Kinetic Monte Carlo method for dislocation migration in the presence of solute

Chaitanya S. DeoPrinceton Materials Institute, Princeton University, Princeton, New Jersey 08540, USA

and Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48105, USA

David J. SrolovitzDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544, USA

Wei Cai and Vasily V. BulatovChemistry and Materials Science Directorate, Lawrence Livermore National Laboratory, Livermore, California 94550, USA

(Received 21 April 2004; published 6 January 2005)

We present a kinetic Monte Carlo method for simulating dislocation motion in alloys within the frameworkof the kink model. The model considers the glide of a dislocation in a static, three-dimensional solute atomatmosphere. It includes both a description of the short-range interaction between a dislocation core and thesolute and long-range solute-dislocation interactions arising from the interplay of the solute misfit and thedislocation stress field. Double-kink nucleation rates are calculated using a first-passage-time analysis thataccounts for the subcritical annihilation of embryonic double kinks as well as the presence of solutes. Weexplicitly consider the case of the motion of ak111l-oriented screw dislocation on a{011}-slip plane inbody-centered-cubic Mo-based alloys. Simulations yield dislocation velocity as a function of stress, tempera-ture, and solute concentration. The dislocation velocity results are shown to be consistent with existing experi-mental data and, in some cases, analytical models. Application of this model depends upon the validity of thekink model and the availability of fundamental properties(i.e., single-kink energy, Peierls stress, secondaryPeierls barrier to kink migration, single-kink mobility, solute-kink interaction energies, solute misfit), whichcan be obtained from first-principles calculations and/or molecular-dynamics simulations.

DOI: 10.1103/PhysRevB.71.014106 PACS number(s): 61.72.Lk, 61.72.Bb, 02.70.Uu, 61.82.Bg

I. INTRODUCTION

The principal mechanisms of crystal plasticity involve themotion of crystal defects of different dimensionality. Thesedefects include point defects(e.g., vacancies, self-interstitials), line defects(dislocations), planar defects(e.g.,twins), and volumetric defects(e.g., resulting from marten-sitic transformations). Plasticity in most metallic alloys is amacroscopic manifestation of dislocation activity, except atvery high temperature and small stress. Dislocations glidealong crystal planes(glide planes), cross-slip, or climb ontoother glide planes and interact with each other and with otherdefects within the crystal. The plastic response of metals istypically the result of the motion of a large number of dislo-cations.

Dislocation dynamics(DD) simulations(e.g., see Bulatovet al.,1,2 Gomez Garciaet al.,3 Tanget al.,4 and Zbibet al.5)model the motion and interaction of a large number of dis-locations and their influence on the mechanical properties ofcrystals. These simulations treat the dislocations as line seg-ments connected through nodes. The dynamics of dislocationmotion is thus reduced to the dynamics of the nodes anddepends on the driving forces(elastic or chemical) actingupon the dislocations. The elastic interactions between dislo-cations occur over large distances and are relatively com-plex. The simulation of the motion of a large number ofdislocations over physically meaningful time scales is a ma-jor computational bottleneck for DD simulations. In alloys,dislocation dynamics is further complicated by the interac-

tions of dislocations with solute atoms(solute atoms interactelastically with dislocations and modify the dislocationcore).

In order to simulate a large number of dislocations forreasonable lengths of time, DD simulations employ simplerules that dictate the motion of the dislocations. These mo-bility rules prescribe the relationship between the velocity ofa dislocation and the forces acting upon it. The forces mayarise from the applied stress, elastic self-interactions,dislocation-dislocation interactions, and, in the case of al-loys, interactions with solute atoms. The relationship be-tween dislocation velocity and force is commonly describedvia power laws6,7 or exponential8,9 relationships. The reli-ability with which the DD simulations mimic the plastic be-havior of the crystal depends on the accuracy of these mo-bility rules.

Quantitative experimental information on the intrinsicmobility of individual dislocations is rare. On the other hand,first-principles calculations and atomic-level simulationssuch as molecular dynamics49–52 (MD) may be utilized toobtain some of the energetics of dislocation motion. How-ever, determination of dislocation force-velocity(mobility)relations from simulations requires tracking the dynamics ofindividual dislocations over relatively long time scaless,msd. Given the inherently small time scales of MD(10−12 s), such simulations are rarely possible today.

Kinetic Monte Carlo (KMC) methods10–12 provide abridge between the large-scale dislocation dynamics simula-tions of crystal plasticity behavior and the atomic-scale

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simulation of dislocations that can be used to determine themobilities of dislocations. These KMC methods use informa-tion on the energetic barriers to atomic hopping or kink mi-gration that control the motion of a dislocation in order toobtain the relationships between the velocity of the disloca-tion and the driving forces acting upon it. This informationcan provide the requisite input to mesoscale dislocation dy-namics simulations.

Several researchers have performed kinetic Monte Carlosimulations of dislocation motion via kink migration. Thesesimulations were based upon the assumption that disloca-tions are an ensemble of pure screw and edge segments.Rates of the fundamental processes in kink migration-controlled dislocation dynamics(i.e., kink pair nucleation,lateral kink migration, and kink-kink annihilation) were de-termined and a KMC procedure was employed to select be-tween these types of events at each simulation step. Thesemodels incorporated the long-range elastic interactions be-tween the kinks,8 rather than simply using the line tensionapproximation employed in most continuum models.13 Linand Chrzan14 performed KMC simulations of the dislocationmigration in Ta as a function of temperature and appliedstress by mapping the edge and screw segments onto a fixedlattice. Caiet al. employed a similar KMC method for simu-lating the motion of two partial dislocations in Si(Ref. 15)and a singlek111l-oriented screw dislocation in Mo.16,17Thescrew dislocation in Mo was allowed to move on all threeavailable{110} glide planes that comprised thek111l {110}slip system. All these studies were focused on dislocationmobility in pure crystals.

