Impact Dynamics of Aqueous Polymer Droplets on Superhydrophobic

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Impact Dynamics of Aqueous Polymer Droplets on Superhydrophobic Surfaces Longquan Chen,*,† Yonggui Wang,‡,§ Xiaoyan Peng,∥ Qing Zhu,⊥ and Kai Zhang*,‡ †

School of Physics, University of Electronic Science and Technology of China, Chengdu 610054, China Wood Technology and Wood Chemistry, Georg-August-Universität Göttingen, Büsgenweg 4, Göttingen D-37077, Germany § Key Laboratory of Bio-Based Material Science and Technology (Ministry of Education), College of Material Science and Engineering, Northeast Forestry University, Harbin 150040, Heilongjiang, China ∥ Affiliated Hospital of Southwest Jiaotong University, Chengdu 610031, China ⊥ Institute of Chemical Materials, China Academy of Engineering Physics, Mianyang 621999, China

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‡

ABSTRACT: Controlling the impact of water droplets on nonwetting surfaces is difficult due to their superior liquid repellency. Herein, we experimentally demonstrate that the impact dynamics of water droplets on superhydrophobic surfaces can be effectively altered by adding a very small amount of high molecular weight polymer. The presence of polymer chains in water enhances droplet−surface interactions and liquid viscoelasticity, which couple with the hydrodynamic effect and thus affects the dynamic behaviors of impinging droplets. Whereas various impact phenomena with altered transitional boundaries in the phase diagram have been observed for droplets of aqueous polymer solutions at low concentrations, only deposition was identified for aqueous polymer droplets with high concentrations. Nevertheless, the polymer addditive does not influence droplet spreading but siginificantly slows down droplet retraction and increases the contact time of rebounding droplets on the superhydrophobic surface. Moreover, the induced non-Newtonian elongational viscosity causes strong energy dissipation, while the elasticity of polymer chains increases the restoring force during the postimpact droplet oscillation. rebound on crop leaves.20 Practically, two strategies have been proposed to restrain the rebound of impinging droplets on solid surfaces. The first is to add surfactants into the working liquid to reduce surface tension and thus enhance the wettability of the liquid on the surface.21,22 However, reducing surface tension also alters the droplet size distribution during the spray23 and lowers the threshold velocity for splashing,24,25 which is another undesired phenomenon compromising spray retention.21,26 The second is to employ polymer additives to develop a remarkable elongational viscosity27 or a high normal stress28 within the impinging droplet. These non-Newtonian rheological effects could cause a large resistance to droplet motion during impact, thereby suppressing the rebound.27,28 Moreover, also the elasticity of polymer chains increases the “strength” of the liquid, which reduces or even prevents splashing.29,30 Compared to surfactants, the idea of using polymer additives is therefore more suitable for practical applications associated with spray deposition. Despite its practical importance, the impact of aqueous polymer droplets on superhydrophobic surfaces has not received much attention until recently. Zang et al. performed

1. INTRODUCTION Superhydrophobic surfaces are surfaces with ultralow surface energy, which are characterized by a water contact angle of 150° or higher and a contact angle hysteresis below 5°−10°.1,2 In nature, many living organisms, such as water strider legs,3 duck feathers,4 and lotus and rice leaves,5,6 show excellent superhydrophobicity. Scanning electron microscopy studies revealed that these superhydrophobic biosurfaces are constructed by micro-, nano-, or micro/nanostructures.1−3,7,8 When a liquid droplet impacts on such a surface, the air trapped in these microscopic structures provides a considerable capillary pressure to prevent the liquid from wetting the surface,9−11 and eventually the droplet can completely rebound off. Experimental investigations have showed that the rebound behavior can even occur at an impact velocity less than 0.1 m/ s.12−15 The superior liquid repellency of superhydrophobic surfaces has attracted a tremendous amount of attention for surface engineers due to its potential applications in selfcleaning coatings,8 hydrodynamic drag reduction,16 antiicing,17 biofouling prevention,18 and corrosion inhibition.19 Controlling droplet deposition is a key technique and difficulty in spray applicationsparticularly for pesticide sprayingsince most plant leaves are highly hydrophobic or superhydrophobic.5,6 It has been reported that more than 50% of the initial pesticide spray is lost due to the undesired drop © XXXX American Chemical Society

Received: July 23, 2018 Revised: September 16, 2018

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DOI: 10.1021/acs.macromol.8b01589 Macromolecules XXXX, XXX, XXX−XXX

