Materials &'ience and Engineering A 152 (1992) 89-94 89

An atomistic study of dislocations and their mobility in a model D022 alloy

M. Khantha, V. Vitek and D. R Pope Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6272 (USA)

Abstract

Since basic features of the plastic behavior of intermetallic compounds are often related to the structure of dislocation cores, we investigate here the cores of [1 i 0] and (1/2)[ 11")] screw dislocations in a model D022 alloy. First the stability of planar faults on {111} and {001} planes is analyzed and possible dislocation dissociations discussed. The only glissile dislocation with a planar core is the Shockley partial 1/61112.] bounding a superlattice intrinsic stacking fault; however, the other superpartial bounding the same fault, 1/3[ 112], is sessile. Under the applied stress an intrinsic-extrinsic stacking fault pair is generated at the latter partial and this configuration may then act as a nucleus for mechanical twinning. Such fault pairs have been observed in Ni3V. Another deformation mode in the model alloy is the slip of (110)-type dislocations but since these dislocations are sessile, this requires thermal activation. The major deformation modes of Ni3V and AI3Ti are, indeed, (112) twinning and, at high temperatures, (001){ 111 } slip.

1. Introduction

The mechanical behavior of non-cubic intermetallic compounds has been investigated in recent years as part of the on-going search for high temperature struc- tural materials since many of these compounds show attractive properties such as high strength-to-density ratio, excellent oxidation resistance and high melting points (see proceedings of the two most recent confer- ences on intermetallics [1, 2]). Examples are com- pounds crystallizing in the hexagonal 9019 structure (e.g. Ti3A1), tetragonal D022 structure (e.g. AI3Ti, Ni3V, AI3V ) and tetragonal L10 structure (e.g. TiAI). How- ever, application of these materials has been severely limited by the lack of ductility at ambient and high tem- peratures and the study of their plastic behavior is, therefore, of primary importance.

The plastic deformation of intermetallic compounds frequently exhibits features such as the strong tempera- ture and orientation dependences of the yield stress, complex slip geometry and anomalous positive tem- perature dependence of the yield stress. Such charac- teristics are all typical of the deformation processes controlled by the core structure of dislocations (for reviews see, for example, refs. 3-7). Hence, atomistic studies of dislocation cores and their behavior under the effect of applied stresses are needed to establish a microscopic understanding of the plastic properties of intermetallic compounds [3, 5, 8, 9]. A typical example is the present understanding of the core structure of screw dislocations in L12 compounds such as Ni3Ai

which led to a detailed model of the anomalous yield behavior in these alloys [10-12]. In this paper we present results of such studies for a model D0:2 alloy with the aim to analyze the potential of both plastic deformation and twinning in this structure.

The unit cell of the D022 lattice is a tetragonal super- lattice derived from the cubic L 1: superlattice by intro- ducing an antiphase boundary (APB) of the 1/21i10] type on every other (002) plane. (For simplicity, the Miller indices for crystallographic planes and direc- tions in the tetragonal superlattice are given here using the familiar f.c.c, notation with the third index referring to the c axis of the tetragonal cell. This also allows us to use the usual Thompson's tetrahedral notation for the Burgers vectors of the dislocations.)The deformation behavior of binary AI3Ti [13-16], Ni3V [17-21] and AI3V [22] (with small amounts of ternary additions) were investigated for temperatures ranging from 300 to approximately 1000 K. These studies indicate that there are two classes of D022 intermetallic compounds with distinct deformation behaviors at room tempera- ture. In one class, which includes AI3Ti and Ni3V, the deformation occurs principally by twinning on {111} planes along the (112) direction with the shear magni- tude equal to (1/(2)°5). This shear does not disrupt the symmetry of the lattice and results in an ordered twin. In the other class of compounds, such as A13V, the deformation is carried out by the motion of (110)-type dislocations on {111} planes. Moreover, additions of titanium to AI3V induce deformation twins in addition to the slip leading to improved ductility [22]. At high

0921-5093/92/$5.00 © 1992 - Elsevier Sequoia. All rights reserved

90 M. Khantha et al. / Dislocations in a model DO2_, alloy

temperatures (010){001} and possibly other slip systems are activated in both classes of compounds [t3, 16, 22].