Since engineering materials are never pure, this study fo-cuses upon dislocation motion in a solid solution alloy at lowtemperature. In particular, we employ a KMC method tosimulate the motion of screw dislocations in a substitutional,solid solution alloy. The dislocation moves by the kinkmechanism—double-kink nucleation, kink migration, andannihilation. The dislocation is constrained to move on aglide plane(i.e., cross-slip and climb are not considered),while the solutes are distributed randomly in three dimen-sions. We first provide a detailed description of the simula-tion model and then apply this model to determine the influ-ence of temperature, applied stress, and solute concentrationon the velocity of ak111l-oriented screw dislocation in Mowith substitutional solute.

II. SIMULATION MODEL

In the present simulation, we model the glide of a dislo-cation on a plane, while the solute is distributed in threedimensions. The dislocation has, on average, a screw orien-tation, but consists of screw and edge dislocation segments.This model does not allow for cross-slip of the screw seg-ments or climb of the edge segments.

A. Kink model of dislocation motion

The dislocation is assumed to move via a kinkmechanism—this involves the nucleation of pairs of kinks ofopposite sign, kink migration, and the mutual annihilation of

oppositely signed kinks. Elastic theory for kink formationand diffusion as a mechanism for dislocation motion wasfirst developed by Eshelby18 and Seegeret al.19–21 and hasbeen reviewed in detail by Hirth and Lothe.8 The MonteCarlo simulation samples a list of different classes of pos-sible kink nucleation and migration events. Kink-kink anni-hilation is considered as a special case of kink migration.

Kinks nucleate in oppositely signed pairs. The kinks on ascrew are of edge type and vice versa. Following Hirth andLothe,8 the nucleation rate for a double kink of width 1b(whereb is the Burgers vector) is

jdk = v expS−2Ek + Wint − teffb

2h/2

kBTD , s1d

wherev is the attempt frequency factor,Ek is the energy ofan isolated kink,Wint is the interaction energy between thetwo kinks in the pair,h is the kink height,kBT is the thermalenergy, andteff is the local resolved stress that includes theapplied stress, the stress field of the solutes, and the selfstress field associated with the screw segments. Hirth andLothe calculate the interaction energyWint depending on theseparation between(edge segments) kinks zk as8

Wint = −mh2

8pzkbSbz

21 + n

1 − n+ bx

21 − 2n

1 − nD , s2d

where the nominal orientation of the dislocation line is alongz,bx andbz are thex andz components of the burgers vector,m is the shear modulus, andv is Poisson’s ratio.

The energy of the isolated kink,Ek, may be obtained fromatomistic simulations. For example, Xu and Moriarty22 usedatomistic simulations employing empirical many-body po-tentials for a kink on ak111l-oriented screw dislocation inMo and found thatEk<1 eV. The entire numerator in theexponential in Eq.(1) is referred to as the kink pairenergy19–21or as the double-kink energyEdk (Ref. 8). Nucle-ation of a double kink in the presence of a solute depends onthe solute-core interactions. In the present study, we assumethat a solute atom is an obstacle to kink migration(i.e., itprovides an additional, localized barrierEb to kink motion).

A kink, once formed, can migrate along the dislocationline until it meets a kink of opposite sign and annihilates.The rate of migration of the kinks depends on the magnitudeof the secondary Peierls barrier,Wm (i.e., the activation en-ergy for kink motion). If the secondary Peierls barrier is verylarge(e.g., in the case of Si) or if kink motion is hindered bya solute atom, the migration rate(or kink diffusivity) is givenby the following expression:8,15

jm = v expS−Wm − teffb

2h/2

kBTD . s3d

Kink mobility in the presence of a solute atom also dependson the solute-core interactions. As in the case of double-kinknucleation, we assume that the solute increases the activationbarrier for kink diffusivity by an amount proportional to thelocal solute barrierEb.

If the secondary Peierls barrier for the kink motion is verysmall in the absence of solute[,0.0005 eV for ak111lscrew dislocation in Mo(Refs. 16,17)], the kink diffusivity is

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very large. In such cases, the kink migration is not thermallyactivated but is characterized by a lattice damping(drag)coefficientBk, which is dominated by phonon drag. In such acase, the kink velocityvk is proportional to the driving forceexperienced by the kink:9

nk =teff

Bkb, s4d

whereteff is the total stress. When the kink is adjacent to asolute atom, we describe the motion of the kinks using Eq.(3) and otherwise use Eq.(4).

B. Geometric description of the dislocation and solute

The simulation cell is represented as a three-dimensional(3D) simple cubic lattice(see Fig. 1), where the dislocationglide plane has normaly (at position y=0). The Burgers

vector bW is parallel to thez direction, as are the screw seg-ments. The spacing in thex andz directions is governed bythe nature of the dislocation being modeled. For example, inmodeling a screw dislocation in Mo, we assume that theBurgers vector is of thek111l type and, hence, thez directionin the simulation lattice corresponds to ak111l direction inthe crystal. The simulation cell lattice spacing in thez direc-tion is b=Î3a0/2, the lattice spacing in thex direction ish=a0

Î2/3, and the spacing between the{110} glide planes isl =a0/Î2. An order parameterfsxi ,0 ,zid is defined over theglide plane and is used to identify the location of the dislo-cation:

fsxi,0,zid = H1, sxi,0,zid is part of slipped glide plane,

0, sxi,0,zid is part of unslipped glide plane.J

s5d

A dislocation line segment lies between any pair of nearest-neighbor(square) lattice points in the glide plane that havedifferent values off.

