Droplet impact dynamics on the surface of super-hydrophobic BNNTs stainless steel mesh | Scientific Reports

Nov 13, 2024

Scientific Reports volume 14, Article number: 27695 (2024) Cite this article

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The ‘gas‒liquid‒solid’ mechanism annealing method was used to create a superhydrophobic boron nitride nanotube (BNNT) stainless steel mesh in a tube furnace at 1250 °C in an NH3 environment. Fe powder was used as a catalyst, and B:B2O3 = 4:1 was used as the raw material. The water droplets on the surface of the superhydrophobic material had a contact angle of approximately 150° and a slide angle of approximately 3°. By using molecular dynamics (MD) simulation technology, a three-dimensional braided physical model of nanodroplets and superhydrophobic BNNT mesh surfaces with the same contact angle and rolling angle was prepared via the function weaving method. The Weber number (We) was used as the entrance point to establish the relationship between macroscale experimental studies and nanoscale MD simulation analysis on the basis of these efforts. A study was conducted on the dynamic behaviour of droplets impacting a superhydrophobic BNNT filter surface. We suggest that the wettability, substrate structure, and impact velocity are connected to the impact dynamic behaviour of droplets on the basis of the data obtained at various scales. The findings demonstrate that when the droplet impact velocity increases, several droplet phenomena—such as impact–rebound, impact–spread–rebound, and impact–spread–breaking–polymerisation–spatter—appear on the substrate surface sequentially. The mechanism of impact behaviour at various scales is explained in light of these events. Furthermore, a better theoretical model is proposed to assess the droplet wetting transition at the nanoscale. This model accurately predicts the boundary Weber number that starts the wetting transition. Moreover, the connections among the impact velocity, spreading diameter, and contact time (or We) are examined. The tendencies found via MD simulations match the outcomes of the experiments. Our discoveries and outcomes broaden our understanding of how droplet impact affects the dynamic behaviour of superhydrophobic surfaces. A scientific foundation for examining the dynamic behaviour of droplets is provided by combining simulations and experiments.

The phenomenon of a droplet impacting a solid surface often occurs in nature. Many superhydrophobic surfaces with a contact angle of 150° or greater and a sliding angle of 15° or less have been effectively created and utilised in various applications, such as self-cleaning, anti-icing, and inkjet printing. In 1996, Onda1 and colleagues used alkyl ketene dimers to create a superhydrophobic surface that resembled water. The contact angle between the surface and the droplet reached a remarkable value of 174°, indicating promising possibilities for applications in oil‒water separation, aerospace, and other related industries. Kumar2 and others suggested that creating hydrophobic surfaces with consistent and organised rough textures is more effective in enhancing the hydrophobic characteristics of materials. Kumar reported a considerable improvement in the wettability of the surface structure when the width of the microgroove was increased from 30 to 70 μm. The observed trend in the experiment was corroborated by theoretical analysis, and the mechanism by which droplets slide on the microgroove surface was elucidated. Xue3 et al. utilised a stainless steel filter mesh with a diameter ranging from 34–380 μm as the substrate. They employed N,N′-methylenebisacrylamide (BIS) as the chemical crosslinking agent, acrylamide (AM) as the precursor, and 2, 2′-diethoxyacetophenone (DEOP) as the initiator to synthesise a superhydrophilic polyacrylamide (PAM) hydrogel. Afterwards, a stainless steel filter mesh was coated with a hydrogel via photoinitiated free radical polymerisation, resulting in a highly hydrophilic and oil-resistant surface.

Despite extensive efforts, the analysis and regulation of the contact angle and sliding angle of superhydrophobic materials remain challenging. Recently, there has been increasing interest in superhydrophobic materials and related topics, as researchers have strived to uncover the underlying reasons for the superhydrophobic phenomenon. Research has demonstrated that when droplets are situated on a rough microscopic surface that possesses superhydrophobic qualities, the droplets do not fully spread over the groove gap on the material’s surface. This behaviour results in the formation of a three-phase system interface with interfacial tension between the gas, liquid, and solid. The droplets achieve an equilibrium condition on the material’s surface because of the interfacial tension of the three-phase system4. Typically, droplets are deposited on the surface of a material to characterise the superhydrophobicity of the surface. Nevertheless, when a water droplet possesses velocity, such as that of a raindrop, it effectively impacts the solid surface. Through the use of advanced high-speed cameras, researchers have discovered that droplets exhibit several dynamic behaviours, including deposition, partial rebound, total rebound, splashing, and spraying, when they collide with hydrophobic solid surfaces under varying impact situations5. Researchers have discovered many flow patterns in small-scale phenomena, such as the effect of droplets, and have undertaken thorough investigations and analyses. The process of droplet impact is influenced primarily by the impact velocity, surface wettability, and surface shape of the solid material. Furthermore, when the velocity of the droplet colliding with the hydrophobic surface reaches a certain threshold, the air trapped in the uneven microgap structure of the surface is rapidly expelled by the droplet, resulting in total wetting of the surface6.

Recently, several studies have examined the molecular dynamics of droplets colliding with superhydrophobic surfaces7,8,9. A high-speed camera was used to determine how the spacing between micropillars and the speed of collision affected the movement of microdroplets on the hydrophobic surface of the micropillars10. The physical process of droplet wetting changing from Cassie to Wenzel on a hydrophobic surface has been studied to explain the main reason for the split of the Cassie–Wenzel wetting states11. Considerable efforts have been devoted to the study of droplet impact dynamics, with a focus on understanding the dependence of the droplet impact velocity on the geometry. The wetting transition criteria of droplets from the Cassie‒Baxter state to the Wenzel state are consistent with the wetting behaviour of multiwalled nanotubes at different impact velocities12. Experiments were used to measure the recovery coefficient (ε) of the droplet’s full return, and two different bouncing processes were found on the basis of the speed of impact. When the Weber number is low (We ≲ 0.2), the droplet deforms and bounces back slightly during impact. ε increases substantially as the impact speed increases. When the Weber number is greater (We ≳ 0.2), the droplet deformation becomes stronger as the impact speed increases, and ε starts to decrease13. However, most of the research thus far has focused on how drops move on the macro- and microscales, and little is known about solid surfaces at the nanoscale14,15,16,17. In recent studies, only the conservative level set method has been used to describe the free energy surface tension model of the interfacial tension that affects how ceramic droplets bounce back when they contact hydrophobic materials18. Nanodroplet impact dynamics can be used in surface engineering and spray cooling, among other applications19,20. Nevertheless, it is presently elucidated by a straightforward interface, while the intricate surface is streamlined to form a flat, grooved, and arranged surface substrate. Although this operation can indeed explain the interface mechanism at the nanoscale, it is inaccurate. Therefore, it is highly important to establish a real hydrophobic material model and understand the molecular dynamics behaviour of droplets in detail. We used the prepared BNNT stainless steel filter material with superhydrophobic properties as a simulation model to study the impact dynamic effect of the liquid on the surface of the BNNT superhydrophobic stainless steel filter.

