On the stationary solutions of Doi-Onsager model in general dimension

TL;DR

This paper investigates phase transitions in the Doi-Onsager model for rod-like molecules in dimensions three and higher, establishing uniqueness of isotropic solutions at low concentrations and analyzing bifurcations to nematic phases.

Contribution

It provides new mathematical results on phase transitions and bifurcations in the Doi-Onsager model with general potentials in higher dimensions.

Findings

Isotropic phase is unique at low concentration.

Bifurcation regime for nematic phases identified.

Method extends Leray-Schauder degree to this context.

Abstract

We give new results of the phase transition of dilute colloidal solutions of rod-like molecules in dimension $D \geq 3$. For the low concentration of particles in a carrier fluid, we prove that the isotropic phase is the unique solution to the Doi-Onsager model with the general potential kernel. In addition, we present the regime of the bifurcation of nematic phases in the class of axially symmetric solutions. Our method is based on a generalization of the classical Leray-Schauder degree we developed for this problem.