THEORY AND MODELING OF MICROSTRUCTURAL EVOLUTION IN POLYCRYSTALLINE MATERIALS: SOLUTE SEGREGATION, GRAIN GROWTH AND PHASE TRANSFORMATIONS DISSERTATION

Transcription

1 THEORY AND MODELING OF MICROSTRUCTURAL EVOLUTION IN POLYCRYSTALLINE MATERIALS: SOLUTE SEGREGATION, GRAIN GROWTH AND PHASE TRANSFORMATIONS DISSERTATION Presented n Partal Fulfllment of the Requrements for the Degree Doctor of Phlosophy n Graduate School of the Oho State Unversty By Nng Ma, M.S. ***** The Oho State Unversty 005 Dssertaton Commttee Yunzh Wang, Advsor Sulman A. Drega Hamsh L. Fraser John E. Morral Bran McHale Advsor Materals Scence and Engneerng Department

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3 ABSTRACT To accurately predct mcrostructure evoluton and, hence, to synthess metal and ceramc alloys wth desrable propertes nvolves many fundamental as well as practcal ssues. In the present study, novel theoretcal and phase feld approaches have been developed to address some of these ssues ncludng solute drag and segregaton transton at gran boundares and dslocatons, gran growth n systems of ansotropc boundary propertes, and precptate mcrostructure development n polycrystallne materals. The segregaton model has allowed for the predcton of a frst-order segregaton transton, whch could be related to the sharp transton of solute concentraton of gran boundary as a functon of temperature. The ncorporatng of nterfacal energy and moblty as functons of msorentaton and nclnaton n the phase feld model has allowed for the study of concurrent gran growth and texture evoluton. The smulaton results were analyzed usng the concept of local gran boundary energy densty, whch smplfed sgnfcantly the development of governng equatons for texture controlled gran growth n T-6Al-4V. Quanttatve phase feld modelng technques have been developed by ncorporatng thermodynamc and dffusvty databases. The models have been valdated aganst DICTRA smulatons n smple 1D problems and appled to smulate realstc mcrostructural evolutons n T-6Al-4V, ncludng gran boundary α and globular α growth and sdeplate development under both

4 sothermal agng and contnuous coolng condtons. The smulaton predctons agree well wth expermental observatons.

5 Dedcated to my wfe Xn Zhao v

6 ACKNOWLEDGEMENT I wsh to thank my advsor, Yunzh Wang for ntellectual support, encouragement and enthusasm, whch made ths thess possble, and for hs patence n correctng both my stylstc and scentfc errors. I thank Sulman A. Drega for dscussng wth me varous aspects of my research work. I thank the expermental support from Hamsh Fraser s group, n partcular, Tom Searles, Sujoy Kar, Jame S. Tley, Megan Harper, Danel Huber and Dhrt Bhattacharyya. I thank John E. Morral for dscussng varous aspects of ths thess. I also wsh to thank support of teammates: Qng Chen, Andrew Kazaryan, Kasheng Wu, Shen Chen, Blle Wang, Weq Luo, Nng Zhou and Ncholas A. Brown. Ths work s supported by the U. S. Ar Force through the Metals Affordablty Intatve program. v

7 VITA August 5, Born Hebe, Chna B.S. Materal Scence and Engneerng Tsnghua Unversty, Bejng, Chna 003..M.S. Materals Scence and Engneerng Oho State Unversty PUBLICATIONS 1. N Ma, SA Drega, Y Wang, Solute segregaton transton and drag force on gran boundares, Acta Materala, 51 (003) N Ma, SA Drega, Y Wang, Computer Smulaton of Texture Evoluton durng Gran Growth: Effect of Boundary Propertes and Intal Mcrostructure, Acta Materala, 5 (004) N Ma, Y Wang, Beta gran growth knetcs n T64, Materals processng and desgn: modelng, smulaton and applcatons: NUMIFORM 004, 71 (004) Q Chen, N Ma, K Wu, Y Wang, Quanttatve phase feld modelng of dffusoncontrolled precptate growth and dssoluton n T-Al-V, Scrpta Materala, 50 (004) 471 v

8 FIELD OF STUDY Major feld: Materals Scence and Engneerng v

9 TABLE OF CONTENTS Page Abstract Dedcaton.... v Acknowledgements.. v Vta.. v Lst of Tables...x Lst of Fgures x Chapters: 1. Introducton Phase feld method Dffuse-nterface theory and antphase boundary.. 7. Mult phase feld method 9.3 Couplng between phase feld and concentraton feld Ansotropy.17.5 Fnte volume method and adaptve mesh Solute segregaton transton and drag force on gran boundares Introducton Segregaton model Dffuson and mgraton Smulatons Statonary boundary Movng gran boundary Moblty transton Dscusson Summary v

10 4. Segregaton and wettng transton at dslocatons Introducton Theoretcal Formulaton Segregaton model Dffuson and mgraton Model System Results and dscusson Statonary dslocaton Movng dslocaton Summary Computer smulaton of texture evoluton durng gran growth: effect of boundary propertes and ntal mcrostructure Introducton Method Msorentaton wth three parameters Phase feld method Intal mcrostructure Results Dscusson Effect of boundary moblty ansotropy Effect of boundary energy ansotropy Interplay between energy and moblty ansotropy Effect of dsperson of texture component Effect of gran sze Effect of number of texture component Texture effect on gran growth knetcs Summary Quanttatve Phase Feld Modelng of Dffuson-Controlled Precptate Growth and Dssoluton Introducton Phase Feld Model Length and tme scales Applcaton to β α transformaton n T-Al-V Thckenng of α Plate Dssoluton of Globular α Summary. 139 x

11 7. Applcaton on T-6Al-4V alloy desgn Introducton β gran growth Sample sze effect Effect of texture Gran boundary α Sdeplate and colony structure Valdaton of consttutve equatons by phase feld smulatons Conclusons Bblography. 168 x

12 Lst of Tables Tables Page Table.1 Comparson of computatonal tme between adaptve and unform mesh....1 Table 5.1. Intal slopes ( 10-4 ) of curves shown n Fgure 5.7(b) for varous combnatons of ansotropy n boundary energy and moblty and spatal dstrbutons of the texture component.106 x

13 Lst of Fgures Fgures Page Fgure.1. Schematc of adapatve multlevel mesh for sland gran shrnkng. The mesh s refned along the gran boundary. (a) plot of sland gran mcrostructure (b) correspondng adaptve mesh (c) Enlargement of (b). 19 Fgure 3.1. Equlbrum solute concentraton profles across a gran boundary obtaned from dfferent models for a system of attractve nteracton between solute and gran boundary. The bulk composton s c =0.04 and the temperature s T=1750K..36 Fgure 3.. (a) Gran boundary segregaton as a functon of temperature durng coolng (sold crcles) and heatng (open trangles) processes and, (b) gran boundary energy as a functon of temperature for a system of bulk composton c =0.00. The segregaton transton s ndcated by the thck sold vertcal lne n (a) Fgure 3.3. Stablty dagram for segregaton transton at gran boundary. Dfferent symbols represent results obtaned under varous combnatons of three terms: (1) ϕ(c,x); () κ d c dκ dc ; and (3). The M-S model excludes dx dx dx all three terms whle the H-S model ncludes only term (1). The current model gb ncludes all three terms. For each case, the crtcal temperature ( T c ) s ndcated by the larger symbols.. 41 Fgure 3.4. Relatonshp between gran boundary energy and bulk concentraton under varous condtons: (A) deal solute at nfnte temperature; (B) regular soluton at nfnte temperature; (C) regular soluton at a T>T c (bulk); (D) regular soluton at T<T c but T> T gb c ; (E) regular soluton at T< T gb c. For curve E, open symbols represent boundary energy obtaned wth decreasng bulk concentraton whle sold symbols represent values x

14 obtaned wth ncreasng bulk concentraton. The nset llustrates the horn structure for curve E. c α and cβ correspond to the mscblty gap at two dfferent temperatures.. 43 Fgure 3.5. Varaton of drag force (a) and solute excess (b) wth gran boundary velocty at a temperature T = 680K, whch s above the segregaton transton temperature (T 0 ). The bulk composton s c = Fgure 3.6. (a) Comparson of the drag forces predcted by the current model wth those obtaned from the M-S and H-S models at T=00K whch s above the transton temperature. (b) Steady state concentraton profles obtaned from the current and M-S models at two dfferent gran boundary veloctes. The bulk composton s c = Fgure 3.7. Varaton of drag force (a) and solute excess (b) wth gran boundary velocty at T = 660K whch s below the segregaton transton temperature but above the bulk mscblty gap. The bulk composton s c = Sold crcles represent values obtaned wth ncreasng velocty whle open trangles represent values obtaned wth decreasng velocty. The nset n (a) shows the low branch of the drag force-velocty plot. 49 Fgure 3.8. Predcton of gran boundary moblty transton as a functon of temperature for dfferent solute contents: c 1 = , c = The gran boundary moblty s normalzed by the value at 1000K. The hysteress loop dsappears at hgher solute concentraton. Sold symbols are for the coolng process whle open symbols are for the heatng process. 51 Fgure 4.1. The equlbrum profles of η, η, and Z -Z(x, y) across the dslocaton core employed n the calculatons. The equlbrum profle of η s obtaned by the phase feld model of dslocatons Fgure 4.. Schematc drawng of the computatonal cell...68 Fgure 4.3. Comparson between solute concentraton profles along a vertcal lne through the center of the dslocaton core obtaned usng and system szes.. 69 Fgure 4.4. Contour plots of solute concentraton around the dslocaton at (a) 100K (above the segregaton transton T 0 ) and (b) 1170K (below T 0 ) wth bulk composton c = Fgure 4.5. Solute concentraton profle along a vertcal lne through the center of the dslocaton core at (a) 100K and (b) 1170K. The dashed lnes n the plots ndcate the center of the dslocaton core...73 x

15 Fgure 4.6. Varatons of (a) relatve Gbbs excess of solute at the dslocaton and (b) excess dslocaton lne energy as functons of temperature durng coolng (sold crcles) and heatng (open trangles) n a system of bulk composton c =0.04. The thck vertcal lne n (a) ndcates the equlbrum segregaton transton temperature...74 Fgure 4.7. (a) Stablty dagrams for segregaton transton at dslocaton (sold crcles and trangles) and bulk mscblty gaps (sold lnes) for coherent and ncoherent systems. (b) Normalzed stablty dagrams and bulk mscblty gaps. The normalzaton factors are respectvely the crtcal temperatures for the bulk mscblty gaps of the coherent and ncoherent systems. After normalzaton, the two bulk mscblty gaps become dentcal (sold lne n (b)).. 77 Fgure 4.8. Contour plots of solute concentraton around a movng dslocaton at 100K (above the segregaton transton temperature T 0 ) under varous veloctes: (a) vd 0 /D=0 10-3, (b) vd 0 /D=1 10-3, (c) vd 0 /D= 10-3, and (d) vd 0 /D= The bulk composton s c = Fgure 4.9. Varaton of (a) Gbbs excess of solute and (b) drag force wth velocty for a movng dslocaton at 100K (above the segregaton transton temperature). The bulk composton s c = Fgure Contour plots of solute concentraton around a movng dslocaton at 100K (below the segregaton transton temperature) wth ncreasng veloctes: (a) vd 0 /D=0 10-3, (b) vd 0 /D= , (c) vd 0 /D=1 10-3, and (d) vd 0 /D= The bulk composton s c = Fgure Varaton of (a) Gbbs excess of solute and (b) drag force wth velocty for a movng dslocaton at 100K (below the segregaton transton temperature). The bulk composton s c = Sold crcles represent values obtaned wth ncreasng velocty whle open crcles represent values obtaned wth decreasng velocty...8 Fgure 5.1. Defnton of Euler angles. X,Y and Z stands for the sample reference system and X,Y and Z stands for the crystal coordnaton system.. 94 Fgure 5.. Markenze dstrbuton (three-parameter system) and D unform dstrbuton (one-parameter system) 94 Fgure 5.3. Dependency of boundary energy and moblty on the msorentaton descrbed by Equatons (5.9) and (5.10).. 98 xv

16 Fgure 5.4. Intal mcrostructures wth varous dspersons of the texture component. (a) Startng mcrostructure before assgnng preferred orentatons; (b) (c) unform, random and clustered dspersons of the texture component, wth lght shade representng grans of preferred orentatons (wthn 5 o from (000) n Euler space) and dark shade representng randomly orented grans Fgure 5.5. Texture evoluton durng gran growth n a system consstng of 7% ntally randomly dstrbuted textured grans under the condton of ansotropc boundary energy and sotropc boundary moblty. τ s reduced tme Fgure 5.6. Texture evoluton durng gran growth n a system consstng of 7% ntally randomly dstrbuted textured grans under the condton of ansotropc boundary moblty and sotropc boundary energy. τ s reduced tme Fgure 5.7. (a) Intal MDFs, (b) temporal evolutons of area fractons of the texture component, (c) temporal evoluton of number fracton of gran boundares between textured grans and (d) temporal evoluton of average gran under varous condtons wth 7% ntal texture component. τ s reduced tme. Crcles, trangles and squares stand for unform, random and clustered ntal dsperson of texture component, respectvely. Dfferent shades of gray of the symbols represent varous combnatons of boundary propertes, wth sold, open and gray standng for energy ansotropy only, moblty ansotropy only and both energy and moblty ansotropy, respectvely. The sold lne n (d) s a straght lne representng schematcally parabolc gran growth knetcs. The three curves correspondng to energy ansotropy only n (b) are re-plotted n (e) to show clearly the ther ntal slops.105 Fgure 5.8. (a) Intal MDFs, (b) temporal evolutons of area fractons of the texture component, (c) temporal evoluton of number fracton of gran boundares between textured grans and (d) temporal evoluton of average gran under varous condtons wth 1.5% ntal texture component. τ s reduced tme. Crcles, trangles and squares stand for unform, random and clustered ntal dsperson of texture component, respectvely. Dfferent shades of gray of the symbols represent varous combnatons of boundary propertes, wth sold, open and gray standng for energy ansotropy only, moblty ansotropy only and both energy and moblty ansotropy, respectvely xv

17 Fgure 5.9. Texture evoluton durng gran growth n a system consstng of 7% ntally clustered texture component under the condton of ansotropc boundary moblty and sotropc boundary energy. τ s reduced tme...11 Fgure Temporal evoluton of gran sze dstrbutons of textured (open crcles) and randomly orented (sold crcles) grans shown n Fg τ s reduced tme and f A s number fracton of gran wth area A. 114 Fgure Texture evoluton durng gran growth n a system consstng of 7% ntally clustered texture component under the condton of ansotropc boundary energy and sotropc boundary moblty. τ s reduced tme Fgure 5.1. Effect of ntal gran sze on mcrostructure evoluton n a system consstng of two honeycomb structures. The sold lnes represent the ntal poston of the boundares between the two honeycomb structures. Mcrostructures showed here are quarters of real smulated systems 118 Fgure 6.1 Interpolaton of atomc moblty across the nterface regon 130 Fgure 6. PFM and DICTRA results for the growth of alpha precptate plate. (a) Growth knetcs; (b) Composton profle of Al; and (c) Composton profle of V. 134 Fgure 6.3. The dffuson path correspondng to Fg.6.b and 6.c durng the growth of alpha precptate plate Fgure 6.4 PFM and DICTRA results for the dssoluton of globular alpha. (a) Growth knetcs; (b) Composton profle of Al; and (c) Composton profle of V. 137 Fgure 6.5 Dffuson path at 1000 s durng the dssoluton of globular alpha Fgure 7.1. Two-dmensonal phase feld smulaton of sample sze effect on mcrostructure evoluton (a) and gran growth knetcs (b). The sample thckness s 19 (n reduced unt) and γ s /γ b s Fgure 7.. Re-plot of expermental data presented n reference [4]. Dfferent symbols stand for dfferent sothermal temperatures: open trangle ~100 C, close trangle ~1150 C, open crcle ~1100 C and close crcle ~1050 C. The two horzontal lnes ndcates the range of D*, the gran sze at whch the growth knetcs devates sgnfcantly from the parabolc growth, determned by Eqs. (1) and () wth γ s /γ b =~ xv

18 Fgure 7.3. Texture evoluton durng gran growth obtaned n a system wth an ntally randomly dstrbuted texture component (n lght shade of gray). Both gran boundary energy and moblty are assumed to have hgh values for hgh angle gran boundares n the smulaton. τ s reduced tme..149 Fgure 7.4. Knetcs of texture-controlled gran growth correspondng to the mcrostructural evoluton shown n Fg Fgure 7.5. Valdaton of equaton (7.7) aganst phase feld smulaton results. The lnear regresson gves A~ Fgure 7.6. Comparson of measured texture fracton and beta gran sze at varous peak temperatures wth predcted values for contnuous heatng condton..153 Fgure 7.7. Characterstc temperature ranges for gran boundary alpha growth at varous coolng rates. The top curve ndcates the startng temperature whle the bottom one ndcates the endng temperature Fgure 7.8. Heat treatment schedule for gran boundary alpha growth knetcs measurement 157 Fgure 7.9. Comparson of measured gran boundary alpha thckness at varous coolng rates wth predcted values. Lnear lne s the model predcton, wth model parameters optmzed aganst experment performed at OSU (red sold crcles). The green squares are expermental data obtaned ndependently on GE samples Fgure Lengthenng (a) and thckenng (b) knetcs of a sngle plate calculated wth PFM usng Thermocalc and Dctra database. 16 Fgure Lengthenng (a) and thckenng (b) knetcs of a sngle plate calculated wth PFM usng Thermocalc and Dctra database Fgure 7.1. Smulated sde plate growth and colony structure. Wthn the sxsded gran, whte phase s alpha phase and black color stands for beta phase..164 xv

19 CHAPTER 1 INTRODUCTION Although materals fabrcaton has long been known, t s only recognzed n the last century that the propertes of a gven materal mght not be prmarly controlled by ts chemcal composton but rather by ts mcrostructures. Materals mcrostructures are structural features that are subject to observaton under a mcroscopy. These features nclude vacancy/solute clusters, dslocatons, nterfaces, precptates, ferroelastc/electrc/magenetc domans, and gran structures, characterzed by ther amount, sze, shape, and spatal arrangement. These structural features usually have an ntermedate mesoscopc length scale n the range of nanometer to mcrometer. The man theme of modern materals scence and engneerng s to optmze mcrostructure for desred propertes through advanced processng. However, our ablty to characterze and predct quanttatvely mcrostructural evoluton and hence to yeld unambguous processng-property relatonshp s rather lmted because of the extreme complexty of mcrostructure and the nonlnear nteracton of ts elements. The progress n fundamental understandng of mcrostructural evoluton could be made through massve expermental effort, but such a tral-and-error approach s becomng more and more mpractcal. By 1

20 means of plausble but non-rgorous approxmatons, analytcal approaches are sutable for only qualtatve predcton. Computatonal smulaton s becomng an attractve alternatve n developng quanttatve processng-mcrostructure-property relatonshps, complementng the tradtonal analytcal and expermental approaches. The advent of quantum mechancs (less than 75 years ago) opened the door to understandng the fundamental nteractons of atomc consttuents of matter and, n turn, to the usage of knowledge to desgn and control materals mcrostructures and propertes through novel processng. In practce, however, t s mpossble to smulate drectly the huge number of atoms wth an objectve of predctng macroscopc propertes. It remans a grant challenge to smulate essental propertes that depend crtcally upon phenomena or processes takng place at very dfferent length and tme scales. Ths challenge could be overcome by uncoverng the elusve connectons n the herarchy of quantum/molecular, atomstc/nano, mesoscopc, and macroscopc scales. Results from smulatons at the smaller length scales could feed naturally nto larger length scale models. In ths study we focus on the phase feld method (PFM), whch has recently emerged as a computatonal approach of choce for modelng many types of complcated mcrostructural evolutons at the meso-scale. The PFM employs a set of felds to descrbe a mult-component and mult-phase mcrostructure. Structural and compostonal non-unformtes such as homo-phase and hetero-phase nterfaces are characterzed by the gradents of the felds. The spatal varaton of the felds s governed by the usual thermodynamc functons wth a gradent term that contrbutes to surface excess quanttes. In ths sense, the PFM s a natural

21 outgrowths of the gradent thermodynamc model datng back to the work by Van der Waals [1], Ensten and[] Ornsten and Zerncke[3] for densty, and by Krvoglaz and Smrnov[4] and Cahn and Hllard [5] for concentraton. Ths approach avods the mathematcally dffcult problem of applyng a boundary condton at an nterface whose locaton s a part of unknown soluton. By avodng explct front trackng, t s achevable to model realstc mcrostructures n two or three dmensons usng PFM. Examples of utlzng the PFM for studyng mcrostructural evoluton can be found for soldfcaton [6-10], sold-state phase transformatons [11] and gran growth [1]. In the current study, we extended the PFM n several applcatons: segregaton, gran growth and dffuson controlled phase transformatons. The novelty clams nclude the ntroducton of a self-consstent solute segregaton model on gran boundares and dslocatons, ncorporaton of the dependence of nterfacal propertes on msorentaton and nclnaton, and drect lnkage of the phase feld model to ThemoCalc thermodynamc database and DICTRA knetcs database for quanttatve smulatons. Varous types of defects and ther nteractons are consdered n ths study ncludng mpurtes, dslocatons, and gran boundares and phase nterfaces.. The thess s organzed as the followng: Chapter 1 gves a bref ntroducton followed by research motvaton and thess outlne. Chapter presents the development and generc features of Phase Feld model. Lterature most mportant and relevant to ths thess was revewed. Attempts were made to explore an effcent fnte volume numercal solver based on adaptve mesh. Examples 3

22 of specfc system wll be gven n the followng chapters. Chapter 3 and 4 are devoted to nvestgate solute segregaton and transton at gran boundares and dslocatons and the correspondng drag effect. A contnuum model of gran boundary segregaton based on gradent thermodynamcs and ts dscrete counterpart are formulated. The model dffers from much prevous work because t takes nto account several physcally dstnctve terms, ncludng concentraton gradent, spatal varaton of gradent-energy coeffcent and concentraton dependence of solute gran boundary/dslocaton core nteractons. It s found that omsson of these terms could result n a sgnfcant overestmate or underestmate of the enhancement of solute segregaton and drag force for systems of a postve mxng energy. The emergng mportance of the model was llustrated by offerng explanng the dependency of frst order gran boundary moblty transton on the overall solute composton n the frst tme. The model has been ntegrated nto the phase feld framework to study the effect of solute drag on growth of gran network. The wettng transton at dslocaton wth respect to the velocty could provde new nsghts nto the phenomena of sharp yeld pont drop and stran agng observed n metal alloys. Chapter 5 provdes nvestgaton of orentaton selecton durng gran growth by computer smulaton n two dmensons. The model characterzes msorentaton n threedmenson wth all three degrees of freedom. Gran boundary energy and moblty ansotropy assocated wth msorentaton angle was also taken nto account. The systems consdered consst of a sngle cube component embedded n a matrx of randomly orented grans n the ntal mcrostructure. The numercal experment was featured by 4

