*Article* **Asymmetric Jetting during the Impact of Liquid Drops on Superhydrophobic Concave Surfaces**

**Chengmin Chen 1,2,3, Hongjun Zhong <sup>3</sup> , Zhe Liu <sup>2</sup> , Jianchun Wang 1,2, Jianmei Wang 1,2 , Guangxia Liu 1,2 , Yan Li 1,2 and Pingan Zhu 4,\***


**Abstract:** The impact of liquid drops on superhydrophobic solid surfaces is ubiquitous and of practical importance in many industrial processes. Here, we study the impingement of droplets on superhydrophobic surfaces with a macroscopic dimple structure, during which the droplet exhibits asymmetric jetting. Systematic experimental investigations and numerical simulations provide insight into the dynamics and underlying mechanisms of the observed phenomenon. The observation is a result of the interaction between the spreading droplet and the dimple. An upward internal flow is induced by the dimple, which is then superimposed on the horizontal flow inside the spreading droplet. As such, an inclined jet is issued asymmetrically into the air. This work would be conducive to the development of an open-space microfluidic platform for droplet manipulation and generation.

**Keywords:** droplet impact; superhydrophobic surface; asymmetric jetting; droplet manipulation

**Citation:** Chen, C.; Zhong, H.; Liu, Z.; Wang, J.; Wang, J.; Liu, G.; Li, Y.; Zhu, P. Asymmetric Jetting during the Impact of Liquid Drops on Superhydrophobic Concave Surfaces. *Micromachines* **2022**, *13*, 1521. https://doi.org/10.3390/ mi13091521

Academic Editor: Giampaolo Mistura

Received: 30 July 2022 Accepted: 6 September 2022 Published: 14 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

## **1. Introduction**

Understanding the dynamics of impacting droplets on nonwetting surfaces is of both scientific and technological importance, such as spraying crops with pesticides [1], spray cooling of hot surfaces [2], 3D inkjet printing of micro/nanostructures [3], shedding virus-laden aqueous droplets away for the anti-pathogen purpose [4], and manipulating droplets with open-space microfluidics [5]. Droplet impact is ubiquitous on solid surfaces with various features, including inclined surface [6], rough surface [7], micro-channels [8], curved surface [9], micro-cellular surface [10,11], heated surface [12], cold surface [13,14], and nanoparticles-coated surfaces [15], to name a few. Varying the surface properties can significantly alter the behaviors and dynamics of droplet impact, such as splashing, spreading, bouncing, jetting, and bubble encapsulation [16–20].

The jetting phenomenon has been previously identified when liquid droplets impact both hydrophilic [21,22] and hydrophobic [23] surfaces. Previous studies showed that jetting is induced by the collapse of the air cavity formed by the deformation of the drop at impact [24–26]. Detailed studies on the mechanism of jetting formation can be found in several recent works [27–29]. Apart from flat hydrophobic surfaces, jetting also occurs on superhydrophobic surfaces with tailored structures, such as ice-leaf-inspired grooved superhydrophobic surfaces [30], superhydrophobic surfaces with anisotropic surface patterning [31], superhydrophobic copper meshes [32], oblique surfaces [24] and artificial dual-scaled superhydrophobic surfaces [33]. The previously reported jetting is symmetric, which is vertical to solid surfaces. However, asymmetric jetting that is inclined to solid surfaces is yet to be observed.

Here, we report asymmetric jetting when water droplets eccentrically impact a macrosized dimple on superhydrophobic surfaces. The jetting velocity depends on the Weber number (We) and the impact position of the droplet, as demonstrated by experimental results. In parallel, numerical simulations reveal the internal flow field, pressure field, and momentum variation of the droplet. The combination of experimental and numerical studies shed light on the dynamics and mechanism of asymmetric jetting during droplet impact. Here, we report asymmetric jetting when water droplets eccentrically impact a macro-sized dimple on superhydrophobic surfaces. The jetting velocity depends on the Weber number (We) and the impact position of the droplet, as demonstrated by experimental results. In parallel, numerical simulations reveal the internal flow field, pressure field, and momentum variation of the droplet. The combination of experimental and numerical studies shed light on the dynamics and mechanism of asymmetric jetting during droplet impact.

surfaces is yet to be observed.

