Temperature dependence of the nematic anchoring energy:mean field analysis

In the mean field approximation, we evaluate the temperature dependence of the anchoring energy strength of a nematic liquid crystal in contact with a solid substrate due to thermal fluctuations. Our study is limited to the weak anchoring case, where the microscopic surface energy is small with respect to the mean field energy due to the nematic phase. We assume furthermore that the physical properties of the substrate can be considered temperature independent in the range of the nematic phase. According to the thermodynamical perturbative approach, the macroscopic surface energy is deduced by averaging the microscopic one, with a density matrix containing only the nematic mean field. We show that the thermal renormalization of the anchoring energy coefficients is proportional to the generalized nematic order parameters.