Quantum mechanical total energy calculations based on the local density functional theory have been carried out to determine the relative stability of L12,

D022 and D023 s t ruc tures of several trialuminides [23]. For A13Ti these calculations were also used to evaluate the elastic constants and estimate the energies of APBs and other planar faults [24]. However, atomistic model- ing of dislocation core structures is presently beyond the scope of such calculations. Typically, atomistic simulation of extended defects such as dislocations need to be carried out for large blocks containing several thousands of atoms. The long-range elastic strain fields associated with dislocations provide the major contribution to the total energy and small energy differences between competing core configurations can be discerned only if the contributions from the long- range fields cancel out. Simpler descriptions of inter- atomic forces are therefore necessary to make such calculations computationally feasible. Indeed, very successful atomistic studies of dislocations, interfaces etc. were made in the past using empirically con- structed model potentials which do not represent specific materials but ensure the mechanical and struc- tural stability of a given crystal structure. Such poten- tials allow the atomistic studies to capture certain fundamental properties of crystal defects in given structures [3, 5, 8, 25]. This is the approach we adopted in this study.

Empirical N-body potentials of the Finnis-Sinclair type [26] have been developed to describe a model A3B alloy crystallizing in the D022 s t ruc ture [27]. A brief description of these potentials is presented in Section 2. Since the dislocation splitting is the most important core phenomenon, these potentials were first utilized to investigate the stability of planar faults on (001) and ( 111 ) planes employing the concept of V surfaces [3, 5, 8, 9, 28] (Section 3). On the basis of these calculations possible dislocation dissociations have been con- sidered and atomistic simulations of core structures of the corresponding partial dislocations carried out (Section 4). The effect of applied shear stresses upon the core configurations and related Peierls barriers have also been investigated (Section 5). Atomistic studies of the core structures have all been performed for screw dislocations since these are the most likely to

possess non-planar cores responsible for high Peierls stresses and the temperature and orientation depend- ences of the yield stress [3-5, 8]. Dislocation configura- tions found in this model agree very well with those observed in Ni3V [19, 29]. The possible manifestations of these core configurations in the mechanical behavior of D022 compounds are then discussed in Section 6.

2. lnteratomic forces and equilibrium properties of the structure

The N-body model potentials for an A3B alloy with the D022 s t ruc tu re have been constructed similarly as previously for alloys of noble metals and L1 E ordered alloys [30, 31]. In this approach, which can to some extent be justified on the basis of the tight-binding method [32], the energy of the i-th atom in a system of N atoms is

1L = - & ) -

E 2/=1 /=1

I/2 (1)

where the first term is the sum of pair interactions between the atoms and the second term is the N-body attractive part of the cohesive energy. Both Vv,h(Rij ) and ¢PT,~,(Rii ) are pair potentials depending on the separation Rii between the atoms i and j. The subscript Tj identifies species of the atom at the site j and it is either A or B. In accordance with the earlier work, potentials VAA, VBB, VA~, (I)AA and ¢PBB have been chosen as cubic splines, and the potential CPA~ has been taken as a geometrical mean of (I)AA and @~. The details of the potential parameters and the method of fitting have been described elsewhere [27].

The structural stability of the D022 phase was tested against other possible superlattice structures such as L12, D023, D019, D03 and A15 which can form at the same composition. The mechanical stability of the D022 phase was also tested for both small and large homogeneous deformations. The equilibrium lattice parameters and elastic constants of the tetragonal D022 phase calculated for this model potential are summar- ized in Table 1; the cohesive energy per atom is 4.18 eV. The elastic constants of this model alloy display similar trends as the elastic constants of Alf f i found by the total energy calculations [24], with the shear modulus c44 having the lowest value.

