GENERALIZED CAHN-HILLIARD EQUATIONS FOR MULTICOMPONENT ALLOYS Alain Miranville Universit´e de Poitiers Laboratoire de Math´ematiques et Applications UMR CNRS 6086 SP2MI Boulevard Marie et Pierre Curie 86962 Chasseneuil Futuroscope Cedex - France Email :
[email protected] Giulio Schimperna Universit` a degli Studi di Pavia Dipartimento di Matematica “F. Casorati” Via Ferrata 1 27100 Pavia - Italy Email:
[email protected]
Abstract: Our aim in this article is to extend to multicomponent alloys the derivation of generalized Cahn-Hilliard equations due to M. Gurtin. The main ingredient in this derivation is the introduction of a balance law for internal microforces, i.e., for interactions at 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 important qualitative feature of two-phase systems, namely, the transport of atoms between unit cells. This phenomenon can be observed, for instance, when a binary alloy is cooled down sufficiently. One then observes a partial nucleation, i.e., the apparition of nucleides in the material, or a total nucleation, the so-called spinodal decomposition: the material quickly becomes inhomogeneous, forming a fine-grained structure where each of the two components 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 introducing a free energy, called the Ginzburg-Landau free energy, of the form 1
ψ = ψ(ρ, ∇ρ) =
α 2 |∇ρ| + f (ρ), α > 0, 2
(1.1)
where ρ is the order parameter (a density of atoms; more precisely, if the material consists of 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 wells characterize the phases of the material; a thermodynamically consistent potential has the following expression (we will refer to it as a “logarithmic potential”): f (s) = 2θc s(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 degree four, of the form f (s) =
1 2 2 (s − β 2 ) , β > 0. 4
(1.3)
∂ρ = −divh, ∂t
(1.4)
Then, one has the mass balance
where h is the mass flux which is related to the chemical potential µ (more precisely, if ρ = ρA , then µ is a generalized chemical potential defined by µ = µA −µB ) by the following postulated constitutive equation: h = −κ∇µ,
(1.5)
where κ is the mobility (one usually assumes that it is a strictly positive constant; it can more generally depend on the order parameter ρ and degenerate). Now, the chemical potential is usually defined as the derivative of the free energy with respect to the order parameter. 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/functional derivative of the free energy with respect to ρ, which gives µ = −α∆ρ + f 0 (ρ).
(1.6)
We finally deduce from these three relations the (classical) Cahn-Hilliard equation ∂ρ + ακ∆2 ρ − κ∆f 0 (ρ) = 0. ∂t
(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 the reader to [El], [Mi4] and [NC2] for reviews on the Cahn-Hilliard equation. Now, noting that this derivation is simple, elegant and physically sound, M. Gurtin makes several objections (see [Gu]): • It limits the manner in which rate terms enter the equations. 2
• There is no clear separation between balance laws and constitutive equations. Such a separation has been one of the major advances in nonlinear continuum mechanics over the past years. • It requires a priori specifications of the constitutive equations; in particular, the constitutive 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, assuming that the system is close to equilibrium. • It is not clear how it can be generalized in the presence of processes such as deformations or heat transfers. • It is not clear whether or not there is an underlying balance law which can form a basis for 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 for internal microforces. This microforce balance reads, assuming that there are no external microforces, divξ + π = 0,
(1.8)
where ξ (a vector) corresponds to the microstress and π (a scalar) corresponds to the microforce (again, if ρ = ρA , then ξ = ξA and π = πA ; see Section 3). The introduction of such a balance law is motivated by the following points: • This microforce balance provides a balance for interactions at a microscopic level, whereas standard forces are associated with macroscopic length scales. • At equilibrium, the requirement that the first variation of the total free energy vanishes yields the Euler-Lagrange equation divξ + π = 0, with ξ = ∂∇ρ ψ and π = −∂ρ ψ (here and below, ∂s f denotes the partial derivative of f with respect to the variable s), which represents a statical version of the microforce balance (1.8), with ξ and π being given constitutive representations. The microforce balance (1.8) can thus be seen as an attempt to extend to dynamics an essential feature of statical theories. • It is believed that fundamental physical laws involving energy should account for the working associated with each operative kinematical process (that associated with the order parameter here). It thus seems plausible that there should be microforces whose working accompanies changes in the order parameter. E. Fried and M. Gurtin express this working through terms of the form ∂ρ ∂t , so that the microforces are scalar rather than vector 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 X
ρi = 1.
