Anchoring transitions in the nematic-substrate system: Study of the Landau–de Gennes model

We study the phenomenon of 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 the anchoring direction in a semi-infinite system, and then the phase boundaries between the homeotropic, planar, and conical anchorings are determined. In the cases that we have studied the quadratic approximation predicts a first order homeotropic–conical transition and a first order or continuous planar–conical transition. It also predicts a simple asymptotic expression for the free energy of a finite system, when the sample thickness is large and the deviation of the director from the anchoring direction is small. This asymptotic formula leads in a natural way to the definition of the geometrical measure of the anchoring strength b̄. However, the quadratic approximation is insufficient to predict the correct behavior of b̄ close to a continuous anchoring transition, although it can serve as a rough estimate of b̄ far from the transition.