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 $\rho$, 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 $\rho$ 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 $\rho =0.23\mathrm{}\pm 0.01$. Maxwell construction yields a phase-change region for $0.19\pm 0.01<\mathrm{}\rho <0.29\mathrm{}\pm 0.01$. 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 $0.20\pm 0.01<\mathrm{}\rho <0.30\mathrm{}\pm 0.01$ with $\frac{P{A}^{\prime }}{\mathrm{NkT}}=1.38\mathrm{}\pm 0.03, A^{\prime }$ being the cross-sectional area of a close-packed system of $N$ rods.