GENERALIZED CAHN-HILLIARD EQUATIONSFOR MULTICOMPONENT ALLOYS

Alain MiranvilleUniversite de Poitiers

Laboratoire de Mathematiques et ApplicationsUMR CNRS 6086

SP2MIBoulevard Marie et Pierre Curie

86962 Chasseneuil Futuroscope Cedex - FranceEmail : miranv@mathlabo.univ-poitiers.fr

Giulio SchimpernaUniversita degli Studi di Pavia

Dipartimento di Matematica “F. Casorati”Via Ferrata 1

27100 Pavia - ItalyEmail: giulio@dimat.unipv.it

Abstract: Our aim in this article is to extend to multicomponent alloys the derivationof generalized Cahn-Hilliard equations due to M. Gurtin. The main ingredient in thisderivation is the introduction of a balance law for internal microforces, i.e., for interactionsat a microscopic level.

Key words: Cahn-Hilliard equation, multicomponent alloys, microforce balance.

Abbreviated title: Multicomponent Cahn-Hilliard equations.

AMS classification scheme numbers: 74A15, 80A22, 35Q72.

1. Introduction.

The Cahn-Hilliard equation is central in materials science, as it describes an impor-tant qualitative feature of two-phase systems, namely, the transport of atoms between unitcells. This phenomenon can be observed, for instance, when a binary alloy is cooled downsufficiently. One then observes a partial nucleation, i.e., the apparition of nucleides in thematerial, or a total nucleation, the so-called spinodal decomposition: the material quicklybecomes inhomogeneous, forming a fine-grained structure where each of the two compo-nents appears more or less alternatively. In a second stage, which is called coarsening,occurs at a slower time scale and is less understood, these microstructures coarsen.

The starting point in the derivation of the Cahn-Hilliard equation consists in intro-ducing a free energy, called the Ginzburg-Landau free energy, of the form

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ψ = ψ(ρ,∇ρ) =α

2|∇ρ|2 + f(ρ), α > 0, (1.1)

where ρ is the order parameter (a density of atoms; more precisely, if the material consistsof two species A and B with densities ρA and ρB such that ρA +ρB = 1, then, for instance,ρ = ρA) and f is a coarse-grain free energy: it is a double-well potential whose wellscharacterize the phases of the material; a thermodynamically consistent potential has thefollowing expression (we will refer to it as a “logarithmic potential”):

f(s) = 2θcs(1− s) + θ(slns+ (1− s)ln(1− s)), 0 < θ < θc. (1.2)

Now, very often, such a potential is approximated by a polynomial, generally of degreefour, of the form

f(s) =14(s2 − β2)

2, β > 0. (1.3)

Then, one has the mass balance

∂ρ

∂t= −divh, (1.4)

where h is the mass flux which is related to the chemical potential µ (more precisely, ifρ = ρA, then µ is a generalized chemical potential defined by µ = µA−µB) by the followingpostulated constitutive equation:

h = −κ∇µ, (1.5)

where κ is the mobility (one usually assumes that it is a strictly positive constant; itcan more generally depend on the order parameter ρ and degenerate). Now, the chemicalpotential is usually defined as the derivative of the free energy with respect to the order pa-rameter. Here, such a definition is incompatible with the presence of ∇ρ in the free energy.Thus, this definition has to be adapted and, instead, µ is defined as a variational/functionalderivative of the free energy with respect to ρ, which gives

µ = −α∆ρ+ f ′(ρ). (1.6)

We finally deduce from these three relations the (classical) Cahn-Hilliard equation

∂ρ

∂t+ ακ∆2ρ− κ∆f ′(ρ) = 0. (1.7)

We refer the interested reader to [C], [CH] and [Gu] for more details.This equation has been much studied and one now has rather complete and satisfactory

results on the well-posedness and the long time behavior of the solutions. We refer thereader to [El], [Mi4] and [NC2] for reviews on the Cahn-Hilliard equation.

Now, noting that this derivation is simple, elegant and physically sound, M. Gurtinmakes several objections (see [Gu]):• It limits the manner in which rate terms enter the equations.

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• There is no clear separation between balance laws and constitutive equations. Such aseparation has been one of the major advances in nonlinear continuum mechanics over thepast years.

• It requires a priori specifications of the constitutive equations; in particular, the consti-tutive equation giving the mass flux in terms of the chemical potential is postulated.

• The chemical potential is given, constitutively, in terms of the order parameter, assumingthat the system is close to equilibrium.

• It is not clear how it can be generalized in the presence of processes such as deformationsor heat transfers.

• It is not clear whether or not there is an underlying balance law which can form a basisfor more complete theories.

In order to (try to) overcome (some of) these drawbacks, M. Gurtin proposes in[Gu] an approach which, compared with other macroscopic theories of order parameters,separates balance laws from constitutive equations and introduces a new balance law forinternal microforces. This microforce balance reads, assuming that there are no externalmicroforces,

divξ + π = 0, (1.8)

where ξ (a vector) corresponds to the microstress and π (a scalar) corresponds to themicroforce (again, if ρ = ρA, then ξ = ξA and π = πA; see Section 3). The introduction ofsuch a balance law is motivated by the following points:

• This microforce balance provides a balance for interactions at a microscopic level, whereasstandard forces are associated with macroscopic length scales.

• At equilibrium, the requirement that the first variation of the total free energy vanishesyields the Euler-Lagrange equation divξ + π = 0, with ξ = ∂∇ρψ and π = −∂ρψ (hereand below, ∂sf denotes the partial derivative of f with respect to the variable s), whichrepresents a statical version of the microforce balance (1.8), with ξ and π being givenconstitutive representations. The microforce balance (1.8) can thus be seen as an attemptto extend to dynamics an essential feature of statical theories.

