Systematic Solution of the Mean Field Equation for Liquid Crystals

Starting with a pairwise, spatially and orientationally dependent intermolecular potential of the Kobayashi-McMillan form, we carry out a systematic solution of the mean field equation for liquid crystals. The mean field equation, presented as first of a hierarchy of BBGKY equations, is first reduced to a set of coupled integrodifferential equations by means of expanding the distribution function f (r, δ) and In f (r, δ) in Legendre polynomials and the reciprocal lattice space. In the first level of approximation, the expansion retains only the lowest-order coefficients, permitting a complete decoupling of the equations. In the second level of approximation, the leading coefficient which couples spatial order to orientational order is included. In the third level of approximation two more higher order coefficients are included. At each level, the free energy functional is evaluated to determine the equilibrium phase at given temperatures and chainlengths of a homologous series. It is shown that the expansion converges very rapidly, the second level of approximation being entirely sufficient. This lends support to our earlier variational calculation which contained only three variational parameters.