Thermodynamic basis for a variational model for crystal growth

Transcription

1 PHYSICL REIEW E OLUME 60, NUMBER 1 JULY 1999 Thermoynamic basis for a variational moel for crystal growth Bayar K. Johnson* an Robert F. Sekerka Carnegie Mellon University, Pittsburgh, Pennsylvania 1513 Robert lmgren University of Chicago, Chicago, Illinois Receive 19 October 1998 ariational moels provie an alternative approach to stanar sharp interface moels for calculating the motion of phase bounaries uring soliification. We present a corresponence between objective functions use in variational simulations an specific thermoynamic functions. We emonstrate that variational moels with the propose ientification of variables are consistent with nonequilibrium thermoynamics. ariational moels are erive for soliification of a pure material an then generalize to obtain a moel for soliification of a binary alloy. Conservation laws for internal energy an chemical species an the law of local entropy prouction are expresse in integral form an use to evelop variational principles in which a free energy, which inclues an interfacial contribution, is shown to be a ecreasing function of time. This free energy takes on its minimum value over any short time interval, subject to the laws of conservation of internal energy an chemical species. variational simulation base on this moel is escribe, an shown for small time intervals to provie the Gibbs-Thomson bounary conition at the soli-liqui interface. S X PCS numbers: j, Ln, 0.70.c, Dv I. INTRODUCTION In computational moeling of crystal growth, stanar sharp interface moels are very cumbersome to solve numerically. This arises because the crystal-melt interface is a free bounary that must be tracke by some numerical algorithm with an accuracy necessary to compute its curvature, on which its local equilibrium conitions temperature an composition epen. lternative moels are being evelope that are more efficient for numerical computations. The phase-fiel moel has receive consierable attention because of its basis in the funamental laws of irreversible thermoynamics, an because phase-fiel simulations can be use to compute crystal shapes quite complex in comparison with those obtaine from numerical solutions to stanar sharp interface moels 1 6. ariational moels are another alternative. In variational moels, one minimizes a free energy, subject to conservation conitions, to etermine the position of the soliliqui interface, an in a separate step solves the iffusion equation for temperature or chemical potential in the bulk. By iterating these two steps, one can compute the time evolution of a soli-liqui interface. ariational simulations for the soliification of a pure material were propose an implemente by one of the present authors 7 Roosen an Taylor 8 an Taylor 9 11 use similar principles to evelop simulations of the evolution of interfaces with crystalline facete surface energies. These simulations were base on a mathematical connection to stanar sharp interface moels. In this paper, a thermoynamic basis for variational moels is presente an generalize to inclue alloy soliification. *Present aress: MEMC Electronic Materials Inc., Saint Peters, MO ariational simulations are well suite to massively parallel computation an o not require such careful tracking of the interface as one woul nee for sharp interface moels to compute the interface curvature by some combination of local curve fitting an interpolation. n avantage of variational simulations from a theoretical stanpoint is that the Gibbs-Thomson interfacial bounary conition results irectly from the minimization of a free energy, which is its unerlying basis. We erive variational moels for the soliification of a pure material an the soliification of a binary alloy. Two versions of the variational moel are erive in each case: a quaratic an a linear moel.. Some notation Consier a volume occupie by two phases e.g., a soli crystal an its liqui melt separate by an interface. We treat a subvolume containe within a surface out that may inclue a portion of the interface. The intersections of the soli, liqui, an interface with are calle S, L, an SL, respectively. imensionless potential fiel u represents a temperature fiel in the case of a pure substance an a chemical potential fiel in the case of a binary alloy. In the variational moels an in the stanar sharp interface moels, u is a solution of the iffusion equation, u u, in the bulk soli an liqui. The ot above a variable enotes partial ifferentiation with respect to imensionless time. Later we consier a coorinate system that is moving in the z irection with nonimensional spee 1 with respect to the crystal, in which case the bulk fiels obey a iffusion equation of the form X/99/601/70510/$15.00 PRE The merican Physical Society

