Density-functional theory of curvature elasticity in nematic liquids. II. Effect of long-range dispersion interactions

The theory of curvature elasticity in nematics described in earlier papers is reformulated and simplified. The extent to which the ‘‘splay,’’ twist, and ‘‘bend’’ constants ($K_1$,$K_2$,$K_3$) of nematic liquid crystals differ from one another and the way they depend upon the degree of alignment (as characterized by the nematic order parameters P${¯}_n$) are determined by the structure of the medium which in turn is characterized by the correlation functions. The direct correlation functions of the isotropic liquid in terms of which the elastic constants are expressed have broad functional similarities for systems with rather different intermolecular potentials. An approximate scheme is proposed to estimate the values of the direct pair correlation function. Numerical results for the elastic constants are presented for a range of hard-core sizes and shapes and dispersional strengths and anisotropies. It is found that the effect of the long-range dispersion interactions on the elastic constants is determined not only by the angle-dependent part of the forces, but, and more significantly, by the coupling between the isotropic part and anisotropic hard-core repulsions through the pair correlation function. The attractive interactions are found to make significant contributions to $K_i$ in ordinary nematics (nematics of rod-shaped molecules) when $x_0$ (length-to-width ratio) is small (≲3.0) and in discotic nematics when 1/$x_0$ is large (i.e., ≳3.0). Qualitative features exhibited by our calculation for both the ordinary and the discotic nematics are in agreement with the experiment.