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Scripta Materialia 64 (2011) 1079–1082

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Effect of microelasticity on grain growth: Texture evolutionand abnormal grain growth

Dong-Uk Kim,a Seong Gyoon Kim,b Won Tae Kim,c JaeHyung Cho,d

Heung Nam Hane and Pil-Ryung Chaa,⇑aSchool of Advanced Materials Engineering, Kookmin University, Seoul 136-702, South Korea

bDepartment of Materials Science and Engineering, Kunsan National University, Kunsan, South KoreacDivision of Applied Science, Cheongju University, Cheongju 360-764, South Korea

dKorea Institute of Materials Science, Changwon 641-831, South KoreaeDepartment of Materials Science and Engineering, Seoul National University, Seoul 151-744, South Korea

Received 13 December 2010; revised 15 February 2011; accepted 15 February 2011Available online 23 February 2011

A phase field grain growth model including elastic anisotropy and inhomogeneity was developed to demonstrate the effect ofmicroelasticity on the grain growth. The mechanical response against an external load was found to control grain growth and tex-ture evolution. In contrast to previous macroelastic descriptions, these results showed that elastically soft grains with higher strainenergy density can grow at the expense of the elastically hard grains to reduce the total strain energy.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain growth; Texture; Abnormal grain growth; Microelasticity; Phase-field model

Grain growth (GG) is one of the most funda-mental phenomena in material physics and remains anunsolved problem despite being studied extensively formore than a century. One of the main reasons whyGG has remained unsolved for such a long time is thatit is a representative system with multi-body interac-tions. Grains in a polycrystal grow or disappear throughmulti-body interactions with their neighbors and thegrowing grains keep encountering new neighbors. More-over, the dependency of the grain boundary energy(GBE) and mobility (GBM) on the misorientation be-tween grains increases the complexity of grain growth.

There is another multi-body long-range interactionthat affects GG when a system is constrained elastically:neighboring grains affect each other via a different elas-tic response. This different elastic response generallyoriginates from anisotropy of the elastic moduli withina single crystal grain. In addition, the strain state insidea grain might not be uniform because a grain is in con-tact with several other surrounding grains with differentorientations. We refer to this inhomogeneous strainstate inside a single crystal grain as “microelasticity”.

1359-6462/$ - see front matter � 2011 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2011.02.022

⇑Corresponding author. Tel.: +82 2 910 4656; fax: +82 2 910 4320;e-mail: cprdream@kookmin.ac.kr

This multi-body interaction, together with anisotropyof the GBE and GBM, makes solving the problem ana-lytically significantly challenging. In this sense, any sim-ple method, such as the mean-field model [1], isobviously not applicable.

To overcome the difficulties and describe the graingrowth behavior as realistically as possible, numericalschemes such as the Monte Carlo Potts model (MC)[2–4], vertex model [5,6], surface evolver [7], front track-ing method [8], cellular automata [9], finite element meth-od [10] and phase field model (PFM) [11–14] have beenadopted. However, most studies on GG [2–14] have fo-cused on ideal grain growth or the effect of GBE andGBM anisotropies on GG. Therefore, to develop deeperunderstanding on GG under elastic stress (e.g. microelas-ticity strongly affects growth kinetics and texture evolu-tion, especially, in thin films [15]), it is important tocouple the GG behavior with the effect of microelasticity.

This paper reports a phase field GG model combinedwith a microelasticity effect including elastic anisotropyand inhomogeneity, along with its applications to tex-ture evolution and abnormal grain growth due to anexternal load.

This study combines the PFMs for grain growthproposed by Kim et al. [16] and for the microelasticityproposed by Steinbach and Apel [17]. The PFM for the

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1080 D.-U. Kim et al. / Scripta Materialia 64 (2011) 1079–1082

microelasticity was extended to include the externally ap-plied strain/stress field and an effective solving methodfor elastically inhomogeneous system was implemented.The model proposed by Kim et al. [16] can consider anunlimited number of grains and orientations without anincrease in computational cost, while that proposed bySteinbach and Apel [17] minimizes the numerical instabil-ity of the interface thickness caused by the distribution ofthe elastic field within the interface region. If isotropicGB energy and mobility are assumed, the phase fieldequation for the grain growth of a polycrystalline micro-structure can be written as follows [16]:

@/q

@t¼ 2M/

S

XQ

r–q

SrSqdFd/q� dF

d/r

!ð1Þ

where the order parameter /q (q = 1,2,3, . . ., Q) givesthe orientation state of a point in a polycrystalline sys-tem containing Q grains, the integer q can be regardedas a number indicating a specific orientation of the grainand the sum of all phase-field values in a point (i, j, k) isconserved as

PQq¼1/qði; j; kÞ ¼ 1. The step function is

sq ¼ 1 if /q > 0, otherwise sq ¼ 0. The number of phasescoexisting in a given point is Sði; j; kÞ ¼

PQq¼1sqði; j; kÞ.

The total free energy function consists of two parts:the grain boundary energy density FGB and elastic freeenergy density Fel:

F ¼Z

XF GB þ F eldX ð2Þ

F GB ¼XQ

r;q¼1

e2

2j r/r � r/q j þx j /r/q j ð3Þ

The parameters M/, e and x, Eqs. (1) and (3) have adefinite relationship with the GBE, r, its width, 2k,and the mobility, m, of the GB. The detailed relationshiphas been reported in Ref. [16]. The elastic energy densityis [17]:

F el ¼ 1

2

Xn

a

/aðeija þ eij

AÞCijkla ðekl

a þ eklA Þ

( )ð4Þ

where eija is the strain field in the a grain and the total strain

eij is defined as eij ¼P

a/aeija , eij

A is the externally appliedstrain, Cijkl

a is the elastic stiffness of the a grain and the

total elastic constant Cijkl is Cijkl ¼Pn

a¼1/a½Caijkl��1

h i�1

.

As the strain fields in each grain are included as variablesas well as the total strain variables, eij, new constitutiveequations are required for the additional strain variables,and the following constitutive equations were used [17]:

rija ¼ rij

b ¼ rij ð5Þwhere rij is the total stress field. The elastic field can besolved by assuming a local mechanical equilibrium suchthat the elastic field relaxes much faster than the phase field:

dFdui¼ rjr

ij ¼ 0 ð6Þ

A high-order approximation method [18,19] was usedto obtain accurate solutions of Eq. (6) with minimal cal-culation time (see Supporting Online Materials for thevalidity and convergence of this method). The solutionof Eq. (6) in the nth order of the elastic constant C0

ijkl,i.e. the displacement field to the nth order, becomes

C0ijkl ¼ Cijkl � C0ijkl where C0

ijkl is the elastic modulus inthe reference state)

buNk ¼ ia�1

kl �kidC0ijkle

klA � ki ð dC0ijklrlÞuN�1

k

n oh ið7Þ

where the symbolbrepresents the value of a function inFourier space, kj is the j-directional component of thewave vector and a�1

ki is a component of the inverse oftensor aki, which is defined as C0

ijklkjkl.A finite difference scheme was used for special discret-

ization of the phase field Eq. (Eq. (1)) and Euler’s methodwas used for the integration in time. The followingnumerical conditions and physical properties were usedin all simulations: GB energy r = 0.3 J m–2; grid sizeDx ¼ 0:5 lm; interface width 2k ¼ 6Dx; phase fieldmobility M/ ¼ rm=e2 ¼ 3� 10�6 m3=Js; time step Dt ¼0:9; Dx2=ð6M/e2Þ; e = 6.98 � 10�4(J m–1)0.5. The elasticconstants used for Cu were: C11 = 168. 4, C12 = 121.4and C44 = 75.4 (in units of GPa). All simulations werecarried out on 512 � 512 � 2 grids, which resembles athin film with columnar grains, though the surface energyand groove dragging were not considered and the periodicboundary condition was imposed in all directions. Theinitial number of grains was 10,054, with different crystal-lographic orientations (Euler angles). Gaussian-shapedstandard distributions [23] were used to generate the tex-ture distribution of interest. The intensity distribution ofa texture component in a material was assumed to have abell-shaped structure, which can be described by themodel distribution. The elastic constants of each grainwere transformed by the rotation matrix based on itsEuler angle. The x, y and z axes correspond to the rollingdirection, transverse direction and normal direction(ND), respectively.