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 2 of 13

dual-scaled superhydrophobic surfaces [33]. The previously reported jetting is symmetric, which is vertical to solid surfaces. However, asymmetric jetting that is inclined to solid

#### **2. Experimental Methods 2. Experimental Methods**

The experimental setup is illustrated in Figure 1. Water droplets were generated from blunt syringe tips. The droplet diameter (*d<sup>l</sup>* , as shown in Figure 1b), which depends on the size of the syringe tips, was about 2.40 ± 0.05 mm. Following its detachment from the syringe tip, the droplet was accelerated by gravity and then impacted the superhydrophobic surface. The impact velocity (*v*0) of the droplet was changed by adjusting the height of the syringe tips, as varied from 0.5 m/s to 0.8 m/s. A copper surface was used in the experiments, on which a hemispherical dimple with a diameter of *d*dimple = 1.2 mm was excavated. The eccentric distance (*d*, as shown in Figure 1b) stands for the horizontal distance between the center of the droplet and the center of the dimple. The relative eccentric distance (*e*) was defined as *e* = *d*/*d*dimple. The surface was cleaned with acetone and ethanol and then coated with candle soot for superhydrophobicity. The candle soot was deposited by exposing the surface to the outer flame of a burning candle for 3–5 s to ensure that the surface areas were fully covered by the candle soot particles. The water contact angle was measured by an Optical Surface Analyzer OSA200 (Ningbo NB Scientific Instruments Co., Ltd., Ningbo, China). The apparent contact angle was about 145.3◦ (Figure S1 in Supplemental Materials), averaged from three measurements. The sliding angle was 8◦ and the contact angle hysteresis (the difference between the advancing angle and the receding angle) was about 2.7◦ . The experimental setup is illustrated in Figure 1. Water droplets were generated from blunt syringe tips. The droplet diameter (*dl*, as shown in Figure 1b), which depends on the size of the syringe tips, was about 2.40 ± 0.05 mm. Following its detachment from the syringe tip, the droplet was accelerated by gravity and then impacted the superhydrophobic surface. The impact velocity (*v*0) of the droplet was changed by adjusting the height of the syringe tips, as varied from 0.5 m/s to 0.8 m/s. A copper surface was used in the experiments, on which a hemispherical dimple with a diameter of *d*dimple = 1.2 mm was excavated. The eccentric distance (*d*, as shown in Figure 1b) stands for the horizontal distance between the center of the droplet and the center of the dimple. The relative eccentric distance (*e*) was defined as *e* = *d*/*d*dimple. The surface was cleaned with acetone and ethanol and then coated with candle soot for superhydrophobicity. The candle soot was deposited by exposing the surface to the outer flame of a burning candle for 3–5 s to ensure that the surface areas were fully covered by the candle soot particles. The water contact angle was measured by an Optical Surface Analyzer OSA200 (Ningbo NB Scientific Instruments Co., Ltd., Ningbo, China). The apparent contact angle was about 145.3° (Figure S1 in Supplemental Materials), averaged from three measurements. The sliding angle was 8° and the contact angle hysteresis (the difference between the advancing angle and the receding angle) was about 2.7°.

**Figure 1.** (**a**) Schematic of the experimental setup for droplet impact, (**b**) details of the solid surface. **Figure 1.** (**a**) Schematic of the experimental setup for droplet impact, (**b**) details of the solid surface.

A high-speed camera (Miro M310, Phantom) was used to record the impact process at 4000 frames per second. The camera was placed parallel to the horizontal surface to obtain the side-view images and videos of impacting droplets. An LED lamp with a diffuser was used for illumination. The recorded images and videos were analyzed by the Phantom camera control software obtained from the camera supplier. A high-speed camera (Miro M310, Phantom) was used to record the impact process at 4000 frames per second. The camera was placed parallel to the horizontal surface to obtain the side-view images and videos of impacting droplets. An LED lamp with a diffuser was used for illumination. The recorded images and videos were analyzed by the Phantom camera control software obtained from the camera supplier.

