Nematic-Isotropic phase transition in Liquid crystals: a variational derivation of effective geometric motions

TL;DR

This paper investigates the nematic-isotropic phase transition in liquid crystals using a variational approach, deriving a geometric motion description involving mean curvature flow and phase behavior separation.

Contribution

It provides a rigorous derivation and justification of the geometric evolution of phase boundaries in liquid crystals at the critical temperature.

Findings

Solution gradient concentrates on a surface evolving by mean curvature flow.

On one side of the surface, the phase tends to nematic governed by harmonic map heat flow.

On the other side, the phase tends to isotropic, with convergence proven via weak convergence and energy methods.

Abstract

In this work, we study the nematic-isotropic phase transition based on the dynamics of the Landau--De Gennes theory of liquid crystals. At the critical temperature, the Landau--De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.