In this work, we focus on the impact dynamics of droplets under a real superhydrophobic surface model. First, by adjusting the impact velocity of water droplets impacting the surface of a superhydrophobic material, the dynamic behaviour of droplet impact on the superhydrophobic surface of BNNTs grown on the surface of a 304 stainless steel filter is studied, and the reasons for the complete rebound of droplets after impact on the superhydrophobic surface at the macroscale are analysed. More importantly, simulating the bouncing dynamic effect of droplets and hydrophobic surfaces at the microscopic scale via the lattice Boltzmann method21 and fluid volume method22 is difficult. Moreover, to match the SEM morphology of macroscopic materials, we used MD simulations to prepare by functional braiding a three-dimensional braided physical model with the same contact angle and rolling angle on the BNNT superhydrophobic surface. To establish the relationship between macroscopic experiments and mesoscopic MD simulations, we studied the dynamic bounce effect of droplets on hydrophobic surfaces. Second, on the basis of the different heights of free landing, the contact time of droplets on the hydrophobic surface at different impact velocities is studied. At the microscale, the wetting mechanism of nanodroplets after they impact the superhydrophobic surface was analysed. A theoretical model is proposed to predict the critical velocity at which a droplet impacts a hydrophobic surface in the Wenzel state. Finally, the influence of the physical parameter We on the contact diameter at the impact velocity is studied. The above work is based on experiments and MD simulations to provide specific numerical analysis to study the dynamic effects of droplets on hydrophobic surfaces.

In our previous studies, we proposed a method for preparing superhydrophobic materials23. The ‘gas‒liquid‒solid’ mechanism annealing process was used to create superhydrophobic BNNT stainless steel filters. First, 304 stainless steel mesh with a mesh size of 120 was utilised as the matrix, and B and B2O3 powders were utilised as the raw materials. Second, the temperature in the tube furnace was raised to 1250 °C at a rate of 5 °C/min and maintained for two hours. Simultaneously, the flow rate of NH3 was increased to 40 mL/min. After the tube furnace temperature was lowered to 300 °C, the power supply was turned off, and the samples were removed and statically dried for 20 min before being examined. The preparation procedure is depicted in Fig. 1.

Flow chart for the preparation process23.

The surface morphology of the samples was observed via a field emission scanning electron microscope (FESEM, S4800) provided by Hitachi Japan. The exact method involves cutting a sample’s white area into 2 × 2 mm pieces, pasting these pieces on conductive adhesive, spraying them with gold on the sample pieces, and then looking at them under an electron microscope. To observe the white portion of the BNNT stainless steel filter via high-resolution transmission electron microscopy (HRTEM JEM-2100Puls), the white area of the filter was cut into small pieces measuring 5 × 5 mm and placed in a clean test tube. Alcohol was added to the test tube for ultrasonic dispersion. Clean tweezers were used to drop the solution onto a 3 mm microcopper mesh. TEM (S4800) analysis was performed after drying at 60 °C for 2 h (Fig. 2b–d).

SEM image of a superhydrophobic BNNT stainless steel filter (sliding angle of 3° and contact angle of 150°)23.

Through the use of a contact angle measuring device (JC2000D1, Shanghai, China), the static contact angle and droplet velocity on the surface of the superhydrophobic BNNT stainless steel filter were determined. Prior to evaluating the wettability, a 2 × 2 mm fragment was sliced from the white section of the BNNT stainless steel filter and affixed to the slide using double-sided tape. The sample was subsequently placed on a contact angle measuring instrument test bench, and 5 μL of water droplets were taken with a microinjection pump and dropped at 4 different positions in the sample. The static contact angle and rolling angle of the water droplets with the BNNT stainless steel filter were measured via the angle measurement method. The average of the four static contact angles was taken as the static contact angle of the BNNT stainless steel filter. The rolling angle is the critical tilt angle of the droplet on the sample surface. Similarly, the contact angle and sliding angle of the superhydrophobic BN nanotube stainless steel mesh surface are 150° and 3°, respectively23 (Fig. 2). The detailed preparation process and characterisation properties of the material are given in the manuscript23.

Similarly, a 2 × 2 mm small piece that meets the measurement standard is cut before the dynamic contact angle of the BNNT stainless steel filter is measured. It is pasted on a glass slide with a double-sided adhesive and then fixed on the contact angle measuring instrument platform (as shown in Fig. 3). At a distance of 0.3 cm from the surface, a microsyringe tip is used to suspend an undropped water droplet for video focusing recording. Finally, the water droplet falls naturally, and its dynamic bounce process on the surface of the BNNT stainless steel filter is recorded.

Droplet impact experiment device diagram.

The droplet in this experiment has a density of \(\rho = 0.994\;{\text{g/cm}}^{{3}}\) and a radius of approximately r = 1.05 mm; its impact velocity is defined by its height during free fall, and it starts off with no velocity. In droplets, the impact dynamic behaviour is correlated primarily with the Weber number (\(We = {{\rho {\text{RV}}^{2} } \mathord{\left/ {\vphantom {{\rho {\text{RV}}^{2} } {\upgamma }}} \right. \kern-0pt} {\upgamma }}\)). Here, the surface tension is 72.75 mN/m, the radius is the droplet’s falling radius, and \(\rho\) is the experimental droplet’s density. The impact velocity of a droplet is proportional to its falling height. Table 1 displays the numerical link between the Weber number We and the impact velocity V0 of the water droplet.

A microscale three-dimensional model of a superhydrophobic surface with a contact angle of 150° and a sliding angle of 3° is built via MD simulation to simulate the dynamic behaviour of droplets impacting superhydrophobic surfaces precisely at the microscale, as illustrated in Fig. 4.

Flow chart for weaving boron nitride nanoribbons.

