Stochastic contribution to the anchoring energy: Deviation from the Rapini-Papoular expression

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Abstract
The effect of a stochastic contribution to the surface energy, coming from the direct interaction between an orienting film and a solid substrate, is considered. It is shown that in the hypothesis in which the nematic liquid crystal orientation coincides with the film orientation, the well known Rapini-Papoular expression for the anisotropic part of the surface energy is modified. More precisely the anchoring strength, connected with the square of the sine of the deformation angle, is renormalized. Furthermore, there appears an additional contribution, proportional to the fourth power of the same quantity, whose coefficient is equal in modulus and opposite in sign. The physical origin of the considered effect is connected with the elastic contribution to the surface energy of the film.