Promoting rebound of impinging viscoelastic droplets on heated superhydrophobic surfaces

The promotion of the rebound of viscoelastic droplets on superhydrophobic surfaces via surface heating can be readily demonstrated by comparing the impact process and phenomena of 0.1 g l⁻¹ PEO droplets with the same velocity of V₀ ≈ 0.29 m s⁻¹ at three different surface temperatures (T = 25 °C, 50 °C and 95 °C). After contacting the surface at 25 °C, the droplet spreads out with a violent capillary wave travelling from its bottom to top, resulting in complex deformations as that of low-viscosity Newtonian droplets [38, 39]. As a consequence, a pancake-like structure with a spire, as shown at ∼3.3 ms in figure 1(a), is formed at the maximal extension. For low-viscosity Newtonian liquids such as water, the spire subsequently punches deeply into the droplet due to its high velocity [38] and thus a cylindrical air cavity is created at the droplet center [40, 41]. With the recoiling of the droplet, the air cavity would be squeezed in the direction normal to the surface, and it may coalesce with the air film or bubble formed below the impinging droplet upon impact, which eventually suppresses droplet rebound on superhydrophobic-like soft surfaces since the liquid wets the surface [42, 43]. By contrast, if polymer chains are added in water, the downward motion of the spire can be significantly damped by the extensional viscosity near the droplet center, inhibiting droplet–surface contact [28]. In our experiments, this damping effect was also identified on superhydrophobic surfaces and it is more pronounced for impinging droplets with high PEO concentrations as the created air cavity is much shallower than of pure water droplets reported in previous studies [40–43]. Meanwhile, numerous viscoelastic filaments were observed near the contact line of the recoiling droplet (illustrated by green arrows at 9.5 ms in figure 1(a)). These filaments are anchored on the superhydrophobic surface and successively pulled out from the droplet once the liquid detaches (supplemental movie S1) [24]. This can be more clearly inferred from the scanning electronic microscopy (SEM) images of polymer residue after impact in figure 1(d), where numerous PEO nanofibers (denoted by green arrows) are deposited around the impact point. With droplet recoiling, they are further stretched, and eventually break up when their maximum limits are reached. Although the droplet can rebound off from the surface, the residual viscoelastic filaments restrict it to resume the spherical shape and be highly lifted up. The spreading characteristics described above were also observed for impinging PEO droplets on heated superhydrophobic surfaces at 50 °C and 95 °C, as evidenced by the snapshots in figures 1(b) and (c) and also the temporal evolution of the droplet contact radius (R_c) in figure 1(e). However, distinct droplet behaviors were found in the retraction stage. At any given time the formed viscoelastic filaments around the recoiling droplet are less on a hot surface than that on a cold surface (see 6.7–9.5 ms in figures 1(a)–(c), and supplemental movies S1–S3), and the corresponding retraction process is relatively faster (figure 1(e)). Consequently, on a hotter surface the droplet bounces off earlier with a shape comparably more close to a sphere, and the maximum height to which it can reach is relatively higher, particularly on the surface at 95 °C (figure 1(c)).

Zoom In Zoom Out Reset image size

Figure 2 plots the determined lower (V_L) and upper (V_U) threshold velocities of droplet rebound for those three aqueous PEO solutions we investigated. It is seen that the range of the impact velocity or equivalently the Weber number for rebound becomes wider either by increasing surface temperature or by decreasing polymer concentration. More specifically, V_L of all aqueous PEO solutions decreases with increasing T or decreasing c_PEO, while V_U shows an increase trend when varying T or c_PEO but less obvious.

