Energy of (110) twist boundaries in AgNi and its variation with induced strain

Acta mater. Vol. 44, No. 6, pp. 2309-2316, 1996

Pergamon

Copyright

0956-7151(95)00352-5

S. M. ALLAMEH, of Materials

S. A. DREGIA

15 March

IN Ag/Ni STRAIN

AND

and P. G. SHEWMON

Science and Engineering, The Ohio U.S.A.

(Received

Elsevier Science Ltd 1996 Acta Metallurgica Inc.

Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00

ENERGY OF (110) TWIST BOUNDARIES ITS VARIATION WITH INDUCED

Department

0

1995; in revised form

State University,

7 September

Columbus,

OH 43210,

1995)

Abstract-The

energy of (110) twist boundaries in the Ag/Ni system, as a function of twist angle, has been calculated using the embedded atom method (EAM). Two major cusps appear at 0” and 54.74” in addition to some less prominent cusps, including 35.26” and 70.53”. The variation of the boundary energy with coincidence strain has been studied. It is shown that calculations with small coincidence strains, up to about 2%, yield reasonable approximations to the energy of the strain-free interface. Perturbing the initial atomic positions in the two crystals significantly reduces the energy associated with some special boundaries, one of which is the 54.74” boundary. This is shown to be a consequence of a reconstruction of the boundary that is not accessible to the unperturbed crystals.

1. INTRODUCTION Most studies of solid-solid interfaces are focused on the relationship between interfacial structure and interfacial energy. Both computer simulations and experimental techniques have been used to study these relations. A number of theoretical models have been developed to predict low-energy boundaries, since these can be preferred in many transformations of multiphase microstructures. Geometrical models, such as the one based on the coincidence site lattice (CSL) [l, 21 and the lock-in model [3] are two examples. The CSL concept has been used to interpret the structure and energy of high-angle grain boundaries. According to this model, CSL boundaries with a high density of coincidence sites are expected to possess relatively low energy. The experimental observation of the presence of energy minima for interfaces where there is a high density of coincidence sites has been reported [&6]. Theoretical models that are based on purely geometrical factors are useful for rationalizing some observations, but they cannot explain/predict other orientation relationships (ORs) because they do not include a description of the energy of the boundaries. Analytical models have been developed to include both the geometry of a boundary and its energy [7,8], and these do show that geometrical coincidence is useful in lowering interfacial energy, but only to the extent that coincidence takes advantage of the periodic atomic interactions at the interface. Atomistic simulation of interfaces is one way to incorporate both geometry and detailed atomic interactions in predicting important interfacial properties.

The embedded-atom method (EAM) [9, lo] has been used to study the structure and energy of interphase boundaries in a number of metallic, f.c.c./f.c.c. systems [l l-161. These studies use a computational cell in the shape of a slab to represent a bicrystal of finite thickness normal to a planar interface. In order to apply periodic boundary conditions, the length and width of the computational cell must correspond to translations in both lattices meeting at the interface. Consequently for interphase boundaries, the two crystals are forced (strained) so that their lattices match/coincide at the edges of the computational cell. The effects of initial and boundary conditions on the calculated structure and energy of interfaces were examined only in a few cases. To illustrate some of these effects, the results of some of the previous EAM simulations are cited here. Early simulations ccqsidered f.c.c./f.c.c. interfaces with a parallel orientation relationship (OR) [ll, 121. Other ORs simulated by the EAM include the study of structure and energy of (100) twist boundaries in the Ag/Ni system by Gao et al. [13]. They simulated (forced) coincidence twist interphase boundaries with twist angles in the range O-45”. The simulations predicted minima in the energy-misorientation curve of which the lowest was at 26.56” (CAg/XNi = 4/5). Experiments on the same system found cusps at 26.56” and 4.4’ twist. Consistent with experimental observations, EAM simulations also predicted an especially low energy for the (111)[ 1iO]Ag (I(100) [ 1iO]Ni interface [ 141. Calculations with the EAM have shown that boundary reconstruction can occur when the initial conditions of the bicrystal are changed. Gumbsch et al. [15] studied the accommodation of lattice

