Static coarsening of titanium alloys in single field by cellular automaton model considering solute drag and anisotropic mobility of grain boundaries

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1 Artcle Progress of Projects Supported by NSFC Materals Scence May 2012 Vol.57 No.13: do: /s SPECIAL TOPICS: Statc coarsenng of ttanum alloys n sngle feld by cellular automaton model consderng solute drag and ansotropc moblty of gran boundares WU Chuan, YANG He *, LI HongWe & FAN XaoGuang State Key Laboratory of Soldfcaton Processng, School of Materals Scence and Engneerng, Northwestern Polytechncal Unversty, X an , Chna Receved September 6, 2011; accepted November 22, 2011 Statc coarsenng s an mportant physcal phenomenon that nfluences mcrostructural evoluton and mechancal propertes. How to smulate ths process effectvely has become an mportant topc whch needs to be dealt wth. In ths paper, a new cellular automaton (CA) model, whch consders the effect of solute drag and ansotropc moblty of gran boundares, was developed to smulate statc gran coarsenng of ttanum alloys n the beta-phase feld. To descrbe the effect of the drag caused by dfferent solute atoms on coarsenng, ther dffuson veloctes n beta ttanum were estmated relatve to that of ttanum atoms (T). A formula was proposed to quanttatvely descrbe the relatonshp of the dffuson velocty of T to that of solute atoms; factors nfluencng the dffuson velocty such as solute atom radus, mass, and lattce type were consdered. The ansotropc moblty of gran boundares was represented by the parameter c 0, whch was set to 1 for a fully ansotropc effect. These equatons were then mplemented nto the CA scheme to model the statc coarsenng of ttanum alloys T 6Al 4V, T17 (T 5Al 4Mo 4Cr 2Sn 2Zr, wt%), TG6 (T 5.8Al 4.0Sn 4.0Zr 0.7Nb 1.5Ta 0.4S 0.06C, wt%) and TA15 (T 6Al 2Zr 1Mo 1V, wt%) n the beta feld. The predcted results, ncludng coarsenng knetcs and mcrostructural evoluton, were n good agreement wth expermental results. Fnally, the effects of tme, temperature, and chemcal composton on gran coarsenng and the lmtatons of the model were dscussed. statc gran coarsenng, cellular automaton model, solute drag, ansotropc moblty, ttanum alloys Ctaton: Wu C, Yang H, L H W, et al. Statc coarsenng of ttanum alloys n sngle feld by cellular automaton model consderng solute drag and ansotropc moblty of gran boundares. Chn Sc Bull, 2012, 57: , do: /s *Correspondng author (emal: yanghe@nwpu.edu.cn) The mechancal propertes of ttanum alloys such as the creep resstance, fracture toughness and crack propagaton resstance are determned by ther gran szes n the beta phase feld [1,2]. However, the grans are prone to statc coarsenng when held n the beta phase feld because of the structural characterstcs of beta-ttanum and relatvely hgh stackng fault energy at hgh temperatures [3]; ths results n a marked decrease n the mechancal propertes of fnal products. Therefore, t s necessary to understand and to control statc coarsenng of ttanum alloys n a sngle phase feld to obtan the expected mechancal propertes. Prevously, many expermental nvestgatons have focused on the statc coarsenng behavors of ttanum alloys n the beta phase feld. Ivasshn et al. [4,5] nvestgated the effect of texture evoluton on the values of the statc coarsenng exponent n and actvaton energy for T 6Al 4V alloy n the beta feld. They found values of n rangng from 0.22 to 0.31 at dfferent temperatures, but there were marked devatons attrbuted to dfferent materals and experments when the results were compared wth those of others. Recently, Sematn et al. [6 8] nvestgated the statc gran coarsenng of T 6Al 4V alloy va a seres of heat treatments at typcal temperatures. They found that the coarsenng process of prmary alpha partcles was controlled by the volume dffuson of solute elements at lower temperatures and the growth exponent was equal to The Author(s) Ths artcle s publshed wth open access at Sprngerlnk.com csb.scchna.com

