Modeling of precipitate-free zone formed upon homogenization in a multi-component alloy

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1 Acta Materala 55 (27) Modelng of precptate-free zone formed upon homogenzaton n a mult-component alloy Ch.-A. Gandn a,b, *, A. Jacot c,d a CEMEF UMR CNRS-ENSMP 7635, Ecole des Mnes, BP27, 694 Sopha Antpols, France b LSG2M UMR CNRS-INPL-UHP 7584, Ecole des Mnes, Parc de Saurupt, 5442 Nancy, France c LSMX, MX-G, Ecole Polytechnque Fédérale de Lausanne, 115 Lausanne, Swtzerland d CALCOM ESI SA, PSE, Ecole Polytechnque Fédérale de Lausanne, 115 Lausanne, Swtzerland Receved 29 June 26; receved n revsed form 2 November 26; accepted 2 November 26 Avalable onlne 8 February 27 Abstract A comprehensve model s presented for the smulaton of mcrostructure evoluton durng ndustral soldfcaton and homogenzaton processng of alumnum alloys. The model combnes on the one hand mcrosegregaton due to long-range dffuson durng soldfcaton and subsequent heat treatment wth, on the other hand, precptaton n the prmary Al phase. The thermodynamc data are drectly obtaned from a CALPHAD (CALculaton of PHAse Dagrams) approach to thermodynamc equlbrum n multcomponent systems. The model s appled to the predcton of structure and segregaton evolutons n a 3 alumnum alloy for typcal ndustral soldfcaton and homogenzaton sequences. It s shown that: () accountng for the nucleaton undercoolng of the eutectc/pertectc structures soldfyng from the melt s essental to retreval of the measured volume fractons of ntergranular precptates; () calculatons of ntragranular precptaton are generally not applcable f long-range dffuson s neglected; () the precptate-free zone can be quanttatvely predcted only based on the couplng between ntergranular and ntragranular precptaton calculatons. Ó 27 Acta Materala Inc. Publshed by Elsever Ltd. All rghts reserved. Keywords: Precptate-free zone; Homogenzaton; Precptaton; Modelng; Alumnum alloy 1. Introducton Precptate-free zones (PFZ) form durng the heat treatment of metallc alloys [1,2]. A PFZ conssts of a regon of the ntragranular prmary Al matrx phase wth no or very lmted amounts of precptates. It s delmted by a precptate-rch zone of the same ntragranular prmary Al matrx phase on one sde and ntergranular precptates at the gran boundary on the other sde. Fg. 1 shows a typcal PFZ observed n a homogenzed 3 alumnum alloy. The mcrostructure s composed of several prmary grans of the Al-fcc phase, also hereafter referred to as the matrx phase, surrounded by ntergranular precptates and contanng ntragranular precptates n the core. The role of * Correspondng author. E-mal address: charles-andre.gandn@ensmp.fr (Ch.-A. Gandn). PFZ on mechancal propertes s clearly attested n the lterature. Its effect has been demonstrated for the fracture toughness of alumnum alloys, more specfcally for Al L alloys (2 seres) [3] and Al Mg Zn alloys (7 seres) [4]. Smlarly, the yeld strength, the ultmate tensle strength and the plastc stran to fracture of a c -strengthened nckel-base superalloy have been lnked to the wdth of the PFZ [5,6]. As revewed by Maldonado and Nembach [5], several explanatons are gven n the lterature for the orgn of PFZ. The man mechansm that prevals s the competton between ntragranular and ntergranular precptatons. Indeed, both transformatons requre the dffuson of the same chemcal solute elements of the supersaturated matrx to take place. The wdth of the PFZ thus depends on the dffuson of the solute elements toward the ntergranular precptates compared to the potency of the ntragranular /$3 Ó 27 Acta Materala Inc. Publshed by Elsever Ltd. All rghts reserved. do:1.116/j.actamat

2 254 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Compostons Volume fractons Sze dstrbutons Mean radus Denstes r O alpha (Al(Fe,Mn)S) ntragranular precptates Precptate Free Zone (PFZ) 1 µm Al6Mn6 Mn( (Al(Fe,Mn) 6 ) and alpha (Al(Fe,Mn)S) ntergranular precptates Fg. 1. Homogenzed mcrostructure of a 3 alumnum alloy. A PFZ forms as a result of solutal nteractons between the ntergranular Al6Mn and alpha precptates and the ntragranular alpha precptates [28]. Note the dfference n sze between the ntergranular alpha precptates compared to the fne ntragranular alpha precptates. The ntergranular alpha precptates form as a result of the transformaton of the coarse ntergranular Al6Mn precptates produced durng soldfcaton, whle the ntragranular alpha precptates nucleate and grow n the prmary Al grans. precptates to nucleate and grow n the prmary Al matrx phase. Several models have been proposed descrbng the development of the PFZ durng ageng [7 9] and homogenzaton treatments [1,11]. Whle beng able to predct comprehensvely the wdth of the PFZ, these models are based on one or several of the followng approxmatons: bnary alloy; stochometrc composton of the precptates; smplfed phase dagrams; sothermal heat treatment; and analytcal solutons of the mathematcal problem of long-range dffuson n the matrx. The ntal condtons for mcrostructure calculatons durng homogenzaton heat treatments are also generally approxmated by smplfed soldfcaton paths. In the present contrbuton we develop, a coupled precptaton homogenzaton model whch does not use any of these approxmatons. A partcle sze dstrbuton (PSD) method for the descrpton of precptaton [12,13] has been coupled wth a pseudo-front trackng (PFT) method for the predcton of segregaton profles and ntergranular precptates durng soldfcaton and heat treatment [14,15]. Both the PFT and PSD methods use drect couplng wth the equlbrum computng program Thermo-Calc [16] and an approprate thermodynamc database [17]. As a result of the nteracton between precptaton and long-range dffuson, the PFZ s predcted. The model thus represents a necessary elementary brck of a through process modelng approach that ams to predct the fnal propertes of a metal part by ntegratng the role of each of the ndvdual thermo-mechancal formng steps on mcrostructure evoluton [18]. Comparson of the model s performed wth measured data for a 3 alumnum alloy soldfed as an extruson bllet usng the drect-chll contnuous castng process and then homogenzed at 6 C [19,2]. 2. Modelng 2.1. Soldfcaton and homogenzaton modelng: the PFT method The evoluton of mcrostructure durng soldfcaton s descrbed wth the PFT method [14,15]. The PFT method permts the calculaton of the evoluton of sold/lqud nterfaces that are governed by ansotropc nterfacal energes and the dffuson of several solute speces. Growth of the prmary Al phase (fcc) from the lqud (l) s descrbed by solvng the dffuson equatons n both phases for each solute ¼r½Dm rxm Š wth 8 2½1; nš and m ¼ fcc;l ð1þ where n s the number of alloyng elements n the multcomponent system, x m the concentraton of element n phase m, and D m s the dffuson coeffcent. The poston and velocty of the nterface beng part of the problem, a solute balance has to be formulated at the fcc/l nterface: D fcc ½rx fcc Š n þ D l ½rxl Š n ¼ ðx l x fcc Þv n 8 2½1; nš ð2þ where v * s the nterface velocty and n * s the normal vector to the nterface pontng towards the lqud. The superscrpt * denotes quanttes taken at the curved fcc/l nterface.

