PRL 119, 108001 2017
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Superpropulsion of Droplets and Soft Elastic Solids
Christophe Raufaste,1 Gabriela Ramos Chagas,2 Thierry Darmanin,2 Cyrille Claudet,1 Frederic Guittard,2 and Franck Celestini1,
1Universite Cote dAzur, CNRS, Institut de Physique de Nice, 06100 Nice, France 2Universite Cote dAzur, NICE Lab, IMREDD, 6163 avenue Simone Veil, 06200 Nice, France Received 7 June 2017; published 8 September 2017
We investigate the behavior of droplets and soft elastic objects propelled with a catapult. Experiments show that the ejection velocity depends on both the projectile deformation and the catapult acceleration dynamics. With a subtle matching given by a peculiar value of the projectilecatapult frequency ratio, a 250 kinetic energy gain is obtained as compared to the propulsion of a rigid projectile with the same engine. This superpropulsion has strong potentialities: actuation of droplets, sorting of objects according to their elastic properties, and energy saving for propulsion engines.
DOI: 10.1103PhysRevLett.119.108001
Droplets are very specific objects owing to their inter facial properties 1. They keep their integrity and shape at rest due to surface tension, but are very deformable and difficult to manipulate since they adhere to regular sub strates. Nevertheless droplet motion can be triggered when the surface wetting properties are inhomogeneous: self propulsion is thus observed when the substrate surface exhibits a gradient of free surface energy 2 or when a droplet itself triggers a contact angle difference between its leading and trailing edges 3,4. Their dynamical properties are at the origin of peculiar behaviors 5 and their vibration modes 6 can be forced in a way to control their motion over a substrate 7,8 and to let them move against gravity 9. The contact with the substrate can also be prevented by triggering the Leidenfrost effect 10,11, which leads to nonsticky drops that move easily 12. In the same vein, the development of superhydrophobic surfaces SHSs allows one to minimize the adhesion with the substrate 13 and droplets can bounce on such surfaces like elastic balls with a velocity restitution coefficient that depends on the relative importance of inertial and capillary effects 14. Recently Boreyko and Chen 15 have studied the coalescence of droplets during vapor condensation on SHSs. The surface energy release associated with the coalescence is partly transformed into kinetic energy and induces a vertical propulsion of the merged drops.
In this Letter we investigate the capillaryinertial behav ior of droplets propelled by a catapult. To summarize, we first consider droplets launched with a simple spring catapult, the plate surface of which is superhydrophobic to prevent any adhesion. We show that the transfer of kinetic energy can be increased by 250 as compared to propelling rigid objects with the same engine. This super propulsion phenomenon is obtained when the catapult is tuned to perform the ejection with a subtle matching between its own dynamics and the one of the projectile deformation. The same behavior is then experimentally
recovered with soft elastic solids, demonstrating the gen erality of the phenomenon. The experimental results are in good agreement with a simple physical model and the optimal ejection is reached with a peculiar value of the projectilecatapult frequency ratio, different from the ones of classical resonant phenomena. Finally, several direct applications are discussed and illustrated in the conclusion.
The experimental setup consists in a catapultlike engine built with a spring of variable stiffness. A similar setup was used by Clanet et al. 16 to emphasize the droplet deformation under strong acceleration. The catapult is initially loaded and maintained at rest with an electromag net. The initial distance between the plate and its equilib rium position can be varied and is typically of a few millimeters. Once the electromagnet is switched on, the plate is subject to a sudden and large acceleration, typically 10 times the gravity. The catapult plate is a SHS of high quality to prevent droplet adhesion. We use two different kinds of SHSs with ultralow hysteresis and sliding angle that are obtained by electropolymerization. The first one is a superhydrophobic and superoleophobic fluorinated poly 3,4ethylenedioxypyrrole 17 and the second one a superhydrophobic fluorinated polyfluorene 18. Their low hysteresis and sliding angles, less than 2.0 degrees, are due to the combination of microstructures and nano structures or nanoporosities with the low surface energy of the fluorinated chains. Indeed we have demonstrated and discussed in Ref. 17 how the presence of nanoporosities is crucial to highly reduce the water adhesion on the surface.
Droplets are propelled with this device and the ejection dynamics is imaged by a highspeed camera with frame rates ranging from 500 fps to 5000 fps. Snapshots of the main steps of a drop ejection are given in Fig. 1a: the drop is represented, first at rest on the SHS, second during the initial acceleration of the plate, third at the takeoff time te, and finally during its flight. For each ejection, a spacetime diagram is built along a vertical line passing through the
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FIG. 2. The figure displays snapshots of three experiments performed with several drop sizes and the same catapult char acteristics f 14 37 Hz and A 14 1.4 mm, see movie M1 19. Left: Position before the motion initiation. Right: Vertical position reached after ejection once all the kinetic energy is transferred into gravitational potential energy. The dashed line indicates the position that would reach the center of mass of a rigid object.
