monte carlo simulation of grain growth with the full spectra of grain orientation and grain boundary...

MONTE CARLO SIMULATION OF GRAIN GROWTH WITH

THE FULL SPECTRA OF GRAIN ORIENTATION AND

GRAIN BOUNDARY ENERGY

N. ONO1{, K. KIMURA{2 and T. WATANABE1

1Department of Machine Intelligence and Systems Engineering, Faculty of Engineering, TohokuUniversity, Sendai 980, Japan and 2Graduate School, Tohoku University, Sendai 980, Japan

(Received 20 March 1998; accepted 19 October 1998)

AbstractÐMonte Carlo simulations of grain growth were performed for two-dimensional polycrystals withcrystallographic orientations being assigned to individual grains in terms of randomly generated Eulerangles. Individual grain boundaries were characterized based on the orientation relationships of grainsacross them and grain boundary energies were given with a continuous function of deviation angle fromthe exact coincidence orientation relationships. The physical and technical signi®cance of the conditions ofthe orientation change in the simulations is discussed in terms of the degree of the probablistic nature ofthe simulation. The simulations showed the following characteristic features: (1) the energy spectra makethe grain size distribution more symmetric; (2) they make the average grain growth rate smaller; and (3)the fraction of low energy boundaries increases with grain growth. These results are discussed with respectto local boundary con®gurations and the nature of the present model. # 1999 Acta Metallurgica Inc. Pub-lished by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The theoretical analysis of grain growth, beginning

with the model of Hillert [1], has been intensivelyconducted with computer simulations since theearly 1980s [2, 3]. With the studies on grain bound-aries and interfaces, a signi®cant amount of knowl-

edge has been obtained on the in¯uence of thecharacter and structure of grain boundaries on theproperties of polycrystalline materials. The control

of grain boundary character is expected to be ane�cacious method for the development of advancedmaterials [4].

It is, therefore, desirable for the simulationof grain growth to take into account a variety ofgrain boundary characteristics as previously done

by Grest et al. [5], Rollett et al. [6], Tavernier andSzpunar [7], Saito and Enomoto [8], and so on.Signi®cant in¯uences of these characteristics onthe evolution of structures, such as the tendency

for abnormal grain growth, have been reportedby the above researchers. However, the smallnumber of grain boundary types used in these

simulations makes it di�cult to compare theirresults with experimentally observed grain growthbehavior.

In the present work, simulations with the MonteCarlo method [3] were carried out, the respective

grains having a speci®c orientation described with aset of assigned Euler angles. This allowed us to in-corporate a continuous spectrum of grain boundary

character described by the orientation relationshipbetween neighboring grains and also that of thegrain boundary energy associated with the relation-ship. It has also become possible to analyze the

results of the simulations in terms of the grainorientation distribution and the grain boundarycharacter distribution. Furthermore, the e�ect of

other factors such as the surface energy and thevolumetric force from an applied magnetic ®eld canbe more easily incorporated in the simulation. The

above features of the present simulation model aresimilar to that used by Carel et al. [9]. In theirtreatment, however, the orientations of grains were

limited to those in the {100} and {111} in-plane tex-tures.In the following, the procedure used to deal with

grain orientation distribution and grain boundary

character will be described ®rst. In grain growthsimulation with the Monte Carlo method, a fewconditions can be set rather arbitrarily. With a con-

tinuous spectrum of grain boundary energy, thepresent simulation distinguishes ®ne di�erences ingrain boundary energies. It is, therefore, considered

worthwhile to examine the physical and compu-tational signi®cance of the simulation conditions.The results of some preliminary computations forthis purpose will be given. Finally, the e�ects of a

grain boundary energy spectrum will be summar-ized and discussed.

Acta mater. Vol. 47, No. 3, pp. 1007±1017, 1999# 1999 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/99 $19.00+0.00PII: S1359-6454(98)00391-7

{To whom all correspondence should be addressed.{Present address: Olympus Optical Co., Ltd, Tokyo

163-09, Japan.