3 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) The concentratons x l and x fcc n each phase are deduced from the phase dagram takng nto account the curvature of the nterface. The soluton of ths problem s obtaned wth an explct fnte volume method usng a fxed grd. The dsplacement of the nterface s handled through state transtons of the fnte volume cells, from l to (fcc + l) and from (fcc + l) to fcc. Further detals of the method can be found n Ref. [14]. The calculatons are drectly coupled wth the phase dagram software Thermo-Calc [16] and the Al-Data database [17] usng the optmzed couplng scheme descrbed n Ref. [14]. As the lqud becomes undercooled for another sold phase, predcton of phase transformatons n the ntergranular regons s started out. In ths approach, the ntergranular regons are consdered as a mxture of lqud and sold phases. The followng assumptons are made: () the composton of the ntergranular lqud s unform when the frst ntergranular precptaton phase starts to form; () the ntergranular regon s locally n thermodynamc equlbrum and all phases have unform concentratons; () back dffuson affects the ntergranular regon n an unform manner. A solute balance s performed over the ntergranular regon, X m, for all the solute elements: Z J fcc C fcc=m 8 2½1; nš Z n dc ¼ X Z dx ðx m C fcc=m x fcc Þv n dc where J fcc ¼ D fcc ½rx fcc Š n s the back-dffuson flux of element, x m the average concentraton of element n the ntergranular regon, C fcc/m the contour of the fcc/m nterface whch separates the prmary Al phase and the ntergranular regon, v * the velocty of the fcc/m nterface, and n * s the normal vector to C fcc/m pontng toward X m. Introducng U fcc we obtan: ð3þ ¼ R C fcc=m J fcc n dc and dscretzng (3), U fcc dt ¼ V m dx m þ dv m ðx m x fcc Þ 8 2½1; nš ð4þ where V m s the volume of X m and the symbol d expresses small ncrements. The mxture composton can be expressed as: x m ¼ Xp m¼1 g m x m 8 2½1; nš ð5þ where p s the number of phases n the mxture, x m the equlbrum concentraton of element n phase m, and g m s the volume fracton of phase m n the ntergranular regon. Introducng Eq. (5) nto (4), one obtans: U fcc dt ¼ V m X p þ dv m m¼1! dg m x m þ Xp g m dx m m¼1!! X p g m x m x fcc m¼1 8 2½1; nš ð6þ Assumng that thermodynamc equlbrum s satsfed n the ntergranular mxture, the temperature, T, can be related to the phase dagram: T ¼ T m=fcc x fcc 1 ;...; x fcc n 8m 2½1; pš wth 8m 6¼ fcc ð7þ where T m=fcc ðx fcc 1 ;...; x fcc n Þ expresses the soldus (or solvus) temperature as a functon of the concentraton n the prmary Al matrx phase. Ths functon s evaluated wth the phase dagram software Thermo-Calc [16] and the thermodynamc database Al-Data [17]. The concentratons of the other phases are gven by the te-lnes: x m ¼ k m=fcc ðx fcc 1 ;...; x fcc n Þx fcc 8m 2½1; pš wth 8m 6¼ fcc and 8 2½1; nš ð8þ where the k m=fcc are partton coeffcents defned wth respect to the prmary Al matrx phase (fcc), whch are also obtaned from Thermo-Calc and Al-Data. The back-dffuson contrbuton, U fcc, results form the resoluton of Eq. (1) for m = fcc only. The calculaton s made wth the same explct fnte volume method as for the prmary Al phase calculaton. If the thermal hstory s known, the evoluton of the system can be descrbed by solvng Eqs. (6) (8) whch form a set of n +(n + 1)(p 1) equatons and comprses np + p + 1 unknowns: the fdx m g, the {dg m} and dv m. By appendng the followng equatons to the system, the problem becomes closed: X p dg m¼1 m ¼ ð9þ dv m k ¼ ð1þ dv m þ dg fcc V m Eq. (1) expresses the proporton of the prmary Al matrx phase formed on the fcc/m nterface, whch s drectly related to the varaton of the mxture volume, dv m, wth respect to the total amount (fcc formed on fcc/m nterface and wthn X m ). By selectng an approprate value for the eutectc dstrbuton parameter, k, t s possble to dstngush the behavors of dvorced and coupled eutectcs. If k s set to 1 (.e., dg fcc = ), eutectc Al wll form only on top of prmary Al (dvorced eutectcs), whereas for k = (dv m = ) t wll be dstrbuted n the ntergranular regon and the prmary Al phase boundares wll reman statonary (coupled eutectcs). Each secondary phase, m, s attrbuted a nucleaton undercoolng, DT m nucl, whch s a parameter of the model. The phase m s ntroduced n the calculaton as the followng condton s satsfed: T þ DT m nucl 6 T m=fcc ðx m 1 ;...; xm n Þ 8m 2½1; pš wth 8m 6¼ fcc ð11þ When the volume fracton of a phase reaches, the phase s wthdrawn from the calculaton. After the lqud has dsappeared from the ntergranular phase mxture, the calculaton s contnued so that evoluton of the ntergranular precptates durng homogenzaton

4 2542 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) can be predcted. The effect of ntragranular dffuson upon growth or dssoluton of ntergranular phases s taken nto account wth the same formalsm as for soldfcaton. Ths process s, however, strongly nfluenced by precptaton takng place n the prmary Al phase,.e., ntragranular precptaton. A precptaton model s therefore coupled to the homogenzaton calculaton Modelng of ntragranular precptaton: the PSD method The PSD method [12,13] s used to track the evoluton of the ntragranular precptates formed n the supersaturated prmary Al matrx phase. It s an extenson of models prevously proposed n the lterature [21,22]. Its formulaton s adapted to handle non-stochometrc precptates formed n mult-component alloys. It s also made compatble for couplng wth the PFT model [14,15] by consderng mass balances for an open system. Indeed, long-range dffuson smulated by the PFT model does modfy the local average composton wth tme, thus changng the precptaton knetcs. The conservaton equaton for the densty of m-phase precptates, N m, growng n a supersaturated prmary Al matrx phase (fcc) can be wrtten m ¼ rðn m v mþ n mþ ÞþS m 8m 2½1; qš where v m+ s the growth rate of the precptates of radus r m at tme t, n m+ s the normal vector to the m/fcc nterface pontng towards the prmary Al matrx phase and q s the number of precptaton phases formed n the prmary Al matrx phase. The superscrpt + denotes quanttes taken at the m/fcc nterface. Assumng the precptates reman sphercal and are surrounded by a steady-state