Solute-mediated interactions between active droplets

Concentration gradients play a critical role in embryogenesis, bacterial locomotion, as well as the motility of active particles. Particles develop concentration profiles around them by dissolution, adsorption, or the reactivity of surface species. These gradients change the surface energy of the particles, driving both their self-propulsion and governing their interactions. Here, we uncover a regime in which solute gradients mediate interactions between slowly dissolving droplets without causing autophoresis. This decoupling allows us to directly measure the steady-state, repulsive force, which scales with interparticle distance as $F\sim 1/r^2$. Our results show that the dissolution process is diffusion rather than reaction rate limited, and the theoretical model captures the dependence of the interactions on droplet size and solute concentration, using a single fit parameter, $l=16\pm 3\mathrm{nm}$, which corresponds to the length scale of a swollen micelle. Our results shed light on the out-of-equilibrium behavior of particles with surface reactivity.

Figure 1

Droplet interactions due to surfactant gradients. (a) Schematic overview of DEP droplet dissolution into swollen SDS micelles, giving rise to radial concentration gradients of SDS monomer (blue line) and micelles (green line) surrounding the droplet surface, compared to the bulk concentration (dashed line). (b) Surface tension of DEP droplets in water decreases as a function of the SDS concentration. (c) Two oil droplets swimming in given initial directions repel one another as a result of their concentration gradients. Circles map their trajectories over time.

Figure 2

Solvent-induced droplet interactions. (a) Frames of a movie showing two oil droplets moving away from each other due to the solute-mediated repulsion after they have been brought into contact using optical tweezers. (b) Schematic drawing shows how overlapping SDS concentration profiles lead to droplet interactions. The red triangle represents the gradient causing the motion. (c) Measurements of interdroplet separation as a function of time allow us to determine their size-dependent velocity $U$ as a function of separation in (d).

Figure 3

Hovering due to solute-mediated interactions. (a) Top panel shows a dissolving DEP droplet through a tilted bright-field microscope. The top feature is the real droplet and the lower feature the optical image on the glass slide. Bottom panel shows $xz$ projections of the same droplets imaged in reflection mode on a confocal microscope. (b) Schematic drawing shows how a solute cloud around a particle can lift it from the glass surface against gravity. (c) Measured droplet hovering heights are plotted as a function of their size. Error bars come from repeated measurements.

Figure 4

Dissolution rate of a DEP droplet in a bath of 5 mM SDS as a function of particle size. Inset shows the log-log representation and reveals a slope of $-1$, consistent with a diffusion-limited dissolution.

Figure 5

(a) Schematic drawing of the effect of surface activity on an external gradient. The solid black line represents an unperturbed external gradient in which the active particle is positioned. Due to the activity of the particle the concentration profile is modified as shown by the solid red line. The difference in concentration between the back and the front of the droplet, indicated by the dashed black and red lines, is modified due to the dissolution of the active particle. (b) and (c) show the solute concentration profiles for a diffusion-limited and a reaction-limited dissolution, respectively, in an external gradient. These profiles are calculated with Eq. (4) with $G_{\text{ext}}=1, c^{*}=1, a=1, c_{\infty }=0, l/a={10}^{-3}$ in (b) and $l/a={10}^3$ in (c). Color map indicates increasing SDS concentration from blue to red. The black lines are isoconcentration lines.

Figure 6

Coupling between the advective flow and the solute gradient leads to self-propulsion. (a) Schematic drawing of the flow around a moving particle, which transports solute from the front to the back, resulting in an external gradient. (b) The autophoresis loop depicts the coupling between the solute profile and the flow from panel (a), which can lead to self-sustained motion.

Figure 7

Rescaling of active interactions. (a) Log-log representation of the speed with which droplets move away from each other as a function of interparticle distance. Color represents the initial droplet size and symbols represent SDS concentration. The data from the hovering experiment are indicated in black. (b) Data shown in (a), normalized using Eq. (10). The black line is a fit with a fixed slope of $-2$.