Isotropic-nematic transition of D-dimensional hard convex-bodies within the effective-liquid approach

Abstract

Density-functional theory within the effective-liquid approximation is applied to the problem of the isotropic-nematic transition of D-dimensional hard convex bodies. It is shown that the free-energy functional factorizes into its radial and angular contributions. Due to this factorization two different versions of the self-consistent equations can be implemented, and it is shown that in D = 3 they coincide with previous theories. In the present work all the formulas are worked out with a particular choice for the angular distribution: the one-order-parameter approximation. The problem of determining the excluded volume of two hard convex bodies is discussed. For hard ellipsoids the Gaussian-overlap approximation is used, whereas an exact formula is given for the excluded volume of two hard spherocylinders. For D = 2 the virial coefficients of the isotropic phase as well as the transition are incorrectly predicted, due to the approximation of the direct correlation function involved. For D = 3 the results are in very good agreement with simulations. Expression and data for the isotropic-nematic transition for D > 3 are also provided. Extensive comparisons with the results of other theories are made throughout. The one-order-parameter approximation is proven not to alter the order of the transition. Finally, it is shown that the present approximation becomes exact in the large-D limit.