How a raindrop gets shattered on biological surfaces

Experiments.

We prepared various biological specimens including bird feathers, insects, and plant leaves (Materials and Methods). Then, a water drop is released to impact these biological surfaces. Here, the drop radius, R, and the impact velocity, U range from 1.1 to 2.0 mm and 0.7 to 6.6 m/s, respectively. The corresponding Weber number, We=ρU2(2R)/γ, ranges from 15 to 2,000, which covers the typical We of rainfall (15, 16). Here, ρ and γ are the density (= 1,000 kg/m3) and the surface tension (= 72 mN/m), respectively. The dynamics of drop impact were captured by a high-speed camera at a frame rate of 5,000 to 20,000 s−1 with a resolution of 1,024 × 672 pixels. Selected drop motions on these biological surfaces are illustrated in Fig. 1.

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Fig. 1.

(A–C) Biological surfaces for drop impact experiments (Left) and sequential events after the impact of a water drop of 1.7 mm in radius on the biological surfaces (from Left to Right). (A) Northern gannet feather, where [U, We] = [4.6 ms−1, 970]. (B) Cracker butterfly wing, where [U, We] = [4.2 ms−1, 810]. (C) Katsura leaf, where [U, We] = [5.4 ms−1, 1,340]. A–C, Center Left Insets represent the microscopic images of each biological surface achieved by a focus-stacking technique. A–C, Center Insets visualize shock waves on a spreading drop in the presence of physical morphology at the microscale. The spreading drop is suddenly ruptured by nucleating multiple holes (Center Right). Finally, the holes grow and coalesce with each other, thereby breaking into smaller satellite droplets (Right). Corresponding videos are included in Movie S1.

Drop Impact on Biological Surfaces.

Fig. 1A shows a drop impact on a bird feather, whose surface is superhydrophobic with surface roughness at different scales (17). Fig. 1A, Center Left Inset shows the hierarchical structure: Microscale barbules emerge from barbs attached to a millimeter-scale rachis. When a drop impacts the feather at a high speed (We ≈1,000), the liquid–air interface of the spreading drop is disturbed and generates hundreds of V-shaped, shock-like waves, as shown in Fig. 1A, Center Inset. A typical shock wave is observed when a compressible fluid experiences density discontinuity and forms a V-shaped wave pattern. In our case, the wave observed is not a classic shock wave since the fluid is incompressible at speeds in our experiments and exhibits uniform density. However, similar V-shaped waves result from discontinuity in film thickness of the spreading drop instead of density variations (Fig. 1, Center Insets). Therefore, even though the fluid is not compressed, we term the wave pattern “shock-like waves” in this present work. Similar shock-like waves have been observed as a wake structure on a liquid curtain behind an external perturbation (18) or an oblique liquid flow in a soap film (19). Then, the spreading drop is abruptly shattered/fragmented (Fig. 1A, Right) shortly after rupturing the liquid film and nucleating holes (Fig. 1A, Center Right). A similar morphological transition of an impacting drop was also observed for other biological surfaces such as insect wings and plant leaves as in Fig. 1 B and C (Movies S2 and S3). All these specimens have hierarchical superhydrophobic structures, where an array of micrometer-sized bumps exists with nanoscale structures (20, 21).

Drop Impact on Artificial Surfaces.

To further investigate details of the shock-like wave structure, we prepared two types of substrate with different wettability. One is a hydrophilic glass surface, and the other is a superhydrophobic surface coated by hierarchical micro- and nanostructures (Materials and Methods). On the smooth hydrophilic substrate, a drop merely spreads by forming a radially expanding rim as in Fig. 2A. On the superhydrophobic glass, a drop spreads, retracts, and bounces at a low-impact speed (Fig. 2B), whereas a drop with a high-impact speed exhibits shock-like surface waves and destructive breakup dynamics as in Fig. 2C.

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Fig. 2.