1007

2. SIMULATION PROCEDURE

2.1. Grain boundary character

The present work deals only with the case of

cubic crystals. The e�ect of initial texture isan interesting subject but is left for futureinvestigation. We therefore generated random or

uniform grain orientation distributions as thestarting structures of the simulation. The grainorientations so obtained were converted to respect-

ive rotation matrices and stored in a table for laterreference.As the orientation of each grain is described with

a speci®c rotation matrix, we can characterize the

boundary between a pair of neighboring grains, Aand B, based on their respective rotation matrices,MA and MB. Namely, the relative rotation matrix

across the boundary R is given as

R �MAMÿ1B : �1�

With this R, the grain boundary is characterizedaccording to the coincidence-site-lattice theory. The

method for this is identical to the characterizationof grain boundaries in real polycrystals from grainorientations measured with the SEM/ECP method

or the like [10].Namely, we tabulate known coincidence relations

for cubic crystals in the form of rotation matrices.

As is usually done, 270 coincidence relations fromS3 to S29 [11] were used. Let us denote them as Ci.For each grain boundary, the above R has 23

other equivalent expressions, Ri, because of the

symmetry of a cubic crystal. They can be calculatedwith the 24 cubic symmetric operation matrices, Si,as follows:

Ri � SiR: �2�Each of these is compared to the above coincidencerelations, Ci, as in the following.

Let D denote the deviation of a relative rotationmatrix, Ri, from an exact coincidence relation, Cj.The value of D is obtained by solving the followingrelation:

Ri � DCj: �3�With the trace dii of D so obtained, the deviationangle from the exact coincidence relation Dy is

given as

Dy � cosÿ1�dii ÿ 1

2

�: �4�

It is considered that deviation as such is compen-sated for by the introduction of extra dislocationson the boundary, and the limit for this compen-sation is given in terms of the Brandon angle [12]

given as

DyC � p12

1��Sp : �5�

Here, S is the S-value of the coincidence relationwe are concerned with.

The ratio of the above deviation angle and theBrandon angle, d � Dy=DyC, is considered to be thedeviation normalized in regard to the character of

the particular coincidence relation. We, thus, calcu-late d for all the combinations of Ri and Cj andchoose a Ci that gives the smallest value of d. If d

so obtained is less than unity, the boundary iswithin the Brandon angle from the coincidence re-lation and characterized as the coincidence bound-

ary. If no combination gives d less than unity, theboundary is a random boundary.In the above procedure, it is considered appro-

priate to let Ci include the case of S1 or nil relative

rotation. If this is chosen, the boundary is a S1ÿor low angle boundary.It is di�cult to take into account the inclinations

of grain boundaries in the coincidence site latticemodel used in the present work, and we thusignored them as was done in previous studies [3, 13].

2.2. Grain boundary energy

In comparison with random boundaries, thearrangement of atoms across coincidence bound-aries is expected to have greater compatibility and

thus to have lower grain boundary energies. It has,in fact, been observed that relations between grainboundary energy and misorientation in tilt and

twist boundaries show cusp-like minima at anglesthat correspond to speci®c coincidence relations[11, 14]. The depth of cusps varies depending on the

type of coincidence relations and other factors suchas temperature, the kind and purity of material,and so on. Among experimental results cited inRef. [14], for example, the energy of the S3ÿ h110itilt boundary in aluminum [15, 16] is less than halfthat of random boundaries, while in other cases thisdi�erence is only a few percent or less.

While all kinds of coincidence boundaries have achance to appear in the present simulation, the ex-perimental or theoretical estimation of their energies

has been done only for a small part of them. In thepresent work, therefore, grain boundary energieswere calculated with a hypothetical function thatsimulates observed energy±deviation angle relations

around coincidence relations as described below.First, the energies for exact coincidence bound-

aries and random boundaries, Ecsl and Ergb, were

assumed to be constant, being independent of theircharacteristics. An exception to this is the case ofthe S1 boundary where Ecsl � 0. The Ecsl=Ergb value

in this simulation thus represents the average depthof energy cusps at coincidence relations above S3.We chose Ecsl=Ergb � 0:8 rather arbitrarily, but it

conserves clear cusps around a few special bound-aries and does not exaggerate the nature of averagecoincidence boundaries too much as indicated inthe experimental results cited earlier.

