Role of liquid viscosity and of air entrapped on the splashing of drops impacting over superhydrophobic substrates

Here we analyze the splash transition of drops of liquids with different viscosities impacting normally over different types of rough superhydrophobic substrates, finding that the velocity for the splash transition, $V^{*}$, increases for increasing values of the liquid viscosity and also when the proportion of air entrapped at the substrate corrugations decreases. Our experimental results also reveal that the amount of air entrapped between the substrate and the wall increases with the value of the relative roughness $\epsilon =\varepsilon /R$, with $\varepsilon$ and $R$ indicating the amplitude of the corrugations and the radius of the drop, respectively. We show that our experimental values of $V^{*}$, as well as those reported in similar contributions in the literature, can be predicted using the splashing model presented by Quintero, Riboux, and Gordillo [J. Fluid Mech. 870, 175 (2019)], once the liquid shear stress at the wall is expressed as a decreasing function of $\epsilon$, namely, of the proportion of air entrapped at the substrate.

Figure 1

Impact with a velocity of $V=2.22\pm 0.02{\mathrm{ms}}^{-1}$ of a water drop of radius $R=1.43\pm 0.03$ mm over a smooth glass substrate (left: the value of the static contact angle $\theta$ is $\theta <\pi /2$ and, therefore, the liquid partially wets the substrate) or over a smooth Teflon substrate (right: $\theta >\pi /2$, hydrophobic substrate).

Figure 2

(a) Front and lateral view of different superhydrophobic substrates of increasing roughnesses. (b) The images show the impact with a velocity of $V=2.95\pm 0.01\mathrm{m}{s}^{-1}$ of water drops of radius $R=1.47\pm 0.02$ mm over the three types of substrates depicted in (a) at $t=TV/R=4\pm 0.03$, with the value of the substrate roughness $\varepsilon$ increasing from left to right. (c) The images show the impact with a velocity $V=2.85\pm 0.02\mathrm{m}{s}^{-1}$ of drops of radius $R=1.42$ mm of a 30% water-sugar mixture with a viscosity of three times that of water over the three substrates in panel (a) at $t=TV/R=4\pm 0.03$ with the value of the substrate roughness increasing from left to right. The images in panels (b) and (c) illustrate that the proportion of air bubbles entrapped between the solid and the expanding lamella increases with the value of $\varepsilon$. The presence of air bubbles can be more clearly appreciated from the processed images at the bottom of panels (b) and (c), where gas regions are visualized in black. Here, $T$ indicates the instant of time after the drop first touches the substrate.

Figure 3

Images showing the impact with different velocities of drops of radius $R=1.44\pm 0.05$ mm of either water (blue background) or of a 30% water-sugar mixture with a viscosity of three times that of water (pink background) over three different types of superhydrophobic substrates with increasing roughnesses. The blue or pink boxes are used to highlight the occurrence of splashing. The value of the advancing contact angle is the same, ${\theta }_{\text{adv}}\approx {158}^{\circ }$, for the cases of Teflon $\varepsilon \approx 0\mu \mathrm{m}$, and for the case of glass covered with three layers of NeverWet, $\varepsilon =60\pm 40\mu \mathrm{m}$, see Table 2. The images are represented at $t=TV/R=0.8\pm 0.08$, with $T$ the instant of time after the drop first touches the substrate.

Figure 4

Spreading of a drop of radius $R=1.45\pm 0.03$ mm of either water (a) or a 30% sugar-water mixture (b) impacting with a velocity $V=2.89\pm 0.05\mathrm{m}{s}^{-1}$ over two different types of superhydrophobic substrates with different rugosities. (c) At the left of each of the images in panels (a) and (b), drops impact against glass coated with several layers of NeverWet, ($\varepsilon =60\pm 40\mu \mathrm{m}$) and, at the right, drops impact over PTFE (Teflon, $\varepsilon \approx 0\mu \mathrm{m}$). For both values of the liquid viscosity, the maximum expansion of the drop is attained for the case of the superhydrophobic substrate with the larger value of $\varepsilon$ (left images). Here, $t=TV/R$, with $T$ the instant of time after the drop first touches the substrate.