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mass concentrations of 0.01−1 wt %. The surface tensions of these liquids were measured with the pendant droplet method using a goniometer (Krüss DSA 30, Germany). The polymer additive decreases the surface tension of water, but only to some extent (the difference is less than 16%, Table 1). The equilibrium (θeq),

droplet impact experiments on superhydrophobic surfaces using water containing polymer additives and nanoparticles.31 They found that the inhibition of droplet rebound at sufficiently high impact velocity may be due to the enhanced friction between polymers/particles aggregates and the structured surfaces rather than the viscoelastic property of the liquids. They further demonstrated that the rebound behaviors can be rejuvenated via particle enwrapping, suggesting the importance of the droplet−surface interaction on the impact dynamics.32 A similar conclusion was also drawn lately by Izbassarov and Muradoglu,33 who conducted numerical simulations of viscoelastic droplets impinging on smooth lyophilic, lyophobic, and superlyophobic surfaces. Instead of rebound inhibiting, the viscoelasticity was found to promote drop rebound in the retracting phase.33 In these works mentioned above, the effort has been mostly devoted to studying the rebound behaviors of aqueous polymer droplets. A general picture of the influence of polymer additives on the droplet impact dynamics is still missing. In this work, the impact of droplets of aqueous polymer solutions at diverse concentrations on an artificial superhydrophobic surface was investigated and compared to that of pure water droplets. The objective is to identify the effects of polymer additives on droplet impact behaviors and recover the underlying mechanism. It was found that the coupling effects of droplet−surface interactions, liquid viscoelasticity, and hydrodynamics determine the dynamic behaviors of impinging droplets as well as the postimpact droplet oscillation.

Table 1. Equilibrium (θeq), Advancing (θa), and Receding (θr) Contact Angles of 4 μL Droplets on Superhydrophobic Surfaces water contact angles (deg) liquids pure water 0.1 g/L PEO 0.5 g/L PEO 5 g/L PEO 10 g/L PEO

γ (mN/m) 71.5 62.0 61.1 61.5 60.2

± ± ± ± ±

0.1 0.2 0.3 0.1 0.5

θeq 154.7 151.2 150.9 150.6 149.8

± ± ± ± ±

θa 1.5 0.9 0.5 1.0 0.7

158.4 159.3 158.2 157.1 156.5

± ± ± ± ±

θr 1.7 2.6 2.9 1.4 1.0

152.6 149.2 144.4 140.9 135.7

± ± ± ± ±

1.8 2.0 1.9 1.3 1.1

advancing (θa), and receding (θr) contact angles of 4 μL liquid droplets on the superhydrophobic surface were also measured with the commercial goniometer and are summarized in Table 1. 2.2. Rheological Characterization. The rheological properties of all liquids were first characterized with a standard rheometer (Physica MCR 301, Anton Paar). Figure 2a shows the dependence of the shear viscosity on the shear rate ranging from 1 to 1000 s−1. It seems that the PEO solution with the lowest concentration (0.1 g/L) still behaves like a Newtonian fluid with a viscosity (∼10 × 10−4 Pa s) slightly higher than water (∼9 × 10−4 Pa s). In contrast, for solutions with c ≳ 0.5 g/L, shear thinning behavior becomes more pronounced with the increase of the polymer concentration. Because of the limited resolution of the rheometer, we could only measure the viscoelastic properties of 5 and 10 g/L PEO soultions at 0.01−20 Hz. As shown in Figure 2b, both solutions behave like elastic liquids as G′ ≫ G″, and the elastic strength increases with c and frequency. We further performed experiments studying the breakup of droplets of these liquids from a steel needle using a high-speed camera (Fastcam SAZ, Photron, Japan) at 80000 fps, which is an alternative approach to characterize liquid viscoelasticity.35,36 Figures 2c and 2d show the snapshots of the pinch-off process of a droplet of water and water containing 0.1 g/L PEO, respectively. The time shown corresponds to the time to pinch-off (tp − t), where tp is the pinchoff time and t is the actual time. As a droplet expands at the outlet of the needle, a curved neck is formed when its volume becomes large enough. The neck thins under the action of surface tension with a velocity increasing with time (see 4.0−0.5 ms in Figure 2c and Figure 2d), and eventually forms a filament connecting the pendant droplet and the needle. For water, the filament subsequently breaks up near the droplet as well as near the needle within few tens of microseconds, forming a daughter droplet, as seen at 0 ms in Figure 2c. For aqueous PEO solutions, however, the formed filament is gradually elongated and thins rather than to rapidly break up. At the time close to pinch-off, a number of small droplets are successively developed on the filament (see last three images in Figure 2e), resembling the periodic “beads-on-a-string (BOAS)” structure. This pinch-off scenario is typically observed in the formation of viscoelastic droplets,37,38 and the more viscoelastic the fluid is, the longer the BOAS structure is (Figure 2f). Figures 2c and 2d also show that at any given time the neck of water is wider than that of aqueous PEO solutions, implying the delay of the pinch-off process by the polymer additive. This can be seen more clearly from the temporal evolution of the minimum diameter (Dmin) of the neck in Figure 2g. By shifting the time coordinate of Figure 2g for a certain time td increasing from ∼0.1 ms for 0.1 g/L PEO solution to ∼82.4 ms for 10 g/L PEO solution, all experimental data of the early necking process can be collapsed onto one master curve, as illustrated in Figure 2h. However, Dmin starts to deviate from the master curve at ∼0.5 ms for 5 g/L PEO solution and ∼1.1 ms for 10 g/L PEO solution due to the formation of the viscoelastic filament.

