kink model for extended defect migration: theory and simulations

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Acta mater. 48 (2000) 3711–3717 www.elsevier.com/locate/actamat KINK MODEL FOR EXTENDED DEFECT MIGRATION: THEORY AND SIMULATIONS M. I. MENDELEV and D. J. SROLOVITZ* Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA ( Received 13 March 2000; received in revised form 22 May 2000; accepted 28 May 2000 ) Abstract—The motion of extended defects in solids is the key to understanding many important physical phenomena. This paper addresses the relationship between defect velocity and driving force. Monte Carlo simulations of linear defects in two dimensions are performed. The force–velocity relationship is found to be non-linear, in disagreement with commonly used models. A new analytical model for the force–velocity relationship is derived that includes the effects of kinks on multiple levels, the non-simultaneity of kink formation, the disappearance of kinks and the non-linear kink force–velocity relationship. The resulting force– velocity relationship is relatively simple, but non-linear. This model is shown to yield excellent agreement with the simulation results over a wide range of driving forces and temperatures, and is readily extendable to a wide variety of defects in two and three dimensions. 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Interface; Mobility; Defects; Kinetics; Theory & modeling; Grain growth 1. INTRODUCTION The fundamental mechanisms by which extended defects move in solids have been of widespread inter- est since the inception of the science of materials as a discipline. While the detailed atomistic mechanisms associated with this type of motion are generally unknown (see, e.g., [1]), certain aspects of the motion do not depend on an atomistic mechanism. These include a general rate theory description of migration and the description of motion in terms of nucleation and propagation of kinks on the extended defect. For a description of the rate theory approach, see [1–4]. It is widely accepted that extended defects such as dislocations and grain boundaries move by the forma- tion of kinks (islands in three dimensions) and migration of kinks [1, 3]. The rates of double kink/island nucleation and how quickly they move depend on such factors as the sharpness of the kinks, the kink heights, the number of atoms involved in the unit step of kink motion, etc. Nonetheless, many features of the intrinsic motion of such extended defects can be determined by investigation based on a generic model that incorporates kink nucleation and motion. The goal of the present work is to determine how the velocity of an extended defect depends on driving force and temperature. We can apply these * To whom all correspondence should be addressed. 1359-6454/00/$20.00 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII:S1359-6454(00)00166-X results generically to such cases as dislocation slip or the motion of grain boundaries, ferroelectric and ferromagnetic domain walls in films or thin slabs. While boundary/domain wall motion in three dimen- sions likely has features that cannot be incorporated in two dimensions, the fundamental aspects of island nucleation and growth are captured. A schematic illustration of the motion of a linear defect via kink nucleation and propagation is shown in Fig. 1. First, kinks are thermally nucleated on the defect in pairs [two kink pairs or islands are shown in Fig. 1(b)]. The individual kinks move up and down the boundary, extending or shrinking islands [Fig. 1(c)] until a pair of kinks of opposite sign meet [Fig. 1(d)]. When the islands produced in this way merge and fill an entire layer, the boundary is again flat [Fig. 1(a)] but has moved to the left by one unit. In reality, the nucleation, motion and merging of kinks go on simultaneously and a multi-level boundary structure such as that shown in Fig. 1(e) is more typical. There have been several analyses of boundary motion based upon this type of picture (see, for example [1, 4, 5]) of the following form. In the absence of a force on the boundary, the equilibrium concentration of islands is c = N L = 1 a exp(2E k /2kT), (1) where N is the number of islands, L is the length of

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Page 1: Kink model for extended defect migration: theory and simulations

Acta mater. 48 (2000) 3711–3717www.elsevier.com/locate/actamat

KINK MODEL FOR EXTENDED DEFECT MIGRATION: THEORYAND SIMULATIONS

M. I. MENDELEV and D. J. SROLOVITZ*Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton

University, Princeton, NJ 08544, USA

( Received 13 March 2000; received in revised form 22 May 2000; accepted 28 May 2000 )

