  We study the phase behaviour of the Zwanzig model of suspensions of hard
rods, allowing for polydispersity in the lengths of the rods. In spite of the
simplified nature of the model (rods are restricted to lie along one of three
orthogonal axes), the results agree qualitatively with experimental
observations: the coexistence region broadens significantly as the
polydispersity increases, and strong fractionation occurs, with long rods found
preferentially in the nematic phase. These conclusions are obtained from an
analysis of the exact phase equilibrium equations. In the second part of the
paper, we consider the application of the recently developed ``moment free
energy method'' to the polydisperse Zwanzig model. Even though the model
contains non-conserved densities due to the orientational degrees of freedom,
most of the exactness statements (regarding the onset of phase coexistence,
spinodals, and critical points) derived previously for systems with conserved
densities remain valid. The accuracy of the results from the moment free energy
increases as more and more additional moments are retained in the description.
We show how this increase in accuracy can be monitored without relying on
knowledge of the exact results, and discuss an adaptive technique for choosing
the extra moments optimally.
