Molecular theory of curvature elasticity in nematic liquids

We extend the generalized van der Waals theory to the case of aligned nematic liquids which are subjected to curvature (i.e., "splay", "twist", and "bend") deformations. The Helmholtz free energy is written as an explicit functional of both the orientational distribution f(Ω) and the spatial variation n̂(R) of the director. The entropy contribution is dominated by the packing statistics of hard anistropic cores and is evaluated by the "y" expansion developed earlier to treat liquid-density repulsive reference systems. The energy contribution involves a mean field averaging of the angle-dependent pair attractions, but with explicit allowance made for the excluded volume correlations associated with the impenetrability of the molecular hard cores. Numerical evaluations of the elastic constants are presented for a range of hard core sizes and shapes and of dispersional strengths and anisotropics. The "energy" and "entropy" contributions are found to be of comparable magnitude, each dominated by terms which go as the square of the usual "P̄ ₂" order parameter. But P̄₂P̄₁ contributions are also significant, accounting for sometimes sizable differences between the splay and bend constants. In particular, the relative magnitudes of these two elastic constants are shown to depend sensitively on the size and shape of the molecular hard core and polarizability. The temperature variations of the elastic constants are similarily accounted for. Finally we derive several recent formulations of curvature elasticity as special cases of the present generalized van der Waals approach.