• It is believed that fundamental physical laws involving energy should account for theworking associated with each operative kinematical process (that associated with the or-der parameter here). It thus seems plausible that there should be microforces whoseworking accompanies changes in the order parameter. E. Fried and M. Gurtin expressthis working through terms of the form ∂ρ

∂t , so that the microforces are scalar rather thanvector quantities (see [FrG1] and [FrG2]).

We now consider a multicomponent alloy composed of N species with densities ρi, i =1, ..., N, N ≥ 2, such that

N∑i=1

ρi = 1. (1.9)

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In order to derive the Cahn-Hilliard equations for such a material, we again consider themass balance

∂ρ

∂t= −divh, (1.10)

where ρ = (ρ1, ..., ρN ) is the order parameter and h = (h1, ..., hN ) is the mass flux whichsatisfies

N∑i=1

hi = 0 (1.11)

and is related to the chemical potentials µ1, ..., µN by the following constitutive equationswhich generalize (1.5):

hi = −N∑

j=1

κij∇µj , i = 1, ..., N, (1.12)

where K = (κij) is a constant positive definite matrix; furthermore, it follows from On-sager’s reciprocity law that K is symmetric (see [O1] and [O2]). Finally, the chemicalpotential µi is defined as the variational derivative of the free energy

ψ = ψ(ρ1, ..., ρN ,∇ρ1, ...,∇ρN )

with respect to ρi, i = 1, ..., N . More precisely, we consider the following generalizedGinzburg-Landau free energy:

ψ =12∇ρ ∗ Γ∇ρ+ f(ρ), (1.13)

where Γ is a symmetric positive definite constant matrix, Γ = (Γij), Γij , i, j = 1, ..., N ,being matrices, f is a coarse-grain free energy which has several local minimizers cor-responding to the phases of the material, and ∇ρ ∗ Γ∇ρ =

∑Ni=1∇ρi · [Γ∇ρ]i (here, [·]i

denotes the ith component, i = 1, ..., N ; see the end of this section for further details onthe notation used throughout the paper). This yields

µi = −[div(Γ∇ρ)]i + ∂ρif, i = 1, ..., N, (1.14)

and we finally obtain the following equations:

∂ρ

∂t= K∆µ, (1.15)

µ = −div(Γ∇ρ) + ∂ρf, (1.16)

where µ = (µ1, ..., µN ), div(f1, ..., fN ) = (divf1, ...,divfN ) and ∂ρf = (∂ρ1f, ..., ∂ρNf). We

can note that, for N = 2 (i.e., for a binary alloy), we recover the classical Cahn-Hilliardequation (see Remark 3.3).

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Models of Cahn-Hilliard equations for multicomponent alloys have been derived, dis-cussed and studied, e.g., in [ChD], [ElG], [ElL], [Ey], [GY], [Ga1], [Ga2], [Ga3], [GaNC],[Gr], [H], [MSW], [Me] and [MoC]. We can note however that the derivation of such modelsessentially follows that of the classical Cahn-Hilliard equation and thus has the drawbacksmentioned above (see nevertheless [AP] for a derivation of nonisothermal models based onan entropy principle). It is therefore interesting to extend the approach of M. Gurtin tomulticomponent alloys.

This article is organized as follows. In Section 2, we review the derivation of thegeneralized Cahn-Hilliard equations for binary alloys. Then, in Section 3, we extend theapproach of M. Gurtin to multicomponent alloys.

Notations and terminology

In the sequel, we will often consider functions defined on the n-dimensional (1 ≤ n ≤ 3)region occupied by the material, with values in RN or Rn×N , N denoting the number ofcomponents (this is the case, e.g., for the order parameter ρ and the mass flux h). In orderto avoid confusion of indices, we will use different symbols for the n-scalar product withrespect to the space variables (denoted by ·) and for the N -scalar product with respectto the components (denoted by :). Scalar products in Rn×N will be indicated by ∗ (e.g.,∇ρ ∗ ∇h). The explicit use of indices such as i, j will always refer to the componentsand never to the space variables. We will not use the Einstein’s convention of repeatedindices (i.e., we will always write the summation symbol). Finally, div will stand for thedivergence with respect to the space variables (the same will hold for the gradient, ∇, andthe Laplacian, ∆) and ·· for the contraction product of matrices in Rn×n.

2. Generalized Cahn-Hilliard equations for binary alloys.

a) Derivation of the equations.

In order to derive the generalized Cahn-Hilliard equations, one starts from the massbalance (1.4) and the microforce balance (1.8). Then, one needs to derive the constitutiveequations relating h, ξ and π to the order parameter ρ (and the chemical potential µ).To do so, one considers the restrictions imposed by the laws of thermodynamics. Moreprecisely, M. Gurtin considers a version of the second law which is appropriate to a purelymechanical theory and, starting from the first and second laws, he ends up with thefollowing dissipation inequality:

d

dt

∫Rψdx ≤ W(R) +M(R),

where R is an arbitrary control volume, W(R) is the rate of working on R of all forcesexterior to R and M(R) is the rate at which energy is added to R by mass transport (see[Gu], Appendix A). The above dissipation inequality states that the rate at which the freeenergy increases cannot exceed the sum of the working and of the energy inflow due tomass transport. One then has

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W(R) =∫

∂R(ξ · ν)∂ρ

∂tdσ,

M(R) = −∫

∂Rµh · νdσ,

where ν is the unit outer normal vector to ∂R. Thus, integrating by parts and notingthat the control volume R is arbitrary, we finally have, owing to the mass and microforcebalance laws (1.4) and (1.8), the (local) dissipation inequality

∂ψ

∂t+ (π − µ)

∂ρ

∂t− ξ · ∇∂ρ

∂t+ h · ∇µ ≤ 0. (2.1)

In the classical Cahn-Hilliard theory, the independent constitutive variables are ρand ∇ρ. Then, µ is given, constitutively, in terms of ρ and ∇ρ, assuming, as alreadymentioned, that the system is close to equilibrium. So, if one wants to consider systemswhich are sufficiently far from equilibrium, it seems reasonable to add µ and ∇µ to thelist of independent constitutive variables.

