- Email: [email protected]
Continuum theory of thermally induced phase transitions based on an order parameter
Physica D 68 (1993) 326-343 North-Holland SDI: 0167-2789(93)E0198-K
Continuum theory of thermally induced phase transitions based on an order parameter Eliot Fried a and Morton E. Gurtin b a Department of Engineering Science and Mechanics,
Pennsylvania State University, University Park, PA 16802-1401, USA b Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213-3890. USA
Received 22 September 1993 Revised manuscript received 16 March 1993 Accepted 16 April 1993 Communicated by M. Mimura
Using balance laws for accretive force and energy in conjunction with constitutive equations restricted so as to be compatible with the second law, we develop a theory for the study of solid-liquid and solid-solid phase transitions where accretion and heat conduction dominate mass diffusion and deformation. Our theory furnishes generalizations of the Ginzburg-Landau equation and the phase-field equations, generalizations that allow for anisotropically induced preferred growth and nonlinear transition kinetics.
1. Introduction In [ 1 ] Chan proposes a model for the kinetics of isothermal, diffusionless first-order phase transitions wherein the state, in the sense of phase, is specified by a scalar order parameter-denoted here equatione2 by q. The basis for this model is a relaxation law #I in the form of a Ginzburg-Landau (1.1)
Pu, = AAy, - w,‘J&,
with /I and 2 positive material constants and ~0 a double-well potential, an equation motivated by assuming that a free energy functional of o, relaxes toward an extremum under an influence proportional to the rate U,at which the order parameter changes. Recognizing that the isothermal nature of this model limits its range of its applicability, Chan suggests that in a more general approach thermal effects be accounted for by enforcing energy balance. To treat capillary and kinetic effects not covered by the classical Stefan model for solidification, Langer [ 41, Fix [ 5 1, Caginalp [ 6 1, and Collins and Levine [ 7 ] introduce a model involving an order
Q(c) = J, W((, V[) dv of a list c = (z t, z 2, . , z n) of time-dependent fields is a set with /I 1constant, determined by writing the variation SQ([), for of n relations of the form piii = -wi({,V~,VZy), compactly supported SC, in the form SQ([) = c:=, J,w,({,V[,V T)Szi dv. *2 Attributed by Chan [ 1] to Landau and Khalatnikov [2]. Cf. White and Geballe [3] for further discussion of such equations. *’ A relaxation law for a functional
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E. Fried, M.E. Gurtin / Thermally induced phase transitions
parameter, called the phase-field. The resulting equations-for phase-field q-consist of an equation
the scaled temperature
327
deviation
czi - Qi = kAu, expressing balance of energy, supplemented
u and
(1.2) by a Ginzburg-Landau
equation
puj=nA~++~(l-u,)(2~-l)-eu,
(1.3)
derived from a relaxation law for a free energy functional at fixed temperature *3 . (Here c, k, I, /3,I, and v are material constants with all but !Zstrictly positive.) By investigating a particular asymptotic limit of the governing equations, the authors of [4-71 derive the Gibbs-Thompson relation used in nonclassical Stefan problems, which establishes a correspondence between the phase-field approach to solidification and other more familiar models. This result is broadly extended by Caginalp [9], who determines the connection between various asymptotic limits of ( 1.2)) ( 1.3) and a hierarchy of Stefan and Hele-Shaw problems#4. To Penrose and Fife [ 161, the mixed use of relaxation laws and balance laws that leads to the phasefield model is physically unappealing; in accord with an approach suggested by Chan [ 1 ] and taken by Hohenberg and Halperin [ 8 1, they consider instead a framework based completely on relaxation laws with entropy as potential, which they use to derive a general class of phase-field models. Our goal here is a fully dynamical continuum theory for thermally induced phase transitions in which the notion of phase is characterized by an order parameter, and for which interfaces between phases are identified with thin transition zones across which the order parameter exhibits large spatial gradients. This theory is to be sufficiently general to allow for the modeling of both solid-liquid and solid-solid transitions where the effects of accretion and heat conduction dominate those associated with mass diffusion and deformation. We share the preference of Penrose and Fife [ 16 ] for a theory based on a consistent application of physical principles. However, we feel that the relaxational approach is overly restrictive, since it limits the manner in which rate terms can enter the basic equations. In addition, the relaxational approach requires an a priori specification of constitutive equations, which are then used in the derivation of local balance laws. We believe that, while variational derivations often point the way toward a correct statement of the underlying balance laws, such derivations obscure the fundamental nature of balance laws as basic axioms in any general dynamical framework that includes dissipation. We therefore base our study on an approach, now standard in continuum mechanics, in which general balance laws, supposedly common to large classes of materials, are distinguished from constitutive equations, which differentiate between particular materials. In this approach the second law is used, in the sense of Coleman and No11 [ 171, to restrict constitutive equations. What distinguishes our theory from other macroscopic theories in which phase interfaces are treated as thin transition zones is how we write the basic laws. Here we are motivated by recent theories”5 that carefully specify and use the manner in which inter-facial stresses expend power. We introduce a
t3 The system ( 1.21, ( 1.3) is a special case of model C of Hohenberg and Halperin [ 81. #’ See also Caginalp and Fife [ 10,111, Fife and Gill [ 12,13 1, Caginalp and Nishimura [ 14 1, Fife [ 15 1. *5 Cf. [ 18-201, in which phase interfaces are modeled as surfaces of discontinuity.
E. Fried, M.E. Gut-tin/ Thermally induced phase transitions
328
vector measure of stress, the accretive stressg6 <, and take as its conjugate kinetic variable the rate 4 at which the order parameter ~1is changing: given any part#’ P of the body,
s NJ
&<.nda
(1.4)
represents the associated expenditure of power #8 . Since the order parameter should be nearly constant away from transition zones, (1.4) should be significant only within such zones. Basic to our theory is a balance law /<.nda
+ Jndu
8P
= 0,
(1.5)
P
in which rc represents internal forces. #9 Since it is the accretion of material of one phase at the expense of another that characterizes phase transitions, we use the adjective “accretive” to describe this system of forces as well as its associated balance law. We believe that ( 1.4) and ( 1.5 ) correctly describe the physics underlying accretion #lo . Because of the complexity of the general thermodynamical theory, we find it advantageous to proceed in stages. In Section 2 we consider accretion alone, ignoring thermal effects. We postulate-as an appropriate version of the second law for this simplified setting-a dissipation inequality Jwdz’< P
J$<.nda,
(1.6)
8P
with v/ the free energy. We consider constitutive
equations giving cy, C, and n as functions of q, U;,and
p = vfj7.
(1.7)
When suitably restricted Y = w^(ul,P),
by ( 1.6) these constitutive
r = !gvI,PL
equations
7c = -_Iy?p(vI~P) - B(yl,P,
take the form o;)u;>
(1.8)
where p is a nonnegative (constitutive) function. (Here and in what follows we use subscripts to denote partial derivatives, e.g., I& (q,p) = (a/ap)@(q,p).) In conjunction with the accretive balance (1.5), the constitutive relations ( 1.8) yield the partial differential equation B(v,~,vi)i
= div(ly^,(v,p))
- F~v,P).
(1.9)
In [ 2 1] we give a more basic development of accretive forces in which the accretive stress is tensorial with corresponding kinetic variable the velocity of level surfaces of ~0. 17 The body is a region f?CR3; P (with unit outward normal n) will always denote a part (subregion) of B. Vectors in R3 and tensors (linear transformations of W3 into itself) are distinguished from scalars using boldface italic type; lower case for vectors, upper case for tensors. 1 is the identity tensor. The gradient, Hessian, Laplacian, and time derivative of c are denoted, respectively, by Vi, V2c, A(, and 4. The derivative of a function v of a scalar variable (not time) is denoted by u’. #8 Gurtin [22 ] and Truskinovsky [23] use a power of this form, but do not introduce a general accretive balance such as (1.5). #9 We also allow for an external accretive force y and an external supply of heat r that contribute terms to our equations. For convenience, however, we omit discussion of these in the introduction. #lo Cf. [ 18-20,24-261, where accretive balances are used in theories that model phase interfaces as surfaces of discontinuity. #6
E. Fried, ME. Gurtin / Thermally induced phase transitions
329
By an appropriate narrowing of the class of constitutive functions, (1.9) reduces to the classical Ginzburg-Landau equation ( 1.1). Granted the status of ( 1.1) as a model for the kinetics of first-order isothermal, diffusionless phase transitions, the generalized Ginzburg-Landau equation ( 1.9) encompasses a broad range of effects germaine to such transitions-but not allowed by ( 1.1 )-including nonlinear transition kinetics and anisotropically induced preferred growth. The relations ( 1.5 ) and ( 1.9) reflect a basic difference between our treatment and those involving relaxation laws. The balance law (1.5) involves no time derivatives; rate terms enter the theory via the constitutive equations for the internal force rr. In this regard we do not view (1.9) as a transport equation for q, but instead as a balance law for force in which a term involving U;arises as a consequence of the dissipative character of the internal force; for us this term is of the same nature as the viscous stress in a Newtonian fluid. In Section 3 we discuss the interaction of accretion and heat conduction. The governing laws are the accretive balance ( 1S) in conjunction with the first two laws: balance of energy Jeda
= +nda
P
8P
+ /a;&ndn
(1.10)
8P
and growth of entropy”” (1.11)
Here E is the internal energy, q is the entropy, 19is the (absolute) temperature, and q is the heat flux. We consider constitutive equations relating the free energy cy = E - Oq, 9, q, C, and R, to 8, g = VB,
(1.12)
q, p, and 4. The resulting partial differential equations furnish a broad generalization of the phasefield equations, one that includes the effects of nonlinear transition kinetics, anisotropic preferred growth, and cross-coupling between kinetic and thermal effects. These equations are, however, quite complicated, and for that reason we turn next to the development of a simpler theory. In Section 4 we develop a model appropriate to behavior near a transition temperature 60. To derive such a model one can formally approximate the partial differential equations of the general theory developed in Section 3; unfortunately, the resulting approximation does not typically preserve the underlying thermodynamic structure. Here we take an alternative approach, one that ensures a consistent thermodynamics”* : we formulate versions of the first two laws for situations in which (t9 - &)/tic is small, and then develop an exact model within this setting. Here balance of energy has the form JEdV= P
-JI-ndu, 8P
The Clausius-Duhem inequality (cf. Truesdell and Toupin [27], sections 256-258). #I2 This approach is taken in [28], Section 8.1; [ 29,301. #I1
(1.13)
E. Fried, M.E. Gurtin / Thermally induced phase transitions
330
and we take, as the appropriate
version of the second law, the dissipation inequality*‘3
JpdZ<--
Jug-nda+
P
t?P
(1.14)
[email protected], 8P
where u is the scaled temperature-deviation (8 - t9a)/&, while g = E - 80~ is a Gibbs function. Granted isotropy and assuming a simple form for the resulting constitutive equations, this formulation leads to the partial differential equations c(p)u
i G(q)
= div(k(v)Vu),
/I$ = I1Ay,- F’(q)
- UC’(V) + $‘(q)u2,
(1.15)
where
F(q) + UC(~) -
;c(du2
(1.16)
is a “coarse-grain ” free energy, c (q ) is the specific heat, and k (9) is the thermal conductivity. To characterize phase transitions, with u = 0 the transition temperature (deviation), we assume that F (q ) + UC (p ), as a function of 9, is a double-well potential with one well furnishing a global minimum when u < 0 and the other when u > 0. By selecting F(v) and G (cp) appropriately and choosing c (a, ) and k (q ) to be constant, ( 1.15) reduce to the phase-field equations ( 1.2)) ( 1.3). Other choices for F (v, ) and G (cp) lead to specializations of ( 1.15 ) in which the values of the order parameter that correspond to the two phases are fixed, while the specific heat and thermal conductivity may vary from phase to phase. Discussion is limited to best illustrate both our treatment of order parameters and the role of the accretive force balance. We do not consider: (a) more than one order parameter, (b) order parameters of a vectorial or tensorial nature, (c) inhomogeneous constitutive equations, (d) effects due to deformation, (e) effects due to mass diffusion. Consideration of (a)-(c) is not difficult and involves no new ideas. A theory involving deformation is presented in [ 2 11, while one encompassing mass diffusion will be the subject of a future work.