dffuson feld, the growth rate for the m-phase precptates, N m, s the soluton of the second Fck s law gven by [23]: m m ¼ Dfcc r m x fcc x mþ x fccþ x fccþ 8m 2½1; pš; 8 2½1; nš ð13þ where x fcc s the average composton of element n the matrx. A local equlbrum at the nterface between the matrx and the precptates s assumed. A thermodynamc calculaton s therefore used to determne the compostons of element at the (m/fcc)-nterface n the m-phase precptates and n the matrx, respectvely x mþ and x fccþ. The dffuson coeffcent of solute element n the prmary Al matrx phase, D fcc, s computed followng an Arrhenus law. The source term enterng Eq. (12) for the m-phase precptates, S m, s modeled by an heterogeneous nucleaton law gven by [24 27]: S m ¼ðN m max N m DGm hom totþzb exp f ðhm Þ 8m 2½1; qš k B T ð14þ where N m max s the densty of heterogeneous stes avalable for the nucleaton of m-phase precptates and N m tot s the actual total densty of precptates. Coeffcent b accounts for the rate at whch solute atoms from the matrx can jon the nucleus. For a bnary alloy, ts expresson s gven by b ¼ 4pðr m Þ 2 D fcc x fcc =ðk fcc Þ 4 where k fcc s the dffuson dstance n the prmary Al matrx phase. For a mult-component alloy t s chosen to consder the element wth a combnaton of a slow dffuson rate n the matrx and a low composton as the lmtng factor for ncreasng the sze of the nucleus. The mnmum product D fcc x fcc over all elements n the prmary Al matrx phase s thus used and one can wrte b ¼ 4pðr m Þ 2 MnðD fcc x fcc =ðk fcc Þ 4. The crtcal energy barrer for the formaton of new m-phase precptates of crtcal radus r m ¼ 2r m=fcc V m =DG m n s gven by DGm hom ¼ð4=3Þprm=fcc ðr m Þ 2, where r m/fcc s the nterfacal energy of the m/fcc nterface, V m s the molar volume of the m-phase and DG m n s the drvng force for nucleaton of the m-phase precptates n the prmary Al matrx phase. The wettng functon s gven by the relatonshp f(h m ) = (1/2)(2 + cosh m ) (1 cosh m ) 2, where h m s the wettng angle of a m-phase nucleus wth ts heterogeneous nucleaton ste [27]. The Zeldovtch s factor accounts for the fluctuaton of the sze of the nucleus due to the emsson of solute atoms from the nucleus back nto the matrx phase. Its estmaton s computed pffffffffffffffffffffffffffffffffffff usng the relatonshp Z ¼ðDG m n Þ2 = 8pV m ðr m=fcc Þ 3=2 N Av k B Tf ðh m Þ, wth T the temperature, k B Boltzmann s constant and N Av Avogadro s number. A balance equaton for the composton of solute element can be wrtten as: X X m r 4 3 pðrm Þ 3 N m ðx m xfcc Þ¼x x fcc 8m 2½1;qŠ; 8 2½1;nŠ ð15þ where x s the average composton of. The precptate composton x m s assumed unform. It s smply gven by the nterface composton x mþ. It can vary wth temperature snce the precptates are not stochometrc. Also notceable s the possblty for the average composton of element, x, to vary wth tme. Ths s useful when consderng the couplng wth the long-range dffuson computed by the PFT model as explaned below. Although t s not used n the present contrbuton, the PSD method was developed for the concomtant nteracton of several famles of precptates n the same prmary Al matrx phase. Ths s made vsble wth the summaton of m-phase precptates n Eq. (15). Several famles of precptates could ether correspond to several m-phases precptatng smultaneously n the same matrx, or to one phase wth several nucleaton parameters (e.g., for applcaton to the nucleaton of a sngle precptatng phase on several famles of heterogeneous stes), or also to a combnaton of both. The drvng force for nucleaton of the m-phase precptates n the prmary Al matrx phase s computed wth an deal soluton thermodynamc approxmaton: DG m n ¼ R gt Xn ¼1 x m 1 x fcc ln x fcc 1 8m 2½1; qš ð16þ

5 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Equlbrum compostons of element at the (m/fcc)- nterface, respectvely n the prmary Al matrx phase and n the m-phase precptates, enterng Eq. (16), x fcc 1 and x m 1, are gven by equlbrum calculatons for known values of the temperature, T, and the average composton of elements, x. The Gbbs Thomson effect s accounted for based on a modfcaton of the solublty product, K m, wth respect to ts equlbrum value, K m 1, also computed usng the equlbrum phase dagram compostons: K m ¼ K m 1 exp 2 r m=m V m wth r m R g T K m ¼ Yn ½x fccþ Š xmþ and K m1 ¼ Yn ½x fcc 1 Š xm 1 8m 2½1;qŠ ¼1 ¼1 ð17þ where x fccþ and x mþ are the composton of elements at the (m/fcc)-nterface n the prmary Al matrx phase and n the m-phase precptates, respectvely. Thermo-Calc [16] and the Al-Data database [17] are used for the calculatons of the equlbrum compostons x fcc 1 and x m 1 enterng both Eqs. (16) and (17) Couplng of the PFT and PSD methods Fg. 2 shows a schematc flow chart of the couplng between the PFT and PSD models. Durng a tme-step, the varatons of the matrx composton due to long-range dffuson are calculated wth the PFT model by solvng Eq. (1) for the prmary Al maxtrx phase (m = fcc) and sent to the PSD model. Nucleaton, growth and coarsenng of the ntragranular precptates are predcted by the PSD model by performng separate calculatons for each cell of the PFT fnte volume grd that s located n the prmary Al phase regon of the calculaton doman. The prmary Al matrx compostons n the fnte volume cells, x fcc, are modfed by the PSD and fed back to the PFT model. The tme-step s determned so as to satsfy the stablty crteron of the explct tme ntegraton scheme used by the PFT model. The PSD model also makes use of a fnte volume method to solve Eq. (12), but wth an mplct tme ntegraton scheme. In addton, teraton procedures are requred to ensure convergence of the average matrx compostons and the nucleaton rate n the PSD model as shown n Fg. 2. Smulaton results of the coupled PFT PSD model consst of tme evoluton of profles n the prmary Al phase for the compostons, sze dstrbutons and volume fractons of precptates, as well as volume fractons of ntergranular phases. As a result, the wdth of a possble PFZ can be calculated. The poston of the PFZ s arbtrarly determned as the locaton n the prmary Al phase where the volume fracton of the ntragranular precptates falls below 1% of the same quantty averaged over the entre prmary Al phase. The calculatons were performed n one space dmenson usng a sphercal calculaton doman of 8.6 lm, whch corresponds to half of the secondary arm Loop on tme-steps PFT model Loop on all cells PSD model spacng reported n Ref. [2]. Axes sketched n Fg. 1 llustrate the coordnate system used as well as the typcal profles deduced from smulatons. 