23 focusng on an mportant but usually neglected factor, mcro texture. The smulaton results are analyzed usng Turbull s theory on gran growth. It s found that even though many ndvdual factors affect texture evoluton durng gran growth, the key factor that really controls the process s the local gran boundary energy densty. Chapter 6 demonstrates a general method for quanttatve phase feld modelng of dffuson-controlled phase transformaton n multcomponent systems. Wth drect nputs from CALPHAD thermodynamc and DICTRA knetc databases, the growth and dssoluton of α precptates n T-Al-V s smulated on expermentally relevant length scale. The results agree well wth DICTRA smulatons. Chapter 7 summarzes the mcrostructure model developed for T-6Al-4V (T64) system. We began our descrpton of the mcrostructural evoluton by examnng the T- Al-V ternary phase dagram. Rch mcrostructure features are observed n ths alloy at room temperature ncludng gran boundary α and sde plate. Mechanstc models on varous mcrostructure features were developed based on the phase feld smulaton. Software package was developed for convenent ndustry applcaton. In Chapter 8, we summarzed the major ponts of current research work and dscussed ts possble extenson n the future. 5

24 CHAPTER PHASE FIELD METHOD The man ngredent of PFM contans a nonconserved order parameter or phase feld to dstngush phases and nterfaces. The phase feld presents dfferent but constant values n varous phases and takes a smooth transton across the nterface wth certan thckness. The purpose of ntroducng ths extra feld varable s to avod trackng the phases and ther boundary explctly, thus render numercal exploraton of mcrostructurally complcated system drectly accessble. However, PFM s not only a mathematcal tool. Its underlyng dffuse nterface nature s drectly lnked to the thermodynamcs of phase transformaton. Ths chapter starts wth brefng Allen-Cahn s ant phase boundary model ncludng only one feld varable [13]. Then, mult phase feld model, couplng between phase feld and concentraton feld, nterfacal energy and moblty ansotropy wth respect to fve-degree freedom, ncorporaton wth real thermodynamc and dffuson database and sharp nterface asymptotc analyss wll be dscussed. 6

25 .1. Dffuse-nterface Theory and Antphase Boundary Antphase boundares (APB) are coherent nterfaces separatng homogeneous phases wth dentcal propertes n crystals wth long-range order. The free energy of the homogeneous phase, f 0, can be descrbed as a functon of long-range order parameter η. For a system wth second-order transton, we assume the equlbrum order parameters as 1 η e and η e. The nterfacal regon comprses a volume n whch the feld varable has values ntermedate between two equlbrums. The total excess free energy of entre nhomogeneous system can be present s as followng: where f ( η) κ F = W f ( η ) + dv η (.1) s the ncrease n f 0 when η dffers from η e and takes the double well potental customarly and W and κ are barrer heght and gradent coeffcent respectvely. By standard methods of varatonal calculus, the mnmum n excess energy occurs when η () r obeys: Applyng boundary condton at nfnte, Eq. (.) becomes: κ W f ( η ) = + const η (.) κ W f ( η) = η (.3) By usng Eq. (.3) to combne terms and changng the varable from r to η, we obtan the excess energy per unt area σ of a planar nterface as: 7

26 8 ( ) [ ] = η η κ σ d f W (.4) An evoluton equaton for the phase feld postulated by requrng that () r η evolve so as to mnmze I, n addton that long range order parameter s not a conserved quantty, s Cahn-Allen equaton δη δ η I L t = (.5) wth L nterfacal moblty coeffcent. Followng the theoretcal analyss of Allen and Cahn, the dervaton of mgraton law of nterface s straghtforward. For smplcty, Fan and Chen conducted the calculaton n cylnder coordnate as: η η η κ η d df W r r r t L + = 1 1 (.6) At steady state, the profle of feld varable s nvarant wth tme, whch leads to: t r r t t r v = = η η η (.7) Substtute Eq. (.7) to Eq. (.6), multply both sdes wth r η and ntegrate from < < r 0, we have + = dr r d df W dr r r r r dr r L v η η η η η κ η (.8) Employng the followng boundary condtons: 0 0 = = r r η η and ( ) ( ) = η η f f 0 (.9) we have

27 v L 0 η r 1 η dr = κ dr r r 0 (.10) Notng that feld varable gradent exsts only around the nterface and nterface thckness s much smaller than radus, R, we have κ v = L (.11) R It can be seen that the nterfacal velocty s proportonal to the curvature of the boundary but surface energy s not ncluded explctly. Moreover, calculatons above set up an unambguous relatonshp between the modelng parameters and the materals propertes. In ths sense, ths work poneered the sharp nterface asymptotc analyss... Mult Phase Feld Method The multple orentatons PFM has been developed along two ndependent lnes: one nvolves multple orentaton felds and the other employs a sngle orentaton feld. Snce only one order parameter was appled n latter case, tremendous computatonal resource savng s expected. However, sngle feld method has ts mathematcal dffculty n handlng 3D crystallography. In the current study, we dscuss the former approach. In multple orentatons PFM, the mcrostructure of an arbtrary polycrystallne materal s descrbed by a set of non-conserved long-range order (lro) parameters η, η,..., η ) wth each of them descrbng a specfc crystallographc orentaton of the ( 1 p grans. The evoluton of the system for each lro parameter s descrbed by the tme- 9

28 dependent Allen Cahn equaton: η t δf = L δη (.1) where L s the knetc coeffcent that characterzes gran boundary moblty and F s the total free energy of the system. In gradent thermodynamcs [5], the total free energy s expressed on a coarse-graned level as: [ f ( ) + κ( ) ] F = η dv (.13) 0 η where κ s the gradent energy coeffcent and f 0 s the local free energy. The exact form of f 0 s not mportant as long as t provdes degenerate mnma correspondng to each gran orentaton, η. A smple form that satsfes ths requrement s f 0 a = P = 1 P P 1 4 b η + η + η η j (.14) = 1 j> 4 where P s total number of lro parameters n the system and values of the phenomenologcal parameters a and b are determned by the gran boundary energy. If we assume the knetc coeffcent L, the gradent coeffcent κ and energy barrer factor b are constant, the above equatons descrbe sotropc gran growth. The dependency of gran boundary propertes on msorentaton s ntroduced by makng L, κ and b msorentaton-dependent under the constrant of constant gran boundary thckness. Therefore, msorentaton s needed to be determned throughout the doman. The way of defnng msorentaton feld s not unque. To be consstent wth the feld approach, we employed followng functon n current study: 10

29 () r P, j = P η η θ, j j j j θ (.15) η η Where θ j s pre-calculated msorentaton angle between grans wth orentatons η and η j. Equaton (8) assgns a constant msorentaton angle to a narrow range of the gran boundary. Practcal dffcultes n defnng msorentaton feld that are assocated wth P the fact that η η j tends to zero n bulk regons s nconsequental to the asymptotc, j analyss, snce the role of msorentaton s sgnfcant only n the neghborhood of gran boundary. Wthout losng generalty, we assume that the energy ansotropy s characterzed by a plateau for hgh-angle boundares and by the Read-Shockley formula for small angle boundares: γ ( θ ) θ θ γ 0 1 ln = θ m θ m γ 0 θ < θ θ θ m m (.16) where θ m s the maxmum angle at whch the Read-Shockley equaton stll holds, and γ 0 s a constant. Correspondngly, the moblty ansotropy s characterzed by: L ( θ ) = 5 θ L < 0 θ θ m θ (.17) m L0 θ θ m Note that the magntude of θ m determnes the degree of ansotropy. 11

30 Another commonly used energy formula begns by defnng sum of lro parameters as unt: P η = 1 (.18) Then there wll be p-1 ndependent η. To restore the mathematcal symmetry between all lro parameters, we adopt a new governng equaton by modfyng Allen Cahn equaton to: η t = P j δf δf L j δη δη j (.19) δf where varatonal dervatve s calculated wthout takng nto account any dependence δη between dfferent lro parameter. Ths treatment creates no mathematcal dfference but adopts more symmetry n ts mathematc expresson. The free energy densty s gven by f 0 = P P = 1 j> W j κ j η η j + η η j η j η (.0) where W j and κ j are symmetrc matrxes related to the heght of double well and gradent energy penalty between phase and j. It s easy to shown that the free energy reduces to Allen Cahn s antphase boundary model at the boundary between phase 1 and : f 0 = W = W = W η η η η ( 1 η ) + η ( 1 η ) ( 1 η ) 1 1 ( 1 η ) + η 1 κ1 η1 η η η κ 1 1 κ η 1 (.1) 1

31 The advantages of new mult phase feld confguraton are two fold: 1. The heght of double well and gradent coeffcent are explctly related to gran boundary msorentaton;. The crossover vales of two lros at the boundary s exactly at 0.5, whch make the boundary profle relatvely ndependent from the free energy formula. It s mportant to control the boundary propertes n cases havng complcated couplng between phase varables..3 Couplng Between Phase Feld and Concentraton Feld Phase transton n a real system often nvolves changes of chemcal composton. In these cases, we need couplng between concentraton felds and order parameters. The order parameters characterze symmetry changes accompanyng the phase transformatons and ther choce can be ether physcal or phenomenologcal. The free energy confguraton on mult feld varable s not trval. For an order-dsorder transformaton, the long-range order (lro) parameters are the default order parameters and the local free energy as a functon of concentraton and lro parameters can be obtaned drectly by the CALPHAD technque [14]. For a reconstructve phase transformaton a Landau free energy expanson wth respect to physcally chosen order parameters can be constructed accordng the symmetry changes durng the phase transformaton [15, 16]. The same approach can be appled to alloys, but then t became evdent that the parameters n the Landau free energy must be made temperature and composton dependent. In order to have a Landau free energy consstent wth the expermental or 13

32 assessed equlbrum free energy data n a multcomponent system, we thus have to face a formdable task to ft the expresson n a multdmensonal space at dfferent temperatures. An alternatve approach s to defne a phenomenologcal order parameter that assumes certan dfferent values for phases of dfferent symmetry. The local free energy s then constructed n such a way that the equlbrum free energy of ndvdual phases can be drectly nserted nto the expresson, and the phase equlbrum relatonshp n the temperature-composton projecton can be sustaned n the temperature-compostonorder parameter space. A convenent choce of such an expresson s due to Wang et al. [17] and has been used wdely n the soldfcaton modelng [18] and recently also for sold state transformatons [19]. Adoptng ths choce, we wrte the local molar Gbbs free energy g m as a functon of temperature T, composton x (=1,,, n-1), and order parameter η: g α β ( T, x, η) = [1 p( η)] g ( T, x ) + p( η) g ( T, x ) q( η) (..) m m m + 3 where p ( η ) = η (10 15η + 6η ) and q ( η) = ωη (1 η). The parameterω s the heght of the mposed double-well hump and, along wth the gradent energy coeffcents κ and ε shown below n Eq.(.3), can be determned from nterfacal energy, σ, and nterface thckness, λ. α g m and g β m are the molar Gbbs free energes of the α and β phases, respectvely. For a chemcally and structurally non-unform system under the assumpton of constant molar volume V m, the total Gbbs free energy G can be expressed by 14

33 G G = V m m 1 = V m n 1 κ ε gm( T, x, η ) + x + η dv (.3) V = 1 The temporal evolutons of feld varables are governed by the tme-dependent Gnzburg-Landau equatons [0] and the generalzed Cahn-Hllard dffuson equatons [1] on the bass of the phenomenologcal Fck-Onsager equatons []: η = M t η δg m δη (.4) n 1 x 1 k = m t j = 1 V M kj δgm ( T, x, η ) (.5) δx j where M η s the moblty of the order parameter and can be drectly related to the nterface moblty n the sharp nterface approach. The parameters M kj are the so-called chemcal mobltes n the volume-fxed frame of reference. In a homogenous phase p (p=α, β), accordng to Andersson and Agren [3], the chemcal mobltes M are related to atomc mobltes p M l (l=1,,n) by: p kj M p kj 1 = V n ( jl x j ) m l = 1 p δ ( δ x ) x M (.6) lk k l l where δ jl and δ lk are the Kronecker delta. In a non-unform system, we assume that the same relaton holds for M kj and M l, the atomc mobltes dependent on the order parameter η by: M ( ) ( ) (.7) α β α η β (1 η ) l = M l + M l M l M l The choce of Eq.(.7) ensures that the atomc mobltes n the nterface regon wll 15

34 have a postve devaton from the smple lnear nterpolaton, and the larger the dfference between atomc mobltes n the two phases s, the more postve the devaton wll be. If the dfference s several orders of magntude, the atomc moblty n most part of the nterface regon wll assume almost the value for the phase that has a hgher atomc moblty (see Fg. 1). We beleve that ths scheme s more reasonable than ether lnear or logarthmcally lnear approxmaton because of sgnfcantly more vacances present n the structure of a sem-coherent or ncoherent nterface. Insertng Eq.(.3) nto Eqs.(.4) and Eq.(.5), we can obtan the followng dmensonless governng equatons η ~ ~ ~ ( ~ gm = M η ε η ) (.8) τ η x ~ n 1 k = τ j = 1 ~ M kj ~ g~ ( x m j ~ ~ κ x j j ) (.9) by ntroducng the followng reduced quanttes: ~ = [ / (x/l), / (y/l)]; g ~ m = g m / g m ; M ~ k = V m M k /M; ~ ε = ε /( g m l ); ~ κ = κ /( g m l );τ = (M g m /l )t; M ~ η = M η l /M where l s the mesh sze, g m and M are normalzaton factors for molar Gbbs free energy and atomc moblty, respectvely. Ths dmensonless verson of the governng equatons s partcularly convenent for numercal calculaton and very useful n rescalng the space and tme for dffuson-controlled phase transformatons..4. Ansotropy 16

35 Energy and moblty ansotropy s an mportant factor n nterface equlbrum and dynamcs. Snce the nterface s dffuse n the phase feld model, careful consderaton for the proper ncorporaton of surface free energy s requred. In general, the phase feld parameters, hump heght ω, gradent coeffcent κ and knetcs factor L, are functons of nclnaton angle, Θ and msorentaton. The expresson of varaton n ansotropc system s more complcated. By applyng varatonal calculus for multvarable, the varaton of ntegral s replaced by δi = W δη ( Θ) f η W ' Θ η ' ( Θ) f ( η) W ( Θ) f ( η) x x Θ η y y (.30) + ' '' κ η + κκ ' Θ η κ + κκ [ η Θ η Θ ] x y x y Θ η y where = η η + η x x y Θ η x and = η η + η y x y. Further extenson of term W ' Θ ( Θ) f ( η) η x n Eq. (.30) s undesred snce the produced terms lke Θ = η η y ( η η ) η x x + x y, whch s easy to dverge as approachng to the bulk. In our code, we Θ calculated W ' ( Θ) f ( ) η x η drectly. The equaton above can be reduced to Eq. () n ref. [4] f one treats W ( Θ) as constant. An addtonal source of ansotropy s ntroduced by lettng the moblty coeffcent depend on nclnaton as well, such 17

36 as L = L( Θ)..5. Fnte Volume Method and Adaptve Mesh Snce the nterface s dffuse n PFM, several grd ponts need to be allocated n ths range. Wthout a fne mesh, n addton to loss of accuracy, a coarsened grd often leads to numercal nstablty. In structured grd, puttng enough grd ponts across nterface s computatonally expensve. Further more, the spatal gradent of phase varable s very small far away from nterface. Therefore, local mesh refnement or adaptve mesh s hghly recommended for purpose of effcency. It may open addtonal wndows for quanttatve phase feld smulaton. Example of utlzng adaptve quadrlateral mesh s gven by Lan et. al. n soldfcaton [5]. Although mplementaton of adaptve mesh, such as trangle mesh, on Fnte Element method has been very successful, grd regularzaton for mnmzng grd skewness and complcated node numberng are stll headache ssues. On the other hand, the mesh generaton wth quadrlateral cells s much more flexble. As llustrated n Fg..1, wth mult-level mesh refnement, the excellent mesh qualty can be acheved around nterface. The grd level can span the smulaton doman wth dfferent length scale, whch renders mult-scale smulaton drectly accessble. In addton, the mesh coarsenng and refnement durng nterface mgraton can be qute effectve and have many varetes. Snce the mesh grd s mult-level, efforts are requred to set up herarchcal data structures, defne the refnement and coarsenng mechansm durng the 18

37 nterface mgraton and apply proper mathematcal algorthm to solve the governng equaton. a b c Fgure.1. Schematc of adaptve multlevel mesh for sland gran shrnkng. The mesh s refned along the gran boundary. (a) plot of sland gran mcrostructure (b) correspondng adaptve mesh (c) Enlargement of (b). There are numerous herarchcal data structurng technques n use for representng spatal data. One commonly used technque that s based on recursve decomposton s the quadtree for D data and Octree for 3D data. Durng to potental applcaton of these 19

38 data structures on data compresson n IT, tremendous efforts have been taken n ths area. A great revew s gven by Hanan n hs book Applcatons of Spatal Data Structures. In current work, the must-equpped technques nvolve Tree Traversals, Neghbor-fndng and Converson. The correspondng nformaton can be found n Chapter,3 and 4 n book we mentoned above. Snce the mesh s non-unform but regular, the so-called Fnte Volume Method (FVM) s appled to solve the Alle-Cahn and Cahn-Hllert governng equaton. The key recpe for FVM s Gauss s dvergence theorem. The volume or area ntegraton can be transformed nto surface or boundary ntegraton by: ( ) da = A η η dl (.31) where the drecton of dl s parallel to the surface norm and ponts outsde of calculaton cell. In ts dscrete form: ( ) da = n = 1 A η η dl (.3) Where n s the total number of edges of the calculated cell. If the mesh s unform, Eq. (.3) becomes: ( ) da = dl ( η ) n = η η (.33) It can be seen that FVM s exactly reduced to fnte dfferent method n ths scenaro. In current study, r η s always perpendcular to dl r snce the rectangular grds, whch tremendously decrease the computatonal demand. Applcaton of adaptve mesh was gven n an sland gran shrnkng system and α 0

39 partcle dssoluton n T64 alloy. The phase feld setup has been dscussed n secton one of ths chapter. The comparson between the adaptve mesh and unform mesh n term of shrnkng knetcs shows exact match. It s nterestng to study the tme savng n utlzng adaptve mesh technque. As shown n table.1, t can be seen that adaptve mesh s more effcent n larger system. Qualtatvely, the tme expense n adaptve mesh s manly due to two parts: mesh management and calculaton. Mesh management s overhead tme relatve to unform mesh. In small system or geometrcally complcated system, the mesh management may compensate a lot of tme saved n calculaton. Thus, adaptve may not gan too much. Wth ncreasng system sze, tremendous tme and memory savng n calculaton shfts favorte to adaptve mesh. It s worthy mentonng that adaptve mesh may fnd tself fttng greatly n dffuson controlled phase transformaton PFM smulaton, because large amount of calculaton needed n calculatng the thermodynamc and knetcs data at each node. system sze adaptve unform tme (s) (s) N/A Table.1 Comparson of computatonal tme between adaptve and unform mesh 1

40 CHAPTER 3 SOLUTE SEGREGATION TRANSITION AND DRAG FORCE ON GRAIN BOUNDARIES 3.1. Introducton Most classcal theores of nterface mgraton are based on systems wth sotropc and unform boundary propertes [6-8]. However, n a typcal expermental mcrostructure one encounters a populaton of gran boundares where the thermodynamc and knetc propertes vary from one boundary to another [8, 9]. In recent smulatons of gran growth wth ansotropc boundary propertes based on the Phase Feld [30-3] and Monte Carlo [3-34] methods, t was shown that ansotropy of boundary energy and moblty can have a profound effect on the morphology and knetcs of the mcrostructural evoluton. In addton to crystallographc ansotropy, gran boundares n practcal materals may exhbt dfferent propertes because of segregatng defects such as dssolved mpurtes, second-phase partcles, or nter-granular wettng flms of second phases. For the case of solute atoms n metallc alloys, Aust and Rutter [35] showed that the rate of

41 gran boundary mgraton could be reduced dramatcally even by small average concentratons, but the effect was much less pronounced for certan hgh-angle boundares wth specal structures. Subsequent experments on a varety of bcrystallne and polycrystallne systems revealed further characterstcs of solute drag, ncludng ts senstvty to boundary speed and the dependence of boundary moblty on temperature and composton [9, 36]. In heatng-coolng experments on doped Al bcrystals [37], for example, the boundary moblty exhbted a transton wth a hysteress that could not be ratonalzed on the bass of classcal solute-drag models. In ths chapter we develop a theoretcal model of solute segregaton and solute drag at gran boundares and nvestgate segregaton profle, segregaton transton, drag force and the correspondng effects on boundary mgraton. The present treatment follows the same formalsm as Cahn s solute-drag theory [38] but apples a more robust thermodynamc soluton model. The orgnal theores of solute drag are founded upon a smplfed model of segregaton n dlute, deal solutons [38-41] under the nfluence of a potental well centered on the gran boundary. Here, to provde a bass for our model, we outlne Cahn's treatment, startng wth the assumed form of solute chemcal potental µ ( x, c) = kt ln c( x) + E ( x) const. (3.1) B B + where x s the dstance from the center of the gran boundary, c(x) s the solute atom fracton, and E B (x) s the solute-boundary nteracton potental, whch may translate wth the boundary but s not otherwse altered n shape or ampltude. Thus, the nfluence of the defectve structure of the boundary s represented by E B (x), and as descrbed by Eq. (1), t s analogous to the effect of an external feld mposed on an deal soluton. 3

42 In a bnary (A-B) substtutonal system, and on the assumpton that composton s vared by atomc exchanges, the equlbrum concentraton profle obeys the followng condton: µ(x) = µ( ) (3.) where µ = µ B µ A s the exchange potental and represents values n the bulk, far away from the boundary. Substtutng Eq. (3.1) nto (3.) yelds the equlbrum solute concentraton at a gran boundary as a functon of bulk concentraton and temperature c ( x) c E( x) = exp 1 c( x) 1 c kt (3.3) and n ths case E( x) = E ( x) E ( x) (3.4) 4 B The segregaton sotherm n the form of Eq. (3.3) follows drectly from the condtons of chemcal equlbrum. Cahn s segregaton profle can be obtaned from Eq. (3.3) by assumng a dlute concentraton throughout the deal soluton,.e., c(x) << 1. The robustness of the segregaton sotherm depends on the complexty of the chemcal potental formulaton. Equaton (3.3) s smlar to the McLean sotherm [4], but n ths case the segregaton s allowed to extend over the range of E(x), not localzed to a sngle mathematcal plane. Furthermore, f ste excluson s allowed, the quantty, [ 1 c(x) ], n Eq. (3.3) s to be replaced by [ c c(x) ], where c* s the fracton of gran boundary stes avalable for segregated atoms at saturaton. Usng the chemcal potental of Eq. (3.1) n a dffuson analyss, Cahn derved the steady-state composton profle across a boundary mgratng wth a constant speed. He A

43 showed that when E(x) s an even functon of x, the drag force s assocated wth the asymmetry of the composton profle across the movng boundary and s nsenstve to the sgn of E(x). For certan combnatons of solute concentraton and temperature, the force velocty relaton becomes a mult-valued functon, where two boundary veloctes are possble at a gven drvng force, suggestng a jerky moton. The same phenomenon was also predcted by the analyses of Lücke and Detert [40] and by Lücke and Stüwe [39]. The early deal-soluton models are convenent for llustratng basc characterstcs of solute drag, but more sophstcated models are needed for comparng theory wth experment. Hllert and Sundman (H-S) [43] and, more recently, Mendelev and Srolovtz (M-S) [44] developed more advanced models for solute drag by ncorporatng dfferent elements of regular soluton theory to account for atom-atom and atom-boundary nteractons. The H-S model was appled to evaluate segregaton and drag force over the entre composton range of a bnary alloy, showng a non-monotonc varaton of drag force wth bulk concentraton. The M-S model was appled to nvestgate the effect of the sgn of E(x) on the segregaton. It was shown that for attractve segregaton, a postve mxng energy enhances the solute drag and a negatve one reduces t. Thus, n contrast to the predcton of Cahn s deal soluton model where the drag force s ndependent of the sgn of E(x), dfferent drag forces were predcted for attractve and repulsve segregaton. Even though both the H-S and M-S models are based on regular solutons, ther treatments of segregaton are sgnfcantly dfferent from one another, as wll be 5