Figure 5

The images show the impact of drops over three of the different types of substrates investigated in this study at two different instants of time, $t=TV/R=3\pm 0.03$ (top part of each image) and $t=10\pm 0.2$ (bottom part of each image), with $T$ the instant of time after the drop first touches the substrate, for a nearly constant value of the Weber number $\text{We}=77\pm 10$ and two values of the Ohnesorge number: $\text{Oh}=3.1\times {10}^{-3}$ for the case of water drops—images at the top—and $\text{Oh}=9.1\times {10}^{-3}$ for the case of the 30% water-sugar mixture. As it is expected for the case of superhydrophobic substrates, the rim retracts after reaching the maximum expansion. Notice also that the bright area in each of the images, which is the signature of the air entrapped in the corrugations, is much larger for the case of the SH0 substrate, which is the one with the largest value of $\varepsilon$—see Table 2.

Figure 6

The experimental values of the critical Weber number for splashing, ${\text{We}}^{*}$, are represented for three values of the Ohnesorge number and decreasing values of the substrate roughness, see Table 3. The general trend is that ${\text{We}}^{*}$ increases if Oh increases or $\varepsilon$ decreases. Since the value of ${\text{We}}^{*}$ is unclear in some cases, we include in panel (a) error bars and in panels (b)–(d) we show experiments carried out using the PTFE substrate revealing that, in this case, the number of droplets ejected increases very smoothly with We, this fact justifying the error bars in panel (a).

Figure 7

Sketch illustrating the definitions of the different dimensionless variables used in the model, $s\left(t\right), v\left(t\right), b\left(t\right), u(r,t)$, and $h(r,t)$ [34], which are defined using as characteristic values of length, time and pressure, $R, R/V$, and $\rho V^2$, with $R, V$, and $\rho$ denoting, respectively, the drop radius, the impact velocity and the liquid density. The value for the dynamic advancing contact angle is taken here as ${\theta }_d=\pi$.

Figure 8

The ratio $C_b={\text{We}}^{1/2}\sqrt{b/8}db/dt$ defined in Eq. (2) is calculated solving the system of Eqs. (3) and (4) subjected to the initial conditions given in Eqs. (8) and (10) for different values of We and two values of $\lambda$ and Oh: $\left(a\right)\text{Oh}=3\times {10}^{-3}$ and $\lambda =0, \left(b\right)\text{Oh}=3\times {10}^{-3}$ and $\lambda =1, \left(c\right)\text{Oh}=9\times {10}^{-3}$ and $\lambda =0$ and $\left(d\right)\text{Oh}=9\times {10}^{-3}$ and $\lambda =1$. The critical Weber number for splashing, ${\text{We}}^{*}$, is calculated as the minimum value of We for which the curves cross the horizontal line $C_b={\text{We}}^{1/2}\sqrt{b/8}db/dt=0.1\pm 0.005$. Notice that ${\text{We}}^{*}$ increases with Oh and $\lambda$, namely, as the shear stress with the substrate increases.

Figure 9

(a) The values of ${\text{We}}^{*}$ calculated following the procedure illustrated in Fig. 8 using either Eq. (8) (continuous lines) or the approximate value in the low-Oh limit in Ref. [14] $t_e=1.05{\text{We}}^{-2/3}$ (dashed lines), are represented as a function of $\lambda$ for four different values of Oh. The value of ${\text{We}}^{*}$ corresponding to the potential flow case, $\text{Re}\to \infty$, is indicated with a horizontal black dashed line. The results in this figure clearly show that ${\text{We}}^{*}$ increases with Oh and $\lambda$, namely, as the shear stress with the substrate increases. (b) The friction factor $\lambda$ is calculated as a function of the values of $\epsilon$ given in Tables 3 and 4 by means of Eq. (7) using ${\epsilon }_0=0.02$ (black continuous line).

Figure 10

Comparison between the experimental values of (a) the critical Weber number and (b) the critical velocity for splashing given or deduced from Tables 3 and 4 and the corresponding values calculated solving the system of Eqs. (3) and (4) complemented with Eq. (7) for $\lambda \left(\epsilon \right)$ using ${\epsilon }_0=0.02$, see also Figs. 8 and 9. Notice that $\lambda =0$ for the case of drops impacting in the Leidenfrost regime and also for the superhydrophobic substrates created by depositing one (SH2), two (SH1) or three (SH0) layers of NeverWet over a glass substrate, see Tables 2 and 3.

Figure 11

Values of the critical Weber numbers as a function of the advancing contact angle ${\theta }_d$ obtained solving Eqs. (3), (4), (8), and (10) for the case in which the term $-2{\text{We}}^{-1}$ in Eq. (3) is replaced by $-(1-cos{\theta }_d){\text{We}}^{-1}$ with $\lambda =1$ and different values of Oh.