The procedure of weaving boron nitride nanoribbons is shown in Fig. 4. We employ two common boron nitride nanoribbons, armchair and zigzag boron nitride nanoribbons, as two fundamental construction components. In particular, we select armchair and zigzag boron nitride nanoribbons with widths of approximately 2.1 nm as the fundamental building components. X denotes the width of the band, and Y denotes its direction along the band in Fig. 4a. A three-dimensional model of a BNNT stainless steel filter was weaved for the first time. The SEM morphology of the generated superhydrophobic BNNT material indicated that hexagonal boron nitride, with a lattice constant of a = b = 2.5 Å and c = 3.6 Å, was utilised as the fundamental structural unit of the woven filter. Every atom has an offset value in the Y direction, which is intended to weave the boron nitride nanoribbons into a network structure. Next, the amplitude A = 0.34 nm and wavenumber \(B = 2\pi /\lambda\) (the wavelength \(\lambda\) is 5 nm) are represented by the sine function Z = AsinBy in the Y–Z plane. A banded bending structure is achieved by the B and N atoms in boron nitride nanotubes, which are perpendicular to the banded structure and fluctuate with their functions. The atoms on the surface are then given a specified twist angle of 15°, rotated in the X‒Y direction, and weaved together to make the model more lifelike. The actual diameter of the 120 mesh 304 stainless steel filter wire modified with BN nanotubes is L = 0.2 mm, the pore size is D = 0.125 mm, and the ratio of the droplet diameter to the pore size is R:D = 16.8 in the impact experiment. Then, the smallest unit of the boron nitride stainless steel filter is woven according to the order of Fig. 4a–d. The distance between the adjacent boron nitride strips in the Y-axis direction is approximately 0.67 nm, which is approximately 4.58 times the distance between the boron nitride layers in the SEM image. Figure 3e shows a boron nitride nanoribbon sheet containing four cells (approximately 8700 atoms). Since the basic assembly bands of zigzag boron nitride nanoribbons and armchair boron nitride nanoribbons are very flexible, the chirality of boron nitride nanoribbons is also very diverse. In accordance with the different combinations of the basic assembly bands, various patterns of boron nitride nanoribbon sheets were constructed, resulting in sheets of woven boron nitride nanoribbons (composed of 22 units).

The relevant forces between various atoms in MD simulations are typically separated into two categories: the bond interaction between the bond extension and the angle bending term (Eqs. 3–5) and the nonbonding interaction between the Coulomb and van der Waals terms (Eqs. 1–2):

where \(U_{{\text{c}}}\), \(U_{V}\) and \(U_{{B{\text{ond}}}}\) represent the potential energy generated by the Coulomb interaction, the potential energy generated by the van der Waals interaction and the potential energy generated by the bond stretching, respectively. \(e\), \(\varepsilon_{0}\), \(q_{j}\), and \(q_{i}\) represent the electronic charge constant, vacuum dielectric constant, partial charge of atom j, and partial charge of atom i, respectively. \(\varepsilon_{ij}\), \(\sigma_{ij}\), \(k_{1}\), \(k_{2}\), \(l_{ij}\), and \(l_{0}\) represent the well depth, the force constants and the instantaneous and equilibrium lengths of the O‒H bond, respectively. The parameters of the interatomic LJ12-6 potential in this paper are shown in Table 2.

The model in this work uses periodic boundaries in all directions to eliminate boundary effects; the type is atomic because of the presence of bond angle connections; the particle position and velocity are updated during simulation via the velocity Verlet algorithm, with a 1.0 fs simulation time step. The extended boron nitride Tersoff potential (BN-ExTeP24) is also employed in this research for the boron nitride interaction by choosing a suitable potential energy and modifying the potential energy parameters. This approach enhances bonding accuracy by precisely describing the important B and N structures and applies to large-scale atomic simulations.

In conclusion, Fig. 5 displays the first model of the MD simulation of droplet impact on a superhydrophobic stainless steel filter surface made of BN nanotubes.

Three-dimensional model image at the nanometre scale: (a) top view and (b) three-dimensional view.

To guarantee that the droplet has adequate room to fully eject, the length of the box’s top and the superhydrophobic BN surface in the Y direction are multiplied by 10. Then, the superhydrophobic BN surface in the Y direction is multiplied by 10 to guarantee that the droplet has ample room to fully unfold. According to Joseph et al.'s proposal25, nanodroplets with a minimum of 2000 molecules can completely eliminate the impact of the size effect on the experiment. The nanoscale droplets used in this study comprise 12,000 water molecules. The impact of the size effect on the experiment can be fully eliminated in this instance. In addition, the Lennard–Jones (LJ) potential energy is utilised to characterise (Eqs. 1–2) the correlation between solid atoms and water molecules, which has been proven by Heinz’s group26. Consequently, the interaction between BN and BN-water in this study is described by the LJ12-6 potential. The contact angle and structural characteristics represent how the surface influences the droplet impact on the BN superhydrophobic surface. Thus, as illustrated in Fig. 6, various wetting angles can be achieved on the BN mesh surface by varying the value.

Effect of the LJ energy parameters \(\varepsilon_{{\text{N - W}}}\) and \(\varepsilon_{{\text{B - W}}}\) on the equilibrium contact angle \(\theta_{{\text{E}}}\).

Equation (5) is primarily used to compute the value of \(\varepsilon_{ij}\). The associated contact angle decreases from 148.8° to 19° and is roughly linearly correlated with \(\varepsilon_{ij}\). In the simulation, the boron nitride interlayer distance is used as the equilibrium interlayer distance, and the LJ12-6 potential is also used to describe the nonbonding interaction for the equilibrium interlayer distance. When the interlayer distance displacement of boron nitride is too large, the LJ12-6 potential cannot capture it. To avoid the influence of nonphysical conditions when the droplets impact the surface of the woven mesh, the truncation radius is set to 2.0 Å in the simulation conditions. The droplet maintains a stable temperature of 300 K at the Nose‒Hoover constant temperature under the NVT ensemble. In this work, MD simulations are performed via the open-source software (Large-scale Atomic/Molecular Massively Parallel Simulator LAMMPS, 23 Jun 2022), and the visualisation software OVTIO (Version 3.8.4) is used to observe the simulation results. In accordance with previous studies27,28,29, Its main advantage is the use of the algorithm particle‒particle‒grid method to calculate the Coulomb force, and the accuracy can reach 99%. In each simulation, the following points should be considered: (1) The initial position of the water droplets is as far from the surface of the BNNT hydrophobic mesh as possible to ensure that the cylindrical water droplets do not interact with the superhydrophobic surface, and the energy minimisation method is used to eliminate the potential energy that may exist in the initial water droplet model before the simulation begins. (2) Since the simulated solid surface is fixed, to simplify the calculation and improve its accuracy and efficiency, the water model adopts the monotonic water mW model30. This model can well describe the interaction force between water molecules30 and well reproduce the surface tension, density and structure between water molecules. Compared with the extended point charge SPC/E, TIP3P, Tip4P/2005 and other models, the coarse-grained water model has higher computational efficiency and can accurately reflect the basic properties of water. When the model is used to simulate the temperature of 300 K under the Nose–Hoover method31, ρ = 0.997 g/cm3 and the surface tension γ = 66 mJ/m2 are obtained under the droplet water model, which are very close to the actual values.

The simulation phase has two stages: The Nose‒Hoover thermostat is used to relax the system at a time step of 100,000 at 300 K under the NVT ensemble in the first stage, when the centroid of the mW water model is fixed at the model’s initial position. A time step of 200,000 is then run in the NVE ensemble to balance the model. The water molecules in the nanodroplets are then given a Z-direction velocity component V0, which causes the nanodroplets to affect the superhydrophobic BNNT mesh surface. In addition, to obtain the simulated contact angle, the mW water model on the X‒Y plane is refined and divided into blocks. The block size is 3 Å in the X and Y directions. The average value of the dump file in the simulation process is used to obtain the average local density of the block and the block (as shown in Fig. 7).