2 1474 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No.13 Smlarly, Bradley et al. [9] and Cotrna et al. [10] analyzed coarsenng knetcs va expermental data from optcal observatons. Recently, varous numercal smulaton technques such as Potts-Monte Carlo models (MC) [11 13], phase feld models [14,15] and cellular automaton models (CA) [16 18] have been gradually used to model the mcrostructure evoluton on the mesoscale [19]. The CA methods have been wdely used n these nvestgatons, and have some advantages [20]. Geger et al. [21] nvestgated statc coarsenng by CA modelng and dscussed the effects of orentaton dfference, actvaton energy, and boundary energy on coarsenng. Subsequently, Raghavan et al. [22,23] and Kugler et al. [24] smulated the topologcal and knetc features of gran coarsenng n polycrystallne materals. However, most of these models neglected the effect of solute drag and the ansotropc moblty of gran boundares on the mcrostructural evoluton durng the coarsenng process [15]. The predcted results, ncludng coarsenng knetcs and topologcal evoluton, generally devated from the expermental results. Solute drag and ansotropc moblty greatly nfluence the coarsenng process. The dffuson velocty of solute atoms n the matrx s nfluenced by ther radus, mass, and lattce type. If the dffuson velocty s substantally less than that of matrx, the solute atoms on the gran boundares wll retard the coarsenng process; the drag effect on coarsenng knetcs n ths case s obvous. Because of the dfferent orentatons of each gran, the gran moblty s dfferent from dfferent grans, whch also exerts an mportant nfluence on gran coarsenng. However, most prevous CA models for gran coarsenng gnore the effects of solute drag and ansotropc moblty, whch leads to a dfference between smulated and expermental coarsenng knetcs. Therefore, the objectve of the present paper was to smulate gran coarsenng by CA models takng nto account solute drag and ansotropc moblty. Frstly, a formula was proposed to quanttatvely descrbe the relatonshp between the dffuson veloctes of T and solute atoms, whch was then used to calculate the effect of solute drag. Secondly, a parameter representng the ansotropy of the gran boundary moblty was ntroduced. Fnally, these equatons were mplemented nto a CA scheme and the CA models were appled to smulate the gran coarsenng; the predcted results were compared wth the expermental measurements. 1 Cellular automaton model 1.1 Statc gran coarsenng models Statc coarsenng s a thermally actvated process that has been wdely studed n the doman of carbon-steel alloys [25]. Usually t takes place by the jumpng of atoms from one gran to another and the mgraton of gran boundares. Atoms located n the gran boundares wth enough hgh energy mght overcome the energy barrers to change nto a new state [26]. Smlarly, n CA models, the cells of boundares transfer from one gran to another wth probablty p. Accordng to the Maxwell Boltzmann dstrbuton law, the probablty p s determned by: Q p Q 0, (1a) Q max Qb exp( ) p RT Q 0, (1b) Qb E exp( ) exp( ) RT RT where Q s the dfference between the total energy of cell and self-dffuson actvaton energy Q b. The total energy of each cell conssts of systemc thermal energy E t, and total boundary energy E b. The thermal energy of cell s expressed as [27]: E RTln r, (2) t where r (0<r<1) s a random number produced by the computer program, and R and T have ther unversal meanngs. The boundary energy, γ j, between two neghborng cells and j (dependent on msorentaton) s assumed to follow the ReadShockley equaton [28]: m, j m, (3) j j j j m 1ln, j m m m where θ m s the msorentaton for hgh-angle gran boundares, usually set as 15 and θ j s the msorentaton angle between two adjacent cells and j; γ m s the boundary energy of the hgh-angle gran boundary, whch s drectly calculated by [29]: b m m, (4) 4π(1 ) where μ s the shear modulus of materal at a specfc temperature, b s the magntude of Burgers vector, s Posson s rato. The total boundary energy of cell s expressed as follows: b N E, (5) j where N s the number of nearest neghbors of cell and s set to 4 accordng to the neghborng rule n ths paper. The mgraton of the gran boundares s regarded as curvature-drven boundary mgraton, n whch the drvng force s suppled by the dfference n total energy between two neghborng grans (cells). Thus, the velocty of the coarsenng gran boundares, v, s expressed as follows: v mf, (6) where m and f are the moblty of gran boundares and drvng force exerted on coarsenng gran boundares, respectvely. The drvng force f j s expressed by: fj E/ R j, (7) where E s the dfference n total energy between two j

