Tae Wook Heo a, Saswata Bhattacharyya a & Long-Qing Chen a a Department of Materials Science and Engineering, The

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1 Ths artcle was downloaded by: [Pennsylvana State Unversty] On: 19 July 2013, At: 12:08 Publsher: Taylor & Francs Informa Ltd Regstered n England and Wales Regstered Number: Regstered offce: Mortmer House, Mortmer Street, London W1T 3JH, UK Phlosophcal Magazne Publcaton detals, ncludng nstructons for authors and subscrpton nformaton: A phase-feld model for elastcally ansotropc polycrystallne bnary sold solutons Tae Wook Heo a, Saswata Bhattacharyya a & Long-Qng Chen a a Department of Materals Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 16802, USA Publshed onlne: 19 Nov To cte ths artcle: Tae Wook Heo, Saswata Bhattacharyya & Long-Qng Chen (2013) A phase-feld model for elastcally ansotropc polycrystallne bnary sold solutons, Phlosophcal Magazne, 93:13, , DOI: / To lnk to ths artcle: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francs makes every effort to ensure the accuracy of all the nformaton (the Content ) contaned n the publcatons on our platform. However, Taylor & Francs, our agents, and our lcensors make no representatons or warrantes whatsoever as to the accuracy, completeness, or sutablty for any purpose of the Content. Any opnons and vews expressed n ths publcaton are the opnons and vews of the authors, and are not the vews of or endorsed by Taylor & Francs. The accuracy of the Content should not be reled upon and should be ndependently verfed wth prmary sources of nformaton. Taylor and Francs shall not be lable for any losses, actons, clams, proceedngs, demands, costs, expenses, damages, and other labltes whatsoever or howsoever caused arsng drectly or ndrectly n connecton wth, n relaton to or arsng out of the use of the Content. Ths artcle may be used for research, teachng, and prvate study purposes. Any substantal or systematc reproducton, redstrbuton, resellng, loan, sub-lcensng, systematc supply, or dstrbuton n any form to anyone s expressly forbdden. Terms & Condtons of access and use can be found at

2 Phlosophcal Magazne, 2013 Vol. 93, No. 13, , A phase-feld model for elastcally ansotropc polycrystallne bnary sold solutons Tae Wook Heo*, Saswata Bhattacharyyay and Long-Qng Chen Department of Materals Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 16802, USA (Receved 30 Aprl 2012; fnal verson receved 23 October 2012) A phase-feld model for modelng the dffusonal processes n an elastcally ansotropc polycrystallne bnary sold soluton s descrbed. The elastc nteractons due to coherency elastc stran are ncorporated by solvng the mechancal equlbrum equaton usng an teratve-perturbaton scheme takng nto account elastc modulus nhomogenety stemmng from dfferent gran orentatons. We studed the precptate nteractons among precptates across a gran boundary and gran boundary segregaton precptate nteractons. It was shown that the local pressure feld from one coherent precptate nfluences the shape of precptates n other grans. The local pressure dstrbuton due to prmary coherent precptates near the gran boundary leads to nhomogeneous solute dstrbuton along the gran boundary, resultng n non-unform dstrbuton of secondary nucle at the gran boundary. Keywords: elastcty; polycrystallne; sold solutons; dffuson; phase-feld model 1. Introducton Phase transformatons of sold solutons nvolve a complcated couplng among a number of dfferent dffusonal processes such as solute segregaton/depleton, precptate nucleaton, growth, and coarsenng. The thermodynamcs and knetcs of these processes are often nfluenced by the elastc nteractons n a mcrostructure [1]. Common nternal defects such as dslocatons, gran boundares, and coherent nclusons are sources of elastc stresses n the sold solutons. For example, for systems wth coherent precptates, elastc stresses arse naturally due to the lattce parameter msmatch between the precptate and the matrx [2 5]. Snce most materals n engneerng applcatons are polycrystallne sold solutons, computatonal approach for predctng phase transformatons n polycrystals would be useful *Correspondng author. Emal: tuh134@psu.edu y Current address: Materals Modelng Dvson, GE Inda Technology Center, Bangalore , Inda. ß 2013 Taylor & Francs

3 2 Phlosophcal Magazne 1469 for desgnng mcrostructures of engneerng materals. However, computatonal modelng of dffusonal processes n a polycrystallne sold soluton s more challengng than those n a unform sngle crystal. There are two man challenges. Frst, the solute-gran boundary nteractons should be consdered. The presence of gran boundares leads to solute segregaton or depleton due to chemcal and/or elastc nteracton between solute atoms and defects. To descrbe the solute-gran boundary nteractons, several phase-feld models have been proposed. Fan et al. employed the phenomenologcal model to nduce the gran boundary segregaton [6]. Cha et al. descrbed the gran boundary as a dstngushable phase and ncorporated the segregaton potental wthn the gran boundary regme [7]. More recently, a phase-feld model was proposed by Gro nhagen and A gren for modelng gran boundary segregaton as well as solute drag effects [8]. The model has been successfully appled to propose the abnormal gran growth mechansm [9], smulate the solute-movng gran boundary n the strongly segregatng system [10], and model the stran energy effects on gran boundary segregaton and solute drag effects [11]. The second challenge s the elastc nhomogenety of a polycrystallne sold soluton. A number of approaches have been proposed to model and compute the nhomogeneous elastcty n polycrystals. Wang et al. developed a method based on the calculaton of equvalent egenstran [12]. Tonks et al. employed a phase-feld model to nvestgate the gran boundary moton takng nto account the nhomogeneous elastcty n a bcrystal [13] where the elastc soluton was computed by the numercal technque n [12]. Km et al. ncorporated elastc nhomogenety to a phase-feld gran growth model and studed gran growth behavor and texture evoluton under an external load [14,15]. We recently extended an teratveperturbaton technque usng the Fourer spectral method [16,17] to model the effects of elastc nhomogenety n polycrystals [18,19]. Phase transformatons and mcrostructure evoluton n polycrystals have prevously been modeled usng the phase-feld approach [20 25]. For example, Jn et al. [26], Artemev et al. [27] and Wang et al. [28] nvestgated the formaton and swtchng of martenstc transformatons n polycrystals. Choudhury et al. analyzed the evoluton of ferroelectrc domans n polycrystallne oxdes [29,30]. The exstng works manly focused on the structural transformaton and assumed the homogeneous sotropc elastc propertes. However, the dffusonal processes n elastcally ansotropc polycrystallne sold solutons have not been extensvely studed usng phase-feld smulatons even though there have been many efforts n phasefeld modelng of precptate reactons n sngle crystallne sold solutons, e.g. N alloys [31 33], Al alloys [34 36], etc., and a few phase-feld smulatons of gran boundary effects on spnodal decomposton wthout consderaton of elastc propertes [37,38]. The man objectve of the present work s to extend and generalze the phase-feld models n [11,39] for descrbng the dffusonal processes n elastcally nhomogeneous polycrystallne sold solutons. We ntegrate the elastcty model for an elastcally nhomogeneous polycrystallne system wth phase-feld equatons descrbng dffusonal processes. A bnary sold soluton s consdered for smplcty. The elastc nteractons assocated wth segregaton and precptatons near gran boundares are dscussed.

