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2008 EPL 82 44001

Please note that terms and conditions apply. Wave propelled ratchets and drifting rafts 1 Mati`eres et Syst`emes Complexes, Universit´e Paris Diderot, CNRS - UMR 7057 Ba?timent Condorcet, 10 rue Alice Domon et L´eonie Duquet 75205 Paris cedex 13, France, EU 2 GRASP-Photop?ole, Institut de Physique B5, Universit´e de Li`ege - B-4000 Li`ege, Belgium, EU 3 Mat´eriaux et Ph´enom`enes Quantiques, Universit´e Paris Diderot, CNRS - UMR 7162 Ba?timent Condorcet, 10 rue Alice Domon et L´eonie Duquet 75205 Paris cedex 13, France, EU PACS 47.55.D- ? Drops and bubbles PACS 05.65.+b ? Self-organized systems PACS 05.45.-a ? Nonlinear dynamics and chaos

Copyright c EPLA, 2008

Introduction. ? The self-propulsion of particles investigate, in the same type of experiment, the behaviour moving in a spatially periodic, assymetrical potential has of self-assembled assymetrical bound states. Each element been widely studied recently. Several physical systems is here individually motionless: the motion comes from have been proposed, aimed at providing models for the the assymetry of their assembly. iiittsssccaaavbdwmmtsnhhttpaasenrrarroossrannaeoodtibvuutytttsatthpnaeiiiiccmckfoocetbtbfttuoahiaoLbnntuueeinpmBrnhlrleloetrrlicgemaoereyteeiirueasdndcyyofosdnsetkoteruwvfbidlctreirofnoaiss[hienoomtbap4fnotso.srhdrmisrtn]maeyeof.aoeubeandsmsB[rtrocaeed5htttieiieldyhncu]zimmeoed.idoptoeemsl)cdhdemoacdhnubirIstrietsn,teolnanFureiiswaatntbtopytsybfeirhlrohnheonstmrasaedtmdmhriacatnhcoc[ierabcebl1sheseoilfrattnponyol,niertita2nmt,uculoigscooeathc]oabennn:r.neoaceiunnsfec[cvneilIr8rttiinmsieiwsnn?hcwannsaqpweg1gpasniegwtamutos0wvhnaooshboeeon]eestonneoh.mnfretsntb-kioracvcflseilaacbeNhjotifyanuhnneslsmamroeay[liecpeei6odnwaotatooaeool,ochn.uoriu7nusrndawqedfasaI]s,sslesl,unsnsitytieulsfz,oibsihwaonydsbewygrslenreseodewiy-[emegdt.o3rdhnsigeagitd]rneFFtane..mfvhoyset,isrodaarseffhpremtmarsrr.ui,iiwraeoiaanlceosw)iornwahdd)pttttenncilaeahahehhuntslIIeinndyyyoaeeeeectttt, llifeaaa1?vnpdn?wisaoae0nmiiroq=hrl=tnrco?Endlulgefiipthoipc-3ile0qixx?wdldydrhcsri.uPpmeepita9iiwpdti[mcsnuebal6ed8/heopr.gdir5,·rgepoitarie1eo1sinh.ddTsanbtd0m,0dcshgTera]hid3sigia.deseloiteshuei,lntkhTaonanerbecutlgmftaahsdhotesttali·lro.inierhhetntecqngmwofhtteageddauegrp?eete?mroeittrsoeadaohtede3lstnsTcpaphnn.ircapi0cfcilosehosaitrlieal.kTlitpi1uhseellsoronbfueathigtean<ffelrodtschereimvasitucpeorx?n,osDithoenemdipfffignena=totocoeeh<iyetniraofrtrbonaer0pc0thiadi1nbmgic.it?=loene0ct.adho5fwahaecg2=smbue3ancl.0utmtiylnhthnmthe9y?fbTfhcsrmreemospNiemsbahtnwrvw.qesacberceg·(iyruoWhariiasm1isfsnessetceoctbt0utu(engonhha?ldre20s,ybpscneem.1t?0iiycmneaa?atsTfbairrakycmimiwre0fthnnnecfitoehmtnµeeetd0air)noelgebgld1g.mloae=ebdedefe=vriwbIsniooecresanin8eedorefe/nntt2fird0crdmosisgv0ttltioae)Himonhhehtafi.ent×ndyagazeeeest-:, drop becomes a self-propelled ?walker? moving on the Bound states. ? When two drops are present on the surface at constant velocity [7]. In the present letter we interface they ?condense? into a stable bound state, a 0 distance dbd separating them. The non-local interaction (a)E-mail: antonin.eddi@univ-paris-diderot.fr between drops is provided by the damped capillary waves they emit. When a drop hits the bath, the collision ? (reduced acceleration) forms a small crater in the surface. When the drop lifts up, the wave created by the shock evolves freely, the Fig. 2: The velocity of 3 couples of droplets as a function edge of the crater forming the crest of a circular wave of ?. The dots correspond to a couple of bound drops with propagating radially. In a bound state, each drop falls on D1 = 0.87 mm and D2 = 0.96 mm, the triangles to D1 = 1 mm a surface disturbed by the circular wave emitted during and D2 = 1.12 mm and the rhombs to D1 = 1.03 mm and previous periods by its neighbour. With two drops of D2 = 1.23 mm. The continuous curves are simple interpolaidentical size, the system self-organizes in such a way tions. The crosses correspond to the reversal threshold ?1. that, during its collision with the bath, the horizontal impulse given to each drop is zero. This equilibrium can be note that, after this transition, the distance dbd between obtained at a distance between drops db0d = ?0 ? , where the two drops becomes larger. Finally, the third threshold ?0 is the wavelength of the surface waves at the forcing is reached at ? = ?3, when the drops begin orbiting around frequency [7], and is an offset due to finite duration of each other. the collision. With more than two drops the condensation Before giving an interpretation of these effects we must leads to the formation of stable rafts with a crystalline first characterize the bouncing of a single droplet. In our lattice of the same periodicity db0d. With drops of identical fixed experimental conditions (i.e. viscosity and forcing size both the bound states and the clusters are motionless. frequency being fixed) the types of observed bouncing are When they are formed of drops of different sizes, they have a function of the droplet diameter D and of the reduced a spontaneous drift motion. acceleration ? [7]. Figure 3a is a phase diagram which