We thus set Z = (ρ,∇ρ, µ,∇µ) and assume that h, π, ξ and ψ depend a priori on Z(in particular, ψ is not restricted to the Ginzburg-Landau free energy (1.1) at this stage).Then, the two basic balance laws (for mass and microforces) take the form

∂ρ

∂t= −divh(Z), (2.2)

divξ(Z) + π(Z) = 0, (2.3)

and the dissipation inequality (2.1) can be rewritten as

[∂ρψ(Z) + π(Z)− µ]∂ρ

∂t+ [∂∇ρψ(Z)− ξ(Z)] · ∇∂ρ

∂t+ [∂µψ(Z)]

∂µ

∂t

+ [∂∇µψ(Z)] · ∇∂µ∂t

+ h(Z) · ∇µ ≤ 0,(2.4)

for every Z. Here, it is possible to choose a field Z such that ∂ρ∂t , ∇

∂ρ∂t ,

∂µ∂t and ∇∂µ

∂t takearbitrary values at some chosen point and time. Thus, since ∂ρ

∂t , ∇∂ρ∂t ,

∂µ∂t and ∇∂µ

∂t appearlinearly in (2.4), it follows that, necessarily,

ψ = ψ(ρ,∇ρ), (2.5)

as expected, and that

µ = ∂ρψ + π, (2.6)

ξ = ∂∇ρψ (2.7)

(see [Gu]), and there remains the dissipation inequality

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h(Z) · ∇µ ≤ 0, (2.8)

for every Z, which yields that there exists a matrix A = A(Z), called mobility tensor,which is, in some sense, positive semi-definite and such that (see [Gu], Appendix B, formore details)

h = −A∇µ (2.9)

and the dissipation inequality (2.8) is satisfied.It now follows from the microforce balance (2.3) and the constitutive equations (2.6)-

(2.7) that

µ = ∂ρψ − div(∂∇ρψ), (2.10)

and we recover (1.6), rigorously this time. We also note that the constitutive equation(2.9), which is of the same form as (1.5), is derived rigorously.

Combining finally (2.2), (2.9) and (2.10), we obtain, for the classical Ginzburg-Landaufree energy (1.1), the following generalized Cahn-Hilliard equation:

∂ρ

∂t+ αdiv(A(Z)∇∆ρ)− div(A(Z)∇f ′(ρ)) = 0. (2.11)

Taking A = κI, κ > 0, I being the identity matrix, we then recover the classical Cahn-Hilliard equation (1.7).

We refer the reader to [Mi1] for the mathematical study of equation (2.11).

b) Generalized Cahn-Hilliard equations in deformable continua.

If the material is subject to macroscopic deformations, then the rate of working of allforces exterior toR,W(R), also includes the working of (standard) forces which accompanythe gross motion of the material. More precisely, we have, assuming that there are noexternal volume forces,

W(R) =∫

∂R(ξ · ν)∂ρ

∂tdσ +

∫∂R

(Sν) · ∂u∂tdσ, (2.12)

where S is the (Piola-Kirchhoff) stress tensor and u is the displacement. We thus obtain,considering the force balance

divS = 0 (2.13)

and proceeding as in the previous subsection, the dissipation inequality

∂ψ

∂t− S ··∇∂u

∂t+ (π − µ)

∂ρ

∂t− ξ · ∇∂ρ

∂t+ h · ∇µ ≤ 0. (2.14)

The fundamental balance laws are thus

∂ρ

∂t= −divh (mass balance), (2.15)

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divξ + π = 0 (microforce balance), (2.16)

divS = 0 (force balance), (2.17)

St(I +∇u) = (I +∇u)tS (momentum balance). (2.18)

We then again need to define the independent constitutive variables. For simplicity,we consider situations in which the deformations are infinitesimal and the displacementgradient is small, i.e., we consider linear elastic phases. In that case, the momentumbalance actually reduces to

S = tS, (2.19)

i.e., S is symmetric.We consider here constitutive equations of the form ψ = ψ(Z), S = S(Z), h = h(Z),

ξ = ξ(Z) and π = π(Z), where Z = (∇u, ρ,∇ρ, µ,∇µ), i.e., we now add ∇u to the listof independent constitutive variables. Requiring then that the constitutive functions beinvariant under infinitesimal rotations, we deduce that they can depend on∇u only throughthe infinitesimal strain E = 1

2 (∇u + t∇u), i.e., we actually take Z = (E, ρ,∇ρ, µ,∇µ).Furthermore, the dissipation inequality (2.14) now takes the form (we omit the dependenceon Z)

∂ψ

∂t− S ·· ∂E

∂t+ (π − µ)

∂ρ

∂t− ξ · ∇∂ρ

∂t+ h · ∇µ ≤ 0, (2.20)

which yields

(∂Eψ − S) ·· ∂E∂t

+ (∂ρψ + π − µ)∂ρ

∂t+ (∂∇ρψ − ξ) · ∇∂ρ

∂t+ ∂µψ

∂µ

∂t

+ ∂∇µψ · ∇∂µ

∂t+ h · ∇µ ≤ 0,

(2.21)

for every Z, from which it follows that ψ only depends on E, ρ and ∇ρ and that

S = ∂Eψ, (2.22)