Zoom In Zoom Out Reset image size

The nontrivial dependence of the threshold velocities for rebound on the surface temperature and PEO concentration can be understood by an interfacial force analysis. The impact process and outcomes in figures 1(a)–(c) and 1(e) suggest that it is the viscoelastic filaments formed near the contact line leading to a large resistance force, slowing down droplet retraction, and thus determining the thresholds for rebound. Modeling each filament as a simple spring–mass–damper system (i.e. a point mass connected to a purely viscous damper and purely elastic spring in parallel), we derive the resistance force arising from stretching from the equation of motion [44], ${f}_{\mathrm{r}}=c\dot {L}+kL+m\ddot {L}$. Here L is the length of the filament and is of the order of the maximum spreading radius of the impinging droplet R_max (figure 1(e)); $\dot {L}$ represents the stretching speed and should be of the order of V₀; $\ddot {L}$ denotes the acceleration rate and scales as ${V}_{0}^{2}/{R}_{0}$; c and k are the damping coefficient and the spring constant, both of which increase with increasing the impact velocity and the PEO concentration [24]; m is the mass of the liquid filament, which can be calculated by $\rho {R}_{\text{filament}}^{2}L$ by assuming the liquid filament takes a cylindrical shape, where ${R}_{\text{filament}}\approx \sqrt{\rho /{c}_{\text{PEO}}}{R}_{\text{fiber}}\approx 9-85\enspace \mu \mathrm{m}$ is the filament radius and R_fiber being the radius of the residual PEO nanofibers measured from the SEM image (figure S4). Substituting typical values of observed viscoelastic filaments (R_max ∼ O(1.0 mm), V₀ ≈ 0.1–0.5 m s⁻¹, m ∼ 10⁻¹⁰–10⁻⁸ kg, c ∼ 10⁻⁴ Ns m⁻¹ and k ∼ 1 N m⁻¹ [24]), one finds that the viscous and inertial forces are at least 2 orders of magnitude smaller than the elastic force, and thus can be neglected. We further assume that the anchoring sites of viscoelastic filaments on the nanostructured superhydrophobic surface are uniformly distributed, then the total resistance force for the recoiling droplet can be expressed as, F_r ≈ 2πnf_r R_max ² ≈ 2πnkR₀ R_max ², with n being the anchoring site number per unit area. It is noted that increasing the number of polymer chains in the solution or the liquid–solid contact area would cause a larger n and F_r. We also point out that the viscous force within the droplet [38, 45], which scales as πμR_max ² V₀/H_max, is another resisting source of droplet motion, where μ is the dynamic viscosity and H_max is the height of the droplet at maximum spreading. However, this term is also much smaller than the elastic force of viscoelastic filaments, and its effects on droplet motion can be ruled out as well. On the other hand, the driving force for droplet recoiling is the capillary force that originates from the impact-induced deformation [46], and it should be proportional to the inertial force of the impinging droplet, i.e. ${F}_{\mathrm{c}}\propto \rho {R}_{0}^{2}{V}_{0}^{2}$.

On the superhydrophobic surface at 25 °C, an impinging droplet is slightly deformed in the spreading stage (R_max ∼ R₀) at low impact velocity, and it only touches the top of microscopic surface structures during the whole impact process (i.e., it keeps an ideal Cassie state and n is almost a constant) [13, 24]. Therefore, balancing F_r with F_c yields the lower threshold velocity for rebound

$$\begin{equation}{V}_{\mathrm{L}}\sim \sqrt{\frac{2\pi nk{R}_{0}}{\rho }}.\end{equation} \tag{ 1 }$$

Based on the above equation, one can find that an increase in PEO concentration leads to a larger n and k, and thus a higher V_L, agreeing well with the experimental data in figure 2. With the increase of the impact velocity, droplet deformation becomes significant, resulting in a larger spreading extent (R_max > R₀). Meanwhile, the liquid would impale into the microscopic surface structures if the wetting pressures exceed the anti-wetting pressure [10, 12, 13]. The wetting pressures are the liquid hammer pressure P_H ≈ 0.2ρCV₀ [47, 48], which lasts only a few microseconds on a square-micrometer-sized region around the liquid–solid contact point, and a dynamic pressure ${P}_{\mathrm{D}}\approx 0.5\rho {V}_{0}^{2}$ [10, 13], which exerts on the droplet–surface contact area for the whole spreading process, where C is the sound speed in the liquid. Although the duration of the liquid hammer pressure is much shorter than the dynamic pressure, it has been demonstrated to play a dominant role for the transition from complete rebound to sticky state during the impact of liquid droplets on structured superhydrophobic surfaces [13, 49–51]. The anti-wetting pressures are the capillary pressures caused by the microprotrusions ${P}_{\mathrm{C}\mathrm{M}}=-2\sqrt{2}\gamma \enspace \mathrm{cos}\enspace {\theta }_{a-f}/{S}_{\mathrm{M}}$ and nanoprotrusions ${P}_{\mathrm{C}\mathrm{N}}=-2\sqrt{2}\gamma \enspace \mathrm{cos}\enspace {\theta }_{a-f}/{S}_{\mathrm{N}}$ with θ_a−f being the advancing contact angle on the flat surface and S_i being the distance between two neighboring protrusions. By comparing these terms, we found that the impalement occurs on microprotrusions at V₀ ≳ 0.02 m s⁻¹ (S_M ∼ 10 μm, θ_a−f ≈ 110°, C ≈ 1 495 m s⁻¹) and on nanoprotrusions at V₀ ≳ 0.40 m s⁻¹ (S_N ∼ 500 nm, figure S1), which enlarges the liquid–solid contact area, and thereby the number of filament anchoring sites n. As a result, the droplet rebound is inhibited above an upper threshold velocity V_U of 0.45–0.52 m s⁻¹ (figure 2), which is slightly higher than the impalement velocity on nanoprotrusions, and V_U increases with increasing c_PEO.

