Nematic equilibria in isosceles triangles: The effects of edge length and apex angle on solution landscapes in a reduced Landau-de Gennes framework

Abstract

We study equilibrium configurations of nematic liquid crystals confined to two-dimensional isosceles triangles, subject to tangent boundary conditions. This toy problem is motivated by the effects of geometrical asymmetry on equilibria in variational problems arising in liquid crystal theory. There are two key geometrical parameters for an isosceles triangle - the triangle edge length and the apex angle. The nematic equilibria are modelled by minimizers of a reduced Landau-de Gennes free energy in this setting. For small edge lengths, we provide a universal, angle-based local classification of nematic equilibria near the vertices as to whether the nematic director exhibits a splay, bend or singular profile depending on the vertex opening angle. In the large domain limit, we demonstrate the existence of multiple competing nematic equilibria -- the three rotated solutions, for which the nematic director bends between a pair of adjacent vertices, and a \emph{trefoil} solution featuring an interior point defect. For acute apex angles, we show that the trefoil solution is stable for small edge lengths. The interior point defect of the trefoil solution migrates to one of the base vertices, as the edge length increases, and is finally expelled giving way to the rotated solutions, if the apex angle is small enough. Our numerical results suggest that there is a unique trefoil solution on the equilateral triangle for all edge lengths, and a unique rotated solution on isosceles triangles with wide apex angles. These results yield interesting insight into how geometrical asymmetry can tailor equilibria and self-assembly processes in confined nematic systems.

Figures (9)

• Figure 1: Geometry of the LC sample: Isosceles triangle $\Delta OMN$ of leg length unity and apex angle $2\alpha\in(0,\pi)$.
• Figure 2: Representative solutions of \ref{['eq:lambda0']} with tangent boundary conditions: the black lines label the nematic director field with $2\alpha = 30^\circ$, $2\alpha = 60^\circ$, $2\alpha = 90^\circ$ and $2\alpha = 120^\circ$. The colour bar corresponds to the scalar order parameter $s=2\sqrt{q_1^2+q_2^2}$.
• Figure 3: An arbitrary splay and bend corner with a corner angle $2\alpha'\in(0,2\pi)$, described in terms of the local polar coordinates $(r',\theta')$.
• Figure 4: Plots of the similarity solution \ref{['eq:similarity']} for different choices of $2 \alpha'$. Defects emerge along $\theta'=0$ when $2\alpha'=90^\circ$ and $2\alpha'=270^\circ$.
• Figure 5: The three rotated solutions and the trefoil solution (top row) for an equilateral triangle, computed using \ref{['phieq']} complemented by the boundary conditions illustrated in the bottom row.