Effect of interfacial surfactant transport on the late-time dynamics of Marangoni stress-driven droplets

Active droplets swim spontaneously due to self-generated Marangoni flows that arise from non-uniform interfacial coverage of surfactant. Most theoretical studies on the phenomena have focused on the early time dynamics near the onset of instability. Specifically, they employ a linear equation of state (EOS) to relate the interfacial tension and the surfactant concentration. This EOS is only applicable for low interfacial coverages, which is typical only for early times, and is thus unable to recover long-time dynamics of active droplets.

In our work, we describe a theoretical framework to investigate the effect of the interfacial transport and kinetics of the surfactant on the dynamics of active droplet propulsion. We solve surfactant transport equations in the bulk and on the droplet surface, and couple the two phenomena through an adsorption/desorption source term. Our framework allows for a nonlinear EOS, which is then used to predict droplet propulsion speed based on self-generated Marangoni flow. We demonstrate the utility of our framework by applying it to model problems and show how it is able to capture the time-dependent motion of active droplets. Specifically, we are able to predict time-dependent velocities and droplet stoppage which are not possible through early-time calculations.