Nematic liquid crystal in the wedge and edge geometry in the case of homeotropic alignment

Nematic liquid crystal confined to a wedge or edge is studied on the assumption that the confining surfaces provide strong and weak homeotropic anchorings, respectively. Both infinite and finite systems are considered. The model based on the Frank-Oseen and Rapini-Papoular formalisms predicts two textures of opposite rotations of the director as in the case of strong anchoring on both surfaces. However, the presence of weak anchoring results in a length scale lambda which characterizes the crossover between the regions close to the apex and far from it. The ratio lambda/b , where b is the extrapolation length, is a function of the opening angle alpha. Both stable and metastable textures are considered and the mechanism by which a texture loses its stability is found. It is related to the formation of a defect-like structure at the surface of weak anchoring whose distance from the apex is lambda(alpha) and the loss of stability is signalled by the divergence of lambda. Only in the limit alpha --> 2tau, the defect-like structure transforms into a defect of strength -1/2 located at a finite distance from the apex.