ONO et al.: MONTE CARLO SIMULATION OF GRAIN GROWTH1008

Grain boundary energy increases rapidly as the

above deviation d � Dy=DyC increases and then

asymptotically approaches the value for random

boundaries, Ergb. Here, the idea of the Brandon

angle suggests that the energy will be almost Ergb at

d � 1. The next equation was assumed to simulate

these:

Egb � �Ergb ÿ Ecsl�f1ÿ exp�ÿad �g � Ecsl: �6�Here, a is a constant whose increase makes cusps

sharper and more localized. We chose a � 3 in

order to reproduce the general features of energy

angle relations in the earlier cited experimental ob-

servations; for example, rising curves from neigh-

boring cusps meet each other nearly at the level of

the average energy of random boundaries.

Figure 1 shows the relation between grain bound-

ary energy Egb=Ergb and the misorientation angle

for h100i twist boundaries calculated with

equation (6) using the above values of Ecsl=Ergb and

a, 0.8 and 3, respectively.

The variable d in equation (6) is the value used in

the characterization of boundaries and thus the

smallest value among those for all coincidence re-

lations. Even when the value was larger than unity

and the boundary was judged to be a random

boundary, equation (6) was used to assign its

energy in order to avoid a discontinuity at d � 1.

When the nearest coincidence relation changes from

one to another, the d values and thus Egb for them

are the same if both of them are of S-value 3 or

larger. However, the Ecsl value for the S1-boundaryis di�erent from others, and thus the Egb for the

same d is also di�erent. Because of this, there is a

discontinuity between the S1-cusp and a neighbor-

ing cusp as seen in Fig. 1. The gap, however, is

only a few percent of the energy and can thus beneglected.

2.3. Simulation model

The simulation of grain growth can be doneusing one of the two major ways of modeling: the

network model initiated by Weaire and Kermode [2]or the site lattice model by Anderson et al. [3, 13].In the present work, the latter was used because ofits ease of implementation.

Namely, a polycrystal was modeled with two-dimensional tiling with hexagonal sites. The size ofthe model was 100 sites by 100 sites and the cyclic

boundary condition was used. The steps of thesimulation were as follows:

1. Generate an initial state by assigning a random

integer to each of the 10 000 sites. The integerrepresents an orientation and is often calledorientation number.

2. Choose an inspection site, A, randomly fromamong the 10 000 sites.

3. Choose another site, B, randomly from among

the six sites neighboring A.4. Do nothing if A and B have the same orien-

tation. Otherwise, calculate the energy changeDE for the change in the orientation of A to that

of B. If DE is not positive, let the orientationchange take place.

5. Repeat the procedure starting with step 2.

When this loop has been repeated by the numberof sites in the model, every site is checked for orien-tation change once on average. This is generally

called a Monte Carlo step (MCS) and is used as ameasure of time lapse in the simulation.The orientation numbers were mapped to

respective rotation matrices. The interaction energy

Fig. 1. A part of the assumed grain boundary energy spectrum.

ONO et al.: MONTE CARLO SIMULATION OF GRAIN GROWTH 1009

of two sites, i and j, was given by the grain bound-ary energy calculated from their respective orien-

tation matrices, Mi and Mj, as described in theprevious section.Let Egb�i,j � denote the interaction energy so

obtained. The energy of the site i was evaluated bysumming it for 18 sites up to the second nearestneighbors

Esite �X18j�1

Egb�i,j �: �7�

Namely, interaction range parameter k was set to 2.This is in accordance with the report that by con-sidering the second nearest neighbors, the results of

simulation becomes more similar to the experimen-tally observed structures of polycrystals than thoseobtained with the ®rst nearest neighbors alone [17].

In the present work, grain orientations are speci-®ed with three Euler angles that can change con-tinuously so that the present model may contain

10 000 di�erent orientations. To reduce the compu-tational load, however, the number of di�erentorientations, Q, was limited to 3000. In the graingrowth simulation with the site lattice model, it has

been reported that when the number is made smal-ler, the aggregation of grains with the same orien-tation occurs more frequently and the Q-value

dependence of grain growth rate becomes conspicu-ous, necessitating the use of Q > 500 to avoid thisartifact [17]. The present Q-value is su�ciently lar-

ger than this.Even with this limitation, the number of probable

grain boundary types is as large as 30002/2. To

reduce the load of computing this many boundaryenergies, simulation up to ten MCS was made witha constant grain boundary energy.

