stress-driven migration of simple low-angle mixed grain...

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Acta Materialia 60 (2012) 1395–1407

Stress-driven migration of simple low-angle mixed grain boundaries

A.T. Lim a, M. Haataja a,⇑, W. Cai b, D.J. Srolovitz c

a Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USAb Mechanical Engineering Department, Stanford University, Stanford, CA 94305, USA

c Institute for High Performance Computing, 1 Fusionopolis Way, Singapore 138632, Singapore

Received 12 October 2011; received in revised form 15 November 2011; accepted 17 November 2011

Abstract

We investigated the stress-induced migration of a class of simple low-angle mixed grain boundaries (LAMGBs) using a combinationof discrete dislocation dynamics simulations and analytical arguments. The migration of LAMGBs under an externally applied stress canoccur by dislocation glide, and was observed to be coupled to the motion parallel to the boundary plane, i.e. tangential motion. Both themigration and tangential velocities of the boundary are directly proportional to applied stress but independent of boundary misorien-tation. Depending on the dislocation structure of the boundary, either the migration or tangential velocity of the boundary can switchdirection at sufficiently high dislocation climb mobility due to the dynamics of dislocation segments that can climb out of their respectiveslip planes. Finally, we show that the mobility of the LAMGBs studied in this work depends on the constituent dislocation structure anddislocation climb mobility, and is inversely proportional to misorientation.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundary migration; Dislocation dynamics simulation; Dislocation boundaries

1. Introduction

Grain boundary migration is one of the most fundamen-tal phenomena underlying microstructural evolution pro-cesses, such as grain growth, recrystallization and plasticdeformation of polycrystalline materials. Because of thecomplexity of grain boundary structure, it has not beenpossible to develop a predictive model for the mobility ofgrain boundaries having arbitrary bicrystallography.Although considerable advances have been achieved in the-oretical modeling and experimental characterization of theproperties and behavior of very simple low-angle grainboundaries, high-angle grain boundaries with special orien-tations, and twin boundaries [1], the structure and mecha-nism of migration for a general grain boundary is not wellunderstood.

However, if the misorientation between adjoining grainsis small, the grain boundary structure is relatively simple; in

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. Alldoi:10.1016/j.actamat.2011.11.032

⇑ Corresponding author.E-mail address: [email protected] (M. Haataja).

particular, a low-angle grain boundary (LAGB) can bedescribed as an array of lattice dislocations [2]. The migra-tion of such boundaries should be describable in terms ofthe collective motion of its constituent dislocations. Thisapproach has previously been employed to investigate themigration of low-angle tilt boundaries, for which paralleledge dislocations that constitute the boundaries are repre-sented as points moving in the plane perpendicular to thedislocation lines (i.e. in two dimensions) [3].

In this paper, we provide a systematic analysis of a muchmore general set of grain boundaries, namely mixed tilt/twist grain boundaries consisting of curved, intersectingdislocations with multiple Burgers vectors in the experi-mentally relevant three-dimensional case. In particular,we focus on the stress-driven migration of a series of(1 0 0)[1 1 0] and (1 1 1)[1 1 0] mixed boundaries, initiallyconsisting of dislocations with the smallest lattice Burgersvector in body-centered cubic (bcc) crystals.

Although almost 60 years have elapsed since the stress-induced migration of a low-angle tilt boundary was firstobserved experimentally by Washburn and Parker [4],

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1396 A.T. Lim et al. / Acta Materialia 60 (2012) 1395–1407

systematic studies of the mobility of low-angle mixed grainboundaries (LAMGBs) are rather limited in both scopeand number. Except for a very recent experimental investi-gation of stress-induced migration of a mixed boundary byGorkaya et al. [5] and molecular dynamics simulations of asmall number of mixed boundaries [6,7], all other theoret-ical and experimental studies of boundary mobility are foreither pure tilt boundaries [8–22], pure twist boundaries[23,24], or a finite selection of tilt and twist coincidence sitelattice boundaries [25]. While an arbitrary LAGB cansimultaneously have both tilt and twist character, the diffi-culty in preparing bicrystals with LAMGBs having well-characterized dislocation structures is one reason for thelack of experimental measurements of mobility for thismost common class of LAGBs.

This difficulty in preparing such bicrystals does notextend to theoretical investigations. Indeed, grain bound-aries can unambiguously be represented by arrays of latticedislocations that can relax into equilibrium LAGB struc-tures. Describing LAGBs in this manner enables the useof discrete dislocation dynamics simulation methods toinvestigate the migration of such boundaries. This is theapproach employed in the present study to determine thestructure, migration mechanism and mobility of LAMGBs(which necessarily consist of non-parallel dislocations thatmay react to form dislocation segments of new Burgersvectors).

In the present work, we drive the migration of nominallyplanar LAMGBs via an externally applied stress to investi-gate how the dynamics of the constituent dislocation struc-ture (i.e. the LAGB structure) controls boundary migrationover a wide range of applied stress and dislocation climbmobility (temperature) for several misorientations usingthree-dimensional dislocation dynamics simulation meth-ods. The main objective of this study is to establish the rela-tionship between the mobility of an LAMGB and itsconstituent dislocation structure, and to develop a predic-tive model for the intrinsic mobility of such LAGBs. Suchquantitative theoretical predictions are only possible forLAGBs (and possibly a small number of high-symmetry,higher-angle boundaries) because of the relative simplicityof such grain boundary structures.

The simulations and theoretical analysis presentedherein demonstrate that simple LAMGBs migrate underan externally applied shear stress by means of the glide,climb, and rearrangement of the constituent dislocations.A common feature to all simple LAMGBs studied in thiswork is that the boundary mobilities are not dislocation-climb limited. In addition, we observe that all LAMGBsin the present study translate with components perpen-dicular (i.e. migration) and parallel to the boundaryplane. At a fixed temperature and stress, the velocity ofthe boundary is independent of the misorientation,although the driving force on the boundary increases lin-early with misorientation; thus, the intrinsic mobility ten-sor for such an LAMGB is inversely proportional to themisorientation. Intriguingly, the magnitude of boundary

motion depends on the temperature at fixed appliedstress, and may even switch direction as a function oftemperature.

The remainder of this paper is organized as follows. InSection 2, we present our simulation method and describehow the LAMGBs are constructed. Section 3 contains themain simulation results: the relaxed structures of theLAMGBs, the dislocation structures of the movingboundaries, as well as the parallel and perpendicularcomponents of the grain boundary velocity as a functionof dislocation climb mobility (temperature), applied stress,and misorientation for both (1 0 0)[1 1 0] and (1 1 1)[1 1 0]mixed boundaries. These results are then analyzed andinterpreted within a theoretical framework developed inSection 4. Finally, the implications of the simulationresults and theoretical analyses to the stress-driven migra-tion behavior of a broader class of LAMGBs are dis-cussed in Section 5.

2. Simulation method

To simulate the migration of a LAMGB, we employed athree-dimensional dislocation dynamics technique thataccounts for dislocation topology changes arising fromgeneral forces and reactions between dislocation segmentsas implemented within the Parallel Dislocation Simulator(ParaDiS) code [26,27]. ParaDiS is initialized by providinga dislocation network that is discretized into a series of lin-ear segments that meet at nodes. The dynamics of the dis-location network are then simulated within the frameworkof isotropic linear elasticity, by calculating the force oneach node arising from external and internal forces onadjacent linear segments, and subsequently evolving theposition of the nodes using a nodal equation of motion.In this work, we assume that the velocity of each segmentis directly proportional to the driving force. In the courseof the simulation, topological changes are accommodatedvia addition and removal of nodes: new nodes are addedand existing nodes are removed when (1) nodes on a dislo-cation line segment become too far apart or too closetogether and (2) during a reaction event when dislocationline segments merge, split or annihilate one another.