The solute atoms are distributed in three dimensions andare represented by a fieldhsxi ,yi ,zid which is one at latticepoints occupied by solute atoms and zero otherwise. Thesolute atoms withinÎ2a0 of the glide plane interact directly

with the dislocation core[this corresponds to five(110)planes in the simulation lattice]. Using the geometric schemeproposed above, we can describe the positions of all disloca-tion segments and solutes in sufficient detail to model dislo-cation motion in the presence of solutes via a kink model.

C. Dislocation self-stress

A dislocation creates a stress field that produces forces onother dislocations and results in interaction between the dis-location and solute atoms. The self-stress of a dislocation isthe stress exerted on a dislocation segment by all other dis-location segments. Many earlier approaches accounted forthe self-stress effect by replacing the dislocation line with asmooth curve with finite line tension(it is the line tensionthat models the self-stress). This simple approximation im-plies that the free energy of the dislocation is simply propor-tional to the dislocation lengthL, with a constant of propor-tionality equal to the dislocation line energy.8,13 Thisapproximation does not account for the true, nonlocal natureof the interaction between dislocation segments and isclearly inapplicable in many circumstances, such as wherethe variations in dislocation line curvature are large. Further,the force on a dislocation line segment, in the line tensionapproximation, is independent of the location of all otherdislocation line segments and only depends on the local linecurvature.

In our model, every dislocation configuration is repre-sented by an ensemble of straight line segments in the geo-metric scheme described above. The stress field associatedwith a segmented dislocation of arbitrary shape can be cal-culated everywhere as a summation of the stress fields asso-ciated with each segment, as described below. The accuracyof this approach depends upon the applicability of an edgeand screw segment decomposition of the dislocation shapeand the length scale of these segments. The computationalwork associated with calculating the stress field along theentire dislocation line increases as the square of the totallength of the dislocation line. Because of this overhead,simulations of dislocation dynamics that properly account fordislocation self-interactions have not been performed untilrecently.23 A similar approach was recently proposed for de-scribing dislocation motion in silicon15 and puremolybdenum16,17 within the kinetic Monte Carlo framework.

We employ the approach proposed by Hirth and Lothe8 tocalculate the stress field of a dislocation configuration(rep-resented by a sequence of straight line segments) as a sum-mation of the stress fields of the individual segments. For ageneral segmented dislocation configuration, the stresses ofthe individual line segments can be transformed as tensors toa common coordinate system and summed to give the totalphysical stress. To calculate the stresses of the individual linesegments, the coordinate system shown in Fig. 2(a) isadopted, where the dislocation line segment lies along thezaxis. Following Hirth and Lothe,8 the stresses at an arbitrarypoint rsx,y,zd due to the segmentszA8 ,zB8d with a Burgers

vector b=sbx,by,bzd are obtained by integrating along thesegment fromzA8 to zB8, yielding

FIG. 1. Illustration of the dislocation-solute model. The orderparameter is defined over the glide plane and is used to identify thelocation of the dislocation. The solute atoms are distributed in threedimensions and are represented by a field(see text for details).

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si j = si jszB8d − si jszA8d, s6d

wheresi jsz8d is obtained as

sxxsz8ds0

= bxy

RsR+ ldF1 +x2

R2 −x2

RsR+ ldG + byx

RsR+ ld

3F1 −x2

R2 −x2

RsR+ ldG , s7ad

syysz8ds0

= − bxy

RsR+ ldF1 +y2

R2 −y2

RsR+ ldG− by

x

RsR+ ldF1 +y2

R2 +y2

RsR+ ldG , s7bd

szzsz8ds0

= − bxF 2ny

RsR+ ld+

yl

R3G + byF 2nx

RsR+ ld−

xl

R3G ,

s7cd

sxysz8ds0

= − bxx

RsR+ ldF1 −y2

R2 −y2

RsR+ ldG+ bz

y

RsR+ ldF1 −x2

R2 −x2

RsR+ ldG , s7dd

syzsz8ds0

= bxF n

R−

y2

R3G + byxy

R3 − bzxs1 − ndRsR+ ld

, s7ed

sxzsz8ds0

= − bxxy

R3 + byF−n

R−

x2

R3G + bzys1 − ndRsR+ ld

, s7fd

s0 =m

4ps1 − vd, R= Îx2 + y2 + sz− z8d2, l = z8 − z.

s7gd

Equations(6) and(7) can be applied to each dislocation seg-ment by suitable rotation and translation of the coordinateaxes in Fig. 2(a) and the total self-stress at any point ob-tained by summing over all segments.

In the geometry of the present simulations, we chooseb=s0,0,bzd and constrain the dislocation to lie only on they=0 glide plane. The dislocation segment lengthsad is b or hfor a screw or edge component, respectively. In the coordi-nate system in which the dislocation segment is centered atthe origin and extends along thez axis, the stress at any pointis given by Eq.(6) with the stresses

sxysz8ds0

=5− bx

x

RsR+ ld, z, z18 , z28,

− bxl

xR, z18 , z, z28,

− bxx

RsR− ld, z18 , z28 , z,

6 s8ad

and

syzsz8ds0

=5bx

y

R− bz

xs1 − ydRsR+ ld

, z, z18 , z28,

bxy

R− bz

l

xR, z18 , z, z28,

bxy

R− bz

xs1 − ydRsR− ld

, z18 , z28 , z.6 s8bd

These expressions for the stresses associated with indi-vidual segments were first used in KMC simulations of dis-location motion by the kink mechanism in silicon15 and puremolybdenum.17 The self-stress associated with all of the dis-location segments and the applied stress and stresses from

FIG. 2. (a) Schematic illustration of the interaction between adislocation segmentszA8 ,zB8d and a pointsx,y,zd in the simulationcell. The stress field due to the dislocation segment is described inthe text.(b) Schematic illustration of the short-range and long-rangeinteractions between the dislocation and the solute atoms. The sol-ute atoms are distributed in three dimensions around the dislocation.