4 1470 T.W. Heo et al Phase-feld model Dffusonal processes n a polycrystallne sold soluton nvolve nteractons between nhomogeneous dstrbuton of solute composton and gran structures. Thus, two types of feld varables are requred to descrbe the thermodynamcs and knetc processes of a polycrystallne bnary sold soluton. One s a conserved feld varable Xð~rÞ for the composton of solute, and the other s a non-conserved feld varable g ð~rþ for the crystallographc orentaton of grans. In the dffuse-nterface descrpton [40], the total free energy F of an nhomogeneous system s descrbed by a volume ntegral as a functonal of a set of contnuous feld varables. We adopted and extended a phase-feld model of Gro nhagen and A gren [8] whch s valdated to be quanttatvely correct by Km and Park [9] for descrbng the solute gran boundary nteractons. We ncorporated the elastc stran nteractons of solute atoms n the presence of gran boundares n an ansotropc bnary sold soluton. The functonal form of the total free energy F of the sold soluton s gven by the followng volume ntegral [20]: Z ( ) F ¼ f nc þ!gð 1, 2,..., g Þþ c 2 ðrxþ2 þ X o ðr g Þ 2 þ e coh dv, ð1þ 2 V where f nc s the ncoherent local free energy densty of a sold soluton, g s a multwell free energy densty functon descrbng the gran structure,! s the potental heght of g, c and o are gradent energy coeffcents of composton Xð~rÞ and gran order parameters g ð~rþ, respectvely, and e coh s the local coherency elastc stran energy densty arsng from a compostonal nhomogenety Thermodynamc energy model The ncoherent local free energy contans both the chemcal and the elastc stran energy of a homogeneous sold soluton. In order to explore the orgn of both contrbutons and develop the ncoherent free energy densty n the presence of gran boundares, let us start wth the Gbbs free energy of a sold soluton. The free energy densty of the sold soluton s represented by the lnear combnaton of the chemcal potentals,.e. the partal molar Gbbs free energy, of all the speces. In the case of a bnary sold soluton, the free energy densty s gven by f nc ¼ X þ h ð1 XÞ, ð2þ where s the chemcal potental of solutes and h s the chemcal potental of host atoms n the sold soluton. To explan the free energy densty of the bnary system, a regular soluton model s consdered as the followng functon: f nc ¼ o þ RT ln X þ ð1 XÞ 2 X þ o h þ RT lnð1 XÞþX 2 ð1 XÞ, ð3þ g where o s the chemcal potental of solute atoms at standard state, o h s the chemcal potental of host atoms at standard state, R s the gas constant, T s the temperature, and s the regular soluton parameter for representng the nteractons among atoms. Followng Cahn [41], the nteracton potental E s

5 Phlosophcal Magazne 1471 addtonally ncorporated to represent pure chemcal nteracton between a gran boundary and solute atoms, and Equaton (3) becomes f nc ¼ o þ RT ln X þ ð1 XÞ 2 þ E X þ o h þ RT lnð1 XÞþX 2 ð1 XÞ: ð4þ In the present model, we specfy the pure chemcal nteracton potental E between gran boundary and solutes as ½ m!gð 1, 2,..., g ÞŠ where m s a parameter determnng the nteracton strength between solute atoms and a gran boundary. Pluggng the nteracton potental n Equaton (4) and rearrangng the equaton, we have the ncoherent free energy densty n the presence of gran boundares as the followng: f nc ¼ o X þ o hð1 XÞþRT½XlnX þð1 XÞlnð1 XÞŠ m!gx þ Xð1 XÞ: ð5þ Generally, the regular soluton parameter contans all the contrbutons assocated wth the non-dealty of the sold soluton. Ignorng all other contrbutons, let us focus on pure chemcal and elastc nteractons due to the atomc sze dfference (or sze msmatch) between solute atoms and host atoms. Hence, the elastc stran contrbuton can be separated from the regular soluton parameter ( ¼ chem þ hom elast ). The elastc stran energy densty (e hom) of a homogeneous sold soluton wth composton X [2] s gven by: e hom ¼ 1 h 2 C jkl" m j "m kl hlð~nþ ~n Xð1 XÞ, ð6þ where C jkl s the elastc modulus, " m j s the msft stran tensor, and hlð~nþ ~n s the average of Lð~nÞ over all the drectons of ~n wth Lð~nÞ ¼n j 0 jkkl 0 n l, j 0 ¼ C jkl" m kl, 1 jk ¼ C jlkn n l, and n s the unt wave vector n Fourer space. The detals of calculaton of hlð~nþ ~n are shown n the Appendx. Therefore, the ncoherent free energy densty s represented by the followng expresson as dscussed n [11]: f nc ¼ o X þ o hð1 XÞþRT½X ln X þð1 XÞlnð1 XÞŠ m!gx þ chem Xð1 XÞþ 1 h 2 C jkl" m j "m kl hlð~nþ ~n Xð1 XÞ, where chem s the regular soluton parameter assocated wth the pure chemcal contrbuton,.e. regular soluton parameter of a hypothetcal sold soluton n whch all the atoms have the same sze (Ths representaton s smlar to Cahn s n [42]). The ncoherent free energy s expressed by the summaton of purely chemcal part and elastc stran energy of a homogeneous sold soluton tself: f nc ¼½f chem m!gxšþe hom, where f chem ¼ o X þ o h ð1 XÞþRT½X ln X þð1 XÞlnð1 XÞŠ þ chemxð1 XÞ. Our total free energy wthout e hom and the excess energy term ( chem Xð1 XÞ) s dentcal to the model of Gro nhagen and Ågren [8]. Wth regard to the msft stran tensor " m j near a gran boundary, the elastc stran s relaxed when a solute atom approaches to a gran boundary due to ts relatvely dsordered structure. Therefore, we model the stran relaxaton by employng the poston (or gran structure)-dependent msmatch as the followng: ð7þ ð8þ " m j ð~rþ ¼"m,b j ð~rþ, ð9þ

