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 $K \neq 4$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2019
- DOI:
- 10.48550/arXiv.1910.05740
- arXiv:
- arXiv:1910.05740
- Bibcode:
- 2019arXiv191005740H
- Keywords:
-
- Mathematical Physics
- E-Print:
- 21 pages, 12 figures