Adhesion of oriented nematic droplets

We propose and analyze a two-dimensional model for the equilibrium of oriented droplets of nematic liquid crystals that may adhere to a rigid substrate, while surrounded by an isotropic environment. We obtain the contact condition at the edge where the liquid crystal, the substrate, and the environment come together. We further develop a fairly general method to arrive at the equilibrium shapes of a drop, which is then applied to the case where the surface tension at the liquid crystal interface is given by Rapini and Papoular's expression. In this case, we also predict the existence of concave equilibrium shapes. Here is indeed the main difference between this method and Wulff's construction, which always yields convex equilibrium shapes for a drop free from adhesion.