3. NATURE OF THE MODEL

As described in the previous section, the presentsimulation contains a few options that may criti-cally in¯uence simulation results. Before going into

a description of the results, therefore, it is con-sidered worthwhile to examine the signi®cance ofoptional settings involved in the simulation. Let ussummarize them in terms of the randomness

involved in the simulation.Grain growth occurs as a result of grain bound-

ary migration that is driven by the reduction of the

system's energy. In the framework of the presentsimulation, this reduction is achieved ®rst by themotion of triple points to reduce the length of those

boundaries with relatively higher energies and alsoby the reduction of the length of individual bound-aries through the reduction of their curvatures. In

the present model, the motions of triple points andboundaries are represented by the change in theorientations of individual sites. The energy changedue to this, DE, is evaluated by applying

equation (7) to the states before and after the orien-tation change. The change is permitted if DE is

negative so that the energy of the system will keepdecreasing.This description does not suggest the use of the

Monte Carlo method, or the introduction of ran-domness in the simulation of grain growth.Namely, one may compute the possible orientation

change that yields the largest energy reduction forevery site in the model and then apply it to everysite at once to obtain a new state. This ``determinis-

tic'' procedure appears to best represent the aboveargument.This procedure, however, results in the stagnation

of structure evolution, including an alternation

between two states. This is considered to occurbecause grain growth or energy change occurs dis-cretely in the present model, which consists of sites

with ®nite dimensions, while that in real materialsoccurs continuously, at least on the scale of obser-vation, through the motion of individual atoms.

The energy change in the model is not smooth and,in certain circumstances, has local energy minima.To avoid the stagnation of structure evolution as

such, randomness or probabilistic elements havebeen introduced in the previous works. Let usexamine each of them.

3.1. Selection of an inspection site

Instead of the deterministic procedure that exam-ines all sites and changes their orientations at once,

in the method usually employed, one site is chosenat random, and its orientation is immediately chan-ged if appropriate. This change can in¯uence the

behavior of the system in the next step. This ran-dom selection of an inspection site makes the sys-tem less vulnerable to stagnation. Namely, the

change of the state of a region heading to a localminimum may be postponed by chance. Changes inthe state of its surroundings that take place instead

may destroy the local minimum.However, in general, this is not always su�cient

to avoid stagnation. Furthermore, the other sourcesof randomness as described later work equally well

so that this random selection is not necessarily theonly possible setting for the grain growth simu-lation with the site lattice model. It has, however, a

merit in the use of computer resources. Because itapplies a possible orientation change of a site im-mediately, it does not need a temporary memory

space for storage of all feasible changes found inone state.

3.2. Selection of a new orientation

After an inspection site has been chosen, thereare several options for choosing its new orientation.

One is to choose it from all considered orientationsof the model. The orientation may be di�erent fromany of those presently in the region. Although thisbehavior may be appropriate in the simulation of


recrystallization, it is out of the scope of the presentwork.

We thus choose a new orientation from amongthose of its neighbors and there are again twooptions. One is to select an orientation from among

those of its neighbors at random and another is tochoose the one that yields the largest energy drop.Apparently, the former method is less deterministic

than the latter, but it alone was insu�cient to avoidstagnation.

3.3. Energy change

For an orientation change, the associated energy

change DE can be either of the following:

DE<0: Based on the premise of the model, theorientation change should be duly permitted.

DE � 0: This change has been permitted inmost of the previous work. Tavernier andSzpunar [7] let the change occur at a probabilityof 0.5. One may argue that this change should

not occur because of the lack of a driving force.In the present work, it was found that the prohi-bition of this change resulted in stagnation when

the full spectrum of grain boundary energy wastaken into account.DE > 0: In the deterministic procedure, this

change should not be permitted. Anderson etal. [3] assumed that this change occurs at a prob-ability of exp�ÿDE=kT � where k is the Boltzman

constant and T is the hypothetical temperature,and examined the e�ect of T. This Arrheniusfunction appears to imply that this change to ahigher energy state is due to thermal activation.

However, the migration of grain boundariesbeing examined occurs through the di�usion ofindividual atoms, a thermally activated process.

It appears unnatural to assume another thermalactivation event for a group of atoms with thesize of a site.

