Transactions on Modelling and Simulation vol 17, 1997 WIT Press, ISSN X

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1 Thermodynamieally consistent phase-field models of solidification processes C. Charach*, P.C. Fife* "CEEP, J. Blaustein Institute, Ben Gurion University of the Negev, Sede-Boqer Campus, Sede-Boqer, 84990, Israel bgu.ac, il b Department ofmathematics, University of Utah, Salt Lake City, fife@math. Utah, edu Abstract Diffuse interface models of nonisothermal solidification of a pure substance in an undercooled melt are reconsidered in the frame of Extended Irreversible Thermodynamics. The gradient terms are allowed not only in the free energy and entropy densities but also in the internal energy density. Requiring the local entropy production to be nonnegative we derive a oneparameter family of field equations for the nonconserved order parameter coupled to the heat conduction. The meaning of the above gradient terms is clarified by considering stable and metastable two-phase equilibria. For nonisothermal growth processes we present some asymptotic results concerning the corresponding sharp interface models. The latter models are explored to analyze the effect of the gradient terms on kinetics of nonisothermal growth. 1 Introduction Continuum models of solidification can be divided into two main groups: the sharp interface models and the models with diffused interfaces. In the former models the transport equations in each phase are stated using irreversible thermodynamics of homogeneous systems. [1]. The balance equations for mechanical and thermodynamical fluxes across the sharp interface then yield an appropriate free boundary problem. In this formalism the surface tension is introduced via momentum balance across a curved interface. It yields the capillarity undercooling (the Gibbs-Thompson effect). In addition to this equilibrium effect there is a kinetic undercooling due to the interracial entropy production. It is present even in a planar case owing to the discontinuity of the chemical potentials across a moving interface. For materials with

2 28 Moving Boundaries IV microscopically rough interfaces the kinetic undercooling is typically a linear function of the growth rate. Models with diffused interfaces stem from the phase-field theories [2-9]. In this approach both the bulk and the interfacial aspects of the problem follow from a single set of nonlinear PDE for the phase field and heat diffusion. Within this scheme the crystallization front is treated as a mobile internal layer in which the phase-field undergoes an abrupt but continuous change. In the present paper the phase field models with a nonconserved order parameter, describing solidification of a pure substance, are reexamined in terms of Extended Irreversible Thermodynamics of inhomogeneous systems [10]. Analysis is restricted to the isotropic case. The gradient terms are allowed not only in the free energy and the entropy densities, but also in the internal energy density. Exploring the heat and the entropy balance equations and requiring nonnegative local entropy production we formulate a family of thermodynamically consistent phase-field models for nonisothermal growth processes. Within these models we analyze the effects of the gradient terms on the interface undercooling. 2 Thermodynamics and thefieldequations. First, let us briefly outline some basic concepts of the phase-field models for solidification of a pure substance [4-8]. Disregarding the difference in densities between solid and liquid the phase is defined in terms of a single nonconserved order parameter (ctystallinity)field,denoted by 0. The homogeneous liquid and solid are identified as states <j>=-\ and 0=+l, respectively. The free energy density, / is assumed to be a function of the temperature T and of the phase field, <j>. For 0 = ±1 this function is the standard one, corresponding to the homogeneous phases. It is then analytically continued for ally's between these extreme values. We assume that f(ty,t) is of the "double-well" type with local minima at 0 = ± 1 for any T. It is also assumed that at the equilibrium freezing point T=T^ the values of these local minima are equal. Furthermore, it is assumed that Here L is the latent heat at T= T ^. The energy and the entropy densities are defined as e = f - Ts, and s =. It then follows that

