Cahn-Hilliard equation

An equation modelling the evolution of the concentration field in a binary alloy.

When a homogeneous molten binary alloy is rapidly cooled, the resulting solid is usually found to be not homogeneous but instead has a fine-grained structure consisting of just two materials, differing only in the mass fractions of the components of the alloy. Over time, the fine-grained structure coarsens as larger particles grow at the expense of smaller particles, which dissolve. The development of a fine-grained structure from a homogeneous state is referred to as spinodal decomposition, while the coarsening is called Ostwald ripening (cf. also Spinodal decomposition).

If the average concentration, $\overline { c }$, of one of the species and the temperature, $T$, lie in a particular region of parameter space, spinodal decomposition does not occur and instead, separation into the two preferred concentrations takes place through nucleation. In this scenario, small randomly spaced regions of a preferred state appear due to localized perturbations and then these regions grow. This is similar to the condensation of water droplets in mist, wherein a growing droplet depletes the water in the mist in its immediate vicinity, the depletion being replenished through diffusion-like processes.

In 1958, J. Cahn and J. Hilliard [a9] derived an expression for the free energy of a sample $V$ of binary alloy with concentration field $c ( x )$ of one of the two species. They assumed that the free energy density depends not only upon $c ( x )$ but also derivatives of $c$, to account for interfacial energy or surface tension. To first order in an expansion, the expression for the total free energy takes the form

\begin{equation} \tag{a1} F = N _ { V } \int _ { V } ( f _ { 0 } ( c ) + \kappa | \nabla c | ^ { 2 } ) d V, \end{equation}

where $N _ { V }$ is the number of molecules per unit volume, $f _ { 0 }$ is the free energy per molecule of an alloy of uniform composition, and $\kappa$ is a material constant which is typically very small. The function $f _ { 0 }$ has two wells with minima located at the two coexistent concentration states, labelled $c _ { \alpha }$ and $c _ { \beta } > c _ { \alpha }$. A similar expression for free energy was introduced much earlier by J.D. van der Waals in [a18].

With the average concentration $\overline { c }$ specified, the equilibrium configurations satisfy the stationary Cahn–Hilliard equation

\begin{equation} \tag{a2} 2 \kappa \Delta c - f _ { 0 } ^ { \prime } ( c ) = \lambda \text { in } V, \end{equation}

\begin{equation} \tag{a3} \frac { \partial c } { \partial n } = 0 \text{ on the boundary } \partial V \text{ of } V. \end{equation}

Here, $\Delta$ is the Laplacian (cf. Laplace operator), $\lambda$ is a Lagrange multiplier associated with the constraint $\overline { c }$ (cf. also Lagrange multipliers), and $n$ is the normal to $\partial V$. In [a9] equations (a2)–(a3) together with the constraint are used to predict the profile and thickness of one-dimensional transitions between concentration phases $c _ { \alpha }$ and $c _ { \beta }$.

By considering the second variation of the free energy at the homogeneous state $c ( x ) = \bar{c}$, one can determine the stability of this state. If $\overline { c }$ is such that $f _ { 0 } ^ { \prime \prime } ( \overline{c} ) > 0$ (the metastable concentrations), which includes those values near $c _ { \alpha }$ and $c _ { \beta }$, then the homogeneous state is stable to small perturbations. If $f _ { 0 } ^ { \prime \prime } ( \bar{c} ) < 0$, then if $\kappa$ is sufficiently small or equivalently, if $V$ is sufficiently large, $\overline { c }$ is unstable with respect to some periodic perturbations. This analysis was performed in [a6], where it was also shown that perturbations of a certain characteristic wavelength of order $\sqrt { \kappa }$ grow most rapidly. Thus, spinodal decomposition is described mathematically. Likewise, when $f _ { 0 } ^ { \prime \prime } ( \overline{c} ) > 0$ and $\overline { c }$ lies strictly between $c _ { \alpha }$ and $c _ { \beta }$, the homogeneous state is stable but does not minimize the free energy if $\kappa$ is sufficiently small (see [a8], [a15]). In [a10] the existence and properties of a critical nucleus are discussed. This nucleus is a spatially localized perturbation of the homogeneous state which lies on the boundary of the basins of attraction of the stable state $\overline { c }$ and the energy minimizing state, and is therefore unstable. Thus, nucleation is accounted for by the free energy proposed by Cahn and Hilliard.