In general, the impact dynamics were characterized by the Weber number defined as the ratio of inertial to surface tension forces (Equation (1)), the Reynolds number defined In general, the impact dynamics were characterized by the Weber number defined as the ratio of inertial to surface tension forces (Equation (1)), the Reynolds number defined as the ratio of inertial to viscous forces (Equation (2)), the Bond number defined as the ratio of gravitational to surface tension forces (Equation (3)) and the capillary number defined as the ratio of viscous to surface tension forces (Equation (4)) [34–37]. Here, *ρ*, *v*0, *d<sup>l</sup>* , *σ*, and

*µ* are the density, the impact velocity, initial droplet diameter, the surface tension and the viscosity of the liquid, respectively. Water droplets were used in experiments, of which the density is 1000 kg/m<sup>3</sup> , the surface tension is 0.072 N/m and the viscosity is 0.001003 Pa·s. Re = /μ (2) Bo = g ଶ/4 (3)

We =

as the ratio of inertial to viscous forces (Equation (2)), the Bond number defined as the ratio of gravitational to surface tension forces (Equation (3)) and the capillary number defined as the ratio of viscous to surface tension forces (Equation (4)) [34–37]. Here, , , , , and are the density, the impact velocity, initial droplet diameter, the surface tension and the viscosity of the liquid, respectively. Water droplets were used in experiments, of which the density is 1000 kg/m3, the surface tension is 0.072 N/m and the

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 3 of 13

$$\text{We} = \rho v\_0^2 d\_l / \sigma \tag{1}$$

ଶ/ (1)

$$\text{Re} = \rho d\_l v\_0 / \mu \tag{2}$$

$$\mathbf{Bo} = \rho \mathbf{g} d\_l^2 / 4\sigma \tag{3}$$

$$\mathbf{\cal Ca} = \mu v\_0 / \sigma \tag{4}$$

#### **3. Asymmetric Jetting Phenomenon** difference from previous observations of droplet impact on flat superhydrophobic surfaces. Figure 2 presents snapshots of a droplet eccentrically impinging the dimpled

viscosity is 0.001003 Pa·s.

The behavior of a droplet impacting the dimpled surface showed a remarkable difference from previous observations of droplet impact on flat superhydrophobic surfaces. Figure 2 presents snapshots of a droplet eccentrically impinging the dimpled surface, where the eccentric distance *d* = 0.5*d<sup>l</sup>* , We = 23.41, Re = 1974.10, Bo = 0.20 and Ca = 0.01. The high values of We and Re indicated that inertial forces dominated over capillary and viscous forces during droplet impact, while the low values of Bo and Ca implied that the influences of gravitational and viscous forces were negligible compared with surface tension forces in this study. In Figure 2a, at *t* = 0 ms, the droplet contacted the dimpled surface, with We = 18.97. After contact, the droplet first spread when the time was less than 1.22 ms (Figure 2b) and it fully covered the dimple at *t* ~ 1.22 ms when an inclined jet was issued from the side of the droplet. The jetting angle (*θ* in Figure 2b and Figure S2 in Supplemental Materials) was about 45◦ relative to the horizontal plane of the surface. Afterwards, the droplet adopted asymmetric morphology during the spreading and contracting processes. As a result, the bouncing direction of the droplet could be well-controlled by changing the impact position around the dimple, as we identified previously [38]. surface, where the eccentric distance *d* = 0.5, We = 23.41, Re = 1974.10, Bo = 0.20 and Ca = 0.01. The high values of We and Re indicated that inertial forces dominated over capillary and viscous forces during droplet impact, while the low values of Bo and Ca implied that the influences of gravitational and viscous forces were negligible compared with surface tension forces in this study. In Figure 2a, at *t* = 0 ms, the droplet contacted the dimpled surface, with We = 18.97. After contact, the droplet first spread when the time was less than 1.22 ms (Figure 2b) and it fully covered the dimple at *t* ~ 1.22 ms when an inclined jet was issued from the side of the droplet. The jetting angle ( in Figure 2b and Figure S2 in Supplemental Materials) was about 45° relative to the horizontal plane of the surface. Afterwards, the droplet adopted asymmetric morphology during the spreading and contracting processes. As a result, the bouncing direction of the droplet could be wellcontrolled by changing the impact position around the dimple, as we identified previously [38].

