Contact time of a bouncing drop

Transcription

1 Contact time of a bouncing drop Denis Richard, Christophe Clanet (*) & David Quéré Laboratoire de Physique de la Matière Condensée, URA 792 du CNRS, Collège de France, Paris Cedex 05, France (*) Institut de Recherche sur les Phénomènes Hors Équilibre, UMR 6594 du CNRS, BP 146, Marseille Cedex, France

2 When a drop impacts a solid without wetting it, it can fully bounce, with a remarkable elasticity 1-3. We measure how long the drop remains in contact with the solid during the shock, a question which is reminiscent of the famous Hertz problem 4 (contact time of a solid ball bouncing on a solid), but raised here for a liquid ball. Our findings may help quantifying the efficiency of so-called water repellent surfaces (super-hydrophobic solids 5 ), and improving the cooling of hot solids, for which drop rebounds are a severe limit 6 (even if the problem can obviously be complicated in this case by temperature effects). The way a water drop of radius R deforms during its impact on a highly hydrophobic solid principally depends on its impinging velocity V. The Weber number W = rv 2 R/g compares the kinetic and surface energies of the drop, noting r and g the liquid density and surface tension; the higher W, the larger the deformations during the shock, as shown in Fig. 1. High speed photographs such as in Fig. 1 allowed us to measure the contact time t. The frame rate could be larger than 10 4 Hz, allowing precise measurements of t, observed to be in the range 1-10 ms. Since the shock is mainly inertial (with a restitution coefficient 2 as high as 0.91), t is expected to be a function of only R, V, r and g, and thus to vary as R/V f(w). For a Hertz shock, for example, the maximum vertical deformation d scales as R(r 2 V 4 /E 2 ) 1/5, where E is the Young modulus of the ball 7. Taking as an equivalent modulus for a drop its Laplace pressure E ~ g/r, and noting that t ~ d/v, we find for a Hertz drop f(w) ~ W 2/5, and a contact time varying as V -1/5 and R 7/5. Figure 2a shows that the contact time does not depend on the impact velocity in a wide range of velocities (between 20 and 230 cm/s), although both the deformation amplitudes and the detail of the intermediate stages largely depend on it. This is similar to a harmonic spring, although oscillations here are far from being linear. Moreover, it confirms that viscosity does not play any significant role. Figure 2b shows that t is mainly fixed by the drop radius, since it is well fitted by R 3/2, in a wide range of drop radii (between 0.1 to 4 mm). Both results can be understood simply by balancing inertia (of order rr/t 2 ) with capillarity (g/r 2 ), which yields t ~ (rr 3 /g) 1/2, of the form stated above with f(w) ~ W 1/2. This time is (slightly) different from a Hertz time because for a solid, the kinetic energy is stored during the shock in a localized region, while it forces in our case a global deformation of the drop (as observed in Figure 1).

3 The scaling for t is the same as for the period of vibration of a drop derived by Rayleigh 8, although the motion here is dissymmetric 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, significantly larger than p/ 2 ~ 2.2 for an oscillating drop 8. Another difference between both systems is the behaviour in the linear regime (We << 1): for speeds smaller than reported in Fig. 2, we observed that t depends on V, and typically doubles when decreasing V from 20 cm/s to 5 cm/s. Both the numerical coefficient and the latter behaviour remain to be understood. Finally the brevity of the contact has an interesting consequence. A drop containing surfactants, which spreads if gently deposited on the solid, can bounce if thrown on it: the contact 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. References 1. Hartley, G.S., & Brunskill, R.T. in Surface Phenomena in Chemistry and Biology, p. 214, J.F. Danielli Ed., Pergamon Press, Richard, D., & Quéré, D. Europhys. Lett. 50, (2000). 3. Aussillous, P., & Quéré, D. Nature 411, (2001). 4. Hertz, H. Journal für die reine und angewandte Mathematik 92, (1881). 5. Nakajima, A., Hashimoto, K. & Watanabe, T. Monatshefte für Chimie 132, (2001). 6. Frohn, A. & Roth, R. Dynamics of Droplets, Springer Verlag, Landau, L.D., & Lifschitz, E.M. Theory of elasticity, 3 rd ed., Pergamon Press, Rayleigh, Lord Proc. Roy. Soc. 29, (1879).

4 a b c Figure 1 Millimetric water drops impacting a super-hydrophobic solid, for different Weber numbers W. W compares the kinetic and surface energies of the drop (W = rv 2 R/g, noting R the drop radius, V the impact velocity, and r and g the density and the surface tension of the liquid). For W of order unity (a), the maximum deformation during the contact becomes significant. For W of order 4 (b), waves develop along the surface and structure the drop. For W of order 18 (c), the drop is highly elongated before taking off and emits droplets. However the contact time is found to be insensitive to the details of the shock (Fig. 2a).

5 Figure 2 Contact time of a bouncing drop as a function of the impact velocity V (a): in the explored interval (0.3 < W < 37), the contact time is found to be independent of V. But it largely depends on R, the drop radius (b). The dotted line indicates a slope 3/2.