2. EXPERIMENTAL SECTION 2.1. Surface and Liquids. The superhydrophobic surface was fabricated by coating glass substrates with cellulose stearoyl ester with a degree of substitution of 3 (CSE3), which was synthesized according to the procedures reported elsewhere.34 Clean glass substrates (27 mm × 76 mm × 1 mm) were horizontally submerged into CSE3 solution with a concentration of 5 mg/mL in ethyl acetate at 60 °C. After the solution cooled to the room temperature, the substrates were rinsed in ethanol and then dried in air. Figure 1 shows the

Figure 1. SEM images of the hierarchically structured superhydrophobic surface. The inset shows the side profile of the coating layer. scanning electron microscopy (SEM) images of the superhydrophobic coating at different magnifications. The surface is constructed with microprotrusions (size of 5−8 μm, left image in Figure 1) decorated with branchlike nanostructures (size of 50−150 nm, right image in Figure 1). These hierarchical structures have some similarities with the microscopic structures of lotus and rice leaves5 and are desirable to achieve superhydrophobicity.1,2 The liquids investigated in the experiments are pure water (18.4 MΩ cm, Millipore Synergy, Darmstadt, Germany) and aqueous solutions of poly(ethylene oxide) (PEO, Aladdin, China) with a molecular weight Mw = 4 × 106 g/mol. The polymer was added in water and agitated with a magnetic stirrer at low angular speeds until it is completely dissolved. We prepared four solutions with concentration c of 0.1, 0.5, 5, and 10 g/L, which corresponds to B


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Figure 2. (a) Plot of the shear viscosity as a function of shear rate for diverse liquids. (b) Plot of the storage modulus G′ and loss modulus G″ as a function of rheology frequency f for 5 and 10 g/L PEO solutions. (c, d) Snapshots of the necking process of water and 0.1 g/L PEO solution, respectively. (e) A magnified sequence of the formation of BOAS structure in (d). (f) The formed BOAS structure in 5 g/L PEO solution. The scale bar in (c)−(f) is 0.3 mm. (g) Minimum neck diameter Dmin versus the time to pinch-off (tp − t) for all five liquids. (h) A log−log plot of Dmin as a function of the relative time minus the delayed time (tp − t − td). The dashed line is the best fit with a power law Dmin = 0.43(tp − t − td)2/3. For aqueous PEO solution at c ≲ 0.5 g/L, the whole necking process is still very similar to water, although the filament exhibits viscoelastic behaviors just before breakup, i.e., formation of the BOAS structure (Figure 2d,e). The neck thins according to a power law Dmin ∝ (tp − t − td)2/3 for ∼0.1 ms in the pinch-off regime (Figure 2h), indicating that the dynamics is dominated by capillary and inertial forces.35 2.3. Droplet Impact Experiment. Liquid droplets with radii R0 = 0.95−1 mm were released from the tip of a steel needle (0.24 mm outer diameter) at different heights using a syringe pump. These droplets were accelerated by gravity up to a velocity of 0.02−2.5 m/s before impact on the superhydrophobic surface placed underneath. The corresponding Weber number, We = ρV02R0/γ, which compares the inertial to capillary force, is in the range of 0.01−99, where ρ, γ, and V0 are the density, surface tension, and velocity of the impinging droplet. The impact process was recorded with the Photron highspeed camera at 15000−30000 fps and analyzed using a MATLAB (MathWorks, Inc.) algorithm.

3. RESULTS AND DISCUSSION 3.1. Impact Outcomes. To reflect the effects of the polymer additive on the impact outcomes, experiments with pure water were conducted on the superhydrophobic surface as reference. Similar to other studies,12,15,39−41 regular deposition, complete rebound, partial rebound, and splashing were successively identified with the increase of the impact velocity, as shown in the impact phase diagram (Figure 3). Although the addition of PEO does not significantly change the surface tension (Table 1) and shear viscosity (Figure 2a) of water, the dynamic behaviors of impinging droplets are markedly altered. Whereas most of the above impact phenomena with altered transitional boundary in the phase diagram were observed for PEO solutions at c ≲ 0.5 g/L, only deposition was identified for PEO solutions at c ≲ 5 g/L. We want to point out that an impinging droplet can cause siginificant substrate deformation if the substrate material is soft viscoelastic42 or the substrate is elastic,43 and the deformation would affect the hydrodynamics C