2. Purely accretive processes. Generalization of the Ginzburg-Landau equation To best describe the manner in which the accretive forces and order parameter enter the theory, as well as the manner in which we use thermodynamics to restrict constitutive equations, we begin with a simple theory that neglects thermal effects. 2.1. Balance law for accretive forces. Second law In the present context the basic physical quantitites,
defined on the body a for all time, are
#I3 This form of the second law, without accretive terms, was introduced
in [ 311.
E. Fried, M.E. Gurtin / Thermally induced phase transitions Q
order parameter,
v/
free energy,
t:
accretive stress,
n
internal accretive force,
Y
external accretive force#14
331
(1.16)
Here there is but one balance law, balance of accretive forces: /~.nda+/nd?i+/~do=O P
aP
(2.1) P
for all parts P and all time, where n is the unit outward normal to LIP. When thermal effects are suppressed, the second law is the assertion that: the rate of energy increase cannot exceed the expended power.
(2.2)
Our basic assumption is that accretive force is conjugate to the rate 4 at which the order parameter is changing, so that $<. n and 4 y represent rates at which the accretive stress C and external accretive force y perform work. We therefore write the second law in the form of a dissipation inequality F P
< Se<-nda f3P
+
Syjydlr
(2.3)
P
to be satisfied for all time and all P. The force a, being internal, does not enter the inequality (2.3). This version of the second law can be derived as a consequence of appropriate versions of balance of energy and growth of entropy under isothermal conditions (cf. the discussion following (3.4) ). Since P is arbitrary, (2.1) and (2.3) have the equivalent local forms divC + 71+ Y = 0, div(yiC) + $r 2 U;, which combine to form the local dissipation @-<.ji
(2.4)
inequality
+ nu; < 0.
(2.5)
2.2. Constitutive equations. Consequences of the second law As constitutive equations we allow the free energy, the accretive stress, and the internal accretive force to depend on the order parameter q and-to model capillarity and transition kinetics-also on p = Vy, and ,#14: w = w^(G%P,a;),
c = ~(%P,io),
R = ic^($%P,@).
(2.6)
We do not write a constitutive equation for the external accretive force y, but instead allow y to be assigned in any way compatible with the accretive force-balance. *13Cf. Hohenberg and Halperin [ 81, who introduce a source term in the Ginzburg-Landau *14 Here and throughout
~(o(x,t),p(x,t),gi(x,t)).
we consider only homogeneous constitutive
equation that corresponds to y. behavior; thus, e.g., the first of (2.6) signifies yl (x, t) =
332
E. Fried, M.E. Gurtin / Thermally induced phase transitions
Given an order-parameter field (D,the constitutive equations (2.6) can be used to compute a constitutive process consisting of a, and the fields y, <, and n; the balance (2.4) 1 for accretive force can then be used to determine the external accretive force y that must be supplied to support this process. The second law remains to be satisfied in all such constitutive processes, a requirement that we use to restrict the constitutive equations. Specifically, we assume that the local dissipation inequality (2.5 ) holds for all constitutive processes. Writing h = (p, we see that, granted (2.6), (2.5) is equivalent to the inequality (&0(4VJ)
+ ic^(V,P,h))h
+ @&Ah)
-&V&h))+
(2.7)
+ @h($%P,h)~ < 0,
and, since we can always find a field p such that 8, h, A, p, and i have arbitrarily prescribed at some chosen point and time, we must have I+?&, = 0 and I&, = f. Therefore the free energy and accretive traction are independent of 4 and related through
values
(2.8) and the inequality (&&,P)
+ %%P,ipHio
d
holds for all 01,p, 4. (Conversely, in all constitutive processes. ) Because of the tacit smoothness form ~C%P,a;)
= -&o(%P)
(2.9)
0 consistency
with (2.8) and (2.9) ensures compatibility
of the quantity
in parentheses
in (2.9), ii can be represented
b)b,
- P(P,P,
with (2.5) in the
(2.10)
with p ( y,p, qi) > 0 a constitutive modulus, giving the internal accretive force as the sum of equilibrium and nonequilibrium parts, --FV (9,~) and -fi(q,p, a;)@. Using (2.8) and (2.10) in the forcebalance (2.4) 1 results in the partial differential equation PCv,p,@)i
= div(@,(yl,p))
which represents
-
@~P,P)
a broad generalization
2.3. Quasilinear constitutive
(2.11)
+ Y,
of the Ginzburg-Landau
response
In this section we restrict attention to response functions and (2.10), but withi(q,p) and n^(qo,p,uj) affine functions and (2.10) yield w^(V,P)
= we(v)
equation.
+ w(v)
.P + ;P .n(a,)P,
n^(V,P,@,
of the form (2.6) consistent with (2.8) ofp and 6 for each fixed v. Then (2.8)
= -@,(ul,P)
-P(v)&
(2.12)
with v/o a “coarse-grain” free energy, j? ((p ) > 0, and where, without loss in generality, we require that A (q ) be symmetric. Further, by (2.8)) A must be independent of q; hence w^(V,P) ~(q,P,uj)
= wo(u1) + w(v) = -wIga?)
.P +
- w’(v)
{P.AP,
.P - P(y)ih
&AP)
= AP + w(q),
(2.13)
E. Fried, M.E. Gurtin / Thermally induced phase transitions
333
and these with (2.12) result in the equation P(V)u, = n. v20, - w&d
+ Y.
(2.14)
When j3 (o, ) is strictly positive and A is positive definite, this equation is parabolic, but neither of these assumptions is a consequence of the local dissipation inequality. Although zo(q) does not appear in the local balance (2.14), it would generally enter boundary conditions. If the material is isotropic, so that /i = ill, and if we require that p(v) = /3 = constant, then (2.14) becomes, assuming that y = 0,
PO;= AM-
w&d,
(2.15)
which is (1.1). The general constitutive equations that produce (2.11)) unlike their quasilinear (2.13) allow for nontrivial anisotropic dependence of @( q,p) and p (q,p, @) on p as dence of /3(q,p, U;) on U;. Studies of (2.15) with wO(q) a double-well potential show p small in an appropriate sense, p corresponds to a vector normal to transitions zones the free energy @(4’,P)
=
we(v) +
@aP1*, p^=PAPI,
specializations well as depenthat, for ;1 and #15. Similarly,
(2.16)
where I,Y~ (rq) is a double-well potential and A(.$) > 0 is small, appears to formally approximate interfaces-treated as surfaces of discontinuity with unit normal s-with interfacial energy proportional to m. #I6 Hence, anisotropic dependence of @(q,p) on p allows for the modeling of anisotropic growth through (2.11). The lack of restrictions on !? (q,p ) also allows for the modeling of nonconvex interfacial energies of a type common to crystalline materials with nonsmooth Wulff shapes”*’ . Further, an analysis of traveling wave solutions of (2.11) demonstrates that this equation accomodates a complete spectrum of transition kinetics unavailable when /3 is independent of &an assumption that leads to linear transition kinetics. Note also that our development ensures consistency with the dissipation inequality (2.3), and hence yields the Lyapunov relation
Jvdv=-Jmwdu;zdv
(2.17)
t3
for the body B whenever
y = 0 on I3 and @c. n = 0 on 823.
#t5 Cf. Rubinstein, Sternberg, and Keller [ 321, Evans, Soner and Souganidis [ 331. #t6 Cf. Gut-tin, Soner and Souganidis [ 341. *17 The use of expressions such as (2.16) to model nonconvex interfacial energies arose in discussions of Gut-tin with D. French ( 1988). When @(q,p) is nonconvex in p the underlying partial differential equation (2.11) is rendered backward parabolic for some solutions; models in W2 for the evolution of interfacial curves with such energies are discussed by Angenent and Gut-tin [35] and Gurtin, Soner, and Souganidis [36].
E. Fried, M.E. Gurtin / Thermally induced phase transitions
334
3. Accretive processes with thermal effects 3.1. Clausius-Duhem inequality Our study of thermal effects begins with the quantities &
internal energy,
79
absolute temperature,
II
entropy,
4
heat flux,
r
external heat supply,
and q, C, n, and y introduced
/{.nda aP
+ /rrdu
(2.17)
previously;
+ /ydo
P
these fields are related through the accretivefirce balance
= 0
(3.1)
P
and balance of energy
JEda=-Sq.nda+Srdv+JlpF.nda+/~).dzl P
P
aP
f?P
(3.2) P
for all time and all P. Along with contributions to the rate of energy increase due to the heat flux q and the heat supply r, in writing (3.2) we also account for the rates at which the accretive traction < and the external accretive force y perform work. For future use, we introduce the free energy y = E-&j.