i=1
3
(1.9)
In order to derive the Cahn-Hilliard equations for such a material, we again consider the mass balance ∂ρ = −divh, ∂t
(1.10)
where ρ = (ρ1 , ..., ρN ) is the order parameter and h = (h1 , ..., hN ) is the mass flux which satisfies N X
hi = 0
(1.11)
i=1
and is related to the chemical potentials µ1 , ..., µN by the following constitutive equations which generalize (1.5): N X hi = − κij ∇µj , i = 1, ..., N,
(1.12)
j=1
where K = (κij ) is a constant positive definite matrix; furthermore, it follows from Onsager’s reciprocity law that K is symmetric (see [O1] and [O2]). Finally, the chemical potential 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 generalized Ginzburg-Landau free energy: ψ=
1 ∇ρ ∗ Γ∇ρ + f (ρ), 2
(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 PN local minimizers corresponding to the phases of the material, and ∇ρ ∗ Γ∇ρ = i=1 ∇ρi · [Γ∇ρ]i (here, [·]i denotes the ith component, i = 1, ..., N ; see the end of this section for further details on the notation used throughout the paper). This yields µi = −[div(Γ∇ρ)]i + ∂ρi f, i = 1, ..., N,
(1.14)
and we finally obtain the following equations: ∂ρ = K∆µ, ∂t
(1.15)
µ = −div(Γ∇ρ) + ∂ρ f,
(1.16)
where µ = (µ1 , ..., µN ), div(f1 , ..., fN ) = (divf1 , ..., divfN ) and ∂ρ f = (∂ρ1 f, ..., ∂ρN f ). We can note that, for N = 2 (i.e., for a binary alloy), we recover the classical Cahn-Hilliard equation (see Remark 3.3). 4
Models of Cahn-Hilliard equations for multicomponent alloys have been derived, discussed 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 models essentially follows that of the classical Cahn-Hilliard equation and thus has the drawbacks mentioned above (see nevertheless [AP] for a derivation of nonisothermal models based on an entropy principle). It is therefore interesting to extend the approach of M. Gurtin to multicomponent alloys. This article is organized as follows. In Section 2, we review the derivation of the generalized Cahn-Hilliard equations for binary alloys. Then, in Section 3, we extend the approach 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 of components (this is the case, e.g., for the order parameter ρ and the mass flux h). In order to avoid confusion of indices, we will use different symbols for the n-scalar product with respect to the space variables (denoted by ·) and for the N -scalar product with respect to 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 components and never to the space variables. We will not use the Einstein’s convention of repeated indices (i.e., we will always write the summation symbol). Finally, div will stand for the divergence with respect to the space variables (the same will hold for the gradient, ∇, and the 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 mass balance (1.4) and the microforce balance (1.8). Then, one needs to derive the constitutive equations 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. More precisely, M. Gurtin considers a version of the second law which is appropriate to a purely mechanical theory and, starting from the first and second laws, he ends up with the following dissipation inequality: d dt
Z ψdx ≤ W(R) + M(R), R
where R is an arbitrary control volume, W(R) is the rate of working on R of all forces exterior 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 free energy increases cannot exceed the sum of the working and of the energy inflow due to mass transport. One then has 5
Z W(R) =
(ξ · ν) ∂R
Z M(R) = −
∂ρ dσ, ∂t
µh · νdσ,
∂R
where ν is the unit outer normal vector to ∂R. Thus, integrating by parts and noting that the control volume R is arbitrary, we finally have, owing to the mass and microforce balance laws (1.4) and (1.8), the (local) dissipation inequality ∂ρ ∂ρ ∂ψ + (π − µ) −ξ·∇ + h · ∇µ ≤ 0. ∂t ∂t ∂t
(2.1)