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the solutes make up the total stressteff which is used tocalculate double-kink nucleation rates[Eq. (1)] and kink dif-fusivities [Eqs.(3) and (4)].

D. Solute dislocation interactions

Solute atoms interact with the dislocation both throughtheir long-range stress field and through their direct interac-tions with the dislocation core. The long-range component ofthe solute-dislocation interaction contributes toteff and henceto the double-kink nucleation and kink migration rates. Thelong-range stress field of the solute atoms is treated by mod-eling them as point sources of dilatation for which

srr = −m

p

dn

r3 , suu = sff =m

2p

dn

r3 , s9d

wheredn is the magnitude of the dilatation. We define thesolute-dislocation interactions as short-range or long-rangedepending on the distanceur u of the solute from the disloca-tion segment[as shown in Fig. 2(b)]. If the solute is far awayfrom the dislocation—i.e.,ur u.Î2a0—the stress field is cal-culated using Eq.(11) and added to the total stress that actson the dislocation segment. If the solute is near thedislocation—i.e.,ur u,Î2a—the solute interacts with the coreof the dislocation. We assume that such a core-solute inter-action provides a barrier ofWm=Eb to the motion of the kink[see Eq.(3)].

E. Double-kink nucleation

Of the unit processes in the kink diffusion model, nucle-ation of kink pairs is the slowest and, hence, plays an impor-tant role in determining both the velocity of the dislocationand the efficiency of the KMC simulation. The energy barrierthat must be overcome in order to nucleate a double kink ona straight screw dislocation is calculated using the followingexpression:8

DE = 2Ek −mb2h2

8pwS1 + n

1 − nD −

sbhw

2, s10d

whereEk is the energy of an isolated kink,w is the double-kink width (kink-kink separation), s is the effective appliedstress(see above), h is the kink height, andm is the shearmodulus. The first term is the energy to form two isolatedkinks on an otherwise straight dislocation and is positive.The second term is the kink-kink interaction[i.e., the secondterm in Eq.(12)] and is always negative. The last term is theinteraction of the kinks with the long-range stress field(thisexcludes elastic interactions between kinks) and can be ofeither sign. The double-kink energy is plotted in Fig. 3(a)for an applied stress of 0.1tP, where tP is the Peierlsstress and all parameters were chosen to correspond to ascrew dislocation in MostP=0.0025m ,m=12.3Ã104 MPa,Ek=1 eVd.24,25 The double-kink energy increases with in-creasing double-kink width at small kink separations and de-creases after a critical double-kink width. The criticaldouble-kink width(i.e., above which the double-kink spac-ing increases and below which it shrinks) is a function of the

applied stress as shown in Fig. 3(b). For any positive, finitestress the critical double-kink width is finite, decreasing withincreasing(positive) applied stress.

If a double kink is formed with a width smaller than thecritical width, the two kinks attract and quickly self-annihilate. The rates of formation and self-annihilationevents are so fast that the simulation would normally wastemost of its time forming and annihilating subcritical doublekinks. Such a problem was encountered in earlier simulationsof dislocation motion.15 The solution to this problem lies innucleating a double kink of finite widthw0 which is at leastthe critical width w* [see Fig. 3(a)], such that the doublekink, once formed, is stable and embryonic nucleation eventsnever occur. Given the importance of double-kink nucleationwithin the kink diffusion model, it is necessary to ensure thatthe calculation of the rate at which supercritical double kinksform is consistent with the real time spent forming and self-annihilating subcritical double kinks.

FIG. 3. (a) Variation of the double kink energyEdk as a functionof the width of the double kinkw (separation between the twokinks) calculated using Eq.(10) and parameters described in thetext. The applied stress is expressed as a fraction of the PeierlsstresstP. (b) Variation of Edk as a function of the widthw (separa-tion between the two kinks) for different values of the Peierls stresstP.

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In order to overcome the double-kink formation and an-nihilation bottleneck, we employ a first-passage-time analy-sis that integrates over the initial evolution of the embryonicdouble kink by treating it as a one-dimensional randomwalker in double-kink width space. This approach yields thedistribution of elapsed times before successful double-kinknucleation. The time distribution is obtained by applying anexact first-passage-time analysis to the one-dimensional ran-dom walk as modeled by a temporally homogenous discrete-state Markov process.12 Details of this method can be foundelsewhere.26 Incorporation of such a double-kink nucleationmodel is key to efficient simulations.

Since the stress that a nucleating double kink feels varieswith position along the dislocation(and time), the criticaldouble-kink width is spatially and temporally nonuniform.The critical double-kink width is also affected by solutes thatinteract directly with the dislocation core through modifica-tion of the energy landscape(i.e., the solutes modify thebarriers for kink motion). This effect is also incorporated inthe consolidated nucleation rate using the first-passage-timeanalysis. Incorporation of both of these effects into the Mar-kov chain analysis for double-kink nucleation accelerates thesimulations by a factor of approximately 104.

F. Kinetic Monte Carlo procedure

We have described the incorporation and parametrizationof the unit processes in the kink model for dislocation mi-gration, including the effects of temperature, applied stress,solute concentration, and solute strength. We now describethe kinetic Monte Carlo implementation of this model fordislocation motion in the presence of solutes. The KMCmethod is similar to theN-fold way [or Bortz-Kalos-Lebowite(BKL ) method10], where a variable-time incrementis employed to incorporate events that occur on widely dif-ferent time scales. A similar procedure was employed by Caiet al. to simulate dislocation motion in pure Mo(Ref. 16)and Si (Ref. 15) and by Scarleet al.27 to study dislocationmotion in silicon.

