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Doctoral Thesis

Finite deformation of polymeric glassescontinuum modelling and molecular orientation

Author(s): Wendlandt, Michael Eric

Publication Date: 2003

Permanent Link: https://doi.org/10.3929/ethz-a-004525234

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Diss. ETH No. 15077

Finite Deformation of Polymeric Glasses:

Continuum Modelling andMolecular Orientation

Dissertation

submitted to the

Eidgenossische Technische Hochschule Zurich

for the degree of

Doctor of Sciences

presented by

Michael E. Wendlandt

Diplom Physiker, Universitat Konstanzborn 26.03.1972

citizen of the Federal Republic of Germany

accepted on the recommendation ofProf. Dr. Ulrich W. Suter, examinerProf. Dr. Paul Smith, co-examinerDr. Theo A. Tervoort, co-examiner

Zurich, 2003

ii

Lord, what fools these mortals be !

William Shakespeare,A Midsummernight’s Dream

fur meine Eltern

iii

iv

Contents

Abstract vii

Zusammenfassung ix

Abbreviations, symbols and notation xi

1 Introduction 1

1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Deformation behaviour of polymer glasses and modelling . . . . . . . . . . 1

1.3 Solid state NMR of polymers . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Scope of the thesis and chapter contents . . . . . . . . . . . . . . . . . . . 5

2 Deformation of polymeric glasses: experiment and modelling 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Continuum modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 The compressible Leonov model . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Extension to strain hardening . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Further extensions: multi-mode approach and strain softening . . . 15

2.2.4 Incorporating post yield rate dependence: the shear activation volume 16

2.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Commercial polymer glasses . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Cross-linked Poly(methylmethacrylate) (xPMMA) . . . . . . . . . . 22

2.4 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Characterization of rubber-like materials . . . . . . . . . . . . . . . 23

2.4.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Uniaxial compression tests: true stress - true strain - plots . . . . . 28

2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.1 True stress-strain plots . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.2 Activation volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.3 Strain hardening modulus . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

v

Contents

3 Determination of segmental orientation by solid state NMR 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Solid-state NMR and nuclear spin interactions . . . . . . . . . . . . . . . . 47

3.2.1 Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Chemical shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.3 Nuclear quadrupol interaction . . . . . . . . . . . . . . . . . . . . . 50

3.3 NMR spectral analysis: an ill-posed problem . . . . . . . . . . . . . . . . . 513.3.1 Analysis of the ill-posedness . . . . . . . . . . . . . . . . . . . . . . 553.3.2 Restoring stability by regularization . . . . . . . . . . . . . . . . . . 56

3.4 The solid-state quadrupolar NMR experiment . . . . . . . . . . . . . . . . 613.4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.1 Kernel analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5.2 Monte Carlo tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Orientation-strain relationships of rubbers and polymeric glasses 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Modelling orientation-strain relations . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Affine approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.2 Pseudo-affine approach . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Materials: 2H-labelled Poly(methylmethacrylate) . . . . . . . . . . . . . . 854.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.2 Powder line shape and CD3 quadrupole tensor . . . . . . . . . . . . 87

4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Conclusions 99

A Rotations 101A.1 Active vs. passive rotations . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.2 Euler angles and rotation group O(3) . . . . . . . . . . . . . . . . . . . . . 102

B Expansion of orientational PDFs in terms of Wigner functions 107B.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.3 Rotational transformation of WME under frame rotation . . . . . . . . . . 110

Bibliography 111

Acknowledgements 117

Curriculum Vitae 119

vi

Abstract

This thesis deals with finite, large strain deformation behaviour of polymeric glasses andstructural consequences of plastic deformation with respect to orientational distributions ofmolecular polymer segments. True stress-strain curves obtained for four different polymericglasses, namely PMMA, PC, PS, and PPO, revealed characteristic features in the post-yield regime at large finite strains, which have not been addressed by similar publishedexperiments and modelling in the past: all polymer glasses, which have been tested in thisthesis, exhibit a strain rate dependent slope of stress-strain curves in the post yield regime,which could not be explained by systematic experimental errors. Based on establishedcontinuum mechanical modelling, an empirical modification of the so-called “compressibleLeonov model” will be suggested, which is capable to describe large strain behaviour inagreement with the experimental data of this thesis.

Orientational distributions of molecular segments as a function of deformation of glassyPMMA can be obtained by improved solid-state NMR techniques. In order to be ableto properly extract orientational distribution functions of molecular segments from solid-state NMR spectra, strategies to overcome the ill-posedness of this inverse problem willbe exploited and shown to work for the reconstruction of two dimensionally orientationaldistributions from a series of 13 combined one-dimensional solid-state NMR spectra.

Results on orientation-strain relations revealed agreement with the so-called affine ap-proach, which is based on classical rubber-elasticity theory. Hence indicating that theconcept of an underlying entanglement network as the origin of segmental orientation indeformed polymer glasses is a proper assumption. These results on the orientation-strainrelations of polymer glasses seem to be in contrast to published data obtained by widelyused methods like birefringence or x-ray diffraction, which suggest the so-called pseudo-affine model to be a proper description originally developed to describe orientation-strainrelation of single crystallites. This likely contradiction can be explained by a typical short-coming of these methods, the unknown size of the structural unit of which the orientationis detected by the experiment. This shortcoming can be overcome by solid state NMR oflabelled molecules.

Further a shear modulus of an underlying entanglement network can be calculated fromthe network density suggested by the best fit of the affine approach to the experimentalorientation-strain data. Continuum mechanical modelling of true stress-strain curves of thesame material, which also implements the concept of an underlying entanglement network,revealed a shear modulus, which is in rough agreement with the results obtained fromNMR experiments.

vii

viii

Zusammenfassung

Diese Arbeit befasst sich mit dem Deformations-Verhalten von stark gepressten Polymer-Glasern und den daraus enstehenden strukturellen Konsequenzen, insbesondere der defor-mationsabhangigen Orientierungsverteilungen der einzelnen Polymer Segmente. “Echte”Spannungs-Dehnungskurven wurden fur vier verschiedene glasformige Polymere aufgenom-men, PMMA, PC, PS und PPO. Diese zeigten eine charakteristische Eigenschaft beistarker Deformation, welche in der vorhandenen Literatur bisher nicht behandelt wurde:alle getesteten Polymer Glaser zeigten eine Variation der Steigung der Spannungs-Dehnungskurven mit der Dehnungsrate, welche nicht durch systematische Fehler des Ex-periments erklart werden konnen. Basierend auf etablierten Kontinuums-Modellen kanneine empirische Modifikation des sogenannten “kompressiven Leonov Modells” eine guteUbereinstimmung von Modell und experimentellen Daten erreichen.

Orientierungsverteilungen von molekularen Segmenten in PMMA (Plexiglass) als Funk-tion der Deformation konnen mit Festkorper-NMR bestimmt werden. Die Rekonstruktionsolcher Orientierungsverteilungen aus experimentellen NMR Spektren is ein bekanntes so-genanntes “schlecht gestelltes” inverses Problem. Um damit verbundene Probleme zuvermeiden wurde ein Regularisierungsalgorithmus entwickelt, der eine verlassliche Rekon-struktion erlaubt.

Das Resultat solcher experimentell bestimmten Orientierungsverteilungen zeigteUbereinstimmung mit affinen Netzwerk Modellen, die fur die Beschreibung von gum-mielastischen Materialien entwickelt wurden. Diese Tatsache zeigt, dass selbst in Poly-mer Glasern die Orientierung der Kettensegmente durch eine Art Netzwerk bewirktwird, welches durch die Verhakung der einzelnen Polymerketten untereinander entsteht.Diese Ergebnisse scheinen im Widerspruch zu bisher veroffentlichten Daten zu stehen,welche eine Orientierungsverteilung von Polymersegmenten bestatigen, die dem einzel-ner Kristallite in einer Kontinuumsmatrix entsprechen. Dieser Widerspruch kann jedochaufgelost werden, da die bisher verwendeten Methoden, wie z.B. Doppelbrechung oderRontgendiffraktion, keine Information enthalten uber die Grosse der strukturellen Einheitderen Orientierungsverteilung man misst. NMR hat diesen Nachteil nicht, da im Prinzipjedes einzelne Molekul gekennzeichnet und fur das Experiment sichtbar gemacht werdenkann.

Weiterhin kann aus den NMR Daten uber die Orientierungsverteilungen ein Modul desvorgeschlagenen Netzwerks ermittelt werden. Vergleicht man diesen mit dem Modul dasin der hier erweiterten Kontinuummodellierung des selben Materials verwendet wird, kannman eine grobe Ubereinstimmung feststellen.

ix

x

Abbreviations, symbols and notation

Abbreviations

1D one-dimensional2D two-dimensionalFWHM full width at half maximumLF laboratory frameMF molecular frameNMR nuclear magentic resonanceODF orientational probability distribution functionPAS principle axis systemPC Poly(Bisphenol A carbonate)PDF probability distribution functionPMMA Poly(methylmethacrylate)PPO Poly(phenyleneoxide)PS Poly(styrene) rate of strain tensorRF rotor frameS/N signal-to-noise ratioSF sample frameSVD singular value decompositionWME Wigner matrix element

Symbols

A Helmholtz free energyα isochoric draw ratio~B0 external magnetic fieldCpq correlation between discrete basis spectra p and qCQ quadrupole coupling constantclmn expansion coefficients of the expansion of an PDF/ODF in terms

of Wigner matrix elements

xi

Abbreviations, symbols and notation

D rate of strain tensorDjk dipolar coupling tensorDl

mn Wigner matrix elementδQ quadrupolar anisotropie parameterε engineering strainε true strainη viscosityηQ quadrupolar assymetry parameterF deformation gradient tensorf functionality of a networkG Gibbs free energyg(θ, φ) distribution densityγ magnetogyric ratio

H Hamiltonian~ Plancks constantI identity

I spin operatorJ relative volume deformationK(ω, θ) integral kernel functionKi discrete basis spectrum, ith column of the discrete kernel matrix Kk Boltzmanns constantL velocity gradient tensorL Langevin functionλ draw ratioµ number of cross-links per unit volumeN number of elastic active chains per unit volumen average number of statistical Kuhn segments per chainR strain hardening stress tensorR rotation operatorS driving stress tensor for a single Leonov modeS(ω) spectrum functionσ true uniaxial stressσn engineering uniaxial stress[σR] reduced engineering uniaxial stressσCS chemical shielding tensorσQ quadrupole tensorT total Cauchy stress tensorT absolute temperature

xii

τ shear stressu(ω), v(θ) singular functionsV ∗ shear-stress-activation volumeV electric field gradientW vorticity tensorξ cycle rank of a networkχg normalized root-square discrepancy between input and output ODFχS normalized root-square deviation of a arbitrary NMR spectrum

from a powder NMR spectrumχc normalized root-square discrepancy between Wigner expansion co-

efficients clmn of an input and an output ODF

Notation

ab,AB dyadic producta · b,A ·B inner productA : B double inner productAT transpositionA−1 inversiontr(A) tracedet(A) determinantIA ≡ tr(A) first invariant of AIIA ≡ 1

2(I2

A − tr(A)) second invariant of AIIIA ≡ det(A) third invariant of AAd = A− 1

3tr(A)I deviatoric part of A

A material’s time derivative of A

A Jaumann (co-rotational) derivative of A

xiii

xiv

1 Introduction

1.1 General

Glasses and especially polymeric glasses as we know them today play a significant role inour world. The glassy state can generally be described as a non-equilibrium state, which iscontinuously relaxing towards thermodynamic equilibrium but reveals solid-like behaviourover a broad range of observation times [1]. Moreover, from a structural point of view aglassy system can be described as a liquid that has lost its ability to flow, thus the systemis completely disordered or amorphous.

In the past 40 years the application of glassy polymers has developed considerably. Poly-carbonate (PC), polystyrene (PS), and polymethylmethacrylate (PMMA) are among themost widely used polymeric solids, offering an attractive compromise between the ease ofprocessability, production costs and final mechanical and thermal properties. The mechan-ical behaviour of these materials, especially the macroscopic deformation at finite strainshas been a topic of discussion for almost a half decade. Numerous molecular and contin-uum models [2,3,4,5] and computer simulation approaches [6,7,8] have been proposed, butstill a thorough understanding of the mechanical behaviour of polymer glasses is lacking. Acomprehensive review about the physics of glasses was published by Haward and Young [9].

Hence, the main aim is to find a proper constitutive description which allows to makequantitative predictions of macroscopic mechanical behaviour if the essential material pa-rameters, which are related to the underlying molecular structure of the material, areknown. An essential requirement for reliability tests of any suggested modelling in general,however, is the availability of accurate experimental data, which is an important focus ofthis thesis.

1.2 Deformation behaviour of polymer glasses andmodelling

It is now generally accepted that the long chain nature of polymers in general and polymerglasses especially plays an important role in their mechanical response at large deforma-tions. There are strong indications that, due to steric hinderance, the covalent chainsorient during plastic deformation, resulting in anisotropic materials with enhanced prop-erties in the drawing direction. The exact nature of this orientation process, however, has

1

1 Introduction

e .

S t r a i n

Stress

Figure 1.1: Schematic sketch of typical stress-strain curves of a polymeric glass obtained atdifferent strain rates ε.

not yet been resolved, which hampers the development of efficient constitutive relationsand, ultimately, the design of new polymeric materials with tailored mechanical properties.

A typical stress- strain curve of an amorphous polymer deformed at three different strainrates (far) below its glass transition is shown in fig. 1.1. Usually a rough distinction can bemade between an elastic response at very small strains and an inelastic, i.e. plastic responseat larger strains. Here, the terms “elastic” and “inelastic” are used to refer to deformationthat results in completely recovery of the original shape (elastic deformation) or in some“permanent” strain (plastic deformation) upon unloading in terms of a time frame of atleast weeks at room temperature. A more precise definition makes no sense, since glasses arenot in a state of equilibrium, but rather relaxing continuously accompanied by a decreasein thermodynamic properties such as enthalpy, volume, and entropy. The transition fromelastic to plastic behaviour is usually known as the onset of yielding and is not a sharptransition, but a gradually transitions over a finite strain interval. However, sometimesthe so-called yield point is defined as the onset of significant plastic flow, usually assignedto the local maximum in stress observed for most polymeric glasses. Yield is classicallydescribed by using yield criteria, of which the pressure and rate-dependent “Von Mises”criterion [10,11] seems to be the most successful.

At larger strains in the post-yield regime, most polymers reveal a drop in stress withincreasing plastic strain, usually called “strain-softening”. The exact physical origin ofstrain softening is not completely clear, but appears to be related to the process of physicalaging [12,13]. Physical aging, the slow approach towards thermodynamically equilibrium,tends to increase the yield stress as a function of aging time [14,15], which is schematically

2

1.2 Deformation behaviour of polymer glasses and modelling

S t r a i n

Stress

i n t r i n s i c s t r a i n s o f t e n i n g

p h y s i c a l a g i n g

Figure 1.2: Schematic sketch of typical stress-strain curves of a polymeric glass with differentaging histories. Increased aging also increases strain-softening.

sketched in fig. 1.2. This increase is reversed by plastic deformation, which is the abovementioned strain softening response, also called “mechanical rejuvenation”.

After strain softening, stress increases again with increasing plastic strain. This char-acteristic feature of stress-strain curves of polymeric glasses is called strain-hardening andis usually described by a rubber-elastic response of an underlying entanglement network,although quantitative agreement seems to be lacking [16,15].

The most promising theoretical description of mechanical behaviour of polymeric glasses,with respect to quantitative agreement between experiments and modelling, can beachieved by a viscoelastic continuum mechanical approach. This phenomenological modelcombines liquid-like and solid-like features since polymers can display all the intermedi-ate range properties between an rubber-elastic solid and a viscous liquid, depending ontemperature and the experimentally chosen timescale. In the viscoelastic continuum de-scription of the mechanical behaviour of polymeric glasses, usually a distinction can bemade between the linear viscoelastic regime at very low strains, the non-linear viscoelasticregime at moderate strains, and the yield behaviour at high strains. The linear viscoelasticdeformation is adequately described using linear response theory, which results in the wellknown Boltzmann single integral representation. The description of the non-linear regimehas been, and still is, an active field of research.

Recently a three dimensional constitutive description of finite non-linear viscoelasticbehaviour has been proposed by Tervoort et al. [5] and extended to imply characteristicfeatures of post yield behaviour, like strain softening and strain hardening [14, 17]. Thisapproach, called the compressible Leonov model, is based on the so-called Eyring concept[18, 11], which describes plastic flow as jumps of certain structural elements over energy

3

1 Introduction

barriers under the combined influence of applied stress and thermal energy. The Eyringconcept is the heart of many treatments of yielding of glassy polymers and results in anon-linear viscosity as a function of stress. In general this concept allows to predict ratedependence of post yield stress at a fixed strain in a realistic way resulting in the Eyringequation, which suggests a linear relation between stress and logarithmic strain scaled bythermal energy and the so-called shear activation volume. The latter is mostly treated asan intrinsic material parameter, hence opening a window into the microscopic deformationbehaviour on a molecular scale, since the Eyring concept connects molecular materialparameters, like the shear activation volume, and macroscopic continuum modelling.

However, it is still doubtful, if the Eyring equation is capable to identify the true in-trinsic material parameters, since experiments of this study revealed a stress and straindependence of the shear activation volume,which is treated as an constant intrinsic materialparameter in the Eyring concept.

1.3 Solid state NMR of polymers

Identification of the underlying molecular process, which takes place during viscoelasticdeformation of polymeric glasses plays a key role for the success of atomistic modelling andcontinuum mechanical description, since in the latter case the Eyring concept is stronglyrelated to the microscopic deformation mechanism on a molecular scale. Consequently,the determination of orientational distributions of molecular segments as a function ofdeformation is of particular interest. Experimental methods that are capable to yieldinformation about orientational distributions on a molecular scale have been reviewedextensively by Ward [11], where the most widely used techniques are birefringence andx-ray scattering. Beside x-ray scattering all applied methods have the disadvantage, thatthey cannot determine complete orientational distribution functions as a function of twopolar angles, but rather are restricted to an analysis in terms of moments [19,20].

Over the last thirty years nuclear magnetic resonance (NMR) has become an importanttechnique for structure determination, especially in systems where standard diffractionmethods fail. The latter are extremely well suited for studying the structure of singlecrystals and generally NMR cannot compete in such systems. In powdered crystalline,non-crystalline and dynamical systems NMR is the method of choice, though. The rapidBrownian motion of the molecules averages out all second-rank NMR interactions, thuscreating the resolution needed to resolve the resonances in complex systems.

Especially solid-state NMR spectroscopy has made important progress during the lastdecade and may now be the most valuable tool for elucidation of orientational order inpolymers. The orientational anisotropy of the interaction tensors, which is not averaged outby rotation of the molecules in a solid, e.g. a glass, leads to a characteristic line shape of thebroad solid-state NMR signal as a function of the orientational distribution of the secondrank interaction tensor itself. Thus, it is possible to determine orientational distributionsof molecular segments, if the fixed spatial orientation of the interaction tensor with respectto a molecular fixed frame is known. The design of an NMR experiment plays an important

4

1.4 Scope of the thesis and chapter contents

role with respect to available information and resolution power. So far one [21, 22] andmulti-dimensional [23, 24] experiments have been applied to yield molecular orientationaldistributions. A basic review about available techniques was published by Spiess [19].

In general multi-dimensional techniques always yield the highest amount of informationbut often are mechanically challenging and less robust over long measuring times. Ansimple robust alternative is to measure a combined series of one-dimensional (1D) NMRspectra. Numerical tests showed that for the case of a specific two dimensional experiment,designed for the detection of orientational distributions, a number of 7 combined 1D spectracan be an acceptable alternative [25]. Moreover, a mechanically robust method allowsto measure over longer time periods, which gains a higher signal-to-noise ratio. Thus, ifdistortion due to the mechanically instabilities of the experiment, e.g. flips of the specimenduring data acquisition, has to be reduced, and signal-to-noise can be increased, a combinedseries of 1D experiments may be a good alternative with respect to the reliability of theexperimental data.

1.4 Scope of the thesis and chapter contents

The object of this thesis is twofold. First, two experimental approaches are developed,which yield accurate experimental data about macroscopic and microscopic mechanicalbehaviour of polymeric glasses. Second, experimental data will be compared to establishedcontinuum mechanical models to prove their validity, and to make suggestions of furtherextensions to gain better agreement with experimental data.

Chapter 2 contents the description of deformation experiments of several polymericglasses together with a theoretical analysis of the results by the application of an extendedcontinuum mechanical model. True stress-strain curves, recorded at constant true strainrate, have been performed from small to large finite strains. Minimizing systematic errorsduring the deformation tests can reveal features of macroscopic mechanical behaviour,especially at large strains, which have not been addressed in the past. Comparing theseexperimental data with continuum mechanical modelling indicates shortcomings of presentconcepts. Especially large strain deformation behaviour of polymeric glasses could notbe modelled on a satisfying level in the past and characteristic features of large straindeformation behaviour of polymeric glasses revealed in this thesis, have not been addressedat all. Consequently an extension of well established continuum mechanical modelling willbe developed.

Information about deformation behaviour on a molecular scale, i.e. information aboutorientational distributions of molecular segments as a function of deformation can be ob-tained by solid state NMR experiments. In contrast to birefringence or scattering methodslike x-ray diffraction, which have been used extensively in the past, NMR line shapes,which are dominated by single spin intramolecular couplings only [22], can be analyzed toyield orientational distribution functions of labelled molecules and are not restricted to ananalysis in terms of moments. In Chapter 3, a mathematical method will be describedand analyzed, which reconstructs orientational distributions of molecular segments from

5

1 Introduction

solid-state NMR spectra via line shape analysis. Reliability and resolution power of thesuggested analysis will be tested simulated data.

Chapter 4 contents results about orientational distributions of molecular segments ob-tained by solid-state NMR spectra as a function of deformation. Further, orientation-strainrelations will be compared to two model approaches, namely the so-called “affine ap-proach”, which is derived from classical rubber-elasticity theory, and the so-called “pseudoaffine approach”, which treats molecular segments of a polymer chain as independently,affine deforming stiff rods.

Finally the main conclusions are summarized in the Summary and possibilities forfuture research are indicated.

6

2 Deformation of polymeric glasses:experiment and modelling

2.1 Introduction

It appears, experimentally, that the large strain behaviour of polymeric glasses, which isthe focus of this thesis, is not influenced by physical aging and strain softening [14, 15].Therefore, in what follows, the experimental stress-strain curves will be compared to thatof fully rejuvenated materials only, i.e. the increased yield stress and strain softeningresponse will be neglected.

Most realistic modelling of experimental stress-strain data of polymeric glasses has beenobtained by continuum mechanical modelling approaches, involving a combination of vis-coelastic and rubber-elastic elements. So far established approaches [26, 5] are successfulin predicting true stress-strain behaviour at small and medium strains up to the yieldpoint, but still lack satisfying validity at large post yield strains in a regime where theso-called strain hardening sets in. In section 2.2 a basic three dimensional non-linear vis-coelastic model, the so-called compressible Leonov model, will be described, together witha modification of this basic model in order to describe post-yield large strain behaviour inagreement with experimental true stress-strain curves.

True stress-strain experiments on glassy polymers have to be performed with great carewith respect to the experimental setup to reduce systematic error sources as effectively aspossible. Especially when strains are large, reduction of friction between the compressedsamples and the testing device and the application of a true strain rate play an importantrole in data acquisition, as described in section 2.4.

Finally in section 2.5, experimental true stress-strain data of glassy polymers obtained incompression mode at different true strain rates are analyzed with respect to the modifiedcompressible Leonov mode.

2.2 Continuum modelling

2.2.1 The compressible Leonov model

Information about the derivation and the mathematically formalism of the so-called “com-pressible Leonov model” was treated in literature in detail [27, 28, 29, 5]. Therefore only a

7

2 Deformation of polymeric glasses: experiment and modelling

summarizing description of the model will be presented in this chapter.The spatial position of an arbitrary material point of a reference configuration is repre-

sented by a column vector ~x1. Postulating now that an unstressed state exists, which isrepresented by the reference configuration, local deformation as a results of a macroscopicdeformation of the material can be described as follows: after a time t, following elastic-plastic deformation relative to the reference configuration, the spatial position of the samematerial point may be written as according to the mapping ~x3 = ~x3(~x1, t). If each elementof the material is again unstressed, the configuration ~x2 = ~x2(~x1, t) results. Hence localdeformation at time t with respect to the reference state ~x1 can be expressed convenientlyin terms of the deformation gradient tensor [30]

F =∂~x3

∂~x1(2.1)

For the case of continuous and differentiable mapping, the deformation gradient F canbe decomposed into an elastic Fe and a plastic part Fp following the chain rule for partialdifferentiation

F =∂~x3

∂~x2· ∂~x2

∂~x1= Fe · Fp (2.2)

In this study the theoretical analysis of polymer glasses is based on the so called “com-pressible Leononv model” [31,5]. This basic three-dimensional model provides a constitu-tive description of viscoelastic behaviour with an elastic volume response. A key featureof this model is the decoupling of the elastic volume response and the isochoric “shape”response of a body under deformation. Assuming isotropic elastic behaviour, this decou-pling is achieved by assuming that the free energy of the system is determined by two statevariables:

The first is the relative volume deformation J at time t,

J =dV

dV0

= det(F) =√

detB (2.3)

where V0 is the volume in the state of reference, V the volume after a time t, and B theleft Cauchy-Green strain tensor,

B = F · FT (2.4)

From kinematics [29] the evolution equation for J follows

J = Jtr(D)I (2.5)

where

D =1

2(L + LT ) (2.6)

is the rate of strain tensor and

8


L =∂~v

∂~x3= F · F−1 (2.7)

the velocity gradient tensor with the velocity, i.e. the rate of position of the materialpoints, ~v = ∂~x3

∂t.

