in Figure 3.66(a) show fivefold, threefold andtwofold symmetries and the observed angles between the corresponding zoneaxes. The observed symmetry elements matched with those of an icosahedronas illustrated in the stereogram shown in (b). An inspection of the sequence ofdiffraction spots along a radial direction of the fivefold pattern in Figure 3.66 showsthat the ratio of the distances from the origin to any two bright spots is an irrationalnumber within reasonable experimental error. For icosahedral quasicrystals, thisirrational number is some power of the golden mean, � = �1+√

5�/2�, which arises




from the geometries of icosahedra, pentagons and decagons. Though translationalsymmetry is not present in these patterns, there exists an inflation symmetry. Forexample, the diffraction patterns in Figure 3.66 can be expanded or contracted bya factor of �3 to yield patterns indistinguishable from the originals.

In view of the observed icosahedral symmetry of the diffraction patterns fromsome quasicrystalline structures, the indexing of these patterns has been carriedout on the basis of six real space vectors defined by vectors, ei, that point fromthe centre to the vertices of an icosahedron (as illustrated in Figure 3.70). Thereciprocal lattice vectors are then defined by G1 =∑6

i=1 niei

ei =1√

1+ �2

⎡⎢⎢⎢⎢⎢⎢⎣

1 � 0−1 � 0

0 1 �� 0 1� 0 −10 1 −�

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎣ i

jk

⎤⎦ (3.85)

where ei are the real space basis vectors, ni are integers and go is a constantwhich determines the scale of the diffraction pattern. Unlike in the case of periodiccrystals where the diffraction pattern can be related to the lattice parameter, a,by a relation of the type 2�/a, no single fundamental length, go, can be chosenfor diffraction patterns from quasicrystals ab initio. This is also evident from thepresence of the inflation symmetry.

The reciprocal space of the icosahedral quasicrystal, instead of having a regularlattice of intensity maxima as observed for crystals, can have peaks arbitrarilyclose to any given peak by taking integer linear combinations of fundamental basisvectors. Thus the reciprocal space of the icosahedral quasicrystal is uniformly

(b)

(a)

1

2 3465

eiN

52346

1

ei⊥

Figure 3.70. Schematic diagrams showing six real space vectors defined by ei that point from thecentre to the vertices of an icosahedron.




dense. Within the Landau theory, one expects a decreasing hierarchy of peakintensities as the number of reciprocal lattice star vectors required to arrive at thepeaks increases. Since the peaks close to any given peak are obtained only bythe higher generation number, the intensities associated with these peaks are veryfeeble and are not distinguishable from the background. Peaks associated withlower indices are strong enough to produce a diffraction pattern with a discreteset of spots/reflections.

For an icosahedral quasicrystalline structure, each reciprocal lattice vectorrequires six indices for indexing as has been expressed in Eq. 3.85. The advantageof describing a three-dimensional quasiperiodic structure using a six index system(which corresponds to a six-dimensional space) arises from the fact that a projec-tion from a higher dimensional periodic lattice points on to a lower dimensionalspace can generate either a periodic or a quasiperiodic lattice, depending on theorientation of the projected space. This point can be explained in a simple mannerby taking the example of a projection from a 2D space to a 1D space.

Let us consider a 2D square lattice which is projected on a set of two perpen-dicular directions, designated as g1 and g2. If the direction g1 is drawn from anylattice point of the 2D structure along a rational direction (defined by tan =m/n,where is the angle between g1 and the X-axis of the 2D lattice), the g1 linewill intersect the lattice point with coordinates (n�m) and will periodically inter-sect lattice points (2n�2m), (3n�3m), etc. resulting in a periodic one-dimensionalstructure. In contrast, if tan is an irrational number, the g1, line starting from theorigin will not intersect any other lattice point. For projecting lattice points of the2D structure on to the line g1, we can arbitrarily select a strip indicated by a pairof broken lines parallel to g1 as shown in Figure 3.71. Projections of lattice pointslying within this strip on g1 produce an array of points which is quasiperiodic.Two segments, short and long, appear along g1, but the sequence L S L L S L SL L S # # # is such that one cannot identify a unit which has a repeated periodicappearance. If the strip width is increased more number of spots will appear on theprojected line g1, making the line uniformly dense as the strip width is enlargedto infinity. However, it can also be seen that two points which are very close onthe projected line will appear only if the strip width is increased to a very largeextent, which means that these points arise from two points (in 2D lattice) whichare widely separated along the g2 direction.

A Fibonacci sequence in 1D can be created by projecting a 2D square latticeon a g1 line which has an inclination, tan = �. In case tan is chosen to be arational quantity, the projection will result in a periodic sequence. A particularset of crystalline approximants, the Fibonacci rational approximants, is obtainedwhen the ratio of two consecutive numbers p and q of the Fibonacci sequence is




g ⊥

g ⏐⏐

Figure 3.71. Schematic diagram indicating that projection of lattice points of 2D structure on to theline g1. This can be done by arbitrarily selecting a strip marked by a pair of dashed lines parallelto g1.

chosen (tan = q/p), where

q/p = 1/1�2/1�3/2�5/3�8/5�13�8�21/13# # # (3.86)

It is to be emphasized that the rational ratio is not restricted to the Fibonacciseries since non-Fibonacci rational approximants have also been observed experi-mentally.

The compositional similarities between quasicrystals and their respectiveapproximants suggest similarities in their local atomic structures, substantiated bysimilarities in physical properties. Approximants are important for studies on theformation and stability of quasicrystals since they are amenable to establishedtheoretical tools. Reversible transformations between quasicrystals and relatedcrystalline approximant structures have been encountered in some cases in whichthe structural relationship between them could also be established.

The projection method for the 1D case can be extended to 2D and 3D and can beused for the construction of both the real lattice and the reciprocal lattice. The 3Dprojected lattice represents the icosahedral quasilattice. Just as the reciprocal latticeof a crystalline structure is generated by a basis of three vectors, the icosahedraldiffraction pattern is generated by a set of six reciprocal lattice vectors because ofits incommensurate nature. This means that all the g vectors in the reciprocal spaceof an icosahedral structure can be expressed in terms of linear combinations ofthe basis vectors of the reciprocal space. In this method, a 6D periodic reciprocallattice is projected orthogonally on to a suitably oriented 3D subspace.




For the icosahedral alloys, characterized by six fivefold axes, the 3D structureresults from the projection of points of a 6D hypercubic lattice that are containedin a 3D acceptance domain (analogous to the strip drawn with a pair of brokenlines for the 2D to 1D projection as shown in Figure 3.71) appropriately orientedwith respect to the 6D lattice. This acceptance domain is a triacontahedron in the3D space which is orthogonal to the physical space. For the icosahedral phase,the orientations of the acceptance domain and the physical space are specified bya 6×6 orientation matrix given by :

M = 1√2�3 +2

∣∣∣∣∣∣∣∣∣∣∣

1 −1 0 � � 0� � 1 0 0 10 0 � 1 −1 −�

−� � 0 1 1 01 1 −� 0 0 −�0 0 1 −� � −1

∣∣∣∣∣∣∣∣∣∣∣(3.87)

The upper three row vectors give the 6D coordinates of the three vectorswhich define the physical space while the lower three rows refer to vectorsspanning the orthogonal space. The analogues of the L and S segments generatedon the projection line from the points lying within the strip are oblate and prolaterhombohedra (Figure 3.69) which are the constituent tiles of the 3D Penrose lattice.The edge length of these rhombohedra, ar, plays a role analogous to the latticeconstant for periodic crystals and is, therefore, called the quasilattice constant.

As discussed in the case of the 2D to 1D projection, rational approximantscan be constructed from the 6D hypercubic lattice by a suitable selection ofthe acceptance domain. Fibonacci rational approximants are obtained when � isreplaced by a rational ratio q/p. Elser and Henley (1985) first demonstrated thatthe cubic -AlMnSi structure can be obtained by a 1/1 rational projection from thesame atomic decoration of the 6D hypercubic lattice used to define the icosahedralphases. They also made the first quantitative assessment of the similarities betweenthe bcc (or close to the bcc structure) in the case of the -AlMnSi phase andthe structure of the icosahedral phase. This analogy brings out the fact that theatomic arrangements of the related crystalline phases can be constructed from thesame building blocks which generate the icosahedral phases. The building blocksmay be taken either as icosahedral clusters of atoms or as a set of two types ofrhombohedral bricks which may also be derived from the decomposition of theicosohedral clusters.

3.9.1 Icosahedral phases in Ti- and Zr-based systemsQuasicrystalline structures have been reported most extensively in Al-based alloys.Icosahedral phase formation is now known to be quite common in Ti alloys




also, with Ti-based icosahedral phases constituting the second largest class ofquasicrystals (Kim and Kelton 1995). Alloys containing Ti and 3d transition metalsfrom V to Ni have received the maximum attention. Quasicrystalline phases havebeen reported in many of these alloys in the rapidly solidified condition. In mostcases, the microstructure produced consists of a finely distributed mixture ofquasicrystalline and crystalline phases. The presence of Si and O in these alloysoften plays a crucial role in stabilizing the icosahedral phase. A representative listof alloys based on Ti and Zr in which the formation of icosahedral phases hasbeen reported is given in Table 3.8.

A number of alloys in which both Ti and Zr are present have attracted con-siderable interest (Kelton et al. 1994, Kim and Kelton 1995, 1996) due to theirstrong tendency for icosahedral phase (i-phase) formation. Some of these (such asTi–Zr–Fe alloys) show localized diffuse scattering and significant diffraction spotshape anisotropy while some others (such as Ti–Zr–Ni alloys) are more ordered,as reflected in the sharp diffraction spots obtained from the icosahedral phases inthese alloys.

Table 3.8. Icosahedral phases in Ti- and Zr-based alloys

Alloy composition(approximate)

Processing/stability

Extent ofphason disorder

Quasilatticeparameter (nm)

Reference

Ti-TM-Si-O RSP/MS Very high 0.47–0.48 Libbert andKelton (1995)

Ti–Zr–Fe RSP/MS Very high 0.485–0.488 Kim andKelton (1995)

Ti53Zr27Ni20 RSP/MS Very low 0.512 Zhang et al.(1994)

Ti53Zr27Co20 RSP/MS Moderate 0.510 Kim andKelton (1996)

Ti41�5Zr41�5Ni17 AI/S Very low 0.517 Kim et al.(1997)

Ti63Cu25Al12 C/MS Low – Koster et al.(1996a)

Zr65−70Cu12−17Ni10−11Al75 C/MS Low – Koster et al.(1996b)

Zr65−70Cu10−15Ni10−13Pd7−10 C/MS Low – Murty et al.(2000a)

Zr65Cu12�5Ni10Al7�5M5 C/MS Low – Murty et al.(2000b,c)

M = Ag, Pd, Au, Pt

RSP, rapid solidification processed; AI, annealed ingot; C, crystallized glass; S, Stable; MS, Metastable.




As can be seen from Table 3.8 the icosahedral phases in Ti-based systems can begrouped into two classes. Those belonging to the first are associated with a smallerquasilattice constant (∼0.48 nm) and show arcing of diffraction spots and diffuseintensity distribution, suggesting the presence of phason disorder to a considerableextent. In contrast, members of the second group have larger quasilattice constants(> 0.51 nm) and exhibit sharp diffraction spots indicating a higher degree of phasonorder.

Out of the Ti-based icosahedral phases listed in Table 3.8, the i-phase in onlyTi41�5Zr41�5Ni17, which forms during annealing of arc-melted ingots, is stable. Inall the other cases, the i-phase is metastable, forming during rapid solidificationand disappearing during subsequent annealing.

The microstructures of most of Ti-based icosahedral phases are similar. Fineparticles of the i-phase are usually found dispersed in the amorphous matrix ofrapidly solidified alloys of compositions listed in Table 3.8. These particles alsoappear during the early stages of crystallization of amorphous alloys. In a numberof observations, i-phase particles have been found to be surrounded by the �-Ti(bcc) phase. Several crystalline approximants of the i-phase are found to coexistin partially crystallized samples. The occurrence and the structure of these phasesare briefly discussed in the following paragraphs.

The -1/1 rational approximant, a large unit cell bcc phase (lattice parameter =1.31 nm), consisting of Mackay icosahedra packed face to face along the <111>cubic direction, is frequently observed in Ti-Mn-Si and Ti-Cr-Si alloys. This phaseis believed to be the appropriate approximant to the Ti-3d TM-Si icosahedralphases.

The �-phase, a face-centred orthorhombic phase with a large unit cell (a =3�20 nm, b = 2�66 nm, c = 1�04 nm) appears frequently with the i-phase in Ti-Mn-Si, Ti-Mn-Fe-Si and Ti-Cr-Si alloys. Based on TEM studies, it has been inferredthat the �-phase can be constructed structurewise from Mackay icosahedra packedwith their vertices aligned along the c-direction of the unit cell.

A large unit cell fcc phase (a = 1.12–1.17 nm), presumably with the Ti2Nistructure, has been found in many Ti-Zr-Fe samples. Usually the fcc phase andthe i-phase are found in different regions of the sample, suggesting that these twophases evolve directly from the liquid phase under different conditions of cooling.

A hcp phase with a = 0.515 nm and c = 0.837 nm has been detected in Ti-Zr-Fe alloys containing relatively low concentrations of Ti. Energy dispersivespectroscopy has revealed the composition of this phase to be Ti41−49Zr21−29Fe26−31

with small amounts of Si (Kim and Kelton 1995). This phase is isostructural withMgZn2 type Laves phases. The observed intensity modulation in the diffractionpatterns from this Laves phase is similar to that associated with the i-phase,suggesting a similarity in the local atomic arrangements in the two.




As mentioned earlier, the i-phase is often seen to be surrounded by the �(bcc)-phase. In addition to the prominent bcc reflections, spots at 1/3 and 2/3positions corresponding to the fundamental bcc reflections are recorded. Theseadditional spots suggest the presence of the �-phase in the �-matrix. The presenceof the �-phase is not unusual in �-Ti alloys which contain sufficient amounts of�-stabilizing elements.

Unlike in the cases of Ti-TM-Si-O and Ti-Zr-Fe alloys, the i-phase formingin the Ti-Zr-Ni system is stable. This has been demonstrated conclusively byforming the i-phase by annealing as cast ingots of a Ti45Zr38Ni17 alloy whichinitially contained only a C14 Laves phase and a hexagonal solid solution phase.The crystalline approximant which forms along with the i-phase in Ti-Zr-Ni is thebcc W-phase (which is a 1/1 rational approximant). The composition range of theW-phase is Ti40−50Zr31−42Ni16−19. XRD peak intensities from this phase are quitedistinct from those pertaining to the -1/1 approximant described earlier. Based onthe powder diffraction data and the number of atoms per unit cell (168.5, estimatedfrom measured density), Kim et al. (1997) have inferred that the W-phase has aBergman type structure similar to that encountered in Al-Li-Cu and Al-Mg-Znalloys.

In this context, it is worth mentioning that two basic cluster types, both havingicosahedral symmetry, are used in describing atomic positions in icosahedralphases. The Mackay cluster, as shown in Figure 3.72 is a double-shell icosahedralcluster, with atoms decorating the vertices of the inner and outer icosahedra andthe midpoints of the edges of the outer icosahedron (Mackay 1962). While the -1/1 rational approximant in Ti-TM-Si-O alloys is based on the Mackay cluster,the W-phase in Ti-Zr-Ni alloys is based on the Bergman cluster which is also adouble-shell icosahedral cluster; however, the midpoints of the faces of the outer

Al

Mn

Figure 3.72. The Mackay cluster – a double-shell icosahedral cluster with atoms decorating thevertices of the inner and outer icosahedra and the midpoints of the edges of the outer icosahedron.




icosahedron are occupied, instead of the edge centres as in the case of the Mackaycluster. The i-phases in these two types of systems can thus be grouped into twodistinct classes, the former containing Mackay clusters and the latter constituted ofBergman clusters. Kim et al. (1997) have classified i-phases based on a correlationbetween the measured quasilattice constant, aq, and the atomic separation, as,calculated from the measured i-phase densities. By this method, all three Bergmantype i-phases, including i-(Al-Li-Cu), i-(Al-Mg-Zn) and i-(Ti-Zr-Ni) have the ratioaq/as � 2, while for i-phases which have Mackay type clusters, aq/as = 1�85.

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Chapter 4

Martensitic Transformations

4.1 Introduction 2604.2 General Features of Martensitic Transformations 261

4.2.1 Thermodynamics 2614.2.2 Crystallography 2664.2.3 Kinetics 2774.2.4 Summary 280

4.3 BCC to Orthohexagonal Martensitic Transformation In Alloys Basedon Ti and Zr 2814.3.1 Phase diagrams and Ms temperatures 2824.3.2 Lattice correspondence 2894.3.3 Crystallographic analysis 2944.3.4 Stress-assisted and strain-induced martensitic transformation 324

4.4 Strengthening Due to Martensitic Transformation 3264.4.1 Microscopic interactions 3294.4.2 Macroscopic flow behaviour 335

4.5 Martensitic Transformation in Ti–Ni Shape Memory Alloys 3394.5.1 Transformation sequences 3404.5.2 Crystallography of the B2 → R transformation 3424.5.3 Crystallography of the B2 → B19 transformation 3424.5.4 Crystallography of the B2 → B19′ transformation 3454.5.5 Self-accommodating morphology of Ni–Ti martensite plates 3474.5.6 Shape memory effect 3524.5.7 Reversion stress in a shape memory alloy 3564.5.8 Thermal arrest memory effect 360

4.6 Tetragonal � Monoclinic Transformation in Zirconia 3624.6.1 Transformation characteristics 3624.6.2 Orientation relation and lattice correspondence 3634.6.3 Habit plane 366

4.7 Transformation Toughening of Partially Stabilized Zirconia (PSZ) 3694.7.1 Crystallography of tetragonal → monoclinic transformation in

small particles 372References 373






Chapter 4

Martensitic Transformations

Symbols and AbbreviationsF : Helmoltz free energyT : TemperatureP: PressureTo: Equilibrium transformation temperatureMs: Martensite start temperature�: Molar volumeas: Surface area�f : Chemical free energy change�a: Applied stress tensor�: Microscopic strain� : Interfacial energy per unit areaEp: Total energy dissipated in plastic flowE′: Elastic modulus of parent phase

Ms��: Stress required for martensitic transformation�i: Principal strain with directioni: Principal distortions in the principal directionBi: Bain strain matrixR: Rigid body rotation matrixS: Total shape strain matrixP: Lattice invariant shear matrixE: Total distortion matrixx: Compositionx: Distance

L, B, T : Plate dimensions (length, breadth and thickness): Twin fractionkB: Boltzman constantMB: Martensite burst temperatureMf : Martensite finish temperature

R: Gas constantTan �: Magnitude of shear

�: Poison ratioeTij: Stress free transformation strainli: Direction cosine of the position vector

259




V : Volume of the inclusionUj (s): Displacement at point xDSA: Degree of self-accommodation

IPS: Invariant plane strainLIS: Lattice invariant shear

b: Burgers vectorM�

s : Temperature above plastic yield starts after martensitictransformation

Md: Stress required for stress-assisted martensite nucleation��: Difference between the flow stresses of ′ and � ′ : Flow stress of ′

� : Flow stress of �p: True plastic strainTxy: Stress required to move a dislocation out of a small angle

boundaryh: Spacing between the dislocations�: Shear modulus: Number of dislocations

12: Geometric slip distances for and ′

�stat: Independent component of flow stress�m: Athermal component of flow stress�∗: Thermal component of flow stressA∗: Activation areaT : Homogeneous deformation matrixAs: Austenite start temperatureAf : Austenite finish temperature� �: Reversion stressT�: Reversion temperature

4.1 INTRODUCTION

Martensitic transformations take place in numerous materials. Evidences of theiroccurrence have been found in several pure metals such as Fe, Co, Hg, Li, Ti, Zr,U and Pu, in many ferrous and non-ferrous alloys and in several oxides and inter-metallic compounds such as ZrO2, BaTiO3, V3Si, Nb3Sn, NiTi and NiAl. Someyears ago, the word “martensite” was used solely to describe a microconstituent inquench hardened steels. Bain (1924) put forward a mechanism for the transforma-tion of the face centred cubic austenite to the body centred tetragonal martensite



Martensitic Transformations 261

in steels in which the structural change was considered to be brought about by ahomogenous deformation of the parent lattice. It was implicit in this descriptionthat the transformation did not involve random or diffusive atom movements andthat it resulted from only small relative displacements of neighbouring atoms. Thefact that a similar mechanism is operative in a large number of solid state phasetransformations has led to a proliferation of the use of the terms “martensite”or “martensitic” in a much wider sphere. As indicated in Chapter 3, martensitictransformations are grouped in the general class of displacive transformations andbelong to that subset which involves the operation of a lattice deformation. Thecharacteristic features of martensitic transformations are described in the followingsection to provide a background for discussions on martensitic transformations inalloys, intermetallics and ceramics based on Ti and Zr.

4.2 GENERAL FEATURES OF MARTENSITICTRANSFORMATIONS

Martensitic transformations are characterized by a number of thermodynamic,kinetic, crystallographic and mechanistic features. Experimental observables per-taining to each of these are needed to unequivocally qualify a transformation tobe martensitic in nature. This section is devoted to a brief discussion on theseaspects.

4.2.1 ThermodynamicsThe first and foremost condition of a martensitic transformation is that the prod-uct phase inherits the composition of the parent phase. For a single componentsystem like a pure metal, the driving force for the transformation to occur can berepresented with the help of Helmholtz free energy (F ) versus temperature (T )or pressure (P) plots. Taking the example of pure Fe, the chemical free energychange, �F , accompanying a transformation from the austenite to the ferrite phase,can be expressed as follows:

�F = −1202+2�63×10−3T 2 −1�54×10−6T 3 cal/mol �200K< T < 900 K�

�F = −1474+3�40×10−3T 2 −2�00×10−6T 3 cal/mol �800K< T < 1000 K��

(4.1)

While the former expression is due to Kaufman and Cohen (1956, 1958), thelatter was proposed by Owen and Gilbert (1960). This change in free energy




1800

1600

1400

1200

1000

800

600

4000 100 200 300 400

Temperature (K)

ΔF F

e α→

γ (c

al/m

ol)

ΔF α→γ

= 1202–2.63 × 10–3T 2

+ 1.54 × 10–6T 3

Fe

500 600 700 800

Figure 4.1. The free energy change accompanying the → � in pure Fe as a function oftemperature.

keeps on increasing with a lowering of the temperature from the austenite/ferriteequilibrium transition temperature, 1183 K (Figure 4.1).

When a multicomponent system is considered, the chemical driving force isgiven by the drop in the free energy of the system as the parent phase transformsinto the product, retaining the initial chemical composition. In this sense, thesystem behaves as if the transformation is occurring in a single component system.This point can be illustrated by taking the example of the Fe–Ni alloy system, thephase diagram and the free energy – composition diagrams which are shown inFigure 4.2(a) and (b), respectively. The equilibrium condition between the austeniteand the ferrite phases can be identified by constructing a common tangent whichlocates the compositions of the two phases in equilibrium. When such partitioningof the alloying element is suppressed by a rapid quench, the composition, xo, can bedefined at which the integral molar free energies of the two phases are equal at To.A composition-invariant transition from austenite to ferrite is thermodynamicallypossible only below this To temperature. The composition dependence of To issuperimposed on the phase diagram in Figure 4.2(a).

Martensitic transformations, like any other first-order transformation, do notstart at To, where �F = 0 but are initiated when some supercooling is provided.The temperature at which a martensitic transformation “starts” is known as the Ms

temperature. The difference, To −Ms, indicates the extent of supercooling required




1200

1100

1000

900

800

700

600

500

400

300

200

100

0

Tem

pera

ture

(K

)

100

x (Ni at. %)30 40 5020

αα + γ

α + γ

γ

Ad

To (calc.)Md

As

Ms

½ (Ms + As)

(b)Fe

Ms

ΔF α

→γ

= F

γ –

Fα′

ΔF γ

→α′

= F

α′ –

Fγ

cal/m

olcal/m

ol

A s

x = 0.30x = 0.25

x = 0.20

x = 0.15x = 0.10

x = 0.05

x = 0

(a)Temperature (K)

12001000800600400200

+600

+400

+200

0

–1200

–1000

–800

–600

–400

–200

–600

–400

–200

0

+200

+400

+600

+800

+1000

+1200

x = 0.35

Figure 4.2. (a) Experimental and theoretical determination of To in the Fe–Ni system. (b) Chemicalfree energy change accompanying the martensite transformation in the Fe–Ni system.

to initiate the transformation. For a number of ferrous alloys, this difference isabout 200 K, while for the alloys based on Ti and Zr, it is much lower (≈50 K).

The requirement of supercooling arises from the necessity of overcoming thefollowing energy components which oppose the transformation: (a) the interfacialenergy between the martensite and the parent matrix, (b) the elastic energy storedin the martensite – parent assembly to accommodate the shape change and thevolume change accompanying the transition, (c) the energy dissipated in plasticdeformation of both the martensite and the parent phases and (d) the driving forcerequired for the rapid propagation of the martensite interface.

Apart from the chemical free energy change, �F , an applied stress can contributeto the driving force for a martensitic transformation. This is amply demonstratedin several alloy systems where the transformation can be induced at a temperaturehigher than the Ms temperature by applying stress. This effect can be attributedto the interaction of the applied stress field with the shape strain involved inthe formation of a martensite plate. Provided the interaction has the correct sign,the formation of a plate will relieve the potential of the applied stress field. Theformation of a small region of martensite in the presence of a stress field willrelease a small amount of mechanical energy, which may be positive or negativedepending on the nature of the stress field and the orientation of the plate.




Considering the driving forces arising due to the chemical free energy changeand the applied stress and the restraining forces associated with the four factorslisted earlier, the following energy balance expression can be written for theinitiation of a martensitic transformation:

v�f +A1� �≥ As� + vA2E′�2 +Ep (4.2)

where the volume and the surface area of the martensite plate are denoted by v andas, respectively, �f is the free energy change per unit volume of the martensite, �a

and � are tensors representing, respectively, the applied stress and the macroscopicstrain associated with the transformation, � is the interfacial energy per unit surfacearea of the plate, E′ is the elastic modulus of the parent phase, Ep is the totalenergy dissipated in plastic flow and in imparting a high velocity to the martensiteinterface and A1 and A2 are dimensionless geometrical factors.

The specific interfacial energy, � , between the martensite and the matrix dependson the extent of coherency at the interface. Since atom transfer across the transfor-mation front (interface) occurs through coordinated and highly disciplined atommovements (like regimented movements during a change in a military formation),the maintenance of coherency at the interface becomes a necessary condition. Thepresence of an array of dislocations at the interfaces arises out of a geometricalnecessity as will be discussed in a later section.

The changes in the shape and in the specific volume associated with the for-mation of a martensite plate of a given geometry in the matrix result in thedevelopment of a strain, both within the plate and in the matrix. The partitioningof the strain between the two phases, however, depends on the respective val-ues of their elastic moduli. The strain so developed is accommodated either byan elastic deformation of the assembly or by a combination of plastic flow andelastic deformation. The latter situation prevails when the accommodation stressdeveloped exceeds the flow stress in either of the phases.

The driving force for the martensitic transformation must exceed the correspond-ing restraining force for the growth of the transformation product. The differencebetween the driving and the restraining forces is utilized in moving the interfacialdislocations. For a conservative movement of dislocations, the Peierls stress andthe other internal stresses opposing their motion need to be overcome. The growthvelocity of martensite interfaces, measured by the rate of change of electricalresistance of samples undergoing a martensitic transformation, has been found tobe about one-third the velocity of elastic waves in the parent phase. This growthvelocity was also found by Bunshah and Mehl (1953) to be essentially constant atall temperatures between 73 and 293 K for both types of martensites which showathermal and isothermal kinetics for overall growth. The facts that the growth




velocity is very high even at cryogenic temperatures and that it is independent oftemperature suggest that the growth process is athermal. Rapid growth is com-monly encountered when the transformation is driven by large driving forces andis thus adiabatic. The interface can, therefore, accelerate rapidly up to its limitingvelocity, which is of the same order as the velocity of crack propagation or of twinformation. Such a growth of an isolated plate can cause plastic deformation in thematrix which, in turn, results in the loss of coherency at the interface and in thenucleation of fresh plates in the adjoining untransformed regions. The growth ofthe primary plate ceases at this point. Once the coherency at the interface is lost,it is not possible to reactivate its motion by changing the driving force (either byheating/cooling or by deformation).

In contrast to the scenario described above, a martensite plate can reach athermoelastic equilibrium when it assumes its full size under a given conditionof temperature and applied stress. This can happen if the driving force (havingchemical as well as mechanical components) exactly balances the restrainingforce arising from the surface energy and the elastic strain energy. The basicrequirements for attaining a thermoelastic equilibrium are, therefore, that the elasticstress limits in the parent and the product phases should be high and that the shapestrain associated with the transformation should be small. As the strain energybuilds up with the growth of a plate, a thermoelastic equilibrium is establishedwhen the plate assumes a certain critical size. In such a situation, the interfaceretains complete coherency and is amenable to movement in either direction,leading to the growth or the shrinkage of the plate, depending on the magnitudeof the driving force.

As pointed out earlier, a supercooling to the extent of To −Ms is needed to inducespontaneous nucleation of martensite plates. Martensite plates can, however, benucleated at temperatures higher than Ms if additional driving force is providedby an applied stress. The influence of such an applied stress on the martensitictransformation can be explained by using a schematic diagram (Figure 4.3) whichwas originally presented by Olson and Cohen (1972). It can be seen that thestress required for martensite formation increases linearly as the temperature risesfrom Ms to Ms��; beyond this point, plastic deformation of the parent phase setsin. In the temperature range, Ms < T < Ms��, the applied stress complementsthe chemical driving force which decreases linearly with increasing temperature.Martensite nucleation in this temperature range is stress-assisted. At temperatureshigher than Ms��, the elastic driving force derived from the applied stress isinadequate to satisfy the requirement for martensitic nucleation. As the appliedstress exceeds the flow stress of the parent phase, plastic deformation causes thecreation of fresh martensite nuclei and the formation of strain-induced plates. Thefeatures shown in Figure 4.3 have been explained in detail in Section 4.3.4.




σ1

σ2

c

b

aA

pplie

d st

ress Stra

in-in

duce

d

trans

form

atio

n

0.2% proof stressof austeniteSt

ress

-ass

iste

d

trans

form

atio

n

Ms T1 M

σ sT2 Md

Temperature

Figure 4.3. Schematic diagram showing the critical stress for martensite formation in a typicalferrous alloy as a function of temperature.

4.2.2 CrystallographyThe crystal geometry associated with martensitic transformations in various sys-tems has been found to be governed by invariant plane strain (IPS) considera-tions which will be discussed in this section. The validity of the IPS criterionin predicting the transformation geometry is so overwhelming that sometimes atransformation is identified as martensitic purely on the basis of this geometri-cal criterion. This approach, however, is currently being questioned since somediffusional transformations have also been shown to exhibit geometrical featurespredictable from IPS considerations. In this section, the essential points concerningthe phenomenological theory of martensite crystallography, developed indepen-dently by Wechsler et al. (W-L-R)(1953) and by Bowles and Mackenzie (B-M)(1954), will be discussed.

The important geometrical features of martensitic transformations are listedbelow:

(1) The formation of a martensite plate in a grain of the parent phase createsupheavals (surface relief) on a polished reference surface of the parent grain.This is illustrated in a schematic drawing in Figure 4.4 which shows themacroscopic shear produced in a parent crystal in which a martensite plateis formed. Observations on the displacement of reference lines drawn on thesurface of the crystal indicate that all reference straight lines are transformed




C

Austenite

Austenite

Mar

tens

ite

B

M1

M4

M3

M2

P

AP1

P2

P3

P4

M

P

D

Figure 4.4. The shape deformation due to formation of a martensite plate. Surface M1M2M3M4

remains plane and tilted about M1M2 and M3M4. The straight line AD marked on austenite istransformed into ABCD, where the segment BC within the martensite plate remains a straight lineafter the transformation. There is no discontinuity at points B and C, which are at the martensite–austenite interface, indicating that the interface is undistorted and unrotated.

into straight lines and all reference planes into planes in the product marten-site. This implies that the transformation strain is linear and, therefore, canbe expressed in the form of a matrix. Such a transformation is describedmathematically as an affine transformation. The fact that no discontinuity isproduced at the interface plane separating the martensite plate and the matrixindicates that the interface plane (habit plane) is an undistorted and unrotatedplane (invariant plane).

(2) The habit plane which is seen to be characteristic of a specific transformationis generally irrational.

(3) A precise reproducible orientation relation is invariably present between theparent and the martensite crystals, as revealed from diffraction experiments.

(4) Martensite plates very often contain a periodic arrangement of internal twins.

The concept of lattice strain which came from the suggestion of Bain (1924)is illustrated schematically in Figure 4.5 wherein the fcc austenite is convertedto the body centred cubic (bcc) ferrite by a single “upsetting” process in whichthe dimensions of the fcc unit cell are altered to those of the bcc unit cell by ahomogeneous deformation of the parent lattice requiring only small shifts in theatom positions. The three vectors chosen to define unit cells in this descriptionare mutually perpendicular before and after the lattice transformation. In general,




ao

a o / √2

c

a

(b) (c)

(a)

(101)A → (112)M

X′2

X2X′1

X1

X3, X′3 [101]A → [111]M

Figure 4.5. Lattice correspondence and lattice deformation for the fcc to bct austenite–martensitetransformation in Fe alloys.

in a homogeneous deformation, it is always possible to select three mutuallyperpendicular vectors (say, X1, X2 and X3) which remain perpendicular afterthe deformation, and these are called the principal axes of deformation. Whena volume of the parent phase, represented by a unit sphere, is subjected to ahomogeneous strain, it is transformed into an ellipsoid. The construction of strainellipsoids (Figure 4.6) illustrates the conditions for the homogeneous strain tohave at least one plane undistorted. When the principal strains associated with thehomogeneous strain are all positive or all negative, the strain ellipsoid does notintersect the unit sphere at all, implying that not a single vector remains undistortedby the homogeneous strain. Such a situation is shown in Figure 4.6(a). For aplane to remain undistorted, the necessary and sufficient conditions are that oneof the principal strains should be zero while the other two should be, respectively,positive and negative. If the principal strains, �1, �2 and �3, are such that �1 is zero,




η3 = l+ε3

η2 = l+ε2

lB

A

A′B′

(b)

X3 X3

X2X2

(a)

Figure 4.6. Deformation of a unit sphere into an ellipsoid by homogeneous lattice strain (Bainstrain) (a) 1�2�3 > 1 and (b) 1 = 1, 2 < 1, 3 > 1. The details are explained in the text.

�2 is negative and �3 is positive (as illustrated in Figure 4.6(b)), the strain ellipsoidwill touch the sphere at the point of intersection of the X1 axis with the unit sphereand will intersect the sphere at two points A′ and B′ on the plane containing theX2 and X3 axes. The planes defined by OA′ × X1 and OB′ × X1 vectors remainundistorted though they are rotated from their original positions, defined by theplanes OA×X1 and OB×X1 vectors.

An examination of the Bain strain necessary for the deformation of the parentlattice into the product lattice reveals that, in general, the Bain strain or latticestrain alone does not satisfy the aforementioned conditions which ensure at leastone undistorted plane. Moreover, the macroscopic shape strains measured from thesurface relief observations in several martensites do not match with the respectiveBain strains. It is because of these two factors that the concept of a second shearwas invoked in the martensite crystallography. While the Bain strain is responsiblefor bringing about the change in the lattice, the second shear, observed as thelattice invariant shear (LIS), superimposed on the Bain strain makes the total shearsatisfy the undistorted plane condition.

The necessity of a second shear can be explained by citing a specific example.The lattice or Bain distortion, B1, necessary for transforming the (fcc) austenite




(with lattice parameter ao) into the (bct) martensite (with lattice parameters a andc) can be expressed in terms of a matrix:

B1 =⎡⎣1 0 0

0 2 00 0 3

⎤⎦ (4.3)

where 1 = 2 = a√

2/ao and 3 = c/ao

Substituting the lattice parameter values for a carbon steel, one finds that atensile strain of 12% in all directions perpendicular to the c-axis (X3-axis, markedin Figure 4.5) and a compression of 17% along the c-axis are required for upsettingthe lattice from fcc to bct. It is obvious that this lattice deformation cannot satisfythe condition for having an undistorted plane. Therefore, the total macroscopicshear, which is experimentally shown to be an IPS, must consist of additionalcomponents which, in conjunction with the lattice shear, satisfy the IPS condition.

The same conclusion was arrived at, before the phenomenological crystallo-graphic theory (W-L-R and B-M) was introduced, through an elegant experimentby Greninger and Troiano (1949). They experimentally determined the magnitudeof the macroscopic shear from observations on the surface relief produced dueto the martensitic transformation in an Fe–22% Ni–0.8% C alloy. They noticedthat the experimentally measured macroscopic shear, when applied to the parentaustenite lattice, did not generate the martensite lattice.

In order to account for the observed difference between the macroscopic strainand the lattice strain, an LIS (either slip or twinning) has been introduced as acomponent of the total strain. The phenomenological theory of martensite crys-tallography is based on the postulate that the habit plane (the interface separatingthe parent and the product phases) is not an atomistically flat plane which remainsinvariant on a microscopic scale during the transformation. Misfits between thetwo structures develop and the accumulated misfits periodically get corrected toestablish an average or macroscopic fit. The essence of the theory can be describedby a set of schematic drawings (Figure 4.7). Let us consider the transformation ofthe two-dimensional lattice shown in Figure 4.7(a) to that shown in Figure 4.7(b),the corresponding unit cells being indicated by thick lines. The required latticestrain which brings about the change in the lattice also produces a shape strain;this is reflected in the rotation of the vector AB in the parent lattice to the vectorA′B′ in the product lattice. The magnitude of the vector A′B′ can be brought backto the magnitude of the vector AB without changing the product lattice by theintroduction of an LIS either by slip or by twinning. The geometries associatedwith these options are illustrated in Figure 4.7(c) and (d), respectively. In the caseof the LIS being provided by slip, the product martensite plate consists of a single




Initial crystal After latticedeformation

Lattice deformationfollowed by slip

shear

Two lattice deformationsleading to twinrelated regions

A

B

A′

B′

A′

B′

A′

B′

(a)

(c) (d)

(b)

Figure 4.7. Schematics showing (a) untransformed crystal, (b) after undergoing a lattice deformation,(c) the additional effect of a slip shear and (d) crystal having alternately twined regions. (c) and (d)show that a combination of lattice deformation and lattice invariant deformations (slip or twin) canmake the habit plane an invariant plane.

variant of the martensite crystal. However, if the LIS is provided by twinning, twotwin-related martensite variants form within a single martensite plate. In order tobring the vector A′B′ into coincidence with the original vector AB, an additionalrigid body rotation is necessary. The total macroscopic shape strain (S), which hasto satisfy the IPS criterion, is, therefore, conceptually divided into components,namely the lattice strain (B) which is responsible for changing the parent latticeinto the product lattice, the LIS (P), which, on being superimposed on the lat-tice shear, establishes an undistorted plane, and a rigid body rotation (R), whichensures that the undistorted plane is unrotated as well, S = RPB. In the case oftwinning as the LIS, it is necessary to satisfy another symmetry criterion. The twotwin-related orientations in the product phase evolve from a single parent phasecrystal. It is, therefore, necessary that crystallographically equivalent lattice strainsare operative in the adjacent regions which transform into a pair of twin-relatedproduct orientations of the product crystals. The formation of such a configurationis also expected from the consideration of symmetry breaking in a phase trans-formation. If the number of symmetry elements of the parent crystal gets reduceddue to a transformation process, there is a general tendency for the restoration




B2

B2

D1

A1

A1

D2 B1

B1

C2

C2

C1

C1

Mirror plane

CB

D

2

1

φ2 φ1

A2

A2

Figure 4.8. Schematics showing restoration of the symmetry in a macroscopic sense through thecreation of a number of crystallographic variants.

of the symmetry in a macroscopic sense through the creation of a number ofcrystallographic variants. This can be illustrated in a two-dimensional construction(Figure 4.8) in which a parent square lattice ABCD is transformed into two equiv-alent rectangular lattices, A1B1C1D1 and A2B2C2D2. When these two rectangularregions are rotated to bring them into coincidence along their diagonals, A1C1 andA2C2, a twin is created where the twin plane is derived from a mirror plane in theparent crystal. In fact, the mirror symmetry of the parent crystal on this plane islost due to the transformation, and the formation of the twinned product crystalstends to restore, at least partially, the lost mirror symmetry. The relative volumesof the two orientations, usually expressed in terms of the ratio of the thicknessesof the adjacent twins, are determined by the requirement of the lattice invariantdeformation necessary to satisfy the IPS condition.

Referring back to the transformation described in Figure 4.5, the lattice (Bain)distortions, B1, associated with the two adjacent twin-related variants can berepresented in the (i1� j1� k1) and (i2� j2� k2) principal axes systems respectively bythe matrices

B′1 =

⎡⎣1 0 0

0 2 00 0 3

⎤⎦ and B′

2 =⎡⎣1 0 0

0 3 00 0 2

⎤⎦ (4.4)

These two matrices can then be expressed in the axis system of the parent crys-tal, (i� j� k), by the standard similarity transformation procedure, which involves




rotations of the axes systems from the basis of the martensite crystal to that of theparent crystal. The Bain strains, B′

1 and B′2, can be represented in the axis system

(i� j� k) as B1 and B2, respectively:

B1 =

⎡⎢⎢⎢⎢⎣1 +2

22 −1

20

2 −1

21 +2

20

0 0 3

⎤⎥⎥⎥⎥⎦ and B2 =

⎡⎢⎢⎢⎢⎢⎣

1 +2

20

2 −1

20 3 0

2 −1

20

1 +2

2

⎤⎥⎥⎥⎥⎥⎦

(4.5)

In order to bring the two adjacent regions into twin-related orientations, it isnecessary to introduce rigid body rotations, �1 and �2, to the regions marked1 and 2, respectively, in Figure 4.8. Therefore, �1 and �2 describe the rotationsof the principal axes of the pure distortions in regions 1 and 2 relative to an axissystem fixed in the untransformed parent phase.

Figure 4.9 shows an arbitrary vector r (represented by the straight line OV)in the parent phase which becomes a zigzag line OA′B′C′D′U′V′ in the twinned

A′B′

C′

1

2

1

D′2

1

2

1 U′

V 2 V′

O

(1–x )

(x )

OV = r

OV′ = r′

Figure 4.9. Schematic appearance of internally twinned martensite minor and major regions, whichundergo the lattice deformation along different but crystallographically equivalent principal axes.




martensite crystal where the fractional thicknesses of the two constituent variants1 and 2 are (1−x) and x, respectively. The vector r is transformed into the vectorr′, the latter being an average of the segments OA′, A′B′, B′C′ � � � � U′V′. Thus r′

is the vector sum OV′ = OA′ + A′B′ + B′C′ + · · · · ·+ U′V′ and can be expressedin terms of the pure lattice distortions and rigid body rotation as

r′ = ��1−x��1B1 +x�2B2�r (4.6)

or r′ = Er (4.7)

where E = ��1−x��1B1 +x�2B2� (4.8)

The total distortion matrix, E, when it operates on any vector in the parentlattice, produces the corresponding vector in the transformed twinned martensite.Since the habit plane is an undistorted and unrotated plane, any vector lying onthis plane will satisfy the following condition:

Er = r (4.9)

The rigid body rotations in the regions 1 and 2, as given by �1 and �2,bring the planes (represented by AC and A2C2 in Figure 4.8) derived from themirror plane into coincidence. A rotation � (�2 = �1�), which gives the relativerotation between �1 and �2, can be defined and the total macroscopic shear canbe expressed as

E = �1��1−x�B1 +x�B2�= �1G (4.10)

where G= �1−x�B1 +x B2

The macroscopic distortion thus has the following three components: �1, arigid body rotation: G, a fraction of the twin shear; and the Bain strain, B1. Thematrix algebra problem then reduces to an eigenvalue problem for vectors in thehabit plane with solutions (if they exist) only for certain values of x, fraction ofthe twin shear (or the magnitude of the LIS). From a knowledge of the latticeparameters of and the lattice correspondence between the parent and the productlattices and of the twinning system, it is possible to predict the indices of themacroscopic habit plane, the orientation relationship and the twin thickness ratio,which are all experimental observables. The success of the phenomenologicalcrystallographic theory is well documented in the extensive literature wherein thepredicted habit planes in different systems have been shown to match closely withthose experimentally determined (Table 4.1).




Table 4.1. Comparison of experimentally determined and theoretically computed crystallographicparameters of martensites in different systems.

System Habit plane Orientation relationship

Exp. Comp. Exp. Comp.

fcc–bcc 0.1656 0.1848 (111)f∧(011)b 0.3 0.54

Fe–30.9Ni 0.7998 0.7823 �112�f∧(011)b 2.2 1.67

0.5771 0.5948 �101�f ∧ �111�b −2�4 −3�62

Breedis and Wayman (1962) Diff 1.8

fcc–bcc 0.1910 0.1562 (111)f∧(011)b 0.86 ± 0.10 0.83

Fe–24.5Pt 0.7599 0.7404 �112�f∧(011)bEjsic and Wayman (1967) 0.6214 0.6537 �101�f ∧ �111�b 4.42 ± 0.10 −4�44

Diff 2.56′

fcc–bcc 0.1642 0.1783 (111)f∧(011)b <1 15′

Fe–22Ni–8C 0.8208 0.8027 �112�f∧(011)b 2 1.9

Greninger and Troiano (1981) 0.5472 0.5691 [101]f ∧ �111�b −2�5 −2�7

fcc–hcp 0.696 0.6968 (001)c∧(001)o 0 2.4

Au–47Cd −0�686 −0�6810 [111]c∧[011]o 0 18′

Liberman et al. (1955) 0.213 0.2250

<15

The restriction regarding the selection of the system of lattice invariant defor-mation in a twinned martensite plate (the twinning plane being necessarily derivedfrom a mirror plane of the parent crystal) does not apply when the LIS occursby slip. Despite the physical difference between the two types of inhomogeneousshear (LIS, slip and twin), the macroscopic strain due to the transformation canbe described by the same formal theory. There is, however, no simple criterionwhich can be used for selecting the slip system operating as the LIS in a givenmartensitic transformation. A comparison of the magnitudes of the LIS requiredfor different possible slip systems to satisfy the IPS conditions reveals that theoperating slip system is often associated with the smallest shear magnitude.

Returning to the general theory, it must be emphasized that the various com-ponents of strain described as the constituent parts of the overall macroscopicdeformation are not to be considered to be associated with separate physical steps.It seems probable that at the interface each atom moves directly to the nearest siteof the product lattice, and the break-up of different shear components as describedin the theory is not real in either a spatial or a temporal sense.




The formulation developed by Bowles and Mackenzie (1954) is slightly differentfrom that due to Wechsler et al. (1953) outlined earlier. Bowles and Mackenzieconsidered that the shape deformation, E, combined with a “complementary” or“invisible” lattice deformation, H, could accomplish the change in structure. Themacroscopic effects of H are not detectable because these are counterbalanced byan opposite lattice invariant deformation, G′, which is assumed to be a simpleshear. The total lattice deformation is thus factorized into an IPS E and a simpleshear H so that

�1B = HE (4.11)

The left hand side of Eq. (4.9) represents the combination of the rigid bodyrotation and the Bain strain. The total lattice strain, �1B, is a product of twoinvariant plane strains and, therefore, satisfies the condition of an invariant linestrain. As mentioned earlier, the lattice invariant deformation G′ produces a shapechange equal and opposite to that due to H (G′ H = I, the unit matrix). Theoperation G′ on Eq. (4.9) leads to

G′B = G′HE = E (4.12)

which is essentially equivalent to Eq (4.7) provided G′ �1 = �1G. The matricesG and G′ represent the same lattice invariant deformation applied, respectively,before and after the rotation �1. Bowles and Mackenzie introduced an additionalvariable, namely the dilation parameter, �. The condition of the habit plane beingan IPS was relaxed and the habit plane was allowed to dilate uniformly to a smallextent (up to about 2%) without permitting any rotation of the vectors lying in thehabit plane.

The success of the phenomenological crystallographic theory in predicting theorientation relationship between the parent and the product structures, the habitplane resulting from the transformation and the magnitude of the lattice invariantdeformation (twin fraction in a twinned martensite plate), has been remarkable, asillustrated in Table 4.1. The fact that there is a geometrical necessity of an LISin most martensitic transformations is reflected in the presence of a fine structureof martensite plates, consisting of internal twins, stacking faults or arrays of dis-locations at the martensite/parent interface. The magnitude of the lattice invariantstrain can be determined experimentally in twinned martensite plates by measuringthe ratio of the thicknesses of adjacent twinned components. Martensite plates inwhich the LIS occurs by slip often contain dislocation debris which decorate theslip plane. Though the operating slip plane and the slip direction can be tentativelyidentified from the geometry and the Burgers vector of these dislocations, themagnitude of the LIS is not measurable from experimental observations.




Martensite morphology is yet another important observable which can be char-acterized through detailed metallographic investigations. The word “morphology”in the context of martensites refers to the shape of individual martensite units andthe nature of the assemblage of a group of neighbouring martensite units. Factorssuch as the minimization of the strain energy, the maintenance of a glissile coher-ent interface and the non-isotropic growth of martensitic products are responsiblefor a large majority of martensite crystals assuming the shape of lenticular platesas in mechanical twinning. However, there are many alloy systems in which themartensite/parent interface is essentially planar and thus the martensite unit isslab-like (rather than than lens-like). There are also examples of what is known asthe single interface transformation, as in Au–Cd (Lieberman et al. 1957) and In–Tl(Basinski and Christian 1954) alloys in which an internally twinned martensitebicrystal can be induced to grow under a temperature gradient imposed on a singlecrystal of the parent phase. In such a situation, it is easier to propagate a singleinterface of the martensite unit which nucleates first than to repeatedly nucleatefresh martensite units.

In a number of ferrous alloys, generally, with low contents of substitutionaland interstitial solutes and exhibiting correspondingly high Ms temperatures, themartensite units are lath-like. If the length, the breadth and thickness of a marten-site unit are designated as L, B and T , respectively, then L = B >> T for platesand L>B> T for laths. Apart from the differences in the relative external dimen-sions, the lath and the plate morphologies differ with respect to the nature of theassembly of the martensite units. While a plate morphology is characterized bygroups of martensite units of differing orientation and habit plane variants, the dis-tinctive feature of the lath morphology is the occurrence of groups of near parallelmartensite units which are separated from each other by small angle boundaries.In some alloys, adjacent martensite laths exhibit a twin relation. The grouping ofmartensite units in some typical patterns is motivated essentially by their tendencytowards self-accommodation. The overall strain energy of the system can be sub-stantially reduced by an appropriate grouping of martensite units as is reflectedin the formation of energetically favourable polydomain morphologies in severalmartensites. The issue of self-accommodation is of primary importance in shapememory alloys and will be addressed in detail in Section 4.5.5.

4.2.3 KineticsBeing a first-order-type transition, martensitic transformations occur by the nucle-ation and growth process. The overall kinetics of the transformation in most casesare athermal. This can be represented in a plot of fraction transformed versustemperature (Figure 4.10(a) and (b)). A transformation on cooling begins at the Ms




Temperature

Mar

tens

ite (

%)

MS

TemperatureM

arte

nsite

(%

)Time

Mar

tens

ite (

%)

Mf Mf

~100%T1

Ms > T1 > Mf

Mb

(a) (b) (c)

100%

Figure 4.10. Kinetics of the martensite transformation as a plot of fraction transformed versustemperature for athermal martensite (a), athermal burst martensite (b) and overall transformationkinetics for isothermal martensite (c).

temperature; the extent of the transformation progressively increases with loweringof temperature and it is completed, finally attaining the complete transforma-tion at the temperature Mf , which is known as the martensite finish temperature(Figure 4.10(a)). In some cases of athermal martensitic transformation, the volumefraction transformed at Ms shows a sharp rise in a burst, as shown in Figure 4.10(b);therefore, the start temperature is designated as burst temperature (Mb). The timetaken to reach the indicated fraction transformed at any given temperature betweenMs and Mf is very short, and longer holding at the same temperature does notresult in further increase in the fraction transformed. In another variety of marten-sitic transformations, overall isothermal characteristics are exhibited in the overalltransformation kinetics (Figure 4.10(c)). In these cases, the volume fraction of themartensitic phase keeps on increasing with time at any given temperature betweenMs and Mf . In such cases, although the growth of martensite units occurs by thecharacteristic athermal movement of glissile interfaces, the process of nucleationis thermally activated. Experimental studies on martensite nucleation have beencarried out quite extensively in systems which show the isothermal behaviour.From such studies, it has been established that a homogeneous nucleation processthrough thermal activation cannot account for the observed nucleation at very lowtemperatures (even at temperatures approaching 0 K). Kaufman and Cohen (1956)invoked the presence of pre-existing martensite embryos to explain the observednucleation phenomenon. The same anomaly is encountered if one considers thenucleation of martensite in an ideally perfect parent crystal. Assuming the shapeof a martensite nucleus to be a thin oblate spheroid and choosing realistic valuesof the chemical free energy change due to the transformation, the surface energy




and the strain energy of the assembly of the nucleus and the matrix, the nucleationbarrier (�F ) can be estimated to be about 5 × 103 eV per nucleation event. Thiscorresponds to about 105 kT at temperatures where nucleation is experimentallyobserved. This indicates that the thermal energy is much too small for homoge-neous nucleation to occur. The postulation of pre-existing embryos which couldact as heterogeneous sites for martensite nucleation was recognized to resolvethis anomaly. In the early stages of the development of the theory of marten-site nucleation, these embryos were conceived as being structurally similar to themartensite phase (such as bcc embryos in fcc austenite in ferrous alloys). However,no experimental evidence in support of the presence of such embryos could beobtained. Though there is a general concurrence on the requirement of heteroge-neous nucleation in martensitic transformations, a precise structural description ofthe heterogeneities at which nucleation occurs is still not available.

Olson and Cohen (1976) proposed a general mechanism of martensite nucleationby faulting of groups of existing dislocations. For the fcc → bcc transformation,martensite nucleation can be considered in terms of the splitting of a group ofdislocations which form a parallel array, one above the other, in the parent phase.Movement of partial dislocations subsequent to dissociation produces a martensitenucleus bounded by a coherent interface.

There is an alternative approach to seeking an answer to the problem of marten-site nucleation. In a number of systems, martensitic transformations are precededby precursor phenomena, usually known as “premartensitic” effects. Elastic mod-uli of the parent phase in several systems “soften” prior to the transformation.Clapp (1973) proposed a “strain spinodal” approach according to which a localizedsoft phonon mode may operate near a lattice defect, resulting in the nucleation ofthe martensite phase. The possibility of heterophase fluctuations aided by elasticinteractions with pre-existing dislocations to produce martensite has also beenconsidered.

There have been only a few investigations in which the growth velocitiesof martensite plates have been experimentally measured. Two types of marten-sites have been encountered, one grows very rapidly and the other at a muchslower pace; the former is termed “umklapp” and the later “schiebung”. From“in situ” monitoring of resistivity, the growth velocity of the martensite interfacehas been determined to be 1100 m/s for “umklapp” and 10−4 m/s for “schiebung”,respectively.

The rapid movement of the martensitic interface is driven by the free energydifference between the parent and the martensitic phases. The interface can bevisualized as being semicoherent in nature.

The movement of the coherent segments is not opposed by any reactive force,while the movement of the interfacial dislocations involves Peierls stress due to




lattice friction and other internal stresses opposing their motion. This movementmay be thermally activated if the interaction of the internal strain fields with thestress fields of dislocations is short range, but usually the internal friction stressacts as a long-range stress and, therefore, an athermal movement of the interfaceis necessary.

The dislocations at the interface which acts as the transformation front havetwo functions: to accommodate the misfit between the lattices of the parent andthe martensite phases and to generate the LIS by their propagation. In fact, thereare three possible criteria for the selection of the interfacial array of mismatchdislocations: (a) the criterion of the minimum interfacial energy, (b) the criterion ofthe minimum force required to move the array and (c) the criterion of the fulfilmentof the requirement of LIS. Since all the criteria cannot always be satisfied bythe same set of dislocations, the third criterion is often chosen for modelling themartensite interface.

The fact that in a majority of systems martensite interfaces propagate veryrapidly in an athermal manner suggests that the relationship between the velocityof the interface and the force causing its movement (which is derived from the freeenergy difference between the parent and the product phases at the transformationtemperature) contains an instability. It is likely that the instability is due to thefact that the driving force required to nucleate a martensite plate is much greaterthan that required for its growth.

4.2.4 SummaryVarious features of martensitic transformations have been briefly presented in theforegoing sections. The experimental observables on the basis of which one canidentify a transformation to be martensitic have also been mentioned in the courseof the presentation. Based on these considerations, the atomistic mechanism ofmartensitic transformations has been conceived to involve jumps of atoms fromthe parent lattice sites to the product lattice sites in a coordinated or disciplinedmanner by maintaining a lattice correspondence. If one could label the atoms alonga vector in the parent lattice, one would observe that the same atoms occupy sitesalong a vector in the product lattice in the same sequence. In the same way, alabelled plane in the parent becomes a similarly labelled plane in the product. Thisis what, as illustrated in Figure 4.5, is known as a lattice correspondence. Themaintenance of a lattice correspondence is possible only if atomic jumps from aspecific lattice site of the parent to a corresponding lattice site of the product arepredestined. This is evidenced from the fact that the chemical order of the parentphase is inherited by the martensite phase. Table 4.2 summarizes the distinguishingfeatures of martensitic transformations.




Table 4.2. Characteristic features of martensitic transformation.

1. Coordinated/disciplined jumps of atoms from parent lattice sites toproduct lattice sites

2. Strict lattice correspondence between the parent and the productlattices

3. Strict orientation relation between the parent and the product lattices4. Occurrence of surface tilts representing the macroscopic shears

associated with martensite plates5. Inheritance of the chemical composition and the state of atomic

ordering from the parent to the martensitic phase6. Transformation through a nucleation and growth process, the

nucleation step being either athermal or thermally activated, whilethe growth process is invariably athermal.

7. The growth of martensite plate by a rapid movement of a glissilecoherent interface in a manner similar to propagation of a shearfront

4.3 BCC TO ORTHOHEXAGONAL MARTENSITICTRANSFORMATION IN ALLOYS BASED ON Ti AND Zr

Pure Ti and Zr transform martensitically from the high-temperature � (bcc) phaseto the low-temperature (hcp) phase on quenching from the �-phase field, pro-vided the cooling rate exceeds a certain critical value. There have been only afew experimental investigations of the critical cooling rate for martensitic trans-formation in these pure metals. In order to suppress the competing diffusionaltransformation (massive �→ transformation), a quenching rate of several hun-dred degrees celsius per second is necessary in the case of these metals (if the totalinterstitial content is less than 200 ppm). Once the massive �→ transformationis bypassed, the same structural change occurs through the martensitic processwith the attainment of the required supercooling at the Ms temperature which isabout 50 K lower than the equilibrium �→ transition temperature.

The Ms temperature for alloys of Ti and Zr is a function of the alloy composition.The Ms temperature of any given alloy is determined by the �- or the -stabilizingtendencies and the amounts of the alloying elements present in it. There are threepossible athermal transformation products in �-quenched dilute alloys of Ti andZr. These are the hcp ′ and the orthorhombic ′′ martensites and the athermal� phase which has a hexagonal crystal structure. The mechanism of the � → �transformation is quite different from those of the �→ ′ or the �→ ′′ martensitictransformations and has been discussed in a separate chapter. The orthorhombicmartensitic ′′ phase, which forms in alloys containing large concentrations of�-stabilizing elements, can as well be considered as a distorted hexagonal phase,




the orthorhombic distortion being introduced by a high level of supersaturation ofalloying elements. The crystallography of the transformation can also be describedin general terms for the bcc to orthorhombic structure, the latter encompassing thehcp structure as a special case.

4.3.1 Phase diagrams and Ms temperaturesThe range of alloy compositions over which the different athermal products, ′, ′′ and �, form in a binary system based on Ti or Zr can be illustrated in aschematic � isomorphous phase diagram (Figure 4.11). The Ms temperature forthe martensitic transformation and the � start temperature, �s, as functions of xB,the atom fraction of a �-stabilizing alloying element, are superimposed on thisschematic phase diagram.

Alloy classes

β-Quenched β + ω β

ωs for β → ω

α′ α″

α″ + β + ω

α + β Alloy

α Alloy

β Alloy

TiZr

x 1 x 2 x 3x ′2

xB Atom fraction of alloying element

α

Tβ/α

T1

T2

T4

T5

Tem

pera

ture

Ms

Mf

} for β → α′, α′′

Figure 4.11. Basis for the classifications of commercial Ti and Zr alloys into alloys, +� alloysand � alloys.




On quenching from the �-phase, alloys of different compositions exhibit dif-ferent athermal transformation products. With reference to Figure 4.11, which isa modified version of Figure 1.18, alloys in the composition range 0 > xB > x1

produce ′ (hcp) martensite, while those in the range x1 > xB > x2 transforminto orthorhombic ′′ martensite, x1 defining the level of supersaturation at whichorthorhombic distortion sets in. The plots corresponding to Mf and �s as functionsof xB intersect at xB = x2. This implies that for alloy compositions xB < x2, themartensitic transformation reaches completion during a quenching operation beforethe �s temperature is encountered and therefore the product is fully martensitic.The temperature gap between Ms and Mf in most of the Ti- and Zr-based alloysis very small (in the range of 25 K, as reported in a few systems). Because ofthis reason, an incomplete martensitic transformation is not frequently observed.In the composition range of x′

2 < xB < x′2, the quenched structure consists of

martensitic plates along with some untransformed �-phase in which the �-phase isfinely distributed. In the composition range of x′

2 < xB < x3, the quenched productcontains a distribution of athermal �-particles in the �-matrix. The reason for theformation of a dual phase �+� structure in preference to a fully �-structure willbe discussed in the chapter on � transformation.

Quenching from the ( + �) phase field results in duplex microstructures.Depending on the temperature of equilibration in the ( +�) phase field, a widevariety of microstructures can be produced. This point can be illustrated by takingthe example of the alloy composition, x4. Table 4.3 lists the product phases in thisalloy when quenched from the different temperatures, T1, T2, T3 and T4 as markedin Figure 4.11 which also shows the tie lines at these temperatures. Figure 4.11could serve as a basis for the classification of commercial Ti alloys into , +�and � alloys. Since Zr alloys are almost exclusively used for structural applicationsin nuclear reactors, concentrated alloys do not find much use due to their highthermal neutron absorption cross-sections. Therefore, such a classification schemeis not in vogue for Zr alloys. However, from physical metallurgy considerations,

Table 4.3. Microstructures produced in the alloy of composition x4

(as indicated in Figure 4.11) on quenching from different solutionizingtemperatures.

Solutionizing temperature Microstructure

T1 Fully martensitic ′

T2 Primary + ′

T3 Primary + ′ + ′′

T4 Primary + ′′ +��T5 Primary +��




the same classification scheme is also applicable for Zr alloys. The classificationscheme is briefly described in the following.

Alpha alloys are those which on equilibration at temperatures close to 873 Kconsist of the single-phase hcp ( ) structure. Alloys which on quenching from the� phase field retain the �-phase with or without a distribution of the �-phase aregrouped as � alloys. The composition range between those of the alloys andthe � alloys is covered by the ( +) alloys. These three composition ranges aremarked in the schematic phase diagram in Figure 4.11. The composition rangescorresponding to these three classes of alloys can be better defined in ternary alloyswhere the stabilities of the and the �-phases are balanced by suitable alloyingadditions. The Ti-Al-V system can be chosen as a good example. The pseudobinaryphase diagram (Figure 4.12) shows an enlarged -phase stability region due tothe presence of the strong -stabilizing element Al. A ternary composition Ti–6%Al–3.5% V marks the limit of the -alloys, while a complete suppression of the� → ′ martensitic transformation on � quenching requires an addition of about15% V in a Ti alloy containing 6% Al. This composition (6% Al, 15% V) marks thelow V limit for the �-alloys. As indicated earlier, the composition range between3.5 and 15% V corresponds to the +� class of alloys. It may be noted that a�-quenching treatment given to the ( + �) alloys produces a fully martensitic( ′ or ′′) structure. On subsequent tempering, the �-phase (and/or the pertinentintermetallic phases) can be precipitated in the tempered martensitic matrix ofthese alloys.

1000

900

800

700

0 1 2 3 4 5 6 7 8 9 10

6.9Al93.1Tl

0V

Atomic percent V 6.9Al83.0Tl10.1V

4 wt% Al vertical section

Tem

pera

ture

(K

)

Figure 4.12. A pseudobinary diagram of Ti-Al-V system.




As mentioned in Section 4.2.1, thermodynamic studies on martensitic transfor-mations essentially involve the determination of the relative stabilities of the parentand the product phases and of the To temperature at which the two phases of iden-tical composition possess the same value of the integral molar free energy. Usingthe values of the free energy change, �F →�, associated with the to � phasetransformation for pure Zr and Ti, Kaufman (1959) determined the To line for theisomorphous binary Ti–Zr system. An outline of this thermodynamic treatment isgiven here to illustrate the special case where both the pure metals involved exhibitsimilar allotropic transformations and have complete solid solubility in both thehigh- and the low-temperature phases.

The free energies for the and the � phases, F and F�, in Ti–Zr solid solutionscan be expressed as

F = �1−x�F Ti −xF

Zr −F ex +RT�x ln x− �1−x� ln�1−x� (4.13)

and

F� = �1−x�F�Ti −xF

�Zr −F�

ex +RT�x ln x− �1−x� ln�1−x� (4.14)

where F ��Ti�Zr are free energies of the pure components and F ��

ex are the excess freeenergies of mixing for the and the � phases; x is the fraction of Zr in the Ti–Zralloy.

The condition for equilibrium between the two phases is given by the followingidentities at any given temperature T :

F

Ti�x = F�Tix� and F

Zrx = F�Zrx� (4.15)

where F �Ti� x and F

Ti� x� are partial molar free energies or chemical potentials ofTi in the solid solution of compositions x and x� representing the compositionsof the and the � phases in equilibrium at T (represented by tie lines drawn onthe phase diagram in Figure 4.13. Equations (4.11) – (4.13) yield

�F →�Ti +RT ln

1−x�

1−x =(F

ex −x�F

ex

�x

)x −

(F�

ex −x�F�

ex

�x

)x� (4.16)

and

�F →�Zr +RT ln

x�

x =(F

ex + �1−x��F

ex

�

)x −

(F�

ex + �1−x��F�

ex

�x

)x� (4.17)

The martensitic transformation being composition invariant, the chemical driv-ing force for the �→ ′ transition for a Ti–Zr alloy of composition x is given by

�F ′→� = �1−x��F →�Ti +x�F

→�Zr +�F →�

ex (4.18)




0

800

700

900

1000

1100

10 20 30 40 50 60 70 80 90 100

Ti ZrAtomic percent Zr

Tem

pera

ture

(K

)

To (calculated)

α + β

β (bcc)

xβ xβ

xαxα

α (hcp)

Ms (Duwez (1951))

Figure 4.13. Ti–Zr phase diagram showing the position of Ms temperature for Ti–Zr alloys.

At the To temperature, �F ′→� = 0, while at the Ms temperature, the value of�F ′→� is adequate to provide the surface and the strain energies necessary forinitiating the transformation. The free energy differences between the allotropes and � of pure Ti and Zr, expressed as �F →�

Ti and �F →�Zr , respectively, have

been evaluated in Section 4.1. In order to determine the excess free energiesof mixing for the and the � phases, F

ex can be expressed in a power seriesexpansion:

F ex = x�1−x��Ao +A1x+A2x

2 +A3x3 +· · · � (4.19)

where the coefficients Ai are temperature dependent. By choosing equilibriumconditions at different temperatures, which means substituting in Eqs. (4.14) and(4.15) the free energy values for different temperatures and the corresponding x

and x� from the phase diagram, several values of the coefficients Ai upto the nthorder term can be determined in a generalized manner; this procedure is knownas analysis of phase diagrams (Rudman 1970).

In the case of the Ti–Zr system, Kaufman (1959) has shown that the approxima-tion of using only the zeroth order term, (F

ex = x�1−x�Ao and F�ex = x�1−x�Bo,




can also yield reasonable values of To. With these approximations, Eqs. (4.14) and(4.15) are reduced to

�F →�Ti +RT ln

1−x�

1−x = �x �2A− �x��2B (4.20)

and

�F →�Zr +RT ln

x�

x = �1−x �2A− �1−x��2B (4.21)

Since the Ti–Zr isomorphous phase diagram shows a minimum, at any giventemperature of the /� equilibrium, two tie lines and correspondingly two sets ofx and x� values, one on the Ti-rich side and the other on the Zr-rich side, can beobtained. Values of B−A, calculated from the Ti-rich and the Zr-rich sides, havebeen found to be in good agreement; these values can be expressed at temperaturesbetween 1100 and 810 K:

�B−A�= −2340+1�26T cal/mol (4.22)

Consequently, Eq. (4.16) can be written explicitly for the Ti–Zr system as

�F ′→� = �1−x��F →�Ti +x�F

→�Zr −x�1−x��2340−1�26T� cal/mol (4.23)

Equation (4.21) can be used for calculating the To temperature as a function ofx (the results are presented in Figure 4.13) and for computing �F ′→� as a functionof T for different compositions of Ti–Zr alloys. Kaufman’s plots for �F →� versusT for different alloy compositions, superimposed with experimentally determinedMs temperature values (Figure 4.14), indicate that a chemical driving force of about50 cal/mol is required for initiating the �→ ′ transformation martensitically. Thiscorresponds to a supercooling (To −Ms) of about 50 K.

A comparison of the chemical free energy requirements for martensitic trans-formations in alloys based on Ti and Zr with those in ferrous alloys indicatesthat this requirement is much smaller in the case of the former (50 cal/mol asagainst 300 cal/mol for ferrous alloys). This suggests that the restraining forcescomprising strain energy and surface energy associated with the martensites in Tiand Zr alloys are considerably smaller compared to those associated with ferrousalloy martensites. This point will be dealt with while discussing the relative valuesof lattice strains.

In binary phase diagrams involving Ti or Zr on one side and a �-stabilizingelement B (such as Mo, Nb, Ta or V) on the other, the Ti/Zr-rich side can beapproximated as a � isomorphous system. The chemical driving force for the




+100

0

–100

+100

0

–100

Zirconium-rich

Titanium-rich

(ΔF α′→β

= 0)

(ΔF α′ → β

= 0)

ΔF α′ → β/ ~40 Cal/mol

Ms

ΔF α′→β/ ~60 Cal/mol

Ms

Ms

Ms

To

To

700 800 900 1000 1100 1200

Temperature (K)

Diff

eren

ce in

free

ene

rgy

ΔF α′

→β (

cal/m

ol)

x = 0.3 x = 0.2

x = 0.5

x = 0.8

x = 0.6 x = 0.7

x = 0.9 x = 1.0

x = 0.4

x = 0.0

x = 0.1

Figure 4.14. The chemical driving force for martensitic transformation in Ti–Zr alloys as a functionof composition and temperature.

�→ ′ transformation in moderately dilute solutions (x < 0�07� x < 0�15� x� <0�20) can be approximately written (Kaufman and Cohen 1956, Kaufman 1959) as

�F ′→� = �1−x��F →�Ti�Zr −xRT ln

x�

x − �x−2x� + �x��2�

�x��2

(F →�Ti�Zr +RT ln

1−x�

1−x

)(4.24)

The calculated To versus x plots and experimental Ms versus x plots (Duwez1951, 1953) are superimposed on the phase diagrams of Ti–Mo and Ti–V systemsin Figure 4.15. The fact that the Ms line lies 25–50 K below the To line is consistentwith the general trend of the thermodynamics of martensitic transformations inthese systems.

Experimental values of Ms temperatures for various alloys based on Ti areplotted in Figure 4.15.




1200

1100

1000

900

800

1200

1100

1000

900

800

Calculated

0 5 10 15 0 5 10 15 20

Atomic percent alloying element

To

To

MsMs

Tem

pera

ture

(K

)

Ti – V Ti – Nb

α

α α

α + β

α + β

α + β

α + ββ

β

β

β

To

To

MsMs

Ti – Ta Ti – W

Sotubility ≈ 0.2 at 973 K

To

Figure 4.15. BCC and HCP phase relations in Ti-based alloys. Calculated To–x curve is comparedwith the observed Ms values.

4.3.2 Lattice correspondenceThe lattices of the parent and the product phases can be related in a numberof ways. The correct choice of the lattice correspondence is generally made byselecting one which involves the minimum distortion and rotation of the latticevectors. The choice made by Burgers (1934), as illustrated in Figure 4.16, showsthat the basal plane of is derived from an {011}�-type plane and that [011]� and[100]� directions transform into [0110] and [2110] directions, respectively. Theclose-packed directions [111]� and [111]� lying on the �110�� plane transform totwo close-packed <1120> directions. The other <1120> directions are derivedfrom <100>� directions.




[111]β// [1210]α

[111]β// [1120]α

[011]β// [0110]α

[011]β// [0001]α

(0001)α

[100]β// [2110]α

Figure 4.16. The distorted closed-packed hexagonal cell (hcp), derived from the parent bcc lattice.

As mentioned earlier, martensitic transformations in these systems result in theformation of either an hcp or an orthorhombic structure, the former being a specialcase of the latter structure with the ratio of lattice parameters b/a= √

3. In general,the orthohexagonal axes system can be used for describing the martensite crystal-lography covering both �→ ′ and �→ ′′ transformations. The lattice correspon-dences between the bcc and the hcp and between the bcc and the orthorhombicstructures are depicted in Figure 4.17. Six crystallographically equivalent latticecorrespondences between the orthohexagonal and the bcc lattices are describedand labelled as variants 1–6 in Table 4.4.

The lattice (Bain) distortion B associated with this transformation is given by

B =⎡⎣ 1 0 0

0 2 00 0 3

⎤⎦ (4.25)

where 1 =√

32

(a /a�

)�2 = a /a� and 3 = 1/2

(c /a�

)The substitution of the lattice parameter values for pure Zr shows that the

lattice strains are approximately 10% tensile, 10% compressive and 2% tensile,




[100]β

[010]

[001]

C

c[010]β[001]β

[100]o

a1

a

a3

a2

Cubic (β lattice)

[111]β

// [1120]α

[111]β

// [1210]α

[011]β// [0110]α

Figure 4.17. The lattice correspondence between bcc and orthohexagonal cells for the bcc → hcptransformation. The primitive hcp cell is defined by the vectors a1, a2 and c and the orthohexagonalcell by a�= a1�, b�= a1 +2a2� and c. The broken lines show the position of the bcc unit cell.

Table 4.4. Correspondence between orthohexagonal and cubic cell (oRc).

Variant [1 0 0]o [0 1 0]o [0 0 1]o

1 [1 0 0]c [0 1 1]c [0 1 1]c

1 [0 0 1]c [1 1 0]c [1 1 0]c

3 [0 0 1]c [1 1 0]c [1 1 0]c

4 [0 1 0]c [1 0 1]c [1 0 1]c

5 [0 1 0]c [1 0 1 ]c [0 1 1]c

6 [1 0 0]c [0 1 1]c [0 1 1]c

respectively, along 1, 2 and 3 directions. It may be noted that the lattice strainsin this case very nearly satisfy the IPS condition. The deviation from the IPScondition arises only from the 2% tensile strain in the direction perpendicular tothe basal plane.




An approximate analysis of the crystallography of the martensitic transformationcan be performed by neglecting the 2% strain in the direction along [011]�

∣∣∣∣[0001] (Kelly and Groves 1970). The Bain distortion, Ba, then reduces to

Ba =⎡⎣ 0�9 0 0

0 1�1 00 0 0

⎤⎦ (4.26)

The construction of the strain ellipsoid for the corresponding distortion is illus-trated in Figure 4.18, which shows that the two vectors OP′ and OQ′ have notbeen distorted by the Bain distortion but have been rotated from their initialpositions OP and OQ, respectively. Since there is no distortion in the directionperpendicular to the plane of the paper, the pair of vertical planes containing thevectors OP and OQ also remain undistorted. This follows from the theorem that aplane remains undistorted if three non-collinear vectors lying in that plane remainunchanged in length or, in other words, if the lengths of two vectors lying in theplane together with their included angle remain unchanged.

The total strain requires a rigid body rotation which rotates either OP or OQto its earlier position. If one chooses the vertical plane passing through OQ as thehabit plane, a clockwise rigid body rotation is required for bringing the undistorted

10% Tensile

O

10% Compression

P′P

Y′Y

Q′Q

X′X

[110]β

[011]β[011]β

Figure 4.18. Strain ellipsoid construction for bcc to hcp lattice deformation in Ti and Zr alloys.




plane to its original position. Let the coordinates of the point Q′ be (x� y). Operationof the Bain distortion, Ba, brings the point Q′ to Q, the latter having the coordinates(0.9 x, 1.1 y). The condition

OQ = OQ′ (4.27)

or x2 +y2 = �0�9 x�2 + �1�1 y�2 yields x/y = 1�05.The habit plan which contains OQ and [011]� will be at an angle tan−11�05�=

46�5� with the [100]� direction. It is interesting to note that even in this case, whereno inhomogeneous lattice invariant deformation has been introduced, the habitplane is not a rational plane. The choice of the other vertical plane passing throughOP as the habit plane gives the second solution which is crystallographicallyequivalent to the former. This is a manifestation of the fact that the plane (100)�,i.e. the vertical plane passing through [011]�, is a mirror plane of the parent bccstructure.

The next step is to determine the orientation relationship between the and the� phases. The approximate Bain strain, Ba, and the rigid body rotation maintainthe (0001) plane parallel to the (011)� plane, while the rotation brings the [111]�direction close to the [211] direction (within 1.5). The orientation relationship,therefore, can be described as

(0001) (011)�; [110� [111��This is widely known as the Burgers orientation relation which should be

differentiated from the Burgers lattice correspondence given by Figure 4.16. Itmay also be noted that the Burgers orientation relation renders the (112)� planenearly parallel to the (1100) plane. This planar correspondence will be shownto be of great significance in deciding the /� interface plane in diffusionaltransformations in Ti- and Zr-based alloys.

Though the lattice strain along the 3 direction is non-zero (about 2% extension),for most of the � → martensitic transformations studied in Ti- and Zr-basedalloys, the predictions of the habit plane and of the orientation relation from thisapproximate analysis are not far from those experimentally observed. This is whythe approximate analysis is quite instructive in arriving at a general understandingof the transformation geometry. In this context, the work of Bywater and Christian(1972) may be cited in which a suitable alloy, Ti–22% Ta, was chosen in whichthe lattice strain along the 3 direction is indeed zero. The absence of internaltwins and of any dislocation substructure in the martensite plates in this alloyexperimentally validated the absence of lattice invariant shear in this case.

If the extension along the [110]� [0001] ′ direction, which is present in the Baindistortion for most of these alloys, is not neglected, no plane remains undistortedon the application of the Bain distortion. An LIS then becomes a necessity to




make the total shear satisfy the IPS condition. The crystallographic analysis of aspecific case is discussed in the following section as an illustrative example.

4.3.3 Crystallographic analysisThe �→ ′ transformation in a Zr alloy (Zr–2.5% Nb) is chosen as the illustrativeexample for demonstrating the steps of the crystallographic analysis.

Lattice deformation: Using the Burgers correspondence and substituting thelattice parameter values of the � and the phases in the Zr–2.5% Nb alloy(a = 0�3211 nm, c = 0�5115 nm and a� = 0�3577 nm) in Eq. (4.23), the latticedeformation matrix can be expressed as

B0�1 =∣∣∣∣∣∣

0�89768 0�0 0�00�0 1�09942 0�00�0 0�0 1�01213

∣∣∣∣∣∣ (4.28)

where B0�1 is the strain matrix of correspondence variant 1 of Table 4.5.This Bain distortion is on the basis of an axes system defined by the principal

strain directions (x along [011]� [100]o, y along [011]�[010]o and z along[011]� [001]o directions). The same Bain distortion can be expressed in the axessystem of the bcc crystal by using the similarity transformation for variant 1.

Bc�1 = ORc B0�1 �ORc�−1 =⎡⎣0�89768 0�0 0�0

0�0 1�05578 −0�043650�0 −0�04365 1�05578

⎤⎦ (4.29)

Table 4.5. Bain strain matrices (Bc) of bcc to hcp transformation in Zr–2.5 Nb alloyin the cubic basis for the six possible correspondence variants of the martensitic phase.

Variant Bain strain Variant Bain strain

1

⎡⎣ � 0 0

0 � 0 �

⎤⎦ 4

⎡⎣ � 0

0 � 0 0 �

⎤⎦

2

⎡⎣ � 0

� 00 0 �

⎤⎦ 5

⎡⎣ � 0 −

0 � 0− 0 �

⎤⎦

3

⎡⎣ � − 0

− � 00 0 �

⎤⎦ 6

⎡⎣ � 0 0

0 � − 0 − �

⎤⎦

�= 0�89768; �= 1�05578; = −0�04365.




In a similar manner, the Bain distortion for any other martensite variant canbe determined, all on the basis of the axes system of the parent bcc lattice: thecorresponding matrices are given in Table 4.5.

The Bain distortion, when applied to a unit sphere of the parent phase, producesan ellipsoid. The intersection of the ellipsoid and the unit sphere defines the locusof the position vectors, r, which remain undistorted on the application of the Baindistortion. The undistorted vectors define the Bain cone which can be obtainedfrom the condition

Br2 = r2 (4.30)

The initial and the final positions of the Bain cone for the variant 1 are shownin the stereogram in Figure 4.18.

Lattice invariant shear: The next step in the crystallographic analysis is toidentify the mode and the system of invariant deformation which can occur eitherby slip or by twinning. The lattice invariant deformation is determined in sucha way that in combination with the Bain distortion it will maintain a plane ofzero distortion. The system of a lattice invariant deformation (simple shear) canbe defined by the shear plane normal, m, and the shear direction, l. For a simpleshear the vectors which remain undistorted in length lie on two planes, K1 and K2,as shown in Figure 4.19; K1 is the shear plane and K2 is the second undistortedplane which makes an angle of 90 ±� with the shear plane, before and after theshear operation, being related to the magnitude of shear by the equation

g = 2 tan � (4.31)

Before shearAfter shear

Ko K ′K2

K1 n1

g

g /2

g /2

αα2

Figure 4.19. Schematic showing that vectors lying on planes K1 and K2 remain unchanged in lengthafter application of a simple shear.




For a habit plane solution to exist, it is necessary that the traces of the undistortedplanes K1 and K′

2 (the plane K2 after the operation of the shear) must intersect theinitial Bain cone, Bi. This is due to the fact that the vectors defined by the pointsof intersection remain undistorted on the operation of either the Bain distortion orthe LIS. Therefore, they are not distorted by the total shear as well. The abovecriterion has been expressed as l′/m′ restriction by Bilby and Crocker (1961).They have given the following two inequalities for examining whether a givensystem of shear qualifies for being an LIS in a given transformation:

m21�1−2

2��1−23�+m2

2�1−23��1−2

1�+m23�1−2

1��1−22�≤ 0 (4.32)

l2121�1−2

2��1−23�+ l22

22�1−2

3��1−21�+ l23

23�1−2

1��1−22�≤ 0 (4.33)

where [m1m2m3] is the direction normal to the shear plane and [l1l2l3] is the sheardirection defined with reference to the axes system defined by the directions ofthe principal stress components 1, 2 and 3.

In case the LIS occurs by twinning, the two twin components necessarilymaintain crystallographically equivalent lattice correspondences. This imposesadditional criteria for the selection of the system of LIS. For type I twinning, thetwinning plane K1 should be derived from a mirror plane of the parent crystal,while for type II twinning, the direction of the twinning shear T

2 should be derivedfrom a diad of the parent crystal. Based on the lattice correspondence for variant1, it can be seen that some of the variants of {100}� and {110}� mirror planesare transformed into {1012} and {1011} planes which qualify to be twinningplanes of the LIS as they satisfy the ‘l’ and the ‘m’ criteria (Table 4.6). Mackenzieand Bowles (1957) classified the transformation on the basis of the operatingtwinning system for the LIS, designating {1012} twinning for class A and {1011}twinning for class B transformations. It is also seen from Table 4.7 that some ofthe mirror plane variants of the parent crystal are transformed into mirror planes ofthe martensite crystal (e.g. (100)� and (011)� transform into (2110) and (0001) ,respectively), and therefore, these planes cannot be twin planes of the martensitecrystal.

In case the LIS occurs by slip, there is no restriction that the shear plane hasto be derived from a mirror plane of the parent crystal. Otte (1970) has examinedthe suitability of a large number of shear systems for the LIS and has identified,on the basis of low values of the magnitude of the required shear, the followingas the most probable shear systems: {1101} < 2113> , {0110} < 2110> .

Predictions of crystallographic theory: Let us first consider the bcc toorthohexagonal transformation in which the LIS occurs by twinning on the




Table 4.6. The result of Bilby and Crocker criterion (1 and m) for all the possible shear systems inthe product hcp martensitic phase for correspondence variant 1.

Sr. No. bcc hcp l m

Direction Plane Direction Plane

1(a) [111] (110) [2113] (1101) <0.0 <0.01(b) [111] (011) [2113] (0110) <0.0 <0.01(c) [111] (101) [2113] (1011) <0.0 <0.01(d) [111] (112) [2113] (1121) <0.0 <0.01(e) [111] (121) [2113] (1211) <0.0 <0.01(f) [111] (211) [2113] (2112) <0.0 <0.02(a) [111] (011) [1210] (0001) <0.0 <0.02(b) [111] (101) [1210] (1011) <0.0 <0.02(c) [111] (110) [1210] (1011) <0.0 <0.02(d) [111] (121) [1210] (1013) <0.0 <0.02(e) [111] (112) [1210] (1013) <0.0 <0.02(f) [111] (112) [1210] (1013) >0.0 >0.03(a) [011] (011) (0110) (0001) <0.0 < 0.03(b) [011] (111) (0110) (2114) <0.0 < 0.03(c) [011] (211) (0110) (2112) <0.0 < 0.03(d) [011] (311) (0110) (3034) <0.0 < 0.04(a) [010] (101) [0111] (1101) <0.0 <0.04(b) [010] (001) [0111] (0112) <0.0 <0.04(c) [010] (100) [0111] (2110) <0.0 <0.04(d) [010] (102) [0111] (2314) <0.0 <0.05(a) [100] (011) [2110] (0001) >0.0 >0.05(b) [100] (010) [2110] (0112) >0.0 >0.05(c) [100] (001) [2110] (0112) >0.0 >0.05(d) [100] (012) [2110] (0114) >0.0 >0.06(a) [113] (110) [1213] (1101) <0.0 <0.06(b) [113] (121) [1213] (1121) <0.0 <0.06(c) [113] (121) [1213] (1010) <0.0 <0.07(a) [113] (110) [1123] (1011) <0.0 <0.07(b) [113] (211) [1123] (1100) <0.0 <0.07(c) [113] (121) [1-23] [1123] <0.0 <0.08(a) [311] (011) [1100] (0001) >0.0 >0.08(b) [311] (112) [1100] (1121) >0.0 >0.08(c) [311] (121) [1100] (1121) >0.0 >0.0

(1101) (101)� plane. The lattice correspondence of the major twin componentis chosen to be that of variant 1. The minor twin component thus turns out to bevariant 3 (as indicated in Table 4.8). The Bain deformation matrices for the majorand the minor twin components of the martensitic plate can be expressed as




Table 4.7. Possible twin planes in the hexagonal product phase derived from mirror planes of theparent bcc phase.

Mirror plane Variant 1 Variant 2 Variant 3 Variant 4 Variant 5 Variant 6

(100) (21 10) (0112) (0112) (0112) (0112) (21 10)Mirror Mirror

(010) (0112) (0112) (0112) (21 10) (21 10) (011 2)Mirror Mirror

(001) (01 12) (2110) (2110) (01 12) (011 2) (0112)Mirror Mirror

(110) (1011) (0001) (0110) (1101) (1011) (101 1)Mirror Mirror

(110) (1101) (0110) (0001) (1011) (1101) (1101)Mirror Mirror

(101) (1101) (1011) (1101) (0001) (0111) (1011)Mirror Mirror

(101) (101 1) (1101) (1011) (0110) (0001) (1101)Mirror Mirror

(011) (0001) (1101) (1101) (1011) (101 1) (0110)Mirror Mirror

(011) (0110) (1011) (1011) (1101) (1101) (0001)Mirror Mirror

B1 =⎡⎣0�8977 0�0 0�0

0�0 1�0558 −0�04370�0 −0�0437 1�0558

⎤⎦ (4.34)

B3 =⎡⎣1�0558 0�0437 0�0

0�0437 1�0558 0�00�0 0�0 0�8977

⎤⎦ (4.35)

The crystallographic problem can be described in terms of a simple schematicconstruction (Figure 4.20). A vector r in the parent crystal is transformed into avector r′ as a result of a total macroscopic shear (E) comprising Bain distortions inthe two twin-related regions and the corresponding rotations, as given by Eqs. (4.6)and (4.7). For r to lie on the habit plane, the following condition needs to besatisfied:

r′ =�1 ��1−x�B1 +x�B3� r = �1Fr (4.36)

where F = �1−x�B1 +x�B3 (4.37)

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1Color

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Martensitic

Transform

ations299

Table 4.8. Crystallographic parameters of martensites in different Zr- and Ti-based alloy systems.

x Structure Ms OR Habit LIS

Tia – hcp 800 B 8, 9, 12 (1011)T

Ti–Vb 0.75 13 hcp 600–270 – – (1011)T

10 Vc hcp – B – (1011)T ) (21 1 3)Twin ratio 1:3

Ti–Nbd 0.25 at.%Nb hcp 871–212 – – –35 Nbe Ortho ( ) 175 Ba – –

Ti–Taf 0–22 Ta hcp – – – (1011)T

22–36 Tag ortho ( ′′) – – – (111)T

Ti–Crh 2.4–5.9 Cr hcp 320–67 – {334} {1011}TW

(1102)TW

5.5–18.7 at.%Cri hcp+ fcc – IC-S (fcc) –4 Moj hcp 973 (1011)T

6–10 Mo 883 (111)T

Ti–Mok 6 Mo hcp 600 – – (1011) (0001)11 Mol hcp 340 B (8, 9, 12)4

11, 12.5 Mom hcp 340 – (3, 4, 4)4

15 Mon hcpTi–Moo 4.3–5.2 mm hcp 300 B {334}{344} (1011)T

5 mmp hcp BTi–Feq 3 Fe hcp+ fcc 370 – {334} (1011)T

Ti–Nir 2–5.45 Ni hcp+w 680–540 – – –Ti–Cus 0.56–8 Cu hcp 740–570 A {1011} (1011)T

Ti–Alt 8 al hcp (fcc) – – – –Ti-Al-Mo-Vu 8Al-1Mo-2V hcp – – – (1011)T

4Al-3Mo-1Vv hcp – – – (1011)T

OR: Orientation relation.aWilliams et al. (1954),Erickson et al. (1969).

bMcMillan et al. (1967). cMenon et al. (1980). dC Hammond (1972). eBaker (1971),Bagaryatski et al. (1958).

f Yamane and vedg (1966). gBywater and Christian (1972). hErickson (1969). iLiu (1956), Oka (1967). jDavis et al. (1979).kKa (1967). lLiu (1956). mGaunt and Christian (1959). nHammond and Kelly (1969). oHammond and Kelly (1969),

Liu (1956).pWeing and Machlin (1954). qLiu (1956), Nishiyama et al. (1966). rBarton et al. (1960). sWilliams et al. (1970a). tArmitage (1965).uErickson (1969). vWilliams and Blackburn (1967).




Beforetransfarmation

r r ′ 1

1

1

2

2

2

Majortwin

fraction

Minor twinfraction

Aftertransfarmation

B′

C′C

FE

Bg

x1

– x

AB

Figure 4.20. Schematic diagram of an internally twinned martensite. The complementary twiningshear transforming original vector, �r, in the cubic crystal to vector �r′ averaged over the two twinregions (major and minor twin fractions).

The rotation matrix, �, for twinning on (1I01) for the variant 1 can be obtainedfrom Euler’s formula (Wayman 1964) as reproduced in the following:

�p1 −p2��q1 −q2�

�p1 +p2��q1 +q2�= U tan

!

2(4.38)

where p1, q1 and p2, q2 are the initial and the final positions of two vectors lyingin the unrotated plane, with the axis of rotation being U and the angle of rotationbeing !; � and ! have been evaluated for the present case as

�=⎡⎢⎣

0�98655 0�04079 0�15831

0�03744 0�99856 0�04079

0�15975 0�03474 0�98655

⎤⎥⎦ (4.39)

and ! = 9�658

Substituting the values of B1, B3 and � in Eq. (4.34), the macroscopic shapedistortion, E, can be expressed in terms of the twin fraction, x. In general, F , whichcorresponds to a fraction of the twin shear is a non-symmetric matrix which canbe expressed as the product of a symmetric matrix, Fs, and a rotation matrix, ".




Furthermore, a rotation matrix � can be found which diagonalizes the symmetricmatrix Fs to Fd. Thus the macroscopic shape distortion, E, can be written as

E = �"�Fd�∗ (4.40)

where Fd is a function of x. In this expression for E, all the matrices exceptingFd are rotation matrices and Fd is a diagonal matrix with diagonal elements 1,2 and 3 (eigenvalues). An axes system with basis d(id, jd, kd) can be defined inwhich the distortion is diagonal (axes system defined by the eigenvectors). In thisaxes system (basis d), the distortion of the rd vector can be written as

r′d = Fdrd (4.41)

Those vectors which remain undistorted satisfy the following condition:

rd = r′d = Fdrd (4.42)

which is equivalent to

�21 −1�Xd + �2

2 −1�Yd + �23 −1�Zd = 0 (4.43)

where Xd, Yd and Zd are the indices of a vector rd which is one of the undistortedvectors. As mentioned earlier, the distorted ellipsoid can intersect the unit sphereon an undistorted plane only if the distortion along one of the principal straindirections is zero. This means that one of the elements is unity and Eq. (4.40)reduces to a quadratic equation in x, the twin thickness ratio. Solving this equation,two crystallographically equivalent values of x are obtained, of which the secondvalue corresponds to (1 − x). From Eq. (4.40), the two x values obtained are0.19535 and 0.80465.

Substituting the value of x in Eq. (4.34), the matrix F is determined to be

F =⎡⎢⎣

0�92544 0�0 0�2716

0�01568 1�05578 −0�02797

−0�03324 −0�04365 1�02254

⎤⎥⎦ (4.44)

The eigenvalues (principal strains) of the matrix F are given by1�1�= 0�92546, 2�2�= 1�000 and 3�3�= 1�07936.

It is, therefore, seen that the condition of zero distortion plane is satisfied for athickness ratio of 4.11903.




The eigenvectors, Vi, corresponding to these eigenvalues and the rotation matrix,�, are given by

� =

⎡⎢⎢⎢⎣

V1 V2 V3

0�99779 −0�01582 −0�06459

−0�06302 −0�53496 −0�84252

−0�02122 −0�84473 0�53477

⎤⎥⎥⎥⎦ (4.45)

Experimental determination of crystallographic parameters and comparisonwith theoretical predictions: Gaunt and Christian (1959) examined six-sided crystalbar specimens (refined by Van Arkel iodide process) for determining the orienta-tion relationship and the habit plane of martensites forming in pure Zr (with a verylow total interstitial content). Since the material deposition in the iodide refin-ing occurs at ∼1573 K, the crystal gets deposited in the high temperature �(bcc)phase which transforms completely into the martensitic ′-phase during cooling.In the absence of the parent �-phase in the specimens examined, the orientationsof large-sized �-grains were determined from the shear ridge markings present onthe sample surfaces. These ridges were found to be traces of the �110�� planes ofthe pre-existing �-phase. The �-orientation thus determined was consistent withthe external shape of the crystal bar.

The orientations of large -grains, recognized by polarized light microscopy,were determined from Laue diffraction patterns using microbeam X-ray. TheBurgers orientation relationship (0001) (110)�; [2110] [111]� was establishedwithin the accuracy attainable (±2). It was not possible to measure the shapedeformation for Zr. The experimentally determined habit plane, as reported byGaunt and Christian (1959), for pure Zr is (0�72, 0�51, 0.46) ±2. This experimentalresult is compared with those obtained from computations based on the Bowles–Mackenzie formulation (1957), the Wechsler–Lieberman–Read formulation (1957)and the approximate analysis (described in Section 4.3.2).

The Burgers (1934) mechanism of transformation implicitly assumed the samecorrespondence as described in Section 4.3.2 and accomplished most of the lengthchanges normal to the [110] direction by means of a shear on the (112�� [111��system; the remaining adjustments were not considered in detail. Since the prin-cipal strain in the [110]� direction is very small, it is attractive to retain Burgersassumption that this direction is exactly parallel to the hexagonal c-axis and thatthe requisite strain is taken elastically at the interface and subsequently relaxedplastically. As has been shown in Section 4.3.2, the approximate analysis of thebcc to hcp transformation can be worked out, neglecting the ∼2% distortion alongthe [110]� direction. The invariant line in the (110)� plane (the shear direction)




is readily found to be ∼[223], and the habit plane, defined by this line and by[110]�, is (334��.

Although the approximate theory does not describe the transformation crys-tallography adequately, it is useful in showing the physical significance of theexperimental results, which tends to be obscured by the mathematics of theBowles–Mackenzie theory. Thus the observations that the orientation relation isclose to the Burgers relation, that the habit plane is nearly normal to the hexago-nal basal plane and that the shape deformation is nearly simple shear are all theconsequences of the small principal strain parallel to the c-axis.

Experimental determination of habit plane and shape strain can be made moreaccurately if the parent �-phase can be retained at room temperature in a metastablestate on quenching, the orientations of several �-grains are precisely determinedby Laue diffraction, and martensite plates are introduced either by stress or byquenching to cryogenic temperatures below Ms. It is desirable to have martensiteplates of sufficiently large size present in a matrix of well-characterized � grainsfor orientation determination by diffraction. Stress-induced martensite plates offeran additional advantage, as they can be induced at room temperature and, therefore,the martensite/matrix habit can be examined without subjecting the sample to anychange in temperature.

Gaunt and Christian (1959) determined the habit plane, the direction and mag-nitude of the macroscopic shear and the orientation relationship in a Ti–12.5%Mo alloy in which stress-induced martensite was produced. Experimentally deter-mined habit plane poles and shear directions are plotted in Figure 4.21(a), whichshows that the habit plane corresponding to the stress-induced martensite is closeto �344�� and the shear direction is close to a great circle through the �111��and �011�� poles. These results, when compared with those obtained in thermallyinduced martensite of Ti–11% Mo (Figure 4.21(b)) bring out the important dif-ference between them. In the case of Ti–11% Mo, the habit plane pole is closeto �334�� and the shear direction lies close to a great circle passing through the�111�� and �001�� poles.

Since the microstructure of martensite was investigated by Gaunt and Christian(1959) only at the light microscope level, information regarding the marten-site substructure was not reported. Subsequently a large number of alloys havebeen investigated using TEM. Table 4.8 summarizes the results of martensitecrystallography as reported by several investigators. It may be noted that withincrease in the concentration of �-stabilizing elements the formation of �344��habit martensite is promoted. The observation showing �344�� habit for marten-site plates in Ti–12.5% Mo in contrast to �334�� habit in Ti–11% Mo has beeninterpreted by Bowles and Mackenzie (1957) in terms of maximum resolved shearstress.




110

100

011

101S

001

H

111

111

010101100

5413

2 5

634126

(a)

100

111

101

110

011

S

S H001 111111

3

B44'1

2

B 2 4 3

1

5

(b)

Figure 4.21. Stereographic projections showing the habit plane and shear directions for (a) stress-induced martensite in Ti–12.5% Mo alloy and (b) thermally induced martensite in Ti–11% Moalloy.

Later investigations by Hammond and Kelly (1969) and Banerjee and Krishnan(1971) have shown that the �344�� habit, which is the macroscopic habit planebetween the twinned martensitic ′-plate and the �-matrix, is made up of two seg-ments, each corresponding to �334�� planes. It will be shown in a later section thatlarge primary plates containing periodic

{1011

}

internal twins generally followthe Bowles–Mackenzie ( −�+) solution under which both the twin componentsmatch the Burgers orientation relation very closely. Each of these twin segmentstend to maintain the IPS condition by forming a habit segment of �334�� type.

4.3.3.1 Morphology and substructureMorphologies and substructures of martensites have been studied in a number ofTi- and Zr-based-alloys. On the basis of the observations made in these studies,martensite morphologies can be classified into several groups which are describedin this section. The underlying principles which control the evolution of a given




type of morphology are also discussed. As has been explained later, the mor-phology of martensite is often linked with the internal structure (substructure) ofmartensite crystals. It is for this reason that the discussion on morphology, whichdeals with the nature of assembly of martensite crystals, and on substructure,which reflects the operation of the LIS, is placed in the same section.

The factors which play significant roles in controlling the morphology of marten-site products are the following:

(1) The symmetry elements of the parent and the martensite crystals which dictatethe polydomain morphology.

(2) The strain coupling between martensite crystals in a given region, resulting inautocatalytic nucleation and self-accommodation.

(3) The sequence of formation of different generations of martensite crystals whichcontrols the progressive partitioning of the grains of the parent phase.

(4) The nature of the operating LIS.

Polydomain morphology. The formation of a polydomain structure of marten-site is clearly a consequence of the fact that the different variants of the martensitephase have equal probability of being generated from a single parent grain. Thestructures of all the variants are related by the symmetry operation of the parentcrystal which is not a symmetry operation of the martensite crystal. The method ofdetermining the number of variants on the basis of group theoretical considerationsis described in Table 4.4. For the present discussion, we can use Table 4.4 whereall the possible variants and their lattice correspondences with the parent crystalsare listed. An examination of the relative orientations of these variants reveals thatmany of them are mutually twin related (Table 4.9).

Strain coupling – self accommodation: The equivalence of strain energy min-imization and IPS deformation in explaining the formation of internally twinnedmartensite plates has been discussed in Section 4.3. The process of the minimiza-tion of the strain energy does not remain confined to only within an individual

Table 4.9. The combination of major and minor twin correspondence variantsrelated by the possible twinning planes in the hcp Zr–2.5Nb alloy.

Twinning Major minor twin fraction

(0112)H 1-6 2-3 3-2 4-5 5-4 6-1(1101)H 1-4 2-5 3-1 4-2 5-6 6-3(1101)H 1-3 2-1 3-4 4-6 5-3 6-5(1011)H 1-2 2-6 3-5 4-1 5-2 6-4(1011)H 1-5 2-4 3-6 4-3 5-1 6-2




plate. In a group of martensite plates, mutual strain coupling can significantlyreduce the overall strain energy. This is known as self-accommodation. A methodfor an approximate estimation of the effectiveness of self-accommodation for dif-ferent groups of martensite plates, developed by Madangopal et al. (1993, 1997),Srivastava et al. (1993, 2000) and Srivastava (1996) is summarized here.

The shape strain associated with ellipsoidal martensite plates causes an elasticstrain field in the surrounding parent phase. The elastic strain fields of the differentvariants then interact with each other in such a manner that a maximum reductionin the strain energy density of the assembly of the plates and the matrix is achieved.Plates belonging to a “cluster” are, therefore, “bonded” by the magnitude ofthe strain energy reduction. The Eshelby expression (1957) for determining thedisplacement Ux at a point x at a distance r caused by an ellipsoidal inclusion isgiven in the (i, j, k) axes system by Figure 4.22.

Uj�x�= VeTijgijk

8#�1−$�r2(4.46)

where V is the volume of the inclusion, $ is the poison ratio, eTij is the stress-free

transformation strain which is the shape strain in a martensitic transformation, and

Point x (r, l1, l2, l3)

TB (e1 )

Centre of interactingstrain fields

A (e1 )T

C (e1 )T

Figure 4.22. Schematic illustration of the interacting strain fields of three point forces. Strain energydensity is determined from the displacement measured at a point far from X.




gijk = �1 − 2$��%lj�lk + %ik�lj − %ij�lk + liljlk�� li� lj and lk being the directioncosines of the position vector of the point x.

The above expression is valid when the martensite plate is assumed to be pointforce and the distance r is large.

The strain components due to the displacement Uj�x� can be derived from thefollowing relation:

eij = �Uij +Uji�

2(4.47)

where Uij = �Ui/�xjThe stress at a point x can be computed from the strain eij by using the following

relation:

�ij = eij +2�eij (4.48)

where the elastic constants = c12 and �= c44.The strain energy at the point x is then given by

Eij = 12�ij�eij (4.49)

Integrating the above expression over the r , l1, l2 and l3 space, the total strainenergy density E′ due to a single martensite plate can be determined to be

E′ = 2[

18#�1−$�

]2

D (4.50)

where D = c1e2ij + c2e

2ji + c3eiiejj + c4eij eji and the coefficients cj are

c1 = 16�0+24�8�& c2 = 12�5+19�8�

c3 = 15�0+6�0�& c4 = 25�0+39�6�

The strain energy density for a group of martensite plates can be computed byconsidering the superimposition of the strain fields due to each of them. Since thedistance between the plates within a group is small compared to r, the strain fieldsof the interacting plates can be considered at the same point (i.e. at the origin). Thenet displacement due to a group of n martensite plates (illustrated in Figure 4.22)is given by

Unj �x�= V�gijk

8#�1−$�r2'�eT

ij� (4.51)




It can be seen that the summation term in the above expression is the averageshape strain matrix of all the interacting martensite plates.

The degree of self-accommodation (DSA) can now be defined as

DSA = E�1� −E�n�

E�1�100 (4.52)

where E1 and En correspond to the strain energy density due to an isolated plateand a group of n plates, respectively. The values of DSA for different possiblegroups of martensite plates can be computed for assessing the DSA achieved indifferent groups. Some of the simplifying assumptions used in the aforementionedformulation for calculating the DSA are the following:

(1) The volume fractions of the different variants interacting in a group have beenconsidered to be identical. Further reduction in the strain energy is possibleby optimizing the volume fractions of different variants in a manner similar tofulfilling the IPS criterion by a suitable adjustment of the twin thickness ratioin an internally twinned martensite plate.

(2) The nature of the interfaces between the variants has not been taken intoaccount.

In spite of these approximations, an evaluation of DSA for different groupsof martensite variants provides a basis for the selection of those variants whichgroup together in a self-accommodating manner. This point will be discussedfurther while comparing the predicted grouping with those frequently encounteredin experiments.

The overall martensite morphology develops through a sequence of transfor-mation events. The first-generation plates fragment the parent grain by spanningacross the entire grain. Martensite plates forming in later generations are pro-gressively smaller as the space available in the fragmented parent phase for thegrowth of plates continuously decreases with the progress of the transformation.The crystallographic constraints such as the adherence to habit planes and theself-accommodation tendency remain operative throughout. This causes nearlyself-similar pattern generation as illustrated in Figure 4.23. Hornbogen (1989,1990) has pointed out the fractal nature of the formation of martensite as shownin Figure 4.23, which illustrates pattern generation through successive stages offragmentation. The development of such a pattern, however, does not occur in thelath martensite structure in which a group of laths stacked in a parallel array arearranged within each packet.




Figure 4.23. A Schematic showing self-similarity pattern in martensitic transformation.

Hcp martensites in Ti- and Zr-based alloys exhibit several morphologies whichcan be broadly classified into the lath and the plate types. In the former, marten-site units are grouped together in parallel arrays within a packet and several suchpackets make up the volume of the parent �-grain. In contrast, the plate mor-phology is characterized by acicular plates forming along different variants ofthe habit plane, continuously partitioning �-grains, as schematically illustrated inFigure 4.24. A more detailed description of these morphologies is given in thefollowing sections.

(b)(a)

Figure 4.24. Schematic diagram showing acicular plates forming along different variants of thehabit plane, continuously partitioning �-grains.




Lath morphology. As mentioned earlier, the lath morphology consists of severalmartensite packets within each parent �-grain. These packets are resolvable underthe light microscope. Under polarized light each packet exhibits a separate contrast,suggesting that the orientation within a given packet is nearly the same. The factthat each packet is made up of a number of laths which are stacked in a parallelarray is revealed from surface relief observations and from the microstructurerevealed by TEM. Figure 4.25 shows the typical lath morphology under differentimaging conditions. Williams et al. (1970a,b, 1973) have reported a lath martensitemorphology in dilute Ti–Cu alloys, while a similar morphology of martensitesin dilute Zr–Nb and Zr–Ti alloys has been reported by Banerjee and Krishnan(1971, 1973). TEM investigations on these martensites have clearly revealed thata group of laths present within a single packet belong to the same orientationvariant, notwithstanding the small angle tilt boundaries between the adjacent laths.Contrast analysis of dislocations constituting lath boundaries has shown that thesedislocations have <c+ a> type Burgers vectors. The misorientation between theadjacent laths has been found to be 1.0–1.5 from the observed spacings of thesedislocations.

The alloys which exhibit the lath morphology are all associated with high Ms

temperatures (above 923 K), and therefore, the transformation is complete and noretained �-phase is left in the quenched product. The determination of the habitplane has, therefore, been carried out in such alloys by assuming the Burgers ori-entation relationship. The reported habit plane indices for pure iodide Ti, �8912��(Williams et al. 1954) and for pure iodide Zr, �334�� (Gaunt and Christian 1959)have been obtained from back reflection Laue patterns taken from individual pack-ets, each containing a group of parallel laths. The orientation of the parent �-grainhas been worked out on the basis of the Burgers orientation relation, with differentvariants being operative in different packets within a prior �-grain. Single surfacetrace analysis of parallel laths seen within the packets has indicated the habit planeindices to be close to �334��. Later investigations based on TEM observationshave confirmed the same indices for the lath martensite habit plane in Ti (Williams1973) and in Zr (Banerjee and Krishnan 1971, 1973) alloys. With increasingadditions of �-stabilizing elements, the Ms temperature of the alloys decreasesand this is reflected in the change in the martensite morphology. In general, dilutealloys with Ms temperatures above about 923 K exhibit a packet morphology, eachpacket containing a stack of laths. Two types of orientation distributions of lathsin a given packet have been reported. In one case, laths belonging to the sameorientation variant are stacked together with a small angle boundary separatingthe adjacent laths, while in the other alternate laths are twin related. The packetsof the latter type with twin-related laths are encountered more frequently in alloys




(b) (c)

(d) (e) (f)

(ii)(g) (i)

(a)

Figure 4.25. (a and b) Optical micrographs showing massive martensite in pure zirconia. TEMmicrograh showing (c) large lath with irregular interface in a Zr-0.5% Nb alloy (d) laths withstraight interface. (e and f) Bright and dark field micrographs showing alternately twin related lathsin a Zr-2.5% Nb alloy. (g) Self-accommodating 3-plate clusters of martensite plates of “indentationmorphology”. (i) General view of microstructure showing the presence of large number of 3-plateclusters when viewed along the <111> direction. (ii) Schematic drawing showing the arrangementof the 3-plate clusters. (iii) SAD patterns from each of the interface regions (A-B, B-C, C-A)separating the individual martensite in a 3-plate cluster. (iv) Keys to SAD patterns obtained fromindividual martensite crystal in a 3-plate cluster (in (iii)) and their orientation with respect totheir neighbouring crystals. (v) Relative orientation of plates A, B, and C are shown in the [011]stereogram. Basal planes and the habit plane traces ( HA, HB and HC) of A, B and C are marked.The �1120�H directions of all the crystals are aligned along the �111�C direction. The habit planepoles A, B and C, as marked, are located on the outer circle and on the dotted great circles GB andGC respectively.




1101

1103 11010002

1101B

A1101 1101

110100021103

A B

C1103 1101 1101C

0002

11011101

00021101A

1103

C

C

0002 11011103

1101

11010002

11011101

1103

11011101

(110)b //(0001)M

(111)b //(1120)M

C

(101)b //(0001)M

(111)b //(1120)M

B

(111)b //(1120)M

(011)b //(0001)M

A

2110111 433 1100

FOIL PLANETRACE334

6(+)/2(–)HB

A /C(0001)B

1101101

1210100

0110

0111433

343334

1103 111

(0001) 0114(+)/2(–)

HA(1101)B

1105

B/C

334

GC

110

343GB

111

343433

101

0111

6(+)/4(–)

HCA/B

1011

343

1103

1102(0001)C

1101110

2201

B,C(1120)

111

1010

2110111

4331100

1210

001

0110

433

334111

1120

0011011

011

1010

(iii)(iv)

(v)

Figure 4.25. Continued

such as Ti–5.3% Cu (Zangvil et al. 1973) and Zr–2.5% Nb (Srivastava et al.2000) in which the Ms temperature is brought down close to 923 K.

In most cases laths contain dislocations and are free of periodic internal twins.This suggests that the lattice invariant shear operative within laths is slip andthat the dislocations present are debris of the slip process. The < c+ a > Burgersvector of these dislocations and the alignment of their line vectors along the tracesof �10l1� planes support this contention. The dislocation structure and the lathinterface structure undergo significant alterations during cooling from the relativelyhigh transformation temperature. It is for this reason that lath boundaries and theinternal dislocation structure characteristic of the martensitic transformation areoften not preserved well in the final microstructure.




Plate morphology. The plate morphology is characterized by martensite plates,parallel sided or lenticular, forming along all the habit plane variants and dividingthe parent phase grains by a sequential partitioning process. The fractal natureof this morphology is evident from the self-similarity of assemblies of martensiteplates forming in successive generations. The first-generation plates span/extendacross the entire grain of the parent phase. The partitioned grain is then amenableto transformation in the next stage. As a consequence, the second-generation platesare shorter in length and further partition the parent grain. The next generationof plates form within these fragmented untransformed spaces and the processcontinues till the transformation is complete (in some cases pockets of the parentphase remain untransformed). The pattern generated by a group of martensite platesbelonging to one generation remains essentially the same though the dimensions ofthe plates decrease in every successive generation, maintaining the self-similarityof the structure.

Detailed crystallographic analysis has been primarily carried out on martensiteplates belonging to the earlier generations mainly because they are large and wellformed and, therefore, better suited for the analysis of orientation relations andhabit plane indices. A survey of the substructure of primary martensite plates inTi and Zr alloys reveals that �1011� twinning is frequently observed within theseplates (Figure 4.26). However, a number of primary martensite plates remain onlydislocated. Quantitative data on the frequency of twinning in primary plates arenot available, but the incidence of twinning is seen to increase with a lowering ofthe Ms temperature. Martensite plates forming in later generations are found to beless frequently twinned. Such a tendency is presumably related to the fact that thelater-generation plates form in a matrix which is substantially work hardened by thestresses resulting from the plastic accommodation of the earlier-generation plates.

(a) (b)

Figure 4.26. Bright field TEM micrographs showing plate martensites: (a) internally twined (thicktwins) martensites and (b) internally twined martensite where the minor twin fraction is very thin.




Hammond and Kelly (1969) were the first to point out that the internallytwinned martensite plates in Ti–Mn alloys could be grouped into two classes.The first group was characterized by a large spacing of thick �10l1� twins. Thethicknesses of the major and the minor orientations were sufficiently large toproduce a clearly resolvable zigzag habit which was made up of habit segments ofthe two twin-related variants present within a single martensite plate. The ratio ofthe thicknesses of the minor and the major twins was seen to be 1:4, a value whichclosely matched with that predicted from the magnitude of the LIS calculatedfrom the phenomenological theory. Similar zigzag habit has also been observedin several Ti and Zr alloy martensites where the thicknesses of the major and theminor twin orientations are large (minor twins thicker than 100 nm).

The second group of plates was characterized by the presence of very thintwins stacked within the plates (Figure 4.26). The average value of the ratio of thethicknesses of the minor and the major twin variants was much smaller comparedto that predicted from the phenomenological theory. This was suggestive of thefact that an additional component of the lattice invariant shear (possibly slip) wasoperative within these plates.

An analysis of habit plane variants of the two types of internally twinned plates(containing thick and thin twins) indicates that there exists an important differencebetween them in terms of the orientations of their minor twin components. InBowles–Mackenzie theory, four non-equivalent habit plane solutions, denoted by( ±�±), are obtained in a class A transformation, where class A corresponds tothe LIS on �1011� planes which are derived from the �110��-type mirror planesof the parent bcc phase. For the lattice correspondence given in Figure 4.16 andfor the lattice invariant deformation on the (1101) plane, the orientation relationscorresponding to the ( −�+) and the ( +�+) solutions, as represented inthe stereogram in Figure 4.27, are obtained by a rigid body rotation around the[0001] direction in either the clockwise or the anticlockwise direction. The mostimportant difference between these two solutions arises from the fact that the(1101) plane on which the LIS acts remains nearly parallel (an angular deviationof 2) to the (011)� mirror plane for the class A ( −�+) solution, whereas the(l101� twin plane gets separated from the (110)� plane from which it is derivedby an angle of about 11 for the class A ( +�+) solution. This means that thetwin plane remains nearly parallel to the mirror plane in the former case evenafter the application of the rigid body rotation. In such a situation both the twincomponents follow the Burgers orientation relation very closely. The growth ofa twinned martensite plate involves the creation of successive twin componentsas the plate lengthens. As a particular variant grows, the combined effect of therigid body rotation and the Bain strain (R.B) tends to establish an invariant planebetween the parent and the growing martensite variant. The deviation of R.B from




1002110

1111011

1011120

1111210

110

1101

0110110

1120

111

1101011

101

1101

0110002

0110110

0210111

100 2110

011

0002

0110

011

0210

111

2110 100

101

101110

1011 1120

111

011

0110

111

11201011

101

2110100

1101

1101210

111

Figure 4.27. Stereographic projection showing the orientation relations corresponding to class A( −�+) and class A( +�+) solution of Bowles and Mackenzie formulation. These two orientationsare derived from the clockwise and anticlockwise rotations along [110]�11�001� .

the IPS condition causes a strain build-up which gets periodically corrected bythe creation of a second variant at regular intervals. In the case of a plate whichbelongs to the ( −�+) solution, both the twin components follow the Burgersorientation relation very closely, and therefore, the R.B combination operatingwithin each twin component nearly satisfies the IPS condition. As a consequence,these two twin components can grow to a significant extent (the thickness of theminor twin variant exceeding 100 nm), maintaining independent habit segmentswith the parent phase.

In the ( +�+) case, the twin plane (l101� does not remain parallel to the(110)� mirror plane subsequent to the rigid body rotation. The major twin orien-tation in this case satisfies the Burgers orientation relation and the IPS conditionquite closely. However, the minor twin variant ( 2) is significantly away fromthe Burgers orientation relation, the (0001) plane being 9 away from (011)�.The IPS condition is, therefore, not obeyed even approximately within the minortwin variant. The growth of such a variant would cause a substantial build-up ofstrain, resulting from the deviation from the IPS condition. This would restrictthe growth of the 2 component. The observed stack of very thin twins in themartensite plates belonging to the ( +�+) solution lends support to this con-tention. Since the requirement of LIS is not met in these plates in which the twinthickness ratio is much smaller than the predicted value of 1:4, the operation ofan additional component of LIS in the form of slip appears essential. A periodicarray of < c+ a > dislocations along a different variant of �10l1� planes within 2 components is suggestive of the additional lattice invariant deformation.




Substructure resulting from atomic shuffles: The phenomenological crystallo-graphic theory discussed so far takes into account the fact that the total macroscopicshape strain ratifies the IPS condition. The necessity of introducing a lattice invari-ant deformation in order to achieve this condition in a great majority of alloys hasalso been explained. The lattice invariant deformation can be manifested either bycreating a stacking of internal twins or by repeated slipping at periodic intervals.The former results in the production of the internally twinned martensite, whilein the latter debris of dislocations left behind are responsible for a dislocated sub-structure. The lattice (Bain) deformation shown in Figure 4.28(a)–(c) transformsthe orthorhombic unit cell whose dimensions (a, b and c) match those of theproduct martensite. However, a Bain strain does not bring all the atoms into theright positions consistent with the hexagonal symmetry of the product phase. Anadditional atomic shuffle is required for achieving the right atomic positions. Thispoint was taken into account in the Burgers model for the bcc → hcp transforma-tion. As shown in Figure 4.28, the bcc → hcp transformation can be accomplishedby a shear which converts the (011)� plane into the (0001) plane and a shuffleof the atoms at O locations in the figure into the positions B or C for obtainingthe hcp stacking. It is this choice in the direction of shuffle (either O → B or O→ C) which allows the formation of domains with different stacking sequences.

Figure 4.29(a) indicates the stacking sequence for a perfect hcp structure whileFigure 4.29(b) shows the stacking sequence in a crystal containing two domains,corresponding to the two alternative shuffle directions. Atoms in the A layers remainunaltered while in Figure 4.29(b) a domain boundary is created by introducingdifferent shuffles in the two halves of the crystal. Such a domain boundary is createdby the passage of a 1

3

⟨1010

⟩partial dislocation. The fault in Figure 4.29(c) can be

converted to that of Figure 4.29(b) by the removal of a C layer of atoms at the arrow, byinsertion of a 1

2 [0001] partial. The resulting reaction 13 [1010] + 1

2 [0001] → 16 [2023]

indicates that the faults in Figure 4.29(c) are associated with a displacement vectorof 1

6

⟨2023

⟩. However, since the fault boundary is created by the random shuffling of

atoms, there is no necessity for partial dislocations to be present.A typical substructure of martensite laths and plates, primary as well as sec-

ondary, both dislocated and twinned, consists of fine striations within the laths andplates. Such a substructure was recorded by Ericksons et al. (1969), Hammondand Kelly (1970) and Knowles and Smith (1981a,b) in Ti–Mn and Ti–Cr binaryalloys. Selected area diffraction from any martensite plate and both shows streak-ing which has been characterized by Banerjee and Muralidharan (1998) by detailedtilting experiments. It has been observed that the plane of diffracted intensity(which causes streaking in diffraction patterns along different zone axes) extendsalong a plane nearly perpendicular to the martensite habit plane (as shown in astereographic projection in Figure 4.30. This observation is also consistent with




[111]

(011)

(211)

70.53°

(a)

(b) (c)

(0001)

57°

A

A

A

A

BCO

A

A

A

Figure 4.28. (a) A basic cell for the hcp structure outlined within five bcc unit cells, before thetransformation. Shear along [111] transforms this cell to a hexagonal unit cell. The choice inthe direction of shuffle to obtain the hexagonal structure. (b) Homogeneous shear transforms thiscell into a hexagonal unit cell and (c) shows a choice (OB or OC) in the direction of a shuffleto obtain the hcp structure (after D. Banerjee and Muraleedharan, 1998).

the fact that the fine-scale substructure observed in Ti martensites shows striationsapproximately perpendicular to the habit plane Figure 4.31. Tilting experimentshave also established from the invisibility criteria that the displacement vector asso-ciated with the observed striations is either 1

3 [1010] or 16 (2023]. This conclusion

has been arrived at on the basis of the observation that the striations are visibleonly with g = n

⟨1100

⟩+m�0001� when n = 0. The micrographs in Figure 4.31




Faulting createdby wrong shuffle

hcp stacking

(a) (b)

Fault with [2023]displacement vector

A

AB

BA

BA

BA

AC

AC

AB

AB

AC

AC

AC

BA

BA

Inse

rtion

of

[

0001

]

Shear by

[1010]

(c)

61

21

31

Figure 4.29. (a) The stacking along the [0001] direction in a hcp structure. The fault in the stackingcreated by alternate choices of shuffle and (b) the fault created by a 1/3 [1010] shear.

(1210)

HABIT(111)

(434)

(0110)

(1120)(100)

(1010)

NbNa

(2110)

(011)(1100)

(011)

← plane of intensity

[4223]

(0001)(011)[1123]

[1213]

[2113]

(1210)

(111)

(0110)

(1120)

(100)(1010)

(111)(2110)

(011)(1100)

Figure 4.30. A stereographic projection of the Ti and the bcc-Ti phases in the Burgers relation-ship (open circles). The direction of the streaking measured from bright field micrographs withdifferent zones and the line drawn through them indicates the plane of intensity distribution resultingin the streaking (after D. Banerjee and Muraleedharan, 1998).




