Droplet impact on asymmetric hydrophobic microstructures

Textured hydrophobic surfaces that repel liquid droplets unidirectionally are found in nature such as butterfly wings and ryegrass leaves and are also essential in technological processes such as self-cleaning and anti-icing. However, droplet impact on such surfaces is not fully understood. Here, we study, using a high-speed camera, droplet impact on surfaces with inclined micropillars. We observed directional rebound at high impact speeds on surfaces with dense arrays of pillars. We attribute this asymmetry to the difference in wetting behavior of the structure sidewalls, causing slower retraction of the contact line in the direction against the inclination compared to with the inclination. The experimental observations are complemented with numerical simulations to elucidate the detailed movement of the drops over the pillars. These insights improve our understanding of droplet impact on hydrophobic microstructures and may be a useful for designing structured surfaces for controlling droplet mobility.

Asymmetric hydrophobic microstructures are often exploited by natural species, such as butterfly wings 37 and ryegrass leaves 38 where they assist liquid roll-off. Surfaces with asymmetric ratchets and spikes allow directing a droplet in a desired direction and such anisotropic surfaces are useful particularly in self-cleaning, water harvesting, 39 and cell-directing. 38,40,41 Here, hydrophobic surface properties are advantageous to increase the mobility of a droplet.
On such hydrophobic surfaces, upon impact, droplets bounce off towards the direction in which the surface structures are oriented. 37,[42][43][44][45] However, the detailed mechanisms of bouncing are not fully understood. In particular, the influence of the surface geometry, i.e., the pitch and height of structural features, and the impact velocity remain to be fully elucidated.
Here, we study droplet impact on asymmetric microstructures experimentally and numerically. We observe the distinct influence of surface geometry and impact velocity on impact behaviour. Moreover, we measure the trajectories of bouncing droplets and investigate the conditions for directional rebound. We observe and discuss differences in receding speeds of the contact line in the direction with the inclination and against the inclination.

Experimental set up
The impact of liquid droplets is observed with a high-speed camera (speedsense, Dantec dynamics) at a frame rate of 6000-8000 s −1 with a spatial resolution of 15 µm. The schematics of the experimental setup are shown in Fig. 1 1000N) at a small flow rate (0.10 µL/s) and the droplet pinches off from the needle with the constant initial radius R 0 = 1.14 ± 0.02 mm. The droplet is accelerated by gravity and hits the substrate with an impact velocity V 0 . The impact velocities are varied by changing the distance from the substrate to the needle H 0 . The impact velocity is estimated from the images just before the droplet hits the substrate. The captured images are shown in Fig. 1(e). The height H 0 is varied from 5 mm to 85 mm, which leads to impact velocities V 0 from 0.25 m/s to 1.3 m/s ( Table 1).
The liquid employed in this study is deionized water. The surface tension of water σ is measured to be 0.072 mN/m with TD 2 tensiometer (LAUDA). In this study, we focus on the droplet motion in the direction of the inclination of the pillars.

Surface preparation
The substrates studied are made from Ostemer 220 (Mercene Labs, Sweden), Off-Stoichiometry-Thiol-Ene (OSTE) resin. 46,47 The resin is suitable for fabricating inclined micro patterns by exposing slanted collimated ultraviolet light. The surfaces are prepared in three steps.
First, a base OSTE layer is prepared on a smooth plastic film. Secondly, inclined micropillars are developed on the base layer by exposing slanted ultraviolet light through a patterned photomask. After cleaning uncured OSTE in an acetone bath, hydrophobic surface modification using 1% w/w fluorinated methacrylate (3,3,4,4,5,5,6,6,7,7,8,8,9,9, The equilibrium contact angles of deionized water are reported in Table. 2. The advancing and receding contact angles are measured using the sessile drop method. 48,49 A droplet with the initial volume of 5 µL is deposited on the surface and it is pumped through the needle at a flow rate of 0.1 µL/s to measure advancing contact angle. For the receding angle measurements, the initial volume is set to 30 µL to perform reliable measurements 49 and the droplet is drained at a flow rate of 0.1 µL/s. The average contact angle for 5 seconds after the contact line starts to move is defined as the advancing (receding) contact angle.

