Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion

TL;DR

This paper demonstrates that degenerate mobility phase field models, specifically the Cahn-Hilliard equation, fail to accurately capture surface diffusion in the long-term, instead exhibiting bulk diffusion effects.

Contribution

It challenges the conventional assumption that degenerate Cahn-Hilliard models represent pure surface diffusion, revealing their limitation in long-time behavior.

Findings

Degenerate Cahn-Hilliard models undergo coarsening due to bulk diffusion.

Surface diffusion is not the dominant long-time mechanism in these models.

Phase field models with degenerate mobility may not accurately simulate surface diffusion.

Abstract

Phase field models frequently provide insight to phase transitions, and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulk diffusion, rather than pure surface diffusion. This reveals an important limitation of phase field models that are frequently used to model surface diffusion.