Cahn–Hilliard-type Equations

Abstract

In this work, a robust and efficient numerical method to simulate Cahn–Hilliard-type equations is presented. The considered equations are of fourth order and contain two nonlinearities: one is the concentration-dependent mobility and the other one is the potential. <br /> Using the fact that Cahn–Hilliard-type equations are a gradient flow, we can derive symmetric time-discrete equations for the flux, even though the mobility is concentration dependent. As time-step scheme, we use the TR-BDF2 scheme. <br /> For spatial discretization, we use H(div)-conformal finite elements, more precisely Raviart–Thomas elements. The introduced discrete gradient leads to the appearance of the inverse of the mass matrix. Since there is no mass-lumping method for Raviart–Thomas elements, this inverse is dense. To circumvent this difficulty, we adapt a technique from the area of mixed methods to our situation. It is based on the introduction of inter-element multipliers to guarantee conformality and yields a block-diagonal mass matrix. This approach is up to fifty times faster than more naïve ones like using a matrix-free method. <br /> The discretization was efficiently implemented in a space- and time-adaptive finite element software, which was written from scratch in C++. We use the software to simulate Cahn–Hilliard-type equations arising in two different physical contexts. <br /> The first one is the decomposition of a binary mixture both for the case of a constant mobility and a double-well potential (shallow quench) and the case of a degenerate mobility and a concave potential (deep quench). The latter is approximated with a variable-quench Ansatz and low temperature. We are able to compute large domains and long times. Theoretical results on the coarsening rate were reproduced accurately. <br /> The second application is thin-film crystal growth. We use a diffuse-interface approximation to the standard BCF model which includes the Ehrlich–Schwoebel (ES) barrier. With this approximation we can handle topological changes. As an example, we consider the pinch-off caused by an instability due to the ES barrier. Thanks to the discretization via the flux, we can easily simulate step trains using skew-periodic boundary conditions.