Nematic ordering in a system of rods

We consider a system of rodlike molecules interacting with a model pair potential $\phi \left(\frac{r}{\sigma }\right)$, where $r$ is the center-of-mass distance and $\sigma$ is an angle-dependent range parameter. It is shown that if the pair correlation function $g$ scales as $g\left(\frac{r}{\sigma }\right)$, then the orientational degrees of freedom decouple from the translational ones to all orders in the density $n$. To lowest order in $n$, our free energy reduces to Onsager's virial-expansion result. The approximation transforms the system into a fluid of spheres interacting with a central potential $\phi \mathrm{}\left(y\right)\mathrm{}$ modulated by an angle-dependent excluded-volume term. The order parameter $Q$ is predicted to have a discontinuity $Q_c$ at the transition point, which is independent of $\phi \mathrm{}\left(y\right)\mathrm{}$ but which slowly increases with the axial ratio $k$ of the molecules. Calculations are done for the hard-rod fluid using the equation of state for the transformed hard-sphere fluid. A transition to an orientationally ordered state occurs at a critical value of the packing fraction ${\eta }_c$, which decreases with $k$. For $k\lesssim 3.5$, ${\eta }_c$ is so high that the system tends to crystallize before the ordered liquid state is reached. Using perturbation theory, calculations are done for a soft-rod fluid, where the transformed system of spheres interacts with a potential $\phi \mathrm{}\left(y\right)=\frac{\epsilon }{y^m}$. The transition temperature as a function of density and molecular shape is obtained, and it is shown that the order parameter obeys the scaling law $Q=Q\left(\eta {\left(\frac{\epsilon }{\mathrm{kt}}\right)}^{\frac{m}{3}}\right)$. For $m=12\mathrm{}$ this leads to good agreement with experiment and shows that most of the features of the transition can be explained with purely repulsive interactions.