A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons

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,withauniquepointdefectatthecentreofthepolygonfor$K \neq 4$\mathrm{.}Theringsolutionisunique\mathrm{.}Forsufficientlylarge$\lambda$, we deduce the existence of at least $\left[K\mathrm{/}2\right]$ 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.