3 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No neghborng cells and j, R j s the radus of the curvature of boundary located by cell j adjacent to located by cell. The growth dstance l of cell at ts boundary towards one of the neghborng cells j durng the tme ncrement t s calculated by: lj ( tt) lj ( t) vjt, (8) where l j (t) represents the growth dstance of a cell at tme t, the ntal value s set to zero. l j (t+ t) s the growth dstance at tme t + t. Kugler et al. [30] gave the defnton of tme ncrement t as the rato of grd sze, d, to the maxmum gran boundary velocty, v max. Accordng to eqs. (6) (8), the maxmum velocty s acheved when boundary energy γ j and gran radus R j are taken as γ m and ntal value R 0, respectvely. Thus the tme ncrement t s expressed by: d d R0 t CT CT vmax m t m E j, (9) where d s grd sze, R 0 s gran sze of ntal mcrostructure, and C(T) s a constant related to temperature. 1.2 Quanttatve descrpton of solute drag The concentraton of solute atoms around the gran boundares s relatvely larger than that n the gran nteror because of the structural defects at the gran boundares. Because of the exstng concentraton gradent, the solute atoms on gran boundares tend to dffuse; smultaneously, the dffuson of T (gran boundary mgraton) takes place durng coarsenng. If the dffuson velocty of solute atoms s smaller than that of T (gran boundary mgraton), the solute atoms wll retard the movement of the boundares. There are a varety of alloyng elements n ttanum, the major ones nclude the alpha-stablzng elements, such as alumnum (Al), and beta-stablzng elements, such as vanadum (V), molybdenum (Mo) and tn (Sn). The dffuson veloctes for these solute atoms n beta ttanum vares wth the atom radus, mass and lattce type. Therefore, t s mpossble and unrealstc to calculate the dffuson velocty of each type of solute atom n beta ttanum. To quanttatvely descrbe the effect of solute drag, the dffuson velocty of dfferent types of solute atoms s related to that of T. For ths purpose, a formula was proposed to descrbe the relatonshp between the dffuson veloctes of solute atoms and T. It s assumed that the dffuson velocty rato s nversely proportonal to the radus and mass ratos of the solute atom to T, leadng to: v h, (10) vt 1 2 where v and v T are the dffuson velocty of solute atoms and ttanum atoms, respectvely. h s a parameter representng the effect of the lattce type of the solute atoms on coarsenng. If the lattce type of the solute atoms s the same as that of beta ttanum (as occurs for V, Mo and Sn), h s set to 1, otherwse, ts value s less than 1. 1 and 2 are the atomc radus and mass ratos of the solute to T, respectvely: R 1 R, (11) T m 2 m, (12) T where R s the radus of the solute atom, R T the radus of the ttanum atom, m the mass of the solute atom and m the mass of the ttanum atom. If the value of / v T s larger than 1, the dffuson velocty of the solute atoms s larger than that of T, whch means the solute drag s neglgble. Otherwse, the dffuson velocty of T wll be retarded by the solute atoms. To ncorporate the solute drag effect nto CA models requres the formulaton of CA transfer rules, whch s dscussed n Secton Ansotropc moblty In conventonal CA models, the assumpton of msorentaton-ndependent gran boundary moblty was made, so that the predcted topology features (sze and sde dstrbuton) dd not reproduce the log-normal dstrbuton observed expermentally nstead the smulated topology dstrbuton of grans appeared more left-skewed [31]. In fact, the gran boundary moblty s dependent on the msorentaton and becomes ansotropc. Therefore, we ntroduced a parameter c 0 (eq. (13)) to represent the effect of ansotropc moblty on gran coarsenng. The moblty of gran boundares m j between two neghborng cells, and j, s calculated by [32]: 3 j m 0 1 0exp j m c, (13) 10 where m 0, a constant, s the moblty of hgh-angle boundares. The parameter c 0 s ntroduced to take nto consderaton ansotropc gran boundary moblty, and s set to 0 and 1 for the sotropc and ansotropc cases, respectvely. 1.4 CA smulaton steps for statc gran coarsenng In the present smulaton, the ntal mcrostructure was obtaned by runnng the CA program wth condtons of homogenous ste-saturated nucleaton and equaxed growth. The state varables ncludng the gran orentaton varable, energy varables (total, thermal and boundary), dffuson velocty varable, locaton varable and property varable are defned for each cell. The cell sze s defned accordng to the average gran sze of the ntal mcrostructures. Values of the orentaton varable, rangng from 0 to 180, are automatcally assgned by the computer program before CA smulaton. Energy and velocty varables are calculated n each step of the CA smulaton and assgned to each cell. The locaton varable s used to specfy whether a cell s v T