Rebound velocity estimation
To investigate the influence of the surface structure and the impact velocity on rebound behaviors, the trajectory of the droplet is calculated. The trajectory of the center of mass is obtained by extracting the surface contour from the images (Fig. 1e, f). Assuming ballistic trajectory after the impact, the horizontal and vertical positions X, Z are described as a function of time T , where The positive X indicates the horizontal direction with the inclination of the pillars and Z is the vertical displacement from the substrate. Here, V x , V z are horizontal and vertical velocities, g is the gravitational acceleration, and X 0 and Z 0 are the horizontal and vertical positions at T = T 0 . Equations 1,2 describe the trajectory well when V x , V z are fitted (see the dash lines in Fig. 1g, h). By performing the fitting procedure, V x and V z are estimated as a function of the impact velocity. Here, X 0 , T 0 are set so that Z 0 = 1.1 R 0 for all configurations. Table 2: List of the surfaces. D, P, H are the width, pitch, and height of the pillars. θ e is the equilibrium contact angle. θ a−A , θ a−W , θ r−A , θ r−W are advancing/receding contact angle in the direction against the inclination and with the inclination, respectively. The advancing/receding contact angles in the direction against the inclination and with the inclination on the flat surface are identical.

Results
Bouncing regimes Figure 2 shows series of images of a water droplet spreading after impact. We observe that the pitch between the pillars P and the impact velocity V 0 determine the droplet behavior.
Three distinctive behaviors are observed. First, the droplet completely rebounds from the surface (1,2 in Fig. 2a). Moreover, the droplet rebounds to the direction with the inclination on P = 30 µm and at high V 0 (case (2) in Fig. 2a). Secondly, the droplet breaks up and part of the droplet remains deposited on the surface while the other part bounces up (3 in Fig. 2a). Finally, the droplet does not bounce and sticks to the surface (4 in Fig. 2a). We refer to the three configurations as "Complete rebound", "Partial rebound", and "Stick".
Figure 2(b) shows the pitch-impact velocity parameter map with the "Complete rebound", "Partial rebound", and "Stick" regions are indicated.
The Cassie-Wenzel transition is responsible for the different behaviors. For "Complete rebound" situations, the grooves between the posts are not wetted and air is trapped underneath the droplet (Cassie state). On the other hand, for "Partial rebound" and "Stick" cases, the grooves are partially or fully penetrated by the liquid (Wenzel state). A semiquantitative model to account for the Cassie to Wenzel transition on an array of pillars was proposed by Bartolo et al. 16 The model estimates the critical impalement pressure on a structured surface. When the hydrodynamic pressure over the surface exceeds the critical pressure, the liquid-air interface makes contact with the basal surface of the substrate and the liquid penetrates into the grooves. Above the critical pressure, the Cassie-Wenzel wetting transition occurs, which also corresponds to the transition from bouncing to nonbouncing. The model estimates the critical pressure as p c ∼ σHD/2P 3 for dense arrays of straight pillars, 16 where H and D are the height and width of the pillars, respectively.
In the instant of droplet impact, the hydrodynamic pressure is p d ∼ ρV 2 0 /2 where ρ is the density of the liquid. The balance p c ∼ p d gives the critical impact velocity for the pitch P as V c ∼ σHD/ρP 3 . The dashed curve in Fig. 2(b) depicts this critical value. We observe that the critical curve separates the complete rebound regime and partial rebound regime reasonably well also for inclined pillars. Beyond the critical impact velocity, "Stick" and "Partial rebound" are observed.

