Tunable Persistent Random Walk in Swimming Droplets

We characterize the motility of athermal swimming droplets within the framework of a persistent random walk. Just like active colloids, their trajectories can be modeled with a constant velocity $V$ and a slow angular diffusion, but the random changes in direction are not thermally driven. Instead, $V$ is determined by the interfacial tension gradient along the droplet surface, while reorientation of the surfactant gradient leads to changes in direction with a persistence time $\tau$. We show that the origin of locomotion is the difference in the critical micellar concentration in the front and the back of the droplet, $\mathrm{\Delta }\mathrm{CMC}$. Tuning this parameter by salt controls $V$ from 3 to 15 diameters $d/s$. Surfactant concentration has little effect on speed, but leads to a dramatic decrease in $\tau$ over 4 orders of magnitude. The corresponding range of the persistence length $\ell =V\tau$ extends beyond the realm of synthetic or living swimmers, in which $V$ is limited by fuel consumption and $\tau$ is set by thermal fluctuations or biological activity, respectively. Our tunable swimmers are ideal candidates for the study of the departure from equilibrium to high levels of activity. We show that their collective behavior exhibits the formation of active clusters of a well-defined size.

Popular Summary

Ordinary oil droplets in water are too big to be pushed around by the random thermal motions of the molecules. Yet, researchers have found that even large droplets can “swim” if they dissolve in a way that causes an asymmetry in their surfactants—compounds at their interface that lower surface tension. This causes a difference in pressure that propels the droplets in a given direction, while reorientation of the surfactants causes the droplets to turn. Several groups have identified different modes of motion in swimming droplets—including straight lines, random trajectories, and even spiraling motion—but a unifying microscopic explanation of these different modes of propulsion is still lacking. Here, we explain the molecular engine of droplet motion that gives rise to their persistent random walk.

As a droplet moves, it leaves a trail of dissolved oil behind it. The surfactants prefer to be in the oily trail than in pure water, and this difference in affinity drives surfactants to flow from the front toward the back of droplets, which in turn propels them forward. In our experiments, adding salt to water diminishes this difference in affinity and therefore slows down droplets in a controllable way. Adding more surfactants changes the orientation of surfactants at the interface and allows us to tune the turning frequency of the droplets. We also observe that droplets organize into dynamical clusters much like bacterial colonies.

Our system expands the experimental range of active matter beyond known artificial and living swimmers and offers physicists a playground to quantitatively tune and test dynamical effects away from equilibrium.

Figure 1

Schematic of the mechanism at the origin of self-sustained droplet motion. Spontaneous symmetry breaking arises due to a higher concentration of monomers in equilibrium with the aqueous solution (on the right) compared to those facing DEP-saturated water (on the left). This effect arises because of the difference in the CMC in the two environments, and it causes a Marangoni flow, shown by the arrows, which is enhanced by droplet motion.

Figure 2

Individual trajectories of $30\mu \mathrm{m}$ diameter droplets, lasting 4 s, swimming in aqueous solutions with increasing bulk SDS concentrations. The motion transitions from ballistic to diffusive, with more frequent turns leading to a slower exploration of space, which is still orders of magnitude higher than that of thermal rotational diffusion.

Figure 3

Surface tension versus SDS concentration at the interface of an air bubble in aqueous [NaCl] solution (filled symbols), a DEP drop in DEP-saturated [NaCl] aqueous solution (open symbols), and a silicone oil drop (350 centiStokes) in pure aqueous solution (crosses). As a function of [NaCl], fits to the von Szyszkowski equation (dashed lines) reveal that the surface tension gradient is independent of salt concentration for a given interface. The CMC decreases with salt for both the aqueous and the DEP-saturated interface, but the difference $\mathrm{\Delta }\mathrm{CMC}$ is progressively smaller.

Figure 4

Instantaneous speed $V$ at a constant $[\mathrm{SDS}]=10\mathrm{mM}$ sharply decreases with [NaCl]. This trend is theoretically captured by Eq. (1) with $\mathrm{\Delta }\mathrm{CMC}$ values independently measured by pendant drop experiments. Above $[\mathrm{NaCl}]=70\mathrm{mM}$, corresponding to $\mathrm{\Delta }\mathrm{CMC}=0.5\mathrm{mM}$, droplets cease to swim. Dashed line is computed from the difference between empirical fits of CMC versus [SDS] [34]).

Figure 5

Normalized autocorrelation of the velocity for each concentration. Each dataset is computed from averaging the $C(\mathrm{\Delta }t)$ of experimental trajectories at a given concentration. Solid lines are fit with an exponential decay following Eq. (2).

Figure 6

Mean square displacement curves, averaged over tens of trajectories, show a decrease in the crossover time $\tau$ as a function of SDS concentration. The model of rotational diffusion in Eq. (3) gives an excellent fit to the data and the gray area corresponds to the 95% confidence interval.

Figure 7

Effect of surfactant concentration on motility. (a) Instantaneous speed $V$, estimated directly from trajectories (circles) or the MSD fits of Eq. (3) (squares), is largely independent of [SDS]. (b) Persistence time $\tau$, from the fits and the values extracted from the velocity correlation function $C(\mathrm{\Delta }t)$, sharply decays with [SDS] with a heuristic power-law fit $\tau =6\times {10}^6[\mathrm{SDS}{]}^{-4}$. (c) The rate of droplet dissolution increases nonlinearly as a function of SDS concentration over the experimental range.

Figure 8

Displacements of the droplet between consecutive time steps (i.e., $\mathrm{\Delta }t=25\mathrm{ms}$), for three surfactant concentrations. Colors represent the density at which this location in the displacement field is hit along the trajectory.

Figure 9

Memory effects in droplets. (a) Droplets leave a fluorescently labeled trail (dashed lines), which diffusively spreads with a diffusion constant consistent with that of DEP-filled micelles, confirming the proposed mechanism in Fig. 1. (b) Active droplets transition from a crystal to a more dilute, disordered cluster over time. The graph shows the concomitant drop in the coordination number $Z=6$, shown in red in the snapshots from the video.