Free energy of a system of hard spherocylinders serving as a simple model for liquid crystals

The free energy for a system of hard spherocylinders with midpoints constrained to random motion in a plane, serving as a zeroth-order approximation to one layer of a smectic liquid crystal or a two-dimensional nematic liquid crystal, has been calculated for spherocylinders with a length-to-width ratio of 5. For , the number density measured in fractions of close-packed density, less than 0.22, the partition function itself is evaluated by means of a Monte Carlo scheme employing 22 500 mesh points and 82 possible angles for 25 particles with periodic boundary conditions. For all the liquid-crystal free energy is calculated by minimizing a function of the hard-disk free energy plus the orientational free energy of a "liquid crystal." The low-density Monte Carlo free energy is found to lie below the liquid-crystal free energy, but can be extrapolated to cross it at =0.230.01. Maxwell construction yields a phase-change region for 0.190.01<<0.290.01. A spline polynominal fit to the entire free energy, which interpolates across the phase-change region, does not give strictly constant pressure, but does imply a phase-change region of 0.200.01<<0.300.01 with PANkT=1.380.03, A being the cross-sectional area of a close-packed system of N rods.