Abstract—The motion of extended defects in solids is the key to understanding many important physicalphenomena. This paper addresses the relationship between defect velocity and driving force. Monte Carlosimulations of linear defects in two dimensions are performed. The force–velocity relationship is found tobe non-linear, in disagreement with commonly used models. A new analytical model for the force–velocityrelationship is derived that includes the effects of kinks on multiple levels, the non-simultaneity of kinkformation, the disappearance of kinks and the non-linear kink force–velocity relationship. The resulting force–velocity relationship is relatively simple, but non-linear. This model is shown to yield excellent agreementwith the simulation results over a wide range of driving forces and temperatures, and is readily extendableto a wide variety of defects in two and three dimensions. 2000 Acta Metallurgica Inc. Published byElsevier Science Ltd. All rights reserved.

Keywords:Interface; Mobility; Defects; Kinetics; Theory & modeling; Grain growth

1. INTRODUCTION

The fundamental mechanisms by which extendeddefects move in solids have been of widespread inter-est since the inception of the science of materials asa discipline. While the detailed atomistic mechanismsassociated with this type of motion are generallyunknown (see, e.g., [1]), certain aspects of the motiondo not depend on an atomistic mechanism. Theseinclude a general rate theory description of migrationand the description of motion in terms of nucleationand propagation of kinks on the extended defect. Fora description of the rate theory approach, see [1–4].It is widely accepted that extended defects such asdislocations and grain boundaries move by the forma-tion of kinks (islands in three dimensions) andmigration of kinks [1, 3]. The rates of doublekink/island nucleation and how quickly they movedepend on such factors as the sharpness of the kinks,the kink heights, the number of atoms involved inthe unit step of kink motion, etc. Nonetheless, manyfeatures of the intrinsic motion of such extendeddefects can be determined by investigation based ona generic model that incorporates kink nucleation andmotion. The goal of the present work is to determinehow the velocity of an extended defect depends ondriving force and temperature. We can apply these

* To whom all correspondence should be addressed.

1359-6454/00/$20.00 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S1359-6454(00 )00166-X

results generically to such cases as dislocation slipor the motion of grain boundaries, ferroelectric andferromagnetic domain walls in films or thin slabs.While boundary/domain wall motion in three dimen-sions likely has features that cannot be incorporatedin two dimensions, the fundamental aspects of islandnucleation and growth are captured.

A schematic illustration of the motion of a lineardefect via kink nucleation and propagation is shownin Fig. 1. First, kinks are thermally nucleated on thedefect in pairs [two kink pairs or islands are shownin Fig. 1(b)]. The individual kinks move up and downthe boundary, extending or shrinking islands [Fig.1(c)] until a pair of kinks of opposite sign meet [Fig.1(d)]. When the islands produced in this way mergeand fill an entire layer, the boundary is again flat [Fig.1(a)] but has moved to the left by one unit. In reality,the nucleation, motion and merging of kinks go onsimultaneously and a multi-level boundary structuresuch as that shown in Fig. 1(e) is more typical.

There have been several analyses of boundarymotion based upon this type of picture (see, forexample [1, 4, 5]) of the following form. In theabsence of a force on the boundary, the equilibriumconcentration of islands is

c =NL

=1a

exp(2Ek/2kT), (1)

whereN is the number of islands,L is the length of

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3712 MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

Fig. 1. Schematic illustration of boundary motion by kink nucleation and propagation, showing a flat boundary(a), one with several isolated islands (b) that grow (c) and merge (d). (e) shows a realistic boundary consisting

of islands/kinks on multiple layers.

the boundary,a is the distance between neighboringsites on the line andEk is the island (double kink)formation energy. The rate at which a kink (islandedge), once formed, moves in they-direction (Fig. 1)in response to a driving forceF is

vy =DkT

F, (2)

whereD is the one-dimensional kink diffusion coef-ficient. The boundary migrates to the left (in Fig. 1)by one unit (i.e., one kink height) when the averageisland grows to a size equal to the inverse island con-centration 1/c ( = L/N). The time required for this is(2cvy)21. Therefore, the net boundary velocity is

vx =h

1/(2cvy)= 2hcvy, (3)

whereh is the step kink or island height. If we equatethe kink height with the density of sites along thesurface, we obtain

vx =2DkT

exp(2Ek/2kT)F. (4)

In this model, the boundary velocity is directly pro-portional to the driving forceF.

