Nematic equilibria on a two-dimensional annulus: defects and energies

TL;DR

This paper analyzes the stability and energy of nematic liquid crystal configurations on a 2D annulus, revealing how defects, geometry, and anchoring influence equilibrium states within Oseen-Frank and Landau-de Gennes theories.

Contribution

It provides a comprehensive stability analysis and constructs defect configurations, offering new insights into defect behavior and energies in nematic annuli.

Findings

Derived stability criteria for defect-free states based on geometry and anchoring.

Constructed and analyzed energies of multiple defect configurations.

Extended stability results to the Landau-de Gennes framework using Mironescu's theorem.

Abstract

We study planar nematic equilibria on a two-dimensional annulus with strong and weak tangent anchoring, within the Oseen-Frank and Landau-de Gennes theories for nematic liquid crystals. We analyse the defect-free state in the Oseen-Frank framework and obtain analytic stability criteria in terms of the elastic anisotropy, annular aspect ratio and anchoring strength. We consider radial and azimuthal perturbations of the defect-free state separately, which yields a complete stability diagram for the defect-free state. We construct nematic equilibria with an arbitrary number of defects on a two-dimensional annulus with strong tangent anchoring and compute their energies; these equilibria are generalizations of the diagonal and rotated states observed in a square. This gives novel insights into the correlation between preferred numbers of defects, their locations and the geometry. In the Landau-de Gennes framework, we adapt Mironescu's powerful stability result in the Ginzburg-Landau framework (P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, 1995) to compute quantitative criteria for the local stability of the defect-free state in terms of the temperature and geometry.