Materials Science and Engineering A283 (2000) 164–171

Two-dimensional simulation of grain growth based on an atomicjump model for grain boundary migration

Byung-Nam KimNational Research Institute for Metals, 1–2–1 Sengen, Tsukuba, Ibaraki 305–0047, Japan

Received 25 October 1999; received in revised form 1 December 1999

Abstract

The kinetics and the topological phenomena during two-dimensional grain growth are studied by computer simulations basedon an atomic jump model for grain boundary migration. The grain boundaries are assumed to be straight. The kinetics show the1/2-power growth law for the average grain size, and the size and the side distributions are time-invariant. In particular, thesimulated side distribution is well consistent with the theoretical prediction. The present simulation follows the Aboav–Weaire lawfor entire topological classes, the Lewis law for intermediate topological classes and the von Neumann–Mullins law forintermediate and high classes. The deviation from the von Neumann–Mullins law for low topological classes is reduced by takingaccount of the effects of curved grain boundaries. The other distinctive results of the simulation are also shown and discussed.© 2000 Elsevier Science S.A. All rights reserved.

Keywords: Grain growth; Atomic jump model; Grain boundary

www.elsevier.com/locate/msea

1. Introduction

Normal grain growth is defined as the uniform in-crease in the average grain size of a polycrystallineaggregate, resulted from the annihilation of smallgrains by grain boundary migration. The driving forcefor the grain boundary migration is the reduction ofboundary energy. A grain boundary with curvaturetends to reduce its area through moving towards thecenter of the curvature. Burke and Turnbull [1] as-sumed that the migration velocity of grain boundary isproportional to the curvature, and proposed the equa-tion for grain growth behavior in a polycrystallineaggregate as

ram−r0

m=Kt (1)

where t is the reaction time, ra is the radius of grain att, r0 is the initial value of ra at t=0, m is the graingrowth exponent (=2) and K is the kinetic coefficientdepending on temperature, mobility and boundary en-ergy. Eq. (1) with m=2 was verified later by manytheoretical models and simulations based on differentanalytical approaches [2–9], and now is a representa-tive equation characterizing the normal grain growthbehavior.

The grain growth behavior obtained by a computersimulation was well characterized by Eq. (1) with m=2. Computer simulation models can be classifiedroughly as deterministic and probabilistic models. Therepresentative of the probabilistic models is the Monte-Carlo method, based on the mechanism of atomicjumps across grain boundaries [2,3]. The advantage ofthe method is the ease of implementing it in two andthree dimensions. However, the drawbacks of themethod are associated with the definitions of length andtime scales, which are not at the atomistic scales. Thecharacteristic length scale employed in the Monte-Carlomethod is much larger than atomic dimensions andcomparable to the length of grain boundary facet.Furthermore, the relation between the Monte-Carlotime and the physical time is unclear. In spite of thesedrawbacks, the Monte-Carlo method gives grain sizedistributions close to the experimentally observed onesand a value of m=2.

In the deterministic models, the change of boundaryenergy provides an expression for the moving velocityof the grain boundary normal to itself, which is propor-tional to the curvature of the boundary. The determin-istic models can be divided into those which employstraight grain boundaries [4,5] and those which employ

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B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171 165

curved grain boundaries [6–8]. In the former models, itis required to assume that the mobility of grainboundary is much larger than that of triple point. Inthe microstructure of straight grain boundaries, a netforce is exerted at the triple points by the surfacetension of the boundaries. The triple points move underthe action of this force. By employing the straight grainboundaries, Soares et al. [4] calculated the driving forceby line tensions, and Kawasaki et al. [5] proposed thevertex model. The advantage of this method is theprovision of a simple description of microstructure.