6 1472 T.W. Heo et al. where ð~rþ s an nterpolaton functon whch s 1 nsde grans and 0 at the center of a gran boundary and ts explct form s ð~rþ ¼ 2 mn mn þ2 max mn max mn ð10þ where ¼ P g 2 g, max s the maxmum value of whch corresponds to the value nsde the bulk, and mn s the mnmum value of whch corresponds to the value at the center of a gran boundary. " m,b j s the msft stran tensor nsde the bulk. We assume dlatatonal stran tensor " c j for " m,b j where j s the Kronecker-delta functon, " c s the composton expanson coeffcent of lattce parameter defned as 1 a 0 ð da dx Þ, and a 0 s the lattce parameter of the reference homogeneous sold soluton of overall composton X 0. The stran s assumed to be fully relaxed when the solute atom comes to the center of the gran boundary. Takng nto account the postondependent msmatch, we rewrte e hom usng Equaton (9), e hom ¼ 1 h 2 C jkl" m,b j " m,b kl hl b ð~nþ ~n ð~rþ 2 Xð1 XÞ, ð11þ where hl b ð~nþ ~n s the average of L b ð~nþ over all the drectons of ~n wth L b ð~nþ ¼n 0,b j jk 0,b kl n l, 0,b j ¼ C jkl " m,b kl, 1 jk ¼ C jlkn n l, and n s the unt wave vector n Fourer space. It should be noted that the prefactor 1 2 ½C jkl" m,b j " m,b kl hl b ð~nþ ~n Š n Equaton (11) ncludes the elastc propertes of a gran nteror only, and the nterpolaton functon ð~rþ s responsble for the relatvely dsordered gran boundary structure resultng n the msft stran relaxaton near a generc gran boundary n our model. The possble varatons of elastc modulus or crystallographc symmetry wthn the gran boundary arsng from the relatvely dsordered gran boundary structure were not addressed n the model. One remarkable thng s that the prefactor s ndependent of gran orentaton even f each gran has ansotropc elastc modulus. Snce " m,b j s a dlatatonal tensor, the frst term C jkl " m,b j " m,b n the bracket s nvarant wth kl Fgure 1. (colour onlne) Profles of L b ð~nþ of (a) a reference gran and (b) rotated gran wth respect to the reference gran n k y k z planes.

7 Phlosophcal Magazne 1473 gran rotaton. In addton, the second term hl b ð~nþ ~n s a scalar quantty where all drectons are equally consdered. Fgure 1 shows an example of the L b ð~nþ profles n k-space for grans of dfferent crystallographc orentatons. If we average L b ð~nþ over all the drectons, the values are the same snce the profles are just mutually rotated. Therefore, all the grans have the same values of the prefactor for dfferent gran orentatons, whch means that the elastc modulus of a reference gran can be used for the computatons of prefactors of all other grans. The term e hom s the elastc stran energy of a homogeneous sold soluton. However, compostonal dstrbuton n a sold soluton s generally nhomogeneous. The elastc stran energy stemmng from the compostonal nhomogenety of the sold soluton s the coherency stran energy (e coh ), e coh ¼ 1 2 C jkl" el j "el kl ¼ 1 2 C jklð" j þ " j " j Þð" kl þ " kl " kl Þ, where " el j s the elastc stran tensor whch s equal to ð" j þ " j " j Þ, " j s the homogeneous stran tensor, " j s the heterogeneous stran tensor, and " j s the egenstran tensor. For an elastcally ansotropc and nhomogeneous polycrystal, the poston-dependent elastc modulus s modeled as the followng [18,19,39]: C jkl ð~rþ ¼ X g 2 g ag m ag jn ag ko ag lp Cref mnop, where g s the gran order parameter, a g j are the components of an axs transformaton matrx representng the rotaton from the coordnate system defned on a gven gran g to the global reference coordnate system, and C ref mnop on the rghthand sde s the elastc modulus n the coordnate system defned on the gven gran and C ref mnop of all the grans are same. The egenstran due to the compostonal nhomogenety s defned by ð12þ ð13þ " j ð~rþ ¼"m j ðxð~rþ X 0Þ, ð14þ where " m j s the msft stran tensor, and X 0 s the overall composton of the sold soluton. To represent the structural nhomogenety of a polycrystal, we also employ the poston (or gran structure)-dependent msmatch (" m j ð~rþ) modeled n Equaton (9). The homogeneous stran " j represents the macroscopc shape change of a system and s defned such that Z " j ð~rþdv ¼ 0: ð15þ V When a system s constraned under a constant appled stran (" a j ), the homogeneous stran s smply equal to the appled stran,.e. " j ¼ " a j. On the other hand,

8 1474 T.W. Heo et al. f the boundares are allowed to relax, the homogeneous stran n an elastcally nhomogeneous polycrystal s computed by [18,19] " j ¼hS jkl kl a þh0 kl h kl, ð16þ where hs jkl ¼hC jkl 1, hc jkl ¼ð1=VÞ R V C jklð~rþdv, j a s an appled stress, hj 0¼ ð1=vþ R V C jklð~rþ" 0 kl ð~rþdv, and h j ¼ð1=VÞ R V C jklð~rþ" kl ð~rþdv. The heterogeneous stran can be expressed by the elastc dsplacement u ð~rþ followng Khachaturyan [2]: " j ð~rþ ¼ j, ð17þ To compute the heterogeneous stran feld, we solve the followng mechancal equlbrum equaton snce the mechancal equlbrum s establshed much faster than the dffusonal processes: r j j ¼r j C jkl ð~rþð" kl þ " kl ð~rþ " kl ð~rþþ ¼ 0, ð18þ where j s the local elastc stress. In order to solve the mechancal equlbrum equaton wth the spatally nhomogeneous elastcty n polycrystals, we employ the Fourer-spectral teratve-perturbaton scheme [16,17]. To apply the method, the poston-dependent elastc modulus (Equaton (13)) s dvded nto a constant homogeneous part C hom jkl (or C hom j ) and a poston-dependent nhomogeneous perturbaton part C nhom jkl ð~rþ (or C nhom j ð~rþ),.e.! C jkl ð~rþ ¼C hom jkl þ X g 2 g ag m ag jn ag ko ag lp Cref mnop Chom jkl ¼ C hom jkl þ C nhom jkl ð~rþ, ð19þ where C nhom jkl ð~rþ s ð P g 2 g ag m ag jn ag ko ag lp Cref mnop Chom jkl Þ. The detals of the general procedure are dscussed n [18,19]. For better effcency, we addtonally employed the Vogt notaton scheme to solve the mechancal equlbrum equaton. The procedure s as follows: (1) Zeroth-order teraton: The elastc modulus s assumed to be homogeneous and solve the mechancal equlbrum equaton to obtan the zeroth-order approxmaton of the elastc dsplacements. The equatons n the Vogt notaton are ~u 0 1 ¼ ½ 1ð ~ 1 0 k x þ ~ 6 0 k y þ ~ 5 0 k zþþ 6 ð ~ 6 0 k x þ ~ 2 0 k y þ ~ 4 0 k zþ þ 5 ð ~ 5 0 k x þ ~ 4 0 k y þ ~ 3 0 k zþš, ~u 0 2 ¼ ½ 6ð ~ 1 0 k x þ ~ 6 0 k y þ ~ 5 0 k zþþ 2 ð ~ 6 0 k x þ ~ 2 0 k y þ ~ 4 0 k zþ þ 4 ð ~ 5 0 k x þ ~ 4 0 k y þ ~ 3 0 k ð20þ zþš, ~u 0 3 ¼ ½ 5ð ~ 1 0 k x þ ~ 6 0 k y þ ~ 5 0 k zþþ 4 ð ~ 6 0 k x þ ~ 2 0 k y þ ~ 4 0 k zþ þ 3 ð ~ 5 0 k x þ ~ 4 0 k y þ ~ 3 0 k zþš, where 0 ¼ C hom j " 0 j, 1 k ¼ p Chom jkl k j k l ( k s reduced to ), ¼ ffffffffffff 1, k ~ ¼ðkx, k y, k z Þ s the recprocal lattce vector, the tlde () represents the Fourer transform.