The Peach–Kohler force (per unit length) [28] isdetermined from the total stress ri on the ith linear segmenthaving Burgers vector bi and line direction n̂i as:

f i ¼ ðri � biÞ � n̂i: ð1Þ

Assuming overdamped dislocation motion, the velocity ofthe ith segment may be written as:

vi ¼Mi � f i; ð2Þ

where Mi denotes a mobility tensor that accounts for theglide mobility of screw dislocations (ms), the glide mobilityof edge dislocations (mg) and the climb mobility of edgedislocations (mc). In this study, we set the glide mobilityof screw and edge dislocations to be equal, i.e. ms = mg

A.T. Lim et al. / Acta Materialia 60 (2012) 1395–1407 1397

(this is reasonable for bcc metals at high temperatures), andconstruct the mobility tensor to interpolate smoothly be-tween glide and climb motion (see Ref. [29] for details):

Mi ¼ mgðI� n̂i � n̂iÞ þMcn̂i � n̂i; ð3Þwhere I is the identity matrix, n̂i ¼ ðbi � n̂iÞ=jbi � n̂ij de-notes the slip plane normal of the ith linear segment,Mc ¼ ½m�2g þ ðm�2c � m�2g Þ sin

2 ai��1=2, and ai ¼ bi\n̂i de-notes the angle between vectors bi and n̂i.

In the ParaDiS implementation of discrete dislocationdynamics, the position, orientation and length of each dis-location line segment evolves with time as the nodes termi-nating the segment at each end advance according to theirrespective equation of motion. To find the position of everynode at each time step, the nodal equation of motion isintegrated numerically using the Euler-trapezoid method.

LAMGBs were constructed in accordance with Frank’sfomula [2] using bcc lattice dislocations. Although theboundaries are free of long-range stresses (as mandatedby Frank’s formula), the dislocation configuration of eachboundary was allowed to fully relax within an isotropicsolid with shear modulus l and Poisson ratio m = 1/3 (thedislocations move as a result of reactions and internal stres-ses between dislocation segments) prior to being subjectedto a constant external stress s that drives the boundary tomigrate. To investigate the correlation between the disloca-tion structure of the boundary and its mobility, we evaluatethe boundary velocity V as a function of applied stresses inthe 0 < s 6 0.05 l range, dislocation climb mobility0 6 mc 6 mg, and misorientation angle h = {2.0�, 4.0�}.The details of the dislocation structure and correspondingboundary parameters for the LAMGBs considered hereare given in Appendix A.

In this paper, we focus on simple LAMGBs, for whichthe magnitude of the lattice Burgers vectors and dislocationspacing in each set are equal, i.e. b1 = b2 = b andD1 = D2 = D. In particular, b1 and b2 are chosen fromthe set of a

2h1 1 1i Burgers vectors, typical of bcc metals.

It is important to note that with such specialized Burgersvectors and dislocation line spacings, it follows (see Appen-dix A) that the orientation of the boundary plane is nolonger arbitrary but, in fact, is constrained by:

m̂ � ðb1 � b2Þ ¼ 0: ð4ÞThe restrictions (i.e. b1 = b2 = b and D1 = D2 = D) on

the LAMGB dislocation structure imply that the boundarynormal m̂ must be perpendicular to either b1 + b2, orb1 � b2. In this work, we will focus on these two classesof simple LAMGBs. Although these boundaries make upa subset within the broader category of general mixedLAMGBs, their dislocation networks are complex enoughto impart intriguing features to the intrinsic boundarymobility, and yet amenable to a thorough analysis. By con-struction, a simple LAMGB initially corresponds to a loz-enge network of b1; n̂1

� �and b2; n̂2

� �dislocation arrays.

However, this dislocation configuration is not energeticallystable, as b1 and b2 segments will react to form a third

segment b3 = b1 + b2 2 ah1 0 0i. This reaction is energeti-cally favorable because b23 < b

21 þ b22, and it transforms

the lozenge network into a hexagonal network also seenin experiments [30].

Regardless of the dislocation structure, the boundary isconstructed so that it lies in the yz-plane in the simulationbox, i.e. x̂ ¼ m̂, with the line directions symmetric aboutthe z-axis (see Appendix B). For each set of boundaries con-sidered here, there is only one component of the externallyapplied stress r0 that gives a net driving force for migrationF\, i.e. r0xk ¼ r0kx with k – x (see Appendix C). For theboundary plane determined by m̂ � ðb1 þ b2Þ ¼ 0, the bound-ary migrates under the stress component r0xz. On the otherhand, for the boundary plane determined by m̂ � ðb1 � b2Þ ¼0, the boundary migrates under the stress component r0xy .

In the present simulations, we examine the stress-induced migration of two series of simple LAMGBs in abcc crystal, one for each case in Eq. (4). Both boundariesare modeled using two sets of a

2h1 1 1i dislocations in the

{1 1 0} slip planes for each boundary.We first consider the m̂ � ðb1 þ b2Þ ¼ 0 case, for which

the Burgers vector and line direction pairs are b1 ¼a2½�1 1 1�; n̂1 ¼ 1ffiffi2p ½0 1 1� and b2 ¼ a2 ½1 �1 1�; n̂2 ¼ 1ffiffi2p ½0 �1 1�.

The corresponding boundary plane is (1 0 0), for whichFrank’s formula (Eqs. (A.3) and (A.4)) yields a rotationaxis û ¼ 1ffiffi

2p ½1 1 0� about which the grains adjacent to this

boundary are misorientated by h = a D�1. This is the(1 0 0)[1 1 0] mixed boundary case. The twist angle of thisboundary is 45�, which corresponds to 50% twist charac-ter. The periodicity of the system along the ŷ and ẑ isLy = Lz = L. For this boundary to migrate, stress mustbe applied along the [1 0 0] direction in the (0 0 1) plane(i.e. r0xz ¼ r0zx).

Next, we consider the m̂ � ðb1 � b2Þ ¼ 0 case, for which theBurgers vectors and line directions are b1 ¼ a2 ½�1 1 1�;n̂1 ¼ 1ffiffi2p ½�1 0 1� and b2 ¼ a2 ½1 �1 1�; n̂2 ¼ 1ffiffi2p ½0 �1 1�. Except forn̂1, the Burgers vectors b1 and b2, and line direction n̂2 arethe same as that for the (1 0 0)[1 1 0] mixed boundary.Accordingly, the rotation axis is û ¼ 1ffiffi

2p ½1 1 0�, while the

boundary plane and misorientation are (1 1 1) andh ¼

ffiffiffi3p

að2DÞ�1, respectively. This is the (1 1 1)[1 1 0] mixedboundary case. The twist angle of this boundary is 54.7�,which corresponds to 66.7% twist character. The periodicityof the system along the ŷ and ẑ is Lz ¼

ffiffiffi3p

Ly . For this bound-ary to migrate, stress must be applied along the [1 1 1] direc-tion in the ð�1 1 0Þ plane (i.e. r0xy ¼ r0yx).