4 1476 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No.13 located n the gran nteror (represented by 1) or on a boundary (represented by 0). The property varable s used to specfy whether a cell belongs to T (represented by 0) or other types of solute atoms (represented by an nteger larger than 0). In each smulaton step, the state varables are updated accordng to the CA rules and the mcrostructural evoluton of CA modelng proceeds. A sngle cycle through the gran coarsenng loop conssts of cell jumpng and mgraton of gran boundares, as llustrated n Fgure 1. (1) Cell jumpng. The cells located on the boundares are selected. Accordng to eqs. (2) (4), the thermal and boundary energy varables for these cells are calculated and used to obtan the total energy (eq. (5)). Then the total energy of each cell on the boundares s compared wth the dffuson actvaton energy; f the dfference between them s larger than 0, the cell jumps wth a transton probablty determned by eq. (1a). Otherwse, the transton probablty s calculated by eq. (1b). At the same tme, a random number r (0 < r < 1) s generated for each boundary cell. If the transton probablty s no less than the random number, these cells start to coarsen. (2) Mgraton of gran boundary. For each jumpng cell, the property varable of ts neghborng cell whch t grows towards s evaluated durng the coarsenng process. If the neghborng cell s occuped by the solute cell representng solute atoms, the movement of the jumpng cell mght be dragged by the solute cell. To ncorporate the drag effect nto CA models, the CA transfer rules are formulated as follows: the dffuson velocty of solute atoms, relatve to that of T (eq. (10)), wll be compared wth that of T; f the dfference s less than zero, the solute atoms wll retard the movement of the boundary cell and the mgraton velocty of T wll be decreased by the dffuson velocty rato; otherwse, the velocty stays the same. The growth dstance of a jumpng cell at tme t could be calculated by eq. (8). The rato of growth dstance to grd sze determnes whether the cell can take place n coarsenng or not. If the value s greater than 1, the state varables of a jumpng cell change nto those of ts neghborng cell and the gran coarsens. At the same tme, the growth length loses ts meanng and s set to zero agan. Otherwse, the growth dstance contnues evolvng accordng to eq. (8). (3) Mcrostructure evoluton. After the above mentoned steps, the state varables of each cell on the boundares and ther neghborng cells are updated, and the mcrostructural evoluton of CA modelng proceeds. 2 Model calbraton In ths paper, the system s dvded nto a square lattce and a perodc boundary condton wth Neumann neghborhood s adopted. The smulaton of gran coarsenng by CA models has been carred out on ttanum alloys of T 6Al 4V, TG6(T-5.8Al-4.0Sn-4.0Zr-0.7Nb-1.5Ta-0.4S- 0.06C, wt%) and T 17 (T-5Al-4Mo-4Cr-2Sn-2Zr, wt%) n the beta phase feld. The predcted coarsenng knetcs were compared wth expermental observatons. 2.1 Applcaton to T 6Al 4V alloy T 6Al 4V s a two-phase ttanum alloy. Its chemcal composton s lsted n Table 1. Table 2 shows the materal parameters used n eq. (10). From eq. (10), the dffuson velocty of each type of solute atom (ncludng mpurtes) n T 6Al 4V alloy could be obtaned; these values are lsted n Table 3. Accordng to above results (Table 3), the dffuson velocty of solute atoms ncludng Al and mpurty atoms such Table 1 The chemcal composton of T 6Al 4V ttanum alloy [33] Solute elements Fracton (wt%) Al V Fe C O N H T Balanced Table 2 alloy The radus and mass of solute atoms n T 6Al 4V ttanum Atoms T Al V Fe C O N H Radus (pm) Mass Table 3 feld The rato of dffuson veloctes of solute atoms to T n sngle Fgure 1 Coarsenng smulaton step. Solute atoms Al V Fe C O N H Rato v / v ) ( T

