| Summary: | <p>We study the interplay between flow, structure, and topology in liquid crystals, in both passive and active materials. Liquid crystals are fluids, but at the same time structurally anisotropic and strongly influenced by their topology. They exhibit long-range orientational order, with characteristics between those of simple isotropic fluids and those of crystalline solids. Active materials extract energy from their surroundings at the single particle level and convert it to mechanical work.</p> <p>First, we consider the elongation of passive nematic droplets suspended in their own isotropic phase, i.e. tactoids, under the influence of an electric field, and show that a tenfold elongation is due to strong anchoring of the directors to the interface. Building on this, we introduce more complicated internal structures in the tactoids, and demonstrate that cholesteric tactoids and two-dimensional blue phase tactoids correspond to a wealth of droplet morphologies. Application of a perpendicular electric field produces structures very similar to chiral rafts, with directors normal to the plane of the tactoid in the centre, but twisting away from the normal at the edge.</p> <p>Secondly, we turn to active liquid crystals. Here the interaction between topology, structure, and flow is perhaps even more pronounced, with motile +1/2 topological defects accompanied by large stresses. We study an active two-dimensional blue phase, and show that it is stable for contractile activity, with the active instability screened by the cholesteric pitch length. Extensile activity, on the other hand, leads to dissociation of the double-twist cylinders and causes active turbulence. An alternative method to screen the active instability is introducing substrate friction. We simulate extensile rod-like active nematics and show that anisotropic friction can give rise to a laning state with antiparallel flow. The transition is strongly sensitive to the strength of the anisotropy. Anisotropic friction is an effective way to control active turbulence and to direct the flow in a well-defined fashion.</p> <p>Finally, we introduce a framework for continuum simulations of active nematics on deformable membranes. We demonstrate that long protrusions driven by motile topological defects are formed in three dimensions on deformable shells. This has potential implications for morphogenetic processes such as the regeneration of <em>Hydra.</em> Our results show that changes in the animal body shape can occur from solely mechanical cues.</p>
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