Doubly degenerate diffuse interface models of surface diffusion

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.

Supplementary document for Engineering and detection of light scattering directionalities in dewetted nanoresonators through Dark-field Scanning Microscopy - 6263498.pdf

“…an additional degeneracy improving the convergence to the sharp-interface limit [3,4], κ(φ) an approximation of the local mean curvature and 𝛼 a small positive constant ( ∼ 10 -6 ). 𝛾(𝒏) is the surface energy density and 𝒏 = -∇𝜑/|∇𝜑| the outward-pointing normal of the surface of the solid phase.…”

“…Numerical simulations are performed exploiting the finite element method (FEM) within the AMDiS FEM toolbox [8,9], similarly to, e.g., [4]. We consider adaptive mesh refinement with a resolution ranging between 5h∼𝜀 and 10h∼𝜀, with mesh size h, within the diffuse interface.…”

Gamma Convergence for the de Gennes-Cahn-Hilliard energy

“…However, the most significant difference is that the regularization removes the mechanism which hopefully confines the values of u to be in [u − , u + ]. Numerical simulations and formal asymptotic analysis in [31] show that the degenerate dGCH equation (1.12)-(1.13) may more closely approximate surface diffusion than others. This exciting discovery motivated us to systematically explore properties of the dGCH model.…”

“…Recently one of the authors (Wise) and his collaborators Salvalaglio and Voigt introduced a new variational diffuse interface model [31], in which they imposed a singularity in the formulation of the energy functional. Rather than formulating the free energy as in (1.1), they define the free energy by including a de Gennes coefficient [16,25].…”

“…The factor 1 g 0 is called the de Gennes coefficient [16,25], or the energy restriction function. The corresponding energy dissipation flow is the degenerate de Gennes-Cahn-Hilliard system (which was called the doubly degenerate Cahn-Hilliard system in [31]).…”