44 dscussed further n secton 3. below. More mportantly, nether model consders the effects of steep composton gradents near the boundary. In addton, M-S model overlooks the possble couplng between composton and the solute-boundary nteracton potental. When the level of segregaton s hgh, steep composton gradents are present near the boundary. Accordng to contnuum (gradent-) thermodynamcs of non-unform systems [5] and ts dscrete counterparts [45, 46], the contrbutons of concentraton gradent to chemcal potental must be ncluded n the condtons for equlbrum and n calculatng the drvng forces for dffuson. In fact, ndependent to solute drag, there have been sgnfcant developments n the thermodynamcs of solute segregaton at statc surfaces and nterfaces based on dscrete regular soluton models [47-51]where all these contrbutons were accounted for automatcally wthn the approxmaton of frst nearestneghbor nteractons. In ths chapter, we develop contnuum and dscrete models for segregaton and segregaton transton at gran boundares to obtan the steady-state concentraton profles under statc and dynamc condtons. The model s also appled to calculate the drag forces, segregaton transton temperatures, and the transton of gran boundary moblty as a functon of temperature. Usng a prototype planar gran boundary n a regular soluton, we llustrate the dstnct terms that must be consdered n gran boundary segregaton, ncludng concentraton gradent, spatal varaton of the gradentenergy coeffcent and the couplng between concentraton and structure n the solute boundary nteracton potental. 6

45 3.. Segregaton Model In general, the chemcal potental of solute n a chemcally unform but structurally nonunform (ncoherent or defectve) system can be expressed as: where 0 B B µ B xs µ = µ ( x) + kt ln c + ( x, c) (3.5) 0 xs µ B s the standard-state value and µ B s the part n excess of the contrbuton from confguratonal entropy of mxng. The varaton of the standard-state value wth poston can be expressed as: 0 B B + 0 µ ( x) = µ ( ) E ( x) (3.6) and, correspondngly, the excess chemcal potental s expressed as B xs xs µ ( x, c) = µ (, c) + ϕ ( x, c) (3.7) B B where ϕ ( x, c), as n the H-S model, takes nto account the poston dependence of the B enthalpy of mxng near a defect. The dfference n solute chemcal potentals near the boundary and n the bulk can be expressed as: B xs xs ( c c ) + [ µ (, c) (, c )] µ µ = E ( x) + ϕ ( x, c) + kt ln (3.8) B B B B B µ B In gradent thermodynamcs of non-unform systems, the nterfacal free energy s approxmated by ntegratng contrbutons from local composton and from the composton gradent [5]. For a planar gran boundary, we express the nterfacal energy γ as follows where f ( x, c) κ dc γ = f o ( x, c) + NV ( x) dx (3.9) dx s the free energy change (per atom) upon formng a unform soluton of o 7

46 composton c, at the locaton x, from consttuents n the bulk reservor of composton c, κ s the gradent-energy coeffcent, and N v s the number of atomc stes per unt volume. Both κ and N v are allowed to depend on poston (or structure) but assumed to be ndependent of composton. By defnton, For convenence, we may also wrte where f c ( x, c) = c( µ µ ) + (1 c) ( µ ) f o µ (3.10) B B ( x, c) = f ( c) + f ( x c) A A f (3.11) 0 c s, xs xs xs xs c 1 c () c = c[ (, c) µ (, c )] + (1 c) [ µ (, c) µ (, c )] + kt cln + (1 c)ln µ (3.1a) B B A s the conventonal chemcal contrbuton [5] that arses wherever c c, and the remander takes the followng form ( x, c) = E ( x) + ce( x) + ϕ ( x, c) + cϕ( x, c) A c 1 c f (3.1b) whereϕ(x,c) = ϕ B (x,c) ϕ A (x,c). Thus, f s ( x, c) s A A ncorporates the structural contrbuton and would not vansh even n a chemcally unform system wth c(x) = c. Therefore, at a gran boundary, the structural ncoherence drves the system away from composton unformty. In Cahn s crtcal pont wettng theory [5], ths structural contrbuton s assumed to be short-ranged, and the ntegraton of f s ( x, c) dependng only on c (0). s replaced by a functon The equlbrum composton profle mnmzes γ and obeys the followng Euler- Lagrange equaton: 8

47 f 1 d dc o NVκ = 0 (3.13) c N dx dx V Equvalently, the equlbrum condton may be stated as a requrement of unform exchange potental, analogous to Eq. (3.) ( N κ ) d c 1 d V dc µ = µ ( x, c) κ = µ dx N dx dx V (3.14) The segregaton sotherm obtaned from Eq. (3.14) has the same form as Eq. (3.3), but the free energy of segregaton s gven by: F seg xs (, c) µ (, c ) = E( x) + ϕ ( x, c) + µ xs d c 1 κ dx N v d ( N κ ) V dx dc dx (3.15) The gran boundary energy n a chemcally unform system s gven by: γ = f ( x, c ) c N ϕ dx (3.16) s ) dx = γ A + ( γ B γ A) c + (1 c NVϕ Adx + where γ A and γ B are the gran boundary energes n pure A and B, respectvely. In a chemcally non-unform system, κ dc [ ] γ = γ A + NV cedx + NV ϕ A + cϕ dx + NV f c + dx (3.17) dx Note that the above analyss of segregaton and boundary energy s qute general, not relyng on any assumptons about a partcular soluton model. To apply the analyss to a gven system, t s necessary to specfy a partcular model for the excess chemcal potental, ncludng structural contrbutons and the gradent-energy coeffcent κ. Followng prevous practce, we use a dscrete, nearest-neghbor regular soluton model V B 9

48 to evaluate the requste parameters. We treat a bcrystal as a stack of homogeneous atomc layers parallel to the gran boundary plane. Wthn each atomc layer the solute s randomly dstrbuted, and the concentraton s allowed to vary from layer to layer. Atoms n the gran boundary layers have a smaller coordnaton number than atoms n the bulk, but for any atom n a gven layer the nearest-neghbors are dstrbuted wthn the same layer and n the mmedately adjacent layers. Thus, the total atomc coordnaton z = z o + z + + z, where ( + ) z s the number of nearest-neghbor bonds n the adjacent layers below (-) and above (+) the th layer, and 0 z s the n-layer coordnaton. Note that, owng to structural non-unformty near the boundary, z and z + are unequal, whch dstngushes the present treatment from prevous models of nterfaces, n whch a fully coherent structure was assumed. However, by conservaton of bonds between adjacent layers, the followng relaton holds n z + = n +1 z +1 (3.18) wth n denotng the number of atoms per unt area n the th layer. In regular soluton regme, the mxng enthalpy s a matter of countng the number of varous types of bonds. For randomly dstrbuted atoms wth nearest-neghbor bond energes ε AA, ε BB and ε AB, we obtan energy of a solute atom n the bulk: E B BB ( c ) ε AB = z c ε + z 1 (3.19) Energy of a solvent atom n the bulk: E A AB ( c ) ε AA Energy of a solute atom at the th layer of a gran boundary: = z c ε + z 1 (3.0) 30

49 31 ( ) ( ) ( ) AB BB AB BB AB L BB L B gb c z c z c z c z c z c z E ε ε ε ε ε ε = (3.1) Energy of a solute atom at the th layer of a gran boundary: ( ) ( ) ( ) AA AB AA AB AA L AB L A gb c z c z c z c z c z c z E ε ε ε ε ε ε = (3.) The segregaton energy s calculated by exchangng a solvent atom at the boundary wth a solute atom n the bulk, as mpled by Eq. (3.15). ( ) ( ) ( ) B A gb A B gb seg c c z c c z c c c E E E E E E Ω + + = + = ) (, ε ε ϕ (3.3) where ( ) AA BB z z E ε ε =, ϕ,c ( )= z z ( )ε 1 c ( ), (3.4) Ω = z ε s the regular soluton parameter, ε = ε AB 1/(ε AA + ε BB ) and z s the total coordnaton number of atoms n the bulk. E and ϕ correspond to the E(x) and ϕ(x,c) terms n Eq. (3.15), respectvely. Usng the deal-mxng entropy employed of the regular soluton model, we can express the chemcal potental dfference between the solute and solvent atoms as follows. ( ) ( ) ( ) ( ) A B c c kt c c z c c z c c E + + Ω + = ln 1, 1 1 ε ε ϕ µ µ (3.5) Thus, the dscrete-model segregaton sotherm s gven by ( ) ( ) ( ) Ω + + = + + kt c c z c c z c c c E c c c c 1 1 ) (, exp 1 1 ε ε ϕ (3.6) From the crteron of bond conservaton, the terms ( ) ( ) c c z c c z ε ε n

50 Eqs. (3.3) and (3.6) can be rewrtten as ε z N N z + 1 N + 1 z N 1 z + N ( c c ) z ε + z + ( c + c c ) (3.7) where z = z + + z. Wth the ad of expresson (3.7), the correspondence between contnuum and dscrete model parameters s summarzed as follows and 0 z κ ( x) = εd (3.8) [ c c ] xs xs µ (, c) µ (, c ) = Ω (3.9) where d 0 s the nter-layer spacng. Thus, the total segregaton energy derved from contnuum gradent thermodynamcs can be vewed as a frst approxmaton of the dscrete counterpart wth frst dervatves approxmatng dscrete dfferences, whch becomes more accurate n the lmt of small gradents. In contrast to prevous treatments of gran boundary segregaton and solute drag n regular solutons, the present treatment ntroduces several new terms self-consstently. Not only are concentraton gradents, κ d c dx, ncluded, but owng to structural nonunformty, the couplng between structure and composton s manfested n the product dκ dc, and n ϕ(c,x). Therefore, under the rght assumptons, the present treatment can dx dx be reduced to prevous results. For example, f we assume ε = 0 (.e., deal soluton), then Ω, κ, dκ/dx and ϕ(x,c) all vansh and Eq. (3.15) reduces to Cahn s deal soluton model or the McLean sotherm [4]. If we assume κ = 0 (dκ/dx = 0) and ϕ(x,c) = 0, but somehow 3

51 keep Ω fnte, then Eq. (15) reduces to the M-S model [44], or the Fowler-Guggenhem model [53]. If we assume κ = 0 (dκ/dx = 0), then Eq. (3.15) reduces to the H-S model [43]. Below, we wll nvestgate the effects of the new terms on the solute concentraton profles across the gran boundary, the correspondng drag forces, and the segregaton transton temperature Dffuson and Mgraton To smplfy the dffuson analyses, we assume N V to be constant from now on. Ths should not alter the results qualtatvely. For a movng boundary, the flux of solute atoms n a reference system movng wth the gran boundary at a velocty, v, s gven by: J µ = c( 1 c) [(1 c) ma + cmb ] NV cvnv x (3.30) where m B and m A are the atomc mobltes of solute and solvent atoms, respectvely. To solate the effects of segregaton thermodynamcs, we assume equal atomc mobltes. Thus, substtutng Eq. (3.14) nto Eq. (3.30), we obtan: J dκ dc d c c( 1 c) m µ ( x, c) κ x N V cvnv dx dx dx (3.31) = The evoluton of solute concentraton profle s obtaned by solvng the dffuson equaton N V dc dt 33 = dvj (3.3) usng the fnte dfference method, wth the flux gven by Eq. (3.31). The steady state s defned by unform flux n the gran boundary reference system. The steady-state

52 concentraton profle s obtaned by evolvng an assumed ntal profle. Hllert [54] and Hllert and Sundman [43] provded a general method to evaluate the drag force based on the argument that the drag force derves from the free energy dsspaton assocated wth solute dffuson durng boundary mgraton. At the steady state and for equal atomc mobltes, the drag force (per unt area) may be expressed as P = ktnv v D ( c c ) dx c(1 c) (3.33) where D s the dffusvty of mpurty atoms. Hllert and Sundman [43] showed that the drag force calculated by equaton (3.33) reduces to Cahn s result under the deal-soluton assumpton, but t s also approprate for use n conjuncton wth the regular soluton. In the current study, Eq. (3.33) s employed for the calculaton of the drag force from the steady-state concentraton profle. Under a small drvng force (small velocty), boundary mgraton s domnated by solute drag, and Eq. (3.33) yelds the followng dscrete form for the dependence of moblty on temperature and composton. v D ( c c ) M = = d0 (1 ) (3.34) P ktnv c c Smulatons We apply the model to study segregaton and segregaton transton at both statonary and movng boundares. The regular soluton constant s chosen to be Ω = z ε = 0.3eV. Two ndependent functons are employed to descrbe the atomc coordnaton profle across 34

53 the gran boundary, whch, wthout loss of generalty, are assumed to have the followng Gaussan forms, The bond conservaton crteron for constant N v gves z = z.0exp[ ] (3.35a) z = z 1.0exp[ ( 0.5) ] (3.35b) z (3.36) + = z + 1 and the n-layer coordnaton s gven by z 0 z z z (3.37) + = For llustratve purposes, we assume z = 1, 0 z = 4 and ± z = 4, whch would be strctly correct for a twst boundary parallel to (001) n an f.c.c. system. We also have chosen 1 ( ε BB ε AA )= 0.1eV Statonary Boundary Based on the segregaton model, the equlbrum solute concentraton profle across a statonary boundary s gven by Eq. (3.6) or equvalently by Eq. (3) wth E seg (x) gven by Eq. (3.15). In ths chapter we solve Eq. (3.6) usng the natural teraton method to fnd the equlbrum segregaton profle. We started wth a homogeneous system of c = c. Then we calculate the total energy of segregaton by Eq. (3.3) and adjust the composton for each layer accordng to Eq. (3.6). Ths process s terated tll the 35

54 composton profles converge to a statonary value H-S M-S Current Cahn 0.3 c x/d 0 Fgure 3.1. Equlbrum solute concentraton profles across a gran boundary obtaned from dfferent models for a system of attractve nteracton between solute and gran boundary. The bulk composton s c =0.04 and the temperature s T=1750K. Fgure 3.1 shows the comparson of equlbrum solute concentraton profles for attractve nteracton (E(x) < 0) obtaned from dfferent models. The current, M-S and H- S models all predct an enhancement of solute segregaton n the case of postve devaton of the regular soluton from dealty. However, such an enhancement s overestmated sgnfcantly n both M-S and H-S models. For repulsve nteracton, the solute enrchment n the system s so low that the dfference among the three models 36

55 becomes nsgnfcant. For a system wth postve mxng energy (ε > 0), phase separaton s expected when alloy composton and temperature are wthn the mscblty gap. If surfaces and nterfaces exst, however, segregaton transton at these surfaces and nterfaces have been predcted when the bulk alloy composton and temperature are outsde the mscblty gap [49, 51, 5, 55]. Below, we explore ths transton at gran boundares usng the same approach and model system dscussed above. A seres of calculatons were performed upon coolng and heatng for a set of dfferent bulk compostons. In the coolng cycle, we start wth a homogeneous alloy of unform solute concentraton at a temperature much hgher than the bulk mscblty gap temperature. The equlbrum solute concentraton profle across the boundary s obtaned by the teraton method. Then the same calculaton s repeated at a lower temperature, usng the equlbrum concentraton profle of the prevous temperature as the ntal condton. The results obtaned are shown n Fg. 3.a by the sold crcles, where the relatve Gbbs excess of solute, Γ Β, s plotted aganst temperature. Note that the relatve solute excess s defned by Γ B = N V d 0 1 c (c c ) (3.38) whch s a meanngful measure of the gran boundary segregaton defned through the Gbbs adsorpton equaton n conjuncton wth the Gbbs-Duhem relaton. A clear transton from low to hgh segregaton s observed for ths case at T =66.8K. 37

56 The same calculaton procedures are repeated for a heatng process, startng wth the last equlbrum concentraton profle obtaned at the end of the coolng process. The results + are gven n Fg. 3.a by the open trangles. A hgher transton temperature, T =668.6K, s predcted for the heatng process as compared to the coolng process, resultng n a hysteress loop. Ths ndcates that the transton from low- to hgh-segregaton or vce versa s a frst-order phase transton. The transton temperature s obtaned precsely by plottng the gran boundary energy n dscrete verson of Eq. (3.17): γ γ A = N V d 0 Ω( c c + c E + c ) + kt c ln c ( 1 c ) 1 c ln 1 c ( z z ) εc (1 c ) + ( c c ) + κ d (3.39) as a functon of temperature for both the coolng and heatng processes ( Fg. 3.b). The gran boundary energy ncreases wth ncreasng temperature (less solute segregaton) but wth dfferent slopes of the γ T curves for coolng and heatng. At the transton temperature gran boundares wth low- and hgh-segregatons should have the same energy. Therefore, the temperature at whch the two γ T curves ntersect defnes the transton temperature, and n ths case, T 0 =665.0K. 38

57 ΓB/NVd T(K) (a) coolng heatng equlbrum (γ-γa)/nvd coolng heatng equlbrum (b) T(K) Fgure 3.. (a) Gran boundary segregaton as a functon of temperature durng coolng (sold crcles) and heatng (open trangles) processes and, (b) gran boundary energy as a functon of temperature for a system of bulk composton c =0.00. The segregaton transton s ndcated by the thck sold vertcal lne n (a). The ear formed by the long dashed lne llustrates another representaton of ths frst order phase transton.

58 By plottng the transton temperatures (marked by small symbols n Fg. 3.3) as a functon of the bulk composton we obtan a stablty boundary above the bulk mscblty gap for the segregaton transton at gran boundares. To nvestgate d c separately the effects of ϕ(c,x), κ and dx dκ dc terms, we calculate the transton dx dx temperatures under varous combnatons of these three terms. The transton temperatures predcted by the H-S and M-S models are smlar to each other but dffer sgnfcantly from that of the current model (Fg. 3.3). The crtcal temperatures (defned by the temperature at whch the sharp transton between low- and hgh-segregaton ends, or n other words, the hysteress dsappears, and ndcated by large symbols n Fg. 3.3) predcted by the M-S and current models dffer from each other even more (~ 1000 K). The crtcal temperature predcted by the M-S model concdes wth the crtcal temperature of bulk 3D system gven by T c = zε k (3.40) whch s ndcated by the long-dashed horzontal lne n Fg. 3, whle the crtcal temperature predcted by the current model s closer to the crtcal temperature of an solated D system, whch s gven by [51, 55] z T c = (3.41) k and s ndcated by the short-dashed horzontal lne n Fg It s nterestng to note dκ dc that the addton of ϕ(c, x) and terms, both of whch are related to the number dx dx 0 ε 40

59 of mssng bonds of gran boundary atoms, lowers sgnfcantly the crtcal temperature, d c but has lttle effect on the transton temperatures, whle the κ term lowers dx sgnfcantly both the transton and the crtcal temperatures T c 3D T(K) Bulk mscblty gap 800 T D M-S model c H-S model 600 add term 3 only add terms &3 only 400 current model c Fgure 3.3. Stablty dagram for segregaton transton at gran boundary. Dfferent symbols represent results obtaned under varous combnatons of three terms: (1) ϕ(c,x); () κ d c ; and (3) dκ dc. The M-S model dx dx dx excludes all three terms whle the H-S model ncludes only term (1). The current model ncludes all three terms. For each case, the crtcal temperature ( T ) s ndcated by the larger symbols. gb c Because of solute segregaton, gran boundary energy s n general a functon of both temperature and solute content. Such nformaton should be very useful n materals process desgn. Fgure 3.4 summarzes the energy of the statonary boundary as a 41

60 functon of temperature and bulk mpurty concentraton for the model system consdered. The results for deal soluton arealso presented n Fg. 3.4 for comparson. At T=, the effect of segregaton vanshes and gran boundary energy becomes a lnear functon of c (sold crcles n Fg. 3.4) for deal soluton because all other terms n Eq. (3.39) vansh except c E term ( z z ) c 1 c ) (. For regular solutons, the extra contrbuton from the parabolc ε makes the γ-c sotherm non-lnear (sold squares n Fg. 3.4) even n the absence of segregaton. For pure system, gran boundary energy usually decreases as temperature ncreases. For mpure systems or alloys, however, segregaton reduces gran boundary energy and as a consequence the gran boundary energy may ncrease as temperature ncreases at fnte temperatures. More nterestngly, the γ-c curve becomes sngular for regular solutons when the temperature s below the segregaton transton temperature. An obvous horn structure s observed below the gran boundary crtcal temperature on the solvent rch sde, whch ndcates the segregaton transton. It s mportant to be aware of the segregaton transton when one analyzes expermental data on γ-c relatons. On the other hand, careful examnatons are necessary n experments to reveal the sngularty n the γ-c plot because segregaton transton occurs n a very narrow temperature or composton range. 4

61 (γ-γa)/nvd E D A B C D E c c α (1) c α () c β (1) c β () Fgure 3.4. Relatonshp between gran boundary energy and bulk concentraton under varous condtons: (A) deal solute at nfnte temperature; (B) regular soluton at nfnte temperature; (C) regular soluton at a T>T c (bulk); (D) regular soluton at T<T c but T> T gb c ; (E) regular soluton at T< T gb c. For curve E, open symbols represent boundary energy obtaned wth decreasng bulk concentraton whle sold symbols represent values obtaned wth ncreasng bulk concentraton. The nset llustrates the horn structure for curve E. c α and cβ correspond to the mscblty gap at two dfferent temperatures. 43

62 3.4.. Movng Gran Boundary Frst, we nvestgate the effect of solute drag at temperatures above the segregaton transton temperature. The relatonshps between the drag forces, the relatve Gbbs excess quantty of solutes and the boundary velocty are shown n Fg.3.5. It can be seen that the drag force ncreases wth ncreasng boundary velocty at small veloctes and t decreases wth ncreasng velocty at large veloctes. The segregaton at the gran boundary decreases gradually wth ncreasng velocty. Qualtatvely, these predctons are very smlar to those predcted by Cahn s deal soluton model and the M-S and H-S models. Quanttatvely, however, the H-S model predcts a hgher segregaton as well as a hgher drag force at both small and hgh veloctes whle M-S model predcts hgher values at small veloctes and lower values at hgh veloctes (as shown n Fg. 3.6a). The comparson of solute concentraton profles of M-S and current model under small and hgh veloctes s shown n Fg. 3.6b. 44

63 P/kTNV vd 0 /D (a) ΓB/NVd vd 0 /D (b) Fgure 3.5. Varaton of drag force (a) and solute excess (b) wth gran boundary velocty at a temperature T = 680K, whch s above the segregaton transton temperature (T 0 ). The bulk composton s c =

64 P/kTNV H-S model M-S model Current model vd 0 /D (a) c v1=0.001 v=0.0 M-S model current model M-S model current model x/d 0 Fgure 3.6. (a) Comparson of the drag forces predcted by the current model wth those obtaned from the M-S and H-S models at T=00K whch s above the transton temperature. (b) Steady state concentraton profles obtaned from the current and M-S models at two dfferent gran boundary veloctes. The bulk composton s c = 0.1. (b) 46

65 Now we consder a boundary wth an equlbrum solute concentraton establshed at a temperature below the transton temperature where a phase transton has occurred, leadng to an equlbrum solute concentraton that s much hgher than that establshed at temperatures above the transton temperature. Assumng that the boundary s now movng at a constant velocty at ths temperature, we evaluate the steady-state concentraton profles and the drag force. Fgure 3.7 shows the drag force and the relatve Gbbs excess quantty of solute atoms of the boundary as functons of boundary velocty. An nterestng observaton n these smulatons s that there exst two breakaway transtons. Smlar to the prevous case, the drag force ncreases and solute concentraton decreases wth ncreasng boundary velocty at small veloctes, but the drag force s much hgher because of the hgh solute concentraton at the gran boundary. When the velocty reaches a crtcal value, a sharp transton from hgh- to low-segregaton takes place, whch leads to a smlar transton for the drag force as well. Therefore, when the temperature s below the transton temperature, the system follows two dfferent paths for segregaton and drag force before and after the transton. For example, t follows the hgh branches when the velocty s low and swtches to the low branches when the velocty s hgh. The low branches n the two plots are very smlar to those predcted n the prevous case when the temperature s above the transton temperature. If we start to decrease the boundary velocty after the transton, the nverse transton occurs at a lower crtcal velocty, leadng to a hysteress loop n both plots (Fg. 3.7). Therefore, as far as the gran boundary phase transton s concerned, changng velocty has a smlar effect as changng temperature. The frst breakaway s obvously assocated wth the gran 47

66 boundary phase transton. It has a much stronger effect on the drag force and solute segregaton at gran boundares and hence should have a much stronger mpact on boundary mgraton. 48

67 P/kTNV vd 0 /D (a) ΓB/NVd vd 0 /D 49 (b) Fgure 3.7. Varaton of drag force (a) and solute excess (b) wth gran boundary velocty at T = 660K whch s below the segregaton transton temperature but above the bulk mscblty gap. The bulk composton s c = Sold crcles represent values obtaned wth ncreasng velocty whle open trangles represent values obtaned wth decreasng velocty. The nset n (a) shows the low branch of the drag force-velocty plot.