3. Expermental L and Arnberg [19,2] and Dehmas [28] provde dentfcaton of the phases defnng the PFZ formed upon the homogenzaton heat treatment of a 3 alumnum alloy of composton Al-.58 wt.% Fe-1.15 wt.% Mn-.2 wt.% S-.8 wt.% Cu. The soldfcaton gran structure s made up of globular dendrtes of the prmary Al matrx phase whch are bounded by Al 6 (Mn,Fe) and a-al(fe,mn) S ntergranular precptates. For smplcty, hereafter the two phases wll be referred to as alpha and Al6Mn, respectvely. Durng the heatng stage of the homogenzaton treatment, alpha precptates appear n the ntragranular regons as a result of precptaton from the supersaturated prmary Al matrx phase. Concomtantly, a eutectod reacton partly transforms the ntergranular Al6Mn precptates nto ntergranular alpha precptates and ntergranular eutectod Al. Ths reacton requres the dffuson of solute speces from the prmary Al matrx phase to the ntergranular regon. Smlar observatons were prevously reported by Alexander and Greer n an Al-.5 wt.% Fe-1. wt.% Mn-.2 wt.% S alumnum alloy [29]. L and Arnberg provde measured data that wll be used hereafter [19,2]. 4. Results Fluxes n the prmary Al phase New local compostons n the prmary Al phase 4.1. Calculaton condtons Interdendrtc phases Loop on the prmary Al phase cells Nucleaton drvng force Iteraton loop Precptate sze dstrbuton New local compostons n the prmary Al phase Fg. 2. Partal flow chart of the model showng the prncple of the couplng between the PFT and PSD methods ntegrated n the tmesteppng algorthm. The model was used to descrbe the evoluton of the mcrostructure n a 3 alumnum alloy havng the same composton as the alloy characterzed by L and Arnberg

6 2544 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) [19,2] (composton n at.%:.283% Fe,.57% Mn, and.194% S), yet neglectng the nfluence of Cu. The nfluence of nterface curvature upon thermodynamc equlbrum at the prmary Al/lqud nterface was neglected n the PFT calculatons. The coolng rate was assumed to be.1 C/s durng soldfcaton and between 1.5 and 3 C/s after the lqud had entrely dsappeared. The eutectc dstrbuton parameter, k, was set to 1,.e., a fully dvorced eutectc structure was assumed. For the homogenzaton/precptaton calculatons, the as-cast materal s ramped up to 6 C at 5 C/h, mantaned for 7 h and cooled down to room temperature at 5 C/h, followng the expermental condtons of L and Arnberg [19]. Dffuson coeffcents are taken from Ref. [3]. Table 1 summarzes the three sets of smulatons presented n ths contrbuton, whle Table 2 lsts the values of all other parameters. The parameters changed are essentally the nucleaton temperature of the ntergranular Al6Mn and alpha precptates. Before undertakng coupled smulatons (AwP and CwP), calculatons usng only the PFT model (A, B and C) and only the PSD model (P) were carred out ndependently. Ths allows for a better understandng of the ndvdual behavor of each model Smulaton of soldfcaton mcrostructure The evoluton of the phase fractons durng soldfcaton s presented n Fg. 3. In calculaton A (black curves), both ntragranular Al6Mn (dotted lnes) and alpha (dashed lnes) were consdered for the computaton of thermodynamc equlbrum n the ntergranular regon, assumng no nucleaton undercoolng of these phases. In ths case, the ntergranular precptates obtaned after soldfcaton and coolng down to room temperature are essentally of the alpha phase, whch s more stable than Al6Mn at low temperature. Ths s clearly not realstc snce observaton (full symbols n Fg. 3) ndcates the opposte: an overall volume fracton of ntergranular precptates of 2.9% n the as-cast state, wth a mnor proporton of ntergranular alpha precptates representng only 5% of the ntergranular phases [19]. It s therefore expected that the condtons for the nucleaton of ntergranular alpha precptates are n realty not met durng soldfcaton. To account for ths behavor, calculaton B, n whch the alpha phase s excluded from thermodynamc equlbrum, was carred out. As shown n Fg. 3, a fnal amount of 3.75% Al6Mn s predcted wth calculaton B, whch s substantally more than the measured total amount of ntergranular precptates (2.9%). Calculatons performed wth coolng rates or dffuson coeffcents modfed by one order of magntude or more dd not allow comng close to 2.9%. It was concluded that nucleaton of ntergranular Al6Mn precptates s lkely to occur long after the lqudus of the Al6Mn phase has been reached. A seres of calculatons based on adjusted nucleaton temperatures for ntergranular Al6Mn and alpha precptates was then carred out untl the measured percentages were approxmately retreved. The obtaned nucleaton temperatures are ndcated n Table 1 (calculaton C) and the results are dsplayed as thck grey lnes n Fg. 3. The soldfcaton paths obtaned n calculatons A, B and C are shown n Fg. 4 only consderng Fe and Mn,.e., slow dffusng elements. When no nucleaton undercoolng s consdered (case A), the Mn content of the lqud and prmary Al matrx phases decrease upon coolng as soon as the lqudus of the frst of the ntergranular precptates s reached,.e., Al6Mn. Ths s obvously due to the formaton of the Mn-rch Al6Mn ntergranular precptates. When a nucleaton undercoolng s ntroduced (case C), the equlbrum Mn concentratons n lqud and prmary Al keep ncreasng untl Al6Mn ntergranular precptates are formed. The end of soldfcaton temperature s then consderably dfferent for calculaton C (639.5 C, open crcles) as compared wth calculaton A (646.4 C) and B (642.8). In calculaton C, the prmary Al matrx phase composton reaches 1.3 wt.% Mn, compared to only.5 wt.% Mn for calculatons A and B. Prmary Al s thus more supersaturated n case C when the nucleaton of the Table 1 Lst of the smulatons wth values of the nucleaton parameters used for the ntergranular Al6Mn and alpha precptates and the ntragranular alpha precptates Smulaton dentfer Nucleaton temperature of ntergranular Al6Mn precptates ( C) Nucleaton temperature of ntergranular alpha precptates ( C) Nucleaton parameters of ntragranular alpha precptates for coupled PFT PSD smulatons Nucleaton parameters of ntragranular alpha precptates for uncoupled PFT PSD smulatons A a a AwP a a r alpha/fcc =.15 J m 2 h alpha =45 N alpha max ¼ 1: m 3 B a C CwP r alpha/fcc =.15 J m 2 h alpha =45 N alpha max ¼ 1: m 3 P r alpha/fcc =.15 J m 2 a No nucleaton undercoolng. h alpha =45 ¼ 1: m 3 N alpha max