µ = ∂ρψ + π, (2.23)

ξ = ∂∇ρψ, (2.24)

and there remains the dissipation inequality

h · ∇µ ≤ 0, (2.25)

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for every Z. As in the previous subsection, it follows from (2.25) that there exists a matrixA = A(Z), in some sense positive semi-definite, such that

h = −A∇µ (2.26)

and (2.25) is satisfied.A choice of free energy, which is consistent with the assumption of infinitesimal de-

formations, reads

ψ = W (E, ρ) +α

2|∇ρ|2 + f(ρ), (2.27)

W (E, ρ) =12(E − E(ρ)) ··C(ρ)(E − E(ρ)), (2.28)

where C is the elasticity tensor (it is a symmetric and positive definite, when restricted tosymmetric matrices, linear transformation which maps symmetric matrices onto symmetricmatrices) and E(ρ), a symmetric matrix, is the stress-free strain at density ρ; we willassume, for simplicity, that E(ρ) is linear in ρ,

E(ρ) = e(ρ− ρ)I, e > 0, (2.29)

where ρ is a constant.Assuming, for simplicity, that A and C are constant, we finally obtain the following

generalized Cahn-Hilliard system:

∂ρ

∂t+ αdiv(A∇∆ρ)− div(A∇f ′(ρ)) +

e

2div(A∇Tr(C(∇u+ t∇u)))

− e2Tr(CI)div(A∇ρ) = 0,(2.30)

div(C(∇u+ t∇u))− 2ediv(ρ(CI)) = 0. (2.31)

We can note that the above system can be decoupled. Indeed, working, for simplicity,with displacements with null average (we note that the displacement is known up to a rigiddisplacement), then we can solve the second (elliptic) equation to obtain

u = G(ρ), (2.32)

where G is linear, which yields, injecting this value into the first equation, the followinggeneralized Cahn-Hilliard equation:

∂ρ

∂t+ αdiv(A∇∆ρ)− div(A∇f ′(ρ)) +

e

2div(A∇Tr(C(∇G(ρ) + t∇G(ρ))))

− e2Tr(CI)div(A∇ρ) = 0.(2.33)

We can note that this equation bears some resemblance to (2.11) and we again referto [Mi1] for the mathematical study of (2.33) (see also [BaP], [BoCDGSS], [BoDS] and

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[P]; in [P], the author recovers the equations derived by M. Gurtin by using a differentapproach, due to I. Muller and I.S. Liu).

c) Inclusion of the kinetics.

For simplicity, we do not take into account the macroscopic deformations in thissubsection. Then, it is also reasonable to add ∂ρ

∂t (i.e., the kinetics; we also recall that theworking of the internal microforces is expressed through terms of this form) to the list ofindependent constitutive variables.

We thus set Z = (ρ,∇ρ, ∂ρ∂t , µ,∇µ) and we assume that ψ, π, ξ and h depend a priori

on Z. It then follows from the dissipation inequality (2.1) that

(∂ρψ+π−µ)∂ρ

∂t+(∂∇ρψ−ξ)·∇

∂ρ

∂t+∂ ∂ρ

∂tψ∂2ρ

∂t2+∂µψ

∂µ

∂t+∂∇µψ ·∇

∂µ

∂t+h·∇µ ≤ 0, (2.34)

for every Z. Therefore, we deduce from (2.34) that ψ does not depend on ∂ρ∂t , µ and ∇µ

and that

ξ = ∂∇ρψ. (2.35)

Then, there remains the dissipation inequality

(∂ρψ + π − µ)∂ρ

∂t+ h · ∇µ ≤ 0, (2.36)

for every Z, from which it follows that there exist constitutive moduli β = β(Z) (a scalar),a = a(Z), b = b(Z) (two vectors) and A = A(Z) (a matrix; it is again, in some sense,positive semi-definite) such that

∂ρψ + π − µ = −β ∂ρ∂t

− b · ∇µ, (2.37)

h = −a∂ρ∂t

−A∇µ (2.38)

and the dissipation inequality (2.36) is satisfied (see [Gu], Section 3.4 and Appendix B, formore details). Finally, we deduce from the two above constitutive equations and the massand microforce balances (2.2) and (2.3) the following generalized Cahn-Hilliard system:

∂ρ

∂t− div(a

∂ρ

∂t) = div(A∇µ), (2.39)

µ− b · ∇µ = β∂ρ

∂t+ ∂ρψ − div(∂∇ρψ). (2.40)

We now assume that the constitutive moduli are constant. We then have, for the clas-sical Ginzburg-Landau free energy (1.1), the following generalized Cahn-Hilliard system:

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∂ρ

∂t− a · ∇∂ρ

∂t= div(A∇µ), (2.41)

µ− b · ∇µ = β∂ρ

∂t− α∆ρ+ f ′(ρ), (2.42)

where, due to the dissipation inequality (2.36), we have the “positive semi-definiteness”condition

βx2 + (a+ b) · yx+ (Ay) · y ≥ 0, ∀x ∈ R, ∀y ∈ Rn. (2.43)

Furthermore, taking the divA∇ of (2.42), in which we inject the value of div(A∇µ) givenby (2.41), we obtain the following generalized Cahn-Hilliard equation:

∂ρ

∂t− d · ∇∂ρ

∂t− div(A∇∂ρ

∂t) + αdiv(A∇∆ρ)− div(A∇f ′(ρ)) = 0, (2.44)

where d = a+ b and A = βA− 12 (atb+ bta). We note that, assuming that A is symmetric,

it follows from (2.43) that A is positive semi-definite.We refer the reader to [Mi1], [Mi2], [Mi3] and [MiR] for the mathematical study of

the models derived in this subsection.

