The critical mass constraint in the Cahn–Hilliard equation

Abstract

When the mass constraint of the Cahn-Hilliard equation in two dimensions is lowered to the order of ${ϵ}^{2/3}$, where $ϵ$ is the interface thickness parameter, the existence of droplet solutions becomes conditional. For interior single droplet solutions, there is a critical value for the mass constraint such that above this value two interior single droplet solutions exist, and below this value interior single droplet solutions can not be constructed. One solution has smaller droplet radius than the other one does. The one with smaller radius is less stable than the one with larger radius. The center of the droplets in these solutions is (almost) the point in the domain that is furthest from the boundary. A critical mass constraint also appears when multiple droplet solutions are sought. Above the critical mass constraint, which now depends on the number of droplets, there exist two multi-droplet solutions. In each solution the radii of the droplets are about the same. However when the two solutions are compared, one has larger droplet radius than the other one does. The locations of the droplets are determined by the solution of a disc packing problem.