wangG11acta-Mean-field statistical simulation of grain coarsening in the presence of particles

Mean-field statistical simulation of grain coarsening in the presenceof stable and unstable pinning particlesG.Wang a ,b ,1,D.S.Xu a ,⇑,E.J.Payton b ,2,N.Ma b ,3,R.Yang a ,ls b ,Y.Wang b ,⇑aInstitute of Metal Research,Chinese Academy of Sciences,72Wenhua Road,Shenyang 110016,People’s Republic of China bDepartment of Materials Science and Engineering,The Ohio State University,2041College Road,Columbus,OH 43210,USAReceived 4October 2010;received in revised form 9February 2011;accepted 1April 2011Available online 22April 2011AbstractGrain coarsening in the presence of second-phase particles is investigated using a mean-field statistical model.With any given initial grain size distribution and average size and volume fraction of pinning particles as input,the model predicts grain growth kinetics and temporal evolution of the grain size distribution.Simulations are performed for both stable and unstable particle populations with sev-eral different initial grain size distributions.It is shown that the kinetics of grain coarsening is sensitive to the initial grain size distribution and pinning force variation as a function of time.Under a constant pinning force (stable particles),the final mean grain size does not vary significantly with different initial grain size distributions,though the kinetics of grain growth differs.On the other hand,a pinning force that decreases with time (associated with,for example,particle dissolution)could induce temporarily abnormal grain coarsening and a wider initial grain size distribution results in an earlier onset of the abnormal grain coarsening and a smaller final grain size.Ó2011Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.Keywords:Grain growth;Zener pinning;Time-dependent drag force;Abnormal grain coarsening;Computer modeling1.IntroductionMany engineering alloys are strengthened by second-phase particles.These particles not only interact with dislo-cations,they can also hinder the motion of grain bound-aries,inhibit grain growth and ultimately lead to a fine grain microstructure.The latter phenomenon is often referred to as “Zener pinning ”and has been studied exten-sively for the case of a stable population of pinning parti-cles [1,2].Second-phase particles may,however,become unstable during certain heat treatment schedules,and theircoarsening or dissolution will have an effect on grain growth behavior.It is therefore desirable to predict grain growth behavior in the presence of time-dependent Zener drag forces.Parallel to the treatment of precipitate growth,coarsen-ing and/or dissolution,there are two approaches to grain growth modeling:(i)mean-field models [3–10]based on Hillert’s treatment [11];and (ii)direct computer simula-tions,e.g.by the Monte Carlo method [12,13],the finite-element method [14]and the phase field method [15–17].The major advantage of the latter techniques is that mini-mal assumptions are needed concerning the detailed micro-structural information about both grains and pinning particles such as their shape,size and spatial distribution,or the changes of these characteristics with time.A serious drawback of the latter group,the direct simulation tech-niques,is that they require massive computer resources,especially in the case of three-dimensional (3-D)simula-tions.A detailed comparison of the strengths and limita-tions of three direct simulation methods for modeling1359-6454/$36.00Ó2011Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.doi:10.1016/j.actamat.2011.04.004⇑Corresponding authors.Tel.:+862423971946;fax:+862423891320.E-mail addresses:dsxu@ (D.S.Xu),wang.363@ (Y.Wang).1Present address:South China University of Technology,Guangzhou 510640,People’s Republic of China.2Present address:Institute for Materials,Ruhr-Universita ¨t Bochum,Bochum 44801,Germany.3Present address:ExxonMobil Research &Engineering Company,Annandale,NJ 08801,USA./locate/actamatActa Materialia 59(2011)4587–4594Zener pinning was made by Harun et al.