TABLE 1. The equilibrium lattice parameters (A) and elastic constants ( 1 (}~ Pa) of the model D022 alloy

a c c/a C ll C l 2 c13 c33 c44 c66

4.0926 8.6566 2.1149 1.718 1.095 0.8903 4.5652 0.6127 0.8159

M. Khantha et al. / Dislocations in a model DO:: alloy 91

TABLE 2. Energies of stable planar faults in the model D022 alloy

Plane Type of Displacement Energy fault vector (mJ m - -~ )

(oo I ) (Ill (111 (111 (111 ( i l l

APB APB, SISF l CSF SISF H APBll

1/2)[110] 71.62 1/2)[] 101 94.84 l/3)[i2il i35.00 1/6)[~l II 96.07 1/6)[112] 8.04 1/2)[i01] 64.70

• e c o o o c o o e o © e e -eQe-e -eOe-o o O o o

oeo • ooooo®o:®:oeo:o:oeo:ooooooo ©

. . . . . . . . . . . . . . . . = . . . . . . . . . . . • oOo • oOo • o O o o-e4

o g o o o Q o o o g o • o O o - o - o ~

• o o O o o o O o o o O o e - o ® o e o ® o ® e O e o l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : oOe•eOeoeOeoeOe-o.eOe,o.eOoooO

• • e g o • ® O ® • ® O e - o . o O o - o - o O e - o o O o • n,,

Fig. 2. The core structure of the C~5 partial bounding the SISF. fault on the ( 1 I 1) plane.

c -[1 To]

- t _ [1 0 ~ - I [0 1 1] [1 1 2] A B

Fig. 1. The Thompson tetrahedron notation for vectors in the ( 11 1 ) plane.

3. Planar faults on low index planes

The energy displacement or Y surfaces were calcu- lated for (001 ) and ( 111 ) planes. The positions of local minima on the 7 surface determine possible planar metastable stacking faults. In these calculations relaxa- tion perpendicular to the fault is allowed but no relax- ations parallel to the fault are permitted. More detailed results of these calculations have been given elsewhere [9, 27] and thus we only summarize here the most important findings. The stacking sequence of {111/ close-packed planes for an ideal c/a ratio of 2 has a six- layer repeat pattern of the type... A I B i C I A 2 B 2 C 2 A l . . . . The displacement vectors and energies of the meta- stable faults found are summarized in Table 2.

The lowest energy fault is the SISF. (superlattice intrinsic stacking fault) fault which will be bounded in the lattice by the Shockley partial with Burgers vector (1/6)[112] (C6 in the notation of the Thompson's tetrahedron, see Fig. 1). The motion of such partials on successive (111) planes produces an ordered twin the energy of which is 4 mJ m-2 in this model. As dis- cussed below, this low energy fault may play a signifi- cant role in the mechanical behavior of D022 alloys.

4. D i s loca t ion core s tructures

Two types of perfect superlattice dislocations with Burgers vectors of very similar lengths, [110] and (1/2)[11i] (2AB and 3C6 respectively in Thompson's tetrahedron notation) can exist on the (1 11) plane.

Based on the results of the y-surface calculations, several different dissociations of these superdisioca- tions are possible (CSF indicates a complex stacking fault).

3Cd = Cd + SISF, + 2Cd (2a)

2 A B = A B + A P B + A B (2b)

2 A B = A d + C S F + d B + A P B + A d +CSF+ fiB (2c)

For the case of screw dislocations the cores of the superpartials involved in these three types of splittings have been studied atomistically. These calculations have been made using the same methods as in a number of previous studies of dislocation cores [10, 27, 33]. Periodic boundary conditions were applied in the direction of the dislocation line and the block contained 7000 moving atoms. While only one of the superpartials was always present in the inner, relaxed, region, the other superpartial was assumed to be posi- tioned in the outer region at the appropriate distance determined by the fault energy and its effect was included via the elastic field. The calculated core struc- tures are presented here using the method of differen- tial displacements [3, 5]. The atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line (and the total Burgers vector). Small circles represent the A atoms and large circles the B atoms of the A3B alloy. Consecutive planes perpendicular to the total Burgers vector are always shown and are distinguished by differently shaded circles. The screw component of the relative displace- ment of the neighboring atoms produced by the dislo- cation is represented as an arrow between them. The directions of the arrows depict the signs of the displacements; their lengths are proportional to the magnitudes of the displacements and they are normal- ized with respect to the magnitude of the Burgers vector of the corresponding superpartial. Rows of arrows of constant length mark planar faults such as APBs.