The second state variable determining the free energy of the system is the isochoric leftCauchy Green strain tensor Be,

Be = J−2/3Be = J−2/3Fe · FTe (2.8)

Further, it can be shown, that the hydrostatic stress is coupled to the volume deformationJ , whereas the deviatoric stress is determined by the isochoric elastic strain Be. Choosinga neo-Hookean relationship [11] the total stress tensor S can be written as

S = K(J − 1)I + GBde (2.9)

where K is the bulk modulus, G the shear modulus, and the superscript “d” denotes thedeviatoric part of the tensor.

Assuming that the volume deformation remains elastic and that the accumulation ofelastic strain (at constant volume) is reduced because of a plastic strain rate Dp, thefollowing evolution equation for Be holds,

˚Be = (Dd −Dp) · Be + Be · (Dd −Dp) (2.10)

where ˚Be is the Jaumann or corotational derivative of Be

˚Be = ˙Be −W · Be − Be ·W (2.11)

with the vorticity tensor W,

W =1

2(L− LT ) (2.12)

In the case of uniaxial deformation it is W = 0. For the further development of thetheory, it is necessary to introduce a constitutive description of the plastic rate-of-straintensor Dp [29, 5].

It is well established, that for materials in general and polymers in specific [11], theplastic shear rate is often well modelled by using a so-called Eyring-flow process [18,32,33].A more detailed description of the Eyring concept will follow in section 2.2.4. The basicidea of this concept is that plastic flow is a molecular process, involving jumps over energybarriers, activated by a combination of applied stress and thermal energy. Following thisconcept, in one dimension the plastic strain rate can be expressed as a function of absolutetemperature T and (shear) stress τ

γp = γp(T, τ) =1

Asinh(

τ

τ0

) (2.13)

9


with A a time and τ0 a characteristic stress, respectively related to the height of theactivation energy barrier for plastic flow ∆G and the so-called “shear activation volume”V ∗

A = A0 exp(−∆G

kT) (2.14)

τ0 =kT

V ∗ (2.15)

where k is the Boltzmann constant, A0 a constant pre-exponential factor involving thefundamental vibration energy, and T the absolute temperature. This allows to define aviscosity, which depends on the plastic strain rate

γp =τ

( τ0 arcsinh(Aγp)

γp)

=τ

η(γp)(2.16)

Extending this equation into a generalized three dimensional form, the plastic strain rateDp reduces the accumulation of isochoric strain and can be expressed as a function of thedeviatoric Cauchy stress Sd by a generalized non-Newtonian flow rule

Dp =12Sd

η(γeq)with γeq =

√2tr(Dp ·Dp) (2.17)

where γeq is the generalized equivalent strain rate, defined such that in case of a shearflow it reduces to the plastic shear rate γp. Complementary to the equivalent strain rate,an equivalent stress τeq can be defined satisfying

γeq =1

Asinh(

τeq

τ0

) with τeq =

√1

2tr(Sd · Sd) (2.18)

Substitution of eq. (2.18) into eq. (2.17) results in a three dimensional Eyring-equation,relating the plastic rate of strain tensor to the deviatoric part of the Cauchy-stress tensor

Dp =Sd

2η(τeq)with η(τeq) = Aτ0

τeq/τ0

sinh τeq/τ0

(2.19)

Equations (2.5), (2.9), (2.10), and (2.19) form a complete three-dimensional set of dif-ferential equations providing a starting point for the modelling of viscoelastic behaviour ofpolymer glasses. From these equations it can be seen that the deviatoric stress responseSd as described in this model (to be called:“a single Leonov mode”) can be depicted as asingle Maxwell model employing a nonlinear relaxation time λ = η/G (cf. fig. 2.1).

In the basic compressible Leonov model, as described in this section, the parameter V ∗

in eq. (2.15) is considered to be an intrinsic material constant. This is at least doubtful,since deformation experiments in this study show, that e.g. the activation volume is afunction of equivalent stress and strain rather than a material constant. In section (2.2.4)

10


G h ( t e q )

D e D p

Figure 2.1: Schematic sketch of the deviatoric stress response of a single Leonov mode.

consequences for the overall true stress-strain behaviour of polymer glasses evolving froma non-constant activation volume will be discussed in more detail.

Before incorporation a non-constant activation volume it makes sense to discuss twofurther extension of the basic compressible Leonov model: incorporation of intrinsic strainsoftening and strain hardening. Especially the interpretation of large strain experimentaldata with respect to strain hardening will be distinctly affected by a non-constant activationvolume.

2.2.2 Extension to strain hardening

A polymeric system consists of covalent bonded chains held together by secondary forces.In the glassy state collective dynamics of the polymer chains is effectively suppressed,leading to a solid-like behaviour. Applying stress to the system, intermolecular forces canbe overcome at the yield point, and large scale segmental motion is initiated, where theprimary bonds survive these segmental jumps and give rise to steric hinderance. At smallstrains up to the yield point, stress activated segmental motion is assumed to be responsiblefor non-linear viscoelastic behaviour. In this sense the yield point is viewed upon as a stress-induced glass transition. The steric hinderance between the polymer chains plays also akey role in the flow behaviour of polymer melts: above a certain molecular weight and ona restricted time scale, a polymer melt behaves like a rubber-elastic solid.

Transferring network theory of chemically cross-linked systems to the rubber-elastic be-haviour of polymer melts, leads to the definition of entanglements. Entanglements areenvisaged as physical knots, which cannot unravel on the time scale of the experiment,similar to chemically fixed knots in a rubber. Applying the classical theory of rubber elas-ticity, measurement of the “stiffness” (plateau modulus of dynamic shear experiments) ofa melt allows to determine the molecular weight between entanglements Me, which is ameasure of the diffuse steric hinderance between covalent chains. It is widely accepted,that entanglements play an important role in the large-strain deformation of polymericsolids as well, e.g. as source of the strain hardening in glassy polymers, which is assumedto be a consequence of deformation of the entanglement network.

Harward and Thackray [34] were the first to incorporate concepts of rubber-elasticityinto the modelling of polymer glasses to account for strain hardening. They assumed that

11


the total stress can be decoupled into a viscoelastic part evolving from segmental motionsand a rubber-elastic part evolving from the rubber elastic response of the entanglementnetwork. This one dimensional model was extended by Boyce et al. [4] into a 3-dimensionalfinite strain formulation, the so-called “BPA-model”. Both approaches implied finite exten-sibility of the rubber-elastic network chains to have a significant influence on the networkresponse. In contrast Haward [16] suggested that polymer chains do not approach a fullystretched condition and hence he proposed a neo-Hookean (Gaussian) relation, which wasalso observed by Tervoort et al [15].

A nearly identical approach is depicted by a Leonov mode, in parallel with a rubber-elastic spring. In this study the basic idea of modelling strain hardening of polymer glassesby adding an elastic spring in parallel to a compressible Leonov mode will be adapted.Hence rubber-elasticity is discussed in more detail in the following. An excellent reviewabout rubber elasticity is given by Treloar [35] and more recently by Ermann and Mark [36],and Boyce [37], hence only a very brief overview about modern theory of rubber elasticitywill be provided here.

Constitutive models of rubber elasticity

In what follows, it will be assumed that Helmoltz free energy A depends only on the leftCauchy-Green strain tensor at constant volume B. The evolution equation of B followsfrom kinematics

˙B = Ld · B + B · (Ld)T = Λ : L (2.20)

where Ld is the deviatoric part of the velocity gradient eq. (2.7), and the fourth ordertensor Λ is defined through eq. (2.20). The reversible (elastic) part of the deviatoricCauchy-stress tensor Rd is then determined by the thermodynamic force M = ∂A/∂B,and Rd is given through Λ [27]

Rd = M : Λ (2.21)

The structure of a rubber elastic material is essentially one of randomly oriented,long chain molecules in a network arrangement due to sparse cross-linking between themolecules; furthermore, the intermolecular interactions are weak, such that stress-strainbehaviour is primarily governed by changes in configurational entropy as the randomly-oriented molecular network becomes preferentially-oriented with stretching. Therefore clas-sical theory of rubber elasticity assumes that the internal energy remains constant uponisochoric deformation. Hence, at isothermal conditions, the thermodynamic force is onlydependent on changes in entropy

M =

(dA

dB

)T

=

(dU

dB

)T

− T

(dS

dB

)T

= −T

(dS

dB

)T

(2.22)

Kuhn and Grun [38] first derived an expression for the configurational entropy of astretched molecular single chain, taking into account the effect of finite extensibility of the

12


chain by using Langevin chain statistics. Considering a single ideal chain they calculatedthe Helmholtz-free energy as a function of draw ratio λ

A = NkT

(r

nlβ + ln

[β

sinh β

])− A0 with β = L−1

( r

nl

)(2.23)

where N represents the number of active chains per unit volume and n the averagenumber of statistical links of length l per chain. Root-mean-square distance of the singlechain is

√〈r2

0〉 =√

n l, draw ratio λ = r/√〈r2

0〉 = r/(√

nl) and maximum draw ratioλmax =

√n i. A0 is an arbitrary constant and L−1(β) the inverse Langevin function

defined by

L−1(β) = coth β − 1

β(2.24)

A useful approximation of the inverse Langevin function is it’s first Pade approximant[40]

L−1(x) ≈ x3− x2

1− x2(2.25)

The (one-dimensional) force on the chain under deformation is derived by differentiationof the Helmholtz-free energy

f =

(dA

dr

)=

kT

lL−1

( r

nl

)(2.26)

For elastic behaviour under isothermal conditions, the work-rate of stress per unit volumeis σλ/λ, so that with σλ/λ = A = (∂A/∂λ)λ, we get the (one-dimensional) stress strainrelation

σ = λ

(∂A

∂λ

)(2.27)

In order to calculate the three dimensional stress of a network from the known Helmholtz-free energy of a single chain eq. (2.26), it is assumed, that the chains are randomly dis-tributed in space and deform in an affine manner. Choosing the mathematically exact way,the strain energy function is found by integrating the stress-strain response of all chainsover the whole unit sphere. This so called “full chain” model [41], has the disadvantagethat it cannot be solved analytically, and even numerically approximation is computation-ally intensive. Alternatively, analytical expressions can be obtained, if the integration overthe whole unit sphere is approximated by sampling only a discrete number of directions,like the “three chain” model of James and Guth [42, 43], or the “eight chain” model ofArruda and Boyce [26]. Even further simplification can be achieved when regarding onlysmall strains which leads to the so called “Gaussian network approximation” where thedistribution of end-to-end distances between two network-points is described by Gaussianchain statistics.

iusing the nomenclature as supposed by Ward and Harward [39]

13


Three-chain model: here all N chains are represented by three independent sets of N/3chains per volume along the axis of the principle stretches. Substituting eq. (2.23) intoeq. (2.27), and r/nl = λ/

√n, we get the principal stresses

σi3−ch =

NkT

3

√nλiL−1

(λi√n

)(2.28)

Eight-chain model: this model considers a set of eight chains connecting the centraljunction point and each of the eight corners of the unit cube spanned by the principle axis.It is assumed that eq. (2.23) holds, multiplied by the number of chains per volume N , andthat the stretch of each of the eight chains can be described using an average chain stretchparameter λchain

λ2chain =

λ21 + λ2

3 + λ23

3=

IB

3(2.29)

where IB is the first invariant of the isochoric left Cauchy-Green strain tensor B. Thethermodynamic force according to the eight chain model becomes

M =∂A

∂B=

(∂A

∂λchain

)(∂λchain

∂IB

)(∂IB

∂B

)(2.30)

where

(∂A

∂λchain

)= NkT

√nL−1

(λchain√

n

)(2.31)(

∂λchain

∂IB

)=

√3

6I−1/2

B=

1

6λchain

(2.32)(∂IB

∂B

)= I (2.33)

Using eq. (2.20) and substituting eq. 2.30 - 2.33 into the general expression for thestress tensor eq. (2.21), the deviatoric stress tensor in the eight chain model reads

Rd8−ch =

NkT

3

√nL−1

(λchain√

n

)1

λchain

Bd = G(IB)Bd (2.34)

Note that the eight-chain model can be described as modified neo-Hookean behaviour (cf.next paragraph) revealing a non-constant shear modulus depending on the first invariantIB of B.

The most simple expression for the Helmholtz-free energy of a single chain follows fromthe so-called

14


Gaussian network approximation: the inverse Langevin function may be expanded in aseries

L−1( r

nl

)=

[3( r

nl

)+

9

5

( r

nl

)3

+297

175

( r

nl

)5

+ ...

](2.35)

For small values of (r/nl), only the first term has to be considered, and the expressionfor the the Helmholtz-free energy for a single chain eq. (2.23) becomes

A =NkT

2(IB − 3) (2.36)

This expression is equivalent to the expression obtained for a network consisting of chainswith a Gaussian distribution of end-to-end distances between two cross-links. Again withthe help of eq. (2.20) and eq. (2.21), the deviatoric stress tensor can be calculated andleads to so-called “neo-Hookean” behaviour

Rdgauss = GBd (2.37)

with the shear modulus G = NkT . For rubber elastic materials the Gaussian approxi-mation holds for small to medium deformations. At large deformations there is a significantdeviation from Gaussian behaviour, appearing approximately when r/nl ≥ 0.4 [37].

Finally, the total Cauchy stress tensor T of the extended continuum model, which iscomposed of the driving stress tensor S and the hardening tensor R may be written as

T = S + R (2.38)

However, since we are dealing with continuum modelling of polymer glasses, the questionarising from the rubber-elastic network contribution is, weather finite extensibility has tobe taken into account for the modelling of polymer glasses or not, i.e. if a Gaussian (neo-Hookean) network response already allows a satisfactory modelling of strain hardening.There is strong evidence that finite extensibility is relevant for chemically cross-linkedpolymer glasses as shown in fig. 2.13 but on the other hand it seems less obvious thatfinite extensibility can ever be observed for a (thermoplastic) glassy polymer [15,16].

2.2.3 Further extensions: multi-mode approach and strain softening

So far, the proposed continuum model does not account for two features observed in mosttrue-stress-strain plots: a smooth transition from elastic to viscous behaviour instead ofthe sharp transition suggested by the model presented in this study, and the so-called“strain softening”, which denotes a drop of true stress with increasing strain succeedingthe yield point before strain hardening set in. Extensive studies have been performed inthe past to incorporate smooth elastic-viscous transitions [5, 44, 45] and strain softeninginto continuum models [17].

The use of single Leonov mode, i.e. a single relaxation time, in the description of the vis-coelastic response of a polymer glass like in section 2.2.1, cannot account for the (non)linear

15


viscoelastic response at small to moderate strains, especially not for the experimentally ob-served smooth transition from elastic to viscous behaviour. Tervoort et al. [5] proposeda multi-mode approach, which incorporates a spectrum of relaxation times presented by18 parallel Leonov modes instead of a single one. This approach is capable of describingrealistic constant strain rate experiments, including strain rate dependent yield behaviour,and enables to give a quantitative description of nonlinear stress-relaxation experiments.

To model strain softening, Govaert et al. [17] adopted the idea of Hasan et al. [14]and introduced a history variable, the so called softening parameter, which influences theviscosity. During plastic deformation the softening parameter evolves to a saturation levelindependent of strain history, which marks the onset of strain hardening succeeding thestrain softening minimum.

However, this study is mainly concerned with deformation behaviour of polymer glassesat large strains investigating strain-rate effects on strain hardening. Consequently thequestion may arise, if modelling of a smooth elastic-viscous transition and strain-softeninghave to be incorporated into the continuum model at all, when the focus is on large strainsonly, succeeding the local softening minimum.

First, a smooth transition between elastic and viscous behaviour is an effect occurringat strains located closely around the yield point, far away from the softening minimum.This assumption is encouraged by continuum models which are capable to model a realisticsmooth transition, like the multi-mode leonov model [5].

Second, it is well known that strain-softening changes with time, a thermo-reversibleprocess called physical aging [12]. Numerous experimental stress-strain data are avail-able [14, 45, 17, 15] obtained from polymers exhibiting different degrees of strain-softeningdependent on their thermal history. All these data have in common, that there is no influ-ence of physical aging on the strain-hardening regime succeeding the local strain-softeningminimum.

Hence both features are believed to have negligible influence on the large strain behaviourof polymer glasses, and therefore, for the sake of simplicity, will not be modelled in thisstudy.

2.2.4 Incorporating post yield rate dependence: the shear activationvolume

Inherent in the theory of rubber elasticity, described in the last section, is the strain-rateindependence of the network response. Hence, effects on true stress-strain curves are onlyrelated to the viscoelastic contribution of the continuum mechanical picture resulting in avertical shift of the post-yield stress, such that after yielding, stress plots at different strainrates always run parallel to each other shifted vertically by a factor depending on temper-ature and the activation volume V ∗ (cf. eq. (2.43) and fig. 2.2). In contrast, experimentaldata in this study, obtained from compression test of different glassy polymers (cf. section2.5), strongly indicate that there is a rate dependence of the strain hardening modulus,i.e. the slope of the true stress-strain curves at large strains, which is not addressed by

16


the presented continuum modelling. Assuming that the total stress can be decoupled ad-ditively into a contribution due to segmental motion and due to the entanglement networkresponse, two approaches are generally possible to account for the rate dependence of thestrain hardening modulus:

First, one could modify the rubber-elastic network response into a viscoelastic networkresponse. Second, modifications could be applied to the viscous response of the compress-ible Leonov mode without changing the additive rubber-elastic network response, e.g. theintroduction of a non-constant shear activation volume into the viscoelastic Leonov mode.

The latter approach turns out to be quite promising with respect to accordance withexperimental data and leads to interesting consequences on the interpretation of largestrain behaviour of polymeric glasses. Therefore it will be explored in more detail in thefollowing section.

Eyring rate theory and application of a non-constant activation volume

The nature of the viscous response in continuum modelling opens a window into the un-derlying atomistic mechanism during a viscoelastic process, since it is directly connectedto the deformation mechanism on a molecular scale. Hence it might play a key-role whentrying to bring light into the relation between atomistic structure, deformation mechanismson a molecular scale, and macroscopic deformation behaviour of polymer glasses.

The basic Leonov mode discussed in section 2.2.1 describes the plastic strain rate γp

and stress-dependent viscosity η(τeq) in eq. (2.13) and eq. (2.19) using Eyring’s molecularrate theory [32, 33, 11]. The basis of this concept is the assumption, that a structuralelement on an atomistic or molecular scale, e.g. a chain, a side chain, or statistical Kuhnsegment in polymers, takes part in an event leading to plastic deformation of a polymerglass by passage over an energy barrier. Such an event involving some structural elementand producing plastic strain will be called a “plastic event”. The process is governed bythe jump probability of such a plastic event over an energy barrier, which, as is knownfrom statistical mechanics, can be described by an Arrhenius-relation

P = P0 exp

(−∆G

kT

)(2.39)

where ∆G is the difference in the Gibbs free energy between the equilibrium and thesaddle point configurations, and P0 the attempt frequency, e.g. the Debye frequency.Eyring described this rate effect in a rate equation, where the energy supplied mechanicallyis expressed by A b τ = V ∗ τ , where b is a displacement vector, A the activation area sweptout during the activated plastic event, and τ the shear stress inducing the deformation.Since V ∗ has the dimension of a volume, it is misleadingly called the shear-stress-activationvolume. The jump probability is increased when external work , e.g. stress, aids the passageover the barrier. Hence the probability for a plastic event is determined by the combinationof applied stress and thermal energy. The Gibbs free energy of eq. (2.39) then becomes

∆G = ∆G0 − V ∗τeq (2.40)

17


where ∆G0 represents the height of the barrier at zero shear stress. As already presentedin eq. (2.13), a rate equation for a reversible, isothermal process can be derived from eq.(2.39) to give a correlation between plastic shear-rate and activation energy

γp = A0 exp

(−∆G0

kT

)sinh

(V ∗τeq

kT

)(2.41)

where the frequency factor A0 is a fundamental rate factor, which may depend on theattempt frequency P0, the strain produced per plastic event, and the density of plasticevents. The hyperbolic-sine term accounts for the back-jumps of the flow unit, whichare not negligible at low stresses. When the argument of the hyperbolic sine function islarge, i.e. when τeq is high and back jumps are negligible, it can be approximated by anexponential function so that sinh(x) ≈ 1

2exp(x) and eq. (2.41) may be written as

γp =A0

2exp

(−∆G

kT

)(2.42)

In the case of an uniaxial deformation test, assuming a constant activation volume ata certain fixed configuration Gamma, V ∗ could be determined from experimental truestress-strain data with the help of eq. (2.42) and eq. (2.40)

V ∗Γ = kT

√3

(dσ

d ln ε

)−1

= constant (2.43)

where σ is the total true stress in deformation direction and ε is the total true strain rate,assuming that after yielding the plastic strain rate approximately equals the total strainrate. The index Γ denotes that this equation was derived for a certain fixed configurationof the system under consideration, hence, in the case of plastic deformation of a polymerglass, Γ may denote the actual state of deformation as a function of stress and strain. Thefactor

√3 comes from the fact that in uniaxial deformation the equivalent plastic strain

rate becomes γeq =√

3 ε.

A constant activation volume would always lead to a vertical shift of post-yield stressas shown in fig. 2.2, i.e. true stress-strain plots at different true strain rates show parallellines after yielding. A good test of the validity of a constant shear activation volume isto plot V ∗, obtained from experimental data via eq. (2.43), as a function of true strain.This should give a straight horizontal line since eq. (2.43) should hold in the post yieldregime independent of strain. Therefore any observable dependence of V ∗ on true strainwould indicate that the shear activation volume V ∗ is indeed a function of the actual stateof deformation and not a constant material parameter.

In general, the concept of a constant shear activation volume is at least doubtful, for itimplies that the activation volume V ∗ is not a function of either stress or strain. Accordingto the original definition of V ∗ as the product of an activation area and a displacementvector b, V ∗ could depend in general on stress and strain thus being a function of the actualstate of deformation.

18


The question is now, how can the activation volume be related to the actual state ofdeformation ? The approaches presented in the following are governed mainly by theaim to get the best agreement between modelling and experimental true-stress-strain dataand are purely empirical relations. Since the activation volume V ∗ is the product of adisplacement vector and the area swept out by a plastic event, a good choice may be alinear relation between V ∗ and the product of the average chain stretch and the averagechange in area during deformation. Following the eight chain model of rubber-elasticity,as already discussed in section 2.2.2, the average chain stretch and the average change inarea are functions of the first and second invariants (IB, IIB) of the Chauchy-Green straintensor B

< λchain >=

√λ2

1 + λ23 + λ2

3

3=

√IB

3(2.44)

< A/A0 >=

√λ2

1λ22 + λ2

2λ23 + λ2

1λ23

3=

√IIB

3(2.45)

Therefore it will be suggested to relate the actual activation Volume to the actual stateof deformation in the following way

V ∗ ∼√

IB IIB

3(2.46)

Alternatively, suggesting a relation of V ∗ on the actual state of deformation throughthe equivalent stress of the entanglement network, rather than through the network strain,even gives a slightly better but on a fitting level nearly indistinguishable agreement ofmodelling and experiment when applying the following purely empirical relation

V ∗ ∼ τneteq (2.47)

These relations would indicate, that at a given fixed strain(

dσd ln ε

)−1is still constant,

i.e. that eq. (2.43) holds, which will be supported by experimental data in section 2.5.Changing the actual state of deformation, i.e. varying the total strain, however, will changethe activation volume. Hence the actual activation volume is altered by the entanglementnetwork response, or in other words: the viscoelastic response is no more independent ofthe rubber-elastic network response, but is altered with strain through a dependence ofthe activation volume on the overall network stretch expressed in invariants of the Cauchy-Green strain tensor B or the equivalent network stress τnet

eq .For uniaxial deformation test, assuming incompressibility, the Cauchy-Green strain ten-

sor B gets

B =

e−ε 0 00 e−ε 00 0 e2ε

(2.48)

in this case eq. (2.46) becomes

19


V ∗ ∼ 1

3

[(e−2ε + 2eε)(2e−ε + e2ε)

]0.5(2.49)

and eq. (2.47) becomes

V ∗ ∼ Gsh(e2ε − e−ε) (2.50)

where Gsh is the modulus of the entanglement network, which contributes to strainhardening.

To clarify the effect of a non-constant activation volume on the large strain behaviour ofthe suggested continuum model, representative simulated data according to V ∗ ∼

√IB IIB

3

will be graphically sketched. Note that on a qualitative level as well as on a fitting level ofexperimental data both approaches of eq. (2.46) and eq. (2.47) are almost indistinguish-able. Figure 2.2 shows simulated true stress vs. λ2−λ−1 plots according to the viscoelasticmodel presented in this study using a strain dependent activation volume like in eq. (2.46).Both cases, a constant (dashed lines) and a non-constant (solid lines) activation volumeV ∗ have been calculated. The upper plots shows the two contributions to the total truestress, i.e. the rubber-elastic entanglement network contribution and the viscoelastic con-tribution due to segmental motion, separately, whereas the bottom plot shows how theyadd together to the total true stress. From this graph it can be nicely seen that a materialwhich exhibits strong strain hardening does not have necessarily a large strain hardeningmodulus Gsh, which is only partly contributing to the stress increase at large strains !

A complete analysis of these suggestions follows in section 2.5.

2.3 Materials

Since this study is focused on the mechanical behaviour of polymer glasses, the choiceof materials can be divided into two sections. First an assortment of typical commer-cial polymers forming a glass at room temperature, which was at least 80K below theirglass transition temperatures . Second chemically cross-linked Poly(methylmethacrylate)(PMMA), which will be investigated with varying cross-link densities.

2.3.1 Commercial polymer glasses

Table 2.1 gives an overview of all polymer glasses used in this study. All polymers wereobtained from Polymer Laboratories, UK, except PSc, PMMAc, PC64K (Aldrich, CH )and PPOc (PolySciences, USA). These polymer glasses cover a wide range of entanglementdensities and are completely amorphous in their structure and in a glassy state at roomtemperature.