3. Results

3.1. Relaxation

When the initial (lozenge) network for each configura-tion is allowed to relax, the simulations show that b1 andb2 react at the intersections, producing new segments withBurgers vector b3 = b1 + b2 and line direction n̂r3 that isparallel to ẑ. The reaction proceeds until the line tension

Fig. 1. Schematic representation of the equilibrium LAMGB constructedusing two intersecting a

2h1 1 1i dislocations, i.e. b1 ¼ a2 ½�1 1 1� and b2 ¼

a2½1 �1 1�, which react to form a new a[0 0 1] segment for b3, and the

coordinate directions of the simulation box for the (1 0 0)[1 1 0] and(1 1 1)[1 1 0] mixed boundary.


forces at the triple junctions as well as the interaction forcesbetween segments balance. A schematic illustration of therelaxed dislocation structure for both (1 0 0)[1 1 0] and(1 1 1)[1 1 0] mixed boundaries is shown in Fig. 1.

While the projection of the dislocation configuration onthe mean boundary plane is a hexagonal network (see Figs.2a and 3a), close examination of each boundary in Figs. 2band 3b shows that the b1 and b2 segments do not remain per-fectly straight. Instead, these dislocation segments weave inand out of the mean boundary plane in order to increase thenet screw character and reduce the total energy of the con-figuration. As a result, the boundary acquires a finite widththat scales in proportion to the periodicity of the dislocationstructure (and hence inversely proportional to the boundarymisorientation). Nevertheless, for the range of misorienta-tions under consideration here, the width of each boundary

−100 −50 0 50−100

−50

0

50

100

y

z

(a)

Fig. 2. Projected views (in units of lattice constant a) of dislocation configuratioplane in the absence of applied stress as well as during migration, (b) the xz-plaThe dislocation configuration during migration is very similar to that in thecomparable to the length of the Burgers vector.

is less than the dislocation core radius. Therefore, the mixedboundaries studied here can be considered to be macroscop-ically flat hexagonal networks, characterized by the anglebetween b1 and b2 segments, i.e. n̂r1\n̂r2 ¼ 2/, at the dislo-cation triple junctions. Nonetheless, for very low misorien-tations, such non-planarity may be significant.

Upon relaxation of the (1 0 0)[1 1 0] boundary, theresulting b3 segment is a pure screw dislocation. The angle

between the b1 and b2 segments is n̂r1\n̂r2 ¼ 106:2�. On theother hand, for the (1 1 1)[1 1 0] boundary, b3; n̂r3

� �is a

mixed segment having Burgers vector b3 = [0 0 1] and line

direction n̂r3 ¼ 1ffiffi6p ½�1 �1 2�, with n̂r1\n̂r2 ¼ 97:77�.3.2. Migration

Simulations were performed in which both boundarieswere subjected to each of the six independent componentsof a symmetric stress tensor (simulation frame). The simu-lations confirm that these simple LAMGBs can migrateonly if the external shear stress is s ¼ r0xz for m̂ � ðb1 þ b2Þ¼ 0 and s ¼ r0xy for m̂ � ðb1 � b2Þ ¼ 0. Under such shearstress, each boundary achieves a steady-state velocity V(measured by averaging the velocities of the dislocation tri-ple junction nodes). For both boundaries, the steady-statemigration of the boundary perpendicular to the meanboundary plane V? ¼ V ?x̂ is coupled to a steady-statetranslation of the grain boundary dislocation network ina direction that is tangential to the mean boundary planeVk, i.e. V = V\ + Vk. The directions of the tangentialvelocity vectors are different for the two boundaries.

For the (1 0 0)[1 1 0] boundary, the tangential velocity isperpendicular to the b3; n̂r3

� �(screw) segment, i.e. Vk ¼

V kŷ. The perpendicular (migration) and tangential velocitiesare shown in Figs. 4 and 5, respectively, as a function of theapplied stress s ¼ r0xz for several values of dislocation climbmobility mc and grain boundary misorientation h. Weobserve that this boundary can migrate by dislocation glide,

100 −1 0 1−20

−10

0

10

20

x

z

(b)

−1 0 10

10

20

30

40

x

z

(c)

n for the (1 0 0)[1 1 0] LAMGB at h = 2.0� misorientation onto (a) the yz -ne in the absence of applied stress, and (c) the xz -plane during migration.

absence of an applied stress. In both cases, the width of the boundary is

−101.5 −50.75 0 50.75 101.5−100

−50

0

50

100

y

z

(a)

−1 0 10

10

20

30

40

50

x

z

(b)

−1 0 120

30

40

50

60

70

80

x

z

(c)

Fig. 3. Projected views (in units of lattice constant a) of dislocation configuration for the (1 1 1)[1 1 0] LAMGB at h = 2.0� misorientation onto (a) the yz -plane in the absence of applied stress as well as during migration, (b) the xz -plane in the absence of applied stress, and (c) the xz -plane during migration.The dislocation configuration during migration is very similar to that in the absence of an applied stress. In both cases, the width of the boundary iscomparable to the length of the Burgers vector.

0 0.005 0.01 0.015 0.02 0.025 0.030

0.002

0.004

0.006

0.008

0.01

0.012

τ

V ⊥

mc = 0mc = 0.2mc = 0.4mc = 0.6mc = 0.8mc = 1.0

(a)

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

mc

V ⊥/τ θ = 2° simulation

θ = 4° simulationv1x = v2xv3x

(b)

Fig. 4. (a) Steady-state (1 0 0)[1 1 0] LAMGB migration (perpendicular to the boundary) velocity V\ (in units of almg) as a function of the applied stress s(in units of l) for several values of dislocation climb mobility mc (in units of mg) at misorientation h = 2.0�. (The error bars indicate the range of velocitydue to numerical fluctuations in the simulation box.) (b) The ratio of V\ to s as a function of mc for h = 2.0� and h = 4.0�. Here, v1x, v2x and v3x denote thedislocation segment velocities along the direction perpendicular to the boundary that would result from the external force alone (see Section 4.2).

0 0.005 0.01 0.015 0.02 0.025 0.03−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

τ

V ||

mc = 0mc = 0.2mc = 0.4mc = 0.6mc = 0.8mc = 1.0

(a)

0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

mc

V ||/τ

θ = 2° simulationθ = 4° simulationv1ycos

−2φ = v2ycos−2φ

v3y

(b)

Fig. 5. (a) Steady-state (1 0 0)[1 1 0] LAMGB tangential velocity Vk (in units of almg) (in the direction perpendicular to the b3 screw segment and alongthe boundary plane) as a function of the applied stress s (in units of l) for several values of dislocation climb mobility mc (in units of mg) at misorientationh = 2.0�. (b) The ratio of Vk to s as a function of mc for h = 2.0� and h = 4.0�. Here, v1y cos�2/, v2ycos�2/, and v3y denote the dislocation segmentvelocities along the boundary plane that would result from the external force alone, corrected by the geometrical factor cos2/ (see Section 4.2).


0 0.01 0.02 0.03 0.04 0.05

−5

0

5

10

15

x 10−3

τ

V ⊥

mc = 0mc = 0.2mc = 0.4mc = 0.6mc = 0.8mc = 1.0

(a)

0 0.2 0.4 0.6 0.8 1

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

mc

V ⊥/τ

θ = 2° simulationθ = 4° simulationv1x = v2xv3x

(b)

Fig. 6. (a) Steady-state (1 1 1)[1 1 0] LAMGB migration (perpendicular to the boundary) velocity V\ (in units of a lmg) as a function of the applied stresss (in units of l) for several values of dislocation climb mobility mc (in units of mg) at misorientation h = 2.0�. (The error bars indicate the range of velocitydue to numerical fluctuations in the simulation box.) (b) The ratio of V\ to s as a function of mc for h = 2.0� and h = 4.0�. Again, v1x, v2x and v3x denotethe dislocation segment velocities along the direction perpendicular to the boundary that would result from the external force alone (see Section 4.3).