3 Moving Boundaries IV 29 Here /^ = e^. Adopting the Extended Irreversible Thermodynamics of inhomogeneous systems [10] we now introduce the gradient terms in the internal energy and entropy densities. For an isotropic medium this implies: (3) The corresponding free energy density/ is then given by where f^^7)=/rr^+6^^\ (4) It then follows that = ), =-r (5) (&'=%&' +/,<% + e^^f^ (6a) Tds ' = de ' - / 0rf0 - t\ V<j)d\/(j) (6b) (6c) We now proceed with a thermodynamically consistent derivation of the field equations for solidification of a pure substance. For the solid-liquid system in mechanical equilibrium the balance equations for the internal energy and the entropy densities are given by &' a?' = -dw<, +5 (7) Here J% and J$ are the energy and the entropy flux densities, respectively, and E>0 is the local rate of entropy production. In the present paper we consider only sufficiently small undercooling of the melt. In this case both e^ and s^ can be treated as constant parameters, while s^ is given by Eqs.(5). Using Eqs.(6b) and Eq.(7) it can be shown that where 5 = 5^ + ^.9(7/7) ^ ^^^.9(7/7) (8)

4 30 Moving Boundaries IV Z, = -(7/7) WW, - 4 ^^ 1 ^ = ^ ^ 4 W^9 ^] ( The first term in Eq.(8) is the scalar dissipation due to the order parameter field 0. The second term in that equation corresponds to the vector energy flux and the conjugated temperature gradient. The last term in Eq.(8) can be associated either with the scalar or with the vector contributions. Following the Curie principle, which prohibits coupling between the dissipations due to the thermodynamic forces of different tensorial character, we split E into two components: Here S =Ss+3y (10) *V(1/T) > 0, (1 la) 0. (1 Ib) The parameter a is an arbitrary real number, representing an intrinsic degree of freedom of the theory. Imposing the linear "force-flux" relations one obtains the following equations for coupled diffusion of the phase field and heat: (12) (13) For e^ =0 the above equations are equivalent to those of Ref. [6], whereas for e^ * 0 and a=0 Eqs.(12)-(13) are reduced to those of Ref [9]. 3 Two-phase equilibrium states. In order to clarify the meaning of gradient terms in entropy and internal energy densities let us considerfirsta two-phase equilibrium at T=T^ In this case, corresponding to a planar interface, the right side of Eq. (12) reduces to the standard Euler-Lagrange equation used in Ref. [3], with /' given by Eq.(4). The corresponding solution of Eq.(12) is denoted by ty(t ^,,z) = \if(z). It can be shown that the parameters e \, e and el define the excess values of the free energy, internal energy and the entropy (per unit area of the interface) of this two-phase system:

5 Moving Boundaries IV 31 (14) (15) E. Here a is the surface tension at T= T ' and I is the area of interface. From Eq. (16) it follows that the gradient term in the internal energy density affects the function a(t), so that the surface tension is not of purely entropic origin. This is in accord with molecular theories of surface tension [11-12], which predict a (T)= kjw(t). Here k is the Boltzman constant and W(T) is some function, accounting for temperature dependence of the liquid structure factor. Let us consider now a metastable equilibrium between solid and liquid at T<T^. It can be shown using Eqs. (12) and (13) that for sufficiently small undercooling the free energy barrier for formation of a spherical solid germ of radius R <, = 2^T^L(T^T) is given by, 7) -/f-1, (17) The above expression differs from the classical result corresponding to 8=0 and g=l Here 8 is the half-width of an interface defined by the standard profile \\r and g is an additional correction given by (18) As follows from Eq.(18) the ratioe^ I Bp affects the value of F ^. It, can be fixed, at least in principle, from the homogeneous nucleation data Such data concerning F^ for some materials are given in [13].

