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, λ of the regular polygon, E K with K edges. We analytically compute a novel "ring solution" in the λ → 0 limit, with a unique point defect at the centre of the polygon for K = 4. The ring solution is unique. For sufficiently large λ, we deduce the existence of at least [K/2] classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of λ 2 , thus illustrating the effects of geometry on the structure, locations and dimensionality of defects in this framework.

Tailored nematic and magnetization profiles on two-dimensional polygons

“…We note that magnetic domain walls are difficult to find with either increasing N or increasing c ∈ (0, ∞). The picture with negative c is more complex -we effectively double the number of stable states for small , compared to the results in [29] for c = 0. These stable states are distinguished by vertex defects for Q and vertex vortices for M, so that the multistability is strongly enhanced with increasing N , for c < 0.…”

“…Boundary conditions are a crucial consideration for confined systems. We impose fixed/Dirichlet tangent boundary conditions for the nematic director on the polygon edges, and these boundary conditions create a natural mismtach for the nematic director at the polygon vertices, making them natural candidates for defect sites [23,28,29]. Tangent boundary conditions are well accepted for confined NLC systems both experimentally and theoretically; see [30].…”

“…Specifically, we numerically compute the energy-minimizing (Q, M) profiles for different values of N , c and , along with bifurcation diagrams for positive and negative values of c that track the energy-minimizing stable and non energy-minimizing (unstable) solution branches. Mathematically, this corresponds to solving a system of four coupled nonlinear partial differential equations, subject to Dirichlet conditions for Q and M. The NLC system (with c = 0) has been well described in [29] on 2D polygons, where the authors demonstrate a unique Ring solution profile with a unique nematic point defect at the center, which is the generic stable solution for polygons except for the square for large . The authors find at least N 2 stable states for small .…”

“…A key question is -how does this picture respond to the NLC-MNP coupling, captured by the parameter c? There are various new solutions for these nemato-magnetic systems, as will be described in the sections below, which are not reported for the c = 0 case in [29]. Some notable findings for positive c concern the coexistence of stable (Q, M) profiles with nematic defects pinned at the polygon vertices and magnetic domain walls along polygonal diagonals and polygon edges, that separate distinct domains of magnetization; along with stable P eppa-(Q, M) profiles.…”

Surface, size and topological effects for some nematic equilibria on rectangular domains

“…Our arguments are quite generic and we can extend them to arbitrary 2D polygons. In [37], the authors studied reduced 2D LdG equilibria on regular polygons and in the ϵ → ∞ limit, they report that the limiting profiles have a unique isotropic point at the centre of the regular polygon, with the exception of the square. We can work with irregular polygons too and we speculate that the location of the zeros or isotropic points are determined by the geometrical anisotropies.…”