Wettability-Independent Droplet Transport by Bendotaxis

We demonstrate “bendotaxis,” a novel mechanism for droplet self-transport at small scales. A combination of bending and capillarity in a thin channel causes a pressure gradient that, in turn, results in the spontaneous movement of a liquid droplet. Surprisingly, the direction of this motion is always the same, regardless of the wettability of the channel. We use a combination of experiments at a macroscopic scale and a simple mathematical model to study this motion, focusing in particular on the timescale associated with the motion. We suggest that bendotaxis may be a useful means of transporting droplets in technological applications, e.g., in developing self-cleaning surfaces, and discuss the implications of our results for such applications.

Figure 1

(a) Schematic diagrams explain the mechanism behind bendotaxis for wetting (left) and nonwetting (right) drops. Black arrows indicate the sign and magnitude of the Laplace pressure within the drop; purple arrows show the direction of decreasing pressure and, hence, motion. (b) Experimental demonstration of bendotaxis for a wetting silicone oil droplet (left) and a nonwetting water droplet (right), each between initially parallel, yet deformable, glass cover slips [17]. While the deformation of the channel is different in each case, the direction of droplet motion is the same. (c) Comparison of final channel shape (red lines) with the initial channel shape (dotted white lines) for the section highlighted by the dashed box in (b).

Figure 2

(a) Schematic of a droplet in a flexible channel of undeformed wall separation $2H_0$, width $w$, and length $L$, with menisci positioned at $x=x_{-}(t)$ and $x=x_{+}(t)$. (b) Top view of a droplet of V50 silicone oil (channel length $L=18\mathrm{mm}$, width $w=5\mathrm{mm}$, wall separation $2H_0=310\mu \mathrm{m}$, thickness $b=300\mu \mathrm{m}$, and droplet volume $V=10\mu \mathrm{L}$). Although the droplet spans the width of the channel, it is not precisely two dimensional. We present data for $t_{0.7}$, the time between the events $x_{+}/L=0.7$ and $x_{+}/L=1$, which corresponds to the time between the third and sixth images here. (c) Raw experimental measurements of $t_{0.7}$, for different wall lengths, $L$. Data are shown for droplets of wetting silicone oil and a nonwetting water-glycerol mix; droplet type is encoded as in the legend of Fig. 3. Different shapes encode channel wall separation as follows: $2H_0=310\mu \mathrm{m}$ (right triangle), $360\mu \mathrm{m}$ (left triangle), $430\mu \mathrm{m}$ (square), $540\mu \mathrm{m}$ (circle), $630\mu \mathrm{m}$ (diamond). The size of each point encodes the approximate fraction of the channel taken by the droplet ($\widehat{V}=\mathrm{\Delta }X/L$), with bins corresponding to $\widehat{V}<0.25$, $0.25\le \widehat{V}<0.35$, $0.35\le \widehat{V}<0.45$, and $\widehat{V}\ge 0.45$.

Figure 3

Collapse of experimental data when rescaled according to Eq. (2). Points correspond to experimental observations (with volume and wall separation encoded by point size and shape, respectively, as in Fig. 2; droplet type is encoded by color, as indicated in the legend). The single set of error bars extends 1 standard deviation away from a particular data point, computed from 20 measurements, and is similar for each experiment [17]. Solid curves show results from numerical solutions of Eqs. (3)–(5) with $\widehat{V}=0.5$, 0.4, 0.3 and $\widehat{V}=0.2$. Also plotted is the asymptotic result Eq. (7) (black dashed line), valid for $\widehat{V}\ll 1$ (corresponding to the upper right corner of this plot).

Figure 4

Influence of dimensionless droplet volume $\widehat{V}$ and channel bendability $\nu$ on the time taken to traverse the final 30% of the channel, $t_{0.7}/{\tau }_c$. Numerical results are shown by varying color, while the bold black curve indicates parameter values for which the edges of the channel touch during the motion, trapping the droplet. (Note that the position of this curve depends on the initial condition; here, $x_{+}^0/L=0.6$.) Positive bendability, $\nu >0$, corresponds to wetting drops, while $\nu <0$ corresponds to nonwetting drops; when $\nu =0$, the channel is effectively rigid and the droplet remains stationary. Schematics illustrate typical configurations, and filled circles correspond to the experimental data presented in Figs. 2 and 3; the outliers with $\nu <0$ are in the slow nonwetting regime.