Bottom-view images of an impacting drop with 1.7 mm in radius on artificial surfaces. (A) Drop impact on a smooth glass for high-impact velocity, where [U, We] are [3.8 m/s, 680]. (B and C) Drop impact on a glass coated by hierarchical superhydrophobic structures (type I) for low-impact velocity (B) and for high-impact velocity (C). Corresponding [U, We] are [0.9 m/s, 40], [3.8 m/s, 680], respectively. Insets in B represent the side-view image of drop impact achieved by a synchronized high-speed camera. Insets in C are magnified images of the local area to clearly show the microbump, the shock-like wave on the microbump, and the nucleated hole followed by the shock-like wave, respectively. For low U, the impacting drop merely spreads, retracts, and rebounds, whereas for high U, hundreds of shock-like waves are generated on the spreading drop (Center Left), and then a number of holes are abruptly nucleated (Center), which grow in time simultaneously (Center Right). Finally, the contact time between the drop and the substrate gets shortened by breaking into smaller satellite droplets (Right). (D) Similar dynamics were observed on a different superhydrophobic surface with regularly spaced microbumps (type III), where [U, We] are [3.8 m/s, 680]. The corresponding videos are in Movies S4 and S5, respectively.

At a high U, hundreds of shock-like surface waves were generated in the presence of microscale bumpy structures as in Fig. 2C, Center Left. A drop spreads and then is suddenly ruptured, as holes are nucleated (Fig. 2C, Center). Eventually, the spreading drop is shattered into smaller satellite droplets (Fig. 2C, Right) as the holes get bigger and coalesce. Similar dynamics were observed on a micropatterned surface (type III) with constant spacing and height of bumps as in Fig. 2D. Therefore, we confirmed that, when a drop impacts hierarchical superhydrophobic surfaces with a high-impact velocity, microscale bumpy structures can perturb the spreading drop to generate the numerous shock-like waves on the liquid–air interface and eventually break into smaller droplets.

Such spreading and fragmentation behaviors greatly lower the residence/contact time of a drop on a solid substrate. For a typical bouncing drop (Fig. 2B) with low-impact velocities, the measured contact time (≃18 ms) almost follows the theoretical capillary contact time, 19 ms, estimated from τ=2.3ρR3/γ (6). However, the shock-induced drop fragmentation leads to a more than twofold decrease in the contact time compared to the theoretical expectation as shown in Fig. 2.

Shape of Shock-Like Waves.

Fig. 3A shows the schematic of how a shock-like wave is generated above a microbump on the surface as a drop spreads. Experimentally, we confirmed that the half angle of the shock-like wave increases with an elapsed time (Fig. 3B, Insets) and decreases with the radial distance from the center to a bump rb. Fig. 3C clearly shows the dependency of ψ on t and rb for various hierarchical superhydrophobic surfaces.

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Fig. 3.

(A) Schematic of the formation of shock-like waves by microbumps, where rb, u, uw, and ψ represent the radial distance of the microbump measured from the drop center, the flow velocity of the spreading drop, the speed of wave propagation, and the half angle of the shock wave, respectively. (B) Sequential images of the shape changes of shock waves after the drop of 1.7 mm in radius impacts an artificial superhydrophobic surface (type I) at U=4.2 m/s (We = 826). Insets show the magnified view of the shock-like wave. (C) Experimentally measured ψ versus t, where the size of symbols is proportional to the microbump location, rb. Hence, the larger symbols mean the farther distance from the center. (D) Half angle, ψ, plotted vs. the theoretical expectation based on Eq. 1.

In analogy to the Mach shock wave, the half angle of the shock-like wave, ψ, might be determined by the distance ratio of uw t to u t; sin⁡ψ=uw/u=1/Ma*. Here, uw is the speed of wave propagation, u is the fluid velocity (18, 19), and Ma is the equivalent Mach number as Ma*=u/uw.