We extend these earlier KMC models for dislocation mi-gration to the alloy situation(i.e., with solutes present). Ini-tially, the glide-plane order parameterf that describes thedislocation position is chosen such that there is a singlescrew dislocation in the glide plane. Solute atoms are distrib-uted at random in three dimensions and represented by thesolute field variableh. The simulation cell is periodic in they andz directions(see Fig. 1). The dislocation propagates inthe x direction on they=0 plane. When the dislocationreaches the +x boundary of the simulation cell, the simula-tion cell is extended in that direction by adding additionallattice points and removing the same number of lattice pointsfrom the region near the −x boundary. The additional glide-plane grid points are created with the appropriatef valuesand solute atoms are randomly placed on the new latticepoints. Instead of calculating the stress on each dislocationsegment due to all other segments in an infinitely repeatedunit cell (in the z direction), the self-stress calculation onlyincludes contributions from the current simulation cell, threeimage cells in the +z direction and three image cells in the −zdirection.

At the beginning of each KMC step,f is evaluated overthe glide plane and analyzed to determine the location ofeach screw(horizontal) and edge(vertical) dislocation seg-ments according to the procedure described by Eq.(5). Thestress fieldteff is computed at the midpoint of each disloca-tion segment. This stress field includes contributions fromthe dislocation self-stress[Eqs. (6), (7), and (8)], that asso-ciated with the solute misfit[Eq. (9)], and the externallyapplied stress.

In the KMC simulations, one event occurs during eachstep. This event is chosen based upon a sampling of all pos-sible events that could occur, weighted by their rates(in theunits of s−1). These events include nucleation of double kinksand the displacement of the individual kinks. The double-kink nucleation ratess jDKd are calculated from the applica-tion of the first-passage-time analysis26 and Eq.(1) takinginto account the local solute environment. The rate of kinkmigration depends on the distribution of solute, through thesolute contribution to the stress and the direct interaction ofthe solute with the dislocation core. If the solute fieldh iszero near the kink(no solute), the migration rates jmd of thekinks is simplynk/b, wherenk is calculated using Eq.(3). Inthe presence of a solute atom, there is an additional activa-tion barrier for kink motion andjm is calculated using Eq.(4). Kink annihilation events are treated as special cases ofkink migration and are parameterized by the same set ofequations[Eqs.(3) and (4)].

An event p is selected from the list of rate dependentevents according to the criterion

oi=0

p−1j i

R, ji ,

oi=0

pj i

R, s11d

where R is given below by Eq.(12) and j1 is a randomnumber from[0,1). Event p involves updatingf over theglide plane to reflect double-kink nucleation, kink migration,or kink-kink annihilation. The normalization factorR in Eq.(11) is the sum of the rates of all possible events,

R= op=1

NDK

jDKp + o

q=1

NK

jmq , s12d

whereNDK is the number of possible double-kink nucleationsites andNK is the number of kink migration and annihilationevents(i.e., twice the number of kinks). In the present KMCalgorithm, the time increment is variable. The time incrementdt is sampled from the exponential distribution of rate-dependent events as

dt = −lnsj2d

R, s13d

wherej2 is a random number between[0,1) andR is the sumof the rates of all the possible events computed using Eq.(12). The procedure described above is repeated for eachKMC step.

The mean dislocation position is calculated as the meanxposition of all the dislocation segments, which are evaluatedusing the values of the parameterf over the glide plane. Thedislocation velocity is computed as the local slope of the

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mean dislocation position versus time. In the present study,dislocation dynamics is simulated over a range of tempera-ture, applied stress, and solute concentration.

III. MOTION OF A Š111‹-ORIENTED SCREWDISLOCATION IN MOLYBDENUM AND ALLOYS

We now apply our method for simulating dislocation mi-gration in the presence of solute to the case of the motion ofa k111l-oriented screw dislocation in bcc molybdenum alloyswith up to 10% of tungsten(a substitutional solute). TheBurgers vector of the dislocation issa0/2df111g (i.e., in thezdirection). Assuming that slip occurs on the close-packed

s110d planes(separated byl =a0/Î2), the kink height is the

distance between successives112d planes orh=a0/Î6. In anisotropic material, the interaction between a pure screw dis-location and a dilatational center of misfit(solute atom) iszero and hence the only long-range interaction between thedislocation and the solute is at the kinks, which have edgecharacter. The difference between the lattice parameters ofMo and W is smallsDa0=0.01 Åd; hence, the strength of thedilatation field of the solute is small. In addition, the stressfield associated with the solute decays quickly(as 1/r3).

In the Mo-W alloys considered here, the main obstructionto dislocation motion from the solute field is that associatedwith the interaction between the solute atoms and the dislo-cation core. Although this value could, in principle, be deter-mined via ab initio calculations(albeit with a very largesimulation cell to avoid solute-solute interactions), it has yetto be determined for the case of W in Mo. Based upon ex-perimental measurements of the heats of segregation of vari-ous solutes to dislocations in bcc Fe,8,28–31 we estimate thesolute-dislocation core interaction energy to be in the rangeof 0.3±0.2 eV. In the simulation results presented below weset this value at 0.3 eV. The solute-dislocation core interac-tion energy simply adds to the secondary Peierls barrier fordislocation motion(i.e., it modifies the activation barrier forthe motion of the kinks).

