Molecular theory for freezing of a system of hard ellipsoids: Properties of isotropic-plastic and isotropic-nematic transitions

The theory for isotropic-nematic transition at constant pressure described in an earlier paper is extended to include isotropic-plastic transition. The transition is located from the structural information about the liquid using the first-principles order-parameter theory of freezing. This theory makes the role of the structure of the medium explicit and the role of the intermolecular interaction implicit. For the plastic phase, order parameters are the coefficients of a Fourier expansion of the spatially varying single-particle density ρ(r,Ω) in terms of the reciprocal lattice of the plastic. For the nematic phase, order parameters are the coefficients of a spherical harmonic expansion of an orientational singlet distribution. The theory predicts that the equilibrium positional freezing (plastic) on fcc lattice takes place for the value of c¯${^}_0$,$0^1$ [=1-1/S(‖${\mathrm{G}}_m$‖), where S(‖${\mathrm{G}}_m$‖) is the first peak in the structure factor of the center of mass] ≃0.67 [or S(‖${\mathrm{G}}_m$‖)≃3.07]. The equilibrium orientational freezing (nematic) takes place when the orientational correlation c¯${^}_2$,$2^0$≃4.45. For a simple model of hard ellipsoids of revolution parametrized by length-to-width ratio $X_0$, we find that the plastic phase stabilizes first for 0.57≲$X_0$≲1.75 and the nematic phase for $X_0$<0.57 and $X_0$>1.75. These values are in reasonable agreement with the computer-simulation results. We also find, in agreement with computer simulation, a remarkable symmetry between the systems with inverse length-to-width ratios.