In this study, two biaxial loading (x and y directions)conditions were considered: displacement (externalstrain) controlled and load (external stress) controlled.Figure 1 shows the texture evolution from an initial ran-dom texture and the variation of (0 0 1) pole figures dueto biaxial strain (B and E) and stress (C and F). Initially,10,054 grains were constructed using the standard distri-bution of the Euler angles corresponding to a randomtexture, which is demonstrated with a (0 0 1) pole figure(Fig. 1A) and microstructure (Fig. 1D). Figure 1B and Eshows the pole figure and polycrystalline microstructureat t = 2500Dt for a displacement controlled loading, andFigure 1C and F shows the same for a stress-controlledloading. For a displacement-controlled case a strongh1 0 0i//ND fiber evolves simultaneously, whereas ah1 1 1i//ND texture emerges from a random texture ina load-controlled system.

The results in Figure 1 are consistent with previousexperimental and theoretical studies [15,20,21], whichassumes that either an equal biaxial strain (isostrainaveraging) or an equal biaxial stress (isostress averaging)is supported on average by the grains. From the view-point of strain energy minimization, the soft h1 0 0i-ori-ented grains with the lowest strain energy density growselectively in isostrain averaging (displacement-con-trolled), whereas the hard h1 1 1i-oriented ones growpreferentially in isostress averaging (load-controlled).In the displacement-controlled case, the situation ob-served in the phase field simulations is different from

Figure 1. (0 0 1) pole figures (A–C) and microstructures of grain growth(D–F): (A and D) for the initial condition, (B) and (E) for a displacement-controlled system where the external strain is 0.003, (C) and (F) for aload-controlled system where the external stress is 100 MPa. (B–F) wereobtained at t = 2500Dt. (G) Orientation color index.

Figure 2. Microstructure and pole figures of a polycrystalline structurecomposed of minor h1 0 0i//ND fiber (0.3% of total grain number) anda major h1 1 1i//ND fiber texture. Microstructure and pole figure (top)in the initial state, (middle) at t = 300Dt during grain growth and(bottom) at t = 500Dt.

D.-U. Kim et al. / Scripta Materialia 64 (2011) 1079–1082 1081

the above scenario for the selective growth of h1 0 0i-ori-ented grains because not all the grains can have the samestrain from the standpoint of microelasticity.

To investigate the competition of two different tex-ture components under an external load (displacement-controlled), a polycrystalline structure was set up with0.3% h1 0 0i//ND fiber in a main h1 1 1i//ND fiber tex-ture. Figure 2A and D shows the microstructure andpole figure of a constructed polycrystal, respectively.Only a strong h1 1 1i//ND fiber was observed, whereasa h1 0 0i//ND fiber was indistinguishable due to its smallquantity. External strain (0.3%) was imposed biaxiallyon the specimen and the evolutions of the microstruc-ture and the pole figure during grain growth were mon-itored. The results are shown in Figure 2B–F. Figure 2Band E shows the microstructure and (0 0 1) pole figure ofthe system, respectively, at t = 300Dt and Figure 2C andF shows them at t = 500Dt. The minor h1 0 0i//ND fibergrains grow abnormally at the expense of the h1 1 1i//ND fiber grains due to the external load.

For Cu, the h1 0 0i//ND fiber grains have a smallerin-plane biaxial modulus than the h1 1 1i//ND fiber[22]. As explained in previous reports [15,20,21], themechanism for the selective growth of elastically softgrains (h1 0 0i//ND fiber components) at the expense

of elastically hard ones (h1 1 1i//ND fiber components)is as follows: from the viewpoint of macroelasticity, iso-strain averaging was assumed, which means that an elas-tically soft grain has lower strain energy density than ahard one. To minimize the total strain energy, softh1 0 0i grains grew selectively at the expense of elasti-cally hard h1 1 1i//ND fiber grains.

