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Knotted fields are an emerging research topic relevant to different areas of physics where topology plays a crucial role. Recent realization of knotted nematic disclinations stabilized by colloidal particles raised a challenge of free-standing knots. Here we demonstrate the creation of free-standing knotted and linked disclination loops in the cholesteric ordering fields, which are confined to spherical droplets with homeotropic surface anchoring. Our approach, using free energy minimization and topological theory, leads to the stabilization of knots via the interplay of the geometric frustration and intrinsic chirality. Selected configurations of the lowest complexity are characterized by knot or link types, disclination lengths and self-linking numbers. When cholesteric pitch becomes short on the confinement scale, the knotted structures change to practically unperturbed cholesteric structures with disclinations expelled close to the surface. The drops with knots could be controlled by optical beams and may be used for photonic elements.

Liquid crystals, when confined to a spherical shell, offer fascinating possibilities for producing artificial mesoscopic atoms, which could then self-assemble into materials structured at a nanoscale, such as photonic crystals or metamaterials. The spherical curvature of the shell imposes topological constraints in the molecular ordering of the liquid crystal, resulting in the formation of defects. Controlling the number of defects, that is, the shell valency, and their positions, is a key success factor for the realization of those materials. Liquid crystals with helical cholesteric order offer a promising, yet unexplored way of controlling the shell defect configuration. In this paper, we study cholesteric shells with monovalent and bivalent defect configurations. By bringing together experiments and numerical simulations, we show that the defects appearing in these two configurations have a complex inner structure, as recently reported for simulated droplets. Bivalent shells possess two highly structured defects, which are composed of a number of smaller defect rings that pile up through the shell. Monovalent shells have a single radial defect, which is composed of two nonsingular defect lines that wind around each other in a double-helix structure. The stability of the bivalent configuration against the monovalent one is controlled by c = h/p, where h is the shell thickness and p the cholesteric helical pitch. By playing with the shell geometry, we can trigger the transition between the two configurations. This transition involves a fascinating waltz dynamics, where the two defects come closer while turning around each other.iquid crystals offer fascinating possibilities for producing materials that are organized at a mesoscopic scale, such as photonic crystals or metamaterials (1). In a seminal paper, Nelson (2) proposed to induce valency into simple spherical particles by coating their surfaces with a nematic liquid crystal shell. In this elegant approach, the spherical symmetry of the particle is broken by the presence of topological defects, which stem from frustrations in the liquid crystal orientational order due to the curvature of the particle surface. These defects, once functionalized, could act as sticky surface patches inducing directional particle bonding through patchpatch interactions, which would make possible the fabrication of complex materials by spontaneous self-assembly. The number of surface defects would set the valence of the particle, whereas their position would determine the directionality of the eventual bonds. The type of defect configuration in the shell, and thus its valence, results from a very subtle interplay between topological constraints and elastic free-energy minimization. For a two-dimensional nematic shell, theory predicts a tetravalent configuration where four defects are organized in a tetrahedral fashion (3). These defects have a winding number s i = 1=2, indicating a π-rotation of n, the average molecular orientation, around the defect (4). This is consistent with...

We introduce the idea of transformation trajectories to describe the evolution of nematic shells in terms of defect locations and director field when the elastic anisotropy and the shell thickness heterogeneity vary. Experiments are compared to numerical results to clarify the exact role played by these two parameters. We demonstrate that heterogeneity in thickness is a result of a symmetry breaking initiated by buoyancy and enhanced by liquid crystal elasticity, and is irrespective of the elastic anisotropy. In contrast, elastic anisotropy--in particular, disfavored bend distortion--drives an asymmetric defect reorganization. These shell states can be both stable or metastable.

We demonstrate that high anisotropy of elastic constants of chromonic liquid crystals leads to a number of spontaneously twisted nematic director fields around colloidal particles in these non-chiral fluids. For spherical colloidal particles with surface inducing degenerate planar nematic ordering we observe that boojum defects at the particles' poles acquire twisted internal structure, extending up to three particle diameters along the rubbing direction of the cell. The twist handedness of the two boojum defects at the poles of the particle can be either the same or opposite, and we can switch the defects handedness by localized thermal microquenching. Numerical simulations confirm that the transitions into the distorted states are induced by lowering of the twist elastic constant, which results in two (meta)stable chiral configurations of the boojums, separated by an energy barrier much higher than the thermal energy. We show that boojum handedness can change the pairwise elastic interaction between the two particles positioned along the rubbing direction from repulsive to attractive.