(3.3)
The second law is here the assertion that the rate of entropy production be nonnegative, a law we impose through the Clausius-Duhem inequality: (3.4) P
ap
P
for all time and all P. Note that for isothermal processes, (3.2) and (3.4) reduce, using (3.3), to (2.3), which, with the accretive force balance, was our starting point for the study of purely accretive processes. The global laws (3.1), (3.2), and (3.4) have local forms div{ + R + y = 0, which may be combined
-divq
+ r + div(@C) + U;y = E’, -div(
to give an alternative
E-<.$-I-X$=-divq+r as well as a local dissipation
5) + i < rj,
(3.5)
form of the energy equation (3.6)
inequality (3.7)
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
335
3.2. Constitutive equations. Consequences of the second law We consider relations in which the constitutive are augmented by q, p = Vq, and 4: v/ = !ahw,P,d,
variables 19and g = Vr9 of classical heat conduction
rl = fWJ,%P,yiA
c = k%&~,P,d,
4 = 4^mL~~P>d~
a = W,M4P,d.
(3.8)
Writing c = (19,g, q,p, U;) for the list of constitutive (3.8), (3.7) is equivalent to the inequality
variables
and h = @ we see that, granted
(3.9) From this we conclude that @ cannot depend on g or 4, that (3.10)
(3.11) an inequality
yielding representations
is(C) = -@&~o),P)
-k(C)
.g -P(C)+,
t(C)
= -K(C)g
-HO&
(3.12)
in which k, p, K, and b satisfy
PK)h2+
V’g.KK)g
+ g.(k(l)
+ 19-‘b(0) h 2 0.
(3.13)
The first of (3.10) is in accord with the relationship between entropy and free energy that arises in the study of rigid heat conductors, while the second is analogous to (2.8). An immediate consequence of (3.10) is the “Maxwell relation”
&W, P,P) = -tsw,
V,P).
(3.14)
The functions rkJ(f9, P,P)
= 4&t
V,P),
a...(C)
= -k(C)
. g - PKk
(3.15)
represent the equilibrium and nonequilibrium parts of the internal force ??([). The results (3.10) and (3.12)r, with (3.1) and (3.15), yield the “Gibbs relations” w=-rt~++.~--Kes(8,~,P)~,
E=lsil+t.h-Res(~,O),P)V);
and using (3.16)~ we can rewrite the energy equation t9rj = -divq
- n,,,,(C)@ + r.
(3.16)
(3.6) as a local entropy balance (3.17)
Finally, note that the specific heat C(f+,%P)
= --1969L9(~,a),P)
= WL9(fl,V,P)
depends not only on 19but also on v, andp.
(3.18)
Unless specified otherwise we assume that ~(29, ~,p)
> 0.
E. Fried, M.E. Gurtin / Thermally induced phase transitions
336
Using (3.10) and (3.12) in (3.5)1 and (3.17),anddetining~ system c(s)&
= -div@(C)
P(i)i
= div(&,(c))
+ tit&+(s). -k(i)
06
.w-
+ (f%,(s)
= (rY,p,p), weareledtothegeneral
- ~~,,,(i))ai
+ r,
(3.19)
+ Y,
which is to be supplemented by (3.12)~ and (3.15)~. If for 29near 130and allp, @(I.?,p,p) is a doublewell potential in which one well furnishes a global minimum for 19< Ba and the other for t9 > tie, then (3.19) describe a model for thermally induced phase-transitions in which the two phases correspond to values of o, at the minima, and 80 is the transition temperature. By analogy to (2.11)) the equations (3.19 )-under suitable constitutive assumptions-should have the ability to model anisotropic growth and nonlinear transition kinetics. The “cross-coupling” present in (3.12) also allows g to effect the accretive force and U;to effect the flow of heat. Note that, because our derivation ensures consistency with the basic laws (3.2) and (3.4), solutions = y = 0 of the partial differential equations (3.19) compatible-with q . n = &f ,n = OondBandr on B satisfy the classical relations A A s B
sdv = 0,
Jq
(3.20)
dt) 3 0.
D
Finally, if we omit the cross-coupling r = y = 0, then (3.19) reduces to c(s)8
= div(K(i)VB)
B(i)ti
= div(Fp(c))
terms in (3.12) by taking k = b = 0, and if we assume that
+ t%__S(s).
00, + th&,,(s)i~
+ P(C)(o’
(3.21)
- &Cc).
These partial differential equations appear to be difficult; in what follows we will develop systems of equations that should be more tractable. 3.3. Quasilinear constitutive response We now restrict attention to response functions of the form (3.8) consistent with (3.10) and (3.12), but with i(79, q,p), Z(O,g, q,p, i), and @(8,g, q,p, U;) aftine functions of (g,p, y’) for each (1.9,q). Granted this assumption, simple calculations similar to those used in Section 2c show that k, 8, K, and b depend at most on 19and ~1,and (Y(s) = y/(ti,yl)
+ W(ti,u,).P
+
;PNNP,
Tits1
= -&+(~,o,)-w9(~,y,)~P-
$P4WP,
i(s)
= A(B)P
E(C)
= -w^,(8,0,)-20p(19,V)).P--k(9,0)).g--P(8,V1)V),
7(C)
= -K(fJ,y,)g-b(8,y,)yj,
+ w(29,vD),
(3.22)
where A (8) is a constant symmetric tensor, while k (6, v), p (6, q), K (8, CJI), and b (19,CJI)satisfy (3.13). The relations (3.22) can be used in (3.19) to give a set of partial differential equations which we will not write explicitly.
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
For an isotropic (3.19) become
material,
li = 11, w = k = b = 0, K = kl; if, in addition,
c(6,q)a
= div(k(fi,y,)W)
P(%ol)u;
= A& - @&U)
337
I is constant,
then
+ 1%&(19,0))@ + P(~S,U,)U;~+ r, (3.23)
+ Y.
As before, solutions with q .n = &f .n = 0 on aD, and r = y = 0 on B satisfy (3.20). The quasilinear theory, although somewhat artificial, does provide a fairly general model consistent with the basic laws of thermodynamics. 3.4. The second law in terms of a Gibbs function Consider again the general theory of Section 3b. Let 190denote a fixed temperature, arbitrary, but in the next section will be a transition temperature. Further, let u=-
6 - @I I9
which here is
(3.24)
and consider the Gibbs function
(3.25)
g = & - &)rJ = ry + (6 -B&j.
Then (3.2) and (3.3) yield the global dissipation inequality (3.26)
~a-JUq.nda+Jurdu+JmE.nda+S~~do. P
P
8P
8P
P
This inequality furnishes an alternative version of the second law: granted (3.2) and (3.3), the inequalities (3.4) and (3.26) are equivalent. An immediate consequence of (3.26) is that A
I
(3.27)
gdv
B
whenever 19 = &J or q. n = 0 on da, co<. n = 0 on dB, and r = y = 0 on 23. By (3.10), g is given by a constitutive equation of the form g =
i?(8,P,P),
(3.28)
and if the specific heat (3.18) is strictly positive at 19 = 190,then (3.29)
&JBB(~o,(o,P) > 0,
&9(Oo,a),P) = 0,
so that g (19,q,p ) has a strict local minimum at 19 = 80, a property that makes the Gibbs function useful for analyzing behavior near tic. Further, modulo an appropriate scaling to dimensionless quantities, for 6 K 1 and#18 u,
U,
i,
4,
r,
<,
n,
y
of order O(6),
*18By (3.25) and the first of (3.161, g = u&j + t: .j - n&, plausible.
i
of order O(S*),
(3.30)
which, granted the first of (3.10), makes this assumption
(SP) ‘($‘d‘d)‘8‘n)J~
= b
~($‘d‘d)‘%‘n)_u = 1~ ‘(p5‘d‘d,‘B‘n)!J ‘(p3’d‘o)‘S‘n)_3 = 3
‘(Q,‘d’d,‘B’n)~
= 2 = h
ULIOJalJ1ur suopenba a+lnlylsuoD .Iap!suoDaM
'Z'P 'O~nA.b+fnu+~.~-~a+n, ‘3 =
A +
bmp-
SUIJOJlEDOI aI.jl U! (9Z.E) pUE (E'P) a]lJMaJ Ut?3aM ‘(Z-p) pUE ‘(8Z.f) ‘I(s*f) %U~Sfl .iCg,,,,aanSaylSOalOAazjlsctv[dIfi hroaql I?z?u.r~Sal~ugu~ S!yl Utql!M Snyl :/fi ql!M paIJ!lUaprh+lql!M (E'Z) UIlOJ 01 aU~ql.UOD(f'j~)pII&?(9Z.f) (1UElSUOD = I'Z) SUO~l~pUO~ ~eu.~laylos~ lapun leyl aioN '(1‘f )
amvjvq an+?.mv aql pue ‘(9z.f) iCi!lvnbauluo,wdtss!p p?qo[8 ay) d
‘np.l
(E’P)
s
fip3j’
+vpwbr-=
amv~vqil8naua ayl uoL~oayl~no aseq a.Ioja.tayl aM
'(z'f) ui swal
a+lal~~t? ayl doJp 01 )nq ‘pa8uoyDun (9z.f)
uo~lew~xo~ddt!luals~suo~r!leyll~adxaola~qeuos~a~swaas
ahoaI 01 so I~IXUSn .IOJ
l!‘(()f'f+) %U~I.I~UOD yJeuIal aylJoMayu~ *3n -8
(Z-P)
=
d
uo!l~unj Ln?luauIa~dwos ayl aDnpo.w;r asp? aM
Of? Of?- 8
-=n
(I’P)
uoy?!Aap a~nle~aduxalayl MOU n I.@M‘MI?I puoDas aylJ0 uo!s.IaAaieydoldde aylse (9~‘f)
*yJoMawegsyl ayl dOIaAap uayl put! ‘I~ELIIS a.n?(Of-f)
ui %uy.waddt!surlalayl yyy~
uylrm
ayE aM
13axaw
ui ‘1x3 uy ‘puss!
hroayl
OQ/(QI - 8)
ycyrn ur suogenl!s 01 ale!ldoldde ql~O.18Ldollua put! aXn?jEq i(XIaUa30 smr2lale~nw.IoJaM :yDt?oldde a~~l~wal~t!ue ayw
aM alaH
*a.wcwls
~~weuApow~ay1
uoyeurrxolddt! %u!lpIsalayl ‘K~a~eunUojun f(61.f)
yms agap
walsh
%u+apun
ayl auasald
lou /Cllwaua% 11~~
utx auo lapour t!
