Resummation of higher-order terms in the free-energy density of nematic liquid crystals

The presence of the surfacelike elastic constant ${\mathit{K}}_{13}$ in the expression of the elastic free-energy density ${\mathit{F}}_2$ for a nematic liquid crystal (NLC) makes the free-energy functional unbounded from below. A discontinuity of the director field has been predicted to occur at the interfaces of the NLC. In recent years two very different theoretical approaches have been proposed to bypass mathematical difficulties related to the ${\mathit{K}}_{13}$ problem. Hinov and Pergamenshchik [Mol. Cryst. Liq. Cryst. 148, 197 (1987); 178, 53 (1990), and references therein; Phys. Rev. E 48, 1254 (1993)] consider the surface director discontinuity is an artifact of theory and make the assumption that the director field must be sought in the class of continuous functions. With this assumption a well defined solution for the equilibrium director field can be found and new phenomena are predicted to occur. Barbero and co-workers [Nuovo Cimento D 12, 1259 (1990); Liq. Cryst. 5, 693 (1989)] expanded the free-energy functional F up to the fourth order in the director derivatives $(—second-order elastic theory$)— and showed that the minimization problem now becomes mathematically well posed. A strong subsurface director distortion on a length scale of the order of the molecular length is predicted to occur by using this approach. The macroscopic consequence of the strong subsurface distortion is an apparent renormalization of the anchoring energy as far as the long-range bulk distortion is concerned.