In order to obtain reliable predictions of the dislocationvelocities in the KMC simulations, it is essential that thekinks do not interact with their own periodic images(i.e., wemust have a sufficient number of kinks on the dislocation toensure that the kink-kink interactions are not dominated byimage interactions). We perform several simulations inwhich we measure the dislocation velocity as a function ofsystem size(lengths of screw dislocation), as shown in Fig.4. These results show that the dislocation velocity increaseswith the system sizeL and then asymptotes to a systemsize-independent value at large system size. This size depen-dence has two origins:(i) kinks interacting with their ownimages and(ii ) the temperature dependence of the equilib-rium thermal kink separation. The minimum system size re-quired to achieve size independent results increases with de-creasing temperature and decreasing stress since loweringTand sA both decreases the rate of double-kink nucleation.The simulation results quoted below were all obtained usingsimulation cells of sufficient size to ensure that the disloca-tion velocity is independent of system size. Each simulation

is run for times long enough to ensure that a steady-statedislocation velocity is achieved(usually for times sufficientfor the dislocation to have traveled 50–100 burgers—at least20 mean solute spacings).

In bcc metals, the core of the screw dislocation relaxesinto low-energy planar configurations. This introduces deepvalleys in the Peierls energy landscape and leads to a largelattice friction to the motion of the screw dislocation. At lowtemperatures, the screw dislocation tends to adopt low-energy configurations and, hence, long screw dislocationsegments predominate at low temperatures. In order to movea screw segment normal to itself, the dislocation core mustfirst be constricted. Thus the energy barrier for the motion ofscrew dislocations and the attendant Peierls stress may beexpected to be very large when compared to the energy bar-rier for the motion of edge dislocations. This can be seen inthe values of the Peierls stress for a screw dislocation in Mo,which has been reported to be between 0.020µ and 0.025µ byseveral researchers24,25,32–34 while the Peierls stress of anedge dislocation is 0.006µ. Thus, there is a large differencebetween the mobilities of screw and edge dislocations. Kinkson a screw dislocation are of edge character and have highmobility, but the nucleation rate of these kinks on the screwdislocation is very small, except at very high stress. Thus, thevelocity of the screw dislocation should be governed prima-rily by the formation rate of the double kinks—i.e., double-kink formation is the rate-limiting step in the motion ofscrew dislocations in Mo.

Figure 5 shows the variation of the dislocation velocitysnDd with inverse temperature for several stresses in therange 0.04øsA/tPø0.16 in pure Mo. The velocity-temperature relationship is Arrhenius, in agreement with theanalytical prediction for dislocation velocity in the kinkmodel.8 In contradiction to the linearizations made in mostanalyses of the kink model, the activation energy is a func-

FIG. 4. Plot of the dislocation velocity as a function of systemsize(lengths of screw dislocationL for the same set of operationalparameters). The results are independent of system size at highersystem sizes.

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tion of the applied stress and decreases from 0.95 to 0.78 eVas the applied stress increases from 0.04tP to 0.16tP. In theanalytical model, the activation energy is the formation en-ergy of a kinkFk, which in our case is 1 eV. Thus, the presentsimulation results are consistent with the predicted activationenergy for dislocation motion(aside from the relatively weakstress dependence).

Figure 6 shows the variation of the dislocation velocity

with applied stress for temperatures in the range 300 KøTø to 600 K. The relationship can be expressed as a powerlaw in which the exponent decreases with temperature from1.89 to 1.08 as the temperature increases from 300 to 600 K.While the present results can also be expressed in terms of anexponential fit, the quality of such a fit is inferior to thepower law fit, especially at high stresses. Many researchers6,7

have reported the exponents in power law fits of the disloca-tion velocity versus the applied stress, with the exponentsvarying from 0.75 to 44. Leiko and Nadgornyi35,36 report anexponent of 7.2 from experiments on single crystals in Mo,while Prekelet al.37,38 report an exponent of 6.4. There arelarge variations in the velocities reported by various re-searchers for screw dislocations in Mo, with differences inthe plane of observation and high initial densities of as-grown dislocations contributing to the uncertainty in data. Inlight of the variability in the experimental observations andthe uncertainty in the double-kink energy and kink diffusiv-ity results employed in the present simulation, we can con-clude that our results are not inconsistent with the experi-mental observations for screw dislocation motion in pureMo.

While analytical predictions of the dislocation velocity inthe presence of solute are possible in the framework of thekink model, such predictions have not been made for thecase of alloys where the main role of the solute is as localobstacles to double kink nucleation and kink motion. Figure7(a) shows the dependence of the dislocation velocity ontemperature for solute concentrations varying from 0(pureMo) to 0.1 and an applied stress of 0.1tP. This figure clearlydemonstrates that the activation energy for dislocation mo-tion is very nearly unchanged by the presence of solute. Theeffect of solute concentration on dislocation velocity at fixedsA is shown in Fig. 7(b) for several temperatures. Increasingsolute concentration decreases dislocation velocity at all tem-peratures. While the rate of decrease is very similar in allcases, the solute is more effective at reducing dislocationvelocity at lower temperature. In order to view the effect ofincreasing solute concentration more clearly, the velocity isplotted asnD /nD

0 in Fig. 7(c), where nD0 is the dislocation

velocity in pure Mo. The fact that the solute is more potent atslowing the dislocations at low temperature is not surprisingin light of the fact that the solute provides a barrier to criticaldouble-kink formation. As the temperature decreases, thisbarrier becomes increasingly difficult to surmount and,hence, the dislocation velocity(which is proportional to therate of double-kink production) is more strongly affected bythe solute as the temperature drops.