[18].However,to the best of our knowledge,there have not yet been any published3-D simulations of grain growth in the presence of dissolving and/or coarsening populations of pinning particles using direct methods,though a2-D phase-field study on Zener pinning with evolving particles has been reported[19].In contrast,mean-field models have the abil-ity to make rapid predictions of grain growth kinetics using a desktop computer,and hence have been widely adopted in industrial applications.In previously published work, however,constant drag force was usually assumed and lit-tle attention has been paid to the case of time-dependent pinning forces thus far.In addition,the influence of the shape of the initial grain size distribution(GSD)on the grain growth behavior in the presence of pinning particles has not been fully investigated,especially for the case of time-dependent pinning forces.This paper investigates grain coarsening in the presence of different populations of pinning particles using a mean-field statistical model while maintaining a specific focus on the effect of time-dependent drag forces and differing initial GSDs on the grain growth behavior.The incorporation of drag forces into a3-D mean-field statistical model of grain growth is introduced in Section2,and the simulation meth-odology and critical parameters are discussed in Section3. Simulation results showing the effects of different initial GSDs and variable pinning forces are presented in Section4 and discussed in Section5.The majorfindings are summa-rized in Section6.2.Incorporation of particle pinning into the mean-field statistical grain growth modelIf grain growth is described in terms of elementary mat-ter exchanges between pairs of interacting grains and it is assumed that(i)grain boundary properties(energy and mobility)are isotropic,(ii)grain boundary velocity is pro-portional to the pressure difference caused by curvature and(iii)the average curvature of grains can be calculated from their volume-equivalent spheres(an assumption that is well founded experimentally according to Patterson and Liu[20]),the following kinetic equation for grain coarsening in single-phase polycrystalline materials may be derived[7–10,21]:dV i dt ¼M cXjA ij2R jÀ2R ið1Þwhere V i and R i are the volume and radius of grain i, respectively,t is time,M is grain boundary mobility,c is grain boundary energy,and A ij is the contact area between grain i and an adjacent grain j.Since the contact area between pairs of adjacent grains must always be equal, i.e.A ij=A ji,Eq.(1)automatically satisfies the volume con-servation condition of the whole system.It should be noted that Eq.(1)is sufficiently general since A ij can be chosen to account for any microstructure in the form of correlations between grain sizes.In the present work,the following for-mulation of A ij has been adopted:A ij¼4p R2iR2jXkR2k,ð2Þwhich can be derived from the assumption of a random ar-ray of grains[17].Following Hillert’s treatment[11],one can obtain the rate of volume change of a grain in a parti-cle-containing material by inserting into Eq.(1)a term to account for the drag of particles on moving grain boundaries:dV idt¼M cXjA ij2R jÀ2R iÆz:ð3ÞHere the drag term,z,due to second-phase particles can be evaluated following Sun et al.[22]or estimated from the classical Zener approximation[1,2]:z¼3f=2rð4Þwhere f and r are the volume fraction and average radius of the pinning particles,respectively.If the pinning particles are assumed to have no effect on the contact area between grains,then the grain growth rate can be described as: dR idt¼M cXjR2j2R jÀ2R iÆzXjR2jð5ÞIt is worth emphasizing that the sign for the z term in Eqs.(3)and(5)must be chosen in each case such that the pinning force is acting against the movement of the grain boundaries.The negative sign holds when1/R i<1/ R jÀz/2and the positive sign holds when1/R i>1/R j+ z/2.In the case that1/R i P1/R jÀz/2or1/R i61/R j+ z/2,dR i/dt must be zero.In this mean-field model the pin-ning force z simply imposes a uniform friction force on grain boundary migration.Grain boundaries can move only when local driving forces exceed this friction force.If z=0,Eq.(5)reduces to the classical Hillert equation[11]:dR idt¼2M cXjR j,XjR2jÀ1R i!¼2M c1R crÀ1R ið6Þand the critical grain size R cr¼R2=R agrees with the result obtained by Lu¨cke et al.[23],where the relationship is de-rived from a two-parameter statistical theory.3.Simulation methodology3.1.Numerical solution of the mean-field statistical grain growth equationTo solve Eq.(5)numerically,instead of using continu-ous distributions,grains are divided into discrete classesof size D R and those having sizes from R ciÀD R=2toR ciþD R=2belong to the i th class centered at R ci.The number of grains in class i can be calculated from the size distribution function(SDF)/i,which represents the4588G.Wang et al./Acta Materialia59(2011)4587–4594statistical frequency of occurrence of grains system.Thus,the following relationshipsX i /i¼1X i /iR ci¼ R:Inserting Eqs.(7)and(8)into Eq.(5) equation for grains in class i that reads:dR ci dt ¼M cXj/jðR cjÞ22R cjÀ2R ciÆz!"#,XjThe SDF must also obey the continuity@/ @t ¼À@@R/dRdt:Eq.(10)may be numerically integrated by the following forms:D/i D t ¼À1D R/idR cidtÀ/iÀ1dR c iÀ1dtdR cidtD/i D t ¼À1D R/idR cidtÀ/iþ1dR c iþ1dtdR cidtwhere D t is time step,D/i is the change of/ithe growth rate dR ci =dt is calculated from Eq.poral evolution of GSD may then beEq.