ayl alew!xolddv 4p1~03
aM
01 '0~ alnleladura!uoy~y,n?~lt?.~eau.~o!~eyaq 01 ale!JdolddE 1apor.ue.~ap!suo~~ou
‘(P)O ale swlal8ururetua.I aql lnq ‘(zg)o s! JaMOd aAga.we ayl :laplo woJynJ0 IOU ale (z-f) uoymba ST (gz’f) LlrlEnbauy uogedrssrp ayl u! urea) yDoa
Maua
aql u! suw
ayl ‘puq
.Iaylo ayl uo :( zq)o
339
E. Fried, M.E. Gurtin / Thermally induced phase transitions
Proceding
as before, we conclude, as a consequence
-@&,WA
E^(%v),P) =
F(WW1
of (4.4)2, that !? is independent
of g and 4, that (4.6)
= @&M%P),
and that E(C) = -~&,$%P)
-k(C)
C(C) = -K(C)g
.g -P(C)&
(4.71
- b(C)$%’
with
BK)h2+ g’K(c)g + g. (k(C) + b(C))h 2 0,
(4.81
[ = (u, g, p,p, 6 ), and h = 6. The basic equations of the theory are the local accretive balance (3.5) 1, the local energy balance (4.4) i, and the thermodynamically restricted constitutive equations (4.6), (4.7). 4.3. Generalized phase-field models The difficulties inherent in the study of solidification have led to an interest in approaches, such as that embodied by the phase-field model, that capture the essential interaction between accretion and heat conduction, but are sufficiently simple to allow analysis of the corresponding initial/boundaryvalue problems. In this spirit, we now consider materials defined by constitutive equations of the form Y = F(q)
+ uG(p)
E = c(v)u
- G(v),
c
- $ c(&u2
+ ; P .AP,
=AP,
a = -F’(q)
- uG’(q)
+ $c’(q)u2 - pa;, (4.9)
4 = K(v)g,
where A is a symmetric tensor, /I 2 0, and K (v, ) is a positive semidefinite tensor. These relations are consistent with the thermodynamic restrictions (4.6)-(4.8) and, by (4.2), yield a Gibbs function of the form g =
F(q) +
$c(p)u2+ ipAp.
(4.10)
Note that we allow the specific heat and conductivity to depend on phase through the dependence of c (a, ) and K ( y ) on v. Note also that we do not allow jI and A to depend on q, and we do not allow 4 to effect the heat flux q, nor g to effect the accretive force a, although such dependencies, properly accounted for, are not ruled out by thermodynamics. Substituting (4.9) into the balances (3.5)i and (4.4)1, and assuming that r = y = 0, we arrive at the system #l 9 c(q)u
i G(q)
= div(K(q)Vu),
ai
= A .V2y, - F’(p)
- uG’(p)
+ $ c’(v))u2,
#I9 For an anisotropic material the interface is probably better modeled by allowing for anisotropic terms of the form (2.16).
dependence
(4.11)
on p through
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
340
which for an isotropic material has the forrng20 C(CD)U4 G(q)
= div(k(a,)Vu),
PO; = LAY,- F’(q)
- UC’(~) + i
(4.12)
c’(q)u2.
The dependence of c (o, ) on o, gives rise to the terms UC’(v)U; and $c’(p ) u2; while we expect these terms to be small in applications near the transition temperature, their inclusion is necessary for consistency with thermodynamics. Since Y is essentially the free energy, to characterize phase transitions, with u the transition temperature, we assume that, as a function of 01,F (q) + uG( v, ) is a double-well potential with local minimizers ~0 (u ) and v, 1(u ) such that 90 (u ) furnishes a global minimum for u > 0 and v, 1(u) does so for u < 0. Roughly speaking, ~0 (u) and o,I (u) then represent values of the order parameter corresponding to the two phases. The term m in (4.12) I represents the change in energy induced by the phase transition, therefore seems reasonable to identify the constant
e = G(vl(O))
(4.13)
- G(qo(O))
with the latent heat offusion. One would generally assume that c (p ), k (a, ), /I, and 1 are strictly positive. If, in addition, c and k to be constant and let F(q)
= ;0(1
and it
-P))~,
G(P)
we take
= eq,
(4.14)
with v > 0, then (4.12) reduce to the phase-field equations ( 1.2)) ( 1.3). One feature of the phasefield model made clear by (4.14) is that the values of the order parameter that correspond to the two phases depend on the deviation u from the transition temperature. Alternatively, if we choose G(v) with G(0) = 0 and G’(q)
(4.15)
= 6[~(1 -q),
but leave F (q ) as specified in (4.14)) then p. (u ) and q I (u ) are fixed at 0 and 1, respectively. these choices and with c and k constant, (4.12) reduce to#21 CU - 6!~(1
- ~)a; = kAu,
Bu,=nAa,+~(l-co)(2v~-v-6Lu).
With
(4.16)
In an initial/boundary-value problem for (4.16) with initial data (DE [0, 1 ] and with either v, E [0, 1 ] on aZ? or Vy, . n = 0 specified on dB, the maximum principle requires that v, E [0,1 ] on B so that the behavior of F (y ) + uG( q) for p outside of [0, 1 ] is irrelevant #22. Returning to the more general equations (4.11)) the underlying thermodynamic structure ensures a global dissipation inequality: for q . n = V;<. n = 0 on dB and Y = y = 0 on L?, (3.26) and (4.4) imply (3.27); thus, by (4.10) and (4.14)#‘23.
920 Cf (6.2) and (6.4) of Penrose and Fife [ 161 for analogous equations with c and k constant. #21 Caginalp and Chen [ 371 work with a modification of the original phase-field model in which ( 1.3) is replaced by an equation that qualitatively resembles the second of (4.16) but ( 1.2) remains unchanged. Such a model is not consistent with the thermodynamic framework utilized here. *22 An observation due to H.M. Soner (private communication). 423 Langer [38] and Penrose and Fife [ 161 give Lyapunov relations of the type (4.17).
E. Fried, M.E. Gurtin / Thermally induced phase transitions
J
V’(y)
+
;c(qdu* + ;p -Ilp)dv
d 0.
341
(4.17)
r3
(In fact, the left side of (4.17) equals the negative of the integral over a of the left side of (4.11) with h replaced by 4.) We also have global conservation of energy as expressed by the first of (3.20) supplemented by (4.9)~. Materials scientists often consider a Stefan-type model #24 for solidification that consists of a heat equation in each phase-with specific heats set equal to zero-in conjunction with interface conditions that neglect kinetics, but allow for latent heat and surface energy. Within the current theory these approximations are consistent with c(q)
(4.18)
= P(V) = 0,
choices that, interestingly, are not ruled out by thermodynamics. The phase-field equations ( 1.3), with c = p = 0, yield, after elimination of u, the Cahn-Hilliard equation *25 a; = KA(vq(l
-v,)(l
-2~)
-nAu,),
(1.2),
(4.19)
where K = k/e*. Thus the Cahn-Hilliard equation, which, in its now classical setting, arises in gradient theories for the diffusion of mass, also arises within this phase-field model of heat conduction under dissipationless kinetics (/3 = 0) and vanishing specific heat. More general versions of (4.19) result upon eliminating u from (4.11), granted that G’(p) is strictly of one sign. Also, because (4.18) is consistent with our thermodynamic framework, the global dissipation inequality (4.17) (with c = 0) remains valid in the present circumstances, yielding an alternative verification of a standard inequality for the Cahn-Hilliard equation.
Acknowledgements This work was inspired by a talk given by W.J. Boettinger during a research workshop on the dynamics of phase interfaces held at the Center for Nonlinear Analysis, Carnegie Mellon University, in March 1992. We are grateful to H.M. Soner for numerous valuable discussions. We also thank R.F. Sekerka for providing us with a copy of Langer’s notes on the phase-field model. This work was supported by the Army Research Office and by the National Science Foundation.
References [ 1 ] S.-K. Chart, Steady-state kinetics of diffusionless first order phase transformations, J. Chem. Phys. 67 (1977) 57555762. [2] L.D. Landau and I.M. Khalatnikov, On the theory of superconductivity, Collected Papers of L.D. Landau (ed. D. Ter Haar) Pergamon, Oxford (1965). [3] R.M. White and T.H. Geballe, Long Range Order in Solids, Academic Press, New York (1979). [4] J.S. Langer, unpublished notes, 1978. tc24The Mullins-Sekerka model [ 391. #Z Cf. Cahn [40,41]. See also Pego [42], where the Mullins-Sekerka model is derived as an asymptotic limit of the CahnHilliard equation, and Bates and Fife [43] and Fife [ 15,451, where the phase-field equations and the Cahn-Hilliard equation are compared.
