Saddle-splay and periodic instability in a hybrid aligned nematic layer subjected to a normal magnetic field

A nematic layer with opposite boundary conditions (unidirectional planar and homeotropic) is considered, having strong anchoring at the planar wall. It is known that a periodic deformation of splay-type can intervene between the aperiodic hybrid alignment and undistorted planar state as one decreases the film thickness. This periodic state occurs due to the fact that the energetic cost for a mixed twist-splay can be lower than the surface energetic cost at the homeotropic wall in the case of undistorted planar alignment. Such a situation may also be achieved for nematics which have a bulk elastic isotropy. In this paper, the saddle-splay elastic constant $K_{24}$ is shown to influence strongly the occurrence of the periodic pattern of splay-type, also in the presence of an external magnetic field normal to the cell plates, and the role of the geometrical anchoring in wedge-shaped cells is discussed