9 Phlosophcal Magazne 1475 (2) Hgher-order teraton: The (n 1)th-order elastc soluton s used to obtan the nth-order elastc dsplacements by solvng ~u n 1 ¼ ½ 1ð ~T n 1 1 k x þ ~T n 1 6 k y þ ~T n 1 5 k z Þþ 6 ð ~T n 1 6 k x þ ~T n 1 2 k y þ ~T n 1 4 k z Þ þ 5 ð ~T n 1 5 k x þ ~T n 1 4 k y þ ~T n 1 3 k z ÞŠ ~u n 2 ¼ ½ 6ð ~T n 1 1 k x þ ~T n 1 6 k y þ ~T n 1 5 k z Þþ 2 ð ~T n 1 6 k x þ ~T n 1 2 k y þ ~T n 1 4 k z Þ þ 4 ð ~T n 1 5 k x þ ~T n 1 4 k y þ ~T n 1 ð21þ 3 k z ÞŠ ~u n 3 ¼ ½ 5ð ~T n 1 1 k x þ ~T n 1 6 k y þ ~T n 1 5 k z Þþ 4 ð ~T n 1 6 k x þ ~T n 1 2 k y þ ~T n 1 4 k z Þ þ 3 ð ~T n 1 5 k x þ ~T n 1 4 k y þ ~T n 1 3 k z ÞŠ, Where T n 1 ¼ C total j ð" 0 j " n 1 j Þ C nhom j " n 1 j. The number of teratons s controlled by the tolerance. qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff In ths work, the teratons were contnuously performed untl the R value of v ½ðunþ1 1 u n 1 Þ2 þðu nþ1 2 u n 2 Þ2 þðu nþ1 3 u n 3 Þ2 ŠdV became smaller than 1: Stran energy contrbuton to thermodynamcs of sold solutons Elastc stran energy contrbutons n an sotropc sold solutons were dscussed n [11]. Let us consder the elastc stran energy of an elastcally ansotropc sold soluton n a sngle crystal. Ignorng the macroscopc deformaton of the entre system (" j ¼ 0), the expresson of coherency elastc stran energy E coh of the entre system n Fourer space s gven by [2] E coh ¼ 1 Z d 3 k ~ 2 ð2þ 3 Bð~nÞ ~Xð kþ ~ 2, ð22þ where k ~ s the wave vector n Fourer space, ~Xð kþ ~ s the Fourer transform of Xð~rÞ ¼X X 0, and Bð~nÞ ¼C jkl " m j "m kl n j 0 jkkl 0 n l ¼ C jkl " m j "m kl Lð~nÞ. The total stran energy of the system s the sum of E hom and E coh, E anso total ¼ E hom þ E coh ¼ 1 Z ½C jkl " m j 2 "m kl hlð~nþ ~nšxð1 XÞdV V þ 1 Z d 3 k h 2 ð2þ 3 C jkl" m j "m kl Lð~nÞ ~Xð kþ ~ 2 : The second term n Equaton (23) can be splt nto two parts: Z 1 d 3 k h 2 ð2þ 3 C jkl" m j "m kl Lð~nÞ ~Xð kþ ~ 2 ¼ 1 Z d 3 k h 2 ð2þ 3 C jkl" m j "m kl hlð~nþ ~n ~Xð kþ ~ þ 1 Z d 3 k 2 ð2þ 3 hlð~nþ ~n Lð~nÞ ~Xð kþ ~ 2, where the frst term of the rght-hand sde s the orentaton-ndependent part and the second term of the rght-hand sde s the orentaton-dependent part of the coherency 2 ð23þ ð24þ

10 1476 T.W. Heo et al. stran energy. Applyng the Parseval s theorem to the orentaton-ndependent part and addng to e hom, t produces the followng: Z 1 h C jkl " m j 2 "m kl hlð~nþ ~n Xð1 XÞdV þ 1 Z d 3 k h V 2 ð2þ 3 C jkl" m j "m kl hlð~nþ ~n ~Xð kþ ~ 2 ¼ 1 Z h C jkl " m j 2 "m kl hlð~nþ ~n Xð1 XÞdV þ 1 Z h C jkl " m j V 2 "m kl hlð~nþ ~n ðx X 0 Þ 2 dv V ¼ 1 h 2 VC jkl" m j "m kl hlð~nþ ~n X 0 ð1 X 0 Þ: ð25þ Thus, the total elastc stran energy of the elastcally ansotropc system can be wrtten as E anso total ¼ 1 h 2 VC jkl" m j "m kl hlð~nþ ~n X 0 ð1 X 0 Þ þ 1 Z 2 d 3 k ð2þ 3 hlð~nþ ~n Lð~nÞ ~Xð kþ ~ 2 : whch s consstent wth Khachaturyan s expresson of the elastc stran energy of a sold soluton [2]. For an elastcally sotropc soluton, the second term of Equaton (26) becomes zero snce Lð~nÞ s equal to hlð~nþ ~n and the frst term reduces to 2 1 þ " 2 c 1 X 0ð1 X 0 Þ ðn three dmensonsþ ð27þ or " 2 c 1 X 0ð1 X 0 Þ, ðn two dmensonsþ where s the shear modulus and s the Posson s rato. Ths means that the elastc stran energy of elastcally sotropc sold soluton s not affected by the compostonal dstrbuton (Crum theorem). Based on the above dscusson, we could confrm that both e hom and e coh are essental components for the elastc stran energy of a sold soluton. The elastc stran energy sgnfcantly contrbutes to the thermodynamcs of spnodal boundares. As Khachaturyan dscussed n [2], only the orentatondependent part of the coherency stran energy affects the coherent spnodal boundares wth respect to the chemcal spnodal boundares snce the homogeneous part of free energy should nclude the orentaton-ndependent part of coherency stran energy. Our model correctly descrbes the spnodal boundary. The local free energy densty can be expressed as f ¼ f chem þ e hom þ e ndep coh þ e dep coh : The second dervatve of (e hom þ e ndep coh ) s equal to C jkl" m j "m kl hlð~nþ ~n Xð1 XÞþ 1 h 2 C jkl" m j "m kl hlð~nþ ~n ðx X 0 Þ 2 ¼ 0: ð26þ ð28þ ð29þ