Remark 2.1: We recover, for β > 0, a = b = 0 and A = κI, κ > 0, the viscous Cahn-Hilliard equation introduced in [NC1] and, for β = 0, the classical Cahn-Hilliard equation(1.7).

Remark 2.2: We can further generalize these models by adding∇∂ρ∂t to the list of indepen-

dent constitutive variables, thus obtaining a theory in which the microstress is dissipative(see [EM]). Furthermore, we can also generalize these models by taking into account thedeformations of the material as in the previous subsection or/and thermal effects (see[MiS]).

3. Generalized Cahn-Hilliard equations for multicomponent alloys.

a) Derivation of the equations.

We assume that the material is composed of N components with densities ρi, i =1, ..., N, N ≥ 2, and we call µi the chemical potential associated with the ith component,i = 1, ..., N . We set ρ = (ρ1, ..., ρN ) and µ = (µ1, ..., µN ).

Following [FrG1] (although it is now more difficult to give a precise physical meaningof these quantities), we introduce the microstress ξ (a matrix), ξ = (ξ1, ..., ξN ), whereξi, i = 1, ..., N , are vectors, and the microforce π (a vector), π = (π1, ..., πN ), whereπi, i = 1, ..., N , are scalars. These quantities are related by the (local) microforcebalance (we again neglect the external microforces)

divξ + π = 0, (3.1)

where divξ = (divξ1, ...,divξN ), or, equivalently,

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divξi + πi = 0, i = 1, ..., N. (3.2)

We can note that, since the microforces are related with microscopic interactions, it seemsreasonable to introduce a microstress ξi and a microforce πi, i = 1, ..., N , associated witheach component.

We thus have the microforce balance (3.1) and the mass balance

∂ρ

∂t= −divh, (3.3)

h = (h1, ..., hN ), where hi is the mass flux associated with the ith component, i = 1, ..., N ,together with the constraints

N∑i=1

ρi = 1 (3.4)

and

N∑i=1

hi = 0. (3.5)

In order to derive the constitutive equations, we again consider the dissipation in-equality

d

dt

∫Rψdx ≤ W(R) +M(R), (3.6)

where now

W(R) =∫

∂R(ξ · ν) :

∂ρ

∂tdσ ≡

N∑i=1

∫∂R

(ξi · ν)∂ρi

∂tdσ (3.7)

and

M(R) = −∫

∂Rµ : (h · ν)dσ ≡ −

N∑i=1

∫∂Rµi(hi · ν)dσ. (3.8)

This yields, integrating by parts, the local dissipation inequality

∂ψ

∂t+ (π − µ) :

∂ρ

∂t− ξ ∗ ∇∂ρ

∂t+ h ∗ ∇µ ≤ 0, (3.9)

or, equivalently,

∂ψ

∂t+

N∑i=1

(πi − µi)∂ρi

∂t−

N∑i=1

ξi · ∇∂ρi

∂t+

N∑i=1

hi · ∇µi ≤ 0. (3.10)

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We then need to define the independent constitutive variables. At this stage, we donot take into account the constraint (3.4) and we take

Z = (ρ1, ..., ρN ,∇ρ1, ...,∇ρN , µ1, ..., µN ,∇µ1, ...,∇µN )

as set of independent constitutive variables. The dissipation inequality (3.10) can thus berewritten as

N∑i=1

(∂ρiψ + πi − µi)∂ρi

∂t+

N∑i=1

(∂∇ρiψ − ξi) ·∂∇ρi

∂t+

N∑i=1

∂µiψ∂µi

∂t

+N∑

i=1

∂∇µiψ ·∂∇µi

∂t+

N∑i=1

hi · ∇µi ≤ 0,

(3.11)

for every Z. This yields that, necessarily,

ψ = ψ(ρ1, ..., ρN ,∇ρ1, ...,∇ρN ), (3.12)

ξi = ∂∇ρiψ, i = 1, ..., N, (3.13)

µi = ∂ρiψ + πi, i = 1, ..., N, (3.14)

and there remains the dissipation inequality

N∑i=1

hi · ∇µi ≤ 0, (3.15)

for every Z, from which it follows that there exist constitutive moduli Aij = Aij(Z)(matrices), i, j = 1, ..., N , such that

hi = −N∑

j=1

Aij∇µj , i = 1, ..., N, (3.16)

and the dissipation inequality (3.15) is satisfied (the matrix A = (Aij) is, in some sense,positive semi-definite). We further assume that the matrix A is symmetric (i.e., Aij =Aji, i, j = 1, ..., N), which is consistent with Onsager’s reciprocity law (see [O1] and[O2]; see also [Ga1] and [Me]).

Finally, we have, owing to the microforce balances (3.2), the following system ofequations:

∂ρi

∂t= div(

N∑j=1

Aij∇µj), i = 1, ..., N, (3.17)

µi = ∂ρiψ − div(∂∇ρi

ψ), i = 1, ..., N, (3.18)

13

where

N∑i=1

ρi = 1. (3.19)

Remark 3.1: We assume, for simplicity, that the Aij are constant matrices. Since therelations hi = −

∑Nj=1Aij∇µj , i = 1, ..., N, and

∑Ni=1hi = 0 have to be valid for all

possible values of the chemical potentials, we deduce that, necessarily,

N∑j=1

Aij = 0, i = 1, ..., N (3.20)

(we recall that the matrix A is symmetric).

Remark 3.2: It is convenient, in view of the mathematical analysis of the problem(see [Ga1], [Ga2] and [Ga3]), to set w = (w1, ..., wN ) = Pµ, where wi = 1

N

∑Nj=1(µi −

µj), i = 1, ..., N , and P is the orthogonal projection from RN onto {y = (y1, ..., yN ) ∈RN |

∑Ni=1yi = 0} (which is the tangent space to {x = (x1, ..., xN ) ∈ RN |

∑Ni=1xi = 1}).