6 32 Moving Boundaries IV Transactions on Modelling and Simulation vol 17, 1997 WIT Press, ISSN X 4 Kinetics of growth: asymptotic results Let us consider now the time-dependent problem of growth into undercooled melt. We address this problem exploring the layer asymptotics method given in [6,8] and using the dimensionless quantities defined in Nomenclature. For sufficiently small undercooling level s^ can be expanded near its equilibrium value. Up to thefirstorder ^m = 4(7m)[l+t(G -6J] (19) According to Eq.(5) and Eq.(19) e^ = (36^(7^), where P* =!-?#, As in [6,8] the small parameter of the problem,e is defined as e^x^fl p(t^). In terms of dimensionless quantities the governing equations can be written as: (20) (21) We now briefly outline the method of [6,8] as applied to Eqs.(20)-(21). The interface F<6r',/') is identified as the set where <t>(x\t')=0. The evolution equations are then written in terms of the local coordinates (r,x): r(x\t') represents a signed distance from x* to F(x',t') with r>0 identified as solid; K represents a generic point on F. The interface velocity component normal to F is defined as v =-d r/d t' The sum of two principal curvatures of F is denoted by K (K>0 if the center of curvature is located on the liquid side of the interface). The inner region, located in the vicinity of F, is defined by stretching the coordinate r : z'=r/e. Expanding the relevant quantities in powers of both in the outer and in the inner regions one can derive the corresponding solutions of the evolution equations in each of the above domains and match them accordingly. It can be shown that in the leading order the phase field in the inner region is given just by the standard profile Q(Qm2')=V(z'), which tends to its limiting values ± 1 in the outer region. In the order O(e) the outer solution yields a limiting free boundary problem, equivalent to the classical Stefan problem, modified as indicated below. Analyzing the phase field equation for the inner region up to the order O(e) yields interfacial undercooling effects. It can be shown [14] that the dimensionless temperature at the interface,op, is given by (22) Here ea=at^c/xl^. It defines the capillarity undercooling (the Gibbs- Thompson effect). The coefficients B and G depend only on properties of

Moving Boundaries IV 33 function f(t,(j>) and the parameters a, j3 and!. Their explicit functional form is rather complicated and will be given elsewhere [14].

7 Moving Boundaries IV 33 function f(t,(j>) and the parameters a, j3 and!. Their explicit functional form is rather complicated and will be given elsewhere [14]. The term with the coefficient B represents the nonequilibrium undercooling effect caused by the heat flux through the interface. It has beenfirstderived in [8] assuming e^ = 0. The terms involving the interface velocity v are associated with kinetic undercooling effects induced by the interface motion. 5 Concluding Remarks Extended Irreversible Thermodynamics provides a natural frame for formulation of phenomenological phase-field models for nonisothermal phase transformations with diffused phase interfaces. In general, the nonlocal contributions have to be accounted not only in the free energy and entropy densities, but also in the internal energy density. In this frame the balance equations and the requirement of nonnegative entropy production yield thermodynamically consistent phenomenological models of phase transformations. The above models involve an intrinsic degree of freedom a, which reflects the nonuniqueness of splitting the local entropy production into contributions induced by the scalar and by the vector thermodynamic forces. In the present communication we demonstrated this feature by considering a rather simplified class of models with a single nonconserved order parameter field assuming isotropy of the medium, and adopting the square gradient approximation for nonlocal terms. The main ingredient of such models is the "double well potential "/($, T). It can be fixed just as in the previous works [2-9]. The models also involve four independent dimensionless parameters e, P, a, and y. Most of the phase field models considered so far adopted ad hoc assumption /3 =0, neglecting the gradient terms in the internal energy density. Ref.[9] allowed for 0 # 0, but ignored the possibility of a* 0. By considering two-phase equilibrium states at T< T^ it has been shown that the gradient terms in the internal energy affect the temperature dependence of the surface tension, as well as the free energy barrier for nucleation in the undercooled melt. Hence, the parameters 8 and P (and hereby y) can be estimated, at least in principle, from the theories like those of Refs.[ll-12] and from the homogeneous nucleation data. To the best of our knowledge reliable data are available only for few materials, such as mercury [13]. The remaining parameters, a and y manifest themselves under conditions of growth. In particular, the influence of the gradient terms in internal energy on the nonequilibrium undercooling at the solidification front depends on the value of the parameter a. Ideally by measuring the nonequilibrium first order undercooling effects as defined by Eq.(22 ) would enable one tofixboth a and 7 We are not aware of materials for which all the data required by thermodynamically consistent phase-field models are available. In spite of expected experimental difficulties it seems that further measurments are the only