7 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Table 2 Values of the parameters and physcal propertes used for the 3 alumnum alloy Descrpton Symbol Value Unt Nomnal composton Fe.58 wt.% Mn 1.15 wt.% S.2 wt.% Dffuson coeffcents D fcc Fe exp( 214,/R g T) m 2 s 1 D fcc Mn exp( 211,5/R g T) m 2 s 1 D fcc S exp( 117,6/R g T) m 2 s 1 Gbbs Thomson coeffcent C fcc Km Molar volume V alpha m 3 Dffuson dstance k fcc m PFT number of cell 4 PFT system sze 8.6 lm PSD number of classes 5 PSD system sze 1 lm All nucleaton parameters are reported n Table 1. Volume fracton of ntergranular precptates [1] Tme [s] 2 Case C: alpha Mn-rch ntergranular precptates s delayed. As a result, the materal can experence more ntragranular precptaton upon subsequent heat treatments Evoluton of ntergranular precptates durng homogenzaton heat treatment The nfluence of the as-cast state on homogenzaton knetcs can be assessed by comparng calculatons A and CnFg. 5a, whch shows the tme evoluton of the temperature and of the volume fractons of ntergranular phases for the two calculatons. The comparson shows that the as-cast state has a consderable nfluence on the evoluton of the volume fracton durng the heatng stage of the homogenzaton treatment. However, once the temperature Measured: Al6Mn ( ), alpha ( ) Case B: Al6Mn Case C: Al6Mn Case A: alpha Case A: Al6Mn Fg. 3. Calculated evolutons of the volume fractons of Al6Mn (dotted lnes) and alpha (dashed lnes) ntergranular precptates durng the soldfcaton of a 3 alumnum alloy. Measurements of the volume fracton of ntergranular phases are shown wth sold symbols [2]. Mn concentraton [wt%] Prmary Al path 655 C 653 C C end at C B end at C A end at C Fe concentraton [wt%] plateau s reached, the same global equlbrum s obtaned and tme evolutons are very smlar. The dscusson hereafter wll be focused on condtons C, whch are the most pertnent for comparsons wth respect to the expermental data, snce they permt us to start the smulaton of the heat treatment from a realstc as-cast state. The evoluton of ntergranular precptates durng the heat treatment can be decomposed nto a seres of four stages whch are labeled from I to IV n Fg. 5a. At the onset of heatng (T < 4 C, stage I n Fg. 5a), ntergranular Al6Mn precptates transform progressvely nto ntergranular alpha precptates by thermal actvaton of S dffuson. In the calculaton, the knetcs of ths phase transformaton s ndeed governed by the dffuson of S n the prmary Al phase, whch s requred to form the lqud path Al6Mn lqudus s reached Fg. 4. Calculated soldfcaton path (equlbrum Fe and Mn compostons n lqud and prmary Al) for a 3 alumnum alloy usng assumptons A, B and C for the nucleaton of ntergranular Al6Mn and alpha precptates (see Table 1). The open symbols ndcate the compostons when the last lqud dsappears.

8 2546 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Al6Mn ntergranular precptates alpha ntergranular precptates Case A Case C 7 Volume fracton of ntergranular precptates [1].3.1 I II III IV Tme [h].4 Al6Mn ntergranular precptates alpha ntergranular precptates Case C Case CwP 7 Volume fracton of ntergranular precptates [1].3.1 I II III IV Tme [h] Fg. 5. Calculated tme evolutons of the volume fracton of ntergranular Al6Mn precptates and ntergranular alpha precptates formed upon an ndustral homogenzaton heat treatment of a 3 alumnum alloy (a) wthout precptaton of ntragranular alpha precptates and consderng no nucleaton undercoolng of the ntergranular precptates (case A, black lnes) or adjusted nucleaton undercoolng of the ntergranular precptates (case C) and (b) wthout (case C) or wth precptaton of ntragranular alpha precptates (case CwP, black lnes). The temperature hstory s also ndcated (thn plan lne). S-rch alpha phase. Between 4 and 6 C (stage II n Fg. 5a), ntergranular alpha precptates transform back nto ntergranular Al6Mn precptates. Ths evoluton s due to the ncrease of the S solublty n the prmary Al matrx. The Fe and Mn resultng from the dssoluton of ntergranular alpha precptates cannot be accommodated n the matrx and ths leads to the formaton of ntergranular Al6Mn precptates. Formaton of ntergranular Al6Mn precptates s also enhanced by ncomng fluxes of Fe and Mn from the matrx, whch are due to the Mn and Fe mcrosegregaton profles nherted from soldfcaton. These nterpretatons are supported by the left

9 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) columns of Fg. 6 whch dsplay the solute profles at dfferent temperatures durng heatng. When the 6 C homogenzaton plateau s reached, the equlbrum volume fracton of alpha precptates s obtaned (.15) and remans constant snce the S solublty n the prmary Al matrx no longer vares (stage III n Fg. 5). Intergranular Al6Mn precptates keep growng due to long-range dffuson of Mn. Equlbrum.12 case Case C.12 Case CwP case CwP.1 (a1) 2 C - 4 C.1 (b1) 2 C - C Composton of Fe [wt%] C 5 C 6 C (start) 6 C (end) Composton of Fe [wt%] C 45 C 5 C 6 C (end) 6 C (start) 2 C-45 C Composton of Mn [wt%] (a2) 5 C 6 C (start) 6 C (end) Composton of Mn [wt%] 1. (b2) 2 C - C 4 C.8 45 C.6 6 C (start) 6 C (end).4 5 C.2 case C 2 C.2 case CwP 2 C Composton of S [wt%].2.1 (a3) 2 C C case C 6 C (start, end) Composton of S [wt%].2.1 (b3) 2 C C case CwP 6 C (start, end) 4 C 5 C 5 C 4 C 45 C 45 C Poston [μm] Poston [μm] Fg. 6. Composton profles of Fe (row1), Mn (row 2) and S (row 3) n the prmary Al matrx phase (fcc) at selected temperatures durng heatng and homogenzaton, calculated wthout consderng ntragranular alpha precptaton (case C, left column) and for a coupled precptaton homogenzaton calculaton (case CwP, rght column). The profles correspondng to the begnnng and end of the sothermal plateau are labeled as 6 C (start) and 6 C (end), respectvely.