Figure 4.31. Micrographs showing three-dimensional domains and their boundaries (after D. Baner-jee and Muraleedharan, 1998).

also suggest that the substructure does not merely consist of planar faults whichextend across the martensite plates and laths but is more closely associated withthree-dimensional domains and their boundaries. The domain structure has a closesimilarity to the antiphase domain structure in ordered alloys though the sub-structure is seen in the martensitic ′-phase which is not ordered. The domainstructure is not isotropic in the sense that the bounding surfaces of the domainsare predominantly along basal planes with some segments being present alongprism and pyramidal planes. Based on the observations made on the details of thesubstructure of Ti alloy martensites, Banerjee and Muralidharan (1998) came upwith a schematic diagram (Figure 4.32) showing the domain structure in relationto a section of a martensite plate. The domains have been named as “stack-ing domains”, the displacement vector associated with the boundaries separating

[0001]

(0001)

Mid

rib

(2110)

Habit plane[1210]

Figure 4.32. A schematic diagram of the domain structure in relation to a section of a martensite plate(after D. Banerjee and Muraleedharan, 1998).




adjacent domains being that of a stacking fault in the fundamental hcp structure(unlike a fault vector in the superlattice of an ordered structure). The presence ofstrain contrast in the images of the domain boundaries has also been established.

The origin of domain formation in a martensitic structure can be considered inthe following manner. When a thin sheet of martensite (of the order of a few unitcells in dimension) forms by the expansion of a shear loop which transforms aspecific set of �110�� planes into the �0001� planes, the required atomic shufflesfor accomplishing the structural change occur in a random fashion. This processcreates a given domain. It is not possible to ascertain whether the shuffle occursat the advancing transformation front or within the product after the manifestationof the shape strain within that small volume. With further propagation of theshear front, contiguous domains form by randomly selecting one of the shuffledirections within a domain. Since there is no change in the nearest neighboursacross a (0001) interface between two adjacent domains, the (0001) surfacesbecome the predominant bounding surfaces of these domains. Short segments ofdomain boundaries form along prismatic and near prismatic planes to close thedomain volume as columns parallel to the basal plane, extending across the platethickness (as shown in schematic drawing in Figure 4.32).

4.3.3.2 Transition in morphology and substructureIn the literature on the martensite in Fe-based alloys, a number of terminolo-gies have been used for describing the morphology and the substructure of themartensite product. Various characteristics, such as the nature of the assembly ofmartensite units which includes variant distribution and stacking pattern, the extentof self-accommodation, the nature of LIS and the transformation (Ms) temperatureare taken into account in grouping the martensite products and assigning suitableterminologies for describing them. Krauss and Marder (1971) have made a surveyand have reported that there is a definite preference for the use of the terms: lathmartensite and plate martensite for the two broad classes of martensites, the char-acteristic features of which are given in Table 4.10. These attributes which havebeen identified essentially from a large body of experimental data on Fe-basedmartensites can as well be adopted for classifying martensites in Ti- and Zr-basedalloys. As pointed out earlier, martensites in these alloys can be broadly groupedinto the lath and the plate types. The facts that based on the same characteristicfeatures, martensites in Ti- and Zr-based alloys can be classified into these twogroups and that a transition is morphology and substructure can be introduced bysuitably choosing the alloy composition suggest that there is something fundamen-tal in this morphological transition. The crystallography of the transformations,fcc → bcc or bct in ferrous alloys and bcc → hcp or orthorhombic in Ti and Zralloys, being quite distinct, there must be some common issues (not related to




Table 4.10. Characteristic features of two classes of martensitic products in Fe-based alloys.

Attributes Lath martensite Plate martensite

Shape of the martensite unit,length, width and thicknessgiven by a, b and c

a > b > c, parallel sided a≈ b � c, acicular or lens-shaped

Nature of stacking of ofmartensite units

A group of laths of nearlyidentical habit planestacked in a parallelarray within a packet:(a) martensite variants,(b) close orientations in agroup and (c) alternate lathsof twin-related variants

Multiple variants presentalong intersecting habitplanes in the samemicroscopic domain

Mesoscopic morphology Packets of martensite plates,several packets filling up theentire austenite grain

Fractal morphology arisingfrom repeated partitioning ofthe austenite grains bymartensite plates ofdifferent habit variants –smaller martensite platesfilling up the partitionedaustenite volume created bymartensite plates of previousgeneration

Internal structure arisingfrom the lattice invariantshear

Generally dislocatedmartensite (only in a limitedcases twinning is observedwithin lath martensites)

Frequently encounteredinternally twinned plates

Transformation temperatureand sequence oftransformation events

Relatively high Ms

temperature dilute alloysRelative low Ms temperatureMove concentrated alloys

Kinetics Slowly growing martensitevolume fraction withincreasing cooling

Rapidly growing martensitevolume fraction withincreasing cooling

Mechanism Martensite laths ofsuccessive generationsweakly strain-coupled

Strong strain coupling ofmartensites of successivegenerations

Type Schiebung martensite Umklapp martensite

the crystallographic symmetries and the magnitude of the (Bain strain) whichdictate the transition in morphology and substructure of martensites. The factorswhich appear to have strong influences on the morphology and substructure are(a) transformation temperature, (b) self-accommodation of a group of martensite




plates (c) plastic flow of the soft parent phase in partially accommodating thetransformation strain and (d) autocatalytic nucleation of martensite plates.

These factors are again interdependent. With decreasing Ms temperature, theflow stress of both the high-temperature parent phase and the low-temperaturemartensitic phase increases substantially. This enhances the strain energy accu-mulation due to the formation of a single martensite plate. The requirement ofself-accommodation, therefore, becomes more important as the Ms temperature islowered.

The morphological transition from the lath to the plate type in several ferrousalloys is accompanied by a change in substructure from dislocated to twinned. Thecoincidence of morphological transition (lath to plate) and substructural transition(dislocated to twinned) has also been noticed in Ti- and Zr-based martensites(Banerjee and Krishnan 1973). It is still difficult to conclude whether the transitionin one is instrumental in bringing about the other transition. Figure 4.33(a) showsthe Ms temperature as a function of alloy content in several binary alloys of Tiand Zr. It is seen that as the Ms temperature is brought down below about 923 K,both morphological and substructural transitions occur. In fact, the correlationbetween the transition in morphology and substructure and the Ms temperature isuniversally valid for Ti and Zr systems whereas the correlation is not so perfectin case of ferrous alloys.

Let us now examine the plausible reasons for a transition in morphology and insubstructure with lowering of the Ms temperature. When a martensite lath formsat sufficiently high temperatures where the matrix �-phase remains relatively soft,the shape strain associated with the formation of a lath is primarily accommodatedby the plastic flow of the matrix. No significant stress field is generated aroundthe first-formed laths and therefore, stress-assisted nucleation of other martensitevariants around the first-generation laths remains quite restricted. The surfacesprovided by the first-formed laths, however, act as favourable nucleation sites fornew generation laths. Those orientation variants are favoured which have a habitplane close to that of the first-formed laths and whose orientations are either closeto or twin related with the former. These tendencies are responsible for creating apacket of parallely stacked laths. As far as the selection of the LIS is concerned, alower critical resolved shear stress for slip compared to that for twinning promotesthe operation of the slip mode at high temperatures. This point is illustrated inFigure 4.33(b).

With the lowering of the Ms temperature, the stress field created by the first-generation plates assists the nucleation of several martensite variants in the vicinityof the former. However, the principle of self-accommodation allows the nucleationand growth of those plates which permit minimization of the stress field associatedwith the assembly of plates. The grouping of variants is invariably controlled by




(b)

Twinning

Slip

Str

ess

Temperature

SlippedTwinned

(a)

Lath-dislocated martensite

Acicular (plate)twinned martensite

Nb

TiZr Ta

Cu

Cr

900

800

700

600

500

400

300

200Zr 5 10 15 20 20 15 10 5 Ti

Atom %

Tem

pera

ture

(°C

)

Figure 4.33. (a) Plots of the Ms temperature versus the alloy content in several binary alloysof Zr and Ti showing the necessary level of alloy additions to bring about a change in the morphologyand the substructure. (b) A schematic diagram illustrating the variation of the CRSS for slip and fortwinning as a function of temperature. The diagram suggests that a lowering of the Ms temperaturewould result in a slip to twin cross-over in the operating mode of inhomogeneous shear.

the principle of self-accommodation, leading to a polydomain morphology withleast stored elastic energy. The elastic bonding of one plate with its neighbouringplates is the primary driving force for the generation of the plate morphology.Low transformation temperatures also promote the occurrence of twinning whichpredominates at least in the primary martensite plates in plate morphology.




4.3.4 Stress-assisted and strain-induced martensitic transformationThe chemical free energy change which acts as the driving force for initiating amartensitic transformation can be supplemented by applied stress and/or plasticstrain. As has been mentioned earlier, the Ms temperature defines the extent ofsupercooling (To −Ms) which is required for the provision of the adequate chemicalfree energy change which can meet the surface and the strain energy requirementsfor martensite nucleation. It is because of this reason that the martensite starttemperature is raised above Ms when external stress or plastic deformation isintroduced in the parent phase. The interrelationships among applied stress, plasticstrain and martensitic transformation have been extensively studied in Fe-Ni-Cand Fe-Ni-Cr-C alloys by Bolling and Richman (1970a,b). They have defined atemperature, M�

s (lying above Ms) below which plastic yielding under appliedstress is initiated by the onset of martensitic transformation and above whichplastic flow of the parent phase precedes the nucleation of martensite. Olson andCohen (1972) have provided a schematic representation of the interrelationshipsbetween applied stress, plastic strain of austensite and the stress required formartensitic nucleation (Figure 4.3) at temperatures above Ms. In the temperaturerange Ms < T < M�

s , the stress required for martensite nucleation increases withtemperature and follows a temperature dependence consistent with the elastic stressrequirement for martensite nucleation; the phenomenon in this temperature rangeis, therefore, described as stress-assisted martensite nucleation. With increase intemperature within this range Ms < T <M�

s , the stress requirement increases in alinear fashion as the driving chemical free energy change decreases approximatelylinearly with temperature.

In the temperature range M�s < T < Md, the stress required for stress-assisted

nucleation exceeds the yield strength of austenite, and therefore, the austenite ini-tially yields by slip, causing extensive dislocation activity. With the accumulationof plastic strain (and dislocation multiplication), localized strain centres are createdwhere martensite nucleation becomes energetically feasible. The yield strengthof austenite marks the lower boundary of the stress required for strain-inducednucleation, as shown in Figure 4.3.

To summarize the overall situation, the chemical driving force is sufficient atMs for the pre-existing nucleation sites or embryos in the austenite to becomeoperative without the application of stress. At temperatures between Ms and M�

s ,in the stress-assisted nucleation regime, such nucleation can occur only with theaid of applied stress. Here the required initiating stress is in the elastic rangebut increases with increasing temperature because of the concomitant decreasein the chemical driving force. At temperatures above M�

s , plastic straining ofthe austenite results in the creation of martensite nuclei in the regime of strain-induced nucleation. With increasing temperature above M�

s , further reduction in




chemical driving force necessitates increased plastic strain in order to producedetectable amounts of transformation. This requires the initiating stress to riseabove the yield stress of the austenite. As a result, plastic flow of the austenite,leading to extensive dislocation multiplication, occurs during the onset of plasticdeformation. The nucleation of martensite becomes favourable at locations wherestrain accumulations exceed a certain threshold value.

The formation of stress-assisted martensite is reflected in the appearance of adeviation from the linear elastic behaviour at a stress value lower than the yieldstress value of the parent phase. Figure 4.34 illustrates the stress–strain plot for a Tialloy in which stress-assisted martensite formation commenced at the point markedby an arrow. As mentioned earlier, martensite plates were introduced by applyingexternal stress in alloys of suitable compositions, e.g. Ti–12.5% Mo (Gaunt andChristian 1959), Ti-V (Menon et al. 1980), in which the �-phase could be retainedin a metastable state by � quenching. In such alloys, the Ms temperature in invari-ably below the room temperature and the martensite formed is primarily of theinternally twinned plate type. TEM examination of these martensites has revealedthe presence of periodic twins within these martensite plates (Figure 4.35). Sincethese martensites contain a fairly high level of �-stabilizers (in order to lower theMs temperature below room temperature), they often exhibit orthorhombic distor-tion, though it is not essential that all stress-assisted or strain-induced martensites

WQFC

1000

800

600

400

200

00.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Engineering strain

+α –β heat-treatedβ heat-treated

Eng

inee

ring

stre

ss (

MP

a)

↓

Figure 4.34. Stress–strain plot for a Ti alloy in which stress–assisted martensite formation com-menced at the point marked by an arrow. While furnace cooled (FC) samples do not exhibit stress-induced martensite formation, water quenched (WQ) samples show deviation from linear elasticdeformation at a low stress value (pointed by an arrow) corresponding to the formation of stress-induced martensite.




(a)

200 nm200 nm

(b)

β

750 nm

OM

OM

OM

(c)

Figure 4.35. TEM micrographs showing martensitic plates with periodic twins.

in Ti- and Zr-based systems will be orthorhombic in nature. It may also be notedthat stress-assisted martensite has been encountered only in a limited number ofZr-based alloys, the Zr-Mo-Al alloy system being an example in which clearevidence of stress-induced martensite has been reported.

4.4 STRENGTHENING DUE TO MARTENSITICTRANSFORMATION

It is now well established that the strengthening due to the martensitic transfor-mation arises from various factors such as solid solution hardening, precipitationhardening, order hardening and hardening due to lattice defects and fine crystallitesize. Several attempts have been made to determine quantitatively the individualcontributions of these factors towards the overall strengthening of martensites.Excellent reviews by Christian (1971) on ferrous martensites and by Warlimontand Delaye (1974) on non-ferrous martensites treat the subject in a comprehensivemanner. Extensive research on the strength of ferrous martensites has demonstrated




that the major contribution to the strength of Fe–C martensites arises from thesolid solution strengthening brought about by C atoms in the supersaturated bodycentred tetragonal phase and that the defect structure introduced by the martensitictransformation is responsible only for a relatively small increase in the strengthlevel. That part of the strength which arises from solution hardening could beimparted to the base metal by introducing C atoms in excess of the solubilitylimit even by some alternative methods such as sputter deposition (Dahlgren andMerz, 1971) and thus cannot be attributed to the martensitic transformation per se.In fact, the strengthening due to a fine-scale precipitation or due to Snoek order-ing of C atoms is also essentially connected with the C supersaturation in themartensite. Apart from solid solution hardening, a substantial part of the strengthof martensites in several Cu-based alloys is derived from the ordering of substitu-tional atoms. In other words, the inheritance of a high level of solute concentrationor an ordered arrangement of atoms from the parent phase is primarily responsiblefor the strengthening of martensites in Fe- and Cu-based alloys. Heat treatmentsinvolving the martensitic transformation have been found to be quite effectivein strengthening Ti- and Zr-base alloys. The +� alloys in these systems areamenable to the formation of the martensitic phase on quenching from the � phasefield. Standard heat treatments of these alloys involve a � quenching treatment fol-lowed by a tempering treatment which results in the best combination of strength,ductility and fracture toughness. Unlike in the case of the Fe–C system, the hcpmartensites in Ti- and Zr-based alloys are neither supersaturated with interstitialelements nor possess any distorted crystal structure. It is, therefore, interesting toexamine the extents to which different factors are responsible in imparting strengthto Ti and Zr alloy martensites.

Banerjee et al. (1978) have investigated the strengthening mechanisms operativein martensites in binary Ti–Zr alloys. The isomorphous Ti–Zr system provides aunique opportunity in which the strengthening contributions of the solid solutionand the martensite substructure can be separated. While � quenching producesthe martensitic ′ structure, a slow cooling from the �-phase field results in theformation of a Widmanstatten structure. The chemical composition (includingthe interstitial content) and the crystal structure corresponding to the and ′

phases so formed being identical, the difference between the flow stress values of and ′ can be considered to arise from the differences in the defect structures ofthese products. The martensitic ′ structure contains more closely spaced interfacesseparating adjacent laths, plates and twins and a higher density of dislocations ascompared to the slow-cooled Widmanstatten structure of the same composition.The ′ as well as the phases in this isomorphous system possess the equilibriumcomposition. Since there is no tendency of phase separation in either of thesephases, the possibility of fine-scale precipitation, usually pertinent to the Fe–C




martensite, can be ruled out. The � → transformation attains completion bothduring the quenching and the slow cooling operations, and thus no interferencefrom the retained parent phase is present. A comparison of the true stress versustrue strain curves associated with the and ′ structures of a given alloy directlyyields the strengthening contribution, �� , which can be attributed solely to themartensitic substructure, �� being defined as the difference between the flowstresses of the ′ and the structures, �� = � ′

−�x, for a given value of the trueplastic strain, �p. The assumption which is implicit in this treatment is that thecontributions of the various pertinent factors to the strength are additive.

In order to quantify the microstructural difference between the Widmanstatten and the martensitic ′ (either dislocated lath or twinned plate morphology)phases, the average spacing between the partitioning interfaces in these has beenmeasured for binary Zr–Ti alloys of different compositions (Table 4.11). While inthe case of the Widmanstatten structure, the average thickness of the plates givesthe mean free spacing between the partitioning interfaces, for the lath martensitestructure, it is the average thickness of the laths and for the internally twinnedplate martensite, it is the average twin thickness. The average packet size for thelath martensite structure has also been recorded as the packet boundaries are theeffective barriers against the passage of slip or twin deformation. This point isdiscussed in a later section.

The value of ��, which is a measure of the strengthening contribution solelyof the martensitic structure, has been found to be strongly dependent on the alloycomposition as seen in Figure 4.36. This is not unexpected in view of the fact thatthe morphology and substructure undergo a drastic change from the dislocatedlath to the twinned plate martensite as the Ti content exceeds a limiting valueof about 15% in Zr–Ti alloys. The sharp change in the �� value at around thiscomposition strongly indicates that the twinned plate martensite structure providesa more effective barrier against dislocation motion. The �� value has also beenplotted against the true plastic strain (Figure 4.37) for different alloy compositions.The sharp rise in �� with increasing plastic strain in the case of twinned plate

Table 4.11. Average size of the and the ′ units in Zr–Ti alloys.

Average width ofWidmanstatten plates((m)

Average spacingbetween partitioninginterfaces ((m)

Average packet sizein the lath structure((m)

Zr–5Ti 81 0.8 60Zr–10Ti 11.5 0.4 25Zr–15Ti 3.1 0.2 –Zr–20Ti 18 0.15 –




40

36

32

28

24

20

16

Δσ (k

g/m

n2 )

12

8

4

0

Zr Ti atom percent

MN

/m2

10 20 30

100

200

300

4000.0100.005

0.0020.015 and 0.020

Transition in morphology andsubstructure lath → plate

dislocated → twin

Figure 4.36. The rapid rise in �� corresponds to the composition range over which the marten-site structure changed from the dislocated lath to the twinned plate type. The values of the plas-tic strain at which flow stress values were measured are indicated.

martensites suggests that this substructure exerts a powerful influence on the workhardening process. In contrast, the work hardening rate remains somewhat lesssensitive to the plastic strain for dislocated lath martensites in pure Zr and inZr–5% Ti and Zr–10% Ti alloys.

4.4.1 Microscopic interactionsWe will now examine the effectiveness of different types of interfaces in offeringresistance to an approaching deformation front (either slip or twin).

4.4.1.1 Lath boundariesIt has been observed (Banerjee et al. 1978) that a slip band can propagate throughthe small angle boundaries separating adjacent laths contained within a packet.Slip deformation can be induced in a thin foil sample by beam heating in the TEM.A dynamic experiment has shown that a set of dislocations can easily penetratethrough the lath boundaries, the path followed by each dislocation being markedby a pair of slip traces which suffers a small angular deviation at the lath boundary.

The stress required to push a dislocation out of a small angle boundary, madeup of an array of parallel edge dislocations lying on a plane normal to the slip




(a) (b)

(c)

900

800

700

600

500

400

300

200

100

00.02 0.04 0.06 0.08

Zr –15% Ti

Plastic strain

Str

ess

(MN

/m2 )

σ (α′)

σ (α′)

σ (α)

σ (α)

σ∗ (α) σ∗ (α)σ∗ (α′)

σ∗ (α′)

σμ (α′)

σμ (α′)

σμ (α)

σμ (α)

0

100

200

300

400

500

600

700

0 0.02 0.04 0.06 0.08

Plastic strain

Zr –10% Ti

Str

ess

(MN

/m2 )

σ (α′)

σ (α)

σ∗ (α)

σ∗ (α′)

σμ (α′)

σμ (α)

0

100

200

300

400

500

600

700

0.02 0.02 0.02 0.020.02

Plastic strain

Str

ess

(MN

/m2 )

Zr – 5% Ti

σ (α′)

σ (α)

σ∗ (α)

σ∗ (α′)

σμ (α′)

σμ (α)

(d)

Zr –20% Ti

Plastic strain

Str

ess

(MN

/m2 )

0.04

100

200

300

400

0.020.00

500

600

700

800

900

0.06 0.08

Figure 4.37. The �� value plotted against the true plastic strain for different alloy compositions.




direction, x, is given by

�xy = 0�35�b2h�1−$�

(4.53)

where h is the spacing between the dislocations, � is the shear modulus and $is the Poisson ratio. The stress �xy can be generated at the head of a pile-up ofn dislocations where n� = �xy, � being the applied stress. Since n, � and L, theaverage lath thickness, are related by

n= #L�

�b(4.54)

Eq. (4.51) reduces to

)2 = 0�35�2b2

2#Lh�1−$�(4.55)

Substituting typical values for L�= 1�m�, h (= 40 nm) and b (= 0�32 nm), themagnitude of the applied stress, �, required to push a dislocation out of the lathboundary can be estimated to be about �/5000. Such a small value of � justifiesthe easy penetration of glide dislocations through the lath boundaries.

TEM of deformed lath martensites has also shown that deformation twins toocan propagate through the lath boundaries which separate laths of nearly identicalorientations. The packet boundaries separating laths of two distinct orientationvariants have been found to be quite effective in stopping slip and deformationtwin bands. It is, therefore, the packet size which is more important in deciding thestrengthening contribution of the martensitic structure in dislocated lath martensitesin Ti- and Zr-based alloys. Since the magnitude of the LIS is rather small in thesealloys, the density of dislocations in these martensites is not very large. A highMs temperature (above 923 K) is also responsible for a partial annihilation andrearrangement of dislocations resulting the LIS and the plastic accommodation.As a result, the strengthening contribution of dislocations in the lath martensitestructure is not very significant.

4.4.1.2 Twin boundaries and plate boundariesA large proportion of the partitioning interfaces in the internally twinned platemartensite structure is constituted of �10I1� twin interfaces. Some of these areboundaries separating adjacent twins within single plates while some others arelocated between contiguous martensite plates obeying twin-related variants of theorientation relation.




Contrary to expectations, the increase in strength associated with the transitionfrom a dislocated lath to a finely twinned substructure in steel martensites is notvery large. Had the twin interfaces in steel martensites acted as effective barriersagainst dislocation motion, the internally twinned plate morphology would havebeen about three times as strong as lath martensites, consistent with an orderof magnitude refinement of the mean free slip length. Experiments have showntwinned martensites to be only 8–30% stronger than lath martensites in ferroussystems. This discrepancy has been explained by Kelly and Pollard (1969) on thebasis of the Sleeswyk–Verbraak mechanism (1961) of the passage of a/2 <111>slip dislocations through the coherent twin interfaces.

The effectiveness of �1011� -type twin interfaces in resisting the motion ofdislocations in a twinned martensite plate in a Ti or a Zr alloy can be evaluatedby considering the interaction of the different variants of slip in the �1010� < 1210> system with a specific �1I01� twin interface. Yoo (1969) has expressedthe dislocation reaction at a twin interface in a general form as

X�hkl�m � X∗�hkl�t ±nbt (4.56)

where X is the Burgers vector of a dislocation gliding on the (hkl)m slip planeof the matrix which, after encountering a twin interface, gets converted to a newdislocation in the twin crystal with a Burgers vector X∗(hkl)t which can glide onthe (hkl)t plane of the twin orientation. While crossing the twin boundary, thedislocation creates a twin dislocation with Burgers vector nbt where bt is the Burg-ers vector of a unit twin dislocation. Using the relevant transformation matricesfor transforming the Burgers vectors and Miller indices of the corresponding slipsystems, the interaction of three variants of the first-order prismatic slip, {1010} <1210> , with a specific (1101) twin interface can be described by the followingdislocation reactions:

1/3�2110�m�0110�m → 1/2�5146�t�1343�t −bt (4.57)

1/3�1120�m�1100�m → 1/3�1120�t�1103�t (4.58)

1/3�1210�m�1010�m → 1/2�1546�t�3143�t +bt (4.59)

Out of these possible interactions, only in the second case can a dislocation withb = 1/3 [1120] propagate through the (1101) twin interface without changing itsBurgers vector. The line of intersection of the (1100) slip plane and the (1101) twin plane defines the [1120]m direction and, therefore, the dislocation line assumesa perfect screw character as it encounters the twin plane; and as a consequenceof this, it is amenable for cross slip on the (101)t plane of the twin crystal. This




mechanism has been illustrated schematically in Figure 4.38(a). Passage of slipdeformation in the other two cases is quite unlikely as the indices of the slipplane and direction are irrational and a twinning dislocation is left behind at theinterface. Thus for these two slip systems, the (1101) twin plane acts as totallyopaque. If one considers that the orientations of martensite crystals are nearlyrandomly distributed, the probability of the twin interface being completely opaquetowards a first-order prismatic slip is 2/3. For the remaining 1/3 probability, theentry of 1/3 [1120] dislocations through a cross slip mechanism (illustrated in

b

(1100)M

(1100)T

ξ

(1101) Twin plane

(a)

(b)

(1011)

(1122)

(1011)(0112)T

(0112)[0111][1123]

(2111)T

21°

Figure 4.38. (a) A schematic showing the mechanism for cross slip on the (101)t planeof the twin crystal (b) Schematics showing the entry of 1/3[1120] dislocations througha cross slip mechanism.




Figure 4.38(b)) is possible. However, the stress required to make the dislocationglide on the new slip plane in the twin segment would be different, depending onthe Schmidt factors for the slip systems in the matrix and the twin segments.

The analysis of the interaction of the most favourable prismatic slip modewith {1011} twin interfaces shows that for a majority (2/3 probability) of theorientations of the stress axis a single twin segment would behave as a plate-shaped crystal strongly bounded by a pair of non-deformable interfaces such thatneither can the slip dislocations penetrate these nor can there be any sliding alongthe coherent twin interface.

The deformation of a thin crystal of thickness in the range of 50–200 nm,bounded by non-deformable twin interfaces, results in the generation of a straingradient and as a consequence an array of geometrically necessary dislocationsis created. Ashby and Johnson (1969) has shown that during the deformationof plastically inhomogeneous materials, geometrically necessary dislocations areproduced and the dislocation density is proportional to /bL where and b are themagnitudes of the shear and the Burgers vector, respectively, and L is the spacingbetween the non-deformable interfaces.

Banerjee et al. (1978) have shown that geometrically necessary dislocationsindeed accumulate in Zr–Ti martensites subjected to plastic deformation. Thesedislocations are seen to be lined up along the twin boundaries and their densityincreases with lowering of the mean free slip length, L.

The possibility of the passage of deformation twins cutting across transformationtwins needs be considered in these hcp alloys in which different deformationtwin systems are operative, particularly at low temperatures and at high strainrates. The most common deformation twin systems in these alloys are {1012} <1011> , {1121} < 2113> and {1122} <1123 > . The possibility of theintersection of different variants of these deformation twins across a stack of(1011) transformation twins can be examined by the invoking the following plasticcompatibility conditions proposed by Cahn (1953): (a) the traces of the crossingtwin and the secondary twin in the composition (K1) plane of the crossed twinshould be parallel and (b) the direction, the magnitude and the sense of the shearshould be identical in the crossing and the crossed twins.

If these plastic compatibility conditions are fulfilled, a deformation twin cannotbe impeded by twin interfaces. From such an analysis, it can be seen that thefollowing twin shears can pass through a set of (1011) transformation twins:(0112) [0111] , (1102) [1101] , (1122) [1123] and (2112) [2113]

These types of intersections have been reported in the microstructure of thedeformed martensite in Zr–15% Ti and Zr–20% Ti alloys (Banerjee et al. 1978).Figure 4.39 illustrates a few cases of intersection of deformation twins and trans-formation twins.




(a) (b)

(c)

Figure 4.39. Examples of intersection of deformation twins and transformation twins.

4.4.2 Macroscopic flow behaviourThe influence of partitioning interfaces, which are mostly impenetrable to dislo-cations, on the macroscopic flow behaviour of Zr–Ti alloys with respect to initialyielding as well as work hardening has been examined in this section. The accu-mulation of geometrically necessary dislocations in the vicinity of partitioninginterfaces, observed in deformed Zr–Ti martensites, points to the fact that the flowbehaviour of martensites can be rationalized in terms of Ashby’s one parameterwork hardening theory for plastically inhomogeneous materials. According to thistheory, the work hardening associated with a given microstructure characterizedby a mean free slip length, , is expected to be governed by geometrically nec-essary dislocations if the density of these is much larger than that of statisticallystored dislocations. Under such circumstances, the stress–strain plot would assumea parabolic shape given by the following relation:

� = �0 +C1EM1/2

(b�

)1/2

(4.60)

where �0 and � are the tensile stresses at strain levels zero and �, respectively,C1 is a dimensionless constant which has been estimated to be 0.35 ± 0.15 for




plate-like barriers (Ashby, 1971), b is the Burgers vector, M is Taylor’s orientationfactor and E is the Young’s modulus of a random polycrystalline sample. Theapplicability of Ashby’s one parameter work hardening theory to the plastic flowbehaviour of Zr–Ti martensites has been demonstrated by the linear nature of theplots of � versus �1/2 (Figure 4.40(a)). It can be seen that each flow curve canbe fitted with two straight line segments, the first segment corresponding to strainvalues less than about 1% and showing a much larger work hardening rate than

(a) (c)

Δσ0

(MN

/m2 )

500

400

300

200

100

00 1 2 3

σ0

800

700

600

500

200

100

0

400

300

0 5 10 15

σ (M

N/m

2 )

20 25 30

Zr(α′)

Zr – 5% Ti(α′)

Zr – 10% Ti(α′)

Zr – 15% Ti(α′)

Zr(α)

Zr – 5% Ti(α)

Zr – 15% Ti(α)

σ1 – σ0

Zr – 10% Ti(α)

ε1

(b)

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 1 2 3 4 5 6 7 8

Slope 0.19

Eσ 1

– σ 0

× 1

03

λ½bε1( ) × 103

21

ε × 10221 [(λ 2) – (λ1) ] (μm)2

–12

–12

–1

Slope 0.12 MN × m 2–3

Figure 4.40. (a) A plot of � versus �1/2, (b) a plot of ��1 −�0�/E vs �b�/�1/2/ and (c) a plotbetween ��0 and �

1/22 −

1/21 �.




the second segment. In martensitic structures, the initial flow showing a high workhardening rate can be attributed to the localized flow of some regions of high stressconcentration, while the second segment corresponds to the parabolic stress–straincurve characteristic of the deformation regime where work hardening is essentiallycontrolled by the short-range interaction between gliding dislocations and thesteadily increasing geometrically necessary dislocations. This has been confirmedby demonstrating a linear relation (Figure 4.40(b)) between the two dimensionlessquantities (�1 −�0�/E and (b�/�1/2, where �1 is the stress corresponding toa plastic strain �1, �0 is the extrapolated value of the true stress at � = 0 (asshown Figure 4.40(b) and is the mean free slip length between the partitioninginterfaces which are impenetrable to dislocations. The parameter corresponds tothe average width of Widmanstatten plates for the slow-cooled structure and tothe average packet size and twin spacing in the lath and the twinned martensite,respectively. Substituting M = 4, a value proposed by Armstrong et al. (1962) forhcp metals, the theoretical slope of a plot of (�1 −�0�/E versus (b�1/�

1/2 canbe estimated to be 0.17 which is quite close to that experimentally obtained. Thisanalysis points to the fact that the enhanced work hardening in internally twinnedZr–Ti martensites is due to the presence of closely spaced partitioning interfaces(which are predominantly {1011} twin boundaries).

The �0 values, obtained form the extrapolation shown in Figure 4.40, closelymatch with the 0.2% yield stress values for both the martensitic ′ and theWidmanstatten structures. In order to examine whether the influence of thegeometric slip distance, , on the yield stress of these structures follows the Hall–Petch relation, �0 values for a given alloy for the and ′ structures, representedby �0( ) and �0�

′�, respectively, can be expressed as follows:

�0� �= �i +K�X�−1/21 & �0�

′�= �i +K�X�−1/22

��0 = �0� ′�−�0� �= K�X�

[

−1/22 −

−1/22

](4.61)

In these equations, �i is the friction stress corresponding to zero strain value,K�X� is the Hall–Petch constant and the geometric slip distances for the andthe ′ structures are 1 and 2, respectively. It is implicit in Eq. 4.59 that theparameter �i has nearly the same values for the ′ and the structures for agiven alloy composition in spite of the difference in the dislocation densitiesin these. The Hall–Petch constant K�X� depends on alloy composition mainlythrough its dependence on the shear modulus, �. Since the shear moduli for Zr andTi are not much different, K�X� is not expected to alter significantly with alloycomposition in the binary Zr–Ti system. This justifies a linear relation between��0 and (−1/2

2 −−1/21 ), as shown in Figure 4.40(c). The slope of this plot has been




found to be close to the reported value of the Hall–Petch constant (Ramani andRodriguez 1970) in Zr alloys at 300 K. The applicability of the Hall–Petch relationfor estimating the increase in �0 due to the introduction of the martensitic structuredemonstrates that the refinement of the grain size factor essentially controls themartensite strengthening in this system.

The flow stress, � , can be expressed in terms of several components as perdifferent criteria chosen, viz., thermally activated and athermal, dependent on orindependent of grain size, plastic strain and solute content. Banerjee et al. (1978)have examined how the martensitic structure influences the different componentsof the flow stress of Zr–Ti alloys. On the basis of the preceding analysis, the flowstress, � , can be expressed as

� = �1 +�stat + K�X�1/2 +C�b�/�1/2 (4.62)

where �i is the friction stress, C represents the constant term C1EM−1/2 (see

Eq. 4.48) and �stat is the -independent component of the flow stress arisingfrom the statistically stored dislocations accumulated during plastic flow. Thecontribution of �stat in the case of Zr–Ti martensites has been shown to be verysmall and can be neglected.

Stress relaxation experiments have been used for measuring the athermal com-ponent, �M, of the flow stress, and the thermal component, �∗, of the flow stresshas been obtained by subtracting �� from the total flow stress, � (from the rela-tion, � = �� +�∗). The values of the activation area, A∗, associated with thethermally activated flow in both Widmanstatten and martensitic ′ structuresfor these alloys have also been determined from the stress relaxation data. Thesalient features of the results arrived at are as follows:

(1) The contribution of the martensitic structure (for both the dislocated lath andthe twinned plate martensites) to strengthening is mainly through an increasein the athermal component, ��, of the flow stress.

(2) The thermal component, �∗�, of the flow stress is essentially independent of

the Ti concentration in these Zr–Ti alloys, suggesting that the strengtheningdue to the substitutional solid solution arises mainly from an increase in thelong-range athermal stress. This is consistent with the results obtained in anumber of Zr- and Ti-based substitutional alloys (Conrad et al. 1972).

(3) The value of �∗ is independent of plastic strain, implying that dislocation–dislocation interactions do not constitute the rate-controlling mechanism ofthermally activated deformation.

(4) The values of the activation area for both the and ′ structures in Zr–Ti alloys remain within a narrow band between 25b2 and 35b2 where b2 is




the Burgers vector. This implies that the rate-controlling mechanism of thethermally activated flow remains unaltered with changes in the Ti content ofthese alloys and with the introduction of the martensite structure. The valuesof the activation area in the case of these martensites closely match with thatreported for annealed Zr containing about the same levels of interstitials (about600 ppm O2 and 60 ppm N2).

Based on the results of stress relaxation experiments, it could be inferred that therate-controlling thermally activated deformation mechanism is not altered by theintroduction of the martensitic structure and even by the addition of substitutionalalloying elements. This is consistent with the findings of Conrad et al. (1972) whohave demonstrated that substitutional alloying elements and /� phase distributiondo not affect the rate-controlling deformation mechanism in Ti-based alloys. Theyhave suggested that a mechanism involving dislocations overcoming the interstitialsolute obstacles is rate controlling during low temperature deformation of Ti alloys.However, there is some controversy over this issue, and an alternative view isthat the Peierls–Nabarro barrier, which is increased by interstitial additions, is therate-controlling obstacle.

The fact that �∗ is not affected by the introduction of the martensitic structureand the extent of plastic strain indicates that �∗ is contained within the frictionstress �i (Eq. 4.59), which can be resolved into two components, �∗ and �j(X).The former is essentially controlled by the interstitial level while the latter, a partof the athermal stress, ��, is independent of and � and depends on the natureand the extent, X, of substitutional alloying additions. The total flow stress can,therefore, be expressed as

� = �∗ +�j�X�+ K�X�−1/2 +C�b�/�1/2 (4.63)

While the increments in the initial yield stress and the work hardening rate dueto the introduction of the martensite structure are given by the terms K�X�−1/2

and C�b�/�1/2, respectively, the increments in the initial yield strength due tointerstitial and substitutional solid solution hardening are reflected in the terms,�∗ and �j(X), respectively.

4.5 MARTENSITIC TRANSFORMATION IN Ti–Ni SHAPEMEMORY ALLOYS

Near-equiatomic Ti–Ni alloys and many of their multicomponent derivatives arewell known for exhibiting the shape memory effect and pseudoelasticity. In these




Ti–Ni-based alloys, three martensitic phases occur. The fully annealed binary Ti–Ni alloys transform from the B2 parent phase directly to the monoclinic B19′

phase. In contrast, thermomechanically treated Ti–Ni alloys transform in two steps,namely the B2 phase to the trigonal R-phase (Goo and Sinclair 1985) and theR-phase to the monoclinic B19′ phase (Wang et al. 1968). Ternary Ti–Ni alloyscontaining a few percentage of Fe or Al undergo the same sequence, B2 → R →B19′, of transformations (Mitsumoto and Honma 1976). Ti-Ni-Cu alloys, in whichCu substitutes Ni by 5–15 at.%, also transform in two steps; but in this case, the B2phase first transforms into an orthorhombic B19 phase which finally transformsinto the monoclinic B19′ phase (Nam et al. 1990). The basic crystallography asso-ciated with these transformations has been discussed in the first part of this section.Some of the important microstructural characteristics of these martensites, whichare essentially responsible for imparting the shape memory property, are also high-lighted. The microstructural feature which is common to all shape memory alloysis such that the martensite plates are spatially organized in a very effective (Saburiand Nenno 1982) accommodating manner. The issue of self-accommodation in themartensite microstructure has been introduced earlier in the context of martensitemorphology. The importance of both macroscopic self-accommodation and micro-scopic self-accommodation in the martensitic microstructure of Ni–Ti alloys, asreported recently by Madangopal (1997), has also been discussed in this section.The shape memory effect is intimately linked with some associated phenomenasuch as thermoelastic equilibrium, pseudoelasticity and thermal arrest memoryeffect. An account of these related phenomena is included in this section mainlyto explain the interrelationship of these in terms of the microstructural responseof shape memory alloys to alterations in the applied stress and temperature.

4.5.1 Transformation sequencesThe transformation temperatures and the sequence of transformation of Ni–Tishape memory alloys are known to be sensitive to the chemical composition of thealloy, the memory imparting heat treatment, the prior cold work and the externallyapplied stress. Differential scanning calorimetry (DSC) and resistivity measure-ments have been mainly used for detecting the transformation sequences in Ni–Tialloys with varying compositions and prior thermomechanical histories. A fullyannealed, near-equiatomic NiTi alloy shows only a single peak in the DSC ther-mogram (Figure 4.41) during either a continuous heating or a continuous coolingexperiment, indicating that both the forward B2 → B19′ and the reverse B19′ →B2 transformations occur in a single step. The same material, however, after beingcold worked to a level of about 15% reduction in thickness, exhibits a two-steptransformation during cooling, the first and the exothermic peaks representing theB2 → R and the R → B19′ transformations. The transformation during heating




170 220 270 320 370

Temperature, T (K)

MsTR

Cooling

AfHeating

As

TR

Af

Ms

Mf

(a)

(b)

1.4

1.0

Hea

t flo

wE

ndot

herm

ic e

xoth

erm

ic

R/R

293

K

1.6

× 10

–3 J

/s

Figure 4.41. DSC thermogram showing transformation temperatures and the sequence of transfor-mations in a Ni–Ti shape memory alloy.

still occurs in a single step, giving rise to a single endothermic peak. The tempera-ture gap between the B2 → R and the R → B19′ transformations increases with anincrease in the extent of the prior cold work. The appearance of the R phase duringthe cooling of cold-worked NiTi alloys can be detected by monitoring electricalresistivity as a function of temperature or during in situ cooling of a sample ina TEM. Electron diffraction patterns, obtained from samples comprising the B2phase at a temperature close to the B2 → R transition, show diffuse intensitywith maxima located at reciprocal lattice positions which divide the B2 reciprocallattice vectors in three equal segments. The intensity maxima at the 1/3 positionsremain very weak and dark field micrographs using such reflections fail to revealany observable contrast. However, as the sample is cooled further, plates of the R




phase appear progressively and these plates get organized in a self-accommodatingconfiguration which is described later. In situ experiments have also indicatedthat R-phase plates can nucleate at relatively small strain centres, such as singledislocations. As these plates grow and gradually fill up the B2 grain, sharp 1/3reflections appear in the diffraction patterns. These reflections can be indexed onthe basis of the trigonal crystal structure of the R phase. On further cooling, theR phase transforms into the B19′ phase. Crystallographic features of these twomartensitic transformations are discussed in the next section.

4.5.2 Crystallography of the B2 → R transformationThe lattice correspondence between the B2 and the R phases is given in Table 4.12which lists the relative dispositions of the four crystallographic variants of theR phase with respect to the parent B2 phase. Self-accommodation is achievedin this transformation by an assembly of the variants which is illustrated inFigure 4.42. While between variants 1 and 2 and between variants 3 and 4 thereexists a relation of {1122}R type compound twinning, that between variants 2 and3 and between variants 1 and 4 is of {1121}R type compound twinning. As shownin Figure 4.42(b), the {1121}R-type twin planes are derived from {100}B2 typemirror planes and the {1122}R type twin planes are derived from {110}B2-typemirror planes of the parent phase. It is also evident from Figure 4.42 that sixkinds of equivalent arrangements are possible (two for each of the <001> axes ofthe parent phase) among the four variants generated from a single parent crystal.Two distinct morphologies, as illustrated in the light micrograph in Figure 4.42,are possible, one characterized by a stack of twin-related bands and the other bya herring bone morphology in which the four variants meet along an invariantline parallel <001>B2 axis. In both cases, the planar interfaces separating adjacentvariants are all twin planes.

4.5.3 Crystallography of the B2 → B19 transformationThe lattice correspondence between the B2 structure and the orthorhombic B19structure is shown in Figure 4.43, in which i′, j′ and k′ represent the orthorhombic

Table 4.12. Lattice correspondence between B2 (parent) and R phases.

Variant [100]R [010]R [001]R (001)R

1 [121]p [112]p [111]p [111]p

2 [211]p �112�p �111�p [111]p

3 [121]p [112]p [111]p [111]p

4 [211]p [112�p [111]p [111]p




(a)

2

1

12

21

141 14

1

12

2414

424

3

2

2

24

3 213

1 4

[100]

[001]

[010]

(110) (010) (110)

4

4

2 3

(b)

Figure 4.42. (a) Schematic of an optical micrograph of self-accommodation of the R phase. Thevariant numbers are assigned on the basis of the lattice correspondence described. (b) Three-dimensional arrangement of self-accommodating R-phase variants.

axes while i, j and k correspond to the cubic axes. The k′-axis can be made parallelto one of the cube axes, and in such a case, the i′ and j′ axes can be oriented in twodistinct manners. Therefore, six possible correspondence variants are present. Therelative orientations of these six variants, numbered 1–6 with respect to the axessystem of the parent B2 crystal, are indicated in Table 4.13. The homogeneousdeformation matrix, T , which is associated with the B2 → B19 transformation isgiven by

T = 1/ao

⎡⎢⎣a 0 0

0 b/√

2 0

0 0 c/√

2

⎤⎥⎦ (4.64)




Ni atoms

Ti atomsk,k ′

j

j ′i ′

i ′

bj′cj′

aj ′ k,k ′

β

Figure 4.43. The lattice correspondence between the B2 structure and the orthorhombic B19structure.

Table 4.13. B2-martensite lattice correspondence.

Variant [100]m [010]m [001]m

1 [100]c [011]c �011�c1′ �100�c �011�c �011�c2 [100]c �011�c �011�c2′ �100�c �011�c �011�c3 [010]c �101�c [101]c

3′ �010�c �101�c [101]c

4 [010]c [101]c �101�c4′ �010�c �101�c �101�c5 [001]c �110�c �101�c5′ �001�c �110�c [110]c

6 [001]c [110]c �110�c6′ �001�c �110�c �110�c




(b)(a)

(100)P

(010)P

(011)P

2 4

6

(101)P

(001)P

(110)P

[100][010]

[001]3

5

2 4

16

Figure 4.44. Self-accommodation model of B19 phases.

where ao is the lattice parameter of the B2 structure while a, b and c are the latticeparameters of the B19 structure. Two types of twin relations, namely (111) [211]O

type I and (011) [011]O compound twins, exist between variants which come incontact along a planar surface. The subscript O refers to the orthorhombic B19structure. Four �111�O-type twin interfaces (derived from parent �110�B2 planes)and six �011�O-type twin interfaces (derived from �100�B2 planes) separate adja-cent variants in the self-accommodating assembly of martensite variants producedin this transformation. Self-accommodation model of B19 phases is shown inFigure 4.44.

4.5.4 Crystallography of the B2 → B19′ transformationThe B2 → B19′ transformation can be conceptually divided into two components,namely the B2→ orthorhombic B19 and the B19→ monoclinic B19′ transfor-mations. As discussed in the previous section, six possible variants of latticecorrespondence are possible between the B2 and B19 structures. The lattice shearwhich transforms the orthorhombic structure into the monoclinic structure is asimple shear acting on the (100)O plane along the [001]O direction. Depending onthe sign of this shear, i.e. whether it is along the [001]O or the �001�O direction,two monoclinic variants can be generated from each of the orthorhombic vari-ants; these monoclinic variants are designated as 1,1′, 2,2′, 3,3′, 4,4′, 5,5′ and6,6′. Therefore, a total of 12 variants can form from a single B2 crystal and theirrelative orientations are also indicated in Table 4.13.

The homogeneous deformation matrix, T ′, which converts the parent B2 crystalinto the monoclinic martensite is given by

T ′ = ao

⎡⎢⎣a 0 c cos�/

√2

0 b/√

2 0

0 0 c sin�/√

2

⎤⎥⎦ (4.65)




where � defines the angle between the a- and the c-axes. Considering the case of aspecific variant, 6′, the Bain deformation B�6′� can be calculated by incorporatingthe necessary axis transformation matrix R�6′�:

R�6′�=⎡⎣ 0 1 1

0 −1 −1−1 0 0

⎤⎦ (4.66)

B�6′� can be expressed as B�6′�= �R�6′�� T� �R�6′��−1 and can be shown to be

B�6′�=⎡⎢⎣i j 0

j i 0

l l k

⎤⎥⎦ (4.67)

where j = √2/4 amo�−b+ c sin��, k = c cos�/2 amo, l = a/ao and i = √

2/4 ao

�b+ c sin��Substituting the following values of lattice parameters (Miyazaki et al. 1983),

ao = 0�3015 nm, a = 0�2889 nm, b = 0�4120 nm, c = 0�4662 nm, and � = B�6′�can be evaluated as

B�6′�=⎡⎢⎣

1�0213 0�0550 0�0

0�0550 1�0213 0�0

‘0�0907 ‘0�0907 0�9582

⎤⎥⎦ (4.68)

Bain deformations for the other variants can be computed using the appropriateaxis transformation matrices.

The LIS in these martensites invariably occurs by twinning. As will be discussedlater, the propagation of twin interfaces has a pivotal role to play in the processof shape recovery. The basis of the selection of the twinning systems and theirexperimental identification are briefly covered here.

In order to satisfy the condition of equivalent lattice correspondences for thetwo twin components of a martensite plate, it is necessary that either the twinplane (K1 plane) is derived from a mirror plane of the parent crystal or the 2

direction is derived from a diad of the parent. Examining all the possible twinsystems, Madangopal and Banerjee (1992) have arrived at the conclusion that onlyfour twin systems, viz., �111�M type I, the conjugate [112]M type II, (011)M typeI and the conjugate [011]M type II can qualify to be the lattice invariant shearsystem in this case. Examples of type I and type II twins in the Ni–Ti martensiteare illustrated in Figure 4.45. Since type I and type II twins are conjugate to eachother, the twinning shear is the same in the two twinning modes. In type I twinning




Figure 4.45. TEM micrographs showing the (a) type I and (b) type II twinning as the lattice invariantshear.

the K1 plane, being of rational indices, will be fully coherent. In contrast, theirrational K1 planes of type II twins generally consist of rational ledges and steps(Figure 4.45). In this respect, type I twin interfaces are energetically favourable,and as a consequence, the mean spacing between the twin interfaces is usuallymuch smaller for type I twins than that for type II twins.

Early TEM observations on the as-quenched Ni–Ti monoclinic martensites haveshown the presence of �111�M type I and/or {001}M compound twins (Otsukaet al. 1971, Gupta and Johnson 1973). Knowles and Smith (1981a,b) have detected<011�M type II twins and have pointed out that the {001}M compound twins donot yield any habit plane solution on the basis of the phenomenological crystal-lographic theory. The presence of {001}M compound twins in Ni–Ti martensiteshas been attributed to the plastic accommodation effect. The operation of differentLIS systems in successive generations of martensites appears to be responsible forthe occurrence of such a variety of internal twins in this system.

4.5.5 Self-accommodating morphology of Ni–Ti martensite platesIt has been shown earlier that self-accommodating arrangements of different orien-tation variants of martensite crystals can minimize the strain energy of the assem-bly. Mutual interactions of the elastic strain fields of individual martensite platesneed to be examined in determining the extent of self-accommodation for a given




Table 4.14. The 24 habit plane variants (HPV) and the matching correspondencevariant-combinations (CVC).

HPV CVC HPV CVC HPV CVC

1�+� 1-2 3�+� 3-4 5�+� 5-61�−� 1-2′ 3�−� 3-4 5�−� 5-6′

1′�+� 1′-2 3′�+� 3′-4 5′�+� 5′-61′�−� 1′-2′ 3′�−� 3′-4′ 5′�−� 5′-6′

2�+� 2-1 4�+� 4-3 6�+� 6-52�−� 2-1′ 4�−� 4-3′ 6�−� 6-5′

2′�+� 2′-1 4′�+� 4′-3 6′�+� 6′-52′�−� 2′-1′ 4′�−� 4′-3′ 6′�−� 6′-5′

The first and second numerals designate the major and minor twin-related regions in the habit planevariant, respectively.

group of martensite plates. In the case of the martensitic transformation from theB2 to the monoclinic (B19′) phase, 12 variants of lattice correspondence designatedas 1,1′, 2,2′, 6,6′, as described in Table 4.14, are possible. Each of these orientationvariants has two possible directions of lattice invariant shear, making a total of 24habit plane variants of martensite plates designated as 1�+�, 1�−�, 1′�+�, 1′�−�,2�+� � � � 6′�+�, 6′�−�. The signs (+) and (−) represent the two equivalent latticeinvariant shears which can operate in opposite directions for a given correspon-dence variant. The correspondence variant combinations present in each type ofmartensite plate (i.e. a given habit plane variant) are listed in Table 4.14.

The DSA accomplished in different groups of martensite plate variants hasbeen examined in detail by Madangopal et al. (1991, 1993, 1994) Madangopaland Banerjee (1992) and Banerjee and Madangopal (1996). Possible groups of 2-,3- and 4-variants have been considered using a simplifying assumption that thevolume fraction of each of the variants in a group is the same. Table 4.15 showsthe computed values of DSA corresponding to several 2-, 3- and 4-variant groupsof martensite plates.

TEM observations on the morphology of the Ni–Ti martensitic structure haverevealed that the primary plates which form first are frequently configured in a“V”-shaped pattern (Figure 4.46(a)) which can be described as a 2-variant group.Primary plates which are large, some of them spanning across the prior B2 grains,are also widely spaced. The identification of component variants of primaryplates making a “V”-shaped pattern has brought out the fact that they invariablybelong to a 2-variant group which corresponds to a high DSA (typically 3�+�and 6′�+� having a DSA value of 83.4%). Frequent occurrence of such two plategroups suggests that the formation of a primary plate promotes the nucleation ofa specific second plate variant in its vicinity in such a manner that the stress field




Table 4.15. The degree of self-accommodation (DSA) for the possible straincoupled plate groups of the (−) habit plane variant of the Ni–Ti martensite.

Coupling axis Plate group DSA

Tension�111�c a. 1�−�+4′�−�+5�−� 34.1

b. 1�−�+1�+�+4′�+� 26.7

c. 1�−�+1�+�+5�−� 25.4

�011�c 1�−�+1�+�+1′�+�+1′�−� 49.0

�001�c a. 1�−�+1′�+�+2�−�+2′�+� 64.4

b. 1�−�+2�−�+3′�+�+4′�+� 85.7

Compression�111�c a. 1�−�+3′�+�+6′�+� 89.9

b. 1�−�+1�+�+3′�+� 71.1

c. 1�−�+1�+�+6′�+� 55.4

�101�c 1�−�+2′�+�+5�−�+6′�+� 50.1

�100�c a. 1�−�+1′�+�+2�−�+1′�+� 69.4

b. 1�−�+1�+�+1′�+�+1′�−� 49.0

Figure 4.46. TEM micrograph showing (a) V-shape configured two variant combination of Ni–Timartensite and (b) three variant Ni–Ti martensite grouping.

created by the first aids the nucleation of the second. The formation of a thirdvariant 6′�−�, as is shown in Figure 4.46(b), results in a further improvement inself-accommodation. Primary plates partition the parent B2 grain progressivelyand a self-similar scaled down structure forms within the partitioned volume insuccessive stages in pursuit of achieving a higher DSA (thereby reducing thestrain energy density of the assembly of martensite plates and the matrix of the




untransformed B2 grain). It is because of the tendency of self-accommodationand of the thermoelastic nature of the martensite plates in Ni–Ti- based systemsthe martensite plates of successive generations remain elastically coupled. In asystem like this, the parent to martensite transformation attains completion belowMf without leaving any untransformed B2 phase. Madangopal (1997) has shownthat the regions left untransformed at a late stage of the transformation processare filled up by tetrahedral groups of three variant martensite plates as shownin Figure 4.47. Detailed crystallographic analysis has revealed that this groupconsists of three internally twinned plates, A, B and C, the correspondence variantspresent in each of the plates being, A: 1, 2′; B: 5′, 6′ and C: 4, 3. The habit planevariants for the plates A, B and C have been identified as 1�−�, 5′�−� and 4�+�,respectively. It is interesting to note that the interfaces between the plates A, B andC are �110�-type mirror planes of the parent B2 crystal. The orientation analysisof three plate clusters has also shown that the A–B, A–C and B–C interfaces are�111�A, �111�A and �111�C type I twin planes (corresponding to the major twincomponent of each of the plates. It has also been reported (Madangopal 1997) thatin some cases both major and minor twin variants of a plate find twin orientationsacross a given intervariant plane (as in the A–B case). In other cases (as in theA–C and the B–C cases) though the major twin orientations across the intervariantinterface are mutually twin related, the minor twin orientations are not twinrelated. In such situations the minor twin orientations taper off as they approachthe intervariant interfaces which remain twin related all along the plane of contact.

The shape memory phenomenon which has been described in the followingsection requires microstructural reversibility of the polydomain configuration ofthe martensitic structure. This can be attained only if the interfaces separatingadjacent variants remain coherent and glissile which ensures that both forward andbackward movements of these interfaces can be induced by suitable changes intemperature and/or in applied stress. The coherency between adjacent variants can

Figure 4.47. TEM micrograph showing self-accommodating tetrahedral group of three variantmartensite plates in Ni–Ti martensite.




be easily established when they are related by relations derived from a mirror planeor a diad of the parent crystal. The intervariant interfaces, A–B, A–C and B–C, inthe group of three variants described earlier satisfy this condition. Microstructuresof Ni–Ti martensites have shown the following types of interfaces preponderantly:(a) (111) type I and [011] type II twins within internally twinned martensite plates(Figure 4.45) and (b) between two adjacent internally twinned plates in whichthe major twin fractions have the same orientation while the minor twin fractionsare not mutually twin related. In such cases, there is no “boundary” betweenthe adjacent plates (Figure 4.48(a)), while the minor twin fractions taper as theyapproach the habit plane making a line contact between non-twin-related variants(Figure 4.48(b)). In case neither of the twin components of a plate have a twinrelation with those of a neighbouring plate, a new “bridge” variant is createdbetween the plates (Figure 4.48(c)). The bridge variant establishes twin relationon either side.

An examination of relative orientations of a given variant with all other variantsbrings out the fact that in a majority of combinations a twin relationship betweenadjacent plates can be established. However, in several cases, tapering of theminor twin fraction or formation of a bridge variant is necessary for makingthe intervariant interface a twin plane. Table 4.16 shows the orientation relations

Figure 4.48. Bright field micrograph showing (a) no boundary between adjacent plate, (b) taperingof minor twin fractions as they approach habit plane making a line contact between non-twin-relatedvariants and (c) neither of twin components of a plate have twin components with neighbour plates,a new bridge variant is created, (d) schematic of (c).




Table 4.16. Results of the 13 unique variant interactions of the 1�+� variant of NiTi martensite.

HPV CR Interface plane/bridge variant group (g)/plate (p)clustering∗

Remarks

1�−� – Boundaryless combination (g-g) Tapering of minor TF1′�+� T (100)c//(100)m�1 (g-g) Tapering of minor TF1′�−� T (100)c//(100)m�1//(100)m�2 (p-p) Single interface

or �011�c // (001)m�1 (p-p) Tapering of minor TF2′�+� T �010�c//(011)m�1 (g-g) Tapering of minor TF2′�−� T (010)c//(0,11)m�1//(011)m�2 (p-p) Single interface derived

from conjugate type 1 mode(0.72 01)c//(0.7211)m�2 (p-p) Tapering of minor TF

3′�−� T (110)c//(111)m�1//(111)m�2 (g-g) Single interface4′�−� T (1 10)c//(111)m�1//(111)m�2 (g-g) Single interface6′�+� T (101)c//(111)m�1 (g-g) Taper twin5�−� T (101)c//(111)m, 1 (g-g) Taper twin3�−� B Possible bridge variant 1′�+� (g-g) **5′�+� B Possible bridge variant 1′�−� (g-g) **3�+� B Possible bridge variant 2′�−� (g-g)4�+� B Possible bridge variant 1′�−� (g-g)

HPV, habit plane variant; CR, crystallographic relationship; T, twin variant; B, bridge variant; TF, twin fraction;∗ the (p-p) and (g-g) combinations are assuming {100}c clustering,∗∗ can also form twin interface by tapering of the major twin fraction.

between the 1�+� variant with 13 other habit plane variants with which intervarianttwin interfaces can be formed.

4.5.6 Shape memory effectThe shape memory effect and the pseudoelastic deformation in alloys exhibitingthe martensitic transformation can be described using a set of hypothetical stress–strain plots corresponding to different temperatures (Figure 4.49). Let us considera tensile test sample of an alloy exhibiting the shape memory effect which isbeing subjected to straining in a hard testing machine, with the stress developedin the sample being continuously recorded. When the testing is carried out ata temperature T1 > Md, the stress–strain plot shows elastic deformation up to ahigh stress value, followed by a limited plastic flow and finally a sudden brittlefracture. The parent B2 phase is stable with respect to any phase transformation atT1, and thus the observed deformation behaviour is consistent with that expectedfrom a brittle ordered intermetallic compound. At the testing temperature, T2�Md >T2 > Af�, the B2 phase is unstable with respect to the stress-induced martensitictransformation which is induced when the stress attains a threshold value indicatedby the point 1 on the stress–strain plot. The plastic flow from 1 to 2 corresponds tothe formation of increasing volume fractions of the martensite. While unloading the




1

43

5 6

82

d

b

a 7 109

Recoverablepseudoplasticstrain

Temperature

Strain

Str

ess

Pseudoplasticdeformation

M f > T3

A f A sM s

M f

Md > T2 > AfT1 > M

d

c

8′

Figure 4.49. A schematic drawing showing shape memory effect and pseudoelastic deformation inalloys exhibiting martensitic transformations.

sample, the stress falls from 2 to 3 in a manner similar to elastic unloading. At thepoint 3, the volume fraction of the stress-induced martensitic starts decreasing andthe stress–strain plot follows the path indicated by 3 → 4. The closed stress–strainhysteresis loop indicates that the stress-induced martensitic structure completelyreverts back to the parent phase during unloading. The non-linear deformationwhich is recoverable on unloading is known as pseudoelastic deformation.

The stress–strain plot at T3�T3 <Mf� shows a deviation from elastic deformationat a fairly low stress value, resulting in a deformation plateau (indicated by 5 →6). Unloading from the point 6 shows an elastic unloading to the point 7. Theplastic strain at the point 7, however, is recoverable by heating the sample toa temperature higher than Af and hence is known as pseudoplastic strain. Therecovery of the pseudoplastic strain starts on heating to a temperature above Af .This strain recovery (shape recovery) process of a pseudoplastically deformedmaterial, when subjected to a heating cycle to go through the parent phase, isknown as the shape memory effect.

In case deformation at T3 is continued beyond the point 6, a second stage oflinear elastic deformation is encountered. The stress rises substantially to reachthe point 8 where a deviation from linearity is noticed. Specimens fracture at 9




after plastic flow occurs to a limited extent. Unloading from a point such as 8′

results in elastic recovery of strain in a linear manner (8′ to 9). On subsequentheating above Af , the strain recovery occurs only to a limited extent (9 to 10).

Components of some shape memory alloys, after being subjected to a numberof straining and thermal cycles (for strain recovery), acquire a two-way memory.In such cases, the component (or the specimen) remains in two states of strain(or two shapes) at two temperatures, one above Af and the other below Mf . Thetwo-way shape memory is also denoted by a strain – temperature cycle, abcd, asshown in Figure 4.49.

The schematic drawing (Figure 4.49) shows in an idealized manner differ-ent phenomena, namely pseudoelastic deformation (or rubber-like behaviour),pseudoplastic deformation, shape memory and two-way memory effects. Inten-sive research on these phenomena during the last four decades has establishedthe sequence of structural changes which occur during these processes and aredescribed in detail in several reviews (Delaey et al. 1974, Schetky 1979, Wayman1980, Tadaki et al. 1988). Physical processes accompanying these individual stepsare not exactly identical for all the shape memory alloys. In spite of the differ-ences between the systems, one can attempt an oversimplified description of theprocesses responsible for the aforementioned phenomena. Such an attempt is madein the following paragraphs.

At a temperature between Af and Md, the latter being the upper temperaturelimit for the stress-assisted martensitic transformation, the austenite to martensitetransformation can be induced when the chemical driving force for the transforma-tion is augmented by the applied mechanical stress. The deviation from the linearelastic behaviour is noticed at a value of stress which is adequate for initiatingthe stress-assisted transformation. With a further increase in stress, the volumefraction of the martensite phase increases, the mechanical work done on the sys-tem by the applied stress being entirely used up for the creation of the metastablemartensitic phase. The martensitic phase forming under this condition remainsin thermoelastic equilibrium, and it is, therefore, possible to reverse the process(i.e. to induce the martensite to austenite transformation) once the stress level islowered. The unloading path, as shown by 2-3-4, consists of pure elastic unloadingfrom 2 to 3 followed by the martensite to austenite reversion from 3 to 4. Theloop, therefore, represents the hysteresis in the stress–strain relation with respectto the stress-assisted martensitic transformation.

Deformation of the fully martensitic structure, at temperatures below Mf , showspseudoplastic flow (from point 5 to 6) which can be completely recovered by thethermal cycle as described earlier. This can be contrasted with the usual plastic flowof metals and alloys by slip in which the slipped regions of crystalline grains areshifted to new positions having identical atomic surroundings. Since the slip process




is not reversible, there is no tendency of the slipped regions to retrace the defor-mation path. The slip mechanism cannot, therefore, explain the recoverable pseu-doplastic deformation. The starting structure, being fully martensitic, consists of anaggregate of different variants of plates which are self-assembled in a manner suchthat the strain energy of the assembly is minimized. It has also been shown that amajority of the intervariant interfaces satisfy a twinning relationship across them.On application of an external stress, some of the orientation variants grow at theexpense of their neighbouring orientations. Under the given system of the exter-nal stress, some variants are favourably oriented for growth while others are not.Though pseudoplastic flow can be accomplished by such a reorientation processwhich does not revert during unloading, the self-accommodation achieved in the vir-gin martensitic structure is disturbed. A thermal cycle through the austenite phasebrings back the self-assembly and in that process nullifies the pseudoplastic strain.

A component of a shape memory alloy, when cycled a number of times throughthe process of pseudoplastic deformation and shape recovery, develops a two-way memory. Such a component assumes two shapes given by two states ofstrain corresponding to two temperatures, one below Mf and the other above Af .The two-way memory arises due to accumulation of residual plastic deformationin the material during the course of repeated thermal cycling. The locked-inresidual stress eventually builds up to such a level that it can drive pseudoplasticdeformation without the introduction of any externally applied stress. The shaperecovery occurs in the usual manner during the heating-up step.

In the foregoing paragraphs, it has been brought out that the martensitic trans-formation plays the pivotal role in bringing about shape memory and relatedphenomena. It is also true that all systems, metallic or ceramic, in which martensitictransformations occur do not exhibit the shape memory effect with a significantextent of shape recovery. It is, therefore, important to identify the criteria whichmust be fulfilled for a system to display shape memory up to a few percentageof pseudoplastic strain. These criteria are as follows. (a) The first and foremostcondition is that the system must display a thermoelastic martensitic transforma-tion. This condition essentially implies that the magnitude of transformation strainis not large enough to cause irreversible plastic flow in either the matrix or themartensitic phase. (b) Martensite plates in the product microstructure must forma self-accommodating assembly. The minimization of the overall strain energyresults in a strain coupling between different martensite plates. (c) Interfacesbetween adjacent martensite plates should be such that they can propagate eitherway without losing atomic registry. Such a condition is met if a great majorityof these interfaces satisfy a twin relationship across these interfaces. (d) Atomicordering of both the austenite and the martensite phases favours the occurrence ofthe shape memory effect, though this is not an essential requirement. The presence




of atomic ordering restricts the number of variants for the martensitic transforma-tion and also increases the threshold stress at which thermoelastic reversibility islost due to plastic deformation.

The criteria which are listed here are based on Ni–Ti, In–Tl, Au–Cd and Cu-based shape memory alloys, which were the systems in which this phenomenon wasreported initially. In recent times, this phenomenon has been observed in severalother alloys such as intermetallics and ceramic systems. A number of Fe-basedalloys including Fe-Ni-C, Fe-Mn-Si, Fe-Mn-Si-Cr-Ni and Fe-Ni-Co-Ti have shownshape memory effect to somewhat restricted recoverable plastic strain values.Currently, the search for shape memory alloys with higher recovery temperaturesand larger hysteresis is being pursued in several laboratories. Intermetallics basedon Ni–Al and Ti–Pd show good promise.

4.5.7 Reversion stress in a shape memory alloyThe performance of a shape memory device depends not only on the maximumlimit of the recoverable strain but also on the stress against which the strainrecovery can occur. The reversion stress �r is defined as the stress developed in asample constrained against recovery during heating to a temperature higher thanAf . Data available on reversion stress and on its dependence on prior plastic strainand temperature have been very limited. The method of the measurement of thereversion stress in shape memory alloys and the criteria which determine the upperlimit or �r have been reported by Madangopal et al. (1988).

The experiments involved (a) deforming fully martensitic tensile test samplesof Ni–Ti shape memory alloys to a plastic strain �p using a hard tensile testingmachine at a temperature below Mf , (b) unloading the sample to just above thezero load level, (c) arresting the crosshead of the testing machine at this position,(d) rapid heating of the sample to a temperature Tr �Tr > Af� by introduction ofa hot water bath and recording the rise in stress as the sample underwent thereversion process and (e) unloading from the maximum stress level, �r, main-taining the sample temperature at Tr. These steps are schematically illustrated inFigure 4.50(a). Thermal stresses introduced due to differential thermal expansionof the sample and the loading assembly have been determined to be 15 MPa, whichis much smaller compared to the measured �r values (100–500 MPa).

The influence of �p and Tr on �r and its saturation value is shown inFigure 4.50(b). Figure 4.51(a) shows �r values attained for different levels of initialplastic strain and the unloading paths (pseudoelastic) after attainment of �r. Theshape of this unloading curve closely matches with the pseudoelastic unloadingcurve (outer envelope of the stress–strain hysteresis loop shown in Figure 4.51(a).This suggests that the thermoelastic equilibrium attained after the developmentof �r in a constrained recovery process is similar to (close to but not identical




0

Rev

ersi

on s

tres

s (M

Pa)

, σr→

Constraint (%), εp →

353 K369 K378 K398 K423 K

0

0

0

00 0 0

87654321

100

200

300

400

500

600

(b)

Equiatomic Ni Ti SMA

Ms ~ 318 KAs ~ 323 K

(a)

T1 < Mf

σr

M d > T2 > A f

Str

ess

(σ)

→

Stress (ε) →

Figure 4.50. (a) Schematic stress–strain plot showing the different steps involved in the measurementof the reversion stress. (b) The variation of the reversion stress with temperature and prestrain.




300

200

100

318 K

313 K

308 K

303 K

322 K

σ r →

10

– Crosshead Stationary only chart movement– Thermal perturbations

Tr ~ 313 K

(a)

Str

ess

(MP

a) →

Str

ess

(MP

a) →

Strain (%) →

Strain (%) →

32.521.510.500

50

100

150

200

250

300

350

(b)

Figure 4.51. (a) The unloading path from � superimposed on the correspondence pseudoelasticloading–unloading loop. (b) The change in � with thermal perturbation.

with) that prevailing in the unloading path of the pseudoelastic stress–strain loop.Therefore, it is surmised that �r is nothing but the stress threshold which marksthe start of the reversion of stress-induced martensite (M) into austenite (B2).The fact that a thermoelastic equilibrium is reached after the attainment of �r

is elegantly demonstrated in an experiment in which growth of martensite phaseand a consequent drop in stress have been recorded as the sample temperature isslightly lowered by spraying cold water on the sample (Figure 4.51(b)).

The important observations listed above point to the fact that on the attainmentof �r, the M → B2 transformation gets arrested and a thermoelastic equilibrium isestablished. The stress value at which the stress-induced martensite starts revertingback to the parent austenite phase is generally designated as ��As� which is marked




σAs

σr

σAs

σMs

σMsσ

ε

500

400

300

200

100

283

MsAs

Tr > As

Af

287 300

Temperature (K) Tr →

Str

ess

(MP

a), σ

Ms,σA

s,σ r

→

310 320 3300

Figure 4.52. The variation of ��As�, ��Ms� and �∗r with temperature.

on the pseudoelastic unloading path in Figure 4.52. This stress value essentiallycorresponds to the stress at which the reverse M → B2 transformation commencesduring the unloading process. On the other hand, as the temperature is raisedabove Af in the constrained condition, pseudoplastically deformed martensite pro-gressively transforms to the austenite (B2) phase and the elastic stress developingin the system (consisting of a mixture of M and B2 phases) gradually rises. Thisreversion stress, however, cannot rise above that corresponding to the M → B2thermoelastic equilibrium. This explains why �r attains a maximum value of �∗

r

which equals ��As�.It is possible to predict �∗

r by using a modified Clausius–Clapeyron equationrelating ��As� and the reversion temperature, Tr:

d�∗r

dTr

= −�Hp

T��(4.69)

where �H is the change in enthalpy for the B2→ M transformation, �� is themaximum pseudoplastic strain, T is the Af temperature and p is the density of theB2 phase.




4.5.8 Thermal arrest memory effectA new phenomenon named thermal arrest memory effect (TAME) reported inNi–Ti shape memory alloys is briefly discussed here. When the martensite toparent (M → P) transformation is arrested for a short duration at a temperature,TA, before completion of the transformation (i.e. As < TA <Af� and cooled downto a temperature below Mf , the martensite to parent (M → P) transformation ishalted during subsequent heating at the same temperature, TA. This means thatthe system remembers the temperature at which the transformation was initiallyarrested in the preceding heat treatment cycle.

The phenomenon can be seen clearly in continuous heating and cooling experi-ments in a differential scanning calorimeter (DSC). DSC thermograms illustratingthe exothermic (P→M) and the endothermic (M→P) transformations are shownin Figure 4.53(a). The steps involved in the experiment are described below.

(Arrest in the martensite to parent M→P transformation) Sequence A(Figure 4.53(b)) Step 1: The M→P transformation is arrested at a temperature

Temperature

4

3

3

4

2

12

1

275 K 360 K 280 K360K

360 K 280 K

Thermal arrestduring M-P

Thermalarrest

Thermalarrest 337.1 K

+

+

+

–

–

–

360 K

BA

Thermal cycle:

1 2 3 4

260 K M→ P

M← P

Thermal arrestduring P-M

A

B

ΔH, h

eat f

low

rat

e (a

rbitr

ary

units

)

NiTi

(a)

(b)

(c)

Figure 4.53. DSC thermogram exhibiting the effect of interrupting the M-P and P-M transformationsin Ni50Ti50 alloy.




TA = 337�1 K, between As and Af . Step 2: On cooling from 337.1 K to a tem-perature below Mf , the parent phase generated in step 1 is again transformed tomartensite. Step 3: During heating in the next cycle, the M→P endotherm splitsinto two, centered at TA. Step 4: During cooling from a temperature above Af , theoriginal exothermic peak for the P→M transformation is obtained.

(Arrest of the parent to martensitic (P→M) transformation) Sequence B[Figure 4.53(c))] Step 1: The P → M transformation is interrupted at TA

(= 314.8 K) between Ms and Mf . Step 2: On heating to a temperature above Af ,the partially transformed martensitic microstructure reverts back to the parentphase. Step 3: On subsequent cooling to a temperature below Mf , the originalexothermic peak is obtained without showing any effect of the thermal arrest instep 1. Step 4: During heating the original exothermic peak for the complete M→Ptransformation is observed.

These observations clearly demonstrate that a thermal arrest during the M→Ptransformation is remembered by the system during the next heating cycle. Suchis not the case for a thermal arrest during the P→M transformation. However,in cold-worked samples, TAME is observed for arrest of both P→M and M→Ptransformations. A detailed study of the influence of various factors like super-cooling, premartensitic R-phase transformation, repeated thermal arrest of TAMEhas been reported by Madangopal et al. (1994). Taking into account all theseobservations, the thermal arrest memory effect has been rationalized as follows:

The split in the M→P transformation, as seen in Figure 4.53(b), step 3, impliesthat the M→P transformation stops for a short duration at the arrest temperatureTA in the previous cycle. As discussed earlier, martensite plates tend to minimizethe elastic strain energy of the system by self-accommodation. Plates belongingto different generations (forming at different stages of the transformation) areelastically coupled. An interruption in the transformation, therefore, results individing the population of martensite plates into two distinct groups. At TA, thetemperature of arrest, the system consists of a mixture of the martensite and parentphases. On subsequent cooling, martensite plates transforming from this austeniticregion form a distinct (second) group. The M→P transformation for these twogroups of martensite does not remain linked during the second cycle. The reversetransformation of the second population occurs first and only after the completionof the transformation of this group, the M→P transformation of the other groupcommences. This sequence results in the introduction of a break in the endothermcorresponding to the M→P transformation.

The thermal arrest memory effect is directly linked with the self-accommodationprocess which binds one generation of martensite plates with the next by elasticcoupling. In case the self-accommodation in a small group of three or four platesis very effective, the magnitude of the unaccommodated strain will be small,




and consequently the extent of elastic coupling between generations of martensiteplates will also be small. Such a situation prevails in Cu-Zn-Al martensites whereself-accommodation of neighbouring martensite plates in a group is as high as98%. TAME is, therefore, observed rather weakly in this system.

4.6 TETRAGONAL � MONOCLINIC TRANSFORMATIONIN ZIRCONIA

4.6.1 Transformation characteristicsPolymorphic transformations in pure zirconia (ZrO2) have been briefly describedin Chapter 1. This transformation, first detected by Ruff and Ebert (1929) usinghigh-temperature X-ray diffraction, has been extensively studied using thermalanalysis, X-ray and electron diffraction, light and electron microscopy and elec-trical resistivity measurements. Wolten (1963) was the first to suggest that themonoclinic to tetragonal transformation is martensitic in nature. The importantcharacteristics of this transformation are summarized in the following.

(1) The high-temperature tetragonal phase cannot be retained on quenching toroom temperature.

(2) The transformation is athermal.(3) The observed growth rate of monoclinic platelets has been found to be consis-

tent with a velocity of the transformation front approaching that of sound insolids (Fehrenbacher and Jacobson 1965). The transformation kinetics, as mea-sured from thermal analysis experiments, show a burst-like behaviour (Maitiet al. 1972) during the reverse transformation (i.e. monoclinic to tetragonal).

(4) The transformation exhibits a large thermal hysteresis (Figure 4.54); theforward and the backward transitions occurring during cooling and heatingexperiments show transition temperatures in the vicinity of 1123 and 1323 K,respectively (Figure 4.54).

(5) The monoclinic to tetragonal transformation is accompanied by the suddenappearance of surface relief which corresponds to the formation of tetragonalplates during heating experiments.

(6) A strict orientation relationship exists between the parent and the productphases, as described in detail in the next section. The product phase, eitherplate-shaped or lenticular-shaped, always forms along specific habit planes.The lattice correspondence derived from the observed orientation relationshipsuggests that the transformation can be accomplished by atomic movements,primarily of O atoms, to an extent smaller than the interatomic distance andminor shifts of Zr atoms.




100

80

60

40

20

0600 700 800 900 1000 1100 1200

Temperature (°C)

Per

cent

tetr

agon

al p

hase

Decreasetemperature

Increasetemperature

Figure 4.54. Monoclinic to tetragonal transformation exhibiting a large thermal hysteresis duringcooling and heating.

All these experimental observations strongly point to the fact that both theforward and the backward transformations can occur by a martensitic mode whenthe cooling or the heating rate exceeds a certain critical value.

4.6.2 Orientation relation and lattice correspondenceThere has been some initial disagreement in literature on the exact orientation rela-tion between the monoclinic and the tetragonal phases (Bailey 1964, Wolten 1964,Smith and Newkirk 1965, Patil and Subbarao, 1967). The experimental difficultiesdue to the high temperature of the transformation have been mainly responsiblefor this discrepancy among results initially reported by these different investiga-tors (Bansal and Heuer 1972). They devised an ingenious experiment in whichsingle crystals of monoclinic ZrO2 were grown from a fluxed melt at temperaturesbelow the tetragonal to monoclinic transformation temperature. These single crys-tal monoclinic samples were heated to temperatures above the As temperature forthe monoclinic to tetragonal transformation and were subsequently cooled downto a temperature between the Ms and Mf temperatures. X-ray diffraction patternswere recorded in the initial monoclinic state, after the first reverse transformationand after cooling down below Ms. A comparison of the X-ray diffraction patternsobtained from the sample in these three stages yielded the orientation relationbetween the two phases. These orientation relations are consistent with the possible




LC A

LC B

LC C

atat

at

atat

at

at

at

ct

cmbm

am

am

am

bm

bm

cm

cm

(a)

(b)

Figure 4.55. (a) Tetragonal and monoclinic structures and (b) three possible lattice correspondencesconsistent with the orientation relation of the tetragonal and the monoclinic structures of ZrO2.

lattice correspondences which can be derived from an inspection of the tetragonaland the monoclinic structures Figure 4.55(a) of ZrO2. Three distinct lattice cor-respondences can be envisaged as illustrated in Figure 4.55(b). If a right-handedscrew convention is adopted for the unit cell axes, three lattice correspondences,designated as types A, B and C, can be defined, depending on which monoclinicaxis, am, bm or cm, is parallel to ct , the c-axis of the tetragonal crystal (suffixes“m” and “t” referring to monoclinic and tetragonal structures).

The type A lattice correspondence gives rise to the orientation relationship:(001)m�100�t and [100]m < 100 >t; which has been experimentally observedonly for the first reverse (monoclinic → tetragonal) transformation (Bansal andHeuer 1974). For the type A lattice correspondence, no surface relief has been




noticed, suggesting that this correspondence is not operative in the case of marten-sites forming on the surface of monocrystalline samples.

The type B lattice correspondence leads to the following two orientation rela-tionships, designated as B-1 and B-2:

B-1 * �100�m�010�t& �010�m�001�t& �001�m 9 to �100�t� i�e� �001�m�100�t

B-2 * �100�m 9 to �010�t& �010�m�001�t& �001�m�100�t� i�e� �100�m�010�t

These orientation relations are observed when the crystal is heated to the fullytetragonal phase field and subsequently cooled down to 1273 K (1000C) where it ispartially transformed into the monoclinic phase. The type C lattice correspondenceresults in the following two orientation relations, C-1 and C-2:

C-1 * �100�m�100�t& �010�m�010�t& �001�m 9 to �001�t� i�e� �001�m�001�t

C-2 * �100�m 9 to �100�t& �010�m�010�t& �001�m�001�t� i�e� �100�m�100�t

These orientation relations have been encountered in monoclinic martensiteplates forming in the later stages of the transformation (during cooling down from1273 K 1000C to room temperature). The directions and magnitudes of the princi-pal distortions associated with the three possible lattice correspondences betweentetragonal and monoclinic zirconia can be determined by analytical geometry, asillustrated in Figure 4.56. The principal distortions, 1, 2 and 3, are definedalong the orthonormal axes parallel to the [100]t, [010]t and [001]t directions,respectively. The problem is two dimensional as one principal axis is obtaineddirectly by inspection (Figure 4.55(b), 3bm). The other two principal distor-tions lie in the plane normal to the 3 direction. The method of determinationof principal distortions shows the changes in the relevant tetragonal plane. The

Y

Yt Ym Ym

XmXmXt

X

sin β

Tetragonal plane (a) Expansion(b) Shear

monoclinc plane

Figure 4.56. Schematic showing various steps of martensitic transformation.




Table 4.17. Computed principal strains for three lattice correspondences (LC).

LC A LC B LC C

1 Direction �0�0�8267�0�5627� �0�0�7383�−0�6745� �0�7860�−0�6183�0�Magnitude 1,0956 0.9337 0.9428

2 Direction �0�−0�5627�0�8267� �0�0�6745�0�7383� �0�6183�0�7860�0�Magnitude 0.9287 1.0897 1.1045

3 Direction �100� �100� �001�Magnitude 1.0128 1.0128 0.9896

distortion of this plane can be factorized into (a) a change in dimensions (expan-sion/contraction) and (b) a change in shape (simple shear). The calculated valuesfor the principal strains for the three lattice correspondences are given in Table 4.17(lattice parameters and principal strains).

4.6.3 Habit planeBansal and Heuer (1974) and Kriven et al. (1981) have analysed the crystallog-raphy of the tetragonal to monoclinic transformation in zirconia for the latticecorrespondences designated as type B and type C. The systems of lattice invariantshear considered for this analysis are the following:

For the type B lattice correspondence:

�1I0�t�001�t� �110�t�1I0�t� �100�t�001�t� �100�t�011�t

For the type C lattice correspondence:

�1I0�t�001�t� �1I0�t�110�t� �10I�t�010�t� �111�t�1I0�t

These shear systems, expressed in terms of tetragonal indices, correspond toslip and twin shears of the monoclinic system. Type A lattice correspondence,being the least likely to be operative in view of the highest lattice strains involved,has not been considered here. Experimental observations support the existenceof the correspondences B and C. There is some evidence to indicate that thecorrespondence B is favoured in the transformation in large grains of the tetragonalphase (Hugo et al. 1988). On the other hand, when the transformation occurs insmall particles or fibres, correspondence C is almost always observed (Muddleand Hanmik 1986). The application of the Bowles–Mackenzie (1954) theory withregard to the correspondences B and C yields solutions for habit plane indices,directions and magnitudes of shape strains, a detailed account of which has beenpresented by Kriven et al. (1981) (Table 4.18).




Table 4.18. Predicted habit planes and shape strains.

Lattice invariantshear system

Habit plane indices(PI)

Direction of shapestrain D1

Magnitude of shapestrain M1

(010 [101] −0�615 −0�067 0.120−0�334 −0�9840.730 0.160∼(212) ∼[061]−0�103 −0�620 0.120−0�988 −0�2810.122 0.716∼(192) ∼[710]

(011) [011] −0�049 −0�990 0.159−0�999 −0�1440.011 −0�000∼(010) ∼[710]−0�976 0.028 0.159−0�219 −1�0000.001 0.010∼(510) ∼[010]

(101)[101] 0.018 −0�979 0.164−1�000 −0�203−0�002 0.011∼(010) ∼[510]0.960 −0�097 0.1640.280 0.995−0�011 0.003∼(720) ∼[010]

(111) [011] 0.056 0.992 0.1700.996 0.126−0�075 0.018∼(010) ∼ [810]−0�978 0.026 0.170−0�206 −0�997−0�013 0.074∼(15102) ∼[010]

(010 [100] −0�383 −0�379 0.052Twin system −0�810 −0�801

0.455 −0�452∼(121) ∼ [121]−0�383 −0�379 0.520.810 −0�801−0�455 −0�452∼(121) ∼[121]

(Continued)




Table 4.18. (Continued)

Lattice invariantshear system

Habit plane indices(PI)

Direction of shapestrain D1

Magnitude of shapestrain M1

(110) [001] −0�178 −0�859 0.115−0�047 0.4941.006 0.134∼(106) ∼[741]−0�855 −0�132 0.1150.483 −0�0760.195 0.966∼(952) ∼[107]

Kelly and Ball (1986) and Kelly (1990) has pointed out that the grouping ofmartensitic variants which results in the formation of the important morphologicalfeatures of zirconia martensites can be explained on the basis of habit planepredictions for different lattice correspondences. For both the correspondences Band C, a number of different LIS systems lead to two groups of habit plane. The firstis in the region of (140)t for correspondence B (or (410)t for correspondence C).Depending on the correspondence adopted, the predicted orientation relationshipis B-2 or C-2 within about 1 and in both the cases twin-related variants arepredicted with a junction plane corresponding to (100)t. The twin relationship,though not precise, is normally less than 0�5 from an exact twin -relationship. Theconfiguration of the twin-related variants illustrated in Figure 4.57 is produced forthe B type lattice correspondence with the LIS on (011)t [211]t. It is to be notedthat a similar configuration with two variants nearly twin related across the (100)m

junction can also be generated when a different LIS system is chosen. The habitplane solutions for these cases are of the types {130}t and {140}t.

The other group of solutions gives a habit plane which, for the correspondenceB, is 2 or 3 from (100)t or (001)t and for the correspondence C is in the vicinityof (106)t (i.e. 3–12 from (001)t or (001)m). The predicted orientation relationshipis B-1 or C-1, respectively, and in both cases the theory predicts variants thatare twin related about (001)m within 0.2–0�7. An example of the twin-relatedconfiguration corresponding to the lattice correspondence B is schematically shownin Figure 4.57. A habit plane solution in the vicinity of (671)m or (761)m is predictedfor a restricted choice of LIS systems, the magnitude of the shear being higherthan those for the two sets of solutions discussed earlier. However, there is onesignificant exception, for correspondence C and for a LIS system of �110�t�I12�ta habit plane.




(130)T or (140)T

(100)T//(001)M

(130)T or (140)T

16°

Near (001)M i.e. (100)T

Near (001)M i.e. (100)T

(a)

(b)

Figure 4.57. The configuration of twin-related variants for B type lattice correspondence andLIS on (011)t [211]t (after P.M. Kelly, 1990).

4.7 TRANSFORMATION TOUGHENING OF PARTIALLYSTABILIZED ZIRCONIA (PSZ)

Phase diagrams of binary oxide systems, discussed in Chapter 1, show that the twotransition temperatures (corresponding to the cubic → tetragonal and tetragonal →monoclinic transformations) are lowered by alloying ZrO2 with CaO, MgO, Y2O3

and rare earth oxides. Partial stabilization of ZrO2, which results in a structurecontaining a mixture of the cubic and the monoclinic (or the tetragonal) phases,leads to an improvement in the thermal shock resistance and the toughness ofzirconia ceramics. Reductions in the thermal expansion coefficient and in thevolume change associated with the tetragonal → monoclinic phase transformationare factors which contribute towards the improved thermal shock resistance ofPSZ (Subbarao 1981) (Table 4.19). The toughening, however, has been attributed

Table 4.19. Lattice parameters (in nm) and thermal expansion coefficients of zirconia as a functionof temperature.

Temperature (C) am bm cm � Temperature (C) at ct

956 (meas.) 0.51882 0.52142 0.53836 81 217′ 1152 (meas.) 0.51518 0.52724950 (calc.) 0.51881 0.52142 0.53835 81 22′ 950 (calc.) 0.51485 0.52692Thermal 1.031 0.135 1.468 1.160 1.608expansion coeff.(nm×10−6 C−1)




to a stress-induced martensitic transformation (Garvie et al. 1975, Porter andHeuer 1977, Evans and Cannon 1986, Kelly 1990). The martensitic transformationhas two beneficial effects in this context. First, the transformation plasticity canpartially relieve the applied stress; in other words, the energy of the transformationcan directly contribute towards an increase in the fracture energy. Second, thestrain associated with the transformed region may help counter the high tensilestresses at the tip of the advancing crack and resist its further propagation.

Thermal processing of PSZ to achieve the maximum strength and toughness hasbeen extensively studied and the results are summarized in several reviews (Pascoeet al. 1977, Nettleship and Stevens 1987, Evans and Heuer 1980, Subbarao 1981,Kelly and Rose 2002).

The principle of microstructural engineering of PSZ for enhancing fracturetoughness is based on distributing the tetragonal (and partly, the monoclinic) phaseparticles in the matrix of the cubic phase. The second phase may appear at grainboundaries during the sintering of zirconia, suitably alloyed for controlling therelative stabilities of the competing phases. Intragrain dispersion of the secondphase occurs either during cooling from a temperature above the solvus line orduring post-sintering heat treatments. Intragranular precipitates assume an ellip-soidal shape and are aligned along {100}c habits in order to minimize the strainenergy of the precipitate-matrix assembly. The optimum size of the precipitateparticles has been reported to be about 0.2 (m, as particles of still smaller sizesretain the tetragonal symmetry while larger particles transform into the monoclinicphase spontaneously.

The role of metastable tetragonal precipitates in a cubic matrix in tougheningPSZ has first been noted by Garvie et al. (1975) and later elucidated by Porter andHeuer (1979). It has been shown that all the precipitate particles within severalmicrometres of a crack possess a monoclinic symmetry, whereas all other particlesare tetragonal. This suggests that the stress field near the crack tip causes theparticles to transform into the monoclinic phase. By this mechanism, the elasticstress field at the crack tip can be dissipated through the formation of stress-inducedmartensite in the metastable tetragonal precipitates which also lose coherency inthe process. Further extension of the crack, therefore, requires the application of anadditional stress. Thus the stress-induced tetragonal to monoclinic transformationwithin the fine precipitates essentially operates as a crack blunting mechanismwhich is schematically illustrated in Figure 4.58.

The retention in the fine (smaller than 0.2 (m) particles of the metastabletetragonal phase, when they are distributed in the cubic ZrO2 matrix, can beexplained in terms of a much larger value of the strain energy associated with thecubic–monoclinic assembly in comparison with that associated with the cubic–tetragonal assembly (Porter and Heuer 1979). This strain energy difference more




Criticalcrack

Figure 4.58. Tetragonal to monoclinic transformation in zirconia with volume expansion resultingin microcracks around the particle; cracks propagating into the particle get deviated leading toincreased fracture resistance.

Table 4.20. Transverse rupture strength and fracture toughness of zirconia.

Transverse rupture strength (MPa) KIC (MN/m3/2�

Tetragonal+ cubic ZrO2 650 ≈7.1Monoclinic+ cubic ZrO2 250 3.7

(overaged at 1400C)Cubic ZrO2 (solution – 245 2.8

annealed at 1850C, 4 h)

than compensates for the difference in the chemical free energy between themonoclinic and the tetragonal phases in the case of fine particles. A compar-ison of the strength and the fracture toughness of zirconia ceramics in threedifferent microstructural states (Table 4.20) clearly indicates that the presence ofthe metastable tetragonal phase is directly responsible for enhancing the fracturetoughness and transverse rupture strength quite substantially. Such a conclusioncan also be arrived at from the observation that a nearly fully tetragonal material,prepared by fine particle technology and comprising a ZrO2–Y2O3 solid solution,shows a high strength at room temperature whereas a ZrO2–CeO2 solid solution,in which the tetragonal phase is stable at room temperature, is relatively weak.The latter material exhibits enhanced strength and toughness at liquid nitrogentemperature, where the tetragonal phase is metastable.

The phenomenon of toughening of PSZ by a stress-induced martensitic trans-formation can be compared with transformation-induced plasticity in TRIP steels,in spite of the fact that in the latter case the transformation occurs in the entireaustenite matrix while in the former only the fine dispersed precipitates take partin the martensitic transformation (Figure 4.58).




4.7.1 Crystallography of tetragonal → monoclinic transformation insmall particles

The mechanism of transformation toughening described in the preceding sectionessentially requires the fulfilment of the following conditions:

(1) The strain energy associated with the transformation should be of such amagnitude that particles below a certain critical size are retained in a metastabletetragonal state.

(2) It should be possible to trigger a stress-induced martensitic transformation inthese fine particles.

(3) The volume change associated with the transformation should be positive sothat it could counter the tensile stresses at the tips of advancing cracks.

(4) The magnitude of the macroscopic shear associated with the transformationshould be large for facilitating the occurrence of the stress-induced transfor-mation. At the same time, ceramics being incapable of tolerating large shearstrains, the overall shear strain associated with a single tetragonal particleshould not be very high. These conflicting requirements can be met if a num-ber of variants of the monoclinic martensite plates forming within a particlebelong to a self-accommodating group. In that case, a major part of the shearstrain associated with a single martensite variant can be neutralized by thoseassociated with the other variants present in the group. The crystallographyof the martensitic transformation in fine particles has been studied by Kelly(1990) mainly to identify the possible self-accommodating groups of marten-site variants which can form within these fine particles (Figure 4.59).

mmt

t

[001]m[001]t

Strain= 4.12%

[100]m[100]t

Strain// [010]

Strain = 0.51% = 0%

[100]m[100]t

Strain// [010]

Strain = 2.65% = 0%

[001]m[001]t

Strain=0.97%

(a) (b)

Figure 4.59. The four possible arrangements of twin-related variants together with the range ofstrain values predicted for the distortion.




It is important to note that TEM studies on PSZ have established (Kelley 1990)that each of the fine tetragonal particles undergoes a stress-induced transformation,leading to the formation of an array of alternate bands, twin related on either (100)m

or (001)m planes. These twins do not correspond to the LIS. Instead, these bandsare twin-related variants which have shape strains that lead to the cancellation ofthe shear component when the two variants are of equal thickness. The large shearcomponent associated with the formation of the first variant creates the “backstresses” which induce the formation of the twin-related variant with the oppositeshear strain. The accommodation of shear strain by the combination of these twin-related variants is responsible for the creation of the zig-zag arrangements of thevariants.

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Chapter 5

Ordering in Intermetallics

5.1 Introduction 3805.2 Theoretical Treatments 383

5.2.1 Alloy phase stability 3845.2.2 Order–disorder transformations 3865.2.3 The ground states of the Lenz and Ising model 3975.2.4 Special point ordering 4015.2.5 Concomitant clustering and ordering 4075.2.6 A case study: Ti–Al system 412

5.3 Transformations in Ti3Al-based alloys 4165.3.1 � → D019 ordering 4165.3.2 Phase transformations in �2-Ti3Al-Based systems 4175.3.3 Structural relationships 4215.3.4 Group/subgroup relations between BCC (Im3m),

HCP (P63/mmc) and ordered orthorhombic (Cmcm) phases 4245.3.5 Transformation sequences 4285.3.6 Phase reactions in Ti–Al–Nb system 432

5.4 Formation of Zr3Al 4365.4.1 Metastable Zr3Al (D019) phase 4375.4.2 Formation of the equilibrium Zr3Al (L12) phase 4395.4.3 �+Zr2Al → Zr3Al peritectoid reaction 441

5.5 Phase Transformation in �-TiAl-Based Systems 4435.5.1 Structural relationship between �2- and �-phases 4435.5.2 Phase reactions 4465.5.3 Transformation mechanisms 451

5.6 Site Occupancies in Ordered Ternary Alloys 4585.6.1 Ordering tie lines 4585.6.2 Kinetic modelling of B2 ordering in a ternary system 4605.6.3 Influence of binary interaction parameters 4625.6.4 B2 ordering in the Nb–Ti–Al system 464

References 465






Chapter 5

Ordering in Intermetallics

List of Symbols�

�k�ij : Short-range order parameter for the ij-pair in the kth coordination

shellB�: Bulk modulus of a superstructure �EF: Fermi energy��k�: The kth volume expansion coefficient for Etot

E�tot: Total electronic energy of a superstructure �

E�coh: Cohesive energy of a superstructure �

E�form: Formation energy of a superstructure ��: A superstructure�: A cluster: Long-range order parameter

J�t�� : Effective cluster interaction for the cluster � of type tj�k�� : The kth volume expansion coefficient for J

k: A wave vector in the reciprocal spacekB: Boltzmann constant

N�EF�: Density of states at the fermi level� �r�: Atomic occupation probability at position rp�i�n : Site occupation operator for a species i at a site n

��k�: Amplitude of the concentration wave with wave vector k�� : Probability of a -point cluster to occur in � configuration�i: A site operator which can take value +1�−1�, for a component A

(B) in a binary alloyT−

i : Instability temperature below which the disordered phase isunstable w.r.t. spontaneous ordering

T+i : Instability temperature above which the ordered phase is unstable

w.r.t. spontaneous disorderingTeq: Equilibrium temperatureT0: Temperature corresponding to equal free energies of two phasesTcs: Conditional spinodal temperature for the ordered phaseV�

0 : Equilibrium volume of a superstructure �

379




��t�� : Correlation function for a �-point cluster of type twij: Interaction potential between ‘ij′ pair of atomsG: Gibbs free energyF : Helmholtz free energy

�H : Enthalpy change for a given processG: Space group of a given crystalline phaseH: Subgroup of a given space group G

kF� kR: Rate constants for the forward and backward reactions,respectively

F� R: Pre-exponential factors for the forward and backward reactions,respectively

5.1 INTRODUCTION

In an alloy which can be approximated as an ideal solid solution, the distribution ofdifferent atomic species in the lattice is nearly statistical. The interaction energies,Vij , between atomic pairs, ij, are such that there is no preference for the formationof bonds between either the like or the unlike atoms. When a solid solution exhibitsa tendency to deviate from the ideal behaviour, one can visualize two alternatives.The like atoms tend to cluster together when there is a preference for the formationof bonds between like atoms. The second alternative is that the atoms arrangethemselves in an ordered array where specific atomic species occupy specificlattice sites. The clustering and ordering tendencies are, respectively, associatedwith a positive and a negative deviation from ideality, as the free energy of therespective systems deviates in the positive or in the negative direction from thefree energy of the ideal solid solution.

The phenomenon of ordering in an alloy can be described in terms of the simpleexample of the ordering of a bcc (A2) structure to a CsCl type (B2) superlattice(Figure 5.1(a)). Let us consider an equiatomic alloy of the elements A and B inwhich the atoms are distributed in the bcc lattice. Considering only the nearestneighbour interactions, one can intuitively see that if VAB, a measure of the attrac-tive interaction between the unlike atoms, is much stronger than the interactionbetween like atoms, VAA and VBB, there will be a tendency for A and B atoms to,respectively, select the body centre and the corner positions of the unit cell of thebcc lattice or the other way round. In case a perfect order is established, all Aatoms will have only B atoms as their first near neighbours and vice versa. Whenthe temperature of such an alloy is raised, the entropy factor plays an increasinglyimportant role and above the order–disorder transition temperature, Tc, the dis-tribution of atoms in the lattice sites becomes random. The competition between



Ordering in Intermetallics 381

B2 B32 D03

(a) (b) (c)

Figure 5.1. bcc-based ordered intermetallic phases which are the ground state superstructures underthe first and the second nearest neighbour (NN) pair approximation.

the entropy factor, which tries to randomize the system at elevated temperatures,and the enthalpy factor (resulting from interatomic interactions), which tends toorganize the atoms in an ordered array, dictates the state of the order. When theformation enthalpy of an ordered structure is very large, the state of order maypersist even up to the melting temperature, and in such cases an order to disordertransition is not encountered in the solid state.

Ti- and Zr-based alloys are essentially based on the hcp and bcc lattices, cor-responding to the two allotropic modifications of these elements. Studies on theorder–disorder transition in these alloys, therefore, include an examination of thepossible superlattice structures based on the bcc and hcp lattices, the determi-nation of ground states (energetically favourable superlattice structures at 0 K)and investigations regarding the mechanism of order evolution in these systems.The interest in order evolution is centred around the question as to whether theprocess occurs in a homogeneous manner by the development of a concentrationmodulation with appropriate wave vectors, followed by a gradual amplificationof such modulations or by a heterogeneous process involving the nucleation ofnearly perfect ordered particles in the disordered matrix, followed by the growthof these particles.

In the context of Ti and Zr alloys, the disorder–order transition has been studiedin detail in only a few systems. The most important among the structural transitionsinvolved in these are: (a) � (hcp)→ D019 in the Ti–Al, Ti–Sn and Ti–Ga systemsin the vicinity of the Ti3X (X = Al, Sn, Ga), and (b) � (bcc) → B2 in the Ti–Aland Zr–Al systems.

Theoretical treatments for the determination of thermodynamic parameters ofordering processes and experimental results which illustrate the essential featuresof these processes have been reviewed in this chapter.




Phase transformations (including phase reactions) involving ordered phases inthe aluminides of Ti and Zr have attracted attention ever since the emergence ofthese intermetallics as possible structural materials. Chronologically Zr3Al wasone of the first intermetallics to be considered as a structural material. The prospectof using stoichiometric Zr3Al as a pressure tube material in pressurized heavywater nuclear reactors (PHWRs), originally conceived by Schulson (1977), wasquite bright because of the exceptionally good resistance of this material againstirradiation creep, which happens to be one of the life-limiting factors under theprevailing service conditions. The presence of atomic order could be regardedas being essentially responsible for the superior irradiation creep resistance ofthis material. The intermetallic was designed on the basis of the maxim thatordering generally reduces the rates of migration of vacancies and interstitials,thereby enhancing the probability of annihilation of these defects. This, in turn,reduces the irradiation creep rate under a given stress. Some other virtues ofthis material, namely, a low thermal neutron absorption cross-section, promisingmechanical properties and good corrosion resistance in water in the temperature,pressure and radiation environment present in the PHWRs make it eminentlysuitable for the proposed application. The reason for which Zr3Al was not finallyaccepted for pressure tube application was the tendency of disordering, leadingeven to amorphization, of this alloy under irradiation and its poor room temperatureductility.

The equilibrium structure of Zr3Al is L12 (Cu3Au type) and this phase evolvesthrough a peritectoid reaction. A classical order–disorder transition cannot bestudied in this system as Zr3Al cannot be congruently disordered, unless radiationdisordering is resorted to. Several interesting aspects of phase transformations,such as the peritectoid reaction between the � and Zr2Al phases to form Zr3Al,the formation of a metastable Zr3Al phase (D019 structure) during precipitationfrom the supersaturated � (hcp) matrix of non-stoichiometric Zr–Al alloys and thecellular reaction leading to the formation of an �+Zr3Al�L12� lamellar aggregate,have been discussed in this chapter.

The story of the development of Ti aluminides is even more exciting. The needfor the development of suitable aerospace materials for use in the temperature rangeof 770–920 K has been felt for bridging the gap between the temperature rangespertinent to conventional high �-Ti alloys on one hand and Ni-based superalloyson the other. The binary intermetallic Ti3Al (�2-phase) has a specific modulus anda stress rupture resistance comparable to those of superalloys but the completeabsence of ductility at room temperature comes in the way of its being acceptedas a structural material. Blackburn and Smith (1978) were the first to demonstrateroom temperature ductility in a Ti3Al-based composition (Ti–24% Al–11% Nb)and this was followed by wide-ranging investigations on phase transformations




in and deformation behaviour of a large number of Ti aluminides. These studieshave revealed a variety of phase transformations in multicomponent Ti aluminides,one of the most exhaustively studied systems being the ternary Ti–Al–Nb. Theconstituent phases, the sequences and mechanisms of transformations and themechanical properties are quite distinct in different composition regimes for theTi–Al–Nb system. Based on these considerations, the transformation processescan be discussed in two distinct groups of alloys, one based on Ti3Al and the otherbased on TiAl (� phase, L10 structure).

The Ti3Al-based alloys can again be categorized on the basis of their �-stabilizercontent. Important alloys belonging to the first category are Ti–24Al–11Nb (Black-burn and Smith 1978) and Ti-25Al-8Nb-2Mo-2Ta (Marquardt et al. 1989). Theycontain �-stabilizers in the range of 10–12 at.%. Alloys with about 14–17 at.%�-stabilizers, including Ti-24Al-(14-15)Nb (Blackburn and Smith 1978) and Ti-24Al-10Nb-3V-1Mo (Blackburn and Smith 1989), can be grouped into the secondcategory. The third category of alloys, containing 25–30 at.% �-stabilizers, hasbeen developed more recently (Rowe 1991, Gogia et al. 1993). Phase reactionsand mechanical properties associated with these three categories of Ti3Al-basedalloys are quite different and, therefore, the development of microstructure inthese groups of alloys has been discussed separately. The phase equilibria andsolid-state phase transformations in �-TiAl alloys, with the Al content varyingbetween 40 and 50 at.%, have also attracted considerable attention in the recentyears. This is primarily because of the fact that �-TiAl-based alloys offer anattractive combination of properties, namely, low density, substantial high tem-perature strength and good creep as well as oxidation resistance. These alloys,at room temperature, usually consist of the �2 (D019) and � (L10) phases. The�2-phase disorders into the hcp �-phase above about 1420 K. The occurrence ofthe eutectoid reaction �→�2 +�, the existence of unique orientation relationshipsconnecting the three phases and the sensitivity of phase transformation modes tothe cooling rate are some of the factors responsible for the development of manyinteresting microstructures in �-TiAl alloys. An attempt has been made in thischapter to rationalize the variety of microstructures reported in �-TiAl alloys interms of some plausible phase transformation schemes.

5.2 THEORETICAL TREATMENTS

In this section, we will present some state-of-the-art theoretical tools to studyphase stability and ordering in alloys. Since most commercial alloys are complexmultiphase mixtures, it is important to have knowledge of all the possible alloyphases that can occur in a given system: terminal solid solutions, possible ordered




intermetallic phases and other disordered solid solutions. Hence, the study of alloyphase stability and order–disorder transformations is a major area of researchtoday.

5.2.1 Alloy phase stabilityMore than 70 years ago, Hume-Rothery (1967) laid the foundations for the studyof alloy phase stability in relation to the electronic structure. He pointed out theconnection between the observed crystal structures and the electron concentration,which is related to the valence electron difference between the constituent elementsand to the composition.

Earlier theories of stability of alloy structures have identified the important roleof the density of states (DOS) curve for correlating the extent of phase stabilitywith the electron concentration (e/a). The DOS is defined as the number ofelectronic states per atom (or volume) with energies between certain values:

N�Eo� = �

4�3

∫ ∫E=Eo

dS� �Ek � (5.1)

where the integration is carried out over a constant energy surface in k-spacebetween Eo and Eo +�E and � is the cell volume. The first attempt in thisdirection was made by Jones (1962) in terms of relative electronic DOS curvesbetween any two simple alloy phases (�1 and �2). He tried to explain the phasecompetition between the �- and the �-phases in the Cu–Zn system in terms ofa rigid band model for the conduction electrons. The gist of the argument wasthat, within the low e/a range corresponding to the �-phase, the total DOS for the�-structure is higher than that for the �-structure which follows it, because of theprominent peak in the DOS curve associated with the �111� zone contact withthe Fermi surface. Thus the conduction electrons can be accommodated within alower total energy than that for the �-phase. As e/a increases, the �110� peakin the DOS curve of the �-phase structure becomes prominent, making it a morefavourable structure. Thus, at alloy compositions at which the Brillouin zoneboundary is just touched, a given crystal structure should be particularly favoured.These compositions correspond to those of Hume-Rothery phases.

Experiments have proved that in many metals, the Fermi surface is very differentfrom that of a free electron sphere, giving birth to the nearly free-electron theory.The deep potential energy wells at the centres of atoms play a very small role.In 1959, the pseudopotential theory (Harrison 1966, 1973) took roots; in this, itwas considered that as a conduction electron passes through the central regionof an ion in a metal, it not only experiences a strong electrostatic attraction tothe nucleus but also a strong repulsion from the core electrons. In some metals




like Na, Mg and Al, these two opposite contributions almost cancel, so that theions in these metals appear to be nearly transparent to the conduction electrons.It thus becomes possible to replace the true potential in the Schrödinger equationby a pseudopotential in which the Pauli repulsion is introduced to cancel out mostof the coulomb potential of the nucleus (which can be treated using perturbationmethods on the free electron theory).

The present status of the electronic structure methodologies is attributed to threeimportant breakthroughs during the last 30 years or so. The first was the intro-duction of Hohenberg–Kohn–Sham density functional theory (DFT) (Hohenbergand Kohn 1964, Kohn and Sham 1965) in the mid-1960s, when it was rigorouslyproved that the complicated and unmanageable many-electron problem can betransformed into an effective one-electron problem which, in turn, can be solvedwithin the so-called local density approximation (LDA) (Jones and Gumarsson1989). The DFT is based on the quantum statistical approach. It does not attackthe many-body problems frontally, but it possesses a certain conceptual simplicityfor which it has emerged as the most successful tool in describing the ground stateproperties of inhomogeneous electronic systems. The second breakthrough camein 1975, when linear methods for solving the one-electron band structure of solidswas introduced by Andersen (1975). The LAPW (linear augmented plane wave)(Andersen 1975, Jansen and Freeman 1984) and the LMTO (linear muffin-tinorbital) (Andersen et al. 1993) methods and associated methods like the ASW(augmented spherical wave) method are the most widely used today for ab initio(i.e. without any a priori assumption regarding interatomic interactions) investiga-tion of materials. The third breakthrough came in 1985, when Car and Parrinello(1985) proposed a recipe for ab initio molecular dynamics (MD) simulation ofatomic aggregates, where “forces” acting on atoms are evaluated within the frame-work of DFT. This DF-MD method has since been extensively used for studyingvarious kinds of clusters, nanoparticles, liquids and amorphous solids, although theunderlying plane wave pseudopotential makes its applications restricted mainly tos–p bonded systems (Payne et al. 1992).

In contrast to ordered solids, the calculation of any physical property of a dis-ordered alloys requires configuration averaging over all possible configurations.Most experiments measure configurationally averaged properties. The configu-ration averaging is usually achieved in the framework of the coherent poten-tial approximation (CPA) (Soven 1967, Taylor 1967). The essence of the CPAis to replace the array of random potentials by an effective energy-dependentcoherent potential (��z�). The scattering properties of ��z� are then determinedself-consistently (in a single site mean-field sense), from the requirement that anelectron travelling in an infinite array of ��z� undergoes, on average, no furtherscattering upon replacement of any ��z� with the actual potential. So CPA is a




self-consistent prescription which allows one to obtain an effective non-randomHamiltonian. CPA, coupled with first principles band structure methods in theframework of LDA, provides a starting point for the ab initio calculation of theelectronic structure of disordered alloys. Though CPA provides reliable results,there are many situations where the single site approximation inherent in CPAbegins to fail: as in cases where clustering effects become important (e.g. in theimpurity bands of split band alloys, like the Zn band in a Cu-rich CuZn alloy),where short-range order dominates leading to ordering or segregation and wherelocal lattice distortion because of size mismatch of the constituents leads to essen-tial off-diagonal disorder (as in CuPd and CuBe alloys).

Recently the augmented space recursion (ASR) method which involves therecursion (Haydock 1980) method in the augmented space (Hilbert space (� ) +Configurational space (�)) (Mookerjee 1973a,b, 1987) is gaining importance tohandle disordered alloys. The recursion method (Haydock 1980) offers an alter-native to the band structure method. This method is suitable for computing thelocal properties which are related to the diagonal elements of the resolvent, whereit offers a large computational advantage over the band structure methods. More-over, the recursion method is based on real space and so does not require latticeperiodicity for its operation, in contrast to the band structure methods. As a result,both ordered and disordered systems as well as systems with broken translationalsymmetry (such as surfaces) can be treated within the recursion method. Theaugmented space formalism, introduced by Mookerjee (Andersen et al. 1993), isa novel and conceptually attractive method for the calculation of the configura-tionally averaged Green function of a disordered material. In this method, onetransforms the Hamiltonian describing a given disordered system to an orderedHamiltonian whose Green function matrix elements correspond to appropriateconfigurational averages of the Green function of the original disordered system!The new ordered Hamiltonian is said to be in an augmented space which can bedescribed as the direct product of the Hilbert space, � , spanned by the originalHamiltonian with a configuration or disorder space, � , which spans all possibleconfigurations of the system.

5.2.2 Order–disorder transformationsThe thermodynamics of solid solutions consists of contributions from electronic,displacive, vibrational, magnetic and configurational degrees of freedom. The the-oretical model presented here assumes that the configurational degree of freedomcan be separated from the other degrees of freedom and that change in the lattereffects can be neglected in certain types of phase transformations or phase changes,called coherent phase transformations. These coherent phase transformations are




defined as those where the basic lattice framework of the crystalline solid solu-tion is conserved, i.e. the two phases differ from each other only in the chemicaldistribution of the constituent elements. Hence chemical rearrangement of atomicspecies is allowed along with displacements from the average lattice positions,but any discontinuous change of the lattice framework, leading to dislocations,incoherent interfaces, free surfaces and grain boundaries, is not allowed.

5.2.2.1 Historical developmentsAs far back as 1925, Lenz and Ising (Temperley 1972) introduced a model to studythe thermodynamics of magnetic systems. They proposed to associate a magneticmoment at each lattice site of a given lattice. These magnetic moments can haveonly two components, up or down, along a given direction, and a moment interactswith only its nearest neighbours. The pairs of parallel moments have an energy−J , whereas pairs of antiparallel moments have an energy +J . It is then quite easyto calculate the energy of a given configuration of a finite system. It is not alwayspossible to obtain explicit expressions for the thermodynamic quantities of interestwhen the size of the system increases to infinity. Various approximations havebeen proposed for the solution of the Lenz and Ising model in three dimensions(3D). The resemblance of this model to a binary substitutional alloy in which eachsite is occupied by either the A or the B component was immediately evident, witha site-occupation operator replacing the magnetic moment. The same analogy canbe extended to interstitial alloys where each interstitial site is either occupied orvacant, disregarding the host matrix. This model, within certain approximations,has been shown to contain the essential ingredients to account for various typesof phase transitions and various topologies of phase diagrams.

The thermodynamic state of a system can be easily obtained by finding theproper expression for the partition function � = ∑

� exp�−E��/kBT�, where �signifies a configuration and the summation is over all possible configurations.The basic problem is then to calculate the partition function and its derivativeswith respect to external fields, from which we can obtain the free energy and allaverages of interest. Except in a few cases, it is not possible to perform thesecalculations. One has, therefore, to make some plausible approximations. The mostuseful ones are the so-called mean field (or molecular field) approximation (MFA)(de Fontaine 1979, Ducastelle 1991).

Mean field theories (MFTs) are, in general, derived from variational principlesand have been shown to suffer from serious drawbacks, particularly close to sec-ond order phase transitions, which are characterized by the fact that long-rangeorder (LRO) parameter is a continuous function of temperature, vanishing at thecritical temperature, Tc. At temperatures near Tc, the singularities of the thermo-dynamic functions are not correctly reproduced. In such cases, it is necessary




to treat the fluctuations in the medium. The celebrated renormalization grouptheory (Ma 1976, Toulouse and Pfeuty 1977) has been developed in the 1970sto account for this critical regime. This theory is specially designed to identifythe universal characteristics of phase transitions. However, at temperatures awayfrom Tc, its practical implementation is not very easy and also not very reliable.Since most of the order–disorder transitions concerning the 3D Lenz and Isingmodel seem to be of the first order, i.e. the LRO parameter is discontinuous atTc, transition occurs before the fluctuations have increased too much in the disor-dered phase. In such a situation, MFTs are expected to give reasonably accurateresults.

The simplest MFT (called the Bragg–Williams approximation) (Bragg andWilliams 1934, 1935, Wilson and Kogut 1974), applied to the Lenz and Isingmodel, easily predicts the occurrence of order–disorder transitions and provides adescription of the disordered and ordered states. This is a single-site approxima-tion in which each atom is assumed to be embedded in a mean field. Due to itssimplicity, this theory is successful in many respects, but quantitatively it is ratherinaccurate. The critical temperature are found to be much too high, the short-rangeorder is rather badly treated, and the phase diagrams can even be qualitativelywrong in the case of frustrated lattices. In this theory, correlation between sitesor the local statistical fluctuation was totally neglected. In view of this defect,Bethe (1935) introduced the concept of short-range order. This is to improve theapproximation from a single site in the mean field to a pair of sites in a molecularfield. As expected, a great difference from the result of the B–W approximationappears near the critical point. In particular, the results showed the existence ofshort-range order above Tc, which decays as the temperature is raised. In addition,the critical point was lowered in comparison to the B–W approximation for thesame value of pair interaction, J2. Generally the better is the degree of approxi-mation, the lower is the Tc, and thus, the Tc is taken as a measure of the degreeof approximation.

Since the exact solution of the 3D Lenz and Ising problem cannot be obtained,the most important source of exact information concerning this problem lies in aseries expansion of the partition function and other derived properties at low andhigh temperatures. An important example of a high-temperature series expansionis due to Kirkwood (1938), who devised an ingenious method to evaluate thepartition function in a systematic way by a series expansion in terms of themoments of probability distributions. Kirkwood evaluated these moments downto the third order.

The series expansion approaches can be classified into two types. In the first,a particular selection of terms which can be readily evaluated is summed toprovide closed form approximations similar to other approximate methods. In this




connection, the method of Rushbrooke and Scions (1955) and that of Yvon (1948)and Fournet (1967) may be cited. In the second kind of approach, the known termsof the expansion are used to assess its asymptotic behaviour. This approach enablesone not only to estimate the true critical point but also to closely investigate thecritical behaviour. These methods involve cumbersome mathematical calculations,without adding much physical insight.

5.2.2.2 Static concentration wave modelLandau and Lifshitz (1969, 1980) developed some very useful concepts relatedto order–disorder transitions based on the MFT. Since their theory was based onabstract group theoretical arguments, it could not be exploited by the metallur-gists initially. However, these concepts have been widely used by some Russianscientists like Krivoglaz (Krivoglaz and Smirnov 1964) and Khachaturyan (1978,1983; static concentration wave (SCW) model). Elsewhere the theory has beenapplied mostly to displacive soft mode transitions. The SCW model is in fact aB–W model in an arbitrary number of LRO parameters and has been derived fromLandau’s general analysis of order–disorder reactions. It is useful for analysingorder–disorder systems in which the ordered phases are crystallographic deriva-tives of their respective parent phases (coherent phases). In the SCW model, thecrystal is modelled by a rigid lattice onto which “average atoms”, or atoms whoseproperties are assumed to be averaged according to the occupancy of particularsites, are superposed. The internal energy is assumed to be derived only frompairwise interactions between atoms, and the entropy is taken to be equal to thesum of the entropies of the average atoms. In this model, the occupation proba-bility, � (r), at a given site, r(p), is expanded in a Fourier series, i.e. it can berepresented as the sum of SCWs whose amplitudes are the Fourier coefficientsand whose wave vectors determine the superstructure period:

� �r� =N∑

h=1

��k�e−ik�h��r�p� (5.2)

where �(k)s are the Fourier coefficients given by

��k� = 1N

N∑p=1

� �r�eik�h��r�p� (5.3)

The first summation is over the N lattice points of the periodic crystal and thesecond is over the N points of the first Brillouin zone. The wave vector k(h) andthe lattice vector r(p) are defined by




r�p� = p�a� (� = 1�2�3; p� are integers, summation implied)k�h� = 2�h�b� (h� = m�/N�, m� = 0, ±1�±2� � � � )

where a� and b� are lattice translation vectors and primitive translation vectors ofthe reciprocal lattice such that a� ·b� = ��.

The concentration wave amplitudes ��k′� corresponding to the wave vector k′

that generates the ordering instability are expressed in terms of normalized (withrespect to c) LRO parameter, n, via the following relation,

��k� = n�c (5.4)

The normalized order parameter (n) is related to the standard order parameter(), as n = /max, where max is the maximum order parameter attainable at agiven composition. For a disordered phase, � (r) is independent of r and is equalto the atomic fraction, c, of the solute.

As an example, we will briefly illustrate the binary L12 ordering in an fcc solidsolution using the SCW model. The L12 structure is one in which all the face-centred positions are occupied by one type of atoms and all the corner positionsare occupied by the other type of atoms.