In the classical Cahn-Hilliard theory, the independent constitutive variables are ρ and ∇ρ. Then, µ is given, constitutively, in terms of ρ and ∇ρ, assuming, as already mentioned, that the system is close to equilibrium. So, if one wants to consider systems which are sufficiently far from equilibrium, it seems reasonable to add µ and ∇µ to the list 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 = −divh(Z), ∂t
(2.2)
divξ(Z) + π(Z) = 0,
(2.3)
and the dissipation inequality (2.1) can be rewritten as [∂ρ ψ(Z) + π(Z) − µ]
∂ρ ∂µ ∂ρ + [∂∇ρ ψ(Z) − ξ(Z)] · ∇ + [∂µ ψ(Z)] ∂t ∂t ∂t ∂µ + [∂∇µ ψ(Z)] · ∇ + h(Z) · ∇µ ≤ 0, ∂t
(2.4)
∂ρ ∂µ ∂µ for every Z. Here, it is possible to choose a field Z such that ∂ρ ∂t , ∇ ∂t , ∂t and ∇ ∂t take ∂ρ ∂µ ∂µ arbitrary values at some chosen point and time. Thus, since ∂ρ ∂t , ∇ ∂t , ∂t and ∇ ∂t appear linearly in (2.4), it follows that, necessarily,
ψ = ψ(ρ, ∇ρ),
(2.5)
µ = ∂ρ ψ + π,
(2.6)
ξ = ∂∇ρ ψ
(2.7)
as expected, and that
(see [Gu]), and there remains the dissipation inequality 6
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, for more 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 = ∂ρ ψ − div(∂∇ρ ψ),
(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-Landau free energy (1.1), the following generalized Cahn-Hilliard equation: ∂ρ + αdiv(A(Z)∇∆ρ) − div(A(Z)∇f 0 (ρ)) = 0. (2.11) ∂t Taking A = κI, κ > 0, I being the identity matrix, we then recover the classical CahnHilliard 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 all forces exterior to R, W(R), also includes the working of (standard) forces which accompany the gross motion of the material. More precisely, we have, assuming that there are no external volume forces, Z Z ∂ρ ∂u W(R) = (ξ · ν) dσ + (Sν) · dσ, (2.12) ∂t ∂t ∂R ∂R 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 ∂u ∂ρ ∂ρ ∂ψ − S ·· ∇ + (π − µ) −ξ·∇ + h · ∇µ ≤ 0. ∂t ∂t ∂t ∂t The fundamental balance laws are thus ∂ρ = −divh (mass balance), ∂t 7
(2.14)
(2.15)
divξ + π = 0 (microforce balance),
(2.16)
divS = 0 (force balance),
(2.17)
S t (I + ∇u) = (I + ∇u)t S (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 displacement gradient is small, i.e., we consider linear elastic phases. In that case, the momentum balance actually reduces to S = t S,
(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 list of independent constitutive variables. Requiring then that the constitutive functions be invariant under infinitesimal rotations, we deduce that they can depend on ∇u only through the infinitesimal strain E = 12 (∇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 dependence on Z) ∂ψ ∂E ∂ρ ∂ρ − S ·· + (π − µ) −ξ·∇ + h · ∇µ ≤ 0, ∂t ∂t ∂t ∂t
(2.20)
which yields
(∂E ψ − S) ··
∂ρ ∂ρ ∂µ ∂E + (∂ρ ψ + π − µ) + (∂∇ρ ψ − ξ) · ∇ + ∂µ ψ ∂t ∂t ∂t ∂t ∂µ + h · ∇µ ≤ 0, + ∂∇µ ψ · ∇ ∂t
(2.21)
for every Z, from which it follows that ψ only depends on E, ρ and ∇ρ and that S = ∂E ψ,
(2.22)
µ = ∂ρ ψ + π,
(2.23)
ξ = ∂∇ρ ψ,
(2.24)
and there remains the dissipation inequality h · ∇µ ≤ 0, 8
(2.25)
for every Z. As in the previous subsection, it follows from (2.25) that there exists a matrix A = 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 deformations, reads ψ = W (E, ρ) +
α 2 |∇ρ| + f (ρ), 2
(2.27)
1 (E − E(ρ)) ·· C(ρ)(E − E(ρ)), (2.28) 2 where C is the elasticity tensor (it is a symmetric and positive definite, when restricted to symmetric matrices, linear transformation which maps symmetric matrices onto symmetric matrices) and E(ρ), a symmetric matrix, is the stress-free strain at density ρ; we will assume, for simplicity, that E(ρ) is linear in ρ, W (E, 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 e + αdiv(A∇∆ρ) − div(A∇f 0 (ρ)) + div(A∇Tr(C(∇u + t ∇u))) ∂t 2 2 − e Tr(CI)div(A∇ρ) = 0, div(C(∇u + t ∇u)) − 2ediv(ρ(CI)) = 0.
(2.30)
(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 rigid displacement), 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 following generalized Cahn-Hilliard equation: ∂ρ e + αdiv(A∇∆ρ) − div(A∇f 0 (ρ)) + div(A∇Tr(C(∇G(ρ) + t ∇G(ρ)))) ∂t 2 2 − e Tr(CI)div(A∇ρ) = 0.