20

2.3 Materials

Figure 2.2: The upper plot shows the two additive contributions to the total true stress as afunction of λ2 − λ−1 according to the model presented in this study. Dotted line: entanglementnetwork contribution, dashed lines: viscoelastic contribution with constant V ∗, solid lines: vis-coelastic contribution with non-constant V ∗(ε). The inset shows the applied activation volumeas a function of

√IB IIB/3 for the constant (dashed line) and non-constant (solid line) case. In

the bottom picture the total stress as a sum of the two contributions is presented.

21


Table 2.1: Commercial polymer glasses used in this study. MW is the weight average molecularweight, MN the number average molecular weight, MW /MN the polydispersity, TG the glasstransition temperature, and ρe the entanglement density in units of entanglements/volume.

Material Notation MW [ gMol ]

MWMN

Tg[K] ρe[ 1nm3 ]

Poly(styrene) PS96K 96000 1.03 363 0.0219PS483K 483000 1.05PS1M 1000000 1.03PS4M 3900000 1.05PSc

Poly(methylmethacrylate) PMMA55K 55600 1.02 388 0.0340PMMA100K 100000 1.04PMMA254K 254100 1.04PMMA772K 772000 1.03PMMA1M 1520000 1.08PMMAc 120000 3

Poly(phenyleneoxide) PPOc 50000 2.5 483 0.10455Poly(Bisphenol A carbonate) PC64K 64000 1.66 418 0.2581

2.3.2 Cross-linked Poly(methylmethacrylate) (xPMMA)

To investigate not only the influence of temporary entanglements but also of permanententanglements, so-called cross-links, the polymer chains in PMMA were linked permanentlyby chemical reaction allowing to vary the densities of cross-link points per volume. Cross-linking was effected by diamines with different contour length. A long cross-linker, like1,12-diaminododecane might be elastically active and contribute to the elastic propertiesof the cross-linked polymer. Further the functionality of the cross-link points, i.e. thenumber of elastic active chains coinciding at a cross-link, would change from four to three.This uncertainty would make the determination of the number of cross-links per volume lessreliable. Therefore possible effects of the length of the diamine were checked by comparisonof different diamines (cf. section 2.4.1).

The amidation reaction was catalyzed by p-toluenesulfonic acid. Weighted quantities ofpolymer, diamine, and catalyst were dissolved in methylene chloride. Amounts of diaminewere chosen such that the final cross-link density varies from approximately thirty timesto one tenth of the equilibrium entanglement density of linear non-crosslinked PMMA,covering the range above and below the entanglement density of non-cross-linked PMMA.The catalyst amounted 2-3 equiv % of the amine. After stirring for five minutes, thesolution was poured into a petri-dish and the solvent was allowed to evaporate at roomtemperature over a period of four days. After pressing the thin film into a mold (cf. section2.4.2) to get the desired shape of the sample, the mold was heated under slight pressure to160oC for a period of four days. Degradation of the material is negligible, since the sample

22

2.4 Experimental

is sealed under pressure such that there is nearly no oxygen present during the reaction.

2.4 Experimental

2.4.1 Characterization of rubber-like materials

The basic features of the stress-strain behaviour of rubber elastic materials have been wellmodelled by statistical mechanics treatments of rubber elasticity [35, 36, 37] as describedin (cf. section 2.2.2). It is well established, that for small strains ( r

nl≤ 0.4) neo-Hookean

behaviour gives a satisfactory description [37], whereas at larger deformation finite extensi-bility has to be taken into account. Further, at very large deformations the effect of straininduced crystallization might also increasingly influence the stress-strain behaviour. Sincewe are only interested in the determination of the cross-link density from the rubber-elasticmodulus, it is sufficient to investigate deformation of a few percent only. Hence, in thissection we will completely stick to the Gaussian treatment, i.e. neo-Hookean behaviour.

Determination of the cross-link density of a glassy polymer by application of rubberelastic models provide, that the sample is in a thermodynamic equilibrium state duringdeformation, i.e. deformation tests have to be carried out well above the glass transitiontemperature Tg of the material. In principle this can be achieved in two ways:

Heating the sample above Tg, or swelling the network in a good solvent such that Tg

will decreases far below room temperature. To avoid problems due to thermal degradationat high temperatures and keep the experimental setup as simple as possible, all sampleswere characterized in an equilibrium swollen state. When a network is subjected to a goodsolvent, the solvent will start to penetrate the network and gain free energy of mixing. Onthe other hand the network is stretched by the swelling, thus swelling costs elastic networkenergy and at an equilibrium state, these two contributions balance each other.

Dealing with swollen samples: The phantom model

It is well established [36] that deformation of swollen samples can be described by the socalled phantom model. This rubber-elastic model, using gaussian chain statistics, assumesthat the chains during deformation are subject only to constraints , that arise directly fromthe connectivity of the network, but not from steric hinderance. The effects of junctionsand chains on one another is of no consequence, and the effect of the macroscopic strainis transmitted to a chain through the junctions to which a chain is attached. This isindeed a very good approximation of the situation in a swollen sample, since chains aresurrounded mostly by solvent molecules which have no influence on the elastic propertiesof the network. The latter assumption is one of the presumptions of the so called Frenkel-Flory-Rehner (FFR) hypothesis [46] of the additivity of the free energy of mixing and theelastic free energy in swollen samples, which was strongly supported by experimental workof McKenna et al. [47].

In uniaxial deformation the deformation gradient tensor (cf. eq. (2.1)) of a swollen anddeformed sample can be written generally as

23


i s o t r o p i cs w e l l i n g

s w e l l r a t i o = ( V s / V d ) 1 / 3

V d , A d

u n i a x i a ld e f o r m a t i o na 1 = aa 2 = a - 1 / 2a 3 = a - 1 / 2

V s , AV s , A s

F

F a

Figure 2.3: Decomposition of Cauchy-Green strain tensor F into a contribution due to pure

dilation and isochoric draw ratio α, i.e. F =(

v2v2c

)− 13 Fα

F =

λ1 0 00 λ2 00 0 λ3

(2.51)

where λi is the macroscopic draw ratio along the i-axis. In order to separate the effectsof distortion from volume changes, λi may be expressed as the product of a pure dilationand an isochoric draw ration α at constant volume (cf. fig. 2.3):

λi = (Vs/Vd)13 αi =

(v2

v2c

)− 13

αi (2.52)

leading to

F =

(v2

v2c

)− 13

α1 0 00 α2 00 0 α3

=

(v2

v2c

)− 13

Fα (2.53)

Here, Vd is the volume during formation of the network, and detFα = 1. The quantitiesv2c and v2 are the volume fraction of polymer during formation of the network, and duringthe elasticity experiment, respectively. Let us consider the relation between true stress σand isochoric draw ratio α according to the phantom model [36]:

σ =F

A=

ξkT

Vd

(v2

v2c

) 13

(α2 − α−1) (2.54)

where F is the force acting on the sample, ξ is the cycle rank of the network, and kTthermal energy. The cycle rank is the number of network chains that must be cut to toreduce the network to a state containing no closed cycles. Other important parameters inthe description of networks are N the number of elastic active chains per unit volume, and

24

2.4 Experimental

µ the number of cross-links per unit volume. Relation between these parameters are (inthe limit of N 1):

ξ = N − µ (2.55)

N =f

2µ (2.56)

where f is the functionality of a network, which is the number of chain-ends merginginto a cross-link. Converting eq. (2.54) into engineering stress σn referring to the area ofthe swollen sample As we get

σn =F

As

=ξkT

Vd

(v2

v2c

) 13

(α− α−2) (2.57)

Experimental data is often analyzed in terms of the reduced stress [σR], which is definedas

[σR] =σnv

−1/32

α− α−2=

ξkT

Vd

(2.58)

providing that v2c = 1, which is always the case in this study. Plotting [σR] as a functionof α−1 allows to study deviations from the applied model very easily, since good agreementis given when the stress-strain line shows nearly no dependence on α−1. Figure 2.4(a) showsthat for various cross-link densities, the phantom model is a reasonable way to describestress strain behaviour of swollen xPMMA samples.

As already mentioned, cross-linking was effected by diamines with different contourlength. Elastic moduli of samples cross-linked with diamines having 6, 8, 10, and 12carbon atoms under the exactly same conditions were indistinguishable within the scatterof the experiment as shown in figure 2.4(b).

Thus the elasticity of the cross-linked PMMA in the swollen state is governed primarilyby the polymer chains and not by the cross-linker. Only for very high cross-link densitiesof [σR] > 4, i.e. if the length of the cross-linking molecule is comparable to the length ofa polymer chain between two cross-links, the elastic properties of the polymer might beinfluenced slightly by the cross-linker and the functionality decreases to three.

It should be noted that the elasticity of a cross-linked polymer is characterized by thecycle rank ξ, which can be determined experimentally independent of the knowledge ofthe functionality of the network. This means, that elastic properties of networks can becompared independent of the functionality. Further the influence of the cross-linker-chains,i.e. the diamine molecules, on other than elastic properties is always weighted by a factor1/3, since the ratio between the number of cross-linker-chains and the number of polymernetwork chains is always constant at this value. Thus we perform consistent experiments bydetermining experimentally an effective cycle rank of the cross-linked polymer, which is theonly parameter determining the elastic properties of the network, whatever the microscopicstructure is.

25


Figure 2.4: (a) Reduced engineering stress - engineering strain curves for xPMMA cross-linkedwith various cross-link densities. Solid lines represent the fits of the phantom model. (b) Reducedengineering stress - engineering strain curves for xPMMA cross-linked with diamines of differentcontour length.

26

2.4 Experimental

Figure 2.5: Flory-Huggins interaction-parameter χc determined via equilibrium swelling experi-ments, as a function of the cross-link density, which was determined independently by stress-strainexperiments of swollen samples.

Another way to determine the cross-link density is to measure the equilibrium swellingrate of the polymer network swollen in excess solvent using the FFR-hypothesis [46] ofthe additivity of the free energy of mixing and the elastic free energy in swollen samples.The problem here is the uncertainty in the solvent-polymer interaction necessary to cal-culate the free energy of mixing. Experiments have shown that this interaction cannotbe described by a constant factor χ, the so called Flory-Huggins interaction parameter,but depends linear on the degree of cross-linking [48]. Figure 2.5 shows the Flory-Hugginsinteraction parameter χc determined via equilibrium swelling experiments as a functionof cross-link density, which was determined independently from stress-strain plots of theswollen material. Obviously the data in this thesis corresponds nicely with the linearrelation, suggested by McKenna et al. [48]

χc = χ0 + αµ (2.59)

Thus it is very questionable if it is possible to get reliable data on the cross-link densityof the network via equilibrium swelling experiments and all data on cross-link densitiesrepresented in this study were determined by analysis of uniaxial compression tests ofswollen samples.

27


2.4.2 Sample preparation

All samples used in this study were produces by compression molding of the polymer melt.Compression molding was done very carefully to avoid two severe problems during process-ing: thermal degradation and introduction of strain due to the non-neglectible viscosityof the polymer melt during molding. Where thermal degradation could be avoided by acareful choice of the processing temperature, the introduction of strain can be minimizedin a twofold manner: first, a very slow compression speed allows the samples to flow intothe preheated mold and reach an equilibrium state bevore temperature is decreased slowly(> 3h) below the glass transition at a very low pressure of less than 0.02 GPa, and second,all samples were annealed for 30 min slightly above their glass transition after they hadreached room temperature in the compression mold. In a final step the surfaces of all sam-ples were polished to yield flat parallel surfaces. Cross-linked samples were processed thesame way, but keeping the processing temperature over a period of 4 days before coolingdown as described in section 2.3.2.

In general two types of shapes of the samples were realized. Fur uniaxial compressiontests the samples had cylindrical geometry with 4mm in diameter and 3mm of height. Forthe plane strain compression tests and NMR experiments, samples were shaped rectangular(4mm x 3mm x 3mm), to fit into the compression setup sketched in fig. 2.8.

2.4.3 Uniaxial compression tests: true stress - true strain - plots

Basic definitions The prefix true in the definition of stress, strain and strain rate isusually used to distinguish from the engineering definitions of the assigned variables, whichrelate measured data to time independent reference values. Infinitesimal true strain dε isdefined as

dε =dl

l(t)(2.60)

where l(t) is the actual length of the sample. Infinitesimal engineering strain dε refersnot to the actual length l(t) of the sample, but to the initial length at the beginning of theexperiment l(t = 0) = l0.

dε =dl

l0(2.61)

Table 2.2 shows the time evolution of the engineering and the true physically definitionof stress, strain and strain rate for a one dimensional deformation experiment.

True deformation and engineering uniaxial deformation

A real experiment usually only gives raw data from the applied force and the absolutelength of the sample in one dimension. Constitutive modelling of experimental deformationbehaviour of polymer glasses requires true stress - true strain data. To convert the raw

28

2.4 Experimental

Table 2.2: Time evolution of the engineering and the true physically definition of stress, strainand strain rate for a one dimensional deformation experiment.

true engineering

stress σ(t) = F (t)A(t)

σn(t) = F (t)A0

finite strain ε(t) = ln( l(t)l0

) ε(t) = l(t)−l0l0

strain rate ε(t) = l(t)l(t)

ε(t) = l(t)l0

data into a true stress - true strain plot, it is necessary to have information not only aboutthe actual force F (t) and the actual length l(t), but also about the actual area A(t) theforce is acting on, i.e. in the ideal case full three dimensional control over the deformationof the sample body is necessary. Since full three dimensional deformation control is oftenrestricted by the experimental setup, one has to implement additional assumptions, suchas conservation of volume, which allow calculation of the three dimensional deformationfrom the measured deformation of one dimension only. Obviously the deviation of thetrue values from the engineering approximation is zero at t = 0 and increases with thetime evolution of the experiment, i.e. with the applied strain. Thus at small strains thedifference between true and engineering stress and strain is often within the scatter ofthe experiment and the engineering quantities are therefore often preferred for the sake ofsimplicity. From medium to large strains this is no more true and true stress-true strainexperiments are indispensable. In this case it is advantageous to measure both the axialand lateral strain of a sample with cylindrical symmetry during the experiment, since onlythe assumption of homogeneous deformation has to be implemented for the calculation oftrue stress - true strain curves.

To visualize the effect on a stress-strain plot, Figure 2.6 shows typical stress-strain curvesof a polymer glass (Poly(methylmethacrylate) deformed at room temperature in compres-sion at true constant strain rate of ε = 0.001. Even for very small strains (> 0.04) fig.2.6(a) shows considerable deviations from the true stress-strain plot. Approximation ofthe true strain by engineering strain is not that sensitive provided that the stress is truestress. Obvious deviations occur in fig. 2.6(b) not before strain values greater than 0.3.This is of course not a surprise, since the ration of true strain to engineering strain equalsln(1 + ε)/ε and the ratio of true stress to engineering stress behaves like 1/ exp(ε) whenplotted against true strain (see Table 2.2).

Hence only the linear regime of the stress strain curve can be approximated by theengineering approach to a satisfying extend. Another even more costly effort is to keep thetrue strain rate constant during the experiment. Imagine two experiments with constantengineering strain rate ε = const. and constant true strain rate ε = const.. Then the ratioof the deformation rates l(t) in the two experiments scales like

29


Figure 2.6: Typical stress-strain curves of a polymeric glass (Poly(methylmethacrylate)) de-formed at room temperature in compression at true constant strain rate of ε = 0.001. (a) truestress curves plotted versus true and engineering strain. Obvious deviations occur at strainsgreater than 0.3.(b) True and engineering stress curves plotted versus true strain. Obvious devi-ations occur at strains greater than 0.04

30

2.4 Experimental

Figure 2.7: Typical true stress - true strain curves of glassy poly(methylmethacrylate) deformedat room temperature in uniaxial compression at various constant strain rates increasing frombottom to top (0.0001 s−1, 0.0003 s−1, 0.001 s−1, 0.003 s−1). Dotted lines represent experimentswith constant engineering strain rates, whereas the solid lines show experiments with constanttrue strain rates of the same value.

l(t)eng

l(t)true

∼ l0l(t)

(2.62)

Hence the deformation rate in a compression experiment with constant engineering strainrate is faster than the true strain rate by a factor l0/l(t) increasing with time. Sincedeformation of polymer glasses beyond the linear regime is very sensitive to strain rate,constant true strain rate experiments are also indispensable to draw reliable conclusionsfrom the strain rate dependence of true stress - true strain curves. Figure 2.7 showstrue stress - true strain curves of a polymer glass, Poly(methylmethacrylate), deformed atroom temperature in uniaxial compression at different strain rates keeping the engineering(dotted lines) and the true strain rates (solid lines) constant.

Experimental setup

All deformation experiments were performed on a mechanical Zwick Z020 tensile and com-pression tester. The mode of deformation was either uniaxial compression or plane straincompression. During the experiment the sample dimension along the compression directionwas monitored by the crosshead of the tensile tester and corrected instantly by the finite

31


S a m p l e

D i e

S t a m p

XY

Z

p l a n e s t r a i n c o m p r e s s i o n u n i a x i a l c o m p r e s s i o nFigure 2.8: Schematic sketch of the experimental setup of plane compression and uniaxialcompression tests. In plane strain compression the deformation in x-direction is constraint.

compliance of the whole machine before data acquisition. The lateral sample dimensionof the cylindrical symmetric samples was monitored by a video extensometer from Mess-physik. The error in the lateral dimension is a little higher, around 5·10−2 from comparisonof data provided by the crosshead and the video extensometer, since the performance ofthe video extensometer is restricted by various factors like resolution, illumination of thesample, or horizontal alignment of the camera. A user-specific software, which was de-veloped by the Zwick company on our demands, was used to keep the true strain rateconstant during the experiment. Note that the correction by the finite compliance of thetest setup is implied permanently during data acquisition, thus giving a real constant truestrain rate. To control the temperature the setup was placed in a climate chamber whenperforming a compression test. Thus true stress-strain experiments at constant true strainrate were performed with full control of two sample dimensions and temperature.

Figure 2.8 shows a schematic sketch of the experimental setup for plane strain anduniaxial compression, which was fitted to the mechanical testing machine. The geometryof the samples for the plane strain compression was rectangular and cylindric for uniaxialcompression. To avoid bulging, or sometimes also called barrelling, of the samples duringcompression, friction between the compression setup and the sample surfaces has to bekept as small as possible. This was achieved by placing a thin sheet of teflon (0.02 mmthickness) in combination with commercial fluid dishwasher (a 6/1 blend of anionic tensidesand soap) on top of all surfaces of the sample, which had contact with the compressionsetup. In this way bulging could be avoided up to the maximum strains applied in thisstudy of about ε = −0.9 (cf. fig. 2.14).

32

2.5 Results and Discussion


2.5.1 True stress-strain plots

True stress-strain plots have been obtained in uniaxial compression at different constanttrue strain rates for the polymers listed in table 2.1, where the total number of obtainedsets, each including 4 − 5 different strain rates, was 35, thus having a reliable numberof statistical significant experiments. Note that all the effects discussed below could beobserved in every set of true stress-strain curves.

Figures 2.9(a) - 2.12(a), show representative plots of PMMA, PPO, PC, and PS togetherwith the best fits (red solid curves) of the suggested continuum model of section 2.2.4including a non-constant activation volume according to eq. (2.46). Figures 2.9(b) -2.12(b) show the experimental activation volume determined from true stress - true strainplots via eq. (2.43) together with the activation volume (red curves) used for the continuummodelling of trues tress - true strain plots in Figures 2.9(a) - 2.12(a). Since the aim of thisthesis is to investigate plastic deformation of glassy polymers at large finite strains. Asdiscussed in section 2.2.3 strain softening has no influence on the large strain behaviourof polymer glasses, thus justifying the comparison of experimental large strain data tocontinuum modelling of fully rejuvenated samples. Hence the regime of fitting was alwayschosen to start after the local strain softening minimum, as indicated by the shaded parts ofthe graphs, which were not taken into account for the modelling. The initial elastic modulusof the modelled curves was determined from the experimental stress-strain curves.

First of all, the most striking feature of these plots is, that, obviously, the experimentalshear activation volume is not independent of the actual state of deformation, but ratherdecreases with increasing strain. Or, in other words, strain hardening, i.e. the slope of thetrue stress-strain curve at a given true strain, increases with increasing strain rate, whereasa constant shear activation volume would lead to parallel curves at different strain rates,i.e. strain rate independent strain hardening.

Moreover, it appears that a quantitative description of the experimental curves at largedeformation can be achieved by the incorporation of a non-constant shear activation volumein the compressible Leonov model.

True stress-strain test have been performed for cross-linked PMMA with varying cross-link-densities in a similar way. Figure 2.13 shows engineering stress curves plotted against(λ− λ−2) for different degrees of cross-link density increasing from the bottom to the topcurve. Any deviation from Gaussian (neo-Hookean) behaviour is reflected by a non-linearstrain hardening response. Obviously there is an influence of finite extensibility at highdegrees of cross-linking.

So far only gaussian chain statistics of infinitely extendible chains has been used for theentanglement network response in the continuum model. Fits of the basic Leonov modelmodified by a non-constant shear activation volume to the experimental data of cross-linked PMMA (not shown) were performed by the application of finite extendible chains(Langevin chain statistics).

As described in section 2.2.2 Langevin chain statistics might be applied by the application

33


Figure 2.9: (a) True stress - true strain plot of PMMA772K obtained in uniaxial compressionat different true strain rates increasing from bottom to top curve. Applied strain rates areε = −0.0001, ε = −0.0003, ε = −0.001, ε = −0.003, ε = −0.01. (b) Experimental activationvolume determined via eq. (2.43) from the plots in (a) as a function of

√IB IIB/3 as suggested

in eq. (2.46). Red curves show the best fit of continuum model simulations with a non-constantactivation volume to the experimental data in the non shaded region of the graph.

34


Figure 2.10: (a)True stress - true strain plot of PPOc obtained in uniaxial compression atdifferent true strain rates increasing from bottom to top curve. Applied strain rates are ε =−0.0001, ε = −0.0003, ε = −0.001, ε = −0.003, ε = −0.01. (b) Experimental activationvolume determined via eq. (2.43) from the plots in (a) as a function of

√IB IIB

3 as suggested ineq. (2.46). Red curves show the best fit of continuum model simulations with a non-constantactivation volume to the experimental data in the non shaded region of the graph.

35


Figure 2.11: (a) True stress - true strain plot of PC64K obtained in uniaxial compressionat different true strain rates increasing from bottom to top curve. Applied strain rates areε = −0.0001, ε = −0.0003, ε = −0.001, ε = −0.003, ε = −0.01. (b) Experimental activationvolume determined via eq. (2.43) from the plots in (a) as a function of

√IB IIB

3 as suggested ineq. (2.46). Red curves show the best fit of continuum model simulations with a non-constantactivation volume to the experimental data in the non shaded region of the graph.

36


Figure 2.12: (a) True stress - true strain plot of PS483K obtained in uniaxial compressionat different true strain rates increasing from bottom to top curve. Applied strain rates areε = −0.0001, ε = −0.0003, ε = −0.001, ε = −0.003. (b) Experimental activation volumedetermined via eq. (2.43) from the plots in (a) as a function of

√IB IIB

3 as suggested in eq. (2.46).Red curves show the best fit of continuum model simulations with a non-constant activationvolume to the experimental data in the non shaded region of the graph.

37


Figure 2.13: Engineering uniaxial stress against (λ − λ−2) for PMMA with varying cross-linkdensities determined at constant true strain rate ε = −0.001 . The bottom curve was determinedwith non cross-linked PMMA.

of the eight-chain model with the help of an additional parameter, the average number ofstatistical Kuhn segments n per elastically active chain. Unfortunately the introductionof this additional parameter caused instabilities in the data fitting, thus leading to non-reproducible fitting results, which will not be discussed further.

Nevertheless, the experimentally obtained behaviour of the activation volume of cross-linked samples can be analyzed with respect to the suggested relations eq. (2.46) and eq.(2.47), which will be discussed in the next section.

One may argue, that the effect observed here are only a result of systematic errors. Butsuch arguments can be invalidated by the following discussion:

Systematic errors

To get accurate and reproducible true stress - true strain curves in compression, exper-iments have to be performed with great care, especially in the regime of large strains.Although systematic errors can never be ruled out completely, the focus of every experi-ment was to minimize systematic errors relative to their impact on the experimental dataas far as possible. Therefore three main sources causing adulterated data will be addressedin the following. Beside using a constant true strain rate instead of a constant absolute de-formation speed (cf. section 2.4.3), great care has to be taken about the finite complianceof the testing machine, lateral friction effects, and effects due to internal heat generationin the tested specimens.

A finite compliance of the testing machine always leads to a shift in the recorded strains,

38


Figure 2.14: Virtually no bulging can be observed at different degrees of compression. The leftmost sample is uncompressed.

Figure 2.15: Slopes of true stress vs. λ2 − λ−1 plots, taken at constant absolute force (solidsquares), and constant strain (open squares).

39


which is a function of absolute applied force during the compression test. To correct forthis a force-displacement curve of a sample of same diameter as the polymer specimens, butvirtually incompressibility relative to the testing machine was recorded. This test samplewas machined from extreme hardened steel having the same diameter as the polymerspecimen but a much smaller height of only 0.2mm, which is only a factor 0.001 of theheight of the whole test setup. Hence giving the deformation of the test setup as a functionof absolute applied force. All compression test where corrected by this curve in real timeduring the deformation test.

Nevertheless the application of such a correction curve still cannot rule out completelyan influence of finite machine compliance. Assuming for example, that the striking featureof strain rate dependent slopes of true stress -strain curves in the strain hardening regimecould be claimed completely on effects which are a function of absolute force, like finitemachine compliance or lateral friction, a contradiction arises: plotting the slope in thestrain hardening regime determined at constant absolute force as a function of strain rate,there should be no effect if friction or finite machine compliance were completely respon-sible for the strain rate dependent slopes observed at constant strain. Figure 2.15 showsthe observed strain rate effect on the strain hardening slopes recorded at constant straintogether with strain hardening slopes taken at constant absolute force. In the plot takenat constant absolute force, strain hardening slopes should be independent of strain rate ifmachine compliance or friction would be the dominating factor of the observed strain ratedependence of strain hardening slopes taken at constant strain. Another argument againstsignificant friction, is to look at the barrelling of the samples at large strains. Figure 2.14shows various samples compressed to different degrees in comparison to an uncompressedsample. Obviously barrelling could be minimized even for large strains using advancedlubrication methods as described in section 2.4.3.