2 706 JOHNSON, SEKERK, ND LMGREN PRE 60 u u u z. F f f xs, 5 For both the variational moels an stanar sharp interface moels, the bulk fiels are etermine by a solution of the iffusion equation, but the moels iffer in the way that the bounary conitions at the crystal-melt interface are treate. B. Stanar sharp interface moels In stanar sharp interface moels, u is continuous at the soli-liqui interface an takes on the value uũf xs f xs K, where ũ is a reference potential that can frequently be taken to be zero, f xs is proportional to an excess interfacial free energy, an K is the imensionless curvature of the interface. Since f xs is assume to epen on the surface orientation, the notation ( f xs f xs )K, which woul be correct in two imensions, is a symbolic representation for ( f xs xs f 1 )K 1 1 (f xs xs f )K, which applies in three imensions. The subscripts represent ifferentiation of f with respect to, an the subscripts 1 an ientify angles between the normal an the two principal axes of curvature for the interface. Equation 3 is often referre to as the Gibbs-Thomson equation. There is also a conservation conition at the crystal-melt interface of the form n u S u L n, where n is the imensionless local normal growth spee, an n is the normal to the interface pointing into the liqui. Equation 4 applies when the thermal conuctivities in soli an liqui are equal, which is an approximation we iscuss later. Equation 1 or, subject to the bounary conitions 3 an 4 on SL with Neumann, Dirichlet, or other suitable conitions on the external bounary, constitute a mathematically well-pose moel for the evolution of crystal shapes. C. ariational moels In variational moels, the process of etermining the evolution the interface position an potential fiel over a short time interval from t to tt, is ivie into two computational steps: i bulk iffusion an ii interface motion. The potential fiel e.g., temperature for soliification of a pure material can change uring both steps. In the first step, the bulk fiels are compute by allowing heat to iffuse for time t without regar to the presence of a soli-liqui interface, an the interface position oes not change. In the secon step, the interface can change position. The bulk fiel changes in the secon step only to account for the change in potential ue to motion of the interface e.g., to account for latent heat in the case of a pure material. The flux of potential associate with the motion of the interface is calle the ifference flux, because it results from a ifference between soli an liqui equilibrium values of a conserve quantity, such as internal energy. In the secon step, the new interface position is foun by the minimization of a total energy, F of the form 3 4 which is minimize over all possible new interface configurations, subject to a conservation constraint. The quantity f is a imensionless bulk energy ensity that is a specifie function of the potential fiel u, an f xs is a imensionless freeenergy ensity of the crystal-melt interface, an can epen on its orientation. The ifference flux plays an important role in the minimization of F. t the beginning of an iteration step, the bulk fiel an the interface position are in a configuration we will call. fter the bulk iffusion step step 1 has been complete, the system is in configuration B, which has interface position an configuration corresponing to but a new potential fiel ue to iffusion for a time t. Configuration B is, therefore, an intermeiate configuration not necessarily representing the physical system configuration at any time. To obtain the actual configuration C one must minimize F with respect to all interface positions an configurations. During the minimizing process, one might try configuration B that iffers from configuration B in the location of the interface, an in the potential fiel. The potential fiel for B is the sum of the potential fiel associate with configuration B an the new potential associate with the ifference flux. The new potential is etermine by taking each volume element that woul change phase if the system were to evolve from B to B an istributing the release potential or withrawing the absorbe potential in the case of melting in a small neighborhoo of approximate imensionless raius t of all volume elements that have change phase. Thus any volume that changes phase can only affect the bulk potential in a local neighborhoo whose size is etermine by t. By istributing potential in a small but finite neighborhoo of material that has change phase, one accounts for the iffusion that woul take place between the time of the phase transformation, assume to be intermeiate to t an t t, an the en of the time step at tt. fter etermining configuration B interface position an new bulk fiel, one can compute F for B, an compare it with that for B. By trying a large number of possible interface configurations, one can fin that configuration that minimizes F, an take that as the upate system configuration C at time t t. This configuration will be configuration for the next iteration. Several variational moels, for which f epens on u in various ways, are escribe in this paper. The erivations presente below treat only the variational step step of the algorihm iscusse above. For each moel, the law of positive local entropy prouction an conservation laws for internal energy an/or chemical species are use to show that F/t. The conservation conition that we will apply to step is use to etermine the ifference flux an can be expresse in the form n u, 7 6