342
E. Fried, M.E. Gurtin / Thermally induced phase transitions
[ 51 G. Fix, Phase field models for free boundary problems, FreeBoundary Problems: theory and applicationsII (eds. A. Fasano and M. Primocerio) Pitman, London (1983). [6] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92 (1986) 205-245. [ 71 J.B. Collins and H. Levine, Diffuse interface model of diffusion-limited crystal growth, Phys. Rev. B 3 1 ( 1985) 6 119-6 122. [8] P.C. Hohenberg and B.I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49 (1977) 435-479. [9] G. Caginalp, Stefan and Hele-Shaw models as asymptotic limits of the phase-field equations, Phys. Rev. A 39 ( 1989) 5887-5896. [lo] G. Caginalp and PC. Fife, Phase-field methods for interfacial boundaries, Phys. Rev. B 33 (1986) 7792-7794. [ 111 G. Caginalp and P.C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math. 48 ( 1988) 506-5 18. [ 121 P.C. Fife and G.S. Gill, The phase-field description of mushy zones, Physica D 35 (1989) 267-275. [ 131 P.C. Fife and G.S. Gill, Phase transition mechanisms for the phase-field model under internal heating, Phys. Rev. A 43 (1991) 843-851. [ 141 G. Caginalp and Y. Nishimura, The existence of traveling waves for phase field equations and convergence to sharp interface models in the singular limit, Q. Appl. Math. 49 ( 199 1) 147- 162. [ 151 P.C. Fife, Pattern dynamics for parabolic PDE’s. Forthcoming. [ 161 0. Penrose and P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 ( 1990) 44-62. [ 171 B.D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 ( 1963) 245-26 1. [ 181 M.E. Gurtin, Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal. 104 (1988) 195-221. [ 191 M.E. Gurtin, A mechanical theory for crystallization of a rigid solid in a liquid melt; melting-freezing waves, Arch. Rational Mech. Anal. 110 (1990) 287-312. [20] M.E. Gurtin and A. Struthers, Multiphase thermomechanics with interfacial structure 3. Evolving phase boundaries in the presence of bulk deformation, Arch. Rational Mech. Anal. 112 ( 1990) 97- 160. [2 1 ] E. Fried and M.E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Physica D, submitted. [22] M.E. Gurtin, On the nonequilibrium thermodynamics of capillarity and phase, Q. Appl. Math. 47 (1989) 129-145. [23] L. Trunskinovsky, Kinks versus shocks, Shock Induced Transitions and Phase Structures in General Media (eds. R. Fosdick, E. Dunn and M. Slemrod) Springer-Verlag, New York ( 199 1). [24] C. Herring, Surface tension as a motivation for sintering, The Physics of Powder Metallurgy (ed. W. E. Kingston) McGrawHill, New York (1951). [25] D.W. Hoffman and J.W. Cahn, A vector thermodynamics for anisotropic surfaces-l. Fundamentals and applications to plane surface junctions, Surface Sci. 31 (1972) 368-388. [26] J.W. Cahn and D.W. Hoffman, A vector thermodynamics for anisotropic surfaces-2. Curved and faceted surfaces, Acta Metall. Mater. 22 (1974) 1205-1214. [27] C. Truesdell and R.A. Toupin, The Classical Field Theories, Handbuch der Physik III/ 1, Springer-Verlag, Berlin ( 1960). [28] M.E. Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rational Mech. Anal. 96 ( 1986) 199-241. [29] F. Davi and M.E. Gurtin, On the motion of a phase interface by surface diffusion, Zeit. angew. Math. Phys. 41 (1990) 782-811. [ 301 M.E. Gurtin and P.W. Voorhees, The continuum mechanics of two-phase elastic solids with mass transport, Proc. R. Sot. London A 440 (1993) 323-343. [ 311 M.E. Gurtin, Thermodynamics and the supercritical Stefan equations with nucleations, Q. Appl. Math.. Forthcoming. [32] J. Rubinstein, P. Stemberg and J.B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989) 116-133. [33] L.C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature. Forthcoming. [34] M.E. Gurtin, H.M. Soner and P.E. Souganidis. Forthcoming. [35] S. Angenent and M.E. Gurtin, Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 ( 1989) 323-39 1. [36] M.E. Gurtin, H.M. Soner and P.E. Souganidis, Anisotropic planar motion of an interface relaxed by the formation of infinitesimal wrinkles. Forthcoming. [37] G. Caginalp and X. Chen, Phase field equations in the Singular limit of sharp interface problems, On the Evolution of Phase Boundaries (eds. M.E. Gurtin and G.B. McFadden) Springer-Verlag, New York (1991). [38] J.S. Langer, Models of pattern formation in first-order phase transitions, Directions in Condensed Matter Physics (eds. G. Grinstein and G. Mazenko) World Scientific, Singapore ( 1986). [ 391 W.W. Mullins and R.F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, J. Appl. Phys. 34 (1963) 323-329.
E. Fried, M.E. Gurtin / Thermally induced phase transitions [ 401 [41] [42] [43]
343
J.W. Cahn, On spinodal decomposition, Acta Metall. Mater. 9 ( 1961) 795-801. J.W. Cahn, On spinodal decomposition in cubic crystals, Acta Metall. Mater. 10 (1962) 179-183. R.L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. R. Sot. Lond. A 422 (1989) 261-287. P.W. Bates and P.C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D 43 (1990) 335-348. [44] PC. Fife, Models for phase separation and their mathematics, to appear in: Nonlinear Partial Differential Equations and Applications (eds. M. Mimura and T. Nishimura) Kinokuniya, Tokyo.
Continuum theory of thermally induced phase transitions based on an order parameter Eliot Fried a and Morton E. Gurtin b a Department of Engineering Science and Mechanics,
Pennsylvania State University, University Park, PA 16802-1401, USA b Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213-3890. USA
Received 22 September 1993 Revised manuscript received 16 March 1993 Accepted 16 April 1993 Communicated by M. Mimura
Using balance laws for accretive force and energy in conjunction with constitutive equations restricted so as to be compatible with the second law, we develop a theory for the study of solid-liquid and solid-solid phase transitions where accretion and heat conduction dominate mass diffusion and deformation. Our theory furnishes generalizations of the Ginzburg-Landau equation and the phase-field equations, generalizations that allow for anisotropically induced preferred growth and nonlinear transition kinetics.
1. Introduction In [ 1 ] Chan proposes a model for the kinetics of isothermal, diffusionless first-order phase transitions wherein the state, in the sense of phase, is specified by a scalar order parameter-denoted here equatione2 by q. The basis for this model is a relaxation law #I in the form of a Ginzburg-Landau (1.1)
Pu, = AAy, - w,‘J&,
with /I and 2 positive material constants and ~0 a double-well potential, an equation motivated by assuming that a free energy functional of o, relaxes toward an extremum under an influence proportional to the rate U,at which the order parameter changes. Recognizing that the isothermal nature of this model limits its range of its applicability, Chan suggests that in a more general approach thermal effects be accounted for by enforcing energy balance. To treat capillary and kinetic effects not covered by the classical Stefan model for solidification, Langer [ 41, Fix [ 5 1, Caginalp [ 6 1, and Collins and Levine [ 7 ] introduce a model involving an order
Q(c) = J, W((, V[) dv of a list c = (z t, z 2, . , z n) of time-dependent fields is a set with /I 1constant, determined by writing the variation SQ([), for of n relations of the form piii = -wi({,V~,VZy), compactly supported SC, in the form SQ([) = c:=, J,w,({,V[,V T)Szi dv. *2 Attributed by Chan [ 1] to Landau and Khalatnikov [2]. Cf. White and Geballe [3] for further discussion of such equations. *’ A relaxation law for a functional
0167-2789/93/$06.00
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Science Publishers
B.V. All rights reserved
E. Fried, M.E. Gurtin / Thermally induced phase transitions
parameter, called the phase-field. The resulting equations-for phase-field q-consist of an equation
the scaled temperature
327
deviation
czi - Qi = kAu, expressing balance of energy, supplemented
u and
(1.2) by a Ginzburg-Landau
equation
puj=nA~++~(l-u,)(2~-l)-eu,
(1.3)
derived from a relaxation law for a free energy functional at fixed temperature *3 . (Here c, k, I, /3,I, and v are material constants with all but !Zstrictly positive.) By investigating a particular asymptotic limit of the governing equations, the authors of [4-71 derive the Gibbs-Thompson relation used in nonclassical Stefan problems, which establishes a correspondence between the phase-field approach to solidification and other more familiar models. This result is broadly extended by Caginalp [9], who determines the connection between various asymptotic limits of ( 1.2)) ( 1.3) and a hierarchy of Stefan and Hele-Shaw problems#4. To Penrose and Fife [ 161, the mixed use of relaxation laws and balance laws that leads to the phasefield model is physically unappealing; in accord with an approach suggested by Chan [ 1 ] and taken by Hohenberg and Halperin [ 8 1, they consider instead a framework based completely on relaxation laws with entropy as potential, which they use to derive a general class of phase-field models. Our goal here is a fully dynamical continuum theory for thermally induced phase transitions in which the notion of phase is characterized by an order parameter, and for which interfaces between phases are identified with thin transition zones across which the order parameter exhibits large spatial gradients. This theory is to be sufficiently general to allow for the modeling of both solid-liquid and solid-solid transitions where the effects of accretion and heat conduction dominate those associated with mass diffusion and deformation. We share the preference of Penrose and Fife [ 16 ] for a theory based on a consistent application of physical principles. However, we feel that the relaxational approach is overly restrictive, since it limits the manner in which rate terms can enter the basic equations. In addition, the relaxational approach requires an a priori specification of constitutive equations, which are then used in the derivation of local balance laws. We believe that, while variational derivations often point the way toward a correct statement of the underlying balance laws, such derivations obscure the fundamental nature of balance laws as basic axioms in any general dynamical framework that includes dissipation. We therefore base our study on an approach, now standard in continuum mechanics, in which general balance laws, supposedly common to large classes of materials, are distinguished from constitutive equations, which differentiate between particular materials. In this approach the second law is used, in the sense of Coleman and No11 [ 171, to restrict constitutive equations. What distinguishes our theory from other macroscopic theories in which phase interfaces are treated as thin transition zones is how we write the basic laws. Here we are motivated by recent theories”5 that carefully specify and use the manner in which inter-facial stresses expend power. We introduce a
t3 The system ( 1.21, ( 1.3) is a special case of model C of Hohenberg and Halperin [ 81. #’ See also Caginalp and Fife [ 10,111, Fife and Gill [ 12,13 1, Caginalp and Nishimura [ 14 1, Fife [ 15 1. *5 Cf. [ 18-201, in which phase interfaces are modeled as surfaces of discontinuity.
E. Fried, M.E. Gut-tin/ Thermally induced phase transitions
328
vector measure of stress, the accretive stressg6 <, and take as its conjugate kinetic variable the rate 4 at which the order parameter ~1is changing: given any part#’ P of the body,
s NJ
&<.nda
(1.4)
represents the associated expenditure of power #8 . Since the order parameter should be nearly constant away from transition zones, (1.4) should be significant only within such zones. Basic to our theory is a balance law /<.nda
+ Jndu
8P
= 0,
(1.5)
P
in which rc represents internal forces. #9 Since it is the accretion of material of one phase at the expense of another that characterizes phase transitions, we use the adjective “accretive” to describe this system of forces as well as its associated balance law. We believe that ( 1.4) and ( 1.5 ) correctly describe the physics underlying accretion #lo . Because of the complexity of the general thermodynamical theory, we find it advantageous to proceed in stages. In Section 2 we consider accretion alone, ignoring thermal effects. We postulate-as an appropriate version of the second law for this simplified setting-a dissipation inequality Jwdz’< P
J$<.nda,
(1.6)
8P
with v/ the free energy. We consider constitutive
equations giving cy, C, and n as functions of q, U;,and
p = vfj7.
(1.7)
When suitably restricted Y = w^(ul,P),
by ( 1.6) these constitutive
r = !gvI,PL
equations
7c = -_Iy?p(vI~P) - B(yl,P,
take the form o;)u;>
(1.8)
where p is a nonnegative (constitutive) function. (Here and in what follows we use subscripts to denote partial derivatives, e.g., I& (q,p) = (a/ap)@(q,p).) In conjunction with the accretive balance (1.5), the constitutive relations ( 1.8) yield the partial differential equation B(v,~,vi)i
= div(ly^,(v,p))
- F~v,P).
