Abstract
When a liquid drop lands on a solid surface without wetting it, it bounces with remarkable elasticity1,2,3. Here we measure how long the drop remains in contact with the solid during the shock, a problem that was considered by Hertz4 for a bouncing ball. Our findings could help to quantify the efficiency of water-repellent surfaces (super-hydrophobic solids5) and to improve water-cooling of hot solids, which is limited by the rebounding of drops6 as well as by temperature effects.
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The way in which a water drop of radius R deforms during its impact with a highly hydrophobic solid depends mainly on its impinging velocity, V. The Weber number, W = ρ V2R/γ, compares the kinetic and surface energies of the drop, where ρ and γ are the liquid density and surface tension, respectively. The greater the value of W, the larger are the deformations that occur during the impact (Fig. 1).
High-speed photography (Fig. 1) enabled us to measure the drop's contact time, τ. The frame rate could be greater than 104 Hz, allowing precise measurements of τ, which we found to be in the range 1–10 ms. As the impact is mainly inertial (with a restitution coefficient2 as great as 0.91), τ is expected to be a function of only R, V, ρ and γ, and thus to vary as R/V.f(W). For a Hertz shock, for example, the maximum vertical deformation, δ, scales as R(ρ2V4/E2)1/5, where E is the Young's modulus of the ball7. Taking a drop's Laplace pressure, E≈γ/R, as an equivalent modulus and noting that τ≈δ/V, we find for a Hertz drop that f(W)˜W2/5 and that the contact time varies as V−1/5 and R7/5.
Figure 2a shows that the contact time does not depend on the impact velocity over a wide range of velocities (20–230 cm s−1), although both the deformation amplitude and the details of the intermediate stages largely depend on it. This is similar to the case of a harmonic spring, although oscillations in the drop are far from being linear. Moreover, this finding confirms that viscosity is not important here.
Figure 2b shows that τ is mainly fixed by the drop radius, because it is well fitted by R3/2 over a wide range of radii (0.1–4.0 mm). Both this result and the finding shown in Fig. 2a can be understood simply by balancing inertia (of the order ρ R/τ2) with capillarity (γ/R2), which yields τ≈(ρ R3/γ)1/2, of the form already stated with f(W)˜W1/2. This time is slightly different from the Hertz time because the kinetic energy for a solid is stored during the impact in a localized region, whereas in our case it forces an overall deformation of the drop (Fig. 1).
The scaling for τ is the same as for the period of vibration of a drop derived by Rayleigh8, and is consistent with a previous postulation9, although the motion here is asymmetric in time, forced against a solid, and of very large amplitude. Absolute values are indeed found to be different: the prefactor deduced from Fig. 2b is 2.6 ± 0.1, which is significantly greater than π/√(√2≈2.2) for an oscillating drop8. Another difference between the two systems is the behaviour in the linear regime (W<<1): for speeds less than those shown in Fig. 2, we found that τ depends on V, and typically doubles when V is reduced from 20 to 5 cm s−1, which could be due to the drop's weight10.
The brevity of the contact means that a drop that contains surfactants, which will spread when gently deposited onto the solid, can bounce when thrown onto it; this is because the contact time is too short to allow the adsorption of the surfactants onto the fresh interface generated by the shock. Conversely, the contact time should provide a measurement of the dynamic surface tension of the drop.
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Richard, D., Clanet, C. & Quéré, D. Contact time of a bouncing drop. Nature 417, 811 (2002). https://doi.org/10.1038/417811a
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DOI: https://doi.org/10.1038/417811a
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