Nematic liquid crystals at rough and fluctuating interfaces

Nematic liquid crystals at rough and fluctuating interfaces are analyzed within the Frank elastic theory and the Landau-de Gennes theory. We study specifically interfaces that locally favor planar anchoring. In the first part we reconsider the phenomenon of Berreman anchoring on fixed rough surfaces, and derive new simple expressions for the corresponding azimuthal anchoring energy. Surprisingly, we find that for strongly aligning surfaces, it depends only on the geometrical surface anisotropy and the bulk elastic constants, and not on the precise values of the chemical surface parameters. In the second part, we calculate the capillary waves at nematic-isotropic interfaces. If one neglects elastic interactions, the capillary wave spectrum is characterized by an anisotropic interfacial tension. With elastic interactions, the interfacial tension, i.e. the coefficient of the leading $q^2$ term of the capillary wave spectrum, becomes isotropic. However, the elastic interactions introduce a strongly anisotropic {\em cubic} $q^3$ term. The amplitudes of capillary waves are largest in the direction perpendicular to the director. These results are in agreement with previous molecular dynamics simul