The present observation that the effects of alloying ondislocation motion are more severe at low temperature thanat high temperature is consistent with experimental observa-tions in bcc alloys.39–42 These experiments show thatdsy/dCs increases with decreasing temperature, wheresy isthe yield stress or the critical resolved shear stress. It is in-teresting to note that these alloys include several cases wherethe solute misfit is very small(as assumed here). The factthat the same type of results are also observed for alloys inwhich the misfit is large suggest that the solute-dislocationcore interactions may be dominant in all of these cases.

Figure 8(a) shows the variation of the dislocation velocitywith applied stress in five Mo alloys at 400 K. Increasing the

FIG. 5. Semilogarithmic plot of the dislocation velocitynD as afunction of the inverse temperature for four different stress valuesfor pure Mo. The corresponding temperature is shown in the barabove the plot. The applied stress is expressed as a fraction of thePeierls stresstP.

FIG. 6. Log-log plot of the dislocation velocitynD as a functionof the applied stresssA at seven different temperatures for pure Mo.The applied stress is expressed as a fraction of the Peierls stress,tP.

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solute concentration does not change the exponent in thevelocity-stress power law, but rather decreases the prefactor.This is consistent with the observations of Leikoet al.,35,36

who show that increasing the impurity concentration in Moleads to smaller dislocation velocities but has little affect onthe stress exponent. Figure 8(b) shows the same data plottedas a function of the solute concentration for different appliedstresses. The effect of solute on dislocation velocity is largeat very small concentration; however, further increase in thesolute concentration produces a much smaller change in thedislocation velocity at all applied stresses. The same data isplotted in Fig. 8(c), with the dislocation velocity normalizedby nD

0 , the dislocation velocity in pure Mo. With this normal-

ization, it is clear that the effect of solute on dislocationvelocity is independent of the applied stress. In other words,stress and solute effects on dislocation velocity are multipli-cative. This is consistent with the experimental data dis-cussed at the end of the preceding paragraph.

IV. DISCUSSION

In the preceding section, we have shown an application ofour kinetic Monte Carlo method for studying dislocation mo-tion in bcc Mo and alloys. The results of the simulations areconsistent with the analytical predictions for pure metals andwith observations in bcc alloys. In light of all of the approxi-

FIG. 7. (a) Semilogarithmic plot of the dislocation velocitynD as a function of the inverse temperature for five different alloys(varyingsolute concentrationCs). The corresponding temperature is shown in the bar above the plot. The applied stress is constant for all thesimulationsssA=0.1d. (b) Semilogarithmic plot of the dislocation velocitynD as a function of the solute concentrationCs for seven differenttemperatures. The applied stress is constant for all the simulationsstP=0.1d. (c) Plot of the normalized dislocation velocity as a function ofthe solute concentrationCs for seven different temperatures. The applied stress is constant for all the simulationsstP=0.1d. The dislocationvelocity is normalized by the dislocation velocity in the pure Mo,nD

0 .

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mations made in this model and in the analytical theory, thisresult is nontrivial. The dislocation velocity and, hence, thestrain rate are Arrhenius functions of temperature and apower law function of the applied stress. While it is notpossible to directly determine the critical resolved shearstress or the yield stress directly from the results of thepresent simulations, we can interpret the results to under-stand how these quantities should vary with such factors astemperature, applied stress, and solute concentration. The ac-tivation energy for the velocity-temperature relationship isclose to the formation energy of a kink and hence the rate-determining step is the formation rate of the double kinks(the diffusivity of the edge character kinks is very fast).

That the dislocation mobility is critically dependent on thenature of double-kink nucleation further underscores the im-

portance of accurately calculating the double-kink nucleationrate. Previous KMC simulations14,15 have madead hocas-sumptions regarding the calculation of the double-kinknucleation rate. The calculation of the double-kink nucle-ation rate by the first-passage-time method considers all pos-sible permutations of double-kink nucleation and annihila-tion during the subcritical double-kink growth. It is becauseof this that we are able to predict dislocation velocities moreaccurately than heretofore possible. As an example, considerthe nucleation of a double kink in the presence of a solute.The nucleation rate of the double kink depends on the dis-tance of the solute from the double-kink nucleation site. Ifthe solute is closer to the double-kink nucleation site, theeffect of the solute on the nucleation will be much stronger.This phenomenon is properly(and automatically) accounted

FIG. 8. (a) Semilogarithmic plot of the dislocation velocity as a function of the applied stress for five different alloys(varying soluteconcentrationCs). All simulations were performed at the same temperaturesT=400 Kd. (b) Semilogarithmic plot of the dislocation velocityas a function of the solute concentrationCs for four different applied stresses at constant temperaturesT=400 Kd. (c) Plot of the normalizeddislocation velocity as a function of the solute concentrationCs for four different applied stresses at constant temperaturesT=400 Kd. Thedislocation velocity is normalized by the dislocation velocity in the pure Mo,nD

0 .

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for in determining the double-kink nucleation rate.The double-kink nucleation rate and, hence, the disloca-

tion velocity depend strongly on the applied stress, as shownin Fig. 6. The kink model should provide an accurate de-scription of dislocation velocity above a critical stress. Hirthand Lothe8 have estimated this critical stress to be of ordertP/6, wheretP is the Peierls stress. In the case of molybde-num, the Peierls stress is thought to be about0.025tP,22,24,25,43wheretP is the shear modulus of Mo. Thisdepends on the width of the kink, which increases as theapplied stress increases. Thus at higher stresses, the assump-tion that the kinks on the dislocation are straight and narrowmay no longer be valid and, hence, the double-kink energycannot be accurately expressed via Eq.(12).