We want to point out that when an impinging droplet approaches the target surface, the compression of the air between them would lead to the buildup of a large lubrication pressure, which deforms the droplet bottom into a dimple shape [46]. As a consequence, an air layer is always entrapped below the impinging droplet after contact with the surface, and it subsequently contrasts into an air bubble under the action of surface tension. Although surface roughness affects the initial contact between the droplet and surface [36], the shapes of the finally entrapped bubbles are barely influenced [52]. The entrapment of the air bubble has been theoretically investigated by Mandre et al [53], and experimentally resolved in finer details by van der Veen et al [54], and Li and Thoroddsen [55] on flat hydrophilic surfaces, and by van der Veen et al [52] and Langley et al [36] on structured superhydrophobic surfaces. The maximum pressure during the bubble formation ${P}_{\mathrm{max}}=0.88\frac{{R}_{0}{V}_{0}^{7}{\rho }^{4}\mathrm{C}{\mathrm{a}}^{1/3}}{{P}_{0{\mu }_{\mathrm{a}}}{R}_{0}^{1/2}\mathrm{S}{\mathrm{t}}^{7/9}}$ [56] serves as another anti-wetting pressure to prevent the liquid impalement, where Ca = μ_a V₀/γ is the capillary number with μ_a being the viscosity of air, and St = μ_a/ρV₀ R₀ is the Stokes number. However, P_max (∼3.0 × 10⁹ Pa) at the upper threshold velocity for rebound is 7 orders of magnitude higher than P_D (∼101–135 Pa) and 4 orders of magnitude higher than P_H (∼1.4–1.6 × 10⁵ Pa). This suggests that the entrapped air bubble would stay below the impinging droplet during the whole impact process, separating the liquid from the surface in a small region near the impact point (which has been experimentally observed in previous studies [36, 52, 56, 57]), and the liquid impalement (either in microprotrusions or in nanoprotrusions) triggered by the hammer pressure should occur at the rim of the entrapped bubble. Indeed, it was found that the residual PEO nanofibers after impact in figure 1(d) highly accumulate at a distance of ∼50 μm from the impact point, which is in the same order of magnitude with the bubble size L_b ∼ R₀St^4/9 ∼ 10 μm [58, 59], indirectly confirming the above analyses.

For impinging droplets on superhydrophobic surfaces at higher temperatures, local heat transfer happens at the solid–liquid contact area, and it is described by the heat conduction equation [60], $\frac{\partial T}{\partial t}=\alpha \frac{{\partial }^{2}T}{\partial {y}^{2}}$, where t is the time, α is the thermal diffusivity of the liquid and y is the distance of the liquid to the solid surface. As the impact process is radially symmetric, the one-dimensional heat conduction equation is thus employed for the analysis. A non-dimensional analysis of the equation provides a characteristic thermo-diffusion length ${L}_{\mathrm{T}}=\sqrt{\alpha \tau }$, in which the characteristic time τ should be on the timescale of droplet impact $\sim {\left(\rho {R}_{0}^{3}/\gamma \right)}^{0.5}$ [46]. We found that L_T is about 25 μm (α = 0.146–0.164 × 10⁻⁶ m² s⁻¹, τ ≈ 3.8 ms), which is comparable to the filament sizes (9–85 μm, figure S4) of diverse polymer solutions. It causes the fast evaporation of the water in viscoelastic filaments, significantly reducing their tensile strength, and thus they can easily break up during droplet recoiling (figures 1(a)–(c)). Consequently, a decrease of V_L but an increase of V_U with increasing T was identified in the experiments, regardless of the PEO concentration of impinging droplets (figure 2).