We first focus on the association of two drops of summarizes the behaviors observed for a single droplet. In diameter D1 and D2 (with D1 < D2). For low values the region B of this diagram, the drop bounces at the forcof ? = ?m/g the bound state they form is observed to ing frequency. When ? is increased the successive jumps translate, the large drop pushing the small one. This is become alternatively large and small, so that the period of the mode 1 shown in fig. 1a. In fig. 2, we have plotted the motion doubles (in region PDB of fig. 3a). The transithe velocity of several bound states as a function of ?; tion to this period doubling strongly depends on the drop?s mode 1 being associated to negative velocities. The bound size. Larger drops do not lift away so easily because their state?s velocity is a function of both diameters of the drops. deformation increases the size of their zone of near contact. Relatively fast translation motions are observed as, e.g., Both the simple bouncing and the period doubling occur V = 3 mm/s for drops with D1 = 1 mm and D2 = 1.12 mm. for larger values of ?. This is related to the deformation

When ? is increased, the velocity of the pair becomes of the drop during its collision with the substrate. This at first larger, then a reversal of the direction of motion is deformation depends on the drop?s size D since it is observed so that the small drop now pushes the large one characterized by the Weber number: W e = (?V 2D)/(2?), (mode 2, fig. 1b). This transition, observed for all pairs the ratio of the kinetic energy of the drop to its surface of drops, occurs at a value ?1 which is a function of the energy. For drops of intermediate size, 0.5 < D < 0.9 mm diameter D1 of the smaller drop (fig. 2). the period doubling can become complete so that the drop

When the two droplets have diameters in between D = touches the surface once in two periods. Correlatively, it 0.5 mm and 0.9 mm, a more complex sequence of behaviors becomes a ?walker? (in the region W in fig. 3a) moving at is observed, characterized by two new thresholds, ?2 and a constant velocity in the horizontal plane. The walkers ?3. Over the value ?2 there is a second reversal in the were already investigated elsewhere [7]. direction of motion (transition to mode 3, fig. 1c). In this We can now return to pairs of interacting drops. They case the large drop pushes again the small one. One can form stable bound states at a well-determined distance Fig. 3: a) Phase diagrams of the droplet?s behaviour as a function of their diameter D and of the reduced acceleration ?.

