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

## TL;DR

This paper models chemically active drops to understand how simple interface properties and surfactant-driven flows can lead to steady, oscillatory, or chaotic self-propulsion, revealing the nonlinear dynamics underlying these behaviors.

## Contribution

It introduces a minimal axisymmetric model incorporating diffusiophoresis and Marangoni effects to explain diverse droplet dynamics, including chaos, based on surfactant advection and flow instabilities.

## Key findings

- Strong advection destabilizes steady propulsion leading to stop-and-start motion.
- Higher advection levels induce chaotic oscillations in droplet movement.
- Diffusiophoresis influences the onset of high-order instability modes.

## Abstract

Individual chemically active drops suspended in a surfactant solution were observed to self-propel spontaneously with straight, helical, or chaotic trajectories. To elucidate how these drops can exhibit such strikingly different dynamics and `decide' what to do, we propose a minimal axisymmetric model of a spherical active drop, and show that simple and linear interface properties can lead to both steady self-propulsion of the droplet as well as chaotic behavior. The model includes two different mobility mechanisms, namely, diffusiophoresis and the Marangoni effect, that convert self-generated gradients of surfactant concentration into the flow at the droplet surface. In turn, surface-driven flow initiates surfactant advection that is the only nonlinear mechanism and, thus, the only source of dynamical complexity in our model. Numerical investigation of the fully-coupled hydrodynamic and advection diffusion problems reveals that strong advection (e.g., large droplet size) may destabilize a steadily self-propelling drop; once destabilized, the droplet spontaneously stops and a symmetric extensile flow emerges. If advection is strengthened even further in comparison with molecular diffusion, the droplet may perform chaotic oscillations. Our results indicate that the thresholds of these instabilities depend heavily on the balance between diffusiophoresis and the Marangoni effect. Using linear stability analysis, we demonstrate that diffusiophoresis promotes the onset of high-order modes of monotonic instability of the motionless drop. We argue that diffusiophoresis has a similar effect on the instabilities of a moving drop.