Definition of the contact angle (\(\theta\)) on the basis of the density contour plot of a static nanoscale droplet on the BNNT surface.

Fitting the density distribution produced the average density profile (\(\theta\)) of the balanced nanodroplets on the surface of the three-dimensional model of the BNNT mesh. The contact angle is represented by the tangent between the top of the screen surface and the droplet surface (two black lines), and the red curve is meant to roughly suit the droplet profile.

According to the macroscopic experimental data in Table 1, the surface impact velocities of the nanodroplets impacting the superhydrophobic BNNT mesh model in the MD simulations are shown in Table 3.

In this work, the dynamic behaviour of droplets from different heights impacting the surface of a BN superhydrophobic mesh is refined into two parts: spreading and retraction. Many articles on hydrophobic mechanisms have reported such phenomena8,32. As shown in Fig. 8, as the macroscopic droplet velocity increases continuously, the droplets exhibit three distinct behaviours during the process of rebound and retraction after hitting the surface of the substrate to reach the maximum spreading diameter: regular rebound, splash, and jet rebound.

Effect of impact velocity on droplet dynamics.

When a droplet with an initial velocity of 0.198 m/s impacts the BNNT superhydrophobic surface, it experiences a regular impact and rebound process. Following the droplet’s interaction with the superhydrophobic BNNT surface via gravity and inertial force, the bottom of the droplet is first compressed. The droplet starts to expand in the X direction over time. The droplet’s form reaches its limit position, and its associated spreading diameter reaches its maximum at 2.84 ms. Next, the droplet moves into the 3.51 ms retraction stage. In general, this process is axisymmetric. At this point, the droplet takes the shape of a multilayered pancake. This shape could be due to the droplet’s upper and bottom surfaces having different velocity distributions after the collision with the superhydrophobic BNNTs, which causes the rebounding droplet to assume a different shape. Moreover, the time required for the retraction phase of the droplet also increases; finally, the droplet completely bounces off the surface of the superhydrophobic BNNTs, reaching the maximum rebound height at 6.81 ms and achieving a complete impact–tilting–retraction–bouncing process. The droplet subsequently repeats the movement process until it reaches stability. This dynamic process is similar to the droplet impact dynamic behaviour of other superhydrophobic surfaces. More intriguingly, as the droplet velocity increases to 0.3130 m/s, the droplet’s flat structure becomes more noticeable, and upon rebounding, it displays an elliptical shape (Fig. 8b); the droplet hits the superhydrophobic surface at velocities of 0.42 m/s and 0.5048 m/s. The droplet’s lower surface separates following tile rebound. The outside contour of the large droplets encircles the partially split small droplets, preventing them from being completely separated. The centres of the large droplets start to create pores at this point. The small droplets return to the inside of the giant droplets during the rebound process. This finding aligns with reports from the literature6,33. The dynamic behaviour of the droplet changes more visibly when the droplet velocity is high enough (0.9899 m/s, 1.4 m/s). When the droplet reaches the maximum spreading diameter at 2.06 ms and 1.65 ms, its structure differs from that of other droplets, with a lower velocity reaching the maximum spreading stage. At this time, the structure of the droplet is similar to the shape of a gear. Notably, during the rebound process, the droplet rebounds and ejects. At this time, the diameter of the separated small droplets is almost the same as the thickness of the maximum spreading diameter, which is consistent with the research results of Reyssat34 et al. As shown in Fig. 8e,f, as the velocity of the droplet impacting the surface of the superhydrophobic BNNTs increases, the spreading diameter of the droplet increases, and the thickness of the droplet decreases.

At the macroscopic level, high-speed photographs of droplets colliding with superhydrophobic BNNTs reveal that as the droplet height decreases, the droplets progressively start to exhibit various phenomena, such as tiling, rebounding, splustering, and fracturing. Zhang35 et al. suggested that the rebound of a droplet might be influenced by droplet size, impact velocity, and surface wettability on the basis of the droplet rebound phenomenon. Deng36 et al. applied a specific pressure before a droplet reached the surface of a hydrophobic substance. The experimental results demonstrate the relationship between the wetting pressure and droplet dynamic behaviour when a droplet strikes a material surface. The water hammer pressure PWH = 1/5ρCV during the droplet’s contact with the hydrophobic surface and the dynamic pressure PD = 0.5ρV2 during the droplet impact process are proposed. This approach has been employed in most studies35,37,38 to explain the mechanism of macroscopic droplet impact. Although helpful in explaining a portion of the droplet-bouncing wetting mechanism, a single wetting pressure cannot fully account for the dynamic physical phenomenon of droplet impact. This work presents a new pressure expression to explain the dynamic behaviour of macroscopic droplet impact on the basis of a macroscopic substrate material model.

Figure 2b shows a SEM image of the superhydrophobic BNNT stainless steel filter. The filter is woven up and down via a single mesh. The filter model is shown in Fig. 9a,b.

(a) Number of mesh elements covered by droplets and (b) three-dimensional simplified schematic of the composite filter.

The projection of the droplet impacting the superhydrophobic BNNT filter is an approximately circular area Aw = πD2/4, the opening area of a single filter is \(A_{lv} = \left( {L + \tau } \right)^{2}\), and the number of mesh elements covered by the droplet impacting the surface of the superhydrophobic filter material is Nw ~ Aw/Alv. When the droplet contacts the mesh of the BNNT hydrophobic material, the compression of the droplet shape is closely related to the water hammer pressure. The water hammer pressure PWH can be expressed as:

where \(\Delta \rho_{w}\) represents the variation in droplet density during the impact process. The mass of the droplet is unchanged during the impact process. Since the droplet’s volume is a function of time t, it can be expressed using \(\rho = {m \mathord{\left/ {\vphantom {m {\text{V}}}} \right. \kern-0pt} {\text{V}}} = {{m\Delta {\text{V}}} \mathord{\left/ {\vphantom {{m\Delta {\text{V}}} {{\text{V}}^{2} }}} \right. \kern-0pt} {{\text{V}}^{2} }}\). The sound velocity C can be used to represent \(\sqrt {{{\partial p} \mathord{\left/ {\vphantom {{\partial p} {\partial \rho }}} \right. \kern-0pt} {\partial \rho }}}\).