3 PRE 60 THERMODYNMIC BSIS FOR RITIONL MODEL where n is the normal growth spee of the interface, an results from a statement of conservation of the quantity for which u is the conjugate potential. For example, if u is a imensionless temperature, the conservation conition is a statement of the conservation of energy accounting for the latent heat of soliification. variational moel in the limit t 0 an a stanar sharp interface moel will both impose the Gibbs-Thomson bounary conition for u at the crystal-melt interface. D. pproximations It is assume that the materials being moele are rigi an of constant ensity, so that transport is purely iffusive i.e., no convection. We aopt a symmetric moel, for which the appropriate transport coefficients in soli an liqui are equal, which greatly simplifies the calculations an is essential for the erivation of the variational moel. For the soliification of a pure material, the thermal iffusion coefficients in soli an liqui can be of comparable magnitue, but are usually not equal. For metals, they iffer typically by a factor of, an for ice an water by approximately 3 1. For chemical iffusion in a binary mixture, the iffusion coefficient in the soli is typically three or four orers of magnitue smaller than that in the liqui, so the assumption of equal iffusion coefficients is incorrect for this case. more appropriate approximation woul be to assume that the iffusion coefficient in the soli is zero. There exist systems for which the iffusion coefficients in soli an liqui are comparable, an to which the mathematical formulation presente in Sec. III may be applie. In the experimental work of Melo an Oswal 13, in which they stuy the irectional growth of a liqui crystal phase into another one, the iffusion coefficients in the two phases iffer by roughly a factor of. nother consieration is the relative importance of the transport coefficient in the soli to the overall behavior of some systems. For a pure material growing freely into a supercoole liqui, one can argue that the soli is nearly isothermal because the temperature graients that occur in the soli are ue solely to the effects of capillarity on the interfacial temperature. lmost inepenent of what value one chooses for the thermal conuctivity in the soli, most of the heat transport that leas to growth comes from the much larger temperature graients in the liqui. The choice of thermal iffusion coefficient for the soli in such cases woul not ramatically affect the behavior of the system. By the same argument, choosing a finite iffusion coefficient in soli shoul not give rise to significant material fluxes, provie that the graients in soli are small compare to those in the liqui. lthough the approximation of equal transport coefficients in soli an liqui limit the applicability of the moel, we claim that one can obtain some meaningful insight into the process of crystal growth, an procee with these issues in min. II. SOLIDIFICTION OF PURE MTERIL We first treat a variational moel of soliification of a pure material from its melt in the absence of convection. Derivations that use the same basic laws can be use to obtain stanar sharp interface moels Energy conservation an entropy prouction The law of conservation of energy can be expresse in the form e t S S e L L e xs j e n out, out 8 where e is the internal energy ensity, j e is the internal energy flux, an n out is the outwar pointing normal to out. The superscripts S, L, an xs hereafter enote, respectively, the soli, liqui, an surface excess of the quantity. Equation 8 can be rewritten by applying the time erivative on the left-han sie to obtain ė S S ė L L e S e L n e xs e xs K n ė xs j e n out, 9 out where K is the interfacial curvature, an the ot ( ) above a variable represents partial ifferentiation with respect to time. The excess internal energy ensity may be a function of the orientation of the interface, an may epen explicitly on time. In bulk soli or bulk liqui no SL ), Eq. 9 can be reuce to ifferential form by using the ivergence theorem on the right-han sie an shrinking to a point, resulting in ė j e. 10 If, instea, one shrinks to a portion of the interface, then shrinks that interfacial area to a point, the conservation conition becomes e S e L n e xs e xs K n ė xs j S e j L e n, 11 where n is the unit normal to S,L pointing into the liqui, an j S e an j L e are the internal energy fluxes in the soli an liqui, respectively. The secon law of thermoynamics positive local entropy prouction for this system can be written s t S S s L L s xs j s n out out s I s Ixs, 1 where s is the entropy ensity, j s is the entropy flux, s I an s Ixs are the local rates of entropy prouction per unit volume in the bulk, an on the interface, respectively. These quantities must always satisfy s I an s Ixs, where the equal sign applies only to a hypothetical reversible process.

