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Computational Materials Science 21 (2001) 69±78                                                                           ...
70                      A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78which is fully understood,...
A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78                             71model does not allo...
72                           A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78   The structure of a...
A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78                   73P …Si ;Sj ;DE;T †            ...
relative magnitudes of the grain boundary energy
DE…Si ;Sj †
J …Si ;Sj † M…Si ;Sj † ;
DE60;     compared with the stored energy. The MC reori-
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Monte Carlo RX


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Monte Carlo RX

  1. 1. Computational Materials Science 21 (2001) 69±78 hybrid model for mesoscopic simulation of recrystallization A.D. Rollett a,*, D. Raabe b a Department of Materials Science and Engineering, Wean Hall 4315, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-2890, USA b Max-Planck-Institut fr Eisenforschung Max-Planck-Str. 1, 40237 Duesseldorf, Germany u Received 17 November 1999; received in revised form 24 August 2000; accepted 10 October 2000Abstract A brief summary of simulation techniques for recrystallization is given. The limitations of the Potts model and thecellular automaton model as used in their standard forms for grain growth and recrystallization are noted. A newapproach based on a hybrid of the Potts model (MC) and the cellular automaton (CA) model is proposed in order toobtain the desired limiting behavior for both curvature-driven and stored energy-driven grain boundary migra-tion. Ó 2001 Elsevier Science B.V. All rights reserved.Keywords: Recrystallization; Modeling; Simulation; Monte Carlo; Potts model; Cellular automaton1. Introduction any point in time is governed by weighting the probabilities for each type or reorientation at- This paper gives a brief summary of mesoscopic tempt according to the desired balance in drivingmethods for simulation of recrystallization and forces.introduces a new approach for modeling the e€ectscurvature-driven and stored energy-driven migra-tion of boundaries. The standard form of the 2. Summary of simulation methodsMonte Carlo (MC) model does not result in alinear relationship between migration rate and The need for computer simulation of recrystal-stored energy. On the other hand, the standard lization is driven by two di€erent considerations.form of the cellular automaton (CA) model used One is the need to be able to make quantitativefor recrystallization does not allow for curvature predictions of the microstructure and properties ofas a driving force for migration. The basis for the materials as a€ected by annealing. Simulation cannew approach is the use of two di€erent methods be used to predict the average texture and grainfor determining orientation changes at each site in size which strongly a€ect mechanical behavior. Ana lattice. The choice of which method to apply at equally important motivation for simulation, however, is the need for improved understanding of the recrystallization phenomenon that is highly * Corresponding author. Tel.: +1-412-268-3177; fax: +1-412- complex from a microstructural point of view. The268-1513. development of strong cube textures in fcc metals E-mail address: (A.D. Rollett). or the Goss texture in silicon steels, neither of0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 7 - 0 2 5 6 ( 0 0 ) 0 0 2 1 6 - 0
  2. 2. 70 A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78which is fully understood, illustrates the impor- of the grain structure, i.e., the vertices or tripletance of being able to model the recrystallization junctions between grain boundaries, and areprocess in considerable detail. Indeed, since the therefore, ecient because only the vertex motionmicrostructural evolution process inherent in re- need be calculated, provided that local equilibriumcrystallization depends on the grain boundary can be assumed at triple junctions. They haveproperties which are sensitive to their crystallo- some limitations when second phases must begraphic character then the importance of simula- considered, however, see also the work of Frosttion techniques that can model the full range of [16]. Humphreys [13] has applied the networkboundary behavior is apparent. model to the nucleation process in recrystallization Computer simulation of grain growth and re- by considering coarsening processes in subgraincrystallization was strongly stimulated in the early networks and also in cases where second phase1980s by the realization that MC models could be particles are present [17].applied to problems of grain structure evolution. The CA and MC methods discretize the mi-By extension of the Ising model for domain crostructure on a regular grid. Both methods havemodeling of magnetic domains to the Potts model, been successfully applied to