Internal heat generation at large strain rates can be ruled out to be responsible for theincreasing strain hardening slope with increasing strain rate, since the effect would be asoftening of the material. Further all deformation speeds were kept below ε ≤ −0.01s−1

and a high surface to volume ratio of 1.6 was chosen for the tested specimen to optimizeheat transfer with the surrounding. Temperature effects on true stress-strain curves due tointernal heat generation have been reported indeed in literature, but starting at strain ratesε ≥ −0.03s−1 using samples with lower surface to volume ratios in the range 0.5−1 [49,50].

2.5.2 Activation volume

A crucial point in the whole treatment of the continuum modelling extended with a non-constant activation volume is the question, if eq. (2.43) holds at any state of deformationbeyond the yield point, i.e. if dσ/d ln(ε) = constant is true for any fixed strain. It is wellestablished that this is true at the yield point itself, but what happens in the post yieldregime ? For all sets of compression tests, i.e. one set contains true stress-strain curvesrecorded at 5 different true strain rates, the true stress at fixed total strain was plottedagainst the logarithmic true strain rate ln(ε) for different discrete fixed values of strainspanning the whole post yield regime. Thus, according to eq. (2.43), a linear data fit can

40


Figure 2.16: Relative standard error of a linear fit to true stress - ln(ε plots of the true stress- true strain plots in figures 2.9(a) - 2.12(a) as a function of true strain. Blue: PC64K, green:PS483K, red: PPOc, black: PMMA772K. The approximate strain interval, which covers theyield-regime is indicated by the hatched area.

be performed to yield the activation volume as a function of total strain.

Figure 2.16 shows the relative standard deviation of the linear fit to the true stress -ln(strain rate) plots as a function of true strain for the samples presented in figures 2.9 -2.12. Obviously, the quality of the fit does not get worse with increasing strain featuring arelative standard deviation of the fits not exceeding 15%, with exception of the curves forPC, which show a worse fit in the strain softening regime around ε ≈ −0.3. This is dueto the fact, that in general the strain rate dependence of true stress-strain plots of PC isquite small compared to PMMA or PPO, hence the dependence on strain rate being muchmore sensitive to experimental errors. Nevertheless, fig. 2.16 indicates, that eq. (2.43) isvalid not only at the yield point, but holds independently of the state of deformation inthe post yield regime.

Another interesting feature can be obtained from true stress-strain data of chemicallycross-linked PMMA. The effect of cross-link density on the strain hardening behaviour wasalready shown in fig. 2.13 and confirmed recently by Melick et al. [51], suggesting thatthe entanglement network response plays a significant role in the deformation behaviourof polymer glasses at large strains. Incorporation of a non-constant activation volume assuggested in this thesis implies that V ∗ is a function of the network response, which itselfis a function of invariants of B or the equivalent network stress τnet

eq (cf. eq. (2.46) andeq. (2.47)). Hence there should be an effect of cross-link density on the behaviour of V ∗

during deformation, since it is supposed to change the network response.

Figure 2.17(a) and 2.17(b) show how dV ∗/d[exp(2ε) − exp(−ε)], respectively

41


Figure 2.17: Change of the activation volume according to the two suggested dependencies of eq.(2.49) and eq. (2.50) on invariants of the Cauchy Green strain tensor or the equivalent networkstress. The dotted lines indicate the corresponding value for the non-cross-linked material. Theinsets show the activation volume V ∗ plotted against the suggested dependencies for three differentrepresentative cross-link densities. Solid black lines are linear fits to the data.

42


dV ∗/d[√

IB IIB/3] change with cross-link density. Obviously there is a correlation whichchanges linearly with increasing cross-link density of the network. The insets demonstratethat the suggested dependence of the activation volume on invariants of B or on τnet

eq ,could be approximately described to be linear. Moreover, according to eq. (2.50), theslope dV ∗/d[exp(2ε)− exp(−ε)] should scale linear with the strain hardening modulus Gsh

of the cross-linked network, which itself should also scale linear with the cross-link density(cf. eq. (2.37)). Such a behaviour is indeed observed in fig. 2.17(a).

Of course the physical meaning of the so-called activation volume is still not a prioriclear and the relations presented in eq. (2.49) and eq. (2.50) are purely phenomenological.But the fact, that there is a strong indication for a dependence of V ∗ on the networkresponse, supports the idea, that segmental motion, which is presented by the viscoelasticLeonov mode, is affected by the total network stretch at large strains, i.e. possibly by thesegmental orientation. Hence it would make sense to treat the network response and theviscoelastic not as completely independent additive contributions in continuum modelling.

Further, the activation volume presents a link between the underlying deformation mech-anism during plastic deformation of polymer glasses on a molecular scale and macroscopiccontinuum treatment. A first mechanical model of the molecular deformation process de-termining the viscosity of the viscoelastic element was presented by Argon [2] and laterrefined by numerous authors [3, 4]. Although these models are not capable to reproducethe experimental determined activation volume in this study, the main scope of findinga proper model for the molecular deformation process of glassy polymers, which agreeswith experimental data, might also have a great impact on the improvement of atomisticsimulations [6, 7, 8].

2.5.3 Strain hardening modulus

Another consequence of the application of a non-constant activation volume is that strainhardening results not only from the elastic response of the entanglement network, but alsofrom the contribution of the viscoelastic part due to segmental motion. These two contri-butions to strain hardening are connected by the activation volume, which is a functionof the rubber-elastic network response. Thus the strain hardening modulus cannot be de-termined directly, i.e. model free, from the experimental data, but only through a best fitof the simulated model. Therefore the values obtained for the strain hardening moduluswith the suggested model of this study, should not be taken literally, since they dependon the suggested behaviour of the activation volume V ∗ (cf. eq. (2.46) and eq. (2.47)),which has purely empirical character in this study. Nevertheless, NMR experiments pre-sented in chapter 4, might be capable to obtain parameters of the underlying entanglementnetwork, like the strain hardening modulus, by molecular orientation-strain relations. Tocompare the suggested continuum modelling of experimental true-stress-strain data withresults from solid state NMR experiments of the same material, average strain hardeningmoduli Gsh and activation volumes V ∗

yield obtained at the yield point for different materialsused in this study together with an error-estimation are determined.

Table 2.3 shows the average strain hardening modulus determined from several fitted sets

43


Table 2.3: Strain hardening modulus GSH , and activation volume V ∗ obtained from at least 4best fits of the simulated viscoelastic model to the experimental data. In the case of PPO thereis no error since only a single set of true stress-strain curves was investigated.

Material Gsh V ∗yield

PMMA 20.77± 3.39 0.95± 0.04

PS 7.01± 2.15 1.79± 0.13

PC 19.31± 1.65 6.25± 0.25

PPO 41.42 2.38

of experimental data, each consisting of true-stress-strain curves recorded with at least 4different strain rates and average values for the activation volume obtained at the yieldpoint. Indeed a rough agreement with results from NME experiments can be observed asdiscussed in chapter 4.

2.6 Conclusions

True stress-strain curves obtained in compression for four different glassy polymers show astrain and strain-rate dependent change of slope in the strain hardening regime, beside thewell known vertical shift of true stress as a function of logarithmic strain rate ln(γ). Atthe yield point, defined as the local true stress maximum preceding strain softening, thisvertical stress shift can be modelled by an Eyring-flow process, as confirmed by numerouspublications in literature. Moreover, in this thesis it could be validated, that the Eyring-concept predicts strain rate dependence of true stress in a proper way independently ofstrain for the whole range of applied post yield deformations in this thesis.

In contrast, the experimentally detected change in slope of true stress-strain curvescannot be modelled by continuum models so far, since they only predict the vertical stressshift with strain rate, but assume strain rate independent slopes of true stress-strain curvesat large strains. This is due to the fact, that plastic flow is considered to be independent ofthe actual state of deformation, i.e. independent of total strain, thus leading to a constanttrue stress in the post yield regime, only vertically shifted with strain rate as describedby the Eyring-flow process. Strain hardening is captured in continuum modelling by anequilibrium rubber-elastic response of the entanglement network, which is added to theviscoelastic behaviour due to segmental motion.

To incorporate the effect of strain rate and total strain on the slope of the true stress-strain curves, it is suggested to introduce a dependence of plastic flow on the actual state ofdeformation, i.e. on stress and strain. This could be achieved by giving up the concept ofan intrinsic constant activation volume, which was treated so far as a material parameter

44

2.6 Conclusions

of the Eyring-equation. Instead this material parameter might be replaced by a parameterdepending on the rubber-elastic response of the entanglement network. Two suggestionwere made, which relate the activation volume V ∗ on the actual state of deformation byintroducing a linear dependence of V ∗ on invariants of the Cauchy-Green strain tensor andthe equivalent network stress.

Both suggestions, indistinguishable on a fitting level, have purely empirical character,but are indeed capable to describe in a realistic way the change of slope of true stress-strain curves as a function of strain rate and actual state of deformation. Further, anactivation volume which is a function of the equivalent network stress seems to be ableto describe the effect of varying cross-link density properly. Following this approach, thestrain hardening modulus of the underlying entanglement network cannot be detectedanymore model-free from experimental data, since strain hardening is now a result of therubber-elastic entanglement network response and a change in the activation volume withstrain. Hence it can be concluded that indeed the activation volume, treated so far as amaterial parameter, depends on the actual state of deformation and is rather a function ofstress and strain, than an intrinsic constant parameter.

45

46

3 Determination of segmentalorientation by solid state NMR

3.1 Introduction

The motivation for the nuclear magnetic resonance (NMR) study of polymeric glasses, wasto get information about average orientational distributions of molecular segments as afunction of deformation. Such orientation-strain relations can be compared to theoreticalmodels (cf. chapter 4) and may help to elucidate the microscopic mechanism of glassydeformation on a molecular level, which still lacks a thorough understanding. Moreovercontinuum mechanical modelling and atomistic modelling might profit from a better un-derstanding of the underlying molecular deformation mechanism.

Solid-state NMR in contrast to NMR of liquids can yield information about orientationaldistributions of molecular segments due to the anisotropic nature of the interaction tensors,which are not averaged out due to rapid molecular motion as in the case of a liquid.

In section 3.2 a basic treatment of solid-state NMR will be presented, followed by adetailed discussion of a proper analysis of solid-state NMR spectra in section 3.3 in orderto reconstruct physical properties, such as orientational distribution function of molecularsegments. A robust NMR experiment will be presented in section 3.4, which is sensitive todetect even low degrees of orientational order, which is the case for most tested specimenin this thesis. Finally, in section 3.5, the suggested analysis of solid-state NMR spectrawill be tested with respect to reliability, and resolution power. This will be achieved bya comparison of several different random “input” orientational distribution functions, andthe “output” orientational distribution function reconstructed from the spectra simulatedwith random noise according to the applied experiment. Conclusion will be summarizedin section 3.6.

3.2 Solid-state NMR and nuclear spin interactions

The principles of Nuclear Magnetic Resonance (NMR) spectroscopy are extensively treatedat different levels in the literature [52, 53, 54, 55, 56]. Hence only a brief description of thebasic technique will be outlined. In a typical NMR experiment a sample is placed in astrong magnetic field ~B0 of a superconduction magnet (magnetic field strength ≈ 7 − 14

Tesla), where a macroscopic magnetization is built up along ~B0. A subsequently resonant

47

3 Determination of segmental orientation by solid state NMR

radio frequency (RF) pulse rotates the macroscopic magnetization through 90o into the

plane perpendicular to ~B0 creating transverse magnetization. The magnetization thenprecesses around ~B0 and induces a voltage in a pick up coil which surrounds the sample.This signal, which also called free induction decay (FID), is than recorded and the NMRspectrum is subsequently generated by a numerical Fourier transform (FT).

NMR of rigid solids offers the important feature of orientation dependent, i.e. anisotropicNMR interactions between the nuclear spins and the magnetic field acting on them. Incontrast to NMR of liquids or dissolved substances, where orientation dependence of anynuclear interaction is mostly averaged out due to rapid translational and rotational motionof the molecules, the anisotropic parts of the interaction tensors remain present in thesolid state. This is due to the fact that molecular dynamics, especially in glassy solids, arepractically frozen on the time scale of the experiment.

The anisotropic nature of the interaction tensors can lead to a huge broadening of thesolid state NMR signal by several orders of magnitude (interactions may range from Hz tokHz to MHz) compared to the narrow line of spin under rapid molecular motion. Solid stateNMR may be used to map the orientational distribution of interaction tensors onto theexperimentally accessible NMR spectrum and, vice versa, allows to extract orientationalinformation from the spectrum. Details about spectrum analysis will follow in section 3.3.

The local magnetic field felt by the nuclei is not the same as the external applied field,but slightly modified due to sources of magnetic fields internal to the sample. The totalHamiltonian capturing the most relevant interactions between the nuclear spin and thelocal magnetic field can be written

H = HZ + HRF + HD + HCS + HQ (3.1)

where

HZ is the Zeeman interaction which accounts for the coupling of nuclear spins with theexternal static magnetic field

HRF represents the coupling of nuclear spins with the applied RF field

HD is the dipole-dipole interaction of nuclear spins with each other via their magneticdipole moments

HCS describes the chemical shift associated with the electronic screening of nuclei

HQ accounts for the coupling between nuclear spins with quadrupole moments and electricfield gradients

The contributions to the total Hamiltonian can be separated into two categories: HZ andHRF depend only on external parameters such as the strength of the applied static and RFfields, whereas the remaining terms are generally anisotropic and depend only on internalinteractions and contain the molecular information of interest. The spatial dependence ofthe hamiltonian can be described by second order tensors, where the Zeeman term HZ is

48

3.2 Solid-state NMR and nuclear spin interactions

dominant in most cases, such that the other terms can be treated by first order perturbationtheory.

In the following sections all interaction terms will be discussed only briefly, where ex-tra stress will be put on the discussion of the quadrupolar term, which dominates theinteraction of Deuterium labelled molecules as used in this study.

3.2.1 Dipole-dipole interaction

In contrast to the weak scalar J-coupling, which describes an indirect coupling of the nucleimediated by electrons, the much stronger dipole-dipole coupling is a direct interactionthrough space between the magnetic moments of two nuclei designated j and k. TheHamiltonian can be written

HD =∑

all j,k pairs

I jDjkI k (3.2)

where Djk represents the dipolar coupling tensor and I j the spin operator for the nu-cleus designated j. In solution the dipole-dipole interaction is averaged out completely byrotational motion, since the interaction tensor is traceless, i.e. it has no isotropic part. Ina solid this is not the case and this interaction is a major cause of line-broadening [52,53].Note that the entire interaction is determined by the geometrical arrangement of the cou-pling partners, since the hamiltonian HD depends on the direction of the internuclearvector between two nuclei relative to the externally applied field [53]. Hence, the dipolarcoupling is the one spin interaction which can provide a means of determining internucleardistances and from these, the geometry and conformation of molecules. Nuclei with I = 1

2

can only have magnetic dipole moments, since their higher (static) multipole moments,magnetic as well as electric, vanish [57].

3.2.2 Chemical shielding

Chemical shielding has become one of the most important foundations of the spectroscopicpower of NMR with respect to structural elucidation. This is due to the relatively simpledependence of the chemical shielding on the local chemical structure around the nucleusunder consideration [56]. In the presence of a magnetic field the electron cloud aroundthe nucleus also generates an additional local field, hence shielding the nuclear spin fromthe externally applied ~B0 field. Diamagnetic currents in the electron orbitals and partialunquenching of their paramagnetism generate these local fields that scale with the staticexternal field ~B0

HCS = γI σCS~B0 (3.3)

where γ is the magnetogyric ratio of the nucleus and σ the chemical shielding tensor.Since the chemical shift tensor is a direct consequence of the electronic structure, it containsinformation on that structure, and can serve at least to verify and falsiy calculations of

49


the binding structure of molecules. Chemical shielding is a time independent, anisotropicinteraction.

3.2.3 Nuclear quadrupol interaction

In general, a proper description of distributions of charge, such as protons in a nucleus, re-quires expanding the charge distribution as a series of multipoles. The zero order multipoleis the total charge, the first order the electric dipole moment and the next not vanishingmoments is the fourth order moment, the electric quadrupole moment Q. The quadrupoleinteraction arises from the electrostatic interaction of the nuclear quadrupole moment Qwith the electric field gradient. Consider for example the quadrupole moment of deuteriumbonded to a carbon atom: here the electric field gradient originates from the electrons inthe C − D bond. The electric field gradient i can be expressed in terms of a traceless,symmetric second-rank tensor V = eq .

Nuclei with I ≥ 1, e.g. deuterium (I = 1), exhibit a quadrupole moment Q, which,subjected to an electric field gradient V, can be described by the following Hamiltonian(for a single spin)

HQ =eQ

2I(2I − 1)~IVI (3.4)

Since local fields felt by the deuterium nucleus are small with respect to the staticmagnetic field, local interactions can be treated in first-order perturbation theory leadingto truncated Hamiltonian. Generally the Hamiltonian for first-order quadrupole couplingas a function of the polar angles (θ, φ), describing the orientation of ~B0 with respect tothe principal axis system (PAS) of the electrical field gradient, can be written as

HQ =δQ

6I(2I − 1)[3 cos2(θ)− 1− ηQ sin2(θ) cos(2φ)][3I2

z − I(I + 1)] (3.5)

where two important parameters were introduced:

asymmetry parameter ηQ :=Vxx − Vyy

Vzz

(3.6)

anisotropy parameter δQ := Vzz

=3

4

eQeq

~(3.7)

Another two terms are also widely usedii

quadrupole constant Cq :=eQeq

~=

4

3δQ (3.8)

quadrupole tensor σQ :=eQ

~V (3.9)

ifirst spatial derivative of the electric field at the site of the nucleusiithe definition of the quadrupole constant confers to the covention used in reference [53]

50

3.3 NMR spectral analysis: an ill-posed problem

For the case of symmetric interaction tensors, i.e. η = 0, the hamiltonian of eq. (3.5)simplifies to

HQ =δQ

6I(2I − 1)[3 cos2(θ)− 1][3I2

z − I(I + 1)] (3.10)

This is due to the elimination of the φ dependence. For this symmetric Hamiltonianeq. (3.10) acting within a basis of quantum-mechanical states |I,m >, the transitionfrequencies expected for a quadrupolar nucleus in a strong magnetic field can be found

ωQ(θ) = ω0 −δQ

2

(2m− 1

I(2I − 1)

)(3 cos2(θ)− 1) (3.11)

where ω0 is the transition frequency in the absence of quadrupole coupling. Hence ω(θ)

is a function of the angle θ between the static external field ~B0 and the unique axis ofthe interaction tensor only. For a nucleus with spin with I = 1, there are two transitions,where the m = 1 → m = 0 transition has a +[3 cos2(θ) − 1] dependence, while them = 0 → m = −1 transition has a −[3 cos2(θ) − 1] dependence. It is crucial to realize,that the anisotropic angular dependence of NMR frequencies does not directly reflect theorientations of nuclear spins, but rather the orientations of interaction tensors, which aregenerally fixed to the molecular segments.

For η = 0, the spectrum of non-oriented isotropic powder samples can be calculated inthe following way: transforming every θ interval into the spectrum, such that the integralintensity of corresponding intervals in θ and ω is equal yields the relation between thespectral intensity S(ω) and the angular distribution P (θ)

S(ω(θ))|dω| = P (θ)|dθ| (3.12)

according to differential calculus, eq. (3.12) can be written as

S(ω(θ)) = P (θ)/|dω

dθ| (3.13)

In a powder sample all possible orientations of the interaction tensor are present withequal probabilities, i.e. P (θ) = constant. Superposing the two transitions, m = 1 → m = 0and m = 0 → m = −1 given by eq. (3.11), and plugging them into eq. (3.12) gives riseto a mirror-symmetric powder pattern shown in fig. 3.1, first calculated by Pake [58].The characteristic maxima of the so-called Pake spectrum result from (integrable) square-root singularities of each transition at ω = −1

2δ combined with some homogeneous line-

broadening [53].


Experimental NMR spectra map a certain set of parameters, describing static and dynamicphysical properties of a specimen, into electromagnetic signals, which are detected as afunction of their energy. Reversing the mapping, i.e. extraction of the parameters of

51


d Q

2 d Qww ( 0 o ) w ( 3 5 . 3 o ) w ( 5 4 . 7 o ) w ( 9 0 o )

S ( w )

Figure 3.1: Quadrupolar powder lineshape for a spin-1 nucleus with η = 0, so-called Pakespectrum [58]. The thin lines show the spectra for each single transition in a I = 1 system andthe thick line shows their superposition, corresponding to the experimentally accessible powderspectrum

interest from the spectrum in not straightforward and involves the solution of so-calledill-posed inverse problems. A necessary but not sufficient condition for the solution ofan inverse problem is the complete knowledge of the solution of the corresponding directproblem, i.e. the mapping of a certain set of parameters into the NMR spectrum has to bewell known. Excellent detailed overviews about dealing with inverse problems, especiallyin NMR spectral analysis, can be found in literature [59, 25, 60]. Hence, only a briefdescription of solving inverse problems, capturing the main features of the reconstructionof orientational distributions of molecular segments from NMR spectra, will be given inthis section.

Formally an NMR experiment can be described by an operator T (ω, θ), that maps a dis-tribution density of a certain material-specific parameter θ onto an experimental spectrums(ω) leading to the following linear equation

s(ω) = T (ω, θ)g(θ) (3.14)

where g(θ) is the distribution density of the parameter θ. T (ω, θ), which contains themathematical description of the experiment, usually has smoothing properties such thateven if g(θ) is dominated by high-frequency oscillations, s(ω) may still be smooth. ForNMR spectroscopy eq. (3.14) very often states as a so-called Fredholm integral equationof the first kind

s(ω) =

∫θ

g(θ)K(ω, θ)dθ (3.15)

52


where ω and θ may assign one- or multidimensional spectral and parameter domains,i.e. θ = θ1, θ2, .., θx and ω = ω1, ω2, .., ωy depending on the nature of the experiment.Hence eq. (3.15) covers both the one- and the multidimensional cases in both domains.For a multidimensional parameter domain the integration is taken over all parameters θ(e.g. orientational angles, distances between nuclei, quadrupolar parameters, etc.) andthe differential has to be read as dθ = dθ1 dθ2..dθx. The square integrable integral KernelK(ω, θ) is a continuous real or complex function and coincides with the spectrum s(ω)resulting from a sharply defined distribution g(θ) = δ(θ − θ0)

iii

The question is now, how can the distribution g(θ) be extracted from the experimentalspectrum s(ω) with the help of complete knowledge of the direct problem described in eq.(3.14), i.e. the complete knowledge of the kernel K(ω, θ) ?

Naturally one would expect the solution of the problem to be given by a simple inversionof the mapping

g(θ) = T (ω, θ)−1s(ω) (3.16)

It is well known though, that Fredholm integrals of first kind are ill-posed in the senseof Hadamard [61]: existence, uniqueness and stability of solutions cannot be taken forgranted !

To put more stress on the actual application to a real NMR experiment, which is adiscrete rather than a continuous problem, lets consider a simple NMR experiment thatyields a single line of a certain shape, the position of which is defined by a single physicallyparameter θ. Coexistence of n sites with different θi generates a NMR spectrum that is asuperposition of all individual contributions, weighted by their relative abundance g(θ). Inthis way a discrete grid θi was introduced. Now we need to discretize eq. (3.15) not onlyin the parameter but also in the spectral domain, since an experimental spectrum s(ω) ismeasured at m discrete frequency points ωj. The linear operator mapping the propertydescribed by the parameter θ into the spectrum is the matrix K(ω, θ), again a function ofboth, the frequency ω and the parameter θ

s(ωj) =n∑

i=1

K(ωj, θi)g(θi)∆(θ)

or

s(ω) = K(ω, θ) g(θ) (3.17)

where s(ω) is the data vector [s(ω1), s(ω2), .., s(ωm)]T , g(θ) the vector[g(θ1), g(θ2), .., g(θn)]T of the relative abundances and K(ω, θ) is an m × n sizedmatrix, which columns correspond to the so-called basis spectra [s0

1, s02, .., s

0n]. A basis

spectrum s0i reflects the spectrum of a parameter vector [δi0, δi1, .., δin]iv, such that the

experimental spectrum is a superposition of all basis spectra s0i , again weighted by the

relative abundances g(θ).

iiihere δ(θ, θ0) is Dirac’s delta function with∫ x2

x1f(x)δ(x− a)dx = f(a) for x1 < a < x2 and zero else.

ivwhere δmn = 1 for m = n and δmn = 0 else (Kronecker delta)

53


Now we come back to the solution presented by eq. (3.16). Obviously such a solutiondoes not exist if K(ω, θ) is singular, i.e. m > n. Generally speaking, the existencecriterion is always violated in experimental data analysis, since experimental noise is notimplemented in K(ω, θ). Hence an obvious choice for the inverse solution of eq. (3.17)seems to be a linear least-squares (LS) fit that minimizes the discrepancy between themeasured and a simulated spectrum according to

min‖Kg − s‖ = ‖Kg − s‖ (3.18)

where ˜g(θ) is a reasonable approximation to the true solution g capturing experimental

noise. Uniqueness is violated by the coexistence of several solutions ˜g(θ)i ( ˜g(θ)i 6= ˜g(θ)j

if i 6= j), with equivalent misfits for a given noisy spectrum s(ω). One way to restoreuniqueness is to select that inverse solution which has the minimal norm out of a setof possible ones. Although this “model free” determination of g(θ) needs no a-prioriassumptions about the shape of g(θ), there are still some severe limitations of the thirdmost problematic criterion of stability.

