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Chirality is a recurrent theme in the study of biological systems, in which active processes are driven by the internal conversion of chemical energy into work. Bacterial flagella, actomyosin filaments, and microtubule bundles are active systems that are also intrinsically chiral. Despite some exploratory attempt to capture the relations between chirality and motility, many features of intrinsically chiral systems still need to be explored and explained. To address this gap in knowledge, here we study the effects of internal active forces and torques on a 3-dimensional (3D) droplet of cholesteric liquid crystal (CLC) embedded in an isotropic liquid. We consider tangential anchoring of the liquid crystal director at the droplet surface. Contrary to what happens in nematics, where moderate extensile activity leads to droplet rotation, cholesteric active droplets exhibit more complex and variegated behaviors. We find that extensile force dipole activity stabilizes complex defect configurations, in which orbiting dynamics couples to thermodynamic chirality to propel screw-like droplet motion. Instead, dipolar torque activity may either tighten or unwind the cholesteric helix and if tuned, can power rotations with an oscillatory angular velocity of 0 mean.

The rheological behaviour of an emulsion made of an active polar component and an isotropic passive fluid is studied by lattice Boltzmann methods. Different flow regimes are found by varying the values of shear rate and extensile activity (occurring, e.g., in microtubule-motor suspensions). By increasing activity, a first transition occurs from linear flow regime to spontaneous persistent unidirectional macro-scale flow, followed by another transition either to (low shear) intermittent flow regime with coexistence of states with positive, negative, and vanishing apparent viscosity, or to (high shear) symmetric shear thinning regime. The different behaviours can be explained in terms of the dynamics of the polarization field close to the walls. A maximum entropy production principle selects the most likely states in the intermittent regime. 0

We study the phase behavior of a quasi-two-dimensional cholesteric liquid crystal shell. We characterize the topological phases arising close to the isotropic-cholesteric transition and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two-dimensional topological phases: finite quasicrystals and amorphous structures, both made up of mixtures of polygonal tessellations of half-skyrmions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilizes heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals and amorphous and heterogeneous structures could be sought experimentally by self-assembling cholesteric shells on the surface of emulsion droplets.

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