Doubly degenerate diffuse interface models of surface diffusion

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.

Automated summary

In this study, we discuss two doubly degenerate Cahn-Hilliard models for isotropic surface diffusion, focusing on degeneracy in the mobility function and a restriction function associated with the chemical potential. Our computational results indicate that the restriction functions provide more accurate approximations of surface diffusion. We also explore a slight generalization of a non-variational model and introduce a new variational and energy dissipative model, both of which can be related to the generalized non-variational model and show convergence to the sharp-interface limit of surface diffusion through formal matched asymptotics.