4 708 JOHNSON, SEKERK, ND LMGREN PRE 60 By applying the time erivative on the left-han sie of Eq. 1 one obtains ṡ S S ṡ L L s S s L n s xs s xs K n ṡ xs j s n out out s I s I xs. 13 Equation 1 can also be reuce in bulk to an equation which is similar to Eq. 10 with an extra entropy prouction term, an on the interface, ṡ j s s I, 14 s S s L n s xs s xs K n ṡ xs j S s j L s ns I xs. 15 B. ariational moels The variational moels to be erive here for the pure material were propose previously 7 an motivate intuitively base on the mathematical structure of the problem. If the heat capacities per unit volume c v in the soli an in the liqui are assume to be equal an inepenent of the temperature, then the internal energy ensity e S,L c v TT m e 0 S,L, 16 where the quantities e S,L 0 are constants for which e L S 0 e 0 L v, where L v is the latent heat of fusion per unit volume. Because ets, one can write s S,L c v lnt/t m s 0 S,L, where s 0 S,L are constants for which s 0 L s 0 S L v /T m. 1. Quaratic objective function 17 Consier a free-energy ensity efine by f met m s, which iffers from the Helmholtz free-energy ensity, f e Ts, in that the temperature that multiplies the entropy is constant, the melting temperature for a planar interface. The surface excess of this energy is simply f m xs e xs T m s xs. By expaning f m(t) about T m, one fins to secon orer, f m f mt f mt m TT m 1 f m T TT T TT m m m. 18 The first term in the expansion is the Helmholtz free energy at the melting temperature. The secon term is zero, because ets, an because the erivative is evaluate at T m. The thir term can be evaluate an is foun to be c v T m TT m. 19 Since f m(t m ) an c v are inepenent of phase, f m is inepenent of phase at any given value of T. Next, consier the thermoynamic laws. Multiplying Eq. 1 by T m, subtracting that prouct from Eq. 8, an using j e Tj s, one obtains t f m S,L out f m xs 1 T m T j e n T m s I s I xs, 0 where S L. For a perturbation of a small portion of the soli-liqui interface, the surface out may be place far enough away that the ifference flux from the motion of SL will not reach out in a short time interval t, leaing to j e n on out. By applying Eq. 0 over a small time interval t an to a small volume that, however, is sufficiently large that j e n on out substituting the nonimensionalize temperature, u(tt m )/(L v /c v ), using the expane f m from Eq. 18 an keeping terms to secon orer in u, Eq. 0 becomes xs 1 m f t u E T m E s I s I xs, 1 where the energy ensity parameter is EL v /(c v T m ), an where length variables have been rescale by a characteristic length l an time by l /, where is the thermal iffusion coefficient. Substituting Eq. 16 into Eq. 9, assuming that j e on out an neglecting the terms containing e xs xs e an ė xs because (e xs e xs )KL v an ė xs / n L v, one obtains u n. By efining the objective function f u / an f xs f m xs /E, Eqs. 1 an can be rewritten in the form of Eqs. 6 an 7, respectively. In aition, the bulk u fiels must satisfy the iffusion equation 1. Therefore, a moel base on Eqs. 6, 7, an 1 is consistent with the first an secon laws of thermoynamics.. Linear objective function In this alternative variational formulation, an objective function f that has a ifferent epenence on u is efine. Using Eqs. 16 an 17, one can write f S f LT/T m 1L v ue, 3

5 PRE 60 THERMODYNMIC BSIS FOR RITIONL MODEL where f ets is the true Helmholtz free energy, an u an E are given in the previous section. Multiplying Eq. 15 by T an subtracting the prouct from Eq. 11, then integrating the result over a portion of the interface, one obtains f xs xs f n K 1 u E n Ts I E xs, 4 where f xs (e xs Ts xs ) is the surface excess Helmholtz free energy, which is assume not to epen explicitly on time. Consier an objective function One can write f u/ u/ in soli in liqui. f t f t u n. 5 Substituting this into Eq. 4, an recalling that s I xs, one fins f t f xs E 1 L u S u, 6 where the time erivative of f xs was move outsie of the integral in the secon term of Eq. 4 to obtain Eq Because the latent heat is istribute isotropically a irect result of the assumption of equal transport coefficients in soli an liqui, the thir an fourth terms on the left-han sie of Eq. 6 reuce approximately to a constant times the ifference in volumes occupie by soli an liqui in the region. ssuming is a sphere of small raius r h, the fractional ifference in volume between soli an liqui is proportional to r h K. By choosing r h R, where R is the raius of curvature of the interface, one can neglect the sum of the thir an fourth terms in Eq Setting the sum of these terms equal to zero is equivalent to assuming that the conjugate variable energy associate with the potential u release because of the phase transformation is approximately equally istribute between the soli an liqui. Thus we obtain the approximate inequality, t f xs f E 0. 