The viscosity of the silicon oil used is µ = 20 × 10?3 Pa · s and the forcing frequency f0 = 80 Hz. The various zones correspond to the different behaviors of a single droplet: in B it bounces at the forcing frequency, in PDB the bouncing has undergone a period doubling, in CB the bouncing has become chaotic through a period-doubling cascade. In region W the droplets become walkers with a spontaneous horizontal displacement at a constant velocity. Region F is the Faraday instability zone.

Panel b) shows, on the same phase diagram, the transition value ?1 for nine bound states. Two droplets forming a bound state are linked by a vertical dashed line. Their diameters are given along the ordinate axis. The abscissa at which they are represented is the value of ?1 for each bound state. Panel c) shows, with the same principle, the three thresholds observed for one single pair of droplets with diameters D1 = 0.57 mm and D2 = 0.75 mm. db0d from each other. Their non-local interaction is due to the surface wave they emit by their bouncing. The observed drift motion of uneven drops signifies that the forces exerted by the droplets on one another are not symmetrical: action appears to be different from reaction.

In order to understand this drift we can first consider the motion of each droplet, then we will return to a global description of the system.

Local behavior. ? The fast camera at a large magnification (see fig. 1) reveals images of the drops forming the ratchet in the three different regimes. In mode 1 both drops oscillate at the same frequency. However, the drops being unequal, neither the lift-off, nor the collision after the free flight, are simultaneous. At each period, the small drop hits the interface later than the large one. Being close to it, the small drop falls on the outer slope of the ridge of the crater formed by the larger drop. It thus receives a forward kick. Correlatively, the crater of the large drop becomes assymetrical by the collision of the small drop.

As a result, at lift-off, the large drop also moves forward.

Both drops thus receive a kick in the same direction and propagate together.

Where does the reversal in the motion direction come from? Just over the transition the small drop has undergone period doubling so that its successive collisions become uneven. One out of two shocks is weak and uneffective. During the other collision, the small drop hits the interface before the large one and repells it (see fig. 1b).

This is confirmed by direct observation with a fast video camera. In fig. 3b, we have used the previous phase diagram, fig. 3a, of the individual droplets to represent the threshold values of ?2 for pairs of droplets. ?2 is systematically slightly larger than the value of ? for which the smaller drop has entered the region of period doubling.

Global considerations. ? The two drops receive energy from the vibration generator but, as the imposed vibration is vertical, it does not directly provide a driving force. The motion is due to the initial asymmetry resulting from the different size of the droplets. This is natural: considered as a whole the two drops do not form an isolated system because they emit waves which propagate away. The waves are carrying away a flux of momentum which can be estimated. Observing the waves far from the bound state, one can assume that they are locally travelling plane sinusoidal waves with surface elevation ? = a cos (?t ? kx), a being the wave?s amplitude, ? its pulsation and k its wave number. Such waves possess an average momentum 1/2(??a2) per unit surface, with ? the fluid density [11,12].

This action reaction effect between the droplets and the waves explains the ratchet motion. However, it should be recalled that there is no exact momentum conservation in our system, because of dissipation. Consequently, it is not possible to make this argument more quantitative. Besides dissipation is needed. The emitted waves are damped by viscosity before reaching the boundaries so that no reflected wave returns to the ratchet. If dissipation vanished, waves would reflect on the borders and accumulate on the whole bath. The droplets would then have a chaotic motion on those waves. In a finite cell the breaking of time symmetry by dissipation thus appears necessary to propulsion. Note that in an infinite system dissipation should not be needed, causality being sufficient to give a direction to the wave propagation.