10 2548 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) composton of Fe s almost reached n the core of the grans at the begnnng of the temperature plateau, whle t requres several hours for the Mn to be unformly dstrbuted n the prmary Al matrx phase (see the two profles at 6 C n Fg. 6a2). After about 2 h at 6 C, the equlbrum volume fracton of Al6Mn precptates (6) s reached. Durng coolng, the alpha phase forms at the expense of the Al6Mn phase due to the decrease of S solublty and lmted dffuson of Mn (stage IV n Fg. 5) Evoluton of ntragranular precptates durng homogenzaton heat treatment The precptaton model was frst used wthout couplng wth the long-range dffuson model consdered n the PFT model. In such a way the result of the PSD precptaton model alone could be analyzed. In ths approxmaton, hereafter referred to as calculaton P, the composton profles of the as-cast state at room temperature (2 C) shown n Fg. 6 are used as the average compostons, x, enterng Eq. (15). These average composton profles reman constant durng the heat treatment appled,.e., upon heatng, holdng and coolng to room temperature. As a consequence, the system s closed locally wth respect to solute transfer at all postons along the composton profles of the prmary Al matrx phase. It s also closed wth respect to the ntergranular area. Ths s clearly an approxmaton whch s released n calculaton CwP presented n the successve secton. It must be stated that despte the average compostons, x, remanng constant, the compostons of the precptates, x alpha, as well as the composton of the matrx, x fcc, evolve as precptaton/dssoluton of the ntragranular alpha precptates phase takes place. Eq. (15) s the mass balance lnkng these compostons wth the sze and densty of the ntragranular precptates, r alpha and N alpha, respectvely. The calculaton predcts the tme evoluton of the sze dstrbuton and densty of the ntragranular alpha precptates for a seres of locatons gong from the centre to the perphery of the gran. The volume fractons of ntragranular precptates for the extreme locatons (centre and perphery) are shown n Fg. 7a as a functon of tme. Even though the calculated volume fractons at these two locatons are consderably dfferent, they both ndcate substantal ntragranular precptaton durng the frst part of the heatng ramp, followed by dssoluton of the ntragranular precptates durng the second part. Once the sothermal plateau s reached, the volume fracton of ntragranular precptates remans constant, ndcatng that equlbrum has been reached. Durng the fnal coolng stage (stage IV), reformaton of ntragranular alpha precptates s predcted by the model. To allow for comparson wth the expermental data of L and Arnberg [19], (reported n Fg. 7a wth symbols), the results of calculatons P were averaged over the gran, takng nto account the sphercal morphology assumpton that was also used for the homogenzaton Volume fracton of precptates [1] Volume fracton of precptates [1].1 Measured ntragranular alpha precptates ( ) Case P Average Center Perphery Tme [h] Measured ntergranular Al6Mn+alpha precptates ( ) Measured ntragranular alpha precptates ( ) Al6Mn+alpha precptates Case AwP Case CwP Case C Case CwP alpha precptates.1 2 Case AwP Tme [h] Fg. 7. Calculated tme evolutons of the volume fracton of ntergranular precptates formed durng the homogenzaton heat treatment of a 3 alumnum alloy wthout (a) and wth (b) couplng of precptaton and homogenzaton calculatons. Correspondng expermental data for the volume fractons of ntergranular precptates [2] and the volume fracton of ntragranular alpha precptates [19] are plotted wth symbols. calculaton. The comparson shows that the volume fracton of the ntragranular precptates s overestmated by the model. Also, the progressve ntragranular precptate dssoluton observed by L and Arnberg durng holdng s not predcted. Ths ndcates that the uncoupled calculaton seems to mss some sgnfcant aspects of the phase transformatons Coupled homogenzaton and precptaton calculatons For a better understandng of the nteractons between short- and long-range dffusons leadng to the formaton of ntragranular and ntergranular alpha precptates, respectvely, the mcrostructure evoluton resultng from couplng homogenzaton and precptaton (calculaton CwP, s detaled hereafter based on the transformaton stages I IV defned prevously.