7 Using the same nonimensionalization of the position an time variables that was use for the quaratic case, Eq. 6 takes the form of Eq. 6 where f is given by Eq. 5, an f xs f xs /E. The minimization in the linear case is also subject to the constraint of energy conservation, Eq. 7. C. Global behavior of f an f m We have ientifie thermoynamic quantities F that ecrease monotonically subject to the constraint of energy conservation. Whether such a quantity ecreases globally for the entire variational algorithm for a thermally insulate system epens on how F behaves when the potential fiel iffuses. Existence of a quantity that ecreases globally in time has certain theoretical avantages 17, though it is not necessary for practical computations. To unerstan this, we return to the variational algorithm presente in the Introuction. t the en of step 1, the intermeiate configuration is B an has the same interface geometry as, but iffers by iffusion of the potential. In the minimization step, we consier a variety of configurations B in an attempt to minimize F. Let C be the final minimizing configuration. Now, since B is one of the caniate minimizers in step, we certainly have F(C)F(B). We shall call F a global ecreaser if we can guarantee that F(C) F(); that is, if F necessarily ecreases from one time step to the next. Clearly, this will be the case if F is ecrease in the iffusion step of the algorithm, for then we have F(B)F() an F is a global ecreaser. Negative entropy, for example, is a global ecreaser. Note that the entropy of bulk soli or liqui is a concave ownwar function of the internal energy, an hence the temperature, for constant specific heat, which we have assume. Thermal iffusion rives an insulate system towar a system average temperature an the total integrate entropy to larger values. Negative entropy is also a global ecreaser when multiple phases are consiere. The isavantage of the entropy in the context of variational algorithms is that it is not continuous at the soli-liqui interface. The Helmholtz free energy is not a global ecreaser, an neither is the linear objective function erive from it. This is why we have introuce the objective function f m, which is a global ecreaser, an which is continuous at the soliliqui interface; the quaratic functional erive from f m inherits these characteristics. III. SOLIDIFICTION OF BINRY LLOY We evelop our variational moel of soliification of a binary alloy by aing the conservation of material species to the previous laws. This moel is also applicable to the isothermal precipitation of a crystal with ifferent composition from that of the solution from which it grows.. Material conservation The law of conservation of material is c t S i S c L i L c xs i j i n out, out 8 where c i is the local concentration number per unit volume of species i, an j i is the local flux of species i. The subscripts i allow for several species, each of which must obey Eq. 8. Fluxes are measure with respect to the center of moles reference frame. It is assume that the total concentration i c i is constant an uniform; therefore, the center of moles reference frame is the same as the frame in which the soli is at rest. Equation 8 reuces in bulk to

6 710 JOHNSON, SEKERK, ND LMGREN PRE 60 ċ i S,L j i an on the interface to c S i c L i n c xs i K n j S i j L i n Hereafter, the specific case of two species, labele 1 an, is treate. It is assume that the total concentration (c 1 c ) is constant an uniform in both phases; therefore, c 1 c an j 1 j in either phase. Ientification of the interface position with the equimolar surface for species 1 provies c xs 1. Because (c 1 c ) is constant, the equimolar surface for species 1 is also the equimolar surface for species, an, therefore, c xs. Because any variation in c brings about a relate change in c 1, it suffices to track c alone. Equation 9 can be written for either species 1 or in bulk soli or liqui in the form ċ j, 31 an Eq. 30 becomes c S c L n j S j L n, 3 where cc an jj. We aopt a moel similar to that propose by Langer 18 for irectional soliification for which the concentration vs chemical potential curves have the same slope in both phases. Consier a portion of the phase iagram for which c L c S c 0 is constant. The miscibility gap is usually not constant, but over a certain range of concentrations, an for some materials, this is a reasonable approximation. We also assume that the effective chemical potential, 1,is a linear function of c in both soli an liqui, where 1 an are the chemical potentials of species 1 an, respectively. One can then write c S,L b 0 c S,L 0, 33 where c 0 S an c 0 L are constants with ifference c 0 c 0 S c 0 L, b is a constant slope, an 0 is the effective chemical potential of the liqui at T 0 an c 0 L. compatible form for the s as a function of c an T is 1 S c S,T, 1 L c L,T, S c S,T 1 b cs c 0 S, L c L,T 1 b cl c 0 L, 34 where an are functions of c an T. long the coexistence curve, chemical potentials for each species are equal, 1 S 1 L, S L, 35 leaing to two equations for the equilibrium values, c S * an c L *, where the star * enotes the value of c S or c L on the coexistence curve. One can solve these equations to fin (c L *c S *)c 0. One also obtains the equation FIG. 1. Phase iagram. The concentrations at the coexistence point are linear functions of the temperature. The reference temperature T 0 locates the values of c 0 S an c 0 L. The slope m in this case is negative. c L *c 0,Tc L *,T, 36 which is the general equation for the