Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

ScienceDirect

Acta Materialia 64 (2014) 443–454

Effect of grain boundary energy anisotropy on highly texturedgrain structures studied by phase-field simulations

Kunok Chang a,b,⇑, Nele Moelans a

a Department of Metallurgy and Materials Engineering, KU Leuven, Kasteelpark Arenberg 44, Box 2450, B-3001 Heverlee, Belgiumb Nuclear Materials Division, Korea Atomic Energy Research Institute (KAERI), 989-111 Daedeok-daero, Yuseong-gu, Daejeon 305-353, South Korea

Received 24 June 2013; received in revised form 7 October 2013; accepted 16 October 2013Available online 27 November 2013

Abstract

Two-dimensional phase-field simulations were performed of grain growth in highly textured materials with equal fractions of twotexture components, denoted as a and b grains, and assuming two values of the grain boundary energies, namely rlow for the boundariesbetween grains of a different texture component and rhigh for boundaries between grains of a similar orientation, resulting in microstruc-tures with alternating a and b grains and stable quadruple junctions. For different magnitudes of the anisotropy in grain boundary energyR ¼ rhigh=rlow, the occurrence of the different types of triple and quadruple junctions and the distributions of the normalized grain size,the number of faces per grain, the normalized grain boundary length per grain and the dihedral angles at grain boundary junctions wereinvestigated.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain growth; Phase-field modeling; Textured microstructure

1. Introduction

In order to predict various macroscopic properties ofpolycrystalline materials, characterization of the micro-structural characteristics is an essential step [1–4]. Forinstance, the average grain size plays an important rolein determining the yield strength [1,2] of a material. Theevolution of grain structures has been simulated fre-quently and has been quantitatively characterized bymeans of the average grain size, the grain size distribu-tion, and the number of faces and their distribution [5–7] for isotropic grain boundary properties. Grain growthin anisotropic systems has been simulated based on aMonte Carlo Potts model [8,9] and using the phase-fieldmethod [10]. In previous studies, the role of anisotropy

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2013.10.058

⇑ Corresponding author at: Department of Metallurgy and MaterialsEngineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44,Box 2450, B-3001 Heverlee, Belgium.

E-mail addresses: kunokchang@kaeri.re.kr (K. Chang), nele.moe-lans@mtm.kuleuven.be (N. Moelans).

in grain boundary energy and mobility in determiningthe grain size distribution, the distribution of the numberof faces per grain, and the misorientation distribution ofthe grain boundaries was evaluated. It was assumed thatthe crystallographic orientations were randomly assigned.Recently, it was also shown that the characteristics of tri-ple- and higher-order junctions are important microstruc-tural features in anisotropic systems in the context ofgrain topology and the misorientation distribution ofgrain boundaries [11].

Highly textured materials are also widely used [12,13].Grain growth of textured materials has been investigatedin terms of the distributions of the grain sizes and crystal-lographic orientations [14,15]. Moreover, Cahn, Holm andSrolovitz have analyzed the stabilities of trijunctions andquadrijunctions in conserved and non-conserved 2-Dtwo-phase microstructures as a function of the ratiosbetween the grain boundary energies of the different typesof interfaces [16,17], where a system with two texture com-ponents can be considered as a non-conserved two-phasesystem. Their study shows that quadruple junctions can

rights reserved.

444 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

become stable for a high degree of anisotropy in grainboundary energy. Different from triple junction dihedralangles, quadruple junction dihedral angles are thermody-namically not fixed, but can vary within a certain rangedepending on the ratios of the grain boundary energies.They are therefore expected to affect the grain growthbehavior and grain boundary network topology consider-ably. However, in general, the microstructural characteris-tics of highly textured materials and the effect of the degreeof anisotropy in grain boundary energy in these systemshave not been studied extensively. Moreover, very littleattention has been paid so far to characterize the differentpossible types of vertices and the distribution of dihedralangles in highly anisotropic systems.

In this paper, we present a systematic study of the roleof degree of anisotropy in grain boundary energy in deter-mining the occurrence of different types of vertices, thedihedral angle distribution, and the distributions of thenumber of faces, grain boundary length and grain sizein highly textured materials for microstructures with twotexture components. The grains belonging to the two dif-ferent components will be labeled respectively as a and bgrains. We will consider the case where rab ¼ rlow 6 raa ¼rbb ¼ rhigh with rab; raa and rbb the interfacial energy ofthe boundaries between a grain-b grain, a grain-a grainand b grain-b grain, respectively, for which quadruplejunctions have been predicted to coexist with triple junc-tions and even become the majority junction type withincertain ranges of the degree of anisotropy [16]. This situ-ation is less common than the opposite case whererab > raa ¼ rbb for the classical metallic systems (Al, Ni,Cu, etc., alloys) with a simple face-centered or body-cen-tered cubic structure. However, in more complex non-cubic structures, such as monoclinic zirconia [18] and fer-roelectric Cd1�xZnxTe (CZT) [19], where multiple twinsystems may exist, mosaic-like grain structures with onlytwo texture components and quadruple junctions wherealmost perfect twin boundaries (with extremely lowenergy) and low-angle boundaries meet, were observed.The case where rab > raa ¼ rbb has been studied before[14,17,20] and it was found that the topological character-istics are quite similar to those of isotropic systems sinceonly triple junctions can be stable. The grain growthkinetics can, however, deviate strongly from thoseobserved for isotropic grain structures. Depending onthe initial fractions of the a and b texture componentsand their initial spatial distribution, the steady-state para-bolic growth regime as derived for isotropic grain growth[21–23] may not be obtained.