Consequently, the water hammer pressure PWH can be expressed by Eq. (7):

where the coefficient \(k^{*}\) denotes the size of \({{\Delta {\text{V}}} \mathord{\left/ {\vphantom {{\Delta {\text{V}}} {{\text{V}}^{2} }}} \right. \kern-0pt} {{\text{V}}^{2} }}\). Through Eq. (7), we speculate that the water hammer pressure in this paper is related mainly to the ratio of the volume change of the droplets during impact. At the macroscopic scale, the relationship between the time t at which the droplet falls and the volume of the droplet is \(t \sim {\tau \mathord{\left/ {\vphantom {\tau {\text{V}}}} \right. \kern-0pt} {\text{V}}}\sim 30 - 50{\mu s}\)39. When the droplet hits the superhydrophobic surface, the relationship between the curvature of the diameter D of the droplet depression and the diameter of a single wire can be expressed as \(K\sim {D \mathord{\left/ {\vphantom {D {\left( {L + \tau } \right)}}} \right. \kern-0pt} {\left( {L + \tau } \right)}}\). Therefore, the droplets are compressed on the hydrophobic surface at a scale ratio of \({D \mathord{\left/ {\vphantom {D {\left( {L + \tau } \right)}}} \right. \kern-0pt} {\left( {L + \tau } \right)}}\). Moreover, the hydrophobic surface compresses the droplets in a vertical upwards scale ratio of \({D \mathord{\left/ {\vphantom {D {\left( {L + \tau } \right)}}} \right. \kern-0pt} {\left( {L + \tau } \right)}}\). We find that the change in the volume \(\Delta {\text{V}}\) of the droplet can be approximated by the scale of \({D \mathord{\left/ {\vphantom {D {\left( {L + \tau } \right)}}} \right. \kern-0pt} {\left( {L + \tau } \right)}}\), as shown in Eq. (8).

Combining Eqs. (6–8) yields Eq. (9):

where V, C and m are fixed values and the water hammer pressure is related to the mesh coverage Nw of the droplets hitting the superhydrophobic surface.

The capillary force on the hydrophobic material surface and the three-phase force of the water hammer pressure balance the dynamic pressure of the drop velocity conversion of the droplet on the hydrophobic surface. The three-phase force size distribution influences the dynamic behaviour of the droplet. An ‘air cushion’ is created by the hydrophobic microstructure of the solid surface, which stores air. The droplet creates an interface between gas and liquid with the ‘air cushion’ when it strikes the solid surface. Capillary pressure, sometimes called anti-wetting pressure, causes the droplet’s bottom structure to distort. Therefore, Eq. (10) can be used to represent the capillary pressure (PC):

where γ is the surface tension (72.75 mN/m) and θA is the contact angle between the droplet and the solid surface.

Deng36 et al. reported that the relationship between the wetting pressure and anti-wetting pressure can be used to categorise the wetting conditions of droplets on hydrophobic surfaces into three categories, as shown in Fig. 10.

Details of the three wetting states of a droplet impacting a superhydrophobic surface: (a) complete wetting PC < PD < PWH; (b) partial wetting PD < PC < PWH; and (c) total nonwetting PD < PWH < PC.

The approximate solutions of PC, PD and PWH at the macroscopic scale for droplets with different impact velocities are obtained via the above expressions, and the relevant conclusions are obtained. When the capillary pressure PC < PD < PWH, the droplets completely wet the material. When PD < PC < PWH, the droplet is partially infiltrated by the droplet in the contact area with the material surface. When PD < PWH < PC, the droplet does not infiltrate the surface of the droplet completely to realise the macroscopic bouncing phenomenon. These findings are consistent with the findings of related investigations36,37,40.

In the MD simulation, the relationship between the shape and impact velocity of the nanodroplets was studied. The dynamic behaviour of the nanodroplets on the surface of the superhydrophobic BNNT mesh is shown in Fig. 11.

Dynamic behaviour of nanodroplets at different velocities on the surface of superhydrophobic BNNT meshes.

Droplet shape changes with varying impact velocity, according to macroscopic and microscopic examinations of droplet impact on the surface of a superhydrophobic BNNT stainless steel filter. By analysing the dynamic behaviour of droplets on the surface of a superhydrophobic arched mesh via macroscopic experiments and MD simulations, it is found that, according to the MD simulation results, the nanodroplets undergo an impact–spread–rebound process. However, no jetting or splashing phenomenon was observed, similar to the actual impact. Because there is air in the macroscopic droplet and no gas in the MD simulation, the air cannot be captured in the microstructure of the superhydrophobic surface spacing. Thus, when a droplet impacts a superhydrophobic surface, it will not form a gas‒liquid interface with the ‘air cushion’ in the microspace. Nonetheless, two stages remain for the droplet in the impact process: ‘spreading’ and ‘shrinking’. The shape of a droplet will change, and it will pierce the microscopic space during the ‘spreading’ stage of its impact on the superhydrophobic surface. Furthermore, a relationship between the impact velocity and droplet permeability is discovered. The impact kinetic energy is transformed into interface energy as the impact velocity increases, is subsequently stored in the microstructure, and is subsequently transformed into kinetic energy via ‘contraction’ kinetic behaviour. The droplet ultimately bounces off the cylindrical mesh structure at the needle tip, which is essentially consistent with the macroscopic experimental phenomenon.

To further study the relationship between the impact velocity of the droplet and its dynamic behaviour, according to the MD simulation droplet impact results in Fig. 10, a schematic of the nanodroplet impacting the surface of the superhydrophobic BNNT mesh at different initial velocities was drawn, as shown in Fig. 12.

Impact of nanodroplets on the surface of superhydrophobic BNNT meshes at different initial velocities. (a: 0.0007 Å/fs; b: 0.001781 Å/fs; c: 0.0041292 Å/fs; d: 0.004936 Å/fs).

In the MD simulation of this paper, the atomic layer distance of BNNTs on the superhydrophobic surface is \(h < 5.5L\) (\(L\) represents the columnar height of the model). Lv41 et al. reported that the wetting state of a nanodroplet impacting a superhydrophobic surface is related to its columnar height. We found that even under the condition of \(h < 5.5L\), the droplet will carry out related dynamic behaviour.

When h < 5.5L, Vw < VNB (Vw represents the instantaneous velocity of the droplet impacting the superhydrophobic surface, and VNB represents the instantaneous velocity of the droplet without bouncing), and both macroscale droplets and microscale nanodroplets do not bounce. In this region, the droplets do not rebound. The droplet structure expands weakly only after it impacts the superhydrophobic surface, and then, the droplets undergo damping oscillation to reach the spherical structure and maintain the Wenzel state.

