An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit

Abstract

We propose a two-dimensional phase field model for solid state dewetting where the surface energy is weakly anisotropic. The evolution is described by the Cahn-Hilliard equation with a bi-quadratic degenerate mobility together with a bulk free energy based on a double-well potential and a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. We show that in contrast to the frequently used quadratic degenerate mobility, the resulting sharp interface model for the bi-quatratic mobility is consistent with the pure surface diffusion model. In addition, we show that natural boundary conditions at the substrate obtained from the first variation of the total free energy including contributions at the substrate imply a contact angle condition in the sharp-interface limit which recovers the Young-Herring equation in the anisotropic and Young's equation in the isotropic case, as well as a balance of fluxes at the contact line (or contact point).

A diffuse domain method for two-phase flows with large density ratio in complex geometries

“…We mention that careful studies for wetting and dewetting phenomena using a phase-field model have been presented in Gránásy et al. (2007), Dziwnik, Münch & Wagner (2017) and Rainer et al. (2019).…”

Pressure-dipole solutions of the thin-film equation

“…The existence of weak solutions to (1.3) and the dynamics of interfaces are well-known to depend strongly on the value of n [5,8,9]. Equation (13) can also be viewed as a special case of the Cahn-Hilliard equation with degenerate mobility (used as a phase-field model for problems in image processing and materials science [28]). An important focus of studies of (1.3) has been on the solutions of initial value problems starting from compactly supported initial data, h = h * (x) defined on some finite interval s − * 6 x 6 s + * .…”

The role of degenerate mobilities in Cahn‐Hilliard models

“…According to the preliminary work [1], considering an anisotropic version of the equation, the asymptotic sharp interface limit subtly dependents on the degeneracy of the mobility. The equation of interest is the Cahn-Hilliard equation (in two space dimensions) with a polynomial double well free energy and different order-parameter dependent, degenerate mobilities.…”

“…The equation of interest is the Cahn-Hilliard equation (in two space dimensions) with a polynomial double well free energy and different order-parameter dependent, degenerate mobilities. According to the preliminary work [1], considering an anisotropic version of the equation, the asymptotic sharp interface limit subtly dependents on the degeneracy of the mobility. Whilst a quadratic degenerate mobility leads to a sharp interface model where bulk diffusion is present at the same asymptotic order as surface diffusion, a bi-quadratic degenerate mobility leads to a sharp interface model where bulk diffusion is subdominant.…”

“…In our preliminary work [1] we introduced the following anisotropic version of the two-dimensional Cahn-Hilliard equation for solid state dewetting…”