Figures 2 and 3 show respectively the core con- figurations of (1/6)[112] (C6) and (1/3)[112] (2C6)

92 M. Khantha et al. / Dislocations in a model DOe: alloy

• o oC)o o o Q o o o ( i ) e o-eOo-®-e(~)o-®-oOoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,~ i ~ t , . . . . . . . .

e O e e . e O e - e . e O e . e . o O e o - e O e - o - o O o o o O . . . . . . . . . . . . . . . . . . . . . . . . . . . . • * , , o , % T ' ~ T ' ~ . . . . . . . . . . .

~ ~ o : : : o . e o : o - e e O e e e O

• e e o c e e G o ® e O o e - e ~ - o - e - e ® o ® o O e © . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , | , g , t . . . . . . . . . . . . .

= o O o e o O o e o O o e - o O o - e - ® O e - o - o O o e - ® O

'= • e e O e e e O ® . e e O e e e O e e e O e e e O e e

[+ + 0]

Fig. 3. The core structure of the 2C 6 partial on the ( l 11 ) plane.

configurations of the A6 and 6B partials involved in the dissociation (eqn. 2(c)). Both these cores are also distinctly non-planar, spread along the (112) direction in the ( 11 i ) plane.

We have also investigated the core structure of the 2AB superdislocation split into two AB superpartials separated by the corresponding APB on the (001) plane. The cores of the AB superpartials are non- planar, spread into two intersecting {111} planes. This

, core configuration has been presented elsewhere [27].

0, o .0. o-0- o-0-o-@- o.@. o-Q. o .@

O l O l ' O ' l - O ' l ' O ' t O t O l

,'o . . . . . . . . . . . . / " . t , . . . . . . . . . .

o @ o @ o . @ - o - @ - o . @ . o @ o •

ZI . o . o . o . o . o . o . o • o • o • o- O-o'+ e-o', e - o - • o • [I 12]

Fig. 4. The core structure of the AB superpartial bounding the APBin the(l 11) plane.

0 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 1 1 - 0 - 0 " 0 " 1 1

O o'. O'- o'-o'- o- o- o. e - o- e - o • e 0 l 0 @ • 0 - t - 0 - @ - 0 • @ - O 1 - ~) - @ - 4 B . @

o 0 o. 0 . . . . . . . . . . . . . . . . . . . . t , • . . . . .

O . l O . l . O . l . O - I - O - l - O + l - O . t . O

~l o o o o o o o o e o e - o e o O - O 0 O . O - O - O . O - O - O - O - O - O ' O

Fig. 5. The core structure of the 2AB superdislocations split according to eqn. (2c) in the (111) plane. Only the cores of the A6 and 6B partials separated by the CSF and a short ribbon of APB are visible in the picture.

superpartials involved in the dissociation (eqn. 2(a)). The projection is along the [112] direction and differ- ently shaded circles represent atoms from consecutive (111) planes. The core of the ( 1/6)[112] partial appears to be planar and spread in the plane of the fault; how- ever, the core of the ( 1/3)[ 112] partial is not planar and possesses a zonal character, i.e. it spreads into several parallel ( 111 ) planes.

Figure 4 shows the core structure of the (1/2)[1i0] (AB) superpartial involved in the splitting (eqn. 2(b)). The characteristic feature is the spreading of this core into the (11 ]) plane along the [i 12] direction resulting in a non-planar configuration. Figure 5 shows the core

5. Effect of applied shear stresses

In order to investigate the Peierls barriers associated with the dislocations studied, we carried out simulation of their movement under the effect of shear stresses applied in the directions of the Burgers vectors of the total screw dislocations in the planes of the dissocia- tion. These calculations were performed by imposing the elastic displacement field corresponding to a given stress, evaluated using the anisotropic elasticity, upon the corresponding blocks of atoms and finding in each case new equilibrium atomic positions. The calculation was always started with a small stress and larger stresses were then built up gradually until a critical shear stress was reached when the dislocation com- menced to move through the lattice. This stress is then identified as the Peierls stress which represents the lattice friction to the dislocation motion. (In the present calculations one of the superpartials is always held at a fixed position outside the relaxed block and thus the area of the associated fault increases when the super- partial present in the relaxed block moves. This implies that the minimum stress at which this superpartial can start moving is equal to 7v/bsp where 7r: is the energy of the corresponding fault and b~p is the magnitude of the Burgers vector of the superpartial studied.)

