On the equilibrium of the layer of a nematic liquid crystal with an inhomogeneous boundary

Abstract

The problem of the equilibrium of the layer of a nematic liquid crystal is considered in the case, in which the surface energy is quadratic with respect to the deviation of the orientation vector from a given direction (Rapini-Papoular model), and with account for the divergence terms in the quadratic internal-energy expansion in medium orientation vector gradients (Frank model). The presence of these terms leads to the appearance of director deviations in the plane parallel to the boundary. It is shown that at an appropriate choice of the undisturbed orientation for the layer there exist two critical values of the wavenumbers in whose vicinities the director oscillation amplitude can become arbitrarily large at even weak boundary disturbances, while in the case of a plane boundary the corresponding nontrivial periodic solutions are possible. The existence of one of these wavenumbers does not depend on the layer thickness, while the second wavenumber can exist when the layer thickness is greater than a certain critical value.