11 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Stage I Durng heatng up to 4 C (stage I n Fg. 5b), thermal actvaton of S dffuson n the prmary Al phase allows for ntergranular Al6Mn precptates to transform nto ntergranular alpha precptates and for ntragranular alpha precptates to nucleate. Ths s clearly vsble n Fg. 8, where the densty and the average radus of the ntragranular precptates have been represented as a functon of the temperature. It can be seen that the ntragranular precptate densty ncreases drastcally between and 4 C. In the vcnty of the ntergranular precptates, the supersaturaton s reduced due to the Fe and Mn mcrosegregaton profles nherted from soldfcaton and coolng, whle the S composton s almost unform due to the hgh value of ts Fourer number for the system consdered (see solute profles n Fg. 6 at 2 C). As a consequence, the densty of ntragranular precptates s substantally lower n ths regon. Ths effect s vsble n Fg. 9, whch shows profles of the volume fracton, densty and average sze of the ntragranular precptates at dfferent temperatures durng heatng. As a result of lmted nucleaton n the depleted outer regon of the grans, a PFZ s ntated. Ths can be seen n Fg. 1, where the sze of the PFZ has been represented as a functon of tme. The ntragranular precptate rad shown n Fgs. 8 and 9, ndcate that at 4 C durng heatng, ntragranular precptate growth s not yet very mportant, although the maxmum of the number densty of the ntragranular precptates has already reached. As a consequence, the matrx s stll very rch n Fe and Mn at 4 C (see Fg. 6b1 and Fg. 5b2). Comparatvely, S s almost fully depleted (Fg. 6b3), thus lmtng further nucleaton of the S-rch ntragranular alpha precptates Stage II Phase transformatons takng place durng stage II are much more complex for calculaton CwP than for calculaton C. In CwP, between 4 and 5 C, rapd growth of Densty of the precptates [μm -3 ] Fg. 8. Evoluton upon heatng of the densty (dashed lnes, case CwP) and mean radus (plan lnes, case CwP) of the ntragranular alpha precptates. Correspondng expermental data are plotted wth thnner lnes usng trangles for the densty and squares for the radus [19] Average radus of the precptates [nm] ntragranular alpha precptates takes place (Fg. 7b). Near 46 C, smultaneous growth of ntragranular alpha precptates and dssoluton of ntergranular alpha precptates can be observed (see volume fracton of ntergranular alpha precptates n Fg. 5b and ntragranular alpha precptates n Fg. 7b att = 46 9 h). Ths phenomenon s due to the dependency of the S solublty n the prmary Al matrx wth respect to Fe and Mn concentratons, combned wth the presence of Fe and Mn segregaton profles. The regons close to the ntergranular precptates are poor n Fe and Mn and the equlbrum concentraton of S n the prmary Al matrx s hgher than n the core (Fg. 6b3, 45 C). Ths leads to the dssoluton of ntergranular alpha precptates, whle, deeper n the prmary Al phase, ntragranular precptates contnue to grow and ncorporate the S released by the ntergranular precptates. The volume fracton of the ntragranular precptates reaches ts maxmum about 2 lm from the gran boundary (Fg. 9a1, 45 C), whch also corresponds to the hghest Mn content along the profle (Fg. 6b2, 45 C). One can note that the phenomenon of smultaneous growth and dssoluton of a same phase obtaned n ths calculaton s a partcularty of mult-component systems whch could not be observed n a bnary alloy. Between 5 and 55 C, the solublty of Mn, Fe and S ncreases (see concentratons at the fcc/mxture nterface at 5 and 6 C n Fg. 6b1 6b3) and ntragranular precptates are partly dssolved (Fgs. 7b and 9a1). Intergranular alpha precptates grow slghtly at the expense of ntergranular Al6Mn precptates due to the release of S by the dssoluton of ntragranular precptates n the prmary Al matrx (see Fg. 5b for 1 h < tme < 11 h). Above 55 C, dssoluton of both ntragranular and ntergranular alpha precptates s observed owng to a general ncrease of S solublty n the prmary Al matrx (Fgs. 5b and 7b). As n calculaton C, ntergranular Al6Mn precptates grow at the expense of alpha precptates Stage III At the begnnng of the temperature plateau, dssoluton of ntergranular alpha precptates contnues due to longrange dffuson of Fe and Mn (see the four profles at 6 C n Fgs. 6b1 and 5b2). Ths leads to further growth of ntergranular Al6Mn precptates (Fg. 5b) and dssoluton of ntragranular precptates n ther vcnty, thus extendng the PFZ (see Fg. 7) Stage IV Durng coolng, ntergranular Al6Mn precptates transform nto ntergranular alpha precptates as n the uncoupled calculaton. Between 6 and 5 C, ntragranular alpha precptates grow slghtly due the solublty decrease Comparsons wth expermental data The volume fractons of ntergranular phases and precptates obtaned wth the coupled calculaton (CwP) can

12 255 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Volume fracton of precptates [1] Densty of precptates [m -3 ] Average radus of precptates [nm] (a1) (a2) 45 C (a3) 45 C 4 C 2 C 45 C 4 C 5 C 4 C 5 C 2 C 5 C 6 C 2 C C 6 C C 6 C 2 C 2 C C Poston [μm] Volume fracton of precptates [1] Densty of precptates [m -3 ] Average radus of precptates [nm] (b1) (b2) 2 C (b3) 4 C 2 C C 45 C 5 C 2 C 4 C 45 C 2 C 45 C 6 C 2 C C 5 C 6 C 2 C C 4 C 5 C 6 C Poston [μm] Fg. 9. Calculated evolutons of profles of (1) volume fracton, (2) densty and (3) mean radus for the ntragranular alpha precptates at selected temperatures of the heatng (a) and coolng (b) sequences of the homogenzaton treatment (case CwP). be compared wth the expermental results of L and Arnberg [19,2] n Fg. 7b. One can note that the agreement wth the measured volume fractons of ntergranular alpha and Al6Mn precptates s consderably better than for calculaton P, where ntragranular precptaton s not coupled wth long range dffuson (Fg. 7a). In partcular, the volume fracton of the ntragranular precptates s better predcted. Furthermore, the progressve dssoluton of ntragranular precptates durng holdng s correctly captured. It was shown to be due to long-range dffuson of Fe and Mn combned wth the transformaton of the ntergranular alpha precptates nto ntergranular Al6Mn precptates. The Mn rejected upon dssoluton of the ntragranular precptates s transported toward the boundary of the grans and used for the growth of ntergranular Al6Mn precptates. Ths effect s also vsble n the Mn profles that are shown n Fg. 6b2 at the begnnng and end of the 6 C temperature plateau. Calculaton CwP can be compared wth calculaton AwP where no undercoolng of the ntergranular precptates was consdered. The volume fracton of precpates obtaned n calculaton CwP (Fg. 7b) shows a consderably better agreement wth the expermental data as compared wth calculaton AwP. Ths s clearly related to the

13 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Volume fracton of Precptate Free Zone [1] Case AwP Case CwP Tme [h] Fg. 1. Calculated tme evoluton of the volume fracton of the PFZ wthout (case AwP, grey lne) and wth (case CwP, black lne) nucleaton undercoolng of the ntergranular precptates. Correspondng expermental data are plotted usng dots [19]. The temperature hstory s also ndcated (thn plan lne). better descrpton of the as-cast state, whch s farly well reproduced n calculaton C (Fg. 3). Although the volume fracton of ntragranular precptates s somewhat overestmated n calculaton CwP, the overall precptaton knetcs s well captured and the devatons are farly systematc. Ths ndcates that the physcal mechansms seem to be properly taken nto account. The same comment apples when comparng the tme evoluton of the total volume fracton of ntergranular precptates n Fg. 7b. Smlarly, the volume fracton of PFZ n the mcrostructure shown n Fg. 1, the dstrbuton of ntragranular precptates upon heatng and holdng shown n Fg. 11 and the electrcal conductvtes shown n Fg. 12 compare well wth the measurements of L and Arnberg [2]. The agreement between experment and smulaton can be consdered as very satsfactory f one bears n mnd the complexty of the system and all the uncertantes assocated wth the physcal propertes and measurements. The measurement of the average volume fracton of ntragranular precptates and ts sze dstrbuton s a dffcult task, especally n presence of a PFZ, snce large varatons are expected between the centre and the perphery of the grans. In such a case, the analyss of a large number of samples s requred to obtan an acceptable expermental error. The systematc dfference between expermental and calculated electrcal conductvtes shown n Fg. 12 could also be due to naccurate coeffcents lnkng the compostons to the conductvty and/or to devatons from the nomnal chemstry. Frequency [%] C 4 C 5 C 58 C Frequency [%] C, h 6 C, 1 h 6 C, 4 h 6 C, 7 h Szes [nm] Szes [nm] Frequency [%] C 4 C 5 C 58 C Frequency [%] C, h 6 C, 1 h 6 C, 4 h 6 C, 7 h Szes [nm] Szes [nm] Fg. 11. Calculated (top, case CwP) vs. measured (bottom) evolutons of the sze dstrbuton of the ntragranular alpha precptates upon (a) heatng and (b) holdng [19].