5 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No as C, N, O and H s much larger than that of T. Ths means that the presence of atoms on boundares hardly retards mgraton of gran boundares durng coarsenng, that s, the drag effect s neglgble. Whle the Fe atoms on the boundares mght retard boundary mgraton because of ther smaller dffuson velocty, the volume fracton s so low that the drag effect s also neglgble. The dffuson velocty of V atoms s close to that of T, whch also results n lttle effect of solute atoms on gran coarsenng. Overall, the drag effects nduced by solute and mpurty atoms on boundary mgraton are neglgble, so that the statc gran coarsenng for T 6Al 4V alloy could be consdered to be an deal process, as confrmed by Gl s expermental results. Recently, Gl et al. [33] have studed the statc coarsenng knetcs of T 6Al 4V at dfferent expermental temperatures and wth dfferent holdng tmes. They concluded that the growth exponent vared from 0.53 to 0.55, close to the deal value of 0.5. Thus the statc coarsenng process for T 6Al 4V n a sngle feld s a pure metal coarsenng. In ths paper, the CA models ncludng the solute drag and ansotropc moblty were appled to smulate the statc gran coarsenng of T 6Al 4V alloy n a sngle feld. The model parameters ncludng self-dffuson actvaton energy, gran sze of ntal mcrostructure, temperatures and tme are taken as the same as those n Gl s work. Fgure 2 (a) (c) shows the relatonshps between gran sze durng coarsenng and tme, obtaned from both expermental and CA predcton. The correspondng lnear regons are shown n Fgure 2 (d) (f). As expected, the smulated results show that the growth Fgure 2 Comparson of expermental and CA predcted statc coarsenng knetcs for T 6Al 4V n sngle feld. (a) (c) descrbe the relatonshp of coarsenng gran sze and tme at 1323, 1373 and 1473 K, respectvely, n=1; (d) (e) descrbe the lnear relatonshp of gran sze and tme at 1323, 1373 and 1473 K, respectvely.

6 1478 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No.13 rate ncreases wth ncreasng temperature, whch s ndcated by the change of slope of curves from Fgure 2(a) to (c). The predcted gran szes ndcate very rapd gran growth durng the ntal stages, whle the growth rate gradually drops wth ncreasng tme. Ths may be because of the decrease n the gran boundary area per unt volume, whch, n consequence, results n a decrease n the nterfacal energy per unt volume [33]. Therefore, the drvng force for growth becomes lower and produces a slower knetc process. However, there are stll some dfferences between the experments and CA smulatons. Ths may be caused by the lmtatons of the model, whch wll be dscussed n Secton 3.3. A comparson of expermental and smulated coarsenng exponents s shown n Table Applcaton to the TG6 and T17 alloys TG6 alloy (T 5.8Al 4.0Sn 4.0Zr 0.7Nb 1.5Ta 0.4S 0.06C, wt%) s a new type of near alpha hgh-temperature ttanum alloy, whch has been developed for arcraft engne applcatons [3]. T17 alloy (T 5Al 4Mo 4Cr 2Sn 2Zr, wt%) s a near beta ttanum alloy. Wang et al. [3] have expermentally nvestgated the statc coarsenng knetcs for these alloys n beta phase feld under sothermal condtons. They found that the coarsenng exponent ranged from for TG6 alloy and for T17 alloy. They also calculated the actvaton energes of gran coarsenng for these alloys. For the TG6 alloy, the value of actvaton energy ranges from to kj /mol, whle for T17 alloy, t s from to kj/mol. These values, ncludng coarsenng exponent and actvaton energy, devate greatly from the deal values (exponent of 0.5 and actvaton energy of 97 kj/mol for beta ttanum). The drag effect exerted by the solute atoms on the gran boundary may cause the decrease n exponent and ncrease n actvaton energy [3]. In ths part, the CA model ncorporatng solute drag and ansotropc moblty s used to smulate the statc coarsenng of these ttanum alloys n the beta feld. Ther chemcal compostons and correspondng materal constants are lsted n Tables 5 and 6, respectvely. The ratos of solute atom dffuson velocty to T dffuson velocty are calculated accordng to eq. (10), and the results are lsted n Table 7. Fgures 3(a) (c) and 4(a) (c) show the relatonshps between gran sze and tme durng coarsenng of TG6 and T17, respectvely; both expermental data and CA predcted data are shown. The correspondng lnear regons are shown n Fgures 3(d) (f) and 4(d) (f), respectvely. From these fgures, we can see that the predcted values of n agree well wth the expermental measurements. The relatve errors of n between smulaton and experment are lsted n Table 8. The maxmum relatve error s 13.2% for TG6 and 20.6% for T17. It s evdent that the values are nfluenced by both the temperature and tme. Table 4 Comparsons of expermental coarsenng exponents wth those smulated usng CA model for T 6Al 4V alloy Temperatures (K) Coarsenng exponent (expermental/predcted) 0.55/ / /0.49 Relatve error (%) Table 5 The chemcal compostons of TG6 and T17 alloys (wt%) Alloy Al Sn Zr Mo Cr Nb Ta S C Fe N O T TG Balanced T Balanced Table 6 The rad and masses of solute atoms for TG6 and T17 alloys Alloy Al Sn Zr Mo Cr Nb Ta S C Fe N O T Radus (pm) Mass Table 7 The ratos of dffuson veloctes between solute atoms and T for TG6 and T17 alloys n sngle feld Alloy Al Sn Zr Mo Cr Nb Ta S C Fe N O Rato ( v / v ) T Table 8 Comparsons of expermental and predcted coarsenng exponents for TG6 and T17 alloys at dfferent temperatures Temperatures (K) Coarsenng exponent TG6 T17 (expermental/predcted) 0.35/ / / / / /0.35 Relatve error (%)

7 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No Fgure 3 Comparson of expermental and CA predcted coarsenng knetcs for TG6 n sngle feld. (a) (c) descrbe the relatonshp of coarsenng gran sze and tme at 1328, 1338 and 1348 K, respectvely, n=1; (d) (e) descrbe the lnear relatonshp of gran sze and tme at 1328, 1338 and 1348 K, respectvely. 3 Results and dscusson 3.1 Effect of temperature and tme on mcrostructure Fgure 5 compares typcal mcrostructures obtaned from CA smulaton (Fgure 5(a) (c)) and experment (Fgure 5(d) (f)) for dfferent coarsenng tmes. It s evdent that the holdng tme exerts a sgnfcant nfluence on coarsenng. The average gran sze ncreases wth ncreasng tme or temperature. Fgure 5(a) (c) shows the mcrostructure evoluton of T 6Al 2Zr 1Mo 1V (TA15) alloy at 1313 K from 10 to 30 mn. The predcted gran sze (dameter) ncreases from about 300 to 600 μm n ths tme. The correspondng expermental mcrostructure evoluton for TA15 s shown n Fgure 5(d) (f). The expermental gran sze ncreases from 410 to 630 μm. The average gran sze s larger than the smulated value for the frst 10 mn, as s shown by comparng Fgure 5(a) and (d). The dfference may be due to the nstablty of the ntal mcrostructure where the number of grans changes greatly wth holdng tme. Wth ncreasng tme, the relatve error n gran sze between experment and smulaton becomes smaller as the gran number becomes relatvely stable. However, the topologcal evoluton of mcrostructure s hardly affected by tme and temperature durng coarsenng. It s found that the growth of grans wth more than sx sdes s at the cost of grans wth less than sx sdes n these fgures, as also shown by Raghavan et al. [22,23] and Chen et al. [26]. Ths agrees well wth von Neumann-Mullns theory that the growth rate of an ndvdual gran strongly depends on the number of gran sdes and remans nvarant wth tme or temperature. The

8 1480 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No.13 Fgure 4 Comparson of expermental and CA predcted coarsenng knetcs for T 17 n sngle feld. (a) (c) descrbe the relatonshp of coarsenng gran sze and tme at 1178, 1188 and 1198 K, respectvely, n=1; (d) (f) descrbe the lnear relatonshp of gran sze and tme at 1178, 1188 and 1198 K, respectvely. gran sde dstrbuton durng coarsenng keeps a normal trend and remans self-smlar (tme-nvarant). Gran sde dstrbuton s an mportant topology ndex whch has been nvestgated prevously [11,23]. Because of statstcal errors or the ntrnsc dsadvantages of the CA method, the predcted sde dstrbuton devates, more or less, from the expermental observaton, but s not dscussed n ths paper. 3.2 Effect of chemcal composton on coarsenng knetcs The chemcal composton sgnfcantly nfluences the statc coarsenng knetcs. In a sngle phase feld, the coarsenng exponent for T 6Al 4V alloy ranges from 0.5 to 0.55 [33], whch s close to the deal value 0.5. But for TG6 and T17 alloys, the values of coarsenng exponent are much smaller than those of T 6Al 4V ttanum alloy. Ths s caused by the solute drag effect. For T 6Al 4V alloy, the solute atoms are manly Al and V. The dffuson velocty of Al atoms s nearly equal to that of T (the rato between them s 0.97), and the dffuson velocty of V atoms s larger than the T dffuson velocty (the rato between them s 1.98). Ths means that these two knds of solute atoms barely retard the movement of gran boundares durng coarsenng. Although the dffuson velocty of the mpurty Fe atoms s lower than that of T, ther volume fracton s so low that the drag effect s also neglgble. Therefore, the coarsenng exponent for ths alloy s close to the deal value 0.5. In comparson, a much larger effect of solute drag on coarsenng exponent of TG6 and T17 alloys s evdent.