(2.33)
We can note that this equation bears some resemblance to (2.11) and we again refer to [Mi1] for the mathematical study of (2.33) (see also [BaP], [BoCDGSS], [BoDS] and 9
[P]; in [P], the author recovers the equations derived by M. Gurtin by using a different approach, due to I. M¨ uller and I.S. Liu). c) Inclusion of the kinetics. For simplicity, we do not take into account the macroscopic deformations in this subsection. Then, it is also reasonable to add ∂ρ ∂t (i.e., the kinetics; we also recall that the working of the internal microforces is expressed through terms of this form) to the list of independent 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
(∂ρ ψ +π −µ)
∂ρ ∂µ ∂µ ∂ρ ∂2ρ +(∂∇ρ ψ −ξ)·∇ +∂ ∂ρ ψ 2 +∂µ ψ +∂∇µ ψ ·∇ +h·∇µ ≤ 0, (2.34) ∂t ∂t ∂t ∂t ∂t ∂t
for every Z. Therefore, we deduce from (2.34) that ψ does not depend on and that ξ = ∂∇ρ ψ.
∂ρ ∂t ,
µ and ∇µ
(2.35)
Then, there remains the dissipation inequality (∂ρ ψ + π − µ)
∂ρ + h · ∇µ ≤ 0, ∂t
(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 ∂ρ ψ + π − µ = −β h = −a
∂ρ − b · ∇µ, ∂t
∂ρ − A∇µ ∂t
(2.37)
(2.38)
and the dissipation inequality (2.36) is satisfied (see [Gu], Section 3.4 and Appendix B, for more details). Finally, we deduce from the two above constitutive equations and the mass and microforce balances (2.2) and (2.3) the following generalized Cahn-Hilliard system: ∂ρ ∂ρ − div(a ) = div(A∇µ), ∂t ∂t µ − b · ∇µ = β
∂ρ + ∂ρ ψ − div(∂∇ρ ψ). ∂t
(2.39)
(2.40)
We now assume that the constitutive moduli are constant. We then have, for the classical Ginzburg-Landau free energy (1.1), the following generalized Cahn-Hilliard system: 10
∂ρ ∂ρ −a·∇ = div(A∇µ), ∂t ∂t µ − b · ∇µ = β
∂ρ − α∆ρ + f 0 (ρ), ∂t
(2.41) (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) given by (2.41), we obtain the following generalized Cahn-Hilliard equation: ∂ρ ∂ρ ˜ ∂ρ ) + αdiv(A∇∆ρ) − div(A∇f 0 (ρ)) = 0, −d·∇ − div(A∇ (2.44) ∂t ∂t ∂t where d = a + b and A˜ = βA − 21 (at b + bt a). 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 CahnHilliard 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 independent 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 the deformations 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 meaning of these quantities), we introduce the microstress ξ (a matrix), ξ = (ξ1 , ..., ξN ), where ξi , i = 1, ..., N , are vectors, and the microforce π (a vector), π = (π1 , ..., πN ), where πi , i = 1, ..., N , are scalars. These quantities are related by the (local) microforce balance (we again neglect the external microforces) divξ + π = 0, where divξ = (divξ1 , ..., divξN ), or, equivalently, 11
(3.1)
divξi + πi = 0, i = 1, ..., N.
(3.2)
We can note that, since the microforces are related with microscopic interactions, it seems reasonable to introduce a microstress ξi and a microforce πi , i = 1, ..., N , associated with each component. We thus have the microforce balance (3.1) and the mass balance ∂ρ = −divh, ∂t
(3.3)
h = (h1 , ..., hN ), where hi is the mass flux associated with the ith component, i = 1, ..., N , together with the constraints N X
ρi = 1
(3.4)
hi = 0.
(3.5)
i=1
and N X i=1
In order to derive the constitutive equations, we again consider the dissipation inequality Z d ψdx ≤ W(R) + M(R), (3.6) dt R where now N
Z
X ∂ρ dσ ≡ W(R) = (ξ · ν) : ∂t ∂R i=1
Z (ξi · ν) ∂R
∂ρi dσ ∂t
(3.7)
and Z M(R) = −
N Z X µ : (h · ν)dσ ≡ −
∂R
µi (hi · ν)dσ.
