Self-propulsion of a droplet induced by combined diffusiophoresis and Marangoni effects

Chemically active droplets display complex self-propulsion behavior in homogeneous surfactant solutions, often influenced by the interplay between diffusiophoresis and Marangoni effects. Previous studies have primarily considered these effects separately or assumed axisymmetric motion. To understand the full hydrodynamics, we investigate the motion of a two-dimensional active droplet under their combined influences using weakly nonlinear analysis and numerical simulations. The impact of two key factors, the Peclet number (Pe$Pe$) and the mobility ratio between diffusiophoretic and Marangoni effects (m$m$), on droplet motion is explored. We establish a phase diagram in the Pe-m$Pe-m$ space, categorizing the boundaries between four types of droplet states: stationary, steady motion, periodic/quasi-periodic motion, and chaotic motion. We find that the mobility ratio does not affect the critical Pe$Pe$ for the onset of self-propulsion, but it significantly influences the stability of high-wavenumber modes as well as the droplet's velocity and trajectory. Scaling analysis reveals that in the high Pe$Pe$ regime, the Marangoni and diffusiophoresis effects lead to distinct velocity scaling laws: U similar to Pe-1/2$U\sim Pe<^>{-1/2}$ and U similar to Pe-1/3$U\sim Pe<^>{-1/3}$, respectively. When these effects are combined, the velocity scaling depends on the sign of the mobility ratio. In cases with a positive mobility ratio, the Marangoni effect dominates the scaling, whereas the negative diffusiophoretic effect leads to an increased thickness of the concentration boundary layer and a flattened scaling of the droplet velocity.