11 Phlosophcal Magazne 1477 Fgure 2. (colour onlne) Schematc dagram of the spnodal regmes. Therefore, the dfference n spnodal regmes between chemcal spnodal and coherent spnodal s also determned by the only second dervatve of (f chem þ e dep coh ), whle the ncoherent spnodal regme s determned by the second dervatve of (f chem þ e hom ), whch s the ncoherent free energy. In the case of an sotropc elastc sold soluton, the coherent spnodal boundares are the same as chemcal spnodal snce e dep coh becomes zero, and the ncoherent spnodal regme s wder than coherent or chemcal spnodal regmes. Moreover, the consolute temperature of the ncoherent spnodal decomposton s hgher than that of chemcal or coherent spnodal decomposton by R ð1þ 1 Þ"2 c. A schematc llustraton of the spnodal regme s shown n Fgure 2 Thus, we could conclude that the elastc stran energy terms defned n our model correctly descrbe the dffusonal processes Dffuson knetcs The temporal evoluton of the compostonal felds X s governed by the Cahn Hllard equaton tþ F ¼rM c r, Xð~r, tþ where M c s the nterdffuson moblty, and F X s the varatonal dervatve of the free energy functonal wth respect to composton. Substtutng the total free energy F (Equaton (1)) wth the expressons n Equaton (8) nto Equaton (30), we obtan the followng ¼rM m!g cr 2 X, ð31þ

12 1478 T.W. Heo et al. The dervatves of e hom and e coh wth respect to X n Equaton (31) are derved usng Equatons (11) and ¼ 1 h 2 C jkl" m,b j " m,b kl hl b ð~nþ ~n ð~rþ 2 ð1 ¼ C jklð" j þ " j " j ð32þ kl ¼ C jkl " el " c kl ð~rþ, We employed the varable nterdffuson moblty for M c n Equaton (30) whch s gven by M c ¼ M 0 cxð1 XÞ, ð33þ where M 0 c s the prefactor whch s equal to D/RT, D s the nterdffuson coeffcent, R s the gas constant, and T s the temperature. To solve the Cahn Hllard equaton wth the composton-dependent dffuson moblty, the numercal technque for the varable moblty n [44] s employed. The Cahn Hllard equaton (Equaton (30)) s solved by the sem-mplct Fourer-spectral method [44,45]. 3. Smulaton results and dscussons 3.1. Preparaton of gran structures and numercal nput parameters Even though the model s applcable to smultaneous gran growth and compostonal evolutons, we wll manly dscuss compostonal evoluton on a statc gran structure for smplcty. To generate gran structures descrbed by multple gran order parameters, we employ the followng local free energy densty functonal for gð 1, 2,..., g Þ n Equaton (1) based on the model n [46] n the present model: gð 1, 2,..., g Þ¼0:25 þ X g þ g þ X X 2 g 2 g 0, ð34þ g g g 0 4 g where s the phenomenologcal parameter for the nteractons among gran order parameters. A constant 0.25 n Equaton (34) s employed to make the value of the functon g equal to 0 nsde the bulk to descrbe zero nteracton potental ( m!g n Equaton (7)) nsde the gran for convenence, whch does not affect the knetcs of the gran structure evoluton. The evoluton of the non-conserved order parameters g s governed by the Allen Cahn relaxaton equaton g ð~r, tþ F ¼ L, g ð~r, tþ where L s the knetc coeffcent related to gran boundary moblty, t s tme, and ð F g Þ s the varaton of the free energy functon wth respect to the gran order parameter felds. The equatons are solved by sem-mplct Fourer-spectral method [45]. The knetc equatons n Equatons (31) and (35) were solved n dmensonless forms. The parameters were normalzed by Dx ¼ Dx l, Dt ¼ LEDt,! ¼! E, ¼ E, f ¼ f E, C j ¼ C j E, ¼, and M 0 El 2 c ¼ M0 c where E s the characterstc energy Ll 2 whch was chosen to be 10 9 J=m 3 and l s the characterstc length whch s taken

13 Phlosophcal Magazne 1479 Fgure 3. Examples of phase-feld smulatons. (a) Two-dmesonal (2D) gran structure, (b) dffusonal process n the 2D gran structure wth dverse overall compostons, (c) threedmensonal (3D) gran structure, (d) dffusonal process n the 3D gran structure wth dverse overall compostons, and (e) the temporal evoluton of a composton n the 3D gran structure when X 0 ¼ to be m. For a reference gran, we used the elastc constants of phase n N Al alloy system used n [31] whch were estmated from [48,49]. The normalzed elastc constants n Vogt notaton were C ref 11 ¼ 195.8, Cref 12 ¼ 144.0, and Cref 44 ¼ Each gran n a polycrystal s elastcally ansotropc snce the Zener ansotropy factor A Z ( ¼ 2C ref 44 =ðcref 11 Cref 12 Þ) s equal to The composton expanson coeffcent " c s chosen to be The dmensonless gradent energy coeffcents c and o are set to be The nteracton parameter m was taken to be 0.5, and the normalzed heght! was chosen to be The terms assocated wth the normalzed chemcal free energy such as, h, and chem were set to be 1.0, 1.0, and 2.0, respectvely. The prefactor M 0 c of nterdffuson moblty n dmensonless unt n Equaton (33) was chosen as The dmensonless grd sze Dx was 0.5, and tme step Dt for ntegraton was 0.1. All the smulatons were conducted wth the perodc boundary condton. Examples of generated gran structures n two (2D) and three dmensons (3D) are shown n Fgure 3a and c. For these random gran structures, crcular (2D) or sphercal (3D) grans were randomly dstrbuted n the system at the ntal stage and the system was relaxed by solvng Equaton (35). Once the gran structure s

