Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility

TL;DR

This paper derives the sharp interface limit of the degenerate Cahn-Hilliard equation in two dimensions, revealing a combined surface and bulk diffusion-driven interface motion, and validates the models through perturbation relaxation comparisons.

Contribution

It provides a detailed asymptotic analysis of the degenerate Cahn-Hilliard equation, highlighting the combined effects of surface and bulk diffusion on interface motion, which differs from previous literature.

Findings

Interface motion driven by surface diffusion and nonlinear bulk diffusion.

Bulk diffusion becomes subdominant for higher degeneracies.

Model validation through relaxation rate comparisons.

Abstract

In this work, the sharp interface limit of the degenerate Cahn-Hilliard equation (in two space dimensions) with a polynomial double well free energy and a quadratic mobility is derived via a matched asymptotic analysis involving exponentially large and small terms and multiple inner layers. In contrast to some results found in the literature, our analysis reveals that the interface motion is driven by a combination of surface diffusion flux proportional to the surface Laplacian of the interface curvature and an additional contribution from nonlinear, porous-medium type bulk diffusion, For higher degenerate mobilities, bulk diffusion is subdominant. The sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model.