Recently, the present author proposed a model forgrain boundary migration by atomic jumps across grainboundaries, in which the moving velocity of grainboundary is proportional to [exp(−1/r)−1] not 1/r,where r is the grain radius [9]. For sufficiently large rcompared to the size of an atom, m is 2, while m\2 forsmall r. In the model, the reduction of boundary energyduring grain boundary migration was regarded as thedriving force for atomic jumps, and an analysis wascarried out on the isolated two-dimensional (2D) andthree-dimensional (3D) grains. In the present study, thebehavior of 2D grain growth is simulated in polycrys-talline materials, by introducing the elementary conceptof the atomic jump model. The grain boundaries arerepresented as straight lines, and the moving directionand velocity of triple points are calculated according tothe model.

2. Behavior of grain growth

When the mobility of triple point is considerablysmaller than that of grain boundary, the moving veloc-ity of triple point will obey the entire grain growth rate,and the geometrical configuration of the triple points ofnon-equilibrium state becomes more important ratherthan the configuration of the grain boundaries. In thiscase, the driving force for grain boundary migration isobtained from the angular relationship among threegrain boundaries composing the triple point. The

configuration of the grain boundaries during graingrowth is then determined by the position of the mi-grating triple points.

2.1. Computing algorithm

Grain boundary migration is assumed to occur in thedirection of the fastest way to reduce the boundaryenergy. To find the moving direction and velocity, atriple point is regarded as a computing unit in thepresent study. It is assumed that the moving directionand velocity of a triple point are determined solely bythe configuration of three grain boundaries composingthe triple point. Although the migration behavior of thetriple point is influenced by the migration of adjacenttriple points, this influence is neglected in the presentstudy for simplicity. For an infinitesimal time interval,this simplification would give a first-orderapproximation.

Let us consider a triple point composed of threestraight grain boundaries, as shown in Fig. 1. Theboundary energy is assumed to be constant indepen-dently of the misorientation angle between adjacentgrains. When triple points X1, X2 and X3 are fixedexcept for triple point O, the total boundary energy ofthe grain boundaries G % is a function of the position(x, y) of O

G %=gL=g %3

i=1

(x−xi)2+ (y−yi)2 (2)

where g is the boundary energy per unit length and L isthe total length of the grain boundaries. Eq. (2) is theequation of a curved surface in the rectangular coordi-nates (x, y, G %). Since the driving force for grainboundary migration DG % is the decrease of theboundary energy with respect to the migration of unitdistance, the driving force is proportional to the slopeof G % at O. The tangent vector with maximum slope atO can be obtained readily from a solid geometry, whichhas components of u and 6 in the x and y directionsrespectively and a slope of g(u2+62)/w, where u, 6and w are the direction cosines of the normal vector atO in Eq. (2). Hence, DG % is represented by

DG %=lg(u2+62)/w (3)

where l is a proportional constant.According to the atomic jump model [9], the net

number of atoms Nn jumping across grain boundariesfor an infinitesimal time of Dt, is given by

Nn

Dt=CN

�1−exp

�−

DGkT

�n(4)

where C is a constant relating to the atomic frequencyand the activation energy, k is Boltzman’s constant, Tis the absolute temperature, N is the number of atomsfacing grain boundaries and DG is the average driving

Fig. 1. Triple point as a computing unit. X1, X2 and X3 are fixed. Themoving direction and distance of triple point O are determinedaccording to an atomic jump model.

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171166

Fig. 2. Processes of (a) the recombination of two triple points and (b)the annihilation of a triangular grain. D is the critical distance forthese processes.

straight grain boundaries makes an angle of 120° eachother. Since triple points X1, X2 and X3, however, movesimultaneously, the equilibrium state would not bereached in polycrystalline materials, as long as the graingrowth continues to occur.

2.2. Computer simulation

During the 2D grain growth, two kinds of topologi-cal changes occur. One is the recombination process oftriple points, and the other is the annihilation processof a triangular grain. In the present simulation, the twoprocesses were incorporated in the following way: Therecombination process occurs when two triple pointscome within the critical distance D, as shown in Fig.2(a). The distance between the triple points after therecombination was set to be 1% longer, in order tostabilize the newly constructed microstructure. At thistime, when any two of three triple points forming atriangular grain come within the critical distance, theannihilation process occurs, as shown in Fig. 2(b). Thecritical distance in the simulation is shorter than 1/20 ofthe initial average grain radius.