68 Moblty Transton In the study of temperature dependence of gran boundary velocty [37], t was shown that the steady state mgraton velocty of tlt boundary n alumnum under a constant small drvng force mght change abruptly wth temperature. Most nterestngly, a hysteress loop was observed durng the heatng coolng cycle when the bulk concentraton s below a crtcal value. In the lterature, ths phenomenon was explaned as a consequence of the separaton of solute atoms from the gran boundary, but ths mechansm faled to explan the hysteress and the relatonshp between the hysteress and the bulk concentraton. Below, we apply the segregaton model developed n ths study to analyze ths phenomenon. The actvaton energy of solute atom dffuson s assumed to be 1.0 ev and all the other parameters are kept the same as prevous calculatons. The resultant plots of gran boundary moblty versus temperature for two alloys of dfferent bulk composton are shown n Fg As can be readly seen, the model predcts successfully both a hysteress and a crtcal alloy composton above whch the hysteress dsappears. The result ndcates that a phase transton underles the moblty hysteress. It s nterestng to note that the hgh- and low-temperature branches have dfferent slopes, ndcatng dfferent apparent actvaton enthalpes for boundary mgraton. Ths s because the actvaton enthalpy contans the contrbuton from the segregaton energy, whch has a larger value at lower temperatures. 50

69 0 - log(m/m1000) c c /T Fgure 3.8. Predcton of gran boundary moblty transton as a functon of temperature for dfferent solute contents: c 1 = , c = The gran boundary moblty s normalzed by the value at 1000K. The hysteress loop dsappears at hgher solute concentraton. Sold symbols are for the coolng process whle open symbols are for the heatng process Dscusson Gradent thermodynamcs and ts mcroscopc counterpart have been used extensvely to analyze segregaton transton at surfaces and nterfaces [49-51, 55]. In ths study, we employ a smlar approach to study gran boundares. In the framework of regular 51

70 soluton approxmaton, the model predcts several nterestng phenomena, ncludng the hysteress for gran boundary segregaton transton wth respect to both temperature and velocty and two breakaways of solute atoms from a movng gran boundary. Smlar behavor s observed for mpurty segregaton at dslocatons and the results wll be presented n next chapter. In all the numercal calculatons, the atomc volume s assumed to be constant. Such an assumpton may not be vald, n partcular, for a random large angle gran boundary. However, t should not alter the qualtatve nature of the results. In contrast to prevous gran boundary segregaton models, three physcally dstnctve terms are ntroduced n the segregaton energy n ths study. They are the d c contrbuton from the concentraton gradent, κ, the contrbuton from spatal dx varaton of the gradent-energy coeffcent, dκ dc, and the contrbuton from dx dx concentraton dependence of the solute gran boundary nteractons, ϕ(c, x). Addton d c of the κ term affects sgnfcantly the extent of the segregaton, reducng the dx ampltude and ncreasng the wdth of the concentraton peak across the gran boundary. Ths s not surprsng because the gradent-energy term n the free energy s non-local and always suppresses any concentraton nhomogenety. As a result, t lowers the transton temperature and crtcal temperature and reduces the drag force. The contrbuton from the dκ dc term s postve, whch reduces the solute segregaton. The addton of dx dx ϕ(c,x) has no effect on the transton temperatures but lowers sgnfcantly the crtcal 5

71 temperature. Physcally, a system wth ϕ(c, x) only s equvalent to a system wth Ω as a varable. Followng the analyss of Wynblatt and Lu [51], t s easy to obtan the transton temperature and crtcal temperature of the th layer: T 0 = E Ω k ln ( c 1 c ) (1 c ) T c Ω = (3.4) k Where Ω and Ω are the regular solute constant at th layer and bulk, respectvely. These equatons clearly show that the transton temperature, T 0, s not senstve to the structure change but the crtcal temperature, T c, does. In hs study of crtcal pont wettng, Cahn [5] analyzed the phase transton behavor at a free surface of a bnary alloy usng gradent thermodynamcs and varatonal analyss. In the gran boundary segregaton model developed n ths study, f we assume the coordnaton defct, ( z z ), s localzed to a sngle atomc layer at x=0, we can derve a smlar analyss. Comparson between our dscrete model and Cahn s analyss was made at dfferent temperature. It was found that our dscrete model predcts quanttatvely dentcal results as Cahn s when the temperature s close to the bulk crtcal temperature, where the gradent of the concentraton profle s small. At temperatures far below the crtcal pont, the contnuum varatonal analyss becomes nvald and fals to predct the phase transtons that are predcted by the dscrete model. Note that the current model on segregaton transton of a statc boundary s smlar to the surface segregaton transton models of Wynblatt and coworkers [49, 51]. The major 53

72 dfference s the number of layers consdered n the formulaton of the structure part (e.g,. z(x)) of the segregaton energy. Furthermore, n current study we have derved the connecton between the contnuous and dscrete models and appled the models to movng boundares for the drag effect and gran boundary moblty transton. It s nterestng to note that the current model predcts a lower segregaton and a smaller drag force at low velocty but a hgher segregaton and greater drag force at hgh velocty as compared to the M-S model. Ths s because the three extra terms ntroduced n the current model have dfferent dependence on velocty and ther nterplay determnes d c the overall behavor of the system. Accordng to the above analyss, both κ and dx dκ dc reduce the segregaton. The contrbuton from ϕ(c, x) depends on the solute dx dx concentraton: t enhances the segregaton when solute concentraton at the gran boundary, c gb < 0.5 and reduces t when c gb > 0.5. When the velocty of the gran d c boundary s small, the contrbuton from κ and dx dκ dc domnates. However, the dx dx absolute values of these two terms decrease monotoncally wth decreasng solute concentraton n the gran boundary regon but that of ϕ(c, x) ncreases. At hgh enough velocty, ϕ(c, x) becomes domnant, resultng n a stronger segregaton and greater drag force. Note that only a sngle flat gran boundary s consdered n the current chapter. In polycrystallne materals, mgraton velocty of gran boundares wll depend on local curvatures. The segregaton energy as well as gran boundary energy and moblty could 54

73 be functons of msorentaton and nclnaton of gran boundares, as well as mcrostructure. Usng the segregaton model developed n ths study, one could study drectly the effects of segregaton and phase transton at arbtrary gran boundares on gran growth knetcs and mcrostructural evoluton usng the phase feld method [56]. Correspondng work s underway Summary A self-consstent contnuum model of gran boundary segregaton and segregaton transton s developed based on gradent thermodynamcs and ts relatons to the dscrete lattce model s derved. The model allows for an dentfcaton of several new physcally dstnctve terms that have been gnored n prevous solute-drag models ncludng concentraton gradent, spatal varaton of the gradent-energy coeffcent and concentraton dependence of solute gran boundary nteractons. The applcaton of the model to a prototype planar gran boundary n a regular soluton under varous condtons provdes consderable nsght nto the contrbutons of these terms to the total segregaton free energy, the equlbrum segregaton profle, the segregaton transton and the correspondng drag force on gran boundary mgraton. Consderaton of the contrbuton from the gradent-energy s crtcal n descrbng the behavor of solute segregaton and drag force at gran boundares. For example, ts omsson could result n a sgnfcant overestmate or underestmate (dependng on the boundary velocty) of the enhancement of solute segregaton and the correspondng drag 55

74 force, and lead to much hgher segregaton transton temperature. The concentraton dependence of the nteracton energy between solute and gran boundary has lttle effect on the segregaton transton temperature but lowers sgnfcantly the temperature and alloy composton of the crtcal pont. The segregaton transton (transton from low- to hgh-segregaton) takes place wth changng temperature, bulk composton or gran boundary velocty. The transton s frst-order wth a hysteress. The segregaton transton wth respect to temperature s responsble for the observed sharp transton of gran boundary moblty wth temperature and ts dependence on bulk composton observed expermentally. Recent advances n computer smulatons have made t possble to study the evoluton of populatons of boundares n complex mcrostructures under varous condtons. The model of gran boundary segregaton developed n ths study can be adopted easly nto phase feld modelng of nterface mgraton. Ths wll enable the study of solute segregaton n a populaton of ncoherent boundares and characterzaton of possble effects on the knetcs of mcrostructural evoluton. 56

75 CHAPTER 4 SEGREGATION AND WETTING TRANSITION AT DISLOCATIONS 4.1. Introducton Segregaton and clusterng of solutes at extended crystallne defects such as surfaces, gran boundares, hetero-phase nterfaces and dslocatons play a key role n mcrostructural evoluton and plastc deformaton of alloys durng ther thermal and mechancal processng (for revews see [57, 58]). The chemcal equlbrum of a system of extended defects s characterzed by gradent thermodynamcs [1-3, 59-6] and ts dscrete counterpart (dscrete lattce model) [63-65]. Professor Hllert has poneered the use of regular soluton model n non-unform systems [64, 65], whch was later on appled extensvely to treat solute segregaton at surface and nterfaces [66-7]. In ths artcle we employ the contnuum model developed n [73] for gran boundary segregaton based on gradent thermodynamcs and the phase feld model of dslocatons developed n [74-76] to nvestgate solute segregaton and wettng transton at dslocatons and the correspondng drag force on dslocaton glde. 57

76 The problem of solute segregaton at dslocatons was frst analyzed by Cottrell [77]. Later the treatment was refned by a number of authors [58, 78-84]. It was found that the elastc nteractons between solute atoms and dslocatons cause the solute atoms to segregate at the dslocatons, formng solute-rch and solute-lean regons called the Cottrell atmosphere. The solute atoms may ether move together wth the dslocaton mposng a drag force or break away from t, dependng on the nterplay between the characterstc tme scales of solute atom dffuson and dslocaton moton. At relatvely hgh temperatures and low stran rates, the solute atoms may repeatedly segregate at and break away from the dslocatons, resultng n a repeated yeldng. Such an effect s beleved to be responsble for the serratons n the stress-stran curve and s often referred to as the Portevn-LeChateler (PLC) effect [85]. Most of these studes have focused prmarly on the long-range elastc solutedslocaton nteractons and gnored the short-range chemcal (or nelastc) nteractons by consderng deal or dlute solutons. If there s a strong nteracton between the solute atoms and the extended defects, solute-rch atmospheres around the defects cannot be treated as dlute soluton anymore rrespectve of how dlute the bulk concentraton s, and the solute-solute nteracton has to be consdered. On the other hand, f the level of segregaton s hgh, large concentraton gradents exst near the defects and ther contrbutons to the chemcal potental (ncludng coherency stran energy) must be ncluded n the condton for equlbrum. In a recent study on mpurty segregaton at gran boundares [73] t has been shown that the consderaton of solute-solute nteractons results n a frst-order segregaton transton at gran boundares, leadng to a 58

77 sharp transton of gran boundary moblty as a functon of temperature. Smlar effects are expected for mpurty segregaton at other defects and have been demonstrated for solute-dslocaton nteractons [86]. Although the early models are convenent for llustratng the basc characterstcs of solute segregaton at dslocatons and ther effect on dslocaton glde, more robust soluton thermodynamcs and solute-dslocaton core nteracton models are needed for capturng more sophstcated phenomena observed n experments. In ths study we consder contrbutons to the chemcal equlbrum from both concentraton gradent and structural dscontnuty. In contrast to most prevous studes, our model ncludes several dstnct terms accountng for the short-range chemcal nteractons, concentraton gradent, coherence elastc stran, and spatal varaton of both the gradent-energy coeffcent and the enthalpy of mxng near a dslocaton. The model s appled to study n detal (a) segregaton profles at both statonary and steady-state gldng straght dslocatons, (b) drag forces at the gldng dslocatons, and (c) wettng transtons wth respect to solute concentraton, temperature and dslocaton velocty. Ths study could serve as a prelmnary step n applyng the wdely used phase feld method to study mpurty segregaton at dslocatons of complcated confguratons. Usng gradent thermodynamcs, the phase feld method can, n prncple, descrbe mcrostructural evoluton n non-unform systems consstng of many types of defects, such as homo- and hetero-phase nterfaces, antphase doman boundares, ferromagnetc and ferroelectrc domans walls, dslocatons, mcro-cracks, to name a few. Snce analytcal treatments were lmted to several smple confguratons, there s an ncreasng nterest recently n applyng computer smulaton technques to study varous phenomena 59

78 that nvolve solute-dslocaton nteractons ncludng clusterng of solutes around dslocatons [86-89], dslocaton mgraton n the presence of solutes [86, 88, 90, 91], stress asssted nucleaton n the vcnty of dslocatons [9-96] and collectve dslocatonmpurty nteractons, e.g., dynamc stran-agng [97, 98]. The recent advances n the phase feld approach to dslocaton plastcty [74-76, 89, 91] provde an opportunty for smulatng the effects of solute-dslocaton nteracton on the evoluton of populaton of dslocatons of arbtrary confguratons such as dslocaton substructures. 4.. Theoretcal Formulaton Segregaton model As mentoned earler, the study of solute segregaton at extended defects requres the consderaton of contrbutons to the chemcal potental from both structural and chemcal non-unformtes. For segregaton at dslocatons consdered n ths chapter we follow the segregaton model developed recently for gran boundary [73] to account for the solutedslocaton and solute-solute nteractons. In ths approach, the chemcal potental of solute n a chemcally homogeneous but structurally non-unform (ncoherent or defectve) system s expressed as: ( ) = 0 ( ) + k Tln c+ xs (, c) µ r µ r µ r (4.1) B B B B where r s an arbtrary pont n the system, k B s Boltzmann s constant, T s temperature, µ 0 B s the standard-state value (.e., a state of pure B n the presence of structural defects 60

79 at a pressure of 1 atm) and µ xs B s the part n excess of the contrbuton from confguratonal entropy of mxng. If the structural non-unformty s assocated wth dslocatons, the varaton of the standard-state value wth poston can be expressed as: µ ( r) = µ ( ) + U ( r) + U ( r ) (4.) 0 0 ch el B B B B ch el where represents a pont far away from the dslocatons, U () r and U () r are, respectvely, the short-range chemcal and long-range elastc nteracton potentals ch between solute atoms and dslocatons. U () r s assocated wth the varaton n coordnaton number of atoms at the dslocaton core and s therefore confned wthn the core regon. It may translate wth the dslocatons but s not otherwse altered n shape or el ampltude. U () r s assocated wth lattce dstorton caused by the dslocatons and B therefore depends on the dslocaton confguraton. Correspondngly, the excess chemcal potental s expressed as B xs xs µ ( r, c) = µ (, c) + ϕ ( r, c) (4.3) B B B where ϕ B ( r, c ) takes nto account the poston dependence of the enthalpy of mxng near a dslocaton because of the varaton n coordnaton number. The dfference n solute chemcal potentals near a dslocaton and far away from t can then be expressed as: ( ) ch el xs xs µ µ = U () r + U () r + ϕ (,) r c + ktln cc + [ µ (,) c µ (, c)] (4.4) B B B B B B B B For a chemcally non-unform system, the exchange potental ncludes contrbutons from concentraton gradent and the assocated coherency elastc stran. The equlbrum condton for both structurally and chemcally non-unform systems thus becomes B B 61

80 1 el µ ( r) µ ( r, c) κ c ( NVκ) c β ( r, c) µ NV = + = (4.5) where µ = µ B µ A and µ = µ B µ A, κ s the gradent-energy coeffcent, N v s the number of atomc stes per unt volume, and β el ( r, c) accounts for the contrbuton from the coherency stran energy assocated wth concentraton non-unformty. The coherency stran energy s measured from a reference state of dentcal concentraton nonunformty, but wth each volume element of unform composton separated nto stressfree portons. Dfferent from a structurally unform system, both κ and N v are functon of poston because of the presence of dslocatons. The contrbutons from the elastc nteractons between solute atoms and dslocatons, and from the coherency stran energy due to concentraton non-unformty can be represented by U el B 1 δe r A, = (4.6) N δc el el () U () r + β ( r c) where E el s the total elastc stran energy of a system of both concentraton and structural non-unformtes. Followng the mcroelastcty theory of Khachaturyan [99] one can wrte a close form for the stran energy of a system consstng of arbtrary compostonal and structural non-unformtes characterzed by the phase felds, φ p (r), V el E el = 1 p, q dg (π ) 3 B pq ~ ~ ( e) φ ( g) φ ( g) p q (4.7) where ~ denotes Fourer transform, * ndcates complex conjugate, g s an arbtrary pont n the recprocal space, and e = g/g. 6

81 pq jkl 0 j 0 kl 0 j B ( e) C ε ( p) ε ( q) e σ ( p) Ω ( e) σ ( q) e (4.8) jk 0 kl l 0 where C jkl are the elastc constants, ( p) 0 0 nhomogenety, ( p) C ε ( p) 1 σ, and ( ) j jkl kl ε s the stress-free stran of type p j Ω e = C e e s the nverse of the Green s jk jkl l functon n the recprocal space. n Eq. (4.7) represents the prncple value of the ntegral that excludes a small volume n the recprocal space, (π) 3 /V, at g = 0 (V s the total volume of the system). Detaled examples can be found n [89]. Equaton (4.7) has been appled successfully to the studes of dslocatonmpurty, dslocaton-precptate and dslocaton-dslocaton nteractons [74-76, 86, 89, 91, 94, 95, 100, 101]. When both p and q ndcate dslocatons, Eq. (4.7) characterzes the self- and nteracton-energy of dslocatons, wth p and q ndexng ndvdual slp systems consstng of slp plane α and slp drecton m α on that plane, e.g., φ () r ( m,r) (,r) p = η α, α. η α, m α s a non-conserved feld that characterzes the spatal dstrbuton of an arbtrary confguraton of dslocatons of slp system (α, m α ) [74-76]. Correspondngly ( p) = b (, m ) n ( ) + b (, m ) n ( ) ε α α α α d (4.9) 0 1 j α j j α where n s the slp plane normal, b s the Burgers vector, and d s the nterplanar spacng of the slp planes. When both p and q represent concentraton nhomogenety, Eq. (4.7) 0 descrbes the coherency stran energy, wth φ ( r) = ( r ) and ( p) p c c ε beng the stress-free stran assocated wth the p-type concentraton non-unformty measured from a reference state of unform concentraton of c. Fnally when p and q ndcate, respectvely, the concentraton non-unformty and dslocaton, Eq. (4.7) characterzes the j 63

82 dslocaton-solute nteractons. Dfferent formulatons of the stress-free stran for dslocaton were employed n [89, 91] n Eq. (4.7) to study solute segregaton. Although a regular soluton model was used n these studes, the short-range solute-dslocaton core nteracton and spatal varaton of the gradent-energy coeffcent and the enthalpy of mxng near a dslocaton were gnored. Substtutng Eqs. (4.1) (4.4) and (4.6) nto Eq. (4.5) yelds the equlbrum solute concentraton at an arbtrary confguraton of dslocatons: seg c c F = exp 1 c 1 c kbt (4.10) where the free energy of segregaton s gven by seg xs xs ch 1 δ Eel 1 F = µ (, c) µ (, c ) + U ( r) + ϕ( c, r) + κ c ( NVκ) c N δ c N V V (4.11) and U ch =U B ch U A ch and ϕ = ϕ B ϕ A. The excess dslocaton lne energy (the lne energy dfference between dslocatons n a system of arbtrary concentraton non-unformty and a system of unform concentraton of c ) s gven by: κ ( ) ( 1 )( ) ( ) E = NV c µ B µ B + c µ A µ A + c + Eel da (4.1) 4... Dffuson and mgraton For a movng dslocaton, the flux of solutes n a reference frame movng wth the dslocaton at a velocty, v, s gven by 64

83 DA D B J = NVc( 1 c) ( 1 c) + c ( µ B µ A) NVc kt B kt v (4.13) B where D A and D B are the self dffusvtes of solvent and solute atoms, respectvely, whch are assumed to be equal n ths study,.e., D A = D B = D. The correspondng solute concentraton profle around a movng dslocaton s then gven by N v c t = dvj (4.14) The steady-state drag force by solute atoms on dslocaton gldng along ξ coordnate can be determned by the energy dsspaton method of Hllert and Sundman [68]: P s ( µ µ ) d = J dξ (4.15) 1 B A ξ v dξ 4.3. Model System The above analyss of mpurty segregaton at dslocatons s qute general, ndependent of any assumptons about a partcular dslocaton confguraton, sold soluton model, atom-dslocaton nteracton potental or lattce dstorton caused by the atomc sze msmatch between the solute and solvent atoms (e.g., dlatatonal or non-dlatatonal). It s applcable to elastcally ansotropc systems wth concentraton-ndependent elastc constants. To apply the analyss to a gven system, however, one needs to specfy a dslocaton confguraton, a soluton model for the chemcal and elastc nteractons, and a dslocaton core structure. For smplcty, we consder an nfntely long straght edge dslocaton (along the z drecton n a Cartesan coordnate system) n a cubc bnary 65

84 substtuton alloy of an f.c.c. structure. We assume that the chemcal free energy of the alloy can be approxmated by a regular soluton model wth nearest neghbor nteractons. Then accordng to [73], xs ( c) µ (, c ) = Z ω( c c) xs µ, (4.16) ϕ ( c, x, y) = Z( x, y) ω(1 c) (4.17) U ch ω BB ω AA ( x, y) = Z( x, y) (4.18) κ(x, y) = ωd 0 Z(x,y) (4.19) where xs xs B µ = µ µ xs A, Z s the coordnaton number of atoms n a perfect crystal, ( ) ω = ω ω + ω s the mxng energy, ω j s the bondng energy between atoms AB AA BB and j, Z ( x, y) = Z( x, y) Z, Z(x, y) s the effectve coordnaton number of atoms n the dslocaton core regon whch vares n the (x, y) plan only, and d 0 s the nter-planar dstance of the {111} planes. Correspondngly the segregaton energy becomes wth Ω= Z ω. seg ωbb ωaa F = Ω( c c) + Z( x, y) + Z( x, y) ω(1 c) 1 δeel ωd + ωd Z( x, y) c ( NV Z( x, y) ) c (4.0) N δc N V 0 0 V The solute-dslocaton nteractons wthn the core consst of two parts: the chemcal (or nelastc) nteracton assocated wth mssng bonds and the elastc nteracton assocated wth lattce dstorton. The elastc nteracton s descrbed by the phase feld mcroelastcty model of nteractons between structural and concentraton 66