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mismatch in the (1 lO)Ag/(llO)Ni interface in a parallel OR. They found that introduction of vacancies on the Ni side lowers the interfacial energy. The removal of a complete row of Ni atoms along [1TO]also lowers the interfacial energy significantly. According to these authors, the minimum interfacial energy corresponds to an interface configuration consisting of a Ni atomic layer with 16% reduction in its planar atomic density, separating ordinary Ni and Ag crystals. Based on this result they concluded that the accommodation of lattice mismatch may lead to interface reconstruction. In the calculations cited above, the initial conditions corresponded to two crystals, stacked one on top of the other, with the atoms in their natural positions. Periodic boundary conditions were imposed by forced coincidence, and the conjugategradient method was used for relaxing the structure and finding the energy minima. Apart from the introduction of vacancies, no perturbation in the initial crystal structure was reported. In most cases the manner of stacking the two crystals, i.e. the initial translational state of the interface, was not specified. Since the final results of calculations are dependent on initial conditions, it is useful to vary the initial conditions. Also, because configurational changes that raise the energy of the system are not allowed in the gradient-based relaxation methods, the system can get stuck in a local minimum of a metastable configuration. To overcome this problem, one may use simulated annealing as a means of exploring more of the phase space to find global energy minima. The method used in this study is more informative than the traditional conjugate gradient method but less time consuming than simulated annealing. The main goals of this study were to calculate the (110) twist boundary energies in the Ag/Ni system; to examine the effect of initial conditions on boundary energy and structure; to develop a method for calculating the energy of irrational. non-coincidence interfaces; and to examine the reconstruction of an internal interface. The initial conditions are varied either by changing the stacking sequence of the two crystals. i.e. by a rigid-body relative displacement of the two crystals, or by randomly perturbing the initial atomic positions in the two crystals. The effect of boundary conditions is studied by determining the variation of interfacial energy as a function of forcedcoincidence strain. A comparison between the results of this study and the experimentally observed ORs in the (1 lO)Ag/( 110)Ni twist boundaries was presented elsewhere [ 171. 2. LATTICE

GEOMETRY OF (110) TWIST BOUNDARIES

To simulate a bicrystal. a computational cell is constructed by joining two crystals along the interface plane. Application of periodic boundary conditions in (_Yand y) directions parallel to the interface

OF TWIST BOUNDARIES

requires that the lattices of the two crystals come into coincidence at the corners of the cell. For an interphase interface, exact coincidence is not generally possible for a given OR, but the size of the cell can be chosen so that near coincidence is achieved. With strain applied to one or both crystals, a forced-coincidence cell can then be produced and the periodic boundary conditions imposed. The orientation of the two crystals relative to each other and the twist angle are determined from the CSL model. The stacking sequence of the two crystals across the interface is important and can lead to very different interfacial energies. One obvious choice is to stack the (1 IO)-terminated Ni crystal on the (1 lO)terminated Ag crystal such that coincident Ni atoms are located in the continuation of the Ag lattice. This is designated here by (Ni/Ag), or nickel on silver. The alternate way is to stack Ag on Ni such that coincident Ag atoms are located in the continuation of the Ni lattice (Ag/Ni). The two configurations are related by a rigid-body translation parallel to the interface plane. Both of these configurations were used. since there is no way to determine (1priori which of the two configurations has a lower energy. The gap between the two slabs was chosen as the average value of interlayer spacings in the two crystals. for atomic layers parallel to the interface. The procedure used to generate (110) twist boundaries in this study is similar to the one used by Gao [18] for (001) twist boundaries. First. one seeks two rectangular cells, one from each lattice, of which the sides are lattice vectors lying in the interface plane. For a (li0) interface, these lattice vectors can be expressed as follows: R~,=K~[llO]+La,,[001]=~[KK2L] & R,, = My

[l lo] + Nu,,[OOl] = 3MMZN]

and their lengths

are given by I?

RA,

=

aAg

RN,= aN2 The misfit between

the two vectors,

F. is defined as

F=21R~i-R,,I R,, +

RA,

The other edges of the 2-D cells. perpendicular first two, are given by

to the

S,, = aNL[IVNM] s,,

= a& [LLK]

and the misfit for these edges has the same value as for the first two. If the two crystals. initially in a

ALLAMEH et al.:

ENERGY OF TWIST BOUNDARIES

parallel OR, are twisted to align the 2-D cells, then the required twist angle is given by 0 = arctan

f arctan(

Further, the two cells can be brought by homogeneous biaxial strains so that have identical lengths. The result is forced-coincidence cell characterized ratio of 4, a misfit F, and containing lattice points, C, given by

in coincidence parallel edges a rectangular by an aspect a number of

C,, = 2L2 + K2 C,, = 2N2 + M2. The reciprocal of C is the fraction of 2-D lattice points that are coincidence points; it measures the density of coincidence points in the interface. The algorithm for generating a twist interface is reduced to finding values of the integers K, L, M and N, subject to the constraint that F and C must not exceed predetermined limiting values. Usually, for a given twist angle, a small coincidence cell (small C) is associated with a large value of misfit. To force coincidence under large misfit, a large strain is introduced in the crystals and that would alter the interfacial energy. On the other hand, to achieve very small misfits, the size of the coincidence cell and Z must be made large, which is computationally cumbersome. A compromise has to be made between a small X and a small misfit. The criterion for limiting the misfit is discussed later. With the choice of a maximum allowable misfit and maximum value of C, the main task is reduced to a systematic search for integers and subsequently calculating the corresponding twist angles. In this manner, the energymisorientation curve can be determined with a high density of discrete points. The thickness of each crystal is chosen to be large enough so that the presence of the free surface and the interface does not affect the energy in its bulk. A thickness corresponding to 32 atomic layers was chosen based on experience; the atomic distortions due to the presence of either the interface or the free surface are of the order of 67 atomic layers in each crystal. For the calculation of the energies of the twist boundaries, both crystals are deformed so that the forced-coincidence cell size has an intermediate value between the initial sizes in the undeformed crystals. In some previous simulations, this intermediate size was chosen by minimizing the work of deforming the two crystalline slabs [12-14, 181. A different approach used in this study is to put all of the strain in one crystal by deforming it to match the other. This approach is useful for determining the variation of calculated interfacial energy with the extent of coincidence strain. Since for the same twist angle the coincidence strain can be varied about zero by varying C, it is possible to use interpolation to obtain the

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energy of the non-coincidence, strain-free interface. This calculation is also useful for determining a limiting value of misfit. In addition to (110) twist boundaries in Ag/Ni, two (110) tilt boundaries were simulated. Here, the ORs are characterized by rotations of 35.26” and 54.74’ about a [ 1 lo] axis common to both crystals, but this axis lies in the interface plane. For these tilt boundaries the interface was parallel to (1 lo),,, and hence also parallel to (11 l),, for 35.26” and (112),, for 54.74”. Tilt boundaries with these ORs have been reported as low-energy interfaces in Ag/Ni in particle rotation experiments [ 191. For the 54.74” tilt boundary the coincidence cell directions are [222]Ag //[400]Ni, with F = 0.62%, and [06s]Ag 11 [07?]Ni, with F = 0.41%. Since, as discussed below, the misfit in both directions is small, it does not matter which crystal is deformed to attain exact coincidence. For the present calculation, all of the coincidence strain is in the Ni crystal. 3. COMPUTATIONAL

PROCEDURE

Atomistic simulations were performed with the embedded-atom method [9], using previously published functions for the interactions of Ag and Ni atoms [lo]. The EAM functions were used to compute the energies and forces associated with atoms in the two crystals, and the total energy is minimized by the conjugate-gradient method [20]. This was an iterative procedure and continued until the force on each atom was reduced to less than 10-4eV/A. The interfacial energy was obtained from the individual atomic energies by summing up the excess energies of atoms near the interface, relative to the energies of bulk atoms. The calculations used bicrystalline slabs joined on a rectangular interface in four different initial configurations: perfect bicrystals in Ag/Ni stacking. perfect bicrystals in Ni/Ag stacking, and two corre-, sponding “jiggled” configurations obtained by randomly perturbing the atomic positions. Atoms in the jiggled bicrystals were displaced in all directions by random displacements not exceeding a predetermined amplitude. This is a reasonable surrogate for the effect of thermal vibration on boundary structure and the resulting configuration should better approximate the equilibrium structure of the boundary. While foi most boundaries this did not change the energy minimum, for boundaries such as 54.74” twist about [l lo], it had a large effect as it nudged the atoms to a lower energy configuration. To examine the variation of interfacial energy with the maximum amplitude of jiggling, the 54.74” twist bicrystal was jiggled with different amplitudes of 0.001, 0.01, 0.1 and 1 A. The amplitude that led to the lowest of the four interfacial energies was chosen as the upper limit for jiggling all other boundaries. The magnitude of the coincidence strain affects interfacial energy, but it is a function of the size of the

2312

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ENERGY

8

l:Q

O’Q

O’Q

O’Q

OF TWIST

ing, we get X49/X64 and the misfit is 3.2%. Therefore, by allowing I: to vary for the 0” interface, it was possible to assess the variation of interfacial energy with coincidence strain.