Violation of the stability is reflected by the fact that close proximity in the spectral dataspace of two spectra s1(ω) and s2(ω) does not result in close proximity of the correspondinggenerating functions g1(θ) and g2(θ). This is a consequence of the smoothing properties ofthe linear mapping operator T (ω, θ) respectively of K(ω, θ) in the discrete case, such thateven a highly oscillatory solution g(θ) can satisfy eq. (3.18) with the minimal norm.

To see this, assume a function g(θ) = g0(θ) + N sin(νθ), which is a solution of eq. (3.15)with right hand member

s(ω) = s0(ω) + N

∫θ

K(ω, θ) sin(νθ)dθ (3.19)

then the norm of difference between the spectra corresponding to g(θ) and g0 is

ρ(s, s0) = ‖s(ω)− s0(ω)‖ = ‖N∫

θ

K(ω, θ) sin(νθ)dθ‖ (3.20)

and the maximum norm of the difference in the solution becomes

ρ(g, g0) = max‖g(θ)− g0(θ)‖ = max‖N sin(νθ)‖ = ‖N‖ (3.21)

Applying now the Riemann-Lesbesgue lemma to eq. (3.20), allows to see, that the num-bers ν and N can always be chosen in such a way that, for arbitrarily small discrepanciesbetween the two spectra of ρ(s, s0), the discrepancy between the corresponding solutionsρ(g, g0) can be arbitrary. Thus revealing the smoothing properties of the kernel.

To overcome this problem first an efficient analysis of the actual inverse problem isnecessary. If the degree of ill-posedness is known, several regularization methods can beapplied by imposing additional restrictions on g(θ), such that one can find two numbers ηand ε(η) so that from

‖s1(ω)− s2(ω)‖ ≤ η (3.22)

54


follows

‖g1(θ)− g2(θ)‖ ≤ ε(η) (3.23)

independent of the choice of g1(θ) and g2(θ), i.e. restoring stability of the inverse problemunder the constraint of additional applied restrictions. How this is practically done will bediscussed in section 3.3.2.

3.3.1 Analysis of the ill-posedness

Singular value expansion (SVE) of the kernel [62] provides a useful tool in the numericalinvestigation of ill-posed problems. Let’s go back shortly to the continuous description ofeq. (3.15). This infinite series expansion into orthogonal functions u(ω) and v(θ) is definedas

K(ω, θ) =∞∑l=1

ηlul(ω)vl(θ) (3.24)

The singular values ηl and the singular functions ul(ω) and vl(θ) are related via∫θ

K(ω, θ)vl(θ)dθ = ηlul(ω) (3.25)

The singular values are positive real numbers and are a generalization of the eigenvaluesof a square matrix. They are unique for a specific integral operator, i.e. there are no twodifferent series that satisfy eq. (3.24), usually ordered such that η1 ≥ η2 ≥ η3 ≥ .. ≥ 0,and related to the norm of the kernel by

‖K(ω, θ)‖2 =∞∑l=1

η2l (3.26)

By projecting the spectrum into singular functions the solution to the linear least-squarefit problem of eq. (3.18) can be written as

g(θ) =n∑

l=1

1

ηl

(ul(ω), s(ω))vl(θ) (3.27)

In a real NMR experiment we deal with a discrete problem described by eq. (3.17). Inthis case the singular value expansion of the kernel is approximated by the singular valuedecomposition (SVD) of the m× n sized matrix K

K = UWV T (3.28)

55


Here U is an m× n sized column-orthogonal matrix, W an n× n sized diagonal matrixwith the singular values wl as its elements and V an n×n sized orthonormal matrixv. Thesolution g(θ) which minimizes eq. (3.18) is then given by

g(θ) =n∑

l=1

1

wl

vl(uTl · s) (3.29)

with n the number of basis spectra in the kernel K and ul, vl columns of U and V .Assuming g1(θ) and g2(θ) to be the distributions producing the spectra s1(ω) and s2(ω),the LS discrepancy in the parameter domain is

‖g1(θ)− g2(θ)‖2 =n∑

l=1

1

w2l

‖vl(uTl · (s1(ω)− s2(ω)))‖2 (3.30)

Small singular values wl thus lead to a dramatic error amplification by even the smallestdifferences in the spectra. The decay of the singular values thus provide a measure for thedegree of ill-posedness. Generally a rough classification is made in the following way: ifwl behaves like l−χ (with χ ≥ 1

2) the problem is modestly ill-posed, if an e−l behavior is

observed the problem is severely ill-posed. The condition number is another indicator ofthe severity of ill-posedness [63]. It is defined as the ratio between the highest and lowestsingular value of a discrete kernel.

Correlations between two basis spectra K(ω, θ1) and K(ω, θ2), i.e. between two columnsof the kernel K, determine the decay of the singular values. If all basis spectra would beorthogonal then all singular values would be equal and the integral kernel K not ill-posed.The correlation between two basis spectra is given by the cosine of the angle between therespective spectravi

Cpq =(Kp, Kq)

(Kp, Kp)(Kq, Kq)= cos(anglepq) (3.31)

In case of fully orthogonal basis spectra the correlation factor is zero. For fully correlatedspectra the correlation factor is one.

3.3.2 Restoring stability by regularization

As mentioned above one needs to introduce additional information and thus additionalrestrictions on g(θ), such that eq. (3.22) and eq. (3.23) are satisfied. In general suchadditional restrictions can be classified into those which apply directly on mathematicallyproperties of the solution g(θ), and restrictions on the setup of the problem like discretiza-tion of data points or application of a truncated SVD:

vnote, matrices U , V and singular values wl are not approximations of the singular functions ul(ω) andvl(θ) and the singular values ηl !

vithe inner product (·, ·) is defined as (φ, θ) =∫ 1

0φ(t)θ(t)dt for functions and (~x, ~y) = (

∑i(xiyi)2)

12 for

vectors

56


Discretization is the most evident way to treat the problem of eq. (3.30) and was alreadydescribed above. Since an experimental NMR experiment is naturally discrete in thespectral domain, discretization (gridding) of the parameter domain leads to the alreadymentioned approximation of the continuous functional of eq. (3.15) described in eq. (3.17).This leads to a solution (eq. (3.29)) of the minimization problem (eq. (3.18)) which canbe obtained by SVD (eq. (3.28)) of the discrete Kernel matrix K. Decreasing the griddensity, highly oscillatory solution g(θ) can be blocked. On the other hand, decreasing thegrid density leads to a faster decay of the singular values wl and therefore to an increasedill-posedness of the problem. Hence, alternatively

Truncation of the series of eq. (3.29) at a certain threshold can gain a better solution. Adenser grid leads to a slower decay of singular values but increases the condition number,i.e. increasing the error amplification caused by terms of higher order of the series. Theseterms are simply neglected by truncation. Ideally one likes to keep the grid as dense aspossible, the balance between grid density and truncation has to be evaluated very carefully.

Non-negativity can restore stability to some extend as well. Since negative values in theparameter domain are often physically not reasonable, e.g. in the case of orientationaldistributions, enforcing a positive result for g(θ) is mostly a good choice.

In general, restrictions which can be mapped directly on mathematically properties ofthe solution g(θ) can be introduced by penalizing the minimization problem of eq. (3.18)with a regularization term R[g(θ)], which maps the violation of the applied additionalrestriction as a function of g(θ) on a scalar number. Thus leading to a constraint versionof the minimization problem, which can be generally written as

‖K(ω, θ)g(θ)− s(ω)‖2 + λ R[g(θ)] → min! (3.32)

Where λ is the so-called regularization parameter, which balances between the qualityof the fit and the impact of the additional restriction. A proper choice of λ is of greatimportance, as can be seen for an additional restrictions, which can be applied to mostphysically problems: smoothness of the solution. If λ is too small there will be no significantinfluence of the additional smoothness constraint. If λ is chosen too large, the solution willbe over-smoothed on the cost of the compatibility of fit and data. A reliable determinationof the degree of regularization, i.e. a proper choice of λ will be discussed at the end of thissection (cf. fig. 3.2).

Since the smoothness constraint is the additional restriction, which regularizes the min-imization problem most effectively for many problems, its application will be discussed inmore detail:

Smoothness of the solution g(θ) is a restriction which can be applied to many problemswithout loosing any physically relevant information. Smoothness assumes that a physical

57


property, presented by the parameter θ, in a neighborhood of space or in an interval oftime presents some coherence and generally does not change abruptly.

If the solution is a function of one variable θ only, i.e. g(θ), Tikhonov regularization[64, 63, 65] is one of the most used approaches to implement smoothness by asking for asmall quadratic norm of (g(θ)− g0(θ)) leading to a regularization term

R[g(θ)] = ‖g(θ)− g0(θ)‖2 (3.33)

where g0(θ) is a rough estimate of the expected solution, such that the regularizationis biased in direction to a solution which is closest to the g0(θ). For modestly ill-posedproblems which are only slightly dependent on the first guess, g0(θ) can be chosen to bezero, such that the minimization is always biased towards the smallest solution. In thiscase, i.e. g0(θ) = 0, the solution of eq. (3.29) is

g(θ) =n∑

l=1

wl

w2l + λ

vl(uTl · s) (3.34)

i.e. the 1wl

-dependence has become wl

w2l +λ

which means a reduction of the effects of the

high-order singular values.However, some more severe ill-posed problems are highly dependent on the choice of the

first guess and need to be regularized by a different smoothness constraint, than biasingthe solution close to an rough estimate. A more intuitive way to imply smoothness is to askfor a small quadratic norm of the second derivative of the solution, leading to the so-calledTikhonov-Philips [66] regularization term

‖K(ω, θ)g(θ)− s(ω)‖2 + λ‖g′′‖2 → min! (3.35)

where g′′ = ∂2g(θ)/∂θ2. This regularization method is not suitable for solutions g(θ) ex-hibiting sharp edges, but is a much better choice if the solution is rather flat and stronglydependent on the rough estimate g0(θ). For distributions exhibiting sharp edges in com-bination with flat regions, more advanced edge preserving smoothness constraints have tobe applied [67].

So far these methods were discussed for the one dimensional case, i.e. the solutionis a function of one variable only. In this study we want to reconstruct orientationaldistribution functions from solid-state NMR spectra as a function of two euler angles(θ, φ). Thus the solution of the ill-posed problem is not a curve, but a 3-dimensionalsurface S = (θ, φ, g(θ, φ)), with g(θ, φ)dθdφ the probability of finding a orientation in theinterval [φ±dφ, θ±dθ]. A measure for the smoothness of a curve is its second derivative, butnow it is necessary to find a measure for the smoothness of a surface. This problem is alsowidely known in a field called computational vision, which is concerned with computationalimage reconstruction by interpolation between noisy, discrete input data. Grimson etal. [68] found a regularization term, which is preferred for the reconstruction of smoothsurfaces [67] and can be derived from invariants of the so-called curvature-tensor (cf. [69],chapter 1 therein), called the quadratic or second order variation of g(θ, φ):

58


R[g(θ, φ)] = ‖g2θθ + 2 g2

θφ + g2φφ‖2 (3.36)

hence putting down the regularization problem to a minimization of the functional

‖K(ω, [θ, φ])g(θ, φ)− s(ω)‖2 + λ ‖g2θθ + 2 g2

θφ + g2φφ‖2 → min! (3.37)

where gxy = ∂2g/(∂x∂y). In section 3.5.2 different orientational distribution functionsand their reconstruction from simulated noisy NMR spectra were tested, to evaluate theaverage error and the power of the quadratic variation regularization method. Indeed, otherregularization terms, like the squared Laplacian R[g(θ, φ)] = ‖g2

θθ + g2φφ‖2 or Tikhonov-

regularization, showed worse results depending strongly on the first guess in the lattercase.

Note, that an analytical solution of the inverse problem by singular value decompositionof the kernel and a linear least square minimization (cf. eq. (3.29) and eq. (3.34)), as inthe case of Tikhonov-regularization, cannot be obtained for non-linear functionals like ineq. (3.37). Here different non-linear minimization algorithms have to be applied.

A reliable determination of the degree of regularization is crucial, since the optimalbalance between the severity of the constraint and the discrepancy between the fit and theexperimental spectrum has to be found by a proper choice of the regularization parameterλ. A good way to visualize the influence of the constraint is to plot the discrepancy‖K(ω, θ)g(θ)−s(ω)‖2 against the regularization term R[g(θ, φ)]. Plotting on a logarithmicscale, such a plot almost always exhibits the shape of the letter “L”, hence called L-Plot.The L-plot is perhaps the most convenient graphical tool to determine the optimal valuefor the regularization parameter λ [70].

Figure 3.2 shows a typical L-curve together with a plot of the regularization parameter λ.Each point of the L-curve presents a solution of the minimization problem for a fixed valueof λ corresponding to the same value for ‖K(ω, θ)g(θ)− s(ω)‖2 as the regularization term.An optimal balance between strong and weak regularization, i.e. between the violation ofthe additional constraint presented by the regularization term and the discrepancy betweenfit and data, is found at the edge of the L-curve. Hence, the optimal value for λ can bedetermined graphically by the intersection between the regularization curve and a verticalline through the edge of the L-curve.

Another well-studied method is the so-called discrepancy method [71], where the mostregularized solution, usually the simplest one, which still provides sufficiently low discrep-ancy between fit and measured data is selected. In practice the optimal value for λ is foundaround the edge for both the L-curve and the discrepancy method, although the precisedefinition of the edge is still in discussion.

Alternatively to using a discrepancy in the spectral domain, it is possible to define adiscrepancy in the parameter domain. This method is known as the Self-Consistent (SC)method [72] and was proven to give reliable values for linear regularization functionals.However, finding the optimal regularization parameter for non-linear regularization func-

59


Figure 3.2: Solid line: generic form of a typical L-plot of the discrepancy ‖K(ω, θ)g(θ)− s(ω)‖2

against the regularization term R[g(θ)]. In principle each single point of the L-curve presentsa solution of the minimization problem with a fixed regularization parameter λ. Dashed line:regularization parameter λ as a function of the discrepancy ‖K(ω, θ)g(θ)− s(ω)‖2. The optimalbalance between discrepancy between fit and data and impact of regularization is found at theedge of the L-curve. Thus the optimal value for λ can be found by a vertical projection of theL-curve-edge onto the red dashed line.

60

3.4 The solid-state quadrupolar NMR experiment

tionals of the kind used in this study (cf. eq. (3.37)) is more stable performed via graphicalanalysis of the L-curve.


The aim of the NMR study of polymeric glasses in this thesis is to get information aboutaverage orientational distributions of molecular segments as a function of deformation. Thisimplies, that the tested specimen in this study yield orientation increasing gradually fromno-orientation, as in the case of an non-deformed powder sample, to a certain maximumdegree of order at the maximum compression ratio. The mode of deformation chosenin this thesis for specimen investigated by NMR, is plane strain compression with smallstrains not exceeding ε ≤ −0.55. This mode of deformation can be classified as “neutralaligning deformation” with respect to the “alignement strength” [73], thus leading to amuch weaker alignment as predicted for uniaxial tension. This indicates, that we aredealing with specimen exhibiting only low degrees of order. Thus a NMR experiment anda proper analysis has to be found, to allow to detect low degrees of order and reconstructthe corresponding orientational distributions of molecular segments, which are most likelyclose to an isotropic distribution.

One- and multi-dimensional solid-state NMR techniques [74, 21, 22, 23, 24, 53, 75] havebeen applied in literature to yield information about oriented polymers. The choice, how-ever, of a proper design of the NMR experiment has to be evaluated with great care. Onedimensional 1D solid state NMR spectroscopy has in principle two major problems com-pared with multi-dimensional NMR methods: overlapping of spectral patterns belonging tothe two transitions m = 0 → ±1 (cf. section 3.2.3) and extremely poor angular resolutionat the characteristic maxima of the spectrum, since at these frequencies it is |dω/dθ| = 0.This results in certain instability of best-fit reconstruction routines of molecular orienta-tions in those critical regions. But on the other hand the method is experimentally muchmore robust over longer time periods, since the experimental setup is simply static, suchthat no instabilities are introduced by flipping the sample during the experiment. Thisallows to gain a better signal-to-noise ratio by longer measuring periods. Further VanBeek et al. [25] showed recently, that a series of at least seven 1D spectra has a signal-to-noise ratio considerably higher than a 2D DECODER experiment, assuming that the totalacquisition time is equal in both experiments.

Our approach is to benefit from the advantages of 1D spectroscopy with the goal toovercome the disadvantages mentioned above as far as possible. Thus our approach hastwo main features:

Each orientational distribution of molecular segments, belonging to one sample, willbe mapped experimentally on a series of 13 different but dependent 1D spectra with ahigh signal-to-noise ratio in the range of S/N ≈ 300− 500. Each single 1D spectrumpresents a certain orientation of the sample with respect to the external magneticfield, equally spaced in a plane perpendicular to the axis of the rotor (cf. fig. 3.5).

61


Table 3.1: properties of some NMR active nuclei important in NMR on polymers [76]

Isotope natural spin I µN Q resonance

abundance frequency

(%) (barn) (MHz/Tesla)1H 99.985 1/2 +2.793 0 42.5762H 0.0148 1 +0.857 +0.00288 6.53213C 1.11 1/2 +0.702 0 10.701

Since these 1D spectra are not independent, practically we are doing a multidimen-sional experiment where each single 1D spectrum represents one dimension.

Reconstruction of the orientational distributions of molecular segments from the ex-perimental spectra will be performed using numerical least square minimization ofthe error between the experimental and the simulated spectra under different com-plementary constraints applied to the distribution function. These constraints, likesmoothness or non-negativity, are necessary to stabilize the numerical fit. In thissense, reconstruction of the orientational distribution of molecular segments with thehelp of several complementary fitting routines, lead to a physically reliable result. Adetailed discussion of the numerical analysis of experimental spectra will follow insection 3.3.

Sample preparation for NMR experiments is best done by labelling the nuclei of interestwith atoms exhibiting low natural abundance. Hence the NMR spectrum from interactionsof the labelled nuclei can be identified easily. Additionally a great difference between theresonance frequencies of the labelled and non-labelled sites compared to the resolutionof the NMR spectrometer has the advantage, that overlapping of spectra resulting fromlabelled and non-labelled sites can be neglected. Properties for some NMR active nuclei,i.e. nuclei with spin I ≥ 1

2, important in NMR of polymers are summarized in table 3.1.

3.4.1 Method

Since the NMR spectrum in solid-state NMR is very broad, i.e. the transverse magnetiza-tion decays quickly, the proportion of the full FID lost in the dead time after a pulse is quitebig resulting in severely distorted spectra. To overcome this problem all 1D NMR spectraare obtained by quadrupolar echo NMR spectroscopy [77]. The applied pulse sequence isshown in fig. 3.3: 90o

X − τ − 90oY − τ − acquire. The solid echo serves to refocus the time

evolution of spins, such that at a time 2 τ after the first 90oX pulse the true FID can be

completely recorded from the echo maximum.To map the orientational distribution of molecular segments on different dependent spec-

tra, 13 1D spectra were obtained for each sample, such that each spectrum corresponds

62


t

9 0 ° X

9 0 ° Y

ttFigure 3.3: Solid-quadrupole-echo pulse sequence. Under the condition of maximum echo,spurious signal distributions resulting from second pulse flip angles, which deviate from 90o areremoved and the nuclear magnetization is refocussed completely.

to a different static orientation of the rotor and sample, which is fixed to the rotor, withrespect to the external magnetic field

−→B 0. The angle between the direction of elongation

of the sample, i.e. yRF respectively ySF , and its projection onto the zLF − xLF plane ofthe LF, is increased stepwise from 0o up to 180o with increments ∆α of 15o.

Figure 3.4 shows stack plots of two typical experimental series of 13 combined one-dimensional (1D) spectra obtained from two samples with different degree of orientationalorder deformed to strains of ε = −0.86 respectively ε = −0.64. Each single spectrum wasobtained at a fixed orientation of the rotor, equally spaced in the plane perpendicular tothe rotor axis in 13 steps of 15o. A completely isotropic powder sample would lead to astack plot of 13 identical 1D spectra, hence the deviation from an isotropic distributioncan be clarified by the envelope curves of a full stack of 13 1D spectra as indicated infig. 3.4 by the two solid lines connecting the maxima of each single spectrum. Obviouslythe suggested NMR experiment is sensitive to the degrees of order observed in this thesis.So far this is an essential condition for the analysis of the spectra, but, however, does notimply that the spectra can be analyzed at all. To clarify the power of the suggested methodfor spectral analysis of section 3.3, various simulated spectra will be tested in section 3.5.

The information about orientational order contained in the NMR line shape is providedby the anisotropic nature of the coupling of the nuclei to their surroundings described bythe interaction tensor. Hence analysis of the NMR spectra primarily yields the orientationaldistribution of the principle axes of the interaction tensor relative to the external field ~B0.Nevertheless we are interested in the orientational distribution of molecular segments ofthe macromolecule relative to unique directions of the ordered specimen. Therefore fivedifferent coordinate systems are relevant for the analysis as described in the next section.

63


Figure 3.4: (Top) Stack plots of two series of a full cycle of 13 1D quadrupolar solid state NMRspectra of deuterated PMMA deformed (far) below the glass transition temperature with differentstrains. Each single spectrum corresponds to a fixed rotor angle, increasing stepwise from 0o upto 180o with increments ∆α of 15o, between the direction of elongation of the sample and itsprojection onto the zLF − xLF plane of the LF (cf. fig. 3.5). The values beneath each single 1Dspectrum denote the actual rotor position. Black: PMMA compressed at ε = −0.86 showing loworientational order, red: PMMA compressed at ε = −0.64 showing medium orientational order.For clarity the envelope of each series is shown. (Bottom) Magnification of the indicated range.Indeed these spectra show a clear deviation from a powder sample which would exhibit a stackplot of 13 identical spectra. Thus indicating that our NMR experiment is sensitive to low degreesof order induced by medium strain plane strain compression in glassy PMMA.

64

3.5 Results and discussion

Reference frames

We introduce five “right handed” orthogonal reference frames to describe the spatial ori-entation of the quadrupole interaction tensor with respect to the external magnetic field−→B 0 as shown in fig. 3.5:

Principal axes system (PAS): the principle axis of the average CD3 quadrupole tensorare fixed to the molecule as follows (see fig. 4.3): zPAS along the C5−C1 bond andxPAS coplanar with the C1− C2− C1 bonds.

Molecular frame (MF): this system is defined in order to describe the orientation of aparticular molecule with respect to the sample system. zMOL along a C1−C1 bondand xMOL along the C5− C1 bond parallel to zPAS.

Sample frame (SF): this system is defined by the direction of plane strain deformation.xSF is the dimension which is kept constant during deformation, ySF is the dimensionof elongation, and zSF is the dimension of compression.

Rotor frame (RF): this system describes the orientation of the sample inside the rotorduring the NMR experiment. xRF is anti-parallel to zSF , and yRF is parallel to ySF .

Laboratory frame (LF): the zLF describes the direction of the external magnetic field−→B 0,

and the xRF is chosen to be coplanar with the plane defined by zLF and zRF .

Euler R(α, β, γ) rotations between the frames are also shown in fig. 3.5. For details onthe notation of Euler rotations see appendix A. Hence the complete rotation of the PASinto the LF, as used in the analysis, is

RPAS→LF = R(π − α, θm, π) R(0,π

2, 0) R([0..2π], θ, φ) R(0,−π, 0) (3.38)

where α is the angle which will be increased stepwise within each series of 1D experimentsbelonging to one sample, and angles (θ, φ) give the information about the orientation ofmolecular segments relative to the SF. Note that the orientational distribution of rotationsfrom the MF into the SF, R([0..2π], θ, φ), has to be integrated over the first euler angle,since the molecular segments show rotational symmetry around the C1− C1 bond.


So far we suggested an NMR experiment which is indeed sensitive to detect order oflow degree and an analysis to reconstruct orientational distribution functions from theNMR spectra of our experiment. Before any analysis method and the corresponding NMRexperiment can be applied to real specimen, the experiment has to be discussed withrespect to its ill-posedness via an analysis of the kernel K(ω, θ), and the suggested methodfor spectral analysis has to be tested with simulated data.

65


Figure 3.5: Right handed orthogonal frames used to describe the orientation of the PAS relativeto the LF. Orientation of molecular segments will be described by two angles (θ, φ), which rotatethe molecular fixed frame (MF) into the sample fixed frame (SF). These angles are identical withthe polar angles describing the orientation of a C1− C1 bond in the sample frame. α describesthe angle between the direction of elongation of the sample, i.e. yRF respectively ySF , and itsprojection onto the zLF − xLF plane of the LF. Here the case α = 0 is shown, where ySF iscoplanar with the zLF − xLF plane.

66


3.5.1 Kernel analysis

In order to analyze the ill-posedness of the experiment a singular value decomposition ofthe kernel K(ω, θ) might be of some use as described in section 3.3.1. Three differentdiscretization in the parameter space were tested, including 3 resolutions, namely 10o, 15o,and 20o, applied to the two euler angles (θ, φ) of parameter space, which describe theorientation of a molecular segment.

Figure 3.6(a) shows plots of the singular values of all three kernel matrices. Followingthe classification of ill-posedness suggested in section 3.3.1, the experiment is modestlyill-posed, except for the high-order singular values, which show strong ill-posedness. Thecondition numbers, shown in the inset of fig. 3.6(a), increase by two orders of magnitudebetween the lowest and highest resolution, thus increasing the ill-posedness significantlywith increasing resolution. Figure 3.6(b) shows an angle plot of the discretization schemewith the highest resolution, exhibiting a pathologic behaviour with many minima and max-ima, indicating a high degree of correlation between the basis spectra. A direct comparisonwith angle plots of 2D NMR experiments, as described recently by van Beek et al. [60],indicates that a series of 13 combined 1D spectra seem to be much more difficult to analyzedue to highly correlated basis spectra.