The glissile nature of the core of the C6 partial (Fig. 2) was confirmed by the stress application. The stress at which the dislocation starts to move was found to be less than 0.005/~, where/t is the shear modulus in the plane of the dissociation, i.e. it is comparable with 7F/bsp. When a shear stress is applied such that the force on the superpartial 2C6 points to the left in Fig. 3, practically no changes of the core take place until the stress reaches 0.025~. This is an appreciably higher stress than yF/bsp and thus the superpartial 2C6 is sessile. At this stress level a C 6 - 6 C faulted dipole, is nucleated one plane below the plane of the SISF n (F~'g. 6(a)). While the 6C partial remains stationary as the stress increases, the C6 partial moves away under the influence of the stress creating the configuration shown schematically in Fig. 6(b). Together with the SISFII, the faulted region represents a superlattice intrinsic-

M. Khantha et al. / Dislocations in a model DOe2 alloy 9 3

o o o O o e o O o o o O e e o O o e - o G o e o O o o

oOo.o-oOo.o-oOo-o-oOo.o-oOo-®-oOo.o-oO

. . :-.. •. • , ,T,, ~ _ ~ , ~ J . X 3 N . I X ~ N J ~ , , ' r . . . . . . . • • e ~ e e - o - G o - ~ e ~ u - u - u g i - ~ - o ~ ' ~ ® o t D o ©

'~ l o O o o-oO®o-®tD.®-o-oOoo-®0oooO® o oO • • o D e o-oOo-o-®Oo • o O o o,oO® • o4D-e •

p,

[1 ~ 0l

(a)

C 2 - - / A2 ~ / B 1 / C 2

O . O - O - O - O - O - O . Q 0 • 0 • 0 •

o' o' o' o'-• o • o • o • • o e - o . e - o . e - o ~ o - q o - ~ qo ~ . q o f t ~ - ~

o g o ' . " O 0 0 , 0 0 0 . O - O - O . Q 0 • 0 • 0

* ° ° ° * ° * " ° * ° " ° . . . . . . . . . . . . . . . . . . . . ° ° 0 . 0 . 0 O O - O - O - O - O • O O - O •

[~ ~

Fig. 7. The core of the AB superpartial, originally dissociated into A6 and 6B partials, under the influence of an applied stress.

B 2 ~ / C 1 / A 1 / ~ B 2

A2 - - / B 1 / C2 ~ / ~ A 2 C5 ~ /11111/~//I///1111/11//11/// 0 2C45

C 1 / A 1 ~ / ~ C 1 C 8 o I I I I I I I I I I I I I I I I o 6C

B 1 B1

A 1 A~

Intrinsic Extrinsic (b) fault fault

Fig. 6. The core configuration of the 2C6 superpartial under the influence of a shear stress. A C6-6C dipole has been nucleated one plane below the plane of the SISF~. Under the effect of the stress the C6 partial moves to the left leaving behind a faulted dipole. Together with the SISF., the faulted region represents an intrinsic-extrinsic stacking fault pair. (a) Calculated structure represented by the displacement map; (b) Schematic representa- tion of the configuration.

extrinsic fault pair [17]. It appears therefore, that under the effect of the applied stress the 3C6 superdisloca- tion splits into three C6 partials according to the reaction

3C6 = C6 + SESFII + C6 + SISF n + C6 (3)

and the C6 partials are separated by intrinsic-extrinsic fault pairs. If the energies of the SISF, and SISFII- SESF]t pair are not too different, the dissociation according to eqn. (3) is, of course, more favorable than that according to (2a) on the basis of the usual b 2 criterion.