0’

2-obobobob . o-2-Q

0

It-0 -10

4. RESULTS

-6

O’Q

Q

I ab 1 . 01 -8

-4

O’Q

0’

Figure 1 shows the two structures associated with Ag/Ni and Ni/Ag stacking configurations for 54.74” twist. The two structures are related by a translation vector parallel to the boundary. The calculated interfacial energies of the twist boundaries are listed in Table 1 based on four initial configurations for each boundary. Figure 2 is a plot of interfacial energy versus twist angle, based on the smallest of the four energy values for each boundary. For these calculations, the coincidence strain was partitioned between the two crystals so as to minimize the total elastic strain energy. The major cusps of energy are located at 0” and 54.74”. There are minor cusps at 35.26’ and 70.53” which are not as deep as the other two. The energies of the two tilt boundaries simulated are listed in Table 2. The free surface energies of the two crystals in these simulations are also listed to show consistency with previous calculations [13]. The variation of interfacial energy with coincidence strain for the parallel OR (0 = 0) is shown in Fig. 3. The energies were calculated with the coincidence strain partitioned either entirely in Ni or entirely in Ag. The curves fit to the two data sets intersect at zero strain, and thus give the same interpolated value for the interfacial energy of the strain-free interface. A parabola has been fitted to the calculated data points for each of the two cases. This figure shows that it is possible to obtain the energy of an incommensurate interface that is not accessible to direct simulation. It also shows that forced-coincidence calculations give

Ib * I 0. n ab I * 01 r 1b * 1 -2

0

2

4

6

I3

IO

00000000

. 0 ?? o. 0 ?? o. 0 ‘0. 0 b 00000000

. ?? o. .o. .o* .o 0

0

0

0

(b) Fig. 1. Projections of atomic positions to show the structure of the 54.74” twist boundary in two stacking configurations:

(a) Ag/Ni and (b) Ni/Ag. The solid circles represent Ag atoms while the open circles represent Ni atoms. The numbers

on the two axes are in Angstroms.

coincidence cell. As an example, for (11O)Ag //(110)Ni with 0” twist, we may match 6 lattice periods of Ag with 7 lattice periods of Ni in both [OOl] and [liO] directions. This results in X36/X49 and a misfit of 0.41%. On the other hand, if we choose 7: 8 match-

Table

BOUNDARIES

1.The

calculated

0 (‘)

Misfit

0 10.02 11.45 19.41 23.52 29.50 35.26 38.64 38.94 44.71 54.74 58.78 59.71 60.50 66.82 64.76 69.41 70.53 72.68 76.74 79.98 84.79 90.00

0.0041 0.0012 0.0148 0.0142 0.0167 0.0113 0.0163 0.0062 0.0042 0.0091 0.0062 0.0037 0.0177 0.0012 0.0136 0.0091 0.0147 0.0041 0.0081 0.0129 0.01 I3 0.0030 0.0142

energies of (110) Ag/Ni

ZAg/ZNi 36149 49166 64189 IS/25 48167 25/33 75/98 51/86 108/147 24133 314 50167 33143 49166 51167 8111 41/57 36/49 43/59 19125 25133 38/51 18125

twist boundaries

Energy Ni/Ag unjiggled

Energy Ag/Ni unjiggied

(mJ/m’)

(mJ/m?

881 1000 1110 1151 1201 1230 1204 1254 1302 1235 1270 1295 1290 1253 1330 1103 1154 1194 1177 1109 1299 1206 1241

869 1010 1084 1127 1173 1242 1295 1305 1226 1267 1270 1222 1213 1234 1166 1103 1148 1161 1150 1100 1299 1272 1242

with different

initial

Energy Ni/Ag jiggled (mJ/m’) 881 999 1110 1151 I201 1230 1222 1254 1302 1235 1270 1295 1247 1253 1182 1103 1156 1155 1177 1108 1299 1206 1242

configurations Energy AgiNi jiggled

(mJ/m2 ) 869 1010 1084 1127 1173 1243 1295 1305 1224 1267 1043 1203 1254 1213 1148 1148 1183 1129 II50 1140 I299 1164 1241

ALLAMEH et al.:

ENERGY OF TWIST BOUNDARIES

Fig. 2. Calculated energies of (110)twist boundaries vs twist angle. The energies are chosen from Table 1 as the smallest of the four values presented for each boundary. a good approximation of the interfacial energy as long as the coincidence strain is below 2%. Under these conditions, the manner of partitioning the coincidence strain is also insignificant. Based on this result, we considered only those twist boundaries with a misfit below 2%. For the 0” twist boundary, the effect of 2% coincidence strain is an error of about 10%. Because the energies in Fig. 2 are for boundaries with both Ag and Ni crystals strained, the systematic error in these values is estimated to be about 5%. The effect of jiggling amplitude on the interfacial energy is shown in Fig. 5. The interfacial energy does not change with small jiggling amplitudes of up to 0.001 A. However, it decreases as the amplitude is increased to 0.01 A. Further increases of the amplitude raise the interfacial energy such that it becomes even higher than for the unjiggled boundary. Based on these results, all boundaries were jiggled with an amplitude of 0.01 A. 5. DISCUSSION The experimental verification of the predictions was performed by crystallite rotation technique and is described in detail elsewhere [17]. Small Ag crystallites were sintered with their (110) face on the (110) face of a monocrystalline Ni thin film at a predetermined twist orientation and the ensemble was held at 350°C. The results of the experiments showed three energy cusps at O”, 35.26”, 54.14” and possibly one at 70.53”. This finding is consistent with the calculations and lends credence to the EAM predictions. The energies of the twist boundaries were calculated for four different configurations as reported in Table 1. For some boundaries, such as O”, 79.98” and 90”, the calculated energy does not depend on which Table

2. The calculated

Strain Fig. 3. Variation of calculated energy with coincidence strain for the (110) boundary with 0’ twist. of the four initial configurations is used. On the other hand, some of the other boundaries yield different energies for different configurations. The energy of the boundary with 8 = 66.82’, despite the large C values of X51/X67, is greatly dependent on the initial configuration of the boundary, and a difference of 20% is seen between the energies of the two stacking configurations. For the 0 = 54.74” boundary, with X3/X4, the energies obtained for both stacking configurations (Ni/Ag and Ag/Ni) are the same. However, the dependence of the energy on the initial configuration becomes evident when the atomic positions are jiggled (displaced) by random amounts not exceeding 0.01 A. The energy associated with this boundary becomes significantly smaller (about 18% less) for the jiggled boundary and a cusp in the energy curve appears at this twist angle. A close examination of this boundary reveals a close link between its structure and its energy. Upon relaxation of the jiggled boundary, some of the Ni and Ag atoms “interdiffuse” to produce a compositional ordering with alternating layers of Ni and Ag. This might be the reason for the low energy associated with this boundary. Figure 4 shows the atomic structures for the two boundaries, where the intermixing of Ag and Ni atoms is shown in the cross-sectional views. The connection between energy and structure is further revealed in Fig. 5. When the jiggling amplitude is 0.001 A, the boundary relaxes to the same structure as does the unjiggled boundary, and the interfacial energy remains the same. When the amplitude restricted is 0.01 A, a major change takes place in the atomic positions and the energy is reduced. The translational symmetry of the interface is also restored to that of the coincidence cell before jiggling. As the amplitude increases to 0.05 8, and then to 0.1 A, the translational symmetry is broken and the

enereies of two tilt boundaries

in the Ag/Ni

svstem

Surface energy Parallel

planes

Parallel

directions

(211)Ag~I(Oll)Ni

[lfi]Ag)/[lOO]Ni

(IIl)Ag#(Oll)Ni

[2fi]Ag

11 [lOO]Ni

2313

Tilt angle 54.74 35.26

Interfacial energy

@J/m’) For For For For

Ag Ni Ag Ni

= = = =

730 1720 620 1720

(mJ/m*) 930 990

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ENERGY

40

’

0

Ag

NI

39

38

‘lo

??

Ag

0

Ni

39

Fig. 4. The relaxed structures of the 54.74 boundary are shown for (a) unjiggled and (b) jiggled initial bicrystals. The structure of the boundary is presented with top view (top right) and side views of the bicrystal to show the atomic positions. The intermixing is observed in the jiggled boundary in the side views.

interfacial energy increases. The effect totally vanishes when the amplitude is allowed to be as large as about 1 A. The boundary structures presented in Figs 5(c)
OF TWIST