When h < 5.5L, VNB < Vw < VBS (VBS represents the instantaneous velocity at which the droplet bounces and is in a viscous state after it impacts the superhydrophobic surface), and the nanodroplets bounce. The nanodroplets that are between the viscosity velocity and the nonbounce velocity are subjected to impact–spread–retraction on the surface. At the macroscale, this process occurs several times before the droplet reaches the Cassie wetting state. Finally, the Cassie‒Baxber model is used to predict the macroscopic contact angle. This angle is basically the same. Moreover, the initial kinetic energy of the macroscopic droplet is insufficient to overcome the energy barrier that hinders surface penetration after the impact behaviour of the macroscopic droplet. Furthermore, at the macroscale, although the droplet can bounce, the droplet is in a viscous state. The shape of the droplet impact‒retraction process is symmetrical along the centreline, and the contact line of the droplet almost does not shrink. The instantaneous contact angle is less than 40°. In this state, the droplet is forcibly embedded and pierced by the surface of the superhydrophobic BNNT stainless steel mesh after impact. At the microscale, the nanodroplets remain in a viscous state throughout the impact process. When the initial velocity of the nanodroplets is constant, the viscous strength of the nanodroplets is related to the size of We. When the We value is greater, the viscous state of the nanodroplets is more obvious. When the We value is smaller, the viscosity state of the nanodroplets is weaker. However, when the We value is small, the nanodroplets do not wet the bottom of the superhydrophobic BNNT mesh surface after hitting the surface of the superhydrophobic BNNT mesh, and no surface rebound phenomenon is observed. This behaviour is observed possibly because at low We values, the initial kinetic energy of the nanodroplets is small, and the microgap and rebound behaviour of the nanodroplets infiltrating the surface cannot be guaranteed. At a high We value, the nanodroplets change from the Cassie state to the Wenzel state, and finally, the droplets remain in the Wenzel state but are also in a viscous state.

When h < 5.5L, Vw > VBS, at the macroscale, the droplets splash on the surface of the superhydrophobic BNNT stainless steel mesh, and the droplets do not have an axisymmetric structure when impact occurs. After impact, the droplets are divided into main droplets and small droplets to rebound at the same time. When the impact velocity is high enough, the droplet structure is broken, and many satellite droplets appear near the main droplet. At the microscopic scale, after the nanodroplet reaches a sufficiently high speed, its spreading diameter increases. When the nanodroplet is in the state of the maximum spreading diameter, its thickness is very small. Then, the liquid is squeezed from the microscopic gap, and the droplet returns to the Cassie state and rebounds.

In conclusion, the impact velocity is strongly influenced by the state of droplet impact on hydrophobic/superhydrophobic surfaces, which is consistent with the findings of Li42 et al.

Cassie–Wenzel coexistence and wetting were observed in the MD simulation of the nanodroplet impact process following the impact of the nanodroplets on the superhydrophobic BNNT mesh surface. Only the nanoscale influence of the monostable Cassie surface is observed at the macroscale. Thus, we hypothesise that the distinct wetting mechanism of the two scales causes the disparity in the bouncing process. Two wetting transition processes for bouncing and viscous droplets at the macroscopic scale were postulated by Bartolo43 and other researchers. A theoretical model of the depinning mechanism was proposed by Bartolo. The critical speed of the droplet–wetting transition in the depinning mechanism is the same as the critical speed of the depinning, and it is not reliant on the microstructure of the contact surface. Nevertheless, the wetting conditions of the nanodroplets influencing the superhydrophobic BNNT sieve surface changed with a change in the critical Weber number, according to the MD simulation used in this work. Therefore, the Bartolo model may no longer apply to nanodroplets. At the nanoscale, the velocity of nanodroplets is still an important factor affecting the wetting state. Therefore, to accurately judge the wetting transition of nanodroplets, based on the analysis of the dynamic behaviour of nanodroplets impacting the superhydrophobic BNNT mesh model, a mathematical model is established to predict the critical velocity of the wetting state transition of nanodroplets at the nanoscale.

First, it is assumed that the initial contact line of droplet wetting is fixed at the corner of the nanopillar. At this time, the vertical component of the capillary force at the top of each column wetted by the droplet is represented by Eq. 11:

Under this condition, the reaction force that stops the droplet’s contact line from beginning to migrate along the column’s top sidewall is known as the capillary force. Consequently, we can express the deceleration of the droplet’s contact line as \(a_{d} \sim {{F_{c} } \mathord{\left/ {\vphantom {{F_{c} } {\rho \Phi_{a} }}} \right. \kern-0pt} {\rho \Phi_{a} }}\), where \(\Phi_{a} = {{\Phi_{0} } \mathord{\left/ {\vphantom {{\Phi_{0} } {N_{p} }}} \right. \kern-0pt} {N_{p} }}\) is the average droplet volume of the sieve column, an is the nanodroplet’s volume, \(\Phi_{0} = {{4\pi r_{0}^{3} } \mathord{\left/ {\vphantom {{4\pi r_{0}^{3} } 3}} \right. \kern-0pt} 3}\) is the number of columns the nanodroplet covers on the sieve’s surface, and \(N_{p} = {{\pi r_{s}^{2} } \mathord{\left/ {\vphantom {{\pi r_{s}^{2} } {p^{2} }}} \right. \kern-0pt} {p^{2} }}\) is the radius of the nanodroplet diffusion at this particular moment. Therefore, \(a_{d}\) can be rerepresented by Eq. (12).

The diffusion radius \(r_{s}\) of nanodroplets is related to time \(t\). Here, we define the diffusion radius as the average diffusion radius of the droplets. Therefore, the average deceleration \(a_{da}\) of the contact line of the nanodroplet is expressed by Eq. (13):

where \(r_{sa}^{2}\) is the square mean of the diffusion radius of the droplet after hitting the surface of the superhydrophobic mesh. The average deceleration \(a_{da}\) of the contact angle of the nanodroplet has a certain relationship with \(r_{sa}^{2}\). To obtain the scaling law of \(a_{da}\), the scaling law of \(r_{sa}^{2}\) is obtained first. Kobayashi44 et al. reported the diffusion dynamics of the collision of nanodroplets in the early spreading stage and proposed a mathematical model describing the relationship between the spreading radius \(R_{s}\) and the spreading time \(\tau\): \({{R_{s} } \mathord{\left/ {\vphantom {{R_{s} } {R_{0} \sim \left( {{{\tau V_{0} } \mathord{\left/ {\vphantom {{\tau V_{0} } {D_{0} }}} \right. \kern-0pt} {D_{0} }}} \right)}}} \right. \kern-0pt} {R_{0} \sim \left( {{{\tau V_{0} } \mathord{\left/ {\vphantom {{\tau V_{0} } {D_{0} }}} \right. \kern-0pt} {D_{0} }}} \right)}}^{1/2}\). \(r_{sa}^{2}\) is approximately replaced by the definite integral and scaled in proportion. The result is represented by Eq. (14):

where \(\tau_{t}\) represents the time from the beginning of the nanodroplet contact line to the end of movement. In the simulation process, we found that \(\tau_{t}\) is independent of the impact velocity of the nanodroplet and the height of the pillar on the model surface. Therefore, it can be reasonably assumed that only the inertial force and capillary force exist at this stage. The constant \(\tau_{t}\) is proportional to the inertial force and capillary force: \(\tau_{t} \sim \left( {{{\rho R_{0}^{3} } \mathord{\left/ {\vphantom {{\rho R_{0}^{3} } \gamma }} \right. \kern-0pt} \gamma }} \right)^{1/2}\). Therefore, \(a_{da}\) can be expressed as Eq. (15).