Among diverse characteristics of bouncing droplets, the contact time t_c (i.e. the residence time of an impinging droplet after contact and before taking off the surface [61]) is a key parameter inferring the droplet–surface interaction during impact [17, 46]. For low-viscosity Newtonian liquids such as water, t_c would increase at an impact velocity lower than a critical value, above which it approaches an asymptotic value of ${\tau }_{\mathrm{c}}\approx 2.6{\left(\frac{\rho {R}_{0}^{3}}{\gamma }\right)}^{0.5}$ [24, 61–63]. In figure 3, we comparatively show the contact time of bouncing droplets of 0.1 g l⁻¹ and 1.0 g l⁻¹ PEO solutions on superhydrophobic surfaces at three different surface temperatures. Evidently, t_c of the same liquid droplets is shorter on hot surfaces than that on cold surfaces at similar impact velocities. By comparison, on superhydrophobic surfaces of same temperature, droplets of high PEO concentrations stay longer than droplets of low PEO concentrations before rebound (figure 3 and figure S5). Similar to low-viscosity Newtonian droplets [43, 63], t_c of aqueous PEO droplets was found to increase with decreasing V₀ at low impact velocities, and this trend stops at a critical velocity (∼0.27 m s⁻¹), which is close to that of pure water droplets (∼0.26 m s⁻¹) [24, 49]. However, at higher velocities, t_c first starts to gradually increase (from a value close to τ_c, figure 3) with V₀, and then an obvious increase is observed at V₀ ≳ 0.35 m s⁻¹, which corresponds to the impalement velocity of liquids into the nanoprotrusions of the superhydrophobic surfaces. By decomposing the impact process into the spreading and retraction stages, one can clearly see that the spreading time (t_s) continuously decreases with the increase of V₀ and the experimental data can still be well described by an empirical power-law for Newtonian droplets [38], ${t}_{\mathrm{s}}/{\left(\rho {R}_{\mathrm{max}}{\enspace }^{3}/\gamma \right)}^{0.5}=\xi \mathrm{W}{\text{e}}^{\beta }$, while the retraction time t_r shows a similar dependence on V₀, T and c_PEO as t_c, where ξ is a coefficient and β is the exponent. This finding further demonstrates that the effects of surface heating on bouncing viscoelastic droplets occur in the retraction stage as a consequence of the reduced droplet–surface interaction.

Zoom In Zoom Out Reset image size

To quantitatively characterize the rebound phenomena, we have measured the restitution coefficient ɛ—another parameter reflecting the droplet–surface interaction in the dynamic conditions [11, 49, 64], and results of those two solutions in figure 3 are plotted in figure 4. Due to their nonspherical shapes (figures 1(a)–(c)), we traced the motion of the center of gravity of impinging droplets, and the restitution coefficient is defined as the ratio of the gravitational potential energy at the highest point of rebound and initial kinetic energy, i.e. $\varepsilon =2g{h}_{\mathrm{max}}/{V}_{0}^{2}$, where g is the gravitational acceleration and h_max is the height of the droplet gravity center with respect to the superhydrophobic surface. Obviously, ɛ is much smaller than 1 and decreases with increasing V₀ or We for all bouncing droplets, implying strong energy dissipation during impact. The maximum restitution coefficient, which is found at the lower threshold velocity for rebound, decreases from ∼0.3 for 0.1 g l⁻¹ PEO solution to ∼0.15 for 1.0 g l⁻¹ PEO solution on superhydrophobic surfaces at 25 °C (figure 4 and figure S6). However, by increasing surface temperature, ɛ becomes larger for each aqueous PEO solution and a value up to ∼0.7 is observed for 0.1 g l⁻¹ PEO solution at 95 °C. Most importantly, we found that the correlation between ɛ and We can be fitted by a linear function, ɛ = A − BWe, with A and B being two coefficients. The linear behavior is more pronounced for impinging droplets on superhydrophobic surfaces at T ≲ 50 °C and for impinging droplet at We ≳ 2.0 on superhydrophobic surfaces at T = 95 °C, regardless of the PEO concentration.