liquius line. The solius line is shifte exactly c 0 from the liquius line. lthough this cannot be true in general, an is not true in the ilute limit as c 1 approaches 0, for example, it can be true for some range of values of c. We choose to work within that range. We assume that the liquius curve is a straight line: take the form c L *c 0 L TT 0 /m, 37 where T 0 is a constant temperature, m is a constant liquius slope, an c 0 L is the equilibrium concentration corresponing to the temperature T 0. Figure 1 is a sketch of the phase iagram for this moel. B. ariational moels The variational moel will incorporate an expane version of the Gibbs-Thomson equation that inclues the effects of the local composition on the melting point. Both quaratic an linear moels will be evelope. The reaer is referre to the literature for erivations of the corresponing stanar sharp interface moels 14,16,19. We use the Kramers free energy per unit volume ªeTs i c i, 38 where a summation over i is implie. In the case of a pure substance, is simply the negative of the pressure. The surface excess, per unit area, of the extensive counterpart to is the surface tension for a binary alloy. 1. Quaratic objective function Consier the energy 0 et 0 s 0i c i, which iffers from the Kramers free energy in that the temperature that multiplies the entropy an the chemical potentials that multiply the concentrations are constant. The surface excess of this energy is simply 0 xs e xs T 0 s xs 0i c i xs. The function 0 will be regare to be a function of T an the i.it is assume that the temperature iffers from T 0 by a small amount T, an that the chemical potentials i iffer from 0i by small amounts i. Expaning 0 about T 0, 01, an 0, one fins

7 PRE 60 THERMODYNMIC BSIS FOR RITIONL MODEL T, 1, 00 0 T T T T T T T T 0 1 1, 39 where the subscript 0 after the partial erivatives an the 00 after imply evaluation at T 0, 01, an 0. The first term in the expansion is the actual Kramers free energy evaluate at T 0 an 0i, which is continuous at the interface if T 0 an the 0i correspon to equilibrium at a planar interface. The next three terms are iniviually zero, because ets 1 c 1 c, an because the erivatives are evaluate at T 0, 01, an 0. The remaining terms can be compute from thermoynamic ientities, an after some simplification, Eq. 39 can be rewritten s T c 1 1 T 1 c c 1 T 1 c T c 1 1. T T 40 Combining the three terms in Eq. 40 that contain erivatives with respect to the i, applying the conition that c 1 c is constant, substituting cc c 1 an 1 into Eq. 40, an using the relations c/ 1 c/ an c/ c/, one obtains an 1 c c 1 c T c T 1 c T c 1 T c T 1 T, 41 4 where 0 an Next, substitute Eq. 41 into Eq. 40, then use Eq. 33 to evaluate c/b an c/t an note that s/t 0 c v /T 0, where c v is the heat capacity per unit volume at constant, tofin 0 T, 00 1 b c 0 T 0 T. 43 By efining u( 0 )/(c 0 /b) an ũ(tt 0 )c v / (T 0 E) 1/, where Ec 0 /b, Eq. 43 becomes 0 T, 00 1 Eu ũ. 44 Multiplying Eq. 1 by T 0, Eq. 8 by 0i, an subtracting them from Eq. 8, one obtains t 0 S,L 0 xs j e T 0 j s 0 j n out T 0 s I s I xs, 45 where S L. One can apply Eq. 45 over a small time interval t an to a small volume which, however, is sufficiently large that j e, j s, an jj on out. xs Substituting 0 from Eq. 44 an noting that s I an s I must be positive, one fins where an t f f xs, f 1 u ũ f xs 0 xs /E.. Linear objective function 46 By multiplying Eq. 3 by, multiplying Eq. 15 by T, an subtracting these from Eq. 11, one obtains S L xs xs K n Ts I xs, 47 where xs is assume not to epen explicitly on time. Integrating Eq. 47 over S,L, one fins S L n xs xs K n Ts I xs. 48 The Kramers free energy has as inepenent variables T, 1, an. Expaning this free energy about some T 0, 01, an 0 to first orer in the variables T, 1, an, one fins for each phase S or L, T, 1, 00 s 0 TT 0 c c 0 0, 49 where the subscript 0 on c an s an the subscript 00 on inicate evaluation at T 0, 01, an 0. Choosing the variables T 0, 01, an 0 in soli an liqui to correspon to values for a flat interface at the melting point leas to S 00 L 00 an ª S L s 0 TT 0 c 0 0, 50

8 71 JOHNSON, SEKERK, ND LMGREN PRE 60 FIG. 3. ariation of interface position. The unperturbe interface position is ientifie by the points x 0. This interface is perturbe to a position x 0 n. Then the variational erivative of F is compute to fin the minimum. FIG.. The frozen temperature approximation for irectional soliification. If the thermal iffusion coefficients in soli an liqui are large, an approximately equal, an if the latent heat of fusion is negligible, then the temperature fiel is constant in time an approximately linear in space in a reference frame that is moving at velocity Ṽ in the z irection. The interface is locate at z I somewhere between the thermal reservoirs, which maintain bounary conitions T h an T c. where 1 S 1 L an S L at the interface an where s 0 s 0 L s 0 S, c 0 c 0 L c 0 S. The efinitions cªc an ª 1 still hol, an so the efinition has been use. Consier the ientity n 1 t S LK t L 1 S t. 51 By the same reasoning presente in Sec. II to neglect a similar pair of integrals, an