How to model surface diffusion using the phase-field approach

TL;DR

This paper investigates the challenges of modeling surface diffusion with phase-field methods, identifies issues with existing approaches, and develops a new model that accurately reproduces sharp-interface equations in 3D systems.

Contribution

It introduces a novel phase-field model for surface diffusion that correctly captures asymptotic behavior and is applicable to three-dimensional systems.

Findings

Existing models fail to produce correct asymptotics.

A new model asymptotically approaches sharp-interface equations.

Model demonstrated on elastically induced Grinfeld instability.

Abstract

It is demonstrated that the description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach previously used in the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. An apparently obvious alternative fails for a complementary reason. Finally, a model is constructed that asymptotically approaches known sharp-interface equations without adding undesired constraints. In order to provide a complete physical example, the model is exhibited for the elastically induced Grinfeld instability with material transport by surface diffusion. The whole analysis is carried out for three-dimensional systems to pave the way for simulations in 3D.