0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

τ

V ||

mc = 0mc = 0.2mc = 0.4mc = 0.6mc = 0.8mc = 1.0

(a)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

mc

V ||/τ

θ = 2° simulationθ = 4° simulationv1zsin

−2φ = v2zsin−2φ

v3z

(b)

Fig. 7. (a) Steady-state (1 1 1)[1 1 0] LAMGB tangential velocity Vk (in units of almg) (in the direction parallel to the b3; n̂r3� �

mixed segment and alongthe boundary plane) as a function of the applied stress s (in units of l) for several values of dislocation climb mobility mc (in units of mg) at misorientationh = 2.0�. (b) The ratio of Vk to s as a function of mc for h = 2.0� and h = 4.0�. Here, v1z sin�2/, v2zsin�2/ and v3z again denote the dislocation segmentvelocities along the boundary plane that would result from the external force alone, corrected by the geometrical factor sin2/ (see Section 4.3).


since both V\ and Vk are non-zero for finite values of s atmc = 0. V\ and Vk are directly proportional to the appliedstress s. For positive values of s, the migration velocity V\is always directed along þx̂, but the tangential velocity Vkswitches direction from þŷ to �ŷ at sufficiently high valuefor mc. Figs. 4b and 5b in turn display the slope of V\ vs. sand Vk vs. s, respectively, for two different values of the mis-orientation h. These results show that both V\ and Vk areindependent of h within the accuracy of our data.

For the (1 1 1)[1 1 0] boundary, the tangential velocity is

parallel to the b3; n̂r3� �

segment, i.e. Vk ¼ V kẑ. The migra-tion and tangential velocities, as a function of the appliedstress s ¼ r0xy for various values of dislocation climb mobil-ity mc and misorientation h, are shown in Figs. 6 and 7,respectively. As with the (1 0 0)[1 1 0] boundary, the velocitydata indicate that the (1 1 1)[1 1 0] boundary can migrate ina pure glide or a combined glide/climb mode, and that bothV\ and Vk for this boundary are proportional to s and inde-

pendent of h within the accuracy of our data. However, incontrast to the (1 0 0)[1 10 ] boundary, where the tangentialvelocity changes sign with mc, it is the migration velocity V\that changes sign with increasing mc in the (1 1 1)[1 1 0] case.For positive values of s, Vk in the (1 1 1)[1 1 0] boundarycase is consistently directed alongþẑ, but V\ switches direc-tion from �x̂ to þx̂ at sufficiently large values mc.

The simulations show that the dislocation configura-tions during migration for both mixed boundaries exam-ined in this work are very similar to their correspondinghexagonal networks observed in the absence of an exter-nally applied stress (see Figs. 2 and 3). Hence, themigrating dislocation configuration may be approximatedas a planar hexagonal network of straight dislocationsegments, for which the velocity due to the driving forceon each segment may be evaluated in order to determinehow the dynamics of each segment in the hexagonal net-work influences the overall migration behavior of theboundary.


4. Analysis

4.1. Relaxation

As discussed in Section 3.1, both the (1 0 0)[1 1 0] and(1 1 1)[1 1 0] mixed boundaries are macroscopically flat(on the scale of a Burgers vector for the range of misorien-tations under consideration here) and may be approximated

as a flat, hexagonal network of straight b1; n̂r1� �

; b2; n̂r2

� �and b3; n̂r3

� �dislocation segments, having a characteristic

angle n̂r1\n̂r2 ¼ 2/. We can estimate the angle / by balanc-ing the elastic line tension between the dislocation segmentsthat meet at each triple junction:

E3n̂r3 ¼ E1n̂r1 þ E2n̂r2:For a straight dislocation segment, the magnitude of the

line tension is E ¼ al4p logðR=r0Þ b

2screw þ b2edge=ð1� mÞ

h i[28],

where R is the simulation box size. Neglecting differencesin the core energy and radius of different dislocation seg-ments, the estimated values for / are shown in Table 1.Comparison of the analytical results with those obtainedfrom simulations shows errors less than 5% for the(1 0 0)[1 1 0] boundaries, and 11% for the (1 1 1)[1 1 0]boundaries.

4.2. Migration of (1 0 0)[1 1 0] boundaries

For the (1 0 0)[1 1 0] boundary, we approximate the dis-location configuration during steady-state migration as thehexagonal network shown in Fig. 1 with / as per Table 1.For this boundary, the driving forces due to s ¼ r0xz on theindividual dislocation segments are (see Eq. (1)):

f1 ¼s2

sin /

� cos /sin /

0B@

1CA; f2 ¼

f1xf1y�f1z

0B@

1CA; and f3 ¼ s

0

�10

0B@

1CA:ð5Þ

This shows that boundary migration (motion perpendicu-lar to the boundary plane) is driven by the forces on seg-

ments b1; n̂r1� �

and b2; n̂r2� �

. Although there is no force

along the migration direction for the b3; n̂r3� �

segment,

this segment is dragged by the other two segments. Theforce analysis shows that the tangential motion of theboundary structure has contributions from all three dislo-

Table 1Comparison between simulation results and theoretical predictions for thecharacteristic angle n̂r1\n̂r2 ¼ 2/ of the hexagonal network.Boundary Simulation (�) Theory (�)

(1 0 0)[1 1 0] 106.2 111.1(1 1 1)[1 1 0] 97.77 88.72

cation segments. The total driving force on the boundaryis (see Eqs. (C.1)–(C.3)):

F ¼F ?F k0

0B@

1CA ¼ shffiffiffi

2p

1

�10

0B@

1CA; ð6Þ

where F\ and Fk are the driving forces for migration andtangential motion (along the y-direction), respectively.

Now, the velocities that would result from the externaldriving forces alone on the individual dislocation segmentsare given by (see Eqs. (2), (3) and (5)):

v1 ¼ C0sC1mg þ C2McC3mg � C4McC5mg þ C6Mc

0B@

1CA; v2 ¼

v1xv1y�v1z

0B@

1CA;

and v3 ¼ mgs0

�10

0B@

1CA; ð7Þ

where Mc ¼ ½m�2g þ ðm�2c � m�2g Þ sin2 a��1=2 and

a ¼ b1\n̂r1 ¼ b2\n̂r2 ¼ arccos½ðcos /þ sin /Þ=ffiffiffi3p�, with

the coefficients Ck(/) as given in Appendix D. The applied

stress drives the b3; n̂r3� �

screw segment to cross-slip, and in-

duces the b1; n̂r1� �


mixed segments to glide as

well as climb out of their respective slip planes.

In addition to the driving force, there are forces betweenthe different segments such that the resulting motion of theboundary is a rigid translation of the constituent dislocationconfiguration. The velocity of the boundary V is equal to

that of the dislocation triple junctions and the b3; n̂r3� �

seg-

ment, but is related to the velocities of segments b1; n̂r1� �


by a geometric coupling factor. In particular,

along the tangential direction (along y), the velocity of thedislocation triple junctions (and the boundary structure as

a whole) is greater than that of segments b1; n̂r1� �

and

b2; n̂r2

� �by a factor of 1/cos2/. While it is not straightfor-

ward to account for all the internal forces between the dis-location segments in this simple theoretical analysis, it isstill possible to explain the general features and trendsobserved in the simulation results of Figs. 4 and 5 usingthe velocities of individual dislocation segments in Eq. (7).First and foremost, the velocity of the boundary must bebounded by the fastest and slowest driven dislocation seg-ments, i.e. v3x < V\ < v1x, and v3y < Vk < v1ycos

�2/. Sincea rigid displacement of the boundary structure is observedin the simulations, the motion of the boundary must be lim-

ited by segments b1; n̂r1� �


, which are driven to

climb out of their respective slip planes. Therefore, it is not


unreasonable to expect that the resulting velocity of theboundary be comparable to the velocity of these segments:

V ?K v1x ¼ C0ðC1mg þ C2McÞs ’ ð0:27mg þ 0:15McÞs ð8Þand

V kK v1y cos�2 / ¼ cos�2 /C0ðC3mg � C4McÞs’ ð0:12mg �McÞs; ð9Þ

with the numerical factors obtained using the theoretical valueof / given in Table 1. Comparison between the simulationdata and the analytical bounds displayed in Figs. 4b and 5bclearly shows that the boundary velocities extracted fromthe simulations are indeed bounded by the analytical expres-sions derived from the driven velocities of the dislocation seg-ments. Furthermore, the simulation data is well approximatedby the analytical upper bounds in Eqs. (8) and (9).

