The Onsager theory for the phase behavior of rodlike molecules is extended to rods with polydisperse length and solvent-dependent attractive intermolecular forces for the first time. This combination of polydispersity effects and solvent effects is critical for developing a quantitatively accurate model for phase separation in solutions of rodlike molecules. We utilize a phenomenological square-well potential to capture these attractive interactions and combine these interactions with the Onsager excluded volume potential. For polydisperse systems, the isotropic-liquid crystalline phase separation is a function of the initial rod concentration, unlike monodisperse systems. We use Newton's method and arclength continuation to compute isotropic/liquid-crystalline phase separation and length fractionation as a function of solvent quality.

We show that the resulting phase diagram can capture the phase behavior of experimental systems featuring rods with attractive interactions simply by fitting the adjustable parameters of the square-well potential. Specifically, we show that the model closely replicates experimental data points for isotropic/liquid crystalline phase separation of single-walled carbon nanotubes (SWNTs) in various super-acids. The model predicts that the isotropic cloud point goes to zero at a certain solvent quality, and this prediction was subsequently confirmed by experiments. This result means that for acids below a threshhold acid strength, the liquid-crystalline phase is present even at dilute concentrations. The model yields critical insights into both the fundamental thermodynamics as well as practical applications of SWNT/super-acid systems.