We can then rewrite (3.17)-(3.18) in the form

∂ρ

∂t= div(A∇w),

w = P (∂ρψ − div(∂∇ρψ))

(here, ∂ρψ = (∂ρ1ψ, ..., ∂ρNψ) and ∂∇ρψ = (∂∇ρ1ψ, ..., ∂∇ρN

ψ)). We can note that, forN = 2, then w = 1

2 (µ1 − µ2, µ2 − µ1), i.e., we recover the generalized chemical potentialsintroduced for binary alloys (up to a multiplication by 1

2 ). If we now assume that Aij =κijI, i, j = 1, ..., N (the κij being constant strictly positive scalars), and that A ispositive semi-definite, we recover the equations given in the introduction (see also [Ga1])for the free energy

ψ =12∇ρ ∗ Γ∇ρ+ f(ρ), ρ = (ρ1, ..., ρN ),

where the matrix Γ = (Γij), Γij , i, j = 1, ..., N , being matrices, is symmetric andpositive definite, i.e., we have the equations

∂ρi

∂t=

N∑j=1

κij∆µj , i = 1, ..., N,

µi = −[div(Γ∇ρ)]i + ∂ρif, i = 1, ..., N,

or, equivalently,

14

∂ρ

∂t= K∆w,

w = P (−div(Γ∇ρ) + ∂ρf),

where K = (κij) (let us stress that the coefficients of K are subject to conditions (3.20); inparticular, K cannot be the identity matrix). We refer the reader to [Ga1], [Ga2] and [Ga3]for the mathematical analysis of such systems (actually, in these references, the authoralso considers elastic effects; see Subsection 3. b) below); in particular, well-posednessresults are obtained. One difficulty (which already appears for the classical Cahn-Hilliardequation) is to prove that the order parameters remain in the physically relevant interval,i.e., that ρi ∈ [0, 1], i = 1, ..., N . This can be proven, e.g., for logarithmic coarse-grainfree energies of the form

f(ρ) = θN∑

i=1

(ρilnρi + (1− ρi)ln(1− ρi)), θ > 0

(see [Gu1], [Gu2] and [Gu3]), but cannot be proven in general. Another possibility consistsin adding a penalization term in the free energy (see, e.g., [BoCDGSS] and [BoDS]). Now,for more general Aijs, it is reasonable to expect results similar to those obtained in [Ga1],[Ga2] and [Ga3]; in particular, the same techniques should apply, with minor modifications.

Remark 3.3: We assume that N = 2. Then, we have, noting that A11 = −A12,

∂ρ1

∂t= div(A11∇(µ1 − µ2)), (3.21)

µ1 − µ2 = ∂ρ1ψ − ∂ρ2ψ − div(∂∇ρ1ψ − ∂∇ρ2ψ). (3.22)

Recalling now that ρ2 = 1− ρ1, we set

ψ(ρ1, ρ2,∇ρ1,∇ρ2) = ψ(ρ1,∇ρ1). (3.23)

It is then not difficult to show that

∂ρ1 ψ = ∂ρ1ψ − ∂ρ2ψ, (3.24)

∂∇ρ1 ψ = ∂∇ρ1ψ − ∂∇ρ2ψ, (3.25)

and we finally have the following equations:

∂ρ1

∂t= div(A11∇(µ1 − µ2)), (3.26)

µ1 − µ2 = ∂ρ1 ψ − div(∂∇ρ1 ψ). (3.27)

15

We thus recover the (generalized) Cahn-Hilliard equations obtained for binary alloys, fora constant mobility A11.

b) Equations in deformable continua.

As in Subsection 2. b), we introduce the displacement u and the stress tensor S. Wethen have, assuming again that there are no external volume forces,

W(R) =∫

∂R(ξ · ν) :

∂ρ

∂tdσ +

∫∂R

(Sν) · ∂u∂tdσ, (3.28)

which, coupled with (3.8) and owing to the force balance (2.13), yields the following dis-sipation inequality:

∂ψ

∂t− S ··∇∂u

∂t+ (π − µ) :

∂ρ

∂t− ξ ∗ ∇∂ρ

∂t+ h ∗ ∇µ ≤ 0. (3.29)

Here, as in Subsection 2. b) and considering again linear elastic phases, the funda-mental balance laws are

∂ρ

∂t= −divh (mass balance), (3.30)

divξ + π = 0 (microforce balance), (3.31)

divS = 0 (force balance), (3.32)

S = tS (momentum balance). (3.33)

Taking then

Z = (E, ρ1, ..., ρN ,∇ρ1, ...,∇ρN , µ1, ..., µN ,∇µ1, ...,∇µN )

as set of independent constitutive variables, we deduce from (3.29) the following dissipationinequality:

(∂Eψ − S) ·· ∂E∂t

+N∑

i=1

(∂ρiψ + πi − µi)∂ρi

∂t+

N∑i=1

(∂∇ρiψ − ξi) ·∂∇ρi

∂t

+N∑

i=1

∂µiψ∂µi

∂t+

N∑i=1

∂∇µiψ ·∂∇µi

∂t+

N∑i=1

hi · ∇µi ≤ 0,

(3.34)

for every Z, which yields that, necessarily,

ψ = ψ(E, ρ1, ..., ρN ,∇ρ1, ...,∇ρN ), (3.35)

16

S = ∂Eψ, (3.36)

ξi = ∂∇ρiψ, i = 1, ..., N, (3.37)