Interestingly, the migration of this boundary can occursimply by grain boundary dislocation glide, and hence themobility is not dislocation-climb limited. The boundary isexpected to migrate in the same direction as the driving forceF\, regardless of the temperature. In contrast to V\, the tan-gential motion Vk of the boundary may be parallel or anti-parallel with respect to the driving force Fk. At low valuesof mc, Vk is anti-parallel to Fk; in the limit that mc! 0,Vk cos�2/C0C3mgs ’ 0.12mgs for F k ¼ �sh=

ffiffiffi2p

. For suf-ficiently large values of mc, Vk switches direction so that itbecomes parallel to Fk. This implies that there exists a thresh-old climb mobility mc (and correspondingly a threshold tem-perature T*) at which Vk = 0 and above (below) which it is(anti-)parallel to Fk even at constant F k ¼ �sh=

ffiffiffi2p

. Finally,we note that both V\ and Vk are independent of the misori-entation h, in agreement with the simulation data.

4.3. Migration of (1 1 1)[1 1 0] boundaries

We analyze the migration of the (1 1 1)[1 1 0] boundaryfollowing a similar line of reasoning as in the case of the(1 0 0)[1 1 0] boundary. The driving forces due to s ¼ r0xyon the individual segments of this boundary are:

f1 ¼s

2ffiffiffi3p

cos /

�ffiffiffi6p

cos /ffiffiffi6p

sin /

0B@

1CA; f2 ¼

f1x�f1yf1z

0B@

1CA;

and f3 ¼sffiffiffi3p

1

0

0

0B@

1CA: ð10Þ

This boundary migrates as a result of the driving forceson all segments, in contrast to the (1 0 0)[1 1 0] boundary(where the force along the direction perpendicular to theboundary on one of the segments is zero). Since there isno external driving force tangent to the boundary plane

on the b3; n̂r3� �

segment, it is carried along by the

b1; n̂r1


� �segments as they move within the

boundary plane. The total driving force on the boundaryis (see Eqs. (C.1), (C.4) and (C.5)):

F ¼F ?0

F k

0B@

1CA ¼ shffiffiffi

3p

1

0ffiffiffi2p

0B@

1CA; ð11Þ

where F\ and Fk refer to the components of the drivingforce for migration and tangential motion (along thez-direction), respectively.

Next, we compute the velocities that would result fromthe external driving forces alone on the individual disloca-tion segments based on Eq. (10):

v1 ¼ D0s�D1mg þ D2McD3mg þ D4McD5mg þ D6Mc

0B@

1CA; v2 ¼

v1x�v1yv1z

0B@

1CA;

and v3 ¼mgsffiffiffi

3p

1

0

0

0B@

1CA; ð12Þ

where Mc ¼ ½m�2g þ ðm�2c � m�2g Þ sin2 a��1=2 and

a ¼ b1\n̂r1 ¼ b2\n̂r2 ¼ arccosffiffiffi2pðcos /þ

ffiffiffi3p

sin /Þ� �

3�,with the coefficients Dk(/) as given in Section D.

The applied stress drives the b3; n̂r3� �

mixed segment toglide in its slip plane, but yields both glide and climb forces

on the b1; n̂r1� �


mixed segments. Neverthe-

less, these segments are held in place relative to one anotherby internal forces between the segments so that the bound-ary undergoes a rigid translation at a steady-state velocityunder a constant applied stress.

As in the (1 0 0)[1 1 0] boundary case, the general fea-tures and trends of simulation results for this boundary,displayed in Figs. 6 and 7, may be understood by consider-ing the velocities of individual dislocation segments, Eq.(12). The tangential velocity (along z) of the dislocation tri-ple junctions (and overall boundary structure) is greater

than that for the b1; n̂r1� �


segments by a fac-

tor of 1/sin2/. The velocity of this boundary is bounded bythe fastest and slowest individual segments, such thatv1x < V\ < v3x, and v3z < Vk < v1zsin

�2/, and is expected

to be comparable with the velocity of segments b1; n̂r1� �


, which are driven to climb out of their respec-

tive slip planes:

V ?J v1x ¼ D0ð�D1mg þ D2McÞs’ ð�0:21mg þ 0:42McÞ ð13Þ

and

V kK v1z sin�2 / ¼ sin�2 /D0ðD5mg þ D6McÞs’ ð0:23mg þ 0:78McÞs: ð14Þ


The numerical factors in these expressions wereobtained by setting / = 88.7� (see Table 1). Comparisonbetween the simulation data and the analytical bounds dis-played in Figs. 6b and 7b clearly shows that the boundaryvelocities extracted from the simulations are indeedbounded by the analytical expressions. The analytical

bounds associated with segments b1; n̂r1� �


closely match the boundary velocities determined fromthe simulations.

As in the case of the (1 0 0)[1 1 0] boundary, our analysisand numerical simulations indicate that the motion of the(1 1 1)[1 1 0] boundary is not dislocation-climb limited,and can switch directions as the climb mobility is varied.For this boundary, however, the tangential motion Vk isalways in the direction of the driving force Fk, while itsmigration V\ displays an intriguing switch in sign uponvarying mc. In particular, V\ is anti-parallel to F\ forlow values of mc; in the limit that mc! 0, V\ �sin�2/D0D1mgs ’ �0.21mgs for F ? ¼ sh=

ffiffiffi3p

, and so the bound-ary migrates in the direction opposite to the driving forceF\. On the other hand, for sufficiently large values of mc,V\ is parallel to F\. This again implies that there exists athreshold climb mobility value mc (and correspondingly athreshold temperature T*) at which V\ = 0 and above(below) which it is (anti-) parallel to F\ for constantF ? ¼ sh=

ffiffiffi3p

. Again, we note that both V\ and Vk are inde-pendent of the misorientation h, in agreement with the sim-ulation data.

5. Discussion

We have presented the first systematic investigation ofstress-induced migration of LAMGBs using a three-dimen-sional discrete dislocation dynamics simulation methodwhich accounts for both topological changes in the disloca-tion network and dislocation reactions. As described inSection 2, the mixed boundaries were constructed fromtwo sets of intersecting lattice dislocations having mixededge/screw character. Within each set, dislocation lineswere initially straight and uniformly spaced. The twomixed boundaries were represented initially by bcc latticedislocations, i.e. a

2h1 1 1i dislocations in the {1 1 0} slip

planes. Each of these boundaries represents a broader classof mixed boundaries corresponding to m̂ � ðb1 � b2Þ ¼ 0,respectively. Upon relaxation, dislocation reactions at theintersections lead to approximately planar hexagonal dislo-cation networks, as observed in experiments [30].