The 2D simulation of grain growth behavior wascarried out for a Voronoi microstructure initially com-posed of 6000 grains under a periodic boundary condi-tion. The moving distance of triple point per unit timestep in Eq. (5) was calculated with CDt=1 and lg=0.0513kT. The simulation was carried out on a PowerMacintosh G3 and stopped when about 90% of theinitial grains had been annihilated.

3. Results and discussion

Fig. 3 shows the microstructural evolution duringgrain growth. These figures show a self-similar develop-ment of the microstructure with an equiaxed shape inthe late stage and a resemblance of the microstructuresto those observed in soap froths in two dimensions [10].In order to investigate the characteristic features of thegrain growth behavior, we examined the reaction timedependence of the average grain size, the distributionsof the grain size and the number of sides, etc.

3.1. Kinetics of grain growth

The angles between grain boundaries at triple pointsare more widely distributed in a Voronoi microstruc-ture than in an equiaxed microstructure. The drivingforce and the grain growth rate of a Voronoi mi-crostructure are resultantly higher than those of anequiaxed one. With an increase in grain size, the ini-tially unsteady state behavior of grain growth in aVoronoi microstructure approaches the steady statebehavior of an equiaxed microstructure. The growth

Fig. 3. Microstructural evolution during grain growth. tn is thenormalized time (= tMgr0

−2).

force per atom. Assuming that DG is invariant for Dt,Nn corresponds to the number of atoms existing in theswept area by grain boundary migration when the triplepoint moves from O to O %, as shown in Fig. 1. If we letb be the inter-atomic distance, N and DG are given byL/b and DG %b/L, respectively. Then, Eq. (4) can bewritten by

DADt

=CLb�

1− exp�

−b

kTDG %

L�n

(5)

where DA(=b2Nn) is the swept area by the grainboundaries. By solving Eq. (5) numerically, we canobtain the moving distance of triple point O for Dt. Forthe fixed condition of the adjacent triple points X1, X2

and X3, the progress of sufficient time would result inthe equilibrium state of the configuration where the

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171 167

behavior enters the region of the steady or scaling statecompletely, when the grains have grown by about 1.5times the initial average radius. The average grain

radius at this stage was set to be the initial grain radiusr0, and the grain growth from this stage was examined.

An example of the reaction time dependence of theaverage grain radius is shown in Fig. 4. The graingrowth behavior is well represented by Eq. (1) withm=2, as illustrated by many other simulations [2–6].The average grain radius ra was calculated by (Aa/p),where Aa is the average grain area. Though a fewmodels [11,12] were proposed showing m"2, theirdriving forces are somewhat different from the presentone. The models and the simulations employing thedriving force inversely proportional to the grain size,always give m=2 [6,13,14]. The driving force in theatomic jump model [9] is also inversely proportional tothe grain size, when the grain size is sufficiently largecompared to the atomic size.

3.2. Size and side distributions

The distribution of the grain radius r at the scalingstate is shown in Fig. 5. The distribution is an averageof five runs. In the case of a Voronoi microstructure,the distribution is nearly symmetric with a peak aroundthe average grain radius ra and a maximum grain radiusof about 2ra. The distribution spreads out with graingrowth, and the grain radius showing a peak becomeslower than ra. The maximum grain radius at the scalingstate is about 2.14ra.