(3.8)
i=1 ∂R
This yields, integrating by parts, the local dissipation inequality ∂ψ ∂ρ ∂ρ + (π − µ) : −ξ∗∇ + h ∗ ∇µ ≤ 0, ∂t ∂t ∂t
(3.9)
or, equivalently, N
N
N
∂ψ X ∂ρi X ∂ρi X + (πi − µi ) − ξi · ∇ + hi · ∇µi ≤ 0. ∂t ∂t ∂t i=1 i=1 i=1 12
(3.10)
We then need to define the independent constitutive variables. At this stage, we do not 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 be rewritten as N X
N
N
∂ρi X ∂∇ρi X ∂µi + + (∂ρi ψ + πi − µi ) (∂∇ρi ψ − ξi ) · ∂µi ψ ∂t ∂t ∂t i=1 i=1 i=1 N X
N
(3.11)
∂∇µi X ∂∇µi ψ · hi · ∇µi ≤ 0, + + ∂t i=1 i=1 for every Z. This yields that, necessarily, ψ = ψ(ρ1 , ..., ρN , ∇ρ1 , ..., ∇ρN ),
(3.12)
ξi = ∂∇ρi ψ, i = 1, ..., N,
(3.13)
µi = ∂ρi ψ + πi , i = 1, ..., N,
(3.14)
and there remains the dissipation inequality N X
hi · ∇µi ≤ 0,
(3.15)
i=1
for every Z, from which it follows that there exist constitutive moduli Aij = Aij (Z) (matrices), i, j = 1, ..., N , such that N X hi = − Aij ∇µj , i = 1, ..., N,
(3.16)
j=1
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 of equations: N X ∂ρi = div( Aij ∇µj ), i = 1, ..., N, ∂t j=1
(3.17)
µi = ∂ρi ψ − div(∂∇ρi ψ), i = 1, ..., N,
(3.18)
13
where N X
ρi = 1.
(3.19)
i=1
Remark 3.1: We assume, for simplicity, that the Aij are constant matrices. Since the PN PN relations hi = − j=1 Aij ∇µj , i = 1, ..., N, and i=1 hi = 0 have to be valid for all possible values of the chemical potentials, we deduce that, necessarily, N X
Aij = 0, i = 1, ..., N
(3.20)
j=1
(we recall that the matrix A is symmetric). Remark 3.2: It is convenient, in view of the mathematical analysis of the PNproblem 1 (see [Ga1], [Ga2] and [Ga3]), to set w = (w1 , ..., wN ) = P µ, where wi = N j=1 (µi − µj ), i = 1, ..., N , and P is the orthogonal projection from RN onto {y = (y1 , ..., yN ) ∈ PN PN RN | i=1 yi = 0} (which is the tangent space to {x = (x1 , ..., xN ) ∈ RN | i=1 xi = 1}). We can then rewrite (3.17)-(3.18) in the form ∂ρ = div(A∇w), ∂t w = P (∂ρ ψ − div(∂∇ρ ψ)) (here, ∂ρ ψ = (∂ρ1 ψ, ..., ∂ρN ψ) and ∂∇ρ ψ = (∂∇ρ1 ψ, ..., ∂∇ρN ψ)). We can note that, for N = 2, then w = 21 (µ1 − µ2 , µ2 − µ1 ), i.e., we recover the generalized chemical potentials introduced for binary alloys (up to a multiplication by 21 ). If we now assume that Aij = κij I, i, j = 1, ..., N (the κij being constant strictly positive scalars), and that A is positive semi-definite, we recover the equations given in the introduction (see also [Ga1]) for the free energy ψ=
1 ∇ρ ∗ Γ∇ρ + f (ρ), ρ = (ρ1 , ..., ρN ), 2
where the matrix Γ = (Γij ), Γij , i, j = 1, ..., N , being matrices, is symmetric and positive definite, i.e., we have the equations N
X ∂ρi = κij ∆µj , i = 1, ..., N, ∂t j=1 µi = −[div(Γ∇ρ)]i + ∂ρi f, i = 1, ..., N, or, equivalently, 14
∂ρ = K∆w, ∂t w = P (−div(Γ∇ρ) + ∂ρ f ), where K = (κij ) (let us stress that the coefficients of K are subject to conditions (3.20); in particular, 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 author also considers elastic effects; see Subsection 3. b) below); in particular, well-posedness results are obtained. One difficulty (which already appears for the classical Cahn-Hilliard equation) 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-grain free energies of the form f (ρ) = θ
N X
(ρi lnρi + (1 − ρi )ln(1 − ρi )), θ > 0
i=1
(see [Gu1], [Gu2] and [Gu3]), but cannot be proven in general. Another possibility consists in adding a penalization term in the free energy (see, e.g., [BoCDGSS] and [BoDS]). Now, for more general Aij s, 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 = div(A11 ∇(µ1 − µ2 )), ∂t
(3.21)
µ1 − µ2 = ∂ρ1 ψ − ∂ρ2 ψ − div(∂∇ρ1 ψ − ∂∇ρ2 ψ).
