Theory of surfacelike elastic contributions in nematic liquid crystals

In a recent paper we carried out a systematic expansion of the free-energy density of nematic liquid crystals (NLCs) in the director derivatives for planar director distortions and small director angles. At any order of expansion, the director distortion is the superposition of a standard long-range bulk director distortion and a very-short-range subsurface distortion. The bulk macroscopic distortion is found to be the same as that which is obtained using the Frank elastic form of the free-energy density and an effective anchoring energy function ${\mathit{f}}_{\mathit{s}}$ which implicitly contains the surfacelike elastic constant ${\mathit{K}}_{13}$ and all higher-order elastic constants. In this paper we generalize this theoretical result and extend it to the case of large director angles using the Gibbs theory of interfacial phenomena. Furthermore we extend the theoretical analysis to the more general case of nonplanar director distortions. An alternative theoretical expression of the first-order free-energy density that does not present mathematical problems, and allows us to study any kind of director distortion in NLC’s, is proposed. In the nonplanar case, both of the surfacelike elastic constants ${\mathit{K}}_{13}$ and ${\mathit{K}}_{24}$ are shown to make explicit contributions to the first-order free-energy density. Recent theoretical and experimental results concerning the elastic behavior of a NLC sample enclosed in a cylindrical cavity are reanalyzed in terms of the present theoretical procedure. Rough estimates of the surfacelike elastic constants ${\mathit{K}}_{13}$ and ${\mathit{K}}_{24}$ are obtained from the analysis of the experimental results. A surface orientational transition, which makes it possible to measure the ${\mathit{K}}_{13}$ surfacelike elastic constant, is predicted to occur at a critical value of the radius R of the cylindrical cavity.