In what follows, we always work in the linear regime and characterize the ejection efficiency by the energy transfer factor14 VeVp2. This coefficient characterizes the gain or loss of kinetic energy as compared to the ejection of a rigid projectile that is expected to takeoff from the substrate with Ve 14 Vp or14 1. We performed several experiments varying the size of the drop while keeping constant the catapult frequency. We can see that in most cases, the droplets are ejected with a velocity larger than the one expected for a rigid projectile and thatdepends significantly on the droplet size Fig. 2. By varying both the droplet radius and the catapult frequency in our experiments, we found that all data collapse by plottingas a function of the dropcatapult frequency ratio f0f Fig. 3. Around f0f3,reaches maximal values close to roughly 2.5. Note thatquantifies only the translational kinetic energy of the system. In fact, the droplets exhibit more or less pronounced oscillations after takeoff empha sizing that some energy is transferred to the projectile to stimulate modes as well.
We also measure the time te at which the droplet is ejected. Again, a unique trend is found when plotting teT as a function of f0f. Most of the values are larger than 0.25, the value expected for a rigid projectile because of the plate deceleration. The smaller the f0f ratio, the longer the contact during the deceleration phase.
We now consider the ejection of hydrogel balls by the same spring catapult. These balls are composed of poly acrylamide, a waterabsorbing polymer, and can be found in any regular flower shop. They are initially dry with a radius of 1 mm and their final radius ranges between 5 mm and 10 mm depending on the hydration time in a water bath. The Young modulus of this material is known to be a decreasing function of the hydration time and typically
Ve
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standing on a SHS initially at rest at t 14 0 see movie M1 19. In the initial stage of the droplet propulsion t1, the droplet deformation is not homogeneous but concentrated in its lower part. In an intermediate stage the deformation reaches the top of the droplet. Later on te the droplet leaves the catapult with a complex and deformed shape. During its flight t3, we clearly see the oscillation modes of frequency f0. b A spacetime diagram is built along the dashed line drawn in image t 14 0. It illustrates the oscillatory motion of the catapult plate and the difference between the maximum plate velocity and the ejection velocity of the droplet.
center of the droplet Fig. 1b. It displays both the plate and the droplet motion as functions of time. We have checked that the plate motion is harmonic. Its period T 14 1f and amplitude A are directly measured from the diagram, as well as its maximum velocity Vp, which corresponds to the plate velocity when it reaches its equilibrium position at T4. The drop ejection velocity Ve and the takeoff time te are also determined using the spacetime diagram. We can also see in Fig. 1b and in movie M1 19 that, during its flight after takeoff, the drop experiences oscillations that we used to measure its eigenfrequency f0. We have checked that these measure ments are in agreement with the surface tension driven oscillation modes of droplets, i.e., f0R32 6, R being the radius of the drop.
Experiments were performed with droplet radii ranging between 0.5 mm and 1.8 mm and catapult frequencies between 20 Hz and 70 Hz. For a given droplet radius and a given catapult frequency, a linear regime is found and characterized by the ejection velocity Ve proportional to the load amplitude A. At low amplitude, the adhesion with the substrate cannot be neglected anymore and the ejection is less efficient. At large amplitude, the velocity is smaller than expected as well, which seems related to the nonlinear response of the drop large deformation andor fragmenta tion to strong solicitations.
a Image sequences of a typical droplet propulsion
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with f0f3 differs from the ones of classical resonant phenomena: respectively, 1 and 2 for driven harmonic and parametric oscillators. This suggests to adopt the most basic approach and to solve the wave equation for the projectile deformation with the peculiar boundary condi tions imposed by the catapult propulsion.
The substrate is initially considered at rest at the position x 14 0. At t 14 0, it initiates a harmonic motion given by Ust 14 A121cos2ft. The velocity Vpt 14 dUstdt reaches its maximum value Vp 14 2fA at T4 Fig. 5. These values correspond respectively to the ejection velocity and takeoff time for a rigid object standing on the substrate without adhesion. The plate decelerates between T 4 and T 2, and recesses after T2. The projectile extends initially from x140 to x142R and is simply modeled as a 1D elastic material. We assume linear elasticity and the wave equation derives from Hookes law 20. Denoting ux;t the displacement of an element at a time t that was initially found at the position x, the wave equation is written as
2u
14 c2 ; 1
Drops
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both droplets circles and hydrogel balls squares. Top: Energy
transfer factorplotted as a function of the frequency ratio f0f.