Zoom In Zoom Out Reset image size

We elucidate the inversely linear dependency of ɛ on We using a simple scaling argument. As discussed above, the main source dissipating the kinetic energy (E_k) of the impinging droplet is the viscoelastic resistance of the formed filaments, and the dissipated energy scales as ${E}_{V-E}=\iint \mathrm{d}{f}_{\mathrm{r}}\enspace \mathrm{d}L\approx {\iint }_{0}^{{R}_{\mathrm{max}}}2\pi nk{R}_{\mathrm{c}}L\mathrm{d}r\enspace \mathrm{d}L\approx 0.5\pi nk{R}_{\mathrm{max}}^{4}$. Since the region over which the liquid hammer pressure applied is much smaller than that of the dynamic pressure, n is thus mainly determined by the wetting of surface microprotrusions under P_D . Making reasonable assumptions that these surface microprotrusions are constructed by columns of nanoparticles and the impalement depth of liquid into the microprotrusions increases with the dynamic pressure, one finds that the actual solid–liquid contact area [65] and thus the anchoring sites of viscoelastic filaments should be positively proportional to the Weber number, i.e. n ∝ We. Combining these correlations with the spreading law of impinging droplets on non-wetting surfaces, R_max ∝ R₀We^0.25 [46], we obtain ${E}_{V-E}\propto k{R}_{0}^{4}\mathrm{W}{\mathrm{e}}^{2}$, and the restitution coefficient can be expressed as

$$\begin{equation}\varepsilon \approx \frac{{E}_{\mathrm{k}}-{E}_{V-E}}{{E}_{\mathrm{k}}}\propto -\mathrm{W}\mathrm{e}.\end{equation} \tag{ 2 }$$

It is noted that droplet rebound only happens in an impact event that the input kinetic energy overweights the viscoelastic dissipation energy in the dynamic process. Therefore, the restitution coefficient would take a general form of ɛ = A − BWe as found in the experimental study for impinging droplets at T ≲ 50 °C. At surface temperature of 95 °C, the formed viscoelastic filaments quickly break up during recoiling, which significantly lessens the resistance force and thus leads to a high ɛ. As such, the linear fitting of the experimental data is compromised, especially for impinging droplets at We ≲ 2.0.

The promoted rebound of impinging droplets will reduce liquid retention on solid surfaces, and thus affect technological processes associated with droplet impact, e.g. pesticide deposition in agricultural spray [15, 16] and heat transfer in spray cooling [66]. To illustrate the influence of surface temperature on the deposition of aqueous PEO solutions on superhydrophobic surfaces, we further performed oblique droplet impact experiments, which are more commonly encountered in real-world applications [17, 46]. As displayed in figure 5(a) and supplemental movie S4, an impinging droplet of 0.1 g l⁻¹ PEO solution with V₀ ≈ 0.20 m s⁻¹ rebounds two times on the inclined superhydrophobic surface (with tilting angle of ∼13°) at 25 °C, and it gets stuck on the surface at ∼3 mm away from the impact point. Although only two times of droplet rebounds were observed on the surface at 50 °C, the droplet is moved ∼5 mm due to its larger restitution coefficient (figure 4(a)) and then starts to roll down rather than stops moving (figure 5(b) and supplemental movie S5). In comparison, the number of droplet rebounds is increased to four times on the superhydrophobic surface at 95 °C, and the corresponding displacement distance along the surface is ∼12 mm (figure 5(c) and supplemental movie S6). Similar droplet behaviors were also found for other aqueous PEO solutions. These phenomena indicate an enhancement of the mobility of viscoelastic droplets on superhydrophobic surfaces by surface heating, which would reduce liquid deposition efficiency.

Zoom In Zoom Out Reset image size