Application of finite element method to solve 2D problems related to nematic surface properties

This paper is devoted to numerical studies of two-dimensional problems concerning surface properties of nematic liquid crystals. We use a finite element method, based essentially on the classic variational approach, to find an approximate solution minimizing the Gibbs free energy of the nematic material under given boundary conditions. Three examples illustrate the performance and versatility of this analysis. Two cases are related to the macroscopic orientation induced by periodic boundary conditions: the first is a saw-toothed substrate in the micrometric range and the second is a microtextured surface. We analyze the bulk planar-homeotropic transition conditions for both of them. In the third case, we study the coupling between the spatial variation of the nematic director and that of the order parameter in the presence of surface-induced distortion.