Bouncing microdroplets on hydrophobic surfaces

Abstract

Intuitively, slow droplets stick to a surface and faster droplets splash or bounce. However, recent work suggests that on nonwetting surfaces, whether microdroplets stick or bounce depends only on their size and fluid properties, but not on the incoming velocity. Here, we show using theory and experiments that even poorly wetting surfaces have a velocity-dependent criterion for bouncing of aqueous droplets, which is as high as 6 m/s for diameters of 30 to 50 [Formula: see text]m on hydrophobic surfaces such as Teflon. We quantify this criterion by analyzing the interplay of dissipation, surface adhesion, and incoming kinetic energy, and describe a wealth of associated phenomena, including air bubbles and satellite droplets. Our results on inertial microdroplets elucidate fundamental processes crucial to aerosol science and technology.

Keywords: aerosols; bouncing; deposition; droplet; microfluidics.

Fig. 1.

Microdroplet experiments. (A) Schematic of experimental setup, in which an aqueous droplet is dispensed and imaged on a high-speed camera. (B) Experimental results plotted within the droplet phase space of Weber $\mathit{We}$ vs. Ohnesorge $\mathit{Oh}$ numbers for a Teflon surface with a contact angle of 110°. For each experimental data point, represented by a circle (water), square (2.5%CaCl₂), or triangle (5$\%$ glycerol), the outcome was classified as a sticking event [e.g., points labeled as (C and D) and shown in the corresponding subfigure] or a partial rebound [e.g., subfigures (E and F)]. Error bars denote measurement uncertainties. The transition line is a guide to the eye separating the experimental outcomes. (C–F) High-speed imaging of droplet impact outcomes: the recorded outcomes are water droplets on a Teflon surface with a static contact angle of 110°. (C) Sticking outcome, with an entrained air bubble shown by red arrow. Observed oscillations indicate that sticking occurs in the underdamped regime. $(\mathit{Oh},\mathit{We})=(0.0197,8.80)$. (D) Sticking outcome, without an air bubble but with oscillations. $(\mathit{Oh},\mathit{We})=(0.0191,12.0)$. (E) Partial rebound, with a small sessile droplet shown by red arrow. $(\mathit{Oh},\mathit{We})=(0.0152,23.7)$. (F) Partial rebound, with a smaller sessile droplet (red arrow). $(\mathit{Oh},\mathit{We})=(0.0166,39.8)$.

Fig. 2.

(A) Schematic of finite-element phase-field simulations for microdroplet bouncing. (B–E) Numerical simulations reproduce the experimentally observed outcomes: (B) Sticking with an air bubble and small oscillations, $(\mathit{Oh},\mathit{We},\theta )=(0.019,2,{110}^{°})$. (C) Sticking with a large maximum spread and large oscillations, $(\mathit{Oh},\mathit{We},\theta )=(0.028,56,{110}^{°})$. (D) Total rebound with a large maximum spread and an air bubble, $(\mathit{Oh},\mathit{We},\theta )=(0.019,27,{110}^{°})$. (E) Partial rebound with a large initial spread and a necking instability, $(\mathit{Oh},\mathit{We},\theta )=(0.011,42,{100}^{°})$.

Fig. 3.

(A) Droplet outcome comparing numerical and experimental results, where the black line indicates the numerical transition from sticking (blue) to bouncing (red) for $\theta ={110}^{°}$. The dotted line shows that the same transition for the larger contact angle $\theta ={120}^{°}$ occurs at lower $\mathit{We}$ (SI Appendix, Fig. S5 for numerical data points and SI Appendix, Fig. S6 for a rescaled plot in the Reynolds vs. Ohnesorge numbers phase space). The data points are the experimental results from Fig. 1. (B) A view of the $(\mathit{Oh},\mathit{We})$ parameter space using simulations with a larger range of values and $\theta ={110}^{°}$. For higher impact velocities (i.e., higher $\mathit{We}$), the droplets again begin to stick, showing the reentrant nature of the transition.

Fig. 4.

Rebounding and sticking in the phase space of Velocity vs. droplet Diameter for poorly wetting surface, $\theta ={110}^{°}$, corresponding to the redimensionalized $\mathit{We}-\mathit{Oh}$ axes in Fig. 3. We keep other parameters constant and plot experimental data for water on Teflon at room temperature as points (subset of Fig. 1 data recorded at the same viscosity and size) and finite-element simulations as lines. The graph indicates three distinct sticking mechanisms, with boundaries derived from the numerical results: dissipative sticking occurs for small droplets whose incoming velocity is fully dissipated by viscosity; adhesive sticking occurs when droplets are underdamped but cannot rebound due to surface adhesion; and high-velocity sticking occurs when kinetic energy is dissipated through larger maximum spreading at high impact speeds. See SI Appendix, Fig. S7, for a nondimensional label of these regions in combination with theory.

Fig. 5.

Experimental measurement of the diameter at maximum spread, $D_{\text{m}}$, normalized by droplet diameter $D$, for $(\mathit{Oh},\theta )=(0.015,{110}^{°})$, $(0.017,{110}^{°})$, $(0.019,{110}^{°})$, and $(0.017,{140}^{°})$, for two different surfaces: Teflon and nanoparticle-coated glass. Both fits are consistent with theoretical prediction in Eq. 7 with different $\theta$-dependent prefactors. These data show that the maximum spread is independent of $\mathit{Oh}$ (SI Appendix, Fig. S2).

Fig. 6.

(A) An illustration of the simple model of two masses, two springs, and a damper to represent a microdroplet impact. (B) The extension $\mathrm{\Delta }x$ is plotted for two representative cases, one where $\mathrm{\Delta }x$ exceeds the bouncing threshold (red) and one where it does not (blue, with lower initial velocity). (C) $\mathit{We}-\mathit{Oh}$ phase space for this model with ($\alpha ,\zeta ,\theta$) = (50,1,110°) . This minimal model reproduces the simulation and experimental results (see SI Appendix, Fig. S7 for a further phase space).

Fig. 7.

(A) Phase space of microdroplet impact experiments on a silicon nanosphere coated glass surface of static contact angle $\theta ={140}^{°}$. (B–E) High-speed imaging of droplet impact outcomes: (B) Sticking with large oscillations, $(\mathit{We},\mathit{Oh})=(2.31,0.0177)$. (C) Partial rebound with a small sessile droplet, $(\mathit{We},\mathit{Oh})=(10.1,0.0172)$. (D) Partial rebound with a medium sessile droplet, $(\mathit{We},\mathit{Oh})=(20.7,0.0166)$. (E) Partial rebound with a large sessile droplet, $(\mathit{We},\mathit{Oh})=(31.6,0.0167)$.