recrystallization andit was then possible to represent grains (domains) solidi®cation. The CA method uses physicallyby regions of similarly oriented (lattice) points [1]. based rules to determine the propagation rate of aThe models have been used, for example, to in- transformation (e.g., recrystallization, solidi®ca-vestigate texture evolution during grain growth tion) from one cell to a neighbor [18±21]. The MCand recrystallization, reinforced by parallel work method as derived from the Potts model (i.e., aon analytical models [2,3]. A critical issue ad- multistate Ising model), has been applied to a widedressed by Humphreys and others [4±8] extension variety of problems in recrystallization [22±24].of grain growth theory is that of coarsening of The kinetic MC model simulates boundary motionsubgrain networks in order to clarify under what via an energy minimization procedure for deter-circumstances one expects to observe nucleation of mining a reorientation probability at each step.recrystallization. This view of nucleation as simply Other methods that are suitable for recrystalliza-non-uniform coarsening (i.e., abnormal subgrain tion simulation include the phase-®eld model [25]growth) is signi®cant for its blurring of the dis- and the ®nite element method [20,21].tinction between continuous and discontinuousrecrystallization, at least for pure metals in whichrecovery is rapid. 3. Current status: limitations of the MC and CA There are a number of current methods of modelsmesoscopic simulation for recrystallization. Thegeometrical, method addresses primarily the ®nal Neither the MC model nor the CA model inmicrostructural state; it can be used to investigate their standard form is entirely satisfactory formicrostructural evolution, provided that one is studying boundary migration under the combinednot concerned the e€ect of grain growth occur- in¯uence of two di€erent driving forces. In grainring in parallel. Such models of recrystallization growth, the driving force derives from the curva-were ®rst elaborated by Mahin and Hanson [9] ture of a boundary, [26] for which the MC model isand then, developed further by Frost and satisfactory. In primary recrystallization, theThompson [10]. Furu [11] and Juul Jensen [12] driving force is the removal of stored dislocationshave recently extended these models to predict by the migration of the recrystallization front, forgrain size and texture development during re- which the CA model is satisfactory. In both cases,crystallization. we expect to ®nd a linear relationship between The second method, based on network models, driving force and migration rate (velocity). Theis an ecient way to represent microstructural actual behavior of the MC model does not giveevolution in discretized form [13±15]. These mod- this for stored energy as a driving force (except forels are ecient because they abstract a key feature low driving force, see below), however, and the CA
  3. 3. A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78 71model does not allow for boundary curvature as adriving force. In the case of the grain growth model imple-mented in the MC model, one writes a Hamilto-nian for the system energy, E, 1XX n NNEˆ J…Si Sj †…1 À dSi Sj †; …1† 2 i jwhere the inner sum is taken over the NN nearestneighbors of site i (typically ®rst and secondnearest neighbors), d is the Kronecker delta func-tion, and J is the energy of a unit of boundary Fig. 1. Plot of transition probabilities for the Potts model as a function of the energy change: note the unit probability for zerobetween elements of indices Si and Sj . To evolve energy change. The energy axis is in units of the interactionthe structure, a site and a new index are chosen at strength (J) nominally for a triangular lattice in which therandom. The element is reoriented to the new in- maximum change is Æ6J.dex with probability, PMC , dependent on thechange in energy, DE, and the local properties of and J ˆ 1. This feature is critical to the grainthe boundary. growth model because it means that kinks or stepsPMC …Si ; Sj ; DE; T † on the boundaries can execute random walks 8 along the boundaries. This, in e€ect, allows > J …Si ; Sj † M…Si ; Sj † ; > DE 6 0; changes in curvature to be communicated along a < J Mmax max ˆ boundary [27]. Modi®cation of the transition > J …Si ; Sj † M…Si ; Sj † > probability from the reference condition decreases : exp…ÀDE=kT †; DE > 0; Jmax Mmax the probability linearly with either boundary en- …2† ergy and/or mobility [5]. The MC model has been used extensively towhere DE is the energy change for the reorienta- model recrystallization in which case a stored en-tion attempt, including both con®gurational and ergy term, Hi , is added to the Hamiltonian.stored energy contributions (see Eq. (3), below), Mis the boundary mobility between elements of in- 1XX n NN X Eˆ J …Si Sj †…1 À dSi Sj † ‡ Hi : …3†dices Si and Sj ; Jmax and Mmax are the maximum 2 i j iboundary energy and mobility, respectively, k isthe Boltzmann constant, and T is the lattice tem- At low stored energies, H < J , and ®nite latticeperature. After