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In experiments, an individual chemically active liquid crystal (LC) dropletsubmerged in the bulk of a surfactant solution may self-propel along astraight, helical, or random trajectory. In this paper, we develop a minimalmodel capturing all three types of self-propulsion trajectories of a drop inthe case of a nematic LC with homeotropic anchoring at LC-fluid interface. Weemulate the director field within the drop by a single preferred polarizationvector that is subject of two reorientation mechanisms, namely, the internalflow-induced displacement of the hedgehog defect and the droplet's rotation.Within this reduced-order model, the coupling between the nematic ordering ofthe drop and the surfactant transport is represented by variations of thedroplet's interfacial properties with nematic polarization. Our analysisreveals that a novel mode of orientational instability emerges from thecompetition of the two reorientation mechanisms and is characterized by aspontaneous rotation of the self-propelling drop responsible for helicalself-propulsion trajectories. In turn, we also show that random trajectories inisotropic and nematic drops alike stem from the advection-driven transition tochaos. The succession of the different propulsion modes is consistent withexperimentally-reported transitions in the shape of droplet trajectories as thedrop size is varied.

Orientational instability and spontaneous rotation of active nematic droplets