14 1480 T.W. Heo et al. Fgure 4. (colour onlne) (a) Smulaton setup n a bcrystal generated by a phase-feld smulaton, and (b) profles of elastc constants (C 11, C 12, C 44 ) wth respect to (x y) coordnate system across a gran boundary when ¼ 60. prepared, the local free energy densty g n Equaton (7) s computed and fxed for the composton gran structure nteracton term ( m!gx). On the prepared statc gran structures, we perform the computer smulatons of the dffuson processes takng nto account the chemcal and elastc nteractons between solute and gran boundares. Some examples of the smulatons wth dfferent overall compostons are shown n Fgure 3b for 2D and Fgure 3d for 3D. To nucleate the precptates, the Gaussan random fluctuatons were ncorporated to the compostonal feld at the early stage. Fgure 3e shows the temporal evoluton of compostonal feld n a 3D gran structure, and the solute gran boundary nteracton,.e. gran boundary segregaton, s clearly shown at the early stage of the smulaton Precptate precptate nteracton across a gran boundary The elastc stress feld generated by a coherent precptate n one gran may nfluence the precptatons n other grans snce the elastc nteractons are long-range. To nvestgate the nteractons between precptates n dfferent grans, we desgned a smple bcrystal as shown n Fgure 4a. We labeled the left-hand sde gran as Gran I and the rght-hand sde gran as Gran II. We can vary the msorentaton between two adjonng grans as well as the locatons of precptates nsde the grans. As an example, the gran orentaton of Gran I s fxed as 0, whle Gran II s orented at an angle of 60 wth respect to Gran I. The dstrbutons of the elastc constants (C 11, C 12, and C 44 ) wth respect to the global reference coordnate system (x-y frame) are plotted across a gran boundary n Fgure 4b. The elastc stress feld generated by a coherent precptate n an elastcally ansotropc sold s strongly orentaton-dependent. Fgure 5a shows contour plots of the computed spatal dstrbutons of the elastc stress felds ( xx, xy, yy ) generated from a sngle coherent precptate n a bcrystal. For comparson, the stress felds n a sngle crystal are also plotted n Fgure 5b. In both cases, the stress felds from the precptate propagate over a long range. However, as one can clearly see, the stress

15 Phlosophcal Magazne 1481 Fgure 5. (colour onlne) Contour plots of elastc stress felds ( xx, xy, yy ) generated from a sngle coherent precptate (a) n a bcrystal (red dashed lne represents a gran boundary) and (b) a sngle crystal. Fgure 6. (colour onlne) Morphology of precptates n a bcrystal. A precptate n Gran I s located at a fxed poston, whle a precptate n Gran II s placed at several dfferent dstances to a gran boundary. A yellow dashed lne represents the locaton of a gran boundary. felds, especally, xx and yy, abruptly change across the gran boundary. It means that precptates n Gran II mght be affected by the dstorted elastc stress felds, and the effects would be more sgnfcant f the precptates are located near the gran boundary. To observe the precptaton reacton under the stress felds near a gran boundary, we ntally ntroduced two crcular precptates of R ¼ 20Dx and montor the temporal evoluton of the precptate morphology. One was embedded n Gran I at a fxed locaton, whle the other was embedded n Gran II at several dfferent dstances to the gran boundary, as shown n Fgure 6, to observe the nfluence of the dfferent levels of stress feld on the precptate. To reduce the overlap of the elastc feld due to the perodc boundary condton, we employ a relatvely large system (512Dx* 512Dx* grds). The solute composton n the matrx was taken to be 0.046, whch s close to one of the equlbrum compostons (X matrx eq X precptate eq ¼ 0:963). ¼ 0:037 and

16 1482 T.W. Heo et al. Fgure 7. (colour onlne) Morphology of precptates near a gran boundary for the cases n Fgure 6 (zoomed mages of Fgure 6). The yellow dashed lne represents the locaton of a gran boundary. Fgure 7 shows the zoomed mages of Fgure 6 n order to clearly capture the morphology of precptates near the gran boundary for several precptate locatons n Gran II. Frst of all, the morphology of the precptate s cubc wth rounded corners, as shown n Fgure 7a. The precptate embedded n Gran II s rotated by 60 wth respect to the precptate embedded n Gran I. The precptates n the confguraton of Fgure 7a do not seem to sgnfcantly affect each other. As the precptate n Gran II becomes closer to the gran boundary (see Fgure 7b e), the nterestng features are captured. The morphology of the precptate n Gran II devates from the perfect cubodal shape (see Fgure 7e). Ths means that the dffuson process assocated wth the precptate n Gran II s nterfered by a bas. Dffuson knetcs s generally affected by the elastc stress feld. The relatonshp between the dffuson flux and the elastc stress feld s gven by [1]: J ¼ cmðf X DrPÞ, where J s the flux, c s the total concentraton, M s the dffuson moblty, F s the drvng force for dffuson except the local pressure effect, X s the mole fracton of speces, D s the pure dlaton durng the atomc jump, and P s the local pressure defned by ½ ð xx þ yy Þ=2Š n two dmensons and ½ ð xx þ yy þ zz Þ=3Š n three dmensons. For convenence, we use the negatve local pressure ( P ¼ ð xx þ yy Þ=2). By the defnton, the postve value of P represents the tensle local pressure and the negatve value represents the compressve local pressure. Consequently, the dffuson knetcs can be sgnfcantly affected by the local pressure felds. Thus, we nvestgated the local pressure dstrbuton near precptates n order ð36þ

17 Phlosophcal Magazne 1483 Fgure 8. (colour onlne) (a) Contour plot of negatve local pressure ( P) generated from a sngle coherent precptate n Gran I n a bcrystal, (b) magnfed plot of P wth the gudelnes of the precptates n Fgure 7e, and (c) contour plot of P generated from two coherent precptates n Gran I and II wth the gudelnes of the precptates n Fgure 7e. to explore the orgn of the deformed shape of the precptate. Fgure 8a shows contour plot of P (¼ ð xx þ yy Þ=2) dstrbuton whch arses due to a sngle coherent precptate n Gran I. In the external area of the precptate, the local pressure along the dagonal drecton of the precptate s more compressve. On the other hand, the local pressure along the normal drecton to a flat nterface of the precptate s more tensle. The local pressure feld elongates to the gran boundary, and t s refracted when t passes though the gran boundary n the same way as the stress felds. As a result, the rregular tensle regme next to the gran boundary s formed n Gran II. In Fgure 8b, the gudelnes of the coherent precptates of Fgure 7e, where the precptate n Gran II s closest to the gran boundary, are shown n the enlarged contour plot n order to observe the effect of the local pressure. Most devaton of the morphology (from the perfect cubodal shape) occurs near the tensle regon,.e. the left hand sde corner of the precptate n Gran II tends to be dragged toward the more tensle regme. Fgure 8c shows the dstrbuton of P when the other coherent precptate s also placed at Gran II. Even though the local pressure feld dsplays the more dstorted dstrbuton due to the superposton of local pressure felds from both coherent precptates, t clearly shows the dstorton of the precptate shape n Gran II toward the tensle regme. It should be mentoned that the shape of the precptate n Gran I s also dstorted snce the local pressure dstrbuton around the precptate n Gran I s nfluenced by the stress feld