A representative function characterizing the size dis-tribution at the scaling state may be the Louat function[15]

f(x)=a1x exp(−a2x2) (6)

and the log-normal function [16]

f(x)=b2

b3p1/2 exp

�−2

(log(x)−b1)2

b32

n(7)

where a1, a2, b1, b2 and b3 are constants and x is r/ra.Aboav and Langdon [17] also proposed the empiricalsize distribution function

f(x)=c1 exp[−c2(x−c3)2] (8)

where c1, c2 and c3 are constants. By fitting Eqs. (6)–(8)to the simulated size distribution, we can find that theAboav–Langdon and Louat distributions are closer tothe simulated size distribution rather than the log-nor-mal distribution. However, in other simulations, thelog-normal or the Louat function showed good consis-tence. For example, the size distribution obtained bythe 3D Monte-Carlo simulation is best described by thelog-normal function, whereas the size distribution de-termined from the cross-sectional area is best describedby the Louat function [18].

Fig. 6 shows the distribution of the number of sidesn at the scaling state. The present simulation is consis-tent with that by Weygand et al. [8], but differs from

Fig. 4. Reaction time dependence of the average grain radius.

Fig. 5. Grain size distribution at scaling state.

Fig. 6. Distribution of the number of sides at scaling state.

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171168

Fig. 7. Reaction time dependence of the second moment m2 of grainside distribution. The average value of m2 is 2.01.

that by Kawasaki et al. [5] mainly in the frequency offive- and six-sided grains. The present distribution ofthe number of sides is well described by the log-normalfunction, as shown in Fig. 6. The average number ofsides na is six.

3.3. Topological aspects

It is generally accepted that grain growth at thescaling state will possess time invariant statistical prop-erties. The most commonly employed one of these maybe the second moment m2 of the side distribution func-tion g(n), which is defined as

m2= %�

n=3

g(n)(n−na)2. (9)

An example of the reaction time dependence of m2 isshown in Fig. 7. It can be seen that m2 is almostconstant just with a small fluctuation, indicating thatthe system has reached the scaling state. The averagevalue of m2 is 2.01. The side distribution predictedtheoretically for m2=2.1 by Carnal and Mocellin(Table 2 of Ref. [19]) is also represented in Fig. 6,which shows good consistence with the present simula-tion. The larger m2-value by 0.09 is not expected to yieldsignificant changes in the theoretical predictions of theside distribution.

The correlation between the number of sides n andthe average sides of its neighbors m(n) was proposed byAboav and Weaire [20,21], so-called, the Aboav–Weaire law

m(n)=6−a+6a+m2

n(10)

where a is a constant close to unity. Fig. 8 shows nm(n)as a function of n. Fitting Eq. (10) to the simulation, weobtain a=1.15 and m2=2.34. It may be noted, how-ever, that the curve generated by a=1.0 and m2=2.0does not have significant differences and also agreeswell with the simulation, as illustrated in Fig. 8. Eq.(10) with a=1.0 and m2=2.0 was observed experimen-tally by Aboav [20]. A slight deviation of the simulationfrom the curve is found for few-sided grains (nB5).

Lewis [22] stated that the average area of the n-sidedgrain An is a linear function of the number n

An=b(n−n0) (11)

where b and n0 are constants depending on the proper-ties of the microstructure. An obtained from the simula-tion is shown in Fig. 9. The linear relationship of Eq.(11) is followed only for intermediate topologicalclasses. The present simulation deviates from Eq. (11)for low and high topological classes.

Another property in the relationship between thegrain area and the topological class is the geometricfactor a, which is the ratio of the kinetic coefficientKA(=apMg/3) in the power growth law

Fig. 8. The average number of sides for the neighbors of an n-sidedgrain m(n).

Fig. 9. The average area of an n-sided grain An normalized by theaverage grain area Aa

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171 169

Aa−A0=KAt (12)

to the kinetic coefficient KM(=pMg/3) in the vonNeumann–Mullins law [23,24]

dAn

dt=KM(n−6) (13)

where M is the mobility of grain boundary and A0 isthe initial average grain area (=pr0

2). Fradkov et al.[25] have shown theoretically that a can be obtainedfrom the following topological relationship for 2Dpolycrystals

nA−6=a(A/Aa−1) (14)

where nA is the average number of grain sides for allgrains in a given area interval A/Aa. The linearity wasfound from both experiments and computer simula-tions. Palmer et al. [26] and Fradkov et al. [27] obtainedexperimentally values of a=1.28 and 1.65 for succinon-

itrile thin film and aluminium foil, respectively. Fan etal. [28] also obtained a similar value (1.27) by computersimulation.