85 non-unformtes descrbed earler. The structural non-unformty assocated wth a dslocaton s descrbed by the equlbrum profle of (,r) η α, m α obtaned from the phase feld model of dslocatons [74-76] n a chemcally unform system (Fg. 4.1). It s determned by the nterplay between the elastc energy and the crystallne energy. The dslocaton core structure s characterzed by the gradent of (,r) η α, m α, whch corresponds to the Burgers vector dstrbuton n the Peerls-Nabarro model of dslocatons [58, 10, 103]. It can be seen clearly from Fg. 4.1 that smlar to the Peerls-Nabarro model, the phase feld model yelds a dffuse dslocaton core η Ζ Ζ(x,y) dη/dx x/d 0 Fgure The equlbrum profles of η, η, and Z -Z(x, y) across the dslocaton core employed n the calculatons. The equlbrum profle of η s obtaned by the phase feld model of dslocatons. 67

86 To approxmate the chemcal nteracton, we have defned an effectve coordnaton profle, Z(x,y), across the dslocaton core n the above equatons. Wthout losng generalty, we assume a Gaussan functon for the profle r Z( x, y) = Z( r) = Z Zm exp w (4.1) where r = x + y, w s the varance of the Gaussan functon, and Z m s the average number of mssng bonds per atom along the center of the core. Choosng w= 1.0d0, the effectve coordnaton profle across the dslocaton core matches approxmately the wdth of x η profle, as shown n Fg The value of Z m n Eq. (4.1) s estmated accordng to the core energy dfference between dslocatons n pure solvent and n pure solute. The current choce of Z m = 0.5 gves 0.1 ev/atom for ths energy dfference. Fgure. 4.. Schematc drawng of the computatonal cell. 68

87 In the elastc energy calculaton, we further assume that the lattce parameter of the soluton as a functon of concentraton follows the Vegard law,.e., ε j 00 = 1 a da dc δ j, where a s the lattce parameter of the host lattce at a stress-free state and δ j s the Kronecker delta. Although the elastc energy (Eqs. (4.7) and (4.8)) s formulated for elastcally ansotropc systems, the choce of the elastc constants: c 11 = 16.84, c 1 = 1.14, and c 44 = Pa n ths study yelds an sotropc elastc meda wth the shear modulus and Posson s rato gven by the Vogt average [58]: µ = Pa and ν = The effect of elastc ansotropy on segregaton wll be nvestgated n future c y (Å) Fgure 4.3. Comparson between solute concentraton profles along a vertcal lne through the center of the dslocaton core obtaned usng and system szes. 69

88 The remanng materal constants used n the calculatons are a A = nm, a B = nm, d0 = a A 3, Z = 1, ω = 49.7 mev, and ωbb ωaa = 0.09 ev. The two-dmensonal computatonal cell (Fg. 4.) conssts of a par of nfntely long straght edge dslocatons (along the z dmenson) of opposte sgns, one at the center and the other at the boundary. The reason for the choce of such a confguraton s for the convenence of carryng out the smulatons usng a perodc boundary condton. Wth the perodc boundary condton appled along both the x and y drectons, the system contans a perodc array of dslocatons rather than a sngle dslocaton n an nfnte crystal. The system sze employed n the smulatons s Increasng the system sze to does not produce any sgnfcant dfference n the segregaton profles as shown n Fg. 4.3, whch ndcates that the effect from the dslocaton-dslocaton nteracton on segregaton can be neglected Results and Dscusson For a system wth postve mxng energy (ω > 0), phase separaton s expected when alloy composton and temperature are wthn the mscblty gap. If extended crystallne defects exst, however, wettng transton of solutes at these defects have been predcted [57, 71, 7, 104, 105] when the bulk alloy composton s outsde the mscblty gap. Below we explore ths segregaton transton together wth the conventonal segregaton at dslocatons usng the approach and model system dscussed above. 70

89 4.41 Statonary dslocaton We start wth a system of unform concentraton c and perform a seres of calculatons upon coolng and heatng at temperatures n the sngle-phase regon of the bulk phase dagram. The equlbrum solute concentraton around the dslocatons at each temperature s obtaned by solvng Eq. (4.10) usng the conventonal teraton method, wth the equlbrum concentraton profles obtaned at prevous temperatures servng as the ntal condtons of new calculatons. The smulatons are termnated when the concentraton profle converges to a statonary value. Fgure 4.4 shows the contours of the equlbrum solute concentraton profles around the dslocaton calculated at two temperatures (100K and 1170K) above the bulk mscblty gap for an alloy of c =0.04. The one-dmensonal (1D) concentraton profles along a vertcal lne across the center of the dslocaton core are shown n Fg Clearly heavy segregaton s confned wthn the core regon at both temperatures. At 100K (Fgs. 4(a) and 5(a)), the peak solute concentraton reaches that s more than 10 tmes of the bulk solute concentraton. It s, however, stll far from saturaton (unty). These results are qualtatvely smlar to the predctons by the Monte Carlo smulatons [90] and the phase feld smulatons [91, 95]. When the temperature s lowered by just 30K (stll above the bulk mscblty gap n the sngle-phase regon) the solute concentraton at the core regon jumps to 0.98 (close to saturaton) (Fgs. 4.4(b) and 4.5(b)). At the same tme the segregaton peak becomes much broader and the concentraton profle resembles a new phase partcle. Ths ndcates that a wettng transton has taken place. 71

90 y (Å) y ( (a) Å) x (Å) (a) (b) Fgure Contour plots of solute concentraton around the dslocaton at (a) 100K (above the segregaton transton T 0 ) and (b) 1170K (below T 0 ) wth bulk composton c = x (Å)

91 c y (Å) (a) c y (Å) (b) Fgure 4.5. Solute concentraton profle along a vertcal lne through the center of the dslocaton core at (a) 100K and (b) 1170K. The dashed lnes n the plots ndcate the center of the dslocaton core. 73

92 Coolng Heatng Equlbrum ΓB/NVd T(K) (a) E/NVd Coolng Heatng Equlbrum T(K) Fgure Varatons of (a) relatve Gbbs excess of solute at the dslocaton and (b) excess dslocaton lne energy as functons of temperature durng coolng (sold crcles) and heatng (open trangles) n a system of bulk composton c =0.04. The thck vertcal lne n (a) ndcates the equlbrum segregaton transton temperature. 74 (b)

93 To further examne the wettng transton behavor, we calculate the relatve Gbbs excess of solute, Γ Β, and the excess dslocaton lne energy as functons of temperature upon heatng and coolng for the alloy. The relatve solute excess s calculated accordng to Nd V 0 Γ B = c (, j) c 1 c (4.), j and the excess dslocaton lne energy s calculated usng Eq. (4.1). The results are shown n Fg There are two remarkable features about the varaton of the solute excess wth temperature at the dslocaton core. Frst, n contrast to the conventonal predcton that contnuous changes n temperature lead to contnuous changes n the amount of segregatng solutes, rather abrupt changes n the solute excess,.e., wettng non-wettng transtons, are observed at certan temperatures. Second, the transton occurs at dfferent temperatures upon heatng and coolng, leadng to a hysteress loop. Ths ndcates that the wettng transton at dslocaton s a frst-order phase transton. The actual transton temperature (T 0 ) s ndcated by the vertcal thck lne wthn the hysteress loop, whch s determned by the crtera that the lne energy of the dslocaton of low segregaton equals to that of hgh segregaton at the transton temperature (Fg. 4.6(b)) [73]. The transton temperatures for alloys of dfferent compostons are plotted n Fg. 4.7(a) along wth the bulk mscblty gap. For comparson, the transton temperatures and bulk mscblty gap calculated for an ncoherent system (wthout consderng the coherency stran energy n the model) are also presented n Fg. 4.7(a). It can be seen that 75

94 the coherency stran energy suppresses both the bulk mscblty gap and the wettng transton temperatures. However, n both cases the wettng transton occurs at temperatures sgnfcantly hgher than the bulk mscblty gap. For the coherent system, the wettng transton ends at c ~ where the sharp transton between the hgh- and low-segregaton as well as the hysteress loop dsappear, whle the transton ends at a hgher bulk concentraton n the ncoherent system. A more careful comparson s made n Fg. 4.7(b) by plottng the normalzed mscblty gaps and wettng transton curves (by the bulk crtcal temperatures) n the same dagram. The normalzed mscblty gaps become dentcal for the coherent and ncoherent systems. It becomes clear that the coherent wettng transton has a wder temperature range between the bulk mscblty gap and the wettng transton curve at the same bulk concentraton c as compared to the ncoherent wettng transton. Qualtatvely, ths s not dffcult to understand because ncorporaton of the coherency stran energy s equvalent to reducng the regular soluton constant, Ω, whch suppresses the bulk mscblty gap as well as the segregaton transton temperature. However, n addton to those terms that contan ω n the segregaton energy (Eq. (4.0)), there are other terms that are ndependent of ω. Therefore, t s antcpated that the change of the regular soluton constant may not affect the bulk mscblty gap and the segregaton transton to the same extent. 76

95 3000 T(K) Incoherent bulk mscblty Coherent bulk mscblty 500 Coherent segregaton transton Incoherent segregaton transton c (a) 0.8 T * Bulk mscblty Coherent segregaton transton Incoherent segregaton transton Fgure 4.7. (a) Stablty dagrams for segregaton transton at dslocaton (sold crcles and trangles) and bulk mscblty gaps (sold lnes) for coherent and ncoherent systems. (b) Normalzed stablty dagrams and bulk mscblty gaps. The normalzaton factors are respectvely the crtcal temperatures for the bulk mscblty gaps of the coherent and ncoherent systems. After normalzaton, the two bulk mscblty gaps become dentcal (sold lne n (b)). 77 c (b)

96 4.4. Movng dslocaton In contrast to the statonary dslocaton, solute concentraton around a movng dslocaton n the steady-state regme s determned by solvng the contnuty equaton, Eq. (4.14) usng the same dslocaton confguraton as shown n Fg. 4.. In order to be selfconsstent wth the calculatons carred out for the statonary dslocaton usng the teraton method, where the change of the solute concentraton profle at the dslocaton does not affect the bulk concentraton c, the system s attached to a reservor of fxed bulk concentraton, c. When the velocty s set to zero, the smulaton results converge to those obtaned usng the teraton method. The smulatons are started wth an equlbrum solute concentraton above the transton temperature determned for a statonary dslocaton. We then assume that the dslocaton moves at a constant velocty at ths temperature and calculate the concentraton profle around t. After a system has reached a steady state, the solute concentraton profle s used as the ntal condton of a new calculaton wth a dfferent velocty. Fgure 8 shows the contour plots of the solute concentraton at the dslocaton obtaned at dfferent veloctes. The solute atmosphere becomes asymmetrcal, wth a greater porton tralng behnd the movng dslocaton. As the velocty ncreases, the sze and the peak concentraton of the atmosphere decreases and the degree of asymmetry ncreases, e.g., the atmosphere elongates n the opposte drecton of the dslocaton moton and contracts along the drecton that s perpendcular to the movng drecton. 78

97 y(å) x (Å) x (Å) y(å) y(å) y(å) x (Å) x (Å) a c b d Fg. 8. Contour plots of solute concentraton around a movng dslocaton at 100K (above the segregaton transton temperature T 0 ) under varous veloctes: (a) vd 0 /D=0 10-3, (b) vd 0 /D=1 10-3, (c) vd 0 /D= 10-3, and (d) vd 0 /D= The bulk composton s c =

98 ΓB/NVd vd 0 /D(10-3 ) (a) P /kbtnv vd 0 /D(10-3 ) (b) Fgure 4.9. Varaton of (a) Gbbs excess of solute and (b) drag force wth velocty for a movng dslocaton at 100K (above the segregaton transton temperature). The bulk composton s c =

99 y(å) y(å) y(å) y(å) x (Å) x (Å) x (Å) x (Å) a c b d Fgure Contour plots of solute concentraton around a movng dslocaton at 100K (below the segregaton transton temperature) wth ncreasng veloctes: (a) vd 0 /D=0 10-3, (b) vd 0 /D= , (c) vd 0 /D=1 10-3, and (d) vd 0 /D= The bulk composton s c =

100 Increasng Decreasng ΓB/NVd vd 0 /D(10-3 ) (a).0 P/kBTNV Increasng Decreasng vd 0 /D(10-3 ) (b) Fgure Varaton of (a) Gbbs excess of solute and (b) drag force wth velocty for a movng dslocaton at 100K (below the segregaton transton temperature). The bulk composton s c = Sold crcles represent values obtaned wth ncreasng velocty whle open crcles represent values obtaned wth decreasng velocty. 8

101 The plots of the relatve Gbbs excess of solute and the drag force as a functon of velocty obtaned from the smulatons are shown n Fg It s found that the Gbbs excess of solute around the dslocaton decreases gradually as the dslocaton velocty ncreases (Fg. 4.9(a)). The drag force frst ncreases wth ncreasng dslocaton velocty at small veloctes and then decreases wth ncreasng velocty at large veloctes (Fg. 4.9(b)). In both plots the varatons are smooth, ndcatng a gradual break-away of the solute atmosphere from a movng dslocaton. When the temperature s below the wettng transton temperature but above the bulk mscblty gap, however, abrupt changes n solute concentraton (Fg. 4.10), relatve Gbbs excess, and drag force (Fg. 4.11) occurs at certan crtcal veloctes. Qualtatvely, the shape changes of the solute atmospheres around the movng dslocaton as a functon of velocty shown n Fg are smlar to those obtaned at a temperature above the wettng transton temperate (Fg. 4.8). The major dfference s wthn the core, where the solute concentraton at the center of the core drops sharply (from to 0.353) when the velocty exceeds a crtcal velocty. From Fg one can see clearly that startng from the equlbrum condton at vd 0 /D = 0, the solute excess at the dslocaton decreases and the drag force ncreases gradually when the velocty ncreases. When the velocty reaches a crtcal value (vd 0 /D ), however, a sharp drop n solute excess and, correspondngly, a sharp drop n the drag force are observed. Although one may refer to the drops as solute breakaway, ths phenomenon has a very smlar nature to the wettng non-wettng segregaton transton takng place at the statonary dslocaton wth varyng 83

102 temperature. For example, the fundamental characterstcs of the varaton of the Gbbs excess at the dslocaton as a functon of dslocaton velocty shown n Fg. 4.11(a) s smlar to that observed for the statonary dslocaton as a functon of temperature (Fg. 4.6(a)). In the former, the role of temperature s played by velocty, whch, n addton to makng the concentraton profle asymmetrc (along the glde drecton), tends to reduce solute segregaton at the dslocaton. When the dslocaton velocty s reduced, an nverse transton occurs but at a lower crtcal velocty (vd 0 /D ), leadng to hysteress loops n both the solute excess vs. velocty and the drag force vs. velocty plots. Note that the current calculatons are performed under constant dslocaton veloctes rather than constant appled shear stresses, the hgh- and low-segregaton and hgh- and low-drag force branches are dfferent from the hgh- and low-velocty branches observed under constant stress condton [38]. After the transton, further ncreases n velocty results n further decreases n the maxmum solute concentraton at the dslocaton (Fg. 4.11(b)), and the dslocaton experences a second solute-breakaway that s normally referred to n lterature. The two breakaways observed here are very smlar to that predcted for mpurty segregaton at mgratng gran boundares [73]. In the case of dslocatons, the frst breakaway may correspond to the sharp yeld pont drop and the second breakaway (and re-segregaton) may correspond to the serraton observed on the stress-stran curves of some metal alloys. Note that n the movng dslocaton calculatons we have chosen to change the bulk alloy composton, c, rather than temperature to change the ntal state of the system from below to above the wettng transton. The purpose of dong so s to 84

103 demonstrate that the wettng transton takes place wth varyng temperature, dslocaton velocty, as well as bulk composton. The above results are obtaned for the specfc alloys consdered. Even though the qualtatve characterstcs of the segregaton transton and the segregaton profles are not expected to change, the quanttatve features of the results wll alter when dfferent approxmatons of the sold soluton model, dslocaton core structure and solutedslocaton core nteractons, and dfferent materal parameters are used. Detaled effect of dslocaton core structures as well as the effect of elastc ansotropy and nondlatatonal msft stran between solute and solvent atoms wll be nvestgated separately n a forthcomng work. 5. Summary The chemcal equlbrum n a system of both statonary and gldng dslocatons s analyzed by a contnuum model of segregaton based on gradent thermodynamcs and a phase feld model of dslocatons. In addton to the long-range elastc solute-dslocaton nteractons, the analyses take nto account several dstnct terms accountng for the shortrange chemcal nteractons, concentraton gradent, coherence elastc stran, and spatal varaton of both the gradent-energy coeffcent and the enthalpy of mxng near a dslocaton. In addton to the well-known phenomenon of formaton of Cottrell atmosphere and ts breakaway from a movng dslocaton, the applcaton of the model to straght edge dslocatons n a regular soluton predcts a frst-order wettng transton 85

104 from hgh- to low-segregaton at the dslocaton core wth changng temperature, bulk composton, and dslocaton velocty. The segregaton transton wth respect to velocty results n an addtonal breakaway. Ths phenomenon could play an mportant role n determnng the deformaton behavor of alloys and provde new nsghts nto the sharp yeld pont drop and the PLC phenomena. 86

105 CHAPTER 5 COMPUTER SIMULATION OF TEXGTURE EVOLUTION DURING GRAIN GROWTH: EFFET OF BOUNDAYR PROPERTIES AND INITIAL MICROSTRUCGTURES 5.1. Introducton There s an mportant nterdependence between ansotropy and preferred orentaton (or texture), not only n the performance but also n the processng of materals. Controllng texture s often necessary for optmzng the physcal propertes of a polycrystallne aggregate, especally when those propertes are hghly ansotropc n the ndvdual crystals. But the extent of texture tself s dependent on the ansotropy n propertes of boundares between crystals, and t may ncrease or decrease as boundares mgrate to grow certan orentatons at the expense of others. The structure and composton of gran boundares and hence ther propertes may vary sgnfcantly from one boundary to another because of the varaton n crystallographc msorentaton and boundary plane nclnaton [9, 106]. The relatve abundance of dfferent types of gran boundares s determned by the degree of texture and by the spatal dstrbuton of texture components 87

106 ntroduced by processes such as deposton, deformaton, transformaton, recrystallzaton and gran growth [107, 108]. Texture development has been a topc of ntense research for more than fve decades [109-11]. Recently, wth advances n expermental characterzaton, e.g. orentaton mappng (OIM) [113], and computer smulatons [31, 33, ], there has been an ncreasng nterest n advancng our fundamental understandng of the mechansms underlyng gran orentaton selecton durng thermal and mechancal processng. In ths study we focus on texture evoluton durng gran growth. Texture development durng gran growth has been studed manly through statstcal modelng and Monte Carlo smulatons [33, , 14-16]. For example, Abbruzzese and Lücke [14] studed nteractons between two texture components by extendng Hllert s statstcal model of gran growth [17]. For smplcty, they consdered two types of gran orentatons (or texture components) A and B wth three types of gran boundares (A-A, B-B and A-B). Accountng for ansotropy only n boundary moblty, wth hgh moblty for A-B and low moblty for A-A and B-B boundares, they have shown that the mnorty component should grow at the expense of the majorty component, leadng to oscllatons n the volume fractons of the two texture components. Such a phenomenon has been observed by Mehnert and Klmanek [118, 119] and by Ivasshn and coworkers [10] n ther Monte Carlo (MC) smulatons. Dfferent from Abbruzzese and Lücke [14], Novkov studed the evoluton of a sngle texture component (A) n a matrx of randomly orentated grans (B)[15]. In Novkov s statstcal analyss, both A-B and B-B boundares were assumed to have hgh energy and 88

107 hgh moblty, and t was concluded that the fracton of texture grans (n ths case, the A grans) decreases unless ther ntal average sze was much larger than that of the matrx B grans. Hwang and coworkers used MC smulatons to study a system smlar to Novkov s, but they allowed for ansotropy only n boundary energy [117]. They observed a marked growth of the texture component even when the textured grans dd not have an ntal sze advantage. Recently Rollett [1] also nvestgated evoluton of a sngle cube-texture component n a matrx of randomly orented grans usng the MC method. He employed two sets of functons for msorentaton dependence of moblty and energy, and found that the texture evoluton was senstve to the moblty functon but not to the energy functon. It s apparent that many factors affect texture evoluton durng gran growth, ncludng the number of texture components and degree of texturng, ntal volume fractons, gran szes and sze dstrbutons of dfferent texture components, and ansotropy n gran boundary energy and moblty. In addton, the spatal dstrbuton of the texture component n the ntal mcrostructure should also be an mportant factor because the spatal dstrbuton affects drectly the msorentaton dstrbuton functon (MDF). To better understand the ndvdual and combned effects of these factors, t seems necessary to dstngush the roles played by each of them and, most mportant, to dentfy the key parameter or parameters that control the texture evoluton. In ths artcle we examne systematcally the ndvdual and combned effects of ansotropy n boundary energy and moblty, and fracton and dsperson, on the evoluton of a texture component n a matrx of randomly orented grans. The smulated systems 89

108 consst of a sngle cube component (wth 5 o spread around the cube orentaton) embedded n a matrx of randomly orented grans wth msorentatons followng the Mackenze dstrbuton. Systems contanng more than one texture component wll be nvestgated n the comng work. The average gran sze and sze dstrbutons of the textured and randomly orented grans are smlar n the ntal mcrostructures. The msorentaton s characterzed n three-dmenson wth all three degrees of freedom. Two ntal fractons (1.5% and 7%) and three dfferent ntal dspersons of the texture grans are consdered (random, unform and hghly clustered). Independent of texture evoluton, extensve work has been done n modelng the effect of ansotropy n gran boundary energy and moblty on the morphology of the polycrystallne mcrostructure and the knetcs of coarsenng durng gran growth [31, 33, 13]. It has been demonstrated [31, 13] that startng wth ether a sngle component texture or a random texture, ansotropy n boundary moblty plays lttle role whle ansotropy n boundary energy strongly modfes both the morphology and the knetcs of gran growth. However, the conclusons drawn from the prevous studes may not apply to the systems consdered n the current study, where the startng orentaton dstrbuton functon (ODF) and MDF wll depend on the fracton and dsperson of the textured grans, whch are both tme-dependent. It wll be shown that when a certan fracton of a texture component exsts n a matrx of randomly orented grans, both boundary energy and moblty ansotropy may have profound nfluences on gran growth n general and texture development n partcular. 90

109 We wll lmt our scope of study to bulk materals n ths artcle. For sheets and thn flms, addtonal factors, not consdered here, may contrbute to orentaton selecton durng gran growth, such as ansotropy n surface energy and flm/substrate nterfacal energy, surface groovng drag and thermal or eptaxal stresses n the flms. Consderaton of these factors s straghtforward n the phase feld model employed, and they wll be examned n a separate artcle. 5.. Method Msorentaton wth three parameters Msorentaton can be calculated from the orentatons of neghbored grans. In the 3D space, the orentaton of a partcular crystallne can be represented by the rotaton between the local crystals wth respect to the sample reference system. There are a large number of possble ways to descrbe ths rotaton such as Eluer Angles, rotaton axs and rotaton angle, Mller Indces and matrx representaton. In prncple, all these representatons are equvalent and can be converted to one another. An easy way s to convert all the representatons to matrces frst. The relatonshp between the dfferent representatons can be obtaned by comparng the correspondng terms n the matrces. It s useful to study the exchange between the Euler angles and rotaton axs and rotaton angle. 91

