Front migration in the nonlinear Cahn-Hilliard equation

Abstract

The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions. The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature. On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka ( J. appl. Phys . 34, 323-329 (1963)). On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem. Similarity solutions of the two-phase Stefan problem should describe boundary layers. Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix.

A numerical scheme for moving boundary problems that are gradient flows for the area functional

“…It contains an interaction parameter, which one may think of as describing the width of the layer between the two regions containing mostly one phase or the other. It has been shown that the two-sided Mullins-Sekerka model arises as a singular limit when the interaction parameter approaches zero [3,61].…”

“…For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area).…”

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

“…It was first formally shown by Pego [36] that, as ε 0, the function w := −ε u + ε −1 f (u), known as the chemical potential, tends to a limit, which, together with a free boundary := ∪ 0≤t≤T ( t ×{t}), satisfies the following Hele-Shaw (Mullins-Sekerka) problem:…”

“…In addition to the reason that the Cahn-Hilliard equation is a good model to describe the phase separation and coarsening phenomena in a melted alloy, it has been extensively studied in the past decade due to its connection to an interesting and complicated free boundary problem which is known as the Mullins-Sekerka problem arising from studying solidification/melting of materials of zero specific heat, which is also known as the (two-phase) HeleShaw problem arising from the study of the pressure of immiscible fluids in the air [36,1,16,13,11,33,32]. It was first formally shown by Pego [36] that, as ε 0, the function w := −ε u + ε −1 f (u), known as the chemical potential, tends to a limit, which, together with a free boundary := ∪ 0≤t≤T ( t ×{t}), satisfies the following Hele-Shaw (Mullins-Sekerka) problem:…”

Convergence of phase-field approximations to the Gibbs–Thomson law

“…Ilmanen [17] considered the corresponding L 2 -gradient flows and proved the convergence of the Allen-Cahn equation to the meancurvature flow, in the varifold formulation of Brakke [5]. Convergence of various other phase field problems to the corresponding sharp interface models have been shown either formally or rigorously [1,6,8,9,18,27,38], sometimes in quite involved weak formulations.…”