In view of the simulation technique, however, itis reasonable to introduce a jump to a higher energystate at a certain probability in order to allow the

system to escape from local energy minima that areinherent in the present discrete model. In this argu-ment, the orientation change with DE > 0 does nothave a physical meaning, and thus the probability

of the change may not necessarily be the one givenby Anderson et al. Furthermore, one may assign acertain probability even to orientation change in the

case of DE<0.The probability of orientation change as a func-

tion of DE is expected to have an e�ect similar to

that of the random choice of an inspection site.Because the e�ect of the probabilistic element ofthis kind has not been well examined, we did not

adopt this in the present work. Because of this, thepresent simulation is not in¯uenced by the magni-tude of DE but depends only on its sign, and thuscannot capture the e�ects of the depth and the

sharpness of cusps in the grain boundary energyspectrum represented by Ecsl=Ergb and a inequation (6).

3.4. Summary of the present simulation conditions

In the previous work, the range of the energy

evaluation included either only the nearest neigh-bors or sites up to the second nearest. This par-ameter is not a probabilistic element, but to take a

wider range into account is expected to make thelocal minima of the state energy less probable andthus make a simulation with more deterministic

conditions possible. It has also been reported thatthe system with square sites is more apt to stagnatethan that with hexagonal sites [3].Including these just mentioned, the conditions of

the present simulation are summarized in Table 1.These are close to those used in most of the pre-vious studies and reported to produce grain struc-

tures similar to experimental observations. Thepresent work is unique, however, in that all grainboundaries have energies more or less di�erent

from each other.The factors discussed in this section lead to the

following dilemma:

. probabilistic elements are not desirable becausethey violate the principle of the simulation thatthe structure evolves so as to reduce the energyof the system;

. if there are insu�cient probabilistic elements,however, the simulation is not possible becauseof the stagnation of structure evolution in local

energy minima.

The ®rst point is considered to be especially im-portant in the present work where the e�ect of

subtle di�erences between boundary energies is con-cerned. In preliminary simulations, it was foundthat the system experienced stagnation due to the

prohibition of orientation change at DE � 0 withother elements being kept as in Table 1. This indi-cates that the present simulation was conducted

with an amount of probabilistic elements close tothe smallest necessary amount.

4. RESULTS

4.1. Initial structures

The procedure used in the present work to makean initial structure has the same physical meaning

Table 1. Summary of the present simulation conditions

Aspect Setting

Site shape HexagonalInspection site selection RandomNew orientation Random from neighborsOrientation change with DE � 0 AlwaysOrientation change with DE > 0 NeverInteraction range 2


as that employed by Mori [18] and Garbacz and

Grabski [19, 20] to calculate the statisticallyexpected fractions of coincidence boundaries.Examination of grain boundary character distri-

bution (GBCD) of the initial structures as describedearlier, therefore, amounts to a recalculation oftheir results. A comparison of the present results

with theirs serves to verify the present method andthe computer programs it employs.Such a comparison is made in Fig. 2. Garbacz

and Grabski used 1=4��Sp

for the Brandon angle.

This is smaller than the present value of p=12��Sp

sothat the fraction of every coincidence boundary isslightly smaller than that in the present work and

that in Mori's study. The present results may havea larger error than these because the number ofdi�erent orientations was limited to 3000 and

because the number of sites examined was100�100. While the present model is two-dimen-sional, Garbacz and Grabski pointed out that

GBCD depended, although slightly, on the arrange-ment of model grains. In view of these factors smalldi�erences between these results as seen in Fig. 2are considered to be insigni®cant.

Figure 3 shows the grain orientation distributionin the direction normal to the model plane afterten MCS of grain growth without considering the

grain boundary energy spectrum. It can be seenthat a uniform grain orientation distribution wasobtained.

4.2. Grain growth behavior

Figures 4 and 5 show relations between average

grain area �A and time lapse t, respectively. Thenumber of grains in the model is given as 10000=�A.From this, we can see the number of grains

decreases to about 50 by 200±300 MCS and, asa result, the growth behavior becomes irregularthereafter. The results obtained from simulationswith a grain boundary energy spectrum show some

scatter in the growth behavior even before this

time lapse, depending on the random number series

used in the respective runs. This indicates that thesize of the present model is not large enough

to average out the di�erence in the random

number series. In the following, therefore, we dealwith the sum or the average of the results of three

runs carried out with di�erent random number

series.

Figures 6(a)±(d) show grain size distributions at50 and 200 MCS with and without the grain bound-

ary energy spectrum, respectively. Here, grain size

D was evaluated with the relation D � 2� ��A=pp

, Abeing the area of a grain in terms of the number of

sites in the grain. The horizontal axis is in log D.

