Singular jets during droplet impact on superhydrophobic surfaces

TL;DR

The study tackles jetting during droplet impact on superhydrophobic surfaces by experimentally varying We and Oh and performing a theoretical analysis to uncover two jetting mechanisms. It introduces a novel HI singular jet, in addition to the established CD singular jet, and develops scaling laws for spreading, spire formation, cavity collapse, and central-film dynamics. A comprehensive regime map classifies outcomes (CD/HI singular jets, Worthington jets, and bouncing) and provides transition criteria such as $D_{max}/D_0 \sim We^{1/4}$ and $H_{th}$ scalings, enabling prediction of when thin-film flow drives singular jets. The findings offer insight into rapid jet formation in high-inertia, high-capillary regimes with potential implications for aerosol generation, inkjet applications, and needle-free fluid delivery, while outlining avenues for future work on complex fluids and additives.

Abstract

Hypothesis: The impact of droplets is prevalent in numerous applications, and jetting during droplet impact is a critical process controlling the dispersal and transport of liquid. New jetting dynamics are expected in different conditions of droplet impact on super-hydrophobic surfaces, such as new jetting phenomena, mechanisms, and regimes. Experiments: In this experimental study of droplet impact on super-hydrophobic surfaces, the Weber number and the Ohnesorge number are varied in a wide range, and the impact process is analyzed theoretically. Findings: We identify a new type of singular jets, i.e., singular jets induced by horizontal inertia (HI singular jets), besides the previously studied singular jets induced by capillary deformation (CD singular jets). For CD singular jets, the formation of the cavity is due to the propagation of capillary waves on the droplet surface; while for HI singular jets, the cavity formation is due to the large horizontal inertia of the toroidal edge during the retraction of the droplet after the maximum spreading. Key steps of the impact process are analyzed quantitatively, including the spreading of the droplet, the formation and the collapse of the spire, the formation and retraction of the cavity, and finally the formation of singular jets. A regime map for the formation of singular jets is obtained, and scaling relationships for the transition conditions between different regimes are analyzed.

Figures (8)