Here, u can be approximated by r/t with r being the radial distance from the drop center. This model for u was suggested in the case of a spreading liquid sheet in the air after the drop impact on a small target object, a so-called drop-impact version of the Savart sheet (22⇓–24). Although a drop spreads on a solid substrate in our case, we confirmed that the relation u≈r/t is in good agreement with experiments due to negligible viscous stress on a superhydrophobic substrate (model validation in SI Appendix, section A). For the wave propagation speed, uw, we assumed that the wave propagates under the capillary action, leading to uw=2γ/(ρh), where γ and h are the surface tension and the film thickness of a spreading drop, respectively (18, 19, 25). Hence, we express ψ assin⁡ψ=1Ma*=2γ/(ρh)r/t.[1]Here, the film thickness, h(r,t), can be approximated as R3/(U2t2)(Ut/r)n, where n=3 in the early stage (23, 25), 2 in the middle stage (23, 24), and 0 in the late stage of a spreading drop (validation in SI Appendix, section B). Based on Eq. 1 along with this film-thickness model, the scattered data in Fig. 3C are collapsed onto a single line in Fig. 3D.

Hole Nucleation Criterion.

Above a critical impact velocity, Uc, a number of shock-like waves form and interfere with each other. As a result, the shock-like waves create the nonuniform film thickness of a spreading drop with a certain amplitude, |η|, as depicted in Fig. 4A, thereby leading to a wrinkled pattern on the spreading drop. Then, we observed the sudden formation of holes/ruptures at the same locations where the shock-like waves are formed during the drop-receding stage (Fig. 2 C and D, Center). For this later drop-receding stage, the film thickness is independent of the radial position; h(t)∼R3/(Ut)2, as delineated above. This shock-like wave decreases and propagates upstream toward the drop center (Fig. 4A and SI Appendix, section C) since uw≫u at the drop-receding stage. Then, the liquid film is ruptured as the trough of the perturbed film touches the top of a microbump (Fig. 4A, Right). This scenario suggests the following threshold criterion, h(t)−|η|≈ϵ, with ϵ being the height of the microbump. Fig. 2 C and D shows that the rupturing hole is nucleated on top of the bump apex, which validates our assumption.

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Fig. 4.

(A) Schematic of shock-induced liquid–film rupture due to superhydrophobic microstructures. (B) Critical time for hole nucleation, thole, versus impact velocity, U. The dashed line represents the spreading time, which is defined as the time for the drop to reach its maximum radius. This spreading-time line is from a quadratic regression fit based on experimentally measured spreading times for R=1.7 mm. (C) Experimental data in B replotted against our theoretical models. The solid line is obtained as the upper limit of thole (Eq. 3), where the slope of the best-fitting line is 1.4. The dashed lines are determined by the lower limit of thole as in Eq. 2, where the slopes of the best-fitting lines are 1.4[1+aρ/(γk)]−1/2. Here, the upper and lower dashed lines represent the maximum and minimum values among our surfaces, which correspond to butterfly I and bird feather, respectively. (D) Critical impact velocity, Uc, for different biological and artificial surfaces, which can be characterized by the height of microstructures, ϵ. (E) Experimental data in D replotted based on Eq. 4, where the slope of the best-fitting line is 1.8.

Hole Nucleation Timescale.

Now, we estimate the hole nucleation time, thole, defined as the time interval between when a drop impacts and when the first hole is formed. Fig. 4B shows that thole decreases with the impact velocity, U. Here, we consider two limit cases. First, the impact velocity is well above the critical velocity (U≫Uc), and holes are nucleated on a spreading drop when the drop-spreading radius is close to its maximum radius (Fig. 2 C and D). In this case, the spreading drop is observed to have highly wrinkled patterns, indicating that the amplitude of the perturbed interface, |η|, is large enough to play a significant role in rupturing the film. To estimate the undulation amplitude, we used the unsteady Bernoulli equation to get the dispersion relation of shock-like waves; |η|=αϵuρ/(γk), where α and k represent a constant coefficient and wavenumber, respectively (see SI Appendix, section D in detail). Then, by using tspread(≈16R/(3U)) (26) for the characteristic timescale of |η| and imposing a geometric threshold condition for the hole nucleation (h(t)−|η|≈ϵ), we obtain the asymptotic solution for thole asthole≃1+aργk−1/2ϵ−1/2R3/2U−1 for U≫Uc,[2]where a=(3α/16)UWe1/4.

