Lattice cluster theory for phase behavior of rectangular mesogens

Abstract

The lattice cluster theory is generalized to study a system of hard, rigid, rectangular mesogens, which may exhibit isotropic liquid and nematic and discotic nematic liquid crystalline phases. A common partition function for all three phases is derived as a functional of the different possible orientational distribution functions. The partition function is expressed in the familiar form of a cluster expansion with a zeroth order uncorrelated mean field portion and corrections originating from correlations between different numbers of mesogens. The entropies for the three phases are evaluated to second order in mesogen density φ by including in the cluster expansion only corrections originating from correlations containing two objects. (A previous paper illustrates the application of the general theory to order φ4 for the packing entropy of the smallest square objects, and work is currently in progress to pursue these technically involved fourth order computations for the present case.) Models with different dimensionality represent the influence of greater orientational freedom and are shown to produce qualitatively different phase diagrams. One model for extra orientational freedom is obtained by considering the single object entropy lost upon ordering, while another model is generated by comparing the d‐dimensional lattice model second virial coefficient with that of the continuous space three‐dimensional object.