Self-Propulsion of a Volatile Drop on the Surface of an Immiscible Liquid Bath

We demonstrate the self-propulsion of a volatile drop on the surface of a bath of an immiscible liquid. Evaporative heat pumping is converted into directed motion through thermocapillary stresses, which arise from the coupling between surface-tension-driven flows and temperature advection. A propulsive force arises from convection-sustained temperature gradients along the drop interface, resulting in a warmer pool of liquid being advected by the hydrodynamic flow in the underlying bath toward the back of the drop. The dependence of the drop speed on the activity source, i.e., the evaporation flux, is derived with scaling arguments and captures the experimental data.

Figure 1

Self-propulsion behaviors of volatile drops on liquid baths. First behavior: (a) the trajectory is erratic with $l\lesssim R$ (top view) when the drop is in contact with air; (b) schematic side view, not to scale. Second behavior: (c) the trajectory is straighter with $l\gg R$ (top view) when a film of liquid 2 covers the drop; (d) schematic side view. (a) Drop of 90% v/v ethanol in water. (c) Ethanol drop. (a),(c) Silicone oil bath with ${\eta }_2=0.097\mathrm{Pa}\mathrm{s}$.

Figure 2

Static symmetric (a)–(c) and propulsive asymmetric (d)–(f) state of an ethanol drop on a bath of silicone oil. (a),(d) Streamlines resulting from 15 sec of PTV (side view). (b),(e) Surface temperature field (top view), with $T^{++}>T^{+}>T^{-}>T^{--}$. (c),(f) Side view schematics. White arrows indicate flows. Blue and red arrows indicate Marangoni stresses associated to base and perturbed flow, respectively. (e),(f) Dashed lines delimit the control volume used for the force balance. (a),(d) $R=4.0\mathrm{mm}$. (b),(e) $R=8.0\mathrm{mm}$. ${\eta }_2=0.097\mathrm{Pa}\mathrm{s}$.

Figure 3

Time evolution of drop speed, evaporation flux, and temperature field. (a) Experimental drop speed $v(t)$ (full circles), with $v_s$ the stationary speed. The solid line results from (5) combined with the evaporation flux $J(t)$ in (b). (b) Experimental evaporation flux $J(t)$ (solid line) and angular extension $\alpha (t)$ (plus symbol) of the cold patch. $J_s$ is the stationary evaporation flux and ${\alpha }_{\max }$ is the maximum angle. $j_{\mathrm{air}}^{\mathrm{diff}}$ is the diffusive evaporation flux of an ethanol disk with radius $R$. (c) Temporal evolution of the surface temperature field (Supplemental Movie 1 [27]). The dashed line delimits the upper side of the control volume in the transient regime. $R=8.0\mathrm{mm}$, ${\eta }_2=0.097\mathrm{Pa}\mathrm{s}$.

Figure 4

Relation between drop speed $v_s$ and evaporation flux $J_s$ in the stationary regime. (a) $v_s$ vs bath viscosity ${\eta }_2$. Lines represent the scaling in (6) combined with the fits in the inset. Inset: lines represent fits of $a{\eta }_2+b$. $R=4.0\mathrm{mm}$: (star symbol), continuous lines and $(a,b)=(4.1\times {10}^{-7}\mathrm{m}{\mathrm{s}}^{-1},9.2\times {10}^{-8}\mathrm{m}\mathrm{Pa})$. $R=8.0\mathrm{mm}$: (“o” symbol), dashed lines and $(a,b)=(3.3\times {10}^{-7}\mathrm{m}{\mathrm{s}}^{-1},7.1\times {10}^{-8}\mathrm{m}\mathrm{Pa})$. (b) Side view schematic. White arrows indicate flows.