Second, when U≈Uc, a single hole appears just before the drop rebounds (Movie S6). Thus, the hole formation time, thole, gets close to the capillary contact time, τ=2.3ρR3/γ (6). In this limit case, a shock-like wave on a microbump has a small amplitude and is dissipated quickly to become quite a smooth interface. By assuming |η|≈0 (i.e., α≈0) in Eq. 2, we get the upper limit of thole asthole≃ϵ−1/2R3/2U−1 for U≈Uc.[3]The dispersed data of thole in Fig. 4B are collapsed between the two theoretical limits (Eqs. 2 and 3) as shown in Fig. 4C. Here, the upper and lower dashed lines for Eq. 2 are from variations in k values in biological samples (Table 1): The lower and upper limits are based on butterfly I and feathers, respectively.

Finally, we can determine the critical impact velocity, Uc. At U≈Uc, a spreading drop starts having a hole formed just before the drop bounces off. It indicates that the hole formation time in Eq. 3 equals to the capillary contact time, τ. Then, we obtained an expression of the threshold impact velocity asUc∼γρϵ.[4]The experimental data in Fig. 4D are well collapsed onto the single line in Fig. 4E based on our theoretical model, Eq. 4. Also, it is worth noting that this model predicts the independence of Uc on R (see SI Appendix, section E in detail).

Drop Fragmentation.

For U>Uc, once holes are nucleated on a spreading drop, the holes rapidly expand to lower its surface energy over hydrophobic nanostructured surfaces. On superhydrophobic surfaces, the holes expand at a speed of vh≈2γ/(ρh), which is known as the Taylor–Culick equation (27⇓–29). In our experiments, the holes rapidly grow at vh∼1 m/s until they coalesce with each other, which finally leads to the bursting of the spreading drop as shown in Figs. 1 A–C and 2 C and D.

Decrease in Contact Time.

We measured the contact distance of a spreading drop, d, defined as the minimum distance between the drop center and the contact line on either the outer rim or a nucleated hole. Also, the contact time, tcontact, was measured as the time interval between when a drop impacts and when the drop bounces off the substrate. For U≤Uc, the contact time is approximate to the capillary contact time, τ=2.3ρR3/γ, and the contact distance smoothly expands and retracts on the surface (Fig. 5A, open circles).

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Fig. 5.

(A) Temporal evolution of dimensionless contact distance d/R on artificial surface I with different impact velocities, U, for R=2 mm. For U<Uc, an impacting drop symmetrically spreads and rebounds without forming a hole. For U>Uc, a rupturing hole starts forming during the drop-receding stage, which leads to the sudden decrease in d (the dashed lines in U=2.1 m/s). For U≫Uc, multiple holes nucleate earlier, thereby leading to the dramatic decrease in d and contact time. (B) Experimental contact time, tcontact, versus impact velocity, U, where the dashed line indicates the critical impact velocity, Uc. (C) Dimensionless contact time, tcontact/τ, versus impact velocity, U, where the contact time of the impacting drop starts to decrease beyond the threshold impact velocity, Uc (dashed line).

For U>Uc, the contact distance decreases discontinuously as several holes are nucleated (Fig. 5A and SI Appendix, section F). Through a series of hole coalescence, the drop bounces off from the surface quickly. Therefore, the contact time is significantly reduced as shown in Fig. 5B. The contact time decreases up to 70%, compared to its capillary contact time, τ (Fig. 5C). The upper and lower limits of tcontact/τ are analytically evaluated by considering the aforementioned characteristics of hole nucleation (SI Appendix, section F).