Spontaneous rotation of active droplets in two and three dimensions

We use numerical simulations and linear stability analysis to study active nematic droplets, in the regime where the passive phase is isotropic. We show that activity leads to the emergence of nematic order and of spontaneous rotation in both two and three dimensions. In 2D the rotation is caused by the formation of a chiral $+1$ defect at the center of the drop. With increasing activity the droplet deforms to an ellipse, and then to a rotating annulus. Growing droplets form extended active arms which loop around to produce holes. In 3D the rotation is due to a disclination which loops away from and back to the surface, defining the rotation axis. In the bulk the disclination loop ends at a skyrmion. Active extensile flows deform the droplet to an oblate ellipsoid, contractile flows elongate it along the rotation axis. We compare our results on rotation in two-dimensional droplets with experiments on microtubule and motor protein suspensions and find a critical radius $\sim 700 \mu m$ above which the spontaneous rotation gives way to active turbulence. Comparing the simulation parameters with experiments on epithelial cell colonies shows that the crossover radius for cell colonies could be as large as $2 mm$, in agreement with experiments.