8 34 Moving Boundaries IV Transactions on Modelling and Simulation vol 17, 1997 WIT Press, ISSN X way to avoid ad hoc assumptions concerning specific values of the parameters involved in the phenomenological phase-field models. 6 Nomenclature A - coefficient in Eq.(22) B - coefficient in Eq.(22) c - heat capacity of liquid at the melting point ECX~ excess internal energy e - internal energy density F0x - excess free energy / - free energy density G - coefficient in Eq.(22) g - function in Eq.( 18) J - heat flux J$ - entropy flux K - thermal conductivity fc - dimensionless thermal conductivity, M^ k - the Boltzman constant L - latent heat A/i - coefficient in Eq.(12) A/2 - coefficient in Eq.(13) RQ - critical radius of a solid germ r - local coordinate normal to F S^x - excess entropy s - entropy density T - temperature! - melting point t - time f - dimensionless time, Kt/cX^ X - length scale of the system jc - spatial coordinate x' - dimensionless spatial coordinate, x/x v - dimensionless velocity of the interface, -dr I dt W- function defining temperature dependence of the liquid structure factor z - coordinate normal to the interface z' - stretched coordinate, r/e Greek symbols a - intrinsic degree of freedom j3 - ratio e^/sp F - interface KT^

9 7 - phase-field kinetic parameter, KsLcM^ X^ e^ e - coefficient of the gradient term in e ES - coefficient of the gradient term in s p- coefficient of the gradient term in/ e - dimensionless small parameter, / ^(T^/xVjL K - sum of two principal curvatures of F $ - phase field *F - dimensionless free energy density, f/l \l/ - the standard profile of the phasefieldat T=T ^ E - EQ- local entropy production density dissipation due just to the phasefield, Eq.(9) Sy - dissipation caused by the vector thermodynamic forces S$- dissipation due to the scalar thermodynamic forces <7 - surface tension Z - unit area of the interface 6 - dimensionless temperature, ct/l &n - dimensionless melting point ivioving Dounuiirieb i v Transactions on Modelling and Simulation vol 17, 1997 WIT Press, ISSN X i - dimensionless entropic contribution to p, e^l / c[e^(t^)] References 1. Caroli, B, Caroli, C, Roulet, B Instabilities of planar solidification fronts, in Solids Far From Equilibrium, (ed. C. Godreche), Cambridge Univ. Press, Cambridge, Cahn, J.W., Milliard, I.E. Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 1958, 28, Allen, S.M., Cahn, J. B. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgies 1979, 27, Collins, J.B., Levine, H Diffuse interface model of diffusion limited crystal growth, Physical Review, 1985, B 31, Caginalp, G, Fife, PC Phase-field methods for interfacial boundaries, Physical Review, 1986, B 33, Fife, PC Dynamics of Internal Layers and Diffusive Interfaces, CBMS- NSF Reg. Conf. Series in Appl. Math., 53, SIAM, Pliadelphia, Langer, J. S. An introduction to the kinetics offirst-orderphase transitions, in Solids Far From Equilibrium, ed. C.Goderche, Cambridge Univ. Press, Cambridge, Fife, P. C., Penrose, O. Interfacial dynamics for thermodynamically consistent phase-field models for solidification, Electronic J. Diff. Equations, 1995, 16, Wheeler, A.A., McFadden, G.B. and Boettinger, W.J. Phase-field model for solidification of an eutectic alloy, Proc. R. Soc, Lond. 1996, 452, ou

10 36 Moving Boundaries IV lo.jou, D. Casas-Vazguez, J, Lebon, G. Extended Irreversible Thermodynamics, Springer-Verlag, Berlin Oxtoby, D.W., Haymet, AD A molecular theory of the solid-liquid interface. II. Study ofbcc crystal-melt interfaces, J. Chem. Phys., 1982, 76, Harrowell, P., Oxtoby, D.W., A molecular theory of crystal nucleation from melt, J. Chem. Phys., 1984, 80, Spaepen, F, Homogeneous nucleation and the temperature dependence of crystal-melt interfacial tension tension, Solid State Physics, 1994, 47, N.Charach, Ch., Fife, P. C, On thermodynamically consistent schemes for phase-field equations (in preparation).

Anisotropic multi-phase-field model: Interfaces and junctions

First published in: PHYSICAL REVIEW E VOLUME 57, UMBER 3 MARCH 1998 Anisotropic multi-phase-field model: Interfaces and junctions B. estler 1 and A. A. Wheeler 2 1 Foundry Institute, University of Aachen,