A conservation law for liquid crystal defects on manifolds

Abstract

The analysis of nematics shells has recently become of great importance, with novel applications ranging from the creation of colloidal materials using DNA strands, to inventing contact lenses capable of changing their dioptre. In this piece, we analyse the orientation of a thin nematic film on the surface of a smooth manifold, specifically the strength of point defects located on the surface and the boundary. We model the orientation by a unit vector field which is orthogonal to the surface normal, in this model defects are points where the localised index is non-zero, this motivates us to derive a generalisation to the Poincare-Hopf theorem, which connects the total index of a vector field to the Euler characteristic of the surface, the sum of the interior angles and the integral of the boundary data. In liquid crystals one of the first models derived is the Oseen-Frank energy density, in this piece we consider the one constant approximation with respect to the curved geometry. The energy density diverges to infinity as one approaches a point defect of non-zero strength. Thus for a given manifold and vector field we derive a lower bound for the rate of energy divergence. This divergence rate is a function of the defect strengths and local curvatures, thus observable configurations of defects must minimise this rate of divergence, whilst also obeying the generalisation to the Poincare-Hopf theorem we derived earlier. Thus using these principle we create a heuristic method to predict the defect strengths of a nematic shell given purely geometrical parameters and boundary data. This is then contrasted with known results from both the experimental and numerical results within the liquid crystal community.

Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

“…We expect the limiting profiles in Section III to be good approximations for materials with small elastic anisotropy as in [15], although the elastic anisotropy will destroy some of the symmetries of the limiting profiles. A further extension concerns arrays of polygonal exclusions [26] or fully 3D systems, which offer exotic possibilities for defect structures such as linked defect lines etc. Our overarching goal is to propose universal theoretical frameworks for solution landscapes of confined partially ordered systems, with multiple order parameters.…”