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due to the friction (which scales as γ(cos θr − cos θa)R02), one obtains VL ≈ γ(cos θr − cos θa)/ρR 0 .44 In the experiments, we observed complete rebound for pure water and aqueous PEO solution droplets at c ≲ 0.5 g/L. As shown in Figure 3, water droplets rebound from the superhydrophobic surface at V0 ≳ 0.04 m/s (We ≳ 0.03), matching well with the above scaling analysis. However, the measured VL for 0.1 and 0.5 g/L PEO solution droplets (∼0.10 and ∼0.14 m/s, respectively) is much higher than the predicted value (∼0.05 m/s). The higher VL for aqueous PEO solution droplets can be attributed to the enhanced droplet−surface interaction.31 When such a droplet spreads on the superhydrophobic surface, PEO chains are adsorbed on the top of the microscopic structures contacting with the liquid. The adhesive interactions between the liquid and deposited PEO macromolecules, in turn, become an additional and dominant force resisting droplet retraction.45,46 Thus, a higher VL is required to promote droplet rebound. The enhanced droplet−surface interaction can be verified by measuring the receding contact angle,46 which decreases with increasing PEO concentration (Table 1). For aqueous PEO solutions at c ≳ 0.5 g/L, the droplet−surface interaction is so strong that viscoelastic filaments are formed near the contact line as a droplet starts to recoil (Figure 4b,c). The zoomed view in Figure 4f shows that these filaments are anchored on the surface and pulled out from the droplet (indicated by yellow arrows), further confirming the above statement. As recoiling proceeds, these filaments become longer and thinner, and eventually break, which dissipates a large amount of energy. To the best of our knowledge, this is the first time that these viscoelastic filaments have been visualized during impact of aqueous polymer droplets on solid surfaces. However, a droplet containing 0.5 g/L PEO can still rebound from the superhydrophobic surface if the impact velocity is sufficiently high (Figures 3 and 4b).

Figure 3. Impact phase diagram for water and different aqueous PEO droplets on the superhydrophobic surface.

of the droplet in turn. In our experiments, the rigid, hierarchical cellulose structures were deposited on 1 mm thickness glass substrates. Therefore, the impact outcomes presented here are irrelevant to substrate deformation. In the following, we introduce these phenomena in details and discuss the corresponding dynamics. Complete Rebound. One special characteristic of superhydrophobic surface is that an impinging droplet can completely rebound from the surface due to its excellent antiwetting property,1,2 as shown in Figure 4a. However, the rebound only happens if the kinetic energy of the impinging droplet is high enough to compensate the energy dissipated during the impact; i.e., the impact velocity should be higher than a lower threshold VL. Otherwise, the droplet deposits on the surface as a pearl with a contact angle larger than 150° (Figure 3). For low-viscosity liquids, droplet rebound is mainly hindered by the friction froce arising from the pinning− depinning of the contact line on the surface. Balancing the kinetic energy (of the order of ρR03V02) with the energy loss

Figure 4. Impact of droplets of diverse liquids on the superhydrophobic surface at low velocities. (a) Pure water, V0 = 0.08 m/s. (b) 0.5 g/L PEO solution, V0 = 0.27 m/s. (c) 10 g/L PEO solution, V0 = 0.33 m/s. (d) water, V0 = 0.61 m/s. (e) 0.1 g/L PEO solution, V0 = 0.61 m/s. (f) The zoomed view of the droplet-surface contact region in (c). The scale bar is always 1 mm. D


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Macromolecules For droplets with higher PEO concentrations, the rebound is completely suppressed (Figures 3 and 4c). At low Weber numbers, an impinging droplet is slightly deformed during spreading and forms an oval shape, when the maximum extension is reached (4.5 ms in Figure 4a). Then, it retracts back, resumes to the sphereical shape, and takes off (4.5−16.3 ms in Figure 4a); i.e., the droplet behaves like an elastic ball. At We ≳ 1, the large inertial force causes strong drop deformation. As displayed in Figures 4d and 4e, upon impact a capillary wave is excited at the bottom of the droplet as denoted by blue arrows. This capillary wave propagates along the surface of the droplet and deforms it into a pyramidal structure with several steps; e.g., four steps are observed at 1.5 ms in Figure 4d,e. With droplet spreading, these staircase steps gradually merge into the one close to the surface. In the meanwhile, one spire is converged on the top of the droplet center (denoted by red arrows in Figure 4d,e). The further downward oscillation of the spire causes an air cavity in the droplet center, and the droplet takes a toroidal shape. If the impact velocity is high enough, the cavity can reach deep into the droplet (3.1 ms in Figure 4d,e). For pure water, the fast recoiling leads the cavity top to seal before it collapses, entrapping an air bubble within the rebounding droplet (denoted by red dotted circle in Figure 4d). This bubble entrapment was observed within a narrow range of impact velocities (0.5 m/s ≲ V0 ≲ 0.68 m/s, 3.6 ≲ We ≲ 6.4), which is consistent with the experimental observations in the literature.47−49 For aqueous PEO solutions, droplet recoiling is slower and the cavity always collapses before it seals, leaving no bubble behind (3.1−4.6 ms in Figure 4e). We also noted that an impinging droplet was always elongated before taking off. As can be seen from Figures 4b, 4d, and 4e, the droplet struggles to leave the surface after spreading. However, a pinning-like state was observed at the late stage of droplet retraction, e.g., 8.6−12.6 ms in Figure 4b, and the droplet is vertically stretched. Figure 5 describes the normalized droplet height just before rebound, HR/2R0, as a function of the Weber number, and representative snapshots of these droplets are also shown. It is evident that droplet height increases with the impact velocity, and water droplets are elongated to a length slightly longer than that of droplets of aqueous PEO solutions. Moreover, the shape of water droplets is more uniform than that of aqueous PEO solution droplets, indicating its weaker pinning effect. The pinning phenomenon can be explained by the impalement of liquid into the structured superhydrophobic surface during droplet impact,12,41 which is determined by the competition between the wetting and antiwetting pressures.9,44,50 The wetting pressures contain a dynamic pressure generated by droplet inertia, PD = 0.5ρV02, and a liquid hammer pressure given rise by the sudden compression of the droplet upon impact,51−53 PH = 0.2ρCV0, where C is the sound speed in the liquid. The liquid hammer pressure only lasts a short period of time (on the order of 2R0/C ∼ 1 μs) on an area of magnitude π(2R0V0/C)2. Then, it gives way to the dynamic pressure. The presure preventing liquid impalement is the capillary pressure, PC = −2 2 γ cos θa−f/S, where θa−f is the advaning contact angle on the flat surface and S is the spacing between the neighboring protrusions of microscopic structures. For hierarchically structured superhydrophobic surfaces, there are two antiwetting pressures: the capillary pressure caused by microprotrusions, PCM ∼ 10 kPa (θa−f ≈ 120°, S ∼ 10 μm as denoted by red arrows in the left image of Figure 1), and that

