Hard-rod fluid: Scaled particle theory revisited

A scaled particle theory is developed for a fluid of hard spherocylindrical rods with fixed orientations. Unlike earlier scaled particle treatments of this system, it maintains thermo-dynamic consistency in that the Maxwell relation $\frac{{\partial \mu }_i}{{\partial \rho }_j}=\frac{{\partial \mu }_j}{{\partial \rho }_i}$, where $\mu$ is the chemical potential and $\rho$ is the number density, is satisfied for all orientations $i$ and $j$ without sacrificing the internal logic or consistency of the scaled particle approach. The resulting expression for the Helmholtz free energy is used to derive the equilibrium thermodynamics of hard spherocylinders that are free to rotate among an arbitrary set of allowed orientations. Numerical results are presented for two special cases: (i) rods with a continuous distribution of orientations and (ii) rods permitted to adopt only three mutually perpendicular orientations. In both instances, the properties of the nematiclike ordered phase and of the anisotropic → isotropic phase transition are determined and compared with experimental data for nematic liquids, as well as with the predictions of other theories of the hard-rod fluid. This theory is shown to reduce to the Reiss, Frisch, and Lebowitz (or Percus-Yevick) theory of hard spheres when the length-to-breadth ratio of the spherocylinders decreases to unity.