Resonant and rolling droplet

When an oil droplet is placed on a quiescent oil bath, it eventually collapses into the bath due to gravity. The resulting coalescence may be eliminated when the bath is vertically vibrated. The droplet bounces periodically on the bath, and the air layer between the droplet and the bath is replenished at each bounce. This sustained bouncing motion is achieved when the forcing acceleration is higher than a threshold value. When the droplet has a sufficiently low viscosity, it significantly deforms : spherical harmonic \boldmath{$Y_{\ell}^m$} modes are excited, resulting in resonant effects on the threshold acceleration curve. Indeed, a lower acceleration is needed when $\ell$ modes with $m=0$ are excited. Modes $m \ne 0$ are found to decrease the bouncing ability of the droplet. When the mode $\ell=2$ and $m=1$ is excited, the droplet rolls on the vibrated surface without touching it, leading to a new self-propulsion mode.

Bouncing droplets have been first studied by Couder et al [1]. With a radius R < 0.5 mm, they may generate local waves on a 50 cSt bath and use them to move horizontally and interact with their surroundings : they detect submarine obstacles, they orbit together or form lattices, they are diffracted when they pass through a slit [2,3,4,5,6]. Bouncing droplets exhibit a wide variety of motions and interactions that present some strong analogies with quantum physics, astronomy and statistical physics.
In contrast with these previous studied, we choose a bath viscosity of 1000 cSt, a droplet viscosity between 1.5 and 100 cSt and a droplet radius R between 0.76 and 0.93 mm. With a bath viscosity at least 10 times larger as the droplet viscosity, the bath deformations are inhibited (capillary waves are fully damped) while the harmonic deformations of the droplet are enhanced. As shown recently in [8], the droplet deformation ensures its bouncing ability. Various modes of deformation may be excited as depicted in Fig.1. Each picture has been constructed from an experimental snapshot of the droplet (the left side of each picture) and the calculated 3D spherical harmonic (the right side). Those modes are analogous to the natural modes of deformation introduced by Rayleigh [10] and may be expressed in terms of spherical harmonics Y m ℓ . The simplest mode that may be used for bouncing is the mode Y 0 2 [8]. We will show that droplets can use non-axisymetric mode Y 1 2 to move horizontally, or more precisely to roll over the bath. This new mode of self-propulsion drastically contrasts with the bouncing walker mode described in [3] since the bath deformations are not necessary for the propulsion.
On a bath vertically vibrated according to a sinusoidal motion A sin(2πf t), the periodic bouncing only occurs when the reduced maximal vertical acceleration of the bath Γ = 4π 2 Af 2 /g is higher than a threshold value Γ th , where g is the acceleration of gravity. Formally, Γ th depends on the forcing frequency f [8], the droplet radius R [7], and the physical parameters of the liquids (density ρ, viscosity ν, surface tension σ). We measure the threshold Γ th as a function of the forcing frequency for FIG. 1: Various deformation modes of a bouncing droplet (ν = 1.5 cSt, R = 0.765 mm) observed with a high-speed camera for various forcing frequencies.
The first line (resp. 2nd, 3rd) displays a mode Y 0 2 (resp. Y 0 3 , Y 0 4 ) with m = 0 (axisymmetry). The forcing frequency is 50 Hz (resp. 160 Hz, 275 Hz) and the reduced acceleration Γ = 0.3 (resp. 2, 6). First and second columns represent snapshots at two different times of the oscillation. The spherical harmonic solution (on the right of each picture) is superposed to the experimental pictures (on the left of each picture).
various droplet sizes and viscosities. For each frequency, we place a droplet on the bath in a bouncing configuration, i.e. Γ > Γ th . The forcing acceleration is then decreased progressively. When the threshold is reached, the droplet cannot sustain anymore the periodic bouncing and quickly coalesces with the bath. Since the droplet and the bath are made from different liquids, a coalescence event locally contaminates the bath. Therefore, droplets always need to be placed at different locations on the bath.