Figure 5. Normalized droplet height before taking off HR/2R0 as a function of the Weber number We. The scale bars in insets are 1 mm.

caused by nanostructures, PCN ∼ 1 MPa (S ∼ 150 nm as denoted by red arrows in the right image of Figure 1). Within the range of impact velocity we investigated (0.02 ≤ V0 ≤ 2.5 m/s), PCN is always higher than PH (6 kPa−0.7 MPa, C ≈ 1495 m/s), PD (0.2 Pa−3 kPa) is always lower than PCM, and PH becomes higher than PCM at V0 ≳ 0.05 m/s. Therefore, upon impact the liquid completely impales the microprotrusions locally around the contact point (region A in Figure 5b) without wetting the nanostructures, while partial impalement occurs during spreading under the impinging droplet (region B in Figure 5b). The combination of these two effects causes the pinning and elongation of the recoiling droplet before rebound. At low We, the impaled area in region A is small and the impalement in region B is shallow. Thus, the impinging droplet can easily get depinned and resume to a spherical shape before taking off, i.e., HR/2R0 ∼ 1 (Figure 5a). At high We, partial impalement in region B becomes deep and the impaled region close to the contact point becomes large. As a result, the pinning effect becomes strong, and the droplet is thereby elongated under the recoiling force. Because the droplet−surface interaction is weaker for pure water than that for PEO solutions, water droplets are elongated to a longer and more uniform shape at the same impact velocity (see Figure 5a). However, when the impact velocity is higher than a upper thresold value, VU, ranging from ∼0.27 m/s (We ≈ 1.2) for 0.5 g/L PEO solution to ∼1.2 m/s (We ≈ 21) for water, the pinning effect becomes so strong that the droplet cannot get depinned from the structured superhydrophobic surface anymore (Figure 3). As a result, no complete rebound was observed. Partial Rebound. The elongated droplet is thermodynamically unstable, and it minimizes its free energy under the action of surface tension, i.e., the Rayleigh−Plateau instability.54 It couples with the pinning effect described above leading to E


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Figure 6. Impact of droplets of diverse liquids on the superhydrophobic surface at high velocities. (a) 0.1 g/L PEO solution, V0 = 1.60 m/s. (b) 0.1 g/L PEO solution, V0 = 2.38 m/s. (c) 0.5 g/L PEO solution, V0 = 2.00 m/s. (d, e) Zoomed view of the droplet−surface contact region in (c). The scale bars are 1 mm.

splashing occurs, is ∼53 and ∼64 for water and 0.1 g/L aqueous PEO droplets, respectively. Deposition with BOAS-like Structures. The droplet− surface interaction becomes extremely strong at impact velocity higher than ∼0.61 m/s for 5 g/L PEO solution (We ≳ 5.8, Figure 3) and ∼0.95 m/s (We ≳ 14.7 5.8, Figure 3) for 10 g/L PEO solution. Following this, thick viscoelastic filaments were deposited on the superhydrophobic surface during drop recoiling (see 3.0−9.1 ms in Figure 6c and the zoomed view in Figure 6d). These filaments are not stable and subsequently break up, forming BOAS-like structures around the deposited droplet (denoted by white arrows in Figure 6e). The BOAS-like structures was also reported in a recent study about sliding droplets of aqueous polymer solutions on structured superhydrophobic surfaces.59 However, the formation process, which takes several to a few tens of milliseconds, has not been resolved with a high time resolution. 3.2. Contact Line Dynamics. Figures 7a and 7b show the temporal variation of the normalized contact radius RC/R0 for diverse impinging droplets at two impact velocities. RC was measured from the side view of the droplet as illustrated in Figure 4a. After the contact with the superhydrophobic surface, the inertial force compels the impinging droplet to spread out with a velocity of the same order as V0. The maximum spreading extent is reached within a few to several milliseconds depending on the impact velocity (indicated by black arrows in Figure 7a,b). Similar to observations on hydrophobic surfaces,27,28 we found that the spreading process is not dependent on the addition of PEO additive. The data of diverse droplets collapse onto one master curve for each impact velocity, suggesting the predominant role of inertial and capillary forces for the spreading dynamics. This finding is reasonable because aqueous PEO solutions are typical shear thinning liquids (Figure 2), and an impinging droplet undergoes strong shearing in the spreading phase. In contrast, the PEO additive siginificantly affects droplet retraction. The deposited PEO chains on the superhydrophobic surface enhance droplet−surface adhesion, and even cause the formation of liquid filaments, when the impact velocity or the PEO concentration is high enough (Figures 4f and 6d). These effects synergetically slow down the movement of the