each reorientation attempt, the temperature, a linear relationship between migra-time is incremented by 1/n Monte Carlo Steps tion rate and driving force is recovered. The mi-(MCS), and a new reorientation is attempted. For gration rate of a recrystallization front will bea material with uniform boundary energy, the unit proportional (in the absence of a net curvature) toboundary energy plays no role in determining the the di€erence in forward and backward reorien-transition probability. Only if a range of boundary tation rates. That is to say, for sites where the onlyenergies and/or mobilities is present in the system change in energy is associated with removing oris the transition probability decreased from the adding stored energy, the migration rate, v, isreference value. given by the following, where J ˆ 1, M ˆ 1, and It is instructive to plot the transition probabil- H ( J.ities at zero temperature, Fig. 1, to illustrate that H Hthe transition probability remains at one for zero v G Pforward À Pbackward ˆ 1 À exp À % :energy change. Discrete values of the energy T Tchange are shown, based on a triangular lattice …4†
  4. 4. 72 A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78 The structure of a CA model is geometricallyvery similar with either a triangular or squarelattice in two dimensions and either nearestneighbors included in the environment (NN, orvon Neumann neighboring), or also next nearestneighbors (NNN, or Moore neighboring). Forsimulation of recrystallization, the switching rule issimple: any unrecrystallized cell (site) that has arecrystallized neighbor cell will switch to the ori-entation of that neighbor. Scaling of boundarymigration rates is then achieved by relating the Fig. 2. Plot of the variation in switching probability in the CAratio of the unit distance (lattice repeat) in the model with a range of driving forces or mobilities present, eachmodel to the unit time step (maximum velocity in of which is scaled by a maximum value. Note that the proba-the model) to a maximum migration rate for the bility goes to zero at zero change in energy.speci®ed experiment. Note that the environment oforientations around the site to be reoriented doesnot a€ect the outcome of the individual reorien- that no reorientation can occur if the energytation step. Also if a range of mobilities is present change is zero or positive in contrast to the MCin the system and/or a range of driving forces, then model.the transition probability is scaled according to a Again, it is important to note that the frame-maximum value, following [20,21]. work of the CA model can be adapted to model curvature-driven grain growth provided that a M…Si ; Sj † DH …Si ; Sj † probabilistic local rule is used that reproduces thePˆ ; …5† Mmax DHmax switching probabilities of the MC model. Even the simultaneous updating of all sites inherent inwhere P is the switching probability, M is a mo- the CA model can be accommodated by de®ningbility that depends on the local boundary char- sublattices and performing updates on each sub-acter and DH is the driving force (spatially lattice in turn (see for example [28, p. 296]). Avariable) which is the di€erence in stored energy square lattice requires two sublattices, a triangularbetween site i and site j. Raabe also introduces an lattice requires three, a square lattice with Mooreupper limit on the switching probability that scales neighboring requires four, and so on. The principlewith the number of trials such that the smaller the for construction of the sublattices is that, all thestatistical variance desired, the lower the upper neighbors of a site must belong to a di€erentlimit for the switching probability. The di€erence sublattice than that of the site itself. The issue re-between conventional CA and the probabilistic mains of how to write a rule for reorientation thatCA models currently used is that in the latter correctly combines both curvature and stored en-approach: (a) sites are reoriented sequentially; ergy as driving forces for boundary motion. This(b) the location of the site to be reoriented is point can be illustrated by attempting to write achosen at random; (c) any given reorientation step switching probability for the combined e€ect ofcompares the transition probability to the output curvature and stored energy driving forces in orderof a random number generator (over the interval to obtain a linear relationship between migration0±1) in order to determine whether or not that rate and driving pressure. Based on the particularparticular step is successful. Fig. 2 illustrates the form presented here, Eq. (6a), the switchingvariation in probability with magnitude of the probability yields the desired limiting behavior ifchange in energy associated with a given change in the grain boundary energy approaches zero, butorientation, with the maximum probability set at does not if the stored energy approaches zero be-unity. This allows for a linear scaling in boundary cause energy neutral switches would have zeromigration rate with driving force. Note, however, probability.