(11),and the average grain radius any time through Eq.(8).3.2.Initial conditions and simulationIt is well accepted that the GSD is often log-normal in experimental observations[24] that the probability density function ofbefitted to:fðRÞ¼1R rffiffiffiffiffiffi2pp expÀðln RÀlÞ22r2"#;where R is the grain radius and thethe mean and standard deviation of ln R,respectively.In this paper,three log-normal GSDs are generated as model input,and l and r are chosen such that each case has the same initial mean grain size but a different distribution.Ta-ble1and Fig.1show the parameters and shapes of the three GSDs,respectively.The bin size D R of the GSD his-togram was set to0.2l m in all simulations.Since the value of M c in most alloys is in the range of0.1–10l m2s–1[25],M c=1l m2s–1was adopted in the present work.The time step was set to D t=0.01s,and the total annealing time that was simulated in each case was3600s. Because the GSD shifts from left to right during grain growth,the bin number was set to be400in order to avoid overflow of the maximum grain size from the right boundary of the GSD.In the case of constant pinning force simulations,the effects of drag term with values of z=0.1,0.2and0.3l mÀ1were investigated.For simulations of unstable particle populations,five decreasing pinning forces are assumed,each of which decrease from1.0to 0.01l mÀ1at a different rate.The shapes of the decreasing pinning force curves are given in Fig.2,where L1refers to linear decrease,L2and L3to the initially relatively slow and then rather rapid linear decreases,and L4and L5to the initially relatively fast and then rather slow linear decreases.4.Results4.1.Grain growth with constant pinning forceThe grain growth behavior of the three systems having GSDs shown in Fig.1with the same initial mean grain size1.The initial log-normal GSDs with the same mean grain size usedto the model.2.Five different shapes of decreasing pinning forces examined in present study.G.Wang et al./Acta Materialia59(2011)4587–45944589which indicates that a larger pinning force results in a rower GSD and a correspondingly larger peak value.3.Grain growth kinetics for the three different initial GSDs constant pinning force z =0.1l m À1.4.Temporal evolution of GSD2with constant pinning 0.1l m À1.5.Grain size distributions at t =3600s for the three initial GSDs constant pinning force z =0.1l m À1.6.Grain growth kinetics of GSD2at different levels of constant pinning force.7.Grain size distributions of GSD2at t =3600s with different constant pinning forces.A small kink exists after the peak evolved GSD.This kink is located where changes sign.This sign switching causes a in the distribution evolution equation,but on the overall grain growth kinetics.4.2.Grain growth with decreasing pinning In most commercial alloys,the pinning not always be stable.For example,a 2are two important types of precipitates in and they may dissolve and/or coarsen at different temperatures.According to Eq.lution and coarsening of these particles decreasing the pinning force in the system,ferent conditions the pinning force may ent rates.The case of constant decreasing rate Fig.8for each of the three initial GSDs It can be seen that starting from the same size,the one having the widest distribution initial growth rate,but the growth slows stages;eventually the system ends up with grain size than those having initially tions.The initial jump of GSD3could be the fact that there are more grains with pairwise stagnation criteria (1/R j +z R j Àz /2)in the case of GSD3than in the The later stage of the growth is different constant pinning forces (shown in Fig.3),initial distribution results in the coarsest ilar behaviors are also found in the cases of the results are not shown here.The growth rates for L1and L4increase smoothly as the pinning force decreases,irrespective to the sudden rate change in the case of L4.It is interesting,though,that a wavy feature is exhibited in the growth kinetics of case L5.In this case,the grain size initially holds constant because the driving force for growth cannot overcome the large pinning force.The average grain size then jumps rap-idly due to the rapid reduction in the pinning force.Finally a sharp rate change in the reduction of the pinning force results in a dramatic rate change in grain growth.Fig.10shows the GSDs at t =3600s with GSD1as ini-tial distribution,obtained for the five cases with decreasing pinning force.It can be seen that the larger initial pinning forces in L2and L3lead to narrower distributions and higher peak values of the SDF than for L4and L5,where the pinning