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arXiv · 1910.05740

A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons

Yucen Han·Apala Majumdar·Lei Zhang
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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.

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Yucen Han, Apala Majumdar, Lei Zhang. 2019-10-13. A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons. https://doi.org/10.1137/19m1293156

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