BOUNDARIES

the rise in the energy of the bicrystal is 4.84 x 10-j eV/atom. Compared to kT. this corresponds to a temperature of 58. K. The geometry of the (110) boundary requires that the two orientations of 0” and 70.53 twist have the same misfit in both [I 1I] and [I 121 directions of the two crystals. This is seen in Table 1. Although the coincidence site density of both orientations is the same, they yield quite different energies. The experimental evidence for the difference in the two boundary energies has been observed in twinning experiments [21] in which the (110) AgiNi boundary with a 70.53” twist is replaced by a (1 IO) boundary with 0’ twist through the nucleation and growth of twins. If the coincidence site density were the main factor determining the interfacial energy, the energies of the two orientations would be about the same. The difference may be explained by the fact that in the parallel OR. the close-packed directions of the two crystals in the boundary are parallel while this is not so for the twin orientation. The calculated energy of the twin boundary (70.53 twist) is about the same as some of its neighboring ORs. The rotation of particles towards 70.53 (from 49’. 58 and 65’), observed experimentally [17] can be explained by the effect of twins in the silver crystals. There are two sets of twins in the Ag particles that have { I I I } boundaries perpendicular to the interface. In the Ag crystallites oriented at 70.53”, one set of twins is oriented at 0’ twist. The interfacial energy associated with these twins (at 0 twist) is much smaller than the interfacial energy associated with the rest of the particle (at 70.53 twist), as shown in Table I. If the Ag particles are not at 70.53’ twist, but a few degrees off. the twins will be a few degrees off 0 During annealing. the twins will rotate towards zero if the particles rotate towards 70.53 Exceptions to purely geometrical criteria for determining low-energy boundaries are found by comparing the energies of the 0” boundary with those of the 54.74’ and the 70.53’ boundaries. Despite its much larger X value, the 0’ boundary has a lower energy than the 54.74L boundary. Also, despite the identical values of C, the 0” boundary has a significantly lower energy than does the 70.53” boundary. This shows the shortcomings of the purely geometrical models: a higher coincidence density (smaller C) does not ensure a lower interfacial energy. In 0” twist. the close-packed atomic rows of Ag and Ni are parallel while this is not true for the 70.53’ twist. Although the density of the lock-in rows is not high, the lower energy associated with the 0’ twist lends credibility to the lock-in model. 6. CONCLUSIONS

Computer simulations with the embedded atom method were performed to calculate the energy of the (110) twist boundaries in the Ag/Ni system. The

ALLAMEH

et al.:

ENERGY

OF TWIST

BOUNDARIES

0

. .

$0 0

0.0

.

??

0

??

1

0.0

0.0

0’0 0’0 0’0 0’ . . .

0.0 0

I

00000’

??

0

2315

0.0

0.0

0.0

‘ooo*oo

0

0’0 0’0 0’0 0’

0

’

-I

0.0

0

. 0 BOO

0

.

. (4

0

??

0.0

0’0

0 ??

.

0

0 0

000

0.0 . (4 L

Jiggling Amplitude (A)

0

??

??

0

0 .

0.0 .

0

.

0

00000000

0 (e)

‘0°0’0’

I* O .

0.0.0)

Fig. 5. Changes in the relaxed structure of the 54.74” twist boundary due to jiggling the initial atom positions. (a) The relaxed structure of the unjiggled boundary. (b) A plot of calculated boundary energy versus jig ling amplitude. (c)
show the variation of the results of the calculations Ag/Ni interfacial energies with twist angle and display two major cusps at 0” and 54.74’, in addition to minor cusps at 35.26” and 70.53” twist. It was found that the presence of large coincidence strains (2 2%) in the bicrystal yields large deviation from the energy of the strain-free boundary. The locations of predicted minima on the energy-0 curve are in good agreement with the results of crystallite rotation experiments. The results of this study show that the variation of energy with the initial conditions can be significant. Therefore, to find the energy minima, it is useful and not cumbersome to vary the initial conditions. Small random displacements of the initial atomic positions in the bicrystal do not change the interfacial energy of most boundaries; however, for some special boundaries, such as 54.74” twist, the energy is greatly lowered by such perturbation of the initial conditions. We expect this effect to be observed at other boundaries with small C. In those cases, the inter-

facial energy is basically made up of the sum of a few numbers. A large change, even in one or several atomic energies, can produce a significant change in the overall interfacial energy. Boundary reconstruction and atom displacements of the order of 1 A scale were obtained by jiggling atoms of the order of 0.01 A. Acknowledgements-This study was sponsored by the National Science Foundation under grant DMR-8915143. We also acknowledge the contributions of the Ohio Supercomputer Center under grant PAS-675.

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OF TWIST

BOUNDARIES

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