Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions Original Research Article

We apply implicit numerical methods to solve the Cahn–Hilliard equation for confined systems. Generic boundary conditions for hard walls are considered, as they are derived from physical principles. Based on a detailed stability analysis an automatic time step control could be implemented, which makes it possible to explore the demixing kinetics of two thermodynamically stable phases over many orders in time with good space resolution. The power of the method is demonstrated by investigating spinodal decomposition in two-dimensional systems. At early times of the decomposition process the numerical results are in excellent agreement with analytical predictions based on the linearized equations. Due to the efficiency of the variable time step procedure it is possible to monitor the process until a stable equilibrium is reached.