Elasticity of nematic phases with fundamental measure theory

In a previous publication [R. Wittmann, M. Marechal, and K. Mecke, Europhys. Lett. 109, 26003 (2015)], we introduced fundamental mixed measure theory (FMMT) for mixtures of anisotropic hard bodies, which shows that earlier results with an empirical parameter are inaccurate. Now we provide a deeper insight into the background of this theory in integral geometry. We study the Frank elastic coefficients in the nematic phase of the hard spherocylinder fluid. The framework of FMMT provides us with the required direct correlation function without additional input of an equation of state. A series representation of the mixed measure gives rise to closed analytical formulas for the elastic constants that only depend on the density, order parameters, and the particle geometry, pointing out a significant advantage of our geometry-based approach compared to other density functionals. Our elastic coefficients are in good agreement with computer simulations and increase with the density and the nematic order parameter. We confirm earlier mean-field predictions in the limits of low orientational order and infinitely long rods.