Equation (4) is valid only if the driving force issmall. If the force is not small, the island concen-tration is not accurately predicted by equation (1). Ifa force is present, the rate at which islands form isnearly unchanged (provided that the island nucleationenergy is large compared with the work done by theforce in forming the island). On the other hand, therate at which islands disappear is strongly decreased.Given a nearly unchanged island formation rate and

a significantly reduced rate of disappearance when aforce is applied, the equilibrium island concentrationis significantly greater than predicted by equation (1).Consideration of equation (3) also suggests anotherproblem. Equation (3) is exact provided that allislands form simultaneously and are uniformly placedalong the boundary such that all islands impinge atthe same time. This is not the case here, where islandsnucleate in random positions at stochastically determ-ined times. The problem is particularly severe at lowtemperatures (kT¿Ek) and large driving forces forwhich the rate of forming new islands is very smalland the island growth rate is large. These observationssuggest that the standard double kink nucleation andpropagation model may not be valid in general. Infact, as shown below, the standard model [equation(4)] is inconsistent with simulation results for thedependence of boundary velocity on applied force.

In this paper, we present simulation results for themotion of a boundary within the framework of a verysimple model, i.e., the Ising model. We demonstratethat at large forces, the standard model fails. Next, wepresent a new derivation of the force and temperaturedependence of the boundary velocity and show thatthis new analytical model is in very good agreementwith the simulation results at both small and largedriving force.

2. ISING MODEL SIMULATION OF BOUNDARYMOTION

The simulations are performed within the frame-work of a two-dimensional Ising model with anapplied field. The energy of the system is written as

E = 2J2O

i,j

sisj + HOi

si, (5)

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3713MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

wheresi is the spin at sitei. In magnetic parlance,Jis the exchange energy andH is the magnitude of themagnetic field. However, in the present case it is use-ful to think of J as scaling the boundary energy(=2J/lattice spacing) andH as scaling the drivingforce for boundary migration. The first summation inequation (5) is over all sites and their (four) nearestneighbors. The system was initialized with all of thesites in the right half of the cell with si = + 1 andthose in the left half of the cell with si = 21.

A simple square lattice within a square simulationcell (Nx = 200, Ny = 200) is employed, as shown inFig. 2. Periodic boundary conditions (PBC) areemployed in they-direction and inverse PBC (Mobiusboundary conditions) are used in thex-direction [i.e.,sites on the right and left edge of the cell interact asper equation (5), but with a spin flip upon crossingthe edge]. This boundary condition is chosen suchthat there is only one domain boundary in the entiresystem, yet the boundary can be driven by a constantfield. The evolution of the model is simulated usingthe Monte Carlo method [6] with spin-flip dynamics(i.e., Glauber dynamics). In this approach, a site ischosen at random and an attempt is made to flip itssign (i.e., s→2s). If the energy changeDE is lessthan zero, this spin change is accepted. IfDE = 0, arandom numberR (0,R,1) is generated. If R,exp(2DE/kT) this spin change is accepted, otherwisethe spin is returned to its original orientation. Notethat time in this model is in units of Monte Carlosteps/site (MCS/site), i.e., 1 MCS/site means weattempt to flip each site on average one time.

Fig. 2. A typical image from a simulation, showing the actual arrangement of kinks on a migrating boundary.This simulation was performed withuHu=0.41J andT=0.25J/k in a 200×200 simulation cell.

The mean boundary position,p, was determined interms of the total magnetization:

p =Nx

2 S121

NxNyO

i

siD. (6)

With H,0, the boundary moves to the left. When theboundary translates by (1/4)Nx, a section of the simul-ation cell of size (1/4)NxNy is removed from the rightside of the model and a section of the same size isadded to the left side. This allows the boundary tokeep moving when it hits the edge of the simul-ation cell.