9 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No Fgure 5 Predcted (a) (c) and expermental (d) (f) mcrostructures of TA15 at 1313 K wth dfferent holdng tme: (a), (d) 10 mn; (b), (e) 30 mn; (c), (f) 60 mn. From Table 6, we can see that the rad and masses of solute atoms, such as Sn, Zr, Mo, Cr, Nb and Ta, are much larger than those of T. Therefore, accordng to eqs. (10) (12), the dffuson velocty rato of these solute atoms to T s less than 1, as shown n Table 7. Ths means that the solute atoms wll retard the boundary coarsenng and cause a sgnfcant drag effect. Therefore, the predcted values of coarsenng exponent for these alloys n a sngle phase feld are n the range of 0.27 to 0.38, whch are smaller than the deal value 0.5 and agrees well wth the expermental values. 3.3 Lmtatons of model In ths paper, a new CA model consderng the solute drag effect and ansotropc gran boundary moblty was developed to smulate the statc coarsenng of ttanum alloys n the beta feld. Ths model could be used to predct the effect of temperature and holdng tme on gran coarsenng. Durng the process, the varaton of coarsenng knetcs and gran sze wth temperatures and tme was nvestgated by the CA model. Compared to earler models, the predcted results agree better wth expermental measurements. Alternatve factors such as stran and stran rate could also greatly nfluence gran coarsenng. Stran nduced by prevous deformaton or durng dynamc coarsenng enhances the total energy of each gran, and can change the coarsenng knetcs or coarsenng exponent. The change of stran rates durng dynamc coarsenng could nfluence the coarsenng process. Sematn et al. [8] have studed the dynamc coarsenng of T 6Al 4V alloy expermentally, and obtaned some conclusons. However, n ths paper, the CA model manly smulates the statc coarsenng phenomenon, where there s no plastc deformaton before ths process. Therefore, we do not consder the effect of stran and stran rate on coarsenng. The study of dynamc coarsenng for duplex ttanum alloys n sngle and two-phase felds by CA modelng wll be our next work. For valdaton, the CA model was used to smulate the statc coarsenng of ttanum alloys T 6Al 4V, TA15, T17 and TG6. The comparsons of predcted wth expermental results showed good agreement. However, the error s obvous. The average relatve errors of coarsenng exponent between smulated and expermental values range from 11% to 13%, and the largest relatve error s up to 20.6% for T17 alloy at temperature 1188 K. There are several factors producng these dfferences between smulated and expermental results. The frst s the ntrnsc dsadvantage of the cellular automaton algorthm. Errors wll nevtably be ntroduced when a complcated contnuous process s dealt wth by dscrete method such as the CA algorthm. The second factor causng errors s the model for rato of dffuson veloctes of solute atoms to that of T. Ths s the largest contrbutor to the smulated errors. A more accurate and complex formula to descrbe the dffuson velocty rato may be needed, whch wll requre many experments to develop, and may be part of our future work. Thrdly, other factors nfluencng gran coarsenng such as the secondphase partcles, precptaton and varous mcrostructural defects are not consdered, whch also create smulated errors. 4 Conclusons In ths paper, a new cellular automaton model, whch ncludes the effects of solute drag and ansotropc gran

10 1482 Wu C, et al. Chn Sc Bull May (2012) Vol.57 No.13 boundary moblty on coarsenng, was developed to smulate statc gran coarsenng. To quanttatvely descrbe the drag effects produced by dfferent solute atoms on coarsenng, the dffuson veloctes of alloy elements were calculated relatve to that of T. A formula was proposed to descrbe the rato of dffuson velocty between T and solute atoms. Some factors nfluencng the dffuson velocty such as atomc radus, mass and lattce type were consdered. The parameter c 0 was ntroduced nto the calculaton of boundary moblty to consder ansotropy of gran boundares. The CA models were used to smulate the gran coarsenng of T 6Al 4V, T17, TA15 and TG6 ttanum alloys, and the predcted results were n good agreement wth experments. 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