Another assumption in the model is that positively andnegatively signed kinks have the same diffusivities. Re-cently, Bulatovet al.44 showed that some asymmetry actuallyexists and that a multiplicity of kink structures is possible.This asymmetry with respective to positive and negativekinks may lead to differences in the kink mobility. Such aneffect is not treated in this model, however, it could be easilyincorporated. However, since no quantitative data exists onthis effect, it was not included in the present model.

Solute-dislocation core interactions can have several dif-ferent origins: solutes may be attracted to the dislocationcore and dragged with the moving dislocation, may be re-pelled by the core, or may present static barriers to or trapsfor dislocation. In the example presented above, we exam-ined a case in which the elastic interactions between dislo-cation and solute are weak(yet still included) and consideredthe main role of the solute to be the interaction of the solutewith the dislocation core such that it increases the barrier tokink motion. Other mechanisms for solute dislocation inter-actions and their effect on dislocation mobility may be easilyinvestigated within the framework of the present model. Inparticular, solutes may create a potential well in the energylandscape sampled by the moving dislocation(attractive bar-rier) which can trap kinks or lead to an increase in the coreenergy of the dislocation.13 The present model provides arobust framework to evaluate the effects of all of these pos-sibilities on a uniform footing. Additionally, several modelsfor long range solute-dislocation interactions have beenproposed,28,45–48although most follow the same approach asthat employed here. The relative effects of the short-rangeand long-range interactions could also be investigated inmore detail than we present above in our example applica-tion.

Finally, we note that there is some uncertainty in severalof the parameters employed in the present calculation of thedouble-kink nucleation rates and the kink migration rates.The values employed here were obtained from first-principles calculations, molecular dynamics simulations, andexperiments.22,24,25,32–34The results can be no more accuratethan those parameters(we have not yet performed a sensitiv-ity analysis on our parameter set).

The KMC simulations are closely connected to the atom-istic first-principles calculations molecular-dynamicsstudies.49–52The value of the parameters determined from theatomistic studies will affect the results of the KMC simula-tions. We have already discussed the manner in which

changes in kink diffusivity can be accomodated in the KMCsimulations. Variation in the double-kink formation energysEdkd will affect the rate of nucleation and hence the timescale of the dislocation motion. Relative variation of thedouble-kink energy and kink migration energy may affect thetransition from a nucleation-controlled motion to kink-mobility-controlled motion of the dislocation.

One of the parameters that is not accurately determined isthe solute-dislocation energyEb. Based upon experimentalmeasurements of the heats of segregation of various solutesto dislocations in bcc Fe,8,28–31 we estimate the solute-dislocation core interaction energy to be in the range of0.3±0.2 eV. Molecular-dynamics studies ofk111l screw dis-location interactions with solute have been performed byFarkaset al.49 for Fe-Cr systems. They have employed anembedded-atom method to simulate the effect of substitu-tional Cr on thek111l screw dislocation and found that theaddition of excessive Cr leads to a decrease in kink mobilityand consequently an increase in the Peierls stress, which isconsistent with the observations of the current study. Theyalso observed a change in the structure of the dislocationcore with increasing solute concentration. Such informationcan be incorporated in the present KMC simulation by vary-ing the strength of the solute-dislocation core energy param-eter sEbd with the change in the solute concentration. Basedon the information provided by MD simulations of disloca-tion core interactions, the KMC model can be changed toincorporate the atomistic events with their kinetic and ener-getic parameters.

The present simulation, with its unique ability to rigor-ously sample relatively long time scales, can be use to pro-vide the fundamental dislocation mobility information formore macroscopic simulation, such as the dislocation dy-namics simulations which are capable of simulating the mo-tion of a large number of dislocations.1,2,5 Such DD simula-tions are used to predict overall plastic response of metals.However, all of the information on the type of material istied up in the dislocation mobility information and a fewother materials parameters. Thus the KMC simulation proce-dure developed here can provide a key link between thesmall-length and -time-scale computational methods thatpredict atomic properties and the larger-scale DD simulationsthat investigate large-scale macroscopic response metals.

V. CONCLUSION

In this paper, we have presented a kinetic Monte Carlomethod for simulating dislocation motion in alloys within theframework of the kink model. The model considers the mo-tion of a dislocation on a particular glide plane while thesolute atoms are distributed in three dimensions. The modelincludes both a description of the short-range interaction be-tween a dislocation core and the solute and long-rangesolute-dislocation interactions arising from the interplay ofthe solute misfit and the dislocation stress field. Double-kinknucleation rates were calculated using a first-passage-timeanalysis that accounts for the subcritical annihilation of em-bryonic double kinks as well as the presence of solutes. Weexplicitly consider the case of the motion of ak111l-oriented

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screw dislocation on a{011}-slip plane in body-centered-cubic Mo-based alloys. Simulations were performed to deter-mine dislocation velocity as a function of stress, temperature,and solute concentration. The dislocation velocity exhibitedan Arrhenius dependence on temperature and a power lawdependence on applied stress. These are consistent withsome existing analytical predictions of the dependence of thedislocation velocity on these parameters. Addition of solutesdecreased dislocation velocity in a manner consistent withexperimental observations. The present model is very generaland can be used to study dislocation motion provided that(a)the kink model is valid and(b) the fundamental properties of

the dislocation and solute are known(i.e., single-kink energy,secondary Peierls barrier to kink migration, single-kink mo-bility, solute-kink interaction energies, solute misfit). Theseparameters can all be obtained from first-principles calcula-tions and/or molecular-dynamics simulations. The methodpresented here provides the critical link between atomisticcalculations and dislocation dynamics simulations(i.e., dis-location mobility as a function of stress, temperature, soluteconcentration). This link is key to the development of fullyintegrated models(atomistics to continuum) of the plasticdeformation of alloys.

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