110 The defnton of Euler angles s shown n Fg The correspondng rotaton operaton s gven by: Z X Z g gϕ g g ϕ1 = Φ (5.1) It s easy to convert Eq. (5.1) nto matrx as followng: ( ϕ Φϕ ) 1 cosϕ1 cosϕ snϕ1 snϕ cosφ = cosϕ1 snϕ snϕ1 cosϕ cosφ snϕ1 sn Φ snϕ cosϕ + cosϕ snϕ cosφ 1 snϕ snϕ + cosϕ cosϕ cosφ 1 cosϕ sn Φ snϕ sn Φ cosϕ sn Φ (5.) cosφ The expresson by the rotaton axs (d 1,d,d 3 ) and the assocated rotaton angle ω can easly be converted to the matrx representaton by followng three steps: transform Z axs to the drecton d, rotate Z through angle ω and perform the nverse of the frst operaton. The resultant matrx s gven by: ( d, d 1, d 3 ) ω (1 d1 )cosω + d1 = d1d (1 cosω) d3 snω d1d3 (1 cosω) + d snω d d (1 cosω) + d snω 1 (1 d 3 3 )cosω + d d d (1 cosω) d snω 1 d 1d3 (1 cosω) d snω d d 3(1 cosω) + d1 snω (5.3) (1 d + 3 )cosω d3 If two notatons descrbe the same system, the correspondng terms of matrces must be equal. Accordng to Eq. (5.3), one can easly get the followng relatonshp: m11 + m + m33 1 ω = a cos (5.4) where m 11, m and m 33 are the dagonal elements of rotaton matrx. 9

111 The defnton of msorentaton s the rotaton angle between two neghbored grans accordng to a common axs. If we rotate one of the studed grans to the sample coordnaton, the calculaton of msorentaton angle s reduced to the system we analyzed above. Thus, msorentaton can be calculated by Eq. (5.4) wth the dagonal elements of followng matrx: ' ' ' ϕ Φ and ( ) where ( ) 1 ϕ 1 ϕ 1 ' ' ' ( ϕ Φϕ ) M ( ϕ Φ ) M = M (5.5) 1 1 ϕ ϕ Φ are the Euler angles of two neghbored grans. In ths chapter, we consder the cubc crystal. Cubc group has 4 proper symmetry operatons. Therefore, the rotaton angle s not unque. By defnton, the least of the 4 angles of rotaton so obtaned s the angle of msorentaton. For the cubc polycrystallne materal wth random orentaton, one can get so-called Mackenze dstrbuton [17, 18]. It s nterestng to make a comparson between the current expressons for the msorentaton wth the prevous D one. As shown n Fg. 5., the fracton of low angle gran boundary s hghly exaggerated by D descrpton and the maxmum msorentaton angle s around 6.5 on contrast to 180 n D system. 93

112 Fgure 5.1. Defnton of Euler angles. X,Y and Z stands for the sample reference system and X,Y and Z stands for the crystal coordnaton system Markenze D f(θ) θ Fgure 5.. Markenze dstrbuton (three-parameter system) and D unform dstrbuton (one-parameter system) 94

113 5... Phase Feld Method The phase feld method (PFM) for gran growth has been developed along two ndependent lnes: one nvolves multple orentaton felds [1, 30] and the other employs a sngle orentaton feld [19, 130]. In the current study, we employ the former approach and characterze msorentatons n 3D wth all three degrees of freedom. Orentatons of grans are represented by Euler angles [131] whose values are assgned to each gran accordng to the desred orentaton dstrbuton. The msorentaton of two adjacent grans s then determned from ther Euler trplets. In the smulaton, we calculate all the possble msorentaton angles from two Euler trplets by symmetry operatons of the crystal. We then follow the conventon [17, 18] to assgn the mnmum msorentaton angle to the gran boundary, known also as the dsorentaton [18, 131]. In ths study we gnore the dependence of gran boundary propertes on nclnaton and rotaton axs. Therefore, the msorentaton angle of a boundary unquely determnes the boundary energy and moblty (see, e.g., Fg. 5.3). Bellow, we gve a bref ntroducton to the phase feld method. Detaled descrptons of the method for smulatons of gran growth phenomena n general and gran growth n ansotropc meda n partcular can be found n [1, 31]. In the multple orentaton PFM, the mcrostructure of an arbtrary polycrystallne materal s descrbed by a set of non-conserved long-range order (lro) parameters η, η,..., η ) wth each of them descrbng a specfc crystallographc orentaton of the ( 1 p grans. The evoluton of the system s descrbed by the tme-dependent Gnzburg-Landau 95

114 equaton: η t δf = L δη (5.5) where L s the knetc coeffcent that characterzes gran boundary moblty and F s the total free energy of the system. In gradent thermodynamcs [5], the total free energy s expressed on a coarse-graned level as: [ f ( ) + κ ( ) ] F = η dv (5.6) 0 η where κ s the gradent energy coeffcent and f 0 s the local free energy. The exact form of f 0 s not mportant as long as t provdes degenerate mnma correspondng to each gran orentaton, η. A smple form that satsfes ths requrement s f 0 a = P = 1 P P 1 4 b η + η + η η j (5.7) = 1 j> 4 where P s total number of lro parameters n the system and values of the phenomenologcal parameters a and b are determned by the gran boundary energy. If we assume the knetc coeffcent L, the gradent coeffcent κ and energy barrer factor b are constant, the above equatons descrbe sotropc gran growth. The dependency of gran boundary propertes on msorentaton s ntroduced by makng L, κ and b msorentaton-dependent under the constrant of constant gran boundary thckness [31]. Therefore, msorentaton s needed to be determned throughout the doman. The way of defnng msorentaton feld s not unque. To be consstent wth the feld approach, we employed followng functon n current study: 96

115 () r P, j = P η η θ, j j j j θ (5.8) η η Where θ j s pre-calculated msorentaton angle between grans wth orentatons η and η j. Equaton (8) assgns a constant msorentaton angle to a narrow range of the gran boundary. Practcal dffcultes n defnng msorentaton feld that are assocated wth P the fact that η η j tends to zero n bulk regons s nconsequental to the asymptotc, j analyss, snce the role of msorentaton s sgnfcant only n the neghborhood of gran boundary. In ths study we assume that the energy ansotropy s characterzed by a plateau for hghangle boundares and by the Read-Shockley formula for small angle boundares [13]: γ ( θ ) θ θ γ 0 1 ln = θ m θ m γ 0 θ < θ θ θ m m (5.9) where θ m s the maxmum angle at whch the Read-Shockley equaton stll holds, and γ 0 s a constant. Correspondngly, the moblty ansotropy s characterzed by [133]: L ( θ ) = 5 θ L < 0 θ θ m θ (5.10) m L0 θ θ m Note that the magntude of θ m determnes the degree of ansotropy and t s assumed to be 0 n the smulatons. The dependences of boundary energy and moblty descrbed by Eqs. (5.9) and (5.10) are shown n Fg

116 1. Gran boundary propertes GB moblty GB energy θ Fgure 5.3. Dependency of boundary energy and moblty on the msorentaton descrbed by Equatons (5.9) and (5.10) Intal mcrostructure The ntal mcrostructure (Fg. 5.4(a)) employed n the current study s obtaned from nucleaton and growth of small crystals out of lqud, usng a phase-feld smulaton wth random fluctuatons of the lro parameters. Both nterface and gran boundary propertes are assumed sotropc at ths stage. Texture s ntroduced nto the soldfed mcrostructure by assgnng orentatons to ndvdual grans,.e., assgnng a lro parameter, η, whch s pcked randomly from a lst of N choces. To ensure a partcular area fracton of texture grans, the orentaton pck lst s composed so that N = Nr + Nt, 98

117 where Nr s the number of orentatons pcked randomly from all of Euler space, and Nt s the number orentatons pcked randomly wthn 5 from the orgn of Euler space. Thus, fxng Nr = 36, we obtaned an ntal mcrostructure wth cube texture fracton of 7% wth Nt = 1, and another mcrostructure wth a fracton of 1.5% by usng Nt = 4. In the smulatons, the area fractons of the cube grans are tracked and reported n the results. For the purpose of studyng the effect of dsperson of the textured grans on mcrostructural evoluton, we have generated ntal mcrostructures wth three dfferent dspersons of the texture grans: unform, random and clustered. These mcrostructures and the correspondng MDFs are shown n Fgs. 5.4(b) (d) and 5.7(a), respectvely. In the mcrostructures, randomly orented grans (R-grans) are shaded darkly whle texture grans (T-grans) are lghtly shaded. The algorthm that we used to produce the ntal dsperson s smlar to the one employed by Modownk and coworkers [17], whch chooses two grans randomly and swtches ther orentatons f the exchange reduces the dfference between the resultng dsperson and the desred one. The MDF s measured n terms of number fracton of gran boundares of a specfc range of msorentaton. Usng fracton of gran boundary arc length gves smlar results. And n all the cases consdered, both T-grans and R-grans have smlar ntal average gran sze and lognormal sze dstrbutons. 99

118 (a) (b) (c) (d) Fgure 5.4. Intal mcrostructures wth varous dspersons of the texture component. (a) Startng mcrostructure before assgnng preferred orentatons; (b) (d) unform, random and clustered dspersons of the texture component, wth lght shade representng grans of preferred orentatons (wthn 5 o from (000) n Euler space) and dark shade representng randomly orented grans. 100

119 The varous dspersons of the T-grans shown n Fg. 5.4(b) (d) can be represented qualtatvely by dfferent fractons of low angle gran boundares n the MDF curves. A hgher fracton of low-angle boundares represents a hgher degree of clusterng of the T-grans. In addton to montorng mcrostructural evoluton and gran growth knetcs, therefore, both the dsperson of T-grans and the MDF are examned qualtatvely through the fracton of low angle gran boundares. Accordng to the spread of the orentatons among the T-grans assumed n ths study, the T-T boundares are low angle gran boundares (<5 o ). Followng the Mackenze dstrbuton, most boundares between T- and R-grans (T-R boundares) and between R-grans (R-R boundares) are hgh-angle gran boundares and essentally have smlar propertes. Therefore, the fracton of the T-T boundares, f T-T, s related drectly to the fracton of low angle gran boundares n the MDF curves and hence represents the degree of clusterng of the T- grans. 101

120 τ = Fgure 5.5. Texture evoluton durng gran growth n a system consstng of 7% ntally randomly dstrbuted textured grans under the condton of ansotropc boundary energy and sotropc boundary moblty. τ s reduced tme. 10

121 τ = Fgure 5.6. Texture evoluton durng gran growth n a system consstng of 7% ntally randomly dstrbuted textured grans under the condton of ansotropc boundary moblty and sotropc boundary energy. τ s reduced tme. 103

122 fθ θ (a) ftexture τ (b) ft-t A τ τ (c) (d) 104

123 ftexture τ (e) Fgure 5.7. (a) Intal MDFs, (b) temporal evolutons of area fractons of the texture component, (c) temporal evoluton of number fracton of gran boundares between textured grans and (d) temporal evoluton of average gran under varous condtons wth 7% ntal texture component. τ s reduced tme. Crcles, trangles and squares stand for unform, random and clustered ntal dsperson of texture component, respectvely. Dfferent shades of gray of the symbols represent varous combnatons of boundary propertes, wth sold, open and gray standng for energy ansotropy only, moblty ansotropy only and both energy and moblty ansotropy, respectvely. The sold lne n (d) s a straght lne representng schematcally parabolc gran growth knetcs. The three curves correspondng to energy ansotropy only n (b) are re-plotted n (e) to show clearly the ther ntal slops Results The evoluton of a random dsperson of textured grans (Fg. 5.4(c)), under ansotropy of boundary energy and moblty, s shown n Fg 5.5 and Fg. 5.6, respectvely. Comparng these results, one can readly see the dfference n the fractons of the textured grans n the two cases. Fgure 5. 7(b) shows quanttatvely the evoluton of area fracton of texture 105

124 for all the cases consdered. Clearly, the fracton of the texture component ncreases when energy s ansotropc, but texture decreases when moblty s ansotropc whle energy s sotropc. Also, the growth of texture s a deceleratng process, whereas texture reducton accelerates wth tme. Unform Random Clustered Energy ansotropy Moblty ansotropy Energy+Moblty ansotropy Table 5.1. Intal slopes ( 10-4 ) of curves shown n Fgure 5.7(b) for varous combnatons of ansotropy n boundary energy and moblty and spatal dstrbutons of the texture component. The ntal dsperson of the T-grans plays an mportant role n determnng the knetcs of texture development. The ntal slopes of the texture evoluton curves n Fg. 5.7(b) are lsted n Table 5.1, and they vary sgnfcantly, dependng on ntal texture dsperson. Snce the ntal slopes for the cases wth moblty ansotropy are very small, and therefore subject to greater error, the comparson s made prmarly among systems wth energy ansotropy. For a unform ntal dsperson, the texture fracton remans almost constant at the early stages of annealng. Also, the random ntal dsperson produces a hgher ntal growth rate of texture as compared to the clustered one. 106

125 The geometrc features of the polycrystallne mcrostructures obtaned for energy ansotropy are also dfferent from those obtaned for moblty ansotropy. The dhedral angles at trple junctons mantan a value of π/3 when boundary energy s sotropc, but they may devate sgnfcantly from π/3 when boundary energy s ansotropc. Low energy boundares appear as lghter shades of gray n Fg Fgure 5.7(c) shows the tme dependence of the number fracton of the T-T boundares (low angle boundares), f T-T, n varous cases consdered. The ntal slopes of all the curves are postve, whch means the formaton of T-gran clusters s nevtable for the assumed ntal fracton of the cube texture component. Even when the startng mcrostructure contans only solated T-grans, coarsenng of the mcrostructure soon produces T-gran clusters. When boundary energy s ansotropc, the fracton of low angle boundares ncreases durng gran growth. When boundary moblty s ansotropc but energy s sotropc, the fracton f T-T ncreases ntally and reaches a plateau before t starts to decrease slghtly towards the end of the smulaton, when the overall texture fracton decreases. Even when the boundary energy s sotropc, ansotropy n boundary moblty alone causes a varaton of f T-T wth tme, and hence a varaton of the MDF wth tme s expected. Ths result s dfferent from prevous observatons [31, 13] on systems wth ansotropc moblty and sotropc energy, where the MDFs where tme nvarant. However, the tme nvarance of MDF was observed n mcrostructures that were ether fully random or else fully textured, whereas the present mcrostructures contan textured grans n a matrx of randomly orented grans. 107

126 Fgure 5.7(d) shows the average gran area as a functon of tme for all the cases studed. The curves are normalzed by ther ntal slope for comparson. It s apparent that n all cases the gran growth knetcs devates from that for sotropc gran growth where the average gran area grows lnearly wth tme. These fndngs are also dfferent from the results obtaned n prevous studes [31, 13] for systems of ether sngle component texture or random texture but no mxture of the two, where t was shown that the moblty ansotropy alone does not alter gran growth knetcs. The non-lnear growth of average gran area seems to be related to the fact that the MDF s no longer tme-nvarant n the current case. Quanttatvely, the devaton from lnear growth knetcs s more pronounced when boundary energy s ansotropc and becomes maxmum when both energy and moblty are ansotropc because low energy boundares have also low moblty accordng to the functons employed (Fg. 5.3). The degree of devaton from the lnear area-tme relaton depends also on the ntal dsperson of the T-grans. To examne the possble effect of the ntal fracton of the T-grans, another set of parallel smulatons were performed wth 1.5% ntal cubc texture. The results are presented n Fg The two ntal fractons evolve smlarly, whch suggests that the results of the smulatons are qualtatvely nsenstve to the ntal texture fracton, n the range examned. 108

127 fθ ft-t θ (a) τ (c) A ftexture τ (b) τ (d) Fgure 5.8: (a) Intal MDFs, (b) temporal evolutons of area fractons of the texture component, (c) temporal evoluton of number fracton of gran boundares between textured grans and (d) temporal evoluton of average gran under varous condtons wth 1.5% ntal texture component. τ s reduced tme. Crcles, trangles and squares stand for unform, random and clustered ntal dsperson of texture component, respectvely. Dfferent shades of gray of the symbols represent varous combnatons of boundary propertes, wth sold, open and gray standng for energy ansotropy only, moblty ansotropy only and both energy and moblty ansotropy, respectvely. 109

128 5.4. Dscusson The smulaton results show that boundary energy ansotropy ncreases texture, whereas moblty ansotropy reduces t durng gran growth, n a system comprsng one texture component n a random matrx. When both energy and moblty are ansotropc, the fracton of the texture component ncreases but at reduced rates. Qualtatvely, these results are ndependent of the ntal fracton and dsperson of the textured grans (Tgrans). Quanttatvely, however, the ntal dsperson of the texture component plays an mportant role n determnng the tme-evoluton of the msorentaton dstrbuton and hence affects the overall knetcs of texture evoluton and gran growth. It s apparent that all the factors consdered n ths study, and many others documented n the lterature, play a role n texture evoluton durng gran growth. However, there s a large number of relevant factors and an even larger number of possble combnatons thereof, ncludng boundary ansotropy, ntal dsperson and fracton of the texture component, the number of texture components present, and ther average gran szes and sze dstrbutons. Therefore, analyss of the results of a gven study can be qute complcated, and dsagreements of dfferent studes can be dffcult to resolve. However, as wll be shown below, a key parameter that can be used as a general bass for analyss s the content of boundary energy densty n a cluster of grans, γ/d, where γ s boundary energy and d s average gran sze n the cluster. The same parameter was ntroduced by Turnbull [134] over half a century ago, to estmate the macroscopc drvng force for gran growth. Here, we use the relatve magntude of the γ/d potental n 110

129 contnguous clusters as a predctor of whch cluster tends to grow and whch tends to shrnk 111

130 τ = Fgure 5.9. Texture evoluton durng gran growth n a system consstng of 7% ntally clustered texture component under the condton of ansotropc boundary moblty and sotropc boundary energy. τ s reduced tme. 11

131 Effect of boundary moblty ansotropy When a texture component exsts n the ntal mcrostructure, T-T boundares wll have, on average, much lower mobltes as compared to T-R or R-R boundares. Thus, f the T- grans form clusters, whch s nevtable durng gran coarsenng wth the ntal texture fractons consdered, the gran sze wthn a T-cluster wll coarsen less rapdly than n surroundng matrx of R-grans. Therefore, as the mcrostructure evolves, the average gran sze n a T-cluster wll be smaller than n an R-cluster. To llustrate ths behavor, we plot the temporal evolutons of the mcrostructure and the gran sze dstrbutons of the T- and R-grans n Fgs. 5.9 and 5.10, for the case of ntally clustered dsperson. From Fg one can see that the ntal sze dstrbutons of the T and R-grans almost concde wth each other. As the mcrostructure evolves, however, the gran sze dstrbuton of the R-grans s shfted sgnfcantly towards larger gran szes, resultng n the splttng of a sngle peak at t = 0 nto two peaks at a later tme. The development of gran sze dsparty s also seen clearly n the mcrostructural evoluton shown n Fg Snce the gran boundary energy s sotropc, a T-gran cluster has a hgher boundary energy densty, γ/d, than the R-gran clusters. Therefore, the R-grans should grow at the expense of the T-grans, and hence, the fracton of the texture wll decrease wth tme. Snce the sze dfference between the T- and R-grans keeps ncreasng, the dfference n the γ/d potental ncreases as well, whch accelerates the shrnkage of the T-gran clusters as shown n Fgs. 5.7(b) and 5. 8(b). 113

132 f A random τ=0 textured τ=0 random τ=500 textured τ= loga Fgure Temporal evoluton of gran sze dstrbutons of textured (open crcles) and randomly orented (sold crcles) grans shown n Fg τ s reduced tme and f A s number fracton of gran wth area A. 114

133 τ= Fgure Texture evoluton durng gran growth n a system consstng of 7% ntally clustered texture component under the condton of ansotropc boundary energy and sotropc boundary moblty. τ s reduced tme. 115

134 5.4.. Effect of boundary energy ansotropy For the case of boundary energy ansotropy, the T-T boundares have lower energes than the other, hgher msorentaton boundares n the system. Snce the average gran sze of the T- and R-grans are smlar n the ntal mcrostructure, a T-gran cluster wll have lower γ/d potental than an R-gran cluster. Therefore, the T-gran clusters should grow at the expense of the R-grans. The preferental growth of texture clusters s qute evdent from the concave T-R gran boundares along the T-gran cluster permeter (Fg. 5.11). Ths result agrees wth the MC smulatons by Hwang et. al. [117]. Note also that the dhedral angles at most of the T-T-R trple junctons devate sgnfcantly from π/3, wth the dhedral angles on the R gran sdes close to π. As grans coarsen, the average gran szes of both T- and R-grans ncrease, and the dfference n the γ/d potentals between the T- and R-gran clusters wll decrease. As a consequence, the growth of the T-gran clusters and hence the ncrease of the texture component should decelerate wth tme as shown n Fg If there are few T-gran clusters n the system such as n the case of unform dsperson of the texture component, the ntal growth rate of texture wll be very small as shown n Table Interplay between energy and moblty ansotropy When both boundary energy and moblty are ansotropc, wth low energy correspondng to low moblty, the effect of moblty ansotropy s to slow down gran growth wthn the 116

135 T-gran clusters and hence lead to a hgher γ/d potental than would be expected from energy ansotropy alone. Ths effect wll reduce the dfference n γ/d potentals of the two types of clusters durng the evoluton process, and t slows down the preferental growth rate of the texture component, as shown n Fg In the extreme case, e.g., when the moblty of the T-T gran boundares vanshes, the γ/d potental of the R-gran clusters wll eventually be lower than that of the T-gran clusters and the texture component wll decrease thereafter. Thus, under the smultaneous and opposng nfluences of energy and moblty ansotropy, the fracton of texture could ncrease or decrease, dependng on the functonal forms of boundary energy and moblty and the spread of msorentaton angles n the T-grans. 117

136 τ = (a) (b) τ = (c) (d) Fgure 5.1. Effect of ntal gran sze on mcrostructure evoluton n a system consstng of two honeycomb structures. The sold lnes represent the ntal poston of the boundares between the two honeycomb structures. Mcrostructures showed here are quarters of real smulated systems. 118

137 Effect of dsperson of texture component Clusterng of T-grans ncreases the fracton of small-angle gran boundares and enhances the effect of boundary energy or moblty ansotropy on texture development. The ntal fracton and dsperson of the T-grans control the ntal and subsequent clusterng of the T-grans durng gran growth. Wth smlar dspersons, decreasng the fracton of the texture component reduces the possblty of clusterng. Ths becomes evdent by comparng the texture development n the systems of 1.5% and 7% texture component. A smaller fracton of texture leads to fewer T-gran clusters and slower knetcs of texture evoluton. In addton to sngle-boundary propertes, the rate of texture development and the overall gran growth knetcs depend quanttatvely on the number of T-gran clusters, as well as ther szes, shapes and nterconnectvty. For example, wth the same ntal area fracton and average gran sze of the texture component, the random dsperson of the texture grans yelds the maxmum growth rate of the texture fracton, f texture, and the clustered dsperson yelds the mnmum growth rate, when boundary energy s ansotropc, as shown n Table 5.1. Preferental growth of the T-gran clusters takes place at the cluster permeter, where the dsparty n γ/d represents an mbalance n the local forces actng at trple junctons. The total permeter length of the T-gran clusters s greatest n the case of random dsperson, where the clusters percolate quckly through the mcrostructure. For unform dsperson the total length of the T-gran cluster 119