  5. 5. A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78 73P …Si ;Sj ;DE;T † and MC reorientation attempts determines the 8
  6. 6. relative magnitudes of the grain boundary energy
  7. 7. DE…Si ;Sj †
  8. 8. J …Si ;Sj † M…Si ;Sj † ;
  9. 9. DE60; compared with the stored energy. The MC reori-
  10. 10. DEmax
  11. 11. Jmax Mmax entations are determined only by con®gurationalˆ
  12. 12. DE…Si ;Sj †
  13. 13. J …Si ;Sj † M…Si ;Sj †
  14. 14. energy such that only the ®rst term on the RHS of :
  15. 15. DEmax
  16. 16. Jmax exp…ÀDE=kT †; DE 0:
  17. 17. Mmax Eq. (1) is used to determine DE. The e€ect of …6a† stored energy is incorporated into the CA model. For any given reorientation attempt, the choice ofThe alternative to this switching probability is to which type to apply is governed by a randomwrite an additive combination of the curvature and number, R on the interval …0; 1†: if the value of R isstored energies as follows, where it is understood less than C=1 ‡ C, a CA reorientation is attempted,that DE and J are dependent on the local boundary otherwise an MC reorientation attempt is and local variations in stored energy. The crucial advantage of this algorithm is that the motion of kinks on boundaries under energy-P …Si ;Sj ;DE;T † neutral changes in orientation is preserved which 8
  18. 18. …DE ‡ J †
  19. 19. M…Si ; Sj † ;
  20. 20. DE 6 0; allows curvature driven motion to be modeled: the
  21. 21. …DE ‡ J †
  22. 22. Mmax proportionality of the migration rate of a bound- maxˆ
  23. 23. ary to the stored energy di€erence across the
  24. 24. …DE ‡ J †
  25. 25. M…Si ; Sj †
  26. 26. :
  27. 27. exp…ÀDE=kT †; DE 0: boundary is also correctly modeled. Note that in …DE ‡ J †max
  28. 28. Mmax the conventional MC model applied to recrystal- …6b† lization, [22], any energy change less than zero results in the same uniform switching probabilityThis approach, however, su€ers from the defect (barring local variations in mobility). Therefore,that the switching probability is not proportional variations in stored energy do not lead to anyto the change in stored energy as the latter ap- appreciable variation in boundary migration rate,proaches zero because the prefactor is dominated although they do a€ect whether or not nucleationby the grain boundary energy term (J). In an ef- can occur homogeneously or heterogeneouslyfort to overcome the diculties of combining the [22,30].two types of driving force in a single switchingprobability model, a di€erent approach has beentaken. 5. Gibbs±Thomson A useful test of the hybrid model is to investi-4. Hybrid algorithm gate the behavior of an isolated (island) grain under the combined e€ects of curvature and stored The new algorithm described here adopts both energy driving forces. When only curvature istypes of evolution and combines them in a single present, the grain will shrink by migration of allhybrid model. The essence of the hybrid model is to boundary segments towards their center(s) ofinterleave the two di€erent types of reorientation curvature, i.e., the center of the grain. The curva-in time. Each sub-model obeys the appropriate ture is inversely proportional to the radius and sorules so that the correct limiting behavior is ob- the boundary velocity accelerates with time, lead-tained in the limit that the frequency of either type ing to a characteristic constant rate of decrease ofapproaches zero. Such a combination of switching area, A, with time [31]. If the e€ect of a storedrules has been used to model, e.g., ferromagnetic energy di€erence, DH , (higher stored energy out-¯uids [29] where it is useful to use spins can move side the grain) is included, then dA=dt is as follows,according to a lattice gas model but spin interac- where M is a mobility as before, c is the graintions are governed by an Ising model, see p.293 et boundary energy and R is the radius of the 2Dseq. [28]. A ratio, C, between the frequency of CA island grain.