For the simulated microstructures, we will verify five ofthe findings in Refs. [16,17]:

1. For isotropic grain growth (rlow ¼ rhigh), quadrijunctionor higher grain junctions are unstable and unlikely everto form. Even though they form, they decomposeimmediately.

2. The qabab type of quadrijunction becomes stable whenRð¼ rhigh=rlowÞP

ffiffiffi2p

. Whenffiffiffi2p6 R 6

ffiffiffi3p

; qabab quad-rijunctions and trijunctions coexist. Although taaa andtbbb trijunctions can be stable, taab and tabb trijunctionsare thermodynamically more favorable.

3. Whenffiffiffi3p

< R; taaa and tbbb trijunctions become unsta-ble. For

ffiffiffi3p

< R < 2; taab and tabb trijunctions coexistwith qabab quadruple junctions.

4. Only qabab quadrijunctions are stable when R P 2.

Following the notation of Refs. [16,17], taab is a triplejunction where 2a and 1b grains coexist and a similar nota-tion is used for the three other kinds of triple junctions.qabab is a quadruple junction where two a and two b grainsmeet according to a checkerboard pattern. Other kinds ofquadruple junctions have been shown to be unstable[16,17].

Furthermore, the distributions of grain sizes, averagenumber of faces per grain, dihedral angles and grainboundary length per grain will be characterized and thefindings will be related to the stability of (a) particulartype(s) of junctions. Since it is extremely complex to deter-mine and classify uniquely dihedral angles in 3-D systems,2-D simulations were performed. Although some of thephenomena present in 3-D systems may not be present in2-D systems, the 2-D simulations can already provide use-ful insights. They may also represent grain growth behaviorin thin films, as that observed for the monoclininc zirconiaor Cd1�xZnxTe (CZT) thin films. Moreover, the resultsfrom our 2-D simulations can be interpreted based on theanalytical theory of Cahn [16] which was also constructedfor 2-D systems.

2. Phase-field model and numerical solution

We adopted the multi-order parameter phase-field graingrowth model of Ref. [5]. According to Ref. [5], a single-phase material is represented by a set of non-conservedorder parameters, which are a continuous function of timeand space:

g1ðr; tÞ; g2ðr; tÞ; . . . ; gQðr; tÞ ð1Þ

Each grain is represented by a unique non-conserved orderparameter. The temporal and spatial evolution of the orderparameters is described by the time-dependent Ginzburg–Landau equation:

@giðr; tÞ@t

¼ �LdF

dgiðr; tÞð2Þ

where the kinetic constant L is related to the grain bound-ary mobility and the free energy F is a function of the orderparameter values and their gradients:

F ¼Z

V

XQ

i

g4i

4� g2

i

2

� �þXQ

i

XQ

i–j

hijg2i g

2j þ

j2

XQ

i

ðrgiÞ2

" #dV

ð3Þ

Table 1The ratios between grain boundary energies R ¼ rhigh=rlow and associatedh1 and h2 values used in this study.

R ¼ rhigh=rlow h1 h2

1.00 1.000 1.0001.39 1.400 0.7001.67 1.300 0.6201.81 1.350 0.6003.10 1.325 0.530

K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454 445

The parameters hij and j in the free energy functionaldetermine the grain boundary energies and magnitude ofgrain boundary energy anisotropy. The grain boundary en-ergy of a boundary between grains i and j is given by thefollowing integral expression [24]:

r¼Z þ1

�1

Xk¼i;j

g4k

4�g2

k

2

� �þhijg

2i g

2j þ

j2

dgi

dx

� �2

þdgj

dx

� �2 !" #

dx; ð4Þ

with x measured perpendicular to the grain boundary. Itsvalue was obtained by numerical integration over the equil-ibrated order parameter profiles across a straight grainboundary. A similar model was used by Tang et al. [25].They showed that this model can reproduce the dihedralangles in highly anisotropic systems with high accuracy.

To solve Eq. (2), a bounding box algorithm [26,27] isimplemented in two dimensions in combination with asemi-implicit spectral method [28]. The use of a semi-impli-cit discretization scheme allows us to take considerablylonger discretized time steps ðDtÞ than is possible with thestandard explicit discretization scheme. The boundingbox algorithm is based on a sparse data structure andhas proven to reduce considerably the computationalrequirements for large-scale grain growth simulations. Incontrast to other sparse data structure algorithms [29,30]developed for phase-field models, the bounding box algo-rithm can be combined with implicit and semi-implicittime-stepping schemes. The time derivative in Eq. (2) is dis-cretized using a first-order semi-implicit scheme [28], inwhich the homogeneous energy part is treated explicitlyand the gradient energy part implicitly:

gnþ1i � gn

i

Dt¼ L ðjr2giÞ

nþ1 þ g3i þ gi � 2hijgi

XQ

j

g2j

!" #n" #;

i ¼ 1; 2; . . . ;Q ð5Þ

Let us define /nðrÞ ¼ g3i þ gi � 2hijgi

PQj g2

j

� �h in. ~/nðkÞ and

take ~gnðkÞ to represent the Fourier transforms of /nðrÞ and

gnðrÞ, respectively. By transforming the partial differential

Eq. (5), we get a sequence of ordinary differential equations

in the Fourier space:

d~gni ðkÞdt

¼ L ~/nðkÞ � jk2~gni ðrÞ

h i; ð6Þ

where k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

1 þ k22

qis the magnitude of k, a vector in the

Fourier space. The left-hand side of Eq. (6) is approxi-mated by a forward Euler scheme, giving:

~gnþ1i ðkÞ ¼

~gni ðkÞ þ LDt~/nðkÞ

1þ LDtjk2ð7Þ

In the bounding box algorithm, a cuboid region is definedaround each grain i and for gi Eq. (7) is only solved for thegrid points within this region. In our study, each orderparameter indicates only one grain in the system. There-fore, the non-physical grain coalescence observed in [5] iscompletely eliminated.

