We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by λ, and the well height denoted by . The WORS are distinctive equilibria reported in [10] for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square centre. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic surface energies on the top and bottom well surfaces, along with Dirichlet conditions on the lateral surfaces. Moreover, the WORS is globally stable for λ small enough in both cases and unstable as λ increases. We numerically compute novel mixed 3D solutions for large λ and followed by a numerical investigation of the effects of surface anchoring on the WORS, exemplifying the relevance of the WORS solution in a 3D context.

We study dilute suspensions of magnetic nanoparticles in a nematic host, on two-dimensional (2D) polygons. These systems are described by a nematic order parameter and a spontaneous magnetization, in the absence of any external fields. We study the stable states in terms of stable critical points of an appropriately defined free energy, with a nemato-magnetic coupling energy. We numerically study the interplay between the shape of the regular polygon, the size of the polygon and the strength of the nemato-magnetic coupling for the multistability of this prototype system. Our notable results include (i) the co-existence of stable states with domain walls and stable interior and boundary defects, (ii) the suppression of multistability for positive nemato-magnetic coupling, and (iii) the enhancement of multistability for negative nemato-magnetic coupling.

We study the effects of elastic anisotropy on the Landau-de Gennes critical points for nematic liquid crystals, in a square domain. The elastic anisotropy is captured by a parameter, L2, and the critical points are described by three degrees of freedom. We analytically construct a symmetric critical point for all admissible values of L2, which is necessarily globally stable for small domains i.e., when the square edge length, λ, is small enough. We perform asymptotic analyses and numerical studies to discover at least 5 classes of these symmetric critical points -the W ORS, Ring ± , Constant and pW ORS solutions, of which the W ORS, Ring + and Constant solutions can be stable. Furthermore, we demonstrate that the novel Constant solution is energetically preferable for large λ and large L2, and prove associated stability results that corroborate the stabilising effects of L2 for reduced Landau-de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of L2, which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

We study the effects of elastic anisotropy on Landau–de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, L 2 , and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of L 2 , which is necessarily globally stable for small domains, i.e. when the square edge length, λ , is small enough. We perform asymptotic analyses and numerical studies to discover at least five classes of these symmetric critical points—the WORS , Ring ± , C o n s t a n t and p W O R S solutions, of which the WORS , Ring + and C o n s t a n t solutions can be stable. Furthermore, we demonstrate that the novel C o n s t a n t solution is energetically preferable for large λ and large L 2 , and prove associated stability results that corroborate the stabilizing effects of L 2 for reduced Landau–de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of L 2 , which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

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