  29. 29. 74 A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78dA ˆ À2pMc ‡ 2pRMDH : …7†dtThus, if the grain boundary energy is small and thestored energy di€erence is large, the grain willgrow. For a linear velocity-force relation, one ex-pects to observe a constant velocity, v ˆ MDp, andtherefore, an accelerating (parabolic) rate ofchange of area. Furthermore, there should be abalance between the two forces at a critical sizegiven by DH ˆ c=Rcrit: (DH ˆ c=2R in the stan-dard, 3D case). Fig. 3. Plot of area of single grain vs time for various ratios of In this hybrid kinetic MC model, the critical reorientation frequencies, showing (linear) shrinkage for cur-size can be estimated as follows for the smallest vature dominated migration and (parabolic) expansion forpossible grain of size equal to a single site. Such a stored energy migration. Each set of points has been ®tted to a second order polynomial for the purposes of extracting the rategrain is surrounded by neighbors of a di€erent of change of area.orientation such that a fraction 1/Q MC reorien-tation attempts will result in absorption of theunit-size grain into one of its neighbors. If the shrinkage. The results show the expected parabolicgrain is strain-free but all its neighbors have stored accelerating growth when stored energy dominatesenergy i.e., are unrecrystallized then any CA re- and linear shrinkage when curvature driving forceorientation attempt will succeed in adding another to the unit-size grain. Subsequent CA reori- The critical size above which a new grain isentations will continue to add more sites to the stable …dA=dt 0† can be estimated as follows. Agrain so that it continues to grow. This suggests single grain shrinking under curvature driven mi-that, in the limit where the critical grain size for gration does so at a constant rate of change (loss)recrystallization is that of unit-size grains, the re- of area [1]. The magnitude of the rate is dependentorientation frequency ratio is approximately on the value of Q because of the de®nition of theC ˆ 1=Q ˆ 0:01. At larger sizes, however, the ratio time step in the classical MC scheme. For Q ˆ 100should be larger. The analogous equation to Eq. (7) used here and a square lattice, the rate is approx-is therefore, the following, where a is e€ectively the imately À0:1 a2 MCSÀ1 , where a is the size of eachproduct of mobility and grain boundary energy (in cell and 1 MCS represents one reorientation stepthe MC model), l is the mobility of a boundary in for each cell. Choosing the grain boundary energythe CA model and C is the reorientation ratio. The to be one sets the mobility, a ˆ 0:016. The ex-mobility, l, is not a free parameter but is deter- pected boundary migration rate under storedmined by the characteristics of the particular form energy driving force, however, should be approx-of MC model and CA model chosen. imately one cell per time step, i.e., l ˆ 1. There-dA fore, for an isolated grain of size one, the ˆ À2pa ‡ 2pRlC: …8† reorientation ratio should be of order C ˆ 0:04 fordt the curvature and stored energy driving forces to Fig. 3 shows the area vs time for a single grain be in equilibrium. Testing the hybrid model at thisof initial radius 20a for various choices of the ra- resolution is impractical of course because statis-tio, C, between MC and CA reorientations. The tical ¯uctuations will obscure the result of interest.simulations were performed on a 50  50 square Therefore, it is more useful to examine larger ini-lattice with ®rst and second nearest neighbors. tial sizes.Note that for ratios close to the critical value any Fig. 3 shows the variation in area with time of¯uctuation of the size will render the grain either an island grain of size R0 ˆ 20 for various storedsuper- or sub-critical leading to either growth or energies as expressed by the reorientation attempt
  30. 30. A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78 75ratio. Small values of C allow the grain to shrink ata constant rate whereas large values lead to para-bolic growth. At intermediate values, the grain isquasi-stationary although any ¯uctuation awayfrom the zero growth size will lead to either con-tinuing shrinkage or growth. In other words, thezero growth size is a metastable size. For thisparticular choice of initial grain size, zero growthoccurs for C $ 0:002. By setting dA=dt ˆ 0 inEq. (8), this leads to an estimate of the mobility ofl $ 0:4, which is not too far from unity, as ex-pected. By di€erentiating the time dependence ofthe area one can obtain initial rates of change of Fig. 5. Initial rate of change of area of an island grain for threearea. Fig. 4 plots the slope of the area vs time at di€erent initial sizes, R0 . Each point represents the average for azero time for three di€erent initial grain sizes and set of simulations under the same conditions (as shown in Fig. 4). The reorientation ratio has been scaled, CH