To simulate grain growth in a highly textured material,we will assume that all grains with i even belong to the avariant, and all grains with i uneven to the b variant. Tointroduce the two different grain boundary energies, withrhigh for the boundaries between grains with a similar orien-tation and rlow for the boundaries between grains of a dif-ferent orientation, hij ¼ h1 is set in Eq. (3) when iþ j iseven and hij ¼ h2 is set otherwise. The hij values and thecorresponding ratio between the grain boundary energiesR ¼ rhigh=rlow used in this study are listed in Table 1. Fur-thermore, we choose L ¼ 1:0; j ¼ 2:0 and the time step istaken Dt ¼ 1:0 and Dx ¼ Dy is 2.0. All lengths and timeswill be expressed in multiples of Dx (voxel) and Dt (timestep).

3. Determination of multiple junctions, grain boundary

length and dihedral angles

We implemented a new methodology to classify the mul-tiple junctions and measure the dihedral angles at multiplejunctions from the voxel-based microstructure representa-tions obtained from the phase-field simulations. Moreover,an improved iterative method to measure grain boundarylength in the voxel-based microstructure representation isdeveloped.

3.1. Classification of the multiple junctions and measurement

of dihedral angles

The procedure implemented to find and classify the mul-tiple junctions and measure their dihedral angles in thevoxel-based microstructure representation is describedbelow and illustrated in Fig. 1.

1. First the position of the grain boundaries and triple,quadruple and higher junctions is determined asdescribed below. Consider, for example, the trijunction(point O) in Fig. 1.(a) Extract the voxels for which

Pig

2i <¼cutoff ! P .

These voxels are all located at or near a grainboundary.

(b) Examine for all voxels of P the four first-nearestand four second-nearest neighbors. It is assumedthat a voxel belongs to a grain of type i ifgi > gj; 8j – i.

Fig. 1. Schematic sketch of a trijunction in a voxel-based microstructurerepresentation. The grain regions of the three adjacent grains are indicatedwith different colors. The voxels labeled A1 . . . A4 have all three differenttypes of grains among their neighbors. The voxel labeled O is taken as theposition of the triple junction. The dihedral angles are determined fromthe intersection of the grain boundaries with the green circle. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

446 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

i. If there are two different types of grains among theneighbors ! Gij (grain boundary between ith grainand jth grain)

ii. If there are three different types of grains among theneighbors ! T ijk (trijunction where ith, jth and kthgrains coexist). If there are multiple voxels that meetthis condition next to each other (e.g. the voxelsA1;A2;A3 and A4 in Fig. 1), the

Pig

2i value is com-

pared for the different voxels Ai and the voxel withthe lowest value is taken as the position of the triplejunction.

iii. If there are four different types of grains amongthe neighbors ! Qijkl (quadrijunction where ith,jth, kth and lth grains coexist). If multiple voxelsnext to each other meet this condition, the sameselection process is performed as for the triplejunctions.

iv. If there are five different types of grains among theneighbors ! P ijklm (pentuple junction where ith, jth,kth, lth and mth grains coexist). If multiple voxelsnext to each other meet this condition, the sameselection process is performed as for the triplejunctions.

v. If there are six different types of grains among theneighbors ! Hijklmn (hextuple junction where ith,jth, kth, lth, mth and nth grains coexist). If multiplevoxels next to each other meet this condition, thesame selection process is performed as for the triplejunctions.

(c) If Gij does not exist, T ijk and T ijl merge into Qijkl.(d) If Gij does not exist, T ijk and Qijmn merge into

P ijkmn.(e) If Gij does not exist, T ijk and P ijlmn merge into

Hijklmn.

2. Draw a circle with its center at the multiple junction, e.g.the green circle around point O in Fig. 1

3. Find the intersection of the grain boundaries with thecircle.

4. Calculate the coordinates of the vectors (black arrow)from the multiple junction to the intersections.

5. Calculate the angles between these vectors.

In our study, we chose the circle radius r ¼ 5 in step 2.We examined the effect of taking various values for theradius of the circle and found that for R ¼ 1:00 (isotropicstructure) the peak of the dihedral angle distribution issharpest for r ¼ 5. We have also verified that the dihedralangle distributions for R > 1:00 did not change much whenr is taken equal to 3, 5 or 7.

3.2. Measurement of grain boundary length

The line length measurement method for voxel-basedmicrostructure representation previously introduced inRef. [31] was based on the straight-line assumption. Thisstraight-line assumption was reasonable for the study inRef. [31] because those authors considered systems withisotropic grain boundary energy and mobility. However,we found that the straight grain boundary assumption isinaccurate for grain structures with anisotropic grainboundary properties. Therefore, we introduce here animproved iterative method to measure the curved grainboundary length in voxel-based microstructure representa-tions generated by grain growth simulations of anisotropicsystems.

An N-segment piecewise linear method (hereinaftercalled N-SPLM) was implemented to measure the grainboundary length. The procedure is shown schematicallyin Fig. 2a and b. If the grain boundary is entirely straight,the line length can be evaluated by the distance between thetwo vertices (namely the distance between A and B inFig. 2a and b). The voxels at the grain boundary regionwere sorted in ascending order of distance from the voxelA. Then, to measure the line length using a 2-SPLM, thevoxel at the median rank was taken (C in Fig. 2b) andand the line length was calculated as the sum of the dis-tances from A to C and C to B. This procedure was per-formed recursively and the curved boundary length wasobtained from the n + 1-SPLM if the difference of thelengths of n-SPLM and n + 1-SPLM is less than 3%.