A rather surprising result of Eq. (14) is that nA is6−a not 3 at A/Aa=0. Using the above reportedvalues of a, we obtain values of nA=4–5 at A/Aa=0,which indicates that most of the vanishing grains be-long to the topological classes of four and five. Indeed,the annihilation of four- and five-sided grains wasobserved experimentally in succinonitrile thin film [26],contrary to the prediction of most computer simula-tions [2–8]. Close inspection of the reported data,however, for example, Fig. 16 of Ref. [28] and Fig. 7 ofRef. [27], reveals that nA has a tendency to be smallerthan that predicted by Eq. (14) in the range of A/AaB0.5. When only the three-sided grains are annihilated,nA should have a value of three.

The dependence of nA on A/Aa obtained in thepresent simulation is shown as solid circles in Fig. 10.In the range of A/Aa\0.7, good linearity is foundbetween nA and A/Aa. Fitting Eq. (14) to the solidcircles of A/Aa\0.7, we obtain the a-value of 1.245,which is similar to those by Palmer et al. [26] and Fanet al. [28] However, obvious deviation is found forA/AaB0.7 in Fig. 10, approaching the nA-value of 3not 6−a with a decrease in A/Aa. Since it was assumedthat only the three-sided grains can be annihilated inthe present simulation, the deviation is considered tooccur.

On the other hand, the smaller values of nA thanthose predicted by Eq. (14) indicate that the shrinkingvelocities of grains of nB6 are slower than the theoret-ical ones. In the atomic jump model [9], the movingvelocity of grain boundary is proportional to [exp(−1/r)−1] not 1/r. Due to its exponential dependency, theshrinking velocity of an isolated spherical grain is lowerat small radius than that predicted by 1/r-dependency.The same thing can be said in the present simulation. Inorder to illustrate this fact more explicitly, we calcu-lated the rate of the area changes dA/dt for an embed-ded n-sided grain.

Let us consider an embedded symmetric n-sidedgrain with straight grain boundaries. The moving veloc-ity of triple point can be calculated by Eq. (5) with thesame values of CDt and lg as the present simulation.Fig. 11 shows the dependencies of dAn/dt on the ratiod/rt for n=3 and 4, where d is the moving distance oftriple point per unit step, and rt is the distance betweenthe triple point and the gravitational center of thegrain. Here, the minus sign of dAn/dt means a shrink-age of the grain. When the moving velocity of grainboundary is inversely proportional to the radius of thegrain, dAn/dt would be independent of the radius,according to the von Neumann–Mullins law. Fig. 11,however, represents that the absolute value of dAn/dtlinearly decreases with increasing ratio d/rt. Namely,

Fig. 10. Dependence of the average number of grain sides nA on thegrain area A/Aa

Fig. 11. Dependence of dAn/dt on d/rt for an embedded symmetricn-sided grain with straight grain boundaries.

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171170

Fig. 12. Growth rate of the average area of n-sided grains.

area with curved boundaries Anc to that with straightboundaries Ans for respective topological classes.

The present author recently simulated the 2D migra-tion behavior of embedded symmetric grains by thefinite element method (FEM) (unpublished research). Inthe FEM simulation, it was found that the equilibriumshape of grain boundary at dynamic state is differentfrom that at static state. The simulated shape of grainboundary is not a circular arc. By using the FEMresults, we can obtain the ratio Anc/Ans for respectiven-sided symmetric grains. Multiplying the ratio to thepresent results of dAn/dt yields better consistence withthe von Neumann–Mullins law, as shown in Fig. 12.

