Autothermotaxis of volatile drops

When a drop of a volatile liquid is deposited on a uniformly heated wettable, thermally conducting substrate, one expects to see it spread into a thin film and evaporate. Contrary to this intuition, due to thermal Marangoni contraction, the deposited drop contracts into a spherical-cap-shaped puddle, with a finite apparent contact angle. Strikingly, this contracted droplet, above a threshold temperature, well below the boiling point of the liquid, starts to spontaneously move on the substrate in an apparently erratic way. We describe and quantify this self-propulsion of the volatile drop. It arises due to spontaneous symmetry breaking of thermal Marangoni convection, which is induced by the nonuniform evaporation of the droplet. Using infrared imaging, we reveal the characteristic interfacial flow patterns associated with Marangoni convection in the evaporating drop. A scaling relation describes the correlation between the moving velocity of the drop and the apparent contact angle, both of which increase with the substrate temperature.

Figure 1

(a) Temporal measurement of apparent contact angle $\theta$ during the lifetime of the droplet for different substrate temperature $T_{\mathrm{s}}$. Time is nondimensionalized using the droplet lifetime $t_f$. (b) Increase in the contact angle $\theta$ of a self-propelling ethanol droplet with $T_{\mathrm{s}}$.

Figure 2

Chronophotography of volatile ethanol droplets of volume $4\mu \mathrm{L}$, deposited on a horizontal heated sapphire plate at temperature (a) $T_{\mathrm{s}}=45{}^{\circ }\mathrm{C}$ and (b) $T_{\mathrm{s}}=60{}^{\circ }\mathrm{C}$. Following a short transient period after deposition, the droplet self-propels (autothermotaxis) until complete evaporation. Trajectories are overlaid on the images and color-coded by instantaneous velocity $U$. Droplets move faster and have an increased tendency to reorient on warmer substrates.

Figure 3

(a) Probability distribution function of velocity $P\left(U\right)$ of ethanol droplets on a warm sapphire plate for different substrate temperatures $T_{\mathrm{s}}$. (b) Autocorrelation function of the orientation angle $\beta$ with increasing $T_{\mathrm{s}}$. Inset: Data collapse when nondimensionalizing time with the maximum droplet radius $R_{\max }$ and the mean velocity $\langle U\rangle$. (c) Comparison of the mean velocity $\langle U\rangle$ of the droplet measured in upright and upside-down configurations. Error bars represent the standard deviation of the velocities. Close match between the two configurations highlights the insignificant role of gravity in the droplet motion.

Figure 4

Infrared imaging of self-propelling droplets: (a) Interfacial thermal map (top view) of a moving ethanol droplet while it evaporates on a sapphire substrate at $T_{\mathrm{s}}=55{}^{\circ }\mathrm{C}$. The bright region close to the contact line is the warmest part of the droplet, whereas the dark region near the apex is the coldest. Red and blue dots corresponding, respectively, to the centroid of droplet footprint and convection cells reveal the spontaneous symmetry breaking of the Marangoni flow. (b) Sketches of the thermal gradient and the resulting Marangoni flows for an evaporating droplet on a heated conducting ($k_{\mathrm{R}}>2$) and on an insulating ($k_{\mathrm{R}}<2$) substrate. (c) Temperature difference between the warmest and coldest part of a self-propelling ethanol droplet for various $T_{\mathrm{s}}$. (d) Variation in interfacial temperature patterns with substrate temperature $T_s=45,55,\text{and}70{}^{\circ }\mathrm{C}$.

Figure 5

(a) Mean droplet velocity $\langle U\rangle$ measured as a function of the apparent contact angle $\theta$ for ethanol and Novec 7100 droplets deposited on a warm sapphire plate. The solid black line is the expected scaling $\langle U\rangle \sim {\theta }^2$, cf. Eq. (1). Error bars represent the standard deviation over multiple runs. (b) Sketch of the Marangoni rolls inside the moving drop, after spontaneous symmetry breaking. See main text for explanations.