From these plots, however, a final classification of the actual power of the regularizedminimization algorithm to reconstruct two-dimensional orientational distribution functionsfrom noisy NMR spectra (cf. eq. (3.32)) cannot be established. Even when experimentsare pretended to be severely ill-posed, their analysis can still give good results, since thesignal-to-noise ratio of the experimental spectra and the expected shape of the distributionfunctions, which are not addressed by the above suggested kernel analysis, play an impor-tant role with respect to the success of the analysis. Nevertheless, for the optimization ofa specific experiment a kernel analysis can be helpful indeed [60].

A reliable classification in terms of relative errors between exact and reconstructed orien-tational distribution functions can be obtained in the following way: generation of differentrandom input distributions, simulation of noisy NMR spectra corrsponding to the appliedexperiment and final reconstruction of the output distribution as described in the nextsection.

3.5.2 Monte Carlo tests

Before proceeding to the results, some details should be mentioned about the practicallyapplication of the suggested analysis of the NMR spectra. Minimization of the regularizedfunctional of eq. (3.37) was performed by a non-linear minimization algorithm with non-negativity constraints provided by Matlab (version 6.5 R12). For further details on thealgorithm refer to the Matlab manual and reference [78]. A proper choice of the optimalregularization parameter was obtained by the so-called L-curve criterion, where the optimalbalance between fit to the data and impact of additional constraints can be determinedgraphically. Figure 3.7(b) shows the L-curve for a typically input distribution displayed in

67


(a)

10º

20º

30º

40º

50º

φ ( o )

φ ( o )

0 30 60 90 120150180

0

30

60

90

120

150

180

(b)

Figure 3.6: (a) Singular value plots of kernels with three different grid resolutions in parameterspace. The inset shows additionally the corresponding condition number. Magenta lines: threelimiting cases of weak (∼ l−1/2), modest (∼ l−2), and strong (∼ e−l) ill-posedness, with l thesingular value index. (b) Angle plot for 10o resolution in the parameter space. Each large blockcorresponds to a value for φ and is subdivided into 19 values for θ.

68


0

0.5

1

φ

θ

−0.9 −0.8 −0.7 −0.6 −0.5−4

−2

0

2 (c)(d)(e)

(f)(g)

(h)

log10

||K*g − s||2

log 10

R[g

]

0

0.5

1

φ

θ

0

0.5

1

φθ

0

0.5

1

φ

θ

0

0.5

1

φ

θ

0

0.5

1

φ

θ

0

0.5

1

φ

θ

(a)

(c)

(e)

(g)

(b)

(d)

(f)

(h)

Figure 3.7: Demonstration of a proper choice for the optimal degree of regularization for a typicalinput orientational distribution function shown in (a). (b) Logarithmic L-plot, which shows theregularization term R[g] as a function of the discrepancy between fit and data ‖K(ω, θ)g(θ) −s(ω)‖2. Each point on the curve corresponds to a different value for the regularization parameterλ, increasing from top left to bottom right. The crosses mark distinct values of λ which correspondto the reconstructed output distribution function shown in (c)-(h). Obviously the value of λ, whichis related to the edge of the L-curve gives the optimal regularization as shown in (f).

69


fig. 3.7(a).vii The crosses in fig. 3.7(b) denote different values for the regularization pa-rameter λ, corresponding to the reconstructed orientational distribution functions (ODFs)sketched in fig. 3.7(c)-(h) with increasing λ from (c) to (h). Here it can be nicely seen,that the optimal regularization can be found on the edge of the L-curve, correspondingto fig. 3.7(f), which gives the best agreement between the rconstructed soultion and theinput. Choosing the regularization parameter λ too small, as depicted in fig. 3.7(c)-(e),leads to a non-satisfying, oscillating solutions revealing the ill-posedness of the minimiza-tion problem, whereas a too large λ leads to over-smoothed flat solutions as depicted infig. 3.7(g)-(h).

Before a regularized minimization algorithm can be applied to real experimental data, ithas to be tested with respect to reliability, resolution power and relative error between realand reconstructed orientational distribution functions. Therefore several two dimensionalinput orientational distribution functions (ODF) have been generated, consisting of one ormore gaussian peaks with varying height, symmetry and width.

According to the applied experiment of 13 combined 1D spectra, corresponding NMRspectra were simulated with several different input distributions and combined with ran-dom noise at different levels to generate a realistic NMR spectrum. Note that the directproblem, i.e. the mapping of ODFs on NMR spectra, has to be completely known from thenecessary parameters for a one-dimensional quadrupolar solid-state echo experiment, i.e.quadrupole coupling, line-broadening, asymmetry and anisotropy parameter, and isotropicchemical shift. These parameters have been determined from powder samples as describedin section 4.3.2. Finally, the nonlinear minimization algorithm described in section 3.3.2,implementing non-negativity of the solution and two dimensional smoothness constraints(c.f. eq. (3.37)), was applied to reconstruct the input ODF from the noisy simulated NMRspectra. The standard deviation χg between the reconstructed output distribution andthe input distribution was used as a numerical criterion for the power of the minimizationalgorithm

χ2g =

∑i(inputi − outputi)

2∑input2i

(3.39)

A proper choice of input ODFs is a crucial point, since tests with simulated data areonly relevant if the tested input distributions can be estimated to be at least similar tothe expected experimental distribution functions. Here our choice will be one and multi-peak gaussian ODFs with varying height, position and width of the peaks. This choice issupported by independent NMR and X-ray scattering experiments [22], which found thatmolecular orientational distributions of partially ordered solid polymers are essentiallyGaussian.

From simulations we know the error χg between input and output ODFs arising duringthe reconstruction of the input ODF. Thus we might be able to classify the input ODFs

viiNote, that for better visualization all contour plots of orientational distribution functions in this chapterare scaled such that the maximum of each distribution is 1.

70


with respect to a certain criterion, such that one can assign a certain error range for adistinct class of input ODFs. Of course the classification has to describe a characteristicproperty of the input ODF, which has significant influence on χg. As we will see below,not the signal-to-noise (S/N), but rather the degree of orientational order is the relevantparameter with respect to the success of the analysis. This is due to the fact, that theS/N of the experiments is too high to affect the analysis in a significant way. Hence inthe parameter space such a criterion to classify input ODFs might be the average width ofthe gaussian peaks of the ODF, which are directly connected to the degree of orientationalorder.

But, however, experimental data is only available in the form of NMR spectra. Hence weshould find a criterion, which classifies the NMR spectra and not the ODFs with respectto the degree of orientational order they reflect. If this can be achieved, error-estimatesobtained from simulated data, can be applied to experimental data via a comparison oftheir NMR spectra.

As shown in fig. 3.4, experimental NMR spectra of deformed glassy polymers in thisthesis, are always close to an isotropic powder distribution at medium strains as applied inthis thesis. Hence we suggest as a proper numerical criterion which classifies NMR spectrawith respect to the degree of orientational order the average standard deviation χS

χ2S =

‖(S0 − S)‖2

‖S0‖2(3.40)

where S0 is the NMR spectrum of a powder, and S is the NMR spectrum of an orientedsample. To summarize, we are now able to classify an experimental NMR spectrum byχS with respect to the orientational order it reflects, and further we know the error χg

of simulated input ODFs with the same value for χS. The error χg of ODFs from NMRspectra with the same χS will be estimated to be similar.

Experimental values of χS were in the range of χS = 0.015 − 0.03, which determinesthe regime which has to be covered by the simulations. Practically, gaussian ODFs wererandomly created and the width of the peaks was varied such that the spectral deviationfrom a powder sample spans the regime χS = 0.01− 0.4.

As mentioned above, two main features may influence the reliability and resolution powerof the reconstruction of ODFs from the NMR spectrum: noise of the experimental NMRspectra described by the signal-to-noise ratio S/N , and degree of orientation described bythe deviation from a powder spectrum χS. The strategy will be to test first the stability ofour analysis against noise, and in a second step to test stability against different degreesof orientational order, but with a fixed signal-to-noise ratio.

Stability against noise can become an important factor in data analysis. Signal-to-noiseratios of experimental spectra were in the range of S/N = 300−500, which is a rather highvalue in solid-state NMR. Nevertheless Figure 3.8 shows the stability of the minimizationalgorithm against decreasing signal-to-noise ratio.

10 different input ODFs were tested, each with a varying signal-to-noise ratio of S/N =

71


0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

5 10 30 100 300 10000

0.2

0.4

S/N

χ g

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

ω

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

ω

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

ω

(a)

(c)

(d)

(e)

(b)

←

←

←

Figure 3.8: Reconstruction of several input distributions as a function of the signal-to-noiseration S/N . (a) Contour plot of a typical input multi-peak distribution function. (b) χg as afunction of S/N . Each grey line represents a certain input distribution with varying noise levelof the corresponding NMR spectrum. Error bars indicate the average error range as a function ofS/N (blue). (c)-(e) Left: output distributions reconstructed from the input distribution shownin (a), right: corresponding NMR spectra with three different noise levels (S/N = 10, 50, and500, marked by arrows in (b) ).

72


5 − 700. Figure 3.8(a) shows a representative input ODF and fig. 3.8(c)-(e) show thecorresponding output ODFs together with the simulated NMR spectra from which theyhave been reconstructed. Signal-to-noise ratios of the three representative points are: (c)S/N = 10, (d) S/N = 50, (e) S/N = 500. Figure 3.8(b) shows the error χg obtainedfor each of the tested input distributions as a function of the S/N of the simulated NMRspectrum, . Obviously no dramatic effect on the reliability of the reconstruction wasobserved above a signal-to-noise of S/N ≥ 50, thus justifying the restriction of furthertests to a value of S/N = 200.

Stability against degree of orientational order and error-estimation is probably a moreimportant factor influencing the stability of the analysis of our experimental NMR spectrathan their rather high signal-to-noise ration. Further, a quantitative error-estimation ofthe reconstructed experimental ODFs will be possible with the help of the simulated data.

Generally, the results obtained by regularized methods is affected by two errors. Oneis a bias caused by the regularization and the other is a statistical error caused by thedata noise. There are indeed approaches of Tikhonov regularization algorithms whichallow a rough error-estimate of the solution of the inverse problem [79, 72], but in generalthe reliability of such an estimation is still doubtful since the bias cannot be estimatedand moreover, these algorithms do not suit for more-dimensional solutions so far. Hencethe most reliable way of an error estimation can be achieved by numerical simulation aspresented in this section.

Reconstruction of 144 different input distributions with fixed signal-to-noise ratio ofS/N = 200, but different noise representations, peak positions and peak widths has beentested using 10o grid resolution in the parameter space. Tests of the kernels with a lowergrid resolution (15o and 20o, not shown) revealed no significant advantage with respect tothe power of the fitting algorithm and are not discussed further.

Figure 3.9(b) shows the discrepancy between input and output distributions χg as afunction of the deviation from an isotropic powder distribution χS. It can be seen fromthese error-curves, that an increasing orientational order, i.e. increasing value of χS, hasthe tendency to increase the error χg, hence to decrease the power of the regularizedminimization algorithm. This is due to the fact, that a sharp distribution with strongorientational order cannot be reconstructed optimal by the application of a two dimensionalsmoothness constraint as suggested in this chapter. Here more advanced edge-preservingsmoothness constraints [67] have to be applied. The error-curve with the strongest errorwas obtained from an non-symmetric two dimensional gaussian peak with a ratio betweenfull-width-half-maximum (FWHM) in θ and φ direction of FWHMθ / FWHMφ = 0.2, thusleading to a distribution exhibiting a rather sharp edge, which was not observed in theexperiments of this study. Experimental values of χS were in the range of χS = 0.015−0.03,where the error χg is indeed at an satisfying low level.

Moreover, in this study we are not only interested in orientational distribution functions,but rather in their expansion in terms of their moments, i.e. Wigner matrix elements.This is due to the fact, that in the case of low orientational order such an expansion

73


0pi/2

pi0 pi/2 pi

0

0.5

1

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

χS

χ g0 pi/2 pi

0

pi/2

pi

φ

θ

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

0 pi/2 pi0

pi/2

pi

φ

θ

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

0 pi/2 pi0

pi/2

pi

φ

θ

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

(a)

(c)

(d)

(e)

(b)

→

→

→

Figure 3.9: Demonstration of the reconstruction of typical input distribution functions withvarying peak width and peak position and signal-to-noise ratio of 200. (a) Projected three di-mensional plot of a typical input distribution. (b) χg as a function of the average deviation χS

from an NMR spectrum of a non-oriented sample. Every grey line presents a different inputdistribution with fixed peak position but varying peak width. The average error range is indi-cated by error bars (blue). (c)-(e) Contour plots of a representative orientational distributionfunctions with decreasing peak width, i.e. increasing orientational order from top to bottom.The corresponding values of χS are indicated by the arrows in (b), left: input distributions, right:reconstructed output distributions.

74


0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

χS

χ c (<D

2 00>)

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

χS

χ c (<D

2 20>)

0 0.1 0.2 0.3 0.40

0.5

1

1.5

2

χS

χ c (<D

4 00>)

0 0.1 0.2 0.3 0.40

1

2

3

4

5

χS

χ c (<D

6 00>)

(a)

(c)

(b)

(d)

Figure 3.10: Relative error of order parameters χc of the same distribution functions shown infig. 3.9 as a function of the average difference χS between an NMR spectrum from an orientatedand a non-oriented powder sample. Error bars indicate the average error range (blue).

allows to characterize a two dimensional orientational distribution function by only a smallnumber of non vanishing expansion coefficients, thus obtaining a numerical criterion fordirect comparison of theory and experiment (cf. section 4.2.1 and appendix B). However,relative errors of the Wigner expansion coefficients cl

mn, can be obtained in a similar wayby expanding the input and the reconstructed output distributions and calculating thestandard deviation χc

χ2c =

‖(clmn(input)− cl

mn(output))‖2

‖clmn(input)‖2

(3.41)

By expansion of input and output ODFs in terms of Wigner matrix elements, an errorestimation of reconstructed order parameters, which are directly connected to the expan-sion coefficients of the Wigner expansion as described by eq. (4.10), can be achieved ina similar way. Figure 3.10 shows relative errors of four different order parameters, which

75


were calculated from the same tested input ODFs shown in fig. 3.9. Here again each singleline presents the order parameter of an input ODF with fixed peak position but varyingpeak width as reflected in the value of χS. Obviously the relative error increases signif-icant at higher order terms < D6

00 >, fig. 3.10(d), but still is satisfying for lower orderparameters < D2

00 >, < D220 >, and < D4

00 >, as shown in fig. 3.10(a)-(c), especially inthe relevant range of χS = 0.015 − 0.03. The error bars determined from these plots arethe basis for the error estimation of experimentally determined order parameters shown infigures (4.6)-(4.9).

3.6 Conclusions

In general, reconstruction of distribution functions of structural parameters, like segmentalorientational order, from NMR spectra reveals a so called ill-posed problem in terms ofexistence, uniqueness and, most important, stability of the problem. Hence interpretationof experimental results is a challenging task, which should be handled with great care.Further, singular value analysis and correlations between basis spectra, which provide ameasure of the degree of ill-posedness of the problem, can be helpful for further optimizationof the experiment.

Applying a nonlinear minimization algorithm, regularized by discretization, non-negativity and two-dimensional smoothness constraints, allows to stabilize the inherentnumerical problems and to reconstruct a wide spectrum of practically relevant orienta-tional distribution functions from noisy NMR spectra on a grid of 10o in the parameterspace. A proper choice for the impact of the regularization, measured by the value of theregularization parameter λ, can be obtained in a reliable way by a graphical analysis ofthe so-called L-plot, which shows the balance between the regularization term R[g] and thediscrepancy between fit and data ‖K(ω, θ)g(θ) − s(ω)‖2. The tested input distributionscover all experimental spectra with respect to signal-to-noise and orientational order mea-sured by χS. The difference between real and reconstructed two-dimensional distributionswas always in an acceptable range.

As indicated by theoretical models, the experimental setup and the mode of deformation,orientational order of polymer glasses, deformed in compression to strains not exceedingε ≥ −0.5, are supposed to exhibit only a low degree of orientational order. This makes theapplied regularization method suitable for the experiments performed in this thesis. Onlyfor distributions with sharp edges, which have not been determined in the experiments,the applied minimization algorithm has to be modified by more advanced, edge preserving,smoothness constraints.

Nevertheless, from these simulations, an estimation of the relative error-bars of theexperimentally determined order parameters can be obtained.

76

4 Orientation-strain relationships ofrubbers and polymeric glasses

4.1 Introduction

So far segmental orientation-strain relations of polymeric solids were mostly obtained byoptical methods like birefringence or scattering methods like x-ray scattering. The disad-vantage of these methods is, that the size of the units of which the roientation is experi-mentally determined is normally unknown and, moreover, in the case of birefringence, thedetermination of orientational probability distribution functions is restricted to an analysisin terms of moments. This fact makes it difficult to relate the obtained orientation-strainrelation to a certain structural element of the polymeric solid on a molecular scale.

Solid state NMR can overcome these disadvantages, since NMR-active labels can beplaced at most desired positions along a polymer chain, and analysis of the spectra alwaysyields full probability distribution functions of the orientation of the labelled molecules. Insection 4.2 two approaches to modelling orientation-strain relations for polymeric glasseswill be described based on classical theory of rubber-elasticity and on the so-called pseudoaffine approach describing usually orientation-strain relations of single crystallites.

Section 4.4 presents results of solid state NMR experiments of fully deuterium labelledPMMA, as decribed in section 4.3, allowing to reconstruct average molecular orientationalprobability distribution functions as a function of deformation, respectively their expansionin terms of so-called Wigner matrix elements, which reduce to Legendre polynomials forsymmetric systems. Moreover, experimental orientation-strain relations were comparedto predictions of the two modelling approaches described in the theoretical part of thischapter.

4.2 Modelling orientation-strain relations

Modelling segmental orientation of polymeric solids as a function of strain will be basedon the so called affine deformation assumption, which was developed in the theory ofrubber-elasticity. Generally speaking, this assumption implies, that certain points alongthe polymer chains of the solid, which will be called affine points, move affinely withmacroscopic deformation, i.e. line elements between those affine points change in the sameratio as the corresponding dimensions of the bulk.

77

4 Orientation-strain relationships of rubbers and polymeric glasses

q

fx

y

z

q '

f '

x

y

z

a f f i n e p o i n t a f f i n e p o i n t

a f f i n e p o i n ta f f i n e p o i n t

Figure 4.1: Macroscopic affine deformation of a polymeric solid: the orientation of an arbitrarysingle polymer chain connecting two affine points is described by polar angles θ, φ and chainextension ratio λc = 1 before, and θ′, φ′ and λc = |~r|

|~r0| after affine deformation

In contrast, the orientation of those chain segments placed along the polymer chainbetween these affine points, do not depend in an affine manner on strain, but will ratherbe described by either Langevin or gaussian chain statistics. Here the choice of properchain statistics depends on the fact, wether finite extensibility of a single polymer chainhas an significant effect on the orientational strain behaviour or not.

4.2.1 Affine approach

Consider macroscopic deformation of a sample which consists of randomly distributedpolymer chains connecting affine points. The situation of a single polymer chain betweentwo affine points is sketched in fig. 4.1. Applying affine deformation of the end-to-endvectors between the affine points, a vector oriented at polar angles (θ, φ) before deformation,yields a different orientation (θ′, φ′) after deformation relative to the sample frame. Thesetwo orientations are related in the affine case as

~r

|~r0|= λc

cosφ′ sinθ′

sinφ′ sinθ′

cosθ′

=

λx cosφ sinθλy sinφ sinθ

λz cosθ

(4.1)

where λx,y,z are the linear extension ratios of the end-to-end vectors and λc = rr0

is thechain extension ratio with the initial end-to-end distance r0. Using the well known relationfor the root-mean-square length of a Gaussian chain r0 = l

√n [35], it is

78


r

nl=

λc√n

(4.2)

where n is the average number of statistical links of length l per chain. By solving thesystem of equations eq. (4.1), following relations can be calculated

λc(θ, φ, λx,y,z) =√λ2

z cos2 θ + sin2 θ(λ2x cos2 φ+ λ2

y sin2 φ) (4.3)

θ′(θ, φ, λx,y,z) = arccos

λz cosθ√λ2

z cos2 θ + sin2 θ(λ2x cos2 φ+ λ2

y sin2 φ)

(4.4)

φ′(θ, φ, λx,y,z) = arccos

λx cosθ√λ2

x cos2 φ+ λ2y sin2 φ

(4.5)

Hence the new orientation (θ′, φ′) and λc in the deformed state are a function of theorientation in the non-deformed state and the macroscopic extension ratios of the sample(θ, φ; λx,y,z).

Since we are interested in the probability distribution function (PDF) of segmental ori-entation, we first have to describe the orientation of segments of a single chain:

Randomly jointed single chain under affine deformation of its end-to-end vector

As mentioned in section 2.2.2, orientational distribution functions of polymer segments of asingle chain, were originally derived for rubber-elastic materials assuming thermodynamicequilibrium and finite extensibility of the polymer chains and tested to be a good approx-imation for the theoretical description of stress-strain curves of rubber-elastic materials atmedium to large strains. It is unclear, if the affine approach is still useful in the descrip-tion of orientation-strain relation for plastically deformed polymer glasses, and if it stillmakes sense to imply finite extensibility of the chains. First, there seems to be evidencethat finite extensibility is relevant for the case of plastic deformation of chemically cross-linked polymer glasses. Figure 2.13 shows engineering stress-strain curves for chemicallycross-linked PMMA, with increasing cross-link density from the bottom to the top curve.The cross-link density η of each individual curve, i.e. the total number of cross-links pervolume []/nm3], determined via stress-strain curves in the swollen state (cf. section 2.4.1),are shown in the inset of the graph. Obviously, there is an upswing in the curves at largedeformation depending on the cross-link density of the material, whereas the non-treatedPMMA (bottom curve) shows no significant upswing. Similar results have also been re-ported in literature [51, 15]. In this study we investigated the orientation behaviour ofchemically cross-linked and non-cross-linked PMMA upon plastic deformation (far) belowand elastic deformation (far) above the glass transition temperature. The affine deforma-tion assumption will be invoked employing Gaussian and Langevin chain statistics, wherethe latter involves finite extensibility of the polymer chains.

79


Replacing the actual molecular structure by an idealized chain of n statistical links, or socalled Kuhn segments, each of length l, a solution of the problem of finding the distributionfunction of angles between the segments and the end-to-end vector ~r of a polymer chainwith finite extensibility, was first presented by Kuhn and Grun [38].

Orientation of segments is most effectively described by rotation of molecular, i.e. seg-mental, fixed frames into a sample fixed frame as described in section 3.4. Hence, since thez-axis of the molecular frame (MF) is parallel to the orientation of a Kuhn segment andthe end-to-end vector ~r of the single polymer chain is parallel to the z-axis of the sampleframe, the fraction of Kuhn segments, i.e. of molecular frames, that have an angle of theirz-axis between cos β and cos β +d cos β with respect to the z-axis of the sample frame, wascalculated using statistical thermodynamicsi

Plang(r

nl, cos β) d cos β =

b

2 sinh(b)exp(b cos β) d cos β (4.6)

where L(b) = coth(b) − (1/b) = rnl

is known as the Langevin function and L−1( rnl

) = bis the corresponding inverse Langevin function. This PDF corresponds to the so calledLangevin chain statistics.

As already discussed in section 2.2.2, L−1( rnl

) can be expanded in a series of (r/nl)α≥1

(cf. eq. (2.35)) leading to gaussian chain statistics in the limit of small strains r/nl 1,i.e. to a gaussian distribution of end-to-end vectors. Thus taking only the first term ofthe expansion, the fraction of Kuhn segments, i.e. of molecular frames, that have an anglebetween cos β and cos β + d cos β with respect to the sample frame can be determined inthe gaussian limit to be

Pgauss(r

nl, cos β) d cos β = 3

r

nl

1

sinh(b)exp(3

r

nlcos β) d cos β (4.7)

Before proceeding to the results of the orientation-strain relations, it is necessary tointroduce a widely used representation of PDF:

NMR determines full orientational distributions of molecular segments as a functionof euler or polar angles, unlike orientation-strain relations in literature that are oftenrepresented in terms of so called “order parameters” or “Hermans orientation parameters”.Here the latter denotes the most characteristic order parameter for probability functionsof symmetric distributions and is related directly to birefringence data [80].

Order parameters are connected mathematically with PDFs by so called “Wigner ma-trix elements” (WME). For more details on WME refer to appendix B. These WME,introduced by Wigner in 1931 [81], are functions of three euler angles and originate froma group theoretical treatment of quantum mechanical angular momentum operators andtheir transformation behaviour under frame rotations.

Every PDF can be expanded in terms of an infinite series of Wigner matrix elementsDl

mn(α, β, γ), since Wigner matrix elements represent a complete orthonormal basis

iit is |~r| ≡ r

80


P (α, β, γ) =l=∞∑l=0

m=l∑m=−l

n=l∑n=−l

clmn Dl

mn(α, β, γ) (4.8)

where the average Wigner matrix elements

〈Dlmn〉 =

∫ 2π

0

∫ 2π

0

∫ π

0

P (α, β, γ)Dlmn(α, β, γ) sin β dβ dα dγ (4.9)

are known as order parameters and are related to the expansion coefficients clmn of the

Wigner expansion as

clmn =

2l + 1

8π2〈Dl

mn〉 (4.10)

Depending on the desired chain statistics, the model has to be developed with averagesingle chain WMEs weighted either with a PDF given by eq. (4.6) or eq. (4.7) accordingto eq. (4.9).