The general equation governing the evolution of a non-equilibrium state $c ( x , t )$ is put forth in [a6] and this is what is now referred to as the Cahn–Hilliard equation:

\begin{equation} \tag{a4} \frac { \partial c } { \partial t } = \operatorname { div } \{ M \operatorname { grad } [ f _ { 0 } ^ { \prime } ( c ) - 2 \kappa \Delta c ] \} \text { in } V, \end{equation}

with the natural boundary conditions

\begin{equation} \tag{a5} \frac { \partial c } { \partial n } = \frac { \partial \Delta c } { \partial n } = 0 \text { on } \partial V. \end{equation}

The positive quantity $M$ is related to the mobility of the two atomic species which comprise the alloy.

Other derivations for the free energy, the equilibrium equations and the Cahn–Hilliard equation may be found in, e.g., [a13], [a14], [a17], [a11].

Further studies of spinodal decomposition as predicted by (a4) in one and higher space dimensions and to various degrees of rigour may be found in [a7], [a14], [a12], and [a16]. Nucleation, beyond the existence of the canonical stationary nucleus for (a4), is discussed in [a3], [a4] and [a19]. The coarsening process is formally described for the one-dimensional version of (a4) in [a14] and is rigorously shown to be exponentially slow in [a1] and [a5]. In higher space dimensions, N. Alikakos and G. Fusco show in [a2] that (a4) predicts Ostwald ripening.

It is thus well-established that the Cahn–Hilliard equation is a qualitatively reliable model for phase transition in binary alloys.

References

[a1]	N.D. Alikakos, P.W. Bates, G. Fusco, "Slow motion for the Cahn–Hilliard equation in one space dimension" J. Diff. Eqs. , 90 (1990) pp. 81–135
[a2]	N.D. Alikakos, G. Fusco, "The equations of Ostwald ripening for dilute systems" J. Statist. Phys. , 95 (1999) pp. 851–866
[a3]	P.W. Bates, P.C. Fife, "The dynamics of nucleation for the Cahn–Hilliard equation" SIAM J. Appl. Math. , 53 (1993) pp. 990–1008
[a4]	P.W. Bates, G. Fusco, "Equilibria with many nuclei for the Cahn–Hilliard equation" J. Diff. Eqs. , 160 (2000) pp. 283–356
[a5]	P.W. Bates, P.J. Xun, "Metastable patterns for the Cahn–Hilliard equation. Part I–II" J. Diff. Eqs. , 111/116 (1994/95) pp. 421–457/165–216
[a6]	J.W. Cahn, "On spinodal decomposition" Acta Metall. , 9 (1961) pp. 795–801
[a7]	J.W. Cahn, "Phase separation by spinodal decomposition in isotropic systems" J. Chem. Phys. , 42 (1965) pp. 93–99
[a8]	J. Carr, M. Gurtin, M. Slemrod, "Structured phase transitions on a finite interval" Arch. Rational Mech. Anal. , 86 (1984) pp. 317–357
[a9]	J.W. Cahn, J.E. Hilliard, "Free energy of a non-uniform system I: Interfacial energy" J. Chem. Phys. , 28 (1958) pp. 258–266
[a10]	J.W. Cahn, J.E. Hilliard, "Free energy of a non-uniform system III: Nucleation in a two-component incompressible fluid" J. Chem. Phys. , 31 (1959) pp. 688–699
[a11]	P.C. Fife, "Models for phase separation and their mathematics" M. Mimura (ed.) T. Nishida (ed.) , Nonlinear Partial Differential Equations with Applications to Patterns, Waves, and Interfaces. Proc. Conf. Nonlinear Partial Differential Equations, Kyoto (1992) pp. 183–212
[a12]	C.P. Grant, "Spinodal decomposition for the Cahn–Hilliard equation" Commun. Partial Diff. Eqs. , 18 : 3–4 (1993) pp. 453–490
[a13]	M. Hillert, "A solid-solution model for inhomogeneous systems" Acta Metall. , 9 (1961) pp. 525–535
[a14]	J.S. Langer, "Theory of spinodal decomposition in alloys" Ann. Phys. , 65 (1971) pp. 53–86
[a15]	L. Modica, "The gradient theory of phase transitions and the minimal interface criterion" Arch. Rational Mech. Anal. , 98 (1987) pp. 123–142
[a16]	S. Maier–Paape, T. Wanner, "Spinodal decomposition for the Cahn–Hilliard equation in higher dimensions. I. Probability and wavelength estimate" Comm. Math. Phys. , 195 (1998) pp. 435–464
[a17]	A. Novick–Cohen, L.A. Segel, "Nonlinear aspects of the Cahn–Hilliard equation" Phys. D. , 10 (1985) pp. 277–298
[a18]	J.D. van der Waals, "The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation in density" Verh. K. Nederland. Akad. Wetenschappen Amsterdam , 1 (1893) pp. 1–56
[a19]	J. Wei, M. Winter, "Stationary solutions for the Cahn–Hilliard equation" Ann. Inst. H. Poincaré , 15 (1998) pp. 459–492