18 1484 T.W. Heo et al. Fgure 9. (colour onlne) Morphology of precptates near a gran boundary for the cases of dfferent crystallographc orentatons of Gran II. The yellow dashed lne represents the locaton of a gran boundary. generated from the precptate n Gran II, whch results n strongly asymmetrc local pressure dstrbuton near the precptate n Gran I. We also conducted smulatons wth dfferent crystallographc orentatons of Gran II, and smlar behavor s observed, as shown n Fgure 9. Morphologcal shapes of two precptates are mutually nfluenced by each other. Ths can be one of reasons for the rregular morphology of precptates near or at gran boundares whch have sgnfcantly mportant mplcatons to the mechancal propertes Precptate gran boundary segregaton nteracton The effects of elastc stran energy on gran boundary segregaton profles have been dscussed n [11] for the case of sotropc elastc modulus. In ths secton, we dscuss the effects of elastc stress generated by the precptates wthn grans on solute segregaton at gran boundares n elastcally ansotropc systems. It s easly expected that the elastc stress feld or local pressure profle along a gran boundary stemmng from the multple coherent precptates n adjacent grans s strongly nhomogeneous. In addton, t would depend on the spatal confguraton of the precptates. Fgure 10b shows the negatve local pressure ( P) profles along the gran boundary (shaded regon n Fgure 10a) n the cases of dfferent gran orentatons of Gran II. Correspondng solute composton profles along the gran boundary are shown n Fgure 10c. In all cases, the solute composton at the locally maxmum compressve regon (shaded n red) s relatvely low, whle the composton at the relatvely tensle (locally mnmum compressve) regons (shaded n blue) tends to be at local maxmum. The solute atoms do not prefer compressve stress regons snce the solute lattce parameter expanson coeffcent s dlatatonal and postve. As a result, the composton profle along the gran boundary s non-unform dependng on the confguratons of coherent precptates nsde grans. Smlar behavors of solute segregaton/depleton near a dslocaton were dscussed n [50].

19 Phlosophcal Magazne 1485 Fgure 10. (colour onlne) (a) Scanned regme for the profles of negatve local pressure ( P) and solute composton. (b) Negatve local pressure ( P) profles along the gran boundary n the cases of dfferent gran orentatons of Gran II, and (c) correspondng solute composton profles along the gran boundary. Fnally, the non-unform dstrbuton of solute atoms at a gran boundary can supply the nhomogeneous dstrbuton of canddate stes for secondary (barrer-less) nucleaton at the gran boundary. We performed the smulatons at hgher matrx composton (X m ¼ 0.12),.e. a supersaturated system, n the presence of prmary coherent precptates nsde grans. We montored the secondary nucleaton process at a gran boundary. Fgure 11a shows the temporal evoluton of the process. New precptates are nucleated along the gran boundary regon wthn dashed lne n the fgure. Dependng on the spatal confguraton of prmary coherent precptates, the secondary nucleaton events along the gran boundary occur at dfferent locatons, as shown n Fgure 11b. For better comparson, the composton profles along the gran boundary are plotted n Fgure 11c. The fgure clearly shows the non-unform dstrbuton of the secondary nucleaton. Ths phenomenon can happen n realstc materals systems. For nstance, upon the contnuous coolng, the nucleaton of the secondary 0 precptates occurs n the presence of the pre-exstng prmary coherent

20 1486 T.W. Heo et al. Fgure 11. (colour onlne) (a) Snapshots of the secondary nucleaton process of precptates at a gran boundary. (b) Secondary nucleaton of precptates at a gran boundary wth dfferent gran orentatons of Gran II, and (c) composton profles along the gran boundary. 0 phase n N alloys, whch results n the bmodal dstrbuton of 0 precptates [51]. Our smulaton results ndcate that the prmary 0 precptates nsde grans can affect the spatal dstrbuton of gran boundary nucleated secondary 0 precptates. In addton, the gran boundary precptate dstrbuton may affect mechancal behavor, e.g. creep rate [52]. 4. Summary We have ncorporated the elastc stran energy contrbuton n the phase-feld model of gran boundary segregaton n an elastcally ansotropc polycrystallne sold soluton. The elastc stran energy of a sold soluton was obtaned by solvng the

21 Phlosophcal Magazne 1487 mechancal equlbrum equaton usng the teratve-perturbaton Fourer spectral method. We nvestgated the elastc nteractons between precptates n dfferent grans, and between precptates and gran boundary segregaton. The elastc stress felds from a coherent precptate nsde gran propagate across a gran boundary whch may affect the shape of precptates n other grans near the gran boundary. Precptates near a gran boundary generate non-unform stress feld or local pressure along a gran boundary, leadng to nhomogeneous gran boundary segregaton whch n turn may nduce non-unform nucleaton of secondary precptates along a gran boundary. We are currently applyng ths model to a number of polycrystallne materals systems nvolvng the dffusonal processes, such as T alloys, N alloys, Zr alloys, etc. Acknowledgements Ths work was funded by the Center for Computatonal Materals Desgn (CCMD), a jont Natonal Scence Foundaton (NSF) Industry/Unversty Cooperatve Research Center at Penn State (IIP ) and Georga Tech (IIP ), and computer smulatons were carred out on the CyberSTAR cluster funded by NSF through Grant OCI References [1] R.W. Balluff, S.M. Allen and W.C. Carter, Knetcs of Materals, John Wley and Sons, New Jersey, [2] A.G. Khachaturyan, Theory of Structural Transformatons n Solds, John Wley and Sons, New York, [3] M. Do, Progr. Mater. Sc. 40 (1996) p.79. [4] P. Fratzl, O. Penrose and J. Lebowtz, J. Stat. Phys. 95 (1999) p [5] Y.U. Wang, Y.M. Jn and A.G. Khachaturyan, J. Appl. Phys. 91 (2002) p [6] D. Fan, S.P. Chen and L.-Q. Chen, J. Mater. Res. 14 (1999) p [7] P.-R. Cha, S.G. Km, D.-H. Yeon and J.-K. Yoon, Acta Mater. 50 (2002) p [8] K. Gro nhagen and J. A gren, Acta Mater. 55 (2007) p.955. [9] S.G. Km and Y.B. Park, Acta Mater. 56 (2008) p [10] J. L, J. Wang and G. Yang, Acta Mater. 57 (2009) p [11] T.W. Heo, S. Bhattacharyya and L.-Q. Chen, Acta Mater. 59 (2011) p [12] Y.U. Wang, Y.M. Jn and A.G. Khachaturyan, J. Appl. Phys. 92 (2002) p [13] M. Tonks, P. Mllett, W. Ca and D. Wolf, Scrpta Mater. 63 (2010) p [14] D.-U. Km, S.G. Km, W.T. Km, J. Cho, H.N. Han and P.-R. Cha, Scrpta Mater. 64 (2011) p [15] D.-U. Km, P.-R. Cha, S.G. Km, W.T. Km, J. Cho, H.-N. Han, H.-J. Lee and J. Km, Comput. Mater. Sc. 56 (2012) p.58. [16] S.Y. Hu and L.-Q. Chen, Acta Mater. 49 (2001) p [17] P. Yu, S.Y. Hu, L.-Q. Chen and Q. Du, J. Comput. Phys. 208 (2005) p.34. [18] S. Bhattacharyya, T.W. Heo, K. Chang and L.-Q. Chen, Model. Smulat. Mater. Sc. Eng. 19 (2011) p [19] S. Bhattacharyya, T.W. Heo, K. Chang and L.-Q. Chen, Comm. Comput. Phys. 11 (2012) p.726. [20] L.-Q. Chen, Annu. Rev. Mater. Res. 32 (2001) p.113.