Although it is not instantly obvious, presentation of orientational probability distribu-tion functions (ODFs) in terms of order parameters has indeed some helpful propertiesconcerning their geometrical interpretation (cf. appendix B). Especially in the case of loworientational order, a description of orientational ODFs in terms of their order parame-ters provides a possibility to describe the PDF by only a small number of characteristicnumbers:

For a completely non-oriented isotropic sample, i.e. a constant PDF, only the firstaverage WME (order parameter) 〈D0

00〉 is one, where all other average WME are zero .This means, that the expansion of a constant ODF in terms of Wigner matrix elementshas only one non-zero term, namely 〈D0

00〉=1. Every deviation from an isotropic ODFresults in additional higher non-zero order parameters, where in the extreme case of aDirac delta ODF all order parameters have the same value one and the expansion in termsof WME has an infinite number of non-zero terms. In general ODFs with low orientationalorder exhibit expansions coefficients which nearly vanish for l ≥ 4.

To calculate the average WME of a single chain, assume a single polymer chain consistingof n Kuhn segments of length l, with an end-to-end vector oriented parallel to the ~z-axis ofthe coordinate frame. Note that this special choice of orientation of the end-to-end vectordoes not introduce any geometrical constraints on the further calculation of average WME,since the extension to many chain system implies integration over the whole unit sphere.

Average WME for a single chain (sc), i.e. the coefficients of the expansion of eq. (4.8)in terms of WME, can be calculated analytically for cylindrical symmetry with the help ofeq. (4.2), eq. (4.6), eq. (4.7), and eq. (4.9) (cf. appendix B): ii

〈Dl00(

λc√n

)〉sc =

∫ π

0

Dl00(0, β, 0)P (

λc√n

, β) sin β dβ (4.11)

ii∫ cos β=−1

cos β=+1f(cosβ) d cosβ =

∫ π

0f(β) sinβ dβ

81


Rotating the end-to-end vector of the single polymer chain away from its position parallelto the z-axis of the SF by polar angles (θ, φ) yields modified average WME. Using rotationaltransformation laws of WME (cf. appendix B) yields

〈Dl0n(

λc√n

)〉scrot = 〈Dl00(

λc√n

)〉sc Dl0n(0,−θ,−φ) (4.12)

So far eq. (4.12) describes the transformation behaviour of average WME for a singlechain rotated by polar angles (θ, φ) and stretched by the extension ratio r/nl = λc/

√n.

Note that affine deformation always refers to affine deformation of the end-to-end vectorsof the chains, but not to the precise motion of segments along a polymer chain connectingtwo affine points.

To get the average WME for a system of many chains under affine deformation the caseof randomly isotropic oriented chains is discussed first for clarity.

Ensemble of many chains under affine deformation

Extension of the results for the average WME of a single chain to a system consisting ofmany chains, i.e. a polymeric solid, in principal requires orientational averaging over allchains in the solid. In the case of macroscopic solids, the number of chains is large enoughto replace the sum over all individual chains by an integration over the whole unit sphere.

Ensemble of isotropic randomly oriented polymer chains have average WME〈Dl

0k〉mciso, that can be calculated straight forward by integration of eq. (4.12) over all

possible orientations (θ, φ) weighted with the same constant probability, i.e. the wholeunit sphere iii

〈Dl0n〉mc

iso =1

4π

∫ π

0

∫ 2π

0

〈Dl0n(

1√n

)〉scrot sinθ dθ dφ

=1

4π

∫ π

0

∫ 2π

0

〈Dl00(

1√n

)〉sc Dl0n(0,−θ,−φ) sinθ dθ dφ

= δl0 δk0 (4.13)

Ensemble of oriented polymer chains, deformed affinely with respect to theirend-to-end vectors have average WME 〈Dl

0k〉mcdef , that can be calculated in a similar way:

Assume affine deformation of the end-to-end vectors of the isotropically distributed poly-mer chains oriented at polar angles (θ, φ) and extension ratio λc = 1 before deformation. Af-ter deformation they are oriented at angles (θ′, φ′) and stretched by λc = r/r0 as describedabove. Therefore the average WME of the deformed sample can be calculated by integra-tion again over the whole unit sphere weighted with constant probability, but replacingthe angles (θ, φ) appearing in the integrand with the angles (θ′(θ, φ, λx,y,z), φ

′(θ, φ, λx,y,z))of the deformed sample keeping the integration over all angles in the non-deformed state

iiinote all orientations have the same probability density 14π sinθ dφ dθ in the isotropic case

82


〈Dl0n〉mc

def =1

4π

∫ π

0

∫ 2π

0

〈Dl00(

λc√n

)〉sc Dl0n(0,−θ′,−φ′) sinθ dθ dφ (4.14)

where φ′, θ′, and λc are functions of the non-deformed angles (θ, φ), i.e. the integrationvariables, and the extension ratio λx,y,z as described in eq. (4.3 - 4.5).

4.2.2 Pseudo-affine approach

The so called “pseudo-affine model” assumes that each polymer segment deforms in anaffine manner itself [80] and was originally derived for a material consisting of independentsingle crystallites [38]. That means, that a polymeric solid consists of stiff independentsegments each deforming affinely. Following the formalism applied for the affine approach,the probability of finding segments that have an angle between cos β and cos β + d cos βwith respect to the end-to-end vector between two affine points is described by “Dirac’sdelta function δ(x− x0)”

P (β) d cos β = δ(cos β − 1) d cos β (4.15)

substitution of this into eq. (4.9), respectively eq. (4.11) yields

〈Dl00〉sc = 1 (4.16)

with this the average WME or order parameters according to the pseudo-affine model(cf. eq. (4.14)) become

〈Dl0k〉mc

def =1

4π

∫ π

0

∫ 2π

0

Dlk0(0,−θ′,−φ′) sinθ dθ dφ (4.17)

Both integrals of eq. (4.14) and eq. (4.17) can not be calculated analytically. There-fore the orientational average was computed by Gaussian spherical quadrature [82], whichmeans a numerical approximation of the volume integral by summing over a set of NS

different orientations with different weights according to the applied sampling scheme S:

1

8π2

∫V

f(Ω)dΩ ≈NS∑j=1

wSj f(ΩS

j ) (4.18)

where dΩ = dα d cos β dγ. In this way the number of sampled orientations can beconsiderably reduced. Usually NS is chosen in the range 21 ≤ NS ≤ 616, dependenton the desired accuracy of the approximation. In this study a reliable balance betweencomputation time and accuracy of the approximation could be achieved with the choiceN = 616. Simulated data of average WME 〈D2

00〉 and 〈D400〉 are shown in fig. 4.2 assuming

n = 9 statistical Kuhn segments along a polymeric chain between two affine points.Note that the pseudo-affine approach has no parameter to adjust for an optimal fit to

the data, unlike the number of stiff Kuhn segments in the affine approach. This is simplydue to the fact, that this model corresponds to the affine model with the choice n = 1.

83


Figure 4.2: Average Wigner functions 〈D200〉 and 〈D4

00〉 calculated with the affine and pseudo-affine approach for plane-strain compression and tension as a function of engineering strain λz.Short dashed line: pseudo-affine approach, long dashed line: affine approach, infinite extensi-bility (gaussian chain statistics), solid line: affine approach, finite extensibility (Langevin chainstatistics).

84

4.3 Materials: 2H-labelled Poly(methylmethacrylate)


One main scope of this thesis is to get insight into the molecular mechanism of deformationof glassy (amorphous) polymers. Detailed NMR studies on global and local molecular orderhave been recently performed on 13C enriched poly(Bisphenol-A-carbonate), (PC), by Utzet al. [83,23,24]. Our approach was to extend the study on another polymer glass (PMMA)and improve the experimental setup to get a more complete picture of deformation ofpolymer glasses.

Deuterium labelled poly(methylmethacrylate) was chosen as a model system for thestudy of molecular orientation in a deformed polymer glass. The advantages of deuteriumlabelling are numerous: NMR spectra of a deuterium label are almost exclusively gov-erned by quadrupolar interaction. Furthermore especially for aliphatic compounds, thequadrupole tensor is axially symmetric about the C − 2H bond within 3o or less. Thiswell-defined tensor orientation is of great value in the interpretation of quadrupolar Dspectra. Moreover the electric field gradient is found to be entirely intramolecular in na-ture, since it originates from the electrons in the C − 2H bond. Thus, in deuterium NMRmolecular, order is monitored through the orientation of individual C−2H bond directions.

All deuterated PMMA used for NMR studies in this thesis was obtained from PolymerSource Inc., Canada with molecular weight Mw = 75400 [g/mol] and polydispersity ofMw/Mn = 1.03.

4.3.1 Structure

The structure of non-oriented isotropic atactic PMMA in the solid state has been investi-gated most accurately by Lovell et al. using wide-angle X-ray scattering (WAXS) [84] incombination with conformational energy calculations [85,86]. Their analysis indicates thatregular conformational sequences persist for 16-20 backbone bonds, supporting predictionsmade on the basis of measurements of characteristic ratio and interpretation of small-anglescattering data [87].

Fig. 4.3 shows a stick and ball sketch of the chemical structure of a racemic dyad ofdeuterium labelled PMMA with staggered trans conformation. Deutrium labelling wasachieved by exchanging the hydrogens of the α-methyl group only by deuterium atoms(green color in fig. 4.3). In the case of fast methyl group rotations (cf. section 4.3.2), theresulting effective quadrupole interaction of the labelled CD3 group is an averaged tensorresulting from the electric field gradient of each C−D bond. This gives an excellent NMRlabel for studying orientation dependence, since the interaction tensor is perfect symmetricalong the C5 − C1 bond with an assymetry parameter practically indistinguishable fromzero, i.e. η = 0 (cf. section 4.3.2).

Table 4.1 shows all denoted rotation angles corresponding to the most likely conformationfound by Lovell et al. [84]. The ester group is approximately coplanar with the C5−C1−C3bonds, so that ω ≈ 0o and χ ≈ 0o. For the backbone bond angles Lovell et al. [84] foundthat θ1 < θ2, such that the all-trans chain forms an arc with the substituents pointingoutwards relieving the steric hinderance of adjacent side-groups. Atactic PMMA used

85


Table 4.1: Rotation angles of the most likely conformation of PMMA as suggested by Lovell etal [84].

angle approx. value

χ 0o

ω 0o

θ1 110o

θ2 128o

φ (+10o, +10o,−10o,−10o)

C 1C 2

C 1C 2

C 3C 4C 4

C 3C 5 C 5

C 1

C 2

O 1O 1

O 2O 2

q 1 ffq 2c

cq 1

w we s t e r

e s t e r

Figure 4.3: Perspective spatial stick and ball sketch of the chemical structure of a racemic dyadof α-methyl deuterium labelled PMMA. The colors of atoms are white: H, green: D, grey: C,and red: O. The backbone is marked by a yellow line. All rotation angles (ω, φ, χ) are shown aszero.

in this study contains approximately 60% syndiotactic, 35% isotactic, and 5% atacticchains. Since scattering data from highly isotactic and syndiotactic PMMA are virtuallyindistinguishable, they further suggest, that in broad terms the conformation remains closeto all-trans whatever the tacticity. The likely departures from all-trans conformation canbe described by a sequence of four rotation angles φ of the C1−C2−C1−C2 bonds andwas found to be (+10o, +10o,−10o,−10o), such that the chains form an arc rather than ahelix.

The proposed structure of regular conformation sequences persisting over a length oftypically 16 backbone bonds, i.e. 8 monomeric units, should not be identified with astatistical stiff Kuhn segment under deformation, since it was determined only in theisotropic non-oriented state.

86


Figure 4.4: Simulated and experimental 2H solid-state NMR spectrum of an non-orientedpowder sample. Solid red line: experiment (S/N ≈ 500), dashed black line: simulation

4.3.2 Powder line shape and CD3 quadrupole tensor

Analysis of the powder line shape of a completely isotropic, i.e. non-oriented sample allowcharacterization of the nuclear interaction as described in section 3.2 . For nuclei with spinI = 1, as in case of a deuterated methyl group CD3, the parameters of the dominatingquadrupolar interaction, like the anistropy parameter δQ and asymmetry parameter ηQ canbe determined.

Thermal energy in a solid can produce rotations of parts of molecules, which modu-late the anisotropic spin interactions and change the line shape. Rapid motion on thetimescale defined by the inverse width of the spectrum δ−1 in absence of motion, leads toan averaged effective interaction tensor σ of the nuclei, which can be characterized by aver-aged anisotropy and asymmetry parametersiv η and δ . Figure 4.4 shows the experimentalpowder spectrum of d3-PMMA detected by a 1D quadrupole-echo pulse sequence [77], asdescribed in more detail in the experimental NMR section 3.4, together with a simulatedpowder spectrum. In the case of CD3 methyl rotations around the C1−C5 bond (see fig.4.3) in d3-PMMA, the motion at room temperature is still in the fast limit, even thoughthe solid is the glassy state. This assumption is supported by the line shape of the powderspectrum and detailed studies of the freezing in of methyl rotations [88,59,89].

The line shape of the powder sample can be simulated, as outlined in section 3.2.3,indeed very accurately choosing an asymmetry parameter ηQ = 0 as shown in fig. 4.4. Thebest fit of the simulation to the experimental data was obtained with an optimal choice

ivnote that η may differ from zero, even if η = 0

87


Table 4.2: CD3 powder pattern: simulation parameters

parameter best fit value

quadrupol coupling constant 52.656 KHz

linebroadening 2.077 KHz

isotropic chemical shift 7.016 KHz

Table 4.3: Relative data scatter for four order parameters obtained from simulated data.

order parameter relative scatter ∆<D><D>

< D200 > 0.12

< D220 > 0.13

< D400 > 0.25

< D600 > 0.5

for the quadrupole coupling constant, the linebroadening, and the isotropic chemical shiftas shown in table 4.2.


Although the focus of this chapter is on results for the expansion coefficients of ODFs interms of Wigner matrix elements, i.e. order parameters, three representative ODFs withdifferent degrees of order are shown in fig. 4.5 together with the corresponding series of13 combined 1D NMR spectra . Figure 4.5(a) shows the reconstructed ODF from a cross-linked sample deformed above Tg with strain ε = −0.835 exhibiting only low degree oforientation, fig. 4.5(b) shows the reconstructed ODF from a cross-linked sample deformedbelow Tg with strain ε = −0.764 exhibiting medium degree of orientation, and fig. 4.5(c)shows the reconstructed ODF from a non-cross-linked sample deformed below Tg withstrain ε = −0.835 exhibiting a higher degree of orientation. These contour-plots revealnicely the symmetry of the orientational order around φ = π/2, θ = π/2 as expected fromthe plane strain deformation experiment.

Figures (4.6)-(4.9) show four characteristic average WME for linear PMMA in the cross-linked and non-crosslinked state deformed above and below Tg. Relative Errors of theexperimental data can be estimated from simulated data by comparison of simulated andexperimental NMR spectra via a numerical parameter χS as described in detail in section3.5.2 and listed in Table 4.3.

Short dashed lines in figures (4.6)-(4.9) show predictions of average WME accordingto the pseudo affine model. The most striking results is the discrepancy between the

88


0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

0 45 90 135 180

Sig

nal [

a.u.

]

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

0 45 90 135 180

Sig

nal [

a.u.

]

0

0.5

1

0 pi/2 pi0

pi/2

pi

φ

θ

0 45 90 135 180

Sig

nal [

a.u.

]

Rotor position in [ o ]

(a)

(b)

(c)

Figure 4.5: Three representative reconstructed ODFs with different degrees of orientationalorder together with a stack plot of the corresponding series of 13 combined 1D NMR spectra. (a)cross-linked sample deformed above Tg with strain ε = −0.835. (b) cross-linked sample deformedbelow Tg with strain ε = −0.764. (c) non-cross-linked sample deformed below Tg with strainε = −0.835. Each single NMR spectrum of the stack plots represents a fixed orientation of therotor equally spaced in 13 steps of ∆15o in the plane perpendicular to the rotor axis (cf. fig. 3.4).A non-oriented sample would exhibit 13 identical spectra.

89


Figure 4.6: Experimental average Wigner functions 〈D200〉 for PMMA compressed below and

above Tg in plane-strain mode as a function of true strain ε = lnλz compared with best fittheoretical calculations. Black lines and filled symbols denote a deformation temperature of 25oC(far) below Tg ≈ 1150C whereas grey lines and open symbols denote a deformation temperature of160oC above Tg. (a) non cross-linked PMMA compressed at different true strain rates below Tg,squares: ε = −0.0001, circles: ε = −0.001, triangles: ε = −0.01. (b)-(d): chemically cross-linkedPMMA with decreasing cross-link densities b) η = 1.10, (c) η = 0.20, (d) η = 0.18. Solid lines:affine approach, infinite extensibility (gaussian chain statistics), long dash: affine approach, finiteextensibility (langevin chain statistics), short dash: pseudo-affine approach. The relative scatterof the data was estimated to be ∆D2

00/D200 ≈ 0.12 (cf. section 3.5.2 and table 4.3).

90





91





92





93


experimentally determined orientation upon plastic deformation as measured by solid-state NMR, and the theoretical pseudo-affine deformation approach for linear and cross-linked PMMA. In contrast, most orientational data in literature, determined either bybirefringence or x-ray scattering, show good agreement with the pseudo-affine approach[80, 38]. This apparent discrepancy might be due to the uncertainty of the size of thestructural units of which the orientation is experimentally detected. Hence, agreement ofexperimental data with the pseudo-affine model suggests that these structural units deformaffinely, without giving any further information about their size. Therefore, if the size ofthese structural units cannot be determined by other means, segmental orientation along apolymer chain between two affine deforming points cannot be resolved by these methods.

Here NMR may be a better choice. All monomeric units along every polymer chain of thematerial used in this study were labelled, i.e. individually visible by NMR. This implies,that in principle, supposing maximum angular resolution, the true average segmental ori-entation could be measured. Of course, the angular resolution of NMR is restricted, whichmakes it questionable if the measured data can be identified as the true orientationaldistribution of the monomeric units. Generally speaking, NMR gives the orientationaldistribution of stiff segments within the resolution of the experiment. Assuming the va-lidity of the affine approach, less resolution leads to an experimentally determined higherorientation compared to the true segmental orientation, and more resolution leads to anexperimentally determined lower orientation approaching the true segmental orientation atoptimal resolution. Obviously the angular resolution of solid-state NMR is better, since themeasured orientation is much less than the predicted pseudo-affine orientation supportedby scattering data.

Solid and long dashed lines in figures (4.6)-(4.9) show fits of the affine model to theexperimental NMR data. Here the solid lines represent fits with gaussian chain statistics,valid for small extension ratios r/nl ≤ 0.4 (cf. section 2.2.2), and the short dashed linesrepresent fits with Langevin chain statistics valid for large strains, when finite extensibilityof the polymer chains becomes relevant. Within the experimental scatter, both orientation-strain relations, calculated with or without implying finite extensibility, are quantitativelyvery close for plane strain compression mode in contrast to tension experiments as sketchedin fig. 4.2, and therefore indistinguishable on a fitting level. This makes it additionallydifficult to differ which fit might give better results.

True stress-strain data determined from compression and tension experiments, however,may give some more information. Presenting true or engineering stress data as a functionof (λ2 − λ−1) respectively (λ − λ−2), would yield a straight line assuming neo-Hookeanbehaviour (cf. section 2.2.2), which corresponds to gaussian chain statistics of infinitelyextendable chains. Any upswing in the curvature of such stress-strain plots would indi-cate that finite extensibility gains relevance at the applied strains. Figure 2.13 indicates,that there is no significant upswing in the bottom curve belonging to non-cross-linkedPMMA, as already mentioned in section (4.2.1). Also stress-strain data taken in tensionof Poly(carbonate) [15] show no indication for finite extensibility.

For rubber elastic material the inverse relation between the number of Kuhn segments nalong an elastically active chain and the total number of cross-link points µ per volume, i.e.

94


the cross-link-density, can be determined with the help of eq. (2.56) for a perfect network.With N = ρ

Mcand n = Mc

M0it is

µ =2

f

ρ

Mc

(4.19)

=2

f

ρ

n M0

∝ n−1

where f is the functionality of the perfect network, N is the number of elastically activechains, Mc their molecular weight, and M0 the molecular weight of one statistical stiffKuhn segment. Figure 4.10 shows the average number of stiff statistical Kuhn segments nas a function of inverse cross-link-density µ−1 determined from best fits of the affine modelto the experimental data as shown in figures (4.6)-(4.9).

Since no distinction of a proper choice for the suitable chain statistic can be madefrom NMR data of compression tests only, as already discussed above in this section,results involving both gaussian and Langevin approaches are presented in fig. 4.10(a)respectively fig. 4.10(b). Data taken from samples compressed below the glass transitiontemperature Tg are presented by filled symbols, whereas samples compressed above Tg yielddata presented by open symbols. A remarkable feature of the affine fits is the fact, thatorientation-strain relations of PMMA deformed below Tg (filled symbols in fig. 4.10) seemsto be only slightly affected by chemically cross-linking of the polymer chains within theresolution of the experiments, whereas the effect of cross-linking on segmental orientationis obvious for PMMA deformed above Tg (open symbols in fig. 4.10) and supports thetheoretical rubber-elastic inverse relation n ∝ µ−1. Actually this is not very surprisingly,since the applied deformations of the glassy samples are too low to detect significantinfluence of cross-link density on orientation. Compare to stress-strain data in fig. 2.13,where significant influence of cross-link-density on large strain behaviour appears only atlarge strains exceeding ε ≤ −0.5.

Supported by stress-strain data shown in fig. 2.13, it will be assumed that gaussianchain statistics, i.e. neo-Hookean behaviour, is the relevant choice for the interpretation oforientation-strain data of non cross-linked PMMA. Hence the average number of statisticalstiff Kuhn segments between two affine deforming points in the glassy state is assumed tobe nglass ≈ 2.1 from a best fit of the affine approach.

In principle it is possible to calculate a shear modulus with the knowledge of n, applyingrubber-elastic models. This shear modulus, determined from orientation-strain relations,might be compare with the strain hardening modulus determined by best fits of continuummodels (cf. section 2.2) to true stress - true strain curves of PMMA, as shown in thefollowing:

Using again gaussian chain statistics, the shear modulus of a rubber elastic material canbe calculated in principle by two basic approaches: the affine network and the phantomnetwork approach. The difference between these two approaches lies in the mobility ofaffine deforming network knots. In the affine network, mobility of the network knots is

95


completely suppressed, and chain segments connecting these knots can move freely with-out any constraints through steric hinderance. The phantom network model was alreadydescribed in section 2.4.1 and suits for very diluted samples only, especially to swollensamples, since steric hinderance is assumed to have only very slight influence. The shearmoduli of these two approaches differ by a factor two, thus providing an upper (affine) andlower (phantom) bound for the true modulus, which is located between those two extrememodels, since mobility of the chains during deformation is neither completely suppressednor completely unconstrained. More sophisticated models [90, 91] describe the transitionfrom affine to phantom like behaviour as a function of strain and dilution of the networkby incorporation of constraints on the movement of junctions and network chains, but willnot be discussed in this study. Since we are dealing with plastic deformation of polymerglasses, the affine gaussian network approach might be a good choice for the relation be-tween strain hardening shear modulus and the average number of statistical stiff Kuhnsegments n

Gaff = NkT (4.20)

=ρ

nM0

kT

with this relation the expected strain hardening shear modulus of PMMA can be cal-culated to be Gaff ≤ 14 MPa using typical values for PMMA like ρ = 1.188 g/cm3,M0 = 100.12 g/mol, and nglass = 2.1 determined by fits of the gaussian affine approach tothe experimental NMR data. Since the molecular weight of one statistical Kuhn segmentM0 is not known, the maximum shear modulus is obtained with the smallest value for M0,i.e. by identification of a statistical Kuhn segment with one monomeric unit.

Comparing this value of the shear modulus of PMMA, obtained from the presentedorientation-strain relations, with the value obtained by best fits of the suggested continuummodel with a non-constant activation volume dependent on the actual state of deformation(cf. chapter 1, Gsh ≈ 20), at least rough agreement can be claimed.

4.5 Conclusions

Series of dependent one dimensional solid state NMR experiments have been performed ondeuterium labelled PMMA to get information on average molecular orientation of segmentsalong a polymer chain. Cross-linked and non-cross-linked linear PMMA samples werecompressed in plane strain compression to different degrees above and below their glasstransition temperature.

Orientation-strain relation for the cross-linked samples, deformed above their glass transi-tion temperature, can be modelled by classic rubber-elasticity assuming affine deformationof the network knots, but non-affine deformation of segments along a polymeric network-chain connecting two network knots. Segmental orientation is usually described by gaussianchain-statistics, or, when finite extensibility of the network chains is observed, by Langevin

96

4.5 Conclusions

Figure 4.10: Average number of statistical stiff Kuhn segments between two affine deformingpoints in PMMA as a function of inverse cross-link-density corresponding to the affine model.Affine fits of n to data from PMMA deformed above Tg at T = 160oC (open symbols) and belowTg at T = 23oC (filled symbols) are shown using gaussian chain statistics (a) and Langevinchain statistics (b). The dashed horizontal lines show the values determined from affine fits ofnon-cross-linked PMMA data.

97


chain-statistics. Here it was not possible to distinguish from the experimental compressiondata only which chain-statistics is the better choice, since in compression mode, even forlarge strains, orientation of segments along a network-chain is nearly independent of thechoice of chain statistics on a fitting level. Nevertheless large strain uniaxial compres-sion tests of the same material indicate strongly, that finite extensibility of the networkchains is not an important factor for non-cross-linked systems, but gains relevance withan increasing cross-link density as shown for linear PMMA, chemically cross-linked withvarious degrees. These results show, that classical rubber-elasticity is capable to describeorientation-strain relation of chemically cross-linked PMMA above its glass transition as-suming affine deforming network knots. The only adjustable parameter in this descriptionis the number of statistical Kuhn segments along a network chain.

Orientation-strain relations of cross-linked and non-cross-linked samples, deformed atroom temperature far below the glass transition, show a behaviour which is in contrastto orientation-strain data obtained by x-ray scattering or birefringence. These scatteringexperiments were always in agreement with the so-called pseudo-affine model, which sug-gests affine deformation of each segment itself along a network chain, independent of thenumber of statistical Kuhn segments along a network chain, thus having no adjustableparameter. In principle this pseudo-affine approach equals the affine approach in the limitof one segment per network chain only.

