Ordering transitions, biaxiality, and demixing in the symmetric binary mixture of rod and plate molecules described with the Onsager theory

The phase behavior of a liquid-crystal forming binary mixture of generic hard rodlike and platelike particles is studied with the theory of Onsager [L. Onsager, Ann. N. Y. Acad. Sci. 51, 627 (1949)] for nematic ordering. The mixture is chosen to be symmetric at the level of the second virial theory, so that the phase behavior of the two pure components is identical. A parameter q is used to quantify the effect of the unlike rod-plate excluded volumes on the phase behavior; a value of $q>\mathrm{}1\mathrm{}$ indicates that the unlike excluded volume is greater than the like excluded volume between the rods or plates, and a value of $q<\mathrm{}1\mathrm{}$ corresponds to a smaller unlike excluded volume. Two methods are used to solve the excluded volume integrals: the approximate L2 model [A. Stroobants and H. N. W. Lekkerkerker, J. Phys. Chem. 88, 3669 (1984)], in which a second-order Legendre polynomial is used; and a numerical method where the integrals are solved exactly. By varying the unlike excluded volume interaction q, the isotropic phase is seen to be stabilized (small q) or destabilized (large q) with respect to the nematic phase for both models. Isotropic-isotropic demixing is also observed for the largest values of q due to the unfavorable contribution of the unlike excluded volume to the entropy of the system. A second-order nematic–biaxial nematic phase transition is observed for small values of q in the L2 approximation, and for all q in the exact calculation; in the latter case the stability of the biaxial phase is enhanced by increasing q, while in the L2 approximation nematic-nematic phase separation is favored. This result is the most striking difference between the two methods, and is in contrast with the results of previous studies. We show that the accuracy of the L2 expansion is particularly poor for parallel and perpendicular particle orientations.