A multiphase Cahn–Hilliard system with mobilities and the numerical simulation of dewetting

Abstract

We propose in this paper a new multiphase Cahn-Hilliard model with doubly degenerate mobilities. We prove by a formal asymptotic analysis that it approximates with second order accuracy the multiphase surface diffusion flow with mobility coefficients and surface tensions. To illustrate that it lends itself well to numerical approximation, we propose a simple and effective numerical scheme together with a very compact Matlab implementation. We provide the results of various numerical experiments to show the influence of mobility and surface tension coefficients. Thanks to its second order accuracy and its good suitability for numerical implementation, our model is very handy for tackling notably difficult surface diffusion problems. In particular, we show that it can be used very effectively to simulate numerically the dewetting of thin liquid tubes on arbitrary solid supports without requiring nonlinear boundary conditions.

Approximation of surface diffusion flow: A second-order variational Cahn–Hilliard model with degenerate mobilities

“…A second-order Cahn-Hilliard model with two degenerate mobilities. A different approach is proposed in [11,9] where another mobility N , in addition to M , is incorporated in the metric used to define the gradient flow, thus possibly changing the overall geometry of the evolution problem. The so-called NMN-CH model proposed in [11,9] reads…”

“…complemented with the introduction of a dependence on u of M and N . Using formal asymptotic expansion, it is shown in [11,9] that with…”

“…In Section 3 we introduce and analyse two SAV schemes incorporating mobilities in the general context of gradient flows of a convex function with respect to a suitable energy . The numerical schemes obtained are simple and endowed with unconditional stability and accuracy properties, unlike standard convex-concave schemes used, e.g., in [11,9], to solve models (M-CH) and (NMN-CH). The final Section 4 focuses on the application of these schemes to Cahn-Hilliard models endowed in the inhomogeneous case.…”

“…A convex-concave splitting for the mobility. In [11,9], a convex-concave splitting strategy was considered for the numerical treatment of (M-CH) and (NMN-CH) based on the gradient structure of a functional J defined as…”

“…The approximation by (M-CH), being of first order only, presents challenges when approaching the pure states 0 or 1 due to the emergence of undesired oscillations and imprecise solution profiles. Furthermore, as demonstrated in [11,9], this approximation leads to numerical volume losses, contradicting the inherent volume-preserving nature of the Cahn-Hilliard equation.…”