A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions

Abstract

The Cahn-Hilliard equation is a fundamental model that describes the phase separation process in multi-component mixtures. It has been successfully extended to many different contexts in several scientific fields. In this survey article, we briefly review the derivation, structure as well as some analytical issues for the Cahn-Hilliard equation and its variants. Our focus will be placed on the well-posedness and long-time behavior of the Cahn-Hilliard equation in the classical setting and recent progresses on the dynamic boundary conditions accounting for non-trivial boundary effects.

Evolution dynamics of thin liquid structures investigated using a phase-field model

“…34 The CH model in two-phase flow with wetting boundary conditions has been investigated extensively. 28,[39][40][41] Yue et al 40,42 provided guidelines to choose the model parameters such as interface thickness, mobility or the CH diffusivity, and wall relaxation through calibration with experiment and sharp-interface limit analysis. Carlson et al 43 systematically studied spontaneously wetting liquid droplets on the substrates with a wall relaxation effect through introducing a phenomenological parameter in the PF model, which quantitatively matches with experimental data.…”

Thermodynamically consistent dynamic boundary conditions of phase field models

“…However, it's not general enough to include the Gal Model and the KL model. Recently, Hao Wu reviewed the derivation and analysis of the classical Cahn-Hilliard equation with static and dynamic boundary condition [58]. In addition, all the above mentioned boundary conditions are valid in flat boundaries where the curvature vanishes.…”