Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals

TL;DR

This paper analyzes the bifurcation structure of the Onsager free-energy functional for liquid crystals in three dimensions, providing explicit bifurcation points, symmetry reductions, and characterizing the nature of the first bifurcation.

Contribution

It introduces a general method for finding eigenvalues and eigenfunctions of the kernel operator, and characterizes the bifurcation diagram for a broad class of interaction potentials including Onsager's.

Findings

First bifurcation is a transcritical bifurcation.

The initial bifurcating solution is uniaxial.

The trivial solution is globally minimal at high temperatures.

Abstract

We study the bifurcation diagram of the Onsager free-energy func- tional for liquid crystals with orientation parameter on the sphere in three dimensions. In particular, we concentrate on a very general class of two- body interaction potentials including the Onsager kernel. The problem is reformulated as a non-linear eigenvalue problem for the kernel operator, and a general method to find the corresponding eigenvalues and eigenfunctions is presented. Our main tools for this analysis are spherical harmonics and a special algorithm for computing expansions of products of spherical harmon- ics in terms of spherical harmonics. We find an explicit expression for the set of all bifurcation points. Using a Lyapunov-Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the theory of invariant polynomials. On the basis of these result, we show that the first bifurcation occurring in case of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global proper- ties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser for high temperatures, that it is not a local minimiser if the temperature is low, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.