The motion of such boundaries was investigated as afunction of the applied stress for several values of disloca-tion climb to glide mobility ratios and misorientationangles. Under an applied shear stress that provides thedriving force for boundary migration, the hexagonal dislo-cation network was found to undergo a rigid translationalong directions perpendicular and parallel to the bound-ary plane. The migration of a mixed boundary can occur

by dislocation glide, and necessarily couples to the tangen-tial motion (parallel to the boundary plane) because theapplied stress that gives rise to boundary migration alsogenerates a force on the dislocation network parallel tothe boundary plane. The tangential motion of the grainboundary dislocations having a net Burgers vector in theplane of the boundary is expected to induce grain boundarysliding, while the migration of the boundary is expected toproduce shear deformation (arising from the tilt characterof boundary) and rotation (arising from the twist characterof boundary) of the crystal lattice within the region tra-versed by the migrating boundary [5,16,31].

In the analysis presented in Section 4, we obtained the-oretical estimates for the configurational parameters andthe translation velocity of the dislocation network of theseLAMGBs. Under the applied stresses that drive boundarymigration, two of the dislocation segments that make up

the dislocation array, i.e. b1; n̂r1� �


, are forced

to climb out of their respective slip planes as they glide,

while one of the dislocation segments, i.e. b3; n̂r3� �

, can

move by pure glide. Although the segments are driven tomove at different velocities, every segment within theboundary moves with the same (i.e. boundary) velocity atsteady-state; this occurs through the internal forcesbetween segments. While the internal forces between thesegments are a priori unknown and not easily evaluatedanalytically, rigorous bounds on the grain boundary veloc-ity were obtained. The bound associated with the climbingdislocation segments provides a good estimate of theboundary velocity as a function of the applied stress, dislo-cation climb mobility and misorientation.

Our simulations and theoretical analyses show that themigration and tangential velocities of each boundary aredirectly proportional to the applied stress and independentof the misorientation h. In general, we expect that thedependence of the grain boundary velocity V on the drivingforce F is given by V ¼MGB � F, where MGB denotes thegrain boundary mobility tensor. Given that V is indepen-dent of h and that F is proportional to h as per Eqs. (6)and (11), this implies that the grain boundary mobility ten-sor is inversely proportional to h, i.e. MGB h�1. Further-more, MGB is expected to be both positive definite andsymmetric (as per Onsager’s reciprocity relations), anddepend only on the constituent dislocation structure ofthe boundary. In three spatial dimensions, MGB containssix independent components, which may be found by eval-uating V arising from each component of F separately.However, it can be seen from Eqs. (C.2) and (C.4) thatthe components of the driving force F for boundary motiondue to the external stress r0 cannot be independently variedto resolve each and every element of MGB; indeed, asshown in Eqs. (C.3) and (C.5), the applied stress s that gen-erates boundary migration also induces a non-zero drivingforce parallel to the boundary plane. Nevertheless, we mayevaluate the effective mobilities M? ¼ V ?=s and


Mk ¼ V k=s, each of which is a linear combination of thecomponents of MGB.

To this end, based on the values of the dislocation glidemobility in bcc Mo obtained from molecular dynamics sim-ulations at 300 K [29], the magnitude of the effective mobil-ity for migration M? at temperatures for whichmc� ms � mg (unlike mc and ms, mg does not vary sub-stantially with temperature) may be estimated to be0.46 m s�1 MPa�1 and 0.37 m s�1 MPa�1 for mixedboundaries (1 0 0)[1 1 0] and (1 1 1)[1 1 0], respectively.Intriguingly, our simulations of the migration of(1 1 1)[1 1 0] boundaries also showed that the direction ofgrain boundary migration depends on the dislocation climbmobility (temperature). In addition, the tangential compo-nent of the velocity of some boundaries (e.g. the(1 0 0)[1 1 0] boundaries here) may also depend on disloca-tion climb mobility and hence temperature. Thus, the direc-tion of boundary displacement may be controlled bytemperature, and to a lesser extent, the constituent disloca-tion structure.

The switch in the sign for the grain boundary velocityarises from the competition between dislocation glide andclimb. If the contributions associated with glide and climbare opposite in sign, then a switch in the sign for the veloc-ity may be observed at sufficiently high temperature. Theo-retically, we can anticipate switch-over for each velocitycomponent and approximate the threshold dislocationclimb mobility mc (and hence temperature T

*), for whichthe change in direction occurs by solving for the value ofmc when V\ = 0 and Vk = 0 within the theoretical frame-work established in Section 4. If 0 < mc � mg, then thereexists a possibility that the switch in sign for the boundarymigration or tangential velocity may be observed underexperimentally relevant conditions.

Geometrically, the switch in the sign for the migrationor tangential velocities may be understood by consideringFig. 8. First, we note that all simple LAMGBs studied hereare not dislocation-climb limited, i.e. V\ and Vk are bothfinite at mc = 0 when F – 0. Furthermore, the boundaryvelocity for mc = 0, V(mc = 0), is in general misaligned withrespect to the driving force F. Now, as mc gradually

(a)Fig. 8. Schematic representation of the geometrical relationship between boundboundary motion, and the boundary plane that gives rise to the switch in sign foIf V(mc = 0) and V(mc = mg) both lie between the boundary plane and its normNote that even when mc! mg, V in general is not parallel to F, i.e. while the mograin boundary mobility tensor remains anisotropic.

increases from zero, V tends to rotate towards F, i.e. theboundary mobility tends to become more isotropic. Theswitch in sign for either the migration or tangential compo-nent of V will depend on the orientation of the boundaryplane relative to the boundary velocity V at mc = mg,V(mc = mg), and the boundary velocity V at zero climb.If the boundary plane lies between V(mc = mg) andV(mc = 0), as shown in Fig. 8a, the switch in sign can occurfor the migration component of V. On the other hand, ifthe normal to the boundary plane m̂ lies betweenV(mc = mg) and V(mc = 0), as shown in Fig. 8b, then theswitch in sign can occur for the tangential component ofV. Finally, if V(mc = mg) and V(mc = 0) both lie betweenthe boundary plane and its normal, then no switch in signfor any velocity component is expected to occur. It shouldbe noted that for an arbitrary (as opposed to simple) LAM-GB, V(mc = 0) may assume a direction that is either paral-lel (perpendicular) to the grain boundary. In such cases, themigration (tangential) velocity is dislocation-climb con-trolled, and no switching behavior is expected to occur.

To make the above discussion more concrete, consider abroader class of simple LAMGBs in a bcc crystal, whichmay be described by two sets of equally spaced, intersectinga2h1 1 1i -type dislocations, and which subsequently react to

form a ah1 0 0i-type segment. Employing the theoreticalframework in Section 4, a systematic sampling of 1536 fun-damental boundary plane orientations relative to the Bur-gers vector pairs that make up the mixed boundary, i.e.b1; b2 2 a2 h1 1 1i and the corresponding boundary planenormals m̂ consistent with m̂ � ðb1 � b2Þ ¼ 0, confirms thatthe b1 and b2 segments are in general driven to simulta-neously glide and climb, while the b3 = b1 + b2 segment isdriven to glide (or cross-slip if it is a screw segment) underan applied stress that induces the migration of the bound-ary. Examination of the driving force on segment b1 (or b2)for every orientation of the boundary plane in the ensembleindicates that a change in sign for the migration or tangen-tial translation is not specific to the (1 0 0)[1 1 0] or(1 1 1)[1 1 0] mixed boundaries we have simulated, but isa general feature that applies to most such mixed bound-aries. In the case of m̂ � ðb1 þ b2Þ ¼ 0, the change in sign

(b)ary velocity V at zero and non-zero climb mobilities, the driving force F forr (a) migration and (b) tangential components of the boundary velocity V.al, then no switch in sign for any velocity component is expected to occur.bility tensor for the individual dislocation segments becomes isotropic, the


for tangential motion occurs for 70% of the boundaries at2:6� 10�4 K mc=mg K 0:21. On the other hand, the changein sign for migration is a generic feature occurring at0:22 K mc=mg K 0:27 for all boundaries in the ensemblesuch that m̂ � ðb1 � b2Þ ¼ 0.