The dashed line represents14 1, the value expected for a rigid 2u projectile, while the model based on the projectile deformation
dynamics is represented by the solid line. Bottom: Same data and t2 analysis for teT as a function of f0f. The dashed line
represents teT 14 0.25, the value expected for a rigid projectile.
The figure summarizes measurements performed with
ranges between 2 kPa and 20 kPa. We have experimentally measured eigenfrequencies between 50 Hz and 200 Hz. Experiments are performed for several catapult frequencies as well and the results are superimposed to the former ones obtained with the droplets Fig. 3. The same trends are recovered forand teT as functions of f0f. Again, the superpropulsion is observed withvalues significantly higher than one. As illustrated in Fig. 4, by tuning the catapult frequency, we can choose the hydrogel ball that we want to propel the most efficiently.
We clearly demonstrate the generality of the phenome non observed for these two different systems. We can therefore infer that the phenomenon is related to the deformation dynamics of the projectile: it is triggered by surface tension for the droplets and by elasticity for the soft balls. The peculiar value of the maximal efficiency reached
a b c
with c the propagation velocity. Three boundary conditions are applied. First, the projectile is initially at rest without any deformation ux; 0 14 0. Second, the projectile dis placement at the bottom is driven by the substrate motion u0; t 14 A121cos2ft. Finally, the local deformation on the top is fixed to zero to account for the free end Neumann boundary condition ux2R; t 14 0. It is important to note that no dissipation term is included in the equation. We have verified that both inertial and capillary effects are dominant against viscous ones by estimating the Reynolds and capillary numbers. Equation 1 can be set under a dimensionless form by taking 1f and 2R as time and length scales: it shows that the dynamics is a unique function of the ratio of two time scales, the catapult acceleration time and the traveling time of a perturbation propagating back and forth inside the object. It is worth noticing that the latter corresponds to the fundamental period T0 14 4Rc of the projectile. In the following, we use the dimensionless parameter f0f 14 TT0 for convenience.
The wave equation is solved numerically for a set of f0f values by a finite difference scheme. Each simulation is stopped when the projectile loses its contact with the substrate ux0; te 14 0, defining the ejec tion time te. The average velocity of the projectile Ve is then computed and used to calculate the energy transfer factor .
The agreement between the model and the experiments is very good without any free parameter Fig. 3. The param eterexhibits a maximum of 2.5 around f0f 14 3.4, very close to the values found experimentally. The theoretical
x2
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elastic balls for several catapult frequencies see movies M2, M3, and M4 19. The two first projectiles are hydrogel balls of eigenfrequency f0 14 90 and 150 Hz, respectively. The rightmost ball is rigid.
The figure displays the simultaneous propulsion of soft
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f0f 14 0.5 the velocity of the plate at t 14 te equals zero and the only contribution to Vete comes from Vcmte. For a rigid projectile e.g., f0f 14 7 at t 14 te, the plate velocity is large but the contribution from Vcmte is not significant at all. The best ejection is therefore obtained in an intermediate situation f0f 14 3.4 where the projectile is propelled with a significant velocity coming equally from the motion of the plate and the center of mass of the projectile. This point is illustrated in Fig. 5b where we plot the velocities Vpte, Vcmte and Vete as functions of the frequency ratio f0f.
In this communication we have evidenced an unexpected behavior of droplets and soft elastic projectiles launched by a catapultlike engine. Depending on the frequency ratio between the dynamics of the projectile and the one of the catapult, the transfer of kinetic energy can be increased by a factor 2.5 as compared to the case of a rigid object. The experiments are in very good agreement with a simple physical model accounting for the deformation dynamics of the projectile. The model emphasizes the generality of this physical mechanism and its close connection to the classical phenomenon of resonance. The superpropulsion could thus be viewed as a oneshot resonance.
Besides the fundamental interest, we envisage direct applications in various domains. As illustrated in the movie M1 19, drops can be sorted by size by tuning the catapult frequency at the desired value. To our knowledge it is also the first evidence of an accurate drop actuation in the vertical direction. In the same vein, we have verified that a simple device can sort objects according to their elastic properties movies M2, M3, and M4 19. Experiments presented in this communication have been realized with SHSs without any significant adhesion. This is not the case for all SHSs and the ejection phenomenon could be used to dynamically characterize the substrates. We are actually performing experiments to quantitatively measure the robustness of the CassieBaxter state against the Wenzel one. Finally, we expect new possibilities to save energy in ballistics technologies or to improve the efficiency of propelling engines by tuning not only the deformation properties of the projectile but also the ones of the propelling engine itself.
