RESUMMATION OF HIGHER-ORDER TERMS IN THE FREE-ENERGY DENSITY OF NEMATIC LIQUID-CRYSTALS

The presence of the surfacelike elastic constant K13 in the expression of the elastic free-energy density F2 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 K13 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. In the first part of this paper we propose a simple and rigorous test based on the general principles of mechanics to establish the internal consistency of these very different theoretical approaches. The second-order elastic theory is found to satisfy this test, while the Hinov-Pergamenshchik model is found to be in contrast with it. In the second part of this paper we make a systematic expansion of the free energy at any order in the director derivatives and we analyze the physical effects of the higher-order contributions that were disregarded by the second-order theory. At any expansion order a strong subsurface director distortion is predicted to occur and its macroscopic effect is shown to be equivalent to an apparent renormalization of the anchoring energy. Therefore, the main qualitative predictions of the second-order elastic theory remain satisfied at any expansion order, although the quantitative behavior of the system is found to be greatly affected by higher-order contributions.