ˆ C Á R0 =20,reorientation ratios between 0 and 0.01 with sev- in accordance with Eq. (5).eral trials for each set of parameter values. Theresults show some scatter in the growth rate butthe growth rate appears to be linearly dependent shown in Fig. 4; each point represents an averageon the reorientation ratio. of the set of trials for a given initial size and re- The relation for the rate of area change above orientation ratio. All the points lie close to a singlepredicts a linear dependence on initial radius with straight line as expected although the trend is sub-a constant intercept corresponding to a ®xed linear for large reorientation ratios.shrinkage rate under the in¯uence of curvature- Again, it is important to note that the conceptdriven migration only. Eq. (8) suggests that the of a critical grain size is not new in this context.contributions to the growth or shrinkage rate Although it is not applicable to the standard formshould all scale by the initial size (at t ˆ 0). Fig. 5 of the CA model as typically used to simulate re-therefore, presents the results of scaling the reori- crystallization, a critical size can be readily de®nedentation ratio, Cscaled ˆ CR0 =20, for the results in the standard form of the MC model as applied to recrystallization model. Holm [32] has investi- gated the range of critical sizes for various lattices as follows. Note that the focus is on minimum size required for an embryonic grain to survive and grow into a new grain. Also, certain sizes and shapes of embryonic grains have special properties owing to anisotropy of the lattice. A similar ex- ploration of the behavior standard MC model for recrystallization for critical sizes could be per- formed. For small values of H =J and with ®nite lattice temperature, similar results may well be obtained (Table 1).Fig. 4. Plot of initial rate of change of area of an island grain vsthe reorientation frequency ratio for three di€erent initials sizes.The larger the initial size of the grain, R0 , the smaller the stored 6. Recrystallization kineticsenergy required to allow the grain to grow. The predictedvariation is approximately linear with stored energy and theratio (stored energy) corresponding to zero growth is linear in In order to verify that the new hybrid modelthe initial radius. simulates primary recrystallization correctly, the
  31. 31. 76 A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78Table 1Critical embryo sizes in the standard MC model for recrystal-lization (Eq. (3)) [32] Stored energy: boundary energy Critical size (lattice sites) 2D, triangular lattice H =J 2 (very large) 2 H=J 4 2 4 H =J 6 1 H =J 6 (any embryo grows) 2D, square lattice, with second nearest neighbors H =J 1 (very large) 1 H =J 2 3 Fig. 7. JMAK plot of fraction recrystallized vs time for site 2 H=J 8 1 saturated nucleation with 20 embryos and a range of reorien- H =J 8 (any embryo grows) tation ratios. Recrystallization proceeds less rapidly for smaller 3D, simple cubic lattice, with second reorientation ratios, i.e., lower stored energy. The slope of the and third nearest neighbors curves approaches two which is the classical value for spatially H =J 3 (very large) random, site saturated nucleation and two dimensional growth. 3 H =J 5 5 5 H=J 8 3 8 H =J 26 1 saturated) embryos. The results indicate that the H =J 26 (any embryo grows) expected Kolmogorov±Johnson±Mehl±Avrami kinetics for 2D growth under site-saturated nu- cleation conditions are followed, Eq. (9). That is tobehavior of a 2D system was modeled on a say, after an initial transient, a slope of two ( ˆ n)100  100 square lattice with ®rst and second is observed for plots of log…ln…1 À F †† vs log(time).nearest neighbors as before. The reorientation ra- The time for the preliminary grain growth stagetio, C, was varied and also the number of new has been subtracted from the total time. Since eachgrains. The model was operated in normal grain embryonic grain is inserted into the microstructuregrowth until a time of 1000 MCS had been accu- with nine sites, i.e., a central site and its nearestmulated and then, the recrystallization simulation neighbors, not all new grains survived. The dis-was started by inserting new grains into the sys- appearance of some new grains gives rise to atem. The resulting microstructures, Fig. 6, show transient at short times, particularly for thethe anticipated growth of new grains into a poly- smallest value of C. This aspect of the resultscrystalline matrix that is also undergoing grain suggests that the prior grain structure played a rolegrowth. The area fraction recrystallized was plot- by supporting heterogeneous nucleation [30].ted in the usual double logarithmic plot, Fig. 7, fortwo values of C and three di€erent densities of (site 1 À F ˆ exp Àfktn g: …9† Fig. 6. Microstructural evolution during recrystallization for site saturated nucleation with 20 embryos and C ˆ 4  10À3 .