4. Simulations: initial microstructure and generalobservations

Two different initial states were generated. From eachstate, simulations were started for the different R-valuesgiven in Table 1. The initial states were generated by an iso-tropic grain growth (h1 ¼ h2 ¼ 1:0) simulation startingfrom 20,000 spherical grains distributed randomly in thesystem which were allowed to evolve until 3289 and 3270grains remained, respectively. The remaining order param-eters were reassigned randomly to these grains, and thegrains labeled as gi with i even were taken as a grains

Fig. 2. Flowchart and schematic drawing of the N-segment piecewise linear method to measure grain boundary length from a voxel-based microstructurerepresentation.

Fig. 3. Representation of the microstructure used in this study as an initialstate for the simulations. It was generated by an isotropic phase-fieldsimulation. At the start 20,000 spherical grains were distributed randomly.The isotropic grain growth simulation was stopped when 3289 grainsremained in the system. The microstructure is visualized by mapping thesum of the squared order-parameter values

PQi g2

i to a gray scale. Thesystem size was 1024� 1024 (expressed in voxels). The a grains and bgrains are shown in cyan and yellow, respectively. (For interpretation ofthe references to color in this figure legend, the reader is referred to theweb version of this article.)

16

18

20

22

24

26

28

30

32

34

36

70 75 80 85 90 95 100

Aver

age

grai

n si

ze

Square root of time steps

σmax/σmin=3.10

Fig. 4. The average grain diameter (expressed in number of voxels) withrespect to the square root of simulation time obtained for R ¼ 3:10. Theaverage grain diameter increases from 17.16 grid points to 35.91 gridpoints and the number of grains decreases from 3235 grains to 722 grains.

K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454 447

and those with i uneven as b grains. From this initial micro-structure, anisotropic grain growth simulations werestarted using the different sets of h1 � h2 values listed inTable 1. One of the initial states is shown in Fig. 3.

For all cases, a regime with steady-state grain growthbehavior could be reached. We checked that the grain sizedistribution and the fraction of the two major vertice typesdid not notably change at the steady state, i.e. the fractionsof trijunctions, quadrijunctions, etc., did not vary morethan 1% over 100 time steps during steady-state growth.As R increases, the rate of grain elimination reduces andlonger simulation times were required to obtain the steadystate. However, different from the growth stagnation

observed for large R-values, namely for R >ffiffiffi3p

, in the pre-vious Monte Carlo simulations [17], we achieved steady-state growth and a steady-state grain size distribution evenfor the large degrees of anisotropy. For R = 3.10, we con-tinued the simulation up to 12,000Dt, but did not observeany stagnation. The average grain diameter as a functionof the square root of time is plotted in Fig. 4. Forffiffi

tp

> 100, the remaining number of grains in the systemhas become too low to obtain reasonable statistics;although on average parabolic growth continues, the aver-age mean grain size becomes more and more scattered. Webelieve that the stagnation observed in the Monte Carlosimulations [17] may have been due to artificial lattice pin-ning. Probably, the dihedral angles at the quadruple junc-tions, which have values around 90�, could not beresolved appropriately on the triangular grid used in Cahnet al.’s simulations.

448 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

The simulated microstructures as observed in the steady-state regime for the different R-values are shown in Fig. 5a–e. Pictures are taken at different times for different R-valuessuch that the average grain size is similar. Although theamount of a and b type of grains is not a conserved prop-erty in the considered system, the fraction of the phasesremains 0.5 throughout the simulations due to the symme-try in grain boundary properties.

5. Effect of anisotropy on the stability of different kinds of

triple and quadruple junctions

The fractions of trijunction, quadrijuntion and higherjunctions, and the different types of them, were determinedfrom the simulated microstructures following the proce-dure described in Section 3 and are listed in Tables 2–4.For each R value, two simulations starting from a differentinitial state were performed and the fractions obtained inthe two simulations were averaged. All quantities in thetables were measured at steady state. The fractions oftwo major vertice types did not notably change in thesteady-state regime, i.e. their fractions did not vary morethan 1% for 100Dt.

Fig. 5. Simulated microstructures obtained during steady-state growth for diffesystem size was 1024� 1024 (expressed in voxels). The a grains and b grainreferences to color in this figure legend, the reader is referred to the web versi

As predicted by Cahn et al. [16,17], we rarely observedquadrijunctions or higher junctions in the microstructurewith isotropic grain boundary properties. As given inTable 2, the fraction of quadri- and higher junctions wasapproximately 1.05% for R ¼ 1:00. Furthermore, accord-ing to Table 3, the fraction of taaa/tbbb type junctions amongall trijunctions was 25% for R ¼ 1, which is also consistentwith the simple probabilistic prediction that the possibilityof forming taaa/tbbb type trijunctions is 2

2�2�2¼ 0:25.

