Ballistics of self-jumping microdroplets

Water-repellent materials ideally operate at very different liquid scales: from centimeter-size for bugs living on ponds through millimeter-size for antirain functions to micrometer-size for antifogging solids. In the last situation, it was recently evidenced that microdrops condensing on a highly nonadhesive substrate can take advantage from coalescence to jump off the material, even if the dynamical characteristics of the jump were not established at such microscales. We demonstrate in this paper that the jumping speed of drops is nonmonotonic with the drop size, showing a maximum around $5\mu \mathrm{m}$ (a size commonly observed in dew), below and above which viscous and inertial effects, respectively, impede the takeoff. We quantitatively describe this optimum in antifogging. We also studied the ballistics of the jumping microdrops, from the height they reached to their behavior at landing; a situation where retakeoff is surprisingly found to be nearly unachievable despite the extreme nonwettability of the material.

Figure 1

(a) Scanning electron micrograph of the nanocones used in this study. The scale indicates 100 nm. (b) Sketch of two nonwetting droplets with radius $r$ merging at a velocity $v$ on hydrophobic nanocones; the jumping velocity of the resulting drop is denoted as $U$. (c) Side-view chronophotography of a jumping drop with radius $R=11.3\pm 0.1\mu \mathrm{m}$ resulting from the coalescence of a pair of droplets with $r=8.9\pm 0.1\mu \mathrm{m}$. Images are separated by 0.125 ms. The drop takes off with a vertical jumping velocity $U=55\pm$ 5 cm/s. (d) Plotting the drop velocity $\dot{z}$ as a function of time $t$ for the experiment of Fig. 1 provides our definition of the jumping velocity $U$: it is taken as the maximum of $\dot{z}$. (e) High-speed photography of a symmetric coalescence of two drops with $r=580\pm 5\mu \mathrm{m}$. Images are separated by 3.7 ms, except the last one which is at 15.5 ms, when the drop reaches its maximum height. The first snapshot (at the top left) shows the beginning of the coalescence while the second one corresponds to takeoff; the measured jumping velocity is $U=7\pm$ 1 cm/s.

Figure 2

Jumping velocity resulting from the coalescence of two droplets. (a) Velocity $U$ for symmetric coalescence ($\varepsilon >0.95$) as a function of the radius $r$ of the two merging drops. Black line shows the speed resulting from energy conservation ($U=1.11U^{*}$) and red dashes correspond to $U=0.22U^{*}$, denoting $U^{*}=\sqrt{\gamma /\rho r}$. The red line shows the velocity $U=(U^{*}/4)\left[\alpha -4.9Oh\right]$ where $\alpha =1-6{sin}^2{\theta }_r(1+cos{\theta }_r)$ and $Oh=\eta /\sqrt{\rho \gamma r}$ is the Ohnesorge number with $\eta$ the water viscosity. The coefficient 4.9 is close to 4, the one predicted in Eq. (1). (b) Jumping velocity of droplets after an asymmetric coalescence; $U$ is normalized by $U^{*}=\sqrt{\gamma /\rho r}$ and plotted as a function of the degree of symmetry $\varepsilon =r^{\prime }/r$. The three sets of data correspond to three ranges for the larger radius: $2\mu \mathrm{m}<r\le 5\mu \mathrm{m}$ (red) corresponding to $0.08\le Oh<0.13$, $5\mu \mathrm{m}<r\le 13\mu \mathrm{m}$ (blue) corresponding to $0.05\le Oh<0.08$, and $13\mu \mathrm{m}<r\le 22\mu \mathrm{m}$ (black) corresponding to $0.04\le Oh<0.05$. The dashed line shows $U/U^{*}={\varepsilon }^{5/2}/2(1+{\varepsilon }^3)$ a function given by momentum conservation and neglecting adhesion and viscous dissipation [32], which provides an asymptotical behavior for the data. Red, blue, and black areas show Eq. (2) drawn with the corresponding colors for each range of Ohnesorge numbers.

Figure 3

Flight of departing drops after takeoff. (a) Vertical trajectory $z\left(t\right)$ of a drop with radius $R=12.9\pm 0.7\mu \mathrm{m}$ after its ejection at a velocity $U=29\pm$ 2 cm/s. Inset: Same plot for a drop with radius $R=287\pm 3\mu \mathrm{m}$ departing at $U=11\pm$ 2 cm/s. The function $z\left(t\right)$ is nicely fitted by a parabola (red solid line). (b) Maximum measured height $H$ reached during flight as a function of radius $R$. The dashed line represents the theoretical height $H=U{\tau }_a-g{\tau }_a^{2}ln(1+U/g{\tau }_a)$. (c) Absolute terminal velocity $V$ as a function of radius $R$. The dashed line corresponds to the velocity given by the equilibrium between gravity and Stokes drag $V=(2/9)\rho g{R}^2/{\eta }_a$.

Figure 4

Phase diagram of droplets at landing. The drop radius and velocity are denoted as $R$ and $V$. Green color indicates bouncing and red color indicates sticking. Dots and squares mean that drops are made from condensation or from a spray, respectively. The solid line expresses the balance between adhesion and inertia [Eq. (5)]. The dashed line represents the threshold of bouncing dictated by viscosity $V^{*}\approx \eta /\rho R$.