The Phenomenological Functions that Characterize the Surface Free Energy Density of Nematic Liquid Crystals: A Microscopic Analysis

In a recent paper, Faetti proposed a new phenomenological expression for the surface free energy density $F_{\rm s}$ of a nematic liquid crystal in contact with an isotropic substrate. $F_{\rm s}$ depend on director n, on the unit vector k orthogonal to the interface and on their gradients at the interface. Five different terms appear in the expression of $F_{\rm s}$. The first term is the standard anchoring energy contribution, the second and third terms are elastic contributions that depend on the surface tangential director gradients, the fourth and fifth terms are new geometrical contributions that account for the effect of a local curvature of the interfaces. The expression of the surface free energy density is characterized by five phenomenological functions $W(n_k),\,K_{13}^*(n_k),\, K_{24}^*(n_k),\,A_1(n_k),$ and $A_2(n_k)$ where $n_k$ is the scalar product n$\cdot$k. In this paper we give the first microscopic calculation of these new surface functions by using a simplified microscopic model of intermolecular interactions. Surface functions $A_1(n_k),\,A_2(n_k)$, and $K_{24}^*(n_k)$ are found to be of the same order of magnitude of the Frank bulk elastic constants, whilst $K_{13}^*(n_k)=0$. The geometric surface functions $A_1(n_k)$ and $A_2(n_k)$ satisfy the simple relation $A_1(n_k)=-n_k^2\,A_2(n_k)$. Both the results $K_{13}^*(n_k)=0$ and $A_1(n_k)=-n_k^2\,A_2(n_k)$ are shown to be a direct consequence of the invariance with respect to the transform ${\bf n\rightarrow -n}$ of the interaction energy between molecules