The vertical axis, on the other hand, is the fractionof grains within the interval of DD � 2. If we plot

this value against log D, the diagram becomessomewhat misleading because the constant interval

DD � 2 becomes relatively smaller in the range

of larger D. To compensate for this artifact, therelative frequency f(D) is calibrated with the size of

Fig. 3. Grain orientation distribution in the normaldirection after ten MCS without a grain boundary energy

spectrum.

Fig. 2. The grain boundary character distribution of an initial structure in comparison with theprevious results for random polycrystals.


the interval in the logarithmic scale to yieldf 0�D� � f �D�=flog�D� DD� ÿ log�D�g.In both cases, with and without the spectrum,

grain size distribution does not show notablechange with grain growth. Without the spectrum,the peak of a distribution tends to appear at a lar-

ger grain size value, while distributions obtainedwith the spectrum are more symmetrical. Anexample of structures corresponding to the latter is

shown in Fig. 7. In the grain growth from 50 to150 MCS, it is seen that a few grains considerablyincrease in size while grain growth is slow in thecenter or right-bottom regions.

4.3. Grain orientation distribution

Figure 8 shows the inverse pole ®gure after grain

growth up to 300 MCS with the grain boundaryenergy spectrum. It is seen that grain orientation

distribution is still uniform and similar results wereobtained without the spectrum. This is as expectedfrom the fact that the initial structures do not have

any preferred orientation and the simulation doesnot involve a factor such as the surface energy thatfavors the growth of grains with particular orien-

tations.

4.4. Grain boundary character distribution

Figure 9 shows the changes of the fractions ofcoincidence boundaries up to S29 with graingrowth. In comparison with the results without agrain boundary spectrum, those with a spectrum

show a clear increase in the special boundarieswith the grain growth. The GBCD diagram at 300MCS with the spectrum is shown in Fig. 10. The

increase in the fraction is conspicuous for most S-values.

Fig. 4. Grain growth behavior with a grain boundary energy spectrum for di�erent random numberseries.

Fig. 5. Grain growth behavior without a grain boundary energy spectrum for di�erent random numberseries.


Fig. 6. The e�ect of a grain boundary energy spectrum on grain size distribution: (a) 50 MCS with agrain boundary energy spectrum; (b) 200 MCS with a grain boundary energy spectrum; (c) 50 MCSwithout a grain boundary energy spectrum; (d) 200 MCS without a grain boundary energy spectrum.


5. DISCUSSIONS AND REMARKS

The e�ects of a grain boundary energy spectrum

on the results of the present grain growth simu-

lation may be summarized as follows:

1. it makes grain growth slower;

2. it makes the fraction of large grains larger and,

thus, makes grain size distribution more sym-

metric;

3. with the spectrum, the fraction of low energyboundaries increases with grain growth.

The ®rst result is rather unexpected given that the

di�erence in the grain boundary energies provides

an additional driving force for the elementary pro-

cesses of grain growth such as the migration of tri-

ple points. In certain circumstances, however, the

spectrum may suppress grain growth in the site lat-

tice model. For example, suppose a change in the

orientation of a site with which the number ofneighbors with di�erent orientations does not

change. When the energy spectrum is not con-

sidered, energy E of the site depends only on this

number so that the energy change, DE, is zero.

With the conditions of the present work, this orien-

tation change is applied. When the energy spectrum

is considered, DE for the change will rarely be zero

because di�erent boundaries have more or less

di�erent energies, respectively. On the average, half

of the DE values in such cases will be positive so

that the frequency of orientation change in such

situations will be half that of the case without the

spectrum.

When the spectrum is not considered, grain

growth takes place everywhere uniformly. This

makes the sizes of the largest grains similar to each

other everywhere. When the spectrum is considered,

on the other hand, growth behavior is di�erent

from one region to another depending on the local

con®guration of boundary characteristics as shown

in Fig. 7. This makes the distribution in the larger

grain size range broader resulting in symmetric

grain size distributions as shown in Fig. 6.

In the same situation, the e�ect of the energy

spectrum amounts to making one state more prefer-

able than another in two states that can otherwise

alternate freely. This makes the simulation system

more deterministic and thus makes the local equili-

brium more probable. It is not clear if or how

Fig. 7. An example of regional di�erence in grain growth rate with a grain boundary energy spectrum.