Simulations were performed with 0.01#uHu/J#0.2for T = 0.50J/k and 0.01#uHu/J#0.5 for T = 0.25J/k.The upper limit onuHu was chosen to ensure that nonew domains were nucleated by thermal fluctuationsaway from the boundary during the course of thesimulation. Fig. 2 shows a typical configuration seenduring a simulation atuHu/J = 0.41 and T= 0.25J/k.

Following an initial transient, the mean boundaryposition was found to be a linear function of time,the slope of which is the mean boundary velocity. Nochanges in the measured velocities were observedupon increasing the simulation cell size (toNy = 500). Fig. 3 shows the boundary velocity versusmagnetic field at both temperatures. At high tempera-ture, the velocity appears to be proportional to theapplied field. However, at higher field, the velocity isnoticeably smaller than such a linear extrapolationfrom low field would suggest. At low temperature,

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3714 MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

Fig. 3. The boundary velocity versusuHu for simulations performed atT=0.25J/k (s) andT=0.50J/k (h). Thedashed lines represent the predictions of the analytical model [equation (16)] and the solid lines are the lin-

earized results from equation (17).

the velocity is also a linear function of driving forceat small field, but exhibits a very pronounced devi-ation from this linear behavior at larger fields. Atintermediate fields, this deviation from the low-tem-perature linear behavior is toward lower velocity, butat large fields is toward higher velocity. The non-lin-ear dependence of the velocity on the driving forceclearly disagrees with the predictions of the standardmodel, equation (4).

3. ANALYSIS OF BOUNDARY VELOCITY

We now derive an analytical expression for theboundary velocity. Boundary motion consists of twodistinct processes; namely, double kink or island for-mation and kink propagation, as shown in Fig. 4(a).Consider the formation of an island on the boundary[the first two images of Fig. 4(a)] which correspondsto flipping one spin (21 or white to +1 or black)adjacent to the boundary (steps 1→2). For that islandto grow, a site adjacent to the one that just flippedmust also change (steps 2→3). The energy changescorresponding to these processes are

DE12 = (3J2J)2(23J+ J)22uHu = 4J22uHu>0(7)

and

DE23 = (2J22J)2(2J22J)22uHu = 22uHu,0,(8)

where the subscripts indicate the processes in the

numbered images in Fig. 4(a) and the inequalities arefor the parametersJ and uHu employed in the simula-tions above. Since the first process (1→2) increasesthe energy of the system, such a change will beaccepted with probability

w12 = exp(2DE12/kT) = exp[2(4J22uHu)/kT].(9)

The second process (2→3) lowers the energy of thesystem and will occur with probability w23 = 1. Thereverse of this step will occur with probability

w32 = exp(DE23/kT) = exp(22uHu/kT). (10)

For the conditions of the simulations reported above(and all reasonable values ofJ, H andT), w12¿w32.

The newly nucleated island in image 2 of Fig. 4(a)can either grow (2→3) or disappear (2→1). We nowcalculate the probability that an island eventually dis-appears. An island consisting of a single atom willalways disappear if we attempt to flip it before its twoneighbors in the new layer [provided the inequalityin equation (7) is satisfied]. The probability that thisoccurs is therefore 1/3, since these three sites aresampled with equal probability. Now, consider theprobability of the island disappearing when its neigh-bors were sampled first (one of the neighbors aresampled first with probability 2/3). Ann-site islanddisappears if all of its sites are considered in orderand, every time, that site is flipped (even though eachflip raises the energy of the system). For example, the

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3715MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

Fig. 4. The formation of an island (a) showing a flat boundary(1), nucleation (2) and growth (3). Ann-site island (b) labeled

as in the discussion of equations (11) and (13).

sites of the island shown in Fig. 4 must be consideredin the order 1, 2, …,n or in the ordern, n21, …, 1for this island to disappear. (The probability of flip-ping spins in the middle of the island is very smallcompared with the probability of flipping spins at theisland edges—i.e., at kink sites.) The probability thatthe first site will be considered before the second is1/2. Thus the probability that the island consisting ofn atoms disappears is equal to

2·S12w32DS1

2w32D…S1

2w32D1

2= S1

2w32Dn21

n21

(there is now32 in the term corresponding to the flip-ping of the last site, since the disappearance of a sin-gle-site island decreases the energy of the system).The factor of 2 on the left-hand side of this expressionaccounts for the island shrinking from the top or thebottom. The probability that an island does not disap-pear is

wsurvive = 12wdisappear

= 1213

223

(11)

FS12w32D + S1

2w32D2

+ …G=

23

12w32

12w32/2.