For the L12 phase, all the �(k) values vanish, except �(000), �(100), �(010) and�(001) (Figure 5.2). The value of �(000) is just the bulk (or solute) concentration,c, and the other three amplitudes are equal. The occupation probability for eachsublattice can then be calculated to give

� �r� = c+�(e−i�100�∗�r + e−i�010�∗�r + e−i�001�∗�r) (5.5)

where � = ��100� = ��010� = ��001� and ��000� = c. The exponents in theabove equation have only two values: −i� and −i2� depending on whether theposition vector r lies in a �200� or a �100� plane with respect to the origin,respectively. The atoms are situated on the fcc lattice such that they lie either(a) in all three �100� planes or (b) in one �100� and two �200� planes, so that� (r) assumes only two values: (1) c+3�, if the atom is in a corner position and(2) c−�, if it is in a face-centred position. There are 1

4N corner atoms and 34 N

face-centred atoms.The internal energy, in the pair approximation up to an arbitrary coordination

shell, is given by

E = N

2

N∑h=1

J�k�h��k�h��∗�k�h�� (5.6)




(001)

(100)

(010)

Figure 5.2. The L12 structure (AB3 stoichiometry) viewed as superposition of three {100} concen-tration waves. Filled (open) circles represent A (B) atoms, respectively.

where the star (∗) indicates the complex conjugate of the amplitude of the corre-sponding concentration wave, and J�k�, the Fourier transforms of the pair inter-actions, are given by1

J�k� = 1/NN∑

p=1

J�r��eik�h��r�p� (5.7)

Hence, for the L12 structure, the internal energy is

E = N

2

(J�000�c2 +3J�100��100�2

)(5.8)

The expression for the configurational entropy is given as

S = kB

N∑p=1

�� r�p�� ln�� r�p��+ �1−� �r�p�� ln�1−� �r�p�� (5.9)

1 The dependence of r and k on p and h are assumed if not shown explicitly.




which, in terms of sublattice probabilities, can be expressed as

S = kB

�∑�=1

NS�� ln �� + �1−�� ln�1−�� (5.10)

where � is the total number of sublattices, while �� and NS are the occupationprobability and the number of atoms on the �th sublattice, respectively. In theSCW model, the internal energy is already given in terms of the order parameter(Eqs. (5.6) and (5.8)) or the amplitude of the concentration wave. The LRO param-eter can, most comfortably, be substituted in the expression for the entropy, as willbe seen now. The expression for the configurational entropy, after substitution ofEq. (5.5), is

S = −NkB

{c ln c+ �1− c� ln�1− c�− 1

2�c−��k�� ln�c−��k��

+ 12�c+��k�� ln�c+��k��

}(5.11)

where kB is the Boltzmann constant. Hence the Helmholtz free energy, F , given by

F = E−TS (5.12)

is obtained as a function of T� c and on substituting the normalizing relation(5.4). The free energy of the L12 ordered phases is given as

FL12�T� c�n� = 12

(J�000�+3J�k��2

n

)�c2 + kBT

4�3 �c�1−n� ln c�1−n��

+3 ��1− c�1−n�� ln�1− c�1−n��

+ �c�1+3n�� ln�c�1+3n��

+ ��1− c�1+3n�� ln�1− c�1+3n�� (5.13)

5.2.2.3 Cluster variation methodThe most powerful approximation, now, is the cluster variation method (CVM)(Kikuchi 1951, Sanchez and de Fontaine 1978, Morita 1984, Fontaine 1994) whichincludes the correlation between a few sites in some clusters. The CVM of sta-tistical mechanics was originally proposed as an improved approximation for thesolution of the Lenz and Ising model. It is, in fact, a hierarchy of approximations




ranging from the simplest Bragg–Williams–Gorsky single-site (one-point) approx-imation to a very elaborate one considering clusters of lattice points of varyingsize, shape and complexity. Therefore the CVM is regarded as a natural way ofgeneralizing the MFT for the solution of the 3D Lenz and Ising model used forthe study of configurational thermodynamics of alloys.

In the CVM, the statistical thermodynamics of alloys is described in terms ofatomic configurations of clusters of lattice sites. The state of partial order in alloysis described in terms of multisite correlation functions, as will be described below.Detailed accounts of the subject are provided in many books and review articlessuch as those due to Ducastelle (1991), de Fontaine (1979, 1994), Morita et al.(1994), Inden and Pitsch (1991), Mohri et al. (1985).

The linkage between quantum mechanics and statistical mechanics leading tothe first principles configurational thermodynamics of alloys comes via the config-urational energy expression which is expanded in terms of effective pair/multisite(cluster) interactions (ECI). These configurationally averaged effective interac-tions are calculated using ab initio methods. Traditionally there have been twoapproaches for obtaining these ECIs. The first approach is to start with theelectronic structure calculations and the total energy determination of some judi-ciously selected ordered superstructures (configurations) of a given alloy systemand expand the energy in terms of ECIs. The ECIs can then be calculated bythe inversion of the expansion – the Connolly–Williams inversion method (IM)(Connolly and Williams 1983, 1984). The other approach is to start with the dis-ordered phase, set up a perturbation in the form of concentration fluctuation andstudy whether the alloy can sustain such a perturbation. This approach includesthe generalized perturbation method (Ducastelle and Gautier 1976), the embeddedcluster method (Gonis et al. 1984) and the concentration wave approach (Gyroffyand Stocks 1983), all of which are based on the CPA. The direct configurationaveraging (DCA) method (Dreyssé et al. 1989) consists of obtaining the effectivepair interactions (EPI) for each random configuration of the disordered alloy usingthe recursion technique and then performing the configurational averaging overa (selected) number of configurations. In the ASR (Mookerjee 1973a,b) alongwith the orbital peeling (OP) (Mookerjee 1987) technique, the recursion is appliedin the augmented space and the averaging is performed over all the possibleconfigurations of the disordered alloy to obtain the EPIs.

The present treatment is based on the article by Inden and Pitsch (1991).Consider a crystalline system with N lattice sites and having M constituents.These constituents can be atomic species like A, B, C (or can be “vacancies” also)such that

∑i Ni = N , where Ni is the number of atoms of type i. A particular

distribution of these constituents on the lattice sites defines a configuration. Aconfiguration is specified by site operators, �n (n= 1�2� � � � �N ), which may take




values (numerical labels) �i (i = 1�2� � � � �M), corresponding to the occupationof the site n by the constituent i. For a M-component system, the site operatorcan take the values �M − 1�� 0� � � � �−�M − 1�, e.g., for a binary system,�n corresponding to the lattice site n can take the two values ±1, depending onwhether the site n is occupied by component A (+1) or by component B (−1). Anyconfiguration is then specified by a N -dimensional vector � = ��1��2� � � � � �N �.In total, there are MN different configurations, as each of the N sites can beoccupied in M different ways. Any function of �n (including �n itself) is called aconfigurational variable.

A second operator p�i�n , called the site occupation operator, which allows to

count the number of sites n, occupied by the same type of atom i, for takingaverages, is defined as follows:

p�i�n =

{1 if an atom of type i occupies site n0 otherwise

The cluster approximation: It is much more convenient to consider the configura-tions of much smaller units called clusters in place of the N -point lattice whichis quite large. A cluster is defined by a set of lattice points 1, 2, …,� and aconfiguration on this cluster is given by �� = ��1��2� � � � � �� (Figure 5.3). On a�-site cluster there are, in total, M��M� �MN ) configurations. The configurationsof the N -point system can then be classified into groups with the same numberof clusters N�� with the configuration ��. The probability, ��, of a �-pointcluster to occur in a configuration �� is given by

�� = N��/N� (5.14)

a

b

c d

Figure 5.3. The irregular tetrahedron (IT) cluster (abcd) approximation for the bcc lattice.




where N� is the number of equivalent �-site clusters contained in the system.These fractions are called cluster probabilities or reduced density matrices. Theyspecify the configuration in the �-point cluster approximation. The thermodynamicfunctions derived by the CVM depend on the probabilities of this cluster and alsoon the probabilities of all the subclusters which can be derived from that of thelargest cluster by partial summations.

The correlation functions: We characterize the atomic configuration in terms ofcorrelation functions (�s) by considering averages over equivalent clusters. An�-point (� ≤ �) correlation function, ��i�� , for the cluster type i, can be defined as

��i�� = 1

N�i��

∑��

�1�2 · · ·�� (5.15)

where the summation is taken over all the possible configurations of the �-cluster of type i. These correlation functions, for a perfectly ordered state, canbe determined by inspection. However, these correlations serve as variables withrespect to which the free energy of the system is minimized to arrive at theequilibrium state of order (or partial order).

In order to determine the stability of a configuration at T = 0 K and in theequilibrium with respect to exchange of atoms, one has to minimize the internalenergy, EG�T�V�!A�!B� � � � �, in the grand canonical scheme, given as a functionof temperature (T ), volume (V ) and chemical potential (!i). The grand canonicalinternal energy (EG) is obtained from the canonical energy, E, by its Legendretransformation:

EG�T�V�!A�!B� � � � � = E�T�V�NA�NB� � � � �+M∑i=1

!iNi (5.16)

Generally, one considers a constant volume system and the configurational partof the internal energy is separated. The configurational energy, EG

c , in the simplestapproach, is expressed in terms of pairwise interactions (Vij) up to an arbitrarycoordination shell as

EGc = 1

2N∑k

∑ij

Z�k�V�k�ij

ij0k +N

∑i

!i i0 (5.17)

where Z�k� is the coordination number for the kth coordination shell and position“0” stands for any position in the crystal taken as the origin.

The CVM entropy (S) is evaluated according to S = kB ln , where is thenumber of arrangements which can be formed for given values of the correlation




functions. In the CVM, we take correlations only up to the basic cluster size(�) and therefore, the thermodynamic probability, , is the number of possiblearrangements of this basic cluster, corrected to take account of the overlappingclusters. We will not derive the expression for the CVM entropy here. Interestedreader is referred to some of the excellent articles mentioned above. In brief, theCVM entropy takes the form

S = −kBN�∑

=1

m a

∑�

�� ln �� (5.18)

where the sum runs over all the subclusters, , of the basic cluster, �. The numberof -clusters per lattice point, m , and the Kikuchi–Barker coefficients, a , are tobe derived by geometrical considerations of the lattice. The cluster probabilities, �� , can be expressed as

�� = 12

[1+∑

′≤

v�� ′� �� ′

](5.19)

where the summation is over all the subclusters, ′, of the cluster, ; v�� ′� � are

a sum of ′-order products of spin variables �, the structure of which dependsupon the symmetry of the cluster in question.

As an example of what has been discussed above, we briefly present here theCVM treatment for the equiatomic B2 (CsCl type) structure which is a bcc-basedsuperstructure (see Figure 5.1). The B2 structure has two sublattices correspondingto the corner and the body-centred positions, respectively. The cluster approxima-tion selected is irregular tetrahedron cluster as shown in Figure 5.3.

Let us represent the basic cluster tetrahedron by (abcd), denoting the fourvertices. In a perfectly ordered B2 structure, the cluster sites (a) and (b) will beoccupied by type “A” atoms (�1 = 1) and sites (c) and (d) will be occupied bytype “B” atoms (�2 = −1). Now the enumeration of different subclusters of thisbasic cluster gives:

(1) point cluster (type 1): �a�/�b�(2) point cluster (type 2): �c�/�d�(3) NN pair cluster (type 1): (ac), (bc), (ad) and (bd)(4) NNN pair cluster (type 2): (ab)(5) NNN pair cluster (type 3): (cd)(6) isosceles triangular clusters (type 1): (abc), (abd)(7) isosceles triangular clusters (type 2): (acd), (bcd)(8) irregular tetrahedron cluster: (abcd).




The subcluster probabilities corresponding to the above clusters are then givenas (here we have used short notations: i ≡ �a, j ≡ �b, k ≡ �c and l ≡ �d),

�1�1 = 1

2 �1+ i��1�1 �

�2�1 = 1

2 �1+k��2�1 �

�1�2 = 1

4 �1+ i��1�1 +k�

�2�1 + �ik��

�1�2 �

�2�2 = 1

4 �1+ �i+ j��1�1 + �ij��

�2�2 �

�3�2 = 1

4 �1+ �k+ l��2�1 + �kl��

�3�2 �

�1�3 = 1

8 �1+ �i+ j��1�1 +k�1�2�+ �ik+ jk��

�1�2 + �ij��

�2�2 + �ijk��

�1�3 �

�2�3 = 1

8 �1+ j�1�1�+ �k+ l��2�1 + �jk+ jl��

�1�2 + �kl��

�3�2 + �jkl��

�2�3 �

�1�4 = 1

16 �1+ �i+ j��1�1 + �k+ l��

�2�1 + �ik+ il+ jk+ jl��

�1�2 + �ij��

�2�2 + �kl��

�3�2

+ �ijk+ ijl��1�3 + �ikl+ jkl��

�2�3 + �ijkl��

�1�4 �

On substituting the above expressions for cluster probabilities in the CVM freeenergy expression (Eqs. 5.17 and 5.18), one obtains free energy (F = E−TS) as afunction of T , !is and set of eight independent correlation functions with respectto which the F is to be minimized.

5.2.3 The ground states of the Lenz and Ising modelThe determination of ground states of the Lenz and Ising model is much easier thanevaluating the free energy of a system at finite temperatures. Many exact resultshave been obtained (Allen and Cahn 1972, de Fontaine 1979, Sanchez et al. 1982,Kanamori and Kaburgi 1983, Ducastelle 1991) when the interactions are shortrange. This is based on a method of linear inequalities that allows the determinationof the most stable ordered structures as functions of the concentration and thestrength of the pair interactions. In practice, it is very difficult to include morethan fourth or fifth nearest neighbour interactions. If the range of interactions isfinite, only a finite number of ordered structures can be found.

The correlation functions characterizing the atomic configurations can onlyvary within a restricted range of numerical values due to certain consistencyconditions. This restricted range of values is called the existence domain or the




configuration polyhedron outside which the values do not define actually existingatomic configurations. The dimensions of the configurational space depend eitheron the number of correlations or equivalently on the size of the largest cluster.These existence domains are most useful for an analysis of ground states. We giveexamples of the calculation of existence domain for the bcc and hcp lattices underthe first and second nearest neighbour pair interactions.

The first step is to select a basic cluster which determines the dimension ofthe configuration space. We select the irregular tetrahedron abcd (Figure 5.3)as our basic cluster which accounts for up to the second nearest neighbour pairinteractions. The configurational variables are then the point correlation functionx1 = �1; pair correlation functions, x2 = �1�2 � x3 = �1�3; 3-point correlationfunction, x4 = �1�2�3; and the tetrahedron correlation function, x5 = �1�2�3�4.Because of the constraint that �ijkl�4 ≥ 0, we are led to the following consistencyrelations expressed in the matrix form as (Ducastelle 1991, Inden and Pitsch 1991)

⎡⎢⎢⎢⎢⎢⎢⎣

1 4 4 2 4 11 2 0 0 −2 −11 0 0 −2 0 11 0 −4 2 0 11 −2 0 0 2 −11 −2 4 2 −4 1

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

1x1

x2

x3

x4

x5

⎤⎥⎥⎥⎥⎥⎥⎦

≥ 0

If these inequalities are taken as equalities, these equations define a simplex inthe five-dimensional configurational space. All the vertices correspond to exist-ing atomic configurations. It is well known that the minimum of energy willoccur for atomic configurations corresponding to the vertices of the configurationpolyhedron. Its projection onto the subspace (x1� x2� x3) defines the configurationpolyhedron (Figure 5.4). The coordinates of the vertices are given in Table 5.1.The corresponding crystallographic structures are shown in Figure 5.1 and thepertinent crystallographic data are tabulated in Table 5.2.

Similarly ground state analysis can be carried out for the hcp lattice underthe octahedron–tetrahedron cluster approximation (Figure 5.5), which includesup to third nearest neighbour pair interactions and gives rise to 14 correlationfunctions defined as follows: x1 = �1, x2 = �1�2, x3 = �1�6, x4 = �3�6,x5 = �1�6�7, x6 = �1�3�6, x7 = �1�2�6, x8 = �2�3�4, x9 = �1�2�3�4,x10 = �1�2�3�6, x11 = �1�2�6�7, x12 = �2�3�6�7, x13 = �1�2�3�6�7, x14 =�1�2�3�6�7�8.

The ground state superstructures for the hcp lattice corresponding to the verticesof the 14-dimensional configuration polyhedron are listed in Table 5.3. A few ofthem are also depicted in Figure 5.5.




1

3

2

4

x 3

x 2

x 1

Figure 5.4. Projection of the configuration polyhedron corresponding to the irregular tetrahedronof the bcc lattice. The vertices designated correspond, respectively, to (1) B2, (2) D03, (3) D03 and(4) B32 ground state ordered structures.

Table 5.1. Coordinates of the vertices of the configurational polyhedron corre-sponding to the minimum in energy under the first and second nearest neighbour(NN) pair approximation.

Configuration x1 x2 x3 x4 x5

Pure A (A2) 1 1 1 1 1A3B, AB3�D03) ± 1

2 0 0 ± 12 −1

AB (B2) 0 −1 1 0 1AB (B32) 0 0 −1 0 1Pure B (A2) −1 1 1 −1 1

For the fcc lattice, the octahedron–tetrahedron cluster approximation (Figure 5.6)takes into account up to the second nearest neighbour pair interactions. Theconfigurational polyhedron corresponding to the octahedron is a simplex in thenine-dimensional space corresponding to all possible multiatom interactions. Allsubclusters equivalent in the fcc lattice are also equivalent in the octahedron. Ifwe project this simplex onto the space (x1, x2, x3) corresponding to point, first andsecond nearest neighbour correlation functions, only the 10 vertices survive. Forthe tetrahedron inequalities the corresponding polyhedron is a prism. This prism




Table 5.2. Crystallographic data for various bcc-based ground state superstructures under the firstand the second NN pair approximations.

Structuresbcc-based

Compositionalformulae

Space groupsymbol (no.)

Wyckoffpositions

Multiplicity Transformedbasis

A2 A Im3m (225) A (a) 2 a = a�100�

B2 AB Pm3m (221) A (a) 1 a1= a�100�B (b) 1 a2= a�010�

a3= a�001�

B32 AB Fd3m (227) A (a) 8 a1= a�200�B (b) 8 a2= a�020�

a3= a�002�

D03 A3B Fm3m (225) A (c) 8 a1= a�200�A (b) 4 a2= a�020�B (a) 4 a3= a�002�

The transformed basis vectors are given in terms of the vectors of the bcc lattice.

(a) Unit cell (b) Octahedron (c) Tetrahedron (d) CuPt (e) A2B

(f) B19 (g) D019 (h) D0a

Figure 5.5. A few ordered structures which are the ground state superstructures under the firstand the second nearest neighbour interaction approximation for the hcp lattice. Large circles are in{00n} planes and small circles are in {00n+ 1

2 } planes. (a) Unit cell of hcp structure, (b) octahedron,(c) tetrahedron, (d) CuPt type, (e) A2B, (f) B19, (g) D019 and (h) D0a.

cuts the octahedron polyhedron. The final polyhedron is shown in Figure 5.7. Ithas 15 vertices out of which only 9 vertices are really distinct. These vertices aredescribed in Table 5.4 and the corresponding superstructures are also shown inFigure 5.8; relevant crystallographic data are listed in Table 5.5.




Table 5.3. Ground state structures of the hcp lattice correspondingto the vertices of the 14-dimensional configurational polyhedron.

Concentration Prototype Designation

A Mg A3AB AuCd B19AB CuTeAB WC Bh

A3B Ni3Sn D019

A3B �-Cu3Ti D0a

A2BA2B B2NdRh3

A2B Si2Zr C49A5B B2NdRh3

A4B3

ab

efc

d

g

Figure 5.6. The tetrahedron (abcd)–octahedron (bcdefg) (TO) cluster approximation for thefcc lattice.

5.2.4 Special point orderingA wide range of phenomena related to order–disorder and magnetic transitions canbe explained using the symmetry properties of the pair potentials (Vij). One suchexample is the Landau theory of continuous phase transitions. The free energy of analloy system for different wave vectors corresponding to appropriate concentrationfluctuations has been shown by Khachaturyan (1978) and de Fontaine (1979) to bea useful parameter in the study of the instabilities associated with ordering phasetransitions. The roles that symmetry and wave vectors, terminating at special pointsof the reciprocal lattice, play can be illustrated as follows. If a symmetry elementof the space group in k-space is located at a point h, the vector representing thegradient, �hV�h�, of an arbitrary potential energy function V�h� at that point mustlie along or within the symmetry element. If two or more symmetry elementsintersect at the point h, one must necessarily have

��hV�h�� = 0 (5.20)




x 3

x 1

x 2

1

28

3

4

5

7

6

9

Figure 5.7. Dual configurational polyhedron for the fcc lattice under the tetrahedron–octahedroncluster approximation. The vertices designated correspond, respectively, to (1) pure A, (2) L12,(3) D022, (4) A5B, (5) Pt2Mo (Immm), (6) A2B, (7) A2B2, (8) L10 and (9) L11.

Table 5.4. Ground state structures of the fcc lattice corresponding tothe vertices of the nine-dimensional configurational polyhedron.

Configurations x1 x2 x3

A 1 1 1

A5B23

13

13

L1212

0 1

D02212

023

Pt2Mo13

19

19

A2B13

013

L10 013

1

A2B2 013

13

L11 0 0 1




L11L10L12

D022 A2B2 A2B (Immm )

Figure 5.8. fcc-based ordered intermetallic phases which are the ground state superstructures underthe first and the second NN pair approximation.

At these so-called special points, the potential energy function, V�h�, representsan extremum regardless of the choice of the pair interaction energies. Thus specialpoints play an important role in the search for lowest energy ordered structures.The special points are, however, quite insufficient for a complete description of theground states which can be treated conveniently in the real space as demonstratedin the last section.

Among the most important applications of special points is the study of the onsetof short wavelength instabilities in alloys. Such instabilities may take place at ahigh supercooling below a first order transition temperature, frequently bearing nosymmetry relation to either the low or the high temperature phases. This particularmetastable ordering mechanism, known as spinodal ordering, has been observedin several alloy systems (e.g. Ni–Mo alloys).

The points which differ by a lattice vector of a reciprocal lattice are consideredequivalent. In the case of simple structures with a single atom per unit cell, it issufficient that two symmetry elements intersect at special points. These specialpoints are listed in crystallographic tables. They are always located at the surfaceof the Brillouin zone. The “star” of a special point vector k is obtained by applyingall the rotations and rotation-inversion of the space group on the vector k. Allthese vectors of a star are also considered equivalent.




Table 5.5. Crystallographic data for various fcc-based ground state superstructures under the firstand the second NN pair approximation.

Structuresbcc-based

Compositionalformulae

Space groupsymbol (no.)

Wyckoffpositions

Multiplicity Transformed basis

A1 A Fm3m (225) A (a) 4 a= a�100�

L10 AB P4/mmm (123) A (a) 1 a1 = 12a�110�

B (d) 1 a2 = 12a�110�

c= a�001�

L11 AB R3m (166) A (a) 1 a1 = 12a�110�

B (b) 1 a2 = 12a�101�

c= a�222�

A2B2 A2B2 I41/amd (141) A (a) 4 a1 = a�010�B (b) 4 a2 = a�001�

c= a�200�

Pt2Mo A2B Immm (71) A (i) 4 a= 12a�110�

B (a) 2 b= a�001�c= 1

2a�330�

A2B A2B B2/m (12) A (i) 4 a= 12a�552�

A (c) 2 b= a�220�B (a) 2 c= 1

2a�110�

L12 A3B Pm3m (221) A (c) 3 a= a�100�B (a) 1

D022 A3B I4/mmm (139) A (d) 4 a1 = a�010�A (b) 2 a2 = a�001�B (a) 2 c= a�200�

Ni4Mo A4B I4/m (87) A(h) 8 a1 = 12a�310�

type B(a) 2 a2 = a�130�c= �002�

The transformed basis vectors are given in terms of the vectors of the fcc lattice.

5.2.4.1 BCC special pointsThe special points of the bcc structure are located at the points " , H , P and Nof the Brillouin zone (Figure 5.9). The corresponding stars are given in Table 5.6(Ducastelle 1991). Here we have a single structure per star.

The domain of stability in the case of nearest neighbour and next nearestneighbour interactions is shown in Figure 5.10. The star

⟨12

12 0⟩

does not appearwhich is understood by noting that the corresponding structure has not beenobtained as a possible ground state which requires up to third NN interactions.The only ground state that does not appear is the D03 structure which can beconstructed by a superposition of �100� and

[12

12

12

]waves.




H

H

Γ

N

kZ

kX

kY

Figure 5.9. The Brillouin zone for the bcc lattice where various special points have been marked.

Table 5.6. The special points and stars of the bcc structure.

k-Vector star Members Brillouin zone points Ordering structure

000 �000� "

100 �100� H B2⟨12

12

12

⟩ [12

12

12

] [12

12

12

]P B32

⟨12

12 0⟩ [

12

12 0] [

12 0 1

2

] [0 1

212

][

12

12 0] [

12 0 1

2

] [0 1

212

]N AB

V1

V2

⟨000⟩ ⟨100⟩

(a)(b)

⟨ ⟩12

12

12

Figure 5.10. The domains of stability of the stars for the bcc lattice in the first (V1) and the second(V2) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively,to (a) 2V1 −3V2 = 0 and (b) 2V1 +3V2 = 0.




5.2.4.2 HCP special pointsThe analysis of the hcp structure is much more complex as it has two atoms perunit cell and the space group of the structure (P63/mmc) is not symmorphic. Thereare six special points for the hcp lattice, viz., " (000), M ( 1

2 00), K ( 13

13 0), L ( 1

2 0 12 ),

H ( 13

13

12 ) and A (00 1

2 ). Using symmetry arguments, it can be shown (Ducastelle1991) that some of these special points are irrelevant for the hcp structure. Thethree relevant special points are " , M and H (Table 5.7). At the points " andM , there are two extremes and we must combine the concentration waves in thetwo sublattices in two different ways. These combinations are obtained from theeigenvectors of the potential energy matrix V��k�. The result is that at " and M ,we have simple modes which may be called acoustic and optical modes as in thetheory of lattice vibrations. If we let the origin of sublattice 1 be at (000) thenthe origin of sublattice 2 is at

(23

13

12

). The stars are tabulated in Table 5.7 and the

structures corresponding to the pure modes are described below (Figure 5.11).

Table 5.7. Important special points and stars of the hcp structure.

k-Vector star Members Brillouin zone points

000 �000� "⟨12 00

⟩ [12 00

] [0 1

2 0] [

12

12 0]

M⟨13

13

12

⟩ [13

13

12

] [13

13

12

]H

V1

V2

⟨000⟩I

⟨000⟩(b)

(a); ⟨ 00⟩1

2 I

⟨ 00⟩12 II

⟨ 0⟩14

14

II

Figure 5.11. The domains of stability of the stars for the hcp lattice in the first (V1) and thesecond (V2) nearest neighbour interaction plane. The lines designated correspond, respectively, to(a) V1 − 2V2 = 0 and (b) V1 +V2 = 0. In the hatched region, V�h� is minimum at 1

414 0, which is

not a special point.




Star 000: The acoustic mode 000I corresponds to the segregation processand the optical mode 000II corresponds to an alternate stacking of A and Btriangular planes (CuPt-type structure)

Star⟨

12 00

⟩: Here we regard

⟨12 00

⟩I

and⟨

12 00

⟩II

as modes corresponding to the CuPttype and MgCd (B19) type structures. Also we obtain the D019 structure byconsidering the acoustic combination of k1 = � 1

2 00�, k2 = �0 12 0� and k3 = � 1

212 0�.

There is no structure corresponding to the optical mode.Star

⟨13

13

12

⟩: There is no structure associated with this star. In the case of nearest

neighbour interactions, V (k) is never a minimum at H . When the second nearestneighbour interaction is introduced, the absolute minimum of V (k) in the range0 < V2 <

12V1 occurs at the point [

⟨14

14 0⟩] that does not correspond to a special

point but rather is a “non-special” point.

5.2.4.3 FCC special pointsThe special points of the fcc structure are located at the points " , X, W and L ofthe Brillouin zone (Figure 5.12). The corresponding stars are given in Table 5.8(Ducastelle 1991).

The domain of stability in the case of nearest neighbour and next nearestneighbour interactions is shown in Figure 5.13.

5.2.5 Concomitant clustering and orderingThe thermodynamic analysis of concomitant clustering and ordering was pre-sented by Kulkarni et al. (1985) and Soffa and Laughlin (1988, 1989). The lat-ter theoretically studied various barrierless reaction mechanism for the transition

L

XΓ

W

kZ

kX

kY

Figure 5.12. The Brillouin zone for the fcc lattice where various special points have been marked.




Table 5.8. The special points and stars of the fcc structure.

k-Vector star Members Brillouinzone points

Ordering structure

000 �000� "

100 �100� �010� �001� X L12, L10⟨1 1

2 0⟩ [

1 12 0] [

12 01

] [01 1

2

]W A2B2[

1 12 0][

12 01

][01 1

2 0]

⟨12

12

12

⟩ [12

12

12

] [12

12

12

]L L11[

12

12

12

][12

12

12

]

V1

V2

⟨000⟩⟨100⟩

(a)(b)

⟨ ⟩12

12

12

⟨1 0⟩12

Figure 5.13. The domains of stability of the stars for the fcc lattice in the first (V1) and the second(V2) nearest neighbour interaction plane. The lines designated (a) and (b) correspond, respectively,to (a) V1 −2V2 = 0 and (b) V1 +V2 = 0.

“fcc disordered solid solution �� → ordered state having L12 structure” byemploying the graphical thermodynamics approach and the SCW model.

Soffa and Laughlin have expounded the idea of ordering and clustering reactionsin constructing the schematic free energy versus composition plots which illustratedifferent situations. The stability/instability of a solution with respect to changeof some parameter, e.g. an appropriate order parameter or the site occupancy, isdetermined by the second derivative of the free energy (F ′′) with respect to thatvariable. A solution is considered stable, critical or unstable according to whetherF ′′ > 0, F ′′ = 0 or F ′′ < 0, respectively. The equation F ′′ = 0 describes the point at




which an instability develops in the solution. While the presence of an instabilitycan be determined by examining the sign of the second derivative of free energywith respect to appropriate order parameter, the equilibrium phase boundariesare given by the principle of common-tangents on the free energy–concentrationcurves for the ordered and the disordered phases. The instability lines on thetemperature (T )–concentration (c) plane are defined as follows:

T−i : the line below which the solid solution is unstable with respect to congruent

orderingT+

i : the line above which the ordered solid solution becomes unstable with respectto spontaneous congruent disordering

Tcs: the line below which spinodal clustering instability develops only after thesystem undergoes ordering to a certain extent (conditional spinodal)

Ts: the line below which the disordered phase becomes unstable with respect tospinodal clustering.

The two equilibrium phase boundaries (Teqs) define the ordered, disordered andthe two-phase regions. The four instability lines T−

i , Ts, T+i and Tcs are categorized

in terms of which phase forms the instability: the disordered (�) or the ordered(�) phase. Accordingly, T−

i and Ts are the instabilities of the disordered phase,being the points at which the disordered phase becomes unstable with respect toordering and clustering, respectively. The other instabilities, T+

i and Tcs, are thepoints at which the ordered phase becomes unstable with respect to spontaneousdisordering and clustering, respectively. The computation of the disordered phaseinstabilities is straightforward and can be done exactly, while the ordered phaseinstabilities are somewhat more complex and can be determined from the graphicalthermodynamic approach.

The relative positions of these instability lines, as schematically illustrated inthe phase diagram (Figure 5.14), identify the conditions under which differenttransformation sequences are possible from thermodynamic considerations.

In the cases where barrierless mechanisms are operative, the phase transforma-tion mechanism will be determined by which of the instability lines have beencrossed on quenching. Much research has been dedicated to the study of complexinstability reactions that result in simultaneous ordering and phase separation inalloys.

Laughlin and Soffa compared these results (Soffa and Laughlin 1989) to phe-nomena observed in several systems involving � → L12 phase transition. It ispossible for a miscibility gap to develop in the L12 phase at a temperature wherethe curvature of the free energy is negative. It is equally possible to have a mis-cibility gap develop in the disordered phase by the development of a negative




C

B

AD

E

B′

Tem

pera

ture

Tem

pera

ture

Composition Composition

Tcs

Ts

–Ti–Ti

Ts

+Ti

(a) (b)

Figure 5.14. The instability diagrams which schematically illustrate the relative positions of allthe four instability lines (T−

i , T+i , Tcs and Ts) superimposed on the phase diagram. The two equilib-

rium phase boundaries are drawn with thick lines. (a) shows ordering instability line (T−i ) at higher

temperatures compared to clustering instability line (Ts) while in (b) Ts is above T−i . These lines

segment the phase diagram into several domains in which different reaction mechanisms are opera-tive. Domain A: �′ → �+� (nucleation and growth). Domain B: �′ → spinodal ordering → �+�.Domain C: �′ → simultaneous ordering and clustering. Domain B′: �′ → spinodal ordering →spinodal clustering. Domain D: �′ → spinodal clustering → spinodal ordering within solute-enrichedregions → �+�. Domain E: simultaneous ordering and clustering.

curvature along that branch of the free energy. Depending on the temperature atwhich these develop, a variety of phase diagrams can be produced. They gaveresults for both first order and second order ordering reaction cases.

In all, five different phase diagrams (shown in Figure 5.15; Table 5.9), alongwith the instability lines that would be obtained in each case are: (a) the classicalphase separation case, (b) the first-order ordering case, (c) a monotectoid case,in which a miscibility gap develops in the ordered phase, (d) one in which amiscibility gap develops in the disordered phase that is metastable with respect tothe ordering reaction and (e) a syntectoid case, in which a miscibility gap developsin the disordered phase at high temperatures and an ordering reaction occurs atlow temperatures.

An alloy of composition co is in a single phase disordered region (Figure 5.15(a))at higher temperatures (above the miscibility gap maximum) and within a




(d)

Iα

β

Composition

Tem

pera

ture

VI+

VII

IV + V

T i+

T i–

Ax By

(e)

VIII

Composition

Tem

pera

ture

VI

I X

VI

β T i–

T cs

T i+

Ax By

(a)

Coherentspinodal

Miscibility gap

Composition

Tem

pera

ture

A

C 1 C 0

X

I

C 2

α2α1

α0

B

Chemicalspinodal

α

α1 + α2

(c)

IV

V

II + IIIT i

–

T i+

Composition

Tem

pera

ture

I

α

β

Ax By

Composition

Tem

pera

ture

IV

V

II + IIII

T i–T i

+

Ax By

α

β

(b)

VII + IXII + III

Figure 5.15. The phase diagrams produced by systematically varying the curvatures of thefree energy curves for the ordered and disordered phases. The phase diagram (c) is not observed in theSCW model. (a) A simple phase separation case, (b) a simple ordering diagram, (c) monotectoiddiagram, (d) ordering with metastable miscibility gap diagram and (e) a syntectoid case. The stabilityregions have also been shown (see Table 5.9, after Soffa and Laughlin 1989).

two-phase ordered region at lower temperatures. The two phases differ from oneanother only in composition but have identical crystal structure, i.e. isostructuraldecomposition, whereas in the cross-hatched region in Figure 15.5(b), the homo-geneously ordered phase is unstable with respect to phase separation. The T+

i isthe instability line with respect to disordering upon heating. In Figure 15.5(c), thecross-hatched region is the region of thermodynamic instability with respect tophase separation.

The Laughlin and Soffa approach has the advantage of simplicity and generality;it does not rely on the assumption of any particular model. It has the disadvantagethat the thermodynamic consistency is not built-in, i.e. there are consistency rulesfor construction of free energy curves that have not been examined.




Table 5.9. The characteristic barrierless reactions operative in different stabil-ity regions as predicted by the SCW model (Simmons 1992).

Region Reaction Region Reaction

I No reaction II No reactionIII No reaction IV � → �

� → �′ +�′′

�′ → �V � → � VI � → �

� → �′ +�′′

�′ → �VII � → � VIII � → �′ +�′′

�′′ → �IX � → �′ +�′′ X � → �′ +�′′

�′′ → � (T > Tconsolute) � → �′ +�′′

(T < Tconsolute) �′′ → �

� ≡ disordered phase and � ≡ ordered phase.

5.2.6 A case study: Ti–Al systemAs an illustration of what has been discussed so far, we will present here a sum-mary of the results obtained on the Ti–Al alloy system using the first principlesconfigurational thermodynamic approach by Asta et al. (1992, 1993). The Ti–Alsystem is an incoherent system as the different stable/metastable ordered phasesthat appear across the full composition range are the superstructures of either thehcp (corresponding to �-Ti) or the fcc (corresponding to Al) lattice. Prior to thework of Asta et al., several ab initio electronic structure calculations had beencarried out on the Ti–Al system (Fu 1990, Hong et al. 1991) to study the ener-getics as well as the structural and mechanical properties of stable and metastablestoichiometric compounds at 0 K. But the effect of variation of temperature andcomposition on these properties was not studied.

Asta et al. (1992) have used the local density based full potential linear muffintin orbital (FP-LMTO) method to calculate the ground state cohesive propertiesof all the stable and metastable phases under the tetrahedron–octahedron clusterapproximation for both fcc and hcp lattices (Table 5.10). The ECI were calculatedusing the Connolly–Williams inversion method and the CVM free energies wereused to determine the phase stabilities as functions of temperature, composition andorder parameter to arrive at the Ti–Al phase diagram depicting the stability regimesof various stable and metastable phases and the pertinent transition temperatures.Their results showed that of all the structures considered, only hcp-Ti, fcc-Al,D019-Ti3Al, L10-TiAl and D022-TiAl3 were energetically stable in agreement withthe experimental results. The remaining phases were found to be metastable. Of




Table 5.10. Cohesive properties of some of the intermetallics of Ti–Al system.

System Lattice parameter(a, (c/a)) (Å)

Bulk modulus(GPa)

Formation energy(kJ/mol)

Al 3.985 (1.000) 84 0.0004.046 (1.000) 82

Ti 2.864 (1.617) 106 0.0002.951 (1.588) 105

Ti3Al 5.649 (0.809) 126 −28�75.761 (0.807) 120–145 −25�3

TiAl 3.935 (1.012) 128 −42�04.005 (1.016) 160–176 −36�50

TiAl3 3.790 (2.240) 118 −41�93.848 (2.234) — −30�7 to −37�4

In the first row of each column are the calculated results as obtained by Asta et al. using the FP-LMTOmethod; in the second row are the experimental results.

these metastable phases, L12-TiAl3 was found to be in close competition withD022-TiAl3 phase. Further, the hcp disordered alloys were noted to be stable ascompared to fcc disordered solid solutions up to 50 at.% Al.

The Al-rich and equiatomic compounds were shown to undergo large structuralrelaxations even though the size mismatch between the atomic constituents wassmall. The calculated CVM phase diagrams for the fcc- and hcp-based Ti–Al alloysshowed that ordered fcc-based alloy phases are more stable than hcp-based alloyphases with respect to corresponding disordered solid solution. Consequently thereis a stronger tendency for atomic ordering in fcc-based alloys which was attributedto the differences in the electronic structures between fcc- and hcp-based alloyphases. At Al-rich compositions, the L10 and the D022 phases were found to bestrongly ordered to well above their experimentally observed melting temperatures.Moreover the phase field of D022-structured TiAl3 was predicted to be muchnarrower than that for L10-TiAl which is in agreement with the experimental phasediagram. The theoretical transition temperature for the D019-Ti3Al phase was about600�C too high; this discrepancy was attributed to the neglect of contributionsfrom vibrational and electronic excitations to the free energy.

The phase relationships in the Ti–Al system have been subjected to many mod-ifications in the recent past, partially because of differences in the impurity levelsin the alloys studied and in the interpretation of the observed microstructures.There have been major discrepancies concerning the extension of the (�-Ti) fieldwith respect to temperature. These modifications are also reflected in the vari-ous calculations published in the literature. Kauffmann and Nesor (1978) usedvery simple descriptions in their calculations: the disordered solution phases were




described as quasiregular solutions and the intermetallic compounds Ti3Al, TiAland TiAl3 were assumed to be stoichiometric. Within the limitations of the modeldescriptions used, the general features of the then accepted phase diagram couldbe reproduced. Murray (1987) used the subregular solution model to describe thedisordered phases and assumed TiAl to be stoichiometric. Gros et al. (1988) cal-culated the Ti-rich part of the system. The liquid and (�-Ti) phases were describedas quasiregular solutions and (�-Ti) and Ti3Al were described in terms of thesublattice model as one single phase undergoing an order–disorder transition. Thecalculated partial Gibbs free energies of Al and Ti in (�-Ti) agreed within 5–12%of the experimental values. Within the constraints used, reasonable agreement wasobtained for the (�-Ti) and Ti3Al phase boundaries.

More complete calculations of the Ti–Al system, taking into account the entirephase diagram and the ranges of homogeneity of the ordered intermetallic phases,were carried out later by Murray (1988). He described the disordered solutionphases as quasiregular solutions, used the Bragg–Williams approximation for theordered Ti3Al and TiAl compounds and assumed TiAl3 to be stoichiometric. Inorder to improve the agreement between the calculated and experimental phasediagrams, Murray found it necessary to relax the conditions given by the Bragg–Williams approximation for the Gibbs free energy for these compounds. Later,Hsieh et al. (1987) and Lin et al. (1988) included the TiAl2 and Ti3Al7 phases inthe calculated phase diagram by using a quasisubregular solution model.

Recently, a thorough thermodynamic assessment of the Ti–Al system has beenundertaken by Zhang et al. (1997) in which three different analytical descriptionshave been used to describe the three different types of phases occurring in thesystem: the stoichiometric compounds, the disordered solution phases and theordered intermetallic compounds which have homogeneity ranges. A least squaretechnique has been used to optimize the thermodynamic quantities pertinent to theanalytical description, using the experimental data available in the literature. Wewill give here a brief account of the approach used by Zhang et al. (1997) for theassessment of the Ti–Al phase diagram in order to illustrate the semi-empiricalmethodology adopted.

For an accurate description of these phases, the Gibbs free energy (G) must beexpressed as an analytical function of the thermodynamical variables of interest,viz., composition, temperature and pressure. The Bragg–Williams model is inad-equate to account for the phase reactions/transitions in this system because of itsinability to account for local correlations or short-range ordering effects. Zhanget al., therefore, have used different models for the ordered and disordered phases.They have considered nine phases in their calculations: the disordered solutionphases, liquid, �-Ti, �-Ti and Al; and the ordered intermetallic compounds Ti3Al,TiAl, TiAl2, Ti2Al5 and TiAl3. The TiAl2 and Ti2Al5 phases have been assumed




to be stoichiometric, while the wide ranges of homogeneity exhibited by Ti3Aland TiAl have been taken into account in the analytical model.

The free energies corresponding to the stoichiometric compounds TiAl2 andTi2Al5 are described by

G = xoTiG

oTi +xo

AlGoAl +�Gf (5.21)

where xoi are the mole fractions of component i and Go

i are the respective referencestates. The term �Gf represents the Gibbs free energy of formation. These Gparameters are either constant or are linear functions of temperature.

The disordered solution phases, liquid, �-Ti, �-Ti and Al are described asrandom mixtures of Ti and Al:

G = xTiGoTi +xAlG

oAl +RT

∑i

xi ln xi +xTixAl�Go +G1�xTi −xAl�� (5.22)

where Go and G1 are the coefficients of the terms of the excess Gibbs energy.This model is called the quasisubregular solution model.

The ordered intermetallic compounds Ti3Al, TiAl and TiAl3 exhibit an apprecia-ble range of homogeneity and are described by the sublattice or Wagner–Schottkymodel. These compounds are considered to consist of two sublattices which areoccupied by Ti and Al atoms only in the perfectly ordered state. The partiallyordered regions are described by assuming the formation of substitutional atomson each sublattice. The analytical description is

G = xTiGoTi +xAlG

oAl +RT

∑i

[n1i lnn1

i +RT∑i

n2i lnn2

i −RT∑i

N 1i lnN 1

i

−∑i

N 2i lnN 2

i

]+�Gf +∑

i

n2i G

2i +n1

Tin1Al�G

1o +G1

1�n1Ti −n1

Al��

+n2Tin

2Al�G

2o +G2

1�n2Ti −n2

Al��+n2Tin

1AlG

12

Here we have xi = n1i +n2

i and Nk = nkTi +nk

Al where nki are the mole fractions of

component i on sublattice k and Nk are the site fractions of sublattice k. �Gf isthe free energy of formation of the perfectly ordered phase at the stoichiometriccomposition. Gk

o are the coefficients of polynomial interaction terms betweenatoms on the same sublattice and G12 is that between substitutional atoms on thedifferent sublattices. The quantities nk

i are calculated by minimizing the Gibbs freeenergy for given xi. The optimization and calculation programmes developed byLukas et al. (1982) have been used for these calculations of the Ti–Al system.




5.3 TRANSFORMATIONS IN TI3AL-BASED ALLOYS

5.3.1 � → D019 orderingThe �2-phase based on the stoichiometry Ti3Al has a D019 (hP8) structure andP63/mmc symmetry. The structure is characterized by an atomic arrangement onthe close packed (0001) planes (Figure 1.20) which ensures that Al atoms haveonly Ti atoms as their first near neighbour. A hexagonal stacking �ABABAB�of such planes produces the D019 structure. An examination of the symmetryrules for a second order ordering process in the context of � → D019 transfor-mation reveals that the first Landau–Lifshitz condition (symmetry elements ofthe product phase having a subset of that of the parent) is fulfilled. This meansthat replacement of atoms in an �-lattice can produce a D019 structure. TheD019 structure can be formed by introduction and amplification of a concentra-tion wave with wave vector star of k

⟨12 00

⟩. This wave vector terminates at the

special point of the reciprocal space fulfilling the second Landau–Lifshitz condi-tion. The third condition, however, is not satisfied as the sum of three variantsof the wave vector star (k1 +k2 +k3) results in a reciprocal lattice vector of theparent hcp structure. Therefore � → D019 transformation is not a candidate fora second order ordering process. With a high degree of supercooling, at T < Ti,Ti being the instability, a short wave length concentration wave with k= ⟨

12 00

⟩,

can amplify at least to a limited extent to generate a partially ordered D019

structure.The formation of the Ti3X precipitates having the D019 structure is seen in a

number of binary and ternary alloy systems such as Ti–Al, Ti–Ga, Ti–Al–Ga,Ti–Al–Sn, Ti–Al–In. It is important to note that the mismatch between the latticedimensions between the parent � and the ordered Ti3X precipitates are different indifferent alloy systems and correspondingly the morphology of precipitates variesfrom one system to the other. The extent of mismatch between the matrix � andTi3X �D019� precipitates in different systems are listed in Table 5.11.

Table 5.11. Matrix–precipitate mismatch along a and cdirections for Ti–Al alloys.

Alloy system Matrix–precipitate mismatch

Along a (%) Along c (%)

Ti–Al 0.83 0.35Ti–Al–Ga 0.99 0.51Ti–Al–Sn 0.20 0.15Ti–Al–In 0.58 0.21




5.3.2 Phase transformations in �2-Ti3Al-based systemsThe development of engineering alloys based on Ti3Al (�2) was initiated with theprimary objective of ductilizing the brittle binary Ti3Al intermetallic by the intro-duction of �-stabilizing elements, particularly Nb. As mentioned earlier, Ti3Al-based alloys can be classified into different groups on the basis of the extent ofadditions of �-stabilizers. The relative stabilities of three phases, namely, the �2

(D019), �o (B2) and O (orthorhombic) phases, play a key role in determining thephase equilibria in these alloys. The thermal history, however, remains as the othermajor factor in controlling the mode and the sequence of phase transformationsand the microstructure developed.

The literature on Ti3Al-based alloys is overwhelmingly dominated by that onternary Ti–Al–Nb alloys. The trends in the phase transformations in Ti3Al-basedalloys are, therefore, discussed in this chapter with reference to the ternaryTi–Al–Nb system. Figure 5.16 shows (a) pseudobinary phase diagram (Ti–27.5%Al with increasing Nb) and (b) the isothermal section at 1173 K of the ternaryphase diagram of the Ti–Al–Nb system (Banerjee and Rowe 1993). Phase fieldsin which a single phase, �2, �o or O, exists are shaded and two- and three-phasefields are marked. The composition ranges associated with the three classes ofalloys, containing different levels of �-stabilizers, which have been studied indetail, are indicated by circles, inscribed with the legends I, II and III. This1173 K isotherm of the ternary phase diagram suggests that alloys belonging toclass I consist of a mixture of the �2- and �o-phases while those belonging toclass II made up of a mixture of the �o- and O-phases. The class III alloys ofintermediate compositions exhibit a three-phase (�2 + �o + O) microstructure.

1200

Tem

pera

ture

(°C

)

O

1100

1000

900

10 20 30Ti–27.5AlAtomic % Nb

βo (B2)α + βoα + α2

α

α2 + βo

α2

O + βo

O

(a)

(b)

50Ti

60Ti

10N

b

30Al

40Al

20Al

25Al

βo + α2

βo + O

20N

b

I

II

III

30N

b

βo

α2

O + βo + α2

Figure 5.16. (a) Pseudobinary phase diagram (Ti–27.5% Al with increasing Nb) and (b) the isother-mal section at 1173 K of the ternary phase diagram of the Ti–Al–Nb system.




Since the compositions of all the three classes of alloys fall nearly along the lineof 25 at.% Al, a pseudobinary section along the Ti3Al–Nb line (Figure 5.16(a))can conveniently be used to show the possible phase reactions in these alloys.A comparison can be drawn between this pseudobinary phase diagram and abinary phase diagram of Ti alloyed with a �-stabilizing element (Figure 6.1).The similarity between them essentially arises due to their �-isomorphous naturewhich restricts the composition range of the �/�2 phase field. Another pointof similarity is reflected in the tendency for the formation of an orthorhombicstructure at relatively high levels of �-stabilizer addition. In the case of thebinary Ti–X (X being a �-stabilizer like V, Mo, Nb) system the orthorhombicphase (�′′, orthorhombic martensite) is an extension of the hcp �′-martensite.In comparison, the orthorhombic O-phase in the Ti3Al–Nb phase diagram canbe considered as an extension of the Ti3Al (D019) phase with an orthorhombicdistortion. This argument will be further elaborated when the crystallography ofthese phases is considered. The essential difference between the binary Ti–X andthe pseudobinary Ti3Al–Nb phase diagrams is due to the fact that there is a strongtendency towards chemical ordering in the latter case. In Ti3Al-based alloys withlow Nb contents, the terminal solid solution experiences a strong tendency to orderinto the D019 structure while alloys more enriched with Nb are under the influenceof a strong tendency towards B2 ordering at elevated temperatures and O-phaseordering at lower temperatures. The presence of these ternary ordering tendenciesis responsible for the introduction of additional phase reactions in this system.

Having discussed the general tendencies related to phase stabilities in Ti3Al-based alloys, let us examine the experimental observations made on this system.Banerjee (1994a,b), in his excellent review, has constructed a diagram in whichexperimental observations on phase transformation processes in this system havebeen summarized. Figure 5.17 is essentially a reproduction of the same diagramin which the data reported in a number of research papers have been assimilated(Banerjee et al. 1988; Kestner-Weykamp et al. 1989, 1990; Bendersky et al. 1991;Kattner and Boettinger 1992; Muraleedharan et al. 1992a,b; Rowe et al. 1992).At low Nb levels (<7.5 at.% Nb), quenching from the �-phase field results in amartensitic transformation, leading to the formation of hcp �′-martensite whichundergoes an ordering process to yield the �2-phase. The �′ →�2 ordering reactioncannot be suppressed on quenching and this is clearly revealed in the observation offine D019 domains within the martensite laths in quenched alloys containing up to7.5 at.% Nb (Figure 5.18(a)). The Ms temperature for the �→ �′ transformation israpidly depressed beyond 7.5 at.% Nb (Strychor et al. 1988). This is because neithercan the required chemical ordering occur during the martensitic transformation norcan the ordered �o-phase martensitically transform into the �2- or the O-phase.The occupancy of lattice sites in the parent (�o) and product (�2 or O) structures




1200

βo to α2/0Widanstatten

1100

1000

900

800

700

600

500

4000 5 10 15 20 25

Nb (Atomic %)

βo to α-type

Temperedmartensite

β o to

α2

Mas

sive

βo to 0Massive

Tem

pera

ture

(°C

)

O

α

α2α2 + β α2 + βo

α + β

α2 + βo + 0

βo + 0

Liquid

Ti3Al

Figure 5.17. Ti–Al–Nb pseudobinary phase diagram summarizing experimental observations onphase transformation reactions in this system.

are not consistent with a martensitic transformation. The �-phase in higher Nballoys (7.5–25 at.% Nb), therefore, is retained on quenching (Strychor et al. 1988),although it exhibits instabilities with respect to B2 ordering and also to a varietyof displacement ordering processes. The quenched-in �-phase shows a domainstructure resulting from the bcc to B2 ordering (Figure 5.18(b)). In addition tothe presence of B2 superlattice reflections, extensive streaking along < 110 >�

and < 112 >� directions is noticed (Figure 5.18(c)). These instabilities have beenattributed to transverse displacement waves which are of the �110�� < 110 >� and�112�� < 111 >� types. Strychor et al. (1988) have pointed out that the former isassociated with a precursor for the � → �′ martensitic transformation while thelatter originates from the tendency for the occurrence of the � → # transition. Atweed contrast observed in the quenched-in �o (ordered �) phase (Figure 5.18(d))is consistent with the presence of transverse displacive waves and this tendency isnoticed over the entire range of Nb concentration from 7.5 to 25 at.%.

A relatively slow cooling from the �-phase induces diffusional phase trans-formations in alloys containing 7.5–25 at.% Nb, leading to a mixture of �2 +�o,




(a) (b)

(c) (d)

Figure 5.18. (a) Fine D019 domains within the martensite laths in quenched alloys containing upto 7.5 at.% Nb, (b) quenched-in �-phase shows a domain structure resulting from the bcc to B2ordering, (c) in addition to the presence of B2 superlattice reflections, extensive streaking along< 110 >� and < 112 >� directions is shown and (d) a tweed contrast as observed in the quenched-in�o (ordered �) phase (after Strychor et al. 1988).

�2 +�o + O or O +�o phases. The parent �o-phase undergoes a decompositioninto either �2 +�o or O+�o phase mixtures which is followed by the subsidiaryphase reaction, �2 → �2 + O or �2 +�o → O. The precipitation of the �2-or theO-phase results in a lath, lamellar or mosaic morphology, depending on the alloycomposition and the cooling rate. The Burgers orientation relationship between thehexagonal or the orthorhombic product and the parent cubic phase is invariablymaintained. The volume fraction of the primary �2-phase (which subsequentlytransforms to the O-phase) decreases with increasing Nb content and remainsconfined to grain boundaries as the alloy composition approaches 25 at.% Nb. Themorphology of the product, at the level of light microscopy, bears a significant




resemblance to that observed in conventional �+� Ti alloys slowly cooled fromthe �-phase field. An increase in the cooling rate results in a refinement in thesize of the packet (which is constituted of a group of laths stacked in a parallelarray) and in the formation of a fine basket weave morphology.

As mentioned earlier, quenching from the �-phase field yields an orderedmartensitic structure (�2) in alloys containing up to 7.5 at.% Nb and leads to theretention of the �o-phase in alloys containing 7.5–25 at.% Nb. Tempering of themartensite results in a rapid growth of the B2 domains (Sastry and Lipsitt 1977)and causes �-phase precipitation and recrystallization to a limited extent (Martinet al. 1980). The retained �o-phase, on ageing, decomposes to #-related structuresat temperatures below about 770 K. The transformation from the B2 structure to avariety of ordered #-structures has been discussed in detail in Chapter 6. Ageingof the quenched-in �o-phase in the composition range of 7.5–11 at.% Nb inducesa massive transformation in the �2-phase. Alloys with higher Nb contents (11–25 at.%) undergo a �o → O transformation on ageing; this can occur either by amassive transformation producing equiaxed O-phase grains or by a Widmanstat-ten precipitation of O-phase plates which exhibit a martensite-like substructure.These two decomposition modes can operate in parallel or sequentially with theequiaxed O-phase grains finally consuming all the O-plates (Bendersky et al. 1991;Muraleedharan et al. 1992a,b). The O-phase subsequently decomposes into theequilibrium �2- and/or the �o-phases.

5.3.3 Structural relationshipsThe phases which are present in Ti3Al–Nb alloys (Nb content up to 30 at.%)are all based on two basic structures: bcc at high temperatures and hcp at lowtemperatures. While the bcc and hcp structures are related by the well-knownBurgers lattice correspondence (discussed in detail in Chapter 4), the ��hcp�→�2

(D019) and the ��A2� → �o (B2) transformations involve replacive ordering inwhich the crystallographic axes of the product remain parallel to those of theparent. A structural description of these ordering processes has been given inearlier sections. With increasing Nb content the hexagonal symmetry of the�2-phase gets distorted to orthorhombic symmetry (O-phase) in a manner quitesimilar to the structural change associated with the transition from hexagonal (�′)to orthorhombic (�′′) martensite. The lattice correspondence between the � andthe �′′ (Bagariatskii) is essentially the same as the Burgers correspondence as canbe seen by changing from the hexagonal to the orthorhombic axes system. The�′′ martensite phase is disordered and has a Cmcm space group with four atomsper unit cell. The O-phase in the ternary Ti–Al–Nb system corresponds to thestoichiometry of Ti2AlNb where Nb atoms occupy distinctive sublattices (Banerjeeet al. 1988, Mozer et al. 1990). The A2BC structure shown in Figure 5.19 is




Figure 5.19. The A2BC structure, a prototype of the structure of the Ti2AlNb phase (O-phase), isclosely related to the D019 structure.

the prototype of the Ti2AlNb phase (O-phase) and is closely related to the D019

structure. Crystallographic data pertaining to the ordered A2BC structure are givenin Table 5.12. In this structure, the atoms can occupy a range of positions withoutdestroying the symmetry and the stacking sequence of the O-phase. The ranges

Table 5.12. Atomic positions of Ti, Al andNb atoms in the orthorhombic O-phase.

Atom Positions (x� y� z)

Ti 0.2310, 0.9041, 0.25000.7690, 0.0959, 0.75000.2310, 0.0959, 0.75000.7690, 0.9041, 0.25000.7310, 0.4041, 0.25000.7310, 0.5959, 0.75000.2690, 0.4041, 0.25000.2690, 0.5959, 0.7500

Al 0.0000, 0.1630, 0.25000.0000, 0.8367, 0.75000.5000, 0.6633, 0.25000.5000, 0.3367, 0.7500

Nb 0.0000, 0.6357, 0.25000.0000, 0.3643, 0.75000.5000, 0.1357, 0.25000.5000, 0.8643, 0.7500




of atomic coordinates for each equivalent position are also given in the table. Theatomic sites of the Ti2AlNb unit cell proposed by Mozer et al. (1990) lie withinthese ranges and this structure can be considered to be a special case of the A2BCstructure with b/a less than

√3 and c/a less than the ideal value.

The similarity between the D019 structure and the O-phase structure can beseen by comparing the basal plane projection of the former with the (001) planeprojection of the latter (Figure 5.20). Comparing the first, second, third and fourthnear neighbour bond lengths in the D019 structure with a non-ideal c/a ratio(close to that of Ti3Al) and the corresponding bond lengths in the Ti2AlNb phase,Singh et al. (1994) have shown that these structures have a close similarity. Itmay be noted that the O-phase structure involves further ternary ordering of the�2-phase with Ti, Al and Nb atoms predominantly occupying three different typesof sites (Banerjee et al. 1988, Mozer et al. 1990). In view of the structural relationsassociated with these phases, the relevant structural changes can be described interms of the following:

(1) distortion of {110}� planes and changes in their interplanar spacings(2) shuffles, or relative displacements of neighbouring {110}� planes, and

B

A CA C

B

A C

B

[011

0]

[010

]

α2 [0001] 01 (001) 02 (001)

[100]

[211

0]

0.578 nm 0.595 nm

1.00

1 nm

0.97

0 nm

Atom Al Ti Nb

Layer A

Layer B.c /2 above

AngleABC

α2 01 02

63° 65°60°

Ti2AlNb

Figure 5.20. A comparison between the D019 and the O-phase structures as elucidated by the basalplane projection of the former with the (001) plane projection of the latter.




(3) chemical ordering that changes the occupancies of Ti, Al and Nb atoms amongthe lattice sites.

The main advantage of describing the phase transitions in this system using acommon framework is that a single thermodynamic potential can be identified asa continuous function of a set of order parameters which describe the three typesof structural change. Bendersky et al. (1994) have considered the group/subgroupsymmetry relationship with respect to the transformations in this system and haveidentified the possible sequence of crystallographic transitions.

5.3.4 Group/subgroup relations between BCC (Im3m), HCP (P63/mmc)and ordered orthorhombic (Cmcm) phases

A continuous phase transition of first or higher order requires that the symmetryof the product phase is a subgroup of that of the parent phase. This is, in fact,the Landau–Lifshitz rule I for a transition to qualify as being of second order(refer to Chapter 2). Usually the low-temperature phase has a lower symmetry thanthe high-temperature phase, and a transition involving a lowering of symmetry isknown as ordering while one associated with an increase in symmetry correspondsto disordering.

The approach pertinent to determining the possible sequence of transformationsis based on successive stages of symmetry reduction. The necessary informationregarding symmetry relations is contained in the space group tables of the Inter-national Tables for Crystallography (1991). On the basis of this information, theindividual steps of a transition can be predicted by considering the maximal sub-group relation between the parent and the product phases in any given step. Asubgroup H of the space group G is called a maximal subgroup of G if there isno subgroup L of G such that H is a subgroup of L. This approach helps in theidentification of all symmetry reduction steps though it is not necessary that eachof these steps must really occur in a transformation sequence.

There are no apparent subgroup relations between structures with cubic andhexagonal symmetries, G1 and G2, respectively, due to the presence of non-coinciding threefold <111> cubic and sixfold [0001] hexagonal symmetry axes.A connection between these symmetries can be arrived at by introducing anintermediate structure with space group Gt which is either a supergroup of boththe structures, G1 and G2, or a subgroup of both the structures. A supergroup,constituting a group union of G1 and G2, does not exist as both the bcc and hcpstructures exhibit very high symmetry. However, a subgroup Gt can be found asthe intersection group of G1 and G2� In particular, considering the disordered bccand hcp structures with Im3m and P63/mmc space groups, respectively, and takinginto account the Burgers orientation relationship, (110)��0001��; [111]��[1120]�,




between them, the intersection group Gt is found to be the orthorhombic spacegroup Cmcm, with its c-axis parallel to the [110]� direction.

The Cmcm space group, with an appropriate choice of Wyckoff sites, canrepresent a structure which is close to hcp but differs from it in symmetry andin the relative positions of atoms on the basal plane (Figure 5.21). The Cmcmstructure can also be considered as the bcc structure distorted by relative shifts ofthe (110)� planes.

Bendersky et al. (1991) have proposed sequences of transitions that connectthe higher symmetry cubic and hexagonal space groups to the lower symmetryorthorhombic space group (Figure 5.22). These sequences include all known equi-librium phases observed in Ti3Al–Nb alloys with Nb contents of less than 30 at.%.The sequences shown in Figure 5.22 show two branches, one starting from the dis-ordered bcc structure and the other from the ordered B2 structure. The directionsof symmetry reduction are indicated by the arrows, and the indices of symmetryreduction between two neighbouring subgroups are indicated by numbers in squarebrackets within these arrows. The index of a subgroup is the ratio of the numberof symmetry elements in a group to that in the subgroup. These indices give thenumber of lower symmetry variants (domains) that can be generated when a tran-sition from a high-symmetry to a low-symmetry structure occurs. Inclined arrowsindicate symmetry changes due to atomic site (Wyckoff position) changes, leavingthe occupancy fixed, i.e. displacive ordering. Vertical arrows, on the other hand,

A

B

C

A B

D CF

E

b

[010]

[100]

fC2

fA2

fS2

fA1

fC1

fS1

a

Figure 5.21. The structure corresponding to Cmcm space group, with an appropriate choice ofWyckoff sites, can represent a structure which is close to hcp but differs from it in symmetry and inthe relative positions of atoms on the basal plane.




[2]

[3]

[2]

[2]

[3]

[3]

[2]

[2]

I4/mmm

P4/mmm Pmmm

Cmmm Cmcm (A20)

Pmmc (B19)

P 63 /mmc (A3)

P 63 /mmc (D019)

Cmcm (A2BC)

Im3m (A2)

Pm3m (B2)

[2]

[2]

[2]

[2]

[2]

[4]

Figure 5.22. The sequences of transitions connecting the higher symmetry cubic and hexagonalspace groups to the lower symmetry orthorhombic space group. The directions of symmetry reductionare indicated by the arrows, and the indices of symmetry reduction between two neighbouringsubgroups are indicated by numbers within these arrows.

indicate symmetry changes due to changes in atomic site occupancy, i.e. chemicalordering. Slight adjustments in site positions and occupancies due to new atomicenvironments will accompany chemical and displacive ordering, respectively.

The sequences of possible structural transitions (Figure 5.22) are arrived atfrom the consideration of maximal subgroup relations. The atomic correspondencebetween these structures are depicted in Figure 5.23. The I4/mmm and Fmmmstructures are obtained by homogeneous straining of the parent cubic lattice, A2(Im3m). Similarly the P4/mmm and Cmmm structures correspond to a homoge-neously distorted B2 structure (Pm3m). While a tetragonal distortion along a cubic<100> direction brings about the first step of the transition from the A2 and theB2 structures, the second step is associated with different distortions along twoorthogonal cubic <011> directions. The overall orthorhombic distortion of thecubic structure, if not accompanied by ordering, is probably unstable for materialswith isotropic metallic bonding. Therefore these structures are not expected toexist as metastable states but rather represent a homogeneous strain accompanyingthe subsequent symmetry reduction by shuffle displacements (from Fmmm andCmmm to Cmcm and Pmma, respectively).

The Cmcm and Pmma space groups which correspond, respectively, to theStrukturbericht A20 (�-U prototype) and B19 (AuCd prototype) structures can beobtained by heterogeneous shuffles of pairs of (110) planes of either a disordered




P 63

/mmc (A3)Im3m (A2)011

100

Ti, Al, Nb

011

100

Al, Nb

Pm3m (B2)

– Upper layer

– Lower layer

b

a

Cmcm (A20)

Al, Nba

b

Pmma (B19)

b

a

a1 a2

a1 a2

Al

Ti, Al, Nb

Ti, Nb

P 63/mmc (D019)

Cmcm (A2BC)

Al

Ti

Ti

Ti

Nb

Figure 5.23. The sequences of transitions showing two branches, one starting from the disorderedbcc structure and the other from the ordered B2 structure.

or an ordered cubic structure. The amplitude of the shuffle displacement wave isreflected in the y-coordinates of Wyckoff positions: 4c (0, y, 1

4 ) for the Cmcmstructure and 2e ( 1

4 , y1, 0); 2f ( 14 , y2, 1

2 ) for the Pmma structure.For some special values of Wyckoff position’s y-coordinates and/or lattice

parameters, a structure of higher symmetry can be generated. Such a high symmetrystructure, hexagonal P63/mmc, is generated from the disordered Cmcm structurewhen the shuffles are such that the Wyckoff position parameter y is 1

3 and the ratioof lattice parameters, b/a, is

√3. Such a symmetry enhancement is not possible for

the orthorhombic Pmma structure due to the chemical order inherited from the B2structure. On the other hand, no thermodynamic barrier exists for the Cmcm (A20)to P63/mmc structural transition as the latter structure is associated with a higherentropy (due to its higher symmetry) while the internal energies, estimated on thebasis of first near neighbour interaction energies, of the two structures are nearlythe same. The structure with the lowest symmetry, the O-phase, has the Cmcmspace group and ternary ordering on the Wyckoff positions 4c1�4c2 and 8g. Thisternary ordering is responsible for making the translations in the O-phase structuretwice as much as those in the binary ordered B19 structure. The O-phase structurecan be obtained by ordering either the Pmma (B19) or the P63/mmc (D019)




structure. The D019 structure can be obtained by binary ordering of the disorderedhcp (A3) structure and could correspond to a metastable intermediate phase.

5.3.5 Transformation sequencesThe group–subgroup relations suggest the following three transformationsequences, designated as A, B and C:

A $ Im3m�A2� �12�−−→ Cmcm�A20� �1�−→ P63/mmc�A3�

�contd� P63/mmc�A3� �4�−→ P63/mmc�D019� �3�−→ Cmcm�O�

B $ Im3m�A2� �2�−→ Pm3m�B2� �12�−−→ Pmma�B19� �2�−→ Cmcm�O�

C $ Im3m�A2� �12�−−→ Cmcm�A20� �2�−→ Pmma�B19� �2�−→ Cmcm�O�

The numbers in the square brackets give the number of variants possible at eachof the transition steps. It must be emphasized here that the A2 → O transformationcan also occur in a single step provided the transformation mode is reconstruc-tive. However, coherent transformations proceed in steps, following one of thesequences mentioned above. Each transient phase may exist over a temperaturerange between the upper and lower critical temperatures (or temperatures of phaseinstability for first order transformations).

The number of variants in each transition is equal to the index of the subgroup(as indicated within the square brackets). For a sequence of transitions the numberof variants of the lowest symmetry phase (with respect to the highest symmetryphase) will be equal to the product of the indices for each step. The creation ofthe expected number of variants at each transition state generates a hierarchy ofdomain structures. It is from the domain structure that the exact sequence of stepsfollowed during the transformation can be identified. The microstructures resultingfrom the sequences, A, B and C, will all consist of the same O-phase but with dif-ferent hierarchies and types of interfaces, and the nature of the interfaces, whetherrotational, translational or mixed, will be determined by the group–subgroup rela-tion. Each symmetry reduction step necessarily leads to the formation of morethan one variant of the lower symmetry phase, thereby restoring the lost symmetrypartially in the macroscopic sense. As mentioned earlier, the number of variants ineach transition is equal to the index of the subgroup, indicated within the squarebrackets in the sequences. For the complete sequence of transitions the number ofvariants of the lowest symmetry O-phase is given by the product of the indicescorresponding to each of the steps involved in the sequence. This means that the




sequence A results in 12 × 1 × 4 × 3�= 144� variants while the sequences B andC result in 48�2×12×2� or �12×2×2� variants.

A variety of domain interfaces can be created between adjacent domains. Adomain interface will remain stress free if it is planar and if the self-strains onboth sides are compatible, so that no discontinuity of displacements occurs atthe interface. Bendersky et al. (1991) have examined the expected distribution ofdomains arising out of the different sequences, A, B and C, of the transformationand have listed the types of interfaces which can form between domains producedin different transitions (Table 5.13).

The microstructural development corresponding to the three paths, A, B andC, has also been predicted on the basis of the domain configuration expected tobe generated in the successive steps of the transformation. The three paths differprimarily as to whether the hexagonal symmetry phases or the B19 phase occurat an intermediate stage of transition. All the three paths finally lead to somewhatsimilar microstructures consisting of domains of the orthorhombic phase, althoughthe microstructure at the intermediate steps are significantly different.

For the transformation sequences A and C the first step, A2 → A20, is adisplacive transition and involves the creation of a polytwin structure. In contrast,the first step of the sequence B, namely, the A2 → B2 transition, involves purechemical ordering and produces only antiphase boundaries (APB). Polytwins areseparated from their neighbours by planar boundaries, while the isotropic APBs,which are usually curved surfaces, separate either interconnected or closed volumedomains.

Table 5.13. List of interfaces between domains in different group/subgrouptransitions.

Group/subgroup Type of interface

Im3m�A2� → Pm3m�B2� Translational APBIm3m → Cmcm�A20� Rotational twins

Translational with stacking faultmixed twin/translational

Cmcm�A20� → P62/mmc�A3� No new interfaceP63/mmc�A3� → P63/mmc�D019� Translational APBP63/mmc�D019� → Cmcm�O� Rotational (compound twins)Pm3m → Pmma�B19� Rotational twins

Translational with stacking faultmixed twin/translational

Cmcm�A20� → Pmma�B19� Translational APBPmma�B19� → Cmcm�O� Translational APB

The interfaces are described by the domain generating symmetry operation of lowest symmetry.




A3 D019 O

OB19B2

B19 O

A2

A20

→→

→

Figure 5.24. Schematics of development of microstructure by the creation of anisotropic pla-nar interfaces (due to the associated stacking fault nature), isotropic APBs (resulting frompure chemical ordering) and secondary polytwin boundaries (as in the case of the D019 → Ophase transition).

The B2 → B19 transition in the sequence B is responsible for the introductionof a polytwin structure. Further development of microstructure by the creation ofanisotropic planar interfaces (due to the associated stacking fault nature), isotropicAPBs (resulting from pure chemical ordering) and secondary polytwin boundaries (asin thecaseof theD019 → Ophase transition) is schematically illustrated inFigure5.24

The transformation paths predicted from group–subgroup relations pertain onlyto congruent phase transformations in which a single phase transforms into anothersingle phase without involving a partitioning of the alloying elements. Such trans-formations in multicomponent systems can occur under equilibrium conditionsonly at special compositions, known as consolute points, or in second order tran-sitions. However, under sufficient undercooling, a partitionless transformation canbe induced. Bendersky and Boettinger (1994, Bendersky et al. 1994) have shownthat the predictions made closely match the observed transformation sequencesin some specific ternary Ti–Al–Nb alloys. The salient features of their experi-mental observations are summarized in the following, mainly in order to illustratehow observations on the domain distribution can lead to the identification of thetransformation path.

5.3.5.1 Transformation sequence in the alloy Ti–25 at.% Al–12.5 at.% NbThe observations made on the needle-shaped transformation product in this alloycorrespond to a situation where a displacive transition takes place in the disordered




A2 structure. Therefore, either of the sequences A or C can be expected to beoperative.

Bendersky and Boettinger (1994, Bendersky et al. 1994) have discussed themorphological features of the needle-shaped product formed in this alloy whencooled from 1373 K at the rate of 400 K/s and the displacement vectors associatedwith the domain boundaries contained within these needles. Even though diffrac-tion patterns obtained from individual needles can be indexed in terms of one ofthe three variants of the D019 structure, the splitting of diffraction spots is indica-tive of the occurrence of orthorhombic distortions in the basal plane of the D019

phase, suggesting the presence of the O-phase. Two types of interfaces, namely,wavy, isotropic APBs with the displacement vectors, R = 1

6 < 1120 > and planarstacking fault-like defects with R= 1

4 �0110�, have been detected within these nee-dles. The presence of the 1

6 < 1120 > APBs points towards the occurrence of thedisordered A3 phase as an intermediate step prior to the D019 ordering. Out of thethree possible variants of the 1

4 < 1010 > faults, only one variant has been actuallyobserved. This deviation from hexagonal symmetry suggests that the preferreddirection (arising from orthorhombic symmetry) existed prior to the appearanceof hexagonal symmetry, which is consistent with the initial step of sequence A.Finally the presence of O-phase domains, related to each other by hexagonalsymmetry, is attributable to the occurrence of the last step of sequence A.

5.3.5.2 Transformation sequence in the alloys Ti–25 at.% Al–25 at.% Nb,Ti–28 at.% Al–22 at.% Nb and Ti–24 at.% Al–15 at.% Nb

In both these alloys, Bendersky and Boettinger (1994) and Muraleedharan et al.(1992a,b) have provided evidences for the occurrence of A2 → B2 ordering priorto the transition to the close packed structure. This means sequence B is expectedto be operative in these alloys which show self-accommodating plate-like domains.The observed lattice correspondence, [001]o��[011]c and [100]o��[100]c (o and creferring to the orthorhombic and the cubic phases, respectively), is consistent withthe group–subgroup relations discussed earlier. This gives six rotational variantsof the orthorhombic phase (either B19 or O), each with its basal (001)o planeparallel to one of the six {110}c planes of the cubic structure. Accommodationof transformation strains requires a small mutual rotation between the contactingvariants by the creation of stress-free interfaces. The measured misorientationbetween adjacent plates has been shown to be consistent with that calculatedfrom the lattice strains (Bain strain) required for the B2 → O transformation. Ananalysis of the displacement vectors associated with the domain boundaries hasrevealed the presence of two types of domain boundaries. The first type, with awavy APB appearance, is associated with the displacement vector R= 1

2 [010]o (orthe 1

2 [100]o vector which is equivalent due to the C-centring of the Cmcm space




group of the O-phase). APBs of this type are expected to form as a result of theB19 → O transformation in which the unit cell parameters, a and b, are doubled.The second type of domain boundaries show a distinct faceted appearance andcorrespond to the displacement vectors, R = 1

4 [012] or 14 [010]. These interfaces

are expected to be produced during the B2 → B19 transition. The homogeneous(Bain) strain corresponding to the B2 → B19 transition is also responsible for theobserved orientations of the six twin variants of the orthorhombic or the hexagonalstructure. In view of these observations, it could be concluded that these alloys gothrough sequence B during relatively fast cooling from 1373 K.

As mentioned earlier, the sequences predicted from the group–subgroup relationsand those observed in the alloys discussed in this section correspond to the situationof congruent (partitionless) transformations. Transformation sequences pertainingto a situation where the cooling conditions allow alloy partitioning and phasereactions are discussed in the following section.

5.3.6 Phase reactions in Ti–Al–Nb systemPhase reactions involving partitioning of the alloying elements can be under-stood using the proposed section (Banerjee et al. 1990) of the Ti–Al–Nb system(Figure 5.16) in the vicinity of the composition Ti2AlNb. This pseudobinary sectionis along the composition line joining Ti3Al and Nb3Al. This section has later beenmodified and enlarged to some extent (Figure 5.25) by Bendersky and Boettinger(1994, Bendersky et al. 1994) by taking into account the following factors:

(1) The disordered�-phase definitely exists between the � and the�2-phases in pureTi3Al and the �-phase field narrows down with increasing additions of Nb.

Atomic % Nb

Tem

pera

ture

(K

)

O

βo

α2

α 2 + O

10 20 30

Ti3Al Nb3Al

1300

1200

1100

900

1000

800

700

β

α2 + β

α 2 + βo

α

α 2 + β o

+ O

Ti–25Al–15Nb

Figure 5.25. The pseudobinary section, enlarged and modified, along the composition line joiningTi3Al and Nb3Al for the phase reactions involving partitioning of the alloying elements of theTi–Al–Nb system in the vicinity of the composition Ti2AlNb.




(2) The � → �o ordering transition qualifies to be a second order transition andtherefore, the � and �o-phase fields need not be separated by a two-phasefield.

(3) The maximum stability of the �o-phase is assumed to be centred around theTi2AlNb composition. This is because the two sublattices of the B2 structureof Ti–Al–Nb alloys are preferentially occupied by Ti atoms and by a mixtureof Al and Nb atoms, respectively. The maximum order is, therefore, likely tobe exhibited at a composition where the atomic fraction of Ti atoms is equalto the sum of the atomic fractions of Al and Nb atoms. The maximum in theordering curve is taken to occur at about 1673 K (Bendersky and Boettinger1989).

(4) The � → �, � → �2 and �2 → O transitions are all first order transitions;therefore, the �, �, �2 and O single phase fields are separated by two-phasefields.

(5) The ternary phase diagram essentially depicts the phase stability regimes ofthe three phases, � (�2), � (�o) and O as shown in Figure 5.25. The two-phasefields, �2 +�o� �o + O and �2 + O, remain separated by a three-phase field,�2 +�o +O.

A variety of phase reactions is possible on cooling an alloy through the three-phase field. These include the eutectoid reaction, �o → O +�2, the peritectoidreactions �o + O → �2 and �o +�2 → O, and the precipitation reactions, �o →�o +O, �o → �o +�2 and �2 → �2 +O. Out of these, the reactions that occur aredetermined by the alloy composition and the relative positions of the three phaseisotherms at different temperatures.

A systematic study of phase reactions in the alloy, Ti–25 at.% Al–15 at.% Nb,which passes through the three-phase field, has been reported by Muraleedharanet al. (1992a,b). The results of this study are summarized here (Figure 5.26) toillustrate the variety of phase reactions possible in such an alloy:

(1) Quenching (cooling rate exceeding 100 K/s) from a temperature above 1383 Kresults in the retention of the �o-phase, the �/�o transition temperaturebeing around 1403 K. The � → �o ordering reaction cannot be suppressed byquenching.

(2) Quenching from temperatures between 1253 and 1383 K produces amicrostructure consisting of equiaxed �2-grains in a �o-matrix. Equilibrationat a temperature below but close to 1253 K, followed by quenching, results ina three-phase microstructure comprising the �2- and �o-phases and a blockyO-phase. The transus temperatures bounding different phase fields for thisalloy have been identified from these observations and are indicated by the




1200

1000

800

600

10 102 103 104

Time (s) →

Tem

pera

ture

(°C

) →

Ti–25Al–16Nb T-T-T curves, continuous cooling

B2 → α2 + B2

α2 + α′2 + O α2 + B2→O

α2 + B2

βB2

α2 + B2 + O

O + B2

B2 → O + B2

10°C/sAC

4°C/sAQ

0.7°C/sFC I

0.1°C/sFC II

0.02°C/sFC III

Figure 5.26. Phase reactions in the three-phase field in Ti–24 at.% Al–15 at.% Nb ternary alloy.

intersections of the dotted vertical line corresponding to the alloy compositionwith the �o/�2 +�o and �2 +�o/�2 +�o +O transus lines.

(3) Continuous cooling at slower rates (10–0.02 K/s) results in the formation ofa primary phase – either O or �2 – depending on the cooling rate. Phasereactions induced under different cooling rates are indicated in the schematicT-T-T diagram shown in Figure 5.27. The kinetics of the �o → �2 +�o and the�o → O+�o reactions are such that faster cooling rates result in the �-phasetransforming directly to the O-phase, skipping the formation of the �2-phase,even though the �2 +�o and �2 +�o + O phase fields are crossed during thecontinuous cooling process.

(4) In case the �2-phase is present as a primary decomposition product, a varietyof secondary reactions such as �2 →�2 +O and �2 +�o → O are possible. Thefine multivariant O-phase plates observed at the periphery of the �2-plates,which satisfy the �2/O orientation relation can be identified as the productof the former phase reaction. On the other hand, the monolithic O-phase rimproduced around the �2-plates corresponds to the product of the peritectoidreaction between the �2- and �o-phases.

The various microstructures that can be produced by continuous cooling in theTi–24 at.% Al–15 at.% Nb alloy under different rates of cooling are shown inFigure 5.27 to illustrate the variety of phase reactions observed.




Figure 5.27. Various microstructures produced by continuous cooling in the Ti–24 at.% Al–15 at.%Nb alloy under different rates of cooling (after Banerjee, 1994a).

Ageing of the �-quenched alloys in the O +�o and the �2 +�o phase fieldsproduces �2- or O-phase precipitates in the form of Widmanstatten laths in a �o-matrix, thin films of the retained �o-phase being left behind at the lath interfaces.Samples aged after subtransus solution treatments show changes with regard tothe primary �2-as well as the metastable �o-phases. Ageing in the O +�o phasefield results in the decomposition of the �o-phase through two different kineticallycompetitive modes which operate simultaneously; the first is a conventional con-tinuous precipitation of the O-phase in the �o-phase while the second results in�o-precipitation within O-phase grains. The mechanism of the �o → O transfor-mation has been discussed in detail in a later section.

Based on the phase diagram (Figure 5.25) discussed in this section, a To versuscomposition diagram (Figure 5.28) can be constructed, To being the temperaturewhere the parent and the product phases have the same molar free energy. Suchdiagrams are useful in rationalizing composition-invariant transformations, whichhave been described in Section 5.3.5. A schematic drawing (Figure 5.29) showingthe free energies of the �/�o, �/�2/O phases as functions of composition hasbeen constructed (Bendersky and Boettinger 1994, Bendersky et al. 1994). Thisdiagram can be viewed as the superposition of the free energy–composition plotsfor ordering transformations in the bcc-based and hcp-based phases.




Atomic % NbTi3Al

1300

1200

1100

1000

900

800

70010 20 30

To

(°C

)

Nb3Al

O

βo (B2)β (bcc)

β →

α β →

B2

B2 →

θ

B2 →

B19

α (hcp)

α2 (D019)α2 → O

Figure 5.28. To versus composition diagram (based on the phase diagram in Figure 5.25) to ratio-nalize composition-invariant transformations.

Atomic % NbTi3Al Nb3Al

β

10 20 30

Fre

e en

ergy

Alloy 1 Alloy 2

α

B19

O

α 2

βo

Figure 5.29. A schematic drawing showing the free energies of the bcc-based �/�o and hcp-based�/�2/O phases as functions of composition.

5.4 FORMATION OF ZR3AL

The equilibrium Zr3Al phase has the L12 (cP4) structure. The availability of severalslip systems in the L12 structure, which is essential for meeting the Von Misescriterion for ductility of a polycrystalline material, made it attractive to considerZr3Al as an intermetallic for structural applications. The equilibrium Zr3Al phasediffers from the equilibrium Ti3Al phase in the following subjects: (a) Unlike Ti3Al(D019 structure), the Zr3Al does not form by chemical ordering of the terminal




solid solution (hcp �-phase). (b) The Zr3Al phase forms through a peritectoidreaction, �+ Zr2Al → Zr3Al (refer phase diagram given in Chapter 1). (c) TheZr3Al phase is a line compound which does not tolerate any significant deviationfrom the stoichiometric composition.

5.4.1 Metastable Zr3Al (D019) phaseMukhopadhyay et al. (1979) explored the possibility of forming a metastable Zr3Alphase with the D019 structure by quenching an alloy of composition Zr–14 at.%(4.6 wt%) Al from the �-phase field. Water-quenched thin (0.5 mm thickness)samples of thin alloy showed a lath martensite structure. Selected area diffractionexperiments have shown the presence of diffuse intensity maxima at D019 superlat-tice positions apart from the hcp reflections. On subsequent ageing, these superlat-tice reflections have been found to intensify and sharpen. Dark-field images withthese superlattice reflections have shown the presence of 5–10 nm microdomainsof the D019 phase distributed in the martensite matrix (Figure 5.30). This obser-vation has clearly established that in a supersaturated Zr–Al solid solution (withhcp structure), a metastable D019 phase can form. This is quite expected as thedevelopment of the D019 structure in an hcp lattice can be achieved by replaciveordering. Figure 5.31 shows how by an introduction of a concentration wave withthe wave vector, k= a

2 1120, the D019 structure can be formed in an hcp lat-tice. As has been mentioned earlier, the wave vector satisfies two of the three

Figure 5.30. (a) Dark-field image showing the presence of 5–10 nm microdomains of the D019

phase distributed in the martensite matrix and (b) corresponding D019 diffraction pattern along the(0001) direction.




Layer 01

23

4

L

A layer

B layer

AlZrAlZr

2d1120

t 1 < t 2

t 1

t 2

1 3 4p = 0 2

0.25

0.5

Al f

ract

ion

Figure 5.31. (a) Atomic arrangement of the D019 structure showing compositions of {1120} planes.(b) Development of concentration wave as a function of time (t) leading to the formation of D019

structure from the parent hcp phase (average composition Ti–25 at.% Al).

Landau–Lifshitz conditions for the A3 → D019 transformation to be a second ordertransition. This is indeed a fit case for a continuous ordering under a high degreeof supercooling.

Figure 5.31 illustrates the evolution of the A3 → D019 ordering by formationand amplification of a concentration wave with k= a

2 1120 in an alloy of stoi-chiometric composition, i.e. Zr–25 at.% Al. Supersaturation of the �-phase to suchan extent is difficult, if not impossible. In the Zr–14 at.% Al alloy, investigated byMukhopadhyay and Banerjee (1979), the attainment of near-stoichiometric com-position has been achieved by a spinodal clustering process which has resulted ina concentration modulation of wavelength of about 20 nm. It is reported that thefirst step of the transformation has been the spinodal clustering which resulted in aconcentration-modulated structure along 1010 direction, the elastically soft direc-tion in the �-phase. The regions getting enriched with Al subsequently undergo anordering process by developing and amplifying a short wave length concentrationwave. The overall process can, therefore, be considered as a superposition of spin-odal clustering followed by continuous ordering. Such a transformation sequencecan occur if the system at the ageing temperature is initially unstable with respectto a long wave length (20 nm) concentration fluctuation and, after attainment ofa concentration within the ordering spinodes, becomes unstable with respect tothe A3 → D019 ordering (within the Al-enriched regions). The condition underwhich such a concomitant ordering and clustering processes can occur is shownin the schematic free energy–concentration plots (Figure 5.32) corresponding tothe disordered (�) and the ordered (�, D019) phases. An alloy of compositionC1 (which lies between the spinodes of free energy–concentration plot for thedisordered �-phase) is unstable with respect to spinodal clustering (concentrationmodulation with wave vector, k = 000). This process leads to the development ofalternate Al-rich regions which in turn becomes unstable with respect to �→ D019

ordering. With the gradual amplification of 121120 concentration wave, the order




x xFree

ene

rgy

→

Composition (% B) →A

A

A

D

T 3

Figure 5.32. A schematic free energy–concentration plot depicting the condition under which con-comitant ordering and clustering processes can occur corresponding to the disordered, �, and theordered, D019, phases.

parameter in these regions increases finally leading to nucleation of Zr3Al parti-cles of D019 structure. These ordered particles decorate the alternate Al-enrichedlayers, as shown in Figure 5.30.

5.4.2 Formation of the equilibrium Zr3Al (L12) phaseThe equilibrium Zr3Al (L12) evolves from the �-quenched structure containing adistribution of fine �2 (Zr3Al with D019 structure) particles in the �-matrix. Theemergence of the equilibrium Zr3Al phase has been found to occur through a cellu-lar precipitation process. The nucleation of the two-phase cells (� + Zr3Al) invari-ably occurs at the lath boundaries of the �-quenched structure. These cells growby propagation of the cell boundaries towards the interior of laths (Figure 5.33).Some of the general characteristics of cellular precipitation can be illustrated bythis example. These are listed as follows:

(1) Nucleation of a cell (two-phase region containing a group of � and Zr3Al(L12) lamellae, each with a specific orientation) invariably occurs at the grainboundaries. In the present example, the starting structure is that of a lathmartensitic �-matrix with a distribution of fine Zr3Al (D019) precipitates.Majority of the lath boundaries are small-angle boundaries separating laths ofclose orientations.




(a) (b) (c)

Figure 5.33. Micrographs showing a cellular precipitation process. (a) The nucleation of the two-phase cells (�+ Zr3Al) invariably occurs at the lath boundaries of the �-quenched structure.(b) Growth of the cells by propagation of the cell boundaries towards the interior of laths. (c) Two-phase (�+Zr3Al) transformation product.

(2) The �-orientation (�1) of the cell matches with one of the neighbouring �-lath(�1), while the Zr3Al (L12) orientation maintains exact orientation relationshipwith �1.

(3) A cell so related grows into the adjacent lath (orientation �2). Similarly a cellwith �2-orientation grows into the �1-lath. Such a process essentially makesthe original flat lath boundary into an undulated boundary which acts as thetransformation front, the propagation of which accomplishes the transformationof the �′ +Zr3Al (D019) region into �+Zr3Al (L12) cells (Figure 5.34).

(4) In general, where a supersaturated �′ phase decomposes into a cellular pro-ducer, the transformation can be described as a phase reaction, �′ → �+�,where � is the precipitating phase.

(5) The process of partitioning of the alloying elements between the two productphases occurs primarily of the transformation front. The diffusion field is,therefore, related to the lamellar spacing of the product structure.

Figure 5.34. Micrograph showing periodic spacing of interfacial dislocations at �/Zr3Al boundariesand faults within Zr3Al lamellae.




m

αα

c

Interfacial Dislocations

Zr3Al

(0002)α // (111)

Zr3Al

Figure 5.35. A schematic drawing showing the misfit dislocations along the interface between �and Zr3Al (L12) phases.

(6) The diffusion distance being somewhat small and the diffusion being enhanceddue to the presence of the interface (transformation front), the kinetics ofcellular precipitations are relatively fast.

(7) In the present case, where the �′ +Zr3Al (D019) structure decomposes into theequilibrium �+Zr3Al (L12) structure, it is imperative that the metastable Zr3Al(D019) precipitates get dissolved ahead of the transformation front leading toan increase in the level of supersaturation in the �′-phase which breaks up intothe equilibrium �+Zr3Al (L12) phase mixture at the transformation front.

As mentioned earlier, the orientation relationship between the � and the Zr3Al(L12) phases is maintained within the cells. The lamellar product, therefore, con-tains primarily two types of semicoherent interfaces characterized by arrays ofmisfit dislocations. The degree of misfit along the c-direction of the � phase beingsmall, the spacing between misfit dislocations is quite large. A schematic drawing(Figure 5.35) shows the misfit dislocations along the interface between � andZr3Al (L12) phases.

5.4.3 �+Zr2Al → Zr3Al peritectoid reactionThe phase diagram (Figure 6.30) shows that the equilibrium Zr3Al (L12 phase)results from a peritectoid reaction, �+Zr2Al → Zr3Al. The Zr2Al (B82 structure)can be viewed as an ordered #-phase, as described in detail in Chapter 6. Dueto the close lattice matching between � and Zr2Al phases, the latter grows in the�-matrix as equiaxed particles as an alloy is cooled in the �+Zr2Al phase field.




←

←

Tp

Tα

1300

1200

1100

1000

900

8000.1 1 10 100 1000

Time (h)

80%

50%

5%

Tem

pera

ture

(K

)

Figure 5.36. The proposed time-temperature-transformation (T-T-T) diagram showing that the rateof the peritectoid reaction is fastest at around 888�C, where a Zr2Al particle of about 3!m diametergets fully transformed within about 8 h.

The average size of Zr2Al particles depends on the rate of cooling through thetwo-phase field and varies from 8–10!m in diameter in larger ingots to about1!m in the smaller ones. When the �+ Zr2Al phase mixture is allowed to reactat a temperature below about 975�C, the two phases undergoes a peritectoidreaction resulting in the formation of the Zr3Al (L12) phase. Schulson (1975) andSchulson and Graham (1976) have studied the peritectoid reaction in detail. Thetime-temperature-transformation proposed by them (Figure 5.36) shows that therate of the peritectoid reaction is fastest at around 888�C, where a Zr2Al particleof about 3!m diameter gets fully transformed within about 8 h.

Since peritectoid reaction requires the presence of two reacting phases, the reac-tion invariably initiates at the interface. The reaction product, in this case, Zr3Al“envelopes” the Zr2Al particles and provides a barrier for the reaction to proceedrapidly. The subsequent growth of the product layer requires diffusion of the fastmoving species (in this case Al) across the product layer. Schulson and his co-workers have clearly shown the presence of the envelopes of Zr3Al phase aroundZr2Al particles and have shown that growth of the product phase occurs throughlong-range diffusion-controlled migration of Zr2Al/Zr3Al and Zr3Al/�-Zr inter-faces in opposite directions. Correspondingly, the transformation time is stronglydependent on the size of the Zr2Al particles in the starting microstructure.




5.5 PHASE TRANSFORMATION IN �-TiAl-BASED SYSTEMS

Two-phase �-TiAl-based alloys offer an attractive combination of properties,namely, low density, as well as good creep resistance, high-temperature strengthand oxidation resistance. It is because of the excellent combination of several desir-able properties that these alloys are being considered as candidate materials forhigh-temperature applications in aerospace industries, particularly in gas turbineengines. Monolithic �-TiAl is very brittle at room temperature. Several attemptshave been made to improve the room temperature ductility of TiAl-based alloysby controlling the alloy composition and by tailoring the microstructure throughphase transformations. The evolution of microstructure in these alloys is somewhatunique and is strongly influenced by the orientation relation between the partici-pating phases. The two-phase lamellar structure, which is exhibited by these alloyswhen appropriately heat treated, is responsible for imparting to them an attractivecombination of mechanical properties. The optimization of the mechanical pro-perties requires a proper control of the sizes and volume fractions of the lamellarcolonies and the interlamellar spacing. Another important aspect of these alloysis their microstructural stability, which determines the service life of componentsmade from these during exposure to elevated temperatures. Phase transformationstudies in these alloys, therefore, address two main issues: (a) the evolution ofmicrostructure through different sequences and mechanisms of transformation and(b) the stability at elevated temperatures of the microstructure so obtained.

5.5.1 Structural relationship between �2- and �-phasesIn binary Ti–Al alloys in the composition range of 38–48 at.% Al, two orderedintermetallic phases, �2-(Ti3Al) and �-(TiAl), coexist in equilibrium at tempera-tures below about 1270 K. The D019 structure of the �2-phase has been describedearlier. The �-TiAl phase possesses a tetragonal L10 structure. The lattice para-meters of the �2-phase (with 38 at.% Al) and the �-phase (with 48 at.% Al) are:

�2 $ a = 0�57 nm% c = 0�46 nm% c/a = 0�803

� $ a = 0�40 nm% c = 0�40 nm% c/a = 1�020

The axial ratio of the tetragonal unit cell of the �-phase being very closeto unity, the structure of the �-phase can be approximated to be cubic and thetransformation between the �2- and the �-structures can be considered as thatbetween an ordered hcp and an ordered cubic structure. Figure 5.37 also showsthe atomic arrangement and the interatomic distances on the basal (0001) plane ofthe D019 structure and on the close packed (111) plane of the L10 structure. It may




(0001) of α2-phase (111) of γ-phase

Al atomTi atom

Figure 5.37. Atomic arrangement and the interatomic distances on the basal (0001) plane of theD019 structure and on the close packed (111) plane of the L10 structure.

be noted that the interplanar spacings of the (0002)�2and the (111)� planes are

very close, the difference being smaller than 0.5%. Again, the nearest neighbourdistances in the close packed planes in these two phases differ only by about 2%.

The near-perfect lattice matching between the close packed planes of the �2

(D019) and � (L10) phases is the determining factor which controls the morphologyof the lamellar morphology of the �2 +� structure. A strict orientation relationbetween �2 and �, which is invariably maintained in the lamellar �2 +� structure,is revealed from the diffraction pattern shown in Figure 5.38. This orientationrelationship, (0001)�2

��{111}�; < 1120>�2��< 110>�, first reported by Blackburn

(1967), conforms to the commonly observed orientation relationship in fcc to hcp

(111)γ

(111)γT

(220)γ

(002)γT

(111)γ

fundamentalα2

fundamentalγT

fundamentalγ

superlatticeα2

superlatticeγT

superlatticeγ

(111)γT

(2200)α2

(2202)α2

(0002)α2

000

(002)γ (220)γT

Figure 5.38. Diffraction pattern showing orientation relation between �2 and �, in the lamellar�2 +� structure. Zone axis: 1120�2

��110�.




structural transitions. The diffraction pattern (Figure 5.38) shows reciprocal latticesections of two twin-related � variants, the twinning plane being the specific{111}� plane which is parallel to the basal plane of �2.

The lamellar aggregate of the �- and �2-phases can also be viewed as a stackingof close packed layers in the cubic ABCABC� � � sequence over a certain distancecorresponding to the thickness of the �-plate, followed by a hexagonal stackingsequence of ABAB� � � which builds up the �2-structure. The diffraction patternshown in Figure 5.38 suggests that a group of neighbouring lamellae containsregions not only corresponding to ABCABC� � � stacking but also to ACBACB� � �stacking, the two �-regions being twin related across the {111} plane. Twin-related�-plates can either remain in contact along the twin plane or may be separatedfrom each other by �2-plates. Single �-plates are also seen to be divided intoordered domains which are generated due to the fact that the [110]� and [011]�directions of the L10 structure are not crystallographically equivalent. The possibleorientation variants of �2 +� structure are shown in Figure 5.39.

(a)

(b)

110

101

011

110

011

OR1

1120

101

110110

OR6

011 011

OR2

110

OR5

101

2110

110

OR3

101

1210

101 011

OR4

011 101

α2

TiAl

= γI (ABC-type stacking)

= γII (ACB-type stacking)

[110]

[101][011]

Figure 5.39. The possible orientation variants of �2 +� structure.




5.5.2 Phase reactionsThe relevant portion of the binary Ti–Al phase diagram, which is based on thework of Perepezko and Mishruda (1993), is shown in Figure 5.40. It can be seenfrom the phase diagram and the transformation pathways indicated by verticaldashed lines that the �2- and the �-phases can evolve through one of the pathwayslisted below:

(1) � → � → �2

(2) � → �2 → �2 +�(3) � → �+� → �2 +�(4) � → �(5) � → �2 +�

The phase evolution sequences reported in Ti–Al alloys containing about 35–50 at.% Al (Jones and Kauffman 1993, Yamabe et al. 1995) are discussed in thefollowing.

5.5.2.1 Ti-34-37 at.% Al; � → � → �2

As these alloys are cooled from the �-phase field to the �2-phase field, one ofthe three transformation processes takes place depending on the cooling rate.

11

24

3

5α2

α + β

α + γα + α2

α2 + γ

β + L α + L

β

α

γ

L

1600

1500

1400

1300

1200

1100

100035 40 45 50

Tem

pera

ture

(°C

)

Atomic % Al

Figure 5.40. A portion of the binary Ti–Al phase diagram depicting the transformation pathways,as indicated by vertical lines, for evolution of �2- and the �-phases.




These are: precipitation of either �- or �2-plates in the �-matrix involving long-range diffusion, composition-invariant massive transformation and diffusionlessmartensitic transformation.

Precipitation of the �-phase in the �-phase matrix produces a basket weavemicrostructure akin to that commonly encountered in (�+�) Ti alloys. In contrast,a faster cooling induces a composition-invariant massive transformation in whichmassive �-grains with jagged boundaries are produced. Al being a strong �-stabilizer, the To for the � → � transformation is about 1523 K. At such a hightemperature the kinetics of massive transformation are expected to be very rapid.It is, therefore, difficult to suppress this massive transformation by any solid statequenching technique.

The � → �2 ordering temperature is also very high (between 1373 and 1473 K)and, therefore, the ordering process immediately follows the formation of the�-phase, which forms either as Widmanstatten �-plates or as massive �-grains.

5.5.2.2 Ti-38-40 at.% Al; � → �2 → �2 +�The alloys which follow the transformation sequences (2) and (21) generally belongto hypo- and hyper-eutectoid compositions, respectively. Before discussing thesetransformation sequences it is worthwhile to consider a few special features ofthe eutectoid reaction � → �2 + �. As has been discussed earlier, the �2-phase(D019 structure) is an ordered version of the �-phase (hcp, A3 structure). Thecompositional proximity of the �-phase of eutectoid composition (39.5 at.% Al)and the equilibrium �2-phase (38 at.% Al) suggests that the extent of supersatu-ration required for a congruent ordering to occur is only marginal. This orderingtransformation, occurring at a temperature in the range of 1373–1473 K, proceedsvery fast without the requirement of any significant undercooling. The formationof the �-phase from the �-phase requires a change in the stacking sequence thoughthe lattice matching of the two structures on (0001)��2

��(111)� planes is excellent.In the case of a composition-invariant � → � transformation the structural changecan be brought about by the passage of Shockley partials on every alternate closepacked plane. This mechanism will be discussed in the next section. Such a latticecorrespondence makes the kinetics of the lengthening of �-plates quite fast butthe kinetics of their thickening is rather slow. The overall kinetics of �-phaseformation is, therefore, much slower than that of the � → �2 transition. In viewof this, it is quite unlikely for the high-temperature �-phase to decompose by apearlitic mode in which the two product phases simultaneously emerge from theparent phase across the incoherent transformation front. The non-pearlitic modesof eutectoid decomposition which can operate in hypo- and hyper-eutectoid Ti–Alalloys are outlined in the transformation pathways (2) and (21), respectively.




An alloy in the composition range Ti-38-40 at.% Al, when cooled from the�-phase field to a temperature below the eutectoid, first crosses the To tempera-ture for the � → �2 ordering. As explained earlier, the equilibrium � → �2 +�eutectoid decomposition is kinetically quite unfavourable in comparison with thepathway �→ �2 → �2 +�. The decomposition of the �-phase in a pearlitic modehas, therefore, not been encountered in experiments. The fact that �→�2 orderingprecedes the formation of �-plates is revealed from the observation that �-platescut across APBs in the �2-phase. A typical time-temperature-transformation dia-gram for alloys containing 38–40 at.% Al has been schematically illustrated inFigure 5.41(a) which depicts the kinetics of the competing processes on a rel-ative scale. Since the precipitation of the �-phase from the �2-phase involvesan incubation period of nearly 100 s, it is possible to suppress the transforma-tion by a rapid quench which produces a metastable �2-structure. On subsequentageing �-precipitation occurs within the �2-grains, producing a lamellar �2 + �structure.

5.5.2.3 Ti-41-48 at.% Al; � → �+� → �2 +�In these hypereutectoid alloys, �-phase precipitation in the �-phase is the primarystep that occurs during slow cooling from the �-phase field. Again the samelamellar �+ � morphology, dictated by the Blackburn orientation relation, isexhibited. Subsequently the �-regions undergo the ordering process during coolingthrough the To temperature for �→�2 ordering. At a somewhat faster cooling rate,the step comprising �-phase precipitation in the �-phase is skipped; the � → �2

ordering occurs first which is followed by the decomposition of �2 into the �2 +�lamellar mixture. During a still faster cooling, the formation of the �-phase canbe completely suppressed, resulting in the retention of a metastable �2-product inthe as-quenched condition (as shown in Figure 5.41(b). However, suppression of�-phase formation by quenching is not possible in alloy compositions close to the�-phase field. A competing �→ � massive transformation comes into play in suchalloys. For example, an alloy containing 48 at.% Al, in the as-quenched condition,shows only a small volume fraction of lamellar �2 +� regions while the majorconstituent of the microstructure comprises large �-grains which contain stackingfaults, twins and sometimes APBs. The nature of these defects has been analysedin detail mainly for ascertaining whether the �→ � transformation goes through anintermediate disordered fcc phase. This point will be taken up later while discussingthe mechanism of the massive � → � transformation, which was first reported byWang and Vasudevan (1992) and later confirmed by Denquin and Naka (1995) andZhang and Loretto (1995). Figure 5.41(c) shows a time-temperature-transformation




(a)

Tem

pera

ture

(°C

)

Log time

Eutectoid

α2 (thermal APBs) α2 + γ (fine γ plates)

α2 → α2 + γ α → α2 + γα → α2

(b)

α2 (thermal APBs) α2 + γ (lamellar)

α2 → α + γ

α /α + γ

α /α + α2

α2 → α2 + γ

α2 → α2

Log time

1200

1090

Tem

pera

ture

(°C

)

Eutectoid

Metastableextension

1500

1400

1300

1200

1100

1000

900

800

Time (s)

Tem

pera

ture

(°C

)

0.1 10

bTα

TO

Tγ

γLamellar

γFeathery

γMassive

100°C/s200°C/s

300°C/s

400°C/s

500°C/s

1000°C/s

1

(c)

Figure 5.41. (a) A schematic time-temperature-transformation (T-T-T) diagram for alloys containing38–40 at.% Al depicting the kinetics of the competing processes on a relative scale. (b) T-T-Tdiagram showing kinetics of (i) � → �+ � and (ii) � → �2 → �2 + � processes and (c) T-T-Tdiagram indicating a kinetic competition between the processes, � → �+� (lamellar or feathery)and � → � (massive).




diagram indicating a kinetic competition between the processes � → �+� and� → � (massive).

5.5.2.4 Ti-49-50 at.% Al; � → �Alloys in this narrow composition range can transform into a single phase �-structure under the equilibrium condition. Two distinct types of product microstruc-ture have been encountered; one is described as Widmanstatten colony and theother as massive � (Yamabe et al. 1995). In the former case, the original �-grainsare divided into � or �-like regions with different orientations. Subsequently theseregions transform into a lamellar microstructure containing primarily �-plates.The mechanism of and the motivation for the creation of �-regions of differentorientations have not been identified so far. The second mechanism of the � → �transformation, namely, massive transformation, operates at faster cooling ratesunder which Widmanstatten colony formation is suppressed.

5.5.2.5 Ti49-50 at.% Al; � → �When alloys in this composition range, after solutionizing in the single phase �-field, are brought to the �+� phase field, �-plates precipitate. The cubic symmetryof the parent �-phase allows the formation of different variants of �-plates alongthe {111}� habit. Some illustrative examples of different transformation productsin �-TiAl-based alloys are shown in Figure 5.42.

The formation of the lamellar microstructure and of the massive �-phase are thetwo most important phase transformation phenomena encountered in the �-TiAlalloys. In addition, discontinuous coarsening of the lamellar structure plays a majorrole in the evolution of microstructure. The suggested mechanisms underlyingthese phenomena are summarized in the following section.

10 μm100 μm

(a) (b)

Figure 5.42. Illustrative examples: (a) massive �m products in Ti–49 at.% Al alloy and (b) two-phaselamellar �2 +� structure observed in light microscopy.




5.5.3 Transformation mechanisms5.5.3.1 Formation of the �2 +� lamellar microstructureThe overall transformation process for the � → �2 + � decomposition can beviewed as consisting of the ordering reaction �→�2 and the precipitation reaction� → �. Ordering precedes or follows precipitation depending on whether thedecomposition temperature is below or above the To temperature for ordering.In both the cases, the ordered �2-superlattice is created on the hcp lattice ofthe �-phase by the required atomic replacements. The lattice correspondencebetween the A3 and D019 structures, in which their respective basal planes remainparallel, allows the formation of only a single rotational domain. However, APBscorresponding to translational domains associated with displacement vectors of thetype 1

3 < 1120 > are formed. Such APBs within the �2-phase are usually knownas thermal APBs. The precipitation of the �-phase from the �-phase involvesboth structural and chemical changes. The former is accomplished mainly by atransition in the stacking sequence from the hexagonal to the cubic type andthe latter requires preferential Al enrichment of the regions where �-precipitatesappear. These precipitates invariably maintain lattice registry with the parent �-crystal and the precipitate plates have a strong tendency to form into a group. Thekinetics of the lengthening of the �-plates along the (0001)��2

habit is much fasterthan that associated with their thickening. Groups of �-plates often appear in thevicinity of grain boundaries and occasionally within the �/�2 grains. The habitplane being the basal plane, only one habit plane of �-variant can form within agiven �2-grain. With the growth of these �-precipitates, every single grain of theparent �/�2 structure gets transformed into a lamellar �2 +� structure with a fairlyuniform interlamellar spacing. The nucleation of �-precipitates in a group and theuniform spacing between the �-lamellae suggest that the formation of �-platesand their growth do not take place individually but in a cooperative manner, theconcentration field ahead of the edge of the plates being responsible for decidingthe lamellar spacing. The most important difference between this mechanism andthat of a cellular growth (as in the case of discontinuous precipitation or pearlitictransformation) lies in the fact that the lamellar product is not separated from theparent phase by an interface (Figure 5.43). This is because of the close latticematching between the advancing �-lamellae and the parent �/�2 matrix acrossthe transformation front. This point is illustrated schematically in Figure 5.43.

Transmission electron microscopy (TEM) studies have revealed some importantfeatures of the nucleation and growth steps of �-phase precipitation. Blackburn(1970) and Sundararaman and Mukhopadhyay (1979) have shown that thenucleation of the �-phase occurs heterogeneously, solely by the dissociation ofa6 < 1120 >�2

dislocations on the basal plane. The stacking faults enclosed by thea6 < 1010 >�2

partials can be considered as single layer nuclei of the �-phase.




→

→

→

(a) (b)

(e)

→

(f)

α2

Alloypartitioningα /α2

Replacementordering

α /γ transformationchemical ordering

+change in stacking

sequence

Transformation front

(d)(c)

Incoherentboundary

No lattice sitecorrespondence

Cellulartransformation

γ

Figure 5.43. Schematics of difference between mechanism of lamellar and of a cellular growth asin the case of discontinuous precipitation or pearlitic transformation.

Preferred sites for such nucleation are grain boundaries, subgrain boundaries andindividual dislocations. The density of these faults has been found to increase asthe cooling rate from temperature above the �2/�+�2 transus is decreased.

The growth of these nuclei to �-plates can be compared with the hcp → fcctransformation in so far as the stacking sequence is concerned. The passage ofa6 < 1010 >�2

partial dislocations over every alternate basal plane produces thecubic stacking sequence. One of the possible modes by which a single layerstacking fault can grow into a �-plate of several tens of nanometre thickness isby the operation of a pole mechanism in which a Shockley partial dislocationis anchored to a sessile [0001]�2

pole dislocation. The rotation of the Shockleypartial will change the stacking sequence, and during rotation the partial climbsalong the pole through a distance equal to twice the interplanar spacing; thus each




rotation causes the product phase to thicken by two atom layers. Mukhopadhyay(1976) has provided some evidences for the presence of spiralling Shockley partialdislocations on the basal plane of the �2-phase.

Partitioning of Al atoms between the �2- and �-phases must occur throughdiffusive jumps in order to allow the �-plates to attain the equilibrium composition.The movement of Shockley partials and the Al enrichment of the faults possiblytake place concurrently, as the latter is expected to reduce the stacking faultenergy. No doubt it is difficult for individual atoms to join the �-lamellae on thebroad face which is a highly stable low-energy planar interface between the �2-and �-lamellae. In contrast, lateral growth of �-lamellae through the movementof Shockley partials on the habit plane by the so called “terrace-ledge-kink”mechanism is a plausible mode in which the appropriate composition change ofa growing lamella is ensured by the transfer of atoms on to kinks which providefavourable sites for atomic attachment.

The fact that a combination of shear and diffusional processes is operative forthe growth of �-lamellae has been clearly revealed by high-resolution electronmicroscopy studies (Huh et al. 1990) on the �2/� interfaces which show a lateralgrowth by the passage of a

6 < 1010 >�2dislocations and by atom probe investiga-

tions of Denquin and Naka (1995) showing the sluggish nature of the diffusionalgrowth. The lamellar �-precipitates formed during fairly rapid cooling exhibitnon-equilibrium partitioning of Al in the �2- and �-phases, the former remainingsupersaturated in Al and the volume fraction of the latter being much larger thanthat corresponding to the equilibrium condition. The compositions of these twophases and their volume fractions approach the equilibrium values on ageing. Itis to be noted that the results of the atom probe experiments indicate the absenceof any concentration gradient around the �2/� interfaces; this suggests that thelamellar growth is controlled by the ledge mechanism rather than by a classicallong-range diffusional process. The growth rate slows down with decreasing Alsupersaturation of the matrix which reduces the driving force for the ledge-kinkmotion on the broad face.

5.5.3.2 Mechanism of the � → � massive transformationTi–Al alloys (48–50 at.% Al) which are quite close to the �-phase field, whenfast cooled from the �-phase field, transform into the single �-phase. This trans-formation takes place without any change in composition and the morphologyof the product phase has several features which suggest that the transformationis of the massive type. Wang and Vasudevan (1992) were the first to report theoccurrence of such a massive transformation in this system and this observationhas been confirmed in a number of more recent studies (Jones and Kauffman1993, Zhang and Loretto 1995, Denquin and Naka 1995, Wang et al. 1998). The




�-phase produced by this massive transformation is generally designated as �m inorder to distinguish this phase from the lamellar �-phase.

The quenched structure of alloys containing 48–50 at.% Al often shows a mixedmorphology with massive �m-regions coexisting with lamellar �2 +� regions. Theorientation of the �2-lamellae in a given region gives the orientation of the prior�-grain, as the �-and �2-structures are related by a unique orientation relation.Attempts aimed at finding out a possible orientation relation between the prior�-grain and the �m-product have not been successful (Denquin and Naka 1995),implying that no specific orientation relation exists between them. The interfacesseparating adjacent grains of the �m-phase and those between �m- and matrix �2-phases have been found to be quite jagged and it is clear that such a morphologycannot result from a shear-like process in which a planar contact between theparent and product phases is expected.

The massive �m-grains consist of a large number of faulted domains. Thedefects present in these domains are of various types: stacking faults lying on allthe four variants of {111} planes, microtwins and APBs which do not adhere toany crystallographic plane. The displacement vectors associated with these APBscorrespond to a

2<011> vectors. A detailed contrast analysis of the defects inthe �m-phase (Wang et al. 1998) has revealed that both 1

2 <110� and 12 <101� unit

dislocations are present in this phase, the latter being linked by highly curvednon-conservative APBs. It has also been shown that wide stacking faults, whichare created by the dissociation of 1

2<101> unit dislocations, lie on {111} planesand are bound by Shockley partials of all possible types.

Hug (1988) has examined the possibilities of the formation of different typesof stacking faults in the L10 structure of �-TiAl and have identified the followingthree types: complex stacking faults (CSF), superlattice intrinsic stacking faults(SISF) and superlattice extrinsic stacking faults (SESF). A CSF can be consideredto be a superimposition of an SISF and an APB. The dislocation dissociationreactions which can result in the formation of these stacking faults are

1

2�110� → 1

6�211�+ �SF�+ 1

6�121�

�101� → 12�101�+ �APB�+ 1

2�101� →

16�112�+ �SISF�+ 1

6�211�+ �APB�+ 1

6�112�+ �CSF�+ 1

6�211�

Because of the extremely high APB energy of �-TiAl, the [101] dislocations areexpected to be present as pairs of narrowly spaced 1

2 [101] dislocations linked by




APBs, each 12 [101] dislocation further dissociating into Shockley partials connected

by a SISF or a CSF. Observations on deformed �-TiAl have shown that theseparations between the partial dislocations bounding SISFs and APBs are lessthan 6 and 4.5 nm, respectively.

In contrast to these narrow separations observed in deformed �-TiAl, Shockleypartials of all types have been found to be dissociated, creating stacking faults ofabout 400 nm width, in the �m-phase. Such a wide separation of Shockley partialsin �m can arise if these stacking faults are inherited from the parent �-phase.However, this possibility is ruled out as faulting in that case would have beenconfined only to that {111} plane variant which is derived from the basal plane ofthe parent �-structure. A second possibility is that these wide stacking faults arecreated during the �→ �m transformation. However, it appears quite unlikely thatthese faults form in the ordered L10 structure either during the transformation orduring subsequent cooling in view of the high APB and CSF energies.

The presence of APBs and three variants with the �m-phase has promptedZhang et al. (1996) to propose the following intermediate step in the massivetransformation: � (massive transformation) → disordered fcc phase (orderingtransformation) → �m. The existence of widely separated dissociated 1

2 <110� unitdislocations and 1

2 <101� unit dislocations and the presence of all possible {111}stacking faults bounded by widely separated Shockley partials are perhaps betterevidence for the occurrence of the intermediate step in which the disordered fccphase is formed. The APBs which are intimately attached to these dislocations areessentially by-products of the ordering reaction from the fcc to the L10 structure.

The question as to whether an orientation relation between the parent hcp andthe product fcc phases exists at the time of nucleation has not been adequatelyanswered. Heterogeneous nucleation of the massive product at grain boundarieshas been observed (Denquin and Naka 1995). It is possible that the nuclei ofthe massive product form at grain boundaries, by creating a low-energy interfacebetween the massive product and one of the grains. The highly coherent characterof this interface will not allow it to propagate within the parent grain. The interfacecan, however, grow into the neighbouring grain with which no special orientationrelation exists. The absence of a specific orientation relation between the �m-andthe adjoining �2-regions (which represent the orientation of the parent �-grain)can, therefore, be explained even if a strict orientation relation does exist in thenucleation stage.

The growth of a massive product occurs by thermally activated short-rangejumps of atoms from the lattice sites of the parent to those of the product. Theatomic jumps take place across the transformation front which is either an inco-herent interface or a terraced surface of good atomic fit, interspersed with ledges.While a curved and jagged interface of the product grains characterizes the former




type, a planar interface is expected in the case of the latter type of transformationfront. The observed curved and generally irregular interfaces between the �m-domains within a colony, between colonies and between adjacent �m and matrix�2-regions suggest that the massive transformation in the Ti–Al system takes placethrough a mechanism involving atomic jumps across incoherent interfaces.

5.5.3.3 Discontinuous coarsening of the lamellar �2 +� microstructureThe stability of the lamellar structure during exposure to elevated temperatures isa matter of serious concern in view of the intended high-temperature applicationsof �-TiAl-based alloys. The driving force for microstructural changes duringsuch exposure can arise either from chemical free energy changes resulting fromchanges in the volume fractions and compositions of the constituent phases orfrom the reduction in the total interfacial energy of the system. In the case of thelamellar �2 +� microstructure (Figure 5.44), a combination of both the factors isexpected to contribute towards microstructural changes during exposure to elevatedtemperatures. This is because the lamellar structure contains a very large areaof interphase interfaces and the constituent phases may not have the equilibriumcompositions and volume fractions at the exposure temperature.

There have been a number of investigations on the microstructural stabilityof the lamellar �2 + � structure, two of the important references in this regardbeing those of Denquin and Naka (1995) and Ramanujan et al. (1996). It has beenshown that coarsening of the lamellar structure takes place in either a continuousor a discontinuous mode. Continuous lamellar dissolution has been shown to be

Figure 5.44. Lamellar �2 + � structure produced on cooling a hypereutectoid Ti–Al alloy in atransformation sequence �→�+�→�2 +�. Orientation distributions are marked on the micrograph(cf. Figure 5.39).




initiated by the formation of steps on the broad faces of lamellae and to progress bysubsequent step migration. While continuous coarsening of the lamellar structureby progressive dissolution of some �-lamellae occurs at temperatures in the vicinityof 1073 K, discontinuous coarsening becomes operative at still higher temperatures.

The discontinuous coarsening process involves the propagation of �- or �2-grain boundaries. When such boundaries move towards the primary fine lamellarstructure, the fine structure is consumed and a much coarser lamellar structuredevelops behind the advancing front. This observation clearly suggests that thegrowing �- or �2-grain is different in orientation from the primary grain (containingthe fine lamellar structure) which gets consumed in the process.

The primary lamellar structure consisting of planar, extremely flat and highlycoherent interfaces results from the �-lamellae through the ledge mechanism. Thesolute supersaturation in the �2/� phase is primarily responsible for providingthe driving force for the thickening of the �-plates. The thickening process slowsdown and becomes more and more difficult with progressive reduction in theextent of supersaturation. The continuous coarsening of the structure requires acompetitive diffusion-assisted growth between two lamellae while maintaining thevolume fractions of the � and �2/� phases unchanged. The driving force for sucha coarsening process essentially arises from the reduction in the total interfaceenergy. However, the highly coherent nature of the interface renders solute flowbetween two lamellae extremely difficult. In contrast, the discontinuous coarseningprocess takes place in the vicinity of grain boundaries where the solubility ishigher than that in the matrix. Grain boundaries are, therefore, endowed withsufficient mobility. Discontinuous coarsening can thus be visualized as a processinvolving dissolution of primary �-lamellae at the advancing grain boundary frontand reprecipitation of coarser �-lamellae behind this front.

The orientation relationship between the �/�2 and the �-lamellae is maintainedin the discontinuously coarsened structure, though the lamellar morphology is con-siderably irregular compared to the primary structure. This essentially means thatthe interfaces between the �/�2 and the �-lamellae are not maintained flat on anatomic scale over a long distance. The formation of exceptionally flat interfaces inthe primary lamellar structure has its origin in the movement of Shockley partials.The irregularity observed in the discontinuously coarsened structure suggests thatpropagation of Shockley partials is not a determining factor in its formation.

Ramanujan et al. (1996) have examined the possibility of refining and stabilizingthe primary lamellar structure by W and B additions to �-TiAl alloys. Theyhave demonstrated that not only is the lamellar spacing reduced but the structurealso becomes more resistant against coarsening when these alloying elements areintroduced. Similarly, Cr and Nb additions have also been found to be beneficialin this context. The increased stability of the lamellar structure in these alloys




could be related to a possible reduction in the interfacial energy or to a higherthermodynamic stability of the �2-phase.

5.6 SITE OCCUPANCIES IN ORDERED TERNARY ALLOYS

The degree of ordering has a very important role to play in determining themechanical properties, particularly ductility of ordered intermetallic compounds. Ithas been demonstrated in several cases that ternary alloy additions cause improve-ment in ductility in several ordered alloys either by dispersion of a ductile secondphase at grain boundaries of the ordered phase or by altering the distribution ofconstituent atoms in the sublattice sites of the ordered structure. The latter processcan be modelled on the basis of the chemical rate theory by computing the rates ofatomic exchanges between different sublattices in a given superlattice. Experimen-tally site preferences in ordered compounds can be determined using the techniqueof atom location by channelling enhanced microanalysis (ALCHEMI). In thissection a model for computing site occupancies in ternary B2 compounds is pre-sented and the predictions of the model are compared with available experimentalresults in the ternary Ti–Nb–Al system.

5.6.1 Ordering tie linesSome of the most commonly studied intermetallic alloys such as FeAl and NiAlpossess the B2 (CsCl) structure which can be described in terms of two interpene-trating simple cubic lattices with lattice displaced with respect to the other by thevector 1

212

12. These two interpenetrating lattices, referred to as the �- and the �-

sublattice, are occupied exclusively by the A atoms and the B atoms, respectively,in a fully ordered AB alloy. In the B2 structure, every �-site is surrounded byeight first near neighbour �-sites and vice versa. The fact that all nearest neighbourbounds are between unlike atoms and that the atomic coordination around boththe sublattice sites are identical made it possible to study the evolution of orderfrom order parameter = 0 to 1, without encountering any cluster frustration.

The occupancies of the two sites � and � in a ternary alloy with constituent atomsA, B and C can be conveniently represented graphically in a ternary compositiontriangle. The composition of the �- and the �-sites can be plotted as two pointsin this triangle as shown in Figure 5.45. The line joining these two points iscalled the ordering tie line (OTL). When the alloy is completely disordered, i.e.the concentration A, B and C in the �- and the �-sublattices is the same, whichcan be represented by a single point (the average composition of the alloy) inthe composition triangle. As the alloy is progressively ordered, the concentrations




B C

A

P

Q

θ

xA, xB, xC

ι

Figure 5.45. The composition triangle showing the composition of �- and �-sites and the order-ing tie line.

at the �- and the �-sites which depart from the average composition can berepresented by two points P and Q in the composition triangle. The line joiningP and Q is defined as the OTL. This representation, first introduced by Hou et al.(1996), must be differentiated from a thermodynamic tie line in a ternary phasediagram. While the latter joins the points corresponding to the compositions of twodifferent phases in equilibrium at a given temperature, the OTL joins the pointscorresponding to the compositions of two sublattices of the same ordered phase.