The present simulation was carried out for the mi-crostructure of straight grain boundaries. However, thealgorithm of the simulation is also available for that ofcurved grain boundaries. Inserting points on grainboundary between triple points, we can calculate themovement of the inserted points in a similar way. Inthis case, only one of the three fixed triple points in Fig.1 is deleted. Such simulation will result in curved grainboundaries and is expected to give better consistencewith the von Neumann–Mullins law, as shown in thesimulation by Weygand et al. [8]

4. Conclusions

The 2D grain growth behavior was simulated byusing the concept of the atomic jump model [9] forgrain boundary migration. The major characteristicfeatures of the model is that the growth rate of a grainwith radius r is proportional to [exp(− l/r)− l] not 1/r,so that the growth rate at small sizes becomes slowerthan the classical predictions. The simulation based onthe atomic jump model shows the scaling behavior ofthe average grain size, that is, the growth as 1/2 powerin time in polycrystalline aggregates. The simulationalso shows the scaling properties of the distributionfunctions of the size and edge number of grains. Inparticular, the simulated distribution of the edge num-ber is well consistent with the theoretical prediction byCarnal and Mocellin [19].

The topological aspects were also examined duringthe grain growth. It is proved that the Aboave–Weairelaw with the second moment of the side distribution,m2=2, which is close to the value 2.01 obtained fromthe simulation, describes well the present microstruc-tural evolution, while a slight deviation is found forfew-sided grains. The Lewis law is followed only forintermediate topological classes.

The simulated average number of grain sides for allgrains in a given area interval deviates from the theoret-ical prediction for grains of small radii. Since thegrowth rate at small sizes is slower for the atomic jumpmodel than for the classical predictions, the increased

for nB6, the moving velocity of grain boundary de-creases with decreasing radius. Due to these characteris-tic features of the atomic jump model employed in thepresent simulation, the deviation from the linear rela-tionship appears in Fig. 10. Meanwhile, the lowereddAn/dt at small grain size is not thought to haveconsiderable influences on the average growth rate ofgrains represented by Eq. (1). Despite the loweredvelocity of small grains, the simulated growth behavioris well described by Eq. (1) with m=2.

The simulated dependence of dAn/dt on n is shown inFig. 12 along with the von Neumann–Mullins law ofEq. (13). The simulation shows an approximately linearrelationship between dAn/dt and n for grains with n\6.However, the slope is slightly smaller for the result inthe present simulation than for that of the von Neu-mann–Mullins law. The deviation of the former resultfrom the latter one is remarkable for few-sided grains(nB6). The von Neumann–Mullins law assumes thedriving force of 1/r-dependency and the equilibriumstate of the angular relationship at triple points, whilethe driving force of [exp(−1/r)−1]-dependency andthe non-equilibrium state at triple points (straight grainboundaries) are assumed in the present simulation. Thismay be the reason why the deviation becomes remark-able at nB6 as shown in Fig. 12.

The deviation from the von Neumann–Mullins lawwas also reported in the vertex model employed straightgrain boundaries [5]. The deviation arises from the factthat the shape difference of the straight grainboundaries from the curved ones increases with de-creasing number of grain sides. For embedded symmet-ric grains with identical rt, the growth rate of theaverage area (dAn/dt) in grains with curved grainboundaries is larger than that with straight grainboundaries for nB6, because the grain boundaries fornB6 are bowed out. Hence, the deviation in Fig. 12can be reduced by multiplying the ratio of the grain

B.-N. Kim / Materials Science and Engineering A283 (2000) 164–171 171

number of grains with lower topological classes reducesthe average number of grain sides for small grains. Thepresent simulation also deviates from the von Neu-mann–Mullins law for grains of lower topologicalclasses, while good consistence is found for grains ofintermediate and high topological classes. The deviationat low topological classes, however, can be reduced bytaking the effects of curved grain boundaries at equi-librium state into account.

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