because it has been assume for this moel that the iffusion coefficients in soli an liqui are equal, the thir an fourth integrals on the right-han sie of Eq. 51 sum to zero in the limit of small. Equation 51 can be substitute into Eq. 48 to give t 1 1 S L xs Ts I xs. 5 The time erivative coul be extracte from the secon term in Eq. 48 to obtain Eq. 5 because we have assume that xs oes not epen explicitly on time. Substituting Eq. 50 into Eq. 5 an efining uª b c 0 0, ũtª s 0 E TT 0 1 mc 0 TT 0, one obtains approximately the inequality 6, where f uũ/ in soli uũ/ in liqui, 55 an f xs xs /E. In aition to the inequality 6, material is conserve. By substitution of c from Eq. 33 into Eq. 8, one fins that the conservation conition for material can be written in the form of Eq. 7 where it has been assume that j on out an c i xs as state previously. n equation for the conservation of energy will not be written for this moel because we will only treat the case of an isothermal system or use the frozen temperature approximation, which is escribe below. For a coorinate system that is translating with spee Ṽ in the ẑ irection, lengths are rescale by D/Ṽ an time by D/Ṽ to obtain Eq. for the bulk iffusion equation. C. The frozen temperature approximation The frozen temperature approximation can be use 18,0 to moel irectional soliification of a binary mixture. The temperature fiel is assume to have a constant graient G in the z irection, an to be translating at spee Ṽ in the z irection. In the moving reference frame, the temperature fiel is, therefore, constant in time. The experimental setup in the irectional growth geometry is sketche in Fig.. heat source in avance of the omain of interest an a heat sink behin the omain of interest translate uniformly with respect to the sample at spee Ṽ. The heat source an sink are arrange so that the soliification front is locate somewhere between them. The temperature fiel can thus be written T(z)G (z z 0)T 0, where z an z 0 are imensional lengths zd/ṽ an z 0 D/Ṽ. The value of ũ in Eq. 54 can thus be written where ũ s 0G D zz 0 Mzz 0, EṼ Ms 0 G D/EṼG D/mc 0 Ṽ In this approximation, the temperature fiel is legislate an the only fiel that must be compute is the compositional, or chemical potential, fiel. One coul relax the frozen temperature conition in the variational moel, an then one woul also nee to solve Eq. 1 for the temperature fiel. In this moel, the only effect that the fixe temperature

9 PRE 60 THERMODYNMIC BSIS FOR RITIONL MODEL fiel has on the computation is that it changes the interfacial bounary conition for the compositional fiel. The frozen temperature approximation is a reasonable approximation that greatly simplifies calculations of irectional soliification shapes, an inclues enough of the important physics to isplay complex cellular morphologies similar to those observe in experiments. In particular, the parameter M is recognize to be precisely the bifurcation parameter that enters morphological stability theory for the limiting case of constitutional supercooling 3. D. Corresponence between variational an stanar sharp interface moels We choose a reasonable scheme for istributing the release heat or solute ue to motion of the interface, an emonstrate that in the limit of small t, the bounary conitions maintaine by the variational moel are the same as those that are impose in the stanar sharp interface moel. The conservation conition, Eq. 7, that constrains the minimization in the variational moel, is not specific about how the release potential from the moving interface shoul be istribute into the volume. sensible way of istributing the latent heat in the case of a pure material was suggeste previously 7 an will be use here. minimization step is use to upate the interface position from time t to time tt, where t is a small time interval. The motion of the interface n t will bring about a change in the potential u in the neighborhoo of the moving interface accoring to Eq. 7. ssuming that the istribution of the release heat or solute is etermine by the iffusion equation, the maximum istance that the release potential can iffuse is of the orer of t. heat kernel can be use to compute the change in the potential fiel resulting from interface motion. The interface is a source of magnitue n t, so the change in chemical potential at x can be compute from ux S,L Gx 1x,t n x 1 tx 1, 58 where the function Gx 1 x,t4t exp n/ x 1x 4t 59 is a heat kernel, an n is the imensionality of the space. One can verify that this choice of u satisfies Eq. 7 by integrating Eq. 58 over the volume, an observing that G is normalize to one. u is negligible for x 1 x t. Therefore, the volume that is affecte by the motion of the interface is small, for small t. By choosing a configuration of the interface an allowing a local normal variation of that configuration near a point x 0 see Fig. 3 one can compute a variation in F, subject to conservation of energy in the form 58. One can etermine the actual interfacial configuration by setting the variation equal to zero, which will be true when F is a minimum. Then, the bounary conition