A density functional analysis of the restricted orientation model of liquid crystals and its implications for theories of orientational ordering

Abstract

We compare the predictions of various density functional approximation schemes as applied to the restricted-orientation, hard, rectangular-parallelepiped model of liquid crystals. These calculations illustrate some features of density functional calculations which may aid in understanding other approximate, mean-field theories of orientational ordering phase transitions and their interfaces. The isotropic–nematic coexistence curve of this model is apparently well described by the so-called third-order y expansion, and the analytical expression for the nonideal free energy derived from the simple y expansion thus provides a benchmark against which to compare the density functional approximation schemes. Using standard relations applicable to inhomogeneous fluids, we show how expansions about the bulk isotropic phase compare more favorably with the ‘‘exact’’ y-expansion results when truncated at third order than do second-order truncations. Thus this model behaves somewhat differently than other hard-particle models of fluids, notably the hard-sphere one. We also examine the possibility of expanding the free energy about the ordered phase to obtain the properties of a disordered phase. An expansion about local values of the density leads to the widely used smoothed-density approximation and a hierarchy of systematic extensions. The latter appear more stable than conventional Taylor expansions about bulk disordered phases.