  32. 32. A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78 777. Scaling relationships Again, we can equate velocities to obtain a scaling relationship. It is useful to examine scaling relationships be-tween quantities such as time and length in the MDH 1 ‡ C model vphysical ˆ v : …15†simulations and their corresponding physical l Cquantities. This is most easily accomplished byconsidering parallel expressions for the same Time scaling is then obtained as the length:velocitycharacteristic quantity. For length, the logical ratio:choice is the critical radius at which a nucleus isstable, i.e., dA=dt ˆ 0 in Eq. (7). The physical length c C2 model tphysical ˆ ˆ t : …16†critical size is given by: 2 velocity M…DH † 1 ‡ C cRphysical ˆ crit: : …10† In order to obtain a scaling relationship for time in DH the simulated recrystallization process for variousThe corresponding expression for the critical ra- values of the reorientation ratio, however, it isdius in the simulations in terms of the areal helpful to consider the KJMA relationship. Thus,shrinkage rate under curvature driven motion, consider the time required for 50% recrystalliza-À2pa, the mobility in the CA model, l, and the tion in the case of site saturated, 2D growth. Re-reorientation ratio, C, is given by: arrangement of the standard form, Eq. (9) above, a where I is the nucleation density and G is theRmodel ˆ crit: : …11† growth rate, lCEquating the critical radii yields the following ex- F ˆ 1 À expfÀIG2 t2 gpression for scaling lengths. gives the following expression for t50% c lC modelxphysical ˆ x : …12† DH a t50% ˆ ln 0:5=IG2 : 2 …17†Since the quantities a and l are intrinsic propertiesof the Monte Carlo and CA models respectively,the independent variable in the model is the re-orientation ratio, C: For a given material withgrain boundary energy, c, and stored energy, DH ,the larger the value chosen for C, the larger thee€ective magni®cation in the model (because ofsmaller Rcrit: in the model), or, alternatively, thelower the spatial resolution of the model. For velocity scaling, one can consider the mi-gration rate of a boundary under a di€erence instored energy. In the physical case, the migration isgiven by, where M is the boundary mobility.vphysical ˆ MDH : …13† Fig. 8. JMAK plot of the fraction recrystallized vs normalized time according to Eq. (15). All the data follow a master curveIn the model, the velocity is scaled by the fraction describing recrystallization kinetics except for the smallestof steps that are governed by the CA rule: value of C: in this case, the low (e€ective) stored energy means that a signi®cant fraction of the embryos do not survive Cvmodel ˆ l : …14† to become new grains thereby retarding recrystallization, 1‡C Eq. (13).