NMR experiments as performed in this study, reveal a much weaker dependence ofsegmental orientation of deformed polymer glasses than predicted by the pseudo-affineapproach. Moreover, the obtained orientation-strain relations of glassy PMMA could bemodelled by an affine approach similar to the case of a rubber-elastic material. Thereforeit can be concluded, that the size of a unit, which deforms affinely with strain, cannotbe identified with a single segment of the network chain in a glassy polymer, since NMRis sensitive to orientation on a molecular level. Further, this result indicates, that thereis indeed some underlying network response of a plastically deformed polymeric glass,since structural segments show orientations-strain behaviour similar to that observed inrubber-elastic materials. The reason why these experiments agree with the pseudo-affineapproach might be due to the fact, that the structural units, of which the orientationis experimentally determined, are approximately identically in size with a network-chainconnecting affinely deforming points. Thus NMR provides a much more sensitive methodfor the detection of molecular orientation in solids.

Identification of the entanglement network response, as described in continuum mod-elling, as the dominating source for segmental orientation allows to calculate a shearmodulus, which might be compared to the strain hardening shear modulus of simulatedcontinuum models. In the case of PMMA, continuum modelling suggest a strain harden-ing modulus of approximately 20MPa, whereas orientation-strain relations obtained bysolid-state NMR suggest a modulus of approximately 13MPa, which is at least a roughagreement of data, obtained by completely different experimental techniques and involvingmodified continuum modelling.

98

5 Conclusions

The results presented in this work can be roughly divided into three parts: 1) Mechanicaldeformation experiments on polymeric glasses and their theoretical description by a con-tinuum mechanical modelling approach, 2) spectral analysis of solid-state NMR spectrausing regularization methods to yield orientational distribution functions of molecular seg-ments, 3) determination of orientation-strain relations for cross-linked and non-cross-linkedPoly(methylmethaycrylate) (PMMA) by line shape analysis of solid-state NMR spectra.

An essential precondition for improvement of any modelling of mechanical behaviour ofpolymeric glasses is the availability of reproducible and reliable experimental data. Truestress-strain curves obtained for four different polymeric glasses, namely PMMA, PC, PS,and PPO, revealed characteristic features in the post-yield regime at large finite strains,which have not been addressed by similar published experiments and modelling in thepast: all tested polymers exhibit a strain rate dependent slope of stress-strain curves inthe post yield regime, which could not be explained by systematic experimental errors. Sofar, the increase of stress at large finite strains, i.e. so-called “strain hardening”, has beenmodelled by a rubber-elastic network response of the underlying entanglement networkleading to the observed strain hardening. From measurements of this study it could beconcluded, that this approach is not capable to describe post-yield behaviour at largefinite strains to a satisfying level. Hence we suggest a modification of the viscoelasticresponse, which was based on the Eyring concept of describing plastic flow as jumps ofsome structural element over energy barriers activated by a combined influence of appliedstress and thermal energy. Applying a dependence on stress and strain, i.e. on the actualmicroscopic configuration, of the so-called shear activation volume, which was treated as aconstant material parameter so far, leads to a much better agreement between modellingand experimental data. Hence it can be concluded, that the viscoelastic response at largestrains is a function the macroscopic total stress and strain, which is in contrast to thewidely used decoupling of viscoelastic and network response.

In order to be able to properly analyze our NMR spectra the mathematically ill-posednesswas acknowledged and strategies, that are applied to similar problems in other fields ofscience, were exploited and shown to work for the reconstruction of two dimensionallyorientational distributions from a series of 13 combined 1D solid-state NMR spectra. Nu-merical testing of the suggested algorithm has shown that materials even with low degreeof orientational order can be analyzed with significant resolution.

Results on orientation-strain relations revealed agreement with the so-called affine ap-proach, which is based on classical rubber-elasticity theory. This is in contrast to published

99

5 Conclusions

data obtained by widely used methods, like birefringence or x-ray diffraction, which suggestthe so-called pseudo affine model to be a proper description of orientation-strain relationof polymeric glasses. The disadvantage of these techniques is that the size of the structuralunit, of which the orientational distribution is experimentally determined is not known.Hence it could be concluded, that such a structural unit cannot be identified with a singlemolecular segment of the polymer chain, but rather with a chain connecting two affinedeforming points of some entanglement network. Moreover this result indicates, that theresolution of solid-state NMR is indeed on a molecular scale, where the resolution of theabove mentioned methods seems to be approximately on the scale of the average distancebetween two knots of some entanglement network. Finally modelling of orientation strainrelation with an affine approach yields the entanglement density of the relevant networkduring plastic deformation, which allows to calculate a shear modulus of the networkresponse. Hence a comparison with the fitted shear modulus of continuum mechanicalmodelling, as suggested in this thesis, could be carried out and revealed at least a roughagreement.

Since these results gain insight into the deformation mechanism of polymeric glasses ona molecular level and extend large strain mechanical continuum modelling, they may behelpful in further improvement of both, continuum and atomistic modelling.

100

Appendix A

Rotations

A.1 Active vs. passive rotations

active rotations First let us describe a 3x3 matrixR which performs always an active rotationin a fixed frame of any vector ~r in the mathematically positive sense (counterclockwise). Thenew vector ~r′ is obtained by:

~r ′ = R~r (A.1)

For instance, an active rotation about the z-axis by an angle φ is performed by the 3x3 matrix

Rz(φ) =

cos(φ) − sin(φ) 0sin(φ) cos(φ) 0

0 0 1

(A.2)

or an active rotation about the y-axis by an angle θ is performed by the 3x3 matrix

Ry(θ) =

cos(θ) 0 sin(θ)0 1 0

− sin(θ) 0 cos(θ)

(A.3)

Thus an active rotation of any vector about the two polar angles (θ, φ) is performed by therotation

Rpolary,z (θ, φ) = Rz(φ)Ry(θ) =

cos(θ) cos(φ) − sin(Φ) cos(φ) sin(θ)cos(θ) sin(φ) cos(Φ) sin(φ) sin(θ)− sin(θ) 0 cos(θ)

(A.4)

For a matrix T , the invariance of the scalar ~r′TT ′~r ′ = ~r T T ~r requires

T ′ = RT R−1 (A.5)

passive rotations rotations of the axis produces new coordinates of a vector:

~r ′ = R−1~r (A.6)

Again for a matrix T , the invariance ~r′TT ′~r ′ = ~r T T ~r requires

101

Appendix A Rotations

T ′ = R−1T R (A.7)

Since the axis form a right-handed orthogonal coordinate system, the real rotation matrix Ris orthogonal and unique

R = R∗ RT = R−1 detR = 1 (A.8)

A.2 Euler angles and rotation group O(3)

a b g y 'y

x '

x

zz '

Figure A.1: Illustration of the rotation defined by Euler angles (α, β, γ). Positive rotations aredefined by the right hand screw sense

The most useful way to parameterize the rotation group O(3) is that of Euler. The conventionused here is that employed frequently by workers in the theory of molecular spectra [92], butdiffers from those of Wigner [81], who uses a left-handed frame of reference. The rotations of theinitial right-handed frame 0xyz into a rotated right-handed frame 0XY Z (see fig. A.1) are to beperformed successively in the order:

1. rotation α(0 ≤ α < 2π) about the original z-axis

2. rotation β(0 ≤ α < π) about the intermediate y’-axis

3. rotation γ(0 ≤ α < 2π) about the new Z-axis

We describe the resulting rotation by the already introduced matrix R, which performs anactive rotation of the frame (axis) in the mathematically positive sense (counterclockwise):

Rz,y′,Z(α, β, γ)frame = RZ(γ)Ry′(β)Rz(α)i (A.9)

inote that the order of rotations is (α, β, γ), i.e. the outermost right rotation is performed first.

102


Note that Euler angles (α, β, γ) are always defined by a rotation of a reference frame 0xyz intoa frame 0XY Z, i.e. Euler angles describe a passive rotation of the system but an active rotationof the frame. It is useful to show that the same passive Euler rotation of the frame around theaxis (z, y′, Z) can be produced dealing with rotations about the axes of the original referenceframe only. This alternative equivalent rotation can be performed successively in the order:

1. rotation γ(0 ≤ α < 2π) about the original z-axis

2. rotation β(0 ≤ α < π) about the original y-axis

3. rotation α(0 ≤ α < 2π) about the original z-axis

hence the following active rotations of the frame are equivalent

RZ(γ)Ry′(β)Rz(α) = Rz(α)Ry(β)Rz(γ)⇓

Rz,y′,Z(α, β, γ)axis = Rz,y,z(γ, β, α)axis (A.10)

Vectors and Cartesian tensors perform the rotation R−1 when the frame is rotated by R.This means, that the rotation of a vector (or system) equivalent to an Euler rotation of the framecan be described by

Rz,y′,Z(α, β, γ)frame = Rz,y,z(γ, β, α)frame ↔ R−1z,y,z(γ, β, α)system (A.11)

Note that R(α, β, γ)−1 = RT (α, β, γ) = R(α)R(β)R(γ), i.e. the order of rotations is reversedin inverted or transposed rotation matrices. Keeping this in mind and using R~e (φ)−1 = R~e (−φ)this means that an arbitrary vector ~r transforms under a Euler rotation (α, β, γ) of the frame as

~r ′ = R−1z,y,z(γ, β, α)system ~r

= R−1z (γ)R−1

y (β)R−1z (α)

= Rz(−γ)Ry(−β)Rz(−α)= Rz,y,z(−α,−β,−γ)system ~r (A.12)

which simply means

Rz,y′,Z(α, β, γ)frame ↔ Rz,y,z(−α,−β,−γ)system (A.13)

i.e., the equivalence between a passive Euler rotation of the frame with angles (α, β, γ) aroundthe axis (z, y′, Z) and a pseudo-active rotation of the system with angles (−α,−β,−γ) aroundthe axis (z, y, z).

Analogous a matrix T rotates under a Euler rotation of the frame according to eq. (A.8)

T ′ = R−1z,y,z(γ, β, α)system T Rz,y,z(γ, β, α)system (A.14)

Using cα := cosα, sα := sinα, etc., this pseudo-active Euler rotation matrix, which we call Ris

103

Appendix A Rotations

Rz,y,z(α, β, γ) = R−1z,y,z(γ, β, α)system

= RTz,y,z(γ, β, α)system

= RTz (γ)RT

y (β)RTz (α)

=

cγ sγ 0−sγ cγ 00 0 1

·

cβ 0 −sβ0 1 0sβ 0 cβ

·

cα sα 0−sα cα 0

0 0 1

=

cα cβ cγ − sα sγ sα cβ cγ + cα sγ −sβ cγ−cα cβ sγ − sα cγ −sα cβ sγ + cα cγ sβ sγ

cα sγ sα sγ cβ

(A.15)

Functions and state vectors need a special treatment. While Vectors and matrix represen-tations of Cartesian tensors rotate active by R, the active rotation of (wave) functions ψ(~r) andstate vectors is effected by a rotation operator D. The following invariance

ψ′( ~r ′) = ψ(~r) (A.16)

requires

Dψ(~r) = ψ′(~r) = ψ(R−1~r) (A.17)

i.e. an active rotation of functions and state vectors is performed by a passive rotation of theirvector argument.

An irreducible representation of the 3-dimensional rotation group was first derived by Wigner.The matrix elements of the finite rotation operator of dimension (2l+1) can be written:

〈lm|D|ln〉 = Dlmn (A.18)

where |lm〉 are eigenfunctions of the angular momentum operator Jz and represent the basisof the representation DL of dimension (2L + 1) of the rotation group. Defining that D(α, β, γ)rotates functions and state vectors active through angles (γ, β, α) around the axis (z, y, z), therotation operator for finite rotations is :

D(α, β, γ) := e−iαJze−iβJye−iγJz (A.19)

the Wigner matrix elements (WME) of eq. (A.18) become

Dlmn(α, β, γ) = e−iαm dl

mn(β) e−iγn (A.20)

where the dlmn(β)are given by

dlmn(β) =

t=min[l−n,l+m]∑t=max[0,m−n]

(−1)t

√(l +m)! (l −m)! (l + n)! (l − n)!

(l +m− t)! (l − n− t)! t! (t+ n−m)!× (A.21)

×(cosβ/2)2l+m−n−2t(sinβ/2)2t+n−m

104


They are orthogonal in the sense

∫ 2π

0

∫ 2π

0

∫ π

0(Dl

mn(α, β, γ))∗ Dl′m′n′(α, β, γ) sinβ dβ dα dγ

=8π2

2L+ 1δ(m,m′) δ(n, n′) δ(l, l′) (A.22)

hence with eq. (A.10) and eq. (A.13) the rotation D(α, β, γ) is equivalent to an Eulerrotation of the frame with angles (−α,−β,−γ) around the axis (z, y, z) or equivalent withangles(−γ,−β,−α) around the axis (z, y′, Z).

Observables, Operators. If A is an observable, i.e. a operator with direct observable realvalues, then the matrix elements of this operator are < ψn | A | ψm >. The corresponding matrixelements in a rotated system, i.e. active rotation of the state vectors | ψm >, are < ψn | D+

AD |ψm >. According to the Heissenberg representation, the transformation can interpreted as anoperator transformation A→ A

′

A′ = Rz,y,z(γ, β, α)system

A = D+AD = D−1

AD (A.23)

Now we deduce the transformation of any observable A equivalent to an Euler rotationof the frame. We know from eq. (A.13) that the active rotation of the state vectors, i.e.the quantum system, equivalent to the Euler frame rotation Rz,y′,Z(α, β, γ)frame must beRz,y,z(−α,−β,−γ)system. According to eq. (A.19) this rotation is performed by D−1, sinceD rotates the system active through angles (γ, β, α). Hence the transformation of any observableA equivalent to an Euler rotation of the frame Rz,y′,Z(α, β, γ)frame must be

A′ = Rz,y′,Z(α, β, γ)frame

A = DAD+ = DAD−1 (A.24)

105

106

Appendix B

Expansion of orientational PDFs interms of Wigner functions

B.1 General

Assume a sample system of molecular orientations, each attached to the z-axis of a molec-ular frame. The orientation of molecular segments can now be described by the rota-tion Rz,y′,Z(α, β, γ)MF of the molecular frame (MF) into the sample frame (SF). ThenP (α, β, γ) dα d cosβ dγ is the fraction of molecular frames within the sample system, who can bedescribed by a rotation of the MF into the SF by Euler rotations between Rz,y′,Z(α, β, γ)MF andRz,y′,Z(α+ dα, β + sinβdβ, γ + dγ)MF . Here the infinitesimal steradian is dΩ = dα d cosβ dγ =dα sinβdβ dγ, who describes an infinitesimal area of constant size on the surface of the unitsphere.

Since Wigner matrix elements form a complete orthonormal basis, any orientational distribu-tion function P (α, β, γ) can be expanded in terms of Wigner matrix elements (WME).

P (α, β, γ) =l=∞∑l=0

m=l∑m=−l

n=l∑n=−l

clmn Dlmn(α, β, γ) (B.1)

where either integer or half integer l can be used. In the following we will always assume integerl. We keep the normalization∫ 2π

0

∫ 2π

0

∫ π

0P (α, β, γ) sinβ dβ dα dγ = 1 (B.2)

The average of the WME can be written

〈Dlmn〉 =

∫ 2π

0

∫ 2π

0

∫ π

0P (α, β, γ)Dl

mn(α, β, γ) sinβ dβ dα dγ (B.3)

taking together eq. (A.22)-(B.3) it follows for the expansion coefficients clmn

clmn =2l + 18π2

〈Dlmn〉 (B.4)

107

Appendix B Expansion of orientational PDFs in terms of Wigner functions

B.2 Symmetries

In most cases we are only interested in orientational distributions of vectors, which have rotationalsymmetry. Hence only two angles are necessary to describe the system properly. Assume the samerotation as above of the molecular frame (MF) into the sample frame (SF). The correspondingnew distribution function P (β, γ) of the z-axis of the MF, which is the vector of interest, can bedetermined by integration of P (α, β, γ) over the first euler angle α.i

P (β, γ) =∫ 2π

0P (α, β, γ)dα

=∑l,m,n

clmn

∫ 2π

0Dl

mn(α, β, γ)dγ (B.5)

using eq. (A.20) and∫ b0 e

−iaxdx = b δa0 it follows

P (β, γ) =∑l,m,n

clmn

∫ 2π

0e−iαmdα dl

mn(β) e−iγn

=∑l,m,n

clmn 2π δm0 dlmn(β) e−iγn

= 2πl=∞∑l=0

n=l∑n=−l

cl0n Dl0n(0, β, γ) (B.6)

hence the average WME is

〈Dl0n〉 =

∫ 2π

0

∫ π

0Dl

0n(0, β, γ)P (β, γ) sinβdβdγ (B.7)

analogous the distribution function and average WME can be calculated for cylindrical sym-metry, i.e. now the z-axis of the MF is symmetric around the z-axis of the SF. Hence only oneangle β is necessary to describe the orientational distribution

P (β) = 4π2l=∞∑l=0

cl00 Dl00(0, β, 0) (B.8)

〈Dl00〉 =

∫ π

0Dl

00(0, β, 0)P (β) sinβdβ (B.9)

So far only the case of isotropy with respect to one (α) or two (α, gamma) euler angles wastreated. Further symmetries of the orientational distribution function arising from the setup ofthe experiment or the processing of the sample lead to even more convenient constraints on theexpansion coefficients clmn [93].

ihere∑

l,m,n ≡∑l=∞

l=0

∑m=lm=−l

∑n=ln=−l

108

B.2 Symmetries

The use of symmetry adapted WME. It is now possible to define symmetry adaptedWME Dl

mn(α, β, γ) which gain a mathematically more convenient way of expanding orientationaldistribution functions of symmetric systems into WME as compared to non symmetric systems(cf. eq. (B.1))

P (β, γ) =l=∞∑l=0

n=l∑n=0

cl0n Dl0n(0, β, γ) (B.10)

where l and m are now positive even integer numbers, and the symmetry adapted WME Dlm0

are defined in the following way:

Dl0n =

Dl

00 if n = 02 Re(Dl

0n) if n 6= 0(B.11)

directional distributions

The rotation of any reference frame (ReF) into a rotated frame (RoF) is performed by the rotationRz,y′,Z(α, β, γ)ReF . To calculate the distribution function of a vector fixed in the reference framewith angles (φ, θ) with respect to the z-axis from a known distribution function of the referenceframe, the following recipe can be applied:

Simply replace the rotation Rz,y′,Z(α, β, γ)ReF which rotates the z-axis of the ReF parallel tothe z-axis of the RoF, by a rotation that aligns the (φ, θ)-axis of the ReF parallel to the z-axis ofthe RoF, keeping the same expansion coefficients of the rotation ReF→RoF.

Consider again for example the rotation of the molecular frame (MF) into the sample frame(SF). This rotation is performed by the rotation Rz,y′,Z(α, β, γ)MF . Now we like to know thedistribution of a vector fixed in the MF with polar angles (φ, θ) with respect to the molecularz-axis. This can be done in two steps:

1. rotate the MF so that the (φ, θ)-axis aligns parallel to the original z-axis of the MF byRz,y′,Z(0,−θ,−φ)MF

2. perform the rotation of the MF into the SF by Rz,y′,Z(α, β, γ)MF using the same expansioncoefficients clmn describing the rotation of the original MF into the SF.

This yields a transformed distribution function P (α, β, γ), which can be integrated over thethird Euler angle in order to obtain the desired directional distribution function Pφ,θ(α, β) fromthe known distribution of the MF.

P (α, β, γ) =∑lmn

clmn

∑k

Dlmk(α, β, γ)Dl

kn(0,−θ,−φ)

=∑lmnk

clmnDlkn(0,−θ,−φ) Dl

mk(α, β, γ)

=∑lmk

(∑n

clmnDlkn(0,−θ,−φ)

)︸︷︷︸

clmk

Dlmk(α, β, γ) (B.12)

109

Appendix B Expansion of orientational PDFs in terms of Wigner functions

Integration over the first Euler angle α yields the orientational distribution function of themolecular vector (φ, θ) as a function of the known expansion coefficients clmn corresponding tothe distribution of the MF.

Pφ,θ(β, γ) = 2π∑lk

cl0kDl0k(0, β, γ) (B.13)

where

cl0k =∑

n

cl0nDlkn(0,−θ,−φ) (B.14)

B.3 Rotational transformation of WME under framerotation

How does does an average WME transform under frame rotation ? Suppose the rotation (α, β, γ)is the result of the successive application of, in that order, (α1, β1, γ1) and (α2, β2, γ2). Then it is

Dlmn(α, β, γ) =

k=+l∑k=−l

Dlmk(α2, β2, γ2) Dl

kn(α1, β1, γ1) (B.15)

In this study we search the average WME of a polymer chain first with end-to-end vectororiented along the ~z-axis of the reference frame, and then stretched and rotated with polar angles(θ, φ). May the WME Dl

mn(α, β, γ) describe the rotation of the molecular frame attached to eachsegment of the polymer chain, into the reference frame S. Then the new WME, describing therotation of molecular frames of the rotated polymer chain into the reference frame S, have toperform two successive rotations

Dlmn(α, β, γ) =

k=+l∑k=−l

Dlmk(α, β, γ) Dl

kn(0,−θ,−φ) (B.16)

generating the average yields

〈Dlmn(α, β, γ)〉 = 〈

k=+l∑k=−l

Dlmk(α, β, γ) Dl

kn(0,−θ,−φ)〉

=k=+l∑k=−l

〈Dlmk(α, β, γ)〉 Dl

kn(0,−θ,−φ) (B.17)

Plugging in the average WME for a single chain 〈Dl0n〉 eq. (4.11) with end-to-end vector parallel

to the ~z-axis of the reference frame S, yields the modified average WME 〈Dl0n〉 of a single chain

with end-to-end vector rotated by polar angle (θ, φ)

〈Dl0n〉scrot = 〈Dl

00〉sc Dl0n(0,−θ,−φ) (B.18)

110

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116

Acknowledgements

I started here at the ETH in 1999 with no clear idea what it will be all about, but for sure,if I would be asked today, I never would have expected how things developed with respectto the scientific work as well as to my private life. Fortunately almost every surprise Ifaced during this time turned towards a positiv experience. Therefore it is a great pleasurefor me to mention the people, who were specially involved with my stay here in Zurich.

First of all I would like to thank Prof. Ueli Suter who convinced me to start my PhD inhis group. I had nearly complete freedom concerning my work and great support wheneverI asked for, which turned this period into a very interesting, fruitful and positive part ofmy life. My greatest teachers, who never gave up and are related to a really huge part ofthis work were Dr. Theo Tervoort and Dr. Jacco van Beek. Unfortunately both of themappeared only in the second half of my PhD-“career”, but the positive impact they had wassimply tremendous. Also I will not forget Prof. Beat Meier, who gave me the opportunityto do my NMR experiments in his lab and Dr. Marco Tomaselli, who had mercy with meand assisted me with the final part of my experiments.

Many thanks go to all my group members, which are Albrecht, Aurora, Brigitte, Hans-Ruedi, Hendrik, Ilya, Jorg, Lera, Maged, Martin, Pascal, Peter, Vikas and our lovelysecretaries Sylvia, Christina and Petra. Especially I would like to mention some peopleoutside of my group, which are Sylwia, Lisa, Ansgar, Andreas, Albrecht, Roland and lastbut not least Rainer, my roommate and the best friend I could imagine to have on my sideduring this PhD, and my dear parents, who are supporting me with love all throughoutmy life. Before it gets too wired I will stop here, but I am sure all people know why Imentioned them without going into detail :o)

Finally there remains only one thing: Ob wichtig oder nur ganz bescheiden, der wahreWert dieser Arbeit wird sich vielleicht nie zeigen, die grosste Entdeckung jedoch, die ichhier machte, sie ist mir zugeflogen wie ein Vogelchen, ganz sanft und sachte . . .

117

118

Curriculum Vitae

Personal Data

Name Michael Eric Wendlandt

Date of birth March 26, 1972

Place of birth Konstanz, Federal Republic of Germany

Nationality German

Education

1979 - 1982 Primary school in Allensbach, Germany

1982 - 1991 Comprehensive secondary school (allgemeinbilden-des Gymnasium) in Konstanz, Germany

1991 General qualification for university entry (allge-meine Hochschulreife, Abitur), awarded with thephysics-medal

1991 - 1992 Civilian service at the Malteser Hilfsdienst in Kon-stanz, Germany

1992 - 1993 Undergraduate studies of physics at the TechnischeUniversitat Munchen, Germany

1993 - 1998 Undergraduate studies of physics at the Universityof Konstanz, Germany

119

Curriculum Vitae

January 1998 - February 1999 Diploma thesis under the supervision of Prof. Dr.Jacob Klein and Prof. Dr. Gunter Schatz at theWeizmann Institute of Sciences in Rehovot, Israel

Februar 1999 Graduation as Diplom Physiker at the University ofKonstanz, Germany

June 1999 - April 2003 Doctoral Candidate, Departement of Materials Sci-ence, ETH Zurich. Advisors: Prof. Dr. Ulrich W.Suter, Dr. Theo Tervoort, Dr. Pierre Robyr.

Professional Experience

1995 - 1998 Professional coach for students in mathematics andphysics

2000 - 2002 Assistant in undergraduate courses in informatics

2000 - 2002 Webmaster in the group of Prof. Suter

Publications

Wendlandt, M.; Kerle, T.; Heuberger, M.; Klein, J. “Phase separation in thin films ofPolymer Blends: The Influence of Symmetric Boundary Conditions”, J. Polym. Sci. Pt.B-Polym. Phys. 2000, 38, 831-837.

Wendlandt, M.; Tervoort, T. A.; van Beek, J. D.; Meier B. H.; Suter, U. W. furtherpublications concerned with this PhD thesis in preparation

120

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