Anchoring of a nematic liquid crystal on an anisotropic substrate

A nematic liquid crystal in contact with a flat solid substrate is studied by means of a mesoscopic Landau-de Gennes theory. It is assumed that the substrate is anisotropic, i.e. the directions x and y in the surface of the substrate are not equivalent, and the only symmetry is the mirror symmetry y ↦ - y. Assuming the simplest form of the bare surface free energy, where only the linear terms in the nematic order parameter are taken into account, we study anchoring directions induced by the interaction of the liquid crystal with the substrate. A phase diagram in terms of the surface fields and the temperature is obtained. Depending on the values of the surface fields we find four types of anchoring: the symmetric planar anchoring, with the director along x, the symmetric tilted anchoring, with the director in the xz plane, the antisymmetric planar anchoring, with the director along y, and the asymmetric tilted anchoring, with the director tilted with respect to all three axes.