The B2 ordering in a ternary system can be described using a single con-centration wave vector along the <111> direction of the cubic lattice and twoindependent order parameters. The two order parameters, 1 and 2, in the SCWapproach (Khachaturyan 1983) are defined as differences between the occupanciesof A and B atoms on the two sublattices in the B2 structure. If xj

i is the occupancyof i atoms on the j sublattice, the order parameters, 1 and 2, can be defined as1 = xB

A −x�A, 2 = xB

B −x�B.

These two order parameters are related to the two independent degrees offreedom of the OTL, its length, l, and orientation, & (Figure 5.45). It can beshown from the geometry of OTL that 1 and 2 are related to l and & as1 = 2l sin &/

√3�2 = −l�cos&+ sin &/

√3�.

From the relationship between l, 1 and 2, it is clear that l corresponds tothe amplitude of the < 111 > concentration wave in the SCW formalism. Withincreasing temperature the value of l decreases and reaches zero at the order–disorder transition temperature, Tc. The length of the OTL is a measure of theextent of ordering in alloy and is geometrically constrained to lie within theternary triangle. Therefore, the maximum ordering is achieved when the OTL isextended to a stage when one of the sublattice composition touches either a pointon the binary edge or a unary corner of the ternary triangle. The second degree




of freedom, &, is a measure of the relative preferences of the three elements, A,B and C on the two sublattices and can be considered as the polarization of the< 111 > concentration wave responsible for the B2 ordering. It should be notedthat & is not equal to zero at Tc and therefore does not characterize the transition.

5.6.2 Kinetic modelling of B2 ordering in a ternary systemThe model proposed in this paper is based on the Bragg–Williams (1934) theoryof order–disorder transition using the MFA. Dienes (1955) has shown that theBragg–Williams theory can be derived from kinetic arguments using chemicalrate theory for estimating the rates of exchanges of atomic species A, B and C inthe two sublattice sites, � and �. There are three possible exchange reactions, asshown below, which contribute to the ordering/disordering processes:

A� +B� = A� +B� (a)

B� +C� = B� +C� (b)

A� +C� = A� +B� (c)

Consider a ternary B2 alloy with the average composition, xi, where xi is theatomic fraction of element i. Let the concentrations at the �- and the �-sublatticesbe x�

i and x�i , respectively. The reaction rate for each of the three exchange

reactions shown above can be written in terms of a difference between the forwardand the reverse reaction rates. As an example the reaction rate, R1 for (a) can beexpressed as

R1 = kF ·x�A ·x�

B −kR ·x�A ·x�

B

where kF and kR are the rate constants for the forward and reverse reactions, whichcan be expressed in terms of Arrhenius type relationship as follows:

kF − F · exp(

−Em −�H/2RT

)

and

kR − R · exp(

−Em +�H/2RT

)

�H is the enthalpy change associated with the atomic exchange process betweenthe right site and the wrong site, as depicted schematically in Figure 3.18 and




Em is the migration energy of the atoms in the disordered state. As shown inFigure 3.18, the activation barrier in the disordered alloy is symmetric while thatfor an ordered alloy is asymmetric with a favourable bias for the exchange processwhich leads to ordering. F and R are the pre-exponential factors including activa-tion entropies associated with the forward and the reverse reactions, respectively.

The enthalpy changes �H1 involved in the exchange reaction (a) can be writtenin terms of the nearest neighbour binary interaction parameters wAB, wAC and wBC

between the unlike atom pairs A–B, A–C and B–C, respectively:

�H1 = 2wAB

(x�Ax

�B −x�

Ax�B

)+2wAC

(x�Ax

�C −x�

Ax�C

)(5.23)

= 2wBC

(x�

Bx�C −x

�Bx

�C

)(5.24)

The binary interaction parameter wij is defined as follows: wij = Vij − 12Vii +

Vjj, where Vii, Vjj and Vij are the bond enthalpies of i–i, j–j and i–j bonds,respectively. The enthalpy changes for reactions (b) and (c) can be written in ananalogous manner as follows:

�H2 = 2wBC

(x�Bx

�C −x�

Bx�C

)+2wAC

(x�Ax

�C −x�

Ax�C

)(5.25)

= 2wAB

(x�Bx

�C −x�

Bx�C

)(5.26)

�H3 = 2wAC

(x�Ax

�B −x�

Ax�B

)+2wAB

(x�Ax

�C −x�

Ax�C

)(5.27)

= 2wBC

(x�Bx

�C −x�

Bx�C

)(5.28)

The sublattice concentrations in a B2 alloy with an average composition xi canbe expressed in terms of the length l and the orientation & of the OTL as follows:

x�A = xA − l sin '/

√3 (5.29)

x�A = xA + l sin '/

√3 (5.30)

x�B = xB + l�cos '+ sin '/

√3�/2 (5.31)

x�B = xB − l�cos '+ sin '/

√3�/2 (5.32)

x�C = xC − l�cos '− sin '/

√3�/2 (5.33)




180

θ (degrees)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

160140120100806040200

l R2 = 0 R3 = 0 R1 = 0

R1 = R2 = R3 = 0

Figure 5.46. Three binary interaction parameters, wij , and the rates of reactions (R1, R2 and R3)plotted as function of l and &.

x�C = xC + l�cos '− sin '/

√3�/2 (5.34)

Using the equations and substituting the values of three binary interactionparameters, wij , the rates, R1, R2 and R3, of all the three reactions (a), (b) and(c) can be expressed as functions of l, & and T . Under the equilibrium conditionat a given temperature R1 = R2 = R3 = 0, as shown in Figure 5.46. The values ofl and & will give the equilibrium position of the tie line and the two terminatingpoints of the OTL will define the fractional occupancies of A, B and C in �- and�-sublattice positions.

5.6.3 Influence of binary interaction parametersThe nature of the interactions, wAB, wBC and wAC, can be either ordering (unlikebonds favoured relative to like bonds, w < 0� or clustering (like bonds favouredrelative to unlike bonds, w > 0). How the orientation OTL is controlled by thenature and relative strength of the interaction parameters is illustrated in thefollowing case studies:

Case I: wAB < 0� wBC = wAC = 0

The OTL, calculated for the composition xA = xB = xC = 0�33, shows the OTLlying parallel to the AB line. Since A–C and B–C interactions are similar and aremuch weaker than the strong A–B interaction, the OTL assumes the orientationas indicated in Figure 5.47(a). This implies that, if the element C is added to




(a) (b)

(c) (d)

B C B

A A

A A

B C B C

0.0 Strongly ordering

Strongly ordering Strongly ordering

0.0 0.0 0.0Stronglyordering

Stronglyordering

Stronglyordering

0.0 0.0

C

Figure 5.47. (a) The OTL, calculated for the composition, xA = xB = xC = 0�33, showing the OTLlying parallel to the AB line for A–C and B–C interactions is similar and much weaker than thestrong A–B interaction; (b) Similar to (a) with strongly ordering tendency between B and C atoms;(c) in a case where the binary B–C and A–C interactions are made strongly ordering and of equalmagnitude (−0.65 eV), the calculated OTL for the alloy composition, xA = xB = xC = 0�33, liesalong the perpendicular bisector of the AB side of the triangle; (d) by keeping the same interactionparameters, if the average composition of the alloy is changed, the OTL slope changes; and forthe equiatomic composition alloy, the stronger A–C interaction causes the OTL to tilt towards adirection nearly parallel to the AC line.

the binary B2 ordered AB alloy, it will occupy the � and �-sublattices in equalfractions:

Case II: wAB = 0� wBC = wAC ≤ 0

In a case where the binary B–C and A–C interactions are made strongly orderingand of equal magnitude (−0�65 eV), the calculated OTL for the alloy composition,xA = xB = xC = 0�33, lies along the perpendicular bisector of the AB side of thetriangle (Figure 5.47(b)). Since both B–C and A–C ordering tendencies are equal,the OTL assumes a symmetrical orientation with respect to the vertice C. Thisimplies that one sublattice is occupied primarily by C atoms while the other isequally shared by A and B atoms.




Keeping the same interaction parameters, if the average composition of thealloy is changed, the OTL slope changes as illustrated in Figure 5.47(c). As thecomposition is made higher in A, the OTL rotates towards A. This is consistentwith the tendency for maximizing the strongly ordering bonds. A similar effect isseen when the average composition is made richer in B:

Case III: wAB = 0� wBC < 0�wAC < wBC

This case is examined with an example in which interaction parameters for theA–B, B–C and A–C pairs are 0, −0�13 and −0�65, respectively. For the equiatomiccomposition alloy, the stronger A–C interaction causes the OTL to tilt towards adirection nearly parallel to the AC line (Figure 5.47(d)). Again with a change inthe average composition the tendency for the OTL to point towards a vertice isnoticed.

5.6.4 B2 ordering in the Nb–Ti–Al systemThe model described has been applied to several alloy compositions in the Nb–Ti–Al system in which experimental data on site occupancy are available. Sincethe length of the OTL is a measure of the order parameter, one can obtainan order parameter (l/lmax) versus temperature plot (Figure 5.48) from whichthe transition temperature, Tc, can be determined. The model described in thepreceding section has been applied for predicting OTLs for the five ternary alloysin the Nb–Ti–Al system. The binary interaction parameters, wij (in kJ/mol), usedfor the calculations are as follows (Banerjee et al. 2001): wTi−Nb = +5�wTi−Al =−70�wNb−Al = −43. For this ternary system, the Ti–Al and Nb–Al interactionsare strongly ordering, whereas the Ti–Nb interaction is weakly clustering. This

0.3

0.25

0.2

0.15

0.1

0.05

012001000 1400800

Nb–40Ti–15Al

To = 1370 K

Temperature (K)

Ord

er p

aram

eter

(l )

Figure 5.48. Plot of an order parameter (l/lmax) versus temperature from which the transitiontemperature, Tc, can be determined.




Ti TiTi

900 K 1200 K

40 Ti

25 Ti

10 Ti

40 Ti

25 Ti

40 Ti

25 Ti10 Ti

Nb Al Nb Al Nb Al

(c)(a) (b)

Figure 5.49. A comparison of the OTLs for this ternary Ti–Al–Nb system in which the Ti–Al andNb–Al interactions are strongly ordering, whereas the Ti–Nb interaction is weakly clustering.

system would, therefore, be close to Case II as described earlier. Experimentaldetermination of TLs for different alloy compositions has been made using theALCHEMI technique and the experimental results reported by Hou et al. (1996) on10Ti, 25Ti and 40Ti, by Banerjee et al. (1987) on Ti–24 at.% Al–11 at.% Nb andon Ti–25 at.% Al–25 at.% Nb are compared with calculated OTLs in Figure 5.49.As can be seen the agreement between the calculated and experimental OTLs isquite good. Possible factors which are responsible for the differences betweencalculated and experimental results are the errors in the interaction energies andthe uncertainties in the equilibration temperatures.

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Chapter 6

Transformations Related to Omega Structures

6.1 Introduction 4746.2 Occurrence of the �-Phase 475

6.2.1 Thermally induced formation of the �-phase 4756.2.2 Formation of equilibrium �-phase under high pressures 4796.2.3 Combined effect of alloying elements and pressure

in inducing �-transition 4816.3 Crystallography 484

6.3.1 The structure of the �-phase 4846.3.2 The �–� lattice correspondence 4856.3.3 The �–� lattice correspondence 488

6.4 Kinetics of the �→ � Transformation 4906.4.1 Athermal �→ � transition 4916.4.2 Thermally activated precipitation of the �-phase 4926.4.3 Pressure-induced �→ � transformation 494

6.5 Diffuse Scattering 4956.6 Mechanisms of �-Transformations 499

6.6.1 Lattice collapse mechanism for the �→ � transformation 4996.6.2 Formation of the plate-shaped � induced by shock

pressure in �-alloys 5046.6.3 Calculated total energy as a function of displacement 5066.6.4 Incommensurate �-structures 5096.6.5 Stability of �-phase and d-band occupancy 516

6.7 Ordered �-Structures 5186.7.1 Structural descriptions 5186.7.2 Transformation sequences in Zr base alloys 5226.7.3 Transformation sequences in Ti base alloys 5306.7.4 Ordered �-structures in other systems 5336.7.5 Symmetry tree 534

6.8 Influence of �-Phase on Mechanical Properties 5366.8.1 Hardening and embrittlement due to �-phase 5366.8.2 Dynamic strain ageing due to �-precipitation 539

References 550






Chapter 6

Transformations Related to Omega Structures

List of Symbols�: Phase with hexagonal close packed crystal structure.�: Phase with body centred cubic crystal structure�: Phase with hexagonal crystal structure�′: Martensite phase with hexagonal close packed crystal structure�′′: Martensite phase with orthorhombic crystal structure

Ms(�′): Martensitic start temperature

Ms(�): Start temperature for athermal omegax: Compositionx: DistanceG: Free energy per mole

�G�→�′��: Free energy difference between the �′, � from the �-phaseT : TemperatureTo: Temperature at which the free energy of the two phases are

equal�S: Difference in the entropy of the two phasesR: Gas constantP: Pressure

a�, c�: Lattice parameters of the �-phasea�: Lattice parameters of the �-phase

K�: Wave vector in the reciprocal space associated with the�-phase

�: Reduced wave vector�: Order parameter: Interplanar coupling parameterU: Displacement vectoro: Critical resolved shear stress: Time constant for �-precipitation�m: Dislocation density�: Macroscopic strain ratel: Mean free spacing : Anti-phase boundary energy

473




f : Volume fraction of the �-particlesb: Burger’s vectorr: Average radius of �-particles�: Yield stressE: Young’s modulus

6.1 INTRODUCTION

The �-phase, which is an equilibrium phase in Group 4 metals at high pressures,forms in several alloys based on Ti, Zr and Hf and also in many other bccalloys as a metastable phase. The occurrence of this phase was first reportedby Frost et al. (1954) who found that aged Ti–Cr alloys containing ∼8% Crwere unexpectedly brittle. Subsequent X-ray diffraction patterns obtained fromthese alloys showed the presence of a second phase which was designated as the�-phase. The formation of the �-phase was extensively studied, initially becauseof its deleterious effects on mechanical properties, later because of its influenceon such physical properties as superconductivity and most importantly because the� → � transformation exemplifies an interesting class of phase transformations.In recent years the structures of several ordered intermetallics based on Ti andZr have been shown to be chemically ordered derivatives of the �-structure. Thelattice collapse mechanism of the �→ � transformation, which will be describedin detail in this chapter, therefore finds a much wider application in the context ofthe formation of these chemically ordered �-structures.

The �-phase was encountered as an equilibrium phase in elemental Ti andZr under static high pressures by Jamison (1963, 1964). The equilibrium phaseboundaries in the pressure–temperature plane have been established from a numberof high pressure studies, an excellent review of which is provided by Sikka et al.(1982). The crystallography of the � → � transition has provided some cluespertaining to the mechanism of this transition. The role of dynamic pressures(shock loading) in the �→ � transition has also been investigated.

The diffuse scattering features associated with the � → � transition haveprompted intense theoretical and experimental activities including inelastic neutronscattering experiments for studying the soft phonon mode behaviour. Pretransitioneffects observed prior to the � → � transformation have attracted the attentionof many research groups and a structural description of the pretransition statehas now emerged which is consistent with phonon dispersion relations, observeddiffuse scattering and high resolution electron microscopy observations.

The electronic basis of the stability of the �-phase has been elucidated by Sikkaet al. (1982). They have shown that the athermal �→ � transformation in alloys



Transformations Related to Omega Structures 475

of Ti and Zr requires a certain level of occupancy of the d-shells. They have alsodemonstrated the equivalence of effects of pressure and alloying for achieving acertain d-shell occupancy which causes the stabilization of the �-phase.

6.2 OCCURRENCE OF THE �-PHASE

As mentioned earlier, the �-phase appears as a metastable phase in several binaryand multicomponent alloys of Ti and Zr which contain the required levels of�-stabilizing elements. In the case of the pressure-induced transformation, theequilibrium �-phase can be generated from the �-phase by the application ofadequate static pressure. The � → � transformation is associated with a largehysteresis and because of this the �-phase generated by the application of pressurecan be retained in a metastable state at ambient pressure and temperature. Theformation of thermally induced as well as pressure-induced �-phase is discussedin the following sections, making use of the relevant phase diagrams.

6.2.1 Thermally induced formation of the �-phaseThe �-phase appears in Ti and Zr alloys when the stability of the �-phase isenhanced by adding �-stabilizing elements like V, Nb, Mo, Ta, Fe, Cr, Ni, etc.Binary phase diagrams with these elements belong to �-isomorphous, �-eutectoidand �-monotectoid systems, as classified in Chapter 1. Figure 6.1(a) shows the Ti(or Zr) rich side of a typical binary phase diagram on which the metastable stabilityregime of the �-phase is superimposed. The Ms temperatures corresponding to the�→ �′ martensitic transformation (Ms(�

′)) and the athermal �→ � transforma-tion, (Ms(�)) are also plotted against the concentration of the �-stabilizing element.The composition, x1, at which the Ms(�

′) and the Ms(�) lines intersect, definesthe composition limit up to which the � → �′ martensitic transformation occurson quenching an alloy from the �-phase field. An alloy having a composition, x,where x2 > x> x1, on quenching from the �-phase field, exhibits the formation ofthe athermal �-phase which inherits the composition of the parent �-phase. Thismetastable product undergoes compositional changes during subsequent ageingwhen �-particles depleted in the �-stabilizing element form, replacing the ather-mal �-particles. The composition, x2, corresponds to the limiting compositionfor the formation of the athermal �-phase. Beyond this composition, the �-phaseis retained on quenching. The composition of the aged �-phase is given by themetastable boundary between the �- and the �+� phase fields. Schematic freeenergy versus composition plots corresponding to the �-, the �- and the �-phases(Figure 6.1(b)) illustrate the relative stabilities of these phases. Construction of




x

Equilibrium α /β Metastable ω /β Ms (α)Ms (ω)

↑T

G

αG

ω

G

β

G↑

bdfecax

x

x

x

T1

T1

a bc

d

298 K

Athermal ω Thermal ω

(a)

(b)

x x x

x1 x2 x3 x

↑

x →

Figure 6.1. (a) Schematic equilibrium phase diagram of binary alloys (Ti–V, Zr–Nb) from Groups4 (solvent) and 5 (solute) (marked by solid lines). Superimposed metastable phase boundaries havealso been shown by broken lines. Cross-over between Ms(�) and Ms(�) curves mark the compositionrange beyond which the �-phase forms in preference to the martensites � (�′) phase upon quenching(shaded region in the figure). (b) Schematic free energy curves for the �-, �- and the �-phases. Theletters ‘a’ and ‘b’ shown in the figures correspond to the compositions of � and � in equilibrium;‘c’ and ‘d’ correspond to the compositions of � and � in metastable equilibrium and ‘e’ and ‘f’correspond to Ms(�

′), Ms(�), respectively.

common tangents which represent the equilibria (stable or metastable) between (i)the �- and the �-phases and (ii) the �- and the �-phases defines the compositionscorresponding to the end points of the tie lines within the �+� and the �+�phase fields.

The composition limits for the formation of the �-phase can be indicated withreference to the schematic phase diagram shown in Figure 6.1(a). The athermal�→ � transformation occurs in alloys with compositions in the range x1<x<x2,while aged � forms on isothermal treatments in the composition range x1<x<x3.




In general, the martensitic �′ or �′′ phase does not coexist with the �-phase,although there are some recent reports which show the presence of �-particles inthe retained �-regions coexisting with martensitic �′-plates.

The competition between the martensitic �→�′ and the �→� transformationshas been studied in Ti–Nb alloys as a function of the alloy composition andof the severity of quench (Moffat and Larbalesiter 1988). They have found thatthe martensitic �′-phase forms in alloys containing upto 25 at.% Nb when thequenching rate is in excess of 300 K/s. Under slower quenching rates (∼0.3–3 K/s)the �-phase precipitates in alloys containing up to 50 at.% Nb. Only alloyscontaining 60 at.% or more Nb have been found to retain the �-phase fully uponquenching. While the lower concentration limit (25 at.% Nb) defines the pointwhere the Ms(�

′) curve interests the Ms(�) curve, the higher concentration limit(60 at.% Nb) corresponds to the point where the boundary of the metastable(� + �) phase field approaches room temperature.

The Zr–Nb alloy system has been studied in detail with reference to the�-transformation. The Ms(�

′) and the Ms(�) curves in this system intersect at a Nbconcentration of about 7 at.%, the Ms(�) curve comes down to room temperatureat nearly 18 at.% Nb and correspondingly the metastable boundary of the (�+�)phase field approaches room temperature at a concentration level of about 30 at.%Nb (Figure 6.2). Table 6.1 shows the composition limits within which athermal andthermally activated �-formation occurs in several alloy systems. The compositionx(�′/�) at which the Ms(�

′) curve intersects the Ms(�) curve in a binary alloysystem can be estimated from the concentrations corresponding to the boundaries

302010Zr Atomic % Nb

1073

673

298

Tem

pera

ture

(K

)

1136 Kβ

18.5

β 1 + β 2

Ms(ω)

Ms(α ′)β + ω

β + ω

α + β893 K

Metastable βα ′

0

Figure 6.2. Binary phase diagram of the Zr–Nb system showing the superimposed Ms(�) andMs(�) lines. Composition ranges in which martensitic �′, athermal � in the �-matrix and metastable�-structure are formed on �-quenching are indicated.




Table 6.1. Comparison of calculated and observed compositions where Ms(�′) and Ms(�)

lines intersect in different Ti- and Zr-based alloys.

Alloy system Onset of athermal �-formation (at.%)compositions at which Ms(�

′) andMs(�) intersect

Maximum solutecontent (at.%) of aged�-formation observed

Calculated Observed

Ti–V 16 14 25–30Ti–Cr 8 6 18–20Ti–Mn 6 4Ti–Fe 5 4Zr–Nb 9 7 30–33

of two phase fields, either the equilibrium (�+�) field or the metastable (�+�)field. The slope of the To versus x plot, where To is the temperature at which thechemical free energies per unit volume of the parent and the product phases areequal and x denotes the composition (in at.% solute), can be related to the changesin the free energy of the solution �G�→�′�� at xo and the changes in the entropyof the solvent, �S�→�′��, due to the change in phase from � to either �′ or � as:

�dTo/dx��→�′��xo

= �G�→�′�xo

/�S�→�′��Ti�Zr (6.1)

The above expression is based on the regular solution model. At low soluteconcentrations

�G�→�′��xo

= RT ln�x�/x�′��+�S�→�′��Ti�Zr �T (6.2)

which reduces to

�G�→�′��xo

= RT ln�x�/x�′�� (6.3)

The second term on the right hand side of Eq. (6.2) is much smaller comparedto the first term. The simplifying assumptions and some data which are used forthe estimation of the composition x (�′/�) at which the quenched product changesfrom the �′ to the �-structure in binary alloys of Ti and Zr are listed below:

(a) The heats of transformation for the � → �′ transformation in Ti and Zr aretaken from Kubaschewski et al. (1993) as 4.2 kJ/g mole for Ti and 3.9 kJ/molefor Zr which yield the values of �S�→�′

as 0.74 for Ti and 0.85 for Zr.




(b) The steep slopes of the �/� transus in the pressure–temperature phase diagramsof pure Ti and Zr indicate that the differences in entropy between the �- andthe �-phases in these metals are very small. In view of this �S�→�′

is takento be approximately equal to �S�→�.

(c) The To versus x plot is assumed to be linear.(d) The composition at which the Ms curves for the � → �′ and the � → �

transformations intersect is taken to be the same as the point of intersection ofthe respective To lines. This implies that the supercooling (To −Ms) requiredfor the two transformations is assumed to be equal. This approximation isvalid as the extents of supercooling required for these transformations are quitesmall due to a relatively small contribution of the strain energy associated withthe nucleation of either of the product phases in these systems.

The composition, x(�′/�) can then be expressed as

x��′/��= T�→�′Ti�Zr −T

�→�Ti�Zr

�dT�→�′o /dx�− �dT�→�

o /dx�(6.4)

The difference between the �/� and the �/� transition temperatures for pure Tiand Zr being about 400 K, one obtains by substituting Eq. (6.1) in Eq. (6.4)

x��′/��= �S�→�400/��G�→�′xo

−�G�→�xo

� (6.5)

Using the Eqs. (6.3) and (6.5) the composition, x(�′/�), at which the quenchedproduct changes from �′ to � has been calculated and the result has been comparedwith the experimental value, wherever available (Table 6.1).

6.2.2 Formation of equilibrium �-phase under high pressuresHigh-pressure resistivity studies on pure Zr first showed a phase transitionat a hydrostatic pressure of about 6.0 GPa (Bridgman 1948a,b). Subsequently,Jayaraman et al. (1963) detected transformations in Ti at 8.0–9.0 GPa and in Zr at5.0–6.0 GPa using the same technique. The transformation was initially thought tobe the �→ � transformation, as the �-phase was supposed to be the denser phaseat that time. Jamison (1963) was the first to identify the high-pressure phase to beone with a simple hexagonal structure (�-phase). Jamison also pointed out the dif-ficulties in determining the equilibrium �/� phase boundary in Ti and Zr, as a largehysteresis is associated with the transformation. It has been observed (Jamison1964) that the �-phase, once formed by the application of a high pressure, doesnot revert back fully to the �-phase on removal of the pressure. The metastable




retained �-phase remains in association with the equilibrium �-phase at ambientpressure in samples which have been subjected to a pressure treatment. A completetransformation of the �-phase into the �-phase requires heating Ti to 380 K andZr to 470 K for several hours after the removal of pressure.

Using the equilibrium transformation data compiled by Kutsar (1975) for the�→ �, �→ � and �→ � transformations, Sikka et al. (1982) have constructeda schematic pressure–temperature phase diagram (Figure 6.3) for the Group 4transition metals. The triple point coordinates (Pt, Tt), the equilibrium �→ � trans-formation temperature at 0.1 MPa, T (�→ �), the equilibrium �/� transformationpressure at 293 K, Po (�/�), and the �→ � reversion temperature at 0.1 MPa, T(�→ �), for Ti, Zr and Hf are indicated in Table 6.2. The hysteresis observed inthe �→ � transition is represented in Figure 6.3 by a pair of broken lines corre-sponding to the completion of the �→ � and the �→ � transitions. The fact that

ωα

α β

α

αω

αω

ω

β

α – ωPo

T(Pt , Tt)

T

Tem

pera

ture

(K

)

Pressure (GPa)293

Equilibrium linesHysteresis lines

Figure 6.3. Schematic pressure–temperature phase diagram of Group 4 transition metals. Equili-brium lines are shown by solid lines and the values of the triple points are given in Table 6.2 for allthe Group 4 metals. The broken line represents the hysteresis in the transformation from one phaseto another. This hysteresis could be due to the presence of interstitial solutes in the pure metals.

Table 6.2. Calculated values of triple points for the Group 4 transition metals.

Ti Zr Hf

Pt(GPa) 9 6 30Tt(K) 1100 975 1800T�↔�(K) 1155 1135 2030P�↔�(GPa) 2 2.2 21T�↔�(K) 380 470 –




Table 6.3. Calculated values of the thermodynamic parameters required for various transformationsin Group 4 transition metals.

Ti Zr Hf

�→ � �→ � �→ � �→ � �→ � �→ � �→ � �→ � �→ �

�V −0�032 −0�29 −0�16(cm3/mol) −0�036 0.18 0.14 −0�09 −0�19 0.1 +0�1 −0�45 0.43

−0�052 −0�2 −0�02

�S 3.68 −1�59 5.18 3.55 −1�25 4.81 3.34 2.93 5.85(J/mol/K) 3.64 2.51

dT /dp 8 90 −24 160 38(K/GPa) −26 27 −7 150 70

−9 112 −26 21−10

the �-phase is partially retained at ambient conditions after a pressure treatmenthas also been depicted in this figure as the � → � hysteresis line intersects thetemperature axis at 0.1 MPa at a temperature higher than room temperature. Addi-tional data regarding the volume and entropy changes associated with differentphase transitions and the slopes of various phase boundaries in the P–T plane arelisted in Table 6.3.

The � → � transition has also been studied in dynamic (shock) experimentsinvolving pressure pulses of microseconds duration. Phase transitions under suchconditions are usually revealed in the observed discontinuities in the plots of shockvelocity (Us) versus particle velocity (Up). McQueen et al. (1970) have noticed thediscontinuities in the Us versus Up plots for Ti (17.5 GPa, 370 K), Zr (26.0 GPa,540 K) and Hf (40.0 GPa, 725 K). The transition pressures under shock loadingconditions have been found to be substantially higher than those reported for staticpressure investigations. An extensive hysteresis in the �/� transition has also beennoticed in shock pressure experiments.

6.2.3 Combined effect of alloying elements and pressure in inducing�-transition

As pointed out in the preceding two sections, the �-phase can be generatedin Ti, Zr and Hf systems either by the addition of adequate amounts of a �-stabilizing alloying element or by the application of pressures sufficient for induc-ing the �-transition. The combined effect of alloying elements and pressure hasbeen studied by only a few investigators (Afronikova et al. 1973, Ming et al.1980, Vohra et al. 1981, Dey et al. 2004). High pressure studies on Zr–Nb,




the Ti–Nb and the Ti–V alloy systems have brought out the following commonfeatures:

(a) The application of pressure promotes the formation of the �-phase in thecomposition range where the �-quenched structure is martensitic (�′). Thecomposition range over which the �-phase is present is, therefore, wider underpressure.

(b) The �-start pressure is lowered with increasing concentration of a �-stabilizingelement up to a certain point beyond which the �-start pressure rises withfurther increase in solute concentration.

The results of high-pressure studies on Ti–V alloys up to a pressure of 25.0 GPaare shown in the pressure–composition phase diagram in Figure 6.4. It may benoted that the boundaries of the different phase fields do not correspond to theequilibrium condition as these results are compiled from X-ray diffraction andresistivity data which were obtained from samples maintained under pressure,whereas transmission electron microscopy results pertain to pressure-treated sam-ples which were brought back to ambient conditions.

The features of the pressure–composition phase diagram in Figure 6.4 could berationalized in terms of a set of hypothetical free energy versus concentration plots,as shown in Figure 6.5. Thus the essential features of experimental observations,such as the extension of the composition range of �-formation and the variationin the �-start pressure with concentration, could be explained on the basis of these

Ti 10 20 30 40 50

10

20

βω

ω + β

α + ω

α

V, Atomic %

Pre

ssur

e, G

Pa

Figure 6.4. Pressure–composition phase diagram of the Ti–V system at room temperature. As couldbe noticed from the figure, addition of V in Ti reduces the pressure for �→� phase transformation.Upon comparing this figure with Figure 6.2, a similarity between the trends in the transitiontemperature and transition pressure could be noticed.




α

α

β

β

ω

α + ω β + ω

P = P1

T = 300 K

Composition →A B

↑F

α

α

β

β

ω

α + β

P = Po

T = 300 K

A Composition → B

↑F

α

α

β

βω

ω

α + ω ω + β

P = P2

T = 300 K

Composition → BA

↑F

α

β

βω

ω

ω + β

P = P3

T = 300 K

Composition → BA

↑F

Figure 6.5. Hypothetical free energy curves as a function of composition and pressure. These curvesare drawn based on the phases observed at various pressures and alloy compositions.

hypothetical free energy–concentration plots. Under the equilibrium condition asystem possessing such concentration dependence of free energies for three com-peting phases would be expected to exhibit an eutectoid phase reaction, as shownin Figure 6.6. Though experimental confirmation of an eutectoid phase reaction ina �-forming system in the pressure–concentration plane is still not available, theobserved metastable Ti–V phase diagram (Figure 6.4) and a generalized versionof the same (Figure 6.6) bring out the basic trends in relative phase stabilities asfunctions of pressure and concentration. Of late the formation of the �-phase wasobserved in alloys upon shock loading (Hsiung and Lassila, 2000). These alloysunder normal conditions do not show the formation of the �-phase. Recently,Dey et al. (2004) have shown presence of � in the Zr–20 Nb alloy upon shockloading. The morphology of the �-phase has exhibited plate-like morphology. Inboth these studies the plane of contact between the �- and the �-phases was the




P 3

P 2

P 1

P 0

A B

βα + β

α

ω

β + ωα + ω

Composition

↑

Pre

ssur

e

↑

Figure 6.6. A generalized version of the phase diagram constructed on the basis of the free energyplots shown in Figure 6.5. This phase diagram suggests the presence of an eutectoid phase reactionin the pressure–composition phase diagram.

{112} plane of the �-lattice. Such observations showed that the �-structure canalso be visualized as a defect in the bcc lattice akin to twinning (Hsiung andLassila, 2000).

6.3 CRYSTALLOGRAPHY

The crystallography of the �→ � transformation has provided the essential clueto the transformation mechanism and is, therefore, a subject which has beeninvestigated in great detail. Soon after the �-phase was first reported, a controversyarose regarding the crystal structure of this phase. The structure of the �-phase isdescribed in the following section in which a summary of the early controversy isalso given. The lattice correspondences between (i) �- and �-phases and (ii) �- and�-phases are discussed subsequently.

6.3.1 The structure of the �-phaseThere has been a serious controversy regarding the crystal structure of the �-phase.Early studies suggested that the �-phase had a complex bcc unit cell with a latticeparameter three times that of the �-phase. In fact, selected area diffraction (SAD)patterns taken from the �+� structure corresponding to all zones excepting the




<100>� and the <111>� zones, invariably show extra diffraction spots whichdivide all bcc reciprocal lattice vectors into three equal segments. It has subse-quently been recognized that the occurrence of all the variants of �-particles whichare distributed in the �-matrix gives rise to composite diffraction patterns whichpossess the symmetry of the matrix.

The �-structure has later been determined to be either hexagonal (belonging tothe space group D1

6H (P6/mmm) (Silcock et al. 1955) or trigonal (belonging to thespace group D3

3D (P3m1) (Bagaryatskiy et al. 1955). Both these structures can begenerated from the parent bcc structure by simple atom movements, as discussedin the following section. The discrepancy between the reported hexagonal andtrigonal structures of the �-phase has also been resolved in a study carried outby Sass and Borie (1972) who have shown that increasing solute (�-stabilizer)concentration causes the sixfold symmetry characteristic of D1

6H to degenerate intothe threefold symmetry characteristic of D3

3D.

6.3.2 The �–� lattice correspondenceThe lattice correspondence between the �- and the �-structures not only establishesthe crystal structure of the �-phase but also provides a clue to the transformationmechanism. The orientation relationship between the two phases has been deter-mined in a large number of investigations and has been unanimously described as

�111��0001��< 110 >� ��< 1120 >�

The relation between the lattice geometries of the two phases can be bestillustrated in a diagram showing the stacking sequence of the (222)� atomic planesof the bcc structure (Figure 6.7(a)). The ABCABC� � � stacking sequence of theseatomic planes generate the open bcc structure with the threefold axes along the<111> direction. The �-lattice can be created by collapsing a pair of (222)� planesto the intermediate position, leaving the next plane undistorted, and collapsing thenext pair of planes and so on (Figure 6.7(b)). When the collapse is complete asixfold rotation symmetry is created around the specific <111> direction alongwhich lattice collapse occurs (Figure 6.7(c)). In case the collapse is incomplete,the trigonal symmetry is not lost, and the resulting �-structure is associated witha trigonal structure with the space group D3

3D. The �- and the trigonal structurescan be formed by introducing a displacement wave in the parent bcc (�) structure.This point is illustrated in Figure 6.7(d) where a displacement wave with the wavevector, K= 2/3 <111>, can create the �-structure when the amplitude of thewave is sufficient for moving {222} planes by a distance 1/2{222}. When theamplification of the wave is smaller, the resulting structure is trigonal. On thebasis of the lattice correspondence, shown in Figure 6.7, the lattice parameters of




5 6

+

–

0 1 2 3 4Dis

plac

emen

t U

p

(d)(c)

2 Layer1 Layer0 Layer

[2110]

[1210]

[2110]

3

21.510

(b)

(0001)

Layer No.

[1210]

ω-hexagonal

[0001]

(a)

[110]β(222)

[101]β

[011]β

β bcc

[111]

Figure 6.7. Planar arrangement of the {222}� atomic planes in the bcc lattice; (a) the ABCABC� � �sequences of atomic plane arrangements could be noticed (planes are marked as 0,1,2,3� � � ). (b) Theplanar arrangement of the (0001)� atomic planes in the �-lattice. As could be observed the �-latticecould be obtained from the bcc lattice by collapsing a pair of planes (1 and 2, as shown in figure)in the middle (marked as 1.5) while keeping the third one (0 and 3) undisturbed. (c) Completelycollapsed {222} planes form a sixfold symmetry whereas an incomplete collapse forms a threefoldsymmetry along the <111> axis. (d) The displacement of {222} planes could also be visualizedin terms of a sinusoidal displacement wave where an upward movement is seen as a positivedisplacement and a downward movement as a negative displacement.

the �-structure (a� and c�) can be expressed in terms of the bcc lattice parameter,a�:a� = √

2 a� and c� = �√

3/2�.a�.A comparison of the lattice parameters of the �-phase computed from the a�

value of the corresponding �-phase and those experimentally observed shows thatthe divergence between the two is negligibly small for the athermal �→ � trans-formation (Table 6.4). A similar comparison between the corresponding latticedimensions of the �-phase and the aged �-phase, which is in metastable equi-librium, reveals that the isothermal � → � transition is associated with a linearcontraction of about 5% (volume contraction ∼15%).

Since there are four sets of <111> directions, there are four possible crystal-lographic variants of the �-structure in a given bcc parent crystal. The matrixrepresentation of the lattice correspondence for all the four variants is shown in




Table 6.4. The calculated and experimentally determined values of the lattice parameters of the�-phase encountered in different alloy systems.

Alloy system Calculated values of Experimentally determined Differencelattice parameters (nm) values of lattice parameters (nm) (per cent)

Zr–Nb a = 0.5003 a = 0.5019 ��a� = 0�31c = 0.3064 c = 0.3089 ��c� = 0�82

Zr–Nb–Cu a = 0.5003 a = 0.5029 ��a� = 0�52c = 0.3064 c = 0.309 ��c� = 0�85

Ti–Nb a = 0.4596 a = 0.4627 ��a� = 0�67c = 0.3031 c = 0.2836 ��c� = 6�43

Ti–V a = 0.4596 a = 0.460 ��a� = 0�08c = 0.28146 c = 0.282 ��c� = 0�19

Table 6.5. Correspondence matrices relating the �-(bcc) and the �-(hexagonal) structures.

Variant Orientation relationship Correspondence matrix �[R]�

1 (111)�� (0001)�; [110]�� [1210]�

⎡⎣ 2 2 1

0 2 12 0 1

⎤⎦

2 (111)�� (0001)�; [110]�� [1210]�

⎡⎣ 0 2 1

2 2 12 0 1

⎤⎦

3 (111)�� (0001)�; [110]�� [1210]�

⎡⎣ 2 2 1

0 2 12 0 1

⎤⎦

4 (111)�� (0001)�; [110]�� [1210]�

⎡⎣ 0 2 1

2 2 12 0 1

⎤⎦

Table 6.5. Within the same orientation variant of the �-structure, associated witha given <111> variant of lattice collapse, three distinct subvariants can be cre-ated, depending upon whether the A, B or C plane remains as the uncollapsedplane. These subvariants are schematically illustrated in Figure 6.8. An �-regionwithin the bcc phase remains coherent with the parent, as seen from the smalldifference in their lattice dimensions. However, given the total number of variantsand subvariants (4×3 = 12), the elastic strains (arising from the atomic displace-ments and the slight change in volume) associated with different particles interactwith one another. This elastic interaction is a controlling factor which governs thearrangement of �-particles in the �-matrix.




Uncollapsed plane

ABCABCAB

CA

A B C

ω -Subvariants

Figure 6.8. Schematic presentation of the translational subvariants of the �-phase produced due tothe random collapse of different {222} planes of the bcc lattice. Regions of the different subvariantsare marked in the figure.

6.3.3 The �–� lattice correspondenceDiffraction experiments have established that the following two orientation rela-tions exist between the �- and the �-phases:Orientation Relation I (OR I)

�0001��0111��<1120>� ��<1101>�

Orientation Relation II (OR II)

�0001��1120��<1120>� ��0001��

On the basis of these observed orientation relations two distinct mechanisms,involving either the direct �→ � route or the two stage �→ �→ � route, havebeen proposed for the transformation from the � to the �-phase. Silcock et al.(1955) first proposed a model for the formation of the �-lattice directly from the �-lattice which leads to the orientation relation described in OR II. The shift in atomicpositions required for the direct �→� transformation is shown in Figure 6.9. The(112 0)� plane is generated from the (0001)� plane by displacement of alternatehexagons on the (0001)� plane by 0.148 nm (for Ti). The atoms marked a, b, cand d shift to positions a′, b′, c′ and d′. In addition, a contraction of ∼4.7% alongthe [1120]� and an expansion of ∼4.5% along the [1100]� directions are requiredfor the generation of the �-unit cell.

Usikov and Zilbershtein (1973) have suggested that the �→ � transformationoccurs via the intermediate �-structure which is unstable at the pressures andtemperatures under consideration. This suggestion is based on the fact that the




d

d′

a′

b

b′

c

c′

[1100]α ⎥⎪[1100]ω

(0001)α⎥⎪(1120)ω

0.7667 nm

0.295 nm

a

[1120]α ⎥⎪

[0001]ω

Figure 6.9. Lattice correspondence between the �- and the �-phases. The (1120)� plane of the�-phase is generated from the (0001) plane of the �-phase by shifting the atoms at a, b, c, d to a′,b′, c′, d′ positions. Contraction along [1120]� and expansion along [1100]� direction is also neededto generate the �-unit cell.

orientation relation observed by them (OR I) can be described in terms of theproduct of the correspondence matrices �R� and �R

� which pertain to the �→ �and the �→� transformations, respectively. They have shown that all the observedorientation relations between the �- and the �-phases, as revealed from diffractionpatterns, are consistent with the correspondence matrix associated with OR I.Later TEM studies (Vohra et al. 1980, 1981) on high-pressure treated Ti–V alloys,subjected to high pressures where alloying addition promotes �-phase formation,have confirmed the presence of the �-phase along with the �-phase in the matrixof �-phase grains. This observation provides a direct evidence of the occurrenceof the �-phase as an intermediate state during the � → � transition. The factthat the stability of the �-phase is enhanced by V additions appears to have beenresponsible for the retention of this intermediate phase.

Orientation relations have also been determined in samples in which the �-phasehas been made to form by the application of shock pressure. Kutsar et al. (1990)have reported that the �/� orientation relation in shock-treated Zr matches withOR II. In some recent investigations Song and Grey (1994, 1995) have observed anew orientation relation, as described below, between the �- and the shock-induced�-phases:

�0001��1011�� 1100��<1123>�

On the basis of this observation they have proposed a different transformationmechanism according to which the �-phase forms directly from the �-phase. Jyoti




(b)

(2.5418)

(2.5418)

(2.5418)

(2.5418)

c d

b a

g

109°

k

i

f

ej

h

(011)β

(c)

(1011)ω

c d

ab

k

j e

hf

i

[1101]

(2.1098)

(2.9533)

[1123]

(– 4219)

(+ 4219)

(a)

[1100][1120]

(2.5725)12

0°

(2.5725)

(2.5725)

[101

0]

(2.5725)

(2.5725)

c d

b a

j

kg

hf

e

i

(0001)α

Figure 6.10. Schematic presentation of (a) the (0001) plane of the �-structure (b) the (011) planeof the � and (c) the (1011) plane of the � structure. Filled circles represent atoms on the planewhereas open circles represents atoms away from the planes. The distance of separation is given inparenthesis (in Å). A reference set of atoms is marked on each projection. As could be noticed verysmall movement of atoms j, k, h and g on (0001)� plane is needed to obtain either the �-planar orthe �-phase. A close resemblance suggests a possible path of the �→ � phase transformation mayhave the �-phase as the metastable phase as an intermediate path. Results are more inclined to direct�→ � transformation (for detail see text).

et al. (1997) have examined SAD patterns belonging to a larger number of zoneaxes for determining the orientation relation between the �- and the �-phasesand have attempted a harmonization of all the conflicting results reported earlierand have demonstrated that nearly all the orientation relations reported earliercan be expressed as subsets of OR I. They have considered the possibility of theformation of the �-phase through an intermediate �-structure and have comparedthe crystallographic results with those expected from the direct formation of the�-phase from the �-phase. A schematic representation of the atomic arrangementspertinent to either of the routes is shown in Figure 6.10. Small variation in thepositions of atoms marked in the figure generates all the three structures indicatinga close resemblance among these structures.

6.4 KINETICS OF THE �→� TRANSFORMATION

The � → � transformation has been found to proceed under two different con-ditions, namely, during quenching from the �-phase field and during isothermalholding at a temperature below about 773 K. The kinetic characteristics of these twocases correspond to an athermal and a thermally activated process, respectively.A comparison between these two modes of �-phase formation is made in thefollowing two sections.




6.4.1 Athermal � → � transitionOn quenching an �-forming alloy from the �-phase field to a temperature belowMs(�), the athermal �→ � transformation can be initiated. The initiation of theathermal �-phase is characterized by diffuse intensity distribution in diffractionpatterns, as is illustrated in Figure 6.11. The athermal nature of the transforma-tion has been established through ultrahigh quench rate (∼11 000 K/s) experi-ments (Bagaryatskiy and Nosova 1962). The facts that the transformation is not

000

110

002

00011

00021

(b)(a)

(c)

Figure 6.11. (a) Diffuse intensity patterns obtained on the selected area diffraction patterns fromthe <110> of the bcc lattice. (b) Schematic drawing exhibiting deviation from the 1/3, 2/3 positionfrom the rel vector [112]. (c) Diffuse intensity distributions in various zones.




suppressible even under such quenching conditions and that it occurs at tempera-tures where the diffusivity of both self and solute atoms is negligibly small suggestthat the transformation can occur without any thermal activation. The diffuseintensity distribution gradually changes to sharp �-reflections as the temperature isprogressively lowered below Ms (�), suggesting an increase in the volume fractionof the �-phase and a growth of individual �-particles. The complete reversibilityof the transformation is yet another evidence for its athermal nature. All theseobservations suggest that this transformation can be classified as a displacivetransformation. Particles of the �-phase that form on quenching do not exhibita plate-like morphology or a surface relief effect which are both characteristicfeatures of a martensitic transformation. It is for these reasons that the athermal�→ � transformation is not categorized as a martensitic transformation in spiteof its athermal and composition-invariant character.

6.4.2 Thermally activated precipitation of the �-phasePrecipitates of the �-phase form in a �-phase matrix on ageing at temperaturesbelow about 773 K, which is the upper cut-off temperature for �-phase forma-tion by ageing in most of the �-solid solutions. The progressive increase in thevolume fraction and the growth of �-particles with an increase in the isother-mal holding time are indicative of a thermally activated transformation mech-anism. Time-temperature-transformation (T-T-T) plots pertaining to isothermal�-precipitation have been experimentally generated in several alloys. Figure 6.12shows two such plots corresponding to Zr–Nb alloys. The characteristic C-shape ofthese T-T-T plots is consistent with the well-known dependence of the incubationperiod on the driving force and the diffusivity which change in opposite directionswith a lowering of temperature.

An in-situ study of �-precipitation under 1 MeV electron irradiation in a high-voltage electron microscope has revealed the real time changes (Figure 6.13)in diffraction patterns and has demonstrated the progressive sharpening of�-reflections from the initial diffuse intensity distribution. This observation isremarkably similar to the evolution of sharp �-reflections from the diffuse intensitydistribution which has been reported to occur in the athermal � → � transfor-mation as the temperature is progressively lowered. The lattice correspondencebetween the �- and the “aged �” structures has also been found to be the sameas that recorded in the case of the athermal �→ � transformation. These featuressuggest that isothermal �-precipitation occurs by an atomistic mechanism whichis essentially the same as that operative in the athermal � → � transformation.The thermally activated component of the overall transformation mainly involvesa solute-partitioning process which accompanies the lattice collapse mechanism




0.2 0.5 1 2 5 10 2 5 102 2103 552 104

Time ~ Min

100

200

300

400

500

600

700

800

900Te

mpe

ratu

re ~

°C

α + β Zr + βNb

ω + β Zr

α + β Zr

β

ωr1 h 6 h 1 day 4 days

β

ω + β Zrα + β Zr

α + β Zr + βNbα BY DILATOMETRY

β/α + β Zr

α + β/α + β Nb

(a)

(b)

100

200

300

400

500

600

700

800

900

0.2 0.5 1.0 2 5 10 2 5 10 2 5 10 2 5 10

Time ~ Min

Tem

pera

ture

~°C

1 h 6 h 1 day

α + β Nb + β Zrα + βNb

β

βI/βI + βII

βI + βII/α + βNb β + βI + βII

ω + β Zr

α + β Zr

α + βNb + β Zr

βω + βZr

α + βNb+ β

4 days

β + βNb + βα + βNb

Figure 6.12. T-T-T diagrams for (a) Zr–12% Nb alloys and (b) Zr–17.5% Nb alloys. Increasing Nbconcentration retards the formation of the �-phase from 3 h at 300�C for Zr–12% Nb to 2 days at300�C for Zr–17.5% Nb alloy.




300 K

450 Kt = 0 s

t = 0 s

60 s 300 s

240 s120 s

Figure 6.13. Selected area diffraction patterns from the �-phase regions after irradiating with 1 MeVelectrons for different time periods. Gradual transformation of the diffuse intensity to the�-reflectionscould be noticed.

(as discussed in a later section). It may be noted from Figure 6.13 that the�-transformation can occur even at ambient temperature due to the condition ofenhanced diffusion obtained under 1 MeV electron irradiation.

6.4.3 Pressure-induced � → � transformationIt has been noticed that the transformation pressure, P�→�

s , determined by differentresearch groups, shows a fairly high scatter for pure Ti (from 2.8 to 9.0 GPa) and forpure Zr (2.2 to 6.0 GPa) (Sikka et al. 1982). This large scatter has been attributed tovariations in the pressure exposure time, the starting microstructure, the impuritycontent and the hydrostatic component of the applied pressure. The influence ofthe exposure time at a constant pressure at ambient temperature on the � → �transition has been systematically studied and the time dependence of the growthof the �-volume fraction has been established. The rate of the �-transformationas a function of pressure shows a peak (Figure 6.14) (Zilbershtein et al. 1975,Vohra 1978) analogous to that observed in nucleation rate versus temperatureplots pertaining to thermally activated martensitic transformations (isobaric). Thetime-dependent nature of the pressure-induced �→� transformation suggests thatthere exists a barrier, possibly that of the nucleation step, which can be overcomeby thermal activation. The nucleation step involves the growth of quasistatic�-embryos to the critical size by thermally assisted diffusion processes. Afterattaining the critical size, the nuclei grow spontaneously in an athermal manner.The observed peak in the rate-versus-pressure plot is consistent with the factthat the driving force increases while the diffusivity decreases with increasingpressure.




x xxx

xx x x

xxxx

2.0 6.0 10.0 2.0 6.0 10.0

1

x

x

x

x

x

x

x

xxx

x

x

x

x

x xx x

2

3

4

0

(a) (b)P (GPa)

Rat

e of

tran

sfor

mat

ion

arbi

trar

y un

it

P MSα ω↑

P MSα ω↑

Figure 6.14. The rate of �-phase transformation with pressure shows peaking in between (a) 6–7 GPain the case of Ti and (b) in between 5–6 GPa in the case of Zr. Theoretically calculated start pressuresfor the �→ � transformation are also shown.

6.5 DIFFUSE SCATTERING

The �-forming systems, in certain composition ranges, invariably exhibit diffuseintensity distribution in electron, X-ray and neutron diffraction patterns. The com-plexity of these patterns, though far from being fully understood, has inducedmany research groups to study this unusual displacive phase transformation. Theessential features of the experimental observations regarding the diffuse intensitydistribution and the soft phonon behaviour associated with the �-transformationare summarized here.

Diffracted intensity from the athermal �-phase is often diffuse, centred awayfrom the “ideal �” positions in the reciprocal space and can show pronouncedcurvature and asymmetry. These effects are predominantly manifested as the alloysystem is moved away from the region of relative �-stability in the phase diagramalong either the concentration or the temperature axis. The nature of the diffuseintensity in electron diffraction patterns is shown in Figure 6.11. The diffractionpattern corresponding to the [110] zone axis (Williams et al. 1973) clearly showsthat the intensity maxima in the diffuse intensity distribution are located at positionsslightly away from those expected for the ideal �-structure (c� = √

3/2.a� and a� =√2.a�, with the lattice correspondence described earlier). For the ideal �-structure

the (0001)� and the (0002)� reflections should occur at 1/3 and 2/3 of the distancebetween the (000) and the (222)� reflections, respectively. The line drawn alongthe <112> direction through bcc spots in Figure 6.11 shows where the reflectionscorresponding to the ideal �-structure should be located. The observed (0001)�reflection is displaced from the ideal position away from the (000) spot, while the




(0002)� reflection is displaced towards the (000) spot. Dawson and Sass (1970)have shown that the extent of this deviation depends on the stability of the �-phasewith respect to the �-phase which, in turn, is decided by the alloy compositionand the departure from the Ms(�) temperature.

The observed diffuse intensity in electron diffraction patterns corresponding to[110]� and [102]� zone axes appears in the form of quasicircular arcs (de Fontaineet al. 1971). In-situ experiments carried out using a cooling stage in an electronmicroscope have shown that the diffuse intensity gradually transforms into sharp�-reflections as the sample is cooled to temperatures sufficiently below the Ms(�)temperature. Reheating of samples has shown a gradual disappearance of the�-reflections and the reappearance of diffuse patterns. The complete reversibility ofthe process has been demonstrated by the experiments of de Fontaine et al. (1971).

The possible occurrence of multiple scattering has been considered for explain-ing the appearance of diffuse streaks, especially the quasicircular arcs. Multiplescattering will no doubt alter the relative streak intensities, but it cannot account forthe presence of the streaks themselves. Examining several zones of electron diffrac-tion patterns and considering the symmetry of the reciprocal lattice, de Fontaineet al. (1971) have constructed a three dimensional model of the diffuse intensity(Figure 6.15) which is distributed on quasispherical surfaces centred around the

002

000

200

110

020

Figure 6.15. Three-dimensional model of the intensity distribution of the diffuse intensity in thereciprocal space of the bcc unit cell. The sphere of diffuse intensity touches the octahedra of {111}faces which surrounds it.




octahedral sites 100, 111, � � � , of the fcc lattice (reciprocal of the bcc �-lattice).These spheres of intensity touch all the {111} faces of the octahedra surroundingthem. Below the Ms(�) temperature, �-reflections appear in the form of discs lyingon {111}∗ reciprocal lattice planes at the points of tangency of the intensity sphereswith the faces of the octahedra. The coordinates of these points are 1/3 <111>,2/3 <111> etc., which can be identified as �-reflections. With lowering of tem-perature the spherical surfaces on which the intensity is distributed become theoctahedra themselves which are seen as streaks along traces of {111} planes.

Neutron diffraction experiments (Sikka et al. 1982) have also shown that withincreasing solute content, the �-reflections become diffuse and are shifted awayfrom ideal �-positions in the reciprocal space. Figure 6.16, which shows the (011)�reciprocal lattice section for a Zr–20% Nb alloy, clearly demonstrates this point.The shifts correspond to an increase in the �-wave vector from q� = �2�/a�)

4.0

3.0

2.0

1.0

0.0–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0

0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.04 8 12

h 3

h1

h′1

h′3

[211]β

[111

] β↑↑

Figure 6.16. Neutron diffraction intensity plots showing distribution of the diffuse intensity on the(011)� reciprocal lattice section of the Zr–20% Nb alloy (Keating and Laplace 1974).




(2/3,2/3,2/3) to q + �q. The magnitude of �q, which is a measure of the extent ofincommensuration with respect to the ideal �-lattice, is seen to be dependent on thesolute concentration and the extent of undercooling below the Ms(�) temperature.However, in some systems, cooling to a temperature as low as 5 K does not changethe diffuse intensity, suggesting the presence of the incommensurate structure evenat such low temperatures. It has been noticed that with the application of pressurethe �-reflections get sharper and the structure tends to become commensurate.It has been noticed that longitudinal phonons having a wave vector K which isclose to that for an “�” phonon (K� = 1/3 [222]) exhibit a sharp central peakflanked by a pair of weak diffuse peaks. The diffuse peaks are representative of thedynamic phonon structure whereas the central peak represents a static phonon. Thisobservation can be interpreted in terms of a mechanism involving the propagationof a large amplitude dynamic phonon. As it propagates, it structurally degeneratesinto two alternate configurations, � and anti-� (described in Section 6.6) whichhave a large difference in free energy (Cook 1973). Large amplitude phonons arethus “pinned” in a metastable state at the low free energy �-positions. The latticecollapse mechanism which is implicit in this interpretation will be discussed indetail in Section 6.6.

The experimental longitudinal (L) and transverse (T) phonon dispersion curvesfor �-Zr are shown in Figure 6.17 (Stassis et al. 1978). The pronounced dip inthe longitudinal [111] branch at the reduced wave vector, �m (= Km d/�) 0.7,which is slightly away from the ideal �, is indicative of a tendency of this mode

25

20

15

10

5 5

10

15

25

20

0.5 0.3 0.1 0.20 0.6 1.0 0.8 0.6 0.4 0.2 0

Reduced wave vector

T

LL L

Tbcc Zr

T = 1423 K

Ene

rgy

(meV

)

H P ΓΓN

[110] [001] [111]

Figure 6.17. Experimental phonon dispersion curves for bcc Zr. A pronounced dip in the longitudinal[111] branch (marked as L) at a wave vector ∼0.7 could be noticed. This wave vector is slightlyaway from the wave vector needed for ideal � which is 0.66 (Stassis et al. 1978).




to soften up. It is to be noted that the softening is not due to the extent thatthe system becomes unstable with respect to the development of this longitudinaldisplacement wave. Stassis et al. (1978) have also observed that the intensity ofdiffuse peaks as well as the extent of their shifts from the ideal �-positions aretemperature dependent.

6.6 MECHANISMS OF �-TRANSFORMATIONS

As discussed in the preceding sections, the �-phase can form in Ti and Zr basealloys in the following situations:

(a) on application of hydrostatic pressure to the �-phase – either by a staticpressure or by a shock pressure

(b) on quenching from the �-phase field(c) on ageing the metastable �-phase(d) on application of shock to the �-phase (in �-stabilized alloys such as

Zr–20% Nb)

The mechanisms involved in �-phase formation in these cases are not the same.However, the lattice collapse mechanism, which converts the bcc structure to thehexagonal �-structure, appears to be a common feature in all these cases and is,therefore, discussed first.

6.6.1 Lattice collapse mechanism for the � → � transformationThe � → � transformation has been studied in detail and well documented inliterature. The key experimental observations of these studies are summarized inthe following:

(1) The � → � transformation exhibits strong pretransition effects in terms ofprofuse diffuse scattering (e.g. Figure 6.11) and lattice softening (Sass 1972).

(2) The � → � transformation cannot be suppressed by quenching. In the�-quenched alloys the resultant �-phase always remains finely divided typ-ically less than 5 nm in diameter. Generally the shape of these �-particlesis ellipsoidal. All the possible four variants (corresponding to four <111>�

directions) of �-precipitates remain equally distributed in the �-matrix.High-resolution electron microscopy (HREM) has confirmed the presence ofall variants.

(3) The �-particles grow on ageing and during their growth, they gradually assumea cuboidal morphology with the cube faces parallel to {100} planes. Isothermal




ageing can also produce ellipsoidal �-phase particles in concentrated alloys.The latter process is usually preceded by phase separation in the bcc �-phase,resulting in solute rich and solute depleted �-regions.

(4) Under application of shock deformation, �-plates have been produced (Hsiungand Lassila 2000, Dey et al. 2004) in the �-phase. The product �-phaseassumes a plate shape and the transformation is reported to have similaritieswith the martensitic transformation.

(5) The �→ � transformation, resulting from thermal ageing, is accompanied bya reduction in the specific volume (typically about 5% (Hickman 1969)).

Based on the above experimental features, the resultant �-phase can be classifiedin the following categories:

(1) Fine ellipsoidal particles athermally produced on quenching. Morphology ofthe particles suggests that there is no volume change associated with the �→�transformation.

(2) Cuboidal or ellipsoidal �-particles (20–50 nm), produced during isothermalageing by a transformation resulting from diffusional alloy partitioning fol-lowed by lattice collapse. The transformation is accompanied by a volumechange.

(3) Plate shaped resulting from the shock pressure induced �→� transformation.

The lattice correspondence between the �- and the �-structures, as described inSection 6.3.2, clearly shows that the �-structure can be created from the �-structureby collapsing a pair of adjacent (222)� planes (e.g. 1 and 2 in Figure 6.7(b)),leaving the “0” and “3” planes undisplaced. This results in a transition from theABCABC � � � stacking of the bcc structure (in Figure 6.7(a), these three layers aredepicted as “0 1 2”.) into the AB′ AB′ stacking of the �-structure (shown as “0 1.53” in Figure 6.7(b)), where B′ planes correspond to the collapsed position locatedmidway between B and C planes. A view along the [111]� direction shows how thethreefold rotation symmetry changes to a sixfold symmetry where the collapse iscomplete (Figure 6.7(c)). The ordered sequence of the displacement of the {111}�type planes (Figure 6.7(d)) which causes the �→� transition can be represented bya displacement wave with wavelength �= 3 d222, the corresponding wave vector,K�, being equal to 2/3<111>∗. The development of the longitudinal displacementwave (Figure 6.7(d)) along the [111] direction can, therefore, formally describethe transformation. The merit of this description lies in the fact that it can beapplied to partial collapse situations by varying the amplitude of the displacementwave, the amplitude being directly related to the order parameter, �. The �→ �




transition in Ti and Zr alloys which occurs on quenching from the �-phase fieldhas been found to be associated with the following attributes:

(a) athermal character;(b) presence of pretransition diffuse intensity associated with structural fluctua-

tions, the diffuse intensity peaks being slightly away from the perfect �-peaks;(c) stability of a two phase microstructure comprising fine particles of the �-phase

distributed in a �-phase matrix immediately below the Ms(�) temperature.The following mechanism of the athermal � → � transition, based on thework of de Fontaine (1970), de Fontaine et al. (1971) and Cook (1973), takesthese essential features into account while developing a Landau theory ofdisplacement ordering for this transition.

As shown in Figure 6.7(d), the �→ � transition can be described in terms ofthe introduction of a displacement wave with the wave vector K� = 2/3<111>∗

in the bcc structure. A planar lattice model, as depicted in Figure 6.18, is very

(222) planesi = 0 1 2 3 4 5 6

Anti-ω

bcc [111](a)

(b)

(c)

+

–

0 1 2 3 4 5 6

Up

Up

+

–

0 1 2 3 4 5 6

ω

(d)

(e)

(f)

Figure 6.18. Two possibilities of the collapse of the {222}� planes are shown: (a) representsarrangement of the {222} planes, viewed on edge, in the bcc lattice. One possibility of collapse isshown in (b), where a two-plane collapse structure is shown (planes 1 and 2, 4 and 5), while in(c) a three-plane collapse structure is shown (marked as 2, 3, 4). The two-plane collapse producesthe �-structure whereas the three-plane collapse produces an anti-� structure. (d) A collapse regionin the bcc lattice showing an �-region marked by an ellipse. This schematic provides a cluefor the ellipsoidal morphology of the �-phase. (e) and (f) are the wave representations of the�- and anti-� structures. The final positions of the planes are shown by broken lines.




useful in visualizing the process. It can be seen from the illustrations in Figure 6.18that the �-structure is produced by the �-wave while a wave with an equal butnegative amplitude (hence, negative �), represented in Figure 6.18(f), producesthe anti-omega (−�) structure in which both B and C layers are displaced towardsthe A layer. Needless to say, the anti-� configuration in which three planes comeclose together due to the collapse mechanism will be energetically unstable. Thefact that the anti-� configuration is not equivalent to the � configuration bothin structure and in free energy suggests that the �→ � transition cannot be of asecond (or higher) order. Cook (1973) has put forward this symmetry argumentto justify the attribution of a first-order character to this transition which has beenexperimentally vindicated by the observation of the occurrence of the two phase�+� structure. Considering the planar lattice, the free energy change associatedwith the lattice collapse can be expressed in the form

�G= 1

2�′ijuiuj +

13�′ijkuiujuk+

14�′ijkluiujukul� � �+ (6.6)

where �′ij , the interplanar coupling parameters, are defined by the appropriate

derivatives of the free energy with respect to the planar displacements which aredenoted by ui, uj � � � and are respectively normal to the planes i, j � � � , numberedin sequence from the original at i= 1. The coupling parameters in Eq. (6.6) are notindependent but are linked by the energy translational and rotational invariancerelations (Leibfried and Ludwig 1961).

Substituting into Eq. (6.6) the deformation

u1 = 0� u2 = −u3 = �a�√

3/12� ui+3 = ui (6.7)

where a� is the lattice parameter and � the Landau order parameter for thedisplacement ordering in the � → � transition, the free energy change can beexpressed after rearrangements in terms of the standard Landau expression

�G= A�2 +B�3 +C�4 (6.8)

Schematic free energy versus order parameter plots (Figure 6.19) can show therelative stabilities of the �- and the �-structures at different temperatures. As isexpected in a first-order transition, the stability of the �-structure is brought aboutby a negative value of the third-order constant (B) which is responsible for loweringthe free energy below zero at T < To(�). With reduction in temperature, themagnitude of the third-order constant (B) increases and as temperature approachesTm, a metastable state appears at a positive �. With further reduction in temperature,the potential well of the metastable state deepens and the minimum appears




To(ω) < T < Tm

η

T < To(ω)

T > Tm

T > Tm

ΔF

T = To(ω)

Figure 6.19. Schematic presentation of the free energy of two (� and �) phases as a function oforder parameter. The asymmetric nature of the free energy curves represents the first-order nature ofthe �-phase transformation. At temperature Tm, a local minimum free energy appears which deepensas temperature of the system reduces in the temperature range Tm>T>To��. In this temperaturerange the system temporarily attains the �-structure with life time larger than the inverse of vibrationfrequency. For temperatures lower than To(�), the �-phase is more stable than the bcc phase.

towards the � value corresponding to ideal �-structure as shown by dotted line inFigure 6.19. As temperature reaches below Tm, a true minimum in free energy isattained and the �-structure becomes stable with respect to the bcc structure. Theminima correspond to the metastable states in the temperature range T�<T<Tmare described in literature as �-like fluctuations. These minima allow the systemto locally attain the structure corresponding to the energy minima and remainthere for a duration longer than the time duration prescribed by the vibrationalfrequency of the atoms. At temperatures higher than To(�), a population of suchlocal fluctuations, which will be present in a metastable equilibrium, is expectedto contribute towards the diffuse intensity distribution associated with �-formingsystems at temperatures in the range of Tm > T > To(�).

The strongly anharmonic behaviour of Zr with respect to the displacement order-ing with the wave vector 2/3 <111>∗ has been demonstrated by first principlescalculations of Ho and Harmon (1990). This point is discussed in the follow-ing section. The same conclusion can be arrived at from the fact that the K�




wave, acting alone, can produce a non-vanishing third-order term, as shown bythe summation rule: K�+K�+K� = �222�, a reciprocal lattice vector.

6.6.2 Formation of the plate-shaped � induced by shock pressurein �-alloys

The lattice collapse mechanism of the �→ � transformation is pure shuffle trans-formation and does not envisage any macroscopic shape strain. When such amechanism is operative, the product and the parent phases can maintain completecoherency along the entire interface. For such a transformation the strain energyminimization by a proper choice of the product morphology is not important.However, when a macroscopic shape strain is superimposed on the �→ � trans-formation, choice of the interface is restricted by the criteria of minimization ofthe interfacial energy and the strain energy associated with the transformation. Forexample, in the martensitic transformation, the strain energy contribution domi-nates and the invariant plane strain (IPS) condition is satisfied, the surface energyterm dictates the selection of habit planes. In several diffusional transformationsalso the invariant line (IL) vector, lying along the habit plane, which definesthe primary growth direction, is essentially due to the minimization of the strainenergy associated with the transformation. In the similar fashion, if there is a vol-ume change involved during the �→ � transformation and if the external stressinvolved has a shear component, the choice of the interface is not completelyrandom. In fact, it has been shown under these circumstances the product �-phasehas plate-shape morphology and habit plane between the �- and the �-phases canbe predicted using phenomenological theory of martensite (Dey et al. 2004). Usingthe phenomenological theory of martensitic transformation, Dey et al. (2004) couldderive the Bain strain and also the macroscopic shape strain matrix. The predictedhabit plane was very close to the experimentally determined (121) habit plane(Figure 6.20(a)–(d)). The observation of {112}� planes as the habit planes of the�-phase is suggestive of the fact that the �→� transformation involves a shearingof the �-lattice along a {112}� plane. A bcc structure can be described by a six-layer packing sequence of {112} planes, as shown in Figure 6.21. Hatt and Roberts(1960) have examined the possibility of generating the �-structure by gliding an{112} plane and have shown that a suitable sequence of glides can indeed producethe � → � transformation. Figure 6.21 shows that a macroscopic shear on an{112} plane along an <111> direction, superimposed with atomic shuffles, canproduce the �-structure. Since the macroscopic deformation is a simple shear, theinvariant plane, which is the contact plane between the two phases, is the shearplane itself. In this case, the lattice invariant shear operates for the macroscopicstrain to satisfy the invariant plane strain condition. It is possible to define a homo-geneous shear on the (112) plane with a magnitude of cot/2 = 0�767, where




Figure 6.20. (a) SAD pattern from a Zr–20% Nb alloy showing the presence of the �-phase. Theoccurrence of diffuse intensity indicates the tendency for the �-phase transformation, (b) plate-shaped morphology of the �-phase, (c) interface between a plate of the �-phase and the �-matrixis lying along a <112> direction and (d) composite SAD pattern showing the presence of a single�-variant.

is defined as the angle between the [112] and the [110] directions. The [110]direction (�2) remains undistorted as the result of application of this shear, whichis applied in such a way that the [112] direction is perpendicular to the plane ofshear. Using the well-known geometrical relation of homogeneous deformation,which is very commonly used in the case of deformation twinning, the magni-tude of shear can be determined. The homogeneous deformation comprising theshear part alone produces a lattice, which is indistinguishable from the �-parentlattice. When atomic shuffles similar to those shown in Figure 6.21 are superim-posed on the homogeneous deformation, the �-structure is generated. The lattice




64 53210

Atomic plane no.

Dis

plac

emen

t

[111]

(111

) β⎥⎪

(000

1)ω

[112

]

[111

][110]

[001

]

[112

]β

βω

[111]

Figure 6.21. Planar stacking of the {112} planes of the bcc lattice (shown by open circles). If theshuffling of the atoms is superimposed during the shearing of the {112} planes, the �-structure(shown by the filled circles) is produced. The orientation relationship between the two structuresremain same as shown in Figure 6.7. The two lattices have also been shown. In order to bring outthe equivalence between the shear model and the wave model, the displacement wave has also beenshown in the inset.

correspondence between the �- and �-lattice remains the same as that encounteredwhen a longitudinal displacement wave is introduced in the �-lattice. This equi-valence, generated by (i) a combination of shear and shuffle and (ii) a longitudinaldisplacement wave, can be appreciated from Figure 6.21. Since any specific {112}plane contains only one <111>� direction, a single orientational variant of the�-structure can be produced within a single {112} plate. In short, the deformationproduced by a shock wave in the bcc lattice develops a tendency for the shear ofthe {112} planes where the formation of the �-structure occurs when the shuffleprocess is superimposed on the shear deformation. The axis of the shear directiondecides the choice of the variant for the <111> direction contained in the specific{112} plane and, therefore, plate-shaped single variants of the �-phase emerge inthe shock deformed �-phase.

6.6.3 Calculated total energy as a function of displacementIt was believed earlier that the �-phase occurred exclusively in Ti, Zr and Hfalloys. However, more recently �-structures have been reported in other systemsas well (de Fontaine, 1988). Nevertheless, these Group 4 metals are the only pure




metals in which the �-phase is observed. This suggests that the stability of the�-phase is directly related to the electronic structure of these Group 4 transitionmetals, all three of which exhibit an allotropic bcc → hcp transition. The fact thatthe collapsed plane in the �-structure has the same two-dimensional structure asthe basal plane of graphite (Group 14) is also indicative of the inherent stabilityof the collapsed plane configuration for these systems with some similarity in theelectronic structure.

Ho et al. (1984) have evaluated the total energy as a function of the displacementcorresponding to the longitudinal 2/3 <111> phonon for Mo, Nb and bcc Zron the basis of first principles calculations. The basic procedure for using totalenergy calculations to study a particular phonon mode includes first principles bandstructure calculations taking as inputs the atomic number, the crystal structure,the lattice parameter and the atomic displacement corresponding to a particular“frozen phonon”. This involves the assumption of the adiabatic condition whichimplies that the motion of ions is so much slower than the motion of electronsthat at each instant of a given ionic displacement the electrons are in the groundstate defined by the instantaneous ionic positions. The large difference betweenthe electronic and the ionic masses makes this assumption quite valid. To study aparticular phonon the ionic positions are fixed at various displacements from theirequilibrium positions in the direction corresponding to the phonon eigenvector(polarization). A fully self-consistent band structure calculation has been performedby Ho et al. (1984) for each frozen-in condition and the total potential energyhas been determined, combining the electronic energy and the ion–ion coulombenergy. The total energy, expressed in terms of the displacement, has been obtainednot only for small displacements where harmonic contributions predominate butalso for large displacements which correspond to structural transitions.

Experimentally determined phonon dispersion curves (Figure 6.22(a)) for the<111> longitudinal branch for three bcc metals, Mo, Nb and Zr (high temperaturebcc phase) have been compared (Stassis et al. 1978). Though these elements areneighbours in the periodic table, their phonon frequencies are remarkably different.The pronounced dip seen near 2/3 <111> for Zr is close to that required forinducing the � → � transition. Frozen phonon calculations for this mode haveshown that the total energy versus displacement curves (Figure 6.22(b)) for Moand Nb are close to quadratic, implying an essentially harmonic behaviour. Thecurve for Zr is strongly anharmonic and the minimum in the energy occurs whentwo of the (111)� planes collapse together to form the �-phase. It is seen thatthe ground state energy of bcc Zr rises with displacement on the positive-� sideand after reaching a maximum drops down to a lower minimum at a displacementof 0.5 corresponding to the ideal �-structure, while on the negative displacement(anti-omega structure) side the energy rises very sharply. The results of the first




(a)

8

7

6

5

4

3

2

0

1

0.2 0.4 0.6 0.8

ξ

ν (T

Hz

)L [111]

kωkm

(b)

–0.2 0 0.5

0.010

0

Displacement of (222) planes

Ene

rgy

(Ry/

atom

)

Mo Nb

Nearlyharmonic

Stronglyanharmonic

Zr

bcc

Ideal ω

Figure 6.22. (a) Phonon dispersion curves for Mo, Nb and bcc Zr showing pronounced dip for Zr atk = k�. (b) Calculated total energy as a function of displacement corresponding to the longitudinal(2/3, 2/3, 2/3) phonon.

principles total energy calculations, therefore, are consistent with the Landau plotsfor the � → � transition which have been discussed in the preceding section.These calculations also reveal the influence of large atomic displacements on thelocal charge density. For Zr it has been shown that a transfer of d-like chargefrom uncollapsed (A type) planes to s-p like charge in collapsed (B and C type)planes occurs which is unlike the calculated behaviour of charges in Nb andMo. Thus s-p like bonding tends to develop in the collapsed planes of Zr (andother Group 4 metals), mimicking the bonding of atoms in the basal plane ofgraphite.

The calculated total energy, as a function of displacement, shows that the bccphase is unstable with respect to the �→ � transformation. However, in pure Tiand Zr this transformation cannot be induced without the application of hydrostaticpressure. This is due to the relative stability of the hcp �-phase in these systems.The phonon description of the bcc → hcp transition has been given earlier inChapter 3. The addition of bcc alloying elements (V, Mo, Nb, Cr, Fe, etc.) to Tiand Zr promote the formation of the athermal �-phase. This is incidental becausethese alloying elements actually destabilize the �-phase with respect to the �-phasebut apparently they do so not as strongly as they destabilize the �-phase withrespect to the �-phase. This point is discussed in Section 6.6.4 where �-phasestability is rationalized in terms of d-band occupancy by electrons.




6.6.4 Incommensurate �-structuresThe lattice collapse mechanism described earlier can account for the first-ordernature and the athermal displacement ordering character of the �→ � transition.Landau plots (�G versus �) at different temperatures (Figure 6.19) can alsoprovide a description of the increase in the incidence of interplanar collapsewith decreasing temperature and the presence of pretransition effects. The perfect�-structure, however, cannot explain why the diffuse peak positions are not locatedexactly at the �-reflections and why a fine dispersion of �-particles in the �-matrixinvariably forms in the as-quenched condition. There have been some approachesinvolving incommensuration of the �-structure which attempt to explain the offsetof the diffuse peaks and the duplex fine particle (�+�) morphology.

Cook (1975) has argued that the offset of the diffuse intensity peaks has itsorigin in the modulation of the “ideal-�” structure by a wave with wave vector,q = Km − K�, where Km, as defined earlier, is the wave vector associated withthe maximum of diffuse intensity. He has used a formalism, as outlined here, inwhich the Fourier transform (denoted by K-representation) of the free energy isconsidered. The displacements, ui, are substituted by a Fourier series:

ui =∑K

U�K� exp�iK�xi� (6.9)

in which xi is the distance from the origin to the ith plane of the undistorted planarlattice and U�K� is the Fourier amplitude given by

U�K�= �1/N�N∑i=1

ui exp�−iK�xi� (6.10)

where N is the total number of planes and K is an allowed wave vector in the firstBrillouin zone. The expression for the free energy change, on the substitution ofEq. (6.10) in Eq. (6.7), becomes

�G= N�F2 +F3 +F4 + � � � � (6.11)

where F2, the purely harmonic portion of the free energy, and the third orderanharmonic term F3 are respectively given by

F2 = 12

∑K

�2�K��U�K��2 (6.12)

F3 = 13!

∑K�K′�K′′

�3�K′�K′′�U�K�U�K′��K +K′ +K′′� g� (6.13)




In these expressions, �2(K) and �3(K) are appropriate K-dependent second-and third-order coefficients, which are essentially the Fourier transforms of thesecond-order and the third-order interplanar coupling parameters: the term ��K+K′ +K′′� g� is a delta function which is zero unless the sum (K+K′ +K′′) isequal to a reciprocal lattice vector, g, along an <111> direction. The complexamplitudes U are proportional (in modulus) to the displacements of the atomicplanes which are essentially Fourier transforms of the second-order and third-orderinterplanar coupling parameters:

�2�K�=∑j

�i�i+j exp�iK�xj� (6.14)

�3�K�=∑j

∑l

�i�i+j�j+l exp�iK�xj��iK�xl� (6.15)

For the trivial solution of zero amplitudes for all K, as is the case for the bccstructure, the free energy change �G, assumes the minimum of value zero. If thesystem behaves in an essentially harmonic fashion (i.e. the contributions of F3,F4, etc. are negligible), there is no other minimum in �G as an increase in theamplitude raises the value of the free energy change parabolically. Body-centredcubic metals such as those belonging to Groups 5 and 6 (e.g. Nb and Mo) exhibitsuch harmonic behaviour (Figure 6.22).

The first principles electronic structure calculations for bcc Zr, discussed inSection 6.6.3, clearly demonstrate a strongly anharmonic behaviour, as revealedby the presence of a deep second minimum away from the zero amplitude firstminimum (Figure 6.22). A similar behaviour is also expected in the cases ofTi and Hf. As mentioned earlier, the harmonic contribution (arising out of thesecond-order force constant, �2, to the free energy change) is minimized whenthe wave vector is Km while the anharmonic contribution (arising from the third-order term) can further reduce the free energy change through interactions betweendisplacement waves of different wave vectors lying in between Km and K�. Thefluctuation thus created results from a competition between the second- and thethird-order force constants.

Cook (1975) has demonstrated this fluctuation mechanism by considering thegrowth of athermal displacement modulation represented by a sinusoidal standingwave

ui = A sin��K�+q�xi�= A cosqxi sin K�xi +A sin qxi cos K�xi (6.16)

with the amplitude A being related to the Landau order parameter, �, by therelation

A= �a�/6 (6.17)




When qxi is small, the first term gives the ideal �-structure. As xi increasesto a value of �/q, the anti-� structure develops. The presence of a spectrum ofwavelengths from K� to Km (= K� + q) will generate wave packets of �- andanti-� structures as shown in Figure 6.23(a). The anti-� structure, however, is

(a)

Long

itudi

nal

disp

lace

men

ts

1 2 3 4

Distance in (111) interplanar units

5 6 7 8

9 10

+

0

–

(b)

2π/Δk

ω ω– ω

(e)(d)(c)

(β + ω )Dual structure

G

AAωA∗

Km K = Km

K = Km

Kω K = Kω

G

ΔGω < ΔGω + β

ΔGω + βΔGω

A

T << To(ω)

G

A

δΔGω,βK = Km

K = Kω

T < To(ω )

Kω < K < Km

Figure 6.23. (a) The accumulated phase difference caused by the difference between the Km wavevector and K� wave vector. The positive displacement forms the �-structure, whereas the negativedisplacement forms the anti-� structure. (b) Flipping of phonon causes the region of anti-� to getconverted into one of the variants of �. (c) Represents a situation where temperature is above To(�).This plot can be taken as the reciprocal space representation of Figure 6.19. A local minimum infree energy allows the system to attain a dual phase structure as it is always energetically morefavourable in comparison to a completely single phase (shown by a line joining the two minima).(d) For temperatures lower than To(�), the �-phase becomes more stable than the bcc phase but forintermediate positions of the amplitude, a dual phase structure is again more stable than a singlephase. (e) For temperatures far lower than To(�), �-phase is more stable with respect to either ofthe phases. However, at these temperatures very low diffusion rates do not allow transformation toreach completion.




not expected to survive due to the high energy associated with it. This causesa “phonon flipping” which converts the anti-� regions into �-regions, resultingin the formation of a row of �-wave packets (Figure 6.23(b)). The spacing d�,between the adjacent �-particles (or wave packets) will, therefore, be

d� = �/q (6.18)

Since the longitudinal <111>� phonon is responsible for the formation of thisstructure, �-particles are lined-up along <111>� directions. Substituting the qvalue, the spacing of �-particles can be determined to be about 2.5 nm which isin close agreement with the interparticle spacing experimentally determined byDawson and Sass (1970).

As mentioned earlier, the athermal �→ � transformation has never been foundto reach completion. Invariably athermal �-particles remain distributed in the �-matrix in the quenched product. Schematic Landau plots can be used for explainingthe stability of this dual phase �+� structure. A possible explanation for suchincomplete transformation of the �-phase to the �-phase can be given using areciprocal space (k-space) representation of free energy as a function of amplitudesand wave vectors associated with the dual phase (�+�).

As discussed earlier, the transformation associated with Km leads to second-order phase transformation and will, therefore, correspond to a minimum harmonicenergy. The plot corresponding to Km is drawn with a curvature smaller thanthat for the K� vector in the vicinity of the origin. At larger amplitudes dueto the influence of third-order term, the free energy of the K� vector becomessmaller. The wave vector associated with dual phase having a metastable statelying between origin (representing bcc structure) and A� (amplitude associatedwith the �-structure) will have lower energy than the single phase as shown inFigure 6.23(c). At extremely low temperatures (Figure 6.23(e)), thermodynami-cally single phase � become more stable in comparison to the dual phase structure.However, at such low temperatures due to kinetic reasons, completion of phasetransformation appears difficult. At intermediate temperatures, that is, at lowersuper cooling, T < To(�) (Figure 6.23(d)), it can be shown that two phase �+�structure corresponds to a lower value of free energy in comparison with that of a�-structure.

Sanchez and de Fontaine (1977) have proposed a different model for explainingthe appearance of curved diffuse intensity streaks characteristic of the � → �transformation. By considering the transformation of the observed diffuse intensitydistribution they have calculated a plausible atomic displacement field which isfound to have a transverse width of a few lattice parameters. The displacementpatterns in the longitudinal direction exhibit a variation in sign and magnitude along




a given <111>� atomic row. Displacements in adjacent rows also vary in a similarmanner but are out of phase. Incommensuration of these types is schematicallyshown in Figure 6.8 which shows that along a given <111>� direction three �-subvariants, designated as �1, �2 and �3, corresponding to the location of theuncollapsed plane on A, B or C atomic layers, respectively, can form.

The basis of the various incommensuration models proposed lies in the natureof the stacking of different subvariants (Borie et al. 1973, Kuan and Sass 1976,Pynn 1978, Horovitz et al. 1978). Borie et al. (1973) have considered a specificsequence of subvariants, namely, �1, �3, �2, �1� � � , to be present along a given<111>� direction. This sequence, which corresponds to a phase shift of 4�/3 in thedisplacement wave from one subvariant to another, can correctly predict the shiftin the diffuse intensity. However, this model does not explain the diffuse intensityobserved near the bcc positions, the presence of the untransformed �-phase andthe reason for the occurrence of the particular ordered sequence of subvariants.The model of Kuan and Sass (1976) assumes the presence of a vacancy whichintroduces a phase shift from one variant to the other. This model suffers from thedrawback that the predicted shift for the (2/3 2/3 2/3)� reflection is opposite tothat observed experimentally. Horovitz et al. (1978) have proposed the presenceof a small �-phase region between successive �-phase variants. These �-layerscan be considered as solution walls which bring about the phase shift from one�-variant to the next. Such discommensurations are quite common in models oflong period structures. In this model the variation in the lattice parameters ofthe �- and the �-phases has also been considered and the peak shift has beenattributed to the phase shift as well as the density shift, the latter arising fromthe small variation in the lattice dimensions of �- and �-regions. If the differ-ence in the lattice constants is very small, the �-cluster is completely locked intothe �-matrix and a commensurate structure is produced as is the case for alloyshaving Ms(�) temperatures above ∼573 K. When the mismatch is large, solutionappears in the ground state, partially unlocking the cluster from the �-matrix. Sucha situation is encountered in alloys with Ms(�) temperatures close to room tem-perature and as a consequence the reflections move away from the commensuratepositions.

HREM has recently been employed for studying the commensurate and incom-mensurate structural configurations in �-phase forming alloys. The �-regions inthe �-matrix can be easily recognised in HREM images as the members of an atompair which is a pair of atoms that are pushed into the collapsed plane come too closeto be resolved in the image along an <110>� direction and they jointly produce arelatively bright dot contrast. How the configuration of the �-regions develop inthe �-matrix is schematically shown in Figure 6.24(a) using the structural relation-ship between the �- and the �-phases. Schryvers and Tanner (1990) have shown




0.285 nm

0.42

5 nm

Z = 0

Z = 1/2

[112]

[001]

[110]

[110][111]

β ω

0.425 nm

Figure 6.24. (a) Schematic presentation of formation of �-lattice from the �-lattice through thecollapse mechanism and (b) HREM image showing the above two phases.

that commensurate �-domains, formed in an aged Ti–8% Mo alloy indeed exhibitbright dots arranged in a rectangular lattice, the short dot spacing (0.285 nm) beingalong either the [111] or the [111] direction and the large dot spacing (0.425 nm)being along [112] or [112], respectively. Tilting experiments have shown that thedifferent variants of commensurate � fill in the entire volume without leaving any




�-region untransformed. Figure 6.24(b) shows a high-resolution image obtainedfrom the Zr–Nb alloy. The presence of the �-region could easily be identified fromthe matrix �-regions. For imaging the structure of incommensurate �-regions inthe �-matrix, as quenched samples of Ti–8 Mo and Ti–15% Mo alloys have beenexamined by Schryvers and Tanner (1990). HREM images of the �-phase whenviewed along [110] direction have shown presence of strings of dots along a linevector slightly deviated from the K�111�. Figure 6.25(a) shows the arrangement ofatoms in regions where �-like defects are observed in the bcc lattice. As could benoticed from the figure, because of such arrangements the line joining the ellipticaldots, representing incommensurate �, form a vector deviating from the <111>direction of bcc. Figure 6.25(b) is a high-resolution image from a region showingdiffuse intensity pattern (shown as inset in the figure) where series of dots canobserved. Joining these dots forms a line which is nearly 10 degree away fromthe <111> direction of the bcc.

(a)

(b)

(110)

γ →α ←β →γ ←

α β γ α[111]

[112]ki

k

10°

[111](c) (d)

0.23 nm

k111

Figure 6.25. (a) Incommensurate �-structure produced by partial collapse of row of atoms causingan angular deviation of 10� from k = �111� of the bcc lattice and (b) high-resolution image takenfrom the [110] zone axis showing short strings of white dots of incommensurate �.




6.6.5 Stability of �-phase and d-band occupancyThe occurrence and stability of the �-phase have initially been explained in termsof the concept of the Fermi surface touching the Brillouin zone (BZ) boundary inthe reciprocal space by treating both sp and d electrons within the framework of thenearly free electron approximation. These arguments are essentially similar to theHume-Rothery rules and are based on the fact that it is costly in terms of energyto add further electrons to an alloy, once the filled states reach the zone boundary.Then the crystal makes a phase change to another crystal structure in whichmore electrons can be filled before the Fermi surface touches the BZ boundary.Luke et al. (1968) have assigned an electron/atom (e/a) ratio of 4.06–4.14 for theoccurrence of the �-phase from experimental data. However, as pointed out byHickman (1969), the range of stability of the �-phase in alloys cannot be predictedfrom the e/a ratio alone. The position of the alloying element in the periodictable is also an important factor. It is to be emphasized that Fermi surfaces intransition metals are far from spherical and that the electronic density of statesdeparts considerably from that given by the free electron theory because of thed-band and the s-d hybridization effects. Therefore, the stability of the �-phase inTi and Zr base alloys cannot be predicted accurately from the free electron theory.

The importance of the d-band occupancy in determining the equilibrium crystalstructure of transition metals has been pointed out by Friedel (1969). Pettifor(1977) has shown that the number of sp electrons remains essentially constant(∼1 e/a) across the transition metal series and it is only the d-band occupancythat changes from one element to the other. It may be noted that the equilibriumcrystal structures of transition metals show the definitive series hcp → bcc → hcp→ fcc as the d-shell is progressively filled with electrons with the exception ofthe magnetic elements Mn, Fe and Co and of La, the element at the beginning ofthe rare-earth series. This suggests that the cohesive energy of transition metals isprimarily governed by the bonding contribution of the d-band (U bond

d ).The equilibrium and the metastable phase diagrams which delineate the com-

position ranges of stability of �-, �- and �-phases in Zr or Ti base alloys areshown in Figure 6.1. Hypothetical free energy-composition diagrams (Figure 6.1)can be converted into free energy versus d-band occupancy plots. With increasingadditions of �-stabilizing elements (V, Nb, Mo and Ta) the d-band gets progres-sively filled, as shown in Figure 6.26. The metastable �-phase appears in a certaincomposition range which is defined by the cross-over points, �N 1

d and �N 2d where

the free energy curves for the �- and the �-phases, respectively, intersect that forthe �-phase. The minimum in the free energy for the �-phase is located at �No

d .This construction reiterates the point that for a composition range correspondingto �N 1

d to �N 2d , the �-phase is more stable than either the �- or the �-phases




β

ωα

ΔN1dA B

α – ωΔFo

Free

ene

rgy

ΔNd (Change in d-band occupancy)ΔN

0d ΔN

2d

Figure 6.26. Energy curve as function of d-band occupancy. The �-phase is more stable in com-parison with �- and �-phases for electron concentration in d-band falling between �N1

d and �N2d.

individually but is metastable with respect to the equilibrium mixture of the �- andthe �-phases.

Sikka et al. (1982) have computed the �Nod values for different �-forming

alloys from the experimentally obtained compositions of the aged �-phase inthese alloys (Table 6.6). It may be noted that the athermal �→ � transformationis thermodynamically favourable in the range �N 1

d –�N 2d where the drop in free

energy is the maximum for the composition invariant �→ � transformation. Thefact that the � → � transformation occurs under high pressures implies that thefree energy gap, �G�−�

o , as indicated in Figure 6.1 is closed by the application ofpressure. This can be explained from the fact that the elements belonging to thefirst half of the transition metal series show large transfer of electrons from thes-band to the d-band on application of pressure. Physically this can be attributed to

Table 6.6. The composition of the �-phase in the pseudo-equilibrium state during ageing and thecorresponding equivalent change in d-band occupancy for various �-forming alloy systems.

Alloy Alloying content in aged � (at.%) �Nod in aged � (electron/atom)

Ti–V 13.8±0.3 0.138±0.003Ti–Cr 6.6±0.2 0.130±0.004Ti–Mn 5.1±0.2 0.153±0.006Ti–Fe 4.3±0.2 0.172±0.008Ti–Nb 9±2 0.090±0.020Ti–Mo 4.3±0.4 0.086±0.008Zr–Nb 10–11 0.100–0.110




the large sp-core in the case of transition metals. Under compression the bottom ofthe s-band (Bs) shows a rapid rise because of its increased kinetic energy as theseelectrons are pushed into the core region where they are repelled by orthogonalityeffects. This increase in Bs reduces the spacing between the s-and the d-bands andthereby gives rise to an increase in the d-band occupancy (Nd). Vohra (1979) hascalculated the extent of the s → d transfer (�Nc

d ) at the transition pressures forTi, Zr and Hf. The calculated values, listed in Table 6.6, show that the applicationof 2.0 GPa pressure on Ti will lead to the transfer of 0.0246 e/a from the s- tothe d-band. This value is not adequate for meeting the requirement of the �→ �transformation as indicated by the �No

d values for different alloys (Table 6.6).

6.7 ORDERED �-STRUCTURES

The displacive transformation mechanism of the �→ � transformation has beendiscussed in the preceding sections of this chapter. It has also been argued thatthe transformation can be viewed as a displacement ordering process which canbe accomplished by the introduction of a periodic displacive wave in the bccstructure. This section is devoted to transformations which involve a combinationof both displacive and replacive ordering. The possible superlattice structureswhich can evolve from the bcc structure are described in Chapter 5. We shall nowexamine the mechanism of the formation of intermetallic phases which can bebest described as chemically ordered �-structures.

6.7.1 Structural descriptionsThe structures of some equilibrium and metastable intermetallic phases are suchthat they can be described as chemically ordered �-phases. The fact that thelattice correspondences between these structures on the one hand and the parentbcc structure on the other, are essentially the same as that between the �- andthe �-structures also validates this description. The possibility of the involvementof the lattice collapse mechanism in the formation of such chemically orderedintermetallic structures has been examined purely from the consideration of thelattice relationships between the parent bcc and the intermetallic structures.

The B82 structure (hexagonal, P63/mmc) of the equilibrium Zr2Al, Ti2Al andZr2Sn phases and of some ternary phases is illustrated in Figure 6.27 in whichthe stacking sequence of this structure is compared with that of the bcc structure.It is clear from this figure that the �→ B82 structural transition would require acombination of the collapse of the layers (1, 2; 4, 5; etc.) and a chemical ordering.The fact that such a lattice correspondence indeed exists in several real systems




ω′ ω″

(a) (b) (d)(c)

6543210

a2

c

a1

B2

1a

2d1

2d2

1a, 2d2

1b, 2d1

P3m1P3m1Pm3m

1b1a1b

Z2 Z1

2a

B82

P63/mmc

2c

2d

Figure 6.27. Formation of the ordered �-structures. (a) The (222) planar stacking of the orderedbcc structure. (b) Onset of the collapse structure producing a lower symmetric phase with the samenumber of Wyckoff points. The structure is named as �′. (c) A structure (�′′ similar to the �′-structure with a higher number of Wyckoff points. (d) The B82 structure is higher symmetry structureattained only after the collapse is complete in the bcc lattice.

has been demonstrated (Banerjee and Cahn 1983, Strychor et al. 1988). Basedon this correspondence the lattice parameters (a and c) of the B82 phase can beexpressed in terms of the lattice parameter, a�, of the �-phase as:

a�B82�= √2a�

c�B82�= 6d222 = √3a�

Substituting the extrapolated values of a� for Zr–33 at.% Al and for Ti–33 at.%Al, the lattice parameters of Zr2Al and Ti2Al work out to be the following:

For Zr2Al : a = 0.4852 nm; c = 0.5942 nmFor Ti2Al : a = 0.4580(3) nm; c = 0.5520(4) nm

While examining the B2→ B82 transformation in ternary Ti–Al–Nb alloys,Bendersky et al. (1990a,b) have encountered a metastable intermediate phase, �′′,which is of a lower symmetry than either of the initial B2 and the final B82

structure. The �′′-phase is characterized by partially collapsed (111)� planes andis reordered relative to the B2 phase. The occurrence of a partially collapsedstructure essentially indicates a strong coupling between the chemical (replacive)and the displacive ordering processes. If the starting structure is B2, as described in




Figure 6.27(a) with the atomic layering sequence A-B-A-B-A-B-A the �-collapseoperation produces a sequence A-B/A-B-A/B-A. Such a structure is designated �′

(Bendersky et al. 1990a,b). The hexagonal c-axis of the ordered �′-structure istwice as long as that of the disordered �-phase.

A diffusional B2→� transition yields a P3m1 crystal structure with four distinctWyckoff sites (1a, 1b, 2d1 and 2d2) but only two distinct site occupancies (as in theB2 structure) (Figure 6.27(b)). Thus the �′-phase is inherently unstable with respectto changes in chemical order that lead to four rather than two (Figure 6.27(c))distinct site occupancies because the inherited B2 occupancies are characteristic ofa higher symmetry than P3m1. To summarize, the �′-structure is the one which canform by the operation of a �-collapse mechanism on the B2 structure inheritingthe chemical order on the collapsed planes while in the �′′-structure the collapsedplanes do not have an ordered arrangement of atoms (Figure 6.27(d)).

The structure to be examined next is the D88 structure (prototype Mn5Si3). Theequilibrium Zr5Al3 and the metastable Ti5Al3 (Bendersky et al. 1990b) phasesexhibit this structure with a hexagonal unit cell which possesses 18 Wyckoffpositions (Table 6.7). The stacking sequence and the lattice correspondence in theD88 structure are compared with those in the B2 structure in Figure 6.28. Thisfigure also depicts the lattice correspondence between these two structures. Basedon this correspondence the lattice parameters of the D88-phase are related to thatof the �-phase (B2 structure) as

a�D88�= √6a�

c�D88�= 6d222 = √3a�

Table 6.7. Wyckoff positions, site occupancy, interplanar spacings and the relative peak intensitiesfor Zr5Al4 (Ga4T5 types) and Zr5Al3 (Mn5Si3) phases.

Prototype structure Wyckoff position/occupancy hkl d-spacing (nm)(�= 0�1544 nm)

Relativeintensity

Ga4Ti5 2(b) 0 0 0 Al 100 0.727 25(Zr5Al4) 4(d) 1/3 2/3 0 Zr 110 0.420 0

6(g1) x1 0 1/4 Zr 200 0.363 62.56(g2) x2 0 1/4 Al 111 0.343 62.5x1 = 0.23, x2 = 0.60 002 0.297 –

Mn5SI3 2(b) 0 0 0 vacancy 100 0.727 0(Zr5Al3) 4(d) 1/3 2/3 0 Zr 110 0.420 66

6(g1) x1 0 1/4 Zr 200 0.363 1006(g2) x2 0 1/4 Al 111 0.343 66x1 = 0.23, x2 = 0.60 002 0.297




[121]

[211]

[111]

(111)

B2 D88 – Zr5Al3

6

55,44

3

22,11

0

Zr atomAl atomVacancy

Layer No.[0001]

[1120]

(0001)

B2

B2

B2B2

D88

D88

[2110]D88

D88

Figure 6.28. Lattice correspondence between the ordered bcc (B2) and the D88 structures. Thecollapse of 222 planes of the bcc structure (designated as 1 and 2 or 4 and 5) at the intermediatepositions (designated as 2,1 or 5,4) would produce the D88 structure.

In the case of Zr5Al3, the substitution of the corresponding a� (= 0.3455 nm)values yields the a and c parameters of the D88 phase as 0.8464 nm and 0.5984 nm,respectively, which are fairly close to the experimentally obtained lattice parame-ters (a = 0.845 nm and c = 0.5902 nm) of the equilibrium Zr5Al3 phase. Figure 6.28shows that the D88 structure can also be generated from the B2 structure by acombination of lattice collapse and chemical ordering. The pairs of {222} planeswhich collapse in the � → B82 and in the � → D88 structure are essentiallysame. In case of the D88-structure an ordered array of vacancies is necessary inthose B2 planes which remain undisplaced during the collapse. This is essentialfor meeting the stoichiometric requirement. The difference between the B82 andthe D88 structures lies in whether the 2(b) positions are occupied by Zr atomsor are vacant (Figure 6.29). The fact that two vacancies for 18 atom positionsper unit cell are required for the formation of the D88 structure necessitates theretention of a high concentration of vacancies in the parent B2 structure and anordered arrangement of these vacancies. If the 2(b) positions are filled with Alatoms, a stoichiometry of Zr5Al4, having a Ga4Ti5 structure, is attained. Benderskyet al. (1990a,b) have reported such a structure in ternary Ti–Al–Nb alloys. Rather




0.497 nm

Zr atom

Al atom

1,2 Layer0 layer

aD = 0.845 nm

[112] // [1120]D88[121] // [1210]D88

0 1Cv →

D88

B82

⊥⊥

⊥⊥

[110] // [2110]B82[101] // [1210]B82

(111)

AtomVacancy

√3aB = 0.847 nm

Figure 6.29. A schematic representation of the basal planes of the D88 and the B82 structures inaccordance with their relationship with the bcc matrix. Difference between the two structures liesin the arrangement of the vacancies on the basal plane. The plan view of the collapsed plane isdrawn on scale at the bottom of the figure. As could be noticed from this figure, the atoms could beaccommodated without introducing any substantial strain.

recently the Zr5Al4 phase with a hexagonal structure has been included in thephase diagram (Massalski et al. 1986).

6.7.2 Transformation sequences in Zr base alloysPhase transformation sequences have been studied in binary Zr–Al (near Zr–25at.%Al) alloys (Banerjee and Cahn 1982, 1983, Banerjee et al. 1997) and in ternaryZr3Al–Nb alloys (Tewari et al. 1999). Rapid solidification was employed forprocessing the alloys used in these studies for suppressing the equilibrium phasereactions. A portion of the binary phase diagram of the Zr–Al system is shown




2000

1900

1700

1500

1300

1100

900

700

500

50%10%

40200

At.% Al

ZrA

l975

1350 →

β

α

L

X

Zr 3

Al

Zr 2

Al

Zr 5

Al 3

Zr 3

Al 2

Zr 4

Al 3

wt% Al

Tem

pera

ture

°C

Zr

Figure 6.30. Zr-rich side of the binary phase diagram of the Zr–Al system showing phase fields ofthe pertinent intermetallic phases.

in Figure 6.30 in which the vertical broken lines indicate the compositions of thealloys for which the transformation sequence is discussed in this section. In allthese alloys (Zr–25% Al, Zr–27% Al, Zr–Al–Nb) the rapidly quenched structureshows a retained supersaturated �-phase. This observation suggests that Al, whenpresent in Zr at a level of about 25%, tends to stabilize the bcc �-phase. The factthat the �-phase field extends considerably at the eutectic temperature (1623 K)supports this contention. It is interesting to note that in more dilute alloys (Alcontent less than about 10%) Al acts as an �-stabilizer and the �/� transitiontemperature increases with Al additions.

The retained �-phase in the aforementioned alloys in the quenched conditionexhibits a tendency towards phase separation, leading to the evolution of a spin-odally decomposed structure (Figure 6.31). The formation of the compositionallymodulated structure and the amplification of such modulation with ageing have




Figure 6.31. (a) Rapidly solidified Zr–Al and Zr–Al–Nb alloys showing modulations along <100>�

directions on ageing at 723 K for 20 h. SAD pattern shows asymmetric intensity spread around 110reflections along [100] and [010] directions. These are consistent with the modulations observed inthe dark-field images (b–e) taken from reflections as marked in the key to SAD.

been demonstrated (Tewari et al. 1999). Typically, a modulation wavelength ofabout 20 nm along the elastically soft <100>� directions has been observed in thequenched alloys. An order of magnitude estimate of the time constant, tc, of thespinodal decomposition can be made using the following relationship (Cahn 1970):

1/tc = 2KM�4m

where M is the atomic mobility, K is the gradient energy coefficient and �m is thewave number of the fastest growing concentration wave. An approximate valueof K is given by

K = NvkBTcoh�2�/�m�2




where Nv is the number of atoms per unit volume, kB the Boltzmann constant andTcoh the coherent spinodal temperature, an upper bound estimate for which is about1500 K for the alloys being considered here. It can be shown that for the spinodaldecomposition to occur within 10−5 s (during the rapid quenching treatment) thediffusion constant, D�MkBT �, needs to be of the order of 10−11 m2/s. This valueis about two orders of magnitude larger than those reported for the diffusion ofseveral solutes in �-Zr at about 1500 K(Douglass 1971). Such anomalous fastdiffusion is not unexpected in these systems for the following reasons:

(1) the retention of excess vacancies in rapidly quenched alloys may be responsiblefor an enhanced diffusivity and

(2) anomalous diffusion can be promoted due to the �-forming tendency in thesealloy systems. This point has been discussed earlier in Section 6.5. Since theformation temperatures of ordered �-phases in these alloys are very high, theconcentration of �-embryos is expected to be large at temperatures in the rangeof 1300–1500 K where the spinodal decomposition appeared to have occurred.An estimation of enhanced diffusivity due to �-like embryos (Banerjee andCahn 1983) yields a diffusivity of 10−11 m2/s in this temperature range.

The spinodal decomposition of the �-phase leads to the formation of a com-positionally modulated structure, the directions of the modulations being alongthe <100>� directions which are elastically soft directions in bcc Ti and Zralloys. Composition modulations along these three mutually perpendicular direc-tions result in the creation of Al-rich cuboids at the nodes of the concentrationwaves. These Al-rich nodes remain separated from neighbouring nodes by Al-depleted �-phase regions. A schematic description of such a structure is given inFigure 6.32.

The next step in the transformation sequence is the formation of either theZr2Al (B82) or the Zr5Al3 (D88) phase in the Al-rich nodes. These two phasesform under equilibrium conditions through a peritectoid and an eutectoid reaction,respectively. The morphological features exhibited by these intermetallic phases,as observed in rapidly solidified alloys such as Zr–27% Al, Zr–25% Al andZr3Al–3% Nb (Banerjee and Cahn 1983, Tewari et al. 1999) and as illustrated inFigure 6.33 clearly indicate that they are not products of the equilibrium phasereactions.

The features based on which the transformation mechanism has been proposedare as follows:

(a) Each cuboidal solute-rich region develops into a single domain of one of theordered structures (either B82 or D88), depending on the alloy composition.




Ageingtemperature1223 K

(c) Zr2Al

100

010

β

Ageingtemperature1123 K

β

Zr2Al

(d)

Zr2Al

Zr3Al

(L12)

β

(b)

010

100

(a)

Zr2Al

D019 in Zr2Al laths

Zr2Al–3Nb Ageing time = 600 s Ageing time = 4 h

Figure 6.32. Schematic representation of the microstructures developed upon ageing rapidly solid-ified Zr–Al–Nb alloys. (a) and (b) represent microstructures developed at temperatures lower thanthe temperature of second peritectoid reaction, as shown in Figure 6.30, whereas (c) and (d) representmicrostructures developed at temperatures higher than the temperature of second peritectoid reaction.

Figure 6.33. Rapidly solidified alloy microstructures showing solute-rich regions with cuboidalmorphology.




(b) All the possible crystallographic variants are present in the quenched structure.This is also revealed by the fact that the superlattice reflections correspondingto the different crystallographic variants are of nearly equal intensity. Thisindicates that though the probabilities of the appearance of different variantsare the same, each individual cuboidal region comprises only a single variant.

(c) The B82 as well as the D88 particles exhibit lattice correspondences which canbe arrived at by the lattice collapse mechanism discussed earlier.

(d) Diffraction patterns show diffuse intensity distributions characteristic of par-tially collapsed �-structures in the Al-depleted regions separating the B82 orD88 particles.

(e) The size of the B82 and the D88 particles correspond to the wavelength of thecompositional modulation.

In the event of the formation of B82 or D88 particles by a classical nucleationand growth process, one would expect homogeneous nucleation of all the variantsto occur under the large supercooling provided by rapid quenching. This wouldresult in the formation of several variants of B82 or D88 particles within a singlecuboidal Al-rich region. Contrary to this, each cuboidal region has been found to beconverted into a single variant particle of the B82 or D88 phase. Taking into accountall the observed features of the transformation as listed earlier, it has been suggestedthat the transformation within each of these cuboids is driven by an instability,leading to the development of a concentration wave and a displacement wave.

The lattice correspondences between the �- and the Zr2Al (B82) phases orbetween the �- and the Zr5Al3 (D88) phases, as described in Section 6.7.1, suggestthat the �→ Zr2Al and �→ Zr5Al3 transformations in stoichiometric alloys canbe viewed as a superimposition of concentration and displacement waves whichcan be described in the following manner.

The periodic displacement wave which collapses pairs of {222} planes (p =1�2�4�5� � � � ) to intermediate positions, keeping the preceding and the succeedingplanes (p= 0, 3, 6 � � � ) undisplaced, can be represented by a stationary longitudinaldisplacement wave (Banerjee and Cahn 1983):

Up = Ad sin Kd ·xp (6.19)

where Up is the displacement of the pth plane and Ad is the amplitude of the dis-placement wave. This is essentially the same as the displacement wave associatedwith the �→ � transformation where in the complete collapse corresponds to Ad

=0.577 d222. Kd is the wave vector (=2�/�; � being the wavelength) and xp isthe distance from the origin to the pth plane. The concentration wave which needs




to be superimposed on the displacement wave for accomplishing the �→ Zr2Al(B82) transformation can be expressed as

Cp = 13

+ 16Ac

6∑n=1

(cos

2n�6�p−1�+ cos

2n�6�5−p�

)(6.20)

where Cp is the concentration of Al on the pth (222) plane, Ac is the amplitudeof the concentration wave and the wave vector, Kc = 2�/6 d222 The �→ Zr5Al3

(D88) transition requires the superimposition of a vacancy concentration wave(Eq. 6.21) on the two waves described by Eqs. (6.19) and (6.20):

Cv = 13

+ 23Av cos

2�3

(6.21)

where Cv is the concentration of vacancies and Av is the amplitude of the vacancyconcentration wave.

The Landau order parameter, �, can be so defined that � = 0 representsthe completely disordered bcc structure while � = 1 corresponds to the fullyordered structure (B82 or D88) in which both the displacement and the replacementordering are fully accomplished. The parameter � can then be related to theamplitudes of the displacement and the concentration waves. Though the � →B82 and the �→ D88 transitions can be viewed as processes which can proceedby progressively increasing the order parameter, it can be easily shown fromsymmetry considerations that they do not qualify to be second-order transitions.

The free energy versus order parameter plots (Figure 6.19) proposed by Cook(1975) for explaining the fine particle morphology and the dual phase structure, asdiscussed in Section 6.6.4, illustrate that at a temperature below To, the instabilitytemperature, the free energy of the system continuously drops with increase in orderparameter. Under such a situation the transformation can occur homogeneouslywithout involving the nucleation of the product phase. The system as a wholecan proceed towards the product structure progressively reducing its free energyall along. The bulk transformation of the Al-rich cuboidal regions into the Zr2Al(B82) and Zr5Al3 (D88) phases suggests that a homogenous transformation occursin these cases. Each of the cuboidal regions picks up a fluctuation corresponding toone of the crystallographic variants and that specific fluctuation grows in amplitudeleading to the formation of the intermetallic structure through a hybrid mechanismconsisting of displacive and chemical ordering. A random selection of the variantof the initial fluctuation results in the appearance of different variants of the orderedstructures with equal probabilities. In order to reach the instability temperature,a high degree of supercooling is essential. The rapid quenching employed in the




experiments discussed here appears to be adequate in bringing about the requisitesupercooling.

The gradual enrichment of the atomic layers corresponding to p = 0, 4, 8 � � �in the smaller Al atoms (and also in vacancies in the case of the D88 structure) isexpected to facilitate the collapse of the adjacent layers p = 0 and p = 1 on onehand and p= 4 and p= 5 on the other. It is, therefore, attractive to envisage thatthe two fluctuations, one in concentration and the other in displacement, wouldamplify simultaneously. It is difficult to determine experimentally whether the twoprocesses are fully coupled or they occur in succession. For a homogeneous trans-formation to occur, it is more likely that the two fluctuations grow simultaneously,one facilitating the other.

The as-quenched structure consists of cuboidal particles separated from eachother by Al-depleted regions within which the �-phase is either fully retained orpartially transformed into the mixed � and athermal �-structure. The size of the�-particles within these Al-depleted regions is much smaller (2–5 nm) than that ofthe cuboidal particles. These fine particles also exhibit the ellipsoidal morphologycharacteristic of the athermal �-phase.

The evolution of the equilibrium structure from the rapidly solidified metastablestructure has been studied in both Zr–27 at.% Al and Zr3Al–3 at.% Nb alloys(Tewari et al. 1999). In the former case the retained �-regions first get trans-formed into a mixed � + � structure. Subsequently the equilibrium Zr3Al (L12

structure) phase forms through a peritectoid reaction between the Zr2Al (B82) andthe �-phases. The phase transformation sequence in the latter alloy has revealeda very interesting transformation sequence. The cuboidal D88 particles, on ageingat a temperature of 1123 K, get converted into the B82 structure within a periodof 600 s. Each D88 particle has been found to be transformed into a single B82

particle, maintaining the lattice correspondence shown in Figure 6.27. This obser-vation can be explained by a mechanism which invokes the diffusion of Zr atomsinto the Zr5Al3 (D88) particles, causing the required changes in the stoichiometryand structure through the elimination of structural vacancies. A rough estimateindicates that a diffusion coefficient of 10−17m2/s is required for the completetransformation of a 20-nm sized D88 particle into a B82 particle within the periodindicated. These B82 particles can be retained in the aged structure if the age-ing treatment is carried out at temperatures between the equilibrium peritectoidtemperatures corresponding to the following reactions:

(1) � + Zr5Al3 → Zr2Al (1523 K)(2) � + Zr2Al → Zr3Al (1248 K)




Ageing at temperatures below the second peritectoid temperature initially pro-duces the Zr2Al (B82 structure) phase; however, subsequently the metastable Zr3Al(D019 structure) and the equilibrium Zr3Al (L12 structure) phases emerge in suc-cession. The orientation relationship between the B82 and the D019 structures hasbeen found to be (0001)H�� {1120}B; < 1I00>H ��<11I00>B where the subscripts Band H, respectively, stand for the B82 and the D019 structures (Tewari et al. 1999).This observed orientation relation shows that the lattice correspondence betweenthe B82 and the D019 structures compares reasonably well with that between the �-and the �-phases observed in the �→ � transition induced by the application ofhigh pressures, as discussed in Section 6.3.3. The B82/D019 structural relationshipsuggests that the D019 structure can be created by the introduction of Zr atomsin the B82 structure. An (1120) plane of the latter from which the (0001) planeof the former emerges is shown in Figure 6.34. The positions marked by greycircles indicate the vacancies in the B82 structure which are filled up by Zr atomsin the D019 structure. The transformation sequence, Zr5Al3 (D88)→ Zr2Al(B82)→Zr3Al(D019), can, therefore, be considered as a process in which an open structureis gradually being filled by an influx of Zr atoms, progressively changing thecomposition and finally attaining a close packed configuration. Such a process canoccur as long as the chemical potential of Zr in the adjoining �-matrix is higherthan that in the intermetallic phase, which finally attains the stoichiometry of Zr3Al.

The metastable D019 structure of the Zr3Al phase subsequently transforms intothe equilibrium L12 structure through a congruent transformation. This transforma-tion can be accomplished by a change in the stacking sequence from the hexagonalABABAB to the cubic ABCABCABC stacking of the close packed atomic layerswhich individually satisfy the Zr3Al stoichiometry. The low-stacking fault energy(Holdway and Staton-Bevan 1986) (2 mJ/m2) of Zr3Al, which can be taken as ameasure of the free energy difference between its L12 and D019 forms, is consistentwith the fact that the metastable D019 phase appears first when Zr3Al precipitatesfrom a supersaturated hcp Zr–Al solid solution (Mukhopadhyay et al. 1979). Thewhole sequence of formation of various phases has been schematically presentedin Figure 6.35.

6.7.3 Transformation sequences in Ti base alloysThe sequences of phase transformations involving ordered �-phases have beenstudied primarily in the ternary Ti–Al–Nb system by Strychor et al. (1988),Bendersky et al. (1990a,b) and Bendersky (1994).

The pseudo-binary phase diagram of the Ti3Al–Nb system shows the com-position range over which the �-phase undergoes a martensitic transformationfollowing which an ordering process leads to the formation of the ordered�2-phase (D019 structure), as discussed earlier. In Ti3Al–Nb alloys rich in




[0001]B//[1120]H

[1100]B//[1100]H CH = 0.493 nmaB = 0.489 nmDistortion = 0.82%

B type atomsA type atomsA type atoms missingin B82

√3 aB = 0.8476 nm

√3 aH = 1.043 nmDistortion = 18.73%

aH

= 0.604 nm

cB

= 0.592 nm

Distortion

= 1.82%

Figure 6.34. Schematic drawing of a prismatic <1120> plane of the B82 structure. A systematicplacement of Zr atoms on this plane can produce the basal plane of the D019 structure (shown bybroken lines). The subscripts H and B, respectively, stand for the D019 and the B82 structures.

Nb, the �→ � transformation is suppressed on quenching from the high tempera-ture �-phase field, and the �-phase undergoes ordering to assume the B2 structure.The quenched B2 structure exhibits a tweed morphology with modulations alongthe <110> directions. The corresponding diffraction patterns show strong streaksalong <110> rel vector with intensity maxima at the 1/2 <110> positions. Itmay be noted here that the modulations observed in Zr3Al-based alloys are alongthe soft <100> directions and have been ascribed to modulations is chemicalcomposition. The <110> modulations appear in several bcc alloys such as Ni–Al,Ni–Ti and so on and have been identified to be the result of an incommensurateshear of the {110} <1I0> type which can as well be described in terms of a trans-verse displacement wave with <110> wave vector and <1I0> polarization vector




Zr3Al – Nb Alloys

Solute partitioning(Nb,Al atoms)

Constitutionalsupercooling

Dendritic morphology

Dendritic region(Nb-lean phase)

Orthorombic phase

Interdendritic region(Nb-rich phase)

β – ZrStabilized/partially

stabilized

ω -collapse + Ordering

B82 phase

~102K/s

Normal solidificationliquid

Rapid solidificationliquid

~105 K/s

Partitionless solidification

(β )

Spinodal decomposition(composition fluctuations)

Solute rich regions Solute lean regions

β – ZrStabilized/partially

stabilizedD88 phase

Ageing treatment

Below peritectoid Above peritectoid

B82 Phase B82 Phase

D019 Phase

L12 Phase

ω collapse + ordering

Figure 6.35. Schematic presentation of the formation of various phases in the Zr3Al–Nb alloys.

(Strychor et al. 1988). It has been shown that the extinction of the streaks andstriations observed under different diffracting conditions is consistent with {110}<1I0> shear strains and that the 1/2<110> satellite reflections, which commonlyoccur at the intersections of the streaks are associated with the transverse displace-ment wave. The observed lattice shears correspond to atomic movements requiredfor the transformation of the bcc lattice into a 2H orthohexagonal martensite lattice.Since resolvable martensite crystals could not be imaged, Strychor et al. (1988)have concluded that such pseudo-hexagonal regions are too fine to be resolved.




In addition to the previously mentioned B2 reflections, diffracted intensity max-ima at �-positions have been observed in quenched Ti3Al–Nb alloys suggestingthe occurrence of a longitudinal displacement wave as well. Reflections that appearat half of the normal distance of �-reflections from the origin have also beendetected. This suggests that the chemical ordering present in the B2 structure isinherited in the �-structure. Ageing of a Ti–27.8 at.% Al–11.7 at.% Nb alloy at673 K has been found to result in a gradual degradation in the tweed structure andan enhancement in the intensities of �-reflections. Dark-field imaging using thesereflections has shown the presence of fine �-like particles. Convergent beam elec-tron diffraction has confirmed the B82 structure of these �-like particles observedin the aged samples.

Bendersky et al. (1990a,b) have shown that the B2 phase in a Ti–37.5 at.%Al–12.5 at.% Nb alloy transforms into the B82 phase via a metastable intermediate�′′-phase. This alloy, when quenched or slow cooled (∼400 K/s) from 1673 Kand 1373 K, exhibits a microstructure containing coarse (10–1000 m) D019 andL12 precipitates in a matrix of the transformed B2 phase. The latter consists ofa mixture of fine domains (∼0.1 m) of trigonal �′′- and hexagonal B82-phases.Prolonged (26 days) ageing at 973 K results in a complete replacement of !′′-domains by B82-domains, which show a substantial coarsening. These observationshave established the B2 → �′′ → B82 transformation sequence in alloy with thenominal stoichiometry of Ti4Al3Nb.

6.7.4 Ordered �-structures in other systemsOrdered �-structures have also been reported in several other alloy systems suchas Cu–Zn (Prasetyo et al. 1976), Ni–Al (Georgopolous and Cohen, 1976, 1981;Reynaud, 1977) and maraging steels (Servant et al. 1987). Diffuse scattering,characteristic of �-related phases, has been observed in all these systems. Table 6.8lists the �-related phases in several alloy systems and some important featuresassociated with them.

Georgopoulos and Cohen (1981) have tested several theories that have beenproposed for the �-transformation and have shown that Cook’s phenomenologicalmodel (1974) based on the Landau theory predicts a much weaker diffuse intensityat the 4/3 <111> rel points than is experimentally observed in an Al–46.2 wt%Ni alloy. In contrast, they have found a satisfactory agreement between the theo-retically calculated and the experimentally measured diffuse scattering intensitieswhen they consider the model of Kuan and Sass (1976) which invokes hetero-geneous nucleation of �-particles around point defects. The requirement, for thisprocess, of about 1% quenched-in vacancies in Zr and Ti alloys has also beenrelaxed due to the suggestion that substitutional atoms too can act as nucleationsites for �-particles.




Table 6.8. Summary of �- and �-related phases observed in various alloys as reported in literature.

Alloy Phases observed in the system(Symmetry of the phase iswritten in the bracket)

Comments

Zr3Al–Nb � (Im3m), B2 (Pm3m), Sequence of formation ofB82 (P63/mmc), D88 (P63/mcm) phases has been established

and shown in the symmetrytree in Figure 6.36.

Ti–Al–Nb �, B2, �′′(P3m1), B82, D88 Sequence of formation ofphases has already beenestablished.

Cu-based �, D03(Fm3m), Ordered �, � Formation of ordered �-phasealloys from D03 phase has been

shown.

Co–Ga �, � High vacancy concentrationleads to the formation of �-phase.

Ni–Al �, B2, B82, �′′ B82 structure in the Ni-richNi2Al phase and �′′-structurein the Ni3Al2 phase have beenobserved.

Maraging �, B2, Fe7Mo6(R3m), �, B82 The formation of these varioussteel phases has been reported.