partial rebound. For water and 0.1 g·L PEO solution, partial rebound occurs at high Weber numbers (We > 10, Figure 3). As shown in Figure 6a, the recoiling droplet is always accompanied by the formation of satellite droplets until it is torn into two parts. When the main part rebounds off, the other part pins on the surface (21 ms in Figure 6a). Occasionally, we found that the main part falls back to the surface, coalesces with the pinned one, and then completely rebounds. For 0.5 g/L PEO solution, partial rebound has been observed at low Weber numbers (We ≳ 1.2), and the formation of satellite droplets happens at We ≳ 11.5. In comparison to water and 0.1 g/L PEO solution, the elongated droplets undergo a typical non-Newtownian process before the breakup: first thinning and draining under the capillary force and then forming a periodic structure of microdroplets strung together, i.e., the BOAS structure as illustrated in Figure 2f. Splashing. If a droplet impacts onto a solid surface at very high velocity, splashing is frequently observed.24,25 Splashing is termed the formation of satellite droplets at the rim of the impinging droplet, and it can be generally classified into three categories,24−26 namely, prompt splash, corona splash, and receding splash. The onset of splash can be described by a dimensionless number K = We1/2Re1/4,25,55 where Re = ρR0V0/ μ is the Reynolds number. It is noted that this parameter does not consider all physical effects on the complex impact phenomenon. For example, recent studies have demonstrated that droplet splashing can be effectively inhibited by reducing the pressure of surrounding air.56−58 Nevertheless, K is sufficient to describe the role of physical properties of the impinging droplet on splash impact at atmospheric pressure and is very useful in engineering applications.24,25 On the structured superhydrophobic surface, we observed receding splashing at V0 ≳ 2.1 m/s (We ≳ 59) for water and at V0 ≳ 2.3 m/s (We ≳ 78) for 0.1 g/L PEO solution. As displayed in Figure 6b, the droplet is deformed into a thin sheet during spreading, and perimetric wavy fingers are developed at the periphery of the sheet (0.6−1.9 ms in Figure 4d). In the retraction phase, these fingers cannot follow the receding motion and thus pinch off, resulting in many small droplets with a domain size of several tens of micrometers (4.0 ms in Figure 4d). The corresponding threshold K, above which F


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Figure 8. A log−log plot of the maximum spreading factor βmax as a function of We for water and diverse aqueous PEO droplets.

4a. It is shown that experimental data of different liquids collapse onto one master curve. This further confirms the dominant role of the inertial and capillary forces on drop spreading. For capillary inertial impact, simple scaling laws have been developed to explain the correlation between βmax and We.24,25 If complete conversion of the kinetic energy of the impinging droplet into its surface energy takes place at the maximum spreading, a power law of βmax = A1We0.5 can be obtained at high Weber numbers,62,63 where A1 is a coefficient. Alternatively, if the droplet shape at the maximum spreading is modeled as a pancake flatterned by the inertial force, the growth of βmax depending on We can be described according to another power law, βmax = A2We0.25,64 where A2 is another coefficient. In this work, we found that the experimental data at We ≳ 40 can be fitted by βmax = 0.39We0.5 (dashed line in Figure 8) and those at We ≳ 10 follows βmax = We0.25 (dotted line in Figure 8). Data in this range could be the beginning of these scaling regimes, suggesting that the theoretical models can only capture part of the experimental results. However, a large deviation from these scaling models is observed at low Weber numbers. In particular, βmax approaches to a limit value of ∼1.05, while these models predict βmax = 0 as We → 0. To explain the complex, nonlinear relationship between βmax and We in Figure 8, we derive an empirical equation based on the modified energy analysis. When a droplet is gently deposited on the superhydrophobic surface, it beads up on the surface with an equilibrium spreading factor βeq ≈ 1.6[1/(1 − cos θeq)2(2 + cos θeq)]1/3, which is ∼1 for θeq = 150° (i.e., R eq ≈ R0 ). However, if the droplet impacts on the superhydrophobic surface with a certain velocity, the kinetic energy causes the droplet to deform by a quantity δ in contrast to its equilibrium radius at the maximum extent (inset in Figure 8), i.e., Rmax = Req + δ, and the change of surface energy scales as γδ2. Thus, the energy balance gives δ = A′We0.5, and the maximum spreading factor can be reformulated as

