A Reduced Landau-de Gennes Study for Nematic Equilibria in Three-Dimensional Prisms

TL;DR

This paper investigates nematic liquid crystal configurations in three-dimensional polygonal prisms using a reduced Landau-de Gennes model, linking 3D critical points to 2D solution landscapes through asymptotic analysis and numerical examples.

Contribution

It introduces a novel approach connecting 3D nematic equilibria with 2D solution landscapes via boundary conditions in a reduced Landau-de Gennes framework.

Findings

Establishes a correspondence between 3D critical points and 2D pathways.

Analyzes multistability in cuboid and hexagonal prisms.

Provides numerical examples illustrating the solution landscape.

Abstract

We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet conditions on the top and bottom surfaces are special in the sense, that they are critical points of the reduced Landau-de Gennes energy on the polygonal cross-section. The choice of the boundary conditions allows us to make a direct correspondence between the three-dimensional Landau-de Gennes critical points and pathways on the two-dimensional Landau-de Gennes solution landscape on the polygonal cross-section. We explore this concept by means of asymptotic analysis and numerical examples, with emphasis on a cuboid and a hexagonal prism, focusing on three-dimensional multistability tailored by two-dimensional solution landscapes.