Full energy expression of a uniaxial nematic phase with spatially dependent density and order parameters: From microscopic to macroscopic theory

We present a microscopic derivation of the full macroscopic energy expression of a spatially bounded uniaxial nematic phase. The surface is described by spatial variations of the density and scalar order parameters of all even orders. The method developed in the paper allowed us to unambiguously separate the surface elastic K-24 and K-13 terms and isotropic and anisotropic surface tension (anchoring). The full energy expression incorporating variations of the director, scalar order parameters, and density is obtained. The macroscopic coefficients are derived in terms of the isotropic and anisotropic fractions of the microscopic intermolecular interaction. An important physical consequence of the obtained formulas, in particular, is that the observed considerable difference K-33-K-11 between the bend and splay elastic constants unambiguously indicates that (i) the intermolecular interaction has a large anisotropic fraction, and thus, the effective constant K-13 and intrinsic anchoring are considerable; (ii) at least some scalar order parameters of order four and higher are essentially nonzero. Relation of the developed theory with the Nehring-Saupe theory and Landau-de Gennes approach is considered.