How merging droplets jump off a superhydrophobic surface: Measurements and model

We investigate how drops merging on a nonwettable surface jump off this surface, for both symmetric and asymmetric coalescences. For this purpose, we design and build a microelectromechanical system sensor able to quantify forces down to the micro-Newton scale at a high acquisition rate (200 kHz). Using this device, we perform direct force measurements of self-propelled droplets coupled to high-speed imaging. Experimental data show that the total momentum of the drop after coalescence mainly depends on the size of the smaller drop. Exploiting this finding, we quantitatively predict the takeoff speed of jumping drop pairs and show how to correct the usual argument based on energy conservation.

Figure 1

(a) Sketch of an asymmetric coalescence between droplets with respective radius $R$ and $r$; the resulting drop jumps at a velocity $U$. (b) Scanning electron microscopy viewgraph of the MEMS designed to measure the forces induced by drops merging on a superhydrophobic surface $S$. (c) Sketch of the MEMS showing the different layers. Gray, red, yellow, and light blue respectively represent silicon, doped silicon, $\text{Au}/\text{Cr}$ layer, and ${\text{SiO}}_2$ layer. (d) High-speed imaging of (top) a symmetric coalescence $\left(R=530\mu \text{m}\right)$ and (bottom) an asymmetric coalescence $(R=530\mu \text{m}$ and $r=300\mu \text{m},\varepsilon =r/R=0.57)$. Snapshot 1 shows the beginning of the coalescence, and snapshot 7 corresponds to takeoff. Images are separated by 0.65 ms, except snapshot 8 shot at 8.8 ms, when the drop reaches its maximum height. (e) Velocity $U$ of jumping drops as a function of the ratio $\varepsilon =r/R$. Colors indicate the volume $\mathrm{\Omega }$ of the large drop. Colored dashed lines show the speed expected from energy conservation, Eq. (1). (f) Same data after normalizing $U$ by $U^{*}=\sqrt{\gamma /\rho R}$, as a function of $\varepsilon$. The solid line shows Eq. (3) drawn without an adjustable parameter.

Figure 2

(a) Force exerted by merging drops on their substrate, for the experiments shown in Fig. 1 (data sampled at 200 kHz). Black and red lines correspond respectively to symmetric and asymmetric coalescence. Time starts with contact $\left(t=0\right)$ and ends at takeoff $\left(t=\tau \right)$. (b) Force integrated over the jumping time $P$ as a function of the momentum $(M+m)U$ of the departing drop. The dashed line has a slope of 1.15. (c) $P$ as a function of the small radius $r$. The dashed line shows a slope 5/2.

Figure 3

(a) Schematic describing the transfer of quasihorizontal momentum of the small drop into quasivertical momentum. (b) Snapshots 1 and 3 of the asymmetric coalescence in Fig. 1 illustrating the retraction time ${\tau }_r$, defined as the time needed for the edge of the small drop to move by its own radius. In this example, we have ${\tau }_r\approx 1.3$ ms. (c) Retraction time ${\tau }_r$ as a function of $r$. The black dashed line shows ${\tau }_r=2\sqrt{\rho r^3/\gamma }$. (d) Force integrated over the jumping time as a function of the measured momentum $mv=mr/{\tau }_r$ of the small drop. The dashed line has a slope 1.15.