6.7.5 Symmetry treeIt is useful to examine the subgroup/supergroup symmetry relations, as given in theInternational Tables of Crystallography (Hahn 1987), that describe transformationsequences involving a combination of displacive and replacive ordering. Benderskyet al. (1990a,b) and Tewari et al. (1999) have constructed a symmetry tree fordisplaying the symmetry relationships between the different phases encountered inthe transformation path. On the basis of these symmetry relations, inferences canbe drawn on the following points: (i) whether a transformation is of the first or ofa higher order; (ii) how many rotational and translational variants can be createdor destroyed in a transformation and (iii) which intermediate phases are likely tobe metastable.

The International Tables of Crystallography list “maximal nonisomorphic sub-groups” which are the space groups of possible product phases that may form ifthe symmetry reduction does not include an intermediate step. This point can beexplained by taking the example of the transition of the B2 structure (Pm3m) to the�′′-structure (P3m1); the latter is not listed as a maximal non-isomorphic subgroupof the former. The symmetry reduction accompanying the B2→ �′′ transition




involves the sequence Pm3m→ R3m → P3m1, and both the steps correspond tolisted supergroup–subgroup relations. The index of symmetry reduction (or incre-ment) is an integer (shown in square brackets in the symmetry tree) which is equalto the number of symmetry elements in the supergroup divided by the number inthe subgroup. If the index is an odd integer then the transition is necessarily ofthe first order (except at an isolated point) and if it is an even integer, then thetransition can be of either the first or a higher order. The index also correspondsto the number of domains/variants that may be created in the product phase froma single crystal of the parent phase.

The first step of symmetry change, Pm3m→ R3m destroys the fourfold symme-try of the B2 structure by effecting a homogeneous rhombohedral distortion alongan <111>B2 direction which is the direction of lattice collapse and is selected asthe threefold axis in the trigonal �-type phase. The second step, R3m → P3m1selects which of the (111) planes collapse to form double layers and which remainundisplaced. The symmetry index [4] corresponding to the first step gives thenumber of rotational variants associated with the rhombohedral distortion whilethe index [3] gives the number of translational subvariants associated with eachrotational variant.

The �′ → B82 transition involves an increase in symmetry: P63/mmc is asupergroup P3m1. Usually an increase in symmetry is associated with an increasein entropy and so the low entropy phase is expected to be the low temperaturephase as well. The isothermal supergroup formation in the �′′ → B82 transition,therefore, suggests that the �′′-phase is metastable. This is also indicated from thefact that the �′′ → B82 transition proceeds from a more ordered �′′-phase to a lessordered B82 phase.

The fact that P63/mmc is not a subgroup of Pm3m indicates that the B2 (Pm3m→ B82 (P63/mmc) transition must be strongly of the first order. The observedtransformation path, B2→ �′′ → B82, traverses a state of minimum symmetry(�′′, P3m1), that is a subgroup of both Pm3m and P63/mmc. This space groupof minimum symmetry is, in fact, the intersection of the parent and the productgroups. It is worth noting that the formation of the �′′-phase as an intermediatemetastable phase provides a continuous structural path for the B2→ B82 transi-tion.The existence of such a continuous path provides the basis for a commonthermodynamic function for all the phases encountered.

The occurrence of vacancy ordering in the D88 structure modifies the symmetrypath, leading to the formation of the D88 phase which has a lower symmetrythan that of the B82 phase. Elimination of structural vacancies results in theconversion of the D88 structure into the B82 structure. The sequence of such phasetransformation which is dictated by symmetry consideration has been shown inFigure 6.36. It can be noticed from the figure that there are multiple choices




Chemically disordered

Im3m (A2:bcc)

Chemically ordered

Chemical ordering

ω -Collapseω -Collapse[4]

ω -Collapse

Ordered ω -Structure

Disordering

Disordered ω

Chemical orderingHomogeneous disordering

Homogeneous distortion

Chemical orderingCompletion of collapse

[2]

[2]

[4]

[3][2] [2], c′ = 2c

Vacancy ordering

Disordering insingle layer

Disordering in single layer

[6]

Homogeneousdistortion

[4]

R3m

Chemical ordering

(Trigonal ω )

[2], c′=2c

P6/mmm(ω Ti)

P63 /mcm(D88:Mn5Si3) P63/mcm

(D88:Ni2In)

R3m

P3m1

P3m1 (ω ′,ω ″ )

Rm3m (B2:CsCl)Fm3m D03:BiF3)

R3m

Figure 6.36. Symmetry tree showing the sequence of the phase transformations of the �-relatedphases.

available on each mode. The transformation from one phase to another is, therefore,dictated by the external parameters.

6.8 INFLUENCE OF �-PHASE ON MECHANICAL PROPERTIES

The mechanical properties of �-Ti alloys undergo drastic changes with the pre-cipitation of the �-phase: a large increase in the yield stress, which is invariablyaccompanied by a significant reduction in ductility, is observed. Gysler et al.(1974) have studied the deformation behaviour of �-Ti–Mo alloys and have shownthat extensive slip localization occurs with �-precipitation.

6.8.1 Hardening and embrittlement due to �-phaseAs mentioned earlier in this chapter, the presence of the �-phase in �-quenchedalloys based on Ti and Zr is often inferred from the high hardness values




16

12

8

4

0 2 4 6 8 10 12

Elongation [%]

c

b

a

Load

[103 N

]

Figure 6.37. Engineering load-elongation curves of the Ti–11% Mo alloy for (a) as-quenchedspecimen, (b) specimen aged at 623 K for 3 h and (c) specimen aged at 623 K for 10 h. Increasein yield stress and sharp drop in ductility upon ageing is directly related with the increase in thevolume fraction of the �-phase.

exhibited by these. A comparison of the stress–strain diagrams (Figure 6.37) ofthe �-quenched Ti–15% Mo alloy after ageing under different conditions clearlyindicates that an ageing treatment which causes �-precipitation can significantlyincrease the yield strength of the alloy though the ductility is severely impaired.Ageing treatments to produce �-precipitation restores the ductility to a limitedextent.

One of the important characteristics of �-embrittlement is that the fracturesurface shows a ductile dimple-type morphology even when the material showsno macroscopic ductility. The microvoid coalescence mechanism, which givesrise to the dimple morphology, appears to be operative on a microscopic scalealong the fracture surface inspite of the occurrence of brittle fracture. Gysler et al.(1974) have studied the microstructures of deformed �-Ti–Mo alloys and havereported that dislocations in the deformed material are primarily arranged alongwell-defined slip bands. Dark-field imaging using �-reflections has revealed thatthese bands are devoid of �-particles. The destruction of �-particles in the vicinityof slip bands essentially implies that these particles get sheared due to the passageof a fairly large number of dislocations through these particles. The stress requiredfor the cutting of �-particles has been estimated by using the following relation(Gleiter and Horbogen 1965)

�"o = 1�02 32G− 1

2 b−2r12 f

12 (6.22)




where the increase in the critical resolved shear stress, �"o, is expressed in termsof the shear modulus, G, of the matrix, the Burgers vector of dislocations, b, theanti-phase boundary energy and the average radius, r, and the volume fraction,f , of �-particles. Gysler et al. (1974) have plotted the yield stress, ��= 2"o�against r1/2f 1/2, taking the experimental values obtained on a Ti–11% Mo alloyand have arrived at a value of to be 300 kJ/m2 (Figure 6.38). Since the anti-phaseboundary energy, , for the �-particles, determines the specific interaction forcebetween these particles and mobile dislocations; this high value of shows thatthe shearing of �-particles is, indeed, a very difficult process. However, the stressrequirement for the alternative mechanism involving dislocations bypassing closelyspaced �-particles has been shown to be even higher than that for particle shearing(Figure 6.38). The observed hardening due to �-precipitates can, therefore, berationalized in terms of the stress required for the shearing of �-particles alongthe slip bands.

The formation of sharp slip bands in �-alloys containing a distribution of denselypopulated, fine, aged �-particles can also be viewed as a consequence of the par-ticle shearing process. Once a set of leading dislocations glide along a slip plane,cutting across the �-particles on their path, a soft �-free channel is created. Subse-quent deformation in these soft channels can occur without dislocations experienc-ing such high resistance on the slip path. Such �-free channels have been observedin deformed samples and from trace analyses of these slip bands it has been shown

900

600

1200

1500

1800

0 1 2 3

r½ f½ [Å½]

σ 0.1

[MN

m–2

]

Figure 6.38. Yield stress (#0�1) of the Ti–11% Mo alloy aged at 623 K as a function of size (r1/2)times volume fraction (f 1/2) of the �-particles. Circles represent the calculated stress for the bypassmechanism.




that in most cases they occur along the {110}� planes and less frequently along the{112}� and the {123}� planes. The Burgers vectors associated with the dislocationsalong these planes have been found to be of the a/2 <111>� type.

The observed decrease in macroscopic ductility with increasing ageing time(Figure 6.37) can be explained qualitatively in terms of an increased tendency forinhomogeneous slip. Plastic flow remains confined within a limited number of slipbands, the points where they mutually interest or they meet the grain boundariesbecoming the sites of crack nucleation. The limitation of plastic flow to narrow slipbands causes high stress concentrations and favours crack nucleation and growthat low macroscopic plastic strains. The local deformation within the slip bands isquite pronounced, as revealed from the large offsets observed at the intersections ofslip bands with other planar features. A survey of the fracture surface topographyof �-alloys as a function of ageing time, when aged at temperatures below about773 K, shows the following trends:

(a) The fracture surfaces of �-solutionized samples, which contain very fine ather-mal �-particles and which exhibit quite high macroscopic ductility (about 10%elongation) and a reasonably homogeneous plastic flow, are characterized bylarge dimples with an average spacing of about 50 m.

(b) As the percentage elongation comes down to about 4–5% as a consequenceof �-precipitation due to ageing, the average dimple size drops down to about1–2 m.

(c) As the macroscopic ductility falls to a negligible level, fracture propagatespredominantly along grain boundaries, as revealed in low-magnification micro-graphs. Each grain boundary facet, at higher magnifications, shows the pres-ence of steps corresponding to offsets created by slip bands and a distributionof fine (1–2 m) dimples. The extensive microscopic flow along the fracturepath is well demonstrated inspite of nearly zero macroscopic ductility.

6.8.2 Dynamic strain ageing due to �-precipitationRecent studies (Banerjee and Naik 1996) on the plastic flow behaviour of a �-Tialloy (Ti–15% Mo) and of an intermetallic (Nb–Ti–Al (Grylls et al. (1998))) in thetemperature range 575–775 K have shown that very pronounced serrated yieldingoccurs in these systems over a certain range of strain rates. Such non-linear plasticbehaviour is often connected with a plastic instability arising out of dynamicinteractions of mobile dislocations with individual solute atoms, their clusters orfine precipitates.

The phenomenon of dynamic strain ageing, leading to serrated yielding, isknown in the literature as the Portevin–Le Chatelier (PLC) effect. The occur-rence of the PLC effect in substitutional alloys has generally been attributed to a




periodic locking and unlocking of dislocations by individual solute atoms. Sincethis process involves long range diffusion of solute atoms, enhanced diffusivitydue to plastic flow has been invoked for explaining the kinetics of the process.Numerous experimental observations, showing the presence of a critical strain,�c, preceding the appearance of serrated yielding, have provided support to thediffusivity enhancement model. The extensive literature on this subject has beenreviewed elegantly by Estrin and Kubin (1995). The occurrence of PLC effects hasbeen treated on the basis of the collective movement of a group of dislocations bySchoeck (1984) and of the propagation of localized deformation bands by Schlipf(1992). There have also been reports in which the locking is considered to beeffected not by individual solute atoms but by clusters of solute atoms (Onoderaet al. 1983, 1984). The dynamic strain-ageing phenomenon due to �-precipitationassumes a special significance, as this is an example which demonstrates that thePLC effect can also result from the dynamic interaction of a group of dislocationsdirectly with precipitates. Experimental evidences (Banerjee and Naik 1996) onthe basis of which this new mechanism of the PLC effect has been established aresummarized in this section.

The stress–strain plots for �-quenched Ti–15% Mo alloy samples tested atdifferent temperatures in the range 455–775 K at a strain rate of �= 1�31×10−4 s−1

are shown in Figure 6.39 which also includes a magnified portion of the load-elongation plot illustrating the characteristic C-type serrated flow, as per theclassification scheme proposed by Brindley and Worthington (1970). The rangesof strain over which the serrated flow behaviour is observed are indicated on theflow curves schematically by short vertical segments representing the extents of theload drop. The temperature–strain rate regime corresponding to serrated yieldingis represented in the strain rate versus reciprocal temperature plot (Figure 6.40).The shaded region in this figure, which represents the regime of serrated flow,is confined within a low temperature and a high temperature limiting lines, theformer being temperature dependent and the latter being athermal.

The characteristic features of dynamic strain-ageing, such as (i) the presence ofa hump in the descending part of the yield stress versus temperature plot and (ii)a negative strain rate sensitivity, are observed under the conditions correspondingto the occurrence of serrated yielding. The critical strain, �c, is seen to decreasewith temperature; this is a behaviour which is known to be normal in the PLCeffect literature.

Tensile specimens unloaded after the occurrence of a few load drops show thepresence of macroscopic deformation bands (known as PLC bands) across thewidth of the specimens. An one-to-one correspondence between the number ofload drops and the number of PLC bands is seen in the early stages of serratedflow. A magnified view of PLC bands shows fine slips bands which are not




5%

50 M

Pa

Str

ess

525 K

455 K

575 K

300 K

675 K625 K

Elongation

0.01 nm

725 K 775 KLo

ad

Strain

5 kg

Strain rate = 1.31 × 10–4 S–1

Figure 6.39. Stress–strain plots for the �-quenched Ti–15% Mo samples tested at different tem-peratures (as indicated alongside of the flow curves) at a strain rate of 1�31 × 10−4s−1. Serratedflow indicated by short vertical lines on the flow curves is observed in the temperature range575–725 K. Arrows are marked at the onset of serrated flow. The inset shows a magnified portionof a load-elongation plot.

perfectly straight and are confined within �-grains. These slips bands which arenot observed outside the PLC bands, create offsets when they intersect each otheror when they meet a grain boundary. TEM examination of PLC bands shows“deformation bands” on a still finer scale. These bands are essentially channelsdevoid of �-precipitates. Figure 6.41 shows PLC bands, deformation bands and�-free channels.

The occurrence of serrated flow and the magnitudes of load drops can becontrolled by changing the stability of the �-phase with respect to �-precipitationat the testing temperature. This has been demonstrated by altering the compositionof the matrix either by giving a prior ageing treatment or by choosing an alloy of adifferent composition. The influence of prior ageing on the amplitude of serrationsin the Ti–15% Mo alloy is shown in Figure 6.42.

Prior ageing at 675 K for 1 h and 168 h which produces a Mo-lean aged �-phasemakes the �-matrix richer in Mo. The magnitude of stress drops is seen to decreasewith increasing ageing time which corresponds to increasing Mo concentration in




0.653

1.31

2.62

6.55

13.1

26.2

65.5

0.01

0.02

0.05

0.1

0.2

0.5

0.0051.2 1.4 1.6 1.8 2.0 2.2

1/T × 103 (K–1)

773 723 673 623 573 523 498 473 453

Temperature (K)

Str

ain

rate

× 1

0 (

s–1)

Cro

ss-h

ead

spee

d (c

m/m

in)

No serrationSerration flow

Serrationflow

Figure 6.40. Strain rate-temperature region in which serrated flow is observed in the Ti–15%Mo alloy.

the matrix. The serrated flow can be fully suppressed by giving a prior ageingtreatment at 875 K for 1 h, which produces �-precipitates and causes further Moenrichment of the matrix. An alloy containing 25% Mo does not show serratedyielding in the entire range of temperature and strain rate scanned in Figure 6.43.

The fact that �-precipitation is responsible for the dynamic strain-ageing phe-nomena in �-forming systems is also revealed from static ageing experiments inwhich the cross-head motion of the testing machine is arrested after the start ofserrated flow and the load is continuously recorded as a function of time. This testis similar to a stress relaxation test (Gupta and Li 1970) normally carried out forthe determination of the activation parameters of plastic flow. During such testsit has been noticed that the stress initially relaxes and subsequently rises. As thestress value reaches the upper envelope of the flow curve, a sudden drop in stressis observed. This is followed by an increase in stress upto the upper thresholdvalue when a second drop in stress occurs. Such repeated load drops continue tooccur, giving rise to a serrated appearance of the stress-versus-time plot. This static




(a)

(d)

(b)

(e)

(c)

(f)

Figure 6.41. Microstructures of tensile specimens unloaded after the occurrence of a few load drops.(a) Microstructure showing the presence of deformation bands within PLC bands, (b) deformationband within a �-grain. Termination of deformation of these bands at the grain boundary can benoticed. (c) and (d) are Bright-field TEM micrographs showing parallel arrangement of dislocationswithin a deformation band. (e) Reduced density of particles within the imaged band and (f) restorationof �-particles within a deformation band after ageing.

0.2%

Strain

Str

ess

20 MPa

T = 575 K(a)

(b)

(c)

(d)

Figure 6.42. The influence of prior ageing treatment on the amplitude of serration is seen from partsof the flow curves at 575 K of Ti–15 Mo alloy, heat-treated at different ageing conditions, (a) 575 Kfor 1 h, (b) 675 K for 1 h, (c) 675 K for 168 h and (d) 875 K for 1 h.




Strain

5%

Str

ess

50 M

Pa

Strain rate = 1.31 × 10–4 S–1

Figure 6.43. Stress–strain curve of the Ti–25% Mo alloy. Absence of serrated yielding could benoticed from the figure.

ageing phenomenon can be understood by considering the different componentsof strain which contribute towards the total strain, �tot, of the sample (�tot = 0,when the cross-head motion is stopped).

�tot = ddt

(�/E+ �p +Cf +M�

)= 0 (6.23)

The modulus, E, of the specimen is given by (Eo+�E), where Eo is the modulusat the start of relaxation, �E the difference between the moduli of the �- and the�-phases and f the volume fraction of the �-phase formed during time t; �p is theplastic strain of the specimen due to relaxation. Since the �→ � transformationis associated with a 14% volume contraction, there is a tendency for contractionof the specimen due to the dynamic �-formation as given by Cf , where C is aconstant related to the volume change due to the transformation. The last termin Eq. (6.23), M� , is the strain contribution due to the elastic deformation of thetesting machine, M, which is related to the stiffness of the machine.

The volume fraction, f , of the �-phase after time t can be written as

f = fo�1− et/"� (6.24)

where " is the time constant of �-precipitation and fo the equilibrium volumefraction of the �-phase. The relation between the plastic strain rate �p and � canbe written as

� = �1 +K log �p/�o




Table 6.9. Input parameters used for simulating the stress-versus-time plotduring on-load ageing (Figure 6.44).

Elastic modulus, Eo 50 000 MPaModulus difference, �E 20 000 MPaMachine stiffness 50 000 N/mmEquilibrium �-volume fraction, fo 10%Time constant for �-formation, 50 s [Figure 6.44]

100 s [Figure 6.44]Threshold stress for formation of 490 MPa∗deformation bands, �1

where K is inversely proportional to the activation volume of the process and �o isthe maximum strain rate at a given temperature at which serrated flow is observed.Substitution of reasonable values (as indicated in Table 6.9) yields a �-versus-tplot (Figure 6.44) showing an initial stress relaxation, followed by a rise in stressand then repeated stress drops, as encountered experimentally.

The oversimplified treatment presented here qualitatively explains the relaxationbehaviour due to the usual plastic flow and the competing rise in stress resultingfrom specimen contraction due to dynamic �-precipitation. When the latter process

530

520

510

500

490

480

470

460

450

480

470

490

500

(a)

(b)

100 200 300 400 500 600 700

Time (s)

T0 =100 s

Str

ess

(MP

a)

T0 = 50 s

Figure 6.44. Computed stress-versus-time plot for static ageing under load experiment. The para-meters used for computation are given in Table 6.9.




dominates, there is a net rise in stress due to on-load static ageing. When thestress finally reaches the threshold value for the load drop, deformation bandsare nucleated. The repeated occurrence of load drops in the stress–time plot can,therefore, be explained in terms of a combination of a stress relaxation processand an �-ageing process.

Repeated yielding is encountered in alloys, in general, during deformationby repeated mechanical twinning, stress-assisted martensitic transformation anddynamic strain ageing (PLC effect). In the �-forming systems discussed here, nodeformation twins or martensite plates are detected within the PLC bands. Further,the serrated yielding process in �-forming systems is thermally activated whereasdeformation twinning and martensite formation are athermal processes. Contraryto the observations made in �-forming systems, the latter processes are favouredat lower temperatures.

Dynamic strain-ageing in an alloy can occur through either of the followingmechanisms:

(a) by solute atom–dislocation interactions to form a solute atmosphere aroundthe dislocations and

(b) by the establishment of chemical and/or displacement order or fine scaleprecipitation.

Both these mechanisms can lead to pinning of dislocations and under thosetest conditions where locking of mobile dislocations and unlocking of pinneddislocations become competitive the alloy exhibits repeated yielding.

In solute atom–dislocation interactions, solute atoms migrate rapidly to forman atmosphere around freely moving dislocations resulting in a drag on them.The stress required to move dislocations thus increases until they are able tofree themselves from the solute atmosphere. The mechanism involving pinning ofdislocations by either interstitial or substitutional solute atoms is not appropriatein the context of serrated yielding in �-forming systems in view of the followingobservations:

a) Serrated flow behaviour can be completely or partially suppressed in thesesystems by stabilizing the �-phase with respect to the �→ � transformation.

b) The upper temperature limit of the strain rate–temperature regime for serratedflow is independent of the strain rate and is dictated by the dissolution temper-ature of the �-phase.

Based on the experimental observations described in the preceding paragraphson serrated yielding in �-forming systems (Ti–Mo alloys, Nb–Ti aluminides),Banerjee and Naik (1996) have proposed the following mechanism.




A sharp load drop can be exhibited in a stress–strain plot when a sudden burstof plastic flow is triggered. It is envisaged that such sudden bursts of plasticflow can occur when deformation bands are created by the shearing of �-particleswithin these bands. Microstructural observations reveal that the localization ofsudden plastic flow occurs in three different length scales. In the finest scale,a single �-free channel is created by the passage of leading dislocations whichfacilitates the passage of an avalanche of trailing dislocations through the same softchannel. A typical �-free channel is seen to be a few nanometres wide. A groupof such channels constitute a deformation band, typically a few m thick. Suchdeformation bands remain confined within the grains, but plastic flow is triggeredin adjoining grains by the nucleation and growth of fresh deformation bands. Inthis manner plastic compatibility between neighbouring grains is maintained. Thedensity of deformation bands decreases as one moves from the centre to the edgeof a macroscopic deformation band (PLC band, typically a few millimetres wide)of inhomogeneous plastic flow.

The sudden plastic flow is, therefore, caused by the creation of �-free channels,the grouping of these channels in deformation bands and the propagation of such adeformation process from one grain to its neighbouring grain. The triggering stepinvolves the movement of an ensemble of dislocations in a material containinga distribution of fine �-particles which are sheared and even destroyed with thepassage of the leading dislocations. It is this shear localization process whichis responsible for the exceptionally poor ductility of a material with the �+aged-� microstructure. As shown by Gysler et al. (1974) extensive plastic flowalong soft �-free channels results in large offsets at grain boundaries leading tothe nucleation of microvoids and subsequently fracture at very low macroscopicstrains. The same microstructural state can exhibit a reasonably high ductility whenthe testing temperature is raised to the range 550–750 K where serrated yielding isobserved. This is possible because at these temperatures the dynamic formation of�-precipitates harden the soft channels, arresting plastic flow within them. In theserrated flow regime, plastic flow within �-free channels cannot continue upto thepoint of microvoid nucleation as the dynamic restoration of �-particles graduallypin the mobile dislocations, the density of which decreases with time between twosuccessive load drops.

The PLC effect can be understood in terms of the density of mobile dislocationsby considering the following expression relating the macroscopic strain rate, �,the mobile dislocation density, �m, the Burgers vector b and the velocity, $, ofmobile dislocations:

�= �mb$ (6.25)




The average velocity of moving dislocations in a material containing a distri-bution of dislocation pinning obstacles can be expressed as

$= l/�tw + tf�≈ l/tw (6.26)

where l is the mean free spacing of the obstacles, tw is the average waiting timefor dislocations at obstacles and tf (<< tw) is the time of flight of dislocationsfrom one obstacle to the next. Schoek (1984) has argued that an instability ofplastic flow (accompanied by a negative strain rate sensitivity) occurs in the neigh-bourhood of a critical value, �p, of the mobile dislocation density (Figure 6.45)where

tw/tp = �m/�p (6.27)

tw being the average value of the waiting time and tp, the average ageing time forpinning a dislocation. The same argument can be invoked in describing the plas-tic flow within the localized deformation bands in �-forming systems. When thekinetics of �-formation are favourable, the time (tp) required to pin dislocations byrestoration of �-particles along the �-free channels becomes smaller than the wait-ing time tw. Under this condition, as plastic flow continues within a �-free channelimmediately after the propagation of an avalanche of dislocations through it, moreand more dislocations become pinned by fresh �-particles. Consequently, thestress required to sustain the strain rate gradually rises. The flow within a channelat a constant strain rate can be schematically represented by a “local” stress–strainplot (Figure 6.46) in which the flow stress, �2, corresponds to the high mobiledislocation density (�2) within a channel immediately after its creation while thethreshold stress for shearing of �-particles is given by �1, with the correspondingmobile dislocation density �1. Following the stress drop, (�1–�2), as �m decreases

σe

ε2

ε1

ρp1ρp2

ρm

Figure 6.45. The change of effective stress (�e) plotted against the density of moving dislocations�m = ��/lb�tw for two constant strain rates �1 and �2 > �1.




ε

σσ1

σ2

ρ1

ρ2

Figure 6.46. Schematic stress (�) versus strain (�) diagram within a localized deformation band;�1 represents the threshold stress required for triggering a deformation band.

Figure 6.47. Multiscale deformation process shown in form of pyramidal structure. Lowest portionof the pyramid shows macroscopic PLC bands in a sample (in the scale of mm) unloaded after theappearance of two load drops in the stress–strain plot. Scanning electron micrograph showing the�-grain structure revealed within the PLC band. Presence of finer deformation bands (in the scaleof 10 s of m) within the grains inside a PLC band could be noticed in the next block. Parallelarrangement of the dislocations within a �-grain and nanometre wide �-free channels were observedin the same region at much finer scale (∼nm) as shown in two upper blocks. In the top block of thepyramid is the lattice collapse mechanism operative at the unit cell level.




from �2 to �1 with increasing degree of pinning, the stress rises and finally attainsthe threshold stress, �1, which triggers either the creation of fresh channels or thereactivation of channels which have earlier experienced a similar burst of disloca-tion mobilization. Successive operation of these processes leads to a serrated flowin which the maximum and the minimum stress envelopes are given by �1 and �2,respectively.

The activation energy of the ageing process can be evaluated from the plotof ln � against 1/T , where �o is the maximum strain rate at the temperatureT where serrated flow is observed. The experimental value of 125 kJ/mol forTi–15% Mo is consistent with that for the diffusion of solute atoms (Mo) in�-Ti (125 kJ/mol). Since the segregation of solute atoms occurs as a precur-sor to displacement ordering for the restoration of �-particles, the thermal acti-vation required for the restoration process is essentially the same as that fordiffusion.

In summary, it is to be emphasized that the aged �-particles, which renderthe �-alloys exceptionally brittle at room temperature, are responsible for a ser-rated plastic flow at temperatures where dynamic �-formation can kineticallycompete with the particle shearing process. As has been pointed out earlier, thegeneral phenomenon of strain ageing is related to a competitive process of pinningand mobilization of plastic flow. While in a majority of alloy systems, soluteatom–dislocation interactions are responsible for causing this phenomenon, in �-forming alloys, pinning of dislocations by fine �-particles and creation of soft�-free channels followed by a burst of plastic flow along these channels, resultin periodic serration in the stress–strain plot. The multiscale deformation dur-ing serrated yielding could be summarized in the form of a pyramid shown inFigure 6.47.

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24, 1307.Usikov, M.P. and Zilbershtein, V.A. (1973) Phys. Status Solidi, 19A, 53.Vohra, Y.K. (1978) J. Nucl. Mater., 75, 288.Vohra, Y.K. (1979) Acta Metall., 27, 1671.Vohra, Y.K., Sikka, S.K., Menon, E.S.K. and Krishnan, R. (1980) Acta Metall., 28, 683.Vohra, Y.K., Sikka, S.K., Menon, E.S.K. and Krishnan, R. (1981) Acta Metall., 29, 457.Willaims, J.C., de Fontaine, D. and Paton, N.E. (1973) Metall. Trans., 4, 2701.Zilbershtein, V.A., Chistotina, N.P., Zharov, A.A., Grishina, N.S., and Estrin, E.I. (1975)

Fiz., Metal Metalloved, 39, 445.




Elsevier UK Code: PTA Ch07-I042145 6-6-2007 5:37p.m. Page:555 Trim:165mm×240mm Integra, India


Chapter 7

Diffusional Transformations

7.1 Introduction 5587.2 Diffusion 560

7.2.1 Diffusion mechanisms 5607.2.2 Flux equations: Fick’s laws 5627.2.3 Self- and tracer-diffusion coefficients in �-Zr and �-Ti 5647.2.4 Self- and tracer-diffusion coefficients in �-Zr and �-Ti 5667.2.5 Interdiffusion 5707.2.6 Phase formation in chemical diffusion 5787.2.7 Diffusion bonding 584

7.3 Phase Separation 5877.3.1 Phase separation mechanisms 5897.3.2 Analysis of a phase diagram showing a miscibility gap 5977.3.3 Microstructural evolution during phase separation in the

�-phase 6037.3.4 Monotectoid reaction – a consequence of �-phase immiscibility 6067.3.5 Precipitation of �-phase in supersaturated �′-phase during

tempering of martensite 6097.3.6 Decomposition of orthorhombic �′′-martensite during

tempering 6167.3.7 Phase separation in �-phase as precursor to precipitation of

�- and �-phases 6187.4 Massive Transformations 623

7.4.1 Thermodynamics of massive transformations 6237.4.2 Massive transformations in Ti alloys 626

7.5 Precipitation Of �-Phase in �-Matrix 6327.5.1 Morphology 6337.5.2 Orientation relation 6427.5.3 Invariant line strain condition 6437.5.4 Interfacial structure and growth mechanisms 6487.5.5 Morphological evolution in mesoscale 655

7.6 Precipitation of Intermetallic Phases 6577.6.1 Precipitation of intermetallic compounds from dilute solid

solutions 6577.6.2 Precipitation in ordered intermetallics: transformation of

�2-phase to O-phase 662



7.7 Eutectoid Decomposition 6707.7.1 Active eutectoid systems 6757.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe system 676

7.8 Microstructural Evolution During Thermo-Mechanical Processingof Ti- and Zr-based Alloys 6837.8.1 Identification of hot deformation mechanisms through

processing maps 6847.8.2 Development of microstructure during hot working of Ti alloys 6877.8.3 Hot working of Zr alloys 6917.8.4 Development of texture during cold working of Zr alloys 7017.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt%

Nb alloy tubes 706References 710



Chapter 7

Diffusional Transformations

List of SymbolsJA�B�v: Atomic flux of species A, B and vacancyc or C: Concentration of solutes

x: DistanceD: Diffusion coefficient or diffusivityD0: Pre-exponential constant for diffusivityQ: Activation energy for diffusionT : TemperatureTm: Melting temperature−→v : Velocity of lattice plane and transformation front

JA� JB: Interdiffusion flux of A and B�: Width of intermediate phase region in a diffusion couple

�G: Total free energy change�Gc: Chemical free energy change��: Surface energy change�W : Mechanical work due to volume change accompanying formation

of a new phase at the interdiffusion couplew: Width of diffusion zoneKp: Penetration constantQp: Activation energy for diffusion-controlled growth

�HXS: Excess enthalpy of a solid solution (deviation from enthalpy ofideal solid solution)

G: Gibbs free energy�A�1(C1): Chemical potential of the component A in the �1-phase having

solute (B) concentration, C1

: Wave length of concentration modulation in spinodallydecomposed solid solution

�G�: Gradient energy of the solid solution with concentrationmodulation

: Fractional change in lattice parameter, a, per unit compositionchange

E: Young’s modulus�: Poisson’s ratioVm: Molar volume

557




g�c : Free energy density of solid solution of composition cR∗: Critical radius of a clusterTc: Critical temperature of the miscibility gapCc: Critical composition of the miscibility gapK: Gradient energy coefficientCij: Elastic constantS: EntropyT ∗

s : Coherent spinodal temperatureTS: Chemical spinodal temperature

Tmono: Monotectoid temperatureg: Diffraction vector excited

��: Distance of a side band from the main reflectionM: Atomic mobility

7.1 INTRODUCTION

Diffusion of atoms plays a dominant role in a variety of phase transformationsin alloys, intermetallics and ceramics. As pointed out in Chapter 2, diffusionaltransformations can be further classified into several types, primarily on the basisof the diffusion distances involved in these. One may ask whether there is anyspecial reason for studying diffusional transformations in Ti- and Zr-based systemsin the context of gaining an understanding of this class of transformations, ingeneral. The answer to such a question is strongly affirmative in view of thefollowing points:

(1) Nearly all types of diffusional transformations are encountered in Ti- andZr-based metallic and ceramic systems.

(2) As will be shown in this chapter, the � → � transformation in Ti- andZr-based alloys essentially follows the Burgers orientation relation, irrespectiveof whether the transformation is diffusional or martensitic. Therefore, thesesystems offer a unique opportunity for making a comparison between the char-acteristic features of transformations occurring by martensitic and diffusionalmechanisms. Crystallographic and morphological details, thermodynamic andkinetic information and a knowledge of the nature of interface structures inrelation to the �→ � transformation are all available for both martensitic anddiffusional transformations. A comparative study of these, therefore, leads toa better understanding of the mechanism of such transformations in general.



Diffusional Transformations 559

(3) The �/� interfaces in Ti- and Zr-based alloys have proved to be considerablymore amenable to detailed TEM investigations than such interfaces in manyother alloy systems.

(4) Diffusion of substitutional atoms in the �-phase exhibits an anomalousbehaviour, due to which some of these transformations are seen to occur atunusually fast rates. Such transformations are somewhat unique and are notgenerally encountered in other alloy systems.

(5) Precipitation reactions in ceramics have been studied most exhaustively inthose based on ZrO2. It is, therefore, very suitable as a prototype system forstudying diffusional transformations in ceramics.

(6) The issue of the participation of shear in several hybrid diffusional–displacivetransformations has attracted renewed attention in current literature. Transfor-mations in some Ti- and Zr-based systems exhibit novel features and providenew insight in connection with such hybrid processes.

The wide variety of diffusional transformations in these systems and the depthof understanding achieved through numerous studies undertaken on them havemade this field quite fascinating as will be illustrated in this chapter. The firstsection of this chapter focuses on the phenomenon of diffusion in the �- andthe �-phases, including anomalous diffusion in the latter, and interdiffusion indifferent alloy systems. The general principles pertaining to the formation of inter-metallic compounds and intermediate phases, either equilibrium or metastable, inthe interdiffusion zones of diffusion couples of dissimilar metals have been dis-cussed. The applications of these general principles in rationalizing the formationof intermediate phases in diffusion couples between Zr or Ti on one side and Al,Ni or Fe on the other have been addressed.

The tendency towards phase separation in the �-phase field is a controlling factorin a number of phase reactions such as spinodal decomposition in the �-phase,monotectoid reactions, tempering of martensite and even in �- or �-precipitation ina �-matrix. All these aspects will be covered under the section on phase separation.

The precipitation of �-plates in the �-matrix has been a subject of a numberof studies, primarily due to the adherence to the orientation relation between thetwo phases and to the simple nature of the �/� interface. Different morphologicalvariations of the �-plates are described and the mechanisms involved in theirformation are discussed. The importance of the invariant line strain in dictatingthe growth direction has been duly emphasized.

Massive transformations occur in a number of alloys based on Ti and Zr. Thefact that short-range atomic jumps across the advancing transformation front isthe operative mechanism for a massive transformation has been well illustrated inTi-based systems. The composition range over which massive transformations are




thermodynamically feasible has also been discussed in this chapter. Precipitationof intermetallic phases from the supersaturated �-phase has been dealt with interms of the precipitate morphology and orientation relation. The precipitationof an O-phase in a B2 matrix has been covered in some detail. O-phase platesof different crystallographic variants intersect at several points where offsets arecreated. The presence of offsets gives an impression that a shear mechanism takespart in the formation of the O-phase plates. The origin of the offsets in a diffusionaltransformation has been explained by invoking the concept of the “lattice-sitecorrespondence” which exists between the matrix and the precipitate phases.

The eutectoid decomposition in these systems can be classified into two distinctgroups, namely, the “slow” and the “active” eutectoid decomposition. While theformer has the characteristic features of pearlitic reactions (as observed in steelsand in Cu–Al alloys), the mechanism associated with the latter involves atomtransport across the transformation front by maintaining lattice correspondencesimultaneously between the parent and the two product lattices.

7.2 DIFFUSION

Diffusion is essentially the mass flow process by which atoms (or molecules)change their positions relative to their neighbours under the influence of thermalenergy and gradients which can be a concentration gradient, a magnetic or anelectrical field gradient, a stress gradient or a combination of these. Diffusionplays a pervasive role in bringing about non-athermal phase transformations inthe solid state and constitutes an irreversible process. There are two types ofapproaches for studying diffusion phenomena: macroscopic and microscopic. In themacroscopic approach, fluxes are the observables. A formal expression for thesefluxes is obtained as functions of thermodynamic forces and of parameters whichare referred to as phenomenological coefficients. In the microscopic approach, thefluxes are estimated by considering atomistic mechanisms wherein the parametersused are the jump frequencies and atomic distances. Unlike the phenomenologicalcoefficients, which are nothing but proportionality constants, these frequencieshave a tangible physical connotation. In fact, a knowledge of the underlying atomicmechanisms is required for a proper description of diffusion phenomena in boththe formalisms.

7.2.1 Diffusion mechanismsIn crystalline solids, the atoms occupy well-defined equilibrium positions.However, they do oscillate about their equilibrium positions with a mean ampli-tude that increases with increasing temperature. There are infrequent occasions




when the relative oscillation of a local group of atoms is such that one of theatoms is ejected out of its original site and gets transplanted in an adjoining equi-librium site. The fraction of an atom’s oscillations which may lead to such a jumpis miniscular, even at temperatures close to the melting temperature of the hostmaterial; however, such jumps do occur and make diffusion possible.

Bocquet et al. (1996) have summarized the possible mechanisms of diffusionin “dense structures” in the following manner:

(1) Exchange mechanisms: One can think of two exchange mechanisms: directand cyclic. In the former, two neighbouring atoms exchange their positions. How-ever, this mechanism is unlikely to operate in dense structures because it involveslarge distortions, leading to unrealistically large activation energies. In the latter, agroup of atoms exchange their positions simultaneously. In this process, the energyinvolved is substantially lower than that associated with direct exchange. However,due to the constraints imposed by the requirement of a collective motion, thismechanism is unlikely to operate. There are, as of now, no experimental evidencesfor the operation of either of these exchange mechanisms in crystalline metallicmaterials. However, in liquid and amorphous metallic materials such cooperativemotions are more likely to operate.

(2) Mechanisms involving point defects: The presence of an optimum numberof point defects, the number being dictated by the temperature, lowers the freeenergy of a crystal. For this reason, under thermal equilibrium a crystal alwayscontains point defects such as a monovacancies, vacancy aggregates (divacancies,trivacancies, etc.) and interstitials. The occurrence of these defects facilitates themovement of atoms without producing very large distortions. It is to be noted thatgenerally the enthalpy of formation of an interstitial is much larger than that of avacancy while the converse applies with respect to the enthalpy of migration ofthese point defects.

In the interstitial mechanism of diffusion, atoms move from one interstitial site toanother. Generally small interstitial atoms, such as those of H, B, C, N, O, diffusethrough the lattice by this mechanism. In the somewhat more complex intersti-tialcy mechanism, atoms move from an interstitial site to a substitutional site andvice versa. This mechanism becomes important under non-equilibrium conditionsbrought about, for example, by irradiation or plastic deformation, when an equalnumber of vacancies and interstitials are created and contribute to diffusion.

The equilibrium vacancy concentration increases with temperature. In metalsand alloys, at temperatures approaching the melting temperature, vacancies occupyabout 1–10 out of 10 000 atomic sites. This small number of vacancies facilitatesthe movement of atoms and, therefore, the process of diffusion. The vacancymechanisms of diffusion constitute the more prevalent operating mechanisms and




involve jumps to nearest neighbour (NN) sites. Jumps to next nearest neighbour(NNN) sites are also to be taken into account, particularly for bcc crystals, wherethe NN and NNN distances do not differ greatly. Apart from monovacancies,vacancy aggregates, particularly divacancies, can also contribute to the diffusionprocess. The role of divacancies becomes more important at high temperaturessince the ratio of the number of divacancies to that of monovacancies generallyincreases with increasing temperature. The vacancy mechanisms are particularlyrelevant while considering the diffusion of substitutional solutes.

(3) Mechanisms involving extended defects: Apart from point (zero-dimensional)defects, in crystalline materials linear (one-dimensional) defects like dislocationsand planar (two-dimensional) defects like grain boundaries, interfaces, surfacesare present. These defects can be considered to be disordered regions in whichatomic migration is relatively easy. This implies that diffusion can occur morereadily via such defects as compared to diffusion occurring in the bulk of thecrystal. Diffusion along these preferred paths is faster and is known as short circuitdiffusion.

In addition to these mechanisms, some less common mechanisms of diffusionhave been invoked for rationalizing diffusion behaviour under special circum-stances. For example, a mixed mechanism has been proposed to account forunusually fast diffusion. In this mechanism, it is envisaged that the fast diffusingsolute dissolves in the host lattice substitutionally as well as interstitially and thatmass transport is dictated by a mixed vacancy and interstitial mechanism involvingvacancy–interstitial pairs. This mechanism is operative in the diffusion of noblemetals in alkali metals (Santos and Dyment 1975).

7.2.2 Flux equations: Fick’s lawsDiffusion takes place by atomic jumps occurring throughout the material andthis has to be related to measurable, macroscopic parameters such as flux (J), orconcentration (c), when one considers mass flow under a concentration gradient.In this context, flux equations, expressed in the form of Fick’s laws of diffusion,are very important.

Fick’s first law can be used to describe mass flow under steady-state conditionsand is identical in form to Fourier’s law for heat flow under a constant temperaturegradient or to Ohm’s law for current flow under a constant electric potentialgradient. Assuming for simplicity that the concentration varies only along thex-direction, this law can be expressed as

J = −D�c�x

(7.1)




where D is the diffusion coefficient or diffusivity and (�c/�x) is the concentrationgradient along the diffusion direction, x. A partial derivative is used to recognizethe fact that the gradient can vary with time. The negative sign on the right-handside of Eq. (7.1) indicates that mass flow occurs down the concentration gradient.The diffusion coefficient, D, depends on the nature of the diffusing species, thematrix in which it is diffusing and the temperature. Under the condition of steady-state flow, the flux, J, is independent of time and remains the same at any cross-sectional plane along the diffusion direction. A schematic representation of theconcentration–penetration profile under steady-state flow is shown in Figure 7.1.When D is independent of concentration, the profile is ideally a straight line.However, when D is a function of concentration, the profile is such that the productD(�c/�x) is a constant. As long as steady-state flow occurs, the profile remainsunchanged over time, in either case.

Fick’s second law is an extension of the first law to conditions of non-steady-state flow. In these cases, at any given instant, the flux is not the same at differentcross-sectional planes along the diffusion direction, x. Apart from this, the fluxchanges with time at the same cross-section; in other words, the concentration–penetration profile changes with time. Fick’s second law for unidirectional massflow under non-steady-state conditions can be expressed as

�c

�t= �

�x

(D�c

�x

)(7.2)

→

→

C 1A

J

J

(b)

(b)

(a)

(a) Steady stateNon-steady state

X

X 1 X 2

dcdx

BC 2

Figure 7.1. Schematic representation of diffusion flux (concentration–penetration profile) under(a) steady-state diffusion and (b) non-steady-state diffusion.




In case D is independent of concentration and thus of x,

�c

�t=D

�2c

�x2(7.3)

In geometrical terms, the interpretation of Eq. (7.3) is straightforward; �2c/�x2

is essentially the curvature of the concentration–penetration profile. Thus, if c isconcave upward in a given region, the concentration increases with time in thatregion.

Empirically it is found that the diffusion coefficient, D, varies exponentiallywith temperature and an Arrhenius type of relationship is valid:

D=D0 exp�−Q/RT (7.4)

where D0 is a pre-exponential constant and Q is the activation energy for diffusion.A plot of ln D versus 1/T should yield a straight line.

7.2.3 Self- and tracer-diffusion coefficients in �-Zr and �-TiSelf-diffusion coefficients of pure Zr and self- and impurity-diffusion coefficientsof a few elements in Ti are plotted in Figures 7.2 and 7.3, respectively. It couldbe seen that diffusion in �-Zr shows an unusual temperature dependence andthe Arrhenius plot indicates a negative curvature which could be due to differentmodifications of the vacancy mechanism. The pertinent factors could be (a) areduction in the self-diffusion enthalpy as a result of lattice softening prior to thetransformation to the �-phase; (b) a temporary trapping of vacancies at impurityatom sites; and (c) rapid diffusion due to the presence of impurity atom–vacancypairs. It has been concluded that the rapid pair formation mechanism controlsself-diffusion in the �-Zr phase (Frank, 1989). Unlike in the case of Zr, Ti exhibitsa normal self-diffusion behaviour in the �-phase.

The large volume of data available on the solute and self-diffusion in �-Ti and�-Zr has been reviewed by Hood (1993). Recently reviewed data pertaining tosome of these diffusivities, published in Defects and Diffusion Forum (1999), arereproduced in Figures 7.4 and 7.5. The main issues addressed in these studiesare: (a) the atomic size effect, (b) the anisotropy of diffusion in the hcp latticeand (c) the effects of impurities. There is a clear correlation between the atomicradius of the diffusing element and the diffusion coefficient in both �-Zr and�-Ti. It is observed that large D values are associated with small radii of impurityatoms. It also appears that diffusion along the a- and c-axes in �-Zr as well as�-Ti is generally different. It has been observed that the diffusion coefficientfor Fe, Cr and Ni along the c-axis in �-Zr is about three times larger than that




1773 1273 973 773

10–11

10–13

10–15

10–17

10–19

10–21

10–23

β-Zr(bcc)

α-Zr(hcp)

T αβ

4 6 8 10 12

D T (

m2 /

s)↑

104/T (K–1) ↑

T (K)↑

Figure 7.2. Arrhenius plots of the tracer self-diffusion coefficient DT in �-Zr and �-Zr(after Frank 1989).

along a direction perpendicular to this axis while for self-diffusion, the ratio ofthe diffusion coefficients parallel and perpendicular to the c-axis is around 1.3(Hood 1974). Accurate measurements with regard to this anisotropic behaviourare very important for Zr in view of its use as a fuel cladding material in nuclearreactors. It should be emphasized in this context that a factor of two in the diffusionanisotropy produces a significant effect on the sink biases and microstructuralchanges through the so-called “diffusion anisotropy difference” (DAD) in materialsduring irradiation (Sizmann 1978).

There is a large scatter in the impurity and self-diffusion coefficients reportedin literature, presumably due to the varying purity of the materials used. It hasbeen realized that single crystals and pure materials are essential for conductingexperiments if reliable and reproducible data are to be obtained.

In general, the vacancy mechanism appears to operate over the whole tempera-ture range in both �-Zr and �-Ti. It has been suggested that monovacancy as well




5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0

1873 1673 1473 1273 1073 873

β(bcc) α(hcp)

Pha

se tr

ansi

tion

P

Ni

CoFe

Mn

Ti

Mel

ting

poin

t

Ti

PP

Fe

Co

Co

NiFe

Mn

Mn

Ni

10–11

10–12

10–13

10–14

10–15

10–16

10–17

10–18

10–10

10–9

10–8

D (

m2 /

s)

104/T (K–1)

T (K)↑

Figure 7.3. Temperature dependence of self- and impurity-diffusion of a few elements in bothparallel (��) and perpendicular (⊥) directions to the c-axis in Ti (after Hood 1993).

as divacancy mechanisms contribute to the overall diffusion, with an increasingparticipation from the latter at higher temperatures (Seeger 1997).

7.2.4 Self- and tracer-diffusion coefficients in �-Zr and �-TiThe self- and tracer-diffusion coefficients for Zr, Ti and Hf, both in the �- andthe �-phases, on a normalized temperature scale, Tm/T (Tm being the meltingtemperature), for some selected bcc metals and impurity-diffusion coefficients forFe, Co and Ni are presented in Figure 7.6. It could be seen that the diffusioncoefficients for Zr, Ti and Hf are larger in the �-phase than in the �-phase andthat this behaviour is more pronounced at low temperatures. It could further be




Ti alloys

0.5

AuCoGaInSiZrHfNiPbSn

1 1.5 2 2.5 3 3.5

103/ T (K)

D (

m2 /

s)

10–12

10–14

10–16

10–18

10–20

10–22

Figure 7.4. Diffusivity of various elements in �-Ti (after Fisher 1999).

D (

m2 /

s)

Cr in ZrCr in Zr–2SnCr in Zr–2.5NbHf in ZrO in ZrZr in Zr–3.2AgHf in Zr–1NbHf in Zr–2.5Nb

0.5 1 1.5 2 2.5 3

103/ T (K)

Zr alloys10–10

10–12

10–14

10–16

10–18

10–20

10–22

10–24

Figure 7.5. Diffusivity of various elements in the �-phase of Zr alloys (after Fisher 1999).

seen that the impurity-diffusion coefficients of Fe, Co and Ni are a few orders ofmagnitude larger than self-diffusion coefficients in �-alloys.

The major deviation from normal diffusion behaviour, generally termed anoma-lous diffusion, observed in several bcc metals and alloys, is characterized by thefollowing features: (a) non-linearity in plots of ln D versus 1/T; (b) very low




Co in β–Zr

Fe in β–Zr

β–Hf

Zr

Ti

Na

K

LiTa

NbVW

MoCr

1.0 1.4 1.8

D (

m2 /

s–1)

Tm / T

10–10

10–12

10–14

10–16

10–18

10–20

Figure 7.6. Comparison of self-diffusion values of selected bcc metals and of very fast impuritydiffusion in bcc metals on a normalized temperature scale (after Petry et al. 1989).

values of the pre-exponential factor, D0; and (c) very low values of the activationenergy, Q.

Extensive work has been carried out with a view to understand the anomalousdiffusion behaviour of �-Zr, �-Ti and of some other bcc metals (Herzig and Kohler1989, Mundy 1992). Several bcc metals, such as V, Nb, Ta, Mo, W, which do notundergo any allotropic transformation, also show slight curvatures in the Arrheniusplots if the temperature range of the study is extended. The anomalous diffusionbehaviour of bcc metals is attributed to the various defect structures present. Someof the models associate these defects with the lattice stability of the bcc phase.The occurrence of the �/� transformation in Ti- and Zr-based systems can berelated to the mechanical instability of the bcc lattice at the transition temperature,arising from a drop in the elastic constant. As experimental evidences for latticesoftening prior to the �/� transition have not been obtained, this model has notfound general acceptance.

A model for anomalous diffusion in the �-phase of Ti- and Zr-based alloys due tothe presence of �-embryos has been proposed by Sanchez and de Fontaine (1978).




B3

B1

B2

C1

C2

C3

V

I

V 1/3 1/2 2/3 I

I2/31/21/3V

C1

C2

C3

B3

B1

B2

Figure 7.7. Saddle point configurations and activation energy plot for diffusion by a vacancymechanism in the bcc structure.

The path of the atom–vacancy exchange process, as schematically illustrated inFigure 7.7, shows that an atom goes through two saddle point configurationswhich resemble a subunit cell cluster of the �-phase. As an atom moves alonga <111> direction to occupy a vacant position, it goes through centroids of tri-angular arrangements of atoms marked B1, B2, B3 and C1, C2, C3. These twopositions mark the atomic arrangements for the two saddle points. As describedearlier, the �-phase can be created from the �-phase by collapsing adjacent �222��layers, keeping the third �222�� layer undisplaced. Therefore, from structural con-siderations, the saddle point atomic configuration can be viewed as an embryonicatomic cluster representing the �-phase. In systems which have a tendency forthe formation of the �-phase, there is a finite probability of encountering such�-embryos in the �-phase; the higher the stability of �, the higher the numberdensity of such �-embryos. The presence of saddle point configurations of acti-vated complexes for diffusion, therefore, enhances the diffusivity. An estimate ofthe population of �-embryos, as a function of temperature, can be correlated withenhanced diffusivity in these systems. At relatively high temperatures, where the�-phase becomes unstable, the population of �-embryos is reduced and as a resultthe diffusivity tends to attain normal values.

Kohler and Herzig (1988) have proposed a model based on soft phonon withoutinvolving the presence of static �-embryos. It is well known that the phononspectra in these systems exhibit soft modes with wave vectors q= 2/3 <111> andq = 1/2 <110>. While the former is associated with the �→ � transformation,the latter relates to the �→ � transformation. Softening of the modes essentiallyleads to a lowering of the energy barrier associated with atomic displacements.




These modes, being of low frequency, contribute to large fluctuations in the dis-placement coordinate, leading to small migration enthalpies and correspondinglyhigh diffusion coefficients. Recently Seeger (1997) has drawn attention to theweakness of this model by pointing out that the soft mode hypothesis is concernedonly with the migration enthalpy and not with the enthalpy of vacancy formation.

7.2.5 InterdiffusionAs discussed in the previous sections, self- or impurity-diffusion relates to masstransfer of a single species under the influence of its own concentration gradientand usually the concerned species only diffuses without effectively altering thecomposition of the host lattice in which it is diffusing.

The situation is different when two blocks of materials with different composi-tions interact with each other at elevated temperatures. The species from both theblocks diffuse into each other. The diffusion process occurring in such a case isreferred to as an interdiffusion or a chemical diffusion process. For simplicity letus consider the two blocks 1 and 2 of solid solutions of the A–B binary systemof different compositions, as shown in Figure 7.8. Let block 1 comprise a solidsolution relatively rich in B and block 2 be made up of a solid solution rich inA. If these blocks, placed in contact with each other, are heated to an elevatedtemperature and held at the temperature for sufficiently long times for diffusion tooccur, atoms of the species A from the A-rich solid solution will diffuse into theB-rich solid solution and B atoms will diffuse from the B-rich solid solution to theA-rich solid solution by a chemical diffusion process. Such a chemical diffusionprocess takes place in order to homogenize the composition. Basically the processoccurs to decrease the free energy of the system and the driving force arises from

A →← B

12

A atom

B atom

A-richsolid solution

B-richsolid solution

Figure 7.8. Diffusion couple between A-rich and B-rich solid solutions of an A–B binary system.




A B2 1

G2 G3 G1

G4

→ →

Mol

ar fr

ee e

nerg

y (G

) →

Concentration of B-atoms (C B) →

Figure 7.9. Schematic molar free energy diagram for the A–B binary solid solution system.

the chemical potential gradient. A schematic molar free energy diagram for theA–B binary solid solution system is shown in Figure 7.9. The compositions of theblocks 1 and 2 are marked on the concentration axis. G1 and G2 are the molar freeenergies at these concentrations. Before the diffusion process starts, the total freeenergy of the system is given by G3. The molar free energy of the homogeneousalloy is given by G4. The A and B atoms will diffuse to lower the free energy toG4. The diffusion process stops with the equilibration of the chemical potentialand the attainment of system equilibrium.

During the interdiffusion process, the species diffuse via the vacancy mech-anism. The resultant concentration profile at the end of diffusion is shown inFigure 7.10. The rate at which each species exchanges with vacancies is its ownintrinsic diffusion coefficient. In general, the intrinsic diffusion coefficient of eachspecies will be different. As atoms jump via vacancies, the flux of atoms andvacancies are equal but in opposite directions. Of the two atomic species A and Bin the A–B binary system, if the intrinsic diffusivity of B is higher than that of A,there will be a net vacancy flux in the direction opposite to that of B atoms. Theintrinsic fluxes of A and B atoms can be calculated by considering their intrinsicdiffusivities and concentration profiles. The fluxes of these species A, B and ofvacancies are represented by vectors and are shown in Figure 7.10. The relationbetween these fluxes is given as

−→JA = −→

JB −−→Jv (7.5)

The net flux of vacancies cause the movement of lattice planes with respect tothe laboratory fixed frame of reference. The vacancy flux, Jv, gives the velocityof the lattice planes and is given as Cov where Co is the number of atoms per unit




CA

O

(a)

(b)

(c)

O

J

JBJvJA

X

JB

JA

Vacanciescreated

Vacanciesdestroyed

OdJ v

dx

Markerplane

Figure 7.10. Concentration profile of A atoms in the A–B binary solid solution system.

volume and is independent of composition. Hence the velocity of lattice planes isgiven as

Co−→v = −→

JA −−→JB (7.6)

This shifting of lattice planes with respect to the laboratory fixed frame ofreference is referred to as Kirkendall effect: such shifting was first observed byKirkendall in his famous experiment on the Cu–Zn system using Mo marker wires.The flux of vacancies requires the creation of lattice sites on one side of the markerplane and destruction of the same on the other side, as shown in Figure 7.10.

As seen from the region, where no diffusion takes place or in other words, asseen from the inertial laboratory fixed frame of reference, the interdiffusion fluxof the component A, JA is equal and opposite to that of the component B, JB, i.e.

JA = −JB (7.7)

The interdiffusion flux is composed of two components, viz., the intrinsicdiffusion flux and the velocity of the lattice planes through which diffusion istaking place.




The nature of the diffusion zone during interdiffusion depends upon the natureof the phase diagram of the system and on the composition of the end members ofthe diffusion couple. If the ranges of composition covered by the starting alloysare within the range of solid solubility, the diffusion zone is of a single phase. If,however, the ranges of composition of the end members of the diffusion coupleencompass intermediate phases, as depicted in the phase diagram, the diffusionzone consists of a number of bands corresponding to the intermediate phases.

The concentration profile in single phase diffusion couple is a smooth sigmoidalcurve between the two compositional limits of the end members of the couple.A typical concentration profile is shown in Figure 7.11; the composition of one endof the couple is c− and that of the other is c+. As discussed earlier in this section,the lattice planes move during interdiffusion. Due to this the original interfacecannot be established for the evaluation of the flux or the diffusion coefficients.

Distance (x ) →

C+

Con

cent

ratio

n (C

) →

C′

C–

Slope = (dc /dx )

Shaded area = x dcC′

C –

Mat

ano

inte

rfac

e ∫

Figure 7.11. Boltzmann–Matano method for evaluation of chemical diffusion coefficient in a binarysystem.




Before the start of diffusion, the flux across the original weld interface is zero.After diffusion commences, fluxes move on both sides of this plane.

The Matano interface is defined as the plane in the diffusion zone across whichthe net flux is zero. This corresponds to the original interface, in general. In acase where there is a molar volume change, a shift is there between the Matanointerface and the original interface. The flux at a given plane with concentra-tion c′ is given as the area under the concentration profile and is expressed as∫x dc. Mathematically the Matano interface is described as

∫ c+c− x dc = 0 where

c− and c+ are concentrations in the right and left endments of the diffusion couple(Figure 7.11).

The interdiffusion coefficient is generally a function of the concentration andhence of the position x. Fick’s second law in such cases is written as

dcdt

= �

�x

(D�c

�x

)(7.8)

The solution of this differential equation cannot be found analytically. Theinterdiffusion coefficients are evaluated using the Boltzmann–Matano method.This method is exclusively employed to evaluate interdiffusion coefficients inbinary as well as ternary systems. According to this, the interdiffusion coefficient,D�c′), at a concentration c′ can be given as

D�c′ = 12t

dxdc

∫ c′

cx dc (7.9)

where t is the duration of the diffusion anneal, dc/dx is the concentration gradientat c′ and

∫ c′cx dc is the area under the concentration penetration profile up to the

concentration c′. The terms used in Eq. (7.9) are indicated in Figure 7.11.As an illustrative example, we consider single phase diffusion in the Zr–Al

system, studied by Laik et al. (2002). The diffusion couples were made withpure Zr on one side and a Zr–2.8 wt% Al alloy on the other side. W markerswere placed at the original weld interface to monitor the movement of latticeplanes. The couple was annealed at 1203 K so that both the end members of thediffusion couple were within the solubility range at that temperature. Figure 7.12shows the microstructure of the resulting diffusion zone. The figure also shows theposition of the W marker at the end of the diffusion process. The correspondingconcentration–penetration profile of Al across the diffusion zone is shown inFigure 7.13 which also marks the position of the Matano interface and the markerplane. The interdiffusion coefficients were evaluated at various concentrations.Intrinsic diffusion coefficients of Zr and Al were also evaluated from the markermovement velocity.




Figure 7.12. Microstructure of diffusion zone between pure Zr and Zr–7.8 at.% Al alloy.

0 50 100 150 200 250

92

94

96

98

100

0

2

4

6

8

Pure Zr Zr–7.8% Al

Distance (μm)

Con

cent

ratio

n of

Zr

(at.%

) Marker positionMatano interface

ZrAl

Con

cent

ratio

n of

Al (

at.%

)

Figure 7.13. Concentration profile across diffusion zone between pure Zr and Zr–7.8 at.% Al alloy.




100 20 30 40 50 60 70 80 90 100

BA (Ti)

L

β

T1

α A2B(Ti2Ag)

AB(TiAg)

1900

1800

1700

1600

1500

1400

1300

1200

1100

900

800

1000

1162

1943

(Ag) γ

2000T

empe

ratu

re (

K )

→

Concentration of B (Ag) in at.% →

Figure 7.14. Phase diagram of A–B (Ti–Ag) binary system with intermediate phases.

The diffusion zone in a multiphase diffusion couple is characterized by bandscorresponding to intermediate phases shown in the phase diagram. Figure 7.14shows the phase diagram of a binary A–B (Ti–Ag) system. It shows terminalsolid solutions along with intermediate phases. Intermediate phases may havesome solubility ranges or may be line compounds. If a diffusion couple madebetween pure A (Ti) and pure B (Ag) is annealed at an elevated temperature,T1, the intermediate phases stable at the annealing temperature appear in thediffusion zone as bands. Figure 7.15(a) shows the layers of intermediate phases in aschematic microstructure of the multiphase system. The concentration–penetrationprofile across the diffusion zone (Figure 7.15(b)) is not continuous and has stepscorresponding to the intermediate phases forming in the system. The step isflat and of a constant concentration for a line compound (A2B). The step has aconcentration range corresponding to the solubility range of the intermediate phase(AB). The concentrations at the phase boundaries generally match well with thosegiven in the phase diagram.

The interdiffusion coefficients at various concentrations are evaluated fromconcentration profiles employing the Boltzmann–Matano method. Generally theinterest in such cases lies in the interdiffusion coefficients in the intermediate phase




A α A2B AB γ B

Distance (x) →

Con

cent

ratio

n of

A (

CA)

→

α

A2B

AB

γ

(a)

(b)

A

Figure 7.15. (a) Layers of intermediate phases in A–B (Ti–Ag) binary system at temperature T1.(b) Schematic concentration profile in A–B(Ti–Ag) binary system at temperature T1.

alone. The interdiffusion coefficient values do not change much with concentrationin the intermediate phase. Hence a single value of the interdiffusion coefficientis calculated for the intermediate phase. A modified Boltzmann–Matano–Heuman(Heuman 1952) method is used to evaluate the interdiffusion coefficient in respectof an intermediate phase. It is expressed as

D = 1

2td

�c

∫ c1/2

c−x dc (7.10)

where d is the width and �c is the solubility range to a pertinent intermediatephase and c1/2 is its average composition as shown in Figure 7.16. An inspectionof Eq. (7.10) shows that the method cannot be used for line compounds since �ctends to zero and D tends to infinity. As discussed in Section 7.2.6, diffusion takesplace to lower the free energy of the system, and the chemical potential or activitygradient is the driving force.

In most situations, activity and concentration have a monotonic relationshipand it is easy to measure the concentration and hence it is usually used to eval-uate diffusion coefficients. Garg et al. (1999) have described thermodynamicinterdiffusion coefficients employing the activity gradient. The method is quiteuseful for the evaluation of diffusion coefficients of line compounds. In a line




d

ΔCC½C

→

x →

C+

C

–

Figure 7.16. Boltzmann–Matano–Heuman method for evaluation of diffusion coefficient in anintermediate phase in a binary system.

compound, even though there is no concentration gradient, there is a finiteactivity gradient and, therefore, thermodynamic interdiffusion coefficients can beevaluated.

As an illustrative example, let us consider the multiphase diffusion in theZr–Ni system (Bhanumurthy et al. 1990). The Zr–Ni phase diagram exhibitseight intermetallic compounds. Diffusion couples made between pure Ni and Zrwere annealed at 1133 K. The microstructure of the diffusion zone, as shownin Figure 7.17, indicated the formation of only five layers of (NiZr2, NiZr,Ni10Zr7, Ni7Zr2 and Ni5Zr) intermediate phases which are stable at this annealingtemperature.

7.2.6 Phase formation in chemical diffusionAs mentioned earlier in Section 7.2.5, the nature of the diffusion zone in chemicaldiffusion depends on the nature of the phase diagram. The product may be asolid solution or an intermediate phase. From thermodynamic considerations alone,all the intermediate phases shown in the phase diagram which are stable at thetemperature of diffusion annealing are expected to appear in the diffusion zone.Figure 7.14 shows the phase diagram of a binary system A–B (Ti–Ag). Diffusionannealing at the temperature T1 would form the intermediate phases A2B andAB which are stable at T1. However, it is observed that sometimes none of thepertinent phases or only a few of these appear in the diffusion zone. For example,Kale et al. (1980) have reported that chemical diffusion in the Fe–Ti system in thetemperature range of �-Fe does not lead to the formation of any of the pertinent




Figure 7.17. Microstructure of diffusion zone in Zr–Ni diffusion couple annealed at 1133 K. Theintermetallic compounds are marked as (1) NiZr2, (2) NiZr, (3) Ni10Zr7, (4) Ni5Zr2 and (5) Ni5Zr.

intermediate phases (TiFe or TiFe2) in the diffusion zone. Bhanumurthy et al.(1990) have reported the formation of a few phases only in the Zr–Ni system.The lack of correspondence in the phases formed in the diffusion zone and thoseshown in the equilibrium phase diagram is due to the fact that quasi-equilibriumconditions prevail during diffusion. Hence the nature of the phases formed inthe diffusion zone also depends on the type of sources of flux, i.e. the type ofdiffusion couple. If the diffusion couple is made between two components whosethicknesses are much larger than the width of the diffusion zone, the couple iscalled an infinite diffusion couple and the sources of both the components areinexhaustible. In such cases, all the intermediate phases depicted in the phasediagram may appear simultaneously as a series of parallel layers in the sameorder as shown in Figure 7.15(a). If, however, the thickness of at least one ofthe components is smaller than the width of the diffusion zone, the couple iscalled semi-infinite and the source of the “thin” component is exhaustible. In suchcases, all the intermediate phases do not form simultaneously. The first compoundformed grows until the exhaustion of the thin component and then the subsequentcompound appears. Let us consider that the component A is thin and exhaustible.At time t= t1, the intermediate phase A2B may form and grow till A is exhausted,as shown in Figure 7.18. Subsequently diffusion between A2B and B will takeplace and intermediate phase AB will form at the interface.




A B

t = 0

A

A2B

B

t = t1

B

A2BAB

t = t 2

Figure 7.18. Sequential formation of intermediate phases in A–B binary system in a semi-infinitediffusion couple.

7.2.6.1 Phase nucleationThe formation of a new phase as a result of diffusion is associated with thefree energy change of the system. The free energy change has two components:(a) chemical free energy change and (b) non-chemical free energy change. Let usconsider the formation of the intermediate phase AB in a diffusion couple madebetween A and B, as shown in Figure 7.19. Let x be the width of the regioncomprising the intermediate phase AB. Due to the formation of AB the originalinterface between A and B is eliminated and two new interfaces between A andAB and B and AB are created.

The free energy of the system decreases when the intermediate phase forms asits free energy of formation is negative. If �Gv is the free energy of formation perunit volume, then the decrease in free energy is given by �Gvx, considering unitcross-sectional area of the couple.

Non-chemical free energy is brought into play due to the creation of newinterfaces and to the mechanical work done consequent to the formation of thenew phase. If �1, �2 and �3 are the interfacial energies for the interfaces betweenA and B, A and AB and B and AB, respectively, then the total interfacial energy

A B A AB B

γ3γ2γ1

Figure 7.19. Formation of an intermediate phase AB in a diffusion couple between A and B.




change per unit area can be given as

�� = �2 +�3 −�1 (7.11)

The total free energy change of the system per unit area can be expressed as

�G= �Gvx+��+�W (7.12)

where �W is the mechanical work done due to volume change.The phase AB would nucleate if �G is negative. The free energy change, �G,

will be always negative if the non-chemical component is negative. In case thenon-chemical component of the free energy is positive, the total free energy changeof the system �G will depend upon the relative magnitudes of the chemical andnon-chemical components of free energy.

In the case of a phase for which the non-chemical component has a large positivevalue, the total free energy change �G will be positive up to a certain thicknessx∗ of the phase. The total free energy change will be negative when the thicknessof the phase is larger than the critical thickness x∗. The phase will nucleate if thiscritical thickness x∗ is of the order of the lattice parameter and will not nucleateif x∗ is much larger than the lattice parameter.

7.2.6.2 Phase growthThe nucleated phase grows if the kinetics of growth is favourable. Let us considerthe growth of the nucleated phase AB in the diffusion couple made between Aand B, as shown in Figure 7.20.

We assume DA �DB; the phase AB forms and grows as the species A diffusesthrough AB into B. The component A, with a flux of J1, reacts with B to formAB as follows:

A�J1 +B → AB (7.13)

A ABJ1

BJ2→ →

Figure 7.20. Growth of an intermediate phase AB formed in a diffusion couple between A and B.




The phase AB decomposes as the species A from AB diffuses into B with aflux J2:

AB → A�J2 +B (7.14)

The net increase in the width of the phase AB can be expressed as

Co

dxAB

dt= J1 − J2 (7.15)

where xAB is the width of the phase AB and Co is as described in Eq. (7.6). Theequation suggests that the growth of this phase depends upon the net flux acrossthe phase.

The conditions under which a subsequent phase nucleates and grows are differentfrom those pertaining to the first phase. Let us consider that the phase AB hasgrown in a diffusion couple between A and B. The driving force for the formationof the next phase A2B will be changed due to the presence of the AB phase.Figure 7.21 shows the free energy values for the AB and A2B phases in the A–Bsystem. After the phase AB is formed, the free energy of the A–AB system is givenby P ′. Hence the driving free energy for the nucleation of A2B is given by P ′Qwhich is lower than the free energy change PQ, associated with the composition-invariant transition from the corresponding solid solution (P) to the intermetalliccompound (Q).

As illustrative examples we consider the formation of intermediate phases duringdiffusion in the two systems Ti–Fe and Zr–Al.

P

P ′

Q

A A2B AB B

Fre

e en

ergy

(G

) →

Concentration of B (C B) →

Figure 7.21. Schematic free energy diagram for the intermediate phases AB and A2B in the binarysystem A–B.




600

800

1000

1200

1400

1600

1800

2000

0 10 20 30 40 50 60 70 80 90 100FeConcentration of Fe (wt%) →Ti

Tem

pera

ture

(K

) →

0 10 20 30 40 50 60 70 80 90 100

Atomic per cent iron

1943

1155

1358

868

← (αTi)17

24.7 51.3

54.1 68.2

1590

1700

1562

75.4 8691.3

(αF

e)

TiF

e 2

TiF

e ←(γ Fe)

Magnetictransition

(βTi)

L

Figure 7.22. Equilibrium phase diagram of Ti–Fe system.

The equilibrium phase diagram of the Ti–Fe system is shown in Figure 7.22.It depicts two intermediate phases, namely, TiFe2 and TiFe. The interdiffusion inthis system has been studied by Kale et al. (1980). They have reported that noneof these phases appears during diffusion annealing of couples comprising pure Feand Ti or even of incremental couples made between pure Fe and the TiFe phase,when annealed in the �-Fe region. But in the case of Fe/Ti–Fe diffusion couplesannealed in the �-Fe range, the formation of TiFe2 is observed. The phases donot grow in the �-Fe range as the flux entering these phases are lower than thoseleaving them, as shown in Table 7.1. The formation of TiFe2 takes place when theFe flux entering the phase is higher than that leaving it (Table 7.1).

Table 7.1. Flux of Fe atoms in Fe–Ti system

Couple Phase Flux atoms (m2/s)

Entering Leaving

�-Fe/�-Ti TiFe2 8�4×1010 1�4×1012

TiFe 1�4×1012 2�0×1014

�-Fe/TiFe TiFe2 1�45×1010 2�86×107




0

–10

–20

–30

–40

–500.0 0.2 0.4 0.6 0.8 1.0

Concentration of Al (mole fraction)

Mol

ar fr

ee e

nerg

y, G

(kJ

/mol

)

α-Zr

Zr2Al3

ZrAl2

ZrAl3

ΔGZ

r 2Al 3

ΔGZ

rAl 2

Figure 7.23. Molar free energy diagram for formation of various phases in Zr-Al system.

The Zr–Al phase diagram indicates the presence of 10 intermediate phases.Interdiffusion in this system has been studied by Laik et al. (2004). They havereported that ZrAl3 forms first and subsequently Zr2Al3 forms in the diffusionzone. The phase which has a higher diffusivity grows first. In the absence ofdiffusivity data, a simple relationship of self-diffusivity with melting temperaturecan be extended to intermediate phases also. Accordingly, ZrAl3, which has alower melting point and hence a higher diffusivity, grows first.

The molar free energies of formation of various phases in the Zr–Al system havebeen plotted in Figure 7.23. When the ZrAl3 phase first forms, a local equilibriumbetween the �-Zr and ZrAl3 is attained and the free energy of the system is givenby the common tangent between the two corresponding curves. It is seen from thefigure that the free energy of formation of Zr2Al3 is more negative compared tothat of ZrAl2, indicating the subsequent formation of Zr2Al3.

7.2.7 Diffusion bondingDiffusion bonding is a technique which employs the solid state diffusion of atomsas a main process for the development of a joint. Diffusion bonding involveskeeping the work pieces to be joined in close contact under moderate pres-sure and heating the assembly at an elevated temperature for a certain duration.




The bonding occurs in three stages. In the first stage, the materials yield and creep.The asperities at the faying surfaces deform and the mating area grows so that alarge fraction of the areas of the faces to be joined comes into contact with eachother. In the second stage, atoms in the contact area diffuse via grain boundaries.This causes a rearrangement of grain boundaries and the elimination of pores inthe bonded zone. In the third and final stage, volume diffusion dominates and thebond is formed.

The success of the bonding process depends on three parameters: (1) bondingpressure, (2) bonding temperature and (3) holding time. The bonding pressureshould be sufficient to make intimate contact at the faying interfaces, deform thesurface asperities and fill the gaps between the mating surfaces. The applied pres-sure breaks the oxide film on the surfaces which, otherwise, hinders the diffusionprocess. It also suppresses the formation of voids due to the Kirkendall effect. Thebonding temperature is generally kept at 0.5–0.7 times the melting temperature ofthe most fusible component in the assembly. The bonding temperature influencesthe interdiffusion across the weld interface and decides the extent of the diffusionzone and the nature of the reaction products formed in it. The optimum thicknessof the bond region maximizes the bond strength.

Diffusion bonding of similar materials results in the formation of a singlephase in the diffusion zone. The width of the diffusion zone, w, follows the timedependence given by the expression

w= Kp t1/n (7.16)

where Kp is the penetration constant and n is the reaction index which is usuallyequal to 2 for the diffusion process. The temperature dependence of the penetrationconstant, Kp, follows an Arrhenius relationship given as

Kp = Ko exp�−Qp/RT (7.17)

where Ko is a pre-exponential factor and Qp is the activation energy for diffusioncontrolled growth. The width of the diffusion zone formed in diffusion bondingcan be estimated from these relations and the bonding parameters, viz., temperatureand time, can be established.

Diffusion bonding of dissimilar materials leads to the formation of intermediatephases in the diffusion zone. These phases are generally brittle in nature and formas bands. Their presence deteriorates the bond strength, and hence the formationof these intermediate phases in the diffusion zone should be avoided.

The formation of intermediate phases can be avoided by using interlayers of dif-ferent materials in between the two parent materials. If diffusion bonding between




A and B leads to the formation of intermediate phases and if C is such that itdoes not form intermediate phases with either A or B in the diffusion zone, theformation of intermediate phases between A and B can be prevented by usingC as an interlayer. Additionally, the interlayers used may help in (a) reducingthe thermal expansion mismatch between the parent materials, (b) inhibiting thediffusion of undesired elements and (c) reducing the bonding temperature and/orpressure. The intermediate phase may have an incubation period for formation andhence its formation can be avoided if bonding is carried out at low temperaturesor for small durations.

As an illustrative example we consider the diffusion bonding of zircaloy-2 withstainless steel. Direct diffusion bonding between zircaloy-2 and stainless steel leadsto the formation of Zr-based intermediate phases in the diffusion zone. Ni fromstainless steel also diffuses into zircaloy-2. Diffusion bonding between zircaloy-2and stainless steel has been studied by Kale et al. (1986) using Fe and Ti asinterlayers. The Ti interlayer has been used in contact with zircaloy-2 as Zr–Ti isan isomorphous system. Similarly the Fe interlayer has been used in contact withTi since in the Ti–Fe diffusion zone intermediate phases do not form, as shownby Kale et al. (1980). It also prevents the diffusion of Ni from stainless steelto zircaloy-2. Diffusion of Ni in zircaloy-2 is undesirable as it enhances hydrideformation. Thus the whole assembly does not form intermediate phases in thediffusion zone and the bonding is good.

Diffusion bonding between zircaloy-2 and stainless steel has also been studiedby Bhanumurthy et al. (1994) using Nb, Cu and Ni interlayers. The assemblyhas the sequence: zircaloy-2/Nb/Cu/Ni/stainless steel. Zr–Nb, Cu–Ni and Ni–Fe,the major components of stainless steel, form solid solutions. The Nb–Cu phasediagram shows the occurrence of intermediate phases. But the temperature and timeof diffusion bonding have been appropriately adjusted such that no intermediatephases form at any interface. Figure 7.24 shows the microstructure of the Zr-2/SSassembly indicating the absence of intermediate phases at various interfaces. Thestrength of the bond is very good.

The use of interlayers in diffusion bonding may pose problems relating to cor-rosion as there are many interfaces in the assembly. In certain applications, the useof interlayers is not permitted due to detrimental physical/chemical service condi-tions. In such cases, direct diffusion bonding can be resorted to and the formationof brittle intermediate phases can be avoided by controlling the temperature andholding time in such a manner that the corresponding incubation period is notcrossed. As an illustrative example, we consider the diffusion bonding of Ti andstainless steel, which has been studied by Kale et al. (1996). Ranzetta and Pearson(1969) have studied the diffusion reaction between Ti and stainless steel and havereported intermediate phase formation at 1223 K. Figure 7.25(a) and (b) show




Figure 7.24. Microstructure of the interface of a Zr-2/SS specimen diffusion bonded at 1173 Kusing several interlayers.

the backscattered electron (BSE) image and composition profile for the Ti/SSdiffusion-bonded specimen at 1223 K for 2 h. At this temperature the diffusionzone shows several intermetallic compounds and the presence of layer structureof �-phase shows inferior mechanical and corrosion properties. Kale et al. (1996)have bonded the samples at 1173 K for a short duration of 2 h to prevent theformation of intermediate phases in the diffusion zone. The bond has been foundto exhibit a good strength of 150 MPa. The diffusion-bonded region has not shownany corrosion in a simulated solution. The joint has good integrity and the leakrate is less than 3×10−9 std cc per s of He.

7.3 PHASE SEPARATION

The tendency towards phase separation in the �-phase of several Ti- and Zr-basedalloys is reflected in the miscibility gap (�1 +�2-phase field) in the correspond-ing �-isomorphous phase diagrams. The occurrence of the monotectoid reaction,�1 →�+�2, is also a manifestation of the phase separation tendency in the �-phasein a number of Ti and Zr alloys containing �-stabilizing elements. Reference maybe drawn to the classification of phase diagrams presented in Chapter 1, where




31

Stainlesssteel

Ret

aine

d β(

Ti,

Fe,

Cr,

Ni)

Sta

inle

ss s

teel

80

60

40

20

00 20 40 60 80

10 μm BSE1 20 kV 20 nA

Titanium

σ-Phase

σ-Phase

CrFeNi

Ti

(a)

(b)

Distance (μm)

Con

cent

ratio

n (a

t.%)

2

31

2

Figure 7.25. (a) Backscattered electron image for the Ti-stainless steel diffusion-bonded spec-imen at 1223 K for 2 h. The micrograph shows �-phase close to steel and also (1) Fe2Ti(Cr),(2) TiCr2 +Fe2Ti(Cr) and (3) FeTi (Cr,Ni) in the diffusion zone. (b) The composition profiles ofFe, Cr, Ni and Ti taken across the region marked at A–B in (a).

the �-isomorphous and the �-monotectoid systems are discussed. It is to be notedthat the excess enthalpy of solution of the � solid solutions of Ti and Zr alloysystems exhibiting the aforementioned classes of phase diagrams is invariablypositive which is a necessary but not a sufficient condition for the existence ofa miscibility gap within a single phase field. The mode of decomposition of asingle phase into the equilibrium two-phase mixture within a miscibility gap is




primarily decided by the consideration of metastability and instability of the alloysystem with respect to composition fluctuations. The decomposition can occureither through a discrete process in which second phase particles nucleate andgrow or by an alternative process in which a concentration fluctuation of infinites-imal amplitude develops homogeneously and is gradually amplified. The formeris known as a heterophase fluctuation or nucleation while the latter involves ahomophase fluctuation or spinodal decomposition. These two processes are illus-trated schematically in Figure 7.26. In the case of a heterophase fluctuation, nucleior energetically stable solute-rich clusters form by thermal fluctuations. It will beshown in the next section that the formation of stable nuclei requires the devel-opment of localized concentration fluctuations of a sufficiently large amplitude.In contrast, when the solid solution becomes unstable, concentration fluctuationswhich are spatially extended but have small amplitudes develop through a processof spinodal decomposition. The thermodynamic conditions for these two distinctmodes of phase separation are discussed in the following section.

7.3.1 Phase separation mechanismsIssues relating to metastability and instability of a solid solution can bebest explained by considering a phase diagram showing a miscibility gap(Figure 7.27(a)) within which the high-temperature �-phase decomposes into amixture of two phases, �1 and �2, which have different compositions but possessthe same crystal structure as the �-phase. Such a phase diagram corresponds to asolid solution with a positive excess enthalpy, �HXS (positive deviation from theideal solution behaviour). The free energy (G) versus composition (c) plots forsuch a binary A–B system at temperatures below Tm, the maximum temperatureof immiscibility, exhibit a doubly inflected shape, as illustrated in Figure 7.27(b).The construction of common tangents to the G–c plots defines the compositionsof the two phases which remain in equilibrium at different temperatures. Theboundary of the two-phase field, also known as the coexistence curve, intersectsthe tie line for a given temperature at compositions given by the points of contactof the common tangent with the two depressions in the G–c plot. The condition ofequilibrium is satisfied between the �1- and the �2-phases, having c1 and c2 atomfractions of B, respectively, as the chemical potentials (given by the intercepts oftangents on the G-axis) of either of the components in the �1- and the �2-phasesare equal at the points where the common tangent touches the G–c plot, as hasbeen illustrated in Figure 7.27(b) for the temperature T1. It is also clear from thisconstruction that the integral molar free energy G(c) of the single phase �-solidsolution between the points c1 and c2 is higher than that of the mixture of the twophases �1 and �2, the free energy G(c) of the �1 + �2 mixture being given by thesegment of the common tangent between c1 and c2.




c6

c4

c3

c2

c3

c3

c1

c1

c2

Stage

Early

Later

Final

2R*

Con

cent

ratio

n, c

cS2

cS1

c4

cS2

cS1

c4

cS2

cS1

Distance, x

(a) “Heterophase”fluctuation

(nucleation)

(b) “Homophase”fluctuation(spinodal)

λ (t1)

λ (t2)

Nuc

leus

Sta

ble

prec

ipita

te

2R (t2)

2R (t3) 2R (t3)

→

→

→

→ →→

→→

→

→→→

→

→→

→

→

→

→

←

←

Figure 7.26. Time evolution of phase separation by two mechanisms are illustrated (time, t1 <t2 < t3): (a) heterophase fluctuation (nucleation followed by growth) is shown by the compositionprofile at a nucleation site; (b) Homogeneous fluctuation (spinodal decomposition) is depicted by thedevelopment of a small amplitude composition fluctuation, subsequent growth of the amplitude andfinally creation of sharp interfaces between the two product phases. Concentrations, ci, are indicatedin Figure 7.27.

Let us now consider an alloy whose composition lies between c1 and c2. If thealloy remains as a metastable single phase on quenching from the �-phase field,its free energy at T1 will be given by the doubly inflected G–c plot shown inFigure 7.27(b). One can examine the decomposition modes of alloys of different




Single phase regionβ Equilibrium

phase diagram(coexistence curve)

Spinodalcurve

Tm

T1

Tem

pera

ture

, T

β1 β2Two-phase regionβ1+β2

c3

c, concentration of solute, B

c1c3 c4 c5 c2 c6cS1 cS2A B

β1(c1) = β2(c2)

Free

ene

rgy,

G

β1

β2

dGdcIdentical

slopes

=dGdcc3 c6

>0<0>0

ΔG>0

ΔG<0

C3–ΔC

C4–ΔC

C3

C4

C3+ΔC

C4+ΔC

d2Gdc2

(b)

(a)

S1

S2

T = T1

μA μA

μβ1(c1) = μβ2(c2)AB

Figure 7.27. (a) Equilibrium phase diagram of a system showing miscibility gap (� → �1 +�2).Chemical spinodal curve (dotted line) is superimposed on the phase diagram. (b) The correspond-ing doubly inflected free energy (G)–concentration plot for a temperature T1. Spinodes (where�2G/�c2 = 0) are indicated by S1 and S2. Insets shown within dotted circles show that a smallamplitude concentration fluctuation is metastable outside the spinodal region (e.g. at c3) whereas thealloy is unstable with respect to concentration fluctuation within the spinodal region (e.g. at c4).

compositions at the temperature T1. The curvature of the G�c curve changes signat the points of inflection, S1 and S2, defined by the following conditions:

(�2G

�c2

)S1

=(�2G

�c2

)S2

= 0 (7.18)




The miscibility gap can be divided into different regimes: those which corre-spond to �2G/�c2 > 0 and those which correspond to �2G/�c2 < 0. The concavityof the G�c plot is upwards in the former case and downwards in the latter.

Let us consider an alloy with a composition given by the point c3 for which�2G/�c2 > 0. Such an alloy is metastable with respect to the development ofconcentration modulations of very small amplitudes. This can be easily shownby constructing a straight line segment joining two points corresponding to alloycompositions given by c3 +�c and c3 −�c. The free energy of the mixture of twophases having compositions deviating from c3 by a small extent, �c, will alwaysremain above the G�c plot in this regime where �2G/�c2 > 0. A tangent drawn atG�c3 intersects the G�c plot at the composition given by c5 . This implies thatthe nucleation of a �-phase particle in the ��c3) alloy becomes thermodynamicallypossible on considerations of the chemical free energy change alone, only if theconcentration of the element B in the nucleus exceeds c5. The chemical free energychange within the volume of the nucleus is given by the vertical drop in the freeenergy from the tangent and is maximized at c6 where the tangent to the G�c curve near the �2-region is parallel to that drawn at c3.

In the regime where the G�c curve is concave downwards �2G/�c2 < 0, anyinfinitesimal fluctuation in concentration will reduce the free energy of the alloy,as shown for the alloy of concentration c4. This means that the alloy is unstablewith respect to concentration fluctuations and, therefore, no concentration barrierexists in the phase separation process. It is also seen that an amplification of theconcentration modulation, which can be represented by an increasing separationof the compositions of the solute-rich and the solute-lean regions, leads to a con-tinuous decrease in the free energy. In the early stages of such a decompositionprocess, known as spinodal decomposition, the solute-rich and solute-lean regionsremain coherent without creating any sharp interfaces between them as shownin Figure 7.26(b) and, spatially speaking, the concentration modulation extendshomogeneously across the entire grain of the parent phase. The rate of spinodaldecomposition is controlled by the interdiffusion coefficient, D, which is nega-tive within the spinodal. The amplitude of the concentration modulation increasesexponentially with time, with a characteristic time constant � = −2/4�2D, where is the wavelength of the modulation (if assumed to be one dimensional). Therate of transformation, therefore, may become very rapid by making as small aspossible – a situation which corresponds to short wavelength modulation or chem-ical ordering. However, this would be an unrealistic effect as spinodal clusteringoccurs in systems exhibiting positive deviations from the ideal solution behaviour,i.e. in systems where like atoms have a tendency to cluster together.

The opposing factors which do not allow the wavelength to decrease below acertain limit are the interfacial energy and the strain energy associated with the




composition modulation. Cahn and Hilliard (1959) and Cahn (1962) have includedthe contributions of the interfacial (gradient) and the strain energies along withthe chemical driving force in working out the composition–temperature domainwithin which spinodal decomposition is possible and the time evolution of thecomposition modulation.

Drawing reference to Figure 7.27, let us consider an alloy of composition c4,which is within the spinodal regime. If this alloy decomposes into two parts withcompositions, c4 +�c and c4 −�c, the chemical free energy change, �Gc, isgiven by

�Gc = 12�2G

�c2��c 2 (7.19)

As mentioned earlier, during the early stages of spinodal decomposition, sharpinterfaces are not created between the A-rich and the B-rich regions. A compositiongradient, which is produced at such diffuse interfaces, also causes an incrementin the energy of the system due to an increase in the number of unlike atomicpairs at these interfaces when compared to the homogeneous solution. This energycomponent, known as the “gradient energy” (�G�), can be expressed in terms of theamplitude, �c, and the wavelength, , of a sinusoidal composition modulation as

�G� = K

(�c

)2

(7.20)

where K is a proportionality constant dependent on the difference in the bondenergies of like and unlike atom pairs.

The strain energy built up in the system due to the development of a compositionmodulation arises essentially from the difference in the atomic sizes of the differentcomponents. The coherency strain energy, �Gs, can be expressed as

�Gs = 2��c 2E′Vm (7.21)

In this expression, is the fractional change in the lattice parameter, a, per unitcomposition change and is given by

= 1a

(�a

�c

)(7.22)

E′ is an elastic constant given by E′ = E/�1 − � where E and � are Young’smodulus and Poisson’s ratio, respectively, for a system with zero elastic anisotropy,and Vm is the molar volume. It is to be noted that �Gs is independent of .




When all the contributions to the total free energy change, �G, accompanyingthe formation of a composition modulation are considered, one gets

�G=[�2G

�c2+ 2K�2

+22E′Vm

]��c 2/2 (7.23)

The condition for a homogeneous solid solution to be unstable and to decomposespinodally is

�2G

�c2>

2K�2

+22E′Vm (7.24)

At the spinodal boundary, the system becomes just unstable, implying thatthe wavelength associated with the fluctuation will be approaching infinity. Thecoherent spinodal boundary, therefore, is given by the condition

�2G

�c2> 22E′Vm (7.25)

The coherent spinodal line remains entirely within the chemical spinodal(�2G/�c2 = 0). The maximum wavelength of the composition modulation that candevelop within the coherent spinodal is given by the condition

�2 = −2K(�2G

�c2+22E′Vm

)(7.26)

Let us now focus our attention on an alloy of composition c3, which is locatedoutside the spinodal but inside the miscibility gap at T1. It is clear that a smallamplitude fluctuation in composition will lead to an overall increase in the freeenergy as all the three components of free energy are positive (�2G/�c2 is posi-tive outside the spinodal). A critical fluctuation which will be thermodynamicallystable will necessarily have a large deviation from the bulk composition. Cahnand Hilliard (1959), in their non-classical nucleation theory, have considered thatthe inhomogeneous solid solution in its metastable state contains homophase fluc-tuations with diffuse interfaces and a composition which varies throughout thecluster. A critical fluctuation is, therefore, characterized by its spatial extension orwavelength, , and its spatial concentration variation. The free energy change asso-ciated with the transformation from a homogeneous system to an inhomogeneoussystem can be expressed as

�G=∫V�g�c −g�co +K��c 2 +2E′�c− co

2�dV (7.27)




where g�c and g�co correspond to the free energy densities of the solid solutionsand co is the composition of the homogeneous solid solution. For the case beingdiscussed, co = c3. Neglecting the elastic free energy term, 2E′�c− co

2, andassuming isotropy, the composition profile c�r of a spherical fluctuation (r beingthe radial distance from the fluctuation centre) is obtained from a numericalintegration (Cahn and Hilliard 1959) as

2Kd2c

dr2+4

K

r

dcdr

= �g

�c

∣∣∣∣c − �g

�c

∣∣∣∣co

(7.28)

with the boundary conditions dc/dr = 0 at the fluctuation centre (r = 0) and ata point far away from it (r → �) where c = co. The critical nucleus can thenbe defined as that fluctuation which remains in an unstable equilibrium withthe matrix. The free energy change, �G∗, associated with the formation of acritical nucleus can be worked out by considering a minimum nucleation barrier(de Fontaine 1982) and is expressed as

�G∗ = 4�∫ �

o

[�g�c∗ +K

(dcdr

)2]r2 dr (7.29)

where c∗ corresponds to the composition of the centre of the fluctuation (nucleus).The free energy for nucleation, therefore, is given by the vertical distance fromthe tangent drawn on the G–c plot at co (co = c3 in Figure 7.27(b)). As theinitial alloy composition, co, is chosen nearer to the spinodal point S1, the soluteconcentration at the centre of the nucleus decreases and the composition profilebecomes more diffuse. Furthermore, with the initial composition approachingthe spinodal point, the spatial extent of the critical fluctuation, R∗, increasesrapidly and finally approaches infinity at co = cS1 . The non-classical nucleationtheory, therefore, predicts a discontinuity in the decomposition mechanism atthe spinodal line. The radius, R∗, of the critical fluctuation outside the spinodalregime and the wavelength, �c, of the spinodally decomposed structure have beenplotted in Figure 7.28 for alloys of different initial compositions for illustratingthe discontinuity at the spinodal line (Wagner and Kampman 1991).

The generalized theory of phase separation proposed by Binder and co-workerstakes into account nucleation in the metastable regime as well as spinodal decom-position in the unstable regime. The theory is essentially based on a clusterdynamics approach in which the attachment and splitting of clusters are consid-ered. A homogeneous supersaturated solid solution, when aged at a temperaturesufficiently high for solute diffusion to occur, will contain microclusters of soluteatoms. A cluster containing i number of atoms is called an i-mer. The theory of the




Coexist.curve

Spinodal curve

Metastable

λc (spinodal)

R* (Cahn–Hilliard)

R* (Binder)

cαe cs

α csβ cβ

e

Concentration, c

BA

To

↑T

↑R*, λ c

Unstable

Figure 7.28. Phase diagram of a binary system with a miscibility gap. The coexistence curve andspinodal curves are also shown. The variation of characteristic length c and R∗, according todifferent theories, with composition is shown.

nucleation of second phase particles in a solid solution has both static and dynamiccomponents. The free energy of formation of an i-mer and the distribution, f(i),of i-mers are treated in the static part. In the dynamic part, the kinetics of thedecomposition of the solid solution, which is described by the given distributionof non-interacting microclusters, are calculated in terms of the time evolution off(i), which finally leads to the determination of the rate of formation of stableclusters, i.e. the nucleation rate. A detailed account of the kinetics of the earlystages of decomposition and a comparison between the theoretical approaches andthe predictions thereof for both classical and non-classical nucleation has beenprovided by Wagner and Kampman (1991).

Binder (1977) has considered the time evolution of f�i� t , the cluster sizedistribution function, in terms of the rates of the following four processes:




(1) splitting of (i+ i′)-mers into i- and i′-mers(2) splitting of i-mers into (i− i′)- and i′-mers(3) coagulation of (i− i′)-mers and i′-mers into i-mers(4) coagulation of i- and i′-mers into (i+ i′)-mers.

Thus the kinetics of the time evolution of f�i� t is given by

d

dtf�i� t =

�∑i′=1

�i+i′�i′f�i+ i′� t − 12

i−1∑i′=1

�i�i′f�i� t

+ 1

2

i−1∑i′=1

�i−i′�i′f�i′� t f�i− i′� t −

�∑i′=1

�i�i′f�i� t f�i′� t

(7.30)

where � and � refer to the rate constants of the respective processes. These tworate constants can be replaced by a single rate constant, W, using the followingexpression which relates the cluster concentration, C�i , in thermal equilibriumwith the metastable matrix, and the rate constants

W�i� i′ ≡ �i+i′�i′C�i+ i′ = �i�i′C�i C�i′ (7.31)

The time evolution of the cluster concentration can be obtained by numericalintegration of Eqs. (7.30) and (7.31), provided inputs such as the initial clusterdistribution, f�i� t = 0 , W�i� i′ and C�i are available. In the absence of thenecessary data for common alloys, solution of the kinetics equations has beenobtained (Mirold and Binder 1977) with some plausible input values. These resultshave provided a profound, though qualitative, insight into the dynamics of clusterformation and growth.

The most important feature of the nucleation theory of Binder and co-workersis that it addresses nucleation in the metastable regime as well as spinodal decom-position in the unstable regime of the miscibility gap. The size of the criticalcluster, R∗, predicted from this theory exhibits a monotonic variation when plottedagainst the composition, as shown in Figure 7.28. The divergency of both thecritical radius, R∗, of the nucleus and the wavelength, c, of the critical fluctuation,inherent in the Cahn–Hilliard theories in the vicinity of the spinodal line, is notseen in the critical cluster size.

7.3.2 Analysis of a phase diagram showing a miscibility gapThe miscibility gap is the simplest system for a phase diagram analysis whichinvolves the determination of thermodynamic quantities from a given phase dia-gram. The fact that a single phase decomposes into a mixture of two phases, all




the three having the same crystal structure, is basically responsible for making thephase diagram analysis very simple. Let us consider the miscibility gap shownin Figure 7.27. The condition of equilibrium between the two phases, �1 and�2, with compositions given by c1 and c2, can be graphically represented by theconstruction of a common tangent. The numerical computation of thermodynamicquantities can be carried out using the method elaborated by Rudman (1970) andoutlined in the following.

The excess integral molar thermodynamic quantity, QXS, is expressed in termsof a power series of the form

QXS = c�1− c N∑i=0

aQi ci (7.32)

where c, as defined earlier, is the atomic fraction of the solute and aQi are thecoefficients of the series for the thermodynamic quantity Q. Assuming that thespecific heat, Cp, is constant over a temperature range, the relative integral molarenthalpy, H, and entropy, S, are expressed by the following equations:

H =H0 +Cp�T −T0 (7.33)

S = S0 +Cp ln�T/T0 (7.34)

where H0 and S0 are the integral molar enthalpy and entropy at the referencetemperature, T0.

Using Eqs. (7.32)–(7.34), the composition and the temperature dependence ofthe excess integral molar free energy of the �-phase is expressed as

G�XS = c�1− c

N∑i=0

aGi ci (7.35)

where aGi = aH0i − Ta

S0i + a

Cpi ��T − T0 − T ln�T/T0)] and aGi , aH0

i , aS0i and a

Cpi

represent the expansion coefficients of the excess integral molar free energy,enthalpy, entropy and specific heat, respectively. It is possible to express the freeenergy as a function of temperature and composition if one can find out the valuesof these coefficients. In a system showing a miscibility gap, the computation ofthese coefficients can be performed by using the following equilibrium conditionwhich equates the partial molar free energies of a given component in the twosolutions corresponding to the compositions c1 and c2 given by the phase boundary(coexistence curve) at a given temperature, T:

��1A �c1� T1 −�

�2A �c2� T1 = 0 (7.36)




��1B �c1� T1 −�

�2B �c2� T1 = 0 (7.37)

Substituting the values of c1 and c2 for different values of T, it is possible toevaluate aGi by a least square analysis.

Menon et al. (1978) have presented the computed excess enthalpy, excessentropy and free energy as functions of compositions for the �-phase in theZr–Nb system. It has been shown by Rudman (1970) that the accuracy of thecalculation of excess thermodynamic quantities does not significantly improve bythe inclusion of more than two terms in the series expansion and in view of thiseach of the excess enthalpy and entropy of the �-phase in the Zr–Nb system hasbeen expressed in the form of a series containing two terms. The values of thecoefficients obtained using the miscibility gap data from the phase diagram are

aH0 = 3352�39 cal/mol� aS0 = −0�6322 cal/mol K

aH1 = 1096�74 cal/mol� aS1 = −1�9094 cal/mol K

Figure 7.29 shows the quantities H�XS and S

�XS for the �-phase as functions of

composition. The asymmetrical miscibility gap in the �-phase seen in the phasediagram is reflected in the H�

XS–c plot. The negative value of S�XS is consistentwith the clustering tendency which is clearly revealed from the positive deviation(positive value of HXS) of the �-phase from the ideal solution behaviour. Free

800

600

400

200

00 0.2 0.4

Concentration, c

(a) (b)

0.6 0.8 1.0 0

βS

xs

(cal

/mol

k)

βH

xs

(cal

/mol

) →

–0.4

–0.2

0

0.2 0.4 0.6 0.8 1.0

Concentration, c

Figure 7.29. Computed (a) excess enthalpy and (b) excess entropy of the �-phase of Zr–Nb alloysas a function of Nb concentration, c.




200

0

–200

–400

–600

–800

–1000

650 K

750 K

1050 K

1450 K

G (c

al/m

ol)

→

0 0.2 0.4 0.6 0.8 1.0

Concentration, c

Figure 7.30. Computed free energy (G) versus Nb concentration (c) plot for the Zr–Nb system atvarious temperatures.

energy versus composition (G–c) plots (Figure 7.30) constructed at different tem-peratures show the doubly inflected shape at temperatures below about 1200 K,implying that the critical solution temperature of the �-phase is nearly 1200 K.Flewitt (1974) has studied the isothermal decomposition behaviour of �-Zr-Nballoys and has experimentally determined the critical solution temperature to beclose to 1250 K.

Modelling of the Zr–Nb equilibrium phase diagram has been developed on aregular solution approximation by Kaufman (1959) and he has obtained a fairlygood agreement with the experimental phase diagram by taking the values ofthe interaction parameters of the �, � and liquid phases to be 33472, 18410 and6276 J/mol, respectively. The lattice stability terms used by Kaufman are given inTable 7.2.




Table 7.2. Lattice stability terms for Zr and Nb.

Element �G�→� (J/mol; T, in K) �G�→L (J/mol; T, in K)

Zr 3.766(1144 − T) 8.368(2125 − T)Nb −3.347(1875 + T) 8.368(2740 − T)

Abriata and Bolcich (1982) have made a critical assessment of experimentalphase diagram data and results of thermodynamic modelling with regard to theZr–Nb system. They have indicated that a reasonable agreement exists between themiscibility gap determined by different investigators and that phenomenologicalthermodynamic calculations are consistent with the experimental phase diagram.However, the ideal situation of coupling phase diagram and experimental ther-modynamic data is still somewhat elusive for want of reliable thermodynamicmeasurements.

The determination of the chemical spinodal line from the G�–c plots(Figure 7.30) becomes possible by locating the points of inflection where thefollowing condition is valid:

�2G�

�c2= 0 (7.38)

The chemical spinodal (line 1) thus determined is shown superimposed on theZr–Nb phase diagram in Figure 7.31. Cook and Hilliard (1965) have derived thespinodal line in a binary phase diagram in terms of the critical temperature, Tc,and the critical concentration, cc ,of the miscibility gap as

cscc cecc

[1−0�422

T

Tc

](7.39)

where cs and ce are the spinodal and the equilibrium concentrations at a temperatureT. This formulation assumes a parabolic free energy function in the vicinity of cc.The spinodal curve (line 2) obtained from Eq. (7.39) is also plotted in Figure 7.31to demonstrate the close agreement between the results obtained from the twoapproaches discussed.

The coherent spinodal is defined by the equality

�2G

�c2+2K�2 +22Y = 0 (7.40)

where � is the wave number (�= 2�/�) of the sinusoidal concentration fluctuation,K is the gradient energy coefficient and Y is a function of the crystallographic




600

800

1000

1200

1400

Tem

pera

ture

(K

)

0.0 0.2 0.4 0.6 0.8 1.0

Atomic fraction of niobium →

α + β1

β

βI + βII

α + βII

1

2

Coherentspinodal

Chemicalspinodal

Figure 7.31. The phase diagram of the Zr–Nb system with superimposed chemical spinodal lines(line 1, determined from G–c plots; line 2, obtained from Eq. (7.39)) and the coherent spinodal line.

direction (associated direction cosines being l, m and n) along which the concen-tration modulation develops. Y is given by

Y = 12�C11 −2C12

[3− C11 +C12

C11 +2�C44 −C11 +C12 �l2m2 +m2n2 + l2n2

](7.41)

where Cij are elastic constants. The fractional changes in the lattice parameter, ,is given by Eq. (7.22).

Hilliard (1970) has shown that

T ∗S −TS = 22Y/S′′ (7.42)

where T ∗S and TS, respectively, indicate the temperatures of the coherent and the

chemical spinodals and S′′ = �2s/�c2, where S is the entropy per unit volume. Usingthe values of entropy obtained from the phase diagram analysis of Menon et al.(1978) and from the reported values of elastic constants (Goasdoue et al. 1972) andlattice parameters (Pearson 1967) the coherent spinodal line has been determinedfrom Eq. (7.40) and has been plotted in Figure 7.31 One of the advantages of thephase diagram analysis as elaborated here is that the spinodal line at temperatureslower than the monotectoid temperature, Tmono, can also be evaluated without usingthe extrapolated phase boundaries.

An analysis of the phase diagram of the Zr–Ta system has also been performedby Menon et al. (1979) who have shown that the free energy hump between the




�1- and �2-phases is much larger in the case of Zr–Ta than that in the case ofZr–Nb; this is consistent with the higher Tc associated with the former system.

7.3.3 Microstructural evolution during phase separation in the �-phaseExperimental investigations on spinodal decomposition in Ti- and Zr-based alloysare only a few in number. Flewitt (1974) has reported the results of a systematicstudy on the decomposition behaviour of the �-phase within the miscibility gap inrespect of the Zr–Nb system. A summary of these results is given in this section.

The maximum of the miscibility gap and Tc for both chemical and strain spin-odals lie at a composition of Zr–60 at.% Nb, which was selected by Flewittfor testing the occurrence of spinodal decomposition in the Zr–Nb system. Onquenching this alloy in iced brine from a temperature above 1250 K, the homo-geneous solid solution is retained in a metastable state, as demonstrated by theresults of X-ray and electron diffraction. On subsequent ageing at temperaturesabove 883 K (the monotectoid temperature, Tmono), fine modulations, as observedin TEM images obtained with the electron beam aligned along an <100> direc-tion, develop and the contrast grows with increasing ageing time. The modulationsare best observed when a g = �h00� beam is excited and these are seen to beperpendicular to g, the excitation vector. The corresponding diffraction patternsshow the presence of side bands, the spacing, �r, of the side band from the main�hkl� reflection (at a distance r from the 000 position in the reciprocal space)giving the dominant wavelength, �, of the concentration fluctuation (Daniel andLispon, 1944):

�= har

�h2 +k2 + l2 �r(7.43)

Wavelengths measured from bright- and dark-field micrographs have beenshown to be consistent with those calculated from side band spacings.

It is to be noted that the same alloy (Zr–60 at.% Nb), when subjected to asomewhat slower quench (such as water quench or oil quench), shows evidences ofconcentration modulations along <100> directions as revealed by the occurrenceof mottling in diffraction contrast images and by the presence of side bands indiffraction patterns. This observation suggests that suppression of the spinodaldecomposition, which is a homogeneous transformation, can be achieved onlypartially by kinetic means, i.e. by restricting the number of diffusive atomic jumps.The fact that the transformation is homogeneous is also testified by the observationthat the modulation develops uniformly in the entire grain right up to the grainboundary.

As has been pointed out in Section 7.3.1, the evolution of non-localized, spatiallyextended concentration fluctuations, the amplitude of which increases gradually




with ageing time, can be defined as a true “spinodal decomposition” in the senseof the Cahn–Hilliard definition (1959). The time dependence of the concentrationc(r,t) at a position r at an instant t is given by the linearized diffusion equation

�c�r� t

�t= M

�2

[(�2G

�c2

)∣∣∣∣c0

+22Yc�r� t −2K�4 c�r� t

](7.44)

where M is the atomic mobility and is related to the interdiffusion coefficient Dby the relation

M = Dnv

∣∣∣∣�2G

�c2

∣∣∣∣c0

(7.45)

As M is always positive, D takes the sign of �2G/�c2 and is thus negative insidethe spinodal regime, giving rise to an uphill diffusion of solute atoms.

The reported values of the wavelength as a function of the ageing time (Flewitt1974) show that the wavelength remains more or less constant at the very earlystages of the decomposition process and at the subsequent stages it follows acoarsening law which can be expressed as �= K1 t

1/3 (Figure 7.32(a)).The rate constant (K1) in the Lifshiftz–Wagner equation is given by

K1 =[

8�DcV 2m

9RT

]1/3

(7.46)

where � is the matrix/precipitate interfacial energy, D is the coefficient of diffusionof the solute in the matrix and Vm the molar volume of the precipitate.

From the plot of logK3 versus 1/T, the activation energy for the coarseningof modulations has been determined to be 64 ± 10 kcal/mol which is in goodagreement with the value of 70 kcal/mol reported by Hartely et al. (1964) forself-diffusion in the �-phase of Nb–Zr alloys.

The morphology of the spinodally decomposed alloy is of interest and deserves aspecial mention, as it is through this distinctive criterion that a spinodal decomposi-tion can be distinguished from a precipitation process. A recent review by Wagnerand Kampman (1991) emphasizes that in view of the current theoretical develop-ments, summarized by Binder (1991), it is rather difficult or even impossible toassess on a thermodynamic basis whether an alloy system is truly quenched into andaged within the spinodal regime of the miscibility gap. It is suggested that the mor-phological criterion can be used for identifying a spinodal decomposition process.

The observed morphology (Flewitt 1974) of the spinodally decomposed Zr–Nballoys of both symmetric (∼60 at.% Nb) and asymmetric compositions comprises




0.0 0.2 0.4 0.6 0.8 1.0 1.2

1/T × 103 K–1

105

106

107

108

Rat

e co

nsta

nt K

1 (

Å /h

)3

3

(a)

1.00.80.60.40.20–300

–200

–100

0

T = 883 K

G (

cal/m

ol)

G

β

G

α

c α

c β1

c β2

Concentration, c

N

P

(b)

Figure 7.32. (a) Rate constant for coarsening of the spinodally modulated structure as a function ofreciprocal temperature in Zr–Nb alloys (Flewitt 1974); (b) free energy–concentration plots for the�- and the �-phase in the Zr–Nb system at the monotectoid temperature 883 K.




a rod-like arrangement of solute-rich and solute-lean regions with the axes lyingalong the <100> directions. The irregularity of the initial morphology, as pointedout by Cahn (1962), arises from the range of wavelengths centred on a dominantwavelength at any instant. The diffracted intensity distribution around the sidebands also indicates the presence of such a spectrum of wavelengths.

Cahn (1962) has pointed out the importance of elastic anisotropy with regardto the final form of the periodic concentration fluctuations. The elastic anisotropyfactor, 2C44/�C11 −C12), is less than unity for pure Nb but the addition of Zr to alevel of about 40 at.% increases this factor to a value larger than unity. Under sucha condition, concentration waves develop as plane waves on all the three {100}planes, so that an interlocking network of <100> rods, alternately enriched anddepleted in solute content, appears in the spinodally decomposed microstructure.The structure becomes more and more interconnected with increasing reactiontime. Lattice defects and grain boundaries play little role in the morphologicalevolution, indicating the homogeneous nature of the transformation.

As the concentration waves get amplified, a situation is reached where thestrain gradient at the interface between the solute depleted and enriched regionsis adequate for nucleating misfit dislocations which are arranged in a hexagonalnetwork built up from three sets of dislocations. Contrast analysis has shown thatthese two sets of dislocations are associated with a<100> Burgers vectors andhave primarily edge characters while the third set comprises a<110> screw-typedislocations. For the Zr–60 at.% Nb alloy the critical wavelength for the lossof coherency has been found to be about 30 nm at 973 K. The wavelength atwhich coherency is lost is seen to decrease with the decomposition temperature,as is expected from the fact that the amplitude of the concentration wave (andcorrespondingly the mismatch between the regions lean and rich in the solute)increases with decreasing reaction temperature.

7.3.4 Monotectoid reaction – a consequence of �-phase immiscibilityThe tendency of phase separation in the �-phase, resulting from the positivedeviation from the ideal solution behaviour, is manifested in the appearance ofmiscibility gaps in several binary Zr and Ti alloys. The occurrence of miscibil-ity gaps in systems like Zr–Nb, Zr–Ta as reported quite sometime back in thephase diagram compilation by Hansen (1958). A number of binary Ti alloy phasediagrams such as Ti–V, Ti–Nb and Ti–Mo were classified some years back as�-isomorphous systems. Recent assessments of many of these phase diagramshave identified the presence of miscibility gaps in the �-phase field.

The interplay of the two-phase transformation tendencies, namely, the �/�transformation of the terminal solid solutions, rich in either Ti or Zr, and the phaseseparation within the �-phase field, is responsible for bringing about monotectoid




phase reactions in these systems. This point will be explained by taking theexample of phase equilibria in the Zr–Nb system. In a later part of this section,other systems showing monotectoid reactions will be discussed.

At temperatures between 883 K, the monotectoid temperature in the Zr–Nbphase diagram, and 1135 K, the temperature at which pure Zr undergoes the �/�phase transformation, the �- and the �-phases remain in equilibrium. The tie linewithin the (�+�)-phase field gives the compositions of the two phases, c� andc�, in equilibrium, which is defined by equating the partial molar free energies ofthe two components in the �- and the �-phases:

��A�c��T = �

�1A �c�1� T (7.47)

��B�c��T = �

�1B �c�1� T (7.48)

The numerical values of the right-hand sides of these equations at various tem-peratures can be obtained from the analysis of the miscibility gap in the phase dia-gram, described in Section 7.3.2. The partial molar free energies for the �-phase fordifferent compositions and temperatures can then be obtained using Eqs. (7.47) and(7.48). Thus G� versus c plots for different temperatures, both above and below Tm,can be obtained. Using this formalism, Menon et al. (1978) have obtainedG–c plotsfor both the �- and �-phases at different temperatures. The plots corresponding tothe monotectoid temperature are given in Figure 7.32(b) as an illustrative example.

Moffat and Kattner (1988) have considered not only the stable equilibria involv-ing the �-, �1- and �2-phases but also the metastable equilibria in respect of the�-, �1- and �2-phases using the CALPHAD approach in which the free energyG�c�T of the given phase is expressed in terms of the lattice stability parametersof pure elements in the different phases and interaction parameters:

G�c�T = �1− c GA�T + cGB�T +RT�c ln c+ �1− c ln�1− c �+ l′c�1− c (7.49)

where GA or GB is the lattice stability of element A or B, respectively, and l′ isthe interaction parameter. These parameters, taken from Kaufmann and Bernstein(1970), are listed in Table 7.3. Figure 7.33(a) shows the calculated phase diagramof the Ti–Nb system. It may be noted that though the shape of the (�+ �)/�transus line is suggestive of a positive deviation from ideality, the extent of thisdeviation is not large enough to cause the appearance of the miscibility gap andthe monotectoid reaction. In contrast, the calculated metastable phase diagraminvolving the �-, �1- and �2-phases shows both the monotectoid reaction and themiscibility gap (Moffat and Kattner 1988). The regular solution parameters usedby Moffat and Kattner (1988) for generating the metastable phase diagram arealso listed in Table 7.3.




Table 7.3. Regular solution parameters for the Ti–Nb stable and metastableequilibria.

Lattice stability parameters

�G�→LTi �T = 20 610−12�134T �G�→L

Ti �T = 19 337−12�510T

�G�→LTi �T = 16 259−8�368T �G

�→�Nb �T = −6276−3�347T

�G�→LNb �T = 16 669−11�715T �G�→�

Nb �T = −1273−0�376T

�G�→LNb �T = 22 945−8�368T �G

�→�Nb �T = −7549−3�723T

�G�→LNb �T = 15 396−12091T

Interaction parameters

l′�� = 13 075 l′�� = 2510l′�� = 13 075

All values of �G and l′ are in J/mol and that of temperature in Kelvin.

β

α

Weight per cent niobium

20100 30 5040 60 10070 80 90

20100 30 5040 60 10070 80 90

Atomic per cent niobium NbTi

400

500

600

700

800

900

Tem

pera

ture

(°C

)

(a)

676°C

425°C

Weight per cent vanadium

Atomic per cent vanadium

50

Tem

pera

ture

(°C

)

500

Ti

200

100To

00

300

400β→α

β→ω

10 20

To

30 40

α

0

600

700

800

900

100040302010

18.51

5.46

β1

β2

94.54

50 60 70 80V

(b)

10090

807060

81.49

10090

Figure 7.33. (a) The calculated phase diagram (Moffat and Kattner 1988) of the Ti–Nb system (solidline). Experimental points for locating the boundaries of �+�/� phase fields are marked. (b) Thecalculated phase diagram (Moffat and Kattner 1988) of the Ti–V system showing the equilibrium�1 → �+�2 reaction at 676 C and the metastable �1 → �+�2 reaction at 425 C. The equilibriumphase diagram (solid line), the metastable phase diagram (dashed line) involving �- and �-phases,To lines (dotted lines) for the �→ � and the �→ � transformations are also shown.




The tendency of phase separation in the �-phase has now been recognized in anumber of systems which have earlier been designated as �-isomorphous systems.The equilibrium and the metastable phase diagrams of the Ti–V system, calculatedby Moffat and Kattner (1988), are superimposed in Figure 7.33(b) to illustratethat two monotectoid reactions, namely, the equilibrium �1 → �+�2 reaction andthe metastable �1 → �+ �2 reaction, can indeed occur in this system. Similarconclusions have also been arrived at for the Ti–Mo system.

7.3.5 Precipitation of �-phase in supersaturated �′-phase duringtempering of martensite

Tempering of martensite is an important heat treatment often employed in heat-treatable alloys of Ti and Zr. Unlike the steel martensites, supersaturation in Ti andZr martensites is with respect to only substitutional alloying elements. Thereforeduring a tempering heat treatment the martensitic phase rejects those substitutionalsolute atoms which remain in excess of the solubility limit at the temperingtemperature. Depending on the nature of the solute element, the precipitating phasewhich separates during tempering can be an intermetallic phase rich in the solute orthe �-phase. A great majority of heat-treatable alloys of Ti- and Zr-based alloys are�+� alloys in which a combination of �- and �-stabilizing elements are added. Inthe as-quenched state, such alloys consist of either a fully martensitic �′ or a mixedprimary �+ martensitic �′ microstructure. On tempering, the martensitic �′-phasedecomposes to produce a distribution of the �-phase. The miscibility gap in the�-phase plays an important role in deciding the phase transformation sequenceduring the tempering of such �+� alloys. This point will now be examined bytaking the illustrative example of Zr–Nb alloys.

The identification of the precipitating phases, the orientation relations betweenthe precipitates and the matrix and the thermodynamics of phase evolution duringtempering in the Zr–Nb system have been reported by Banerjee et al. (1976),Menon et al. (1978) and Luo and Weatherly (1988). It has been observed thatthe precipitation process is very sluggish at temperatures lower than 773 K (about100 K below the monotectoid temperature of 883 K). The volume fraction of theprecipitate phase has been found to be quite small, consistent with what is predictedfrom the equilibrium phase diagram. At temperatures below the monotectoid tem-perature, the equilibrium-precipitating phase is the Nb-rich �2-phase (containingabout 85 at.% Nb) which can be distinguished from the �1-phase (Zr–20 at.% Nb)from an accurate determination of the lattice parameter of the precipitating phase.

The most striking observations have been the following:

(1) The phase which precipitates in the temperature interval of 773–883 K is the�1-phase and not the equilibrium �2-phase (Figure 7.34).




(a)

(c)

(b)

(d)

mm

m

Figure 7.34. Precipitation along the twin boundaries in internally twinned martensite plates inZr–2.5 Nb (a) �2-phase precipitation at the tempering temperature of 773 K. (b) and (c) �1-phaseprecipitation at the tempering temperature of 823 K, the dark-field image (c) is taken with a {110}�1

reflection. (d) Diffraction pattern corresponding to (b) and (c) showing superimposition of reciprocallattice sections of two twin-related �- and the precipitate �1-orientations.

(2) At temperatures below 773 K and above 883 K, precipitation of the equilibriumphases, �2 and �1, respectively, takes place during tempering of the Zr–2.5 Nbmartensite.

(3) On tempering, the Zr–5.5% Nb martensite initially reverts back to the parent�-phase via a composition-invariant process; subsequently the reverted �-phasetransforms into a structure consisting of Widmanstatten �-plates in a �1-matrix.

These observations have been rationalized in terms of calculated free energy–composition �G–c plots. The �G–c plots in Figure 7.35 represent the free energycurves for the �- and the �-phases at 850 K, a temperature slightly below themonotectoid temperature. At such temperatures two common tangents can beconstructed, one of which (line A in Figure 7.35) touches the �- and the �1-curves




–300

–250

–200

–150

–100

–50

0

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

A B

K

Gα

G

β

β1

Composition, cc 2

G (

cal/m

ol)

ΔG

αβ (c 2, c n)

•

•

••

P(c p)

Q

R

S(tangent to β2)

T < T mono

c 1

AC

E

D

c 5c

K

B

c 1 c 3 c n c 4

↓

↓

Figure 7.35. Free energy–concentration plots for the �- and the �-phases in the Zr–Nb systemat 850 K showing that a metastable equilibrium (represented by the common tangent A) can beestablished between the �- and the �1-phases. See text for transformation sequences for alloys havingcompositions c and c5.

while the other (line B) touches the free energy curves corresponding to the �- andthe �2-phases. This implies that a metastable equilibrium between the �- and the�1-phases is feasible at such temperatures. The lines A and B merge at Tmono,the monotectoid temperature where a single line touches the free energy curvescorresponding to the �-, the �1- and the �2-phases. At a temperature slightlylower than Tmono the line A remains above the line B, implying that the structureconsisting of � and �1 is metastable (i.e. associated with a higher value of freeenergy) with respect to the equilibrium �+�2 structure.

On the basis of these �G–c plots the sequences of transformation during tem-pering of alloys of different compositions can be rationalized. The common tangentbetween the G� and G�1 curves touches the former at a composition c1 while theG� and the G� curves intersect at a composition c2. For an alloy with composi-tion c, where c1 < c< c2, the supersaturated �′ martensite decomposes through ametastable step consisting of the �+�1 structure before attaining the equilibrium�+�2 structure. These two steps correspond to free energy changes shown by thefree energy drop from C to D for the �′ → �+�1 decomposition followed by thedrop from D to E corresponding to the �+�1 → �+�2 transformation (see insetof Figure 7.35). It is the evident that the driving forces for the �′ → �+�1 and




the �′ → �+�2 reactions are not much different. The strong preference for theformation of �1-precipitates during tempering at temperatures lower than but closeto the monotectoid temperature can be understood in terms of the compositionalbarriers associated with the nucleation of the �1- and the �2-precipitates.