Figure 7. Temporal evolution of the normalized contact radius RC/R0 for five different droplets after the impact on the superhydrophobic surface at V0 = 0.24 m/s (a) and at V0 = 0.74 m/s (b). (c) Plot of the contact time tC as a function of We.

receding contact line. As a result, at the same impact velocity, aqueous PEO droplets retract slower than water droplets, and the retraction velocity decreases with the increase of PEO concentration. On the other hand, the droplets of the same liquids retract faster at a lower impact velocity. As described above, the presence of PEO macromolecules slows down droplet retraction and thus should increase the contact time of rebounding droplets on the superhydrophobic surface. Here, the contact time (tC) refers to the time that a droplet is in contact with the surface before rebound off.10 As shown in Figure 7c, tC is indeed longer for aqueous PEO solution droplets, and it increases with c. tC has a value of 10− 18 ms, which is still in the same order of magnitude with the characteristic capillary inertial time, τ = (ρR03/γ)0.5.10,60,61 Figure 7c also shows a nontrivial dependence of tC on We. It is obvious that tC increases with the decrease of impact velocity at We ≲ 0.6 (V0 ≲ 0.2 m/s) for all bouncing droplets due to the influence of the gravity.61 At higher We, tC of water droplets first approaches a plateau value of ∼2.6(ρR03/γ)0.5 and then starts to increase at We ≳ 10 (V0 ≳ 0.82 m/s) as a result of the pinning effect we discussed in the previous section. A similar phenomenon was also reported on other artifacial superhydrophobic surfaces.12 In contrast, for 0.1 and 0.5 g/L PEO solutions, droplet−surface interactions become stronger with increasing impact velocity and so does tC. 3.3. Maximum Spreading Factor. The dependence of the maximum spreading factor, βmax = Rm/R0, on the Weber number for all impinging droplets is summarized in Figure 8, where Rm is the maximum spreading radius defined in Figure

βmax = (R eq + δ)/R 0 ≈ βeq + A′We 0.5

(1)

where A′ is a coefficient. As shown in Figure 8, the above equation does well capture our experimental data with a fitting parameter A′ ≈ 0.22 (the solid line Figure 8). 3.4. Postimpact Oscillation. The postimpact oscillation is an indispensable phenomenon during droplet impact on solid G


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Figure 9. Temporal evolution of the flatness factor ζ for diverse droplets after impact at different velocities: (a) a water droplet at V0 = 0.14 m/s, (b) a 0.5 g/L aqueous PEO droplet at V0 = 0.11 m/s, (c) a 10 g/L aqueous PEO droplet at V0 = 0.10 m/s, and (d) a 10 g/L aqueous PEO droplet at V0 = 0.81 m/s.

oscillation system,49,67−69 which consists of mass, spring, and damper. The motion of the droplet can thus be described by the oscillation equation for underdamped harmonic oscillator70

surfaces. It determines surface coverage area and thus is critical for various applications associated with droplet solidification, e.g., design of anti-icing strategies for aircraft.65 On the superhydrophobic surface, we found that depositing droplets of water and aqueous PEO solutions at c ≲ 0.5 g/L oscillate in an underdamped manner. The droplet height H (defined in Figure 1b) or equivalently the dimensionless flatness factor, ζ = H/R0, gradually approaches to an equilibrium value (Heq or ζeq, see Figure 9). Similar oscillations were also observed for depositing droplets of aqueous PEO solutions at c ≳ 5 g/L at an impact velocity lower than ∼1.5 m/s (We ≲ 37) and 1.1 m/ s (We ≲ 19), above which ζ oscillates around a value that positively increases with time as a result of the slow drop retration caused by the formation of viscoelastic filaments. Figure 9a−c compares the oscillation behaviors of pure water and aqueous PEO solution droplets after the impact at low velocities. A water droplet oscillates at a frequency of ∼60 Hz on the superhydrophobic surface (inset in Figure 9a), which is about half that of a freely oscillating droplet with

ζ = ζeq + A e−(c /2m)t sin(ωt + φ)

(2)

where m is the droplet mass, c is the damping coefficient, ω=

k m

−

2

( 2cm )