**Figure 2.** Snapshots showing a drop impacting the dimpled surface at We = 23.41. (**a**) Start of droplet impact at *t* = 0.00 ms. (**b**) Satellite droplets issued at the end of the Jetting. (**c**) Droplet spreading to its maximum diameter. (**d**,**e**) Asymmetric morphology of the droplet during retraction. (**f**) The rebound of the droplet from the surface. **Figure 2.** Snapshots showing a drop impacting the dimpled surface at We = 23.41. (**a**) Start of droplet impact at *t* = 0.00 ms. (**b**) Satellite droplets issued at the end of the Jetting. (**c**) Droplet spreading to its maximum diameter. (**d**,**e**) Asymmetric morphology of the droplet during retraction. (**f**) The rebound of the droplet from the surface.

To understand the jetting phenomenon, we plotted the jetting velocity in variation with the Weber number We and relative eccentric distance *e* in Figure 3. The jetting velocity was determined using image analysis by which the change in the position of the jetting tip was divided by the time interval between two successive frames of the captured video. The jetting velocity increased as the Weber number increased. The highest jetting

velocity reached about 4.6 m/s when We = 33.02. At different values of We, the jetting velocity increased at first and then decreased, and the maximum jetting velocity occurred at *e* = 0.7–0.8 (Figure 3). At We = 15.20 and 18.97, no jetting occurred when *e* was smaller than ~0.60. velocity reached about 4.6 m/s when We = 33.02. At different values of We, the jetting velocity increased at first and then decreased, and the maximum jetting velocity occurred at *e* = 0.7–0.8 (Figure 3). At We = 15.20 and 18.97, no jetting occurred when *e* was smaller than ~0.60. velocity increased at first and then decreased, and the maximum jetting velocity occurred at *e* = 0.7–0.8 (Figure 3). At We = 15.20 and 18.97, no jetting occurred when *e* was smaller than ~0.60.

To understand the jetting phenomenon, we plotted the jetting velocity in variation with the Weber number We and relative eccentric distance *e* in Figure 3. The jetting velocity was determined using image analysis by which the change in the position of the jetting tip was divided by the time interval between two successive frames of the captured video. The jetting velocity increased as the Weber number increased. The highest jetting

To understand the jetting phenomenon, we plotted the jetting velocity in variation with the Weber number We and relative eccentric distance *e* in Figure 3. The jetting velocity was determined using image analysis by which the change in the position of the jetting tip was divided by the time interval between two successive frames of the captured video. The jetting velocity increased as the Weber number increased. The highest jetting velocity reached about 4.6 m/s when We = 33.02. At different values of We, the jetting

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 4 of 13

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 4 of 13

**Figure 3.** The jetting velocity under different conditions. **Figure 3.** The jetting velocity under different conditions.

Figure 4 shows the jetting angle, which increased as the Weber number increased for We < 30. At a fixed value of We, the jetting angle first increased then decreased with the increase in *e*. When the values of We were close to each other, a slight difference in the jetting angle was observed, as shown for the datasets with We = 21.90 and 23.22 in Figure Figure 4 shows the jetting angle, which increased as the Weber number increased for We < 30. At a fixed value of We, the jetting angle first increased then decreased with the increase in *e*. When the values of We were close to each other, a slight difference in the jetting angle was observed, as shown for the datasets with We = 21.90 and 23.22 in Figure 4. We < 30. At a fixed value of We, the jetting angle first increased then decreased with the increase in *e*. When the values of We were close to each other, a slight difference in the jetting angle was observed, as shown for the datasets with We = 21.90 and 23.22 in Figure 4.

Figure 4 shows the jetting angle, which increased as the Weber number increased for

**Figure 4.** The jetting angle under different conditions. **Figure 4. Figure 4.** The jetting angle under different conditions. The jetting angle under different conditions.

#### **4. Numerical Simulations of Asymmetric Jetting**

To unveil the mechanism of asymmetric jetting, we employed Fluent 2020 to simulate the velocity, momentum, and pressure inside the impacting droplet for a deep understanding of the interaction between liquid droplets and solid surfaces.