(3.22)
Recalling now that ρ2 = 1 − ρ1 , we set ˜ 1 , ∇ρ1 ). ψ(ρ1 , ρ2 , ∇ρ1 , ∇ρ2 ) = ψ(ρ
(3.23)
It is then not difficult to show that ∂ρ1 ψ˜ = ∂ρ1 ψ − ∂ρ2 ψ,
(3.24)
∂∇ρ1 ψ˜ = ∂∇ρ1 ψ − ∂∇ρ2 ψ,
(3.25)
and we finally have the following equations: ∂ρ1 = div(A11 ∇(µ1 − µ2 )), ∂t
(3.26)
˜ µ1 − µ2 = ∂ρ1 ψ˜ − div(∂∇ρ1 ψ).
(3.27)
15
We thus recover the (generalized) Cahn-Hilliard equations obtained for binary alloys, for a constant mobility A11 . b) Equations in deformable continua. As in Subsection 2. b), we introduce the displacement u and the stress tensor S. We then have, assuming again that there are no external volume forces, Z Z ∂ρ ∂u W(R) = (ξ · ν) : dσ + dσ, (3.28) (Sν) · ∂t ∂t ∂R ∂R which, coupled with (3.8) and owing to the force balance (2.13), yields the following dissipation inequality: ∂u ∂ρ ∂ρ ∂ψ − S ·· ∇ + (π − µ) : −ξ∗∇ + h ∗ ∇µ ≤ 0. ∂t ∂t ∂t ∂t
(3.29)
Here, as in Subsection 2. b) and considering again linear elastic phases, the fundamental balance laws are ∂ρ = −divh (mass balance), ∂t
(3.30)
divξ + π = 0 (microforce balance),
(3.31)
divS = 0 (force balance),
(3.32)
S = t S (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 dissipation inequality: N
N
∂E X ∂ρi X ∂∇ρi (∂E ψ − S) ·· + (∂ρi ψ + πi − µi ) + (∂∇ρi ψ − ξi ) · ∂t ∂t ∂t i=1 i=1 N X
N
N
(3.34)
∂∇µi X ∂µi X + ∂µi ψ + ∂∇µi ψ · + hi · ∇µi ≤ 0, ∂t ∂t i=1 i=1 i=1 for every Z, which yields that, necessarily, ψ = ψ(E, ρ1 , ..., ρN , ∇ρ1 , ..., ∇ρN ), 16
(3.35)
S = ∂E ψ,
(3.36)
ξi = ∂∇ρi ψ, i = 1, ..., N,
(3.37)
µi = ∂ρi ψ + πi , i = 1, ..., N,
(3.38)
and there remains the dissipation inequality N X
hi · ∇µi ≤ 0,
(3.39)
i=1
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, N X hi = − Aij ∇µj , i = 1, ..., N,
(3.40)
j=1
and the dissipation inequality (3.39) is satisfied. Finally, we have, owing to the microforce balance (3.31), N X ∂ρi = div( Aij ∇µj ), i = 1, ..., N, ∂t j=1
(3.41)
µi = ∂ρi ψ − div(∂∇ρi ψ), i = 1, ..., N,
(3.42)
div(∂E ψ) = 0,
(3.43)
where N X
ρi = 1
(3.44)
i=1
and, assuming, for simplicity, that the Aij are constant matrices, N X
Aij = 0, i = 1, ..., N.