Consider two limiting cases. If H= 0, w32 = 1 and theisland survival probability is wsurvive = 0. In the caseof large uHu (still assuming that w12,w32), w32<1and wsurvive<2/3. This means that only single-siteislands will disappear ifuHu is large.

The net average number of islands that are formedand survive through one MCS/site is

nsurvive =w12

Nx

wsurvive =23

w12

Nx

12w32

12w32/2. (12)

We now determine the average rate of growth ofan island. The island will grow by one atom [towardsthe bottom of Fig. 4(b)] when the (n+ 1)th atom isconsidered. If thenth atom is considered, the islandwill shrink by one atom with probabilityw32. Theprobability that any particular atom will be con-sidered in one MCS is equal to 1/(NxNy). Taking intoaccount that the island can grow in two directions,the mean island growth rate can be written as:

visland =2

NxNy

(12w32). (13)

Islands can disappear by shrinking to zero size orby impingement with another island (i.e., in this casethe total number of islands decreases by one). Suchtwo-island coalescence events occur most frequentlywhen a layer is nearly complete. Lett be the totalnumber of MCS necessary for the boundary toadvance by one lattice spacing. During this time,nsurvivet new islands form and disappear (bycoalescence) during the nextt time steps. In steadystate, the rate of island disappearance is equal to therate of island formation, as determined in equation(12). Therefore, the change in size (length) of allislands in timet is

Lt = Oti = 1

(nsurviveτ2nsurvivei)visland (14)

+ Oti = 1

nsurvivevislandi,

where the first term corresponds to the change in thetotal length of the islands that were formed prior tothe current time period and the second term gives thetotal length of new islands. Sincet is the timerequired for the boundary to advance by a single lat-tice spacing, we can write Lτ = Ny. Substituting equa-tions (12) and (13) into equation (14) yields:

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3716 MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

τ2 = (NxNy)234

12w32/2w12(12w32)2. (15)

We can now determine the overall velocity of theboundary by recalling that the boundary moves a dis-tance of one lattice spacing in timet and convertingtime from MCS to MCS/site. This yields

v =NxNy

τ= (12w32)!4

3w12

12w32/2, (16)

where we substitute in fort by using equation (15).In the limit that uHu¿kT, this reduces to

v = 4!23

exp(22J/kT)uHukT

. (17)

In this limit, the velocity is proportional to the drivingforce (2uHu). This is consistent with the commonassumption that boundary velocity is proportional todriving force for boundary migration [7].

In Fig. 3, we compare the predicted velocity versusapplied field [equation (16)] with the simulationresults. Clearly, the predicted dependence of theboundary velocity on driving force [equation (16)] isin very good agreement with the simulation data atboth small and large driving forces and at both lowand high temperatures. Equation (16) quantitativelycaptures the essential non-linearities in the boundaryvelocity–driving force relation. On the other hand, thelinearized velocity–driving force relation [equation(17)] is valid only for very small uHu. Based on thevery good agreement achieved between the predictedand simulation results, we conclude that equation (16)is an excellent description of boundary velocity as afunction of driving force (H), temperature (T) andboundary energy (i.e., line tension,J).

4. DISCUSSION

Comparison of the standard expression for theboundary velocity–driving force relation [equation(4)] with that derived above [equations (16) and (17)]suggests that the standard model is simply a lineariz-ation of that presented here [equation (16)] for smalldriving force. Recasting the energetics used herein interms of those used in equation (4), we identifyF;2H, D;1/2 and Ek;4J. Using these identities, wefind that the two predictions for the velocity at verysmallH [equations (4) and (17)] are absolutely ident-ical except for a constant factor of√8/3<1.63. Thissmall discrepancy is because the standard model [equ-ation (4)] does not account for the simultaneous exist-ence of kinks on more than one layer [as seen inFig. 2(e)].

