Impact Forces of Water Drops Falling on Superhydrophobic Surfaces

A falling liquid drop, after impact on a rigid substrate, deforms and spreads, owing to the normal reaction force. Subsequently, if the substrate is nonwetting, the drop retracts and then jumps off. As we show here, not only is the impact itself associated with a distinct peak in the temporal evolution of the normal force, but also the jump-off, which was hitherto unknown. We characterize both peaks and elucidate how they relate to the different stages of the drop impact process. The time at which the second peak appears coincides with the formation of a Worthington jet, emerging through flow focusing. Even low-velocity impacts can lead to a surprisingly high second peak in the normal force, even larger than the first one, namely when the Worthington jet becomes singular due to the collapse of an air cavity in the drop.

Figure 1

(a) Experimental setup: a water drop of diameter $D_0$ impacts on the superhydrophobic quartz plate at velocity $V_0$. (b) Numerical results for a drop impact dynamics for $D_0=2.05\mathrm{mm}$ and $V_0=1.2\mathrm{m}/\mathrm{s}$: $t=$ (i) 0 ms (touchdown), (ii) 0.37 ms, (iii) 2.5 ms, (iv) 3.93 ms, (v) 4.63 ms, and (vi) 5.25 ms. The left part of each numerical snapshot shows the dimensionless local viscous dissipation rates ${\dot{\tilde{E}}}_d^{\text{local}}$ (see Supplemental Material, Eq. S8 [48]) on a ${\log }_{10}$ scale and the right part the velocity field magnitude normalized with the impact velocity. The black velocity vectors are plotted in the center of mass reference frame of the drop to clearly elucidate the internal flow. (c) Spreading diameter $D(t)$ and impact force $F(t)$ on the substrate as function of time: comparison between experiments and simulations ($\mathrm{We}=40$). The insets show representative snapshots at specific time instants overlaid with the drop boundaries from simulations in orange, revealing good agreement again. $F_1\approx 5.1\mathrm{mN}$ and $F_2\approx 2.3\mathrm{mN}$ are the two peaks of the normal force $F(t)$ at $t_1\approx 0.37\mathrm{ms}$ and $t_2\approx 4.63\mathrm{ms}$, respectively. $t_m$ is the moment corresponding to the maximum spreading of the drop and $t_3$ represents the end of contact ($F=0$). See Supplemental Movie S1 [48].

Figure 2

Characteristic times as functions of We. The times $t_1$, $t_m$, $t_2$, and $t_3$ are normalized by the inertial timescale ${\tau }_{\rho }=D_0/V_0$ [in (a)] or by the inertial-capillary timescale ${\tau }_{\gamma }=({\rho }_d{D}_0^3/\gamma {)}^{1/2}$ [in (b)]. The black dashed and solid lines represent $t_1\approx 0.3{\tau }_{\rho }$ and $t_1\approx 1/6{\tau }_{\rho }$, respectively. The gray dashed lines show the best straight line fits to the experimental data, $t_m\approx 0.20{\tau }_{\gamma }$, $t_2\approx 0.44{\tau }_{\gamma }$, and $t_3\approx 0.78{\tau }_{\gamma }$.

Figure 3

Dimensionless peak forces, (a) ${\tilde{F}}_1$, (b) ${\tilde{F}}_2$ as functions of We. The variation of the second peak force ${\tilde{F}}_2$ with We divides the parameter space into four regimes: I. Capillary, II. Singular jet, III. Inertial, and IV. Splashing. (c) Evolution of the normal force $F(t)$ of an impacting drop for the case with highest ${\tilde{F}}_2$ ($\mathrm{We}=9$). Note again the outstanding agreement between the experimental and the numerical results, including the various wiggles in the curve, which originate from capillary oscillations. The insets show representative snapshots at specific time instants. (d) Numerical snapshots at the instants of the second peak force ($t_2$) for $\mathrm{We}=$ (i) 2, (ii) 6, (iii) 12, (iv) 40, (v) 100 and (vi) 300. The left part of each numerical snapshot shows the dimensionless local viscous dissipation rates ${\dot{\tilde{E}}}_d^{\text{local}}$ (see Supplemental Material, Eq. S8 [48]) on a ${\log }_{10}$ scale and the right part the velocity field magnitude normalized with the impact velocity. (e) Experimental drop geometry at $t_2$ for $\mathrm{We}=40$ (along with the drop contour from numerics in orange) to illustrate its spreading diameter $D_2$, height $h_2$, retraction velocity $v_2$, jet diameter $d_j$, and jet velocity $v_j$. (f) Comparison of the second peak force ${\tilde{F}}_2$ with its theoretical prediction ${\tilde{F}}_2^{\text{theory}}={\tilde{v}}_j{\tilde{v}}_2/{\tilde{D}}_2$ [Eq. (2)].