Expansion and retraction dynamics in drop-on-drop impacts on nonwetting surfaces

Impacts of liquid droplets on other stationary droplets on a surface are ubiquitous in numerous applications such as agricultural sprays, inkjet printing, and rain impact on surfaces. We experimentally study the maximum expansion diameter and retraction rate in drop-on-drop impacts on superhydrophobic surfaces. We identify an inertial-capillary and a viscous regime for the expansion phase and we interpret the results using two distinct models. In the inertial-capillary regime, the first model predicts that the maximum diameter is set by an effective capillary length due to the deceleration upon impact. We introduce an effective diameter, velocity, and Weber number that allow the accurate determination of the maximum diameter in drop-on-drop impacts. We use our model to predict the transition to the viscous regime and rationalize the maximum diameter in this case with an energy balance. In our second model, we use an energy balance in both regimes and accurately predict the maximum diameter with a unified expression. We finally show that the retraction phase is a no-memory phenomenon and only depends on the volume of the coalesced droplet. We identify capillary and viscous regimes for the retraction and accurately model the retraction rate in each regime. Our approach provides a framework to characterize the dynamics of multiple-drop impacts.

Figure 1

(a) Schematic of the experimental setup. A high-speed camera films the impact of a droplet on a second droplet gently deposited on the surface. The surface is superhydrophobic and the scanning electron microscopy image in the inset shows the nanograss texture. (b) Schematic of the typical phases of drop-on-drop impacts: The droplets coalesce and expand until they reach a maximum diameter $D_{\max }$ corresponding to a thickness $h$, then retract and bounce off. (c),(d) Snapshots of the high-speed video of drop-on-drop impacts. (c) Corresponds to the inertial-capillary regime $[D_i=D_s=2.3\mathrm{mm},V=1.10\mathrm{m}{\mathrm{s}}^{-1},\mu =8.9.{10}^{-4}\mathrm{Pa}\mathrm{s}\left(\mathrm{water}\right)]$, while (d) corresponds to the viscous regime ($D_i=D_s=3.4\mathrm{mm},V=1.23\mathrm{m}{\mathrm{s}}^{-1},\mu =0.34\mathrm{Pa}\mathrm{s}$). Videos are available in Ref. [55].

Figure 2

(a) Snapshots of the high-speed video of the impact of small water droplet on a bigger one (first row: $D_i=2.3\mathrm{mm},D_s=3.5\mathrm{mm},V=1.6\mathrm{m}{\mathrm{s}}^{-1}$) and vice versa (second row: $D_i=3.8\mathrm{mm},D_s=2.0\mathrm{mm},V=0.98\mathrm{m}{\mathrm{s}}^{-1}$). Both impacts are in the inertial-capillary regime. Videos are available in Ref. [55]. (b) Comparison of predicted (solid line) droplet thickness at maximum expansion and experimentally measured (symbols) thicknesses for various drop size ratios, velocities, and viscosities. The solid line is a linear fit ($y=1.09x$, $R^2=0.899$). (c) Comparison of predicted and experimentally measured maximum diameter for the same data. The solid line is a linear fit ($y=0.91x$, $R^2=0.887$).

Figure 3

Maximum expansion diameter results for drop-on-drop impacts. The $x$ axis is the impact parameter $P$ and the $y$ axis is the product of the normalized maximum diameter by the Reynolds number to the power 1/5. Two distinct regimes are observed. In the inertial-capillary regime ($P<1$, the solid line has a slope of 0.25 with a coefficient of determination $R^2=0.897$), $\frac{D_{\max }}{D_t}$ varies as $\mathrm{W}{\mathrm{e}}^{1/4}$ and does not depend on viscosity. In the viscous regime ($P<1$, dashed line), $\frac{D_{\max }}{D_t}$ varies as the $\mathrm{R}{\mathrm{e}}^{1/5}$. The shown data are our results for size ratios of 1 and 2 as well as single drop data from literature. All data collapse on a single master curve.

Figure 4

Maximum expansion diameter results for drop-on-drop impacts with an alternative model. The $x$ axis is the modified interpolation function from Laan et al. and the $y$ axis is the product of the normalized maximum diameter by the Reynolds number to the power $-1/5$. The solid line is a linear fit ($y=1.95x$) with $R^2=0.898$.

Figure 5

(a) Retraction results for drop-on-drop impacts. The $x$ axis is the Ohnesorge number and the $y$ axis is the product of the retraction rate by the inertial-capillary timescale. Two distinct regimes are observed. In the inertial-capillary regime $\dot{\epsilon }{\tau }_i$ is constant and does not depend on impact velocity, viscosity, and size ratio. In the viscous regime $\mathrm{Oh}>0.5$, the retraction rate scales as the inverse of the viscous-capillary timescale. (b) Snapshots of a drop-on-drop impact of water droplets of the same size ($D_i=D_s=2.3\mathrm{mm},V=1.25\mathrm{m}{\mathrm{s}}^{-1}$). (c) Snapshots of a single water drop impact ($D_i=2.9\mathrm{mm},V=0.98\mathrm{m}{\mathrm{s}}^{-1}$). The contact line moves exactly the same as the last case in the retraction phase. (d) Snapshots of a drop-on-drop impact of water droplets of the same size ($D_i=D_s=2.3\mathrm{mm},V=0.71\mathrm{m}{\mathrm{s}}^{-1}$). Videos are available in Ref. [55].