Torus-like solutions for the Landau–De Gennes model.

Adriano Pisante

doi:10.5802/afst.1676

Abstract

In this note we report on some recent progress [10, 11, 12] about the study of global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the related biaxial escape phenomenon. Then, under suitable assumptions on the topology of the domain and on the Dirichlet boundary condition, we show that smoothness of energy minimizing configurations yields the emergence of nontrivial topological structure in their biaxiality level sets. Then, we discuss the previous properties under both the norm constraint and an axial symmetry constraint, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis. In addition, we show that singularities may appear due to energy efficiency and we describe precisely the asymptotic profile around singular points. Finally, in an appropriate class of domains and boundary data we obtain qualitative properties of the biaxial surfaces, showing that smooth minimizers exibit torus structure, as predicted in [16, 24, 25, 39].

Highlights

Nematic liquid crystals are mesophases of matter intermediate between crystalline solids and isotropic fluids
For a nematic liquid crystal filling a region Ω ⊆ R3 the key feature of a orientational order can be modeled in different ways, depending on the choice of parameters
In order to discuss the properties of liquid crystal configurations, following [12] we find convenient to modify the usual definition of the biaxiality parameter as follows

Summary

Introduction

Nematic liquid crystals are mesophases of matter intermediate between crystalline solids and isotropic fluids (see [17]). Nematic molecules typically have elongated shape, approximately rod-like, and the interaction between them yields a local (mean) orientational order. For a nematic liquid crystal filling a region Ω ⊆ R3 the key feature of a (mean) orientational order can be modeled in different ways, depending on the choice of parameters. In order to discuss the properties of liquid crystal configurations (such as isotropic/nematic phase transition, biaxial escape), following [12] we find convenient to modify the usual definition of the biaxiality parameter as follows. The goal here is to account on some results from [10, 11, 12] on the regularity of minimizers and the emergence of topological structure in the corresponding biaxial surfaces {β ◦ Q( · ) = t}, t ∈ (−1, 1), which in the model case of a nematic droplet are expected to be of torus type [16, 24, 25, 39]

Energy functionals and Lyuksyutov constraint

Regularity of energy minimizing configurations

Topology of minimizing configurations

Axially symmetric minimizing configurations

Asymptotic behaviour at axially symmetric singularities

Torus solutions vs split solutions