Doubly Degenerate Diffuse Interface Models of Surface Diffusion

TL;DR

This paper introduces and compares two doubly degenerate Cahn-Hilliard models for isotropic surface diffusion, demonstrating that restriction functions improve accuracy and establishing their convergence to the sharp interface limit.

Contribution

The paper presents a new variational, energy dissipative model for surface diffusion and compares it with a non-variational model, highlighting the benefits of restriction functions.

Findings

Restriction functions improve approximation accuracy.

Both models converge to the sharp interface limit.

The variational model is energy dissipative and related to the non-variational model.

Abstract

We discuss two doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion. Degeneracy is introduced in both the mobility function and a restriction function associated to the chemical potential. Our computational results suggest that the restriction functions yield more accurate approximations of surface diffusion. We consider a slight generalization of a model that has appeared before, which is non-variational, meaning there is no clear energy that is dissipated along the solution trajectories. We also introduce a new variational and, more precisely, energy dissipative model, which can be related to the generalized non-variational model. For both models we use formal matched asymptotics to show the convergence to the sharp interface limit of surface diffusion.