Although according to the predictions of Cahn et al.[16,17] quadruple junctions are expected to become stableonly for R P

ffiffiffi2p

, the fraction of quadri- and higher junc-tions was approximately 30.0% of the total vertice countin the simulations for R ¼ 1:39 <

ffiffiffi2p

, which is alreadymuch higher than for R ¼ 1:00. Moreover, from the con-secutive images in Fig. 6a and b, it is clear that most ofthe quadrijunctions can exist for a considerable time andcan thus be considered as being stable. The theoretical pre-dictions of Cahn et al. assumed, however, straight grainboundaries [16,17], whereas in real microstructures mostgrain boundaries are curved, which may slightly affect thestability of the junctions. Moreover, the value of 1.39 isvery close to

ffiffiffi2p� 1:41. In agreement with the theoretical

rent degrees of anisotropy, namely for R ¼ 1, 1.39, 1.67, 1.81 and 3.10. Thes are shown in cyan and yellow, respectively. (For interpretation of theon of this article.)

Table 2Fractions of the trijunctions, quadrijunctions, quinquejunctions andsexajunctions for different R values measured from the simulatedmicrostructures.

R ¼ rhigh=rlow Tri (%) Quadri (%) Quinque (%) Sexa (%)

1.00 98.94 1.03 0.02 0.001.39 69.70 30.27 0.03 0.001.67 30.18 69.78 0.04 0.001.81 15.98 83.62 0.32 0.083.10 0.82 98.82 0.23 0.14

Table 3Fractions of the two different types of the trijunctions at given R values.For each R value, two simulations starting from a different initial statewere performed and the fractions obtained in the two simulations wereaveraged.

R ¼ rhigh=rlow taaa/tbbb(%) taab/tabb(%)

1.00 25 751.39 5 951.67 2 981.81 0 1003.10 0 100

Table 4Fractions of the three different types of quadrijunctions at given R values.For each R value, two simulations starting from a different initial statewere performed and the fractions obtained in the two simulations wereaveraged.

R ¼ rhigh=rlow qaaaa/qbbbb(%) qaaab/qabbb(%) qabab(%)

1.00 6.98 55.81 37.211.39 0.00 3.60 96.401.67 0.00 0.46 99.541.81 0.00 0.47 99.533.10 0.00 0.50 99.50

Fig. 6. Highly magnified part of the simulated microstructures obtained attwo different time steps for R ¼ 1:39. The a grains and b grains are in cyanand yellow, respectively. A red spot indicates a trijunction and a greenspot a quadrijunction. Comparison of (a) and (b) shows that thequadrijunctions on the left and in the center do not immediatelydecompose into trijunctions. On the other hand, the quadrijunction inthe right top part of Fig. 6a at 6500Dt has split into two trijunctions inFig. 6b at 6520Dt. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 7. Highly magnified part of the simulated microstructures obtained attwo different time steps for R ¼ 3:10. The a grains and b grains are in cyanand yellow, respectively. A red spot indicates a trijunction and a greenspot a quadrijunction. The two trijunctions seen in Fig. 7a at 8500Dt havemerged into one quadruple junction in Fig. 7b. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the webversion of this article.)

K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454 449

prediction, both types of triple junctions are stable. Themajority of the triple junctions is, however, of the typetaab=tabb, since rab < raa ¼ rbb and therefore grains of dif-ferent types have the tendency to alternate, as was alsoobserved in previous Monte Carlo simulations [17].

In the simulations with R ¼ 1:67, the major kind of ver-tices was the qabab type quadrijunction. There were also stilla considerable number of triple junctions. This is in agree-ment with the theoretical predictions of Cahn et al. [16],since, for

ffiffiffi2p6 R ¼ 1:67 6

ffiffiffi3p

, it is expected that bothtypes of trijunctions and the qabab type of quadrijunctionare stable. As in the previous case, the fraction oftaaa=tbbb type of triple junctions is quite low, namely lessthan 4%, since a and b grains have the tendency to alter-nate for rab < raa ¼ rbb.

When R ¼ 1:81 >ffiffiffi3p

in the simulations, more than 81%of the vertices are quadrijunctions and more than 99% ofthem are of qabab type. The major type of triple junctionsis taab=tabb. This is in agreement with the predictions ofCahn et al. [16] for

ffiffiffi3p

< R < 2.

If R ¼ 3:10 > 2:0, more than 98% of the junctions areabab-typed quadrijunctions in the simulations. Triple junc-tions can form; however, they appear to be unstable. InFig. 7a, for example, one can see two trijunctions locatedclose to each other in a simulated microstructure forR ¼ 3:10. However, they merge into one quadrijunctionwithin less then 20Dt as shown in Fig. 7b. This is in agree-ment with the prediction of Cahn et al., according to whichthe qabab quadrijunction is the only stable vertice type whenR P 2.

In general, our observations thus show good agreementwith the trends predicted by Cahn et al. [16,17]. The junc-tion types that are unstable in the predictions by Cahnet al.’s analysis are always found to be a minor junctiontype in the simulations, although in some cases we can stilldetect them in the system.

Fig. 8. The distribution of the dihedral angles at the scaling regime obtained in the simulations for different R-values.

450 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

6. Distributions of dihedral angles, number of faces,

normalized grain size and normalized grain boundary length

6.1. Dihedral angles

The equilibrium dihedral angles at triple junctions arefixed and can be calculated easily as a function of the mag-nitude of the grain boundary anisotropy from an interfacialtension balance (Lami’s theorem). The taaa=tbbb type of tri-ple junction is accordingly expected to have all three dihe-dral angles equal to 120� and the taab=tabb type of triplejunction is expected to have two angles equal top� arccosðR=2Þ and one angle equal to 2 arccosðR=2Þ.Therefore, a distribution with sharp peaks at the possibleequilibrium dihedral angles is expected if only triple junc-tions are present. The equilibrium dihedral angles at qua-druple junctions, however, are not uniquely determinedby an interfacial tension balance. Assuming grain bound-aries are straight and behave independently from the restof the grain boundary network, Cahn calculated a lowerand higher bound between which the dihedral angles at a

stable quadruple junction may vary, with the lower boundgiven by Ul P 2 arccosðR=2Þ and the higher bound byUh 6 p� Ul. As discussed before and shown in Fig. 6, inconnected grain structures it is possible that quadruplejunctions with an angle outside this range become stableas well, although it is a minority event. Consequently,one cannot easily deduce how the degree of anisotropy willaffect the distribution of dihedral angles when quadruplejunctions become stable.