Fig. 8. Grain orientation distribution in the normaldirection after 300 MCS with a grain boundary energy

spectrum.Fig. 9. Change in the fraction of special boundaries with

grain growth.


much the present results su�er from such an arti-

fact. However, the symmetric grain size distri-butions obtained, which resemble those observedexperimentally, would suggest that the application

of the energy spectrum made the present simulationmore realistic.The previous grain growth simulations with the

anisotropy of grain boundary energy were con-ducted with a number of di�erent orientations less

than a hundred. They adopted a rather arbitrarysetting on the fraction of low energy boundaries,the ratio of energies for special and other bound-

aries, and so on [5±8]. It is, therefore, di�cult tocompare the results of those simulations with thepresent results. In the simulations by Rollett et

al. [6], for example, the abnormal grain growth wascaused by the introduction of the anisotropy of

grain boundary energies. The spectrum used in theirwork, however, is considered to be too exaggerated.The energy spectrum used by Grest et al. [5] is simi-

lar to the present one and was reported to have asimilar e�ect on the grain size distribution. Theseresults suggest that, in the grain growth from tex-

ture-free initial structures, the grain boundaryenergy spectrum alone would not cause abnormalgrain growth.

The third of the above listed points, the increasein the fraction of low energy boundaries, agrees

with the simulation results reported by Saito and

Enomoto [8]. This is considered to be a naturalresult of the algorithm that computes a state withthe smallest energy. However, the simulation results

as such do not agree with experimental observationsthat the fraction of special boundaries decreaseswith grain growth [21]. With regard to this disagree-

ment, Aust and co-workers have pointed out that,other than the e�ect of the grain boundary energy

spectrum, kinetic and geometric factors should beconsidered and that their relative signi®cance isstrongly in¯uenced by the purity of material [22, 23].

In normally pure materials, for example, low Sboundaries have mobilities higher than otherboundaries and thus tend to disappear earlier as

indicated by the recent geometric analysis by Lin etal. [24]. The present simulations as well as those inRef. [8] do not incorporate a mobility spectrum,

and it may not be relevant to compare them withgrain growth behavior in such conditions.

Within the scheme of the present simulations,however, a mechanism that could cause the prefer-ential disappearance of low energy boundaries has

been suggested.A typical example of this is shown in Fig. 11

where the boundaries A and B are S1 and S19b,respectively. Because of the lower energies of theseboundaries, they cannot form the typical Y-shaped

Fig. 10. Grain boundary character distribution at 300 MCS.

Fig. 11. An example of structure change when low energy boundaries are preferentially eliminated.Numbers in the ®gures are deviations from respective exact coincidence relations.


triple points but take the con®guration of an arcspanning a wedge formed with other boundaries.

Such boundaries decrease in length by simply mi-grating toward the tip of the wedge and then disap-pear.

Grain growth behavior as a whole is the result ofcompetition between the overall trend for reducingthe system's energy and the motion of individual

boundaries controlled by local con®gurations asseen in this example. We may not simply expectthat boundaries with lower energies should continue

to survive even from the energetic viewpoint alone.Higher mobilities for low energy boundaries as indi-cated by Aust and co-workers will accelerate theoperation of this mechanism. Also, the local beha-

vior as such is considered to be sensitive to thecharacteristics of the model as summarized inSection 3, and its in¯uence might be underestimated

in the present simulations and also in those bySaito and Enomoto.In the above, the conditions or the settings of

simulations have been summarized from the view-point of the dilemma between the degree of thedeterminancy of the system and the local energy

minima. It is considered desirable to examine theindividual conditions in more detail in order toavoid the local energy minima without disturbingthe physically valid process of the smooth decline

of the system's energy. This is especially importantwhen we attempt to conduct grain growth simu-lations that capture the ®ne di�erence of boundary

energies as implemented in the present work.

AcknowledgementsÐThe contributions from YujiYasuhara, graduate student, Tohoku University (now atSega Enterprises, Ltd), in the fundamental program codesfor the present simulation are gratefully appreciated. Theauthors are also grateful for the valuable comments of thereferees which facilitated revision of this manuscript.

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monte carlo simulation of grain growth with the full spectra of grain orientation and grain boundary...

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