Large impact velocities suppress the splashing of micron-sized droplets

Here we investigate the transition from spreading to splashing of drops with radii $R$ varying from millimeters to tens of microns impacting onto a smooth and dry partially wetting substrate at normal atmospheric conditions. Experiments show that the smaller $R$ is, the larger the impact velocity $V$ for the drop to splash needs to be but also that splash is inhibited if ${\mathrm{We}}_{\lambda }=\rho V^2\lambda /\sigma \gtrsim 0.5$, with $\sigma$, $\rho$, and $\lambda$ indicating the interfacial tension coefficient, the liquid density, and the mean free path of gas molecules. This result has been validated for two different values of the Ohnesorge number ${\mathrm{Oh}}_{\lambda }=\mu /\sqrt{\rho \lambda \sigma }$, with $\mu$ indicating the liquid viscosity, defined using only the material properties of the liquid and of the surrounding gaseous atmosphere. The underlying reason for this a priori unexpected finding results from the fact that the thin liquid film ejected after the drop touches the substrate is, under many practical conditions, $H_t\simeq \sigma /\left(\rho V^2\right)$ Riboux and Gordillo [Phys. Rev. Lett. 113, 024507 (2014)]. Then, for sufficiently large values of $V$, the thickness of the lamella becomes similar to the mean free path of gas molecules, i.e., $H_t\approx \lambda$, and, under these conditions, the splash of the drop is inhibited because the lift force causing the liquid to dewet the partially wetting solid is negligible. The spreading to splashing and the splashing to spreading transitions observed experimentally as the impact velocity is increased and the radii of the droplets is above a certain threshold value are very well predicted by the theory in G. Riboux and J. M. Gordillo [Phys. Rev. Lett. 113, 024507 (2014)] and J. M. Gordillo and G. Riboux [J. Fluid Mech. 871, R3 (2019)] once the aerodynamic lift force is set to zero for $H_t/\lambda \lesssim 2$, i.e., when ${\mathrm{We}}_{\lambda }\gtrsim 0.5$.

Figure 1

The final shape of a impacting drop onto a smooth solid substrate is circular if the drop spreads over the solid (a) or it is star shaped if the drop splashes (b).

Figure 2

(A) Sketch of the different experimental setups used to produce droplets with radii varying from millimeters to tens of microns. These droplets impact against a dry and smooth silicon substrate at normal atmospheric conditions. (a) Quasistatic generation of millimeter-sized drops, $R\sim 1.15$ mm ($\mathrm{Oh}=7\times {10}^{-3}$), from a needle placed at different heights from the impacting surface in order to change the impact velocity, $V$, within the range $0.96<V<2.98$ m/s. In this case, the high-speed camera (FASTCAM, SA-X, Photron) is operated at $10000$ frames per second (f.p.s.) and the spatial resolution is $\sim 10$ $\mu \mathrm{m}$/pix. (b) Generation of micron-sized droplets by means of the device described in Ref. [22]. The radii of the droplets produced in this way vary within the range $R\sim 79$–330 $\mu \mathrm{m}$ ($\mathrm{Oh}\sim 13\times {10}^{-3}$–$27\times {10}^{-3}$), whereas the impact velocity varies within the range $V\sim 1$–16 m/s. The high-speed camera (FASTCAM, SA-X, Photron) is operated at $100000$ f.p.s. and, in this case, the spatial resolution is $\sim 8$ $\mu \mathrm{m}$/pix. (c) A continuous ink-jet printer (CIJ) (Kishu Giken), is used to produce droplets with radii $R\sim 45$–69 $\mu \mathrm{m}$ ($\mathrm{Oh}\sim 29\times {10}^{-3}-36\times {10}^{-3}$) and impact velocities vary within the range $V\sim 12.2$–18.0 m/s. In this case, the drop impact process is recorded using a high-speed camera (Kirana, Nobby Tech) operated at $500000$ f.p.s., obtaining images with a spatial resolution of $1.5\mu \mathrm{m}$/pix. (B) Images obtained using each of the three setups depicted in (A). (a) A drop of radius $R\sim 1.15$ mm impacting at $V\sim 3$ m/s ($\mathrm{We}\sim 365$) splashes and disintegrates into tiny droplets. (b) Splashing of a drop of radius $R\sim 200\mu \mathrm{m}$ impacting at $V\sim 6$ m/s ($\mathrm{We}\sim 254$). (c) Spreading of a drop of radius $R\sim 65\mu \mathrm{m}$ impacting at $V\sim 18$ m/s ($\mathrm{We}\sim 742$). The limited spatial resolution does not exclude the unlikely possibility that submicron-sized droplets are produced in the experiments.