(3.45)
j=1
Remark 3.4: A classical free energy is the following generalized Ginzburg-Landau free energy: ψ=
1 ∇ρ ∗ Γ∇ρ + f (ρ) + W (E, ρ), 2 17
where W (E, ρ) =
1 (E − E(ρ)) ·· C(ρ)(E − E(ρ)), 2
C being again the elasticity tensor (it is also a symmetric and positive definite, when restricted to symmetric matrices, linear transformation which maps symmetric matrices onto symmetric matrices) and E(ρ) being the symmetric stress-free strain at density ρ. Such models are studied in [Ga1], [Ga2] and [Ga3], for mobilities of the form Aij = κij I, i, j = 1, ..., N , the κij being constant (strictly positive) scalars; again, it is reasonable to expect similar results for more general Aij s. Remark 3.5: Again, when N = 2, we recover the equations derived for binary alloys, for a 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 ,
∂ρN ∂ρ1 , ..., , µ1 , ..., µN , ∇µ1 , ..., ∇µN ) ∂t ∂t
as set of independent constitutive variables (we neglect, for simplicity, the macroscopic deformations). Therefore, the dissipation inequality (3.10) now gives
N X
N
N
N
∂ 2 ρi X ∂ρi X ∂∇ρi X ∂µi (∂ρi ψ + πi − µi ) + + (∂∇ρi ψ − ξi ) · ∂ ∂ρi ψ 2 + ∂µi ψ ∂t ∂t ∂t ∂t ∂t i=1 i=1 i=1 i=1 N
N X
∂∇µi X + hi · ∇µi ≤ 0, + ∂∇µi ψ · ∂t i=1 i=1 (3.46) for every Z, which yields that ψ = ψ(ρ1 , ..., ρN , ∇ρ1 , ..., ∇ρN ),
(3.47)
ξi = ∂∇ρi ψ, i = 1, ..., N,
(3.48)
and there remains the dissipation inequality N X
N
∂ρi X (∂ρi ψ + πi − µi ) + hi · ∇µi ≤ 0, ∂t i=1 i=1
(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
N N X ∂ρj X hi = − aij − Aij ∇µj , i = 1, ..., N, ∂t j=1 j=1
(3.50)
N N X ∂ρj X − bij · ∇µj , i = 1, ..., N, ∂ρi ψ + πi − µi = − βij ∂t j=1 j=1
(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: N N X ∂ρj ∂ρi X − = div( Aij ∇µj ), i = 1, ..., N, aij · ∇ ∂t ∂t j=1 j=1
µi −
N X
bij · ∇µj =
j=1
N X
βij
j=1
∂ρj + ∂ρi ψ − div(∂∇ρi ψ), i = 1, ..., N, ∂t
(3.52)
(3.53)
together with the constraint N X
ρi = 1.
(3.54)
i=1
Furthermore, it follows from (3.5) and (3.50) (which should hold for all possible values of the order parameters and the chemical potentials) that N X
aij = 0, j = 1, ..., N,
(3.55)
i=1 N X
Aij = 0, i = 1, ..., N
(3.56)
j=1
(we recall that the matrix A is symmetric). It finally follows from the dissipation inequality (3.49) the “positive semi-definiteness” condition N X i,j=1
βij xi xj +
N X
(aij + bji ) · yj xi +
i,j=1
N X
(Aji yi ) · yj ≥ 0,
i,j=1
(3.57)
∀xi ∈ R, ∀yi ∈ Rn , i = 1, ..., N. Let us stress that the above relation might not be true if the constitutive moduli are not constant. In that case, the positivity condition coming from (3.49) has a more complicated formulation. 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
˜ − (µ1 − µ2 )) ∂ρ1 + h1 · ∇(µ1 − µ2 ) ≤ 0, (∂ρ1 ψ˜ − div(∂∇ρ1 ψ) ∂t for every fields, where ψ˜ is as in Remark 3.3, which yields, in particular, the existence of constitutive moduli b (a vector) and β (a scalar) such that µ1 − µ2 − b · ∇(µ1 − µ2 ) = β
∂ρ1 ˜ + ∂ρ1 ψ˜ − div(∂∇ρ1 ψ). ∂t
(3.58)
1 We can note that these constitutive moduli depend a priori on ∂ρ ∂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
(3.59)
˜ + ∂ρ1 ψ˜ − div(∂∇ρ1 ψ), which yields, owing to (3.58),
(b11 −b21 )·∇µ1 +(b12 −b22 )·∇µ2 +(β11 −β12 −β21 +β22 )
∂ρ1 ∂ρ1 = b·∇(µ1 −µ2 )+β . (3.60) ∂t ∂t