• Figure 1: (A) Schematic diagram of the experimental setup for the impact of droplets on the super-hydrophobic surface. (B) Optical photos of the super-hydrophobic surface. (C) Schematic diagram of the super-hydrophobic surface. (D) SEM images of the super-hydrophobic surface, inset: a magnification of the surface.
• Figure 2: Two types of singular jets during droplets impact on the super-hydrophobic surface. (A) Singular jet induced by capillary deformation (i.e., CD singular jet). The corresponding experimental setting is ${\text{We}}={5.92}$, ${\text{Oh}}={2.08}\times {10}^{ -3}$. (B) Singular jet induced by horizontal inertia (i.e., HI singular jet). The corresponding experimental setting is ${\text{We}}={27.70}$, ${\text{Oh}}={1.92}\times {10}^{-2}$. In each panel, the first row is experimental images and the second row is the schematic drawing. Video clips of these processes are available as Supplementary Material as Movies 1 and 2.
• Figure 3: (A1) Dimensionless maximum spreading diameter as a function of the Weber number. The red solid line is ${{{D}_{\max }}}/{{{D}_{0}}}={0.82}{{\text{We}}^{{1}/{4}}}$ (i.e., Eq. (\ref{['eq:01']})). (B1) Formation and collapse of the spire. A schematic diagram of the critical state before and after the spire formation is shown in the dashed box, where ${{t}_\text{top}}$ is the time for the formation of a center liquid column; ${{U}_\text{cap}}$ is the velocity of the capillary wave; ${{U}_\text{top}}$ is the velocity of the top surface of the spire; ${{D}_\text{top}}$ is the top diameter of the spire; ${{H}_{\text{spire}}}$ is the height of the spire. (B2) Dimensionless top diameter of the spire as a function of the Ohnesorge number. The solid line is ${{D}_\text{top}}/{{D}_{0}}={2.0}{\text{Oh}}^{{1}/{3}}$ (i.e., Eq. (\ref{['eq:05']})). (B3) Dimensionless height of the spire as a function of the dimensionless droplet thickness ${H}/{{{D}_{0}}}$. The solid line is ${{{H}_{\text{spire}}}}/{{{D}_{0}}}={1.53}{H}/{{{D}_{0}}}$ (i.e., Eq. (\ref{['eq:07']})). (C1-C2) Two geometrical models to estimate the droplet volume. (C1) The cylindrical model: the cavity is small and the droplet volume can be estimated as a cylinder. (C2) The toroidal model: the center liquid film is thin and the droplet volume can be estimated as a toroidal. (D1) Dimensionless droplet thickness ${H}/{{{D}_{0}}}$ as a function of Weber number. The red solid line is ${H}/{{{D}_{0}}}={1.14}{\text{We}}^{-1/2}$ (i.e., Eqs. (\ref{['eq:12']})), and the red dash line is ${H}/{{{D}_{0}}}={0.44}{{\text{We}}^{{-1}/{8}}}$ (i.e., Eq. (\ref{['eq:18']})), in which the coefficients are fitted from experimental data based on Eqs. (\ref{['eq:11']}) and (\ref{['eq:17']}), respectively. (D2) Dimensionless maximum cavity diameter ${{{D}_\text{cav,m}}}/{{{D}_{0}}}$ as a function of Weber number. The red solid line is ${{{D}_\text{cav,m}}}/{{{D}_{0}}}={0.079}{\text{We}}$ (i.e., Eq. (\ref{['eq:15']})), and the red dash line is ${{{D}_\text{cav,m}}}/{{{D}_{0}}}={0.53}{\text{We}}^{{1}/{4}}$ (i.e., Eqs. (\ref{['eq:20']})), in which the coefficients are fitted from experimental data based on Eqs. (\ref{['eq:14']}) and (\ref{['eq:19']}), respectively.
• Figure 4: (A) Cavity diameter as a function of time ${{t}_\text{c}}-t$, where ${{t}_\text{c}}$ is the collapse time. The circles and the triangles are experimental data of GMW (58.3wt%) and water respectively. Rhombus and inverted triangles are data reported by Bartolo et al Bartolo2006 and Chen et al Chen2017, respectively. The solid line is ${{D}_\text{cav}(t)}={0.04}{{\left( {{t}_\text{c}}-t \right)}^{{1}/{2}}}$. (B) The jet velocity as a function of the dimensionless jet radius. Data reported in the literature Bartolo2006 is also included for comparison. The solid line is ${{U}_\text{jet}}{{[{\sigma} /({\rho} {{R}_{0}})]}^{-1/2}}={0.678}{{\left( {{R}_\text{jet}}/{{R}_{0}} \right)}^{-1}}$. (C) Dimensionless jet velocity ${{{U}_{\text{jet}}}}/{{{U}_{0}}}$ as a function of Weber number. The black solid line is ${{{U}_\text{jet}}}/{{{U}_{0}}}={6187}{{\text{We}}^{-3}}$, and the black dash line is ${{{U}_\text{jet}}}/{{{U}_{0}}}={113.6}{{\text{We}}^{-{9}/{8}}}$, in which the coefficients are fitted from experimental data; Dimensionless jet radius ${{{R}_\text{jet}}}/{{{R}_{0}}}$ as a function of Weber number. The black solid line is ${{{R}_\text{jet}}}/{{{R}_{0}}}={1.47}\times {{10}^{-4}}{{\text{We}}^{{5}/{2}}}$, and the black dash line is ${{{R}_\text{jet}}}/{{{R}_{0}}}={6.68}\times {{10}^{-3}}{{\text{We}}^{{5}/{8}}}$, in which the coefficients are fitted from experimental data.
• Figure 5: Regime map for singular jets during droplet impact. There are four phenomena, including two types of bouncing, the CD singular jet, the HI singular jet, and the Worthington jet. The green, blue, orange, and red lines respectively represent the critical condition of “bouncing” transforms into “CD singular jet”; “bouncing” transforms into “HI singular jet”; “CD singular jet” transforms into “HI singular jet”; “HI singular jet” transform into “Worthington jet”.