We can now consider the flux of momentum due to each of the droplets. At low forcing acceleration, observation Fig. 4: Two top views of the same ratchet. Left: below ?1 with a fast camera shows that both drops oscillate at the the larger drop emits a wave of larger amplitude. Thus the forcing frequency f0. Because of the difference of their bound state propagates to the right. Right: over ?1, the smaller masses the waves emitted by the two drops have different droplet has undergone a period doubling and emits a Faraday amplitudes, the larger the mass, the larger the amplitude. wave of larger amplitude: the bound state propagates to the The reaction, resulting from the emission of momentum left. by the waves, pushes the bound state, the large droplet being behind.

With the increase of ?, the bouncing of the smaller drop displacement of the whole cluster is observed. The nature undergoes a period-doubling transition while the larger of the motion depends on the symmetry of the system. The drop continues bouncing at the forcing frequency. global shape of the small agregates is dominated by the

After period doubling, the small drop begins to emit trend to form a triangular lattice. It is combined with a surface waves of frequency f0/2. This is the Faraday trend for an aggregate formed of a given number of particle frequency, which is the least damped by the system, to minimize its outer perimeter. because of the proximity of the Faraday instability thresh- The simplest possibility is the case of three drops. old. Independent measurements enable us to measure the This type of bound state, with either a small drop and amplitude of the waves. In the typical situation shown two large ones or the reverse situation, present the same in fig. 4, the amplitude of the wave emitted by the small first reversal. If we consider larger aggregates, in first droplet is approximately five times greater than the ampli- approximation, each drop located at the periphery emits tude of the wave emitted by the larger drop. Thus, the a wave, which can only propagate in the free surface. The assymetry of the wave emission is reversed and the flux of reaction to the emission of the wave by each droplet placed momentum carried away by the emitted wave is larger on at the periphery will be perpendicular to the local facet or the side of the small drop. The resulting reaction pushes if it is located at a vertex, along the bisector of the wedge the bound state, the small drop being behind. Figure 3b under which it ?sees? the free surface. If the direction confirms that ?1 corresponds to the value for the period of emission passes through the center of mass of the doubling of the smaller drop. aggregate, the reaction will generate a drift, if not it will

When the two droplets have diameters in the narrow create an angular momentum and the cluster will rotate. range between D = 0.5 mm and 0.9 mm, at a value ?2 Figure 5 shows two rotating aggregates. For drops which the distance dbd between the two droplets increases and have diameter D in between D = 0.5 and 0.9 mm, they are correlatively the direction of motion changes again. The able to undergo a transition to a subharmonic bouncing. plot in fig. 3c confirms that ?2 corresponds to the value At the transition, the drops take various phases relatively for the period doubling of the larger drop. In this case, to the forcing frequency. The mutual distance between two both droplets emit waves at the Faraday frequency. The drops depends on whether they bounce in phase or with distance dbd changes accordingly to reach db1d = ?F ? , opposite phases. The aggregate thus becomes disordered where ?F is the wavelength of the surface waves at before reorganizing in a more complex crystalline structure the Faraday frequency (see fig. 1c). Finally, the third with two typical lengths that will be discussed elsewhere. threshold ?3 is reached where the drops begin orbiting.

This corresponds to the situation when one of the drops achieves a complete period doubling and enters the W region of the phase diagram (see fig. 3c).

Conclusion. ? In our experiment each bouncing drop is a mobile wave source. If isolated, it is either motionless or can move at a constant velocity by breaking of symmetry. When several wave emitters are present

Drifting and rotating aggregates. ? The existence simultaneously on the surface they interact and form of aggregates has been previously investigated [7,13,14]. bound states and organized clusters. Here we have shown However, we show here a new behaviour relying on that these systems, when formed of uneven droplets, are the same physical effect as the spontaneous motion of spontaneously mobile by reaction to the waves they emit a two-droplets bound state. When the drops forming outwards. Such a mean of propulsion by reaction to the the aggregate are of uneven size, a slow spontaneous emission of surface waves was one of the mechanisms laboratories have been financially helped by the COST action P21 and by the ANR 06-BLAN-0297-03. [1] Julicher F., Adjari A. and Prost J., Rev. Mod. Phys.,

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Boudaoud, M. Rossi, N. Vandewalle and S. [15] Sun S. M. and Keller J. B., Phys. Fluids, 13 (2001) Dorbolo for stimulating discussions. Exchanges between 2146.