The distributions of the dihedral angles as obtainedfrom the simulations for different magnitudes of the anisot-ropy R are shown in Fig. 8a–d. For each case, it is shownhow the dihedral angles at the triple and quadruple junc-tions separately contribute to the total distribution of dihe-dral angles.

When R ¼ 1 (Fig. 8a), there are only triple junctions andthe dihedral angle distribution peaks at 120�, as expected inthe case of isotropic grain boundary properties [32,33].

When R ¼ 1:39, most (70%) of the junctions are still tri-junctions (see Table 2) and most of these are of the taab=tabb

type. The contribution from the trijunctions in Fig. 8b

Fig. 9. Grain geometry used to verify the equilibrium dihedral angles attriple junctions obtained in the simulated systems. The energy of the grainboundary between the yellow and cyan grains is 1.39 times higher thanthat of the grain boundary between the yellow and red grains and thatbetween the yellow and cyan grains. The blue lines indicate the tangentialto the grain boundary at the triple junction. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the webversion of this article.)

Fig. 10. Highly magnified region of the microstructure obtained forR ¼ 1:39 showing the typical triangular grains with non-equilibriumdihedral angles present in this microstructure. The a grains and b grainsare in cyan and yellow, respectively. (For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version ofthis article.)

σσσσσ σ

σσσσ Δ

ΔΔΔΔ

Fig. 11. The distribution of the number of faces at the steady-state regimefor R ¼ 1:00 (red), R ¼ 1:39 (blue), R ¼ 1:67 (green), R ¼ 1:81 (brown) andR ¼ 3:10 (navy). The grain size distributions were plotted at 1500Dt fromthe initial state for R ¼ 1:00; 1:39, at 2000Dt from the initial state forR ¼ 1:67 and at 3000Dt from the initial state for R ¼ 1:81; 3:10. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Fig. 12. R ¼ 3:10. Highly magnified region of the microstructure. The agrains and b grains are in cyan and yellow, respectively, and a green spotindicates a quadrijunction. A disappearing 3-sided grain is indicated with aviolet circle. As a result, two 4-sided grains transform into a 3-sided grain,whereas only one 5-sided grain transforms into a 4-sided grain. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454 451

(blue curve) shows two peaks, one around 70–80� andanother around 150�, which deviates from the equilibriumangles 91:95� and 134:03� calculated from Lami’s theoremfor the taab=tabb type of triple junction and R ¼ 1:39. To ver-ify that the deviation of the triple junction angles from theirequilibrium value is not due to the modeling approach or anumerical artifact, a grain growth simulation was per-formed considering a system with three grains as shownin Fig. 9. The grain boundary energy of the boundarybetween the yellow and cyan grains is 1.39 times higherthan that of the boundaries between the yellow and redgrains and the yellow and cyan grains. In this simulation,dihedral angles at the triple junction were equal to 90�

and 135�, which are very close to the equilibrium angles(the deviation is less than 2%). It was also shown by Tanget al. [25] that the implemented phase-field model is able toreproduce dihedral angles in highly anisotropic systemsaccurately. Therefore, we conclude that the observed devi-ation from equilibrium in the simulations for R ¼ 1:39 isbecause of topological restrictions imposed by the sur-rounding grain network. A closer look at the

microstructure shows that the taab=tabb triple junctions aremost often at rather small triangular grains, which oftenexist in twos with two triple junctions and a grain bound-ary in between, as shown in Fig. 10. Highly curved grainboundaries at these triangular grains would thus berequired to obtain the equilibrium dihedral angles in thetriple junctions. Moreover, the large driving force toshorten the high-energy a–a or b–b grain boundary con-necting the two triangular grains also prevents the dihedralangles at the triangular grains from obtaining their equilib-rium values. Consequently, the surfaces of the triangulargrains are only weakly convex during grain evolution andthe inner angles of the two triangles are an angle D sharperthan expected, i.e. 92� � D, while the two outer angles arean angle D=2 larger than expected, i.e. 134�+D=2. Thereis a wide spread of the value of D depending on the sur-rounding grain boundary network. The distribution of

Fig. 13. The overall steady-state normalized grain size distributions and contributions from each topological class for different R-values. The data werecollected from two different sets of simulations to obtain more rigorous statistics.

452 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

the quadruple junctions (green curve) peaks at 90�, but theangles vary within a wide range. In the total distribution,the peaks from the trijunction angle distribution are stillpresent; however, the peak around 67:4� is smeared outdue to the contribution from the quadrijunction angles.

For R ¼ 1:67 (Fig. 8c), almost 70% of the junctions areqabab junctions. The overall distribution of the dihedralangles in Fig. 8c has accordingly a broad peak around90� (red line). The curve of the triple junction dihedralangles shows two peaks, one around 50� and a broad peakaround 160�. Also in this case the taab=tabb triple-junctionangles of the evolving microstructure deviate from theirequilibrium values and are determined by the topologicalrestrictions imposed by the surrounding network. On aver-age, the deviation of the triple-junction angles from their

equilibrium values is larger than for the case R ¼ 1:39 sincethe driving force to shorten the high-energy boundary inbetween two triangular grains is larger. For R ¼ 1:81 andR ¼ 3:10, quadrijunctions are dominant (see Table 2) andthe dihedral angle distribution has a wide peak at 90�, asshown in Fig. 8d and e.