138 permeter s mnmum at the begnnng, but t gradually outgrows the case of clustered dsperson. Therefore, the growth rate of f texture for the case of unform dsperson s mnmum at the begnnng and gradually ncreases and eventually exceeds the case of clustered dsperson. However, the relatvely slow growth of texture n the case of ntally clustered grans s due n part to the compactness (convexty) of the clusters and ther beng well separated from one another. Therefore, the quanttatve dfferences shown by the smulatons of dfferent ntal dspersons may not be unversally applcable, but the results show nonetheless that the detals of ntal dsperson have an mportant nfluence on the knetcs of texture development Effect of gran sze It should be emphaszed that the man concluson on the effect of boundary energy ansotropy n ncreasng texture s vald only for the specfc ntal mcrostructures consdered. If the T- and R-grans had dfferent average gran szes to start wth, the gran boundary energy densty (γ/d) n the T-gran clusters could be hgher than that n the R- gran clusters and hence the texture component could decrease rather than ncrease even f the boundary energy s ansotropc. Ths has been confrmed by the smulaton results shown n Fg. 5.1, where two regular honeycomb mcrostructures of dfferent gran szes are consdered. In the two parallel smulatons, Fg. 5. 1(a)-(b) and (c)-(d), the rato of gran boundary energy of the two honeycomb mcrostructures has been fxed to, whle the rato of the gran szes of the two honeycomb mcrostructures have been chosen as 4 10

139 and 1., respectvely. Therefore, n the former the texture component (the smaller grans that have lower boundary energy) has a hgher γ/d potental, whle n the latter the texture component has a lower γ/d potental. As a consequence, the texture component decreases n the former and ncreases n the latter. Such a stuaton could also be reached from coarsenng of T- and R-grans of smlar ntal sze dstrbuton f the moblty of the T-T boundares s much lower than that of the T-R and R-R boundares, as has been dscussed earler n Secton The man concluson drawn on the effect of boundary moblty ansotropy on texture evoluton,.e., boundary moblty ansotropy reduces texture component durng gran growth, however, holds rrespectve of gran szes. Suppose that the average gran sze of the T-grans s larger than that of the R-grans n the ntal mcrostructure. The T- gran clusters wll grow at the expense of the surroundng R-grans at the begnnng. However, snce grans n the T-gran clusters wll coarsen much slower than those n the R-gran clusters, eventually the average gran sze of the R-gran clusters wll exceed that of the T-gran clusters, leadng to a lower γ/d potental for the R-gran clusters. Therefore, the texture component wll eventually shrnk under moblty ansotropy Effect of number of texture component In the current study, a sngle texture component embedded n a matrx of randomly orented grans s consdered. The smulaton results show monotonc ncrease or decrease of the texture component, whch s dfferent from prevous work on a mxture of 11

140 two texture components [118-10, 14] where an oscllaton n the fractons of the two texture components has been predcted. However, ths oscllaton n texture can also be explaned by applyng the same argument of the γ/d potental. For example, n an ntal mcrostructure consstng of two texture components of smlar average gran szes and sze dstrbutons but dfferent volume fractons, the texture component of smaller volume fracton wll be embedded n a matrx of the major component and hence has more A-B (hgh moblty) boundares per unt volume. As the mcrostructure evolves, the average gran sze of the mnor component wll exceed that of the major component and hence yeld a lower value of γ/d. Then the mnor component wll grow at the expanse of the major component at an acceleratng rate tll a moment when all the orgnal matrx grans of greater values of γ/d are consumed and only locally small pockets of lower values of γ/d left (because grans do not have the same sze). Then these small pockets wll become the new mnor component of lower γ/d and the process wll be reversed Texture effect on gran growth knetcs If the entre mcrostructure has ether random or sngle-component texture, the γ/d potental becomes unform. In ths case t has been shown prevously [31, 13] that gran growth n systems of sotropc boundary energy but ansotropc moblty behaves exactly the same as n an sotropc system: the MDF s tme-nvarant and the average gran area ncreases lnearly wth tme. Ths s not true anymore for systems consstng of mxtures of R- and T-grans where γ/d potental s non-unform. As analyzed earler, the fracton of 1

141 the texture component wll ether ncrease or decrease n ths case, leadng to a tmedependent MDF. Accordng to [135], the gran growth knetcs wll devate from the parabolc law derved for sotropc systems. For systems of both boundary energy and moblty ansotropy, hgh-energy boundares wll be elmnated frst durng gran coarsenng. Because the low energy boundares have low moblty as well, whch accelerates the selectve elmnaton process of the hgh-energy boundares, the negatve devaton from lnear growth knetcs become more severe (Fg. 5.7(d)). For systems wth moblty ansotropy only, the fracton of low moblty T-T boundares decreases as T-gran clusters shrnk, as shown n Fg In ths case, an acceleraton of the overall gran growth s expected. Ths behavor can also be found n Fg 5. 8(d) Summary The ndvdual and combned effects of ansotropy n gran boundary energy and moblty, and dsperson and fracton of texture component on texture evoluton durng gran growth are nvestgated by computer smulatons usng the phase feld method. The systems consdered consst of a sngle cube component embedded n a matrx of randomly orented grans n the ntal mcrostructure. Even though all the factors consdered affect texture evoluton, t s found that the key parameter that controls texture evoluton s the gran boundary energy densty characterzed by γ/d. The fracton of the texture component ncreases when γ/d of the texture clusters s smaller than that of the 13

142 randomly orented grans, and decreases when γ/d of the textured grans s greater than that of the randomly orented grans. The ndvdual factors consdered affect texture development through ther nfluences on the γ/d potentals of the textured and randomly orented grans. Ther ndvdual and combned effects on the γ/d potentals and hence on texture evoluton and gran growth knetcs are analyzed and are found agree well wth the smulaton results. Clusterng of textured grans s nevtable durng gran coarsenng. The degree of clusterng depends on the ntal dsperson of the textured grans. Clusterng causes an ncrease n the fracton of low angle boundares n a system and, hence, alters the msorentaton dstrbuton functon (MDF). As a consequence, ntal mcrostructures of dfferent degrees of clusterng of textured grans are assocated wth dfferent tmeevolutons of the MDF and hence dfferent gran growth knetcs. 14

143 CHAPTER 6 QUANTITATIVE PHASE FIELD MODELING OF DIFFUSION-CONTROLLED PRECIPITATE GROWTH AND DISSOLUTION 6.1. Introducton Dffuson-controlled phase transformatons n mult-component systems have tradtonally been smulated by usng the sharp nterface approach and the smulatons are usually lmted to 1D because of the necessty of front trackng and determnaton of local-equlbrum te-lnes at the nterfaces [136]. In contrast, the phase feld method (PFM) [ ], also known as the dffuse nterface approach, avods trackng the movng nterface and local equlbrum calculaton by treatng heterogenetes as varatons of ntroduced contnuous feld varables, and because of ths convenence t has become a method of choce to study mcrostructure evolutons due to but not lmted to phase transformatons n D and 3D. Recently, there has been an ncreasng nterest n usng ths method to treat real bnary and multcomponent alloys by lnkng bulk free energes n the phase feld models to crtcally assessed thermodynamc databases [14, 19, ]. However, there have been few attempts to fully ncorporate both 15

144 thermodynamc and moblty databases nto the PFM and verfy the knetc results aganst that of the establshed sharp nterface model, whch s apparently a necessary step towards quanttatve smulaton of mcrostructural evoluton. Another mportant ssue that has not been addressed properly so far s how to break the nherent length scale lmt n a quanttatve phase feld modelng where materal specfc free energy and nterfacal energy data are used [143]. In ths chapter, we ntend to develop a multcomponent phase feld model that can make drect use of assessed thermodynamc and moblty databases and smulate phase transformatons on real length and tme scales. Applcaton examples are gven for the β α transformaton n the T-Al-V system. Smple geometres wll be used to facltate comparson of phase feld predctons wth that from commercal software DICTRA [136] based on sharp nterface approach. Wth ths crucal valdaton work done, complex geometres and real mcrostructure evoluton wll be consdered n a forthcomng publcaton. 6.. Phase Feld Model To descrbe phase transformatons n an n-component system usng the PFM, we need n- 1 concentraton felds and a set of order parameter felds as well as the dependence of local free energy on them. The order parameters characterze symmetry changes accompanyng the phase transformatons and ther choce can be ether physcal or phenomenologcal. For an order-dsorder transformaton, the long-range order (lro) 16

145 parameters are the default order parameters and the local free energy as a functon of concentraton and lro parameters can be obtaned drectly by the CALPHAD technque [14]. For a reconstructve phase transformaton, lke bcc (β) to hcp (α) n T, a Landau free energy expanson wth respect to physcally chosen order parameters can be constructed accordng the symmetry changes durng the phase transformaton [15, 16]. The same approach can be appled to alloys, but then t became evdent that the parameters n the Landau free energy must be made temperature and composton dependent. In order to have a Landau free energy consstent wth the expermental or assessed equlbrum free energy data n a multcomponent system, we thus have to face a formdable task to ft the expresson n a multdmensonal space at dfferent temperatures. An alternatve approach s to defne a phenomenologcal order parameter that assumes certan dfferent values for phases of dfferent symmetry. The local free energy s then constructed n such a way that the equlbrum free energy of ndvdual phases can be drectly nserted nto the expresson, and the phase equlbrum relatonshp n the temperature-composton projecton can be sustaned n the temperature-compostonorder parameter space. A convenent choce of such an expresson s due to Wang et al. [17] and has been used wdely n the soldfcaton modelng [18] and recently also for sold state transformatons [19]. Adoptng ths choce, we wrte the local molar Gbbs free energy g m as a functon of temperature T, composton x (=1,,, n-1), and order parameter η: g α β ( T, x, η) = [1 p( η)] g ( T, x ) + p( η) g ( T, x ) q( η) (6.1) m m m + 17

146 3 where p ( η ) = η (10 15η + 6η ) and q ( η) = ωη (1 η). The parameterω s the heght of the mposed double-well hump and, along wth the gradent energy coeffcents κ and ε shown below n Eq.(), can be determned from nterfacal energy, σ, and nterface thckness, λ. α g m and g β m are the molar Gbbs free energes of the α and β phases, respectvely. For a chemcally and structurally non-unform system under the assumpton of constant molar volume V m, the total Gbbs free energy G can be expressed by G G = V m m 1 = V m n 1 κ ε gm( T, x, η ) + x + η dv (6.) V = 1 where κ and ε are the gradent-energy coeffcents for concentraton and order parameter nhomogenetes, respectvely [144, 145]. The temporal evolutons of feld varables are governed by the tme-dependent Gnzburg-Landau equatons [0] and the generalzed Cahn-Hllard dffuson equatons [1] on the bass of the phenomenologcal Fck- Onsager equatons []: η = M t η δg m δη (6.3) n 1 1 k = m t j = 1 V x M kj δgm ( T, x, η ) (6.4) δx j where M η s the moblty of the order parameter and can be drectly related to the nterface moblty n the sharp nterface approach. The parameters M kj are the so-called chemcal mobltes n the volume-fxed frame of reference. In a homogenous phase p (p=α, β), accordng to Andersson and Agren [3], the chemcal mobltes M are p kj 18

147 related to atomc mobltes p M l (l=1,,n) by: M p kj 1 = V n ( jl x j ) m l = 1 p δ ( δ x ) x M (6.5) lk k l l where δ jl and δ lk are the Kronecker delta. In a non-unform system, we assume that the same relaton holds for M kj and M l, the atomc mobltes dependent on the order parameter η by: α β α η β (1 η ) l = M l + M l ( M l ) ( M l ) M (6.6) The choce of Eq.(6) ensures that the atomc mobltes n the nterface regon wll have a postve devaton from the smple lnear nterpolaton, and the larger the dfference between atomc mobltes n the two phases s, the more postve the devaton wll be. If the dfference s several orders of magntude, the atomc moblty n most part of the nterface regon wll assume almost the value for the phase that has a hgher atomc moblty (see Fg. 6.1). We beleve that ths scheme s more reasonable than ether lnear or logarthmcally lnear approxmaton because of sgnfcantly more vacances present n the structure of a sem-coherent or ncoherent nterface. 19

148 Fgure 6.1 Interpolaton of atomc moblty across the nterface regon. Insertng Eq.(6.) nto Eqs.(6.3) and Eq.(6.4), we can obtan the followng dmensonless governng equatons η ~ ~ ~ ( ~ gm = M η ε η ) (6.7) τ η x ~ n 1 k = τ j = 1 ~ M kj ~ g~ ( x m j ~ ~ κ x j j ) (6.8) by ntroducng the followng reduced quanttes: ~ = [ / (x/l), / (y/l)]; g ~ m = g m / g m ; M ~ k = V m M k /M; ~ ε = ε /( g m l ); ~ κ = κ /( g m l );τ = (M g m /l )t; M ~ η = M η l /M where l s the mesh sze, g m and M are normalzaton factors for molar Gbbs free energy and atomc moblty, respectvely. Ths dmensonless verson of the governng equatons s partcularly convenent for numercal calculaton and very useful n rescalng the space and tme for dffuson-controlled phase transformatons. 130

149 6.3. Length and tme scales Snce the nterface s dffuse n phase feld approach, enough ponts have to be allocated at the nterface to guarantee the accuracy and stablty of the numercal calculaton. On the other hand, the physcal thckness of nterfaces s n the order of nanometers or even Ångstroms, whch confnes the total length scale of a fathfully quanttatve phase feld modelng n a unform mesh to a few or a few tenth of mcrometers. Two schemes could be adopted to treat mcrostructures n tens or hundreds of mcrons: 1. Apply adaptve mesh as we dscussed n Chapter ;. Apply thn nterface approach (scalng). In ths work, we focus on the latter one. In ths approach, we have to make the nterface more dffusve by adjustng certan model parameters and at the same tme keepng fxed the drvng forces of the process n order to man the quanttatve nature of the smulaton. The smplest scalng problem can be found n Allen-Cahn s antphase boundary modelng. If the gradent coeffcents ncreases ξ tmes,.e. ε =ξ ε and κ =ξ κ, a larger nterface thckness λ =ξλ wll be obtaned. Accordng to Allen and Cahn [145], the velocty of the moton of the η profle s proportonal to εm η. To keep the same velocty for a more dffusve nterface, we should then use a scaled moblty of order parameter M η = M η /ξ. Usng the same number of mesh ponts to dscretze the nterface, we can now use a larger mesh sze l =ξl and meanwhle the knetcs of antphase boundary s nvarant. The scalng for dffuson controlled phase transformaton s not a trval problem due to the couplng between concentraton and phase felds. At thermodynamc equlbrum state, the varatonal calculus gves: 131

150 g e ( x η) = g const, (6.9) x x = and dη ε d x We defne a local excess energy functon F as = g η (6.10) F = g Consderng F = 0, t s straghtforward to obtan c therefore, Eq. (6.10) become: n 1 ( ) x, = η g x (6.11) 1 ( ) e x df g η η x,η d = (6.1) d η df ε = dx dη (6.13) Multplyng both sdes wth dη dx and ntegratng respect to η, Eq. (6.13) results n: dη ε = F (6.14) dx It can be seen that functon F we proposed here s practcally the energy pump n Allen- Cahn s analyss dscussed n Chapter. It s nterestng to examne the exact form of F F = [1 p( η)] g α m ( T, x ) + p( η) g β m ( T, x ) + q( η) n 1 = 1 g e x x (6.15) Then the energy hump has an addtonal postve energy contrbuton from concentraton, whch lmts the nterfacal thckness. If the free energy form s gven, the statonary concentraton and phase feld profle can be solved numercally n general. Therefore, one 13

151 can evaluate the proper smulaton parameters under certan length scale proposed Applcaton to β α transformaton n T-Al-V The Gbbs free energes as a functon of temperature and composton were adopted for the β and α phases n the T-Al-V system from a T-base thermodynamc database developed by [146] usng the CALPHAD technque. Based on ths thermodynamc nformaton and wth help of DICTRA, a set of self-consstent parameters descrbng the atomc moblty of T, Al, and V n the two phases were obtaned by assessng the expermental dffusvty n the ternary and ts consttuent bnary systems [147]. After drectly ncorporatng these thermodynamc and knetc data nto Eqs. (6.7) and (6.8), model parameters related to nterface propertes,.e.ω, κ and ε, were ftted to assumed nterfacal energy σ=0.5 J/m (typcal value for ncoherent phase nterfaces) and nterface thckness λ = m (smlar n order to gran boundary thckness) by performng a one-dmensonal phase feld dynamcal relaxaton of a system wth sharp nterface and equlbrum composton. In ths fttng procedure, a grd sze of 10-9 m has been chosen, whch means that 5 dscrete nodal ponts have been used to sample the hype-surface of total Gbbs free energy. For smplfcaton, the composton gradent coeffcents κ were set to zero as ths wll not affect the knetc results. Under these condtons, we obtaned ω = 30 kj/mol and ε = J m. Takng l =10-9 m and V m =10-5 m 3 /mol as well as the normalzng quanttes g m = 50 kj/mol and M =

152 m /s/j, we had ~ ~ ε = 1. and τ = 50000t. Fnally we chose M = 6 to warrantee a dffusoncontrolled process. η Thckenng of α Plate Fgure 6. PFM and DICTRA results for the growth of alpha precptate plate. (a) Growth knetcs; (b) Composton profle of Al; and [148] Composton profle of V. We consdered a α precptate growng wth a planar nterface nto supersaturated β at 1173 K. The ntal thckness of the α plate was put to 0. µm, and ts composton set to equlbrum value: 11.3 at%al, at%v. The ntal composton of beta was at%al and 3.6 at%v. The total system sze was chosen as 10 µm. The phase feld smulaton was performed wth 500 grd ponts and the grd sze of 10-9 m n one dmenson. In order to descrbe the desred system sze, the length scale of the phase feld 134

153 modelng must be ncreased by 0 tmes, whch means the actual tme must be scaled up by 400. Sharp nterface smulaton on real length and tme scale has also been carred out by usng DICTRA. The two results are compared n Fg.6.. It s clear that the phase feld modelng results are n good agreement wth that of DICTRA smulaton. As expected and clearly shown n Fg. 6.a, the thckenng of the plate follows the parabolc law ntally. The gradual slow down of knetcs n the later stage s due to the soft mpngement (see Fg. 6.b and 6.c). It s worth mentonng that the local equlbrum comes out automatcally n the phase feld method wthout need of explct calculaton. Because the dffusvtes of Al and V are dfferent n both phases, the local equlbrum te lnes keep changng accordng to flux balance durng the transformaton process. Ths s more obvous f the two composton profles are supermposed n the equlbrum phase dagram of the system T-Al-V at 1173 K. As can be seen from Fg. 6.3, frst the system fnds a te lne away below the equlbrum one, and then a seres of te lnes approachng and slghtly passng the equlbrum one. If annealng tme s long enough and homogenety can be reached wthn each phase, the equlbrum te lne wll be assumed eventually. Ths nterestng feature, whch may be puzzlng at frst sght for people used to bnary systems, s ubqutous n ternary and hgh order systems, and t has been well captured n our phase feld smulaton. 135

154 Fgure 6.3. The dffuson path correspondng to Fg.6.b and 6.c durng the growth of alpha precptate plate. The system descrbed above corresponds to the growth of gran boundary alpha precptates except that n realty sdeplates wll form shortly ether by nterface nstablty [149, 150] or sympathetc nucleaton mechansm [151] and nhbt the further growth of the layer of gran boundary alpha. However, the predcted ntal thckenng knetcs should be applcable f no gran boundary dffuson enhancement,.e. the collector plate mechansm for Al and rejector plate mechansm for V [15], s nvolved. 136

155 6.4. Dssoluton of Globular α Fgure 6.4 PFM and DICTRA results for the dssoluton of globular alpha. (a) Growth knetcs; (b) Composton profle of Al; and [148] Composton profle of V. We consder now the dssoluton of a globular α that was orgnally at equlbrum wth the beta matrx at 1173 K and s rased nstantly to 13 K. For phase feld smulaton, a mesh of 500x500 and a quarter of a crcle stuated at the lower left corner are used, and are expected to account for an actual system n 100x100 µm. In ths case, as ponted out n Secton 3, we cannot apply the mesh sze l =10-9 m and scale t up 00 tmes because any partcle on the nanometer scale wll experence strong Gbbs-Thompson effect and hence ts dssoluton knetcs be affected sgnfcantly. In order to represent the realty on mcron scale, where the Gbbs-Thompson effect s neglgble, we need to have an energetcally equvalent base system on a larger length scale,.e. one wth same chemcal potental dfferences and nterfacal energy but wth more dffuse nterface. For the T-Al- V system, the mesh sze can be ncreased by 50 tmes (l = m) whle mantanng the 137

156 same nterfacal energy by settng ω = 100 J/mol and ~ ε ' = Accordngly, we have ~ M ' = 300. After performng smulaton on ths base system, a straght scalng on length η (by 4 tmes) and tme (by 16 tmes) was carred out to match the actual sze 100x100 µm. The results are compared wth DICTRA smulaton n Fg.6.4. Apparently, an excellent agreement between them has been obtaned. It s nterestng to note that the dssoluton process n ths case s not smply a reverse of growth process, and the dssoluton knetcs does not followng the parabolc law, see Fg.6.4c. The local equlbrum at the nterface s obtaned mplctly n the phase feld method. Agan, the system found dfferent te lnes from the one across the orgnal composton of α phase. Fgure 6.5 depcts the dffuson path at 1000 s n the T-Al-V Gbbs trangle where two sets of te lnes at 1173 and 13 K are drawn. 1173K 13K Fgure 6.5 Dffuson path at 1000 s durng the dssoluton of globular alpha. Two sets of te lne at 1173 and 13 K are drawn. The alloy concentraton s gven by small trangle. 138

157 6.5. Summary We have devsed a scheme for quanttatve phase feld modelng of mult-component dffuson-controlled phase transformatons, n whch both thermodynamc and knetc data from exstng databases can be nserted drectly nto the phase feld model and the length and tme of the dynamc system can be readly scaled. Applcatons to phase transformaton between α and β n smple geometres have been proved very successful by comparng results wth that of DICTRA smulatons. The powerfulness of phase feld method reles on ts capablty to handle arbtrarly complex geometres and ths shall be demonstrated n a forthcomng publcaton. Some assumptons have been made n fgurng out the model parameters n the method n order to have a volume dffuson controlled process. Dfferent assumptons may have to be evoked f solute drag, solute trappng or nterface-controlled process s consdered, for example, a larger value of nterface moblty M η may result n solute drag and massve transformaton. 139

158 CHAPTER 7 APPLICATION ON TI-6AL-4V ALLOY DESIGN 7.1 Introducton Snce the ntroducton of ttanum and ttanum alloys n the early 1950s, these materals have n a relatvely short tme become backbone materals for the aerospace, energy, and chemcal ndustres. The combnaton of hgh strength-to-weght rato, excellent mechancal propertes, and corroson resstance makes ttanum the best materal choce for many crtcal applcatons. Today, ttanum alloys are used for demandng applcatons such as statc and rotatng gas turbne engne components n advanced aerospace structural applcatons. The use of ttanum has expanded n recent years to nclude applcatons n nuclear power plants, food processng plants, ol refnery and bomedcal mplant. An mportant characterstc of ttanum-base materals s the reversble transformaton of the crystal structure from alpha (α, hexagonal close-packed) structure to beta (β, body-centered cubc) structure when the temperatures exceed a certan level. 140