force decreased dramatically at the beginning of the simulation.5.DiscussionIn this paper,grain growth in particle-containing mate-rials was simulated using a mean-field statistical model8.Grain growth kinetics for three different initial GSDs with linearly decreasing pinning force L1.9.Grain growth kinetics of GSD1with different decreasing pinning forces.10.Grain size distributions of GSD1at t =3600s with different decreasing pinning forces.G.based on the pair interactions of neighboring -pared with the early mean-field models [6–9],the model used here can capture more accurately the contributions from surrounding grains to the growth pressure because it does not rely on the calculation of a critical grain size,and it can be applied to cases with either constant or vari-able pinning forces.Although three different log-normal distributions with the same initial mean grain size are assumed as input here,the model may in principle accept any GSD as input.For simplification,the properties of all boundaries are set to be isotropic in the paper,but it is possible to extend the model to the cases with anisotropic boundary properties [21]or non-monotonic annealing temperatures.The kinetic equation in the present model is only an approximate solution for the evolution of the GSD.Recently,a rigorous and specific formula for the growth rate (volume change)of a grain in a 3-D microstructure was presented based upon the von Neumann relation [26]:dV dt ¼À2p M c L ðD ÞÀ16X ni ¼1e i ðD Þ !;ð13Þwhere L (D )is a natural measure of the linear size of do-main D and P ni ¼1e i ðD Þis the summation of the length of triple line e i over all n triple lines of D .For a spherical grain,the mean width of D ,L (D ),is twice its diameter [26],thus Eq.(13)can be written as:dR dt ¼M c 112R X ni ¼1e i ðD ÞÀ2R !:ð14ÞComparing the above equation with Eq.(5),one can see that both equations can be reconciled if the following expression holds:X n k ¼1e k ðD i Þ¼12R 2i 2X R j =XR 2j Æz :ð15ÞSpherical grains cannot fully fill space,and thus cannot have triple lines or fulfill any other topological constraints of a 3-D space-filling network.Therefore,the present sta-tistical model may be regarded as an approximation of the exact formula given by Eq.(13)based on two assump-tions:(i)the polycrystalline microstructure is described by arrays of spherical grains,and (ii)the equivalent value of P nk ¼1e k ðD i Þof spherical grains can be calculated by Eq.(15).One can also see that this value is affected by both the neighboring grain size and the mean-field pinning force.However,it should be mentioned that when the pinning force is zero,this mean-field model reduces to the classical Hillert grain growth model [11]and yields the Hillert distri-bution that tends to disagree with experimental observations.The grain growth processes of the three different initial GSDs considered here exhibit different kinetic features although they have the same initial grain size.Under con-stant pinning forces,the average grain sizes of all GSDs increase with time but the rates of increase decline towardzero with time (Figs.3and 4)when the pinning force even-tually balances the driving force in Eq.(3).The simulation results also indicate that with identical pinning force and initial average grain size,wider GSDs produce larger grain sizes (Fig.3).This effect arises because a wider distribution has a larger grain size discrepancy and therefore a larger average value of |1/R j À1/R i |.It can be seen from Eq.(5)that the driving force for grain growth is a function of GSD,so should be the final grain size.Thus,the Zener equation should only be regarded as a zero-order estima-tion of the stagnant grain size.The Zener size,R Z =4r /3f =2/z =20l m,is close to our calculated values shown in Fig.3.The distribution shapes converge as the grain size stag-nates (Figs.4and 5).This is not surprising because the local driving forces (local grain size differences)in all the cases are balanced by the same constant and uniform pin-ning force at stagnation.For a given initial GSD,the sim-ulation results show that a larger pinning force leads to a slower growth rate (Fig.6)and a narrower final GSD (Fig.7).In the case of a large pinning force,boundary motion is only possible between pairs of grains that are at both sides of the distribution.Since the absolute value of the driving force is much higher for small grains,the dis-tribution narrows as the large grains approach the stagnant size,at which point there is little driving force for their con-tinued growth.However,the GSD is still wider than the initial one,which means that when the drag force is large,fewer grains are outside the stagnation range for a given starting