Figure 3

This figure represents the range of experimental values of $R$ and $V$ investigated here for each of the three different experimental setups sketched in Fig. 2, with $D$ indicating the diameter of the capillary tube used in the device described in Ref. [22]. Empty symbols indicate that the droplet spreads over the substrate and blue ones that the droplet splashes. Orange and green symbols indicate the experimental conditions represented at the top and bottom rows of Fig. 4, respectively. The symbols in red indicate the values of $R$ and $V$ corresponding to the experimental images in Fig. 2.

Figure 4

The top sequence of images shows the effect of increasing the drop impact velocity for the case of droplets with radii $R/\lambda \approx 2000$, where a spreading-splashing-spreading transition is observed. The sequence at the bottom, where no splash is observed in spite of the impact velocities are even larger than those in the images at the top, corresponds to the case of drops with $R/\lambda \approx 800$ generated by means of a CIJ printer. The experimental conditions corresponding to the images at the top and bottom rows are indicated in Fig. 3 using, respectively, orange and green symbols.

Figure 5

Sketch and definition of the different variables used in the analysis in main text, with $F_L$ the lift force, $H_t$ and $V_t$ denoting the thickness and velocity of the edge of the expanding liquid sheet and $V_v$ the vertical velocity. All these variables are functions of the dimensionless ejection time $t_e=T_{e}V/R$, defined as the instant at which the thin liquid sheet is ejected tangentially to the substrate. The angle $\alpha$ refers to the wedge angle formed by the advancing rim and the substrate. The flow field entering into the lamella at the top right sketch is represented in a frame of reference moving with the velocity at which the drop wets the substrate, namely $V\dot{a}=Vda/dt$ with $a=\sqrt{3t}$ and $t=VT/R$ the dimensionless time. The expression for the thickness of the boundary layer $\delta$ in Eq. (9) was deduced in Ref. [12] in the moving frame of reference, where there exists a stagnation point, also represented in the sketch.

Figure 6

The boundary separating spreading and splashing conditions obtained experimentally is compared with the theoretical predictions in a $R/\lambda \text{−}{\mathrm{We}}_{\lambda }$ plane, with ${\mathrm{We}}_{\lambda }=\rho V^2\lambda /\sigma$ and $\lambda =68\times {10}^{-9}$ m. The white region indicates spreading, whereas splashing is represented in blue for the case of ethanol (${\mathrm{Oh}}_{\lambda }=\mu /\sqrt{\rho \sigma \lambda }=0.91$) and in yellow for the case of water (${\mathrm{Oh}}_{\lambda }=0.46$). The thicker continuous line and the dash-dotted black line are used to respectively indicate the predicted spreading-splashing transition for the cases of ethanol (continuous line) and water (dash-dotted line). The vertical thin line indicates ${\mathrm{We}}_{\lambda }={\mathrm{We}}_{\lambda 0}=0.5$. The experimental data point corresponding to the spreading of a water droplet of radius $R=25\mu \mathrm{m}$ impacting with a velocity of 50 m/s on a smooth substrate is taken from Visser et al. [5, 6].

Figure 7

The condition expressed by Eq. (17) with $K=0.33$ are represented using striped lines in the phase diagrams corresponding to (a) ethanol drops in air at normal atmospheric conditions, for which ${\mathrm{Oh}}_{\lambda }=0.91$ and (b) water drops in air at normal atmospheric conditions, for which ${\mathrm{Oh}}_{\lambda }=0.46$. The present theoretical results cannot be applied to predict whether the drop spreads or splashes in the stripped area, which represent conditions under which the thickness of the boundary layer is similar to that of the ejected lamella. The experimental data point corresponding to the spreading of a water droplet of radius $R=25\mu \mathrm{m}$ impacting with a velocity of 50 m/s on a smooth substrate is taken from Visser et al. [5, 6].