When a shear stress was applied so as to move the configurations shown in Figs. 4 and 5 (eqns. 2(b) and 2(c) respectively) either to the left or to the right, no appreciable movement of the dislocations occurred until very high stress levels, usually in excess of 0.05/x, were attained. This demonstrates that the AB super- partial as well as the A6 and 6B partials are sessile and their motion is possible only with the help of thermal activation. Additionally, when a shear stress was applied so as to move the A6 and 6B partials in Fig. 5

to the left, the configuration shown in Fig. 7 resulted for stresses of the order of 0.02/~ and higher. In this, the Ad partial remained stationary while the 6B partial moved towards it annihilating the CSF on the (111) plane. Faulted dipoles are then formed on two adjacent (111) planes that contain a generalized stacking fault which is, presumably, stabilized by the applied stress. While the extent of these dipoles increases with increasing stress, the whole configuration is entirely immobile.

The Peierls stress of two AB superpartials separated by an APB on the (001) plane was also calculated. The AB superpartials possess non-planar cores and are very sessile moving only at stresses higher than 0.05/~.

6. Discussion

Screw dislocations with the Burgers vectors [1]0] and (1/2)[ 112] studied here have all been found to be sessile. Hence, their motion is either impossible or can only occur with the aid of thermal activations via a double-kink formation mechanism. Slip on the [110](001) system is, indeed, activated in Ni3V and AI3Ti at high temperatures and in this case the yield stress rapidly increases with decreasing temperature [13, 16]. The sessile nature of the cores of 1/2[l i0] (AB) superpartials, found in the present study, is a natural explanation for the strong temperature depen- dence of the yield stress for this slip system.

However, twinning is the major deformation mode in both Ni3V and AIsTi, and the present calculations, indeed, suggest that it is likely to be the most common deformation mode in DOn compounds in general. The glissile nature of the C6 partials suggests that the formation of an ordered twin by the propagation of C6 partials on successive (111) planes is likely. The 2C6 partial is however sessile. As shown in Fig. 6, the dis- location configuration described by eqn. (2(a)) trans- forms under the effect of the applied stress into that described by eqn. (3). An SISFI]-SESFll pair of faults on (111) planes is formed during this process. This

94 M. Khantha et al. / Dislocations in a modeI DO,_, alloy

configuration may serve as a nucleus for mechanical twinning which may proceed, for example, by forma- tion of faulted dipoles on successive (111 ) planes and their subsequent extension via the motion of C6 partials.

If the twin nucleus is formed as a result of external loading by transforming the 2C6 partial into two Cd partials, the stresses needed are rather high and local stress concentrations will be necessary for this to occur. However, the dislocation configuration described by eqn. (3) may occur spontaneously if the energies of the SISFII and SISFIrSESF~ pair are similar, and thus the twin nuclei may always be present in the D022 struc- ture. A systematic study of dislocation splittings in Ni~V was carried out by Escaig and co-workers [17-21] who correlated electron microscope observa- tions with various suggestions of possible dislocation dissociations. A configuration observed in both undeformed and deformed specimens of Ni3V at room temperature is the 3C6 screw superdisiocation dis- sociated into three C6 partial dislocations bounding an intrinsic-extrinsic stacking fault pair [19]. Recently such a configuration has also been observed by Francois et al. [29] in Ni3V deformed at 680 °C. This observed configuration is, of course, the same as that shown in Fig. 6 (and described by eqn. (3)) which was found by the atomistic modeling described in this paper. Its presence in the undeformed Ni3V suggests that in this material the nuclei needed for the onset of mechanical twinning are always present. Nonetheless, as seen in the previous section, the growth of twins via formation of faulted C 6 - 6 C dipoles requires a rather high stress and thus local stress concentrations will still be needed.

Acknowledgments

Numerous discussions with Dr. J. Cserti during the course of this work are gratefully acknowledged. This research was supported by the US Air Force Office of Scientific Research grant AFOSR-89-0062.

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