22 1488 T.W. Heo et al. [21] W.J. Boettnger, J.A. Warren, C. Beckermann and A. Karma, Annu. Rev. Mater. Res. 32 (2002) p.163. [22] L. Granasy, T. Puszta, T. Bo rzso ny, G. Toth, G. Tegze, J.A. Warren and J.F. Douglas, J. Mater. Res. 21 (2006) p.309. [23] H. Emmerch, Adv. Phys. 57 (2008) p.1. [24] N. Moelans, B. Blanpan and P. Wollants, Comput. Couplng Phase Dagr. Thermochem. 32 (2008) p.268. [25] I. Stenbach, Model. Smulat. Mater. Sc. Eng. 17 (2009) p [26] Y.M. Jn, A. Artemev and A.G. Khachaturyan, Acta Mater. 49 (2001) p [27] A. Artemev, Y. Jn and A.G. Khachaturyan, Phl. Mag. A 82 (2002) p [28] Y.U. Wang, Y. Jn and A.G. Khachaturyan, Acta Mater. 52 (2004) p [29] S. Choudhury, Y.L. L, C.E. Krll III and L.-Q. Chen, Acta Mater. 53 (2005) p [30] S. Choudhury, Y.L. L, C. Krll III and L.-Q. Chen, Acta Mater. 55 (2007) p [31] J.Z. Zhu, Z.-K. Lu, V. Vathyanathan and L.-Q. Chen, Scrpta Mater. 46 (2002) p.401. [32] J.Z. Zhu, T. Wang, A.J. Ardell, S.H. Zhou, Z.-K. Lu and L.-Q. Chen, Acta Mater. 52 (2004) p [33] T. Wang, G. Sheng, Z.-K. Lu and L.-Q. Chen, Acta Mater. 56 (2008) p [34] D.Y. L and L.-Q. Chen, Acta Mater. 46 (1998) p [35] V. Vathyanathan, C. Wolverton and L.-Q. Chen, Phys. Rev. Lett. 88 (2002) [36] V. Vathyanathan, C. Wolverton and L.-Q. Chen, Acta Mater. 52 (2004) p [37] H. Ramanarayan and T.A. Abnandanan, Acta Mater. 51 (2003) p [38] H. Ramanarayan and T.A. Abnandanan, Acta Mater. 52 (2004) p.921. [39] T.W. Heo, S. Bhattacharyya and L.-Q. Chen, Sold State Phenom (2011) p [40] J.W. Cahn, J. Chem. Phys. 28 (1958) p.258. [41] J.W. Cahn, Acta Metall. 10 (1962) p.789. [42] J.W. Cahn, The Mechansms of Phase Transformatons n Crystallne Solds, Sdney Press, Bedford, 1968, p.1. [43] J.W. Cahn, Acta Metall. 9 (1961) p.795. [44] J. Zhu, L.-Q. Chen, J. Shen and V. Tkare, Phys. Rev. E 60 (1999) p [45] L.-Q. Chen and J. Shen, Comput. Phys. Comm. 108 (1998) p.147. [46] L.-Q. Chen and W. Yang, Phys. Rev. B 50 (1994) p [47] S.M. Allen and J.W. Cahn, Acta Metall. 27 (1979) p [48] S.V. Prkhodko, J.D. Carnes, D.G. Isaak and A.J. Ardell, Scrpta Mater. 38 (1998) p.67. [49] S.V. Prkhodko, J.D. Carnes, D.G. Isaak, H. Yang and A.J. Ardell, Metall. Mater. Trans. A 30 (1999) p [50] S.Y. Hu and L.-Q. Chen, Acta Mater. 49 (2001) p.463. [51] Y.H. Wen, J.P. Smmons, C. Shen, C. Woodward and Y. Wang, Acta Mater. 51 (2003) p [52] X.-J. Wu and A.K. Koul, Metall. Mater. Trans. A 26 (1995) p.905. Appendx. Calculaton of hlð~nþ ~n By defnton, hlð~nþ ~n s the average of Lð~nÞ over all the drectons of ~n where Lð~nÞ ¼n j 0 jkkl 0 n l, j 0 ¼ C jkl" m kl, 1 jk ¼ C jlkn n l, and n s the unt wave vector n Fourer space. The mathematcal expresson of hlð~nþ ~n s gven by [2] hlð~nþ ~n ¼ 1 I Lð~nÞd, ða1þ 4

23 Phlosophcal Magazne 1489 R 1 rk where d s the sold angle element. Snce ð 4 3 r3 k Þ 0 4k2 dk ¼ 1, Equaton (A1) becomes hlð~nþ ~n ¼ r3 k Z rk 0 4k 2 dk 1 4 I Lð~nÞd, where r k s an arbtrary radus of a sphere n k-space. Applyng the defnton of the sold angle (d ¼ sn dd), we obtan hlð~nþ ~n ¼ r3 k Z 2 Z Z rk Lð~nÞk 2 sn dkdd ¼ r3 k Z Z Z sphere r¼r k Lð~nÞd 3 k: Therefore, hlð~nþ ~n becomes the sphercal average. In [2], the Debye cutoff radus (k d ) defned by the relaton 4 3 k3 d ¼ ð2þ3 v 0 where ð2þ3 v 0 s the volume of the frst Brlloun zone was chosen for the radus of the sphere, and hlð~nþ ~n was calculated by hlð~nþ ~n ¼ 1 Z Z Z Lð~nÞd 3 k ¼ h 1 Z Z Z LðnÞd 3 k: ða4þ 4 3 k3 d sphere r¼k d ð2þ 3 v 0 sphere r¼k d It should be noted that we can also take any r k wthn the frst Brlloun zone. Thus, we can take the ntegraton over the sphere whose radus s equal to =Dx for the sphercal average. Hence, hlð~nþ ~n can be computed as hlð~nþ ~n ¼ ð Dx Þ3 Z Z Z sphere r¼=dx Lð~nÞd 3 k ¼ 1 V k Z Z Z sphere r¼=dx LðnÞd 3 k: n the numercal calculatons where V k represents the volume of sphere whose radus s equal to =Dx. ða2þ ða3þ ða5þ