Rebound velocity
As seen in the previous sections, the rebound behavior in the horizontal direction depends on the surface structure and the impact velocity. This section investigates the directional behavior within the rebound regime. Impact velocity (m/s) (4) Stick (4) Complete rebound The vertical rebound velocity V z in Fig. 3(b) increases with the impact velocity up to ∼ 0.25 m/s. Larger V z is observed for P = 40 µm compared to P = 30 µm. This is likely because of the higher level of hydrophobicity, which is indicated by the larger equilibrium contact angle on P = 40 µm (see Table. 2). As a result, the droplet moves in the direction with the inclination up to 1.3 mm (Fig.3c). The directional displacement is observed only for V 0 > 0.5 m/s and on P = 30 µm. This is similar to the observation made by Li et al. 45 They also observed a larger horizontal displacement on arrays of inclined cones with a smaller spacing.
It is noticeable that the expansion phase until the droplet reaches the maximum deformation is symmetric on the inclined hydrophobic pillars (see the snapshots at 5 ms in  The Ohnesorge number in this study is 3.5 × 10 −3 , which is reasonably low.
Contrary to the first expansion, the retraction immediately after the initial expansion can be asymmetric. The asymmetric retraction is responsible for the observed asymmetric bouncing. The retraction is governed by how the contact line detaches from the surface structures.
The underlying mechanism of the asymmetric receding speed is in the wetting of the asymmetric microstructure. Figure 5 shows the schematic model of the receding contact line on the asymmetric microstructure. The key factor is that a part of the inclined sidewall is wetted. The wetting on asymmetric microstructured surfaces was also illustrated by Guo et  Fig. 5a), the apparent receding angle θ r−W is θ r + β ∼ 133 degrees. Therefore, the contact line smoothly recedes on the sidewall.
On the other hand, the apparent receding angle in the direction against the inclination (θ r−W ) -when the contact line moves down on the sidewall -should be θ r − β, i.e. as small as 13 degrees (Fig. 5b). The contact line is then pinned at the obtuse corner until the liquid detaches from the sidewall. This pinning delays the receding in the direction against the inclination. Note that the liquid inertia helps the interface detach from the sidewall, so the apparent receding angle in the experiments is not as small as θ r − β.   Fig. 6a). The spreading is nearly symmetric until 0.2 ms, then the contact line starts to recede (see Fig. 6b). During the retraction in the direction against the inclination, the liquid is arrested on the sidewall of the structures. While the contact line is pinned at the obtuse corner, the liquid phase can not detach from the sidewall (see the inset in Fig. 6a). The apparent contact angle has to decrease below 60 degrees before the contact line detaches from the sidewall (see Fig. 6c). On the other hand, during the retraction in the direction with the inclination, the apparent contact angle does not become lower than 80 degrees. Consequently, the retraction is faster in the direction with the inclination than in the direction against the inclination.
This difference is responsible for the directional motion in the direction with the inclination.
Moreover, the numerical simulations with different impact velocities are consistent with our experimental observation. Figure 6(d) shows the horizontal displacement of the center of the mass of the droplet with different impact velocities. The larger the impact speed, the faster the horizontal motion becomes. As seen in Fig. 6(b), the retraction distance is larger for a higher impact velocity. Here, the longer retraction distance of the contact line is responsible for the stronger effect of the pinning, which leads to the larger displacement.