GSD and hence limited coarsening occurs to alter the GSD.The situation becomes more complicated when the pin-ning force changes with time.It can be seen from Fig.8that the three initial GSDs lead to different growth kinetics.In contrast to the cases of constant pinning force,where the growth rates decrease continuously (Fig.3),the growth rates increase continuously within the simulated annealing time as the pining force decreases monotonically.More interestingly,the narrower distribution GSD1has a more sluggish growth initially but grows more rapidly at later stages,ultimately yielding coarser grains.The sluggish ini-tial growth rate is due to the limited number of grains with high enough driving force for growth,i.e.the pairs of grains at the both sides of the distributions and outside the bounds set by the Zener drag.When growth is not stag-nant and the coarsening is most prominent at the tails of the distribution,abnormal grain coarsening may occur until a new balance is reached between the driving force and the drag force.When pinning forces with non-constant decreasing rates are assumed,the kinetic curves exhibit some interesting characteristics,such as a flat region during the first stage due to the excessively large pinning force and a rapid jump at some later time when the pinning force is reduced sud-denly (Fig.9).It can be observed from Figs.8–10that both the pinning force in the system and the initial size distribu-tion influence the shape of the GSD.In particular,Fig.84592G.Wang et al./Acta Materialia 59(2011)4587–4594shows that the narrower initial GSD(e.g.the largerfinal grain size when the decreasingis applied.In the case of GSD1,as the decreases,there is a relatively sharp increaseof grains with sizes outside the stagnationgrains are unleashed)at certain given2100s),and hence the grain growth ratetwo cases from then on.This is illustratedwhich the comparison of GSDs at t=2100sferent initial GSDs is given.Novikov[6,7]studied grain growthdecreasing pinning force and analyticallyreduction in pinning force can lead toof large grains.In order to check thean amplified growth during the processeswe calculate Y¼RdR max=dtÀR max dR=dt.growth rate of the largest grain is higher thanof average size,indicating amplified growth of large grains,which is related to abnormal(secondary recrystallization type)grain growth[27].Fig.12shows the change in Y as a function of time for the systems with GSD1and GSD3 subject to decreasing pinning forces L1,L2and L5(only the positive part of Y is shown here).It can be seen that amplified growth of large grains does takes place temporar-ily when the pinning force decreases,and this temporary abnormal grain coarsening is induced at an earlier time in the case of a wider initial distribution(GSD3in Fig.12).This is a result of the interplay between the char-acter of the initial GSD and the decreasing rate of the pin-ning force.For example,a wider distribution contains a larger number fraction of grains having sizes far from the critical grain size.Grains that are much smaller and larger than the critical grain size and are adjacent to each other higher the possibility for abnormal grain coarsening.Both the pinning force and the initial GSD(and its temporal evolution)determine the character of abnormal grain coarsening(i.e.the intensity and the time at which it occurs)during the heat treatment process.Ultimately,this results in different growth kinetics for different initial GSDs when growth occurs in the presence of an unstable popula-tion of second-phase particles.While the parameter Y does indicate that abnormal grain growth is possible for a few cases,it occurs only over a short time and with a small value of Y.This is related to the fact that the GSDs shown in Fig.10do not contain any truly abnormal grains that are typically an order of magni-tude larger than the average.On the other hand,the pin-ning force is distributed uniformly throughout the system in the mean-field model.Under such a circumstance,a true abnormal grain growth may never occur.Dennis et al.[28] studied abnormal grain growth in an Al–3.5wt.%Cu alloy and found that the fraction of pinning phase was much lower at abnormal grain boundaries than at normal bound-aries in thefine-grained matrix into which the abnormal grains were growing.Since the mean-field model can be applied easily in cases with more complicated variation of the pinning force,three additional types of unstable particles that could be encoun-tered in practice were considered.Thefirst is coarsening in which the volume fractions of particles are constant and their mean size is proportional to the cube root of time; the second is size-dependent dissolution,where the volume fractions of particles follow f=f0exp(Àkt n)(where f0is the initial volume fraction and k and n are constants)and the average particle radius can be estimated by r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2Àatp(where r0is the initial radius and a is con-stant)[29];and the third considers multiple populations of precipitates,where one population disappears com-pletely,leading to a step function in f/r.For real systems, the detailed expressions of f and r can be obtained byfitting the experimental measurements and the change of z with time can be calculated by Eq.