Numerical modeling of phase separation on dynamic surfaces

Abstract

The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn–Hilliard equation posed on a time-dependent surface. Only elementary tangential calculus and the embedding of the surface in R3 are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. A hybrid method, finite difference in time and trace finite element in space, is introduced and stability of its semi-discrete version is proved. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the equation. Thus, the method is capable of solving the Cahn–Hilliard equation numerically on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. We assess the approach on a set of test problems and apply it to model spinodal decomposition and pattern formation on colliding surfaces. Finally, we consider the phase separation on a sphere splitting into two droplets.