  33. 33. 78 A.D. Rollett, D. Raabe / Computational Materials Science 21 (2001) 69±78This shows that recrystallization times scale as the Referencesboundary velocities, Eq. (15), and not according tothe time scaling given in Eq. (16), i.e., [1] M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Acta metall. 32 (1984) 783. [2] G. Abbruzzese, K. Lucke, Acta Met. 34 (1986) 905. CH 1 ‡ Ct50% …CH † ˆ t50% …C†: …18† [3] H.J. Bunge, U. Khler, Scripta metall. et mater. 27 (1992) o 1 ‡ CH C 1539. [4] A.D. Rollett, W.W. Mullins, Scripta metall. et mater. 36Fig. 8 shows the same set of data as in Fig. 7 but (1996) 975.with the times scaled according to Eq. (18). All the [5] A.D. Rollett, E.A. Holm, in: T.R. McNelley (Ed.), Proceedings of Recrystallization 96, ReX 96, 1997, p. sets are scaled to the same curve except for [6] F.J. Humphreys, in: T.R. McNelley (Ed.), Proc. Recrys-the smallest value of C, for which not all the em- tallization-96, ReX96, 1997, p. 1.bryos survive to become new grains, thereby re- [7] B. Radhakrishnan, G.B. Sarma, T. Zacharia, Scripta mat.ducing the e€ective value of the nucleation density, 39 (1998) 1639.I, in Eq. (17). [8] F.J. Humphreys, Acta mater. 45 (1997) 4321. [9] K. Mahin, K. Hanson, J. Morris, Acta Met. 28 (1980) 443. [10] H. Frost, C. Thompson, Acta Met. 35 (1987) 529. [11] T. Furu, K. Marthinsen, E. Nes, Mater. Sci. Technol. 6 (1990) 1093.8. Summary [12] D. Juul Jensen, Scripta metall. et mater. 27 (1992) 1551. [13] F.J. Humphreys, Scripta metall. et mater. 27 (1992) 1557. A new approach to mesoscopic simulation of [14] C. Maurice, F.J. Humphreys, in: H. Weiland, B.L. Adams,recrystallization has been described that models A.D. Rollett (Eds.), Proceedings of the Grain Growth in Polycrystalline Materials III, TMS, 1998, p. 81.the Gibbs±Thomson e€ect correctly. The simula- [15] V.E. Fradkov, L.S. Shvindlerman, D.G. Udler, Scriptation incorporates two types of reorientation pro- metall. 19 (1985) 1285.cess, one based on curvature-driven migration and [16] H.J. Frost, C.V. Thompson, D.T. Walton, Acta based on stored-energy driving forces. By mater. 40 (1992) 779.allowing for two types of reorientation, it is pos- [17] F.J. Humphreys, in: H. Weiland, B.L. Adams, A.D. Rollett (Eds.), Proceedings of the Grain Growth in Polycrystallinesible to obtain the correct limiting behavior for Materials III, TMS, 1998, p. 13.both types of driving force for migration. The [18] M. Rappaz, C.-A. Gandin, Acta metall. mater. 41 (1993)necessity for this arises from the non-local nature 345.of the e€ect of curvature in the MC model. The [19] H.W. Hesselbarth, I.R. Gbel, Acta metall. 39 (1991) 2135. okinetics of primary recrystallization for site-satu- [20] D. Raabe, Mater. Sci. Forum 273±275 (1998) 169. [21] D. Raabe, Phil. Mag. A 79 (1999) 2339.rated conditions are correctly modeled. The hy- [22] D.J. Srolovitz, G.S. Grest, M.P. Anderson, Acta metall. 34brid approach can be generalized to three (1986) 1833.dimensions and should also be applicable to the [23] A.D. Rollett, D.J. Srolovitz, M.P. Anderson, R.D. Doh-kinetics and microstructural evolution of phase erty, Acta metall. 40 (1992) 3475.transformations. [24] P. Tavernier, J.A. Szpunar, Acta metall. mater. 39 (1991) 557. [25] L.-Q. Chen, Scripta metall. et mater. 32 (1995) 115. [26] C. Herring, Phys. Rev. 82 (1951) 87. [27] P. Sahni, D. Srolovitz, G. Grest, M. Anderson, S. Safran,Acknowledgements Phys. Rev. B 28 (1983) 2705. [28] B. Chopard, M. Droz, Cellular Automata Modelling of Discussions with Professors R. Swendsen, Physical Systems, Cambridge University Press, Cambridge,J. Rickman and Drs. E. A. Holm and R. A. LeSar 1998. [29] V. Sofonea, Europhys. Lett. 25 (1994) 385.are gratefully acknowledged. This work was sup- [30] D.J. Srolovitz, G.S. Grest, M.P. Anderson, A.D. Rollett,ported primarily by the MRSEC Program of the Acta metall. 36 (1988) 2115.National Science Foundation under Award [31] W.W. Mullins, J. Appl. Phys. 27 (1956) 900.Number DMR-9632556. [32] E.A. Holm, personal communication, 1996.