When the contact line deceleration \(a_{da}\) is known, the maximum displacement of the contact line can be approximately expressed as \(\Delta X \sim \left( {V^{2} - V_{0}^{2} } \right)/a_{da}\), the contact line is reduced from the initial velocity V0 to 0, and the height of the mesh protrusion is expressed as L. When \(\Delta X = L\), the nanodroplet contacts the lowest position of the substrate model, and the wetting state of the nanodroplet begins to change. At this time, the impact velocity corresponds to the critical velocity of the wetting state of the nanodroplet. It can be expressed by Eqs. (16, 17)

Given the above description of the three-dimensional model, the wavelength of the model is established by a sine function. The contact area of the actual droplet impacting the sieve cylinder has a certain degree of error in the simulation. To improve the accuracy of the mathematical model, according to the number of boron nitride nanoribbons composed of a single unit of the model, the influence factor \(\xi \sim {A \mathord{\left/ {\vphantom {A 4}} \right. \kern-0pt} 4}\sin {\text{p/Ly}}\) is used to reduce the overall error. Therefore, it is represented by Eq. (16).

The results are shown in Fig. 13 to further verify the accuracy of the model. The critical impact velocity of the droplet conversion wetting state is determined for the existing model structure parameters and MD simulation data, and the simulation data are consistent with the model prediction.

Verifying the critical impact velocity of the hydrophobic surface: (a) model structural parameters and (b) various contact angles (the dotted line represents the model prediction (red represents the predicted value of Eq. (17), and green represents the predicted value of Eq. (18)), and the symbol represents the MD simulation data).

The total contact time, spreading time and rebound time of the droplets at different V0 values impacting the surface of the superhydrophobic BNNT stainless steel filter are shown in Fig. 14.

Contact time of droplets with different impact velocities on the surface of a superhydrophobic BNNT stainless steel filter at the macroscale.

Interestingly, we found that the total contact time, spreading time and rebound time are related to the velocity V0 of the droplets hitting the superhydrophobic surface from the height of the sieve column h < 5.5L, which is inconsistent with the research theory of Li37 et al. The droplet velocity V0 < 0.31 m/s in this work is unrelated to the overall contact time, spreading time, or rebound time of the droplet after it impacts the superhydrophobic surface. The length of the extension time is unaffected by a droplet impact velocity of 0.31 m/s < V0 < 1.17 m/s. The total contact time and rebound time following droplet impact on the superhydrophobic surface increase as the droplet size increases. The total contact duration, expansion time, and rebound time of the droplet start to decrease at a droplet velocity of V0 > 1.32 m/s. The dynamic behaviour of the droplet impacting the superhydrophobic surface across various time intervals is depicted in Fig. 8. The droplet is in a viscous condition at V0 < 0.31 m/s, a rebound state at 0.31 m/s < V0 < 1.17 m/s, and a splash and spray state at V0 > 1.32 m/s. Therefore, we conclude that the total contact time, spreading time and rebound time of the droplets after they hit the surface of the superhydrophobic BNNT stainless steel filter are related to the initial velocity and wetting state of the droplets.

In the MD simulation, the total contact time, spreading time and rebound time of the nanodroplets at different impact velocities and the surface of the superhydrophobic BNNT mesh are shown in Fig. 15.

MD simulation of the contact time of nanodroplets at different initial velocities on the surface of the superhydrophobic BNNT mesh.

The experimental results show that when h < 5.5L and V0 < 0.001102 Å/fs (V0 < 0.31 m/s), the total contact time, spreading time and retraction time of the nanodroplets impacting the surface of the superhydrophobic BNNT mesh are unaffected by the nanodroplet velocity V0. When h < 5.5L and 0.001102 Å/fs < V0 < 0.0041292 Å/fs (0.31 m/s < V0 < 1.17 m/s), the total contact time and rebound time increase with increasing V0, and the spreading time decreases with increasing V0. When h < 5.5L and V0 < 0.0046824 Å/fs (V0 > 1.32 m/s), the total contact time and rebound time of the nanodroplets decrease with increasing \(V_{0}\), and the corresponding spreading time increases with increasing V0. This result is consistent with the conclusions drawn from the droplet impact experimental data. Therefore, the data from the above fitting curve also show a general similarity between the trends of the MD simulation and the real experiment.

The dynamics of droplet impact on a superhydrophobic surface are often studied by measuring the change in contact diameter between the droplet and the surface over time7,45,46. In the above section, the effects of different impact velocities of droplets on the wetting state of droplets at the macro- and microscales are studied in detail. In this section, we collate the data of the change in the contact diameter of droplets on the superhydrophobic surface at different impact velocities, as shown in Fig. 16.

(a) Variation trend with time of the contact diameter of droplets at different impact velocities on the surface of superhydrophobic BNNTs. (b) Variation trend of Dmax/D0-We1/4 at the macroscale (Illustration: the final morphology of droplets after impact).

According to Fig. 8, droplets with varying impact velocities exhibit dynamic behaviours such as spreading–retraction, splashing, and ejection once they contact the superhydrophobic BNNT surface at the macro scale. The data curve presented in Fig. 16 illustrates how the spreading diameter of the droplets varies over time following their impact on the superhydrophobic BNNT surface. Initially, the spreading diameter increases, but it subsequently decreases. These findings are consistent with previous research findings on the subject37,47,48. After the droplets at different speeds reach the superhydrophobic BNNT surface, their maximum spreading diameter reaches the maximum spreading diameter with time, their maximum spreading increases with increasing droplet impact velocity, and their spreading dynamic behaviour consumes much less time than dynamic behaviours such as retraction/ejection/splashing do. According to the trend of the curve, the curves of the rapid retraction/splash/ejection stage and the rapid spreading stage of the droplet almost completely mirror the symmetry trend.

