The Radial Hedgehog Solution in the Landau-de Gennes Theory: Effects of the Bulk Potentials

TL;DR

This paper investigates equilibrium configurations of nematic liquid crystals in spherical droplets using a sixth-order Landau-de Gennes potential, revealing new biaxial phases, stability effects, and bifurcation behaviors compared to the traditional fourth-order potential.

Contribution

It proves the existence and uniqueness of radial hedgehog solutions with a sixth-order potential and identifies new biaxial critical points influenced by the higher-order bulk potential.

Findings

Radial hedgehog solutions exist for all temperatures and sizes.

The sixth-order potential stabilizes biaxial critical points.

Discovery of a novel bulk biaxial critical point driven by the sixth-order potential.

Abstract

We study equilibrium configurations in spherical droplets of nematic liquid crystal with strong radial anchoring, within the Landau-de Gennes theory with a sixth-order bulk potential. The sixth-order potential predicts a bulk biaxial phase for sufficiently low temperatures, which the conventional fourth-order potential cannot predict. We prove the existence of a radial hedgehog solution, which is a uniaxial solution with a single isotropic point defect at the droplet centre, for all temperatures and droplet sizes, and prove that there is a unique radial hedgehog solution for moderately low temperatures, but not deep in the nematic phase. We numerically compute critical points of the Landau-de Gennes free energy with the sixth order bulk potential, with rotational and mirror symmetry, and find at least two competing stable critical points: the biaxial torus and split core solutions, which have biaxial regions around the centre, for low temperatures. The size of the biaxial regions increases with decreasing temperature. We also compare the properties of the radial hedgehog solution with the fourth-order and sixth-order potentials respectively, in terms of the Morse indices as a function of the temperature and droplet radius; the role of the radial hedgehog solution as a transition state in switching processes; and compare the bifurcation plots with temperature, with the fourth- and sixth-order potentials. Overall, the sixth-order potential has a stabilising effect on biaxial critical points and a de-stabilising effect on uniaxial critical points and we discover an altogether novel bulk biaxial critical point of the Landau-de Gennes energy with the sixth-order potential, for which the bulk biaxiality is driven by the sixth-order potential.