In this study, however, the macroelastic explanationmentioned above could not be applied to grain growthdriven by an external load (displacement-controlled).Figure 3A and B shows the distribution of the local strainenergy density at t = 300Dt and 500Dt. Unlike the mac-roelastic description, the strain energy density was local-ized in each grain and showed an inhomogeneousdistribution even in a single grain. This suggests thatthe elasticity in a polycrystalline structure has a micro-elastic nature. In particular, the strain energy densitystored in the soft h1 0 0i//ND fiber grains is larger thanthat in the hard h1 1 1i//ND fiber grains, which is in con-trast to the macroelastic mechanism based on isostrainaveraging. As shown in Figure 3C, the abnormal growthof elastically soft grains with a higher strain energy den-sity was accompanied by the decreasing total strain en-ergy of the system, which appears to be a paradox.

To explain the above paradoxical results quantita-tively, consider a fiber grain (b grain) embedded in a ma-trix grain (a grain) subjected to external biaxial strain e(see Fig. S1 in Supporting Online Materials). If the sys-tem is assumed to be elastically isotropic for convenience,the total strain energies stored in the a and b grains aregiven under plane strain conditions as follows (seeSupporting Online Materials for a detailed derivation):

Figure 3. Distribution of the local strain energy density at differenttimes (A at 300Dt and B at 500Dt) and variation of the total strainenergy with increasing time (C).

1082 D.-U. Kim et al. / Scripta Materialia 64 (2011) 1079–1082

Qel;tota ¼ pEaðAa

1RaÞ2

2 1þ mÞð1� 2mð Þ 1þ RRa

� �2 !(

� 1� f2ð1� 2mÞ RRa

� �2 !

þ 4fð1� mÞ RRa

� �2

lnRRa

)ð8aÞ

and

Qel;totb ¼ pEbðAb

1RaÞ2

2ð1þ mÞð1� 2mÞRRa

� �2

ð8bÞ

where Qel;toti is the total strain energy stored in grain i,

Aa1¼

ð1�2mÞEaþEb

ð1�2mÞEaþEbþDEðR2=R2aÞ

e, Ab1 ¼

2ð1�mÞEa

ð1�2mÞEaþEbþDEðR2=R2aÞ

e, DE ¼Ea � Eb, Ei is the Young’s modulus of grain i and m is

the Poisson’s ratio. The effective Young’s moduli ofthe Cu films with h1 0 0i//ND and h1 1 1i//ND fiber tex-tures is reported in Ref. [22]. The ratio of the effective

Young’s modulus, Eh111ieff =Eh100i

eff , was approximately

�1.525. Using this ratio, the growth of an elasticallysofter grain at the expense of the harder grain reducesthe total strain energy of the system as shown in Figs.S2 and S3 (see Supporting Online Materials), eventhough it has a larger stored strain energy density.

This study examined the effect of an external load ongrain growth using a phase field model including elasticanisotropy, inhomogeneity and microelasticity. Owingto the elastic anisotropy of each grain and the elasticinteractions with its neighbors, an external load had astrong influence on grain growth and texture evolution.A strong h1 0 0i//ND fiber was developed in Cu from aninitial random texture by biaxial external strain, whereasa h1 1 1i//ND fiber texture evolved under biaxial exter-nal stress. The external strain induced abnormal graingrowth of the h1 0 0i//ND fiber grains in Cu with themain h1 1 1i//ND fiber texture. In contrast to previousmacroelastic descriptions [15,20,21], the soft h1 0 0i//ND fiber grains with a higher stored strain energy den-sity grow abnormally at the expense of hard h1 1 1i//NDfiber grains, which induces a decrease in the total strain

energy. These results suggest the possibility of tailoringthe texture by controlling the external load.

This study was supported by a National Re-search Foundation of Korea Grant funded by the Kor-ean Government (MOEHRD, Basic ResearchPromotion Fund) (KRF-2008-314-D00218) and par-tially supported by the Korea Science and EngineeringFoundation (KOSEF) funded by the Korea Govern-ment (MOST) (R0A-2007-000-10014-1). It was also par-tially supported by the Technology InnovationProgram, 09010020, “Development of Highly Func-tional Porous Metallic Materials”, funded by the Minis-try of Knowledge Economy (M.K.E., Korea) and thePriority Research Centers Program through the Na-tional Research Foundation of Korea(NRF) fundedby the Ministry of Education, Science and Technology(2010-0028287).

Supplementary data associated with this article canbe found, in the online version, at doi:10.1016/j.scriptamat.2011.02.022.

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