is the angular freqency, and k is the spring

constant; A and φ are two coefficients. We further fitted the maxima of the oscillation amplitude (red circles in Figure 9) with an exponential function, ζ = ζeq + Ae−αt (red lines in Figure 9), from which the damping coefficient and spring constant were determined using c = 2mα and ÉÑ ÄÅ Å c 2Ñ k = mÅÅÅÅω 2 + 2m ÑÑÑÑ, respectively. As summarized in Figure ÑÖ ÅÇ 10, c is typically ∼10−4 N s/m and k has a value of 0.2−1.6 N/ m, which are in the same order of magnitude as those of droplets of low-viscosity Newtonian fluids.49,67 However, both c and k of aqueous PEO solution droplets are always larger than those of water droplets, and they increase with either higher PEO concentration or larger impact velocity. The latter finding is very different from the oscillating Newtonian droplets, which were found to be nearly constants on surfaces with similar wettability.49 The effect of PEO dissolved in water on the postimapct oscillation as well as different oscillation behaviors of aqueous PEO solution droplets and Newtonian droplets, i.e., the dependence of c and k on the impact velocity, can also be interpreted by the dynamic viscoelasticity of aqueous polymer solutions. In the oscillation theory, the damping coefficient is a parameter measuring the energy dissipation of the oscillation system and the spring constant characterizes the ability of the system to restore force or energy.70 For Newtonian fluids including water, liquid motion is resisted by the viscous forces within oscillating droplet, particularly by the elongational viscosity, which is 3 times the shear viscosity, and the capillary

( )

f0 = ( 2 /π ) γ /ρR 0 3 ≈ 122 Hz .66 This is because the traveling distance of the oscillation wave on a droplet suspended on the superhydrophobic surface is around twice of that on a freely suspended droplet in air. The oscillation can last for ∼1.5 s, which is about 3−7 times of that on the hydrophobic and hydrophilic surfaces.49 With the presence of PEO additive in water, droplet oscillation becomes faster and decays much earlier. For example, the oscillation frequency increases to ∼62 Hz for 0.5 g/L aqueous PEO solution droplets and ∼70 Hz for 10 g/L aqueous PEO solution droplets, while the oscillation duration shortens to ∼1.0 s (Figure 9b) and only ∼180 ms (Figure 9c), respectively. A similar trend was also identified by increasing the impact velocity (Figure 9c,d). To quantitatively analyze the postimpact droplet oscillation, we adopted the approach employed in most relevant studies to model the oscillating droplet as a single-degree-of-freedom H


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velocity is sufficiently high. Quantitative analysis of the contact line motion reveals that the dissolved PEO chains do not affect droplet spreading due to the shear thinning property of aqueous polymer solutions. The spreading dynamics is dominated by capillary and inertial forces, and the maximum spreading factor can be well described by an empirical equation obtained from a simple energy analysis. However, the presence of PEO chains significantly slows down droplet retraction and consequently increases the contact time of rebounding droplets. It was also found that liquid viscoelasticity siginificantly affect the postimpact droplet oscillation. The analysis of the oscillation using the standard model for underdamped oscillators suggests the strong effect of elongational viscosity on the damping coefficient, while the spring constant is influenced by the liquid elasticity.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (L.C.). *E-mail: [email protected] (K.Z.). ORCID

Longquan Chen: 0000-0002-6785-5914 Figure 10. Plot of the damping coefficient c (a) and the spring constant k (b) as a function of We. The dotted lines in the (a) and (b) indicate c and k for oscillating water droplets.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We acknowledge Duyang Zang and Shiji Lin for the rheology measurements. This work was supported by the National Natural Science Foundation of China (Grant 11772271), the National Young 1000 Talents Plan, the Young 1000 Talents Plan of Sichuan province, and the Sichuan Province Science Foundation for Youths (Grant 2016JQ0050). L.Q.C. and Q.Z. are also grateful to the Presidential Foundation of China Academy of Engineering Physics (Grant YZ2015010).

force is the only restoring force. Thus, one would expect that c and k of impinging Newtonian droplets on similar wettable surfaces should be independent of the impact velocity. This has been confirmed by a number of experimental works in the literature.49,67,69 In comparison, the polymer chains in water do not only increase its shear viscosity but also induce highly elongational viscosity, which markedly dissipates the kinetic energy of oscillating droplets. The elongational viscosity is caused by stretching the flexible PEO chains under the elongation stress, and it increases with the elongation rate.27,28 As a result, c of aqueous PEO droplets is higher than that of water droplets and increases with impact velocity as shown in Figure 10a. On the other hand, the stretched PEO chains generate another restoring forcethe elastic forcefor the oscillation, which leads to a higher k. The higher the impact velocity is, the longer the polymer chains are stretched and the stiffer the liquid oscillator becomes.

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REFERENCES

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4. CONCLUSIONS In summary, we systematically investigated the impact dynamics of pure water droplets and droplets containing PEO macromolecules on superhydrophobic surfaces constructed from hierarchically hydrophobic structures. It was found that droplets of aqueous PEO solutions at c ≲ 0.5 g/L exhibit similar dynamical behaviors as water droplets during impact, and various impact outcomes, including regular deposition, complete rebound, partial rebound and receding splashing, were well identified. However, PEO chains dissolved in the solutions enhance the droplet−surface interaction and liquid viscoelasticity and thus alter the transitional boundary of these phenomena in the phase diagram. With respect to aqueous PEO solutions at c ≳ 5 g/L, the strong droplet− surface interaction causes all impinging droplets sticking on the superhydrophobic surface, and the adsorbed viscoelastic filements lead to the formation of the beads-on-a-string-like structures around the depositing droplets, when the impact I


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Impact Dynamics of Aqueous Polymer Droplets on Superhydrophobic

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