159 Ths allotropc behavor, whch depends on the type and amount of alloy contents, allows complex varatons n mcrostructure and more dverse strengthenng opportuntes than those of other nonferrous alloys such as copper or alumnum. Therefore, a broad range of propertes and applcatons can be served wth a mnmum number of grades. Ths s especally true of the alloys wth a two-phase, α+β, crystal structure. Wth others compared, the T-6Al-4V alloy domnates structural applcatons and s a benchmark alloy. The sx weght percent Al stablzes the α phase to provde sold soluton strengthenng. It contans a favorable balance of propertes wth moderately hgh tensle strength, good fatgue strength, wth ntermedate fracture toughness. Reasonable propertes are retaned up to about 350 C (660 F). The hgh cost of ttanum alloy components may lmt ther use to applcatons for whch lower-cost alloys, such as alumnum and stanless steels. The relatvely hgh cost s often the result of the ntrnsc raw materal cost of metal as well as fabrcatng costs. Reducng the fabrcatng costs reles very much on fndng unambguous relatonshps between the heat treatment and mechancal propertes. However, such capacty s stll weak. The Metals Affordablty Intatve (MAI) program offers a great opportunty to develop a set of functonal tools for ndustry use. Ths program nvolves multple teams workng on dfferent areas such as experment desgn and measurement, mechancal propertes modelng usng neural network, mcrostructure modelng and deformed texture modelng. The modelng tools form a herarchy structure. For example, results from mcrostructure modelng wll feed nto mechancal propertes model to relate the 141

160 processng wth the product propertes fnally. Our focus here s the mcrostructure modelng and tools development. In partcular, we wll address the formaton and evoluton of followng mcrostructural features: β gran, gran boundary α, sdeplate and colony structure. 7. β Gran Growth Consderng modelng mcrostructural evoluton durng thermal and mechancal processng of ttanum alloys, gran growth durng beta annealng seems to be the smplest case that one can easly get hs hands on. Ths turns out not to be true because gran growth n these alloys s always complcated by the presence of many ntrnsc and extrnsc factors that alter the growth knetcs, ncludng surface groovng and precptate pnnng, ansotropy n gran boundary energy and moblty and the correspondng effects of texturng. Whle some of the studes n the lterature showed smple parabolc growth [153, 154], others revealed complcated behavors such as oscllaton n growth rate [155], and strong devaton from the parabolc law [156, 157]. Under the support of the Ar Force sponsored Metals Affordablty Intatve (MAI) program, we have been workng on developng a predctve model for gran growth durng beta annealng n T-6Al-4V. It has been shown that under ndustral condtons gran growth n T alloys does not follow the parabolc law reported n the lterature, even wth the elmnaton of some obvous extrnsc factors that cause the devaton from parabolc growth. In ths artcle we dscuss two factors that nfluence 14

161 gran growth durng beta annealng: the effects of sample sze and texture, and ntroduce the approprate equatons that account for these effects Sample Sze Effect Thermal grooves wll form when gran boundares ntercept wth free surfaces. Extra force s needed to move a gran boundary away from the thermal groove. Ths mposes a pnnng force that may cease gran growth when the average gran sze reaches the smallest dmenson of a sample. Beck et. al. [158] frst observed ths phenomenon and referred to t as the sample sze effect. Usually, people are concerned about sample sze effect when dealng wth gran growth n thn flms [159] and tend to gnore the effect when studyng bulk samples. In fact, the sample sze effect s an mportant factor that one should consder n beta gran growth because the fast growth rate at the annealng temperature n these alloys results n exceptonally large gran szes. For example, the beta grans grow from 6 µm up to.5 mm n dameter after hours annealng at 100 C [156]. Such large gran szes are frequently observed under ndustral heat treatment condtons. The sample sze effect has been studed extensvely n lterature. Burke developed the frst gran growth equaton that accounts for the groovng drag effect [160]. Assumng a lnear dependence of gran boundary velocty on drvng force and a constant pnnng force from surface groovng, he showed that gran growth wth surface groovng drag could be descrbed by the followng equaton: 143

162 dd 1 = K 1 dt D D f (7.1) where D s the average gran sze, K s a temperature-dependent rate constant (a product of gran boundary energy and moblty) and D f s the lmtng gran sze. Burke s model suggests that gran growth devates substantally from the parabolc law when D s approachng D f. Assessng D f from measured gran growth data, he showed that the model predcton agreed well wth expermental observatons. To make the model truly predctve, however, one needs to evaluate D f ndependently wthout expermental measurement. Based on the Gbbs-Thompson equaton, Mullns [161] proposed a theory that relates the lmtng gran sze due to thermal groovng to sample thckness. Assumng surface dffuson to be the mechansm for groovng he showed that under normal gran growth condton, D f 3γ s = a (7.) σγ b where σ s a geometrcal factor (~5 for normal gran growth), γ s and γ b are surface and gran boundary energy, respectvely, and a s sample thckness. One can see that the lmtng gran sze s lnear proportonal to the sample thckness. Combnng Eqs. (7.1) and (7.), one can predct gran growth behavor wth sample sze effect based on materal parameters and sample geometry. More drect studes on sample sze effect were carred out wth the ad of computer smulatons. Frost et. al. [159] modeled gran growth n thn flms usng front trackng method. Ther smulaton results showed explctly the scalng between average 144

163 gran sze and flm thckness n stagnant gran structures. Novkov [16] analyzed the pnnng effect n 3D bulk samples usng a statstc model. He observed a sgnfcant slowdown of growth when the average gran sze approached the sample thckness. In both studes, however, the pnnng force from thermal groovng s an nput parameter. To study the surface groovng effect wthout any a pror assumptons about the drag force, we developed a phase feld model for gran growth n systems wth free surfaces. The mcrostructure evoluton and correspondng gran growth knetcs obtaned n a twodmensonal (D) smulaton are shown n Fg It can be readly seen that thermal grooves form when gran boundares ntercept wth sample surfaces (Fg. 7.1(a) and (b)). A clear devaton of the gran growth knetcs from parabolc growth (the ntal lnear regon n Fg. 7.1(c)) s observed when the average gran sze reaches about 1/9 of the sample thckness,.e., D * =1.4 n reduced unt. Ths result agrees wth calculatons from Eqs. (7.1) and (7.). The expermental results obtaned n [4] are re-plotted n Fg. 7.. It can be seen clearly that the gran growth knetcs starts to devate sgnfcantly from the parabolc law when the average gran sze reaches about 1/6 of the sample smallest dmenson (6 mm). Usng γ s /γ b =~3 (note that γ s /γ b should be, n prncple, a functon of temperature) n Eq. (7.) and substtute the value obtaned, D f = 1.~1.8a, nto Eq. (7.1), we fnd when the average gran sze reaches about 0.13~0. of the sample thckness, the smallest dmenson, the growth knetcs starts to devate sgnfcantly (e.g., more than 10%) from the parabolc growth. Ths predcton agrees well the expermental observaton (Fg. 7.). 145

164 τ=100 (a) τ=400 (b) 1000 D (n normalzed unt) D * τ(n normalzed unt) Fgure 7.1. Two-dmensonal phase feld smulaton of sample sze effect on mcrostructure evoluton (a) and gran growth knetcs (b). The sample thckness s 19 (n reduced unt) and γ s /γ b s

165 6 5 4 D (mm ) tme(s) Fgure 7.. Re-plot of expermental data presented n reference [4]. Dfferent symbols stand for dfferent sothermal temperatures: open trangle ~100 C, close trangle ~1150 C, open crcle ~1100 C and close crcle ~1050 C. The two horzontal lnes ndcates the range of D*, the gran sze at whch the growth knetcs devates sgnfcantly from the parabolc growth, determned by Eqs. (1) and () wth γ s /γ b =~ Effect of Texture Two types of texture, {110}<11> and {001}<110>, have been observed n forged T- 6Al-4V alloys [155]. In ther expermental work, Sematn et. al. [157] showed sgnfcant dfference n gran growth knetcs between two texturally dfferent but otherwse mcrostructurally dentcal samples, whch suggests that texture may play an mportant role n beta gran growth. Ivasshn et. al. [155] observed a dscontnuty n the growth knetcs. 147

166 They attrbuted the phenomenon to the oscllaton of two beta texture components durng beta gran growth. Despte the large body of lterature on texture-controlled gran growth, a quanttatve knetc model that accounts for texture effect s stll unavalable. Ths may be due largely to the fact that texture evoluton s determned by the nterplay among many parameters, ncludng the number of texture components, ntal fracton, spatal dstrbuton and gran sze dstrbuton of each texture component, and the dependences of gran boundary energy and moblty on msorentaton. To nvestgate the ndvdual and combned effects of these factors on texture evoluton and gran growth knetcs, we developed a D phase feld model and carred out a seres of smulatons wth dfferent combnaton of the factors. The detals have been dscussed n a separated work [11]. A typcal example of the texture evoluton durng gran growth obtaned from the smulatons s shown n Fg. 7.3, where grans of dark shade have random orentatons and grans of lght shade are textured grans. Startng from random spatal dstrbuton of both the textured and randomly orented grans of smlar sze dstrbutons, and assumng that hgh angle gran boundares have hgh energy and hgh moblty, we observed a substantal ncrease of the texture component (Fg. 7.3) durng gran growth. The correspondng gran growth knetcs s presented n Fg. 7.4, whch shows a clear devaton from the parabolc growth. 148

167 τ=100 τ=1000 Fgure 7.3. Texture evoluton durng gran growth obtaned n a system wth an ntally randomly dstrbuted texture component (n lght shade of gray). Both gran boundary energy and moblty are assumed to have hgh values for hgh angle gran boundares n the smulaton. τ s reduced tme. 800 D (n normalzed unt) τ(n normalzed unt) Fgure 7.4. Knetcs of texture-controlled gran growth correspondng to the mcrostructural evoluton shown n Fg

168 To develop a knetc equaton characterzng texture-controlled gran growth, below we present a smple statstcal analyss. Based on a mean feld approxmaton, the texturecontrolled gran growth knetcs may be descrbed by the followng equaton: 1 dd = Kθ f ( θ ) dθ dt (7.3) D where θ s the dsorentaton angle (the mnmum angle among all the symmetry related msorentatons), K θ s a rate constant and f(θ) s the length fracton of gran boundares of a partcular dsorentaton. If the mobltes of low-angle gran boundares are much smaller than that of random hgh-angle gran boundares, ther contrbutons to gran growth can be neglected. Thus, Eq. (7.3) could be smplfed further as 1 dd = K h f m dt (7.4) D where K h s the rate constant for hgh-angle gran boundares and f m s the fracton of moble gran boundares. In materals wth heavy texture, f m s usually much smaller than unty, whch suggests that the present of texture wll slow down sgnfcantly the gran growth knetcs as compared to a texture-free system. Consderng the fact that an mmoble boundary may anchor moble boundares that ntersect wth t, we may approxmate f m as: f =1 (7.5) m Af l where A s a postve constant related to the spatal dstrbuton of low-angle gran boundares and f l s the fracton of the low-angle gran boundares. In hs statstcal analyss, Markenze [163] suggested that low-angle gran boundares seldom ext n a 150

169 system wth fully randomly orented grans. Therefore, the amount of low-angle gran boundares n a system s drectly related to the ntensty and spatal dstrbuton of texture components. For the sake of smplcty, though not necessary, we assume a random dstrbuton of the texture components. In ths case, the fracton of low-angle gran boundares can be obtaned by the probablty of occurrence of two adjacent texture grans, e. g., f l f T = ( ) (7.6) where f T s the area fracton of the th texture component. Combnng Eqs. (7.4), (7.5) and (7.6), the gran growth knetcs can be expressed as a functon of the fractons of exstng texture components 1 ( f ) dt dd = K h 1 A T (7.7) D The only unknown varable n Eq. (7.7) s the fracton of texture. It can be obtaned ether from computer smulatons or from expermental measurements. Fgure 7.5 shows a plot of D K h dd dt 1 vs. ( ) f T for the smulaton results presented n Fg The lnear relatonshp observed confrms the valdaton of Eq. (7.7) n the partcular case consdered n the smulaton. In ther expermental work, Ivasshn et. al. [164] measured the gran growth as well as texture evoluton data n T-6Al-4V durng beta annealng under contnuous heatng condtons. Predctons made usng the model are shown to compare well aganst ths set of data, as shown n Fg

170 In summary, we revewed the effects of sample sze and texture on gran growth knetcs of T-6Al-4V durng beta annealng. It s shown that these effects could result n sgnfcant devaton of the knetcs from the parabolc law. Based on expermental observatons and computer smulaton studes, gran growth models that account for these factors are ntroduced and valdated aganst expermental observatons and computer smulatons K/K h f T *f T Fgure 7.5. Valdaton of equaton (7.7) aganst phase feld smulaton results. The lnear regresson gves A~

171 ftexture T( C) Calculated Measured D(µm) Calculated measured T( C) Fgure 7.6. Comparson of measured texture fracton and beta gran sze at varous peak temperatures wth predcted values for contnuous heatng condton. 7.3 Gran Boundary α Consttutve equatons for gran boundary alpha growth n T-6Al-4V under contnuous coolng condtons were developed based on expermental observatons and theoretcal models n lterature, and Phase Feld and Dctra smulatons. The model parameters were optmzed usng expermental data obtaned at OSU on Gleeble 1500 and valdated aganst expermental measurements conducted ndependently on samples heat treated at GE Arcraft Engne. In prncple, gran boundary alpha precptaton n T-based alloys s smlar to 153

172 gran boundary ferrte formaton n steels. The latter has been studes ntensvely n recent years [ ]. Smlar work can also be found for T-based bnary alloys [ ]. Gran boundary alpha usually has a coherent nterface wth the matrx gran on one sde of the boundary and a sem-coherent or ncoherent nterface on the other sde. Ths has been found to be generally true for varous alloy systems [17-176]. A revew of nucleaton of gran boundary allotromorphs n steels was gven recently by Aaronson [177]. The treatment of Aaronson s based on classc nucleaton theory on gran boundary proposed by Cahn [178, 179]. The nfluence of crystallography on gran boundary nucleaton was studed by Johnson et al. [180]. The theory has been appled to T-based bnary alloy n the work of Menon and Aaronson [171]. Once nucleaton has set the orentaton relatonshp, the lengthenng of gran boundary precptates proceeds by growth along the boundary and concurrent thckenng. Accordng to expermental observatons n ndustral samples, however, nucleaton and lateral spreadng of gran boundary alpha n T-6Al-4V take place rather quckly and the thckness of gran boundary alpha s largely determned by the thckenng knetcs. Accordng to Zener [148] and Atknson et. al [166], the thckenng of gran boundary allotromophs as a dffuson-controlled process follows parabolc knetcs. Expermental studes also revealed that both thckenng and lengthenng obey the parabolc law. Snce the growth knetcs of gran boundary alpha n many systems s somewhat too fast to be volume-dffuson controlled, t s n general beleved that gran boundary dffuson plays an mportant role. Both Phase Feld and Dctra methods were appled to smulate the thckenng 154

173 knetcs of gran boundary alpha n T-6Al-4V. We assume that a layer of gran boundary alpha of neglgble thckness develops rght after the temperature drops below the beta transus. Durng further coolng or sothermal holdng, gran boundary alpha contnues to grow tll the moment when sdeplates develop. Both methods recovered the parabolc growth law [181] wth assessed knetc and thermodynamc databases for T-V-Al. A seres of smulatons usng both Phase Feld and Dctra methods were conducted at dfferent temperatures. Based on both theory and smulaton results, an sothermal gran boundary alpha growth model was develop: δ ()= t AT ( 0 T ) exp Q RT t (7.8) where δ s gran boundary alpha thckness, A s a constant, T 0 s beta transus temperature, Q s actvaton energy and R s gas constant. Usng the prncple of addtvty [18], the gran boundary alpha thckness under contnuous coolng condtons can be obtaned by summatons of many short sothermal processes wthn tme duraton t at temperature T Q δ = A( T0 T ) exp () t (7.9) RT t Please note that mportant nputs to the above models are the beta transus temperature and sdeplate startng temperature (for contnuous coolng) or sdeplate startng tme (for sothermal holdng), whch have to be obtaned separately from thermodynamcs database and experments. Based on the CCT dagram (Fg. 7.7) establshed for T-6Al- 4V by Matre [183] and confrmed by recent studes [184, 185], the dependence of the beta transus temperature and the maxmum transformaton rate temperature (sdeplate 155

174 startng temperature) on the coolng rate can be obtaned. Assumng that the growth of gran boundary alpha stops when sdeplates start to develop, one can dentfy the temperature range for gran boundary alpha growth under varous coolng rates, as shown n Fg Fgure 7.7. Characterstc temperature ranges for gran boundary alpha growth at varous coolng rates. The top curve ndcates the startng temperature whle the bottom one ndcates the endng temperature. A seres carefully desgned experments on gran boundary alpha growth under contnuous coolng condtons have been conducted at OSU for model development. The samples are taken from a rolled plate from Howmet Corporaton. The dmensons of the samples are (mm). Heat treatments were performed on Gleeble 1500, followng the schedule shown n Fg The mcrostructures developed were characterzed by SEM and the stereology procedures were used to measure the gran boundary alpha 156

175 thckness β treatment Gran boundary α Fgure 7.8. Heat treatment schedule for gran boundary alpha growth knetcs measurement Gran boundary alpha thcknesses under three dfferent coolng rates were measured. The thckenng knetcs follows the followng equaton δ = ( T 0 T ) exp 0849 RT t (7.10) where δ s n µm, t s n second, T s n K and R = J/mol-K. The actvaton energy n Eq.( 7.10) s about 15% less than that of T self-dffuson n bulk, whch agrees wth the expermental measurement by Aaronson [177]. The obtaned model was further valdated aganst a set of data from samples heat treated at GE (forged sample wth dmensons of ~3 ~6mm). The agreement between the expermental measurement and model predcton s reasonably good, as shown n Fgure

176 3.0.5 Measured thckness(µm) OSU data GE data predcted values(µm) Fgure 7.9. Comparson of measured gran boundary alpha thckness at varous coolng rates wth predcted values. Lnear lne s the model predcton, wth model parameters optmzed aganst experment performed at OSU (red sold crcles). The green squares are expermental data obtaned ndependently on GE samples. A smlar model can be developed for sothermal treatments f a TTT dagram for sothermal transformaton, n partcular, the startng tmes of sdeplates at dfferent temperatures are avalable. In summary, a gran boundary alpha growth model for T-6Al-4V under contnuous coolng condtons has been developed. The model predcted successfully gan boundary alpha thckness for two ndependent sets of expermental data. Wth more expermental data becomng avalable, the model wll be further refned and enhanced. A smlar model for sothermal processes s readly developed when the startng tmes for sdeplate formaton at dfferent temperatures become avalable. 158

177 7.4 Sdeplate and Colony Structure Quanttatve studes of both lengthenng and thckenng knetcs of sdeplate have been conducted. The mechansm of edgewse growth of sdeplates was explored by Zener and later on by Hllert n steels [148, 186]. Buldng on Zener s work Hllert ncorporated the effects of thermodynamcs ncludng capllarty n a rgorous fashon. For a volume dffuson controlled reacton, the edgewse growth rate of a plate s gven by: v D 1 1 r = Ω c 1 r r (7.11) where Ω s the super-saturaton, r c s the crtcal radus correspondng to zero growth r c rate and 1 s Gbbs-Thompson factor. r Assume that the stable state obtaned at the maxmum growth velocty r = rc, above equaton was translated nto: vr 1 c = Ω (7.1) D 8 Krkaldy [187] further approxmated rate expresson for Wdmanstatten growth of the form Q / RT v ~ D e T (7.13) Krkaldy [187] found that hs expermental observaton n low alloy steel fts very well wth Eq. (7.13)

178 Sdewse growth of a plate s less understood than ts edgewse growth. Two dfferent growth mechansms have been suggested n the lterature. Accordng to Aaronson [188], the coherent broad face could only grow perpendcular to tself by nucleaton and mgraton of ledges. Enomoto [189] solved the mgraton of a tran of steps usng fnte dfference scheme. He found that the thckenng knetcs follows parabolc law,.e., the thckness s proportonal to the square root of tme. On the other hand, our phase feld modelng of the dffuson controlled growth of an solated sdeplate also showed a parabolc thckenng knetcs, whch s n agreement wth the work of Agren [190]. Accordng to these studes, one can assume the followng consttutve equaton for the thckness of a free thckenng sdeplate Q / RT 1/ λ = D e (7.14) 0 Tt Snce the sdeplate spacng s comparable wth the dffuson dstance n the colony structure of T64, soft mpngement between the adjacent advancng plates must be taken nto account to descrbe the thckenng knetcs of a sdeplate n a colony structure. Usng mean feld approxmaton we have: 1 Q RT d = D e / 1/ λ λ 0 Tt 1 dt (7.15) s where s s the sdeplate spacng. In our earler work of phase feld modelng of sdeplate growth a morphologcal nstablty mechansm was assumed for the ntaton of sdeplates from gran boundary alpha. Accordng to Mullns and Sekerka [191], the spacng between the sdeplates s gven by 160

179 s = γv RT D c v c c 1/ m 0 π 3 (7.16) where V m s the molar volume, D s dffuson coeffcent, c 0 s equlbrum solute concentraton n the prmary phase, c s the solute concentraton n precptate phase, c s s the solute concentraton n local equlbrum at nterface and v s the velocty of nterface. All these parameters are obtaned at the moment of nucleaton of sdeplate, whch can be determned by the CCT dagram for gran boundary alpha growth dscusson n prevous secton. Qualtatvely, the sdeplates wll nucleaton at lower temperatures at larger coolng rates, leadng to smaller spacng as well as lower dffuson rate. Therefore, one would expect smaller sdeplate thcknesses at larger coolng rates. s Valdaton of consttutve equatons by phase feld smulatons Phase feld smulatons were carred out for both an solated sngle sde and a group of sdeplates n a colony structure for T64 to valdate the proposed consttutve equatons. Fg shows the thckenng and lengthenng knetcs of a sngle sdeplate. It can be seen that the lengthenng velocty s constant and thckenng knetcs s parabolc wth respect to tme. Fg llustrates the average thckness of an aggregate of sdeplates n a colony structure as functons of tme. Snce the overlap of the dffuson felds among neghbored sdeplates, the average thckness of the sdeplate n a colony structure s smaller than that of the free growng sdeplate n long tme tendency and the thckenng 161

180 knetcs of multple sde plate can be ftted nto Eq. (7.15) very well L(µm) w(µm) t(s) (a) t(s) (b) Fgure Lengthenng (a) and thckenng (b) knetcs of a sngle plate calculated wth PFM usng Thermocalc and Dctra database. 16

181 w(µm) sngle sdeplate sdeplate colony (a) t(s) (b) Fgure Lengthenng (a) and thckenng (b) knetcs of a sngle plate calculated wth PFM usng Thermocalc and Dctra database. In order to study the colony structure formed by competton between dfferent sde plates, a mult feld model was developed (Chapter.). The model has been successfully appled to a system wth one sx-sded gran. The smulaton was ntated wth a layer of gran boundary alpha. After quench the system to the low temperature,.e. 800 C, sde plates wth varous orentatons develop automatcally followng the nterface nstablty mechansm. Eventually, the colony structure was formed by the competton between dfferent sde plates (as shown n Fg. 7.1). 163

182 Fgure Smulated sde plate growth and colony structure. Wthn the sx-sded gran, whte phase s alpha phase and black color stands for beta phase. 164