in the limit can be etermine by taking the limit of this configuration as t approaches zero. The interface position will be ientifie by x 0 an the interface configuration by x 0 n, where is a function efine on x 0, n is the unit normal vector pointing into the liqui, an is a small parameter. The variation F using the linear form for f given in Eq. 5, can be compute as follows: F f S,L fxs. 60 Evaluating the volume part of this variation first, one fins f 1 S uũ L uũ S,L uũ 1 S S,L G L S,L G S,L uũ, 61 where is a ifferential element of area. The secon term in Eq. 60 can be compute to be f xs S,L S,L fxs f xs K, 6 where it has been assume that f xs () oes not change explicitly uring the variation. In the limit as t 0, the range of G becomes small compare to the raius of curvature of the interface, an the volume integrals in Eq. 61 cancel. Setting F to fin the minimum configuration, one fins uũf xs f xs K, 63 which is the Gibbs-Thomson conition, Eq. 3, as it is written for the stanar sharp interface moel. Therefore, for small t, the two moels shoul give similar results. ariational simulations have been teste by showing that the simulation prouces solutions that agree with some analytical solutions foun by solving stanar sharp interface moels. simulation 4 of the process of irectional soliification using the vibrational algorithm preicte the critical conitions for morphological stability that agree which analytical solutions for the critical conitions calculate from sharp interface moels 3. In another implementation of a variational simulation, parabolic enrites were compute for which the tip curvatures an Peclet numbers agree with the values preicte by Ivantsov 5. I. CONCLUSIONS In this paper, variational moels for soliification are erive by using the principles of nonequilibrium thermoynamics. Laws of conservation an entropy prouction lea to the equations that are summarize in Sec. I. ariational prin-

10 714 JOHNSON, SEKERK, ND LMGREN PRE 60 ciples are obtaine for two ifferent physical systems: the free growth of a pure material into a supercoole melt, an the irectional soliification of a binary mixture. In both cases, two ifferent expressions for the relevant free-energy ensity are foun: a quaratic form, an a linear form. The quaratic form oes not require the neglect of terms such as the secon term in Eq. 6, an is also associate with a global minimizer for the entire variational algorithm. ariational moels ha been use previously on the basis of their mathematical connection to stanar sharp interface moels. The erivations that are presente here provie a irect link between those moels an irreversible thermoynamics. These variational moels provie an alternative approach for moeling the process of soliification. Several moeling assumptions that can limit their applicability were necessary. The system is assume to be symmetric in that the transport coefficients are equal in soli an liqui. For thermal transport this is reasonable, but for the iffusion of material, this approximation is inappropriate in most cases. The frozen temperature approximation can be use to simplify the calculation for the binary mixture. computational avantage of the variational approach over stanar sharp interface moel is that the curvature oes not nee to be compute irectly. Using a simulation base on the variational moel 7,4, one can employ relatively coarse meshes to compute the evolution of crystal shapes. nother avantage is that the variational approach incorporates thermoynamic laws more irectly through the minimization of an energy, as oppose to assigning a bounary conition whose value was etermine separately by such a minimization. CKNOWLEDGMENTS The authors woul like to acknowlege funing from the National Science Founation uner Grant Nos. DMR an DMR R.. was supporte by NSF Career Grant No. DMS S. L. Wang et al., Physica D 69, Wheeler, W. J. Boettinger, an G. B. McFaen, Phys. Rev. 45, Wheeler, B. T. Murray, an R. J. Schaefer, Physica D 66, G. Caginalp an W. Xie, Phys. Rev. E 48, R. Kobayashi, Bull. Jpn. Soc. In. ppl. Math 1, H. M. Soner, rch. Ration. Mech. nal. 131, R. lmgren, J. Compu. Phys. 106, Roosen an J. Taylor, J. Comput. Phys. 114, J. Taylor, Bull. m. Math. Soc. 84, J. Taylor, cta Metall. Mater. 40, J. Taylor, SIM J. Control Optim. 31, W. H. Giet, Principals of Engineering Heat Transfer an Nostran Co., Princeton, F. Melo an P. Oswal, Phys. Rev. E 47, S. ngenent an M. E. Gurtin, rch. Ration. Mech. nal. 104, S. ngenent an M. E. Gurtin, rch. Ration. Mech. nal. 108, B. K. Johnson an R. F. Sekerka, Phys. Rev. E 5, F. lmgren, J. Taylor, an L. Wang, SIM J. Control Optim. 31, J. S. Langer, Rev. Mo. Phys. 5, R. F. Sekerka an W. W. Mullins, J. Chem. Phys. 73, J. S. Langer an L.. Turski, cta Metall. 5, K. Brattkus an S. H. Davis, Phys. Rev. B 38, J. inals, R. F. Sekerka, an P. P. Debroy, J. Cryst. Growth 89, R. F. Sekerka, Phase Interfaces: Morphological stability, in Encyclopeia of Materials Science an Engineering, eite by Michael B. Bever Pergamon Press Lt., Oxfor, B. K. Johnson, Ph.D. thesis, Carnegie Mellon University, G. P. Ivantsov, Dokl. ka. Nauk. SSSR 58,