In practice, the predicted value of mc=mg may be muchlarger than the range of values realizable under typicalexperimental conditions, as is the case with the prototypi-cal (1 0 0)[1 1 0] and (1 1 1)[1 1 0] LAMGBs. Althoughthe actual change in sign for the boundary velocity maynot be directly observable, our finding has important con-sequences for interpreting experimental measurements ofgrain boundary mobility for such boundaries. For exam-ple, changing the temperature will lead to changes in boththe dislocation glide and climb mobility, and hence theratio of these two mobilities. This implies that the bound-ary mobility will not, in general, be a simple Arrheniusfunction, but will have contributions from the activationenergies of both the dislocation climb and glide mobilities.

In the work of Gorkaya et al. [5], the grain boundarymisorientation decreases during the stress-induced bound-ary migration. This occurs because finite-sized grains rotateby expelling dislocations through the free surfaces in theseexperiments. In the present simulations, the grain bound-ary structure is periodic such that no free surfaces are pres-ent; hence, no dislocations can be expelled and there can beno misorientation change. However, based upon the resultspresented above, we can evaluate the influence of suchchanges in misorientation on stress-driven boundarymigration in finite-sized grains. The simulations and analy-sis show that the grain boundary migration velocity V\ isindependent of misorientation and hence will not changeif dislocations were expelled (as long as the structure ofthe dislocation network is preserved). However, the drivingforce for boundary migration F\ scales proportionally tothe misorientation h and will therefore change if disloca-tions are expelled. This implies that if h decreases duringboundary migration, the boundary mobility will increaseas 1/h. While the present simulations say nothing aboutdislocation egress through a free surface, a qualitative senseof that rate may be deduced from the tangential velocity ofthe dislocation structure that makes up the grain boundary.

Finally, in the present work, the boundaries were alsoassumed to be free of impurities and “extrinsic” or redun-dant dislocations that are not geometrically necessary (andthus do not contribute to the overall misorientation of theboundary). Previously, we have shown, both numericallyand analytically, that the presence of such extrinsic disloca-tions can significantly alter the mobility of simple low-angle tilt boundaries [3]. Similarly, it is well known thatthe interaction of grain boundaries with solutes can alsosignificantly alter boundary mobility [32–36]. The analyti-cal framework employed in this work can be used to quan-titatively assess these effects on grain boundary mobility.While the present approach is limited to low-angle grainboundaries, it does allow for a much more mechanisticand structurally aware understanding of boundary migra-

tion than is possible for high-angle boundaries, where theboundary structure is much more complex (and boundarymotion is not easily describable in terms of discrete disloca-tion dynamics), and for which the coupling of externallyapplied stress to the boundary structure is not known.

6. Conclusion

Dislocation dynamics simulations and theoretical analy-ses of the stress-driven migration of LAMGBs have beenemployed to provide a detailed understanding of the influ-ence of grain boundary structure on boundary motion.Unless impeded by other crystal defects such as impuritiesand extrinsic dislocations, simple LAMGBs can move bydislocation glide. Concomitant with the stress-inducedmigration of such boundaries is the motion parallel tothe boundary plane. The results presented here showedan intriguing reversal in the direction of boundary migra-tion or tangential motion as a function of dislocation climbmobility, and hence, temperature. Theoretical analysis ofover 1500 grain boundaries suggests that this type of phe-nomena is general for this class of boundaries.

Appendix A. Construction of LAMGBs

Whereas a simple low-angle symmetric tilt boundarymay be constructed using just one set of dislocations, aLAMGB must contain at least two sets of intersecting dis-locations [2]. In the present study, we modeled LAMGBs

using two sets of straight dislocations b1; n̂1� �

and

b2; n̂2

� �that are uniformly spaced with line spacings D1

and D2, respectively. The intersecting dislocation linesuniquely define the boundary plane, which is perpendicularto the boundary normal vector:

m̂ ¼ n̂1 � n̂2n̂1 � n̂2�� : ðA:1Þ

For such a dislocation array to represent a grain bound-ary, it must be free of long-range stresses. This is achievedby relating the parameters of the dislocation array (such asthe Burgers vector, dislocation line direction, and disloca-tion spacing of each set) to the five macroscopic degreesof freedom for a grain boundary through Frank’s formula.For example, the dislocation line directions must be relatedto the Burgers vectors and boundary plane normal m̂ asfollows:

n̂1 ¼b2 � m̂b2 � m̂j j

and n̂2 ¼m̂ � b1m̂ � b1j j

: ðA:2Þ

If this condition is satisfied, then the dislocation arrayrepresents a boundary between grains that are misorientedabout the axis:

û ¼ b1 � b2jb1 � b2j; ðA:3Þ


at an angle that is inversely proportional to the dislocationspacing in each set:

h ¼ jb1 � b2jb2 � m̂j j

� 1D1¼ b1 � b2j j

m̂ � b1j j� 1D2: ðA:4Þ

As the above procedure is equally valid for a twist bound-ary, the Burgers vectors and line directions of dislocationsused to construct a mixed boundary must be judiciously cho-sen so that the boundary has both twist and tilt character, i.e.m̂ is neither parallel nor perpendicular to û. If the anglebetween m̂ and û is v, the extent of twist character may bespecified in terms of the deviation of m̂ from pure tiltcharacter, i.e. p/2 � v; this is also known as the “twist angle”[5]. Alternatively, the fraction of twist and tilt character for amixed boundary may be determined by resolving m̂ into twist(cosv) and tilt (sinv) components and making use of the trig-onometric identity sin2v + cos2v 1. Therefore, the frac-tion of twist and tilt for a mixed boundary may bedetermined as cos2v and sin2v, respectively. Both descrip-tions are employed.

In the present study, we focus on simple LAMGBs, forwhich the magnitude of the lattice Burgers vectors and dis-location spacing in each set are equal, i.e. b1 = b2 = b andD1 = D2 = D. In this case, Eq. (A.4) yields b2 � m̂j j ¼m̂ � b1j j. This implies that sinj2 = sinj1, where j1 and j2

are the angles between m̂ with b1 and b2, respectively.Hence, j2 = j1 or j2 = p � j1 and this implies Eq. (4).

Hence, once the Burgers vectors b1 and b2 are chosen,Eqs. (A.1), (A.2) and (4) are employed to determine the dis-location line directions n̂1 and n̂2, and the boundary planenormal m̂ self-consistently. The restrictions on the parame-ters of the constituent dislocations imply that the boundarynormal must be perpendicular to either b1 + b2, or b1 � b2.It should be noted that m̂ cannot simultaneously fulfill bothconditions; otherwise the boundary loses its mixed charac-ter and becomes a pure twist boundary, since m̂kb1 � b2implies m̂kû by Eq. (A.3).