In addition, after the nanodroplet impacts the surface of the superhydrophobic filter, the droplet diffuses on the surface. When the droplet reaches a certain spreading diameter, it begins to shrink and rebounds under specific conditions. These phenomena indicate that the dynamic behaviour of droplets is related to the Weber number We, material surface wettability and other factors. Antonini49 et al. proposed a certain quantitative relationship between the dimensionless maximum spreading diameter (\(\beta_{\max } = {{D_{\max } } \mathord{\left/ {\vphantom {{D_{\max } } {D_{0} }}} \right. \kern-0pt} {D_{0} }}\)) of a droplet impacting the surface and the Weber number, that is, \(\beta_{\max } \propto We^{\alpha }\), where \(\alpha\) is a function of the contact angle θ. To confirm the correctness of this theory, Antonini et al. conducted a series of droplet impact experiments on surfaces with different wettabilities. The experimental results show that when millimetre droplets impact a superhydrophobic surface with a contact angle of 160°, the dimensionless maximum spreading \(\beta_{\max }\) of the droplets satisfies the relationship \(\beta_{\max } \propto We^{0.4}\) at a Weber number of \(We < 200\). The above research results are compared with the model data in this paper in Fig. 16b. On the surface of the superhydrophobic BNNT mesh with a contact angle of θ = 150°, the dimensionless maximum spreading diameter \(\beta_{\max }\) of the droplet impacting the superhydrophobic surface is basically consistent with \(\beta_{\max } \propto We^{0.4}\), and Yarin50 proved through experimental exploration that the Weber number We = 8.13 and that the droplet ejection phenomenon is consistent with the experimental results in this paper. Li37 experimentally proved that when We = 40.11 (We0.4 = 10.756), droplet spatter occurs, which corresponds to the experiment in this paper. We found that the main reason for the splashing of droplets at high Weber numbers is the imbalance of surface tension and bubble pressure inside the droplets. This finding also confirms that the spreading process of droplets on superhydrophobic surfaces is determined by capillary action and inertia51,52.

To investigate the correlation between the impact velocity and the contact diameter of nanodroplets via MD simulation, this research presents MD simulation data of different velocity impacts, as illustrated in Fig. 17.

(a) Change trend with time of the contact diameter of nanodroplets at different impact velocities on the surface of the superhydrophobic BNNT mesh model. (b) Change trend of Dmax/D0-We1/4 at the microscale.

The contact diameter between the surface of the superhydrophobic model and the nanodroplet increases with the impact velocity. At the early stage, an increasing trend was observed in the tiling diameter over time. After reaching its maximum value, the nanodroplet contact diameter decreases to 0 with increasing duration to 0. According to the slope of the curve in Fig. 17a, the trend of the contact diameter between the nanodroplet and the superhydrophobic surface in the spreading stage is greater than that in the shrinking stage. We then created a schematic of the Dmax/D0-We1/4 relationship of the nanodroplets (Fig. 17b). The fitting curve shows good agreement between the data. These findings agree with the real experimental findings.

In this work, by combining molecular dynamics simulations and experimental data analysis, we use the Weber number as the research medium to explore the dynamic behaviour of droplets impacting the surface of a superhydrophobic BNNT stainless steel filter. Factors such as substrate wettability, impact velocity and contact time are considered.

Firstly, the macroscopic experimental data show that with increasing impact velocity of the droplet, different phenomena, such as impact rebound, impact spread–rebound, impact spread–rebound, and impact spread–breaking–polymerisation spatter, appear successively on the substrate surface. An analysis of the droplet impact phenomenon reveals that the dynamic behaviour of a droplet is related mainly to the wetting pressure (dynamic pressure PD, water hammer pressure PWH) and anti-wetting pressure when the droplet hits a superhydrophobic surface with a rough structure. The MD simulation results agree well with the results of the macroscopic experiments. When h < 5.5L and Vw < VNB, the macroscopic droplets and the microscopic nanodroplets do not bounce. When h < 5.5L and VNB < Vw < VBS, the nanodroplets bounce. When the nanodroplets are between the viscous and nonbounce speeds, they impact the surface. Spreading–shrinking and other processes. When h < 5.5L and Vw > VBS, the droplets splash on the surface of the superhydrophobic BNNT stainless steel filter, and the droplets with impact behaviour do not exhibit an axisymmetric structure. On this basis, a suitable theoretical model is proposed for the nanoscale judgement of the droplet–wetting transition. This model reveals the nanoscale wetting transition mechanism to predict the critical Weber number triggering wetting transition conditions. The prediction agrees well with the MD simulation results.

Second, the influence of the contact time after the droplet impacts the surface of superhydrophobic BNNTs is analysed at the macroscale. When the column height h < 5.5L, the impact velocity affects contact time. When the droplet velocity V0 < 0.31 m/s, the total contact time, spreading time and rebound time after the droplet impacts the superhydrophobic surface are independent of V0. When the droplet velocity is 0.31 m/s < V0 < 1.17 m/s, the total contact time and rebound time of the droplet after it impacts the superhydrophobic surface increase with increasing V0, and the expansion time is unaffected by V0. When the droplet velocity V0 > 1.32 m/s, the total contact time, expansion time and rebound time of the droplet begin to decrease. At the nanoscale of the MD simulation, the simulation data are consistent with the macro results.

Finally, the influence of the impact velocity of the droplet impacting the surface of the superhydrophobic BNNTs on the contact diameter is explored. At the micro- and macroscales, the maximum spreading diameter of the droplets after they impact the surface of the superhydrophobic BNNTs increases with increasing impact velocity, and the spreading diameter of the droplets changes with time. The spreading diameter first increases but then decreases. The maximum contact diameter data conform to Dmax/D0-We1/4.

Therefore, studying the dynamic behaviour of droplets impacting hydrophobic/superhydrophobic surfaces can help in understanding the wetting mechanism of superhydrophobic surfaces. Our findings and related results not only deepen the understanding of the dynamic behaviour of droplet impact at the nanoscale but also provide a relevant theoretical basis for the subsequent preparation of superhydrophobic surfaces. More importantly, establishing an experimental study between nanoscale and macroscopic experiments through molecular dynamics is valuable.

Data will be made available on request. Please contact the author Zhang Lie, email: [email protected].

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We would like to express our gratitude to all those who have helped us during the writing of this thesis.

These authors contributed equally to this work: Lie Zhang, Liang Li, Shuzhi Li, Bo Yuan, Xiaoxia Han and Zhenxin He.

Rocket Force University of Engineering, Xi’an, 710025, Shaanxi, China

Lie Zhang, Yongbao Feng, Liang Li, Shuzhi Li, Bo Yuan, Xiaoxia Han & Zhenxin He

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Lie Zhang: Writing-original draft, Methodology, Data curation. Yongbao Feng: Conceptualization, Methodology Validation, Formal analysis. Liang Li: Software, Investigation, Project administration. Shuzhi Li: Writing-review and editing, Supervision, Formal analysis. Bo Yuan: Software, Formal analysis Resources. Xiaoxia Han: Experiments, Discussion, Revising. Zhenxin He: Funding acquisition, Writing-review and editing.

Correspondence to Yongbao Feng.

The authors declare no competing interests.

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Zhang, L., Feng, Y., Li, L. et al. Droplet impact dynamics on the surface of super-hydrophobic BNNTs stainless steel mesh. Sci Rep 14, 27695 (2024). https://doi.org/10.1038/s41598-024-75825-z

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Received: 25 May 2024

Accepted: 08 October 2024

Published: 12 November 2024

DOI: https://doi.org/10.1038/s41598-024-75825-z

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