As discussed earlier in the context of phase separation mechanisms, the chemicalfree energy change, �G��c2, cn), accompanying the nucleation process (where c2

and cn are the compositions of the supersaturated �′-martensite and the �-nuclei,respectively), can be estimated by constructing a tangent to the G� curve at c2. Thevertical distance between the tangent (marked CK in Figure 7.35) and the G� curvegives the value of �G�� (c2, cn) for a �-nucleus composition, cn. The tangent CKintersects the G� curve at a composition c3; therefore, the Nb concentration in the�-nucleus must be higher than c3. One can visualize the following two essentialsteps for the nucleation of a �-particle in the �′-matrix: (a) clustering of Nb atomsenriching a local region and (b) crystallographic transformation of this Nb-richregion from hcp to bcc. The sharp rise in the free energy of the �-phase withan increase in Nb content will preclude the formation of �-regions substantiallyenriched with Nb. Alternatively the formation of �-nuclei with Nb concentrationsclose to but higher than c3 is thermodynamically feasible. This route of nucleationwill involve much smaller composition and free energy barriers. Such �-nucleiwill continue to get enriched with Nb and the free energy of the nuclei will followthe G� curve. During the process of Nb enrichment the �-nuclei composition willattain the level, c4, where the common tangent between the G� and G�1 curvesmeets the G�1 curve. A metastable equilibrium between the matrix �- and the�1-phase is established at this point. A further Nb enrichment of the �-nuclei toapproach the composition level corresponding to the �2-phase is hindered by thepresence of a large free energy barrier in the G� curve. The magnitude of thisbarrier is given by the vertical distance NP (as shown in Figure 7.33) where thetangent to the G� curve at N is parallel to the common tangent touching the �1- and�2-arms of the G� curve. For a supersaturated martensite with a compositionc > c2 (Figure 7.36), tempering at 8508 K can cause a reversion from �′ to �.The thermodynamic feasibility of a composition-invariant �′ → � transformationis shown for a �′-composition c5 by the vertical drop in the free energy from P toQ. In a subsequent step, the �-phase decomposes into a mixture of �+�1 phasemixture (metastable) and finally into the equilibrium �+�2 structure. The overalltransformation can then be described by the following scheme:

�′�c5 → ��c5 → �+�1 → �+�2�

Drops in free energy for this transformation sequence are shown by the verticalsegments, PQ, QR and RS. With decreasing temperature, the �1-arm of the G�




(a) (b)

1 μm

Figure 7.36. Light (a) and TEM (b) micrographs showing �-grain boundary allotriomorphs atcoarse �-grain boundaries and fine �-plates within the �-grains in the Zr–5.5% Nb martensitetempered at 823 K. The formation of the coarse �-grain structure indicates reversion of �′-martensiticinto � which subsequently decomposed into a structure consisting of �-plates within �-grains and�-allotriomorphs at �-grain boundaries.

curve goes up with respect to the G� curve and below a certain temperature limit(T< (Tmono− 100 K)); it is not possible to construct a common tangent between theG� curve and the �1-portion of the G� curve which implies that the establishmentof a metastable equilibrium between the �- and the �1-phases is precluded. In suchcases, direct nucleation of the �2-phase takes place during tempering. Obviouslythis process involves the overcoming of a very large free energy barrier and hencethe reaction is expected to be very sluggish as is indeed the case.

Luo and Weatherly (1988) have studied the precipitation behaviour of theZr–2.5% Nb alloy and have confirmed that both �1- and �2-precipitates form inthis alloy during tempering. The rationale of the formation of bcc precipitates oftwo widely varying compositions (�1: Zr–20% Nb and �2: Zr–85% Nb) at tem-peratures close to but below the monotectoid temperature has been corroboratedby their studies. They have also pointed out that the precipitate phase forming at773 K is exclusively �2 while both �1- and �2-precipitates form at 873 K. For thelatter treatment, homogeneously nucleated precipitates within the martensite plateshave been identified to be of the �2-type and the twin boundary nucleated precipi-tates are invariably of the �1-type. The crystallography of precipitation of �1- and�2-phases has been studied by Banerjee et al. (1976) and Luo and Weatherly(1988). Figure 7.37(a) and (b) depicts the key for the superimposed diffractionpatterns of twin-related �-crystals and of �1-precipitates at twin boundaries. Thelatter study has convincingly demonstrated that the orientation relation followed bythe needle-shaped �2-precipitates (Figure 7.37(b)), which form within the marten-site plates, is slightly different from the Burgers relation which is obeyed bythe �1-precipitates. In both the cases, the basal plane (0001)� remains parallel to{110}�-type planes, but the angle, �, between the [0110]� and the [110]� directions




(a)

Matrix α

(0111)αT

(1101)αT

(1010)αM

(0002)αM(011)β1

(011)β1

Twin αPrecipitate β1

(b)

Figure 7.37. (a) Key of the diffraction pattern shown in Figure 7.34(d). The �1-precipitate bearsBurgers orientation relationship with both the twin �-orientations. (b) Twin boundary �1-precipitatesand needle-shaped �2-precipitates within martensite plates in the tempered Zr–2.5 Nb alloy.

is different for �1 (Burgers relation, � = 5�26 ) and for �2 (� = 3�8 –4�1 ) phases.They have also shown that the orientation relation obeyed by the needle-shaped�2-precipitates is consistent with the invariant line strain (ILS) condition proposedby Dahmen (1982) and that the growth direction of these needles is very close tothe ILS direction.

Let us now consider the crystallography of twin boundary precipitation. It hasbeen observed by Banerjee et al. (1976) and Luo and Weatherly (1988) that�1-precipitates forming at {1011} twin boundaries adhere to the Burgers orienta-tion relation. The two �1-variants that nucleate at the twin boundaries maintainthe parallelity between the [111]� and [1120]� directions and the twin plane lieswithin 1.5 of the (011)� plane. The common close-packed direction in the twophases remains unrotated and gets only slightly contracted. Since a {110}�-typemirror plane of the �-phase is nearly parallel to a {1011}�-type twin plane, theplate-shaped precipitates along the twin boundary can maintain nearly equivalentorientation relations with respect to both the twin-related crystals. The orientationof �1-precipitates at the twin boundary is, therefore, the same as that of the parent�-phase from which the twinned martensite plate has formed. The nucleation ofsuch an orientation will require a very small surface energy, as the �1-nucleuscan maintain coherency on both sides of the twin plane. This low surface energybetween the �- and �1-crystals along the {1011}��{110}�1 plane will promotespread of the �1-crystal along the twin boundary. As a result, thin plate-shapedprecipitates will form along the twin plane. �1-precipitates of this specific variant,




(b)

ααT

ααT

α

→→

(110)β1

(110)β1

(a)

0.5 μm

0.5 μm

Figure 7.38. TEM morphology (a) and schematic drawing (b) showing the morphological evolutionof �1-precipitates at

{1011

}�

twin boundaries of Zr–2.5 Nb martensite during tempering. Theorientation of the twin boundary �1-precipitates is the same as that of the parent �-phase. � and �Tare

{1011

}twin related.

nucleated independently, will also join up to produce lamellae of �1-precipitatesalong twins. Figure 7.38 illustrates the morphology of such �1-precipitates. Fewsystematic investigations on microstructural changes occurring during temperingof Ti alloy martensites have been made. The close similarity between the crys-tallography of the �/� transformation in Zr and Ti alloys suggests that a similarmechanism of �-phase precipitation along twin boundaries of the �′-martensitewill control the morphological development of the two-phase structure during thetempering of internally twinned Ti martensites also.

In several alloy systems of Ti and Zr, the precipitation of intermetallic phasesoccurs during tempering of martensites. A few examples of intermetallic phaseprecipitation have been discussed in Section 7.6.




7.3.6 Decomposition of orthorhombic �′′-martensite during temperingThe tempering of hcp �′-martensites in alloys belonging to the �-isomorphous and�-monotectoid systems, as discussed in Section 7.3.5, results in the precipitationof the �-phase. In contrast to this, tempering of the orthorhombic �′′-phase leadsto the following decomposition process:

�′′ → �′′enriched +�→ �+�

Davis et al. (1979) have studied the decomposition process in detail in theTi–Mo system and have shown that the �′′-phase can decompose either by adiscrete precipitation or by a continuous spinodal process. Some of the importantresults reported by them are summarized here.

In the Ti–4% Mo alloy, the microstructural features of the martensite are sug-gestive of the presence of both hcp �′- and orthorhombic �′′-phases. On tempering,six variants of precipitates appear in some of the plates while in some other plateslarge areas are covered by only two variants. The periodicity in the precipitatedistribution in the latter gives rise to satellite reflections in the diffraction pat-terns. Such a periodic distribution of a pair of nearly orthogonal variants appearsto originate as a consequence of elastic strain interactions between the variants.The elastic strain imposed by the pair introduces an orthorhombic distortion inthe matrix which, in turn, suppresses the formation of four other variants locally.In Ti–4% Mo precipitate free zones are invariably present at the boundaries ofmartensite laths. Therefore a continuous mode of decomposition cannot be postu-lated in this alloy.

In alloys containing 6 and 8% Mo, the tempering treatment results in a phaseseparation in the orthorhombic �′′-phase (Figure 7.39). The observed compositionmodulation extends right up to the lath boundaries, suggesting the operation ofa spinodal decomposition process within the �′′-martensite. As the concentration

Figure 7.39. Microstructure of the tempered orthorhombic �′′-martensite in Ti–8% Mo showingphase separation (after Davis et al. 1979).




modulation amplifies, the regions rich in Mo transform to the �-phase while theregions depleted in Mo transform to the hcp �-phase.

The continuously changing compositions of the enriched and depleted regionsof the �′′-martensite with the progress of tempering are reflected in the rotationof the habit plane of modulation as recorded by Davis et al. (1979). The rotationof modulations continues in response to the changing elastic forces resulting fromvarying lattice parameters of the depleted and enriched regions of �′′. Finallythe appropriate orientation relation between the equilibrium �- and �-phases isestablished.

The Ti–8% Mo martensite reverts, at least partially, to the parent �-phase of thesame composition when heated rapidly to temperatures above 798 K. Precipitationof �-plates subsequently occurs within the matrix of reverted �.

Davis et al. (1979) have proposed a free energy–composition relationship in thissystem which can rationalize the aforementioned experimental observations. Sincethe transition from hcp � to orthorhombic �′′ is observed in a continuous mannerwith increasing solute content, the free energy–composition plot for the �- and the�′′-phases can be drawn with a single line with two points of inflection located atc2 and c4, as shown in Figure 7.40. The point (c3) of intersection of G��′′ andG�� curves defines the composition limit up to which martensite formation isfeasible at this temperature. Below the composition c1 only the hcp �′-martensitecan exist while between c1 and c3 both �′- and �′′-martensites can form. At c2

there exists a point of inflection which marks the composition beyond which theorthorhombic �′′-martensite undergoes spinodal decomposition during temperingif the tempering temperature is below the spinodal line.

c 1 c 2 c 3 c 4

Mo concentration →

Fre

e en

ergy

G (α′)G (α″)

G (β)

Ti

Figure 7.40. Free energy–Mo concentration diagram at T1 for Ti–Mo alloys showing doublyinflected (dashed line) plot for the orthorhombic �′′-phase. Points of inflection of the G��′′) plot arelocated at c2 and c4 (after Davis et al. 1979).




The tempering reaction in such a situation initiates by developing a homoge-neous composition modulation, the amplitude of which increases with the durationof tempering. The regions depleted in Mo gradually enter the composition rangewhere hcp �′ is more stable and, therefore, these regions transform into that phase.The �′′-regions enriched in Mo gradually reach compositions beyond c3 where thehigh stability of the �-phase induces a �′′ → � transformation.

The martensite phase, �′′, can revert back to the parent �-phase when thetempering temperature exceeds T0 corresponding to the alloy compositions.

7.3.7 Phase separation in �-phase as precursor to precipitationof �- and �-phases

The precipitation of the metastable �-phase and the equilibrium �-phase takesplace subsequent to phase separation in the �-phase in those alloy systems inwhich the tendency for �-phase separation is present. The Zr–Nb system can onceagain be used for illustrating this point.

Flewitt (1974) has shown that the supersaturated Nb-rich �-phase, having acomposition within the coherent spinodal of the Zr–Nb system, on ageing attemperatures below the monotectoid temperature initially undergoes a spinodaldecomposition following which a metastable hcp phase (designated as �′) precip-itates within the regions enriched in Zr. The strain field contrast associated withthese �′-precipitates and the diffuse intensity in diffraction patterns reveal thatthese precipitates maintain at least partial coherency and a Burgers orientationrelation with the matrix. Menon et al. (1978) have considered the possibility of theformation of �-phase by a composition-invariant structural transition in the Zr-rich�-regions present in a spinodally decomposed �-alloy. The computed G�−c andG�−c plots corresponding to 673 K show that once the �-phase is separated intothe �1- and �2-phases, a composition-invariant transformation from �1 to �′ isthermodynamically feasible, considering only the chemical free energy change.The drop in free energy, �G (�1 → �′) shown in Figure 7.41 corresponds to thecomposition-invariant �1 → �′ transformation.

If a supersaturated �-alloy is aged in a region outside the coherent spinodal,the decomposition into a mixture of �1- and �2-phases will still occur by adiscrete nucleation process and the system will further reduce its energy by themonotectoid decomposition of the �1-phase into the equilibrium �+�2 mixture.The thermodynamic analysis of Menon et al. (1978) has pointed to the followingtwo possible sequences of the transformation of the �-phase in the Zr–Nb system:

(1) �′ → spinodal decomposition → �2 +�1; �1 → �′ +�2 → �+�2.(2) �′ → discrete precipitation → �+�1; �1 monotectoid → �+�2.




–500

–400

–300

–200

–100

0

G

β

G

α

U

ΔG

β–α

0 0.05 0.10 0.15 0.20

c →

c

G (

cal/m

ol)

→

T = 673 K

β1

V

c s

Figure 7.41. Free energy–concentration plots for the �- and the �-phases showing the possibility ofthe compositionally invariant transformation of the �1-phase into the �-phase, the �1-phase resultingfrom a prior spinodal or discrete phase separation in the �-phase. The line U represents the commontangent to the �1- and �2-arms of the G�–c plot and the line V is the common tangent to G� andG�2 curves.

While rationalizing the relevant observations, Flewitt has suggested two separateG–c plots for the �′- and the �-phases. The computed G–c plots obtained byphase diagram analysis show that the �′-phase is nothing but the �-phase with Nbsupersaturation and that its free energy can be expressed on the same G�−c plot.

At temperatures below 773 K, the formation of the metastable �-phase mayalso intervene in the sequence of phase transformations. The thermodynamics of�-precipitation has already been discussed in Chapter 6. The metastable equilib-rium (represented by G–c plots in Figure 7.42) which can be established betweenthe �1-phase and the �-phase leads to the appearance of the metastable (�1 +�)phase field separating the single phase fields associated with the �- and the�1-phases (Figure 7.42). The G�, G� and G� versus c plots demonstrate that attemperatures much below 773 K where a common tangent between the G� curve




Concentration

Free

ene

rgy

G

α

G

ωG

β

c a

c bc c

Figure 7.42. Free energy–concentration plots showing metastable equilibrium between the �- andthe �-phase.

and the �1-arm of the G� curve cannot be constructed, a metastable equilibriumrepresented by a common tangent between the G� curve and the �1-arm of the G�

curve can be drawn. In this context, it may be noted that three common tangentscan be constructed between the G� and the G� curves, the tangents touchingthe G� curve at three points denoted by ca, cb and cc (cc > cb > ca), close to thethree extrema of the G� curve. It can be seen from this construction that thereis no driving force for the nucleation of the �-phase in the composition rangeca < c < cb and, therefore, phase separation in the �-phase must proceed to suchan extent that the Zr concentration in some regions exceed the limit given by ca.The nucleation of �-particles, therefore, can start only after the �-phase separationprocess is nearly complete.

The sequences of transformations of the �-phase in a monotectoid system suchas Zr–Nb, Ti–Nb can then be predicted from the computed G–c plots correspondingto the equilibrium and metastable phases as follows:

(1) � → �1 + �2 within the miscibility gap at temperatures above Tmono. Thereaction initiates in a spinodal mode within the coherent spinodal regime andby a heterogeneous nucleation mode outside this.

(2) �→ �1 +�2 →�′ +�2 →�+�2 at temperatures where the �-phase is unstable.(3) �→�1 +�2 →�+�2 →�+�2 at temperatures below Tmono where the�-phase

is metastable.

Many of the Ti alloy systems which were earlier known as �-isomorphoussystems have now been redesignated as �-monotectoid systems. Phase separationin the �-phase, therefore, has a pronounced influence on the subsequent phase




transformations in these systems. The arguments which are used for predicting andexplaining the phase transformation sequences in Zr–Nb alloys can be extended toa number of Ti-based alloys which exhibit a strong phase separation tendency inthe �-phase field. Two illustrative examples of the Ti–Mo and the Ti–Cr systemsare discussed here to bring out the importance of the phase separation tendencyof the �-phase in dictating the subsequent phase reactions.

The current version of the Ti–Mo phase diagram shows a monotectoid reaction(Figure 1.7) at 968 K and the critical temperature of �-phase separation is 1117 Kat 50 wt% Mo. The corresponding free energy–composition plots for the �-phaseat temperatures below the critical temperature will, therefore, be doubly inflected,the miscibility gap being defined by the compositions at which the commontangent touches the G� plot with the spinodes defining the chemical spinodalline. The calculated coherent binodal (miscibility gap) and coherent spinodallines (Figure 7.43) have recently been reported by Furuhara et al. (1998). Theyhave observed that the retained �-phase in a Ti–40 wt% Mo alloy when agedat 773 K produces uniformly distributed �-precipitates within the �-grains, whilenon-uniform grain boundary nucleated (allotriomorphs and side plates) �-platesare formed when the ageing temperature is 873 K or higher. They have attributedthe uniform nucleation of �-plates within the grain to the occurrence of a phase

700

800

900

1000

1100

1200

20Ti 40 60 80 Mo

Mo (mass%)

Tem

pera

ture

T (K

) β

α

β + β′

Chemicalspinodal (cal)

Chemicalbinodal (cal)

Figure 7.43. Calculated phase diagram (Furuhara et al. 1998) of the Ti–Mo system showing amonotectoid transformation, phase separation within �-phase and chemical spinodal lines.




separation within the �-phase prior to the nucleation of �-plates. Though clearevidences in support of �-phase separation, spinodal or otherwise, followed by thenucleation of either �- or �-precipitates within the phase-separated matrix havenot been reported in �-monotectoid systems of Ti, such a sequence is expectedfrom thermodynamic considerations.

The Ti–Cr system, which is known to belong to the class of �-eutectoid systems,exhibits a strong tendency for phase separation. Analysis of thermodynamic data byMurray (1987), Kauffman and Nesor (1978), Menon and Aaronson (1986), Prasadand Greer (1993), Sluiter and Turchi (1991), Mebed and Miyazaki (1998) haveindicated the presence of a miscibility gap, the location of which has been foundto be somewhat different by different investigators. Figure 7.44 shows a super-imposition of calculated binodal, chemical spinodal and coherent spinodal on thephase diagram of the binary Ti–Cr system. In case the formation of the equilibriumLaves phases, �-, �- and �-TiCr2, is suppressed during ageing of an alloy withinthe coherent spinodal, the alloy is expected to develop concentration modulations.Time-dependent evolution of such modulations by phase field modelling and by

1400

1200

1000

800

1600

Ti 0.2 0.4 0.6 0.8 Cr

C14

C15

C36

1

2

3

Tem

pera

ture

(K

)

X Cr

β

α + βα

Figure 7.44. Equilibrium phase diagram of the Ti–Cr system with superimposed metastable binodaland spinodal lines corresponding to �-phase separation. Line 1 shows the metastable equilibriumboundary representing �1 +�2 phase separation when the formation of the intermetallic phases issuppressed. Line 2 indicates the calculated chemical spinodal. Line 3 is the calculated coherentspinodal for concentration modulations along <100> (after Menon and Aaronson 1986).




TEM experimental observations has been demonstrated by Mebed and Miyazaki(1998). The quenched metastable �-phase, during ageing at 673 K, decomposesinto solute-rich �2- and solute-lean �1-phases, with modulations occurring alongthe elastically soft <100> directions. In the early stages of decomposition, the�1-regions in this compositionally modulated �-matrix become amenable for atransformation either into the �- or the �-phase while the �2-phase, which getsricher in the �-stabilizing element Cr, remains untransformed. Fine scale mod-ulation being responsible for the nucleation of �- or �-precipitates, a uniformdistribution of these precipitate phases is generated unlike the grain boundaryprecipitation which occurs in homogenous �-grains.

7.4 MASSIVE TRANSFORMATIONS

A massive transformation is a solid–solid phase transformation in which the prod-uct phase inherits the composition of the parent phase but unlike the case ofa martensitic transformation, the growth of the product phase is controlled bydiffusional jumps of atoms from the parent to the product lattice sites acrossthe transformation front. The composition-invariant massive transformation in atwo or a multicomponent system can, therefore, be compared with a diffusionalpolymorphic transformation of a single component system. Needless to say, a trans-formation of a two-component system in a composition-invariant process cannottake place under the equilibrium cooling condition (except in the case of a congru-ent transformation). A moderately rapid cooling from the high-temperature phasewhich provides the necessary supercooling is, therefore, a necessary prerequisitefor initiating as well as propagating the transformation. Massive transformationshave first been reported in Cu-based alloys by Massalski (1958, 1970). Systematicstudies on massive transformation in Ti-based alloys have been carried out byPlichta and co-workers (1977, 1978, 1980) and it is through these studies thatsome of the special issues connected with massive transformations of Ti-basedalloys have been identified. A brief account of these are given here.

7.4.1 Thermodynamics of massive transformationsThermodynamics of massive transformations can be conveniently discussed withreference to a typical eutectoid phase diagram in which the terminal solid solutionhas an extensive solubility of the solvent (Figure 7.45(a)) and the eutectoid com-position, ce, is not far from the point showing the maximum solubility, cs, of theterminal solid solution. The corresponding free energy–composition (G–c) plotsfor the �- and the �-phases at a temperature T1 are shown in Figure 7.45(b). The




T1

T2

α+β

α

β+γ

α+γ

β

To

C s C e

(a)

C 1

α+γ

T2

Gα

Gβ

Gγ

I

J

K

C 5

(c)

C 3C 1 C oC 4

T1

E C D

A

B

Gαc

Gβ

Gα

(b)

Figure 7.45. (a) Eutectoid phase diagram of a binary system which can exhibit a massive transfor-mation. (b) G–c plots corresponding to Ti for the �- and the �-phase. Massive transformation atc3 and c4 will be driven by a change in free energy from A to C and B to D, respectively. G�c

corresponds to the G–c plot for the �-phase in the presence of the capillarity effect. (c) The drivingforce for the massive transformation at T2 for an alloy with concentration c5 is shown by the dropIJ. The equilibrium condition is denoted by the point K.

free energy curves for the two phases intersect at the composition co. This meansthat the To temperature for the composition-invariant � → � transformation forthe alloy of composition co is located at T1. The composition dependence of theTo temperature has been schematically shown in Figure 7.45(a) by a dashed line.There are a number of Ti-based binary systems, such as Ti–Si, Ti–Au and Ti–Ag,which exhibit similar phase diagrams and in which massive transformation fromthe �- to the �-phase has been reported by Plichta et al. (1978, 1980).

The change in free energy due to a composition-invariant �→ � transformationis given by the vertical drop from the G� curve to the G� curve; this is shown in




Figure 7.45(b) by vertical segments AC and BD for two compositions c3 and c4,respectively. From the consideration of the change in integral molar free energy,such a transformation from � to � is possible at T < To for the entire compositionrange 0 < c < co. This composition range can be divided into two parts, namely,0<c< c1 and c1 <c< co, the former conforming to the single phase �-region, andthe latter in the �+� region of the phase diagram at the selected transformationtemperature T1.

Massive transformations are initiated by diffusional nucleation of the productphase at grain boundaries and/or other inhomogeneities in the parent phase. Homo-geneous nucleation of massive product is rather difficult because the massiveproduct, once it nucleates heterogeneously, grows at a very high speed towardscomplete transformation without allowing the creation of sufficient undercoolingfor homogeneous nucleation. Let us now consider the nucleation process in thesingle phase (0 < c < c1) and the two-phase (c1 < c < co) regions of the phasediagram.

As discussed earlier, the free energy change during nucleation is given by thevertical distance from the tangent drawn on the G–c curve of the parent phase.Application of this criterion, which can be easily shown by construction of paralleltangents on the G� and G� curves, demonstrates that the composition of the criticalnucleus cannot be the same as that of the matrix. For example, free energy of thealloy having a solute concentration, c3, in the �-phase at the temperature T1 isgiven by the point A on the G� curve. E is the point at which a tangent to the G�

curve can be drawn parallel to the tangent at point A on the G� curve. Thereforethe free energy change for nucleation is maximum when the �-nuclei have acomposition given by the point E which is different from c3, the compositionof the parent �-phase. Therefore, nuclei of composition indicated by point E areexpected to form predominantly. During the growth process due to the kineticconsideration the composition shifts from E to C to establish the compositioninvariance criterion. In case the free energy composition plot for the �-phase israised due to the capillarity effect to occupy the G�c curve (drawn with dashedline), it is possible to visualize a situation where a composition-invariant nucleationcan occur. The product �-phase exhibits a tendency for further transformation. Itmay be noted that in the composition range, c1 < c < co, the product �-phase atT1 does not remain in equilibrium and exhibits a tendency for decomposition intoa mixture of the �- and �-phases.

At a temperature, T2, �→ � massive transformation can occur if the formationof the equilibrium �-phase can be suppressed. The drop in free energy for an alloywith composition c5 is shown by the segment IJ in Figure 7.45(c); the free energycorresponding to the equilibrium �+� structure is indicated by the point K onthe tangent common to G� and G� plots.




7.4.2 Massive transformations in Ti alloysThe study of a large number of binary Ti-based eutectoid systems (Plichta et al.1978, 1980) has revealed that only in Ti–Ag, Ti–Au and Ti–Si the �→�m massivetransformation occurs. In other systems, such as Ti–Fe, it is difficult to suppress theformation of the equilibrium �-phase. A limited solubility of the solute elementsin the �-phase and a sharp drop in the To temperature with alloy addition arethe factors responsible for the non-occurrence of massive transformation in suchsystems.

Plichta et al. (1978, 1980) have shown that the morphology of the �-quenchedproducts in the composition ranges, Ti–4.7 to 13.5 at.% Ag, Ti–1.8 to 4.4 at.%Au and Ti–0.68 to 1.10 at.% Si can be easily recognized as the massive �m-phasebased on the following features:

(1) Irregular grain shapes with jagged boundaries, as shown in Figure 7.46(a)characteristic of a massive transformation product, are observed. In cases where�m is coexistent with martensitic �′ (Figure 7.46(b)), they can be distinguishedon the basis of their morphologies.

(2) Energy dispersive analysis has established that the �m-grains inherit the chem-ical composition of the parent �-phase. The retention of the solutes in excess oftheir solubility limits has also been demonstrated by second phase precipitationwithin the �m-regions during subsequent ageing treatments.

(3) Unlike the martensitic �′, the �m-phase does not exhibit any specific habitplane with respect to the parent �-phase. The substructure, though containing ahigh density of dislocations, does not show any similarity with either dislocatedlath on internally twinned plate martensites.

The alloy composition ranges over which massive transformation has beenobserved are primarily in the hypoeutectoid regions of the respective phase dia-grams, though in a few cases the massive transformation could be induced evenin hypereutectoid compositions (e.g. Ti–13.5 at.% Ag). This is possible in systemsin which the eutectoid composition is not far away from the maximum solubilitylimit of the equilibrium �-phase (as shown in Figure 7.45(a).

Thermal analysis experiments using continuous cooling have provided valuableinformation regarding the thermal arrest temperature as a function of the coolingrate, the enthalpy of transformation and the growth rate of massive �m-phase.A typical cooling curve is shown in Figure 7.46(c) which shows the arrest tem-perature and the duration over which the massive transformation takes place. Thegrowth rate, G, of a massive transformation product can be expressed approxi-mately as

G d/tg (7.50)




(a) (b)

(c)

5 43 2

1

Ti – 17.5 w/o Ag1 – 120°C/s2 – 355°C/s3 – 530°C/s4 – 120°C/s5 – 2420°C/s

0 500 1000 1500

Time (ms)

1000

900

800

700

600

Tem

pera

ture

(°C

)

t g

50 μm 100 μm

Figure 7.46. (a) Light micrograph showing irregular grain structure of the massive transformationproduct in the Ti–17.5 at.% Ag alloy. (b) Light micrograph showing a mixture of massive andmartensitic product in the alloy. (c) Cooling curves exhibiting thermal arrest due to the massivetransformation at different cooling rates in Ti–17.5 at.% Ag. (after Plichta et al. 1978, 1980).

where d is the grain diameter of the product �m-grains and tg is the duration ofthermal arrest, as shown in Figure 7.45(c). Although �m-grains are very irregularin shape, they can be better approximated as equiaxed than as acicular for themeasurement of d.

The thermal arrest temperature versus cooling rate data obtained for some ofthe Ti–Au and Ti–Si alloys have shown an initial decrease in temperature withincreasing cooling rate followed by a Ms plateau. Microstructural examination ofthe samples cooled at different rates has confirmed that martensitic transformationbecomes operative when the critical cooling rate is exceeded.




An estimate for enthalpy of transformation, �HT(�→ �m), for massive trans-formation can be made using the fact that the rate of heat generation, a= dq/dt,which matches the rate of heat extraction during the thermal arrest is proportionalto the rate of transformation. This can be written as

�HT ��→ �m �dfdt

=Q= Cs

dTs

dt(7.51)

where Cs and Ts are the heat capacity and temperature of the specimen, Cs beingdependent on the volume fractions, X� and X�m

, of the � and the �m-phases atany instant of the transformation:

Cs = X�C�+X�mC�m

= (1−X�m

)C�+X�m

C�m(7.52)

Based on the thermal analysis experiments, Plichta et al. (1978) have deter-mined thermodynamic data on massive transformations in Ti alloys. Table 7.4lists the values of enthalpy and free energy changes and To temperatures for a fewrepresentative alloys.

The experimentally measured growth rate, as a function of undercooling belowTo, for different alloys has yielded the activation enthalpy of the growth process.Burke and Turnbull (1952) have expressed the growth velocity, G, for an interface-controlled reaction as

G= �kT

hexp

(�SDb

R

)(−�F��→ �m

RT

)exp

(−�HDb

RT

)(7.53)

where � is the width of the boundary, k is Boltzmann’s constant, h is Planck’s con-stant, �SDb and �HDb are the activation entropy and enthalpy for diffusion acrossthe transformation front and �F (�→�m) is the free energy change accompanyingthe � → �m transformation. A plot of log (−GT/�F (� → �m))versus reciprocalof the absolute temperature therefore yields the value of �HDb which has beenfound to lie between 50 and 93 kJ/mol.

Table 7.4. Values of To, �HT(� → �m) and �G(� → �m) for representativeTi alloys.

Alloy composition (at.%) To �HT(�→�m)(J/mol)

�G(�→ �m

Ti–6.5% Ag 1136 −2870±460 −2870+2�55TTi–2.6% Au 1128 −3035±420 3035+2�72TTi–1.1% Si 1140 −2470±500 2470+2�18TTi–47.55% Al 1325 −3712 −3712+2�31T




Massive transformations in alloys (except in cases of equilibrium congruenttransformations) result in the formation of metastable phases. Therefore, for amassive transformation to occur, it must compete successfully with other reactionsleading to equilibrium products. Usually the massive products nucleate at a muchslower rate than the equilibrium precipitate phase(s). However, the growth kineticsof the massive transformation are usually several orders of magnitude faster thanthose of precipitation reactions involving solute partitioning. This is primarilydue to the fact that the growth of a massive product occurs by diffusive atommovements only across the transformation front and does not involve long-rangevolume diffusion. Plichta et al. (1978) have reported the measured growth rates of�m in Ti alloys which are in the range of 1�5 × 10−5–2 × 10−4 m/s. These valuesare two to three orders of magnitude higher than the estimated growth velocitiesof the equilibrium �-precipitates evolving from the parent �-phase.

Massive transformation has recently been studied in detail in titanium alu-minides. Wang et al. (1992), Wang and Vasudevan (1992) and Veeraragha-van et al. (1999) have shown that in binary Ti–Al alloys with Al contentbetween 46.5 and 48 at.% the massive �–� transformation can be induced byrapid cooling. Figure 7.47(a) and (b) shows the transformation products of the�-phase of Ti–46.5 at.% Al alloy obtained by furnace cooling and water quench-ing. The continuous cooling transformation diagram of this alloy as shown inFigure 7.47(c) depicts that a cooling rate exceeding 300 C/s is required to suppressthe lamellar transformation product completely. The experimentally measuredmassive start temperature, �S

M, has been found to be weakly dependent on thecooling rate.

Some of the open questions regarding the mechanism of massive transforma-tions have been addressed in the recent work on Ti–Al massive transformation.The first and foremost question is whether a strict orientation relationship existsbetween the parent and the product crystals during the nucleation and the growthstages of massive transformation. Recent results from TEM and orientation imag-ing microscopy experiments (Wang et al. 2002) have clearly shown that in acolony of the product �m-crystals, though some of the �m-crystals maintain the�/� orientation relation: (0001)�� (111)� observed in lamellar product (discussedin Chapter 5), many of them do not have low index orientation relations withthe parent �. Presence of twin relation between �m-crystals within a “colony”,however, is frequently encountered. By arresting the transformation at early stagesit has been established that the nucleation step at grain boundaries and graincorners invariably involves formation of �M-crystals which maintain orientationrelationship with one of the contacting �-grain but the growth of �m-crystals inthe orientation-related �-grain remains considerably restricted. In contrast, theincoherent boundary rapidly propagates into the grain with which no low index




(a) (b)

(c)

1500

1400 T o

T o

T E

1000°C/s500°C/s

400°C/s

300°C/s

200°C/s

100°C/s

1300

1200

1100

1000

900

8000.1 1

Time (s)

Tem

pera

ture

(°C

)

10

γLs

γMs

Figure 7.47. (a) Light and (b) TEM micrographs showing the massive product �m in the �2-matrixin the Ti–46.5 at.% Al alloy sample quenched from the single �-phase field. (c) shows contin-uous cooling transformation diagram which depicts the onset of the � → lamellar �L (�s

L) andof the � → massive �M (�s

M) transformations under different rates of cooling in the same alloy(after Wang et al. 1992, Veeraraghavan et al. 1999).

orientation relation exists. Structurally incoherent interphase boundaries whichoften exhibit faceting are the interfaces which propagate at the fastest rate inthe parent �-grain to accomplish � → �m massive transformation. The absenceof �/�m orientation relation during the growth stage is further evidenced fromthe observation that an interphase boundary of a growing �m-grain can advanceinto more than one parent grain without any change in orientation. Micrographscapturing several features of the nucleation, the growth and the interface structureassociated with the �→ �m transformation are shown in Figure 7.48.




(a) (b)

(c)

(d)

Figure 7.48. (a) Light micrograph showing �m-crystals formed at the grain boundary of � (sub-sequently transformed into �2) in the Ti–46.5 at.% Al alloy sample in which the transformation isarrested by a special heat treatment. (b) A colony of �m-crystals, mutually twin related (90 and150 ), is nucleated at an �-grain boundary in Ti–48 at.% Al–2 at.% Cr. (c) The growing �m/�2 inter-face boundaries are free of dislocations and other defects like misfit compensating ledges. (d) The�m-crystal which bears usual orientation relation with �2 (I) but no low index orientation relationwith �2 (II) grows into the �2 (II) grain (after Wang et al. 1992).




7.5 PRECIPITATION OF �-PHASE IN �-MATRIX

A great majority of the �+� alloys of Ti and Zr are mechanically processed inthe �+� phase field and are heat treated in such a manner that the �-phase is pre-cipitated in the �-matrix. Such a solid state precipitation reaction often producesthe �-phase precipitates in the form of plates which maintain strict orientationrelation and habit plane with the �-matrix. A comparison can be drawn betweenthese plate-shaped precipitates and the proeutectoid ferrite plates which form insteels. A considerable amount of attention has been paid to the mechanism offormation of plate-shaped precipitates of the product phase where the transforma-tion takes place through a thermally activated diffusional process. The presenceof features like orientation relationship, habit plane and sometimes surface reliefsuggests that a lattice shear mechanism takes part in the overall mechanism lead-ing to the formation of such �-plates. This viewpoint has been questioned anda controversy has persisted in the literature for several decades. In recent years,attempts have been made to resolve this controversy and, in the opinion of thepresent authors, evidences collected in this regard in Ti- and Zr-based alloys canindeed play a major role in clarifying many of the contentious issues. These alloysare particularly suitable for making a comparison between martensitic and diffu-sional transformations in view of the fact that the “lattice correspondence” and“atomic site correspondence” between the parent �- and the product �-phases areremarkably similar for both martensitic and diffusional transformations. Crystallo-graphic features such as orientation relations, habit planes and interface structuresassociated with these transformations can be compared with an aim of examin-ing whether the transformation mechanisms have characteristic imprints on theseexperimental observables.

The feature which is often used to distinguish a martensitic from a diffusionaltransformation is the presence of a surface relief effect in the former. The origin ofsuch surface relief is believed to be related to the invariant plane strain associatedwith the martensitic transformation (Bilby and Christian 1956). In contrast to this,Liu and Aaronson (1970) have presented experimental evidence that the forma-tion of hcp � (Ag2Al) precipitate plates in Al–Ag alloys in a typical diffusionaltransformation is accompanied by the appearance of surface reliefs. In some recentarticles, it has been recognized that diffusional transformations can exhibit surfacerelief despite the fact that lattice correspondence does not exist between the par-ent and the product lattices in such transformations. The presence of atomic sitecorrespondence across the transformation front in diffusional transformations canpreserve the shape deformation and produce a surface relief effect. The structureof the interface between the parent and product phases therefore attracts specialattention for examining the presence of atomic site correspondence in diffusional




transformations. These aspects of �-phase precipitation in the �-phase matrix inTi- and Zr-based alloys are discussed in this section.

7.5.1 MorphologyThe principal morphologies of �-plates in the �-matrix can be classified accordingto the Dube et al. (1958) morphological classification scheme, originally introducedwith regard to the formation of proeutectoid ferrite plates in steels. Morphologicaldescriptions, crystallographic and interfacial features and formation sequences ofeach of these morphological types are detailed in the following.

Grain boundary allotriomorphs (GBA) are the plates which form along the high-angle grain boundaries of the parent �-phase. Usually these are the first plates toappear in the course of �-precipitation (Figure 7.49).

Allotriomorphs nucleate at and grow preferentially along grain boundaries in amanner similar to the “wetting” of the grain boundary surface by the emerging �-phase. The interfacial energies between the �-plate and the two adjacent �-grainsmust be low for making the nucleation kinetics of GBAs favourable. This isachieved by establishing the Burgers orientation relationship between the �-platewith one of the �-grains while the boundary with the other grain is usuallyirrational. Furuhara et al. (1988) have shown in a Ti–6.6 at.% Cr alloy that theBurgers-related interface maintains partial coherency as reflected in the presenceof a periodic array of structural ledges with a uniform spacing of approximately8 nm and a height of 2–3 nm. The schematic of a Burgers-related bcc/hcp interface,projected on to the (0001)�� (011)� plane, contains one biatomic structural ledge.The terrace on which there is a close atomic fit is parallel to the (1100)�� (211)�plane and the structural ledge is associated with a Burgers vector of a/12 [111]�(details discussed in Section 7.5.4).

The interface of the �-allotriomorph which is not related to the adjacent�-grain by a Burgers relation shows a set of widely spaced ledges, relatively high(approximately 8 nm), following an irregular path and variably spaced. Two finersets of linear defects, both uniformly spaced, are also observed and these have beenidentified as misfit dislocations. This observation suggests that the non-Burgers-related �/ � orientation also tends to maintain coherent facets which are separatedby misfit correcting ledges.

Furuhara and Maki (2001) have shown that morphologically indistinguishable�-precipitates along a relatively straight prior �-grain boundary belong to the samecrystallographic variant. The selection of variant is made in such a manner thatthe

⟨1120

⟩�� 111�� direction remains nearly parallel to the grain boundary plane.

This is in agreement with the proposition that, in general, the low-energy facetslike the �0001�� 110�� and

{1010

}�� 112�� make the least possible angle with

the grain boundary plane.




(A)

(B)

Figure 7.49. (A) Grain boundary allotriomorph of �-phase along �-grain boundaries. (B) Twoorientations of �-crystals appearing alternately along a �-grain boundary. Orientations of the two�-crystals, their basal planes being parallel are rotated with respect to one another by 10.5 aroundthe normal to the basal plane (after Bhattacharyya et al. 2003).

All these conditions put quite stringent restrictions on the possible �-variantsthat can be precipitated at �-grain boundaries. The conclusions arrived at inthese investigations are based on TEM observations made on a scale of a sev-eral micrometres. In a recent work, Banerjee et al. (2003) have extended thescale of observation by studying the orientation distribution of grain boundaryallotriomorphs (�-plates) along a grain boundary extending over a distance ofabout 10 mm in a compositionally graded Ti–8Al–XV sample prepared by thelaser deposition technique. The strongly columnar growth morphology of �-grainswith a composition gradient has allowed in this case the study of �-precipitation




along a long length of a grain boundary. Some of the important observations aresummarized here:

(1) Most of the grain boundary �-precipitates exhibit Burgers orientation relationwith one of the �-grains (referred to as �1-grain).

(2) It is frequently observed that alternate �-precipitates belonging to two differentcrystallographic variants which share the same (0001) plane are lying parallelto the same �011� plane of the �I-grain. These two variants differ in their⟨1120

⟩directions which are parallel to two different �111�� directions on the

same �011� plane of �1. These two orientations shown in Figure 7.49 arerelated to each other by a rigid body rotation of nearly 11 around the axisperpendicular to (0001)�� 110�� plane.

(3) Out of the six possible �110� planes, the basal plane of the grain boundary�-precipitate preferentially chooses the �110� plane which is closest to thegrain boundary plane.

(4) Grain boundary �-precipitates exhibit a tendency towards maintaining theminimum possible misorientation from the Burgers orientation relation withthe adjoining �2-grain by the selection of a suitable orientation variant. Insome rare instances, where the grain boundary plane permits, Burgers orien-tation relation is established with both the adjoining grains. Such a precipitategeometry can be compared with those of �-precipitates along internal twinsof �′-martensite plates (Section 7.3.5).

(5) Occasionally �-precipitates are encountered along the �-grain boundarieswhich do not obey the Burgers orientation relation with either the �1- or the�2-grains. Figure 7.49 shows the distribution of �-orientation along a �-grainboundary as imaged by orientation imaging microscopy; the orientation of thetwo �-grains, �1 and �2, and that of the �-precipitates are indicated by thepole figures placed alongside.

Widmanstatten side plates form either by nucleating at grain boundaries or bybranching out from GBAs as shown in Figure 7.50. The plates which nucleateat grain boundaries are designated as primary side plates while those created bybranching of GBAs are known as secondary side plates. Side plates are often foundto grow in a group resulting in the formation of a colony of parallel �-plates.Unnikrishnan et al. (1978) have shown in a Ti–6 wt% Cr alloy that the growth of agroup of �-side plates can be treated in terms of the colony growth kinetics whichis usually applicable to cellular transformations. However, important differencesexist between the growth of a group of � side plates and that of a colony ofcellular reaction products (lamellar eutectoid or cellular precipitates) and these arelisted below and are illustrated in Figure 7.51:




(a)

(b)

Figure 7.50. Widmanstatten side plates: (a) bright- and dark-field TEM images of primary platesnucleating from the grain boundaries. (b) SEM images of secondary side plates formed by branchingof grain boundary allotriomorphs.

βbβa

αbβb

βa

αa

(a) (b)

Figure 7.51. Schematic drawing for making a comparison between (a) growth of a colony of sideplates and (b) growth of a nodule consisting of several plates in cellular precipitation.




(1) The growing tips of a group of side plates do not push the grain boundary asthey propagate into a parent grain unlike in the case of a cellular transformation.Side plates maintain a fairly strict orientation relation with the grain in whichthey grow and the parent phase retained in the intervening space betweenadjacent �-plates has the orientation of the untransformed �-phase lying aheadof the transformation front.

(2) The solute partitioning between the �- and the �-phases in the product takesplace through a lattice diffusion process. A group of side plates which havethe same orientation relation and habit plane grow by simultaneous movementsof their tips as the diffusion field through the�-phase ahead of the transformationfront forces them to do so. The growth pattern is also consistent with the edgewisegrowth of plates by the movement of growth ledges at the tips of the plates.

Intragranular plates Figure 7.52 constitute the third morphological variety of�-plates and nucleate in the interior of the �-grains. These plates also obey fairlystrict orientation relations. Intragranular plates can form either in an isolatedmanner or in a group. The former continuously partition the prior �-grain intosmaller and smaller volumes with fresh generations of plates appearing. The self-similarity of the structure in decreasing scale is consistent with a fractal description(Figure 7.52). The formation and growth of a group of parallel intragranular plateswhich remain stacked in a parallel fashion within a given packet give rise tothe “basket weave” morphology (Figure 7.53). A variation in the packet size ofthis structure, which can be induced by changing the alloy composition and the

10 μm

Figure 7.52. Fractal morphology of intragranular �-plates which continuously partition the parent�-grains causing a reduction in the size of the plates with every successive generation.




Figure 7.53. Basket weave morphology of intragranular �-plates which appear in a parallely stackedgroup within a colony.

extent of supercooling, is responsible for bringing about a significant change inthe appearance of the microstructure in the optical microscope scale (typically formagnifications ranging from 50× to 500×).

A detailed study of the morphology of the intragranular �-plates in Ti–Cr alloysby Menon and Aaronson (1986) has shown that these plates can be classifiedinto two types: “normal plates” which form at temperatures T > 873 K and “blackplates” forming at temperatures T < 873 K. While normal plates exhibit not soperfect habit planes, the slender black plates are nearly perfect. The latter isnamed so for their dark etched appearance in optical microscopic investigations.The formation of two distinct morphologies of �-plates in Ti–Cr alloys has beenrationalized in terms of the G–c diagram of the system. Menon and Aaronson(1986) have shown that the �-phase in the Ti–Cr system is associated with a strongclustering tendency which is reflected in the miscibility gap and a monotectoidreaction in the metastable phase diagram (Figure 7.44) when the formation ofequilibrium intermetallic phases is suppressed. The precipitation of the �-plates, farabove the eutectoid temperature, is governed by the equilibrium set up between �and �1 (Figure 7.54(a)), while at temperature below the eutectoid, �-plates remainin equilibrium with the �2 matrix (Figure 7.54(c)). At temperatures close to theeutectoid, as shown in Figure 7.54(b), a metastable equilibrium between � and �2 ispossible in addition to the stable �/�1 equilibrium. The difference in morphologyof normal plates and black plates, therefore, arises due to a significant differencein the lattice parameters of �1- and �2-phases which remain in contact with thegrowing �-plates. As shown in the G–c plot (Figure 7.54(c)) corresponding toT = 850 K, a growing �-plate can establish metastable equilibrium with the Cr-rich�2-phase and not with the Ti-rich �1-phase. It is, therefore, expected that during




T = 1000 K

Gα

Gβ

–500

–1000

–1500

G (

J/m

ol)

Ti 0.2 0.4 0.6 0.8 Cr

Ccr

α–β1

T = 950 K

Gα

Gβ

Ccr

Ti 0.2 0.60.4 0.8 Cr

–1000

–500

0

α+β1

α+β2

G (

J/m

ol)

T = 850 K

Gα

Gβ

Ccr

α+β2

G (

J/m

ol)

Ti 0.2 0.4 0.6 0.8 Cr

α+β2

–1000

–500

0

Figure 7.54. G–c plot for the Ti–Cr system at (a) 1000, (b) 950 and (c) 850 K.

the growth of �-plates at T < 873 K a rim of Cr-rich �-phase surrounds the growingplate even in alloys which are Ti-rich (as in the cited case for Ti-6.6 at.% Cr).

Similar observations have been recorded in Zr–7 at.% Nb and Zr–10 at.% Nb(Banerjee et al. 1988) where �-precipitates forming in the �-matrix during isother-mal decomposition establish local equilibrium with the �1- and �2-phases at 974and 823 K, respectively. In the latter case, an enveloping rim of the �2-phase hasbeen observed around the �-precipitates.




Table 7.5. Microstructure evolutions dictated by stable and metastable equilibriumin the Zr–Nb and the Ti–Cr systems.

Transformation process Sequence Operating equilibrium

Tempering of martensitic�′ in Zr–Nb

�′ → �+�1

(800 K< T < 893 K)�/�1 Metastable

�′ → �+�2 �/�2

(T < 800 K ) Stable

Phase separation– �→ �1 +�2 �1/�2

precipitation → �1�� +�2 �1/�in �-Zr-Nb → �+�2 �2/�

�-precipitation �→ �+�n �1/�in �-TiCr (T > 873 K) Stable

�→ �+�b �2/�T < 873 K Local metastable

�n: normal plates (�); �b: black plates (�).

A comparison can be drawn between the precipitation of �-phase from supersat-urated �′ Zr–Nb martensite (Section 7.3.5) and the precipitation of �-plates fromthe �-phase in the Zr–Nb and the Ti–Cr alloys. In all these cases, a competitionbetween �/�1 and �/�2 equilibrium dictates the course of the transformationprocess. Table 7.5 summarizes key observations on the formation of equilibriumand metastable precipitates in the Zr–Nb and the Ti–Cr systems which can berationalized in terms of the stable and metastable equilibrium between the �-phaseon one side and the �1- and �2-phases on the other.

All the morphologies of �-precipitates forming in the �-matrix discussed so farare produced when the � → � transformation occurs directly and not mediatedthrough the �-phase. Unnikrishnan et al. (1978) have shown that two distinctmorphologies of �-precipitates appear in the alloy of the same composition (Ti–Cr)when it is subjected to the following heat treatment sequences:

(1) �-solutionization is followed by rapid cooling to the isothermal reaction tem-perature (973 K) where the �-precipitation is allowed to proceed.

(2) �-solutionization is followed by water quenching to retain the �-phase at roomtemperature and subsequently ageing at the same reaction temperature (973 K).

Figure 7.55 (a) and (b) show the product �-precipitates forming during the heattreatments (1) and (2), respectively. The difference in the morphologies of the�-products has been rationalized (Unnikrishnan et al. 1978) in terms of heteroge-neous nucleation of �-precipitates on fine �-particles present in the �-quenchedalloy subjected to the heat treatment (2). In a recent study by Ohmori et al. (1998),




(c) 60 μm

(d) 50 μm (e) 50 μm

(a) 60 μm(b) 25 μm

Figure 7.55. (a) and (b) Group of parallel Widmanstatten �-plates growing from �-grain boundariesduring isothermal transformation of � in Ti–6% Cr at 725 C. These �-plates have nucleated directlyfrom the �-matrix (compare schematic drawing of Figure 7.51). (c)–(e) Group of �-plates of differentvariants nucleated from �-particles have formed within the �-grain in Ti–6% Cr in the �-quenched,followed by ageing at 725 C. (d) and (e) show �-particles and �-plates imaged with respectivereflections in the dark field. (a), (b) and (c) are optical micrographs while (d) and (e) are TEMmicrographs.

it has been demonstrated that �-laths nucleate at �/� interfaces (Figure 7.56) andduring their growth consume the �-particles. The �/�/� orientation relationshiphave been established to be

�1100 ��1010 ��211 �� 1120 ��0001 ��011 �� 0001 ��1210 ��111 �

The terrace plane between �- and �-phases is (1100)��(1010)�, which is theminimum misfit low index plane between the two phases.




Figure 7.56. High-resolution micrograph showing the �/�/� orientation relation.

7.5.2 Orientation relationThe orientation relation between the parent � and the product �-phases in adiffusional transformation has been studied in a number of Ti ( Furuhara et al.1988) and Zr (Banerjee et al. 1988, Perovic and Weatherly 1989) alloys. TheBurgers orientation relation has been found to be operative approximately inalmost all cases. More accurate measurements reported in recent papers, however,have shown that a small deviation from the Burgers relation occurs and that thisdeviation is important in the context of the operation of the invariant line straincondition in diffusional transformations in these systems.

As discussed in Chapter 4, the lattice correspondence operative in the �→ �′

martensitic transformation of Ti- and Zr-based alloys is given by the followingcrystallographic relation:

�001��2110�� 110��0001�� 110��0110��

This lattice correspondence, illustrated in Figure 4.20, is known as the Pitsch–Schrader relationship (Pitsch and Schrader 1958). The Burgers orientation relation-ship(Burgers1934)canbeobtainedfromthisbyrotating thehcpcrystalby5.26 aboutthe [0001]� direction clockwise (or counterclockwise) in order to bring the [1210]�direction nearly in coincidence (within 1.5 ) with the [111]� direction (or the [1120]�




direction to within 1.5 of the [111]� direction). This rotation, as has been shown inChapter 5, establishes the IPS condition when the 2% Bain strain along the directionperpendicular to the basal plane is neglected. The Burgers orientation relation whichis usually described as (110)�� (0001)�; [111]� nearly �� (within 1.5 )[1210]� alsobrings the (112)� plane to be appropriately parallel to (1010)�plane. The importanceof these two planes being parallel can be seen in Section 7.5.4 where the atomicmatching between the two phases along these planes is considered.

In some recent experiments on diffusional phase transformations in Ti- andZr-based alloys, an accurate determination of the �/� orientation relation hasyielded the following results:

[101]�� [1011]�; [111]�� [1210]; [110]� inclined to [0001]� by ∼ 1.5 .

This Potter (1973) orientation relationship can be established only if the smallangle between the [110]� and [0001]� directions is precisely determined. Since acareful determination of the angle between the directions [0001]� and [110]� hasnot been attempted in a great majority of earlier studies, it is difficult to assess inhow many experimentally determined cases the reported orientation relationshiptruly corresponds to the Burgers or the Potter type, the latter being predicted fromthe condition of ILS.

7.5.3 Invariant line strain conditionThe morphological development of precipitate plates or laths in a diffusionaltransformation can be rationalized in terms of the hypothesis proposed by Dahmenand co-workers (1982, 1984, 1986). They have shown that the product phase inmany diffusional transformations grows as a lath or a needle parallel to a vector,known as an invariant line, which remains unchanged in length and directionduring the course of the phase transformation.

If the overall transformation is described in terms of a linear homogeneousdeformation, A, which is composed of a pure lattice (Bain) deformation, B, and arigid body rotation, R�, then A can be expressed as

A = R� ·B

Using the Pitsch–Schrader correspondence for the diffusional �/� transforma-tion (which is the same as in the case of the martensitic transformation), B can beexpressed as

B =∣∣∣∣∣∣1 1 11 2 11 1 3

∣∣∣∣∣∣ (7.54)




where 1 = �3/2 1/2a�/a�; 2 = a�/a�; and �3 = 12c�/a�, the lattice parameters of

the �- and the �-phases being given by a� , c� and a�, respectively. The matrixrepresentation of the rigid body rotation R�, which brings back the undistortedvectors to their original positions, is given by

R� =∣∣∣∣∣∣

0�997894 0�002106 0�0648240�002106 0�997894 −0�064824

−0�064824 0�064824 0�995789

∣∣∣∣∣∣ (7.55)

The condition for a vector, X, in the parent lattice to remain invariant on theapplication of the homogeneous strain, A, is

AX = X

Equation (7.56) is satisfied if det∣∣∣A −�I

∣∣∣= 0 and one of the eigen values �is equal to 1. The predicted invariant line is extremely sensitive to the latticeparameters ratios, a�/a� and c�/a�. Substituting the values of the lattice parametersof the �- and the �-phases, Perovic and Weatherly (1989) have shown that theinvariant line for the �/� transformation in the Zr–2.5 wt% Nb alloy lies very closeto the [212]� direction, which is located at the intersection of the trace of (101)��(1011)� and the cone of unextended vectors generated by the lattice strain, B. Thisis shown in the stereographic projection in Figure 7.57 in which the elliptical coneof unextended vectors is represented by a thick line. The rigid body rotation, R�,can be decomposed into two rotations, the first one corresponding to that involvedin rotating from the Pitsch–Schrader to the Burgers orientation while the secondone rotates the Burgers to the Potter orientation. These two rotations are markedschematically in Figure 7.57.

Dahmen and co-workers (1982, 1984, 1986) have suggested that laths or needlesof the product phase in a diffusional transformation grow along the invariant lineof the transformation with the orientation relationship being determined by therestrictions imposed by the invariant line criterion. In transformations where theproduct phase has a plate morphology with a well-developed habit plane, the habitplane must contain the invariant line while the selection of the other vector fordefining this plane is made on the basis of minimization of the interfacial energy.

It is attractive to consider a few interesting parallels that can be drawn betweenthe habit planes of products of diffusion controlled and martensitic transformations.In the case of martensitic products, the habit plane is an invariant plane, i.e. allthe directions lying on the plane are invariant as far as the macroscopic “average”habit plane and the total shape strain within the plate as a whole are concerned.In the microscopic scale, the habit plane of a martensitic plate will contain either




100

110

111

101

212

111

110

010 011 001 011 010

111101 111

100

110 110

[001]//[2110]

[110]//[0001]

X2

X1

[110]//[0110]

Figure 7.57. Stereographic projection showing the direction of invariant line strain [2 1 2]�.

a row of dislocations marking the directions along which the lattice invariant slipplanes intersect the habit plane in a dislocated martensite or a row of lines alongwhich the twin planes meet the habit plane in a twinned martensite. In the lattercase, the macroscopic habit is made up of zig zag segments which meet along theinvariant line. In contrast, the habit plane in a diffusional transformation does not,in general, satisfy the invariant plane strain condition though at least one vectoralong the habit plane remains invariant. It is along this direction that laths or platesresulting from a diffusional transformation grow and this direction is often markedon the habit plane by a row of nearly equispaced dislocations, the line vectors ofwhich lie parallel to the invariant line.

The interface structure of �-laths in the �-matrix has been studied in Zr–Nballoys by Perovic and Weatherly (1989) and Banerjee et al. (1997). In the Zr–2.5 wt% Nb alloy, in which the volume fraction of the �-phase is rather small,both studies have shown the presence of arrays of parallel 1/3 <1123> (or <c+a>) dislocations at the interfaces (Figure 7.58). These dislocations, which areexceptionally straight and maintain a spacing of about 6–8 nm depending on thealloy composition, have been found to lie along the common (101)��(1011)� plane.




100 nm

(a)

α β

Figure 7.58. Interfacial dislocations at the �/� boundaries in the Zr–2.5 Nb alloy.

Such dislocations can glide as the interface migrates. In addition to these, some1/3 <1120> (or <a>) dislocations have been observed to lie both parallel to andacross the <c+a> dislocations. The density of the <a> dislocations has beenfound to increase with increasing rotation of the �/� interface from the flat facetswhere only one set of <c+a> dislocations is present. Because of the irregularityof the �/� surface in the Zr–2.5 wt% Nb alloy an accurate determination of thehabit plane has not been possible. However, the line vectors of the <c+ a>dislocations have been identified to be the direction of the ILS.

In a Zr–20 wt% Nb alloy, �-laths distributed in the �-matrix have been foundto be quite amenable for the determination of the habit plane and also for thecharacterization of the dislocation structure at the �/� interface. Laths of the�-phase formed in this alloy on isothermal treatment at 823 K are typically 100×200 nm in cross-section, with the length varying from 200 to 1000 nm. Severalvariants of �-laths are often encountered in a single field of view; a typicalexample is depicted in Figure 7.59(a). Orientations of all the variants match theBurgers relation quite closely. The �/� interfaces of these laths have invariablybeen found to contain arrays of parallel, equispaced dislocations of <c+ a>type (Figure 7.59(b)). The spacing between adjacent dislocations varies from 8 to10 nm. The habit plane of these laths, as defined by the plane containing the lengthand the width directions (the plane perpendicular to the thickness direction), hasbeen found to lie between the {103}� and {113}� poles (Figure 7.58(c)) and theline vectors of the <c+a> dislocations at the interface are along the <334>�

directions which match closely with the invariant line. The fact that the line vectorsof such �/� interfacial dislocations are parallel to the long direction of the �-laths




100 nm

(a)

(c)

(b)

100 nm

β

α

110

111112

001

122

111

221

130

131

110

111

013

113001

(112)//(1100)

HA trace

HC trace

HB trace

CL

212

(111)

//(1120)

(121)//(1010)C

(110)//(0001)

DA

LA

LB

DB

DC

311(211)//(1010)

HC

HC

HA

βHC

βHA

(001)(0001)C

Figure 7.59. (a) Several variants of �-precipitate laths in the �-matrix. All variants exhibit arrays ofequispaced dislocations lined up along the length of the precipitates. (b) Interfacial (�/�) dislocations,with line vectors parallel to the long direction of the precipitates. (c) Stereogram showing that threevariants of orientation relationship are operative for three habit variants of precipitates shown in (a).

is demonstrated in the electron micrograph in Figure 7.60(a) in which two �-laths,designated as A and B, are shown. The �/� interface at the side face of theselaths is seen to be parallel to the foil plane examined in TEM. For the lath Athis face is not retained within the sample sectioned, whereas this face is retained




B

A

200 nm 100 nm

(b)(a)

Figure 7.60. (a) Interfacial dislocations lying parallel to the foil plane on the �/� interface of plateB, the line vector of dislocation lines being parallel to the invariant line. (b) Equispaced <c+a>dislocations at �/� interfaces lying along the long direction of this precipitate plate.

within the foil for the lath B. In the latter case <c+ a> dislocations are seenalong the long direction of these laths and their line vector remains parallel to theinvariant line. The foregoing observations on the morphology, orientation relationand crystallography of �-laths forming from the �-phase in a diffusional processcan be summarized as follows:

(1) The orientation relation between the �- and the �-crystals is very close to theBurgers relation.

(2) The habit plane as defined by the plane containing the length, l, and the breadth,b, directions is an irrational plane. Habit plane poles of different variants arefound to lie between the {103}� and {131}� poles. For all the variants of laths,habit planes remain close to the �/� conjugate pair {0110}��{112}�.

(3) Interfacial dislocations with <c+ a> Burgers vector remain aligned alongthe invariant line and are arranged in a parallel array with a spacing of about8–10 nm (Figure 7.60(b)).

(4) The long direction of laths matches closely the invariant line direction<433>�.

7.5.4 Interfacial structure and growth mechanismsDetailed studies have been made on the nature of the structure of the �/� interfacein Ti–Cr alloys (Furuhara et al. 1991). Both diffraction contrast and phase contrastTEM experiments have been carried out for deciphering the interfacial structure.It is through these studies that the atomic movements necessary for the structuraltransformation from the bcc to the hcp phase during the diffusional process havebeen identified. As discussed earlier, two types of Widmanstatten �-plates areencountered in the Ti–Cr system, namely, the normal �-plates which maintainnear equilibrium with the �1-phase during growth and the black plates which form




in contact with the �2-phase. Let us first discuss the �-normal �-interface structurewhich has the following characteristics:

(1) As in the case of Zr–Nb alloys, normal �-plates obey the Burgers relationalmost exactly:

[1120]��[111]�(0001)��(011)�(1100)��(211 �

(2) The broad face of the plate has an irrational habit plane close to (11,11,13)�. This habit plane results from the uniform arrangement of structuralledges which step down along the lattice invariant line (approximately [335]�)with (112)��(1100)� terraces. The concept of structural ledges introduced byHall et al. (1972a,b) and Rigsbee and Aaronson (1978) envisages that theseare steps, one to a few atom planes high which, when spaced regularly onan interphase interface, reduce the misfit along the rational interface directionnormal to the ledge. How structural ledges reduce the misfit on a bcc/hcpinterface is shown schematically in Figure 7.61.

(3) A set of misfit dislocations with the Burgers vector a/2 [110]��c/2 [0001]�exists along the lattice invariant line on the broad face of �-plates with a

α (hcp)

β (bcc) [111]β // [1120]α

b = a /12[111]β

[112]β// [1100]α

L

Figure 7.61. Schematic representation of structural ledges showing the compensation of misfit at�/� boundaries.




spacing of about 12 nm. These dislocations have a sessile character with respectto the migration of the broad face, and thus the growth of �-plates mustaccompany diffusional jumps of atoms across the interface.

The interfacial structure of the bcc/hcp interface as deduced from TEM obser-vations and from modelling of the structure of the two phases across the planeof good fit, i.e. (1100)�/(211)� contains two different kinds of linear defects atthe boundary: structural ledges and misfit compensating ledges. Aaronson andco-workers have distinguished between these two types of linear defects on thebasis of the presence of extra half atomic layers in case of the misfit compensatingledges: such extra half layers are not present in the structural ledges. However,Christian (1994) has pointed out that steps, ledges and interface dislocations canall be described in terms of transformation dislocations of Burgers vector bt whichin the case of structural ledges are smaller than the smallest lattice translationvector. It may be noted that both the direction and the magnitude of bt may beirrational.

Furuhara et al. (1991) have shown that the broad face of an �-lath containsa set of misfit dislocations, with c-type Burger vector. These dislocations, whichhave their line vectors parallel to the invariant line direction, �533��, loop aroundthe �-laths. Similar observations have been reported in Zr–Nb alloys by Zhangand Purdy (1994) and Banerjee et al. (1997). It has been hypothesized (Zhang andPurdy 1993a,b) that the optimum orientation relation is that which minimizes theinterfacial misfit in the habit plane. The lines along which the two lattices matchbest are defined as O-lines (based on the O-lattice description of interfaces byBollman 1970). The habit plane is, therefore, selected to be the plane containing anarray of O-lines with the largest spacing between these lines. The comparison ofexperimental observations on and geometrical analysis of habit planes of �-platesproduced in a diffusional transformation from the �-phase has established thatthe habit plane (broad face of �-laths) is characterized by a single set of misfitdislocations whose line vectors are coincident with the invariant line vector andwhose spacing is equal to the calculated distance between the O-lines.

Experimental and theoretical work on the nature of �/� interfaces in diffusionaltransformation products of Ti- and Zr-based alloys has proved beyond doubtthat there exists a good coherency between the bcc matrix and the hcp productacross the interface. This is in agreement with diffusional transformations inother systems involving hcp/fcc (Howe et al. 1987) or bcc/fcc (Furuhara et al.1995a,b) transformations. The presence of coherency at the transformation front issuggestive of an atomic site correspondence which should be distinguished fromlattice correspondence, an essential feature of displacive transformations. Duringthe growth of a plate or lath in a diffusional transformation, individual atoms cross




the transformation front by thermally activated diffusional jumps, maintaining thecorrespondence of atomic sites between the two phases. Such a growth processleads to a macroscopic shape change due to the transformation. This is reflectedin the observation of tilting of originally flat surfaces and a change of directionof fiducial lines (scratches) inscribed on the surface of the parent phase sample.Such a change was earlier known to be the distinguishing feature of a martensitictransformation, but there is now good evidence that shape changes can occur insome diffusional transformations as well.

It is worthwhile to draw a comparison between the interfaces/transformationfronts in martensitic transformations and those encountered in the �/� diffusionaltransformation. Both types are shown to be irrational and usually partially coherent.Fully coherent irrational interfaces are rare, but they are found in some martensitictransformations in Ti alloys, e.g. Ti–22 at.% Ta (Bywater and Christian 1972).In general, the partially coherent interfaces of martensite plates are usually irra-tional because of the invariant plane strain requirement and the fact that the latticeinvariant shear is a simple shear. However, these interfaces should be necessarilyglissile and should conserve the number of atoms during propagation. In contrast,partially coherent irrational interfaces in diffusional transformations are essentiallyepitaxial and are made up of terraces of rational interfaces, separated by steps orledges. A new layer of the product phase forms by the migration of a growth ledge.This process does not conserve the number of atoms. In general, diffusional trans-formations involve non-conservative movements of transformation, dislocations atthe transformation fronts.

Ledges of any height that are not an intrinsic part of the interface structuremay be regarded as growth ledges or transformation dislocations. Isolated stepson a coherent interface or steps on a faceted high-index interface are examplesof such ledges. Aaronson and co-workers have distinguished between structuraland growth ledges, the former being an intrinsic part of a high-index interface, asshown in Figure 7.61. However, the distinction between them is not always verysharp as the roles they play in misfit compensation and in the growth process mayoften merge. Superledges with large Burgers vectors, though unstable due to theirhigh self-energies, are encountered on the transformation front in some cases ofdiffusional transformation.

The growth of an �-lath in Ti–Cr alloy has been schematically illustrated inFigure 7.62, which has been taken from the model proposed by Furuhara et al.(1995a,b). Growth ledges, which have been observed both on the broad face andthe side facet of the lath, are responsible for thickening and widening of thelath. Each growth ledge on the side facet contains a misfit dislocation with theBurgers vector, a/2, [110�= c/2 [0001]� on its riser. This dislocation is in a sessileorientation and, therefore, it must climb for the motion of the growth ledge riser.




Bi-atomicstructuralledge

[1100]α//[211]β

Misfit-compensatingc-type ledge

b = a /18[1120]α

EdgeSidefacet

[335]

Primary growthdirection

Broadface

Figure 7.62. Schematic illustration of the interfacial structures of �-plates growing in �; both thebroad face and the side facet are shown.

Since diffusion and partitioning of alloying elements accompany such a growthprocess, non-conservative motion of dislocations at the transformation front isfacilitated. The thickening of a lath means addition of new layers on the broad face,which can be accomplished by a ledge growth mechanism as shown in Figure 7.62.The broad face of the lath is formed by the coalescence of growth ledges on thissurface. This riser plane of growth ledges on the side facet contains the structuralledges and is thus semicoherent. The migration of the growth ledges should occurby the kink on ledge mechanism in which diffusing atoms get attached to the kinkplane. Since these ledges do not coalesce to the extent necessary for forming acomplete layer on the side facet, the broad face remains as a stepped interfacewith steps along the lattice invariant line. These steps are nothing but the misfitcompensating c-type ledges.

The ledge growth mechanism in a diffusional transformation requires (a) atomicdiffusion to effect the partitioning of the alloying elements, (b) some mechanismby which new ledges are formed on the newly created layer and (c) maintenanceof the semicoherent interfacial structure. In a steady state, these processes must




all conform to the overall rate that is determined by the process which consumesthe major part of the driving force.

The ledge growth mechanism described here can be compared with that ofcrystal growth from the vapour phase. In the latter case, atoms from the vapourphase condense on the terraces of a growing crystal, diffuse over the surfaceuntil they either evaporate back into the vapour or find a kink in a step wherethey ultimately join the crystal lattice. In the case of a solid–solid diffusionaltransformation, the growth, as described earlier, involves attachment of atoms atkinks in steps on solid–solid interfaces. The diffusion of atoms along the partiallycoherent interface, however, is not quite favourable. In martensitic transformations,such as the fcc → hcp transformation, the evidence suggests that a step can glideas a unit with such a high velocity that separate kink motion seems unlikely. Indiffusional transformations, a kink on a superledge can undoubtedly provide afavourable site for attachment of freshly arriving atoms to join the lattice sites ofthe product crystal.

The growth velocity of the interface by ledge growth mechanism can be mod-elled using the Jones and Trivedi (1971) treatment. Let us consider an interfacewith perfectly coherent terraces separated by an incoherent ledge at a spacing of� as shown in Figure 7.63. The coherent areas can be considered to be entirelyimmobile with attachment of atoms to the growing phase occurring at the ledge.Cahn et al. (1964) expressed the growth rate of the boundary in the directionnormal to the coherent areas as

G= av

�(7.56)

where a is the height of the riser, � is the average distance between the ledgesand v is the lateral migration rate of the ledges. Jones and Trivedi have made a

Va

V

G

λ

Figure 7.63. Schematic representation of the ledge growth mechanism. a: height of the riser, :average spacing of ledges, V : velocity of ledge perpendicular to the growth direction, G: rate ofinterface movement along the direction normal to partially coherent areas.




detailed treatment of v in which interface reaction kinetics are again taken intoaccount. For reasons presented later, the interface kinetic coefficient, �, will beassumed infinite, yielding the relationship.

v= D�x�� −x�

�a�x�� −x

��

(7.57)

where = a constant which can be likened to a diffusion distance and which is acomplicated function of the Peclet number, p = va/ 2D, which in turn varies in acomplicated manner with the supersaturation. Combining Eqs. (7.56) and (7.57):

G= D�x�� −x�

�x�� −x

��

(7.58)

It may be noted that the ledge height, a, does not appear in the equation whichis due to the fact that the velocity of a ledge is inversely proportional to its height.Jones and Trivedi (1975) have also investigated the course of the overlappingof the diffusion fields of the adjacent ledges. Figure 7.64 schematically showsthe overlapping of two diffusion fields. It can be noticed that the diffusion fieldextends more on the broad face at the bottom of the riser than the face at the topof the riser. Therefore, the diffusion field of the trailing ledge affects the diffusionfield of the leading ledge move than the latter affects the former. Modelling ofthis effect is rather complex because the overlapping diffusion fields cannot besimply superimposed, as that would alter the concentration gradient along the riser

6.05.04.03.02.01.00.0–1.0–2.0–3.0–4.0–5.0–6.0Length/ledge height

Figure 7.64. Reduced isoconcentration curves about two ledges whose diffusion fields overlap.Numbers on the x-axis correspond to distances in multiples of the ledge height.




itself. The concentration gradient must be calculated carefully so that all portionsof the riser migrate at the same rate. However, such calculations can be done usingthe confirmed mapping technique and it is possible to evaluate the concentrationgradient on which the migration rate of each ledge depends. It is found that theconcentration gradient for the trailing ledge is higher than the leading edge, whichmeans that the trailing ledge will catch up and merge with the leading edge.

7.5.5 Morphological evolution in mesoscaleThe transformation of the �-phase, either in an isothermal condition within the�+� phase field or during continuous cooling, produces a variety of microstruc-tures. Since the properties of commercial Ti alloys are strongly influenced by themicrostructure in mesoscale, a description of the microstructure evolution resultingfrom the �/� diffusional transformation is given here.

During continuous cooling from the �-phase field, the �-phase appears first asGBAs. As mentioned earlier, �-allotriomorphs are invariably related to one of theadjoining �-grains by the Burgers orientation relation. The �/� interface of theallotriomorphs which tend to propagate more towards the Burgers-related �-grainsundergo a morphological instability, leading to the development of periodic pro-trusions (Figure 7.50) designated as saw tooth morphology. The invariant linedirection corresponding to the operating variant of orientation relation is chosento be the growth direction of the protrusions which develop into � side plates.The combined influence of the stress field and concentration field ahead of the�/� interface is responsible for the growth of a group of �-laths of identicalorientation (same as that of the allotriomorph at a nearly equal spacing). Thefully grown region of such parallel laths define a colony, a microstructural featurewhich has a strong bearing on mechanical properties such as yield strength andfracture toughness.

The allotriomorphs cannot grow into the ��2 grain with which it does nothave a Burgers relation. However, a set of �-laths with an orientation Burgersrelated to �2 can get sympathetically nucleated on the surface of the allotriomorph.They can grow in a similar manner into the �2-grain forming a colony. With asomewhat slow cooling rate, the entire �-grain volume is gradually filled up by anumber of such colonies of �-laths which nucleate from grain boundaries.

The fact that �-laths growing into the �1-grain bear the same orientation ofthe allotriomorph is demonstrated in Figure 7.49 where two colonies are seen togrow from two allotriomorph orientations which share the same basal plane butare rotated by an angle of 10.5 (as described earlier in Figure 7.49). The invariantline directions for these two variants make an angle of 87 which is reflected inthe large angle between their long axes (primary growth direction).




With increasing cooling rate the volume fraction corresponding to the colonymorphology decreases and the remaining part of the �-grains transforms into a“basket weave” morphology which is characterized by the presence of laths ofdifferent orientations in the same region.

As has been demonstrated through many examples in the book, phase transfor-mation and microstructural evolution in commercial titanium and zirconium alloysare extremely complex. Traditional models that characterize microstructural fea-tures by their average values without capturing the anisotropy and spatial variationmay not be sufficient to quantitatively define the microstructure and hence to estab-lish a robust microstructure–property relationship. Recent progress in computersimulation of complex microstructures using the phase field method (Wang andChen 2000) offers a unique opportunity to rigorously and realistically address theproblem. Extensive efforts have been made in integrating thermodynamic mod-elling and phase field simulation to develop computational tools for quantitativeprediction of phase equilibrium and spatiotemporal evolution of microstructuresduring thermal processing that account explicitly for precipitate morphology, spa-tial arrangement and anisotropy. Figure 7.65 shows an example of side plateformation in Ti–6Al–4V obtained from phase field simulations (Wang et al. 2005).Side plates, the �-phase lamellae growing off grain boundary � upon cooling, arethe major microstructural constituents of many �/� Ti alloys. Two mechanismshave been proposed for the initiation of side plates from grain boundary �: sym-

Figure 7.65. Simulated micrographs produced by the phase field simulation of formation of the�-plates in the �-matrix. A competition between formation of side plates and of basket weavestructure in the 2D simulation is shown. (After Wang et al. 2005). (a) t= 5 s, (b) t= 10 s, (c) t= 15 s,(d) t = 20 s.




pathetic nucleation (Aaronson and Wells 1956) and interface instability (Mullinsand Sekerka 1963). In the current example, the latter mechanism was assumed.Two-dimensional phase field simulations were carried out at 1123 K, with a start-ing microstructure of a thin layer of grain boundary � that is in contact withsupersaturated �-grains. A random fluctuation of the �/� interface position wasintroduced by a random number generator (as shown in Figure 7.65(a)). The inter-facial energy between � and � is assumed to be anisotropic, with the one of theinterface parallel to the growth direction being 1/3 of that of the one perpendicularto the growth direction. The time evolution of the initial fluctuations in interfaceposition into a colony of side plates is shown in Figure 7.65(b)–(d). It is readilyseen that the spatial variation and shape anisotropy in precipitate microstructureare well captured by the simulation.

7.6 PRECIPITATION OF INTERMETALLIC PHASES

Precipitation of intermetallic phases can occur in Ti- and Zr-based alloys either inthe supersaturated �′-matrix or in the matrix of a metastable intermetallic phase.The former situation is encountered in systems containing �-stabilizing alloyingelements, such as Al, Sn and Ga, and also a large number of Ti–X and Zr–Xbinary eutectoid systems in which the solubility of the element X in the �-phaseincreases sharply with temperature. The precipitation of an ordered intermetallicphase from a disordered solid solution can go through several intermediate stages,depending on the clustering and/or ordering tendencies of the solid solution and thepresence of coherent intermediate structures between the parent and the equilibriumprecipitate phase. Section 7.6.1 is devoted to a discussion on some examplesof such precipitation reactions. The formation of an ordered precipitate phasefrom a parent phase which is also chemically ordered has been exemplified in Tialuminides. The crystallography of such transformations, which has been recentlystudied in detail, brings out some important aspects of the diffusional growthmechanism. This aspect has been elaborated in Section 7.6.2. The formation ofintermetallic phases through cellular growth mechanisms is covered in Section 7.7.

7.6.1 Precipitation of intermetallic compounds from dilute solid solutionsThere are a number of Ti- and Zr-based alloys in which the equilibrium constitutionis a mixture of the �-phase and an intermetallic compound. Many of these systemsexhibit an eutectoid reaction, essentially due to the fact that the solubility of thealloying element is higher in the �-phase than in the �-phase. Supersaturated�-solid solution can be formed in such systems by quenching an alloy either from




the �-phase field or from the �-phase field, the latter giving rise to a higherlevel of supersaturation in the martensitic �′-phase. On ageing, precipitation ofintermetallic phases, usually the equilibrium phase richest in the solvent (Ti orZr), occurs.

The process of the precipitation of intermetallic phases in �-solid solution hasbeen studied in a number of binary and multicomponent alloys of Ti and Zr. As anillustrative example, we will first discuss the precipitation of ZrCr2 in the �-phasedilute Zr–Cr alloys. Two structural variants of ZrCr2 have been reported. Bothare topologically close-packed Laves phase structures: C14 (hexagonal, MgZr2

type) with a= 0�5079 nm and c = 0�8279 nm and C15 (cubic, MgCu2 type) witha= 0�721 nm, the cubic form being the equilibrium structure at temperatures lowerthan 1000 C.

Mukhopadhyay and Raman (1978) have reported that in an alloy having acomposition of Zr–2 wt% Cr, �-quenching cannot fully suppress precipitation ofZrCr2 which appears along closely spaced rows in the �-matrix (Figure 7.66(a)).Such a microstructure suggests that these rows of precipitates trail the advancingtransformation front at which the �-phase rejected the � and the ZrCr2 precipitatesimultaneously. Such a process is thermodynamically possible under a conditionwhere the extent of supercooling is sufficiently high to cause direct eutectoidreaction even in this hypereutectoid alloy.

Ageing in the temperature range of 350–550 C has resulted in the formation ofbimodal size distribution of precipitates, larger precipitates at �-lath boundariesoften being connected to form stringers. The presence of coarser precipitates eitherat lath boundaries or along dislocation is suggestive of enhanced solute diffusionalong the defects being responsible for a faster growth of precipitates.

The ZrCr2 precipitation reaction has also been studied in a heat treatment involv-ing �-solutionizing followed by isothermal holding at the reaction temperaturein the range of 700–800 C. The resulting microstructures suggest that the ZrCr2

phase is the first phase to emerge from � followed by the formation of �-lathswhich remain supersaturated with solute (Figure 7.65(b)). Fine scale precipitationwithin the �-laths occurs at a subsequent stage.

The orientation relation between the � and the ZrCr2 phase has been determinedfrom superimposed diffraction patterns of � and ZrCr2 (Figure 7.66(b)) to be thefollowing:

The most commonly used Zr alloys in nuclear industry are zircaloys whichcontain tin, iron, chromium and in some cases nickel (as shown in Table 7.6).Zircaloys are essentially �-Zr-Sn solid solutions containing fine intermetallic pre-cipitates. The distinct types of intermetallic particles are observed in zircaloy-2,namely, Zr2(Fe,Ni) and Zr(Cr,Fe)2. The former has a body-centred tetragonalstructure (space group 14/mmm, D17

4n) while the latter is a lower phase with a




(a) (b)

(c) (d)

m m

Figure 7.66. Precipitation of ZrCr2 precipitates in the �-matrix (a) shows rows of precipitateparticles which appear to have formed along a trail following the advancing transformation front.(b) Coarse ZrCr2 particles forming at lath boundaries followed by fine scale precipitation within thematrix. (c) Diffraction pattern showing superimposed reciprocal lattice sections of �-matrix (M) andZrCr2 precipitates (P). (d) Key to the diffraction pattern.

C14 hexagonal structure. Usually precipitate particles forming at grain boundariesof zircaloy-2 are larger (typical diameter 3�m) and are of the Zr2(Fe,Ni) typewhile intragranular precipitates (typical diameter ≤ 1�m) are of both Zr2(Fe,Ni)and Zr(Cr,Fe)2 types. Presence of stacking faults within the latter is the distin-guishing feature of the Zr(Cr,Fe)2 particles. Mechanical and corrosion propertiesof zircaloys-2 and 4 are strongly influenced by the size and the distribution ofthe ordered intermetallic particles which form during the thermal and mechanicalprocessing of zircaloy components.

ElsevierUK

Code:

PTA

Ch07-I042145

6-6-2007

5:37p.m.

Page:660

Trim:165mm×

240mm

Integra,India

Font:Times

F.Size:11/13pt

Margins:

Top:19mm

Gutter:19mm

Width:128mm

Depth:40Lines

1Color

Recto

660P

haseT

ransformations:

Titanium

andZ

irconiumA

lloys

Table 7.6. Chemical composition of zirconium alloys for nuclear application.

Alloy element Zircaloy-2 Zircaloy-4 Zr–1Sn–1Nb Zr–2.5Nb Zr–1Nb Excel Zr–Nb–Cu Ozhennite

Sn (wt%) 1.2–1.7 1.2–1.7 0.9–1.1 – – 3.5 – 0.2Fe (wt%) 0.07–0.2 0.18–0.24 0.1 <650 ppm <500 ppm – – 0.1Cr (wt%) 0.05–0.15 0.07–0.13 – <100 ppm – – – –Ni (wt%) 0.03–0.08 – – <35 ppm <200 ppm – – 0.1O (ppm) 900–1300 900–1400 900–1300 900–1300 <1000 – 900–1300 –N (ppm) <80 <65 <65 <65 <60 <70 <65 –Nb (wt%) – – 0.9–1.0 2.4–2.8 1.0 0.8 2.4–2.8 0.1Cu (wt%) – – – – – – 0.3–0.7 –Mo (wt%) – – – – – 0.8 – –H (ppm) <25 <25 <20 <5 <15 – <25 –C (ppm) <270 150–400 – <125 <200 – <270 –C1 (ppm) <20 – <0.5 – – – – –P (ppm) – – – <10 – – – –

(a) (b) (c) (d) (e) (f) (g)

(a) Fuel tube and plug material for BWR and PHWR; (b) fuel tube and plug material for PWR and PHWR; (c) experimental fuel tube alloy; (d) pressure tube materials for PHWR;(e) fuel tube and end plug in VVER; (f) experimental pressure tube alloy; (g) garter spring material for PHWR.All the alloys contain A1 < 75 ppm, B and Cd < 0.5 ppm, Co and Mg < 20 ppm and Hf <50–150 ppm.




The search for zirconium alloys suitable for use as a cladding material in highburn-up fuel has resulted in alloys in which both �-forming Nb and intermetallic-forming Sn and Fe are added. A typical composition of such an alloy isZr–1Nb–1Sn–0.4Fe which exhibits better corrosion resistance and high radiationstability. These alloys are also suitable under conditions of partial boiling ofcoolant water as they do not show modular corrosion. Unlike zircaloys or binaryZr–1Nb alloys, these Zr-Nb-Sn-Fe alloys do not show a transition in the kineticsof irradiation-induced growth and creep up to a neutron fluence of 5×1026 nm−2

(neutron energy E > 0.1 MeV).The microstructure of these alloys are characterized by a distribution of inter-

metallic phase particles in the �-matrix. The total volume fraction of precipitatesin the Zr–1Nb–1Sn–0.4Fe alloy thermomechanically processed for cladding tubeapplication is about 3–3.5. The majority of precipitates have a globular shapewith diameters in the range of 0.1–0.2�m though a limited number of precip-itates are as large as about 1�m. X-ray microanalysis of precipitates extractedon a carbon replica, as reported by Nikulina et al. (1996), has shown that theprecipitates have an average composition of about 40Zr–40Nb–30Fe. Diffractionanalysis, however, has shown that two types of intermetallic precipitates, namely,the (Zr,Nb)3Fe phase having an orthorhombic structure and the Zr(Nb,Fe)2 phasehaving a hexagonal structure, are parent (Figure 7.67). It is interesting to note thatthe intermetallic precipitates in this alloy are susceptible for a change due to irra-diation. When irradiated to a fluence of about 4×1025 nm−2, the precipitate phases�-Nb-Zr and Zr3�5Fe appear at the expense of those present in the pre-irradiatedcondition.

Intermetallic Ti5Si3 precipitation in Ti–Si alloys has been studied in detail byFlower et al. (1971). Heterogeneous nucleation of precipitates on dislocations

Figure 7.67. Precipitation of Zr(Nb,Fe)2 and (Zr,Nb)3Fe precipitates in the �-matrix in the Zr–1Nb–1Sn 115 M and 0.4 Fe alloy.




has been reported. These precipitates being an important hardening agent, severalcommercial Ti alloys with Si addition have found extensive applications.

The Ti–2% Cu alloy shows formation of Cu-rich Guinier Preston zones in theearly stages of precipitation. During continued ageing, these zones lose coherencyand undergo an in situ transformation into the Ti2Cu phase (Williams et al. 1970).

7.6.2 Precipitation in ordered intermetallics: transformation of �2-phaseto O-phase

Some cases of diffusional transformations have recently been studied systemati-cally in the context of microstructure evolution in structural intermetallics. It isinstructive to examine these transformations as examples of diffusional transfor-mations which follow martensitic crystallography. Some key issues related to thecrystallography of these transformations and to the mechanism of atomic move-ments are discussed in this section with reference to the specific example of thetransformation of the �2-phase to the O-phase in Ti–Al–Nb alloys.

The transformation from the �2-phase (DO19 structure) to the O-phase(orthorhombic structure) during isothermal ageing, investigated by Muralidharanet al. (1995), has revealed several special features as listed below:

(1) The parent and the product phases are both ordered.(2) The transformation is associated with only a small lattice distortion, the three

principal strains being positive, negative and zero, fulfiling the IPS conditionwithout involving any lattice invariant strain.

(3) The parent and the product phases differ in composition and the productgrows with isothermal ageing; both these features indicate the involvement ofdiffusion and thermal activation in the transformation process.

An alloy with the composition, Ti–28.5 at.% Al–13 at.% Nb, chosen for thisstudy, on quenching from the �-phase field, followed by annealing at 1198 K for24 h, produces equiaxed grains of the �2-phase. On subsequent ageing in the �2 +Ofield, plates of the O-phase are precipitated. The lattice correspondence betweenthe parent �2 and the product O-phase structures is shown in Figure 7.68 whichindicates that the O-phase structure has the same site occupancy for Ti, Al andNb atoms as in the case of the parent �2-structure but possesses an orthorhombicsymmetry. The dimensions of the �2 and the O-phase unit cells along the principalstrain directions, [100]O, [010]O and [001]O, are as follows:

a�2 = 0�578 nm; b�2 = 1�001 nm; c�2 = 0�467 nmaO = 0�595 nm; bO = 0�970 nm; cO = 0�467 nm




B

A CA C

B

0.578 nm 0.595 nm

1.00

1 nm

0.97

0 nm

[010

]α2[0001] O [001]

[100]

[011

0]

[211

0]

Atom Al Ti Nb

Layer A

Layer B.c /2 above

AngleABC

α2 O

63°60°

Figure 7.68. Structure of �2- and O-phases projected along the <001> orientation. The small circlesindicate atoms in a plane 1

2 above or below along the direction of projection. Note that the O-phasecontains Nb atoms randomly distributed in Ti sites.

The �2-structure is described here in terms of an orthohexagonal basis so thatthe correspondence between the parent and the product lattices is given by

[100]�2��[100]O; [010]�2��[010]O; [001]�2��[001]O

The principal lattice distortions can then be evaluated to be

along [100], 1 = 1�029411along [010], 2 = 0�969031 andalong [001], 3 = 1�000000

The construction of a strain ellipsoid in accordance with the phenomenologicaltheory of the martensitic transformation (Figure 7.69) shows the two invariant linesOA and OB which are generated by the lattice strain from OA′ and OB′, respec-tively. A rigid body rotation by an angle, !, either clockwise or counterclockwise,brings back OA or OB to their original positions. Since the distortion along the �3

direction, perpendicular to the plane of the paper, is zero, the planes containing the




[010][0110]α2

[2110]α2[100]O

BB′

AA′

θ = 1.73°Hab

it for

+ve

solut

ionHabit for

-ve solution

φ2 = 42.95° φ1 =44.68°

Figure 7.69. Habit plane determination for the �2 to O transformation.

�3 direction and OA or OB are the two habit plane solutions. The indices of theundistorted vectors on the basis of the parent lattice are given by <1K0> where

K = ±(

1−21

�2 −1 2

)1/2

= ±0�989 (7.59)

The rigid body rotation, ", is given by

" = #1 −#2

where #1 and #2 are the angles made by the undistorted vectors OA and OBbefore and after the application of lattice strain; their values are

#1 = tan−1�0�989 = 44�68

#2 = tan−1�0�931 = 42�95

" = 1�73




The indices of the habit planes which satisfy the IPS condition are determinedby two vectors which remain invariant on the application of the combination of thelattice strain and the rigid body rotation. The indices of the plane containing [001]and [1 ± 0.9890] are {1 ± 1.7510} which work out, in the hexagonal four-indexsystem, to be (83110) and (81130). This plane in the orthorhombic basis is {470}O.

Three crystallographically equivalent O-phase variants, A, B and C, satisfy thelattice correspondence by making the [100]O axis parallel to one of the directions,[2110]�2

[1210]�2or [1120]a2

. The small rigid body rotation of 1.73 in clockwiseand counterclockwise directions for each of the correspondence variants generatesa total of six orientation variants designated as A+, A−, B+, B−, C+ and C−.Muralidharan et al. (1995) have experimentally determined the habit planes foreach of these variants and the results obtained are presented in a stereogram inFigure 7.70. It is to be noted that the angles between the different habit variantsare 30 , 60 and 90 , as indicated in Table 7.7. The excellent agreement betweenthe experimentally determined and theoretically predicted habit planes clearlyestablishes the fact that the crystallography of the precipitation of the O-phase inthe �2-matrix satisfies the IPS condition.

(0110)(110) B+ (110) C–

(1120)(130) A–

(130) B+

(110) A–(110) C+

(1010)

(130) C+

(2110)(130) B–

(110) A+(110) B–

(1100)

(1210)

(130) A+ (130)C–(0110)

(1120)

(1010)

(2110)

(1100)

(1210)

(0001)

C+ B-

A+

C–

A–

B+

(31180) (38110)

(81130) (83110)

(11380)(11830)

Figure 7.70. Experimentally determined habit planes for �2 to O transformation for all the sixvariants.




Table 7.7. Angular relationship between O-phase variants.

1 Habit planes at 30 A−, B+; A−, C+;B−,C+;B−, A+; C−, A+; C−, B+

2 Habit plane at 60 A−, B−; A−, C−; B−, C−;A+, B+; A+, C+;B+, C+

3 Habit plane at 90 A−, A+; B−, B+, B+; C−, C+

The development of the morphology of intersecting O-phase plates andthe grouping of O-plate variants during the thermally activated growth (bothlengthening and thickening) process have been found to be controlled by theself-accommodation of the participating variants. The manner in which self-accommodation is accomplished is described in the following by considering afew interesting examples.(a) Interactions of O-plate variants with habit planes at 30 : Figure 7.71(a) showsthe interaction of pairs of variants, C+/A− and C+/B−. These variants, duringthe process of growth, come in contact along (1100)�2 and (1210)�2 planes whichare nearly parallel to {110}O and {130}O planes. These plates do not terminateat the intersection point, and in such a situation the non-terminating plate tapersdown in thickness. Combinations of three plates, each making a 30 angle with theneighbouring plates, are also encountered. The interfaces separating such platesare also of {110}O and {130}O types. Figure 7.71(a) shows how three plates meetat a point. Plate configurations like this suggest that the stress fields of the threeplates interacted at the nucleation stage, leading to a low strain energy (high degreeof self-accommodation) grouping of plates.(b) Plates with habit planes at 60 : Interactions of such plates lead to the propa-gation of one plate across the other. Plates are physically displaced in a manneranalogous to that observed in the case of the intersection of twins or martensiteplates, the offset created due to the displacement being related to the macroscopicshape strain associated with the intersecting plate. Such an interaction can alsoresult in the nucleation of a third variant as shown in Figure 7.71(b) where A+�B+and C+ variants meet. It is to be noted that the region where three variantsoverlap undergoes a retransformation into the �2-phase. Selected area diffractionand microdiffraction analyses have established that the orientation of the retrans-formed �2-region is not the same as that of the parent �2-grain. The retransformed�2-orientation is obtained by a 3.5 rotation of the parent �2-orientation aroundthe [0001] axis. As a consequence of this rotation, the interfaces created betweenthe three O-phase variants and the retransformed �2-region are again parallel to{470}O��{83110}�2 planes and are perpendicular to the original habit planes of therespective variants.




(a) (b)

(c) (d)

Figure 7.71. Interactions of O-plates: (a) at 30 , (b) at 60 , (c) at 90 , (d) combined features.

The �2-phase is also observed at the centre of the coarsened star-shaped three-plate group containing, for example, the A+�B+ and C+ variants having theirhabit planes at 60 to one another. It remains unresolved whether the “retransfor-mation” of the central region into the �2-structure occurs during the early stagesof plate formation or at the coarsening stage.(c) Plates with habit planes at 90 : Variants such as A+ and A−�B+ and B−and C+ and C− meet each other at 90 (Figure 7.71(c)). A fresh nucleation ofO-plates of other crystallographic variants inevitably occurs. The new variantswhich nucleate are at 30 to one of the intersecting plates. The freshly nucleatedvariants share common interfaces with those plates which are at 30 to them. The60 star configuration with the retransformed �2-phase is produced as a result ofthis interaction as shown in Figure 7.71(b).




The strong elastic interactions among the different variants of the O-phase plates,as illustrated in these examples, can be rationalized in terms of self-accommodationof plates – a feature commonly encountered in martensitic structures. An exami-nation of the average shape strain associated with a given group of variants canreveal the extent of self-accommodation that can be achieved in such a group.The groups in which the average shape strain is considerably reduced are thosewhich appear more frequently. This analysis, however, assumes equal volume ofeach of the variants, neglects the presence of the matrix and, most importantly,ignores the anisotropy of strain around a plate-shaped precipitate and, therefore,the spatial arrangement of plates in a given group. Table 7.8 shows elements ofshape deformation matrices for individual O-plate variants referred to a commonset of principal axes, as defined in Figure 7.52. Six variants of O-plates can begrouped in 57 different ways taking two, three, four, five or six variants in a singlegroup. A consideration of the symmetry of the transformation reduces the numberof groups to 17. Average shape strains corresponding to these groups are presentedin Table 7.9. It is seen that the group numbers 9, 10 and 17 yield an averageshape strain which results in almost no distortion of the parent. Experimentalobservations have clearly pointed out that it is these groups that preferentiallyevolve during the nucleation and also during the growth processes. Stress couplingbetween the variants, therefore, is the motivating factor for the frequent associa-tion of these variants in a group. It may be noted from Table 7.9 that while thegroups of variants whose habit planes lie at 30 and 60 , respectively, yield anegligible average shape strain, those whose habits lie at 90 are associated withlarge values of the dilatation as well as the shear component (group number 5).The interactions of plates in this group may lead to fresh nucleation of additionalvariants, resulting in the formation of groups with a more favourable averageshape strain.

Another important factor which controls the grouping of O-plate variants isthe nature of the interfaces separating the plates in a group. A recent analysisby Bendersky and Boettenger (1994) shows that if the variants in a group are

Table 7.8. Elements of shape deformation matrices for individual O variants.

Variant �11 �12 �13 �21 �22 �23 �31 �32 �33

A+ 1.0293 −0�0018 0 −0�0018 0.9691 0 0 0 1A− 1.0293 +0�0018 0 +0�0018 0.9691 0 0 0 1B+ 0.9825 −0�0252 0 −0�0252 1.0159 0 0 0 1B− 0.9857 −0�0270 0 −0�0270 1.0127 0 0 0 1C+ 0.9857 0.0270 0 +0�0270 1.0127 0 0 0 1C− 0.9825 +0�0252 0 +0�0252 1.0159 0 0 0 1




Table 7.9. Elements of average shape strain matrices associated with different groupings of Ovariants.

Group Typical variant �11 �12 �13 �21 �22 �23 �31 �32 �33

number combinations

1 A−�A+ 1.0293 0 0 0 0.9691 0 0 0 12 A−�C+ 1.0075 +0�0144 0 +0�0144 0.9909 0 0 0 13 A−�B+ 1.0059 −0�0117 0 −0�0117 0.9925 0 0 0 14 A−�B− 1.0075 −0�0117 0 −0�0126 0.9909 0 0 0 15 A+�A−�B− 1.0148 −0�0090 0 −0�0090 0.9836 0 0 0 16 A+�A−�C+ 1.0148 +0�0090 0 +0�0090 0.9836 0 0 0 17 A+�C−�C+ 0.9992 +0�0168 0 +0�0168 0.9992 0 0 0 18 A+�A−�C− 1.0137 +0�0084 0 +0�0084 0.9847 0 0 0 19 A+�C+�B− 0.9992 0 0 0 0.9992 0 0 0 110 A+�C+�B− 0.9992 0 0 0 1.000 0 0 0 111 A+�A−�B+�C− 1.0059 0 0 0 0.9925 0 0 0 112 A+�A−�B+�C+ 1.0067 +0�0005 0 +0�0005 0.9917 0 0 0 113 A+�A−�B−�C+ 1.0075 0 0 0 0.9909 0 0 0 114 A+�A−�B−�C− 0.0067 −0�0005 0 −0�0005 0.9917 0 0 0 115 A+�A−�B−�B+ 1.0067 −0�0131 0 −0�0131 0.9917 0 0 0 116 Except B− 1.0019 +0�0005 0 +0�0005 0.9965 0 0 0 117 All 0.9992 0 0 0 0.9992 0 0 0 1

oriented with respect to each other in such a way that they come in contactalong a common {110} or {130} twin plane, stress-free interfaces across whichno discontinuities or displacements exist are created. Needless to mention, suchlow-energy interfaces readily form during the growth of plates. It may be notedthat in group number 9 which consists of A+�B+ and C+ (or A−�B− andC−) variants, such stress-free interfaces are not possible between the variants. Insuch a situation, retransformation into the �2-phase occurs in the region wherethe plates come in contact. The retransformation is, therefore, motivated by twofactors: first, self-accommodation of the variants in the group and second, thecreation of low-energy interfaces between the retransformed �2-grain and the threeplates.

Ternary Ti–Al–Nb intermetallics of compositions close to Ti2Al–Nb andTi4AlNb3 remain in the two-phase field consisting of the B2 and the orthorhom-bic phases at a temperature of 973 K. Both the alloys, on rapidly cooling from1673 K, produce an ordered B2 structure, the degree of ordering being very lowin the case of the latter alloy. Ageing for an extended period (26 days) hasbeen shown to result in the following sequences of transformations (Benderskyet al. 1991):




(1) For Ti2AlNb: Complete transformation of the retained high-temperature B2phase into a highly faulted orthorhombic phase occurs. This is followed byrecrystallization of the faulted grains into fault-free orthorhombic grains andfinally precipitation of the bcc phase in the orthorhombic matrix.

(2) For Ti4AlNb3: Precipitation of the equilibrium orthorhombic phase occurs inthe weakly ordered B2 matrix.

Bendersky et al. (1991) have proposed a tentative pseudobinary phase diagramfor interpreting the sequence transitions which occur in these alloys.

7.7 EUTECTOID DECOMPOSITION

Eutectoid decomposition – a solid state phase reaction involving diffusion – isknown to occur in a large number of alloy systems. In the context of a binarysystem, this reaction entails the decomposition, on cooling, of a high-temperaturephase into a mixture of two low-temperature phases. The transformation mecha-nism as well as the morphology and the substructure of the decomposition productsmay vary from system to system and depend on the reaction temperature. For agiven system, the appropriate reaction path is determined by the nucleation andgrowth processes that are kinetically favoured by the system. Eutectoid reactionscan occur in higher component systems also (Rhines 1956). For example, in aternary eutectoid reaction, the decomposition of a single phase may lead to thesimultaneous evolution of three phases. In the present text, however, only binaryeutectoids will be considered.

Alloy eutectoids can be divided into two classes: those which originate throughthe allotropy of the base metal and those which are associated with the decom-position of an intermediate phase (Spencer and Mack 1962). Eutectoids in Fe-,Ti- and Zr-based alloys are examples that belong to the first category. Eutectoidreactions involving the decomposition of intermediate phase are quite numerousin Cu-based alloys and in many of these cases, the parent phase is an electroncompound or nearly so (Spencer and Mack 1962).

During an eutectoid transformation in a binary alloy, the two low-temperatureequilibrium phases do not necessarily precipitate directly from the high-temperature phase; the formation of intermediate products comprising transitionphases is not uncommon. Even when the equilibrium phases form, the extentof solute partitioning dictated by the equilibrium diagram is rarely achieved inthe initial stages. Several modes of eutectoid decomposition, involving differentreaction mechanisms and engendering different types of microstructure, have beenobserved.




Pearlite reactions, bainite reactions, massive reactions and proeutectoid pre-cipitation reactions are the most important reactions in the context of eutectoidsystems. In alloys of near-eutectoid compositions, the first two are the most com-monly observed modes of eutectoid decomposition and give rise to characteristicmicrostructures: pearlite contains a lamellar distribution of the decompositionproducts while the state of aggregation of these product phases is essentially non-lamellar in bainite (Franti et al. 1978, Aaronson 1978). On the basis of Hillert’s(1962) suggestion pearlite may be regarded to form by a cooperative growth of thesolute-lean and the solute-rich phases which grow side by side in a synchronousmanner. In bainite, on the other hand, non-cooperative growth occurs and the twoproduct phases grow alternatively rather than synchronously.

As mentioned in Chapter 2, the eutectoid reaction occurs in a large number ofbinary Ti–X and Zr–X alloy systems, where X indicates a �-stabilizing element.The decomposition in these cases can be represented as �→ �+� where � is anintermetallic phase of the type TimXn or ZrmXn . In many of these systems, theeutectoid compositions correspond to low levels of solute concentration so that itis possible to obtain structures with large volume fractions of the eutectoid producteven in dilute alloys.

Figure 7.72(a) shows a schematic binary eutectoid phase diagram in whichmetastable boundaries between different phase fields are extended. For example,the extension of the �/�+� boundary, denoted by the broken line AA′, essentiallyrepresents the composition of the �-phase which remains in metastable equilibriumwith the �-phase at temperatures below the eutectoid temperature, Te. This pointis illustrated in the schematic free energy–composition plot (Figure 7.72(b)) forthe �-, �- and �-phases at a temperature below Te.

The temperature–composition regimes over which the parent �-phase tends toseparate either the �- or the �-phase or simultaneously both �- and �-phases canbe identified from the phase diagram with the extrapolated metastable boundaries.This can be illustrated with the example of an alloy with the composition givenby co. If this hypoeutectoid alloy is rapidly cooled down from the �-phase fieldto a temperature T1 the �-phase tends to separate the �-phase, the reaction beingdriven towards the establishment of a metastable equilibrium between the �- and�-phases. It is to be noted that at T1 there is no thermodynamic driving force forthe nucleation of the �- and the �-phases. In contrast to this situation, if the alloyis rapidly brought down to a temperature T2 which is below both the �/�+�and the �/�+ � extrapolated boundaries, the parent �-phase can nucleate boththe �- and the �-phases simultaneously. This is the essential condition for theeutectoid decomposition. Since the �- and the �-phases which can simultaneouslynucleate at T2 differ in composition, the nucleation of one of the phases facilitatesthe formation of the other phase in the vicinity of the former. Fine lamellae of




Composition →

P1

P2

Tem

pera

ture

→

α

E ' E '

Co Ce

B '

Te

T1

T2

α + β E

β

γ

β + γB

(a)

Composition, c

Free

ene

rgy,

G α

β

γ

(b)

T1

Figure 7.72. (a) Phase diagram showing a eutectoid reaction. Metastable extensions of phaseboundaries are drawn by dashed lines. (b) Free energy–concentration plots for �� and � at T1

corresponding to the phase diagram shown in (a).

the two phases are nucleated one after another, resulting in the formation of acolony. Such a colony continues to grow by the movement of the colony bound-ary (advancing transformation front) and eventually boundaries of neighbouringcolonies impinge on each other, marking the completion of the �→ �+� trans-formation. It is to be noted that while the equilibrium eutectoid decompositionoccurs at a fixed composition, given by the point E in Figure 7.72(a), the reactionat lower temperatures can occur over a composition range shown by the shadedportion of the phase diagram.




Not much work appears to have been carried out on the mechanisms of eutectoiddecomposition in Zr alloys although this reaction is a very common feature ofZr–X systems where X is a �-stabilizing element. The Ti–X eutectoids, whichare nearly as numerous, are relatively better explored. Aaronson et al. (1957,1958, 1960) have studied the Ti–Cr eutectoid in detail and have found that inhypoeutectoid alloys eutectoid decomposition occurs by a bainite or non-lamellarmode while in hypereutectoid alloys a lamellar structure does form, though onlyat large undercoolings. TEM investigations of Williams (1973) have shown thatin the Ti–Cu system, the decomposition of alloys of near-eutectoid compositions,during continuous cooling, leads to the formation of a lamellar structure. Frantiet al. (1978) have carried out a survey of eutectoid decomposition in a largenumber of Ti–X systems. Microstructural studies, using mainly light microscopyand supplemented by the determination of T–T–T curves, undertaken in thiswork have generated a large volume of useful information. They have foundthat in hypoeutectoid Ti–Bi, Ti–Co, Ti–Fe, Ti–Mn, Ti–Ni, Ti–Pb and Ti–Pdalloys a bainitic reaction leading to the formation of a non-lamellar dispersion ofparticles of the intermetallic phase, �(TimXn), in proeutectoid � occurs. However,in hypoeutectoid Ti–Cu alloys both bainitic and pearlitic (giving rise to a lamellarstructure) modes are observed. The occurrence of the pearlitic mode has beennoted to be more frequent in near-eutectoid Ti–X alloys: pearlite is the onlyeutectoid decomposition structure in these cases when X stands for Co, Cr, Cu andFe. But in the Ti–Ni, Ti–Pd and Ti–Pt systems, only the bainitic mode has beenobserved even in alloys of near-eutectoid compositions. It has been found that theT–T–T curve for the initiation of the pearlite reaction is, in general, displaced tolonger times with decreasing eutectoid temperature. However, the correspondingcurve for bainite reaction initiation does not show such a regular, straightforwarddependence.

It is well known that in Fe–C alloys of near-eutectoid as well as hypoeutectoidcompositions, pearlite is the primary product of the eutectoid decomposition ofaustenite when the reaction occurs at temperatures not far below the eutectoidtemperature. At lower reaction temperatures, the bainite reaction predominates.There is overlap of pearlite and bainite over quite a wide range of intermediatetemperatures. The results of Franti et al. (1978) clearly indicate that the situation isquite different in a large number of Ti–X alloys of hypoeutectoid compositions –the bainite reaction is the only mode of eutectoid decomposition. This differencehas been attributed (Franti et al. 1978) to differences in the morphology and thedistribution of the proeutectoid phase in the two cases. It has been pointed out thatpearlite evolves through cooperative growth and the principal barrier to pearliteformation pertains to the “initiation process” involving the formation of viablenuclei of the two constituents phases and the evolution of their growth patterns




into the cooperative mode required for the development of the pearlite colony.Appreciable areas of disordered interphase boundary between the parent and theproeutectoid phases are needed for the evolution of pearlite because such disor-dered interface regions have “sufficient flexibility of shape” to permit the initiationof cooperative growth. Once the initiation barrier is overcome, pearlite growthcan occur rapidly because of the short diffusion paths involved. In hypoeutectoidFe–C alloys, over wide ranges of C content and reaction temperature, the pre-dominant morphology of the proeutectoid ferrite phase is of the grain boundaryallotriomorph type. An appreciable fraction of the interfacial area of these grainboundary allotriomorphs has a disordered structure – a situation that facilitatespearlite formation and makes pearlite the major product of eutectoid decomposi-tion. The pearlite reaction occurs so rapidly that the evolution of side plates fromthe grain boundary allotriomorphs and the formation of intragranular plates aresuppressed. Only at very large undercoolings does an appreciable population ofWidmanstatten plates with partially coherent interfaces form. When this happens,pearlite formation becomes difficult because of lack of disorder at the interfacesand the bainitic mode of decomposition becomes dominant. In hypoeutectoid Ti–Xalloys, however, the situation is quite different in that side plates and intragran-ular plates constitute the predominant proeutectoid morphology even at smallundercoolings. The partially coherent boundaries of these plates are unsuitable forinitiating pearlite formation. However, the ledges on these boundaries may act asnucleation sites for isolated precipitates of the intermetallic phase, �. The bainitereaction is, therefore, preferred. In the singular case of the Ti–Cu system, wherepearlite does form in alloys of hypoeutectoid composition, it has been suggested(Franti et al. 1978) that the kinetics of pearlite nucleation are so rapid that thereaction can start at some grain boundary allotriomorphs in the very short periodavailable before the possibility of pearlite formation is foreclosed by the evolutionof closely spaced side plates from the allotriomorphs.

The observation that the occurrence of a pearlite structure is more common innear-eutectoid alloys than in their hypoeutectoid counterparts has been rationalizedalong the following lines (Franti et al. 1978). In alloys of near-eutectoid composi-tions, the driving forces for the precipitation of the �- and the �-phases are morenearly equal than in the hypoeutectoid case. Therefore, even if �-allotriomorphsprecipitate first, there is a reasonable chance that at suitable undercoolings, inter-metallic phase precipitation can compete with side plate formation at the allotri-omorph boundaries. Precipitation of the �-phase at such locations can provide anopportunity for the initiation of the process of pearlite formation. Thus systems inwhich the pearlite reaction predominates near the eutectoid composition are thosewhere the nucleation of �-precipitates at �/� interfaces can occur rapidly. In fact,it has been seen (Franti et al. 1978) that in near-eutectoid Ti–Cu, Ti–Co and Ti–Fe




alloys, the T–T–T curves for the initiation of the proeutectoid � and the pearlitereactions almost coincide.

It is likely that many of the features of the Ti–X eutectoids are pertinent toZr–X eutectoids also. However, detailed experimental confirmation in this regardis yet to come.

7.7.1 Active eutectoid systemsIn Ti–X and Zr–X eutectoid systems, the relative “activity” of the eutectoid decom-position process varies significantly from system to system. “Active” eutectoidsystems (Jaffee 1958) are those in which the �-phase decomposes rapidly intothe �- and the �-phases; the latter is not necessarily the equilibrium intermetal-lic phase pertinent to the system. In alloys of near-eutectoid compositions, this�-phase decomposition occurs so fast that it cannot be suppressed even by arapid �-quenching. The state of aggregation of the decomposition products in suchalloys is lamellar, so that a fine, pearlite-like microstructure is obtained on rapidcooling. This structure may be too fine to be resolved by light microscopy and itsdetails can be seen only by techniques like transmission electron microscopy. Onsubsequent ageing at temperatures below the eutectoid temperature, discontinuouscoarsening may occur and the fine lamellar structure may be replaced by a coarselamellar structure. The coarsening reaction is driven by (a) the chemical freeenergy change associated with the replacement of the metastable phases resultingfrom incomplete partitioning of the alloying elements in the rapidly formed finelamellae by the equilibrium phases and (b) the reduction in the surface energy dueto a significant annihilation of interfaces separating the product phase lamellae.

In a “sluggish” eutectoid system, the eutectoid decomposition of the �-phase isslow and it is possible to retain this phase metastably at low temperatures even onslow cooling.

On the basis of observations made so far on Ti–X and Zr–X eutectoid systems,it has been suggested in a qualitative manner (Jaffee 1958, Mukhopadhyay et al.1979) that active eutectoid decomposition occurs in those systems where theeutectoid temperature is high, the eutectoid composition is solute lean and theintermetallic phase resulting from the reaction is rich in the base metal. Theformation of a fine lamellar structure, resolvable by TEM, on rapidly quenchingalloys of near-eutectoid compositions from the �-phase field, has been observedin the systems Zr–Cu, Zr–Fe, Zr–Ni (Mukhopadhyay 1985) and Ti–Cu (Williamset al. 1970). In all these cases, the aforementioned criteria for the occurrence ofactive eutectoid decomposition hold good.

A survey of the reported eutectoid temperatures and compositions correspondingto binary Ti–X and Zr–X eutectoid systems also reveals that, in general, thelatter are associated with higher eutectoid temperatures and eutectoid compositions




leaner in the solute as compared to the former (Massalski et al. 1992). Whetherthese facts imply that active eutectoid decomposition is more common in Zr–Xthan in Ti–X eutectoids is a question that is yet to be resolved.

In the following sections, the observations made on the active eutectoid decom-position in near-eutectoid Zr–Cu and Zr–Fe alloys will be reviewed.

7.7.2 Active eutectoid decomposition in Zr–Cu and Zr–Fe systemActive eutectoid decomposition in a Zr–Cu alloy of near-eutectoid composition hasbeen studied by Mukhopadhyay et al. (1979) and their observations and inferenceswill be discussed at some length here. These workers have attempted to provide amorphological and crystallographic description of the decomposition product andhave also proposed a possible mechanism of the transformation. It has alreadybeen mentioned that a similar decomposition reaction has been observed in near-eutectoid Ti–Cu alloys (Williams et al. 1970) The eutectoid reactions in thesesystems are �→�+Zr2Cu and �→�+Ti2Cu, respectively. Both the intermetallicphases are isostructural with MoSi2 (Pearson 1967). The lattice parameters area = 0�32204 nm� c = 1�11832 nm for Zr2Cu and a = 0�2944 nm� c = 1�0782 nmfor Ti2Cu so that the values of the axial ratio are 3.47 and 3.66, respectively.Pearson (1972) has pointed out that the values of the c/a ratio for MoSi2-typephases are clustered in two ranges: 2.4–2.5 and 3.2–3.7. In the former, each atomhas a CN 10 environment while in the latter (e.g. in Zr2Cu and Ti2Cu) the CN 10polyhedron is distorted so as to produce an approximately CN 8 environment. Itcan be mentioned here that a CN 8 environment also exists in a phase with the bccstructure (e.g. �-Zr or �-Ti). A comparison of the �-Zr and the Zr2Cu structure isshown in Figure 7.73.

The microstructural observations made on a �-quenched, near-eutectoid(Zr–1.6 wt% Cu) alloy could be summarized as follows (Mukhopadhyay et al.1979). The quenched material consists of several colonies with lamellar structure,consecutive lamellae of the �-phase being separated by ribbon-like features. In agiven cell, the �-lamellae have a single crystallographic orientation and so havethe ribbons. In some regions, the latter are straight and parallel to the {1011}�-typeplanes while in some other regions in the same colony these have a “wavy” appear-ance (Figure 7.74(e) and (f)). In the transition region, a one to one correspondencebetween the straight and the wavy segments exists. The average colony size andthe average interlamellar spacing are about 2.0 and 0.1�m, respectively. Someof these features are illustrated in Figure 7.60. The ribbons give rise to extraspots in the selected area diffraction patterns and these spots could be indexedin terms of the Zr2Cu structure in an approximate manner in the sense that anexact matching between the observed and the calculated interplanar angles is notobtained. This suggests that the structure of the observed second phase is similar




β unit cella = 3.609 Å

β' unit cella = 3.40 Å

C

Zr2Cu unit cella = 3.22 Å; c = 11.18 Å

ZrCu

a

a

a

Figure 7.73. Unit cells of �, metastable �′ and equilibrium Zr2Cu phase.

but not identical to that of Zr2Cu. The matching is better in regions with wavyribbons than in regions with straight ribbons. Assuming the phase constitutingthe ribbons to be the Zr2Cu phase, the following planar and directional matchingwith the �-phase could be obtained: (0001)��(013)Zr2Cu; [1100]�� [031]Zr2Cu. Thisorientation relationship (Figure 7.75) is consistent with that observed betweenthe �- and the Ti2Cu phases in the Ti–Cu system (Williams 1973) It is alsofound that the flat faces of the ribbons correspond to one of the following Zr2Cuplanes: (110), (110) (103), and (103). It could be seen that habit plane variantssuch as (110) and (103) are quite similar with respect to atomic arrangement onthese planes and interplanar spacing though small differences in interatomic andinterplanar distances do exist because of the difference in the values of a andc/3 for the Zr2Cu unit cell. No interfacial dislocations are observed on the flatfaces of the ribbons, implying that the coherency strains are not large enoughto warrant the generation of misfit dislocations. However, considerations of themismatch between the {1011}� planes – the habit planes of the ribbons in termsof the �-structure – on the one hand and (110) and (103) planes of Zr2Cu onthe other indicate that interfaces separating � and Zr2Cu crystals should, in alllikelihood, be semicoherent and, therefore, misfit dislocations should be present




(b) (c)

(d) (e) (f)

(a)

0.5 μm 0.5 μm 0.5 μm

0.2 μm0.2 μm0.2 μm

Figure 7.74. Microstructure of active eutectoid product in Zr–1.6 wt% Cu (a) period and straightlamellae of �+�′ structure resembling internally twinned martensite structure. (b) Straight lamellaedegenerating into wavy lamellae. (c) and (d) Bright and dark (�′-reflection) field image of �+�′

structure. (e) and (f) Bright- and dark-field images show one to one correspondence between straightand wavy lamellae.

1013112

112

002 002

114

004 004

116

1102

1211

006

116 114 110

110

000

112 114

112 114

116

1211

006

116

1102

0111

0111

α-Reflections

Zr2Cu Reflections

1013

Figure 7.75. Diffraction pattern showing orientation relation between the �- and the �′-phases.




at such interfaces. It appears that the observed second phase in the �-quenchedalloy has neither the exact structure nor the exact stoichiometry of the equilibriumZr2Cu phase, presumably because the partitioning of the Cu atoms into the twoconstituent phases of the decomposition product remains incomplete during therapid cooling process.

While attempting to work out a possible mechanism of the observed activeeutectoid decomposition, Mukhopadhyay et al. (1979) have pointed out that theZr2Cu unit cell could be constructed by stacking three bct cells, one above theother (Figure 7.62). In fact, the Zr2Cu structure could be built by introducing astate of order in a disordered bcc structure (designated as �′ in such a mannerthat there would be approximately a threefold increase in the unit cell dimensionsalong the c-axis. The lattice parameters of the bcc phases � and �′ are not muchdifferent and it could be considered very likely that a one to one correspondence,implying the parallelism of the corresponding orthogonal axes, exists between the�- and �′-lattices. Making use of such a correspondence, the indices of planes anddirections in either of the two bcc structures could be expressed in terms of indicesreferred to in the Zr2Cu structure. Using such relationships and the observed latticecorrespondence between the � and the Zr2Cu phases, it has been possible to obtainthe orientation relation between the parent �- and the �-phases. It is seen that thelattice correspondence between these phases is the Burgers correspondence. It isalso noticed that all the observed {1011}�-type habit plane variants of the secondphase lamellae are derived from {110}-type mirror planes of the �′-phase. Further,the (1102)� plane is derived from the (001)� plane. It could be seen in suitableselected area diffraction patterns that the (1102) spot of the �-phase and the (002)spot of the �′-phase are near coincident and that superlattice spots subdivide theline joining the central spot and the (002)�′ spot into three equal segments. Thisindicates that the metastable phase constituting the ribbons has attained a certainextent of long-range order and this has resulted in an approximately threefoldincrease in the dimension of the �′ unit cell in the [001] direction.

In the light of the aforementioned considerations, it has been suggested thatduring �-quenching, the �-phase decomposes into a mixture of the �-phase anda partially ordered phase derived from �′. Regions containing straight and wavylamellae in the same colony possibly correspond to different levels of Cu enrich-ment and, therefore, to different degrees of long-range order. The crystallographicdescription of the lamellar product structure with respect to the parent �-structureis illustrated schematically in Figure 7.76. The presence of a lattice correspon-dence suggests that it might be possible for the lamellar, two-phase decompositionproduct to form from the �-phase essentially by cooperative atom movements.However, the development of partial order within the second phase lamellae wouldnecessitate solute transport to these regions by diffusion.




Tran

sfor

mat

ion

front

α

(0001)α

(110)β′

(110)β

(0001)αβ′

(Zr 2

Cu)

Figure 7.76. Lattice correspondence between the parent �- and the product �- and �′-structuresacross the transformation front.

The interfaces between the parent �-phase and the �-phases lamellae would besimilar to martensite interfaces and would thus be glissile. The boundaries separat-ing the �-phase and the second phase lamellae would also be expected to be glissilein view of the one to one lattice correspondence between the �- and the �′-phases.The propagation of these second type of interfaces across the parent �-regionwould make the �-lattice shrink by about 5% to yield the �′-lattice and would,at the same time, cause a certain degree of long-range order to be established inthis newly formed �′-lattice. This ordering would require the long-range migrationof Cu atoms within the untransformed phase, ahead of the transformation front.Since the solubility of Cu in �-Zr is much smaller than that in �-Zr (Massalskiet al. 1992) such a process would not be unlikely. The release of Cu atoms froman advancing �-lamella would render the untransformed �-phase region ahead of




it enriched in Cu. The composition gradient built up in this manner would resultin a continuous flow of Cu atoms towards the �-regions ahead of the adjacentlamellae of the metastable second phase. Thus morphology wise and also from thepoint of view of the mechanism of the distribution of Cu atoms in the two productphases, the mode of decomposition would be akin to a pearlite reaction. However,in a typical pearlite reaction, the transformation front is incoherent and no latticecorrespondence exists across this interface (Hornbogen 1972). In the Zr–Cu case,the presence of wavy lamellae suggests that at a certain stage in the progress ofthe reaction, the crystallographic requirements are relaxed and the transformationfront ceases to be coherent, making the reaction tend to resemble a normal pearlitereaction.

