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Individual chemically active drops suspended in a surfactant solution wereobserved to self-propel spontaneously with straight, helical, or chaotictrajectories. To elucidate how these drops can exhibit such strikinglydifferent dynamics and `decide' what to do, we propose a minimal axisymmetricmodel of a spherical active drop, and show that simple and linear interfaceproperties can lead to both steady self-propulsion of the droplet as well aschaotic behavior. The model includes two different mobility mechanisms, namely,diffusiophoresis and the Marangoni effect, that convert self-generatedgradients of surfactant concentration into the flow at the droplet surface. Inturn, surface-driven flow initiates surfactant advection that is the onlynonlinear mechanism and, thus, the only source of dynamical complexity in ourmodel. Numerical investigation of the fully-coupled hydrodynamic and advectiondiffusion problems reveals that strong advection (e.g., large droplet size) maydestabilize a steadily self-propelling drop; once destabilized, the dropletspontaneously stops and a symmetric extensile flow emerges. If advection isstrengthened even further in comparison with molecular diffusion, the dropletmay perform chaotic oscillations. Our results indicate that the thresholds ofthese instabilities depend heavily on the balance between diffusiophoresis andthe Marangoni effect. Using linear stability analysis, we demonstrate thatdiffusiophoresis promotes the onset of high-order modes of monotonicinstability of the motionless drop. We argue that diffusiophoresis has asimilar effect on the instabilities of a moving drop.

Nonlinear dynamics of a chemically-active drop: from steady to chaotic self-propulsion