Statistical Mechanics of Rodlike Particles. III. A Fluid of Rigid Spherocylinders with Restricted Orientational Freedom

A scaled particle treatment is applied to a fluid of rigid spherocylinders of radius $a$ and cylindrical length $l$ , with cylindrical axes permitted to point in only three mutually perpendicular directions. Singling out one of these, by means of some arbitrary device such as an infinitesimal applied field, and denoting by $s$ the fraction of rods in each of the directions perpendicular to this “preferred” direction in any given configuration of the system, an expression is derived for the configurational Helmholtz free energy $A_c$ as a function of $s$ , and an equation determining $\mathrm{s̄}$ , that value of $s$ which minimizes $A_c$ , is obtained. At densities below some critical value ${\rho }_c$ , the only solution to the latter equation is $\mathrm{s̄\hspace{0.17em}=\hspace{0.17em}}\frac{1}{3}$ , but for ${\mathrm{\rho \hspace{0.17em}>\hspace{0.17em}\rho }}_c$ , an anisotropic solution is also possible. A first‐order anisotropic–isotropic transition is observed and the transition densities, free volume fractions, and entropy change are determined at two values of $a$ and for ratios $\mathrm{l\hspace{0.17em}/\hspace{0.17em}a}$ from 2 through 18. Thus, that transition which Zwanzig observed in a fluid of very long rods $\mathrm{(l\hspace{0.17em}/\hspace{0.17em}a\hspace{0.17em}\to \hspace{0.17em}\infty )}$ is also observed in fluids of relatively short rigid rods. The transition free volume fractions, the entropy of transition, the degree of order $S$ in the anisotropic phase at the transition, and the relative density change ${\mathrm{\Delta \rho \hspace{0.17em}/\hspace{0.17em}\rho }}^{\mathrm{aniso}}$ are all seen to increase with increasing $\mathrm{l\hspace{0.17em}/\hspace{0.17em}a}$ , and values obtained for the latter two quantities are compared with experimentally determined values. The isothermal compressibility and the expansivity of the system are also calculated at free volume fractions (densities) near the anisotropic→isotropic transition, but no pretransition effects are observed.