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

TL;DR

This paper investigates how a continuous distribution of rod lengths affects isotropic-nematic phase separation in the Onsager theory, revealing conditions for complex phase coexistence including three-phase regions.

Contribution

It demonstrates that length polydispersity can induce three-phase coexistence in hard rod systems, even with single-peaked distributions, and analyzes the effects of distribution tails on phase behavior.

Findings

A finite upper cutoff is necessary for well-posed phase analysis.

The cloud curve exhibits a kink indicating a three-phase I-N-N region.

Long rods dominate phase behavior at large cutoffs, causing phase separation at any nonzero density.

Abstract

We analyse 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 behaviour, 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 large cutoff, both in good agreement with the numerical results.