A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons

Abstract

We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions, in a reduced two-dimensional Landau-de Gennes framework, discussing their relevance in the full three-dimensional framework too. We work at a fixed temperature and study the reduced stable equilibria in terms of the edge length, $\lambda$ of the regular polygon, $E_K$ with $K$ edges. We analytically compute a novel "ring solution" in the $\lambda \to 0$ limit, with a unique point defect at the centre of the polygon for $K\ne 4$. The ring solution is unique. For sufficiently large $\lambda$, we deduce the existence of at least $\left[K/2\right]$ classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of ${\lambda }^2$, thus illustrating the effects of geometry on the structure, locations and dimensionality of defects in this framework.

arXiv comment21 pages, 12 figures

References in corpus (9)

• Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond
• Construction of a pathway map on a complicated energy landscape
• Instability of point defects in a two-dimensional nematic liquid crystal model
• Topology and Bistability in liquid crystal devices
• Pathways connecting two opposed bilayers with a fusion pore: A molecularly-informed phase field approach
• The Well Order Reconstruction Solution for Three-Dimensional Wells, in the Landau-de Gennes theory
• Bifurcation analysis in a frustrated nematic cell
• Tailored Morphologies in 2D Ferronematic Wells
• Polymeric nematics of associating rods: phase behavior, chiral propagation and elasticity

Cited by in corpus (8)

• Solution landscape of a reduced Landau-de Gennes model on a hexagon
• Defect patterns of two-dimensional nematic liquid crystals in confinement
• Tailored Nematic and Magnetization Profiles on 2D Polygons
• Solution Landscapes of the Simplified Ericksen--Leslie Model and its Comparison with the Reduced Landau--de Gennes Model
• Colloidal smectics in button-like confinements: experiment and theory
• Hierarchies of Critical Points of a Landau-de Gennes Free Energy on Three-Dimensional Cuboids
• Exact and approximate solutions for elastic interactions in a nematic liquid crystal
• Bayesian Parameter Identification in the Landau-de Gennes Theory for Nematic Liquid Crystals