We thank T. Frisch, J. Mathiesen, and S. Dorbolo for fruitful discussions. This work was partially supported by CNPq, Conselho Nacional de Desenvolvimento Cientifico e TecnologicoBrazil Process NO. 20228020144.
Corresponding author. Franck.Celestiniunice.fr
1 P.G. de Gennes, F. BrochardWyart, and D. Quere,
Capillarity and Wetting Phenomena: Drops, Bubbles,
Pearls, Waves Springer, New York, 2004.
2 M. K. Chaudhury, and G. M. Whitesides, Science 256, 1539
1992.
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a Plate velocity Vpt dotted line, center of mass
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velocity in the reference frame of the plate Vcmt dashed lines, and ejection velocity Vet solid lines as functions of time for three frequency ratios. b V p , V cm , and V e at the ejection time te as functions of f0f. All velocities are rescaled by the maximal plate velocity Vp 14 2fA.
prediction for teT is also very satisfactory. For small f0f values, the model predicts a saturation value te 14 T2 corresponding to the recessing time of the plate, which is not observed in the experiment. This slight disagreement is probably due to the simplicity of the 1D model geometry compared to the spherical shape of droplets and hydro gel balls.
This model helps to reach a physical interpretation of the phenomenon. The key point is that the ejection velocity at te is the sum of two different quantities: first the velocity of the plate, Vpte, second the velocity of the center of mass of the projectile in the frame of reference of the plate, Vcmte. The velocity Vpt is initially positive and an increasing function between t 14 0 and t 14 T4 at which it reaches the maximum Vp before decreasing. The projectile is initially in a compression phase, so that the velocity V cm t is negative and decreases with time to reach a negative maximum. It then enters an extension phase reaching a positive maximum. The transition between the compression phase Vcm0 and the extension phase Vcm0 is related to both T and T0. It ranges between T0 more rigid projectiles and T2 softer projectiles. The superpropulsion phenomenon is therefore obtained when a subtle matching between T and T0 is obtained for the maximization of Vete 14 VpteVcmte with an ejec tion time te at which the reaction force on the plate is 0. This is illustrated in Fig. 5a where we plot V p t, V cm t, and V e t for three different cases. For a soft material e.g.,
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3 F. Domingues Dos Santos and T. Ondarcuhu, Phys. Rev. Lett. 75, 2972 1995.
4 Y. Sumino, N. Magome, T. Hamada, and K. Yoshikawa, Phys. Rev. Lett. 94, 068301 2005.
5 A. Frohn and N. Roth, Dynamics of Droplets Springer, Berlin, Heidelberg, 2000.
6 H. Lamb, Hydrodynamics Cambridge University Press, Cambridge, England, 1932.
7 S. Daniel, M. K. Chaudhury, and P.G. de Gennes, Lang muir 21, 4240 2005.
8 X. Noblin, R. Kofman, and F. Celestini, Phys. Rev. Lett. 102, 194504 2009.
9 P. Brunet, J. Eggers, and R. D. Deegan, Phys. Rev. Lett. 99, 144501 2007.
10 J. G. Leidenfrost, De Aquae Communis Nonnullis Qualita tibus Tractatus Ovenius, Duisburg, 1756.
11 D. Quere, Annu. Rev. Fluid Mech. 45, 197 2013.
12 H. Linke, B. J. Aleman, L. D. Melling, M. J. Taormina, M. J. Francis, C. C. DowHygelund, V. Narayanan, R. P. Taylor,
and A. Stout, Phys. Rev. Lett. 96, 154502 2006.
13 D. Vollmer and H.J. Butt, Nat. Phys. 10, 475 2014.
14 D. Richard, C. Clanet, and D. Quere, Nature London 417,
811 2002.
15 J. B. Boreyko and C.H. Chen, Phys. Rev. Lett. 103, 184501
2009.
16 C. Clanet, C. Beguin, D. Richard, and D. Quere, J. Fluid
Mech. 517, 199 2004.
17 T. Darmanin and F. Guittard, J. Am. Chem. Soc. 131, 7928
2009.
18 G. Ramos Chagas, X. Xie, T. Darmanin, K. Steenkeste,
A. Gaucher, D. Prim, R. MealletRenault, G. Godeau, S. Amigoni, and F. Guittard, J. Phys. Chem. C 120, 7077 2016.
19 See Supplemental Material at http:link.aps.org supplemental10.1103PhysRevLett.119.108001 for the movies M1, M2, M3, and M4.
20 L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Vol. 7 of Course of Theoretical Physics, 2nd English ed. Pergamon, New York, 1970.
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