The linearized kinetics, as expressed by equation(17), should be valid at small driving forces, i.e.,whereF/kT = 2uHu/kT¿1. Examination of Fig. 3 sug-gests that the linearized velocity expression should

not be used beyondF/kT<0.3. Expanding the velo-city versus force relation in equation (16) to secondorder in uHu/kT demonstrates that the correction toequation (17) is (12uHu/kT). This term causes theinitial negative deviation from linearity in Fig. 3 (atboth small and large temperature). This higher-orderterm has contributions from theH dependence of thekink velocity, the island survival time and the rate ofproduction of islands. In the standard model, theHdependence of the kink velocity is only included tolowest order and the other two effects are independentof H (which is reasonable only at largeH).

At still larger driving forces, the expansion of thevelocity in uHu/kT becomes fruitless. WhenuHu islarge, the kink velocity [see equation (13)] and theisland survival time [see equation (11)] both go toconstant values, while the rate of production of newislands increases as exp(2uHu/kT). Therefore, at largeuHu the boundary velocity should increase quicklywith uHu. This is consistent with the simulation resultsshown in Fig. 3 at low temperature. A similar increaseis not seen in the high-temperature data because thosesimulations were terminated at smalleruHu to ensurethat no new domains were nucleated.

The present analysis of the force–velocity relation-ship for extended defects was performed in twodimensions for the case where the interaction betweensegments of the defect is very short-ranged—i.e., thesurface tension limit. These are reasonable assump-tions for grain boundaries and related defects (e.g.,antiphase boundaries) in film or thin plate appli-cations, where the system is pseudo two-dimensional.In cases where the system is fully three-dimensional,the above analysis has to be extended to include two-dimensional island formation (terraces) bounded bysteps and kinks on steps. This is one level greaterin complexity than considered here. Nonetheless, thesame approach is readily extendable to three dimen-sions. The dimensionality of the present analysis ismore appropriate for one-dimensional defects, such asdislocations or disclinations. However, in these cases,the interactions between different segments of thedefect are long-ranged (elastic) and the simplesurface/line tension model used herein is not strictlyvalid. While such effects are very important when theamplitude of the roughness of the dislocation line islarge or when kinks are very close together, thesurface/line tension model used here is adequate formany applications [2].

5. CONCLUSIONS

We have presented both Monte Carlo simulationsof and analytical results for the force–velocityrelationship for linear defects in two dimensions. Thesimulation results agree with the most widely usedformulation (i.e., velocity proportional to force) onlyin the limit of very small driving force. On the otherhand, the simulations and new analytical predictionsare in excellent agreement. As the driving force is

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3717MENDELEV and SROLOVITZ: KINK MODEL FOR EXTENDED DEFECT MIGRATION

increased, the velocity changes from being pro-portional to the driving force to a sub-linear depen-dence. At large driving force, the velocity increasesrapidly (exponentially) with driving force. The suc-cess of the new, analytical model is attributable toinclusion of the effects of kinks on multiple levels,the non-simultaneity of kink formation, the disappear-ance of kinks and the non-linear kink force–velocityrelationship.

Acknowledgements—The authors wish to thank Professors J.P. Hirth, B. S. Bokstein and L. S. Shvindlerman for usefuldiscussions during the performance of this work. This workwas supported by the Division of Materials Science of the

Office of Basic Energy Sciences of the United States Depart-ment of Energy, Grant DE-FG02-99ER45797.

REFERENCES

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2. Mott, N. F.,Proc. Phys. Soc., 1948,60, 391.3. Sutton, A. P. and Balluffi, R. W.,Interfaces in Crystalline

Materials. Clarendon Press, Oxford, 1995.4. Hirth, J. P. and Lothe, J.,Theory of Dislocations. Wiley-

Interscience, New York, 1982.5. Friedel, J.,Dislocations. Pergamon, New York, 1964.6. Binder, K.,Monte Carlo Simulation in Statistical Physics.

Springer-Verlag, Berlin, 1979.7. Upmanyu, M., Smith, R. W. and Srolovitz, D. J.,Interface

Sci., 1998,6, 41.