Uniaxial versus Biaxial Character of Nematic Equilibria in Three Dimensions

TL;DR

This paper analyzes the behavior of nematic liquid crystal minimizers in three dimensions, showing convergence to harmonic maps, the coexistence of uniaxial and biaxial regions near singularities, and the instability of purely uniaxial configurations at low temperatures.

Contribution

It provides new insights into the low-temperature limit of Landau-de Gennes minimizers, including convergence results, the structure of biaxial and uniaxial regions, and stability analysis of uniaxial states.

Findings

Minimizers converge uniformly to harmonic maps away from singularities.

Existence of biaxial points and uniaxial regions near singularities.

Uniaxial minimizers are unstable at low temperatures.

Abstract

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to the low-temperature limit. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) there exist both a point of maximal biaxiality and a nonempty Lebesgue-null set of uniaxial points near each singular point of the limiting harmonic map (this improves the recent results of \cite{contreraslamy}); (iii) estimates for the size of "strongly biaxial" regions in terms of the reduced temperature $t$. We further show that global LdG minimizers in the restricted class of uniaxial $\Qvec$-tensors cannot be stable critical points of the LdG energy for low temperatures.