µi = ∂ρiψ + πi, i = 1, ..., N, (3.38)

and there remains the dissipation inequality

N∑i=1

hi · ∇µi ≤ 0, (3.39)

for every Z. It thus again follows from (3.39) that there exist matrices Aij , i, j = 1, ..., N ,such that A = (Aij) is symmetric and, in some sense, positive semi-definite,

hi = −N∑

j=1

Aij∇µj , i = 1, ..., N, (3.40)

and the dissipation inequality (3.39) is satisfied.Finally, we have, owing to the microforce balance (3.31),

∂ρi

∂t= div(

N∑j=1

Aij∇µj), i = 1, ..., N, (3.41)

µi = ∂ρiψ − div(∂∇ρi

ψ), i = 1, ..., N, (3.42)

div(∂Eψ) = 0, (3.43)

where

N∑i=1

ρi = 1 (3.44)

and, assuming, for simplicity, that the Aij are constant matrices,

N∑j=1

Aij = 0, i = 1, ..., N. (3.45)

Remark 3.4: A classical free energy is the following generalized Ginzburg-Landau freeenergy:

ψ =12∇ρ ∗ Γ∇ρ+ f(ρ) +W (E, ρ),

17

where

W (E, ρ) =12(E − E(ρ)) ··C(ρ)(E − E(ρ)),

C being again the elasticity tensor (it is also a symmetric and positive definite, whenrestricted to symmetric matrices, linear transformation which maps symmetric matri-ces onto symmetric matrices) and E(ρ) being the symmetric stress-free strain at den-sity ρ. Such models are studied in [Ga1], [Ga2] and [Ga3], for mobilities of the formAij = κijI, i, j = 1, ..., N , the κij being constant (strictly positive) scalars; again, it isreasonable to expect similar results for more general Aijs.

Remark 3.5: Again, when N = 2, we recover the equations derived for binary alloys, fora constant mobility A11.

c) Inclusion of the kinetics.

We now add the kinetics to the list of constitutive variables, i.e., we take

Z = (ρ1, ..., ρN ,∇ρ1, ...,∇ρN ,∂ρ1

∂t, ...,

∂ρN

∂t, µ1, ..., µN ,∇µ1, ...,∇µN )

as set of independent constitutive variables (we neglect, for simplicity, the macroscopicdeformations). Therefore, the dissipation inequality (3.10) now gives

N∑i=1

(∂ρiψ + πi − µi)

∂ρi

∂t+

N∑i=1

(∂∇ρiψ − ξi) ·

∂∇ρi

∂t+

N∑i=1

∂ ∂ρi∂t

ψ∂2ρi

∂t2+

N∑i=1

∂µiψ∂µi

∂t

+N∑

i=1

∂∇µiψ · ∂∇µi

∂t+

N∑i=1

hi · ∇µi ≤ 0,

(3.46)for every Z, which yields that

ψ = ψ(ρ1, ..., ρN ,∇ρ1, ...,∇ρN ), (3.47)

ξi = ∂∇ρiψ, i = 1, ..., N, (3.48)

and there remains the dissipation inequality

N∑i=1

(∂ρiψ + πi − µi)∂ρi

∂t+

N∑i=1

hi · ∇µi ≤ 0, (3.49)

for every Z, from which it follows that there exist constitutive moduli βij = βij(Z)(scalars), aij = aij(Z) (vectors), bij = bij(Z) (vectors) and Aij = Aij(Z) (matrices;we again assume that the matrix A = (Aij) is symmetric), i, j = 1, ..., N , such that

18

hi = −N∑

j=1

aij∂ρj

∂t−

N∑j=1

Aij∇µj , i = 1, ..., N, (3.50)

∂ρiψ + πi − µi = −N∑

j=1

βij∂ρj

∂t−

N∑j=1

bij · ∇µj , i = 1, ..., N, (3.51)

and such that the dissipation inequality (3.49) is satisfied.We now assume, for simplicity, that the constitutive moduli are constant. We thus

obtain the following equations:

∂ρi

∂t−

N∑j=1

aij · ∇∂ρj

∂t= div(

N∑j=1

Aij∇µj), i = 1, ..., N, (3.52)

µi −N∑

j=1

bij · ∇µj =N∑

j=1

βij∂ρj

∂t+ ∂ρiψ − div(∂∇ρiψ), i = 1, ..., N, (3.53)

together with the constraint

N∑i=1

ρi = 1. (3.54)

Furthermore, it follows from (3.5) and (3.50) (which should hold for all possible values ofthe order parameters and the chemical potentials) that

N∑i=1

aij = 0, j = 1, ..., N, (3.55)

N∑j=1

Aij = 0, i = 1, ..., N (3.56)

(we recall that the matrix A is symmetric). It finally follows from the dissipation inequality(3.49) the “positive semi-definiteness” condition

N∑i,j=1

βijxixj +N∑

i,j=1

(aij + bji) · yjxi +N∑

i,j=1

(Ajiyi) · yj ≥ 0,

∀xi ∈ R, ∀yi ∈ Rn, i = 1, ..., N.

(3.57)

Let us stress that the above relation might not be true if the constitutive moduli are notconstant. In that case, the positivity condition coming from (3.49) has a more complicatedformulation.

