Comparison of phase-field models for surface diffusion

TL;DR

This paper compares different phase-field models for surface diffusion, highlighting issues with existing models, introducing two new models that better approximate sharp-interface equations, and demonstrating their improved performance through stability analysis and simulations.

Contribution

The paper introduces two new phase-field models for surface diffusion that accurately reproduce sharp-interface limits without additional constraints.

Findings

Existing models may fail to produce correct asymptotics.

New models show improved stability and accuracy.

Numerical simulations demonstrate superior performance of the new models.

Abstract

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 from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Linear stability of a planar interface is investigated for the resulting phase-field equations and shown to reduce to the desired limit. Finally, numerical simulations of the standard and a more sophisticated model from the literature as well as of our two new models are performed to assess the relative merits of each approach. The results suggest superior performance of the new models in at least some situations.