6.2. Number of faces

The distributions of the number of phases obtained fordifferent R-values are plotted in Fig. 11. As predicted bythe former works [37,38], the distribution of the numberof faces has a peak at f ¼ 6 for the isotropic case(R ¼ 1:00) in our study. For all other cases(R ¼ 1:39; 1:67; 1:81; 3:10), the distribution of the number

σσσσ σ

σσσ

Fig. 14. The distribution of the normalized grain boundary length at thescaling regime for R ¼ 1:00 (red), R ¼ 1:39 (blue), R ¼ 1:81 (green) andR ¼ 3:10 (brown). The distributions were taken at 1500Dt after the initialstate for R ¼ 1:00 and 1.39 and at 3000Dt after the initial state for R ¼ 1:81and 3.10. E represents the grain boundary length of each grain boundaryand hEi the average grain boundary length. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the webversion of this article.)

K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454 453

of faces has a peak at f ¼ 3. The distributions of the num-ber of faces have a very similar shape when R ¼ 1:67; 1:81and 3:10, where the majority of the junctions are quadru-ple. There is a peak at f ¼ 3, but there is also a large num-ber of grains with four sides. For R ¼ 1:39, where bothtriple and quadruple junctions are present, the distributionis much broader and the number of 3-, 4- and 5-sidedgrains is comparable even though the distribution peakedat f ¼ 3.

Since the dihedral angles at quadrijunctions are notthermodynamically fixed [17], it is not intuitive how torelate the distributions of the dihedral angles and the ver-tex types with that of the number of faces. In the trijunc-tion dominant system, the hexagonal grain (6-sided) isthe most frequent type of grains, and the second mostfrequent type of grain is the pentagonal grain (5-sided)[37,38]. Therefore, we can conclude that trijunctions seemto favor 5- and 6-sided grains and quadruple junctions 3-and 4-sided grains. Cahn et al. predicted the presence ofthe 3-sided and 4-sided ZIC grain which has zero inte-grated curvature in quadrijunction-dominant systems[17].

As shown in Fig. 12a and b, if one 3-sided grain is elim-inated (the grain surrounded by the violet circle), the sur-rounding 4- or 5-sided grains also lose one of their sides(yellow). As a result, two 3-sided grains (yellow) and one4-sided grain are generated from the removal of one 3-sided(cyan) grain. This appears to be a reasonable explanationwhy the number of 3-sided grains is higher than that of4-sided grains for R P 1:39.

We also note that, while 2-sided grains do not exist inthe isotropic case, there is a small fraction of 2-sided grainspresent in anisotropic systems with quadruple junctions.Two-sided grains were also noticed in phase-field simula-tions of the evolution of conserved two-phase systems withquadruple junctions [36].

6.3. Normalized grain size

The normalized grain size distribution for different mag-nitudes of anisotropy is plotted in Fig. 13a–e. The individ-ual contributions from the different topological classes areplotted as well. The distributions are plotted at one partic-ular time step in the steady-state regime; however, it wasverified that the distributions did not change considerablyonce steady-state growth is reached. According toFig. 13a, for R ¼ 1:00 the three distribution profiles ofthe major topological classes (f ¼ 5; f ¼ 6 and f ¼ 7)form a distribution with a plateau as was also pointedout by Kim et al. [7] for systems with isotropic grainboundary energy and mobility. For increasing magnitudeof the anisotropy R, the contributions from grains withthree and four sides become more significant (seeFig. 13a), and affect the shape of the grain size distribution.As can be seen in Fig. 13d for R ¼ 1:81, peaks, mainlyformed by the contributions from the 3- and 4-sided grains,are initiated on the plateau. When R ¼ 3:10 (see Fig. 13e), abimodal grain size distribution has clearly developed, withthe left peak mainly coming from the 3-sided grains and theright peak from the 4-sided grains. Although it is generallyaccepted that a bimodal grain size distribution is a signifi-cant evidence of abnormal grain growth [34,35], in ouranisotropic grain growth simulation, the bimodal grain sizedistribution is observed without implementation of anyabnormality. It is because the 3- and 4-sided grains formthe major two peaks of the bimodal distribution. More-over, this distributions is invariant in time.

6.4. Normalized grain boundary length

The distributions of the normalized grain boundarylengths are plotted in Fig. 14 for different R-values. Thedistributions obtained for R ¼ 1:39; 1:81 and 3.1, where aconsiderable number of quadruple junctions are present,are much broader than that obtained for the isotropic case(R ¼ 1:00).

7. Conclusion

Two-dimensional phase-field simulations were per-formed of grain growth in highly textured materials withequal fractions of two texture components and assumingtwo values of the grain boundary energy, namely rlow forboundaries between grains of a different orientation andrhigh for boundaries between grains of a similar orientation.For R ¼ rhigh=rlow > 1, the simulated structures consistedof alternating a and b grains. Different from the growthstagnation observed for large R-values in previous MonteCarlo simulations, we achieved steady-state growth evenfor high degrees of anisotropy (i.e. large R-value). Quadru-ple junctions were already stable from R ¼ 1:39, which is aslightly lower value than the R ¼

ffiffiffi2p

predicted by Cahnet al. However, in general, our observations were consistentwith the predictions of Cahn et al. [16,17].