Since this relation has to be valid for all values of the order parameters and the chemical potentials, 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 such relations if the constitutive moduli in (3.50)-(3.51) were not constant. We thus obtain the equations (we can note that equation (3.62) below follows from (3.52), (3.55) and (3.56)) ∂ρ1 ∂ρ1 −a·∇ = div(A11 ∇(µ1 − µ2 )), ∂t ∂t
(3.62)
∂ρ1 ˜ + ∂ρ1 ψ˜ − div(∂∇ρ1 ψ), (3.63) ∂t where a = a11 − a12 , i.e., we again recover the generalized Cahn-Hilliard equations derived for binary alloys (in the case of constant constitutive moduli). PN −1 ∂ρi PN −1 N Remark 3.7: More generally, noting that ∂ρ i=1 ∂t and hN = − i=1 hi , we ∂t = − deduce from (3.49) that µ1 − µ2 − b · ∇(µ1 − µ2 ) = β
N −1 X
˜ − (µi − µN )) ∂ρi + (∂ρi ψ˜ − div(∂∇ρi ψ) ∂t i=1 20
N −1 X i=1
hi · ∇(µi − µN ) ≤ 0,
(3.64)
˜ 1 , ..., ρN −1 , ∇ρ1 , ..., ∇ρN −1 ) = ψ(ρ1 , ..., ρN , ∇ρ1 , ..., ∇ρN ). We for every fields, where ψ(ρ 1 deduce from (3.64) that there exist constitutive moduli (which depend a priori on ∂ρ ∂t , ..., ∂ρN −1 ∂t , ∇µ1 , ..., ∇µN ; we again assume that they depend at least continuously on these arguments) ˜bij (vectors) and β˜ij (scalars), i, j = 1, ..., N − 1, such that
µi − µN −
N −1 X
˜bij · ∇(µj − µN ) =
N −1 X
j=1
∂ρj ˜ i = 1, ..., N. (3.65) + ∂ρi ψ˜ − div(∂∇ρi ψ), β˜ij ∂t j=1
Furthermore, it follows from (3.53) that
µi − µN −
N X
N −1 X
(bij − bN j ) · ∇µj =
j=1
(βij − βN j − βiN + βN N )
j=1
∂ρj ∂t
(3.66)
˜ i = 1, ..., N. + ∂ρi ψ˜ − div(∂∇ρi ψ), Comparing (3.65) and (3.66), we deduce that N X
(bij − bN j ) · ∇µj +
j=1
N −1 X
N −1 X ∂ρj ˜bij · ∇(µj − µN ) = + βN N ) ∂t j=1
(βij − βN j − βiN
j=1
+
N −1 X
(3.67)
∂ρj β˜ij , i = 1, ..., N, ∂t j=1
and, since this relation has to be valid for all values of the order parameters and the chemical potentials, we obtain (again, for constant constitutive moduli in (3.50)-(3.51)) bij − bN j = ˜bij , i, j = 1, ..., N − 1, N −1 X
˜bij , i = 1, ..., N.
biN − bN N = −
(3.68)
(3.69)
j=1
Summing (3.68) over j, we have, owing to (3.69), N −1 X j=1
bij −
N −1 X
bN j = bN N − biN , i = 1, ..., N,
j=1
which yields N X j=1
bij =
N X
bN j , i = 1, ..., N,
j=1
21
(3.70)
PN i.e., the sum j=1 bij is independent of i. We call b this common value. Setting b = (bij ), we then have PN µ i i=1 . 1 ) b ∗ ∇w = b ∗ ∇(N µ − . N . PN i=1 µi (3.71) PN b · ∇ i=1 µi . 1 , = b ∗ ∇µ − . N . PN b · ∇ i=1 µi PN b · ∇ i=1 µi . = 0, we can where w is as in the previous subsection. Noting now that P . . PN b · ∇ i=1 µi rewrite (in view of the mathematical analysis of the problem) (3.52)-(3.53) in the form ∂ρ ∂ρ −a∗∇ = div(A∇w), ∂t ∂t w − b ∗ ∇w = P (B
∂ρ + ∂ρ ψ − div(∂∇ρ ψ)), ∂t
(3.72)
(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 of ∂ρ1 ∂ρN the kinetics (i.e., ∇ ∂ρ ∂t = (∇ ∂t , ..., ∇ ∂t )) to the list of independent constitutive variables. We can also consider deformable continua or/and thermal effects (see also [AP] for a different approach for thermal effects based on an entropy principle). For instance, if one includes both the macroscopic deformations and the kinetics, then, combining Subsections 3. 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. [AP] H.W. Alt and I. Pawlow, Thermodynamical models of phase transitions with multicomponent order parameters, in Trends in applications of mathematics to mechanics, D.P.M. Marques and J.F. Rodrigues eds., Pitman Monographs and Surveys in Pure and Applied Mathematics 77, Longman, New York, 1995. [BaP] L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity, Research Report IBS PAN, RB/2, Warsaw, 2004. 22
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