Isotropic-nematic phase equilibria in the Onsager theory of hard rods with length polydispersity

We analyze the effect of a continuous spread of particle lengths on the phase behavior of rodlike particles, using the Onsager theory of hard rods. Our aim is to establish whether "unusual" effects such as isotropic-nematic-nematic (I-N-N) phase separation can occur even for length distributions with a single peak. We focus on the onset of I-N coexistence. For a log-normal distribution, we find that a finite upper cutoff on rod lengths is required to make this problem well posed. The cloud curve, which tracks the density at the onset of I-N coexistence as a function of the width of the length distribution, exhibits a kink; this demonstrates that the phase diagram must contain a three-phase I-N-N region. Theoretical analysis shows that in the limit of large cutoff, the cloud point density actually converges to zero, so that phase separation results at any nonzero density; this conclusion applies to all length distributions with fatter-than-exponentail tails. Finally, we consider the case of a Schulz distribution, with its exponential tail. Surprisingly, even here the long rods (and hence the cutoff) can dominate the phase behavior, and a kink in the cloud curve and I-N-N coexistence again result. Theory establishes that there is a nonzero threshold for the width of the length distribution above which these long-rod effects occur, and shows that the cloud and shadow curves approach nonzero limits for a large cutoff, both in good agreement with the numerical results.