A simplified analysis has been carried out with a view to compare the observedreaction rates with those estimated. It has been found that the observed and theestimated growth rates are more or less of the same order at temperatures notfar below the eutectoid temperature. This analysis emphasizes the fact that solutepartitioning between the two product phases would be quite significant in spiteof the extremely fast reaction rates. This is possible due to rapid anomalousdiffusion in the �-phase and the exceedingly small interlamellar spacing in thedecomposition product.

To summarize, Mukhopadhyay et al. (1979) suggest that the rapid eutectoiddecomposition in near-eutectoid Zr–Cu alloys, resulting in the formation of alamellar aggregate of the �-phase and a partially ordered, bcc, metastable phase(structurally similar but not identical to Zr2Cu), occurs through the following steps.(1) Partitioning of solute atoms takes place ahead of the transformation front bybulk diffusion. (2) Cooperative atom transport occurs across this front, leading tothe formation of lamellae of the �-phase and a metastable second phase which hasan ordered bcc structure. During this process, the front remains coherent and thelamellae strictly adhere to {1011} habits. (3) Subsequently the transformation frontbecomes incoherent, facilitating the partitioning of Cu atoms and the attainmentof the stoichiometry and state of order characteristic of the equilibrium Zr2Cuphase.

It has been observed that an active eutectoid decomposition, leading to theformation of a lamellar structure on rapid �-quenching occurs in near-eutectoidZr–Fe alloys also (Mukhopadhyay 1985). The microstructural features associatedwith this decomposition reaction are quite similar to those pertaining to the Zr–Cucase and will be described briefly here. Rhines and Gould (1962) have suggestedthat in the Zr–Fe system the intermetallic phase richest in Zr is Zr4Fe and that theeutectoid reaction involves this phase: �→ �+Zr4Fe. However, Buschow (1981)has found that in conventionally melted Zr–Fe alloys, the Zr4Fe phase does notoccur and that Zr3Fe is the first intermetallic phase to form on the Zr-rich side.




Thus the eutectoid reaction in the Zr–Fe system is �→�+Zr3Fe. The Zr3Fe phasehas a base-centred orthorhombic structure (Re3B type) with a = 0�3326 nm� b =1�0988 nm and c = 0�8807 nm.

A Zr–4 wt% Fe alloy, on being rapidly cooled from the �-phase field, exhibitsa pearlite-like lamellar structure. In each colony, �-lamellae are separated byribbon-like features which are straight in some regions and wavy in some others.As in the Zr–Cu case, a one to one correspondence exists between the straightand the wavy segments. In some instances, the second phase lamellae have afragmented appearance. This phase (constituting the ribbons) gives rise to spots inselected area diffraction patterns and can be imaged in the dark field by using thesereflections. A limited amount of analysis has shown that these extra diffractionspots can be indexed in terms of the Zr3Fe structure. Similar features have beenobserved in a rapidly quenched Zr–3 wt% Fe alloy, except that in this case thecolonies have a lath-like, and not equiaxed, shape. Moreover, some of these lathsdo not have a lamellar structure and appear to contain only the �-phase. In eitheralloy, ageing at temperature below the eutectoid temperature causes the lamellarstructure to be replaced by a coarse distribution of globular Zr3Fe particles in the�-matrix. Some of these features are illustrated in Figure 7.77.

As discussed earlier, the mechanism proposed for the active eutectoid decom-position in near-eutectoid Zr–Cu alloys makes use of the fact that the Zr2Custructure is related to and can be derived from the bcc structure in a simplemanner. A similar situation exists in respect of the Ti–Cu system. In the case ofZr3Fe, however, such a simple structural relationship is not apparent. Neverthe-less, from considerations of reaction kinetics and post-reaction microstructure,the decomposition process does appear to be very similar in all these threecases.

(b)(a)

Figure 7.77. Microstructures of active eutectoid product in Zr–3 wt% Fe.




7.8 MICROSTRUCTURAL EVOLUTION DURINGTHERMO-MECHANICAL PROCESSING OF Ti- ANDZr-BASED ALLOYS

Bulk metal working of Zr and Ti alloys comprises a series of steps, each of whichhas important roles to play in determining the microstructure of the finished prod-uct. It generally involves two stages of hot working following the melting stepand subsequently a few steps of secondary working operations for achieving suit-able microstructure, specified shapes and desired mechanical properties (McQueenand Bourell 1988, Weiss and Semiatin 1999). Amongst these steps, not workingin the form of forging, extrusion or rolling is very important, as in addition tobreaking the cast microstructure and shaping the ingot, this step is responsiblefor providing a chemically homogeneous product with a suitable microstructurefor subsequent processing. In industry, Ti and Zr alloys are hot worked in the �,� or in the (�+�) phase regime, depending on alloy composition and specificmicrostructural/mechanical property requirements. A difference in crystal struc-ture, coupled with a very high diffusivity in the �-phase (2–3 orders of magnitudehigher than that in the �-phase), promotes the processes of recovery, recrystalliza-tion and diffusional deformation at lower temperatures and higher strain rates inthe �-phase as compared to the �-phase. Consequently, the constitutive deforma-tion behaviour of these phases under hot working conditions are different and thisdictates the evolution of microstructure in the product. In general, the importantstructural changes occurring during hot working are recovery and recrystallization,leading to the annihilation of various crystal defects and softening of the materialwhich dynamically balance strain hardening. In addition to the dynamic recrys-tallization (DRX) and dynamic recovery(DRV) processes, superplastic deforma-tion, in which large-scale structural modifications are not involved, has also beenreported in a number of Ti alloys and in a few Zr alloys (McQueen and Bourell1988). During DRV, structural changes result primarily from the annihilation andrearrangement of dislocations into low-energy arrays. These mechanisms lead tothe formation of low-angle sub-boundaries, dislocation arrays which have a lim-ited capability for migration. The nucleation process in DRX involves dynamiccreation and propagation of interfaces which separate highly deformed areas fromnewly formed, relatively strain-free grains. The growth begins with migration ofthis interface when it assumes the configuration of a large angle boundary. Thenucleation mechanism may be different, depending on the strain rate. At low strainrates nucleation is by boundary bulging and at high strain rates the nucleationresults from conversion of subgrain walls into high misorientation boundaries as aresult of the accumulation of glide dislocations. During superplastic deformation,the major contributors to superplastic deformation are grain boundary or interface




sliding, grain rotation and grain switching, and consequently large-scale structuralmodifications, as seen in DRX, are not possible. Although the dislocations are notinvolved in a major way during plastic deformation, they play a significant rolein the accommodation process. However, minor structural modifications such asagglomeration of second phase particles is possible during the grain switching pro-cess. Structural changes can also occur during the intervals of hot working in theabsence of stress (static) as well as during cooling after hot working (McQueen andBourell 1988). These static modifications in structure brought about by recoveryand recrystallization are important in deciding the final mechanical properties ofboth the alloy systems. Because of these structural changes during hot deformationand different types of phase transformations occurring in Ti alloys as has beenmentioned earlier, there is extensive opportunity for thermomechanical processing(TMP). A large volume of research has been carried out to understand variousaspects of TMP of Ti alloys including the recently developed Ti aluminides. It hasbeen found that besides processing parameters (temperature, deformation strainrate and amount of deformation), factors like initial microstructure, alloy compo-sition and physical properties such as stacking fault energy and diffusivity have aprofound effect on the evolution of the microstructure (McQueen and Jonas 1975).

7.8.1 Identification of hot deformation mechanisms through processingmaps

In order to understand the microstructural evolution, the underlying hot deforma-tion mechanisms must be identified by studying the constitutive behaviour of thework piece (relationship between stress, strain rate and strain) under hot work-ing conditions. This is necessary as each of the hot deformation mechanisms isdistinctly different from the others in terms of the permitted ranges of strain rateand temperature for its operation, its dependence on initial microstructure andassociated strain rate sensitivity (m= � ln �/� ln $) of stress. In addition to thesehot deformation mechanisms, Zr and Ti alloy systems are known to exhibit strainlocalization leading to non-uniform microstructure during deformation processing.Thus control of microstructure and avoidance of non-uniform microstructure areof paramount importance in optimizing the final mechanical properties. There areseveral approaches of materials modelling to study the constitutive behaviour aswell as for predicting hot deformation mechanisms, namely, study of the shape ofthe stress–strain curve, kinetic analysis of the deformation process and construc-tion of processing maps (Prasad and Seshacharyulu 1998a,b) with identificationof different processes operating in different domains. Amongst these approachesmentioned, the construction of processing maps has been found to be quite usefulin various alloys (Prasad and Seshacharyulu 1998a,b, Prasad 1990). The process-ing maps are developed using the principles of the dynamic materials model which




considers the work piece being subjected to hot deformation essentially as a dissi-pator of power (Prasad and Seshacharyulu 1998a, Prasad 1990). The constitutivebehaviour describes the manner in which the power is converted at any giveninstant into two forms: most of it through a temperature rise (G content) anda smaller part through a microstructural change (J co-content). The factor thatpartitions power between these two processes is the strain rate sensitivity of flowstress which is unity for an ideal linear dissipator. At a given temperature, strain Jis given by (Prasad and Seshacharyulu 1998a,b, Prasad 1990)

J = �$m/�m+1 (7.60)

where $ is the strain rate. The efficiency of power dissipation () of a workpiece can be obtained by comparing its power dissipation through microstructuralchanges with that occurring in an ideal linear dissipator (m= 1) and is given by

�= 2m/�m+1 (7.61)

The variation of with temperature and strain rate gives the power dissipationmap. Over this frame is superimposed a continuum instability map in which adimensionless instability parameter, %($) defined as (Kalyanakumar 1987),

%�$ = � ln�m/m+1 � ln $

+m≤ 0 (7.62)

is plotted as a function of strain rate and temperature. The instability map is devel-oped on the basis of extremum principles of irreversible thermodynamics appliedto large plastic flow as given by Zeigler (1963). Flow instabilities are predicted tooccur when %($) assumes negative values. The processing maps exhibit domains inwhich shows local maxima where specific microstructural mechanisms operateas well as regimes where flow instabilities like adiabatic shear bands or flow local-ization occur. To illustrate the concept, a contour map (isoefficiency contours ona strain rate temperature frame) for commercial pure Zr is shown in Figure 7.78.The processing map shows two domains, one corresponding to DRV and the otherrelated to the process of DRX (as marked). These domains have been identifiedon the basis of the examination of the deformed microstructure (conditions $, T ;Figure 7.79) of samples quenched from the processing temperature (Chakravartty1992). In the following, some efforts made so far to understand the microstruc-tural evolution in Ti- and Zr-based alloys using various methodologies, especiallyconstruction of processing maps, are highlighted.




650 690 730 770 810 850

2

1

0

–1

–2

–3

28

11

15

19

24

28

41

37

32

1524

37

41

DRV

DRX

Temperature (°C)

Log

stra

in r

ate

Strain = 0.4Zirconium- 2 (β-quenched)

Figure 7.78. Processing map for zirconium for a strain of 0.4 showing domains of dynamic recrystal-lization (DRX) and dynamic recovery (DRV). The number against each contour indicates percentageefficiency.

Figure 7.79. The optical micrograph of �-Zr deformed at 800 C and 0.1 s−1 within the DRX domainto a strain of 0.7 showing dynamically recrystallized grains.




7.8.2 Development of microstructure during hot working of Ti alloys7.8.2.1 �-alloysPrasad and Seshacharyulu (1998b) have studied the hot deformation characteristicsof commercially pure �-Ti over wide ranges of strain rate (0.001–100 s−1) andtemperature 873–1123 K by constructing processing maps. DRX has been observedin the temperature range of 873–1123 K and strain rate range of 0.001–0.1 s−1.The occurrence of DRX has been found to be sensitive to the O content, alowering of which results in increasing the strain rate and temperature of theDRX domain. In addition to DRX, DRV has also been reported in higher purityTi. Weiss and Semiatin (1999) have recently reviewed the deformation behaviourof �-Ti alloys and have observed that during deformation below the �-transustemperature, �-alloys undergo DRV in which the rate of hardening by generationof dislocations is balanced by the rate of softening due to dislocation annihilation,resulting in a steady-state flow curve. The deformed microstructure shows bothelongated and equiaxed morphology of the �-phase with substructure within.Under similar deformation conditions a near-�-Ti alloy IMI 685 with equiaxed(�+�) morphology has been reported to exhibit superplasticity (Gopalkrishna et al.1997). The deformed microstructure of this micro-duplex alloys has been foundto contain partially recrystallized � and plate-like � in a transformed �-matrix.However, the same alloy, with a lower O content and with an acicular morphologyhas been found to show a domain of �–� spheroidization at strain rates lowerthan 0.001 s−1 in the temperature range 1123–1233 K. The constitutive behaviourof these alloy systems has been studied above the �-transus temperature and ithas been observed that the shape of the stress–strain diagrams are of steady-statetype and the apparent activation energy associated with the process of deformationis in the range 180–220 kJ/mol which is close to the activation energy for self-diffusion. Thus it appears that deformation in the �-phase is controlled by dynamicrecovery (Weiss and Semiatin 1999). Large-grain superplasticity and DRX havealso been reported in �-alloys during deformation at temperatures in the �-phasefield, respectively, at lower and higher strain rates (Prasad and Seshacharyulu1998b). It has been noted that some � Ti alloys like IMI 685 have very narrowprocessing window because of the occurrence of a flow instability at strain rates aslow as 0.1 s−1 and higher over a wide range of temperatures. Therefore, in practiceprocessing steps are carefully designed for successful forging and microstructuralcontrol.

7.8.2.2 �+� alloysThese alloys offer considerable scope for obtaining a variety of microstructures byTMP (McQueen and Bourell 1988). Depending on the detailed thermo-mechanical




treatment, lamellar, equiaxed or bimodal microstructures can be obtained. Thedeformation characteristics of the two-phase alloys have been studied in great detailover wide ranges of temperature and strain rate, for different initial microstruc-tures. In an early work (Shakanova et al. 1980) on Ti–6Al–4V and Ti–6Al–2Mo–2Cr alloys, the evolution of microstructure during hot deformation was studiedover the temperature range of 1123–1373 K and strain rate range of 0.0001–1 s−1

for two different initial microstructures, namely, the (�+ �) equiaxed and thetransformed �-structures. The �-phase was observed to have undergone polygo-nization during deformation over the entire range of temperatures studied, butas the deformation temperature was increased (T > 1193 K), DRX became thedominant deformation mode at lower strain rates. Increasing the temperature ofdeformation enhanced the process of DRX. Both polygonization and DRX werealso observed in the �-phase in which subgrains were formed. DRX and polygo-nization was recorded on deformation in both the initial microstructures but theDRX process was dominant in the case of equiaxed morphology. These alloysalso exhibited superplasticity with equiaxed morphology on deformation at lowerstrain rates (≈0.0001 s−1) and at temperatures in the range 1198–1223 K. Sastryet al. (1980) carried out detailed TEM investigation of hot deformed Ti–6Al–4Vmaterial at a strain rate of 0.05 s−1 and at temperatures above 850 C and reportedthat both DRX and DRV occurred as evidenced by a hexagonal network of dis-locations in the �-phase, formation of small equiaxed �-grains and absence ofshearing in �-phase. Recently, Seshacharyulu et al. (2000, 2002) studied the effectof initial microstructures (equiaxed and �-transformed) and impurity content onthe hot working behaviour of Ti–6Al–4V by constructing processing maps over awide range of strain rate and temperature. The processing map for the equiaxedmorphology exhibited two domains, viz., a domain of fine grain superplasticityin the (�+�) region and another domain of DRX in the �-phase field. The rate-controlling accommodation process during superplastic deformation was identifiedto be cross slip in �-phase which was located at the triple junction of �-grains.The activation energy for the process of DRX in the �-phase was close to that forself-diffusion in �. During deformation in the (�+�) phase field , the �-platesof the transformed �-structure assumed equiaxed morphology by the process ofglobularization. The primary �-grain size varied linearly with the Z parameter(Zener Hollomon parameter Z = $ exp [Q/RT], where $ is the strain rate, T is thetemperature in K and Q is the apparent activation energy) in a manner similar tothat observed during DRX. The deformation in the �-phase field resulted in DRXwith characteristics similar to that observed in the initial equiaxed microstructure.Chen and Coyne (1976) carried out isothermal forging experiments on Ti–6Al–4Vand identified DRX as the rate-controlling deformation process on the basis ofthe apparent activation energy obtained by kinetic analysis. Weiss et al. (1986)




carried out detailed TEM investigation of the hot deformed microstructure of aTi–6Al–4V alloy with Widmanstatten morphology and reported that �-lamellaebreak up during deformation either heterogeneously by forming intense shearbands or homogeneously by formation of a dislocation substructure. Dependingon the extent of localized shear strain, the �-lamellae may sever completely orpartially. These sheared regions, during the course of deformation, will either form�/� grain boundaries or will get converted to a structure in which �-regions areseparated by �-phase by its complete severance by percolation of �-phase. Duringthe homogeneous deformation, however, dislocations rearrange to form recoveredsubstructure leading to the formation of subgrains which divide the lamellae intosmaller units. The �-phase then penetrates along the �/� sub-boundaries.

7.8.2.3 �-alloysAlthough the �-alloys are cold workable and can be recrystallized at temperaturesas low as 1073 K to obtain fine grains, the initial bulk working of the ingot is doneat temperatures above their �-transus followed by secondary working either aboveor below the transus (Weiss and Semiatin 1998). Numerous hot working studieshave been conducted on �-alloys both in the single phase � as well as two-phase(�+ �) regime. The various aspects of thermo-mechanical processing of �-Tialloys have been reviewed by Weiss and Semiatin (1998). These alloys were foundto exhibit discontinuous yielding followed by almost a steady-state flow behaviour,formation of serrated �-grain boundaries on deformation and the presence of sub-boundaries ranging from medium- to high-angle boundaries during deformationin the �-phase. The deformation in the �-field appeared to be controlled bydynamic recovery with an apparent activation energy which matched with that ofself-diffusion. Similar conclusion was drawn by Robertson and McShane (1998)who examined the effect of two initial microstructures, viz., �-transformed andequiaxed (�+�), on the deformation behaviour of Ti–10V–2Fe–3Al alloy at lowerstrain rates. Balasubrahmanyam and Prasad (2001, 2002) have characterized thehot deformation behaviour of two �-alloys, Ti–10V–2Fe–3Al and Ti–10V–4.5Fe–1.5Al over a wider range of strain rates and temperatures using processing maps.They found that the processing of �-alloy (Ti–10V–4.5Fe–1.5Al) showed a singledomain in the temperature range 1023–1173 K and strain rate range 0.01–0.1 s−1.On the basis of the microstructural features, the variation of grain size of thedeformed grain structure (predominantly equiaxed) with temperature and tensileductility variations, it was concluded that the prevailing deformation mechanismin the domain was DRX. The processing maps of the metastable �-alloy (Ti–10V–2Fe–3Al) on the other hand showed two domains during deformation in the similartemperature and strain rate ranges. In the temperature range 923–1023 K and atstrain rates lower than 0.1 s−1, the material deformed superplastically exhibiting




large ductility. The accommodation process was identified to be dynamic recoveryfrom the nature of stress-strain diagram, which was of steady state type, and fromthe apparent activation energy value, which matched with the activation energyfor self-diffusion in �-Ti. At temperatures higher than 1073 K and strain rateslower than ∼0.1 s−1, the alloy exhibited large grain superplasticity (LGSP) with adeformed microstructure containing stable subgrain structure within large �-grains.LGSP was also reported to occur during hot deformation of �-Ti alloys by Griffithand Hammond (1972).

7.8.2.4 Ti-aluminidesTitanium aluminides offer a combination of properties such as low density, highstrength to weight ratio, high modulus and good elevated temperature strengthproperties which make them excellent candidate material for gas turbine engineand airframe structure in advanced aircrafts. The aim during primary ingot break-ing of these materials is to prevent catastrophic fracture during shape change aswell as to provide suitable homogeneous microstructure to enhance hot worka-bility during secondary operations (Semiatin et al. 1992, 1998). One of the mainlimitations of these materials is their limited workability during conventional metalworking processes such as forging, rolling and extrusion because of the genera-tion of various internal defects. The importance of primary hot working can beemphasized by taking a typical example of a near-�-TiAl alloy. This variety ofaluminides, with aluminium content more than 46%, does not have single phase�-field as in the case of conventional titanium alloys. This coupled with significantcoring of the dendritic arms that result from the peritectic solidification and pres-ence of ordered phases even at very high temperatures (> 1573 K) make primaryingot breakdown a technological challenge. The hot deformation mechanisms ofthese materials during secondary processing have been found to be a strong func-tion of the starting microstructure, deformation temperature and strain rate in amanner similar to that observed with conventional Ti alloys. The major issues ofthermo-mechanical processing of Ti-aluminides have been reviewed by Semiatinet al. (1998). A considerable amount of effort has been directed to identify thesuitable hot working conditions and the associated deformation mechanisms toobtain defect-free products for a variety of single phase �2, single phase �, two-phase (�2 +�) and near-� alloys by developing workability maps as well as bymicrostructural observation of hot deformed materials. Semiatin et al. (1998) andHuang et al. (1995, 1998) studied the deformation behaviour of cast Ti–24Al–11Nbover a range of temperature and strain rate and identified three different regimesof deformation: a regime of warm working where deformation led to distortionof �-laths, wedge cracking and cavity formation, a region of globularization of�-laths in the two-phase (�2 +�) and a domain of hot working in the single phase �.




Sagar et al. (1994, 1996) studied the hot deformation behaviour of a similar alloyin two different initial microstructural states employing dynamic materials model.They identified a domain of DRX of �2-phase in both the starting microstructures.In addition, they have also observed a second domain of DRX and a domain ofsuperplastic deformation of �-phase for different combinations of strain rate andtemperature. The flow behaviour and microstructural development during forgingof super �2 (Ti-25Al-10Nb-3V-1Mo) have been studied by Huang and co-workers(1995, 1998). It was found that while both DRV and deformation heating wereinvolved in microstructural evolution during hot deformation of (�2 +�) alloys atrates lower than 10 s−1, the dominant deformation mechanism was DRX at rateshigher than 10 s−1. However, both DRX and DRV coupled with certain amountof grain boundary sliding were responsible for flow softening during deformationin the �-phase field at both high and low rates of deformation. In view of veryrestricted workability of large lamellar microstructure of aluminides, several TMPmethods are being increasingly employed to break down the initial microstructureto smaller units (Martin 1998, Sun et al. 2002). For microstructural refinementof orthorhombic Ti–22Al–27Nb, Martin (1998) used near-isothermal processesinvolving rolling at temperatures just below the �-transus followed by a thermaltreatment to obtain a fine grain metastable �-phase. The lamellar microstructure ofnear-� Ti–Al, which does not have an overlying �-phase, is broken down by extru-sion in the �2 +� phase field by dynamic recrystallization. Subsequently furthermicrostructural refinement is achieved by multiple isothermal forgings. Recentlythe technique of hot-die incremental deformation has been used with intermittentrecovery to breakdown the large �2 +� lamellae into fine �2 +� microstructure(Pan et al. 2001). Although the hot deformation mechanisms that are involvedin conventional Ti alloys are also present in the titanium aluminide alloys, thewindow of hot working of Ti-aluminides in the as-cast condition is very restrictedbecause of the ease of intergranular fracture, cavitation and flow localization evenat higher temperatures. An initial thermo-mechanical treatment to break up thelamellae into fine microstructure is therefore essential for good workability, withthe added capability of achieving even superplasticity.

7.8.3 Hot working of Zr alloysThe number of commercial alloys of Zr is considerably smaller than that of Ti.Zr-based alloys are chosen primarily for the manufacture of some critical structuralcomponents in nuclear reactors. As has been indicated earlier, these alloys arefabricated by processing routes, involving a melting stage, hot working in twostages and cold working with or without annealing. It has been observed byChedale et al. (1972) that the scale of the final microstructure and texture aredetermined primarily in the second hot working stage which is extrusion. It has




been shown that the extrusion speed and temperature are two important parameterswhich have significant effect on the development of texture, microstructure andmechanical properties of the finished products (Chedale et al. 1972, Konishi et al.1972). It is therefore essential to optimize them in order to arrive at the desiredfinal properties.

7.8.3.1 � and near-�-Zr alloysJonas and co-workers (Lutton and Jonas 1972, Abson and Jonas 1973) studiedthe high-temperature deformation behaviour of �-Zr and �-Zr-tin alloys in com-pression in detail and examined the microstructural development by TEM. Theyconcluded dynamic recovery to be occurring in the temperature range 898–1098 Kand strain rate range 10−4 to 3 × 10−3 s−1 on the basis of the observed shape ofstress–strain diagram, kinetic analysis performed and detail TEM investigation ofdeformed microstructure. Although the deformation was carried out at or abovethe recrystallization temperature (static), no recrystallized grain has been observedin the deformed material. The grain shape in general was elongated and wasfound to contain sub-boundaries. The mean size and perfection of these boundariesdecreased with decreasing temperature, increasing strain rate and increasing tincontent. On the other hand, Ostberg and Attermo (1962) carried out forging ofZircaloy-2 (a near-�-alloy) at 1073 K and at a strain rate of 0.4 s−1 and observedthat the original Widmanstatten �-plates were broken up into new grains whichwere equiaxed. They concluded that DRX was involved in the modification ofthe microstructure. It may be noted that the deformation of Zircaloy-2 at 1073 Kis in the �-phase and deformation at temperatures higher than 1083 K and up to�-transus will be essentially in two-phase (�+�). Garde et al. (1978) studied hottensile deformation behaviour of zircaloy-2 in the two-phase (�+�) field at var-ious strain rates. The microstructure evolved during deformation at temperaturesaround 1123 K and lower strain rates (< 10−3 s−1) was characterized by equiaxed�-grains separated by �-phase. The phenomenon of grain boundary sliding hasbeen found to be responsible for the occurrence of equiaxed grain structure. Whenthe deformation rates are increased (≈0.1 s−1 and at the same temperature), elon-gated �-grain structure separated by a thin film of � is usually seen. Ostberg andAttermo (1962) have observed similar elongated morphology of � separated by�-films on forging at 1173 K and 0.4 s−1 (estimated strain rate). A few �-plateshave been found to change from single crystals to fragmented structure of newgrains/subgrains as evidenced by difference in contrast under polarized light. Theformation of new grains from the original �-plates into smaller units is sugges-tive of the occurrence of dynamic recrystallization process in the �-phase of thetwo-phase (�+�) mixture. Transgranular deformation mechanisms involving dis-locations are important at these rates of deformation to produce the elongated




morphology. These observations clearly indicate the effect of strain rates on themode of deformation and evolution of microstructure. As regards the effect of ini-tial microstructure, an equiaxed grain structure has been found to exhibit better hotductility than a Widmanstatten microstructure. In (�+�) material of zircaloy-4,the nickel-free variety of zircaloy-2, a hot working activation energy of 155 kJ/molwas reported in the temperature range 840–970 C and at strain rates greater than3×10−3 s−1 (Rosinger et al. 1979). This value of activation energy was similar tothat obtained in �-phase. Zr and zircaloy-2, when deformed in the �-phase field atlow rates ≈ 10−3 s−1, exhibit superplasticity and high ductility (Garde et al. 1978,Bocek et al. 1976).

Chakravartty et al. (1991, 1992a,b,c) studied the hot deformation characteristicsof commercial pure Zr (containing approximately 1000 ppm of oxygen) with a�-quenched starting microstructure in both �- and �-phase fields using processingmaps generated by carrying out compression testing in the strain rate range of10−3–102 s−1 and over the temperature range of 650–1050 C. The processingmap revealed (Figure 7.78) two safe domains and a regime of instability: (a) adomain of DRX in the temperature range of 730–850 C and strain rate rangeof 10−2–0.1 s−1 with its peak efficiency of 40% at 800 C and 0.1 s−1 which wasconsidered as optimum hot working parameters, (b) a domain of dynamic recoveryat temperatures lower than 700 C and strain rates lower than 10−2 s−1 and (c) aregime of flow instability at strain rates higher than 1 s−1 and temperatures above670 C. The characteristics of dynamic recrystallization were found to be similarto those of static recrystallization regarding the sigmoidal variation of grain size(or hardness) with temperature, although the DRX temperature was much higherthan the static recrystallization temperature 873 K. Within the domain of DRX,the grain size increased with increase of temperature and decrease of strain rate(Chakravartty 1992, Chakravartty et al. 1992b). The manifestation of instabilitywas in the form of localized shear band (Figure 7.80). During hot working, theregime of instability should be avoided.

The processing map of Zr in �-phase showed only a single domain in thetemperature range 925–1050 C and at strain rates lower than 0.1 s−1 with thefollowing features: (a) the efficiency of power dissipation is high (≈ 60%) with acorresponding high strain rate sensitivity of flow stress (0.45), (b) the stress straincurves show steady-state behaviour with no flow softening and (c) flow stressvalues are generally low. On the basis of these informations, the domain has beeninterpreted to represent LGSP in �-Zr (Chakravartty et al. 1992a,b,c).

Hot working characteristics of zircaloy-2, with �-transformed initial microstruc-ture, have been investigated by Chakravartty et al. (1992b) using processing mapsin the temperature range of 650–950 C. In this range, zircaloy-2 exists eitheras predominantly �-phase or as a mixture of � and �. The processing maps




125 μm

Figure 7.80. The optical micrograph of Zirconium samples deformed at 700 C and 100 s−1 showingadiabatic shear band across which a variation of microstructure can be seen. The instability parameterassumes negative value under the conditions of deformation.

(Figure 7.81) exhibited three distinct safe domains in the temperature range studiedwith a discontinuity at around 850 C. The domains occur at 800 C and 0.1 s−1,650 C and 10−3 s−1 and 950 C and 10−3 s−1 which have been characterized tobe the domain of DRX, the domain of DRV and the domain of superplasticity(marked SPD), respectively. It has been reported that the initial �-transformedmicrostructure is converted to equiaxed grains of � during the process of DRX.Besides the domains of stable flow, at strain rates higher than 1 s−1 and temper-ature of 700–850 C, zircaloy-2 exhibits microstructural instability in the form ofadiabatic shear band.

A comparison of the processing maps of zircaloy-2 and commercially pure Zrin �-quenched condition reveals that the maps are strikingly similar. The DRXdomain of both the maps extend over similar strain rate and temperature ranges,although the peak efficiency (38%) for DRX for zircaloy is slightly lower thanthat of Zr (42%). The addition of tin to Zr as in zircaloy-2 lowers the stackingfault energy of Zr (240 mJ/m2) significantly (Sastry et al. 1974) while the diffusioncoefficient is not greatly altered (1�6 × 10−16 m2/s in �-Zr while 3 × 10−17 m2/sin Zr–1.03Sn alloy) (Chakravartty 1992, Chakravartty et al. 1992b). A simplemodel was used to explain the DRX characteristics of both Zr and zircaloy-2(Chakravartty 1992). Of the two competitive processes involved in DRX, namelyformation of interfaces (the nucleation step which involves dislocation generationand their simultaneous recovery) and migration of a large-angle boundary, it wasfound that the process of formation of interface is the rate-controlling step inboth these materials. The thermal recovery by diffusion-controlled climb plays




2

1

0

–1

–2

–3650 710 770 830 890 950

3530

25

25

30

35

38

38

44

49

56 64

30252118

11

11

72

DRX

DRV

SPD

Temperature (°C)

Log

stra

in r

ate

Strain = 0.4Zircaloy- 2 (β-quenched)

Figure 7.81. Processing map of �-quenched zircaloy-2 for a strain of 0.4 showing various domains:(a) domain of dynamic recrystallization (marked DRX), (b) domain of dynamic recovery (markedDRV) and (c) domain of superplasticity (marked SPD). The number against each contour indicatespercentage efficiency.

an important role in both these materials (Chakravartty 1992, Chakravartty et al.1991). The characteristics of dynamic recovery and flow instability in both thesematerials were also similar (Chakravartty et al. 1991). The variation of grainsize with temperature and strain rate, observed within DRX domain of zircaloy-2, would permit microstructural control. In the domain of superplasticity, theefficiency value was high and the overall deformed grain structure was equiaxed.

In order to validate the processing maps of zircaloy-2, extrusion trials havebeen carried out on full size billets and ingots used for making various zircaloy-2products for nuclear reactor use. Two different temperatures (720 and 800 C) andvarious extrusion ratios and ram speeds were chosen (Chakravartty et al. 1991).The experimental extrusion conditions were located on both the processing map(Figure 7.82) and instability map (Figure 7.83). Most of the extrusions carried outat 800 C are within the DRX domain and the extrusions carried out at 720 C fallin the instability regime. The extrusion carried out at 800 C showed dynamicallyrecrystallized grains while the microstructure of extruded material depicted signsof flow localization (Chakravartty et al. 1992a,b,c).




1

2

0

–1

–2

–3650 680 730 770 810 850

Temperature (°C)

DRX

39

3235

39

3228

2420

2824

20

13

59

13

DRV

Zircaloy- 2 (β-quenched)

Extrusion at 720°CExtrusion at 800°C

Log

stra

in r

ate

Figure 7.82. Extrusion conditions superimposed on processing map of �-quenched zircaloy-2 for astrain of 0.4. The extrusions carried out at 800 C are within the DRX domain while those carriedout at 720 C are outside the DRX domain.

7.8.3.2 �+� alloysJonas and co-workers (Jonas et al. 1979, Choubey and Jonas 1981, Rodriguezet al. 1985) studied flow properties of Zr–2.5Nb alloys by constant true strainrate compression testing in the strain rate range 10−4 to 1 s−1 and at tempera-tures from 750 to 1000 C. The alloys showed considerable flow softening in the(�+�) phase region and steady-state flow behaviour in single phase �-region.The strain rate sensitivity was found to increase from 0.18 during deformation inthe (�+�) phase field (800–875 C) to 0.22 in the �-phase field (900–1000 C). Themechanism involved during hot deformation, however, has not been identified.Zr–2.5Nb alloys have been found to exhibit superplasticity in the temperature range720–850 C and at strain rates in the range 10−2–10−4 s−1 (Nutall 1976, Holm et al.1977, Rosinger et al. 1979). Superplasticity has also been reported in cold workedand stress relieved Zr–2.5Nb pressure tube material during tensile deformationin the temperature range of 650–800 C at strain rates lower than 10−3 s−1 (Singhet al. 1993). The development of microstructure during high-temperature tensiledeformation of Zr–2.5Nb pressure tube (hot extruded and cold worked) has beenstudied by Perovic et al. (1985) using TEM. At deformation temperatures lowerthan 600 C, the starting lamellar morphology is retained but the original disloca-tion structure is destroyed and is replaced by sharp sub-boundaries which fragment




2

1

0

–1

–2

–3650 680 730 770 810 850

Temperature (°C)

Log

stra

in r

ate

0.00

0.00

–13 SAFE

–88–63–38

–25

–13

Strain = 0.4Zircaloy- 2 (β-quenched)

Extrusion at 720°C

Extrusion at 800°C

Figure 7.83. Extrusion conditions superimposed on instability map of �-quenched Zircalloy-2 fora strain of 0.4. The extrusion carried out at 800 C are in the stable flow regime while those carriedout at 720 C are within the instability regime.

the elongated �-plates. When deformation is carried out at temperatures lower than�-transus, the initial lamellar structure changes over to an approximately equiaxedstructure of �- and �-grains. In this temperature region (>700 C and < �-transus)during tensile deformation of Zr–2.5Nb in equiaxed (�+�) microstructure, Nutall(1976) observed that initial equiaxed morphology is maintained even after largedeformation (>500%) and suggested superplasticity to be the deformation mech-anism. The development of microstructure during hot extrusion of Zr–2.5Nb hasbeen studied by Chedale et al. (1972) and Perovic et al. (1985). The temperatureand extrusion ratio have been found to be the two major factors that influence themicrostructural evolution in the billets. The extrusion of the �-quenched billet at780 C results in a microstructure in which (�+�) grain structure has been alignedby working. The �-constituent is present in the form of lenticular plates which arejoined together by sub-boundaries.

The hot deformation behaviour of two-phase (�+�) alloys, Zr–2.5Nb and Zr–2.5Nb–0.5Cu in the temperature range 650–1050 C and at strain rates in the range0.001–100 s−1 have been studied by Chakravartty et al. (1995, 1996) using pro-cessing maps generated by carrying out compression testing. Two different initial




microstructures of Zr–2.5Nb alloy, namely equiaxed (�+�) and �-transformed(Chakravartty et al. 1992a,b,c), were studied. In both these materials, a domainof DRX has been identified in the processing maps by noting variation of theefficiency values with temperature and strain rate, nature of flow behaviour and bycarrying out detailed metallographic investigations using both optical and trans-mission electron microscopy (Figure 7.84). While a steady-state flow behaviourhas been observed in the equiaxed (�+ �) material, with a peak efficiency of45% at 850 C and 0.001 s−1, the �-transformed material exhibits stress–straincurves with continuous flow softening and higher peak efficiency (>50%) undersimilar deformation conditions. Deformation of equiaxed (�+�) structure in theDRX domain results in the development of equiaxed �-grains distributed in amatrix of �. A majority of these grains have been found to be further subdi-vided into subgrains separated by twist or tilt boundaries. In the DRX domain of�-transformed material, speroidization of deformed microstructure occurred as aresult of the shearing of �-platelets followed by globularization (Figure 7.85). Thisobservation is similar to that observed by Weiss and co-workers (1986) duringtwo-phase deformation of (�+�) titanium alloys. A simple model has been usedto analyse the characteristics of DRX by considering the rates of nucleation andgrowth of recrystallized grains (Prasad and Ravichandran 1991). Calculations ofthese two rates show that they are nearly equal and that the nucleation of DRX isessentially controlled by mechanical recovery involving cross slip of dislocations(Chakravartty et al. 1996). Tensile tests carried out over a range of temperaturesand at a strain rate of 0.001 s−1 showed a peak in ductility within the DRX domain.The optimum hot working temperatures for equiaxed (�+�) and �-transformedpreform microstructures are 850 and 750 C, respectively, while the optimum strainrate is 0.001 s−1 for both. It may be noted that the DRX temperature 850 C, ofZr–2.5Nb, is higher than in �-Zr (800 C) and this increase is expected in viewof the back stress generated by the long-range elastic interactions of the solute.Further, in comparison with the processing map for �-Zr, the map for Zr–2.5Nbshowed that the DRX domain has shifted to lower strain rates by two ordersof magnitudes (from 0.1 s−1 in �-Zr to 0.001 s−1 in Zr–2.5Nb). The lowering inDRX strain rate has been attributed to an increase in the probability of recoveryor a decrease in link length of dislocation or both. In view of likely increaseof stacking fault energy and solid solution strengthening caused by niobium, itis possible that both these contributions are responsible for the lowering of thestrain rate for DRX (Chakravartty et al. 1996). Figure 7.86 shows a TEM pho-tograph of the dynamically recrystallized Zr–2.5Nb which exhibits the extensiveformation of twist boundaries due to enhanced cross slip of screw dislocationsand its role in forming the interfaces. Kapoor and Chakravartty (2002) studied




2

1

0

–1

–2

–3

3

8 14

19

24

29

35

40

45

650 710 770 830 890 950

(a)

Temperature (°C)

Log

stra

in r

ate

Strain = 0.4Zr–2.5Nb (E)

38

35

–3

–2

650 710 770

49

45

52

42

–1

0

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21

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17

24

830 890 950

28

31

(b)

Zr–2.5Nb (Q)

Temperature (°C)

Log

stra

in r

ate

Strain = 0.4

Figure 7.84. Processing maps for (a) �+� equiaxed starting microstructure and (b) �-transformedstarting microstructure of Zr–2.5Nb for a strain of 0.4. That the maps show a single domain inthe (�+�) phase filled for a strain rate lower than 5× 10−1 s−1. The number against each contourindicates percentage efficiency.




1.2 μm

β

α

α

Figure 7.85. TEM micrograph of �-transformed microstructure of Zr–2.5Nb deformed at 800 C and0.001 s−1 to a strain of 0.3, showing initial stages of dynamic recrystallization involving fragmentationand shearing of �-plates followed by globularization.

0.4 μm

βα

α

Figure 7.86. TEM micrograph of equiaxed (�+�) microstructure of Zr–2.5Nb on deformation at800 C and 0.001 s−1. The dynamically recrystallized �-grains are found to contain twist boundariesin the form of hexagonal network of screw dislocations. Enhanced cross slip of screw dislocationsresults in the formation of interface required for dynamic recrystallization.

deformation behaviour in the �-phase field of Zr–2.5Nb alloy and reported LGSPat temperatures greater than 950 C and at strain rates lower than 0.01 s−1.

The instability map generated revealed that Zr–2.5Nb alloy undergoesmicrostructural instability at temperatures lower than 700 C and strain rates higherthan 1 s−1 and the manifestation of the instability has been in the form of flow




localization in both the initial microstructures and the severity of flow localizationhas been found to increase with rate of deformation (Chakravartty et al. 1996)).

In Zr–2.5Nb–0.5Cu, copper lowers the � to � transition temperature and par-titions entirely to �-phase. It is expected therefore that copper modify the defor-mation characteristics of �-phase alone in the alloy. The deformation behaviourhas been studied in the temperature range of 650–1050 C encompassing both the(�+�)and �-phase fields (transformation temperature for this alloy being 870 C).The hot deformation characteristics of this material is similar to that of Zr–2.5Nbin the (�+�) phase deformation in terms of flow behaviour and occurrence ofDRX domain. The domain of DRX is seen with its peak efficiency at about 750 Cand 0.001 s−1 (Chakravartty et al. 1995). In addition to the DRX domain, anotherdomain was found in the �-phase field of this material at 1050 C and 0.001 s−1

which has been characterized to be that of LGSP (Chakravartty et al. 1995). Theinstability map reveals that microstructure instabilities will occur at temperaturesgreater than 800 C and strain rates higher than 30 s−1.

7.8.3.3 �-alloysThe deformation characteristics of �-Zr-Nb alloys (containing 10–20% Nb) havebeen investigated by Jonas and co-workers (1979) in the temperature range from725 to 1025 C and with strain rate range 10−5–10−1 s−1 to assess flow propertiesof the metastable �-phase in the two-phase (�+�) field of Zr–2.5Nb alloy. Theflow curves obtained on Zr–Nb alloys exhibited flow softening, and magnitude ofthis effect decreased as the test temperature is increased. The occurrence of flowsoftening has been attributed to the disintegration or over-ageing of Nb clustersduring straining.

7.8.4 Development of texture during cold working of Zr alloysThe overall fabrication schedule which consists of one or two hot working stepsfollowed by a series of cold working steps determine the microstructure of thefinished product. The development of crystallographic texture during the processof TMP can again be divided in two parts – the first being produced during hotworking operation while the second resulting from the cold working and inter-mediate annealing steps. The anisotropic crystal structure of �-phase makes itquite sensitive to texture development which contributes significantly towardsanisotropy of mechanical and other physical properties of the finished product.Although similar deformation characteristics of Ti and Zr result in similar texturaldevelopment during TMP in their alloys, the texture development in Zr alloyshas attracted greater attention owing to the fact that the texture plays a signifi-cant role in-service performance of Zr alloys in irradiation environment. A briefaccount of texture development of zirconium alloys during deformation processing




and its importance in mitigating some life-limiting degradation mechanisms inactual components is given here.

One of the prime concerns in deformation processing is the development of afavourable crystallographic texture which is usually represented in terms of thedistribution frequency of important crystallographic planes (e.g. basal or prismaticplanes) with reference to the axial, radial and circumferential directions for tubularproducts and to the rolling, normal and transverse directions for plate products.As both slip and twinning contribute to plastic deformation of Zr, the deformationtexture is strongly dependent on the process variables such as rate of deformation,temperature, extent of deformation and state of stress (Tenckhoff 1988). Thesevariables in turn dictate available slip/twin systems and their respective criticalresolved shear strength. The development of texture with increasing degree ofcold work is due to the lattice rotations caused by a number of deformationsystems. Twinning causes spontaneous lattice rotations through large angles thoughthe resulting strain is small. Consequently at low values of plastic strain, thecontribution of twinning is more significant in texture development. After largeplastic strains, texture development is mainly controlled by slip deformation whichtends to rotate the basal pole by ≈ 20 –40 from the radial normal directiontowards the circumferential/transverse direction, provided the major compressivedeformation is along the former (Tenckhoff 1988).

In the production of zircaloy tubing, tube reduction processes, such as pil-ger milling involving triaxial stresses and strains, are commonly employed. Thereduction in the cross-section (RA) is achieved by reduction in the wall thickness(RW) and the diameter (RD). Depending on the value of the ratio Q (= RW/RD),different crystallographic textures are developed in pilgered tubes. A high wallthickness to diameter reduction (Q > 1) produces a texture with the basal polealigned parallel to the radial direction while a deformation with Q < 1 leads toa texture with the basal poles aligned parallel to the circumferential direction.Cold rolling of sheets also produces a similar texture with the basal poles beingabout 20 –40 away from the normal direction. Texture development in differenttypes of metal-forming operations in zirconium alloys is schematically illustratedin Figure 7.87.

The crystallographic texture of zirconium alloys influences the physical,mechanical and corrosion properties both out-of-pile and in-pile. Under neutronirradiation, single crystals of �-zirconium shrink along the c-axis and dilate on thebasal plane with the volume remaining more or less constant. This means a tubewith a predominant circumferential basal pole texture will grow along the axialdirection under irradiation. In order to minimize such growth, the fraction of basalpoles along the axial direction needs to be increased. Similarly the creep strengthalong the hoop and axial directions of the cladding can be tailored by appropriately




Type ofdeformation

Bodydimensioninitial–final

ElementSchematic

(0002)pole figure

T

u

b

e

r

e

d

u

c

t

i

o

n

Sheetrolling

Wiredrawing

RD AD

TD

ADRD

TD

ADRD

TD

ADRD

TD

R A

T TD

AD

TD

RD

AD

TD

AD

TD

AD

TD

RD> 1

RW

= 1RDRW

< 1RWRD

Figure 7.87. Various textures developed during cold working by pilgering, sheet rolling and wiredrawing operation involved in fabrication of zirconium alloy components.

modifying the texture. Anisotropy of plastic deformation and resistance to stresscorrosion cracking also depend on the texture. By far the most important appli-cation of texture control is in achieving a favourable orientation distribution ofbrittle hydride platelets in cladding tubings with a strong radial basal pole texture.




The development of texture in a two-phase Zr–2.5Nb alloy has recently beenstudied in detail by Kiran Kumar et al. (2003). The main findings of this workare summarized below to emphasize the role of �-phase (though small in vol-ume fraction yet present as continuous matrix phase) in controlling the develop-ment/evolution of texture. Two different microstructures of this alloy, namely,martensitic single phase hcp structure obtained by rapid quenching from the�-phase and two-phase �+� structure obtained by solutionizing in the �-phasefollowed by annealing in the �+� phase field, were considered. The initial bulkcrystallographic textures were similar in these structures. On cold rolling, while thesingle phase �-structure showed considerable textural changes, the bulk texture oftwo-phase structure remained virtually unchanged. The Taylor-type deformationmodel which considers different combinations of slip-twin systems and constraintscould explain adequately the texture evolved in single phase material. However, itcould not explain the lack of textural development in two-phase �+� structure. Inthe two-phase structure, microtexture observations showed that �-plates remainedapproximately single crystalline even after cold working, while the continuous�-matrix underwent significant orientational changes as evidenced by relative hard-ening estimated by X-ray peak broadening. In addition to these, the aspect ratio of�-plates remained unchanged with cold rolling suggesting the absence of effectivemacroscopic strain in hcp �-phase. On the basis of these observations, a simplemodel, which considered that the plastic deformation was solely concentrated inthe matrix �-phase within which the embedded �-plates were subjected to in planerigid body rotation, was proposed. The salient feature of this model is depictedin the Figure 7.88. The schematics of Figure 7.88(a) explain the alignment of�-plates floating in continuous �-phase matrix along RD and on the rolling plane(i.e. in plane rigid body rotation) on deformation. The nature of this alignmentof �-plates on deformation has been substantiated by microstructural observationswhich showed �-plates did not align in any specific direction on the other twocross-sections LT and ST and there was no dimensional changes of �-plates inany cross-section (particularly in the rolling plane). In order to test this model,random in-plane rigid body rotation(s) were simulated on the starting texture. Thiswas accomplished by imparting random rotations commensurate with the rangeof angular realignment necessary for �-plates (Figure 7.88(b)). An orientationdistribution function (ODF) of two-phase material clearly shows the absence oftexture development on deformation involving in-plane rigid body rotation. How-ever, a simultaneous combination of in-plane rigid body rotation and deformation(Figure 7.88(c)) could not reproduce the texture and thus was found unsuitable forexplaining the lack of textural development in �-plates.




(0 2 2 1)

β-Matrix

α-Plates

Cold rolling

<3 2 1 2>

TD

RD

(0 1 1 4)<2 1 1 0>

(0 2 2 1)<2 1 1 0>

0 90

90

0

90

90

0

90PHI2=40

PHI2=0 PHI2=0 PHI2=10 PHI2=20 PHI2=30

PHI2=40 PHI2=50 PHI2=60

PHI2=10 PHI2=20 PHI2=30

PHI2=50 PHI2=60

90 90 90 0 090 090 090 90

(0 1 1 4)<2 1 1 0>

(0 1 1 4)<2 3 4 4>(0 1 1 2)

<1 4 4 4>

(b) (c)

(a)

Figure 7.88. The in-plane rigid body rotation model: (a) schematic diagram illustrating rotation(s)on the rolling plane and along RD, of �-plates in a continuous �-matrix. An imaginary unit cellrepresented few of the �-plates, both before and after deformation. It is to be noted that the shadesshown in these unit cells are just to give a sense of necessary rotation and do not correspondto the indicated crystallographic planes. The schematic overestimates �-fraction, but otherwisedimension tolerances were kept in consideration for 60% reduction. (b) The simulated, with randomin-plane rigid body rotation, ODF of the two-phase material. Absence of texture development in(b) corresponds to the experimental observation in two-phase structure. (c) Simulated ODF for 20%deformation (by unidirectional rolling) simultaneously with in-plane rigid body rotation or randomND rotation. Random ND rotations were applied after small incremental strain (2%). The ODF failsto capture lack of texture development typical of deformed �-plates in two-phase material.




7.8.5 Evolution of microstructure during fabrication of Zr–2.5 wt% Nballoy tubes

Although the hot working step imparts large deformation and provides a suitablemicrostructure by operation of one or more of the dynamic softening processesidentified in titanium and zirconium alloys, the final product properties are con-siderably influenced by several other factors involved in TMP subsequent to hotworking. These are rate of cooling and associated phase transformation after hotworking, ageing, cold working and annealing. A variety of TMP techniques havebeen employed in several titanium alloys to tailor microstructure for obtainingspecific mechanical properties, the details of which are well documented and areunderstood (Fleck et al. 1984). From these investigations, it appears that a knowl-edge of phase transformation and an understanding of the softening (recovery andrecrystallization) processes that occur statically are crucial in imparting desiredmechanical properties to the finished product. In order to highlight this, it is nowproposed to give an account of a study on the evolution of microstructure duringfabrication of Zr–2.5 wt% Nb alloy pressure tubes for application in PHWRs.

The fabrication schedule of Zr–2.5Nb pressure tubes, as largely practiced,involves hot working of the ingot in several stages either by forging or by extru-sion followed by a single cold working step like tube drawing which introducesthe required 25% cold work and shapes the final product within the dimensionaltolerances specified. The last hot working step, hot extrusion, is carried out in the(�+�) phase field with an accurate control of the temperature, the deformationrate (ram speed) and the extrusion ratio. The volume fractions of the two phases,and consequently their compositions, the aspect ratios of the �- and �-grains inthe (�+�) fibrous microstructure and the crystallographic texture of the productare essentially determined by the process parameters of the last hot extrusion step.Figure 7.89 shows some typical microstructure of the hot extruded tube in both lon-gitudinal and transverse directions as revealed by SEM and TEM. SEM photographshows that the two-phase (�+�) structure is elongated in the direction of extru-sion. The �-phase (bright constituent) stringers are discontinuous and separate theelongated �-phase units. The details of �-grain structure, however, are not clearlyrevealed in SEM. TEM micrographs of both longitudinal and transverse directionreveal that the elongated �-units are essentially made of cluster of nearly equiaxedfine �-grains separated by thin �-phase film. This observation suggested the oper-ation of DRX in Zr–2.5Nb alloy during hot extrusion. In addition to high-angleboundaries in the structure, some of the equiaxed grains have also been found tocontain sub-boundaries. The combination of high strength with a good ductility andtoughness of Zr–2.5Nb pressure tubes is essentially derived from the fine (�+�)fibrous microstructure consisting of elongated �-grains containing about 0.5Nband the �-phase (15–20Nb) stringers primarily located at �-grain boundaries. Such




Figure 7.89. SEM and TEM micrographs of two-phase (�+ �) microstructure of hot extrudedtube: (a) elongated morphology in the longitudinal section (SEM), (b) serrated morphology in thetransverse section (SEM), (c) and (d) dynamically recrystallized equiaxed �-grains within �-stringersin longitudinal and transverse sections, respectively (TEM).

a structure cannot remain stable if it is subjected to a cold working–full annealingcycle which tends to coarsen the structure into a distribution of equiaxed �- and�-grains, resulting in a substantial drop in the ultimate tensile strength.

In recent years, there has been a sustained effort towards improving the resis-tance to irradiation-induced growth (one of the life-limiting factors of pressuretubes in PHWRs) through the modification of microstructure of the pressure tubematerial (Srivastava et al. 1995). However, the challenge was to qualify the prod-uct in terms of microstructure, dislocation density, crystallographic texture andshort-time mechanical properties as specified for the conventionally processedmaterial. This has been achieved by incorporating cold working by drawing or pil-gering in two steps (instead of one in the conventional process) with an optimizedintermediate annealing treatment (Figure 7.90). Srivastava et al. (1995) studiedthe evolution of microstructure and of tensile properties during each fabricationstep of the modified route and compared it with that of the conventional route.The development of microstructure in both the routes was identical during extru-sion in the two-phase field as the �- and �I-phases dynamically recrystallizedand the �I-phase layers were sandwiched between elongated �-phase (stringers).




Autoclaving400°C/72 h

II – Pilgering25% reduction

Vacuum annealing550°C/6 h

I – Pilgering50–55% reduction

Vacuum stress relieving480°C/3 h

Modified routeConventional route

β-Quenching1000°C/30 min

β-Quenching1000°C/30 min

Hot extrusionExt. ratio 11:1

800°C

Hot extrusionExt. ratio 8:1

800°C

Cold drawing25%

Autoclaving400°C/24 h

Figure 7.90. A comparison of fabrication routes of Zr–2.5Nb pressure tube material.

The elongated �-units were essentially made up of a number of nearly equiaxed�-grains. The �-phase precipitated within the �I during cooling after hot extrusionfollowing either route. However, in the modified route, the aspect ratio of both thephases was lower, presumably because of lower extrusion ratio employed and the�I volume fraction was higher (with lower niobium content) owing to faster cool-ing rate subsequent to extrusion. A typical microstructure in radial–axial sectionis shown in Figure 7.91(a) which shows elongated �-grains separated by �-phase.Subsequent to hot working, in the conventional route, 20–25% cold work is givento produce a dislocation density of the order of 1014 m−2 which results in optimumcombination of tensile strength and in reactor creep behaviour (Figure 7.91(b)).




(a) 0.7 μm

0.2 μm

0.8 μm 0.4 μm

0.3 μm(b)

(c)

(d) (e)

Figure 7.91. Transmission electron micrograph of Zr–2.5Nb alloy: (a) as-extruded microstruc-ture showing elongated morphology of �-phase separated by �-phase stringers. (b) First pilgeredmicrostructure illustrating the very high dislocation density. The dislocations are concentrated pri-marily at the �/� interface. (c) Incomplete recrystallization of the �-stringers as evidenced by thepresence of a substantial number of dislocations after annealing at 500 C for 6 h. (d) Coarseningof the �-grains caused by redistribution and agglomeration of �-phase. The �-phase is located atthe triple junctions of �-grains after annealing at 600 C for 1 h and (e) completely recrystallized�-lamellae obtained after annealing at 550 C for 3 h. The lamellar morphology of the two phases isnot altered by this annealing treatment.




The purpose of intermediate annealing treatment prior to second pilgering in themodified route is to annihilate all the cold work introduced in the first pilgeringstep so as to get a microstructure and flow properties similar to that obtained afterextrusion. Retention of this structure is necessary for obtaining optimum tensileproperties at service temperature of 310 C. In this investigation, the annealingtemperature was varied between 500 and 650 C and annealing duration variedbetween 1 and 6 h. It was observed that the elongated two-phase structure could beretained only if the annealing temperature is kept below 575 C. Annealing at lowertemperature, for example, at 500 C for 6 h resulted in recovery of �-stringers asevidenced by the presence of sub-grain boundaries. In addition to these, a highdensity of unrecovered dislocations was also noticed (Figure 7.91(c)). The tensilestrength of the resultant structure was higher than the as-extruded material. Incontrast to this, annealing at higher temperature, e.g. 600 C, even for 1 h resultedin complete recrystallization and morphological modifications of both �- and �-phases (Figure 7.91(d)). The annealed microstructure comprised essentially ofnearly equiaxed �-grains, and �I-phase was noticed primarily at the boundariesand trijunctions of �-grains. The significant coarsening of the structure accompa-nied by morphological changes resulted in lowering of tensile strength to a levelbelow that of the hot extruded material. An annealing treatment at 550 C for 6 hcaused nearly complete recrystallization of phase without altering the elongatedmorphology of the �+�I microstructure (Figure 7.91(e)). A comparison of tensileflow behaviour of this annealed material with that of as-extruded material revealedthat the flow stresses at different levels of plastic strains are nearly the same. Onthe basis of the observed similar tensile flow properties and morphology of thetwo-phase structure, it was suggested that annealing at 550 C for 6 h would be theoptimum condition for achieving the desired results.

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Sizmann, R. (1978) J. Nucl. Mater., 69–70, 386.Sluiter, M. and Turchi, P.E.A. (1991) Phys. Rev. B, 43, 12251.Spencer, C.W. and Mack, D.J. (1962) Decompositions of Austenite by Diffusional Processes

(eds V.F. Zackay and H.I. Aaronson) Inter Science Publishers, New York, p. 549.Srivastava, D., Dey, G.K. and Banerjee, S. (1995) Metall. Mater. Trans., 26A, 2701.Sun, J., He, Y.H. and Wu, J.S. (2002) Mater. Sci. Eng., A329–331, 885.Tenckhoff, E. (1988) Deformation Mechanisms–Texture and Anistropy in Zirconium and

Zircoaloy, ASTM STP 966, Am. Soc. Test Mater Philadelphia, PA.Unnikrishnan, M., Menon, E.S.K. and Banerjee, S. (1978) J. Mater. Sci., 13, 1401.Veeraraghavan, D., Wang, P. and Vasudevan, V.K. (1999) Acta Metall., 47, 3313.




Wagner, R. and Kampman, R. (1991) Materials Science and Technology – A Comprehen-sive Treatment (eds R.W. Cahn, P. Haasen and E.J. Kramer) Phase Transformations inMaterials, Vol. 5, VCH, Weinheim, p. 213.

Wang, Y. and Chen, L.Q. (2000) Methods in Materials Research (eds E.N. Kaufman)John Wiley & Sons Inc., New York, p. 2a.3.1.

Wang, Y., Ma, N., Chen, Q., Zhang, F., Chen, S.L. and Chang, Y.A. (2005) JOM, 57(9), 32.Wang, P. and Vasudevan, V.K. (1992) Scr. Metall. Mater, 27, 89.Wang, P., Vishwanathan, G.B. and Vasudevan, V.K. (1992) Metall. Trans. A, 27, 89.Wang, P., Veeraraghavan, D., Kumar, M. and Vasudevan, V.K. (2002) Metall. Trans. A.,

33, 2353.Weiss, I. and Semiatin, S.L. (1998) Mater. Sci. Eng., 243, 46–65.Weiss, I. and Semiatin, S.L. (1999) Mater. Sci. Eng. A, 236, 243.Weiss, I., Welsch, G.E., Fores, F.H. and Eylon, D.E. (1986) Metall.Trans., 17A, 1935.Williams, J.C. (1973) Phase Transformation in Ti alloys: A Review (eds R.I. Jaffee and

H.M. Burte) Titanium Science and Technology, Vol. 3, Plenum Press, New York,p. 1433.

William, J.C., Polonis, D.H. and Taggart, R. (1970) The Science, Technology and Appli-cation of Titanium (eds R.I. Jaffee and N.E. Promisel) Pergamon Press, Oxford, p. 733.

Zeigler, H. (1963), Progress in Solid Mechanics, Vol. 4, (eds I.N. Sneddon and R. Hill)Wiley, New York, p. 93.

Zhang, W.Z. and Purdy, G.R. (1993) Acta Metall. Mater., 41, 543.Zhang, W.Z. and Purdy, G.R. (1993) Philos. Mag., 68, 291.



Chapter 8

Interstitial Ordering

8.1 Introduction 7208.2 Hydrogen in Metals 721

8.2.1 Ti–H and Zr–H phase diagrams 7228.2.2 Terminal solid solubility 725

8.3 Crystallography and Mechanism of Hydride Formation 7288.3.1 Formation of �-hydride in the �- and �-phases 7288.3.2 Lattice correspondence of �-, �- and �-phases 7298.3.3 Crystallography of � → � transformation 7308.3.4 Crystallography of � → � transformation 7358.3.5 Mechanism of the formation of �-hydrides 7378.3.6 Hydride precipitation in the �/� interface 7378.3.7 Formation of �-hydride 739

8.4 Hydrogen-Related Degradation Processes 7418.4.1 Uniform hydride preciptiation 7428.4.2 Hydrogen migration 7438.4.3 Stress reorientation of hydride precipitates 7458.4.4 Delayed hydride cracking 7468.4.5 Formation of hydride blisters 747

8.5 Thermochemical Processing of Ti Alloys by TemporaryAlloying with Hydrogen 753

8.6 Hydrogen Storage in Intermetallic Phases 7548.6.1 Laves phase compounds 7548.6.2 Thermodynamics 7568.6.3 Ti and Zr-based hydrogen storage materials 7568.6.4 Applications 761

8.7 Oxygen Ordering In �-Alloys 7648.7.1 Interstitial ordering of oxygen in Ti–O and Zr–O 7648.7.2 Oxidation kinetics and mechanism 769

8.8 Phase Transformations in Ti-Rich End of the Ti–N System 772References 780






Chapter 8

Interstitial Ordering

List of Symbols�a

ij: External stresseT

ij: Stress-free transformation strainVH/Vhyd: Partial molar volume of hydrogen/hydride

xRB: Correspondence matrix relating chosen coordinate system (xi) and

that of the �-bcc (B)

xRF: Correspondence matrix relating chosen coordinate system (xi) and

that of the �-fct (F)S�SB: Strain in x and B coordinate systemP� PB: Lattice invariant shear in x and B coordinate system

FB: Macroscopic shape strain in B coordinate systemHB: Habit plane in B-coordinate systemSH: Lattice strain in orthohexagonal coordinate systemJ : Flux of hydrogen atomsD: Diffusivity of hydrogen in metalcx: Hydrogen concentration at point x

Q∗�V ∗: Heat and volume of transparent of hydrogen in metalP: Hydrostatic pressure

CBFT: A threshold hydrogen concentration for blister formationc∗

TSS: Hydrogen concentration corresponding to the terminal solidsolubility

Te: Eutectoid temperature in the Ti–H system�Ec: Configurational internal energy

Pij� ij: Number of ij pairs and corresponding interaction energiesH�M : Concentration of dissolved hydrogen in metal

pH2: Partial pressure of hydrogen

�Hs: Enthalpy of solution�Gs: Strain energy density associated with hydride precipitates

�Gaccs : Accommodation strain energy

�Gints : Strain energy due to applied stress field, residual stress and stress

field due to other inclusionsCT

o : Equilibrium concentration of hydrogen in solution at TCgrowth

o /Cdisso : Hydrogen concentration during growth/dissolution of hydride

Wacc

el : Strain energy due to elastic accommodationW

acc

p : Strain energy due to plastic accommodation

719




rp: Plastic zone radiusp: Hydrostatic pressure within the hydride precipitateK: Bulk modulus�: Ratio of bulk moduli of hydride and matrix�: Poisson’s ratio : Shear modulus

8.1 INTRODUCTION

The formation of interstitial compounds in a metal matrix can be treated as aprocess of interstitial ordering whereby interstitial solutes, which remain distributedrandomly in the interstitial sites, assume an ordered array with a lowering of thetemperature of the system. An excellent example of interstitial ordering is theformation of Fe4N in the fcc lattice of dilute Fe–N alloys (Jack 1948). In thestructure of Fe4N, the N atoms occupy the octahedral holes at the body-centredpositions of the fcc lattice in such a way that each N atom is separated fromits neighbours by the maximum possible distance. However, the stoichiometricFe4N alloy does not exhibit an order-to-disorder transformation at an elevatedtemperature as other phase reactions come into play.

The bct martensite forming in the Fe–C system can also be interpreted as havingan interstitially ordered structure. The occupation of the octahedral interstices inthe bcc lattice causes a distortion of the lattice to produce a tetragonal symmetry.Zener (1946) has pointed out that such an ordered configuration of C atoms isproduced below a critical temperature. Khachaturyan (1978, 1983) has presenteda theory of interstitial ordering by considering the interstitial solid solution as asystem of interacting particles in the periodic field of the host lattice, and hasdiscussed the formation of various types of superlattices.

The formation of interstitial compounds, TixI and ZrxI, where I stands for H,N and O, in the metallic matrix of titanium- and zirconium-based alloys can betreated as interstitial ordering. In this chapter, we will examine the mechanismand to a limited extent, the kinetics of the formation of interstitial compoundssuch as hydrides, nitrides and oxides of Ti and Zr. Emphasis will be laid onthose transformations in which the interstitial compound phase maintains a latticerelationship with the parent phase. As we proceed, we will encounter situationsin which interstitial ordering is accompanied by changes in the dimension andstacking sequence of the parent lattice. The crystallography of such transformationshas been discussed in detail to examine whether the phenomenological theory ofmartensite crystallography remains valid in these cases.



Interstitial Ordering 721

The formation of hydrides in Ti and Zr is discussed in the first part of thischapter. As will be elaborated later, the transformation of the � (hcp) phase intothe �-hydride (fct) phase is an excellent example of a hybrid transformation inwhich the lattice of the metal atoms undergoes a displacive transformation whilethe movement of H atoms is by diffusion. As in the case of shear transformations,the hydride precipitation process is strongly influenced by external stress–a phe-nomenon which has serious implications in respect of the in-service degradationmechanisms of alloys based on these metals.

Oxygen atoms dissolved in the �-phase of Ti and Zr occupy the octahedralinterstitial sites of the host hcp lattice. Ordering of O atoms has been reported atcompositions near the O/metal ratios of 1

3 and 12 . The structures and transformation

behaviours of interstitial alloys of such compositions are discussed in a subsequentpart of this chapter. The corrosion rate of zirconium alloys in service in nuclearreactors is essentially controlled by the growth of oxide layer on zirconium alloycomponents. The mechanism and the kinetics of oxide growth are also outlined inthis chapter.

The case of the formation of Ti2N and TiN phases in titanium alloys is chosen asexamples of nitrogen ordering. The crystallography of the phase transformationsinvolved in the formation of titanium nitride coatings is discussed in the lastsection of this chapter.

8.2 HYDROGEN IN METALS

Hydrogen is soluble to some extent in almost all metals. When present in excessof a certain limit, H can have a damaging effect on the mechanical propertiesof a metal, particularly its fracture properties. These effects have been studiedextensively in common metals such as Fe, Cu and Ni in which H is present eitherin solution or is precipitated as gaseous bubbles. In another class of metals suchas Ti, V, Zr and Nb which have a strong chemical affinity for H, metal hydridesare precipitated when the H content exceeds the solubility limit. In these metalsH solubility is quite high at elevated temperatures, but decreases sharply withlowering of temperature leading to hydride precipitation (Dutton 1976).

Dissolution of H gas in a metal involves ionization of H atoms, with the electronsentering the conduction band of the host metal and the ions being positionedat interstitial sites. The equation of dissolution of hydrogen in metals can berepresented as

1

2H2 → HM (8.1)




Table 8.1. Selected hydrogen solubility parameters for Ti and Zr.

Phase Enthalpy of solution�Hs (kcal/mole)

Maximumconcentration (at.%)

H solubilitytemperature (K)

�-Ti (hcp) −10�8 6.72 573�-Ti (bcc) −13�9 >50 >873�-Zr (hcp) −12�2 5.9 823�-Zr (bcc) −15�4 >50 >1023

where the subscript ‘M’ refers to H dissolved in metals. Sievert’s law for theconcentration of dissolved H, H�M is

H�M = �pH2�1/2 exp�−�G/RT� (8.2)

where �G is the standard free energy change for the reaction, pH2 is the par-tial pressure of H and R and T have their usual meanings. The correspondingenthalpy change, �H , has two components: (a) the enthalpy of dissociation ofthe H molecule (about 52 kcal/mole) and (b) the enthalpy of solution �Hs of Hatoms in the metal lattice. Depending on the sign of �Hs, metals are classified asendothermic and exothermic occluders of hydrogen (Dutton 1976). This classifi-cation serves to distinguish the stable phase formed when the solid solubility limitis exceeded. The endothermic occluders are in equilibrium with the H gas, often inthe form of internal gas bubbles, and the exothermic occluders are in equilibriumwith a precipitated solid hydride phase. Ti and Zr are exothermic H occluders, theenthalpy of solution and the maximum solid solubility limit of H in the � (hcp)and in the � (bcc) phases being indicated in Table 8.1.

8.2.1 Ti–H and Zr–H phase diagramsThe equilibrium phase diagrams of the Ti–H and Zr–H systems are shown inFigures 8.1(a) and (b) in each of which the metastable �-hydride phase field isalso marked. The two equilibrium hydride phases are the �-hydride phase and the-hydride phase. In both the systems the �-hydride phase results from an eutectoiddecomposition of the �-phase, as indicated in the phase reaction

� = �+� (8.3)

The equilibrium �-hydride (fcc) phase is stable in the composition range of 56.7–66.0 at.% H (Zuzek 2000). In this phase H atoms occupy some of the tetrahedralinterstices at ( 1

414

14 ) and equivalent positions. Only a very small fraction of H

atoms are located in octahedral holes. The volume misfit of �-hydride with respectto �-Ti and �-Zr is about 24% and 17.1%, respectively.




Hydrogen (Wt%)2

Hydrogen (at.%)

Tem

pera

ture

(K

)

473

73

–27

Ti

0

373

273

173

0.5 1

α-Ti α-Zr

β-Ti

β-Zr

1.5

10

973

873

773

673

573

1173

1073

0

573 K

0.15

20 30

1.33

40

2.5 3 3.5 4 4.5

ε

2.16

50

δ

60

(a)

0.8

823 K

Tem

pera

ture

(K

)

273

Zr

0 0.2

573

373

473

673

773 0.07

0.4 0.6

~0.659

200

1073

973

873

1173

127310 30 40

ε1.43

1.41 1.2 1.6 1.8 2

50 60

δ

Hydrogen (at.%)

Hydrogen (Wt%)

(b)

Figure 8.1. (a) Ti–H and (b) Zr–H phase diagrams.

The tetragonal �-hydride phase has a c/a ratio less than unity and a stoichiometryclose to TiH2 and ZrH2. H atoms occupy the interstitial sites of tetragonal symmetryat ( 1

414

14 ) or equivalent points. There are two such sites per Zr/Ti atom. While the

radius of the interstitial site in the �-hydride structure is 0.0378 nm that in the-hydride structure is 0.043 nm (in the case of Zr hydrides). The atomic radius ofH is about 0.040 nm which matches closely with the sizes of the interstitial sitesin both � and � hydrides.

The metastable �-hydride phase has a fct structure with a c/a ratio greaterthan unity and a stoichiometry close to ZrH and TiH. The lattice parameters ofthis phase depend on the alloy composition and the thermal treatment. Typically,the �-hydride phase in a dilute Zr–H alloy has lattice parameter values of a =0�4596 nm and c = 0�4969. H atoms occupy ordered tetrahedral interstitial sites ina fct Zr lattice.

The hcp � phase of Ti and Zr can take a certain amount of H in solid solution.There are two types of interstitial sites in the hcp structure – tetrahedral andoctahedral (Figure 8.2); their radii are 0.0364 and 0.068 nm, respectively, in �-Zr.For titanium the sizes are 0.0343 and 0.0619 nm, respectively. Neutron diffractionexperiments have shown that H atoms occupy tetrahedral interstitial sites in the �phase of Ti and Zr (Narang et al. 1977). There are four such sites of tetrahedralcoordination at ( 2

313

16 ), ( 2

313

56 )(00 1

3 ) and (00 23 ). Darby et al. (1978) in a theoretical

study, have calculated the relative energy of H in tetrahedral and octahedral sitesin the hcp �-phase assuming that the energy terms most strongly dependent on




a√3

2√2

(b)

a0

A B

3√a 0

C

(c)

Tetahedral intersticesMetal atoms

a

Metal atomsOctahedral interstices

(a)

a√2

a

Figure 8.2. (a) Octahedral, (b) tetrahedral interstitial sites in hexagonal close packed structures and(c) a single interstitial layer described in terms of a hexagonal net with transalation of

√3ao.

site symmetry are those pertaining to the electronic and protonic electrostaticcontributions. Their calculations have shown that the tetrahedral site is preferredover the octahedral site in agreement with experimental observations.

There have been several attempts to predict H solubility in the �-phase. Usinga strain energy-based model, Sinha et al. (1970) have obtained a saturation sol-ubility of 62 at.% H in �-Zr at room temperature. However, the experimentallyobserved maximum solubility limit is about 6 at.% at the eutectoid temperature(823 K) in the Zr–H system. In a thermodynamic study (Northwood and Kosasih1983), it has been assumed that hydrogen atoms occupy only alternate sites inthe �-phase and that they oscillate between the two interstitial sites with an esti-mated vibration frequency of 1�21 × 1013 Hz. Sinha et al. (1970) have used thisconcept in modifying their strain energy model but the results have still shown asolubility limit much higher than the experimentally observed value. ConsideringH as a quantized oscillator, a solubility limit of 0.492 at.% at room temperaturehas been obtained, which is in good agreement with the observed value (Sinhaet al. 1970).




8.2.2 Terminal solid solubilityThe maximum hydrogen concentration, which can be retained in solid soltuionwithout forming hydride precipitate, is called terminal solid solubility (TSS) (Panet al. 1996). The amount of hydrogen in excess of solid solubility precipitatesas one of the hydride phases depending on the cooling rate and the hydrogenconcentration (Northwood and Kosasih 1983). Similar to any solid state phasetransformation process, the free energy of hydride precipitation has three compo-nents, viz. chemical free energy, interfacial energy and strain energy. The drivingforce of chemical free energy change associated with the transformation of matrixto hydride is primarily opposed by the strain energy arising from the densitydifference between the matrix and hydride precipitate. The interfacial energy isnegligibly small when hydride precipitates are fine (<1 m in diameter with aaspect ratio of ∼20). The strain energy, �Gs, can again be expressed as the sum�Gs = �Gacc

s + �Gints (Puls 1990). While �Gacc

s represents strain energy densityassociated with the accommodation of individual hydride platelets, �Gint

s is due tothe change in the accommodation strain energy in the presence of an externallyapplied stress or residual stress or stress field due to other inclusions.

Experimentally determiend values of TSS of hydrogen in zircaloy-2 are shownin Figure 8.3 (Singh et al. 2005). TSS obtained during heating corresponds to thedissoltuion of hydrides and hence termed as terminal solid solubility for dissoltuion(TSSD). On the other hand, TSS determined during cooling corresponds to the

700

600

500

400

300

Tem

pera

ture

(K

)

TSSD (Heating)

TSSP (Cooling)

α-Zr + hydride

α-Zr

0 100 200 300

CH (ppm)

Figure 8.3. Terminal solid solubility of hydrogen in Zircaloy-2 obtained during heating (TSSD) andcooling (TSSP) using dialatometry.




precipitation of hydrides and is termed as terminal solid solubility for precipitation(TSSP). As can be seen in the Figure 8.3, TSSP is higher than TSSD. Theexperimental results on TSS of hydrogen in hydride forming metals are analysedin terms of a hypothetical equilibrium solvus cT

o , the concentration of hydrogen insolution in equilibrium with hydride,

cTo = A exp �−�Hs/RT� (8.4)

where A is a constant and �Hs is the enthalpy of solution with respect tothe hydride (Puls 1990). The equilibrium TSS defined in Eq. (8.4) cannot bedetermiend experimentally as hydride precipitation is always associated with alarge value of strain energy of accommodation. The accommodation strain energyis reported to be dependent on the size of the precipitate and its tendency to grow ordissolve leading to hysteresis behaviour between the TSS obtained during heatingand cooling. During nucleation the accommodation energy is fully elastic, whereasfor a growing precipitate the accommodation energy is partly due to elastic andpartly due to plastic deformation of the matrix and precipitate. Usually interfacialfree energy is neglected and in that case chemical free energy released due tophase transformation must provide for the strain energy associated with elasticand plastic deformation. The strain energy stored in the material due to elasticdeformation can be recovered but the strain energy spent in plastic deformationis partly locked in and around the defects and is partly dissipated in the form ofsound, heat, etc.

Thus, the TSS expression for the precipitate growth and dissolution duringcooling down and heating up, respectively, in terms of the stress-free solvusdefined by Eq. (8.4) is given as (Puls 1990):

cGrowtho = cT

o exp(�W

acc

el +Wacc

p �/RT)

(8.5)

and

cDisso = cT

o exp(W

acc

el /RT)

(8.6)

The expression for strain energy associated with elastic, Wacc

el , and plastic, Wacc

p ,accommodation have been obtained for the case of spherical, misfitting inclusionshaving isotropic positive or negative misfit strains (Puls 1990):




Wacc

el

V hyd

= p2

2�K�1−��+ �2

y

��

{13

(rp

a

)3

− 16

}(8.7)

Wacc

p

V hyd

= �2y

��

{(rp

a

)3

ln(rp

a

)− 1

3

(rp

a

)3

+ 13

}(8.8)

where rp = plastic zone radiusa = radius of the hydride precipitateV hyd = partial molar volume of hydridep = Kmatrix�eT�1−�� = hydrostatic pressure within the hydride precipitateK = bulk modulus� = Khyd/Kmatrix

� = �1+��/4�1−��, where � is the Poisson’s ratio� = E/2�1+��and � is solution to

2��eT

�y

�� +�−�� = exp{

K�eT

2�y

�1−��−1}

(8.9)

Influence of stress on TSS: The partial molar volume of hydrogen in solution in�-zirconium is positive and hence, in the presence of a tensile hydrostatic stress,the solid solubility is enhanced. Also, the hydride precipitation in these alloys isassociated with positive misfit strains. Thus, in the presence of tensile stress, thetendency to hydride precipitation is also enhanced. The equilibrium concentrationof dissolved hydrogen cT

P can be obtained corresponding to the aforementionedsolvus by multiplying the right-hand side of Eqs. (8.5) and (8.6) by the followingfactor (Puls 1990).

exp[(

pV H −V hyd�aije

Tij

)/RT

](8.10)

which represents the change in strain energy of the system, when one moleof hydrogen in solution under the hydrostatic pressure p, is precipitated underan external stress of �a

ij with the stress free transformation strain eTij . While V H

represents partial molar volume of hydrogen atoms in solution, V hyd is the specificvolume of hydride precipitate per mole of hydrogen. It turns out that for zirconiumalloys, the two interaction energies are nearly equal (pV H ∼ V hyd�

aije

Tij) and under

uniform applied stress, TSS changes negligibly. Thus, TSSP and TSSD do notchange significantly in the presence of stress.




8.3 CRYSTALLOGRAPHY AND MECHANISM OF HYDRIDEFORMATION

As mentioned earlier, three types of hydride phases, namely, the �-phase (fct,nearly equiatomic, ZrH, TiH), the �-phase (fcc, ZrH1�5� TiH1�5) and the �-phase(tetragonal with c/a < 1, ZrH2, TiH2) are encountered in Ti–H and Zr–H systems.The kinetics of the formation of these hydride phases are considerably different.While the � → � and the � → � transformations are so rapid that they cannot besuppressed by quenching, the other two hydride phases can form only on ageing.Increasing H contents and slower cooling rates promote the formation of � and �hydrides.

8.3.1 Formation of �-hydride in the �- and �-phasesThe metastable �-hydride phase, which has an fct structure (c/a >1.0 ) can coexistwith the � as well as with �-hydride phases. The formation of the �-hydride phaseis favoured by a fast cooling rate from a temperature where H is taken into thesolid solution (typically, above about 625 K for Zr containing about 40 ppm H).Northwood and Kosasih (1983) have suggested a shear mechanism for the � → �transformation. The temperatures at which �-hydride can form are too low forany significant self-diffusion of Zr (or Ti) atoms to occur. Interstitial diffusion ofH atoms within the time scale of the transformation is still possible. The overall� → � transformation process can, therefore, be viewed as a shear transformationof the hcp lattice of �-Zr/Ti with an accompanying H redistribution and interstitialordering. The morphological features (Figure 8.4(a); Dey et al. (1982) Dey andBanerjee (1984)) of �-hydride precipitates in the �-matrix include plate shape withspecific habit plane and internal twinning which suggest an invariant plane strain

(a)

200 nm 300 nm

(b)

Figure 8.4. Micrographs showing �-hydride in (a) �- and (b) �-phases.




transformation. The crystallography of the transformation is, therefore, analysedin terms of the phenomenological theory of martensitic crystallography.

The �-hydride phase can also form in the matrix of �-stabilized alloys (Flewittet al. 1976, Dey et al. 1984). The � → � transformation also exhibits severalfeatures of shear transformations such as low transformation temperatures (atwhich self-diffusion of Zr or Ti is negligible), a plate morphology and internaltwinning of hydride precipitates.

In �+� alloys, H is partitioned between the two phases. Since the solubility ofH in the �-phase is much more than that in the �-phase, precipitation of hydridesin � + � alloys is encountered more frequently in the �-phase and along �/�boundaries than in the �-phase.

8.3.2 Lattice correspondence of �-, �- and �-phasesBased on the observed orientation relations between the �-hydride and the �- andthe �-phase matrix, Dey et al. (1984) have proposed the following lattice corre-spondence in respect of these phases (as shown in Figure 8.5).

X1 � 001��1120��101��

X2

X1

X3

[1100]α½[121]γ

[110]β

[110]γ

[1210]α

[111]β

½[101]γ

[1120]α

[001]β

[0001]α½[111]γ

[110]β

Figure 8.5. Lattice correspondences between �-, �- and �-hydrides.




X2 � 110��1100��121��

X3 � 110��0001��111��

It may be noted here that while studying the precipitation of �-hydride from the�-phase, Weatherly (1981) has noted �-hydride precipitates of two distinct classes:

(a) Type I �-hydride precipitates with{1017

}�

habit plane and the orientationrelationship (111)��(0001)�; [110]��[1210]�.

(b) Type II �-hydride precipitates with{1010

}�

habit plane and the orientationrelationship (001)��(0001)�; [110]��[1210]�.

The lattice correspondences for both type I and type II �-plates could be regardedas crystallographically equivalent if one considers the � to � transition to have twocomponents: the first being the � to � transition in accordance with the Burgerscorrespondence and the second being responsible for converting the � structureto the �-hydride structure. The next two sections have, therefore, been devotedto crystallographic descriptions of the � → � and the � → � transformations,respectively.

8.3.3 Crystallography of � → � transformationIn the crystallographic analysis presented here the orthogonal basis vectorsx1, x2, x3 are taken to be parallel to the directions [001]�, [110]� [110]�, respec-tively. These vectors are shown in Figure 8.5 which depicts how the (110)� planecan be distorted to the (111)� plane. The correspondence matrices relating thechosen axial system (xi) on one hand and the bcc (�) and fct (�) axial systems onthe other, represented by xR

B and xRF, respectively, are given by

xRB =

⎡⎣0 1 1

0 1 11 0 0

⎤⎦ and xR

F =⎡⎣1 1 1

0 2 11 1 1

⎤⎦ (8.11)

The distortion associated with the � → � transformation can be conceptuallydivided into two components, S1 and S2. While S1 is responsible for straining the(110)� plane to match the dimensions of the (111)� plane and for adjusting theinterplanar spacing of the former phase to be equal to that of the latter, the secondcomponent, S2, is necessary for effecting the necessary change in the stackingsequence. The principal distortions, �1, corresponding to S1, can be worked outfrom the lattice correspondence shown in Figure 8.5, to be the following:




A

B

A

B

A

B

A

B

A

B

C

A

[110]β [001]β(110)β

BCCDistorted by S1

(a)

Distorted by S2

(b)

HCP

(0001)α (111)γ

FCT

(c)

Figure 8.6. Simple shear acting homogeneously on every atomic plane.

Along x1 � �1 = a�/2√

2a�

Along x2 � �2 = a�/2√

3a�

Along x3 � �3 = √2a�/

√3a�

The shear, S2, which changes the AB AB � � � type stacking of the distorted(110)� planes (distorted by S1) to the ABC ABC � � � type stacking of the (111)�

planes, is a simple shear acting homogeneously on every atomic plane (Figure 8.6).The shear S2, being a pure shear, can be expressed as:

S2 =⎡⎣1 0 0

0 1 0�35350 0 1

⎤⎦ (8.12)

The total lattice distortion matrix can, therefore, be expressed as S = S1 ·S2 whereboth S1 and S2 are on the basis of the axial system, xi as indicated in Figure 8.5.

In order to consider a specific example, the lattice parameters of the �-hydridephase and of the �-phase in a Zr-20% Nb alloy are substituted for obtaining thelattice strain matrix, S1, which can be expressed as:

S1 =⎡⎣0�9734 0 0

0 1�1322 00 0 1�0856

⎤⎦ (8.13)




110

110

100

110 110

311

301

Habit planepole

(110)trace

Before S1

After S1Undistortedcones

Rigid bodyrotation

Habit planetrace

001

B′

B

Direction of S1

Y

A A′

a,b before S2 and S1A,B after S2A,B after S2 and S1

For S2 = tan 11°ab = A′B′∧ ∧

Figure 8.7. Stereographic projection showing lattice distortion, S1, and the shear, S2 involved inthe � to � transformation.

When S1 is applied to a unit sphere, the set of vectors which remain undistortedgenerate an elliptical cone. The positions of this cone before and after the appli-cation of S1 are represented in the stereographic projection shown in Figure 8.7.It can be seen that the superimposition of a certain fraction of the shear S2 canlead to the fulfilment of the invariant plane strain (IPS) condition for a planedefined by the undistorted vectors a and b as shown in Figure 8.7. The pole ofthis irrational plane is located within the stereographic triangle defined by thepoles (311)�, (100)� and (301)�. This habit plane prediction matches well withthat experimentally determined for the �-hydride precipitates forming within the�-phase matrix in a Zr-20% Nb alloy.

The Wechsler, Liebermann and Read (WLR) methodology of martensite crys-tallography is applied for the � → � transformation for the precise determinationof the habit plane and the magnitude of the lattice invariant shear. The lattice strainS = S1 ·S2 represented with reference to the axial system, xi, can be transformedinto the fct axial system by employing a similarity transformation:




SB = xRBS �BRx�−1 =

⎡⎣0�9088 0�2234 0

0�1768 1�3090 00 0 0�9734

⎤⎦ (8.14)

As mentioned earlier, a superposition of the S2 shear on S1 can generate the�-hydride lattice. If the direction of S2 is reversed, a second variant of the �-hydridecrystal is created. These two �-orientations have a twin relation between them.These two shears, denoted as S+

2 and S−2 are along the following two directions:

S+2 � �110��110��111��121��

S−2 � �110��110��111��121��

Consistent with this fact, the lattice-invariant shear (LIS), has been selected tobe (110)� [110]�, the lattice-invariant shear, P with reference to the axial systemdefined by the shear direction, d, the shear plane normal, n, and the vector, dxn,can be expressed as:

P =⎡⎣1 0 g

0 1 00 0 1

⎤⎦ (8.15)

where, g is the magnitude of LIS. The LIS matrix, PB for a shear along (110)�

[110]� represented in the bcc axial system, will be

PB =⎡⎣0 0 0

0 1 00 0 1

⎤⎦ + g

2

⎡⎣1 1 0

1 1 00 0 0

⎤⎦ (8.16)

The macroscopic shape strain, FB, is then given by

FB = PB ·SB

=⎡⎣ 0�904 0�2137 0

0�1814 1�2991 00 0 0�9734

⎤⎦ + g

⎡⎣0�5427 0�5427 0

0�5427 0�5427 00 0 0

⎤⎦ (8.17)

The IPS condition can be met by adjusting the value of g. In the present case,the magnitude of the lattice invariant shear, g, has been found to be 0.1699 and




substituting this in Eq. (8.17) the average macroscopic shape strain matrix can bedetermined to be

FB =⎡⎣0�9962 0�1215 0�0

0�0892 1�2069 0�00�0 0�0 0�9734

⎤⎦ (8.18)

The strains in three principal directions could be determined by finding theeigen values of the macroscopic strain matrix FB and these are given by:

�1 = 0�9734� �2 = 1�0 and �3 = 1�2131

where �1, �2 and �3 are the strains along the three principal axes. It could bethus seen that �1 < 1, �2 = 1 and �3 > 1 which is the necessary and sufficientcondition for IPS (Lieberman et al. 1957, Wayman 1964). Therefore, an invariantplane strain condition in � → � hydride transformation could be achieved whenthe magnitude of the shear is 0.1699 for the LIS system given by shear S+

2 .Furthermore, applying the WLR matrix methodology (Lieberman et al. 1957,

Wayman 1964), the habit plane or the interface plane (HB) for the correspondencevariant given in Figure 8.5 and for the LIS given by shear S+

2 corresponding totwo opposite shear directions are found to be the same:

HB�+� = �−0�0850� 2�9968� 1�0000� and HB�−� = �−0�0850� 2�9968� 1�0000��

Following the method of comptuation described in WLR theory (Liebermanet al. 1957), the other relevant crystallographic parameters for the given shearsystem were calcualted and the results are presented in Table 8.2.

Table 8.2. The computed crystallographic parameters of � → � transformation, for correspondencevariants shown in Figure 8 and first shear system.

Crystallographic parameters Computed values

Positive solution Negative solution

Shape strain matrix

⎡⎣ 1�0001 −0�0059 −0�0020

−0�0059 1�2098 0�07000�0025 −0�0873 0�9709

⎤⎦

⎡⎣ 0�90376 −0�07302 0�06930

0�07084 1�05375 −0�05101−0�05749 −0�04362 1�04140

⎤⎦

Shape strain shear direction[0�0089 −0�3163 0�9486

] [0�0089 −0�3163 −0�9486

]Shape strain direction

[−0�0262 0�9230 −0�3839] [

0�0262 −0�9230 −0�3839]

Magnitude of shape strain 0.2398 0.2398




It is to be noted that the LIS chosen here are the same as S+2 and S−

2 and that theLIS magnitude of (0.1699) corresponds to nearly half of the shear S2. This meansthat a �-plate which consists of two twin related �-variants (one correspondingto S+

2 and the other to S−2 shear) with the ratio of their thicknesses equal to 1:3

will satisfy the IPS condition. Experimentally, a twin thickness ratio of 1:3 isoften observed in �-plates containing (111)� twins, the twin plane being derivedfrom the parent (110)� plane. The habit plane solutions obtained from the WLRanalysis have been found to be of the (130)� type which also match closely withthe experimentally determined habit plane indices. From these observations it isclear that the phenomenological theory of martensite crystallography can be usedfor the prediction of crystallographic observables such as the habit plane and themagnitude of lattice-invariant shear (or ratio of twin thicknesses) in the context ofthe precipitation of �-hydride plates in the �-phase matrix (Srivastava et al. 2005).

8.3.4 Crystallography of � → � transformationThe crystallography of the � → � transformation and the internal structure of�-hydride precipitates in the �-matrix have been studied in detail by Weatherly(1981), who has pointed out that there are two distinct orientation relationshipsand habit planes exhibited by �-hydride precipitates in the �-matrix. These aregrouped as type I and type II, the crystallographic details of which are given inTable 8.3.

The hcp �-structure can be transformed into the fcc �-structure by altering thestacking sequence from the ABABAB to the ABCABCABC type .

The lattice strain required for transforming the hcp �-lattice to an fcc lattice(an imaginary precursor to the �-hydride phase) consists of a small dilation of the(0001)� planes (to increase the c/a ratio from 1.593 to the ideal value of 1.633) anda simple shear to change the stacking sequence. This fcc structure (with a latticeparameter of 0.4575 nm can be transformed into the fct �-hydride structure (a =0�4596 nm and c = 0�4969 nm by a second homogeneous deformation. The overalllattice strain, referred to the axes [1210]�, [1010]� and [0001]� is

Table 8.3. Orientation relationship and habit planes for Type I andType II �-Hydride precipitation in �-Zr.

�-Hydride Habit plane Orientation relationship

Type I {1010 }� [110]��[121 0]�; (001)��(0001)�

Type II {1017 }� [110]��[1210]�; (111)��(0001)�




SH =⎡⎣1�0048 0 0

0 1�0592 0�33560 −0�0471 1�0308

⎤⎦ (8.19)

It may be noted here that the �/� correspondence observed for the type II�-hydride precipitates is crystallographically equivalent to that depicted inFigure 8.5. The fact that the lattice strain along one of the principal axes isless than 0.5% allows one to neglect the strain along the [1210]� direction foran approximate analysis which can be reduced to two dimensions. Weatherly(1981) has shown from the construction of the unit circle and of the deformedellipse that the IPS condition is satisfied for habit planes which are indicated bythe undistorted vectors (AB) and after rigid body rotation (A′B′) in Figure 8.7which corresponds to the [1210]� zone axis. As seen in this construction, twopossible habit planes are predicted, both having a normal lying along the [1210]�

direction. The habit plane normal for the first solution lies approximately 12�

from the [0001]� pole and a rigid body rotation of 3� is required to bring theundistorted plane back to its original orientation. For the second solution, thehabit plane normal lies 16� away from a (1010)� pole and a rigid body rotationof 20� is required. The experimentally observed habit of (1017)� makes an angleof about 14� with (0001)�, indicating the validity of martensitic crystallographyin the case of �-hydride precipitation in �-Zr.

�-hydride plates of type I which exhibit the{1010

}type habit plane have

been found to follow one of the two orientation relations listed in Table 8.3. For�-plates corresponding to the first orientation relation, (001)��(0001)�, internaltwins are seen along two sets of �110�� planes which are obliquely inclined tothe habit plane. In contrast, for �-plates following the second orientation relation,(111)��(0001)�, the twin planes are parallel and perpendicular to the habit plane.

The transformation mechanism associated with �-plates exhibiting{1017

}�

habit planes (type II), involves a simple shear of 30� on a{1010

}�

plane ina < 1210>� direction. It is, however, interesting to note that the lattice corre-spondences for the �-plates of type I and type II could be regarded as crystal-lographically equivalent if one considered the � to � transition to occur in twoconceptual steps: the first step being the � to � transition in accordance withthe Burgers correspondence followed by the second step comprising the � to� transition as discussed in the preceeding section. A stereographic projectionshowing the approximate orientation relations between � and � through an inter-mediate � orientation brings out the equivalence of two lattice correspondences(Figure 8.7). It could be seen from this stereographic projection that for the typeI plates, the (110)� plane which is derived from the (0001)� plane becomes the




(111)� plane, whereas for the type II plates, other �011�� planes are converted to�111�� planes.

8.3.5 Mechanism of the formation of �-hydridesThe �-hydride phase precipitates form in both the �- and the �-alloys at highquenching rates. Alloys containing hydrogen to a level exceeding the terminalsolubility, will invariably produce �-hydride precipitates. The charcteristic fea-tures of �-hydride precipitates, namely, surface relief on polished surfaces, strictorientation relationship and habit plane, operation of the invariant plane strain con-dition, periodic internal twinning and rapid transformation kinetics suggest that thetransformation involves a lattice shear. The fact that the transformation occurs atsufficiently low temperatures where self-diffusion of zirconium (or titanium) atomsis negligible also suggests that the transformation of the �- or the �-lattice intothat of �-hydride does not involve diffusive jumps of the metal atoms. However,hydrogen migration which is known to be very fast even at such low temperatures(<550 K) is necessary for the attainment of the right stoichiometry and interstitialordering for the formation of the �-hydride structure. Considering all these factors,the precipitation of �-hydride in either the �- or the �-matrix can be approximatelydescribed as a hybrid transformation involving a shear transformation of the parentZr-lattice with the accompanying process of hydrogen partitioning and interstitialordering. In this way, the �-hydride precipitation can be grouped in the class ofBainitic transformation.

In view of the close lattice correspondence of the �-, the �- and the �-structures,the fact that hydrogen is a strong �-stabilizer and the crystallographic observationsreported by Dey et al. (1982), Dey and Banerjee (1984) and Srivastava (1996,2005), it is attractive to envisage that the �-hydride formation in the �-phase occurthrough an intermediate �-step. It is possible that as hydrogen segregation occursin certain localized regions within the �-phase, these microscopic regions firsttransform into the �-structure which finally goes over to the �-hydride structure toaccommodate the large concentration of hydrogen atoms. Since direct experimentalevidence in support of this step wise transformation, � → � → �, has not beenobtained, the size of the embryonic �-phase appearing in the path of the � → �transformation cannot be ascertained.

8.3.6 Hydride precipitation in the �/� interfaceTransmission electron microscopic examiantion of electropolished thin foils of�+� titanium alloys has revealed the presence of an interface phase at �/� inter-faces (Rhodes and Williams 1975, Unnikrishnan et al. 1978). This interface phasehas been found to possess fcc structure with a lattice parameter of 0.44 nm. It hasalso been observed that this fcc phase is present only when the �-phase has evolved




(a) (b)0.3 μ0.3 μ

Figure 8.8. �-hydride precipitates enveloping � precipitates in the �-matrix.

through a diffusional transformation from the parent �-phase and not along themartensitic �′ interface. The observation of the fcc phase in a wide variety of �/�titanium alloys suggests that its origin is fundamentally connected with the natureof the bcc → hcp transformation. Later experiments (Banerjee and Arunachalm1981, Banerjee and Williams 1983) have conclusively proved that the �/� interfacephase is �-hydride which forms during the electropolishing operation (Figure 8.8).

The detailed crystallographic analysis reported by Banerjee and Arunachalm(1981) has shown that the �-Hydride phase bears two distinct types of orientaitonrelationship with the �-phase hydride plates at the �/� interface are designatedas type I and type II, based on the following orientation relationship they followwith the �-phase:

Type I � �110��1100��001��0001��

Type II �111��0001��110��1120��

Morphologically these two types of �/� interface hydride can be distinguished asthe type I hydride forms a monolithic layer along the interface while the type II pre-cipitates are usually finely divided and are stacked along the faults in the �-phase.

Let us examine the possible causes for the formation of a �-hydride envelopearound the �-plates in a �-matrix. The large difference in the solubility of hydrogenin the �- and the �-phases is one of the contributing factors. During electropolishingof samples, hydrogen ingress occurs to a considerable extent and once the limitof solubility is exceeded, hydride precipitation commences. A higher solubility ofhydrogen in the �-phase is expected to provide a preponential path for hydrogen




entry into the sample through the �-phase region. The �-phase region in contactwith the �-phase, therefore, attains the limit of solubility at the earliest instance.The other factor which is responsible for the preferential �-hydride formation atthe �/� interface is the reduced nucleation barrier for �-hydride at such locations.This point is explained considering the crystallographic mismatch associated withthe type I and the type II �-hydride at the �/� interfaces.

Let us consider first type II hydrides which form along the basal planes ofthe �-phase. The mismatch along the habit plane is a uniform dilation of 3.3%and a large dilation of 8.3% perpendicular to the habit plane. The presence ofc-dislocations along the �/� interface, necessary for compensating misfits on{1100

}�� 112�� interface planes, facilitate accommodation of the large dilation

perpendicular to the habit plane of hydrides which nucleate at a periodic intervalsin the �-side of �/� interfaces.

The accommodation of the type I hydride at the interface is not immediatelyobvious. Banerjee and Arunachalam (1981) have argued that the 17% dilationin the direction normal to the habit, which offers the main constraint againstthe nucleation of type I hydride, can be accommodated if the �/� interface isatomistically rough. The presence of ledges on the �/� interface appear to beresponsible for inducing nucleation of type I hydride phase.

The formation of hydrides at the �/� interfaces in samples undergoing a � → �diffusional transforamtion can be visualized as follows. As the � → � transfor-mation proceeds, partitioning of alloying elements occurs. Since the difference insolubility of hydrogen in the �- and the �-phase is large, the �-regions at theinterface becomes supersaturated with hydrogen resulting in hydride precipitation.Since hydrogen solubility drops down drastically at temperatures below 573 K,hydride precipitation occurs only at such low temperatures in samples containingvery low hydrogen level (<50 ppm).

8.3.7 Formation of �-hydrideThe �-hydride of Zirconium is the equilibrium phase with fcc structure havinglattice parameter a = 0�4775 nm. As mentioned earlier, hydrogen atoms occupysome of the tetrahedral interstitial sites � 1

414

14� and equivalent positions in the fcc

lattice.The radius of the site is about 0.0378 nm and there are two such sites perzirconium atom (Northwood and Kosasih 1983).

A number of studies involving electron microscopy, dilatometry and X-raydiffraction have shown that the metastable �-hydride phase is favoured under afast quenching rate while with ageing at temperatures in the range of 150–250�C�-hydride precipitates transform into �-hydride precipitates in an irreversible man-ner. During continuous cooling from an elevated temperature, the formation ofthe equilibrium �-hydride precipitates occur preferentially when the cooling rates




are slow, particularly during the nucleation stage. The volume change associatedwith the formation of the �-hydride from the � Zr phase is about 17.2%. This issubstantially larger than that for the �-phase formation in � Zr (∼12.3%). Theformation of the �-hydride is also promoted as the hydrogen level in the alloy isincreased. This is reflected in the time–temperature–transformation diagrams forthe formation of �- and �-hydrides in Zr–H alloys of different hydrogen concen-trations (Figure 8.9).

10–2 10–1 10 10 102 103 104

Time (sec)

0

473

673

0

473

673

0

473

673

Tem

pera

ture

(K

)

(a)

(b)

(c)

αZr

δ

γ

αZr

δ

γ

δ

cH = 640 ppm

cH = 500 ppm

cH = 50 ppmαZr

γ

Figure 8.9. Time–Temperature–Transformation diagram for �- and �-hydride formation in Zr alloysfor hydrogen concentrations of (a) 640 ppm, (b) 500 ppm and (c) 50 ppm.




The orientation relationship between the �- and the �-hydride phase has beenfound by Northwood and Kosasih (1983) and Une et al. (2004) to be the following:

�111��0001�� 110��1120��

8.4 HYDROGEN-RELATED DEGRADATION PROCESSES

Zirconium alloys which are extensively used in structural components in nulcearreactors absorb hydrogen during service. A stringent control of hydrogen level ismaintained in these alloys during the entire processing route starting from the pro-duction of zirconium sponge followed by alloy melting and fabrication. In fact, recentspecifications of some components such as pressure tubes allow a maximum of 5 ppmof hydrogen in the finished products. It is the in-service absorption and accumula-tion of hydrogen which cause serious degradation of mechanical properties of thesecomponents. The problems resulting from hydrogen ingress into zirconium alloys getfurther aggrevated due to migration of hydrogen leading to localization at points oftensile stress concentration and at relatively cooler regions where the terminal solubil-ity is substantially lower. The severity of the hydrogen-related degradation processescan be recognized from the fact that they are responsible for limiting the service lifeof several zirconium alloy components in nuclear reactors.

Deuterium and/or hydrogen is picked up by zirconium alloy components in anuclear reactor from the following sources:

(1) Deuterium/hydrogen evolved in the corrosion reaction

Zr +2D2O → ZrO2 +4D

(2) Radiolytic decomposition of D2O/H2O

D2O → D+OD

(3) Moisture present in the fuel – UO2 pellet which is encapsulated in the claddingtubes made of zirconium alloys;

(4) Hydrogen added in the coolant water for scavenging nascent oxygen evolveddue to radiolytic decomposition

2OD → D2O+O

(5) Hydrogen/deuterium produced in the corrosion reaction between zirconiumalloy components and the moisture present in the annulus gap between calan-dria tubes and pressure tubes in pressurized heavy water reactors.




Hydrogen ingress in zirconium alloy components occurs along the entire surfacearea of these components which remain in contact with coolant or fuel. Themobility of hydrogen at operating temperatures of nuclear reactors is quite high andtherefore, apart from embrittlement due to uniform hydride precipitation severalother dynamic processes, such as stress reorientation of hydrides, localized hydrideblister formation at cold spots and delayed hydride cracking can result in seriousdegradation of the material.

8.4.1 Uniform hydride precipitationAs long as hydrogen remains in solid solution, it does not cause any deleteriouseffect on mechanical properties of zirconium alloys. As the hydrogen concentra-tion exceeds the TSS limit, precipitation of hydride plates occurs in the matrix.Because of the strain associated with the hydride formation, hydride precipitatesacquire a plate shape. The bainite-like transformation mechanism is responsiblefor sheafs of contiguous hydride plates as shown in Figure 8.10 (Singh et al.2006). A uniform distribution of the embrittling hydride phase manifests in theform of reduction in tensile ductility, impact and fracture toughness. However,for a significant reduction in these properties, certain minimum volume fractionof the brittle hydride phase is required.

The degree of embrittlement strongly depends on the hydride plate orientationwith respect to the direction of applied tensile stress. The control of crystallographic

Figure 8.10. Sheafs of contiguous hydride plates formed in Zr–2.5 Nb alloy.




texture of zirconium alloy components is, therefore, of utmost improtance incombating hydride embrittlement. The factors which control the orientation ofhydride plates are as follows:

(a) Crystallographic texture of the major � phase in the fabricated componentsdictates the orientation of hydride plates which exhibit a microscopic habit of{1017

}�

(Northwood and Gilbert 1978) making an angle of about 14� withthe basal plane of the hcp � Zr and a submicroscopic habit of (0001)� (Uneet al. 2004).

(b) Presence of applied or residual stress determine the plane of hydride precipi-tation which occurs on the plane normal to the tensile stress direction.

By suitable control of fabrication parameters of tubular products, particularly theQ-factor which is defined as the ratio of the wall thickness reduction to thediameteral reduction, the desired crystallographic texture can be introduced. Sincea high Q-factor orients the basal poles in the radial direction, thin walled tubesare pilgeried with a high wall thickness reduction. Hydrides forming in such tubesremain primarily circumferential which are least deleterious as far as supportinga hoop stress in the tube is concerned. In tube burst test, the total circumferentialelongation, which is a measure of ductility, has been found to be significantlyreduced with an increase in the volume fraction of hydrides. In view of theimportance of unfavourably oriented hydride plates in thin walled tubes, the ratioof ‘radial’ hydride lengths and total hydride lengths given by a parameter calledFn is monitored as a part of technical specifications. The value of Fn is alwayskept low for the acceptance of such tubes for fuel cladding applications.

In tubular products of larger wall thickness, such as pressure tubes, the orien-tation of uniformly distributed hydride precipitates changes across the thicknessof the wall. This variation is primarily due to the variation in microtexture andmicrostructure and residual stress across the wall thickness. Though microstructureand texture of pressure tubes are carefully controlled for pressure tubes giving dueconsiderations of hydride precipitate orientations, acceptance criteria usually donot include the Fn number specification.

8.4.2 Hydrogen migrationHydrogen migration at operating temperatures (∼300�C) of nuclear reactors isresponsible for restructuring of hydride precipitates. Phenomena such as stressreorientation of hydrides, formation of hydride blisters and delayed hydride crack-ing which are serious causes of material degradation are all outcome of in-servicehydrogen migration. Studies on hydrogen migration have revealed that hydrogenmigrates down the concentration and temperature gradient and up the tensile stress




gradient in zirconium (and also in titanium) alloys. The general diffusion equation,in one dimension, considering all the three gradients is given by (Northwood andKosasih 1983):

J = −Dcx

RT

[RT

d ln cx

dx+ Q∗

T

dT

dx−V ∗ dP

dx

](8.20)

where J is hydrogen flux, D hydrogen diffusivity in metal, (= 0�217 exp−�8380/RT� mm2/s), cx hydrogen concentration at point x, Q∗ and V ∗, heat andvolume of transport of hydrogen in metal and P is the hydrostatic stress.

Two distinct cases, namely, thermal migration and stress directed migration canbe considered using the generalized diffusion equation for hydrogen migration.In absence of a stress gradient, thermal migration, governed by the concentrationand the temperature gradients, occurs. Let us consider a case where the bulkhydrogen concentration is more than the TSS of hydrogen at Tmax, the maximumtemperature of the sample. Imposition of a temperature gradient on such a samplewill introduce a concentration gradient. This is due to the fact that hydrogen atomsparticipating in the diffusion process are those which remain in solid solutionand not those which are locked up in the hydride phase. At the low temperatureend, more of hydride phase will be precipitated and level of hydrogen in solidsolution will be given by TSS at that temperature. This will result in setting upof a hydrogen flux from the high temperature to the low temperature end of thesample. There exists, however, a critical hydrogen concentration called thermalmigration threshold for a given thermal boundary condition, below which thereis no net migration of hydrogen. At this concentration of hydrogen, a dynamicequilibrium is established in which the hydrogen flux due to concentration gradientexactly balances the opposing flux due to thermal gradient. This condition can bedescribed as:

J = −Dcx

RT

(RT

d ln cx

dx+ Q∗

T

dT

dx

)= 0 (8.21)

Solving this equation, one can find out the threshold hydrogen concentration,which is also known as blister formation threshold, as the formation of hydrideblisters at cold spots can occur when this threshold is exceeded. This phenomenonis discussed in detail in Section 8.4.5.

The migration of hydrogen can also occur when only a stress gradient is present.Hydrogen has a tendency to migrate up the tensile stress gradient. Such stress-directed migration has serious implications in the processes like stress inducedhydride reorientation and delayed hydride cracking.




As in the case of thermal migration, stress-directed migration of hydrogen isalso associated with a threshold hydrogen concentration, cth

x , which is determinedby the dynamic equilibrium given by the following equation:

J = −Dcx

RT

(RT

d ln cx

dx+V ∗ dP

dx

)= 0 (8.22)

The stress-directed migration is manifested into processes such as stress-inducedreorientaion of hydride precipitates and migration of hydrogen to locations of hightensile stress concentration leading to delayed hydride cracking. These processesoften encountered during service are sometimes responsible for limiting the servicelife of a zirconium alloy component in a nuclear reactor.

8.4.3 Stress reorientation of hydride precipitatesAs mentioned earlier, the crystallographic texture of zirconium alloy componentsis carefully controlled so that the hydride precipitates are oriented favourably withrespect to the major stress direction. The initial distribution of hydride precipitatesmay undergo appreciable changes due to the dissolution and reprecipitation ofhydride plates during service. Since the TSS of hydrogen increases exponentiallyas the temperature is raised from ambient to the reactor operating temperature, alarge fraction of hydride precipitates dissolve on heating. Subsequently, they mayprecipitate under stress during cooling in grains which have basal planes nearlyperpendicular to the major tensile stress direction.

A threshold value of stress must be exceeded for the reorientation to occur. Thethreshold stress increases with the yield strength of the material but decreases withincreasing solutionizing temperature (Northwood and Kasasih 1983). Experimen-tal evaluation of threshold stress can be made using specially designed taperedtensile specimens (Figure 8.11(a)), which when strained along its axis produces agradient in tensile stress along its length (Singh et al. 2004). The stress level atwhich reorientation of hydrides can be observed gives the threshold stress value(Figure 8.11b). It has been reported that the degree of reorientation is insensitive totime under stress and to small variation in texture, grain size and dislocation den-sity (Northwood and Kosasih 1983). The threshold stresses at the typical reactoroperating temperature of 300�C for hydride reorientation are reported to be about180–250 MPa and 90–120 MPa for cold worked and stress relieved Zr–2.5 Nband cold worked and stress relieved zircaloy-2, respectively (Lietch and Puls1992). For the pressure tubes, if the hoop stress in service exceeds the thresholdstress value, reorientation of hydride plates can cause a shift of preference fromcircumferenial to radial orientation of hydrides.




22

38

63

91

(a)

(b)

All dimensions in mm

37 22

134.7 142.3 152.6 157.5Inside diameter of the tube

160.7 168.7

600 μm

Figure 8.11. (a) Tapered gage tensile specimen. (b) Montage showing hydride orientation as afunction of stress (MPa).

8.4.4 Delayed hydride crackingDelayed hydride cracking is a form of localized hydride-embrittlement phe-nomenon, which in the presence of a tensile stress field manifests itself as a sub-critical crack growth process. It is caused due to hydrogen migration up the tensilestress gradient to the region of stress concentration. Once the solid solubility limitis exceeded in the region of localized stress concentration, brittle hydride platesprecipitate normal to the operating tensile stress and these plates continue to growtill a critical size is formed. At this stage, hydride plates of critical size cracks underconcentrated stress leading to the growth of the crack. This crack growth is delayedby the time required for hydrogen to reach the crack tip and form hydride plateletof critical size. Hence this phenomenon is called delayed hydride cracking (DHC),which is characterized by the crack growth rate called DHC velocity (VDHC).




Based on the important processes occurring, the region ahead of a crack tip fora sample or a component of a zirconium alloy experiencing DHC can be dividedinto three zones (Singh et al. 2002a). These are process zones where the stress� is larger than the yield stress, �o, of the material, a reorientation zone where�o > � > �th, the threshold stress for reorientation of hydride being �th and amigration zone where � < �th. The processes which operate in these zones areschematically illustrated in Figure 8.12.

Experimental observations of the DHC phenomenon have revealed the followingimportant features (Singh et al. 2002a):

(1) DHC crack initiation is associated with an incubation period and fracturesurface is usually marked with striations (Figure 8.12(b)).

(2) The initiation of DHC is associated with a threshold stress intensity factor,KIH, below which the DHC crack growth velocity, VDHC, is too small to bedetected;

(3) KIH is practically independent of material strength;(4) For a given material and for KIH < KI < KIC, VDHC is independent of the

applied stress intensity factor, KI (where KIC is the fracture toughness of thematerial);

(5) VDHC increases with an increase in the strength of the material;(6) VDHC increases with temperature due to increased hydrogen diffusivity and

solubility at higher temperatures.

Though softening of material causes an increase in the process zone size, theincreased hydrogen diffusivity has a dominating effect on VDHC. The variation ofVDHC with temperature is reported to exhibit an Arrhenius type relationship, theactivation energy for DHC being in the range of 40–70 KJ/mole which correspondsto the sum of the activation energy of hydrogen diffusion (30 kJ/mole) and theenthalpy of mixing of hydrogen in zirconium (38 kJ/mole).

8.4.5 Formation of hydride blistersIn the presence of a thermal gradient, hydrogen migrates down the temperaturegradient. Once the solid solubility at the localized low temperature spot (hereforthcalled cold spot) is exceeded, hydride precipitation starts. A sustained thermalgradient allows continuation of hydrogen migration from the high temperatureto the low temperature region as a steady state concentration gradient persistsunder such a condition. The volume fraction of the hydride phase at the coldspot increases to near 100% and a semiellipsoidal module of hydride forms at thecold spot. Since the transformation of zirconium metal into hydride is associated




rp

Distance (r)

Processzone

Reorientationzone

Migrationzone

Modified stress distributionelastic + plastic

σth

σo

σy σy = K

√2πrS

tres

s (σ

)

(a)

80 μm

(b)

Figure 8.12. (a) Schematic representation of three zones ahead of a sharp crack. (b) Fracture surfaceshowing striations resulting from stepwise growth of delayed hydride cracks.




with an increase in volume, a bulge appears on the surface of the cold spot. Thishydride bulge, due to its appearance, is called a hydride blister.

The phenomenon of blister formation attracted special attention of the nuclearengineering community after the failure of a zircaloy-2 pressure tube in a pressur-ized heavy water reactor, PHWR (Pickering - Unit 2) in Canada. In the PHWRdesign, several hundred pressure tubes are arranged in a lattice in a large vesselcalled calandria (Figure 8.13). Each of these pressure tubes which contains a seriesof fuel bundles pass through calandria tubes, the gap between a pressure tube andthe surrounding calandria tube is maintained by a set of spacers known as gartersprings. Pressure tube carries coolant water at around 300�C whereas the calan-dria tube remains in contact with moderator water maintained at a temperatureof 60–80�C. Pressure tubes have a tendency of sagging because of the deflectiondue to the fuel load and due to the in-reactor creep. When the sagging of thepressure tube results in the establishment of a contact between a pressure tube anda calandria tube, the contact points become cold spots (as shown in Figure 8.14(a)and (b)) which are susceptible to hydride blister formation.

Figure 8.13. Lattice arrangement of pressure tubes and calandria tubes in pressurized heavy waterreactors.




Presure tube Calendria tube80°C

300°C

Garter spring

(a) Before creep deformation

(b) After creep sagging

Contact point

80°C

300°C

OD of the pressure tube

O

380 μm

Region III

Region II

Region I

Circumferential direction

Radial direction

SY

Y - S

TR

ES

SE

S

VIE

W

: –54.10852R

AN

GE

: 110.4911

110.5

98.73

86.98

75.22

63.46

51.71

39.95

28.19

–7.080

4.677

16.43

–18.84

–30.59

–42.35

–54.11RO

TX

EM

RC

–NIS

A/D

ISP

LAY

MA

R/05/01 15:46:26R

OT

Y

RO

TZ

0.0

0.0

0.0

Y

X

(c)

(d)

Figure 8.14. (a) and (b) show pressure tube/calandria tube contact creating cold spot on the power.(c) A section of hydride blister showing both the radial and circumferential hydrides. (d) Computedstress field inside the blister and the matrix surrouding it.




The parameters which influence the growth rate of hydride blisters are thedifference in temperature between the bulk of the material and the cold spot,thermal conductivity at the contact point and the hydrogen content of the material.Larger the temperature difference and sharper the thermal gradient, higher theblister growth rate. A higher hydrogen level in the bulk of the material no doubtpromotes the growth of blisters.

Metallographic examination of the section of a hydride blister along theradial-circumferential plane of a pressure tube shows three distinct regions(Figure 8.14(c)), viz. the core region where the hydride volume fraction is nearly100%, the reoriented hydride region where a large fraction of hydride plates arereoriented from their original habit to directions radial to the semiellipsoidal blisterand the region where hydride precipitates retain their original habit. The stressfield present around a growing blister is responsible for reorienting hydrides inthe region adjacent to the blister (Figure 8.14(d)) (Singh et al. 2003).

Two distinct types of blister morphology have been encoutnered, namely (a) asingle convex hump; and (b) a nearby circular ring which is made up by joininga series of independently nucleated blisters (Singh et al. 2002b). The former typeis formed when all the hydrogen present in the alloy is first brought into solidsolution and the cold spot is created subsequently. Under this condition, blisternucleation occurs right at the cold spot. In any case, these blsiters cannot growbeyond a size where their periphery touches the temperature corresponding tothe TSSP of hydrides. The latter type (ring morphology) is formed when thecold spot is maintained during the heating-up operation. As a consequence, thehydride precipitates which remain undissolved in the vicinity of the cold spot act asnucleating centres for blister formation. As the hydrogen flux is directed towardsthe cold spot, hydrogen atoms are arrested at the point where undissolved hydrideprecipitates are present. The ring of blisters, therefore, mark the temperaturecontour corresponding to the TSS limit for hydride dissolution under the imposedthermal gradient. In this context it may be emphasized that the TSS limit forprecipitation and for dissolution of hydrides in �-zirconium are different and aredenoted by TSSP and TSSD, respectively. While TSSP is the controlling factorfor the single blister morphology, the ring type blister formation is governed byTSSD (Singh et al. 2002b).

For the nucleation of blisters, a minimum hydrogen concentration, cBFT, calledblister formation threshold is required. However, cBFT is dependent on the TSS,c∗

TSS, at the cold spot temperature. The relationship of these concentrations with thebulk and the cold spot temperatures can be derived from the steady state conditionin absence of stress, where the net flux J of hydrogen is given by Eq. 8.21.




This partial differential equation can be reduced to

1cx

dcx

dT+ Q∗

RT 2= 0 (8.23)

Separating the variables and on integrating between the bulk dissolved hydrogenconcentration, cb and c∗

TSS at the cold spot temperature, T ∗ we get

∫ c∗TSS

cb

dcx

cx

= −Q∗

R

∫ T∗

Tb

dT

T 2(8.24)

which reduces to

cBFT

c∗TSS

= exp(

−Q∗

R

�Tb −T ∗�Tb ·T ∗

)(8.25)

The ratio of the two concentrations cBFT/c∗TSS, increases with an increase in the

cold spot temperature, T ∗, as is shown in Figure 8.15 where the bulk temperature,Tb is maintained at 573 K (Singh et al. 2002b). Since cTSS depends on the directionof approach of temperature (i.e. TSSP and TSSD), cBFT will also vary dependingon whether it is measured during a heating or a cooling operation.

50

40

30

20

10

0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

250 300 350 400 450 500 550

Tcs (K)

TSSDTSSPc BFT

/c TSS

c T

SS (

ppm

)

c B

FT/c

TS

S

Figure 8.15. Hydrogen concentration corresponding to blister formation threshold (cBFT) as a func-tion of the cold spot temperature, Tcs.




8.5 THERMOCHEMICAL PROCESSING OF Ti ALLOYSBY TEMPORARY ALLOYING WITH HYDROGEN

Titanium and most of its commercial alloys have a high solubility for hydrogen,being capable of absorbing upto about 6 at.% hydrogen at 600�C. Since hydrogenentry in titanium alloys is reversible, temporary alloying with hydrogen can bevery effectively used in stabilizing the �-phase which in turn assists in refinementof the microstructure and in improvement of fabricability of these alloys. Since thealloy chemistry is changed during the processing operation, the technique is knownas thermochemical processing. The basic principle of this processing technique isdiscussed here as an example where hydrogen-induced phase transformations havebeneficial effects. After the processing is completed, hydrogen can be extractedfrom the alloy by vacuum annealing.

The concept of constitutional solution treatment can be explained using apseudo-binary phase diagram for Ti–6Al–4V with hydrogen (Figure 8.16). Sincehydrogen stabilizes the �-phase, it is possible to solutionize Ti–6Al–4V into asingle phase �-solid solution by hydriding at a temperature above the eutectoidtemperature (Te) of 800�C. This hydrogenated alloy (containing 0.5–1 wt% H) oncooling transforms into either �+TiH or Orthorhombic martensite (�′′) dependingon the cooling rate. The latter on subsequent ageing decomposes into a mixture of�+TiH. The large volume expansion associated with hydride formation results inan accumulation of high density of dislocations in the matrix. When this material isdehydrogenerated, a fine �-phase forms by the decomposition of the hydride to amixture of � and spherodized �-phase. The low transformation and dehydrogena-tion temperatures and strain in the matrix are responsible in substantial refinementof the microstructure and consequent improvement in mechanical properties.

1273

1073

873

10 20 30 40

Hydrogen (at.%)

Tem

pera

ture

(K

) β

β + X

X

α + β

α

Hydrogen (wt%)0.80.40.2 0.6 1.0 1.2

α + X

Figure 8.16. Ti-6V-4A1 + H pseudo binary phase diagram.