Remark 3.6: We assume that N = 2. Then, recalling that ρ2 = 1− ρ1 and h1 + h2 = 0,we can rewrite (3.49) as

19

(∂ρ1 ψ − div(∂∇ρ1 ψ)− (µ1 − µ2))∂ρ1

∂t+ h1 · ∇(µ1 − µ2) ≤ 0,

for every fields, where ψ is as in Remark 3.3, which yields, in particular, the existence ofconstitutive moduli b (a vector) and β (a scalar) such that

µ1 − µ2 − b · ∇(µ1 − µ2) = β∂ρ1

∂t+ ∂ρ1 ψ − div(∂∇ρ1 ψ). (3.58)

We can note that these constitutive moduli depend a priori on ∂ρ1∂t , ∇µ1 and ∇µ2; we

assume that they depend at least continuously on these arguments. Now, we deduce from(3.53) that

µ1 − µ2 − (b11 − b21) · ∇µ1 − (b12 − b22) · ∇µ2 = (β11 − β12 − β21 + β22)∂ρ1

∂t

+ ∂ρ1 ψ − div(∂∇ρ1 ψ),(3.59)

which yields, owing to (3.58),

(b11−b21)·∇µ1+(b12−b22)·∇µ2+(β11−β12−β21+β22)∂ρ1

∂t= b·∇(µ1−µ2)+β

∂ρ1

∂t. (3.60)

Since this relation has to be valid for all values of the order parameters and the chemicalpotentials, it finally follows that, necessarily,

b = b11 − b21 = b22 − b12,

β = β11 − β12 − β21 + β22,(3.61)

i.e., these constitutive moduli are constant; we can note that we would not obtain suchrelations if the constitutive moduli in (3.50)-(3.51) were not constant. We thus obtain theequations (we can note that equation (3.62) below follows from (3.52), (3.55) and (3.56))

∂ρ1

∂t− a · ∇∂ρ1

∂t= div(A11∇(µ1 − µ2)), (3.62)

µ1 − µ2 − b · ∇(µ1 − µ2) = β∂ρ1

∂t+ ∂ρ1 ψ − div(∂∇ρ1 ψ), (3.63)

where a = a11−a12, i.e., we again recover the generalized Cahn-Hilliard equations derivedfor binary alloys (in the case of constant constitutive moduli).

Remark 3.7: More generally, noting that ∂ρN

∂t = −∑N−1

i=1∂ρi

∂t and hN = −∑N−1

i=1 hi, wededuce from (3.49) that

N−1∑i=1

(∂ρi ψ − div(∂∇ρi ψ)− (µi − µN ))∂ρi

∂t+

N−1∑i=1

hi · ∇(µi − µN ) ≤ 0, (3.64)

20

for every fields, where ψ(ρ1, ..., ρN−1,∇ρ1, ...,∇ρN−1) = ψ(ρ1, ..., ρN ,∇ρ1, ...,∇ρN ). Wededuce from (3.64) that there exist constitutive moduli (which depend a priori on ∂ρ1

∂t , ...,∂ρN−1

∂t , ∇µ1, ..., ∇µN ; we again assume that they depend at least continuously on thesearguments) bij (vectors) and βij (scalars), i, j = 1, ..., N − 1, such that

µi − µN −N−1∑j=1

bij · ∇(µj − µN ) =N−1∑j=1

βij∂ρj

∂t+ ∂ρi ψ − div(∂∇ρi ψ), i = 1, ..., N. (3.65)

Furthermore, it follows from (3.53) that

µi − µN −N∑

j=1

(bij − bNj) · ∇µj =N−1∑j=1

(βij − βNj − βiN + βNN )∂ρj

∂t

+ ∂ρi ψ − div(∂∇ρi ψ), i = 1, ..., N.

(3.66)

Comparing (3.65) and (3.66), we deduce that

N∑j=1

(bij − bNj) · ∇µj +N−1∑j=1

(βij − βNj − βiN + βNN )∂ρj

∂t=

N−1∑j=1

bij · ∇(µj − µN )

+N−1∑j=1

βij∂ρj

∂t, i = 1, ..., N,

(3.67)

and, since this relation has to be valid for all values of the order parameters and thechemical potentials, we obtain (again, for constant constitutive moduli in (3.50)-(3.51))

bij − bNj = bij , i, j = 1, ..., N − 1, (3.68)

biN − bNN = −N−1∑j=1

bij , i = 1, ..., N. (3.69)

Summing (3.68) over j, we have, owing to (3.69),

N−1∑j=1

bij −N−1∑j=1

bNj = bNN − biN , i = 1, ..., N,

which yields

N∑j=1

bij =N∑

j=1

bNj , i = 1, ..., N, (3.70)

21

i.e., the sum∑N

j=1bij is independent of i. We call b this common value. Setting b = (bij),we then have

b ∗ ∇w =1N

b ∗ ∇(Nµ−

∑N

i=1µi

.

.

.∑Ni=1µi

)

= b ∗ ∇µ− 1N

b · ∇

∑Ni=1µi

.

.

.b · ∇

∑Ni=1µi

,

(3.71)

where w is as in the previous subsection. Noting now that P

b · ∇

∑Ni=1µi

.

.

.b · ∇

∑Ni=1µi

= 0, we can

rewrite (in view of the mathematical analysis of the problem) (3.52)-(3.53) in the form

∂ρ

∂t− a ∗ ∇∂ρ

∂t= div(A∇w), (3.72)

w − b ∗ ∇w = P (B∂ρ∂t

+ ∂ρψ − div(∂∇ρψ)), (3.73)

where a = (aij) and B = (βij). Again, by adapting the techniques used in [Ga1], [Ga2]and [Ga3], one can study the well-posedness of (3.72)-(3.73).

Remark 3.8: Again, we can further generalize these models by including the gradient ofthe kinetics (i.e., ∇∂ρ

∂t = (∇∂ρ1∂t , ...,∇

∂ρN

∂t )) to the list of independent constitutive variables.We can also consider deformable continua or/and thermal effects (see also [AP] for adifferent approach for thermal effects based on an entropy principle). For instance, if oneincludes both the macroscopic deformations and the kinetics, then, combining Subsections3. b) and 3. c), one ends up with the system given by the coupling of (3.52)-(3.53) with(3.43) and the constraint (3.44).

References.

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