#### *4.1. Model Validation* indicated in Section 2, and We = 10.23. The mesh step was set to 0.02 mm, as the grid

*4.1. Model Validation* 

In validating our numerical model, we used the same water droplet properties as indicated in Section 2, and We = 10.23. The mesh step was set to 0.02 mm, as the grid independence study suggested that the simulation was accurate enough when the mesh step is 0.1 mm in size or finer (Figure S3 in Supplemental Materials). independence study suggested that the simulation was accurate enough when the mesh step is 0.1 mm in size or finer (Figure S3 in Supplemental Materials). Figure 5 contrasts the shape of the droplet from simulations (Figure 5a–d) and experiments (Figure 5e–h), in which the time was normalized by the capillary time ,

understanding of the interaction between liquid droplets and solid surfaces.

To unveil the mechanism of asymmetric jetting, we employed Fluent 2020 to simulate the velocity, momentum, and pressure inside the impacting droplet for a deep

In validating our numerical model, we used the same water droplet properties as

*Micromachines* **2022**, *13*, x FOR PEER REVIEW 5 of 13

**4. Numerical Simulations of Asymmetric Jetting** 

Figure 5 contrasts the shape of the droplet from simulations (Figure 5a–d) and experiments (Figure 5e–h), in which the time was normalized by the capillary time *τcap*, as defined in Equation (5). Both numerical and experimental results were consistent with each other. For example, at the normalized time of about 0.2 the droplet covered the dimple, and at the normalized time of about 1.6 the droplets took off from the dimpled surface in both experiment and simulation conditions. as defined in Equation (5). Both numerical and experimental results were consistent with each other. For example, at the normalized time of about 0.2 the droplet covered the dimple, and at the normalized time of about 1.6 the droplets took off from the dimpled surface in both experiment and simulation conditions. ଷ/ (5)

$$
\pi\_{cap} = \sqrt{\rho d\_l^3 / \sigma} \tag{5}
$$

**Figure 5.** The validation of the simulation model. (**a**–**d**) Experimental results of a droplet impacting a dimpled surface with the normalized time of 0.00, 0.21,1.10 and 1.62, respectively. (**e**–**h**) Simulation of a droplet impacting a dimpled surface with the normalized time of 0.00, 0.26, 0.91 and 1.64, respectively. **Figure 5.** The validation of the simulation model. (**a**–**d**) Experimental results of a droplet impacting a dimpled surface with the normalized time of 0.00, 0.21,1.10 and 1.62, respectively. (**e**–**h**) Simulation of a droplet impacting a dimpled surface with the normalized time of 0.00, 0.26, 0.91 and 1.64, respectively.

#### *4.2. Simulation Parameters*

*4.2. Simulation Parameters*  To simulate a water droplet with a diameter of 2.4 mm that eccentrically impacts the superhydrophobic dimpled surface, the 3D simulation domain was set as 6 mm × 6 mm × 8 mm (length × width × height). The mesh step was 0.02 mm, which was accurate enough from the grid independence study. The unstructured mesh was used in the dimple region, and the structured mesh was used in the air region. The fluid was an incompressible Newtonian fluid. The surface tension of the droplet was 0.072 N/m. The Weber number of the impacting droplet was 21.90. The water contact angle of the solid surface was set to 180° for the removal of any adhesion between the droplet and solid surface. The volume of fluid (VOF) model was used for tracking the two-phase interface. No-slip boundary To simulate a water droplet with a diameter of 2.4 mm that eccentrically impacts the superhydrophobic dimpled surface, the 3D simulation domain was set as 6 mm × 6 mm × 8 mm (length × width × height). The mesh step was 0.02 mm, which was accurate enough from the grid independence study. The unstructured mesh was used in the dimple region, and the structured mesh was used in the air region. The fluid was an incompressible Newtonian fluid. The surface tension of the droplet was 0.072 N/m. The Weber number of the impacting droplet was 21.90. The water contact angle of the solid surface was set to 180◦ for the removal of any adhesion between the droplet and solid surface. The volume of fluid (VOF) model was used for tracking the two-phase interface. No-slip boundary condition was applied to the solid surface.

#### condition was applied to the solid surface. *4.3. Simulation Results*
