Mechanisms and shape predictions of nematic disclination branching under conical confinement

Liquid crystals (LCs) are self-organizing anisotropic viscoelastic soft materials that flow like viscous liquids and display anisotropies like crystals. When a nematic liquid crystal is confined to a capillary tube with strong anchoring conditions, disclination defects of higher (+1) and lower (+1/2) topological charges can coexist, connected through a defect branch point. The shape of the +1/2 disclination lines emanating from the branch point are functions of confinement and bulk elasticity. Previous work shows that nematic liquid crystals under cylindrical confinement display a radial (one +1 line)-to-planar polar (two +1/2 lines) defect texture transition through the nucleation and uniform motion of a disclination branch point. Here we present analysis, scaling and modeling based on a non-linear non-local nematic elastic equation that shows that a branch point also can be generated from disclinations in a liquid crystal confined to different conical geometries with homeotropic anchoring conditions. The cone aperture increases the bending stiffness but decreases the curvature of the disclination. These competing effects lead to a decrease in the total disclination curvature, increase in elastic energy and volume of the branching region. The results are summarized into power laws and integrated into a shape/energy diagram that reveals the effects of confinement and its gradient (cone angle) on disclination shape selection. These new findings are useful to assess the Frank elasticity of new nematic liquid crystals and to predict novel defect structures in complex confinement, including biological microfluidics and mesophase fiber spinning.