14 2552 Ch.-A. Gandn, A. Jacot / Acta Materala 55 (27) Electrcal conductvty [m Ohm -1 mm -2 ] Besdes beng evdence of the mportance of consderng ntragranular precptaton and ntergranular phase transformatons as a whole, one of the major results of ths analyss s the drastc effect of the soldfcaton stage. The nucleaton temperature of the ntergranular phases formed as ntergranular precptates durng soldfcaton and coolng to room temperature s found to play a key role n the determnaton of the tme evoluton of the mcrostructure and PFZ formaton. 5. Conclusons Case C Case CwP Tme [h] Fg. 12. Tme evoluton of the electrcal conductvty wthout (case C, grey lnes) and wth (case CwP, black lnes) ntragranular alpha precptaton n the prmary Al matrx phase. Correspondng expermental data are plotted usng crosses [19]. The temperature hstory s also ndcated (thn plan lne). The relaton for the calculaton of electrcal conductvty, r [19]: 1/ r ¼ :267 þ :32w fcc Fe þ :33wfcc Mn þ :68wfcc S. A methodology has been proposed to couple soldfcaton, homogenzaton and precptaton calculatons wthn a comprehensve smulaton approach. The model was appled to the predcton of mcrostructure formaton n a 3 alumnum alloy. The man fndngs can be summarzed as follows. Good agreement between calculated and expermental volume fractons of ntergranular precptates for the materals n the as-cast state could only be obtaned after havng ntroduced a nucleaton undercoolng for Al 6 (Mn,Fe) (Al6Mn) and assumng that condtons for a-al(fe,mn) S (alpha) ntergranular precptates to nucleate n the ntergranular regons are not met durng soldfcaton. Ths result ndcates that the ntergranular precptates n 3xxx alloys are lkely to be formed under condtons that are far from thermodynamc equlbrum. A good predcton of the amount of precptates and sze of PFZ obtaned after a homogenzaton heat treatment s only possble f the as-cast state s correctly descrbed. Ths ncludes the large varatons of the matrx concentratons between the centre and the perphery of the grans whch s an mportant aspect of the as-cast state. Ths result, whch was already ponted out n Ref. [18],s not surprsng snce the as-cast state defnes the ntal supersaturaton of the prmary Al matrx phase pror homogenzaton. The evolutons of ntergranular and ntragranular precptates are closely coupled phenomena whch cannot be consdered separately n a homogenzaton model. Only coupled calculatons succeeded n predctng the measured amount of alpha precptates. Ths s due to the nteracton between short- and long-range dffuson of the same speces requred for the formaton of ntergranular and ntragranular precptates. Sophstcated mcrostructure parameters such as the dstrbuton of the ntragranular alpha precptates or the tme evoluton of the ntragranular composton profles can be quanttatvely predcted by the model. As a result of couplng between ntergranular and ntragranular precptaton the formaton of a PFZ can be descrbed. Currently, the descrpton of ntergranular precptates s lmted to the type of phase and volume fractons. For the extruson processes followng castng and homogenzaton, the sze dstrbuton s a key parameter whch should thus be addressed n further nvestgatons. One should also menton that n ts current form the coupled homogenzaton precptaton model requres large computatonal resources. Efforts are requred to make the present model more effcent and practcal for ndustral applcatons. Acknowledgements Ths research has been conducted as a contnuaton of the VIRCAST project (Vrtual Cast House, Growth Project, OFES Grant ). The authors would lke to thank the European Communty and the Offce Fédéral de l Educaton et de la Scence, Bern, Swtzerland, as well as the companes Alcan (France), Alcan (Swtzerland), Calcom-ESI (Swtzerland), Corus (The Netherlands), Elkem (Norway), Hydro (Norway) and Hydro (Germany) for the fnancal support provded wthn the VIRCAST project. The authors also gratefully acknowledge Professor L. Arnberg, NTNU, Trondhem, Norway for provdng the expermental data. References [1] Porter D, Easterlng K. Phase transformatons n metals and alloys. second ed. New York, NY: Chapman & Hall; [2] Phlbert J, Vgnes A, Bréchet Y, Combrade P. Métallurge du Mnera au Matérau. Pars: Masson; [3] Itoh G, Kanno M, Hagwara T, Sakamoto T. Acta Mater 1999;47: [4] Dumont D, Deschamps A, Bréchet Y. Acta Mater 24;51: [5] Maldonado R, Nembach E. Acta Mater 1997;45: [6] Krol T, Bather D, Nembach E. Acta Mater 24;52: [7] Jang H, Faulkner RG. Acta Mater 1996;44: [8] Jang H, Faulkner RG. Acta Mater 1996;44: [9] Starnk MJ. Mater Sc Eng A 25;39:26 4. [1] Dons A-L. J Lght Met 21;1: [11] Hakonsen A, Mortensen D, Benum S, Pettersen T, Furu T. TMS meetng, Seattle, WA, Lght Metals; 22. p