454 K. Chang, N. Moelans / Acta Materialia 64 (2014) 443–454

We found that the presence of quadruple junctions has alarge effect on the steady-state shape of the distributions ofthe grain sizes, the number of faces, the dihedral angles atjunctions and the grain boundary length. The main findingsare the following:

� Since quadruple junctions are thermodynamically notfixed, the distribution of the dihedral angles has a broadpeak around 90� for R P 1:67, when most of the junc-tions are of the qabab quadruple type, whereas a sharppeak at 120� is generally obtained in the isotropic case.� For 1 < R < 1:67, when both triple and quadruple junc-

tions are stable, the distribution of the dihedral anglescan have multiple peaks coming from the different typesof triple and quadruple junctions. However, due to geo-metrical restrictions imposed by the surrounding grainboundary network, the triple junctions generally deviatefrom the equilibrium dihedral angles.� The distribution of the number of faces peaks at f ¼ 6

for R ¼ 1, where only triple junctions are present, andat f ¼ 3 for R P 1:67, when the majority of the junc-tions are of the quadruple type. Once R P 1:67, verysimilar distributions are obtained for simulations withdifferent R-values. For R ¼ 1:39, where both triple andquadruple junctions are present, a broad distributionis found, which, however, peaks at f ¼ 3.� For a high degree of anisotropy, where only quadruple

junctions are stable, a bimodal grain size distributionis obtained, but does not lead to abnormal grain growth.

Acknowledgements

We sincerely appreciate the CREA-financing of KULeuven, grant CREA/12/012 on phase-field modeling ofthe morphological evolution during phase transitions inorganic materials. Parallel simulations were run on theHigh Performance Computer cluster of VSC (VlaamsSupercomputer Centrum).

References

[1] Hall EO. Proc Phys Soc 1951;64B:747.[2] Petch NJ. J Iron Steel Inst 1953;174:25.

[3] Iwai H, Naoki S, Matsui T, Teshima H, Kishimoto M, Kishida R,et al. J Pow Sou 2010;195:955.

[4] Wang L, Zhang J, Gao Y, Xue Q, Hu L, Xu T. Scripta Mater2006;55:657.

[5] Chen LQ, Yang W. Phys Rev B 1994;50:15752.[6] Srolovitz DJ, Anderson MP, Sahni PS, Grest GS. Acta Metall

1984;32:793.[7] Kim SG, Kim DI, Kim WT, Park YB. Phys Rev E 2006;74:061605.[8] Rollett AD, Srolovitz DJ, Anderson MP. Acta Metall 1989;37:1227.[9] Gruber J, Miller HM, Hoffmann TD, Rohrer GS, Rollett AD. Acta

Mater 2009;57:6102.[10] Kazaryan A, Wang Y, Dregia SA, Patton BR. Acta Mater

2002;50:2491.[11] Johnson OK, Schuh CA. Acta Mater 2013;61:2863.[12] Horn JA, Zhang SC, Selvaraj U, Messing GL, Trolier-McKinstry S. J

Am Ceram Soc 1999;82:921.[13] Seabaugh MM, Kerscht IH, Messing GL. J Am Ceram Soc

1997;80:1181.[14] Ivasishin OM, Shevchenko SV, Semiatin SL. Scripta Mater

2004;50:1241.[15] Abbruzzese G, Lucke K. Acta Metall 1986;34:905.[16] Cahn JW. Acta Metall 1991;39:2189.[17] Holm EA, Srolovitz DJ, Cahn JW. Acta Metall 1993;41:1119.[18] Gertsman VY. Interface Sci 1999;7:231.[19] He ZB, Stolitchnova I, Settera N, Cantonib M, Wojciechowski T,

Karczewski G. J Alloys Compd 2009;484:757.[20] Poulsen SO, Voorhees PW, Lauridsen EM. Acta Mater 2013;61:1220.[21] Ma N, Kazaryan A, Dregia SA, Wang Y. Acta Mater 2004;52:3869.[22] Kazaryan A, Patton BR, Dregia SA, Wang Y. Acta Mater

2002;50:499.[23] Holm EA, Hassold GN, Miodownik MA. Acta Mater 2001;49:2981.[24] Moelans N, Blanpain B, Wollants P. Phys Rev B 2008;78:024113.[25] Tang M, Reed BW, Kumar M. J Appl Phys 2012;112:043505.[26] Vanherpe L, Moelans N, Blanpain B, Vandewalle S. Phys Rev E

2007;76:056702.[27] Vanherpe L, Moelans N, Blanpain B, Vandewalle S. Comp Mater Sci

2011;50:2221.[28] Chen LQ, Shen J. Comp Phys Commun 1998;108:148.[29] Vedantam S, Patnaik BSV. Phys Rev E 2006;73:016703.[30] Gruber J, Ma N, Wang Y, Rollett AD, Rohrer GS. Model Simul

Mater Sci Eng 2006;14:1189.[31] Chang K, Krill CE III, Du Q, Chen LQ. Model Simul Mater Sci Eng

2012;20:075009.[32] Neumann JV. In: Metal interfaces. American Society for Testing

Materials, Cleveland; 1952. p. 108.[33] Mullins WW. J Appl Phys 1956;27:900.[34] Thompson CV, Frost HJ, Spaepen F. Acta Metall 1987;35:887.[35] MacLaren I, Cannon RM, Gulgun MA, Voytovych R, Popescu-

Pogrion N, Scheu C, et al. M J Am Ceram Soc 2003;86:650.[36] Fan D, Chen LQ. Scripta Mater 1997;37:233.[37] Saito Y. Mat Sci Eng A 1997;A223:114.[38] Gill S, Cocks A. Acta Mater 1996;44:4777.