Table 8.4. Thermochemical treatment cycles.

Designation Treatment sequences

Constitutional solutiontreatment

Hydrogenate at T > Te → cool to T < Te →dehydrogenate

�-quench followed byhydride-dehydride

�-solution treatment → water quench → hydrogenate atT < Te and dehydrogenate at T < Te

Hot isostatic pressing orvacuum hot pressing

Hydrogenate to increase �-volume fraction → hotdeformation → dehydrogenate

High Temperaturehydrogenation

Hydrogenate in � → cool to room temperature →dehydrogenate at T < Te

A variety of thermochemical processing cycles, as listed in Table 8.4, have beentried. Froes and Eylon (1990) have summarized the effects of these thermochemicaltreatments on the microstructure and mechanical properties of titanium alloys.

8.6 HYDROGEN STORAGE IN INTERMETALLIC PHASES

8.6.1 Laves phase compoundsHydrogen storage materials (HSMs) are usually intermetallics containing inter-stices with a suitable binding energy for hydrogen which allows its absorptionor desorption near room temperature and atmospheric pressure. A promising can-didate for hydrogen storage is the class of Laves phase compounds having theformula unit of AB2. Laves phase include fcc (MgCu2), hexagonal C14 (MgZn2)and di-hexagonal C36 (MgNi2) structures. Since they transform from one to anotherduring heating and cooling (typically C14 at high temperatures and C15 at lowtemperatures), hydrogen absorption and desorption can essentially be consideredas a phase transformation process. The C15 structure is an fcc-based structurecontaining six atoms (two formula units) in the primitive unit cell, while C14and C36 structures are hexagonal structures containing 12 and 24 atoms in theprimitive unit cell, respectively. The interstitial sites occupied by hydrogen inAB2 Laves phases are tetrahedral sites formed by two A and two B atoms (2A2Bsite) or by one A atom and 3 B atoms (1A3B site) or four B atoms (4B site).There are 17 tetrahedral sites per formula units: twelve 2A2B sites, four 1A3Band one 4B site for both C14 and C15 structures. While in C15 structures, A orB atoms within each type of sites (2A2B, 1A3B and 4B) are locally equivalent,this is not the case for C14 structure. Depending on the local environment of thetetrahedral interstice, 12 2A2B sites and 4 1A3B sites per formula units in the




(a)

Y

XZ

Active

(b) (c)

2

4

7

3

8

69

1

5

Figure 8.17. (a) B2 crystal structure, (b) Crystal structure of Laves phase compounds C14 (hexa-gonal) open circles stand for A atoms and solid circles for B atoms and (c) the C15 Laves structure,where the larger and the smaller circles represent A (Zr) and B (X = V, Cr, Mn, Fe, Co, Ni) atoms,respectively. Here, for example, a 2A2B site is formed by the atoms labeled 2, 4, 7 and 8; a 1A3Bsite by atoms 1, 6, 8 and 9; and a 4B site by atoms 6, 7, 8 and 9.

C14 structures are further sub-grouped into six 2A2B (I) sites, three 2A2B(k2)sites, 1.5 2A2B(h1) sites, 1.5 2A2B(h2), one 1A3B(f) site and three 1A3B (k1)sites. To depict the interstitial sites formed by A and/or B atoms, a ball stickmodel of the fcc structure is given in Figure 8.17. The 2A2B sites has thelargest interstitial hole size and the 4B site has the smallest hole size (Hong andFu 2002).

The volume change associated with hydrogen absorption could be as large as+20%. Such a large volume change during hydrogen absorption and desorptionresults in cyclic loading and unloading of the matrix which may lead to disinte-gration of the host lattice.




8.6.2 ThermodynamicsMost of the thermodynamic properties of a HSM can be obtained from pressure–composition isotherms (PCI). Figure 8.18(a) illustrates a typical PCI diagram(Sinha et al. 1985). The points of intersections of mildly sloping plateau with thesteeply sloping portion of an isotherm define the phase boundaries �/��+�� and(�+��/�, where � is the solid solution of hydrogen in the host alloy and � is thehydride of the alloy. The plateau (mildly sloping region) pressure for formationof hydrides (Pf ) is higher than that corresponding to dissociation of hydrides (Pd)due to hysteresis effect. The degree of hysteresis is expressed by the logarithmicfunction (1/2) RT 1n �Pf/Pd� which is the free energy loss per mole of atomichydrogen in completing a hysteresis loop. With increase in temperature, plateaupressure increases. Equilibrium pressures in the plateau region and its slope areimportant in deciding the application of HSM. Using Vant Hoff equations:

ln Pi =�Hi

RT− �Si

T

where i stands for f (formation) or d (dissociation) of hydrides, the enthalpy offormation and dissociation can be obtained from PCI by plotting plateau pressureagainst the inverse of absolute temperature. The negative of the slope of such aplot will yield enthalpy change value and the intercept will yield the change inentropy values.

8.6.3 Ti- and Zr-based hydrogen storage materialsIn general, a good HSM for automobile applications should have dissociationtemperature and corresponding equilibrium vapour pressure comparable to thatof the ambient. Most of the developed hydrogen storage alloys can be classifiedin two groups, viz. AB5 and AB2 types. LaNi5 is a typical example of the AB5

type hydrogen storage alloys, which is mainly used in Ni-metal hydride batteries.Hydrogen storage capacity of AB5 alloys is about 1.5 wt% (Taizhong et al. 2004).In the AB2 type alloy, A = Ti or Zr and B = Ni, V, Mn Cr etc. These alloysusually possess the structure of Laves phase. The AB2 type Laves phase alloyshave been regarded as promising HSMs because of high hydrogen storage capac-ity. Both Ti- and Zr-based AB2 have been explored for applications as HSM andare briefly discussed in the next sections. Zr-based Laves phase alloy hydridesare very stable with very low dissociation pressure for hydrogen and hence arenot suitable for automobile application. On the other hand, Ti-based Laves phasealloy hydrides are unstable but their plateau characteristics are not suitable forautomobile applications. Thus for both Ti- and Zr-based AB2 type alloy, substitu-tion of both A- and B-type element are being used to modify the plateau pressure,its slope and hysteresis to suitable values.




0.50.30.2

30°C ZrCr1 – γ Fe1 + γ

Pre

ssur

e (N

m)

0 1 2 3(a)

(b)

0.1

0.10.01

1 10 100 1000

TiFe

Com

posi

tion

(HM

)

Time (min)

0.2

0.3

0.4

0.5

0.6

Ti0.8 Zr0.2 FeTi0.9 Zr0.1 Fe

Ti0.99 Zr0.01 Fe

Ti0.9 Zr0.1 Fe(No H.T.)

Figure 8.18. (a) Typical pressure–composition isotherms for ternary Zr–Cr–Fe alloys. (b) Influenceof Zr addition on the hydrogen absorption curves for TiFe alloys.




8.6.3.1 Ti-based hydrogen storage materialsAmong the Ti-based systems, TiFe, TiNi2, TiNi, TiMn2, TiCr and TiV inter-metallic compounds were investigated for hydrogen storage and purification. Oneof the significant problems associated with TiFe was the difficulty in activation.Substitution of Nb or Ta or V for Fe and substitution of Nb or Cu for Ti inTiFe is reported to accelerate activation. The TiFe alloys, to which Fe2O3 withCu were added for Ti, comprised of TiFe matrix and precipitates of Fe2Ti and theoxide (Fe7Ti10O3) phases. The interface between the matrix and the precipitateswere suggested to be active sites for the hydriding reaction and as entrance sitesfor hydrogen to diffuse into the alloy. Addition of small quantities of Zr and Nbenhances the ease of activation (Figure 8.18(b)).

TiMn alloys have been investigated due to their easy activation, good hydriding–dehydriding kinetics, high hydrogen storage capacity and relatively low cost.TiMnx (with x = 1 to 2) have C14 type hexagonal crystal structure. The hydrogenstorage capacity of Ti–Mn hydrogen storage alloy was reported to be 1.5–1.8 wt%.There are two major categories of Ti–Mn HSM, i.e. TiMn2 and TiMn1�5. Themain problem with the Ti–Mn alloys is their high equilibrium plateau pressure,which limits their practical application. Another shortcoming of these alloys isthat hydrides in these alloys are characterized by large hysteresis effects. V isreported to reduce the hysteresis. Homogenization treatment helps in obtainingclear plateau pressure. Several studies have focused on the roles of solid–gasand electrochemical reactions. The activation of the alloys plays a key role inhydrogen absorption process, since it defines the reaction rate of the hydrogenwith the metal and its incorporation in its structure. The hydrogen storage capacityof HSM is a function of the crystal structure–lattice parameter and void size.By suitable alloying addition, void size can be manipulated resulting in increasein hydrogen storage capacity. In late nineties, it was reported that TiV with abcc structure absorbs more hydrogen than conventional intermetallic compounds.Zr substitution in Ti–Cr–V system decreases the equilibrium pressure and thehysteresis of the PCI. However, it increased the slope of PC isotherm and reducedthe hydrogen storage capacity by forming ZrCr2. By selecting suitable compositionrange in the quarternary Ti–Zr–Cr–V system, formation of this phase can beavoided (Figure 8.19). The atomic radius of zirconium (1.62 Å) is larger thanthat of Ti (1.47 Å) and its electronegativity (1.4) is smaller than that of Ti (1.6).Generally, a metal hydride becomes stable with increasing difference between theelectronegativities of hydrogen (2.1) and the metal atom. By adding Zr, therefore,a decrease of the hysteresis and the plateau pressure, and an increase in the widthof the plateau and the hydrogen storage capacity due to an increase in latticevolume is expected. Under heat treated condition, the hydrogen storage capacityof Ti0�16Zr0�05Cr0�22V0�57 is reported to be 3.55 wt% (Table 8.5).




Ti

VCr

at.% at.%

at.%

Figure 8.19. Composition range showing the largest effective hydrogen storage capacity in theTi–Cr–V alloy system.

Table 8.5. Hydrogen storage capacities of the composition controlled and heattreated Ti–Zr–Cr–V alloys.

Composition H-storage capacity Effective H-storage capacity

H/M wt% H/M wt%

Ti0�12Zr0�05Cr0�45V0�26 1.50 2.90 0.85 1.65Ti0�12Zr0�05Cr0�45V0�26 1.44 2.74 0.77 1.47Ti0�12Zr0�05Cr0�45V0�26 1.50 2.87 0.79 1.51Ti0�12Zr0�05Cr0�45V0�26 1.73 3.34 1.07 2.07Ti0�12Zr0�05Cr0�45V0�26 1.67 3.18 0.93 1.77Ti0�12Zr0�05Cr0�45V0�26 1.57 2.99 0.86 1.63Ti0�12Zr0�05Cr0�45V0�26 1.71 3.24 1.02 1.92Ti0�12Zr0�05Cr0�45V0�26 1.87 3.35 1.13 2.14

8.6.3.2 Zr-based hydrogen storage materialsVarious studies performed so far on Zr-based AB2 type HSM have been directed toincreasing the equilibrium hydrogen vapour pressure in the Zr-based Laves phasealloys without markedly reducing the absorption capacity by partial substitutionof A or B type elements by other elements. The A element can be substitutedby Ti while the B element by Cr, Mn, Fe, Co, Ni and Cu. The reduction inhydride stability is possible due to any or all of the following effects: the reductionin the size of the hydrogen occupation site, the change in chemical affinity forhydrogen and the electronic factor. The reduction in binding energy of hydro-gen in C15 Zr X2 ( X = V, Cr, Mn, Fe, Co, Ni) is shown in Figure 8.20. Most




V Cr Mn Fe Co Ni–90

–60

–30

0

30

Bin

ding

ene

rgy

of H

(kJ

/mol

H)

2A2B site

1A3B site

4B site

Figure 8.20. Binding energy of hydrogen in C15 ZrX2 (X = V, Cr, Mn, Co, Ni).

research has been concentrated on the ZrMn2 type and ZrCr2 type pseudobina-ries. Elemental substitution in pseudo-binaries can be classified according to thefollowing:

(1) A element substitution, primarily with Ti: e.g. Zr1−xTixCr2 and Zr1−xTixMn2.(2) B element substitution with a different transition element: e.g. Zr(Mn1−x(Fex)2,

Zr(Cr1−xFex)2.(3) Hyperstoichiometric substitution of B element: e.g. ZrMn2 +x, ZrMn2Fex,

ZrMn1 + xFe1+y

(4) Any combination of the above: e.g. Zr1−xTixMnFe and Zr1−xTix(Fe1−yMny)z

alloys.

The hydrogen absorption characteristics of Zr-based stoichiometric AB2 HSMsuch as ZrMn2, ZrV2, ZrFe2, ZrCo2 and ZrCr2 were first studied by Shaltiel’sgroup (Shaltiel et al. 1977). The hydrides of the first two alloys were observed tobe too stable to be of practical significance. The stoichiometric ZrMn2 hydride, forexample, exhibits a dissociation pressure of 0.007 atm at 323 K. ZrMn2 crystallizesin the C14 (MgZn2) structure. The same structure is observed for ZrMn2 alloys inwhich Zr has been partially substituted by Ti and Mn by Fe. Since the hydride inZrMn2 is stabilized primarily by interaction of hydrogen with Zr, it was suggestedthat the vapour pressure of hydride might be increased by having a system that was




substoichiometric in Zr. Investigation of ZrMn2+x binaries, ZrMn2Fex ternariesand a series of related systems have established that the dissociation pressure andthe hydrogen capacity of these alloys depend on the Zr:Mn:Fe ratio in the latticeand the dissociation pressure was manifold higher than ZrMn2 without signifi-cantly reducing the hydrogen storage capacity (Sinha et al. 1985). Though thesealloys were found to absorb large quantities of hydrogen, the hydrides formedwere too stable to be of practical significance. ZrMn1+xFe1+y, where Zr is partiallysubstituted by Mn and/or Fe and Zr1−xTixMnFe, where Zr is partially substitutedby Ti are reported to be more attractive than LaNi5 because of favourable com-bination of properties such as lower cost, rapid kinetics of hydrogen sorption andlower endothermal nature of dehydrogenation. In comparison with Mn and Fe,the hyperstoichiometric Ni and Co is reported to be more effective in augmentingthe decomposition pressure of ZrMn2 hydride, although their presence reducedhydrogen storage capacity of the alloy. Sinha et al. (1985) reported that the mul-ticomponent Zr-based alloy having C14 structure appear to be very promisingHSM. The ZrCrFe1+x alloys, in particular, have many attractive properties. Thevapour pressure of ZrCr2 hydride is augmented by several orders of magnitudewithout significantly reducing its hydrogen storage capacity when 50% Cr isreplaced by Fe and in addition, hyperstoichiometric Fe is present in the lattice.The ZrCrFe1�5 alloy, for example, exhibits a dissociation pressure of its hydrideof about 2 atm at 23�C. The hydrogen storage capacity of this alloy is comparableto that of LaNi5. The effectiveness of alloying elements in destabilizing ZrMn2

hydride is reported to be in the following order: Cr < Mn < Fe < Ni < Co. Sinhaet al. (1982) reported that 20% substitution of Zr by Ti raises the equilibriumpressure five fold in (Zr1−xTix)Mn2. The change in PCIs for hydrogen absorp-tion and desorption in ternary ZrCr1−yFe1+y alloy is shown in Figure 8.21 withincreasing Fe content (Park and Lee 1990). Substitution of iron by chromiumraises the two phase plateau pressures for hydride formation and dissociation.The extent of hysteresis was also observed to increase. The overall trend in PCIsof Zr0�9Ti0�1Cr1−yFe1+y for the ranges 0 < y < 0.4 is shown in Figure 8.22 (Parkand Lee 1990). Unlike Ti-free ZrCr1−yFe1+y alloy, well-defined plateaux with lowslopes are formed in the two phase region for the whole composition range y. Itis reported that 10% substitution of Zr by Ti in ZrCr1−yFe1+y alloy yield the bestof the Laves phase alloy for various applications due to their low hysteresis andlow slopes.

8.6.4 ApplicationsIn the recent years, hydrogen has received worldwide attention as an alternativeenergy carrier which can substitute petroleum in internal combustion engines. Thepossible applications of HSMs are in automobile industry, moderators in nuclear




(a)

Y0.4

30°CZr0.9 Ti0.1 Cr1 – y Fe1 + y

10

1

0.12 3

H/AB2

PH

2 (a

tm)

0 1

0.30.20.10

(b)

Zr0.8 Ti0.2 Cr1 – y Fe1 + y

30°C

PH

2 (a

tm)

10

0.1

1

2 3

H/AB2

0 1

0.40.20

Y

Figure 8.21. Pressure–composition isotherms for Zr–Cr–Fe–H2 systems in which Zr has beenpartially replaced by (a) 10% Ti and (b) 20% Ti at 303 K.

reactors, heat pumps, refrigerators, getters, etc. With the development of fuelcell technology and the widening of hydrogen application fields, methodology ofhydrogen storage has attracted more and more attention. Three systems of interestare glass microspheres, liquid hydrogen and metal hydrides. The disadvantages forglass microspheres are that charging requires high hydrogen pressure and temper-atures (573 K) while for discharging the microspheres have to be heated to 473 K.On the other hand, liquid hydrogen requires complex handling techniques, a safecryogenic container and involves cost of liquefaction. For a refrigeration cyclehaving 33% efficiency for refrigeration, the refrigeration will require 10 KWh/kgof hydrogen, which accounts for nearly 25% of heat of combustion of hydrogen.In addition to several advantages, the unique feature of metal hydride is the equi-librium existing between the hydrogen and the metal. On cooling, hydrogen isreabsorbed into the metal, in contrast to two other systems in which, hydrogenonce librated, remains in the gas phase. However, several of their properties areinsufficient to preclude any large scale engineering applications, e.g. hydrides canbe contaminated by oxygen, nitrogen, water vapour and carbon-dioxide, all ofwhich lead to loss of hydrogen storage capacity. The driving force for current




Oxygen (wt%)

20

(α-Ti)Tem

pera

ture

(K

)

6730

Ti

873

1073

1273

1155 K

Ti3O

5 10 15

Ti2O

Ti 3O

2

1993

10

1473

1673

18731943(β-Ti)

2073

2273

24730

~2158 K

20 30

L

40

higher magneli phases

25

α-T

iO

30 35 4540

{

~1523 K

γ-TiO

β-TiO

50

β-T

i 2O

3

2115 K

Tin O2n-1

60

TiO

2 (rutile)

2143 K

70

β-Ti1–xO

α-Ti1–xO

Oxygen (at.%)

(a)

0 10 20 30 40 50 60 70

Oxygen (at.%)

0 5 10 15 20 25 30 35

Oxygen (wt%)

3073

2873

2673

2473

2273

2073

1873

1673

1473

1273

1073

873

673

473

Tem

pera

ture

(K

)

(α3"Zr)

(α'Zr)

(α1"Zr)

(α2"Zr)

(α4"Zr )

1136 K

6.6

6.7

6.9

7.4

8.6 10.54.12.0

1.9 2243

2128

5.5%, 2403 K~2338 K

~1798 K

~1478 K

~1243 K

~773 K

22

23.5

26.0

αZrO2–x

βZrO2–x

γZrO2–x

~2983 K

~2650 KL

L + G

β(Zr)

(b)

Figure 8.22. (a) Ti–O and (b) Zr–O binary alloy phase diagrams.




research on HSM is to attain weight reduction, cost reduction, optimizing temper-atures, heat transfer to and from hydride bed, kinetics of hydrogen sorbtion andother parameters. As an example, the requirement for the fuel storage system ofthe hydrogen automobile are (Sinha et al. 1982) listed below:

(1) A minimum delivery rate of 1.25 kg/h to meet the suburban driving cycledefined by the Society of Automobile Engineers;

(2) A high energy density (energy per unit mass);(3) A low storage volume such that it can be accommodated in a car;(4) A high degree of safety during charging and discharging;(5) A reasonable charging time;(6) No necessity of an external energy source to start-up; and(7) An energy as small as possible for hydrogen retrieval.

Hydrogen storage in metals is of special interest, since certain metals, alloysand intermetallic compounds react rapidly with hydrogen in a reversible mannerat nearly ambient temperatures and pressures. The safety of the hydride tanks arecomparable to that of petrol tanks, which is widely accepted in the community.The energy to weight ratio of the metal hydrides are about five to ten times greaterthan that of lead-acid battery, although they are still 10–20 times less than theenergy to weight ratio of petrol. The important properties of hydrides for afore-mentioned applications are hydrogen storage capacity, dissociation temperature,vapour pressure of dissociation, ease of activation (is a function of temperatureand pressure at which absorption and desorption begins), hysteresis and enthalpyof absorption and desorption.

8.7 OXYGEN ORDERING IN �-ALLOYS

8.7.1 Interstitial ordering of oxygen in Ti–O and Zr–OThe oxygen atoms dissolved in �-Ti and Zr occupy the octahedral interstitial sitesof the hcp host lattice. These interstitial sites are surrounded by six metal atoms asshown in Figure 8.2. Dagerhamn (1961) have first reported that the incorporationof oxygen atoms in �-Ti and Zr causes a somewhat peculiar variation of thelattice parameters with oxygen content and oxygen ordering occuring at and nearby the stoichiometric compositions of O/Ti = 1

3 (Holmbeog 1962) and O/Zr = 12

(Hirabayashi et al. 1974). Taking into account the formation of interstitially orderedsuperstructures, the partial phase diagrams of the Ti–O and the Zr–O systems havebeen constructed (Jostons and McDougall 1970, Hirabayashi et al. 1974) which




have been incorporated in the recently compiled binary phase diagrams (Massalski1992) as shown in Figure 8.22.

The ordered hcp phases in Ti–O and Zr–O systems are present in wide com-position ranges and are produced by second-order ordering of interstitial oxygenatoms. The octahedral holes in the hcp lattice form a simple hexagonal latticewith a = ao and c = co/2 where ao and co are the lattice parameters of the hcpmetal lattice which remains unchanged through the ordering transformation. Theordered structures so produced can be described as a stacking of interstice layernormal to the c-axis on which oxygen atoms are distributed. A single interstitiallayer can be conveniently described in terms of a hexagonal net with translation of√

3ao, as shown in Figure 8.2(c). Oxygen atoms can occupy A, B and C positionswhich have their coordinates at (0,0), ( 2

3 , 13 ) and ( 1

3 , 23 ), respectively. If only A

positions are filled by oxygen atoms, the plane is designated as �A� plane, If Aand B positions are filled, the plane is (AB) and when all A, B and C positionsare filled, the interstitial plane is indicated as (ABC). Vacant oxygen layers aredenoted by (V ).

The arrangement of oxygen occupation is not identical in the Ti–O and theZr–O systems. While in the Zr–O system oxygen atoms in fully ordered state aredistributed regularly in every layer of interstice plane with the spacing of co/2,those in the Ti–O system are positioned in every second layer with the spacingco. This seems to be reasonable due to a larger spacing for Zr (co = 0�515 nm)compared to that of Ti (co = 0�468 nm).

As the oxygen concentration changes, the layerwise distribution of oxygeninterstitials alters and a series of closely related structures are produced. For theTi–O system, the oxygen arrangements in Ti6O, Ti3O and Ti2O can be describedby the following sequence of interstital and vacant layers:

Ti6O : �A��V��B��V�, Ti3O : �AC��V��BC��V�, Ti2O : �ABC��V��ABC��V�where V denotes a vacant interstice plane and A, B, C are oxygen inter-

stice planes as described earlier. These structures, as schematically illustrated inFigure 8.23, are closely related to each other and the structural change can beviewed to occur continuously as a function of composition. This point can beillustrated by plotting the occupation probabilities, p, q and r as a function ofcomposition, where p, q and r are defined for the sites A at Z = 0 and B at Z = 1

2 ,the sites C at Z = 0 and 1

2 , and the sites A at Z = 12 and Z = 0, respectively

(Yamaguchi et al. 1970). This shows that the oxygen distribution in the compo-sition range, 1

6 ≤ OTi ≤ 1

3 , the structure changes by continuous filling of C sites byoxygen atoms in the �AC��V��BC��V� � � � layered sequence.

The oxygen ordering in the Zr–O system is somewhat complicated. The stackingsequences of the oxygen layers are basically expressed as �A��B��C� � � � at O

Zr ≤ 13

and �AC��B��AC��B� at OZr ≤ 1

2 . The former is isomorphous with �-Ni3C, being




(d) α'

Co

2Co

C

B

A

√3 ao

A

B

(a) Ti6O (b) Ti3O (c) Ti2O

Figure 8.23. Schematic illustration of packing sequence of various Ti-oxides.

called Zr3O1−x or ZrOx type. The latter is denoted as the ZrO2 type, being aderivative structure of �-Fe2N in which the occupation probabilities, � for thesites A, B and C are given as 1 ≥ pApB > pC . Another type of superstructuresknown as ZrO with the sequence �A��B��A��B� is found at O

Zr ≥ 13 . This structure

is isomorphic with Ni3N. In addition to the three superstructures, a series of longperiod structures with various types of stacking sequences are encountered near thestoichiometric composition O

Zr = 13 . Hirabayashi et al. (1974) have listed the long

period stacking structures of interstially ordered Zr–O alloys at compositions nearOZr = 1

3 . It is observed that with an increase in the oxygen content, the probabilityof hexagonal stacking increases continuously. The preference for ‘hexagonal’stacking in hyperstoichiometric compositions, O

Zr ≥ 13 can be explained in terms

of strain ordering. If the ‘cubic’ stacking of the oxygen layers were extendedover the hyperstoichiometric region, the nearest neighbour pairs of oxygen atomsalong the c-axis with the distance co

2 would inevitably be created by introductionof excess oxygen atoms in any vacant site. On the other hand, the hexagonalstacking can contain the excess atoms without forming the closest pairs upto thecomposition O

Zr = 12 , since the same kind of octahedral holes remain unoccupied

at the stoichiometric composition OZr = 1

3 (in the (A) (B) (A) (B) stacking, C sitesremaining vacant everywhere.

The nearest neighbour distances of the oxygen atom pairs along the c-axis are3co2 and co for cubic and hexagonal stackings, respectively, in the Zr–O system.

The probability of finding oxygen pairs with interatomic distance of co increaseswith the O

Zr ratio as shown in Figure 8.24.




1/6 1/3 1/21.0

0.8

0.6

0.4

0.2

0.00.1 0.2 0.3 0.4 0.5

Composition O/Zr

Zr–O

Co

1.0

0.8

0.6

0.4

0.2Pro

babi

lity

of O

–O p

airs

Pro

babi

lity

of h

sta

ckin

g

Figure 8.24. Plot of probability of finding Oxygen atom pairs with interatomic distance of co versusOZr ratio.

The order–disorder transformation in the Ti–O and the Zr–O systems occur intwo steps, the disordered �-phase first orders into the �-phase in which oxygenatoms are distributed in a regular array of interstitial planes (interplanar ordering),but oxygen atoms within the plane remain randomly distributed. Since oxygeninterstitial planes are placed on every second layer in the Ti–O system and onevery layer in the Zr–O system, the �′-phase in these two cases are structurally notthe same. The second ordering steps results in a periodic distribution of oxygenatoms within these planes (intraplanar ordering). The sequence of transformations,i.e. �′ → � can be shown in the order parameter versus temperature plot for aTi–O alloy ( O

Ti = 0�32), (Figure 8.25) which shows that the interplanar disordering,�′ → � occurs at T2 while the intraplanar disordering is complete at T1.

The disordering process of the Zr–O alloy ( OZr = 1

3 ) involves changing of stackingsequences of long period superstructures (diminishing periodicity with increasingtemperature) and randomization of the ordered arrangement of oxygen atoms oninterstitial planes. The overall transformation, �′′ → � remains to be of the second-order type, the order parameter continuously dropping as the transition temperature(Tc) is approached.

The domain structure formed due to oxygen ordering has been found to besimilar to the B2 ordering (Hirabayashi et al. 1974). This is because only twoantiphase domain configurations as shown in Figure 8.26 are possible in the Ti–Osystem depending on whether oxygen atoms occupy either the (000) plane orthe (00 1

2 ) plane. At the antiphase boundaries which have been drawn paralleland normal to the c-axis (in Figure 8.26) for simplicity, the interstitial oxygen




1.0

0.8

0.6

0.4

0.2

0

SΙ

SΙΙ

T1 T2

∨ ∨Deg

ree

of o

rder

O/Ti = 0.32

200 400 600 700

Temperature (K)

Figure 8.25. Order parameters versus temperature plot for OTi = 0�32.

APB↓↓

O

Ti←

←

C↑

Figure 8.26. The domain structure formed due to oxygen ordering in Ti–O system.

layers are displaced by a vector c/2 keeping the metal lattice identical across theboundaries.

The phenomenon of oxygen ordering is known to have strong influences onseveral properties such as hardness, electrical resistivity, thermoelectromotive forceand magnetic susceptibility, an account of which is summarized by Hirabayashiet al. (1974)

As the oxygen level increases towards the equiatomic TiO composition, the�-phase appears in the microstructure. The �-phase refers to the NaCl type struc-ture which exists over a wide range of compositons including the stoichiometric




TiO. While in the �′- and the �′′-structures titanium atoms are placed in a hcplattice, the �-structure can be regarded as an fcc arrangement of titanium atoms inwhich oxygen atoms occupy octahedral interstitial positions with high equilibriumconcentrations of vacancies in each sublattice. The � → � phase transformation,studied by Jostsons and McDougall (1970), has shown several interesting fea-tures such as strict adherence to the orientation relation and habit plane, shapechange and close relation between the two structures. The usual hcp/fcc rela-tion, (0001)�′ ��(111)�; [1120]�′ ��[110]� has been observed between the �′ and the�-phase. Chill cast samples of Ti–O (35–44.5 at.% O) show grains of parallel�′ plates in the �-matrix. The � → �′ transformation produces martensitic typesurface relief effect. The observed transformation characteristics are consistentwith the fact that the two lattices are fully coherent at their interfaces.

Specimen of Ti–O alloys ( 31–34 at.% O) can be heated from the �-phase tothe �+� phase field at temperatures above 1450�C. A heating and cooling cyclein which the �/� + � transus is crossed results in the formation of �-plates inthe �-matrix during heating and subsequent � → � → �′ transformation duringcooling.

8.7.2 Oxidation kinetics and mechanismTwo forms of oxidation have been recognized for zirconium alloys. These areuniform and localized forms of oxidation. Under most conditions of temperatureand environment, oxidation of zirconium alloys by coolant water or steam resultsin growth of uniform oxide films, especially in early stages. However, in someisolated regimes of temperature and environment, viz. in 300�C boiling or oxy-genated water and in high temperature (>450�C) and high pressure (>5 MPa), alocal form of corrosion occurs.

The growth of oxide layer as a result of the corrosion reaction between zirco-nium alloys and water can be described in terms of two distinct stages, usuallyknown as pre- and post-transition stages. The pre-transition stage is characterizedby a decreasing rate of weight gain which corresponds to cubic or quadratic kinet-ics relation between the weight gain and time (Figure 8.27) expressed in terms ofeffective full power days (EFPD) of operation of a nuclear reactor. The deviationfrom usual parabolic growth kinetics in this stage is attributed to the fact that therate controlling process of the diffusion of oxygen ions through the oxide layer isnot through lattice diffusion but through grain-boundary diffusion. The oxide layergrows in the early stages maintaining epitaxy. The ratio of the oxide volume tothat of the parent (Pilling–Bedworth ratio) is 1.56 for the Zr/ZrO2 system. Thus, asthe oxide grows, the stress build-up due to the volume expansion accompanyingoxidation induces a fibrous texture of oxide crystal, which minimizes the com-pressive stress in the plane of the oxide layer. The presence of the compressive




10

20

30

40

50

60

70

Max

. H p

icku

p (m

g dm

–2)

10

20

30

40

50

60

Max

imum

oxi

de th

ickn

ess

(μm

)

1 2 3 4 5

Time (EFPD) × 1000

Break away

Zr–2.5 Nb oxidation

H pick-up Zr–2.5 Nb

Zr-2 M pick-up

Zr-2 oxidation

Figure 8.27. Long term in-reactor, oxidation and hydrogen pick-up behaviour of zircaloy-2 andZr–2.5 Nb pressure tubes, showing parabolic and then accelerated linear oxidation and hydrogenkinetics in zircaloy-2. A low and uniform rate of corrosion and hydrogen pick-up is seen in Zr–2.5 Nballoy.

stress is also a factor responsible for the stabilization of the tetragonal phase in theZrO2 layer. An examination of the metal/oxide interface reveals that a thin layer(∼10 nm thick) of amorphous oxide is present right on the metal surface. Finecrystallites of zirconia are distributed in the amorphous matrix. The volume frac-tion and size of the crystallites increase with the distance from the metal surface.The growth of the oxide crystals finally results in the formation of a columnarstructure. The intercrystalline boundaries provide the diffusion path of O−2 ionsfrom the environment to the reaction front at the metal/oxide boundary. The oxi-dation process and the nature of the oxide layer on a zirconium alloy sample areschematically shown in Figure 8.28

As the oxide layer grows, the compressive stress at the outer layer of oxide is notsustained and consequently the tetragonal phase becomes unstable and transformsinto the monoclinic phase. Such a transformation causes the formation of a fineinterconnected porosity in the oxide film which allows the oxidizing water to comein contact with the metal surface. With the development of an equilibrium pore andcrack structure in the oxide layer, the oxidation rate becomes effectively linear,a characteristic feature of post-transition oxidation behaviour. Alloying elements,particularly tin, niobium, and iron present in the �-solid solution strongly influenceboth the kinetics and the mechanism of oxide growth in zirconium alloys.




Electrons travel bysurface conduction

Iron oxide

Thin ZrO2 layer

MonoclinicZrO2

Intermetallicprecipitate(Zr – Fe – Ni)

≈2 μ

m

O2 + 4e– → 2O2–

O 2-Diffusion

along crystallite boundaries

ZrO2 crystals

Zr+202– → ZrO2 + 4eAmorphous

oxide

Metallicconduction

Columnaroxide{

Zircaloy - 2

→

Figure 8.28. Schematic diagram showing the mechanism of the oxidation process and the oxidefilm structure on zircaloy.

Non-uniform oxide formation, usually referred to as nodular corrosion, is thelimiting design consideration in boiling water reactors (BWR). Several mecha-nisms have been proposed for nodule nucleation which can occur in various sitessuch as �-Zr grain-boundaries and in areas at which the continuous dense oxidelayer ruptures in the initial stages due to the presence of large intermetallic pre-cipitates. The best microstructure for resisting nodular corrosion in the operatingcondition of a BWR consists of a distribution of fine intermetallic precipitates(dia <0.15 micron). It may be noted here that precipitate size has an oppositeimpact on corrosion rate in the pressurised water reactor (PWR) conditions wherefine precipitates (<0.1 micron) increase the uniform corrosion rate. In additionto metallurgical factors, water chemistry has a strong influence on the corrosionprocess in zirconium alloys. The control of water chemistry required in differentreactor systems is achieved by suitable additions of lithium hydroxide, boric acid,hydrogen/deuterium, oxygen, iron and zinc. In PWR, boric acid is added for reac-tivity control. The pH of the coolant is adjusted by addition of lithium hydroxide




which renders the coolant slightly alkaline, in order to reduce the corrosion ratesof structural materials (stainless steels and Inconels) in the primary heat transportcircuit and thereby inhibit deposition of corrosion products on the fuel cladding.Radiolysis of water produces oxidizing species which enhances oxidation rate ofzirconium alloys in radiation environment. The build-up of these can be suppressedby adding hydrogen in the coolant water. Dissolved hydrogen concentration inPWR is maintained at a level of 2.2–4.5 ppm with a view to enhancing recombina-tion with oxygen radicals formed by radiolysis. Dissolved oxygen in pressurisedheavy water reactor (PHWR) is kept between 10–50 ppb and it has been observedthat the corrosion of zirconium alloys rises to an exceptionally high value at highoxygen concentrations. The BWR coolant contains higher level of oxygen, typi-cally 200–400 ppb. Hydrogen addition in BWR condition is not very effective dueto the segregation of hydrogen in the steam phase.

8.8 PHASE TRANSFORMATIONS IN TI-RICH END OF THE TI–NSYSTEM

The Ti-rich end of the binary Ti–N phase diagram (Figure 8.29) shows the�-stabilizing nature of N leading to an extended phase field of the �-hcp Ti–Nsolid solution, the �-Ti2N phase and the �-TixN1−x phase. While the �-phase ispresent only in a narrow composition range, the �-phase is found in a wide rangeof compositions from 30 to about 53% N. The �-Ti2N phase possesses a tetragonal(antirutile) structure and the �-phase has a cubic NaCl structure. The equilibriumphase diagram shows several invariant phase reactions, the eutectoid, � → �+ �being important in the phase selection using reactive evaporation deposition pro-cess to form titanium nitride coatings. The other solid state invariant reactioninvolves the � (Ti2N) and the � (TiN) phases which react to form the �′-phase(Si2Th structure) through possibly a peritectoid reaction. The congruent transfor-mation, � → � at 1373 K is an interesting example of a polymorphic transformationbetween two interstitially ordered structures.

A metastable tetragonal phase �′, with ordered N atoms, forms during ageingof supersaturated dilute Ti–N alloy (N content well below the stoichiometry ofTi2N). The sequence of transformation processes that occur during the evolutionof the �′-phase, studied in detail by (Sundararaman et al. 1989), serves as a goodexample of an interstitial ordering process in which interstitial sites in the parentmetal lattice are progressively filled up with interstitial solute atoms leading to thegeneration of several intermediate structures.

Nitrogen occupies octahedral sites in hcp �-titanium. The size of the octahedralinterstice in the Ti lattice is not sufficient to accomodate a solute atom without




(ε-Ti2N)

Nitrogen (at.%)

Nitrogen (wt%)10

1943(β-Ti)

Tem

pera

ture

(K

)

773

Ti0

1773

1273

10

(α-Ti)

5

8

2773

2273

3773

3273

0

2293 K1.9 4.0

10

L

2623 K7.0

20

1073 K

15

1373 K

11

δ'

20

10

30

(δ-TiN)

40

3563 K

50

(a)

0 10 20

Nitrogen (wt%)

706050403020100

Nitrogen (at%)Zr

3373

3173

2973

2773

2573

2373

2173

1973

1773

1573

1373

1173

973

Tem

pera

ture

(K

)

L

ZrN

(α-Zr)

(β-Zr)

1136 K

2128 K ~2153 K

~2258 K

953.00.8

(b)

Figure 8.29. (a) Ti–N and (b) Zr–N binary alloys phase diagrams.




dispalcing neighbouring atoms. The stress field created by introduction of N atomsin the interstitial sites of hcp �-Ti tends to get minimized by establihsing anordered arrangement of interstitial sites.

The octahedral interstitial sites in an hcp structure are shown by small circleswhile the positions of the host Ti atoms by large open circles (Figure 8.30(a)),a and c being the lattice parameters of Ti. Sundararaman et al. (1989) proposedthe following scheme of filling these interstitial positions by N atoms in the Ti–Nsystem:

(a) The distance between the two adjacent basal plane layers of the interstitiallattice (shown by dashed lines in Figure 8.30) being small (= c

2 ), only alter-nate interstitial layers are progressively filled by N atoms with increasingconcentration of N.

Ti

N

B

A

X

Y

X

Y

C

C'

B'

A'

√3a√3a

a

c

Figure 8.30. (a) An interstitial sublattice is outlined inside the metal atom lattice. (b) Orderedconfiguration of nitrogen interstitials in the interstitial sublattice. For clarity, the host lattice is notoutlined. A, A,′B, B′ and C, C ′ refer to three types of interstitial sites in alternate layers. Note thenitrogen occupancy in alternate layers. Open circles represent nitrogen interstitials and shaded onesthe forbidden sites.




(b) A single layer of the basal plane of the interstitial site, as schematically shownin Figure 8.30(b), can be considered to have three types of sites designated asA, B and C. The nearest distance between two sites with same designation,A-A, B-B or C-C, is

√3a.

(c) The complete filling of A sites generates the Ti6N structure, of A and B (or Aand C) sites generates Ti3N and of A, B and C sites produces Ti2N structure.

(d) Such progressive filling of interstitial sites on alternate basal planes can accountfor the dilute interstitial TiNx alloys (x < 1

2 ).

The configurational internal energy, �Ec, of TiNx alloys can be expressed as

�Ec = PNV �NV +PNN �NN +PVV �VV (8.26)

where Pij and �ij represent the total number of pairs and interaction energiesinvolving species, i and j, respectively. The subscripts N and V stand for nitrogenatoms and vacant interstitial sites, respectively. The interaction energy enteringthe Eq. 8.26 can be considered as effective bond energies between the speciesconcerned, mediated by the host lattice. The coordination numbers for first (Z1) andsecond (Z2) near neighbours, their respective distances and the interaction energiesare presented in Table 8.6. Using a number of simplifying assumptions such as(a) vv = 0, (b) ij is not concentration dependent and substituting physicochemicaldata from Balasubramanian and Kirkaldy (1985), Sundararaman et al. (1989)have estimated the configurational internal energies, ER and EO, of the random

Table 8.6. Number (Zi) and distance (di) of first and second nearest-neighbours in a hcp-basedinterstitial solid solution for the proposed ordering scheme.

Ordering First-(subscript 1) and second-(subscript 2)nearest neighbours

Z1 d1 N-N Z2 d2 N-N

All A sitescompletely filled�X = 1/6�

2 c �′NN 6 a

√3 �2

NN

A and B sitescompletely filled�X = 1/3�

3 a �′NN 2 c �2

NN

A, B and C sitescompletelyfilled �X = 1/2�

6 a �′NN 2 c �2

NN

Remarks: c = 4.68, a = 2.959 , a√

3 = 5�1094.




0.1 0.2 0.3 0.4 0.5

X →

–ΔE

–4.5

–4.0

–3.5

–3.0

–2.5

–2.0

–1.5

–0.5

0.0

0.5

1.0

1.5

–20.0

–16.0

–12.0

–08.0

–04.0

0.0

4.0

8.0

12.0

–ER

–Eo

Eo ,

ER (

KJ/

mol

) →

ΔE (

KJ/

mol

) →

Figure 8.31. Variation of internal energy with composition for random (ER) and ordered (EO) solidsolutions.

and ordered state of N in Ti–N interstitial solid solution as a function of x(Figure 8.31). It is shown that both configurations reaches a minimum energy atx = 0�2. The effect of ordering of N atoms in the intersitial sites on the lowering ofthe configurational energy, as revealed from �E versus x plot is maximum at x =1/6 corresponding to the ordered stochiometric superstructure Ti6N. Sundararamanet al. (1989) have also shown that the activity ratio, ao

N/aRN where ao

N and aRN

corresponds to the activities of N in ordered and random configuration in Ti–Ninterstial solution (dilute), shows two minima when plotted against x. These twominima corresponds with x = 1

6 and 13 and are separated by a maximum. Such

plots with double minima suggest a phase separation tendency which results inthe formation of Ti6N and Ti3 N regions. For 0 < x < 1

6 , the solution prefers tobe ordered as dictated by the repulsive interactions along the c-axis, and at x = 1

6 ,the solution being fully ordered, a minimum at the Ti6N composition is exhibited.For x values marginally exceeding 1

6 , one can visualize the alloy as the dilutesolution of nitrogen in Ti6N. The strong and replusive first neighbour interaction




is reported to be responsible for interstitial ordering and the stability of Ti6Nsuperstructure (Sundararaman et al. 1989). For x > 1

6 , second-nearest neighbourinteraction comes into play rendering Ti6N phase unstable. With increasing Nconcentration, filling of alternate B and C sites in alternate layers bring backthe dominant first neighbour interactions and the nitrogen activity in the orderedsolution is lowered once again. Thus we get a minimum at x = 1

3 in which casethe order is again partially restored.

Nitrogen atoms in a hexagonal titanium lattice have been found to distort thelattice as shown in Figure 8.32. The distortion of the host lattice can be resolvedinto two components, one along the c-direction and the other one in direction per-pendicular to it. The c-component of the distortion results in a monotonic increaseof the lattice parameter c (Figure 8.33) whereas the distortion component along thebasal plane periodically distorts the titanium atoms towards the central nitrogenatom (Figure 8.32). In other words, a nitrogen atom serves as a contraction centreas far as the basal plane is concerned; while it serves as an expansion centre alongthe c-direction. This is brone out by the lattice parameters “a” and “c” variationwith composition, as depicted in Figure 8.33. The c-axis exhibits a monotonicincrease with nitrogen content while the lattice parameter “a” initially increasesand then reaches a saturation value. Associated with such a local distortion, thereis an elastic stress field around each interstitial atom which can drive a martensitictransformation of the ordered hexagonal structure to a tetragonal one. The volumechange associated with the structural change in the case of the Ti–N system issmall enough to be accommodated by a lattice shear. It is also clear that any such

Ti

NC

C

C

A B

Figure 8.32. Schematic illustration of the second neighbour active interaction. Note that it ismediated by the intervening metal atoms and that an interstitial atom serves as a contraction centrein the basal plane. The arrows point to the direction of displacement of the metal atoms.




1.62

1.58

4.72

4.68

4.76

4.80

2.96

2.95

2.94

0.0 0.05 0.11 0.176

x

a

C

C/a

5 10 15

N, wt. %

C /a

a,Å

C,Å

Figure 8.33. Lattice parameter variation with composition for Ti–N system.

homogeneous crystal lattice distortion must preserve the ordered configuraiton ofnitrogen atoms inherited from the parent lattice. Sundararaman et al. (1980) haveexperimentally shown the existence of a martensitic transformation as one of theprecursor steps in the precipitation process of Ti2N in Ti–N alloys. The salientfeatures of the precipitation reaction as experimentally observed by Sundararamanet al. (1980, 1983) are briefly presented with a view to augment the theoreticalsupport concerning the martensitic transformation:

(a) There exists a specific orientational relation between the �-phase and the TiNx

phase. (1010��011�TiNx

13 < x < 1

2 ; 1210��011�TiNx;

(b) The TiNx precipitates always from on a definite habit plane;(c) The transformation is very rapid; it initiates during quenching;(d) The volume change associated with the transforamtion TiNx is small;(e) The principal strains along three mutually perpendicular directions are: �1210� =

4�88%; �0001� = 5�22%; �1010� = 1�28%




The above values nearly satisfy the requirement of an invariant plane straincondition for a martensitic transformation. The magnitude of lattice invariantshear is, therefore, quite small. The above mentioned lattice correspondence isconsistent with the observed orientation relation between the � and the TiNx

phase (Figure 8.34). The interaction of strain fields in the �-matrix, results inthe formation of the observed mottled contrast (Figure 8.35(a)). The induction ofthe lattice transformation of the host lattice (hcp Ti) and the interstitial orderingof N atoms provides yet another example of a coupled displacive–diffusional(interstitial) transformation.

[1210]α[011]Ti2N

(010.378)Ti2N

(0001)α

(1010)α ||(011)Ti2N

(100)Ti2N

(100)Ti2N

Figure 8.34. Schematic representation of orientation relation between �-(hcp) ordered solid solutionand tetragonal Ti2N. The orientational relation is (1010��011�Ti2N, [1210��011�Ti2N.

0.2 μm

(b)(a)

2.5 nm

[013]

[100]

Figure 8.35. (a) Quenched microstructure exhibiting mottled or tweed contrast typical of the initialdecomposition stage of supersaturated Ti–N solid solution. (b) High resolution structure imageshowing the presence of � and �′ (isostructural with the Ti2N phase) regions separated by coherentinterfaces.




REFERENCES

Balasubramanian, K. and Kirkaldy, J.S. (1985) Calphad, 9, 103.Banerjee, D. and Arunachalam, V.S. (1981) Acta Metall., 29, 1685.Banerjee, D. and Williams, J.C. (1983) Scr. Metall., 17, 1125.Dagerhamn, T. (1961) Acta Chem. Scand., 15, 214.Darby, M.I., Read, M.N. and Taylor, K.N.R. (1978) Phys. Status Solidi (a), 50, 203.Dey, G.K. and Banerjee, S. (1984) J. Nucl. Mater., 124, 219.Dey, G.K., Banerjee, S. and Mukhopadhyay, P. (1982) J. Phys. C4, 43.Dutton, R. (1976) Metall. Soc. CIM, 16, 16.Flewitt, P.E., Ash, P.J. and Crocker, A.G. (1976) Acta Metall., 24, 669.Froes, F.H. and Eylon, D. (1990) Hydrogen Effects in Material Behaviour (eds N.R.

Moody and A.W. Thompson) TMS, Warrendale, PA, p. 261.Hirabayashi, M., Yamaguchi, S., Asano, H. and Hiraga, K. (1974) Proc. Int. Conf. on

Order–Disorder Transformations in Alloys (ed. H. Warlimont) p. 266.Holmberg, B. (1962) Acta Chem. Scand., 16, 1245.Hong S. and Fu C.L. (2002) Phys. Rev.(B), 66, 99.Jack, K.H. (1948) Proc. Roy. Soc., A195, 34.Jostsons, A. and McDougall (1970) Phys. Status Solidi, 29, 873.Khachaturyan, A.G. (1978) Prog. Mater. Sci. 22, 1.Khachaturyan, A.G. (1983) Theory of Structural Transformations in Solids, Wiley,

New York.Lieberman, D.S., Read, T.A. and Wechsler, M.S. (1957) J. Appl. Phys., 28, 532.Leitch, B.W. and Puls, M.P. (1992) Metall. Trans., 23A, 797.Massalski, T.B. (1992) Binary Alloys Phase Diagram, 2nd edition, ASM International

Materials Park, OH, pp. 2066–2067, 1652–1655, 1792–1793, 2078–2079, 1659–1660,1799–1800.

Narang, P.P., Paul, G.L. and Taylor, K.N.R. (1977) J. Less-Common Metals, 56, 125.Norwood, D.O. and Gilbert, R.W. (1978) J. Nucl. Mater., 78, 112.Norwood, D.O. and Kosasih, U. (1983) Int. Metals Rev., 28(2), 92.Pan, Z.L., Ritchie, I.G. and Puls, M.P. (1996) J. Nucl. Mater., 228(2), 227.Park, J.M. and Lee, J.Y. (1990) J. Less-Common Metals, 160, 259.Puls, M.P. (1990) Metall. Trans. A, 21A, 2905.Rhodes, C.G. and Williams, J.C. (1975) Metall. Trans. 6A, 1071, 2103.Shaltiel, D., Jacon, I. and Davidov, D. (1977) J. Less-Common Metals, 53, 117.Sidhu, S.S., Heaton, Le Roy, Campos, F.P. and Zauberis, D.D. (1963) Am. Chem. Soc., 39Singh, R.N., Kumar, N., Kishore, R., Roychaudhary, S. and Sinha, T.K. (2002a) J. Nucl.

Mater., 304, 189.Singh, R.N., Kishore, R., Sinha, T.K. and Kashyap, B.P. (2002b) J. Nucl. Mater., 301, 153.Singh, R.N., Kishore, R., Sinha, T.K., Banerjee, S. and Kashyap, B.P. (2003) Mater. Sci.

Eng. A, 339, 17.Singh, R.N. Kishore, R. Singh, S.S., Sinha, T.K. and Kashyap, B.P. (2004) J. Nucl. Mater.,

325, 26.




Singh, R.N., Mukherjee, S., Gupta, A. and Banerjee, S. (2005) J. Alloys Compd., 389, 102.Singh, R.N., Lala Mikin, R., Dey, G.K., Sah, D.N., Batra, I.S. and Stahle, P. (2006) J. Nucl.

Mater., 359, 208.Sinha, V.K., Shetty, M.N. and Singh, K.P. (1970) Trans. Faraday Soc., 66, 1981.Sinha, V.K., Pourarian, F. and Wallace, W.E. (1982) J. Less-Common Metals, 87, 283.Sinha, V.K., Yu, G.Y. and Wallace, W.E. (1985) J. Less-Common Metals, 106, 67.Srivastava, D. (1996) Ph.D Thesis, Indian Institute of Science, Bangalore, India.Srivastava, D., Neogy, S., Dey, G.K., Banerjee, S. and Ranganathan, S. (2005) Mater. Sci.

Eng. A, 397, 138.Sundararaman, D., Terrance, A.L.E., Seetharaman, V. and Raghunathan, V.S. (1980)

Titanium 80, Science and Technology, Proc. Fourth Int. Conf. on Titanium, Vol. 2, TheMet. Society of AIME, New York, p. 1521.

Sundararaman, D., Terrance, A.L.E., Van Tendeloo, G. and Van Landuyt, J. (1983),J. Phys. Status Solidi (A), 76, K-109.

Sundararaman, D., Raju, S. and Raghunathan, V.S. (1989) J. Phys. Chem. Solids, 50, 1101.Une, K., Nogita, K., Ishimoto, S. and Ogata, K. (2004) J. Nucl. Sci. Technol. 41, 731.Taizhong, H., Zhu, W., Jinzhou, C., Xuebin, Y., Baojia X., Naixin, X. (2004) Mat. Sci.

Eng. (A), 385, 17.Unnikrishnan, M., Menon, E.S.K. and Banerjee, S. (1978) J. Mater. Sci., 13, 1401.Wayman, C.M. (1964) Crystallography of Martensitic Transformation, Mcmillan,

New York.Weatherly, G.C. (1981) Acta Metall., 29, 501.Yamaguchi, S., Hiraga, K. and Hirabayashi, M. (1970) J. Phys. Soc. Japan 28, 1014.Zener, C. (1946) Trans. AIME, 167, 550.Zuzek, E. and Abriata, A. (2000) Phase Diagrams of Binary Hydrogen Alloys, Monograph

Series on Alloy Phase Diagrams, Vol. 13, ASM International, Materials Park, OH.






Chapter 9

Epilogue






Chapter 9

Epilogue

The preceding eight chapters have illustrated the wide variety of phase transfor-mations encountered in Ti- and Zr-based systems and have demonstrated that thewhole subject of phase transformations in condensed matter can be studied usingexamples taken from these systems. It has also been shown that the various con-cepts and formalisms introduced in the study of phase transformations in alloyscan be applied to intermetallics and ceramics as well.

In this concluding chapter, let us examine some general trends in the phasetransformations in Ti- and Zr-based alloys. As has been discussed in earlier chap-ters, solid–solid phase transformations in these systems are essentially governedby the competition between the different allotropes, �, � and � that are struc-turally related through unique lattice correspondences, the Burgers relation for�/� and that involving the collapse of a set of adjacent {222} planes for �/�. Infact, the structural relationships between these phases are so strong that the samerelationships remain valid for both displacive and diffusional transformations.

An inspection of phase diagrams of binary and ternary alloys reveals that in afairly large number of Ti, Zr–transition metal systems, the liquidus temperaturedrops down considerably with alloy additions. This enhanced stability of theliquid phase makes it possible to amorphize these alloys under non-equilibriumprocessing conditions. The chemical short-range order present in some of theseliquid phases which are stabilized to sufficiently low melting temperatures tendsto form clusters with icosahedral symmetry. Such tendencies are reflected in theformation of quasicrystalline phases on crystallization of some of the amorphousalloys in these systems.

Another strong tendency which is responsible for several phase reactions inthese systems is the clustering tendency in the �-phase. Spinodal decomposition,phase separation, monotectoid decomposition, metastable phase reactions duringtempering of some martensites are all consequences of the clustering tendency inthe �-phase in a number of alloy systems.

Tendencies for chemical ordering in the �-, �- and �-phases, present in severalbinary and multicomponent systems, are responsible for the formation of orderedderivations of these phases. Important examples of such chemical ordering arehcp → D019, bcc → B2 and � → B82. Symmetry relationships between the dis-ordered parent structures and their ordered derivatives play a dominant role indictating the path of the transformation sequence in several alloy systems.

785




The hcp structure is a special case of the orthorhombic structure in which theratio of lattice parameters b/a = √

3. With the introduction of some alloyingelements beyond certain limits, the hcp structure is distorted to an orthorhombicsymmetry (b/a �= √

3). This tendency of orthorhombic distortion is also present insome ordered intermetallics having the D019 structure which undergoes a transitionto the O-phase (orthorhombic) following the same lattice correspondence as thatprevailing between hcp and orthorhombic structures.

We will briefly discuss these tendencies which determine the phase transforma-tion sequences in Ti- and Zr-based systems.

Pressure–temperature phase diagrams of single component systems comprisingGroup IVA metals, Ti, Zr and Hf show the three phases, ��hcp�, ��bcc� and� (hexagonal), in the solid state. The most interesting aspects of these phases aretheir unique lattice correspondences and the relationship of their stability regimeswith their electronic density of state. Introduction of alloying elements and/orchange in external variables such as pressure and temperature can induce a changein the electronic structure which in turn is responsible for bringing about the phasetransition.

Out of the three allotropes, the �-phase with the bcc structure is associatedwith the highest symmetry. This is also the first phase to appear from the liquidphase on cooling. Let us, therefore, consider the �-phase as the reference stateand examine how this bcc structure transforms into the hcp �- and the hexagonal�-structures. As has been elaborated in Chapters 1 and 4, the �-phase experiencesa tendency to undergo a transition into the �-phase as the �/� transformationtemperature is approached. The lattice correspondence between � and � is suchthat a distortion of a {110}� plane can introduce a sixfold symmetry, and a rathersmall distortion along the <110>� direction can establish the right c/a ratioof the product hcp structure. In addition to this lattice strain, an atomic shuffleis necessary for bringing the atoms in every alternate basal planes in the rightposition. The �-phase in some alloys cannot maintain the hexagonal symmetryand distorts into an orthorhombic structure.

In addition to the tendency for the bcc to orthohexagonal transition, the �-phaseexperiences another type of lattice instability which can be described as a longitu-dinal displacement wave of wave vector k = 2/3 <111>. The introduction of thisperiodic displacement in the �-lattice generates the �-structure as has been dis-cussed in Chapters 1 and 6. Variations in external conditions such as pressure andtemperature and chemical composition shift the relative stabilities of these phasesand the corresponding transformations can be induced. These two transformationtendencies lead to either an orthohexagonal structure or to the �-structure. Thesetendencies are present not only in dilute alloys but also in several intermetalliccompounds as has been elaborated in Chapters 4–6.



Epilogue 787

The �-phase in several Ti and Zr alloys also exhibits a tendency for phaseseparation as reflected in the presence of a miscibility gap in the �-phase fieldand a monotectoid reaction of the type �1 → � + �2. This phase separation orclustering tendency is instrumental in giving rise to a number of metastable stepsin the phase transformation sequence of both �- and �-alloys.

Chemical ordering of �-, �- and �-phases can produce a host of ordered inter-metallic structures in binary and ternary alloys of Ti and Zr. A comparison betweenseveral equilibrium and metastable intermetallic structures in these systems vis-à-vis the �-, �- and �-structures reveals that the former can be construed as orderedderivatives of the latter. The mechanism of formation of these intermetallic phasesfrom the parent �-phase, therefore, involves displacive atom movements (eitherfrom bcc to orthohexagonal or from bcc to �) in conjunction with replacive (chem-ical) ordering. The path associated with such transformations can be determinedconsidering the symmetry changes associated with the displacive and replaciveordering processes.

In spite of the diversity of solid-state phase transformation processes in Ti- andZr-based alloys, an understanding of the mechanism involved can often be gainedby taking into the account the fact that the structures are invariably closely relatedand that the transformations are often guided by the following general tendencies:

(1) bcc → orthohexagonal lattice distortions,(2) bcc → � transformation by the shuffle dominated lattice collapse process,(3) phase separation tendency of the �-phase in several systems containing

�-stabilizing elements and(4) chemical ordering tendencies of the �-, �- and �-phases leading to the forma-

tion of their respective superlattice structures.

Coming to the issue of the formation of amorphous and quasicrystalline struc-tures, the stabilization of the liquid phase by alloying Ti and Zr with varioustransition metals is reflected in the plunging of the liquidus temperature in therespective phase diagrams. In case the formation of the equilibrium intermetallicphases is suppressed by a non-equilibrium processing technique at a temperaturebelow the glass transition temperature, the amorphous phase becomes the nextbest alternative for the system to adopt. The presence of the chemical short-rangeorder in the liquid phase in compositions close to those of relevant intermetal-lic compounds is a prerequisite for bringing about vitrification in these systems.The nature of clusters is undoubtedly very important in promoting the stabilityof the amorphous phase as is seen from the formation of bulk glasses in a hostof ternary and quaternary Zr–Al–transition metal alloys. These clusters, some ofwhich have icosahedral or decagonal symmetries, have a tendency to retard the




kinetics of crystallization, thereby making the corresponding alloys amenable forglass formation even at a relatively slow cooling rate.

In order to identify different general tendencies of phase transformations, somerecent experimental techniques have been found to be extremely convenient. Toillustrate this point let us cite the examples of studies carried out on miniaturecompositionally graded samples (Banerjee et al. 2003).

Phase transformation studies on a given binary or multicomponent alloy systeminvariably necessitate the preparation of a series of alloys of different composi-tions and thermal analysis, phase analysis and microstructural characterization ofdifferent microconstituents after suitable heat treatments. This has been the generalpractice till recently; now it is possible to produce compositionally graded sampleswith desired gradients by laser melting of controlled powder feeds emanating frommultiple nozzles. Figure 9.1 illustrates the operating principle of such a laser-processing unit. In this process, either prealloyed powder or elemental powderscan be taken as the powder feed. When powders of different compositions are fedfrom separate feeders with a precise control of feed rates at a small size moltenpuddle, the possibility of making samples with graded composition opens up. Theprocess control computer continuously controls the position (x� y� z) of the hotspot (molten puddle) with respect to the component/sample being processed andsimultaneously controls the feed rates of powders from different nozzles. Sucha manufacturing process (which is essentially an extension of rapid prototypingprocess) has the potential not only of producing components of intricate shapes,but also of preparing samples of desired compositional gradients.

Glove box

CO2 lasersystem

Powderfeed

system

Stagecontrol

Figure 9.1. Experimental set-up for production of compositionally graded samples.



Epilogue 789

Suitable slices of one such graded sample can be prepared and given appropriateheat treatments for inducing different phase transformations. Microscopic exami-nation of graded samples by a scanning electron microscope (SEM) attached withmicroanalysis and orientation imaging facilities can yield morphological, crystal-lographic and compositional information on the microconstituents at a level ofabout 0.1 �m resolution. Such studies can be complemented by TEM studies bypicking up samples from related areas by focussed ion beam sectionizing (FIBS)technique. Nanoindentation experiments on small sample areas can further sup-plement the investigation by providing mechanical properties data from each ofthe microconstituents.

Recent work reported on Ti–V and Ti–Mo alloys have shown how these tech-niques can be gainfully employed for phase transformation studies on a systemfrom a single compositionally graded sample (Banerjee et al. 2002, 2003).

The composition profile of a graded Ti–V sample of 25 mm length producedby laser melting of prealloyed powders of different compositions is shown inFigure 9.2. Several longitudinal slices of such a sample after subjecting to differentheat treatments such as (a) as cast, (b) �-quenched and (c) �-solutionized followedby air cooled show a variety of microstructures as described in the following.

The �-quenched sample shows the presence of lath martensite in regions corre-sponding to a V level of 0–3 at.%, plate martensite in the region of 3–11 at.% V

25

Ti

∼ 25 mm

Ti–25 at.% V

20

15

10

5

0#1 #2

#3 #4#5

#6

#7#8

#9#10

#11

#12

#13

#14

10 15

Distance (mm)

Com

posi

tion

(at.%

V)

20 25 300 5

Figure 9.2. Variation of composition of a Ti–V compositionally graded sample.




and near complete �-stabilization in regions containing more than 12% V. TEMinvestigations have revealed that a distribution of fine �-plates in the compositionrange of 12–15% V.

Microstructures of the as-cast sample match closely with those correspondingto �-solutionizing and air cooling. This points out that the laser-processing tech-nique on a sample size, like the present one, essentially produces a normalizedstructure. Microstructures of the compositionally graded samples (both as-cast and�-solutionized followed by air cooled) show a gradual change in the morphologyof Widmanstatten �-plates and monoatomic increase in the volume fraction of theretained �-phase with increasing V content (Figure 9.3). While the �-plates arelong and slender in alloys containing up to about 5 at.%, they are short and stout inalloys more enriched with V. Though the as-cast condition is not expected to yieldequilibrium microstructure, the volume fractions of the �- and the �-phases andtheir respective compositions are seen to approach equilibrium values particularlyin samples lean in V. With an increase in the V level, as the �/� + � transustemperature comes down, the departure from the equilibrium values of volumefraction and composition is larger, suggesting a more incomplete partitioning ofalloying elements in the �- and the �-phases. The reduction in the length ofthe plates is also consistent with a slower growth kinetics of the plates and anincreased gap in the composition of � and �. In regions corresponding to V levelof about 5–8 at.%, the presence of a bimodal size distribution of �-plates can beattributed to a two-stage decomposition process – primary plate forming when thetemperature of deposited alloy drops below the �/�+� transus and the secondaryplates subsequently appearing in the retained �-phase during later heating cycles.

Crystallographic information from these samples at a mesoscopic level (typi-cally at a magnification of 2000×) obtained from orientation imaging microscopy(01 m) has yielded an orientation distribution map for the matrix � and the differentvariants of the �-phase (Figure 9.4). As per the Burgers relationship, six equivalentvariants of planar matching exist, where the basal (0001)� plane is parallel to oneof the six variants of the 110� plane of the �-phase. Again for a given variant ofplanar matching (say (0001)��(110)�), there are two distinct options of directionalmatching, namely [1120]��[111]� and [1210]��[111]�. OIM has identified all the12 orientation variants and provided quantitatively the area fractions occupied byeach of them. The orientation distribution information presented in pole figures(Figure 9.4) corresponding to {110}�, {111}�, (0001)� and 1120� poles have one-to-one correspondence with the 110� poles and the �111�� poles matched with the�1120�� poles. Observations like this not only validate the Burgers relationship,but also show special disposition of orientation variants in specific locations. Ithas been shown recently (Bhattacharyya et al. 2003) that �-phase forming at �grain boundaries (described in Chapter 7 as “Grain Boundary Allotriomorphs”)



Epilogue 791

10 μm

10 μm

Ti–1.8 at.% V Ti–3 at.% V

Ti–6.8 at.% VTi–5 at.% V

Ti–10 at.% VTi–8 at.% V

Figure 9.3. Laser-processed (as-cast) sample with composition gradient shows variation in theaspect ratio of �-plates and in the volume fraction of the �-matrix (lighter shade) with increasingV contents.

often correspond to a stack of two alternate orientation variants which have thesame planar matching (say (0001)��(110)�) with directional matching alternat-ing between [1120]��[111]� and [1210]��[111]�. According to the nomenclatureintroduced in describing martensite variants (Chapter 4) these are designated as(�−�+) and (�+�+), respectively.




Figure 9.4. Distribution of different orientation variants of �-plates in �-matrix as revealed fromorientation imaging microscopy.

Microstructural examinations of a second slice of the same compositionallygraded Ti–V bar – which has been given the �-quenching treatment – reveal thepresence of lath martensite in the region corresponding to 0–3% V, plate martensitein the region having 3–11% V and nearly complete �-phase stabilization in theregion of V > 12%.

Microstructural observations on a finer scale from selected regions have becomefeasible due to the availability of FIBS technique. TEM examination of samplespicked up from regions of interest reveals the following:



Epilogue 793

g = 1012

500 nm2 μm

Figure 9.5. Parallel stacking of �-laths with small angle boundaries laths.

(1) Regions of the laser-processed sample in the composition of 1–2% V showWidmanstatten �-laths, contiguously stacked with small angle boundariesbetween the adjacent laths. In addition to the interfacial dislocations, an arrayof discrete fine �-plates are seen (Figure 9.5) along these interlath boundaries.These fine plates again exhibit an internal substructure akin to martensiticsubstructure (Figure 9.6). Based on this observation, Banerjee et al. (2003)have proposed that the interlath plates indeed form by a martensitic process.This is possible because under the non-equilibrium cooling condition prevail-ing during laser processing, partitioning of V in the �- and the �-phases is notcomplete. Under such a situation, the metastable �+�/� transus can intersectthe Ms line making the martensitic transformation possible within the �-layerat the interlath boundary.

(2) In regions having average V content above 5%, primary and secondary �-platesare seen to be distributed in the �-matrix (Figure 9.7) which exhibits a mottledstructure. Diffraction patterns corresponding to the �-matrix clearly indicatethe presence of both �- and �-reflections. For the region corresponding toan average composition of 5% V, the �-matrix (containing 15% V) showsa distribution of fine �- and �-particles. With further V enrichment of the�-matrix (17.5% V), only �-particles are seen to be present. This is consistentwith the general trend of enhancement of the stability of � in comparison tothat of � as the �-phase is enriched with V.

(3) In regions containing average V content above 20%, the �-phase is retained.However, the �-phase with 20% V has been found to be amenable to stress-induced � → � transformation as revealed from nanoindentation experiments.

Experimental results on the microstructure of laser-processed compositionallygraded sample of about 1 in. length are presented here to illustrate that some ofthe major trends of phase transformations of titanium (and also zirconium) alloyscan all be identified from such a small sample. Some of the recent experimental




g = 1010

g = 1011g = 1011

g = 0002

200 nm

100 nm

Figure 9.6. Vanadium-enriched interlath regions exhibiting martensite-like substructure.

techniques have made it possible to study a whole range of phenomena in a singlesample. Metallurgists for many years have been using Jominy test bars for achiev-ing graded quenching rates in a single sample. The present trend of combinatorialmaterials science can explore, in a similar manner, a wide range of phase transfor-mations in a single compositionally graded sample. Such experimental techniqueswill no doubt complement the recent theoretical development of first principlepredictions of phase stability of alloys.

In this concluding chapter, it is worthwhile to draw a comparison betweenthe observed crystallographic features of diffusional, martensitic and mixed-modetransformations encountered in titanium and zirconium alloy. As mentioned ear-lier, an approximate � − � Burgers orientation relationship is a dominant fea-ture in all types of transformations discussed, namely, martensitic transformation,Widmanstatten �-precipitation, -hydride precipitation and active eutectoiddecomposition. This statement, however, is being made by not distinguishing the



Epilogue 795

Figure 9.7. �-plates in V-stabilized �-matrix containing dispersion of �-particles.




Burgers and the Potter relations, which differ by a rotation of 15�. The rangeof transformations discussed involves shear, diffusional and mixed-mode mecha-nisms, and it is attractive to examine the applicability of the invariant line strain(ILS) and the invariant plane strain (IPS) criteria in determining the habit planesof advancing transformation fronts associated with these transformations.

The macroscopic habit plane for the martensitic plates and laths can be preciselypredicted on the basis of the IPS criterion. The interfaces of a martensite lath consistof arrays of �c+a� dislocations, suggesting the operation of the �1011�� 1123��

slip mode as the lattice invariant shear (LIS). Since the LIS is accomplished bythe passage of these dislocations, it is necessary for the line vectors of thesedislocations to be along the invariant (IL) vector. The lath boundary structure,however, is more complex, as in the lath martensites, the parent �-phase is fullyconsumed and adjacent plates come in contact, resulting in a superimposition oftheir interface structures.

In twinned martensitic plates for both (� − �+) and (� + �+) solutions, forthe two twin-related variants to remain in coherence with the parent �, it isgeometrically necessary for them to meet along the IL vector. The same argumentis also valid for a group of three martensite crystals that form the indentationmorphology. These self-accommodating groups of crystals meet along a line thatis the IL. The line of intersection of habit plane segments corresponding to twoadjacent twin-related martensite variants within a twinned plate following the(�−�+) solution of the Bowles–Mackenzie analysis matches the ILS direction.Martensite plates that obey the (�+�+) solution contain a stack of thin twins,which again intersect the habit planes along the ILS direction. The geometriesof the habit planes in these cases are schematically illustrated in Figure 9.8(a)and (b).

The habit planes of Widmanstatten laths/plates have also been found to beirrational. However, habit plane predictions based on the IPS criterion do not matchthose experimentally observed. Two types of Widmanstatten products, namely,single crystal laths and internally twinned plates, have been encountered. The lathinterface exhibits an array of dislocations with �c + a� Burgers vectors, wheredislocation line vectors are parallel to the ILS direction. In fact, the growth directionof the laths is marked by the line vector of these interfacial dislocations, which liealong all four surfaces parallel to the long axis of the laths. It is also seen that thehabit plane poles are located in the vicinity of the pole, where [112]� and 1010��

come in coincidence, consequent to the operation of the Burgers relation. Thisobservation is consistent with that of Furuhara and Aaronson (1991) who reportedthat the habit of � (hcp) plates in the matrix of � (bcc) Ti–Cr alloys lies close to�130�� and �131�� poles, which are located near the [112]�� 1010�� directions.They also invoked a periodic occurrence of structural ledges, which accounts for



Epilogue 797

(433)β (433)β

IL

(1011)α

Martensite Class A (α−ω+) thick 1011 twins

(a) (d)

IL

(101

1)α

IL

<c+a>

Habit (301)β-(301)β

α Precipitate in β

(b) (e)

Martensite Class (α+ω+) thin 1011 twins α Precipitate in β

β(112)β(211)β

IL

Average habit(301)β-(311)β

(101

1)α/

/(11

0)β

Transformationfront

β′α

Active Eutectoid Decomposition β→α+β′

IL

Habit(301)β-(311)β

(111

)γ

γ Hydrie in β

(c) (f)

IL

(101

1)α/

/(11

0)β

α α α

Figure 9.8. A schematic illustration of habit planes/transformation fronts in various types of diffu-sional, martensitic and mixed-mode processes. The role of the IL vector in all processes is revealed.

the rotation of the average habit interface from the terrace plane (112)��1010��

on which patches of good atomic fit across the �/� interface exist.The criterion of the selection of the habit plane in a diffusional transformation

has been discussed in Banerjee et al. (1997) and references therein. There is ageneral agreement that one of the vectors that defines the habit plane is the ILSdirection. The second vector fulfils one of the following conditions:

• a vector that remains unrotated,• a direction of low strain,• a direction of easy slip in the parent or product phase,• a direction favoured due to the elastic anisotropy of the matrix–precipitate

assembly and• a direction along which structural ledges are aligned.




In a recent article, Zhang and Purdy (1993a,b) have shown that using an exten-sion of the concept of the O-lattice to a system containing invariant lines, acomplete Burgers vector balance can be accomplished at the �/� interface of theZr–2.5Nb alloy by a single set of dislocations with the Burgers vector [010]� anddislocation spacing of 10 nm. This prediction remains valid for Zr–20Nb also, asthe lattice parameters of the Zr–20Nb alloy are not significantly different thanthose for the Zr–2.5Nb alloy. The observed interfacial structure (a single set ofparallel dislocations along the ILS direction) (see Figure 7.91) and the exper-imentally determined habit plane indices are consistent with the predictions ofZhang and Purdy (1993a,b) who have shown that the habit plane is given by anO-lattice plane associated with a plane of the reference lattice, which contains theBurgers vector of the dislocations. The habit plane envisaged in this case differsfrom that predicted from the atomistic model (see Section 7.5.4) proposed in asimilar system of Ti–Cr alloys, though the indices of the macroscopic habit planeand the orientation relation are quite similar in the two cases. Matching of theatomic planes across the �/� boundary in Ti–Cr alloys has been achieved by theintroduction of biatomic structural ledges with Burgers vector 1

6�a� and of misfitcompensating �c�-type ledges. It appears that in the case of the �/� interface inZr–Nb alloys, the misfit along the �c� and �a� components is compensated bya single set of �c + a� dislocations lying along the ILS direction (Figure 9.8(c)and 7.91).

Examples of internally twinned Widmanstatten �-plates have been reported inBanerjee et al. (1997). These plates exhibit �1011� twinning and a habit plane polelying between �130�� and �131��. This habit does not match that predicted fromthe IPS consideration. A detailed analysis of the macroscopic facets that constitutethe average habit plane could not be performed with sufficient accuracy. However,the major facet appeared to be close to the [112]��[1010]� planes. It is, therefore,quite attractive to consider that the macroscopic habit plane is made up of steps of[112]��[1010]� planes on which good atomic fit can be established. The directionalong which two adjacent twin variants meet on the habit plane has been found tobe the ILS direction (Figure 9.8(d)).

The crystallography of precipitation of the -hydride plates in either the �- or �-matrix obeys the IPS criterion. As mentioned earlier, this transformation involvesa shear transformation of the � or � zirconium lattice, with the accompanyingprocess of hydrogen partitioning. For such a transformation, the phenomenologicaltheory of martensite crystallography provides an accurate prediction of the habitplane. The habit plane retains the direction of the IL, which is defined by the lineof intersection of the habit plane and the twin plane (Figure 9.8(e)).

Active eutectoid decomposition of the parent �-phase occurs through a coop-erative growth of the �- and metastable �′-phases. Based on crystallographic



Epilogue 799

observations, it can be inferred that the Burgers relation exists between � and �,while the �/�′ relationship is based on the parallelism between their respectivecube axes. The line along which the �/� interface (which is parallel to �1011��)intersects the transformation front is the IL vector (Figure 9.8(f)).

Crystallographic descriptions of different types of transformation products, asschematically illustrated in Figure 9.8, bring out the importance of the IL vector indefining the habit plane (or the transformation front). The tendency of maintainingat least one undistorted vector on the advancing transformation is not unexpected.It is increasingly being realized that either partial or full coherency is maintainedin a great majority of solid-state phase transformations, both martensitic anddiffusional. This point has been amply demonstrated in the case of transformationsin titanium- and zirconium-based alloys. The operation of Burgers or near Burgersorientation relations has an overwhelming influence on all types of transformationsin titanium- and zirconium-based alloy systems. The diffusional transformationsare, however, distinguishable from displacive transformations from the nature ofthe habit plane. While the habit plane fulfilling the IPS criterion is a necessity indisplacive or diffusional-displace (e.g. formation of -hydride) processes, the habitplanes in pure diffusional transformations are determined by the strong tendencyof maintaining regions of good fit between the parent and the product phases.Since in the latter case interfaces often deviate from being planar, a rotationof the boundary is achieved by rotating the interface around the invariant linevector. A typical �-lath formed through a diffusional process possesses an interfacecharacterized by an array of �c +a� dislocations, which are all aligned along theIL vector.

Even for the active eutectoid decomposition, which involves partitioning ofsubstitutional alloying elements, the two product phases appear to grow in such amanner that the crystallographic registry of the parent is maintained simultaneouslywith two product crystals. The velocity of interface motion is, therefore, limitedby the requirements of chemical diffusion for attaining the appropriate extentof alloy partitioning and for the ordering processes within one of the productphases.

In the preface of the book, we have highlighted the diversity of phase transfor-mations in titanium- and zirconium-based alloys and subsequently have justifiedthat the whole subject of phase transformations can be covered using examplestaken from these alloys. Finally, we are only emphasizing that this wide rangeof transformations are driven by only a few trends of structural and chemicalinstabilities. It is also noted that different transformation mechanisms leave theirimprints on the crystallographic and morphological features of the transforma-tion products and it is through these experimental observables one deciphers therelevant operating mechanisms.




REFERENCES

Banerjee, S., Dey, G.K., Srivastava, D. and Ranganathan, S. (1997) Metall. Trans., 28A,2201.

Banerjee, R., Collins, P.C., and Fraser, H.L. (2002) Metall. Mater. Trans. A, 33, 2129.Banerjee, R., Collins, P.C., Bhattacharayya, D., Banerjee, S. and Fraser, H.L. (2003) Acta

Mater., 51, 3277.Bhattacharyya, D., Viswanathan, G.B., Denkuberger, R., Furrer, D. and Fraser, H.L. (2003)

Acta Mater., 51, 4679.Furuhara, T. and Aaronson, H.I. (1991) Acta Metall. Mater., 39, 2857.Zhang, W.Z. and Purdy, G.R. (1993a) Acta Metall. Mater., 41, 543.Zhang, W.Z. and Purdy, G.R. (1993b), Philos. Mag., 68, 291.

Elsevier UK Code: PTA Index-I042145 5-6-2007 9:33a.m. Page:801 Trim:165mm×240mm Integra, India


Index

� alloys 21, 22, 284, 687� alloys 21, 284, 504–506, 539, 567,

689–690, 701�-hydride 120, 721, 722, 728, 729, 730, 733,

736, 737, 738, 794, 798�-hydride (fct) phase 721�-hydride 722, 723�-hydride 722, 723, 728, 739�-phase immiscibility 606–609�-phase 5, 10, 21, 26, 49, 189, 217, 223,

447, 529, 656, 683�-phase 5, 9, 13, 21, 22, 25, 43, 48, 50,

52–53, 134, 145, 283, 310, 384, 421, 447,475–484, 486, 488, 512, 523, 559, 599,609–615, 632, 655, 688, 739, 753, 787

�-phase 6, 7, 9, 10, 12, 13, 26, 35, 36, 43,44, 50, 52, 251, 284, 490, 492–494, 504,512, 516, 529, 536, 544

�-precipitation 492, 536, 537, 539–540�-stabilizers 21, 22, 154�-stabilizers 21, 25, 53, 174, 325, 383, 417�-stabilizing elements 21–23, 148, 175,

251, 281, 303, 310, 417, 475, 516,587, 609

�-transformed 688, 689, 693, 694, 698,699, 700

�-transition 9, 12, 106, 419, 474, 481–484,491–495, 520

�+� alloys 21, 22, 284, 327, 609, 632,687–689, 696, 729

�–� lattice correspondence 485�–� lattice correspondence 488�–�-isomorphous systems 27�/� interface 559, 646, 647, 650, 657, 737,

738, 799�2-phase 382, 383, 418, 420, 421, 423, 434,

443, 445, 447, 448, 451, 453, 458,662–670

�-eutectoid systems 27, 622�-isomorphous systems 27, 29, 30, 606,

609, 620�-TiCr2 33, 62

�-TiCr2 33, 62�-TiCr2 33, 62, 622

A15 (cP8, Cr3Si type) 54–62ab initio phase diagram calculations 16Accommodation strain energy 719,

725, 726Accommodation stress 264“Active” eutectoid decomposition 560Affine transformation 103, 267Aged � 45, 50, 475, 476, 492, 538, 547Ageing 44, 50, 53, 421, 435, 448, 453, 475,

529, 533, 539, 546, 603, 658, 706,728, 772

Allotrope 5, 286Amorphous alloys 162, 163, 165, 176–179,

183, 204, 205, 207, 209, 210, 250Amorphous phase 160, 161, 164, 165, 175,

176, 178, 182, 183, 185–187, 191, 192,203, 207, 208, 210, 217, 220–229,232, 233

Anomalous diffusion behaviour 13, 568Antiferroelectric 109Antiferromagnetic 109Applied stress 263–265, 324, 331, 340, 350,

354, 355, 370, 725, 727, 747Athermal 43, 44, 49–52, 101, 105, 107, 109,

110, 264, 265, 277, 278, 280–283, 338,339, 362, 474–477, 486, 490–492, 494,495, 500, 501, 508–510, 512, 517, 529,539, 540, 546, 560

Athermal � → � transition 50, 475, 486,491, 501, 512

Athermal martensitic 43, 278Athermal nucleation 110Atomic layer stacking 55Atomic shuffles 111, 316, 320,

504, 505Atomic site correspondence 632, 650Austenite 47, 121, 153, 260–262, 267, 269,

270, 279, 324, 325, 354, 355, 358, 359,371, 673

801



802 Index

Autocatalytic 305, 322Avrami exponent 193, 197, 202, 204, 211

B2 (cP2, CsCl type) 54, 60, 61B82 141, 156, 210, 441, 518, 519, 521, 525,

527, 528, 529, 530, 533, 535, 785Bain distortion 269, 272, 292–296, 298Bain strain 45, 269, 273, 274, 276, 293, 294,

314, 316, 321, 431, 432, 504, 643Bainite-like transformation 742Bainitic transformation 121, 122, 737Ball milling 226, 228, 229Basket weave morphology 421, 637, 656BaTiO3 111, 113, 114, 260BCC special points 404Bf (oC8, CrB type) 54, 60Black plates 638, 648Blister formation threshold 744, 751Boiling water reactors 771Bridge 351Bulk Metallic Glasses 157, 158, 171, 205Burgers 45, 46, 276, 289, 293, 294,

302–304, 310, 312, 314–316, 332,334, 336, 339, 420, 421, 424, 538,539, 547, 558, 606, 613, 614, 618,633, 635, 642–644, 646, 648–651,655, 679, 730, 736

Burgers correspondence 294, 421, 679,730, 736

Burst 278, 362, 547, 550, 743Burst-like behaviour 362

C14 (hP12, MgZn2 type) 54, 59, 61C15 (cF24, Cu2Mg type) 54, 59, 61C16 (tI12, CuAl2 type) 60C32 (hP3, AlB2 type) 60Cellular precipitates 635Ceramics 55, 62, 63, 90, 92, 106, 107, 261,

369, 371, 372, 558, 559Chemical bonding 12Chemical diffusion 555, 570, 578, 799Chemical free energy 202, 238, 261, 263,

264, 278, 287, 324, 371, 456, 580, 592,593, 612, 618, 675, 725, 726

Chemical potential 98, 129, 130, 132, 152,285, 395, 530, 571, 577, 589

Chemical spinodal 558, 594, 601, 602,621, 622

Classification 21, 27, 28, 57, 90–93, 101,105–107, 111, 115, 283, 284, 540, 587,633, 722

Clausius–Clapeyron 8, 359Cluster approximation 394, 395, 396, 398,

399, 412Cluster variation method 392Coherency 65, 99, 240, 264, 265, 350, 370,

504, 593, 606, 614, 618, 633, 650,662, 677

Coherent phase transformations 386Coherent spinodal 99, 525, 558, 594, 601,

618, 620, 621, 622Cold rolling 702Composition-invariant transformations 91,

110, 435Composition modulation 525, 593, 594,

616, 618Composition-induced destabilization of

crystalline phases 220Compressive 290, 702, 769, 770Concentration modulation 99, 381, 438, 557,

592, 602, 603, 622Concentration wave 99, 117, 120, 183,

379, 390, 391, 392, 393, 406, 416,437, 438, 459, 460, 524, 525, 527,528, 606

Concomitant clustering and ordering 407Configurational energy 393, 395, 776Constitutional solution treatment 753Continuous or homogeneous transitions 97Cooling rate 48, 128, 141, 142, 144–151,

153, 158–160, 165, 171–173, 175, 180,182, 205, 281, 383, 420, 421, 433, 434,446, 448, 450, 452, 626, 627, 629, 655,656, 708, 725, 728, 739, 753

Coordination factor 17Coordination numbers 59, 775Correlation functions 393, 395, 397,

398, 399Correspondence matrices 487, 489, 730Correspondence variant 294, 343, 348, 350,

665, 734Corrosion 157, 586, 587, 659, 661, 702, 721,

741, 769, 771, 772



Index 803

Coupled transformation 122Cross slip 332, 333, 688, 698Crystallization 92, 110, 111, 145, 157, 158,

160, 170, 172, 175–178, 181, 182,184–189, 191–197, 200–205, 207,209–211, 213, 220, 250

Crystallization kinetics 193, 200, 210, 211Crystallographic texture 701, 702, 704, 706,

707, 743, 745Crystallography 44, 50, 266, 269, 270, 282,

290, 292, 303, 320, 340, 342, 345, 366,372, 418, 424, 474, 484, 534, 613–615,648, 657, 662, 665, 720, 721, 728–730,732, 735, 736

Cu-Zn-Al 362

d-band occupation 12d-band 13, 18–20, 508, 516–518D019 54, 55, 59, 61, 62, 227, 228, 229, 377,

382, 383, 407, 412, 413, 416, 417, 418,420, 421, 423, 427, 428, 430, 431, 436,437, 438, 439, 440, 441, 442, 444, 447,451, 785, 786

D88 54, 60, 520, 521, 525, 527, 528, 529,530, 535

Deformation twins 331, 334, 546Degradation Processes 741Degree of self-accommodation 308, 666Delayed hydride cracking 742, 743, 744,

745, 746Dendritic 137, 141, 145, 146, 150, 173, 175,

189, 203, 212, 690Density functional theory 14, 15, 385Density of states 18, 19, 384, 516DHC 746, 747DHC velocity 746Diad 296, 346, 351Differential scanning calorimetry 340Diffraction effects 51Diffuse Scattering 51, 112, 249, 471, 474,

495, 499, 533Diffusion anisotropy difference 565Diffusion bonding 555, 584, 585, 586Diffusion mechanisms 157, 555, 560Diffusion zone 559, 573, 574, 578, 579, 584,

585, 586, 587

Diffusional 43, 48, 50, 65, 101, 103–107,109–111, 120–122, 129, 135, 148, 149,185, 204, 266, 281, 293, 419, 453, 500,504, 520, 558–560, 623, 625, 632,642–645, 648, 650–653, 655, 657, 662,683, 738, 739, 779

Diffusional transformation 103, 120, 558,632, 643, 650, 785, 799

Dilation parameter 276Discontinuous coarsening 450, 456, 457, 675Discrete transformations 97, 109, 110Dislocations 49, 121, 264, 276, 279, 280,

310, 312, 315, 316, 327, 329, 331–335,337–339, 342, 387, 441, 451–455,537–540, 546–548, 550, 562, 606, 626,633, 645, 646, 648–652, 661, 677, 683,684, 687–689, 692, 698, 710, 739, 753

Displacement ordering 98, 156, 419,501–503, 509, 518, 550

Displacement 43, 101, 111, 114, 120, 431,451, 506–508, 786

Displacement vector 316, 317, 319, 431, 432,451, 454, 473

Displacement wave 43, 101, 120, 419, 427,485, 486, 499, 500, 506, 510, 513, 527,528, 531, 532, 533, 786

Displacive 43, 101, 109, 113, 120–122, 386,389, 419, 425, 426, 429, 430, 492, 495,518, 519, 525, 534

Displacive transformation 43, 100, 101,103–107, 111, 120, 154, 261, 492,518, 559, 650, 721

Distorted hexagonal 46, 281Divacancies 561, 562Domain structure 319, 419, 428, 767Dynamic recovery (DRV) 683, 685, 686,

687, 691, 694Dynamic recrystallization (DRX) 683, 687,

694, 698, 701, 706Dynamic strain ageing 539, 540,

542, 546

Effective full power days (EFPD) 769Elastic energy 263, 323Electron concentration (e/a) factor 17Electron diffraction 51, 52, 118, 174, 175,

341, 362, 495, 533, 603



804 Index

Electron to atom (e/a) ratio 24Electronegativity factor 17Electronic structure 13–15, 24, 384, 385,

386, 393, 412, 413, 507, 510Ellipsoidal inclusion 306Embrittlement 181, 182, 471, 536, 537,

742, 746Enantiotropic 92Euler’s formula 300Eutectic crystallization 111, 185, 187, 192,

194–197, 202–204Eutectoid Decomposition 22, 27, 28, 65, 91,

111, 153, 186, 447, 448, 556, 560, 670,671, 672, 673, 674, 675, 676, 681, 682,722, 794, 798, 799

Exchange mechanisms 561Exothermic 182, 184, 187, 189, 194, 207,

210, 340, 360, 361, 722Extended defects 562

FCC special points 407Fe24Zr76 191Fe30Zr70 191Fe40Zr60 191Fe–B 157, 162, 177, 187, 320, 356Fe-Ni-Co-B 162Ferroelectric 100, 106, 107, 111,

113, 114Ferromagnetic 94, 107, 116, 162Ferrous 47, 260, 263, 277, 279, 287, 320,

322, 326, 332FeZr2 191FeZr3 191Fick’s laws 555, 562First principles 11, 14, 17, 18, 117, 386, 393,

412, 503, 507, 510First-order transformation 94, 97, 101, 105,

128, 262First-order transitions 5, 93, 98, 107, 115Flow localization 685, 691, 695, 701Flow softening 691, 693, 696, 698, 701Flow stresses 260, 328, 710Fractal morphology 321, 637Fractal 308, 313, 637Fracture 64, 145, 226, 327, 352, 353, 370,

371, 537, 539, 547, 655, 690, 691, 721,742, 747

Gas atomization 150Gd–Co40–50 163Gd–Fe32–50 163Geometrically close packed (GCP)

structures 57Glass formation in diffusion couples 215, 220Glass formation 134, 157, 158, 160, 161, 165,

168, 171, 205, 206, 213, 215, 220, 226Glass transition 158, 161, 181–185, 193, 194,

200, 205, 209, 210, 218Glass-forming abilities (GFAs) 157, 158,

160, 171Glissile 105, 120, 121, 277, 278, 350, 651, 680Gliding 332, 337, 504Glissile interface 105, 120, 121, 278Grain boundary allotriomorphs 633, 664,

674, 690Grain-boundary diffusion 769Ground states 377, 381, 397, 398, 403Group number 18, 24, 26, 668, 669Group/subgroup relations 5, 377, 424Growth 49, 97, 98, 101, 105, 109, 110, 113,

116, 120, 122, 128, 129, 135–137, 139,140, 144–146, 155, 158, 162, 164, 165,172, 175, 177, 185, 188, 189, 191–194,196, 197, 199, 200, 202–206, 208, 211,212, 223, 225, 264, 265, 277–280, 308,314, 315, 322, 355, 358, 362, 381, 421,442, 451–453, 455, 457, 492, 494, 499,504, 510,527, 539, 547, 559, 581, 582,585, 597, 614, 623, 625, 626, 628–630,634, 635, 637, 639, 641, 648, 650–653,655, 657, 658, 661, 666, 668, 669–671,673, 674, 681, 683, 698, 702, 707, 721,726, 746, 747, 751, 769

Growth ledges 637, 651, 652

Habit plane 45, 109, 110, 270, 293, 303, 314,348, 366, 368, 666, 732, 799

Habit plane variants 277, 313, 314, 348, 350,352, 677, 679

Hafnium (Hf) 4Hall–Petch constant 337, 338Hall–Petch relation 337, 338Hardening 326, 327, 329, 335, 336, 337, 339,

341, 536, 538, 604, 662, 683, 687HCP special points 406



Index 805

Heterogeneous 165, 169, 171, 175, 183, 213,221, 222, 279, 381, 426, 451, 455, 533,620, 625, 640, 661, 689

Heterogeneous nucleation 165, 169, 171, 175,213, 222, 279, 455, 553, 620, 640, 661

Heterophase fluctuation 101, 279, 589Hexagonal, H net 55Homogeneous deformation 102, 103, 267,

268, 343, 345, 505, 643, 689, 735Homogeneous nucleation 165, 170–173, 206,

278, 279, 527, 625Homogenization treatment 758Homophase fluctuation 589, 594Hume-Rothery phases 384Hydride blister 742,743, 744, 747, 749, 751Hydride blister formation 742, 749Hydride precipitation 717, 721, 726, 727,

736, 742, 747, 794Hydrogen absorption 754, 755, 758, 760, 761Hydrogen charging 225Hydrogen ingress 738, 741, 742Hydrogen migration 737, 743, 744, 746, 747Hydrogen storage capacity 756, 758, 761,

762, 764Hydrogen storage materials 55, 62, 754, 756,

758, 759, 760, 761, 764Hydrostatic 215, 479, 494, 499, 508, 719,

720, 727, 744Hysteresis 9, 10, 50, 353, 354, 356, 362, 475,

479–481, 726, 756, 758, 761, 764

Icosahedral phases 248–251Incommensurate �-structures 509, 515Inhomogeneous shear 45, 47, 49, 275Instability diagrams 410Instability temperature 98, 107, 115, 528Interaction energies 14, 380, 403, 427, 465,

727, 775Interdiffusion 127, 128, 204, 224, 225, 555,

557, 559, 570, 572, 573, 574, 576, 577,578, 585, 592, 604

Interface instability 146, 657Interface phase 737, 738Interfacial energy 206, 238, 263, 264, 280,

456, 458, 504, 580, 592, 604, 644,657, 725

Interfacial equilibrium 129

Interfacial structure 555, 648, 650, 652, 798Interlayers 225, 585, 586Intermetallics 17, 54, 55, 57, 61, 90, 92, 106,

107, 111, 231, 261, 356, 382, 474, 558,662, 669, 754

Intermetallic compounds 22, 153, 218, 225,231, 414, 458, 559, 587, 657, 758, 787

Intermetallic Phases 26, 33–35, 38, 43,53–55, 57, 59, 61, 62, 160, 162, 164,175, 188, 284, 384, 414, 443, 518, 525,560, 615, 638, 657, 658, 676, 754

Internal twinning 728, 729, 737Internal twins 267, 276, 293, 304, 312, 316,

347, 635, 736Interstitial ordering 719, 720, 728, 737, 764,

772, 779Interstitially ordered superstructures 764Interstitials 231, 232, 236, 339, 382, 561, 765Intraplanar ordering 767Intrinsic diffusivities 571Invariant line strain 276, 555, 559, 614, 642,

643, 796Invariant line 276, 302, 342, 504, 555, 559,

614, 642, 643, 644, 645, 646, 648, 650,655, 663, 679, 796, 798

Invariant plane strain (IPS) 45, 47, 110, 260,266, 504, 632, 728, 737, 796

Invariant plane strain condition 45, 110, 504,645, 734, 737, 779

Invariant plane strain transformation 45, 47,276, 632, 651, 732, 734, 779

Invariant plane 45, 47, 110, 260, 267, 276,314, 504, 532, 644, 651, 654, 728,732, 734

IPS condition 270, 272, 275, 291, 294, 304,315, 643, 662, 665, 733, 735, 736

Irrational 244, 246, 267, 333, 347, 633,648–651, 732

Isomorphous 22, 26–30, 282, 285, 287, 327,418, 475, 586–588, 606, 609, 616,620, 765

Isothermal kinetics 194, 196, 202, 264

Kagome net or K net 55Kirkendall effect 572, 585



806 Index

L12 54, 55, 59, 61, 201, 228, 382, 390, 391,392,402, 408, 409, 413, 436, 439, 440,441, 442, 529, 530, 533

L1o (tP4, AuCu type) 54, 59, 61La78Ni22 163Lamellar 111, 382, 420, 440, 441, 443, 444,

448, 450, 451, 452, 453, 454, 456, 457,629, 635, 671, 673, 675, 676, 679, 681,682, 688, 691, 696, 697

Lamellar eutectoid 635Lamellar microstructure 450, 451, 691Lamellar structure 443, 450, 456, 457, 673,

675, 676, 681, 682, 697Lath martensite 48, 308, 310, 320, 328, 329,

331, 332, 437Lath morphology 48, 277, 310Lattice collapse mechanism 471, 474, 492,

498, 499, 504, 509, 518, 527Lattice correspondence 45, 103, 289, 305,

342, 363, 368, 431, 485, 488, 515, 786Lattice distortion matrix 731Lattice-invariant shear (LIS) 45, 260, 269,

271, 293, 295, 296, 312, 314, 315, 346,347, 366, 368, 504, 651, 732, 733,779, 796

Lattice parameters 7, 46, 51, 227, 290, 345,427, 443, 512, 520, 602, 613, 644, 723,786, 798

Lattice shear 44, 46, 120, 270, 271, 345, 532,632, 737, 777

Lattice-site correspondence 560Lattice strain dominated 111Lattice strains 45, 46, 100, 106, 111, 269,

270, 271, 431, 663, 731, 786Laves phase 55, 59, 60, 62, 235, 250, 251,

622, 658, 754, 756, 759, 761Linear muffin tin orbital (LMTO) 9, 11, 15,

19, 385, 412Liquid/solid interface 129, 134, 135, 146Local density approximation (LDA) 11, 14,

15, 16, 385, 386Localized hydride-embrittlement 746Long-range order (LRO) 387

Mackay cluster 251, 252Macroscopic flow behaviour 335Macroscopic shape deformation 109

Macroscopic shape distortion 300, 301Macroscopic shear 44, 266, 270, 274, 298,

303, 372, 504Macroscopic strain 264, 270, 275, 504, 547,

704, 734Macrosegregation 141, 143–145Martensite interfaces 264, 280, 680Martensite 26, 30–32, 35, 43–49, 106,

110–112, 153, 155, 260, 261, 263–267,269–271, 273–281, 283, 287, 290,293, 295, 296, 302–310, 313–317,319–329, 331–340, 347–352, 354,355, 358–362, 365, 368, 370, 372,418, 421, 437, 504, 532, 546, 559,609–618, 626, 635, 640, 645, 651,666, 680, 720, 732, 735, 753

Martensite phase 26, 43, 44, 48, 279, 280,305, 354, 355, 357, 421, 618

Martensitic transformation 25, 26, 43–45, 47,49, 100, 101, 105, 106, 109–111, 113,114, 155, 260–266, 270, 275–279,281–285, 287, 288, 290, 292, 293,306, 312, 324, 326, 327, 339, 342,348, 352, 354–356, 370–372, 418,419, 447, 475, 492, 494, 500, 504,530, 623, 627, 632, 642–644, 646,651, 653, 663, 777–779

Martensitic 25, 26, 34, 35, 38, 43–47, 49, 64,100, 101, 105, 106, 109–114, 141, 148,149, 149, 153–155, 260–266, 270,275–285, 287, 288, 290, 292, 293, 297,302, 304, 306, 312, 319, 320, 322, 324,326–328, 331, 337–340, 342, 348, 350,352–356, 361–363, 368, 370–372, 418,421, 439, 447, 475, 477, 482, 492, 494,500, 504, 530, 546, 559, 609, 623, 626,627, 632, 642, 643, 644, 651, 653, 658,662, 663, 668, 704, 729, 736, 738, 769,777–779

Massive 48, 106, 109, 110, 185, 205, 217,221, 223, 281, 421, 447, 448, 450,453–456, 559, 623–626, 628–630, 671

Massive transformation 106, 109, 110, 185,205, 421, 447, 448, 450, 453–456, 559,623–626, 628–630

Matano interface 574Maximum resolved shear stress 303Mean field 94, 385, 388



Index 807

Mean free slip length 332, 334, 335, 337Mechanically driven systems 226Mechanism 101–104, 228, 451–453,

499–518, 560–562, 589,648, 737

Melt extraction 150Melt spinning 150, 151, 173, 191Metallic glasses 110, 134, 157, 158, 162,

164, 165, 171, 176–179, 181, 183–185,187, 193, 205, 207, 210

Metal–metal glasses 181, 187, 188, 204Metal–metalloid glasses 187, 204Metastable 23, 35, 53, 92, 129, 215, 403,

410, 502, 611, 671, 793Metastable Zr3Al 337, 382, 437, 441, 530Mf 47, 48, 110, 278, 283, 350, 353–356,

360, 361, 363Microsegregation 141–144Microstructural evolution 235, 555, 603, 656,

683, 684, 685, 691, 697Minor orientations 314Mirror plane 272, 274, 275, 293, 296, 314,

315, 342, 346, 351, 356, 614, 679Miscibility gap 29, 31, 35, 52, 54, 98, 99,

409, 410, 555,558, 588, 589, 596, 597,598, 599, 601, 609, 621, 622, 787

Misfit compensating ledges 650Mixed diffusive/displacive

transformation 120Mixed Mode Transformations 115Modes of crystallization 185, 200Molar free energy 131, 285, 435, 571, 589,

598, 625Molecular dynamics (MD) 16, 235, 385Monoclinic 14, 41, 54, 62–67, 113, 340, 345,

347, 348, 362–366, 369–372, 770Monotectoid reaction 29, 31, 32, 35, 53,

187, 555, 559, 587, 606, 607, 609,621, 787

Monotropic transitions 92Monovacancies 561, 562Monte Carlo (MC) 14, 16Morphological stability 135, 138, 139Morphologies 48, 51, 172, 173, 277,

304, 309, 343, 500, 626, 633,638, 640

Morphology and substructure 48, 304

Ms 44, 47, 48, 101, 110, 112, 113, 140,154, 155, 262, 263, 265, 277, 278,281–283, 287, 288, 303, 310, 312,313, 320, 322, 324, 325, 331, 361,363, 418, 627

Muffin Tin (MT) 15Multilayered structures 237m-ZrO2 63

Nearly-free-electron (NFE) 18Neutron irradiation 235, 237, 702Ni10Zr7 578Ni5Zr 578Ni5Zr2 579Ni–B 162, 187NiZr 578NiZr2 578Nodular corrosion 771Non-ferrous 260, 326Non-collinear 292Non-Equilibrium Phases 23, 26, 43Nucleation 97, 98, 101, 105, 109, 110,

113, 116, 120, 122, 146, 158, 162, 164,165, 167–172, 175, 185, 188, 192–194,196, 197, 199, 200, 202, 206, 208,211–213, 217, 222, 223, 225, 226,229, 265, 277–279, 305, 322, 324,325, 348, 349, 381, 439, 451, 452,455, 479, 494, 527, 528, 533, 539,547, 580, 582, 589, 592, 594–597,612–614, 618, 620–623, 625, 629,630, 633, 640, 657, 661, 666–668,670, 671, 674, 683, 694, 698, 726,739, 740, 751, 771

O-phase 417, 418, 420, 421, 422, 423, 427,428, 431, 433, 434, 435, 555, 556, 662,665, 666, 786

Octahedral and tetrahedral voids 57Omega phase 49Order–disorder 235, 380–382, 384, 386, 388,

389, 401, 414, 459, 460, 767Order–disorder transformation(s) 235, 384,

386, 767,Ordered �-Structures 421, 518–536Ordering tie lines 458



808 Index

Orientation relations 45, 50, 103, 175, 192,267, 293, 302, 303, 310, 313, 314, 363,443, 454, 488, 613, 635, 642, 679,798, 799

Orientation relationship 45–47, 50, 51,103, 175, 192, 274, 276, 293, 302,303, 310, 362, 364, 365, 368, 383,420, 424, 440, 441, 444, 457, 485,530, 629, 632, 633, 641–644, 677,730, 735, 737, 738, 741

Orientational relation 778Orientational variant 506Orthohexagonal 281, 290, 296, 532, 663Orthorhombic 46, 54, 66, 111, 113, 200, 282,

345, 418, 427Orthorhombic �′′-martensite 281, 283, 418,

421, 616–618Oxidation 383, 443, 769, 770, 772

Paraelectric 113Partial dislocations 279, 316, 452,

453, 455Partial molar volume of hydrogen 719, 727Partially crystalline alloys 171Partially Stabilized Zirconia (PSZ) 65,

369–371, 373Partitionless polymorphic solidification 161Partitionless solidification 110, 131,

133–135, 156, 160, 161, 185, 205Peierls–Nabarro 339Peritectoid reaction 27, 141, 187, 382, 433,

434, 437, 441, 442, 529, 772Perovskite structure 113Phase diagrams 7, 8, 11, 13–15, 17, 18, 24,

26–29, 43, 53, 54, 65, 98, 129, 133, 134,160, 161, 163, 218, 232, 282, 286–288,369, 387, 388, 410, 413, 414, 418, 475,479, 516, 587, 588, 606, 609, 624, 626,722, 764, 765

Phase rule 7, 92Phase separation 43, 52, 53, 99, 115, 185,

210, 327, 409, 411, 500, 523, 555, 559,587, 603, 606, 609, 612, 616, 618, 620,621, 787

Phase separation in �-phase 43, 52, 53,559, 587, 589, 603, 606, 609, 618,620–622

Phase stability 13, 14, 16–18, 24, 27, 52, 62,148, 237, 240, 284, 383, 384, 433, 508

Phase transformation (transition) 5, 8, 21, 62,89–92, 94, 105–107, 109, 115, 120, 128,140, 145, 147, 148, 152, 153, 157, 176,186, 195, 271, 285, 352, 361, 369, 382,383, 386, 409, 417–419, 430, 443, 450,474, 495, 512, 529, 530, 535, 558, 560,606, 607, 609, 619, 621, 623, 643, 656,684, 706, 721, 725, 726, 753, 754,769, 772

Phenomenological theory of martensitecrystallography 266, 270, 504, 720,735, 729

Phonon dispersion curves 498, 507PHWR 382, 706, 707, 749, 772Pilger milling 702Plasma rotating electrode 150Plastic accommodation 313, 331, 347, 719Plastic deformation 178, 263, 265, 324, 325,

334, 354–356, 561, 683, 684, 688, 691,702–704, 726

Plate morphology 49, 277, 309, 313, 328,644, 729

Plate-shaped � 504–506Plateau pressures 761Point defects 231, 232, 233, 235, 533, 561Polarized light microscopy 302Polydomain 277, 305, 323, 350Polymorphic crystallization 110, 175, 178,

185, 187, 189, 191, 194, 197, 202, 204, 205Polymorphic transformations 65, 92, 110,

223, 362Polymorphism 4, 5Polymorphous transformation 5Portevin–Le Chatelier (PLC) effect 539Post-solidification transformations 140precipitate particles 370, 659Premartensitic 279, 361Pressure-induced � → � 494Pressure–composition isotherms (PCI) 756,

758, 761Primary crystallization 177, 187–189,

192–194, 196, 202–204Primary plates 304, 313, 348, 349Principal axes of deformation 268Principal strains 46, 268, 301, 366, 662, 778Processing maps 699



Index 809

Promotion energy factor 17Pseudoelasticity 339, 340Pseudoplastic 353–355, 359Pt–Sb 162

Quantum Monte Carlo (QMC) 14, 16Quantum structural diagrams (QSD) 17Quasicrystalline structures 241, 243Quenched-in nuclei 165, 172, 192, 197,

199, 203

R phases 103, 342Radiation-induced amorphization 229, 235Rapid or nonquenchable 105Rapid solidification 36, 128, 139, 150, 152,

153, 155, 159, 160, 169–171, 228, 231,250, 522

Rapid solidification processing 36, 150–153Recovery 346, 353, 684, 692, 710Reconstructive transformations 104Recrystallization 145, 199, 421, 670, 683,

684, 692, 710Resistivity 181, 279, 340, 341, 362, 479,

482, 768Reversion stress 260, 356, 359Rigid body rotation 45, 103, 271, 274, 276,

292, 314, 635, 643, 644, 663, 704,705, 736

Rupture 371, 382, 771

Schiebung 279Schmidt factors 334S–d electron transfer 7, 11, 12, 20Second (or higher order) 5, 502Second-order transitions 93, 115, 528Self-diffusion 176, 564, 568, 604, 688, 689,

690, 728, 729Self-similarity 241, 313Self-accommodation 277, 305, 306, 308,

320–323, 340, 342, 345, 347, 349, 350,355, 361, 362, 666, 668, 669

Serrated flow 540, 542, 545, 546, 550Shape memory effect 339, 340, 352, 353,

355, 356Shape strain 263, 265, 269–271, 303, 306,

308, 316, 320, 322, 366, 373, 504, 644,666, 668,733, 734

Shear band 693, 694Shear direction 103, 295, 296, 302, 303, 506,

733, 734Shear instability 213, 215Shear moduli 337Shear modulus 25, 47, 112, 215, 260, 331,

337, 538, 720Shear plane 295, 296, 504, 733Shear transformation 111, 120, 721, 728,

729, 737, 798Shock pressure 471, 481, 489, 499, 500, 504Shock wave 8–10, 506Shuffle dominated 111Similarity transformation 272, 294, 732Single surface trace analysis 310Sintering 370Site Occupancies 458, 520Size factor 17, 23, 29, 338Slip mode 322, 334Sluggish or quenchable 105Softmode 111Solid solubility 27, 28, 65, 150, 152, 285,

573, 722, 725–727, 746, 747Solid State Amorphization 212, 215, 226,

228, 229Solidification 36, 110, 128–142, 144–148,

150–156, 159–161, 163, 169–171, 175,185, 205, 206, 228, 231, 250,522, 690

Solubility limit 66, 152, 327, 609, 629, 721,722, 746

Solute partitioning 121, 131, 175, 492, 629,637, 670, 681

Sp-band 18Special point ordering 401Specific volume of hydride precipitate 727Spinodal clustering 98, 99, 115, 116, 409,

438, 592Spinodal decomposition 99, 156, 524, 525,

559, 589, 592, 593, 595, 597, 603, 604,616, 617, 618, 785

Spinodal ordering 98, 99, 403Splat quenching 150, 151, 175Square, S net 55Stacking domains 319Static concentration wave 389Static concentration wave model 389Stereographic projection 316, 644, 732, 736



810 Index

Stiffness moduli, C11, C44 and C12 25Strain coupling 305, 306, 355Strain energy density 166, 306, 307, 308,

349, 719, 725Strain energy 47, 99, 166, 238, 265, 277,

305, 347, 370, 479, 504, 592, 666, 719,724, 727

Strain hardening 683Strain-induced martensitic 324Strain rate sensitivity 540, 548, 684, 685,

693, 696Strain spinodal 279, 603Strain-induced plates 265Strengthening mechanisms 327Strengthening 22, 326–328, 331, 338, 698Stress-assisted 48, 265, 322, 324–326,

354, 546Stress-assisted martensite 260, 324–326Stress directed migration 744, 745Stress free transformation strain 259, 306,

719, 727Stress reorientation of hydrides 742, 743Stress–strain 325, 335, 337, 352–354, 356,

358, 537, 540, 547, 548, 550, 684, 687,690, 692, 698

Structural ledges 633, 649, 650, 652, 796,796, 798

Structural relaxation 158, 176, 180, 181, 183,184, 193, 412

Structure maps 17Substitutional 23, 27, 30, 177, 277, 327, 338,

339, 387, 415, 533, 539, 546, 559, 561,562, 609

Substructure 45, 48, 49, 293, 303–305, 313,316, 317, 319–322, 327–329, 332, 421,626, 670, 687, 689

Supercooled liquid 137, 140, 160, 163, 164,182, 207, 208, 211–213

Supercooling 47, 99, 129, 133, 137, 138,154, 161, 213, 214, 262, 263, 265, 281,287, 324, 361, 403, 416, 438, 479,527–529, 623, 638, 658

Superlattice 99, 109, 117, 227, 320, 380,381, 419, 437, 451, 454, 458, 518, 527,679, 720

Superplasticity 685, 691, 695, 701

Supersaturation 120–122, 236, 282, 283,327, 438, 441, 447, 453, 457, 609,619, 654, 658

Surface relief 266, 269, 270, 310, 362, 364,492, 632, 737, 769

Symmetry 5, 6, 15, 51, 52, 55, 58, 99–101,114, 117, 176, 185, 214, 241, 243–245,251, 271, 272, 305, 316, 370, 386, 396,401, 403, 406, 416, 421, 422, 424–429,431, 450, 485, 495, 496, 500, 502,519, 520

Symmetry tree 471, 534, 535Sympathetic nucleation 655Syncretist Classification 105

TEM 45, 49, 173, 189, 191, 196, 220, 250,310, 331, 451, 648, 675, 692, 789

Tempered 284Tempering 284, 327, 421, 559, 609–613,

615–618Tempering of martensite 555, 559, 609, 615Temporary Alloying 753Tensile 145, 157, 270, 290, 291, 335, 352,

356, 370, 372, 540, 689, 692, 696–698,707, 708, 710, 727, 741–746

Terminal solid solubility 717, 719, 725, 726Terminal solid solubility for precipitation

(TSSP) 725, 726Terminal solubility 737, 741Tetragonal 12, 54, 55, 63, 65, 66, 113,

362–368, 372Tetragonal phase (t-ZrO2� 63Tetragonal shear constant 12Texture 691, 701, 702, 704, 707, 743,

745, 769The C11b (tI6, MoSi2 type) structure 60Thermal arrest memory effect 340, 360, 361Thermal cycles 354Thermal expansion coefficient 369Thermal migration 744, 745Thermal shock resistance 369Thermal stability 176, 181, 208, 210Thermo-Mechanical Processing 683,

689, 690Thermochemical processing 753, 754Thermodynamic interaction parameter 26



Index 811

Thermodynamics 8, 92, 93, 128, 129, 133,215, 220, 261, 288, 386, 387, 393, 408,609, 619, 623, 685, 756

Thermoelastic equilibrium 265, 340, 354,356, 358, 359

Thermoelastic 265, 340, 350, 354–356,358, 359

Thermomechanical processing (TMP) 684Thin Film Multilayers 237Thin twins 314, 315Threshold stress 356, 548, 550, 745, 747Threshold value of stress 745Ti-V 11, 27, 29, 30, 32, 33, 53, 288, 325,

482, 483, 489, 606, 609Ti2AlNb 421, 422, 423, 432, 433, 670Ti2Al–Nb 669Ti2Cu 662, 676, 677Ti2N 39, 721, 772Ti45Zr38Ni17 251Ti4AlNb3 669, 670Ti5Al3 520Ti–Ag 576, 578, 624, 626Ti–Al 27, 29, 30, 34, 35, 284, 377, 381, 383,

412, 413, 414, 415, 416, 430, 433, 446,453, 464

Ti–Al–Nb 383, 417, 421, 430, 432, 433, 519,662, 669

Ti-Al-V 284Ti–Au 383, 624, 626, 627Ti–Be 160, 161, 188Ti–Bi 673Ti–Co 673, 674Ti–Cr 27, 29, 30, 33, 34, 250, 316, 474, 621,

622, 638, 639, 640, 648, 651, 673,758, 798

Ti-Cr-Si 250Ti–Cu 27, 157, 160, 163, 179, 232, 310, 673,

674, 675, 676, 677, 682Ti–Fe 161, 582, 583, 586, 626, 673, 674Tight-binding (TB) 18, 19, 215Ti–H 27, 717, 719, 722, 723, 728Ti–Mn 27, 250, 314, 316, 673, 758Ti-Mn-Si 250Ti–Mo phase diagram 621Ti–Mo 27, 29–32, 53, 288, 536, 537, 546,

606, 609, 616, 621

Ti–N 27, 29, 30, 39, 220, 232, 339, 340, 458,464, 477, 482, 606, 607, 620, 673, 772,774, 776–778

TiN 39, 40, 721, 772, 778Ti–Ni Shape Memory Alloys 339Ti–Ni 27, 220, 232, 339, 340, 673Ti–O 27, 764, 765, 767, 769Ti–Pb 673Ti–Pd 356, 673Ti–Pt 673Ti–Si 624, 626, 627, 661Titanium (Ti) 4, 7, 9, 21, 24, 626, 629, 656,

687, 690, 691, 698, 706, 720, 721, 723,737, 738, 744, 753, 754, 769, 772, 777

Titanium nitride 721, 772Ti–V 27, 29, 30, 32, 33, 53, 288, 325, 482,

483, 489, 606, 609Ti–X 24, 26–29, 43–46, 48, 53–55, 57, 59,

61, 62, 148, 418, 657, 671, 673–676Ti-Zr 27, 29, 30, 249–252, 285–287,

327, 758Ti-Zr-Fe 249–251Ti-Zr-Ni 249, 251, 252Topologically close packed (TCP)

structures 57Toughening 369–372Toughness 22, 65, 145, 327, 369–371, 655,

706, 742, 747Tracer-diffusion 555, 564, 566Transformation-induced plasticity 371Transformation sequences 141, 257, 340,

377, 409, 428, 429, 430, 432, 447, 530,621, 786

Transformation temperatures 47, 323,340, 729

Transformation twins 49, 334Transient nucleation 168–170Transition metals 4, 7, 8, 15, 18, 20–22,

24–26, 29, 157, 162, 163, 177, 225, 231,249, 480, 507, 516, 518

Transmission electron microscopic 737Triangular, T net 55TRIP steels 371Triple point 8–10, 218, 219, 480True plastic strain 260, 328True strain 328, 696True stress 328, 337



812 Index

TSS 719, 720, 725, 742, 745, 751TSSD 725, 726, 727, 751, 752TSSP 726, 727, 751, 752Twinning 270, 271, 274, 275, 277, 295, 296,

300, 313, 322, 333, 342, 346, 355, 445,484, 505, 546, 702, 728, 729, 737, 798

Two-way shape memory 354Type I twins 296, 345, 346, 347, 350, 351,

730, 735, 736, 738, 739Type II twins 346, 347, 351

Umklapp 279Undistorted 45, 46, 267–271, 274, 292, 293,

295, 296, 301, 485, 505, 509, 644, 664,732, 736

Unrotated 45, 241, 267, 271, 274, 300, 614

Vacancy mechanism 561, 562, 564, 565,566, 569, 571

Vapour pressure 756, 759, 760, 761, 764Variant 146, 351, 527Viscosity 157, 158, 161, 165, 168, 170, 171,

175, 177, 184, 206, 210Vitrification 128, 157, 158, 160, 164, 165,

171, 172, 175, 192–194, 208, 215,221–223

Wechsler, Liebermann and Read (WLR)methodology 732, 734, 735

Weldments 145Widmanstatten 49, 141, 148, 149, 327, 328,

337, 338, 421, 435, 447, 450, 610, 635,648, 674, 689, 692, 693

Widmanstatten side plates 635WLR 732, 734, 735WLR theory 734

X-ray diffraction 165, 362, 363, 474, 482,739

Yield strength 324, 339, 537, 655, 745

Zener anisotropy ratio 25–26Zigzag habit 314Zircaloys 23, 30, 658, 659, 661Zirconia 62, 63, 65, 227, 362, 365, 366,

368–371, 770

Zirconium (Zr) 4, 7, 10, 23, 24, 656, 661,701, 702, 706, 720, 721, 727, 737, 739,741–745, 747, 751, 758, 769–772

Zr alloys 4, 23, 24, 30, 47, 111, 120, 128,145, 150, 152, 171, 172, 283, 284, 287,313, 320, 327, 338, 381, 475, 501, 525,587, 604, 658, 673, 683, 691, 692, 701

Zr-Nb-Sn-Fe 661Zr(Cr,Fe)2 658, 659Zr(Nb,Fe)2 661Zr2(Fe,Ni) 658, 659Zr–2.5% Nb 24, 294, 312, 613Zr2Al 38, 39, 141, 156, 210, 382, 437, 441,

442, 518, 519, 525, 527–530, 584Zr2Al, Ti2Al 518Zr2Cu 211, 212, 222, 676, 677, 679, 681, 682Zr2Ni 174, 175, 188, 189, 200, 202, 204Zr3 (Fe, Ni) 190Zr3Al 30, 38, 39, 55, 61, 227, 228, 382,

436,439, 440, 441, 442, 523, 525,529, 530, 531

Zr3Fe 35, 191, 192, 200, 202, 681, 682Zr4Fe 681Zr5Al3 38, 141, 520, 521, 525, 527, 528, 529Zr5Al4 521, 522Zr76Fe12Ni12 200Zr76Fe16Ni8 170–172, 200Zr76Fe20Ni4 200, 204Zr76Fe24 189, 191, 194, 200, 202Zr–Al 29, 30, 38, 39, 116, 140, 156, 207,

208, 210, 381, 382, 437, 522, 523, 530,574, 582, 584

Zr–Co 163, 220Zr–Cr 23, 658, 758ZrCr2 62, 658, 758, 760, 761Zr–Cu 157, 160, 163, 207, 218, 220, 222,

223, 232, 556, 675, 676, 681, 682Zr–Fe 29, 30, 35, 36, 163, 172, 187, 195,

200, 220, 232, 249, 250, 251, 676,681, 682

Zr–H 29, 30, 39, 41, 717, 722, 723, 724,728, 740

Zr–Nb 11, 23, 29, 30, 35, 52, 53, 153,154, 155, 310, 477, 481, 492, 515,586, 599, 600, 601, 603, 606, 607,609, 618, 620, 621, 640, 645, 649,650, 661, 701, 798



Index 813

(Zr,Nb)3Fe 661Zr–Nb system 30, 35, 599, 601, 603, 607,

609, 618Zr–Ni 157, 160, 163, 164, 172, 200, 204,

218, 220, 221, 225, 232, 249, 251, 252,578, 579, 675

Zr–Ni–Cu 160Zr–Ni–Fe 163Zr–O 29, 30, 41, 63, 764–767ZrO2 polymorphs 62, 63

ZrO2 62, 63, 65, 66, 363, 770ZrO2–CaO system 66ZrO2–MgO system 66ZrO2–Y2O3 system 67Zr–Sn 29, 30, 36, 38, 658Zr–Ta system 52, 53, 602Zr–Ti 310, 328, 334–338, 586, 692Zr–X 24, 26, 27, 29, 43–45, 48, 53–55, 57,

59, 61, 62, 148, 657, 673, 675, 676




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Phase Transformations: Examples from Titanium and Zirconium Alloys