(1.9)
In [ 2 1] we give a more basic development of accretive forces in which the accretive stress is tensorial with corresponding kinetic variable the velocity of level surfaces of ~0. 17 The body is a region f?CR3; P (with unit outward normal n) will always denote a part (subregion) of B. Vectors in R3 and tensors (linear transformations of W3 into itself) are distinguished from scalars using boldface italic type; lower case for vectors, upper case for tensors. 1 is the identity tensor. The gradient, Hessian, Laplacian, and time derivative of c are denoted, respectively, by Vi, V2c, A(, and 4. The derivative of a function v of a scalar variable (not time) is denoted by u’. #8 Gurtin [22 ] and Truskinovsky [23] use a power of this form, but do not introduce a general accretive balance such as (1.5). #9 We also allow for an external accretive force y and an external supply of heat r that contribute terms to our equations. For convenience, however, we omit discussion of these in the introduction. #lo Cf. [ 18-20,24-261, where accretive balances are used in theories that model phase interfaces as surfaces of discontinuity. #6
E. Fried, ME. Gurtin / Thermally induced phase transitions
329
By an appropriate narrowing of the class of constitutive functions, (1.9) reduces to the classical Ginzburg-Landau equation ( 1.1). Granted the status of ( 1.1) as a model for the kinetics of first-order isothermal, diffusionless phase transitions, the generalized Ginzburg-Landau equation ( 1.9) encompasses a broad range of effects germaine to such transitions-but not allowed by ( 1.1 )-including nonlinear transition kinetics and anisotropically induced preferred growth. The relations ( 1.5 ) and ( 1.9) reflect a basic difference between our treatment and those involving relaxation laws. The balance law (1.5) involves no time derivatives; rate terms enter the theory via the constitutive equations for the internal force rr. In this regard we do not view (1.9) as a transport equation for q, but instead as a balance law for force in which a term involving U;arises as a consequence of the dissipative character of the internal force; for us this term is of the same nature as the viscous stress in a Newtonian fluid. In Section 3 we discuss the interaction of accretion and heat conduction. The governing laws are the accretive balance ( 1S) in conjunction with the first two laws: balance of energy Jeda
= +nda
P
8P
+ /a;&ndn
(1.10)
8P
and growth of entropy”” (1.11)
Here E is the internal energy, q is the entropy, 19is the (absolute) temperature, and q is the heat flux. We consider constitutive equations relating the free energy cy = E - Oq, 9, q, C, and R, to 8, g = VB,
(1.12)
q, p, and 4. The resulting partial differential equations furnish a broad generalization of the phasefield equations, one that includes the effects of nonlinear transition kinetics, anisotropic preferred growth, and cross-coupling between kinetic and thermal effects. These equations are, however, quite complicated, and for that reason we turn next to the development of a simpler theory. In Section 4 we develop a model appropriate to behavior near a transition temperature 60. To derive such a model one can formally approximate the partial differential equations of the general theory developed in Section 3; unfortunately, the resulting approximation does not typically preserve the underlying thermodynamic structure. Here we take an alternative approach, one that ensures a consistent thermodynamics”* : we formulate versions of the first two laws for situations in which (t9 - &)/tic is small, and then develop an exact model within this setting. Here balance of energy has the form JEdV= P
-JI-ndu, 8P
The Clausius-Duhem inequality (cf. Truesdell and Toupin [27], sections 256-258). #I2 This approach is taken in [28], Section 8.1; [ 29,301. #I1
(1.13)
E. Fried, M.E. Gurtin / Thermally induced phase transitions
330
and we take, as the appropriate
version of the second law, the dissipation inequality*‘3
JpdZ<--
Jug-nda+
P
t?P
(1.14)
[email protected], 8P
where u is the scaled temperature-deviation (8 - t9a)/&, while g = E - 80~ is a Gibbs function. Granted isotropy and assuming a simple form for the resulting constitutive equations, this formulation leads to the partial differential equations c(p)u
i G(q)
= div(k(v)Vu),
/I$ = I1Ay,- F’(q)
- UC’(V) + $‘(q)u2,
(1.15)
where
F(q) + UC(~) -
;c(du2
(1.16)
is a “coarse-grain ” free energy, c (q ) is the specific heat, and k (9) is the thermal conductivity. To characterize phase transitions, with u = 0 the transition temperature (deviation), we assume that F (q ) + UC (p ), as a function of 9, is a double-well potential with one well furnishing a global minimum when u < 0 and the other when u > 0. By selecting F(v) and G (cp) appropriately and choosing c (a, ) and k (q ) to be constant, ( 1.15) reduce to the phase-field equations ( 1.2)) ( 1.3). Other choices for F (v, ) and G (cp) lead to specializations of ( 1.15 ) in which the values of the order parameter that correspond to the two phases are fixed, while the specific heat and thermal conductivity may vary from phase to phase. Discussion is limited to best illustrate both our treatment of order parameters and the role of the accretive force balance. We do not consider: (a) more than one order parameter, (b) order parameters of a vectorial or tensorial nature, (c) inhomogeneous constitutive equations, (d) effects due to deformation, (e) effects due to mass diffusion. Consideration of (a)-(c) is not difficult and involves no new ideas. A theory involving deformation is presented in [ 2 11, while one encompassing mass diffusion will be the subject of a future work.
2. Purely accretive processes. Generalization of the Ginzburg-Landau equation To best describe the manner in which the accretive forces and order parameter enter the theory, as well as the manner in which we use thermodynamics to restrict constitutive equations, we begin with a simple theory that neglects thermal effects. 2.1. Balance law for accretive forces. Second law In the present context the basic physical quantitites,
defined on the body a for all time, are
#I3 This form of the second law, without accretive terms, was introduced
in [ 311.
E. Fried, M.E. Gurtin / Thermally induced phase transitions Q
order parameter,
v/
free energy,
t:
accretive stress,
n
internal accretive force,
Y
external accretive force#14
331
(1.16)
Here there is but one balance law, balance of accretive forces: /~.nda+/nd?i+/~do=O P
aP
(2.1) P
for all parts P and all time, where n is the unit outward normal to LIP. When thermal effects are suppressed, the second law is the assertion that: the rate of energy increase cannot exceed the expended power.
(2.2)
Our basic assumption is that accretive force is conjugate to the rate 4 at which the order parameter is changing, so that $<. n and 4 y represent rates at which the accretive stress C and external accretive force y perform work. We therefore write the second law in the form of a dissipation inequality F P
< Se<-nda f3P
+
Syjydlr
(2.3)
P
to be satisfied for all time and all P. The force a, being internal, does not enter the inequality (2.3). This version of the second law can be derived as a consequence of appropriate versions of balance of energy and growth of entropy under isothermal conditions (cf. the discussion following (3.4) ). Since P is arbitrary, (2.1) and (2.3) have the equivalent local forms divC + 71+ Y = 0, div(yiC) + $r 2 U;, which combine to form the local dissipation @-<.ji
(2.4)
inequality
+ nu; < 0.
(2.5)
2.2. Constitutive equations. Consequences of the second law As constitutive equations we allow the free energy, the accretive stress, and the internal accretive force to depend on the order parameter q and-to model capillarity and transition kinetics-also on p = Vy, and ,#14: w = w^(G%P,a;),
c = ~(%P,io),
R = ic^($%P,@).
(2.6)
We do not write a constitutive equation for the external accretive force y, but instead allow y to be assigned in any way compatible with the accretive force-balance. *13Cf. Hohenberg and Halperin [ 81, who introduce a source term in the Ginzburg-Landau *14 Here and throughout
~(o(x,t),p(x,t),gi(x,t)).
we consider only homogeneous constitutive
equation that corresponds to y. behavior; thus, e.g., the first of (2.6) signifies yl (x, t) =
332
E. Fried, M.E. Gurtin / Thermally induced phase transitions
Given an order-parameter field (D,the constitutive equations (2.6) can be used to compute a constitutive process consisting of a, and the fields y, <, and n; the balance (2.4) 1 for accretive force can then be used to determine the external accretive force y that must be supplied to support this process. The second law remains to be satisfied in all such constitutive processes, a requirement that we use to restrict the constitutive equations. Specifically, we assume that the local dissipation inequality (2.5 ) holds for all constitutive processes. Writing h = (p, we see that, granted (2.6), (2.5) is equivalent to the inequality (&0(4VJ)
+ ic^(V,P,h))h
+ @&Ah)
-&V&h))+
(2.7)
+ @h($%P,h)~ < 0,
and, since we can always find a field p such that 8, h, A, p, and i have arbitrarily prescribed at some chosen point and time, we must have I+?&, = 0 and I&, = f. Therefore the free energy and accretive traction are independent of 4 and related through
values
(2.8) and the inequality (&&,P)
+ %%P,ipHio
d
holds for all 01,p, 4. (Conversely, in all constitutive processes. ) Because of the tacit smoothness form ~C%P,a;)
= -&o(%P)
(2.9)
0 consistency
with (2.8) and (2.9) ensures compatibility
of the quantity
in parentheses
in (2.9), ii can be represented
b)b,
- P(P,P,
with (2.5) in the
(2.10)
with p ( y,p, qi) > 0 a constitutive modulus, giving the internal accretive force as the sum of equilibrium and nonequilibrium parts, --FV (9,~) and -fi(q,p, a;)@. Using (2.8) and (2.10) in the forcebalance (2.4) 1 results in the partial differential equation PCv,p,@)i
= div(@,(yl,p))
which represents
-
@~P,P)
a broad generalization
2.3. Quasilinear constitutive
(2.11)
+ Y,
of the Ginzburg-Landau
response
In this section we restrict attention to response functions and (2.10), but withi(q,p) and n^(qo,p,uj) affine functions and (2.10) yield w^(V,P)
= we(v)
equation.
+ w(v)
.P + ;P .n(a,)P,
n^(V,P,@,
of the form (2.6) consistent with (2.8) ofp and 6 for each fixed v. Then (2.8)
= -@,(ul,P)
-P(v)&
(2.12)
with v/o a “coarse-grain” free energy, j? ((p ) > 0, and where, without loss in generality, we require that A (q ) be symmetric. Further, by (2.8)) A must be independent of q; hence w^(V,P) ~(q,P,uj)
= wo(u1) + w(v) = -wIga?)
.P +
- w’(v)
{P.AP,
.P - P(y)ih
&AP)
= AP + w(q),
(2.13)
E. Fried, M.E. Gurtin / Thermally induced phase transitions
333
and these with (2.12) result in the equation P(V)u, = n. v20, - w&d
+ Y.
