Effect of a product on spontaneous droplet motion driven by a chemical reaction of surfactant

We focus on the self-propelled motion of an oil droplet within an aqueous phase or an aqueous droplet within an oil phase, which originates from an interfacial chemical reaction of surfactant. The droplet motion has been explained by mathematical models, which require the assumption that the chemical reaction increases the interfacial tension. However, several experimental reports have demonstrated self-propelled motion with the chemical reaction decreasing the interfacial tension. Our motivation is to construct an improved mathematical model, which explains these experimental observations. In this process, we consider the concentrations of the reactant and product on the interface and of the reactant in the bulk. Our numerical calculations indicate that the droplet potentially moves in the cases of both an increase and a decrease in the interfacial tension. In addition, the reaction rate and size dependencies of the droplet speed observed in experiments were well reproduced using our model. These results indicate the potential of our model as a universal one for droplet motion.

Figure 1

(a) Typical example of the concentration field $\tilde{c}(\tilde{r},\theta )$ (color map) and stream line (red arrows). A white circle at the center indicates the droplet, and a blue arrow in the circle indicates the direction of motion. (b) Surfactant profile in the bulk $\tilde{c}\left(\tilde{r}\right)$. The red solid line shows the concentration at the front [along the red solid line in (a)], and the blue broken line shows that at the rear [along the blue broken line in (a)]. Data were obtained from $\rho =0.7,{\tilde{v}}_0=5$.

Figure 2

(a) Droplet speed ${\tilde{v}}_d$ dependence on $\rho$ for lower $K_2$ (broken lines with open symbols) and higher $K_2$ (solid lines with filled symbols). (b) Droplet speed ${\tilde{v}}_d$ against $K_2$ with various $\rho$. Broken lines with open symbols indicate $\beta <0.5$, and solid lines with filled symbols indicate $\rho \ge 0.5$. Dashed vertical line indicates $K_2^c\sim 10$.

Figure 3

Differences between the maximum and minimum values of surfactants at the interface, $\mathrm{\Delta }{\mathrm{\Gamma }}_1$ (filled symbols with solid lines) and $\mathrm{\Delta }{\mathrm{\Gamma }}_2$ (open symbols with broken lines), against $K_2$ for various values of $\rho$.

Figure 4

(a) Trajectory of the droplet motion. The aqueous droplet is composed of ${\mathrm{BrO}}_3^{-}$ (0.4 M) and ${\mathrm{H}}_2{\mathrm{SO}}_4$ (1.8 M). The color of the trajectory indicates the time. (b) Time series of speed. (c) Average speed of droplet motion as a function of concentration of ${\mathrm{H}}_2{\mathrm{SO}}_4$. The error bars indicate standard deviation of speed in time.

Figure 5

Schematic illustration of the mechanism of droplet motion.