Appendix B. Simulation geometry

Regardless of the combination of a2h1 1 1i Burgers vec-

tors chosen, the boundaries can be constructed to havethe same orientation within the simulation box. To ensurethat the boundary is parallel to the yz-plane of the rectan-gular parallelepiped simulation cell, the simulation coordi-nate system is defined as follows:

x̂ ¼ m̂ ¼ n̂1 � n̂2n̂1 � n̂2�� ; ŷ ¼

n̂1 � n̂2n̂1 � n̂2�� ; and ẑ ¼

n̂1 þ n̂2n̂1 þ n̂2�� :

In this coordinate system, the dislocation line directionshave reflection symmetry about ẑ and may be written as:

n̂1 ¼ 0 ny nz� �

and n̂2 ¼ 0 � ny nz� �

; ðB:1Þ

with ny > 0 and nz > 0. The periodicity of the lozenge net-work is Ly and Lz along the ŷ and ẑ, respectively, such that:

LyLz¼ n1y

n1z: ðB:2Þ

Using Eqs. (A.2), (A.3), (B.1) and (B.2), we find that:

b1yLy ¼ b1zLz;and that the components of b1 and b2 are related as follows:b2x = ±b1x, b2y = �b1y, and b2z = b1z. Consequently, if:b1 ¼ bx by bz

� �; where bx – 0 and

bz ¼LyLz

by – 0; ðB:3Þ

b2 and b3 have the following components:

b2 ¼�bx � by bz� �

if m̂ � ðb1 þ b2Þ ¼ b1x þ b2x ¼ 0bx � by bz� �

if m̂ � ðb1 � b2Þ ¼ b1x � b2x ¼ 0

(;

ðB:4Þ

b3 ¼2 0 0 bz½ � if m̂ � ðb1 þ b2Þ ¼ b1x þ b2x ¼ 02 bx 0 bz½ � if m̂ � ðb1 � b2Þ ¼ b1x � b2x ¼ 0

: ðB:5Þ

Regardless of the orientation of the boundary plane, itcan be shown from Eqs. (B.1), (B.3) and (B.4) that

b1 � n̂1 ¼ b2 � n̂2, which means that dislocation b1; n̂1� �

has

the same character angle as dislocation b2; n̂2� �

; the char-

acter angle for b1 and b2 varies from one boundary toanother. When these react in the boundary to form

b1; n̂r1

� �; b2; n̂r2


� �dislocation segments,

the resulting dislocation line directions will once again havereflection symmetry about ẑ:

n̂r1 ¼ 0 nry nrz� �

; n̂r2 ¼ 0 � nry nrz� �

; and

n̂r3 ¼ ẑ ¼ 0 0 1½ �; ðB:6Þ

with nry > 0 and nrz > 0. From Eqs. (B.3)–(B.6), it can sim-

ilarly be shown that b1; n̂r1� �

will also have the same char-

acter angle as b2; n̂r2� �

, while b3; n̂r3� �

is a pure screw

segment for m̂ � ðb1 þ b2Þ ¼ 0, but a mixed segment form̂ � ðb1 � b2Þ ¼ 0. Finally, the straight length segments of

b1; n̂r1

� �; b2; n̂r2


� �within the hexagonal net-

work, which we denote as L1, L2 and L3, respectively, canbe found by setting L1 = L2 due to symmetry consider-ations, and noting that the configuration must fit withinthe dimensions of the unit cell Ly � Lz, which remainsthe same before and after reaction:

L1 ¼ L2 ¼1

2

Lynry

and L3 ¼1

2Lz � Ly

nrznry

�:

Appendix C. Migration of simple LAMGBs induced bystress

To find the stress tensor components that couple to themigration of a simple LAMGB, we consider the most gen-


eral form for the externally applied stress r0 and denote the

Peach–Kohler force on segments b1; n̂r1� �

; b2; n̂r2

� �and

b3; n̂r3

� �to be f1, f2 and f3, respectively. From a mechanis-

tic perspective, a boundary moves when subjected to theapplied stress r0 due to a driving force per unit area:

F ¼ 2ðf1L1 þ f2L2 þ f3L3ÞLyLz

; ðC:1Þ

with the force for migration F\ being the component of Fperpendicular to the boundary plane (the x-direction in ourcase). For each set of boundaries considered here, it will beshown that there is only one component of r0 that gives anon-zero F\.

For the boundary plane determined by m̂ � ðb1 þ b2Þ ¼ 0,the boundary moves due to the driving force:

F ¼ 2Lz

�r0xzbx�r0xzby

r0xxbx þ r0xyby

0B@

1CA: ðC:2Þ

This implies that the boundary migrates under the stresscomponent r0xz ¼ r0zx, for which the Peach–Kohler forces onthe segments normal to the boundary plane are:f1x ¼ f2x ¼ �r0xzbxnry and f3x = 0, and which result in anet driving force:

F ¼F ?F k0

0B@

1CA ¼ � 2r0xz

Lz

bxby0

0B@

1CA: ðC:3Þ

We note that the force on segment b3; n̂r3� �

normal to the

boundary plane vanishes. This implies that the other twosegments must “drag” the third one in order for the bound-ary to migrate.

On the other hand, for the boundary plane determinedby m̂ � ðb1 � b2Þ ¼ 0, the driving force on the boundary aris-ing from the applied stress r0 is:

F ¼ 2Ly

r0xybx

�r0xxbx � r0xzbzr0xybz

0B@

1CA: ðC:4Þ

Therefore, the boundary migrates under the stress com-ponent r0xy ¼ r0yx, for which the Peach–Kohler forces on thesegments normal to the boundary plane are f1x ¼ f2x ¼r0xybxnrz and f3x ¼ 2r0xybx, and which result in a net drivingforce

F ¼F ?0

F k

0B@

1CA ¼ 2r0xy

Ly

bx0

bz

0B@

1CA: ðC:5Þ

In contrast to the first case, the external stress now givesrise to non-zero driving forces normal to the boundaryplane for all the dislocation segments.

Appendix D. Coefficients Ck(/) and Dk(/)

The coefficients Ck and Dk for the driven velocities ofsegments b1; n̂r1


� �for the (1 0 0)[1 1 0] and

(1 1 1)[1 1 0] LAMGBs, respectively, are as follows:

C0¼ ½4ð2� sin2/Þ��1 D0¼ ½6ffiffiffi2pð5þ2cos2/�2

ffiffiffi3p

sin2/Þ��1

C1¼ 2cos/ D1¼ffiffiffi2pð5

ffiffiffi3p

cos/�6sin/ÞC2¼�3cos/þ cos3/þ4sin/ D2¼

ffiffiffi2pð11

ffiffiffi3p

cos/þffiffiffi3p

cos3/

�12sin/�6cos2/sin/ÞC3¼ð�2cos2 /þ sin2/Þcos/ D3¼�cos/ð21þ9cos2/�11

ffiffiffi3p

sin2/ÞC4¼ð3�cos2/� sin2/Þcos/ D4¼�cos/ð9þ3cos2/�

ffiffiffi3p

sin2/ÞC5¼ð1þcos2/� sin2/Þsin/ D5¼ð21þ9cos2/�11

ffiffiffi3p

sin2/Þsin/C6¼ð3�cos2/� sin2/Þsin/ D6¼ð9þ3cos2/�

ffiffiffi3p

sin2/Þsin/:

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Stress-driven migration of simple low-angle mixed grain boundaries1. Introduction2. Simulation method3. Results3.1. Relaxation3.2. Migration

4. Analysis4.1. Relaxation4.2. Migration of (1 0 0)[1 1 0] boundaries

5. Discussion6. ConclusionAppendix A. Construction of LAMGBsAppendix B. Simulation geometryAppendix C. Migration of simple LAMGBs induced bystressReferences

stress-driven migration of simple low-angle mixed grain...

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