Receding contact angle measurements
Here, we demonstrate that the receding contact angle measured with the sessile drop method is consistent with the droplet impact behavior. The receding contact angle in the direction against the inclination (θ r−A = 84 degrees) is smaller than in the direction with the inclination (θ r−W = 100 degrees) for P = 30 µm (Fig. 7a). This is consistent with the fact that the receding speed in the direction against the inclination during the droplet impact is slower (Fig. 7d). This difference leads to directional rebound. Meanwhile, the receding angles on the surfaces with P = 40 µm and 60 µm are similar for the two directions (Fig. 7b, c).
The symmetry in the receding contact angle is consistent with the symmetric receding speed during the droplet impact (Fig. 7e, f).
The pitch in the direction perpendicular to the direction of the inclination of the surface structures is potentially responsible for the difference between P = 30 µm and P = 40 µm where the droplet rebounds in both cases. The effect of the pinning described in Fig. 5 is effective only when the pillars are sufficiently dense along the contact line so as for the pinning to be effective enough to delay the receding. This implies that the pinning site is dense enough only for P = 30 µm but not for P = 40 µm in our experiments. Moreover, the mechanisms in Fig. 5 are undermined for "Partial rebound" and "Stick" cases since the grooves between the posts are filled with water. Therefore, the directional behavior is not expected for "Partial rebound" and "Stick" droplets.
(a) There is however a trade-off between the density of the pinning corners and the hydrophobicity of the surface. As the pitch decreases (P → 0), the number of pinning sites along the contact line increases, but the surface becomes less hydrophobic, as the solid-air ratio increases. An additional degree of freedom of the surface to enhance the directional rebound, is the pitch P 2 in the direction perpendicular to the direction of the inclination (see Fig. 8). Since, it is desirable to increase the number of pinning site while keeping the surface hydrophobic, structures with P > P 2 and a reasonably large static contact angle could enhance the directional rebound.
The directional rebound mechanisms proposed in previous studies are different in certain D H # Figure 8: Another pitch in the direction perpendicular to the inclination of the surface structures.P 2 aspects than in this study. Lee et al. 42 proposed that the stored surface energy between the inclined structures is responsible for the directional rebound. Note that the height of their structures is in the order of 1 mm, whic is two orders of magnitude higher than in this study (and the aspect ratio of the surface dimension is large). For such surfaces, the droplet can fully penetrate the pillar arrays during impact. A similar large penetration is observed by Li et al. 43 The two different situations are schematically shown in Fig. 9. Lee et al. 42  where the rebound mechanisms are described in Fig. 9(a), smaller displacement and velocity are observed. Li et al. 45 reported that on inclined cone structures with H ∼ 300 µm and P ∼ 300 − 400 µm , the displacement is 0.7 mm for V 0 = 0.71 m/s and 2.0 mm for V 0 = 1.73 m/s. The horizontal rebound distance is similar to this study.
On the other hand, for a very tall structure with H > 1 mm and P < 500 µm , where the rebound may follow the scenario in Fig. 9(b), a larger rebound velocity is observed.
Lee et al. 42 reported the horizontal velocity and displacement on thin-spike structures. The horizontal velocity is 0.09 m/s and displacement distance is 9.1 mm for V 0 = 1.13 m/s.
Similarly, Li et al. 43 reported a large horizontal velocity and horizontal displacement of 0.062 m/s and 5.5 mm, but the impact velocity information is missing. The difference in the mechanisms could be responsible for the larger displacement since the surfaces described in Fig. 9(b) are capable of harnessing droplets with larger impact speed and therefore able to store larger surface energy before the rebound. It is worth noting that Lee et .al. 42 also pointed out that the bouncing mechanisms on superhydrophobic nanostructures and hairy spike structures are different. While the droplet rebounds after the full retraction on the superhydrophobic nanostructures, the droplet rebounds by upward capillary forces on their hairy spike arrays.

Conclusions
We studied the droplet impact on asymmetric microstructures. Directional rebound was observed only for dense microstructures and at high impact speeds. The retraction phase and the detailed wetting of the sidewalls of the inclined structures govern the rebound. The wetting of the sidewall leads to a slower receding speed in the direction against the inclination. The contact line can be pinned at the obtuse corner when receding in the direction against the inclination, while in the other direction the contact line recedes continously.
The receding contact angles on the asymmetric pillar structures correlate with the droplet rebound behavior. The directional rebound is found only on surfaces with asymmetric receding contact angles in the direction of the pillar inclination. Numerical simulations provide further detailed visualizations of the two phase interface near the pillars and confirm our experimental observations. We hope that insights gained in this study will be useful for tuning surface structures for directional transport of liquid drops.