(2.14)
When j3 (o, ) is strictly positive and A is positive definite, this equation is parabolic, but neither of these assumptions is a consequence of the local dissipation inequality. Although zo(q) does not appear in the local balance (2.14), it would generally enter boundary conditions. If the material is isotropic, so that /i = ill, and if we require that p(v) = /3 = constant, then (2.14) becomes, assuming that y = 0,
PO;= AM-
w&d,
(2.15)
which is (1.1). The general constitutive equations that produce (2.11)) unlike their quasilinear (2.13) allow for nontrivial anisotropic dependence of @( q,p) and p (q,p, @) on p as dence of /3(q,p, U;) on U;. Studies of (2.15) with wO(q) a double-well potential show p small in an appropriate sense, p corresponds to a vector normal to transitions zones the free energy @(4’,P)
=
we(v) +
@aP1*, p^=PAPI,
specializations well as depenthat, for ;1 and #15. Similarly,
(2.16)
where I,Y~ (rq) is a double-well potential and A(.$) > 0 is small, appears to formally approximate interfaces-treated as surfaces of discontinuity with unit normal s-with interfacial energy proportional to m. #I6 Hence, anisotropic dependence of @(q,p) on p allows for the modeling of anisotropic growth through (2.11). The lack of restrictions on !? (q,p ) also allows for the modeling of nonconvex interfacial energies of a type common to crystalline materials with nonsmooth Wulff shapes”*’ . Further, an analysis of traveling wave solutions of (2.11) demonstrates that this equation accomodates a complete spectrum of transition kinetics unavailable when /3 is independent of &an assumption that leads to linear transition kinetics. Note also that our development ensures consistency with the dissipation inequality (2.3), and hence yields the Lyapunov relation
Jvdv=-Jmwdu;zdv
(2.17)
t3
for the body B whenever
y = 0 on I3 and @c. n = 0 on 823.
#t5 Cf. Rubinstein, Sternberg, and Keller [ 321, Evans, Soner and Souganidis [ 331. #t6 Cf. Gut-tin, Soner and Souganidis [ 341. *17 The use of expressions such as (2.16) to model nonconvex interfacial energies arose in discussions of Gut-tin with D. French ( 1988). When @(q,p) is nonconvex in p the underlying partial differential equation (2.11) is rendered backward parabolic for some solutions; models in W2 for the evolution of interfacial curves with such energies are discussed by Angenent and Gut-tin [35] and Gurtin, Soner, and Souganidis [36].
E. Fried, M.E. Gurtin / Thermally induced phase transitions
334
3. Accretive processes with thermal effects 3.1. Clausius-Duhem inequality Our study of thermal effects begins with the quantities &
internal energy,
79
absolute temperature,
II
entropy,
4
heat flux,
r
external heat supply,
and q, C, n, and y introduced
/{.nda aP
+ /rrdu
(2.17)
previously;
+ /ydo
P
these fields are related through the accretivefirce balance
= 0
(3.1)
P
and balance of energy
JEda=-Sq.nda+Srdv+JlpF.nda+/~).dzl P
P
aP
f?P
(3.2) P
for all time and all P. Along with contributions to the rate of energy increase due to the heat flux q and the heat supply r, in writing (3.2) we also account for the rates at which the accretive traction < and the external accretive force y perform work. For future use, we introduce the free energy y = E-&j.
(3.3)
The second law is here the assertion that the rate of entropy production be nonnegative, a law we impose through the Clausius-Duhem inequality: (3.4) P
ap
P
for all time and all P. Note that for isothermal processes, (3.2) and (3.4) reduce, using (3.3), to (2.3), which, with the accretive force balance, was our starting point for the study of purely accretive processes. The global laws (3.1), (3.2), and (3.4) have local forms div{ + R + y = 0, which may be combined
-divq
+ r + div(@C) + U;y = E’, -div(
to give an alternative
E-<.$-I-X$=-divq+r as well as a local dissipation
5) + i < rj,
(3.5)
form of the energy equation (3.6)
inequality (3.7)
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
335
3.2. Constitutive equations. Consequences of the second law We consider relations in which the constitutive are augmented by q, p = Vq, and 4: v/ = !ahw,P,d,
variables 19and g = Vr9 of classical heat conduction
rl = fWJ,%P,yiA
c = k%&~,P,d,
4 = 4^mL~~P>d~
a = W,M4P,d.
(3.8)
Writing c = (19,g, q,p, U;) for the list of constitutive (3.8), (3.7) is equivalent to the inequality
variables
and h = @ we see that, granted
(3.9) From this we conclude that @ cannot depend on g or 4, that (3.10)
(3.11) an inequality
yielding representations
is(C) = -@&~o),P)
-k(C)
.g -P(C)+,
t(C)
= -K(C)g
-HO&
(3.12)
in which k, p, K, and b satisfy
PK)h2+
V’g.KK)g
+ g.(k(l)
+ 19-‘b(0) h 2 0.
(3.13)
The first of (3.10) is in accord with the relationship between entropy and free energy that arises in the study of rigid heat conductors, while the second is analogous to (2.8). An immediate consequence of (3.10) is the “Maxwell relation”
&W, P,P) = -tsw,
V,P).
(3.14)
The functions rkJ(f9, P,P)
= 4&t
V,P),
a...(C)
= -k(C)
. g - PKk
(3.15)
represent the equilibrium and nonequilibrium parts of the internal force ??([). The results (3.10) and (3.12)r, with (3.1) and (3.15), yield the “Gibbs relations” w=-rt~++.~--Kes(8,~,P)~,
E=lsil+t.h-Res(~,O),P)V);
and using (3.16)~ we can rewrite the energy equation t9rj = -divq
- n,,,,(C)@ + r.
(3.16)
(3.6) as a local entropy balance (3.17)
Finally, note that the specific heat C(f+,%P)
= --1969L9(~,a),P)
= WL9(fl,V,P)
depends not only on 19but also on v, andp.
(3.18)
Unless specified otherwise we assume that ~(29, ~,p)
> 0.
E. Fried, M.E. Gurtin / Thermally induced phase transitions
336
Using (3.10) and (3.12) in (3.5)1 and (3.17),anddetining~ system c(s)&
= -div@(C)
P(i)i
= div(&,(c))
+ tit&+(s). -k(i)
06
.w-
+ (f%,(s)
= (rY,p,p), weareledtothegeneral
- ~~,,,(i))ai
+ r,
(3.19)
+ Y,
which is to be supplemented by (3.12)~ and (3.15)~. If for 29near 130and allp, @(I.?,p,p) is a doublewell potential in which one well furnishes a global minimum for 19< Ba and the other for t9 > tie, then (3.19) describe a model for thermally induced phase-transitions in which the two phases correspond to values of o, at the minima, and 80 is the transition temperature. By analogy to (2.11)) the equations (3.19 )-under suitable constitutive assumptions-should have the ability to model anisotropic growth and nonlinear transition kinetics. The “cross-coupling” present in (3.12) also allows g to effect the accretive force and U;to effect the flow of heat. Note that, because our derivation ensures consistency with the basic laws (3.2) and (3.4), solutions = y = 0 of the partial differential equations (3.19) compatible-with q . n = &f ,n = OondBandr on B satisfy the classical relations A A s B
sdv = 0,
Jq
(3.20)
dt) 3 0.
D
Finally, if we omit the cross-coupling r = y = 0, then (3.19) reduces to c(s)8
= div(K(i)VB)
B(i)ti
= div(Fp(c))
terms in (3.12) by taking k = b = 0, and if we assume that
+ t%__S(s).
00, + th&,,(s)i~
+ P(C)(o’
(3.21)
- &Cc).
These partial differential equations appear to be difficult; in what follows we will develop systems of equations that should be more tractable. 3.3. Quasilinear constitutive response We now restrict attention to response functions of the form (3.8) consistent with (3.10) and (3.12), but with i(79, q,p), Z(O,g, q,p, i), and @(8,g, q,p, U;) aftine functions of (g,p, y’) for each (1.9,q). Granted this assumption, simple calculations similar to those used in Section 2c show that k, 8, K, and b depend at most on 19and ~1,and (Y(s) = y/(ti,yl)
+ W(ti,u,).P
+
;PNNP,
Tits1
= -&+(~,o,)-w9(~,y,)~P-
$P4WP,
i(s)
= A(B)P
E(C)
= -w^,(8,0,)-20p(19,V)).P--k(9,0)).g--P(8,V1)V),
7(C)
= -K(fJ,y,)g-b(8,y,)yj,
+ w(29,vD),
(3.22)
where A (8) is a constant symmetric tensor, while k (6, v), p (6, q), K (8, CJI), and b (19,CJI)satisfy (3.13). The relations (3.22) can be used in (3.19) to give a set of partial differential equations which we will not write explicitly.
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
For an isotropic (3.19) become
material,
li = 11, w = k = b = 0, K = kl; if, in addition,
c(6,q)a
= div(k(fi,y,)W)
P(%ol)u;
= A& - @&U)
337
I is constant,
then
+ 1%&(19,0))@ + P(~S,U,)U;~+ r, (3.23)
+ Y.
As before, solutions with q .n = &f .n = 0 on aD, and r = y = 0 on B satisfy (3.20). The quasilinear theory, although somewhat artificial, does provide a fairly general model consistent with the basic laws of thermodynamics. 3.4. The second law in terms of a Gibbs function Consider again the general theory of Section 3b. Let 190denote a fixed temperature, arbitrary, but in the next section will be a transition temperature. Further, let u=-
6 - @I I9
which here is
(3.24)
and consider the Gibbs function
(3.25)
g = & - &)rJ = ry + (6 -B&j.