(4).The change in drag force11.Grain size distributions at t=2100s for three different initial GSDs with linearly decreasing pinning force L1.12.Plots of Y¼RdR max=dtÀR max dR=dt as a function of time for systems having GSD1and GSD3with decreasing pinning force,L1, L5shown in Fig.2(only the positive axis of Y is shown here).G.with time is plotted in Fig.13,and the GSDs at t=3600s for the system initially having GSD1are given in Fig.14.It is shown that different decreasing rates of pinning force lead to different grain growth kinetics.Therefore,our mean-field model can deal with complex cases,and should therefore be of practical help to metallurgists.6.ConclusionsA mean-field statistical model of grain growth was developed to predict the growth kinetics and evolution of GSDs in systems with both stable and unstable populations of pinning particles.The model is able to consider any arbi-trary variation of pinning particle population as function of time during grain coarsening.For systems with stable particles and hence constant pinning force,the results dem-onstrate that larger pinning forces result infiner grain sizes and narrower GSDs,while for systems with unstable populations of pinning particles,the decreasing pinning forces result in complicated grain growth kinetics deter-mined by the interplay between the character of the initial GSD and the decreasing rate of the pinning force.Abnor-mal grain coarsening was observed temporarily after a rapid reduction in the pinning force,but the uniform distri-bution of pinning force among all moving boundaries assumed in the mean-field model prevented the prediction of extended abnormal grain growth as seen in secondary recrystallization.AcknowledgementsThe support of the Ministry of Science and Technology of China under Grant No.2006CB605104and2011CB606404, the National Natural Science Foundation of China under Grant No.50631030(G.W.,D.S.X.and R.Y.),GE Aviation (E.J.P.,M.J.M.and Y.W.),the US Air Force Office of Scientific Research under Grant No.FA9550-09-1-0014 (M.J.M.and Y.W.)and the US National Science Founda-tion under Grant No.CMMI-0728069and Grant No. DMR1008349(Y.W.)is gratefully acknowledged. References[1]Smith CS.Trans AIME1948;175:15.[2]Manohar PA,Ferry M,Chandra T.ISIJ Int1998;38:913.[3]Anderson I,GrongØ,Ryum N.Acta Metall1995;43:2689.[4]Rios PR.Acta Mater1997;45:1785.[5]Humphreys FJ.Acta Mater1997;45:5031.[6]Novikov VYu.Scripta Mater1997;37:463.[7]Novikov VYu.Scripta Mater1999;40:91.[8]Di Nunzio PE.Acta Mater2001;49:3635.[9]Maazi N,Rouag N.J Crys Growth2002;243:361.[10]Di Nunzio PE.Phys Rev B2003;68:115432.[11]Hillert M.Acta Metall1965;13:227.[12]Miodownik M,Holm EA,Hassold GN.Scripta Mater2000;42:1173.[13]Zollner D,Streitenberger P.Scripta Mater2006;54:1697.[14]Couturier G,Doherty R,Maurice C,Fortunier R.Acta Mater2005;53:977.[15]Suwa Y,Saito Y,Onodera H.Scripta Mater2006;55:407.[16]Moelans N,Blanpain B,Wollants P.Acta Mater2006;54:1175.[17]Moelans N,Blanpain B,Wollants P.Acta Mater2007;55:2173.[18]Harun A,Holm EA,Clode MP,Miodownik MA.Acta Mater2006;54:3261.[19]Fan D,Chen LQ,Chen SP.J Am Ceram Soc1998;81:526.[20]Patterson BR,Liu Y.Metall Trans A1992;23:2481.[21]Abbruzzese G,Lu¨cke K.Acta Metall1986;34:905.[22]Sun N,Patterson BR,Suni JP,Weiland H,Allard LF.Acta Mater2006;54:4091.[23]Lu¨cke K,Brandt R,Abbruzzese G.Mater Sci Forum1995;204–206:19.[24]Underwood EE.In:DeHoffRT,Rhines RF,editors.Quantitativemicroscopy.New York:McGraw-Hill;1968.p.149.[25]Feppon JM,Hutchinson WB.Acta Mater2002;50:3293.[26]MacPherson RD,Srolovitz DJ.Nature2007;446:1053.[27]Thompson CV,Frost HJ,Spaepen F.Acta Metall1987;35:887.[28]Dennis J,Bate PS,Humphreys FJ.Acta Mater2009;57:4539.[29]Wang G,Xu DS,Ma N,Zhou N,Payton EJ,Yang R,et al.ActaMater2009;57:316.13.Drag force z as a function of time for three different typesunstable particles:coarsening particles,size-dependent dissolvingand mixture of dissolving and stable particles.14.Grain size distributions at t=3600s of GSD1for three differenttypes of unstable particles:coarsening particles,size-dependent dissolvingparticles,and mixture of dissolving and stable particles.59(2011)4587–4594。