We study the anchoring and anchoring transitions in the nematic-substrate system using the Landau-de Gennes formalism. The free energy functional is expanded around the bulk tensor nematic order parameter up to the second-order terms. This quadratic approximation is used to find an explicit condition for anchoring directions in a semi-infinite system. A finite system is also studied and a geometrical measure of the anchoring strength, similar to the extrapolation length, is defined. Then we study anchoring transitions between the homeotropic, planar, and conical anchorings. The quadratic approximation predicts a first order homeotropic-conical transition and a first order or continuous planar-conical transition. It also predicts that the geometrical measure of the anchoring strength is always positive and does not diverge even at a continuous anchoring transition.
A.Poniewierski and A.Samborski         JOURNAL of CHEMICAL PHYSICS, Vol. 105 ,7632, (1996)