Then (3.2) and (3.3) yield the global dissipation inequality (3.26)
~a-JUq.nda+Jurdu+JmE.nda+S~~do. P
P
8P
8P
P
This inequality furnishes an alternative version of the second law: granted (3.2) and (3.3), the inequalities (3.4) and (3.26) are equivalent. An immediate consequence of (3.26) is that A
I
(3.27)
gdv
B
whenever 19 = &J or q. n = 0 on da, co<. n = 0 on dB, and r = y = 0 on 23. By (3.10), g is given by a constitutive equation of the form g =
i?(8,P,P),
(3.28)
and if the specific heat (3.18) is strictly positive at 19 = 190,then (3.29)
&JBB(~o,(o,P) > 0,
&9(Oo,a),P) = 0,
so that g (19,q,p ) has a strict local minimum at 19 = 80, a property that makes the Gibbs function useful for analyzing behavior near tic. Further, modulo an appropriate scaling to dimensionless quantities, for 6 K 1 and#18 u,
U,
i,
4,
r,
<,
n,
y
of order O(6),
*18By (3.25) and the first of (3.161, g = u&j + t: .j - n&, plausible.
i
of order O(S*),
(3.30)
which, granted the first of (3.10), makes this assumption
(SP) ‘($‘d‘d)‘8‘n)J~
= b
~($‘d‘d)‘%‘n)_u = 1~ ‘(p5‘d‘d,‘B‘n)!J ‘(p3’d‘o)‘S‘n)_3 = 3
‘(Q,‘d’d,‘B’n)~
= 2 = h
ULIOJalJ1ur suopenba a+lnlylsuoD .Iap!suoDaM
'Z'P 'O~nA.b+fnu+~.~-~a+n, ‘3 =
A +
bmp-
SUIJOJlEDOI aI.jl U! (9Z.E) pUE (E'P) a]lJMaJ Ut?3aM ‘(Z-p) pUE ‘(8Z.f) ‘I(s*f) %U~Sfl .iCg,,,,aanSaylSOalOAazjlsctv[dIfi hroaql I?z?u.r~Sal~ugu~ S!yl Utql!M Snyl :/fi ql!M paIJ!lUaprh+lql!M (E'Z) UIlOJ 01 aU~ql.UOD(f'j~)pII&?(9Z.f) (1UElSUOD = I'Z) SUO~l~pUO~ ~eu.~laylos~ lapun leyl aioN '(1‘f )
amvjvq an+?.mv aql pue ‘(9z.f) iCi!lvnbauluo,wdtss!p p?qo[8 ay) d
‘np.l
(E’P)
s
fip3j’
+vpwbr-=
amv~vqil8naua ayl uoL~oayl~no aseq a.Ioja.tayl aM
'(z'f) ui swal
a+lal~~t? ayl doJp 01 )nq ‘pa8uoyDun (9z.f)
uo~lew~xo~ddt!luals~suo~r!leyll~adxaola~qeuos~a~swaas
ahoaI 01 so I~IXUSn .IOJ
l!‘(()f'f+) %U~I.I~UOD yJeuIal aylJoMayu~ *3n -8
(Z-P)
=
d
uo!l~unj Ln?luauIa~dwos ayl aDnpo.w;r asp? aM
Of? Of?- 8
-=n
(I’P)
uoy?!Aap a~nle~aduxalayl MOU n I.@M‘MI?I puoDas aylJ0 uo!s.IaAaieydoldde aylse (9~‘f)
*yJoMawegsyl ayl dOIaAap uayl put! ‘I~ELIIS a.n?(Of-f)
ui %uy.waddt!surlalayl yyy~
uylrm
ayE aM
13axaw
ui ‘1x3 uy ‘puss!
hroayl
OQ/(QI - 8)
ycyrn ur suogenl!s 01 ale!ldoldde ql~O.18Ldollua put! aXn?jEq i(XIaUa30 smr2lale~nw.IoJaM :yDt?oldde a~~l~wal~t!ue ayw
aM alaH
*a.wcwls
~~weuApow~ay1
uoyeurrxolddt! %u!lpIsalayl ‘K~a~eunUojun f(61.f)
yms agap
walsh
%u+apun
ayl auasald
lou /Cllwaua% 11~~
utx auo lapour t!
ayl alew!xolddv 4p1~03
aM
01 '0~ alnleladura!uoy~y,n?~lt?.~eau.~o!~eyaq 01 ale!JdolddE 1apor.ue.~ap!suo~~ou
‘(P)O ale swlal8ururetua.I aql lnq ‘(zg)o s! JaMOd aAga.we ayl :laplo woJynJ0 IOU ale (z-f) uoymba ST (gz’f) LlrlEnbauy uogedrssrp ayl u! urea) yDoa
Maua
aql u! suw
ayl ‘puq
.Iaylo ayl uo :( zq)o
339
E. Fried, M.E. Gurtin / Thermally induced phase transitions
Proceding
as before, we conclude, as a consequence
-@&,WA
E^(%v),P) =
F(WW1
of (4.4)2, that !? is independent
of g and 4, that (4.6)
= @&M%P),
and that E(C) = -~&,$%P)
-k(C)
C(C) = -K(C)g
.g -P(C)&
(4.71
- b(C)$%’
with
BK)h2+ g’K(c)g + g. (k(C) + b(C))h 2 0,
(4.81
[ = (u, g, p,p, 6 ), and h = 6. The basic equations of the theory are the local accretive balance (3.5) 1, the local energy balance (4.4) i, and the thermodynamically restricted constitutive equations (4.6), (4.7). 4.3. Generalized phase-field models The difficulties inherent in the study of solidification have led to an interest in approaches, such as that embodied by the phase-field model, that capture the essential interaction between accretion and heat conduction, but are sufficiently simple to allow analysis of the corresponding initial/boundaryvalue problems. In this spirit, we now consider materials defined by constitutive equations of the form Y = F(q)
+ uG(p)
E = c(v)u
- G(v),
c
- $ c(&u2
+ ; P .AP,
=AP,
a = -F’(q)
- uG’(q)
+ $c’(q)u2 - pa;, (4.9)
4 = K(v)g,
where A is a symmetric tensor, /I 2 0, and K (v, ) is a positive semidefinite tensor. These relations are consistent with the thermodynamic restrictions (4.6)-(4.8) and, by (4.2), yield a Gibbs function of the form g =
F(q) +
$c(p)u2+ ipAp.
(4.10)
Note that we allow the specific heat and conductivity to depend on phase through the dependence of c (a, ) and K ( y ) on v. Note also that we do not allow jI and A to depend on q, and we do not allow 4 to effect the heat flux q, nor g to effect the accretive force a, although such dependencies, properly accounted for, are not ruled out by thermodynamics. Substituting (4.9) into the balances (3.5)i and (4.4)1, and assuming that r = y = 0, we arrive at the system #l 9 c(q)u
i G(q)
= div(K(q)Vu),
ai
= A .V2y, - F’(p)
- uG’(p)
+ $ c’(v))u2,
#I9 For an anisotropic material the interface is probably better modeled by allowing for anisotropic terms of the form (2.16).
dependence
(4.11)
on p through
E. Fried, h4.E. Gurtin / Thermally induced phase transitions
340
which for an isotropic material has the forrng20 C(CD)U4 G(q)
= div(k(a,)Vu),
PO; = LAY,- F’(q)
- UC’(~) + i
(4.12)
c’(q)u2.
The dependence of c (o, ) on o, gives rise to the terms UC’(v)U; and $c’(p ) u2; while we expect these terms to be small in applications near the transition temperature, their inclusion is necessary for consistency with thermodynamics. Since Y is essentially the free energy, to characterize phase transitions, with u the transition temperature, we assume that, as a function of 01,F (q) + uG( v, ) is a double-well potential with local minimizers ~0 (u ) and v, 1(u ) such that 90 (u ) furnishes a global minimum for u > 0 and v, 1(u) does so for u < 0. Roughly speaking, ~0 (u) and o,I (u) then represent values of the order parameter corresponding to the two phases. The term m in (4.12) I represents the change in energy induced by the phase transition, therefore seems reasonable to identify the constant
e = G(vl(O))
(4.13)
- G(qo(O))
with the latent heat offusion. One would generally assume that c (p ), k (a, ), /I, and 1 are strictly positive. If, in addition, c and k to be constant and let F(q)
= ;0(1
and it
-P))~,
G(P)
we take
= eq,
(4.14)
with v > 0, then (4.12) reduce to the phase-field equations ( 1.2)) ( 1.3). One feature of the phasefield model made clear by (4.14) is that the values of the order parameter that correspond to the two phases depend on the deviation u from the transition temperature. Alternatively, if we choose G(v) with G(0) = 0 and G’(q)
(4.15)
= 6[~(1 -q),
but leave F (q ) as specified in (4.14)) then p. (u ) and q I (u ) are fixed at 0 and 1, respectively. these choices and with c and k constant, (4.12) reduce to#21 CU - 6!~(1
- ~)a; = kAu,
Bu,=nAa,+~(l-co)(2v~-v-6Lu).
With
(4.16)
In an initial/boundary-value problem for (4.16) with initial data (DE [0, 1 ] and with either v, E [0, 1 ] on aZ? or Vy, . n = 0 specified on dB, the maximum principle requires that v, E [0,1 ] on B so that the behavior of F (y ) + uG( q) for p outside of [0, 1 ] is irrelevant #22. Returning to the more general equations (4.11)) the underlying thermodynamic structure ensures a global dissipation inequality: for q . n = V;<. n = 0 on dB and Y = y = 0 on L?, (3.26) and (4.4) imply (3.27); thus, by (4.10) and (4.14)#‘23.
920 Cf (6.2) and (6.4) of Penrose and Fife [ 161 for analogous equations with c and k constant. #21 Caginalp and Chen [ 371 work with a modification of the original phase-field model in which ( 1.3) is replaced by an equation that qualitatively resembles the second of (4.16) but ( 1.2) remains unchanged. Such a model is not consistent with the thermodynamic framework utilized here. *22 An observation due to H.M. Soner (private communication). 423 Langer [38] and Penrose and Fife [ 161 give Lyapunov relations of the type (4.17).
E. Fried, M.E. Gurtin / Thermally induced phase transitions
J
V’(y)
+
;c(qdu* + ;p -Ilp)dv
d 0.
341
(4.17)
r3
(In fact, the left side of (4.17) equals the negative of the integral over a of the left side of (4.11) with h replaced by 4.) We also have global conservation of energy as expressed by the first of (3.20) supplemented by (4.9)~. Materials scientists often consider a Stefan-type model #24 for solidification that consists of a heat equation in each phase-with specific heats set equal to zero-in conjunction with interface conditions that neglect kinetics, but allow for latent heat and surface energy. Within the current theory these approximations are consistent with c(q)
(4.18)
= P(V) = 0,
choices that, interestingly, are not ruled out by thermodynamics. The phase-field equations ( 1.3), with c = p = 0, yield, after elimination of u, the Cahn-Hilliard equation *25 a; = KA(vq(l
-v,)(l
-2~)
-nAu,),
(1.2),
(4.19)
where K = k/e*. Thus the Cahn-Hilliard equation, which, in its now classical setting, arises in gradient theories for the diffusion of mass, also arises within this phase-field model of heat conduction under dissipationless kinetics (/3 = 0) and vanishing specific heat. More general versions of (4.19) result upon eliminating u from (4.11), granted that G’(p) is strictly of one sign. Also, because (4.18) is consistent with our thermodynamic framework, the global dissipation inequality (4.17) (with c = 0) remains valid in the present circumstances, yielding an alternative verification of a standard inequality for the Cahn-Hilliard equation.
Acknowledgements This work was inspired by a talk given by W.J. Boettinger during a research workshop on the dynamics of phase interfaces held at the Center for Nonlinear Analysis, Carnegie Mellon University, in March 1992. We are grateful to H.M. Soner for numerous valuable discussions. We also thank R.F. Sekerka for providing us with a copy of Langer’s notes on the phase-field model. This work was supported by the Army Research Office and by the National Science Foundation.
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342
E. Fried, M.E. Gurtin / Thermally induced phase transitions
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