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We investigate the turbulent dynamics of a two-dimensional active nematic liquid crystal con- strained on a curved surface. Using a combination of hydrodynamic and particle-based simulations, we demonstrate that the fundamental structural features of the fluid, such as the topological charge density, the defect number density, the nematic order parameter and defect creation and annihilation rates, are simple linear functions of the substrate Gaussian curvature, which then acts as a control parameter for the chaotic flow. Our theoretical predictions are then compared with experiments on microtubule-kinesin suspensions confined on toroidal active droplets, finding excellent qualitative agreement., Comment: 6 pages, 4 figures

We confine a nematic liquid crystal with homeotropic anchoring to stable toroidal droplets and study how geometry affects the equilibrium director configuration. In contrast to the case of cylindrical confinement, we find that the equilibrium state is chiral-a twisted and escaped radial director configuration. Furthermore, we find that the magnitude of the twist distortion increases as the ratio of the ring radius to the tube radius decreases; we confirm this with computer simulations of optically polarized microscopy textures. In addition, numerical calculations also indicate that the local geometry indeed affects the magnitude of the twist distortion. We further confirm this curvature-induced twisting using bent cylindrical capillaries.

Nematic order on curved surfaces is often disrupted by the presence of topological defects, which are singular regions in which the orientational order is undefined. In the presence of force-generating active materials, these defects are able to migrate through space like swimming microorganisms. We use toroidal surfaces to show that despite their highly chaotic and non-equilibrium dynamics, pairs of defects unbind and segregate in regions of opposite Gaussian curvature. Using numerical simulations, we find that the degree of defect unbinding can be controlled by tuning the system activity, and even suppressed in strongly active systems. Furthermore, by using the defects as active microrheological tracers and quantitatively comparing our experimental and theoretical results, we are able to determine material properties of the active nematic. Our results illustrate how topology and geometry can be used to control the behaviour of active materials, and introduce a new avenue for the quantitative mechanical characterization of active fluids. Topological defects in a turbulent active nematic on a toroidal surface are shown to segregate in regions of opposite curvature. Simulations suggest that this behaviour may be controlled — or even suppressed — by tuning the level of activity.

We stabilize nematic droplets with handles against surface-tension-driven instabilities using a yield-stress material as outer fluid and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle-splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ~20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional -1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy., Comment: 23 pages, 4 figures, PNAS (2013)

The breakup of a liquid jet into spherical droplets via the Rayleigh–Plateau instability is a common and fundamental part of fluid mechanics. However, teaching this instability in a laboratory setting is challenging, requiring sophisticated methods to generate and study the jet dynamics. Recently, toroidal droplets were shown to break into one or more spherical droplets in the thin-drop limit via the Rayleigh–Plateau instability. We propose a simple experimental setup to generate toroidal droplets that break up on the order of tens of seconds, allowing for easy video capture using a basic CCD camera. With this setup, it is possible to quantify the Rayleigh–Plateau instability in a pedagogical laboratory setting. In addition, the role of curvature on jet breakup can be explored using thick toroidal droplets. We envision this setup as a powerful teaching tool for one of the most fundamental fluid dynamics processes.