FIG. 2:
Evolution of the bouncing threshold acceleration Γth with respect to the forcing frequency. The red, green and blue bullets correspond to droplets with viscosity ν =1.5, 10 and 100 cSt respectively, and with a constant radius R = 0.765 mm. Depending on the forcing frequency, various modes are observed, that may be related to spherical harmonics Y m ℓ .
In Fig.2, the threshold acceleration Γ th is represented as a function of f , for droplets with viscosities ν =1.5, 10 and 100 cSt. The threshold acceleration for the 100 cSt droplet increases with the frequency in a monotone way. By opposition, the 1.5 cSt curve is characterized by regularly spaced local minima that, in some way, correspond to a resonance of the system : a minimal energy supply is required to sustain the periodic bouncing motion. The first minimum, at f = 50 Hz, corresponds to Γ th =0.25, a value significantly less than the 1 g minimal threshold required by inelastic bouncing objects on a vibrated plate. Modes m = 0 and ℓ =2, 3 and 4 are observed for forcing frequencies corresponding to minima in threshold acceleration curve. The higher the frequency, the higher-order the excited mode, and the higher the corresponding threshold acceleration. We note that the asymmetric mode Y 1 2 occurs at a local maximum of the threshold curve.
The bouncing droplet may be considered as an oscillating system analogous to the damped driven harmonic oscillator : surface tension is the restoring force and viscosity the damping process. The dimensionless ratio between both is the Ohnesorge number Oh = ν √ ρ/ √ σR, which is equal to 0.012, 0.078 and 0.775 for a droplet viscosity of 1.5, 10 and 100 cSt respectively. When Oh ≪ 1, the viscous damping is negligible and resonance is im-portant. Damping increases as Oh gets closer to 1. As seen in Fig.2, the bouncing droplet reacts to an increased damping in the same way as the harmonic oscillator : (i) the resonance frequency slightly decreases, shifting the whole threshold curve to the left, (ii) the required input Γ th increases, and (iii) extrema tend to disappear. At 100 cSt (Oh = 0.775), the damping is fully active and the threshold curve increases monotonically with the frequency : no more resonance is observed. This highviscosity behaviour has already been observed by Couder et al [1] for 500 cSt droplets bouncing on a 500 cSt bath. Those authors proposed to model the threshold curve by Γ th = 1 + αf 2 where α depends on the droplet size among others. This equation is obtained by only considering the motion of the mass center of the droplet and the squeezing of the air film between the droplet and the bath, without considering the droplet deformation. This model only fits high viscous regimes in contrast with the model developed in [8]. Boundaries between those zones correspond to maxima in the threshold curve. Moreover, those particular frequencies may be obtained by multiplying the Rayleigh natural frequencies (Eq.1) by a factor 1.15. Arrows indicate the Rayleigh frequencies, which cannot be directly related to inflections in the threshold curve.
In 1879, Lord Rayleigh [10] described natural oscillations of an inviscid droplet. Since those oscillations are due to surface tension, natural frequencies scale as the capillary frequency f c = σ/M , where M = 4π/3ρR 3 is the droplet mass. More exactly, the dispersion relation prescribes the natural "Rayleigh" frequency f R related to a ℓ-mode: The function F (ℓ) may vary (frequencies are shifted by a multiplicative factor), depending on the way the droplet is excited [11,12]. For free oscillations, Eq.(1) is degenerated according to the m parameter. In Fig. 3, threshold data obtained for various droplet sizes collapse on a single curve by using the Rayleigh scaling. Moreover, at the bouncing threshold, the vertical force resulting from the droplet deformation exactly balances the gravity. One may define a characteristic length L = g/f 2 c corresponding to the free fall distance during the capillary time 1/f c . As shown in Fig. 3, the threshold amplitude A th = Γ th g/(4π 2 f 2 ) scales as the length L, whatever the droplet size. The minimum value of A th f 2 c /g does not vary significantly with the mode index ℓ. In Fig. 3, the natural frequencies defined in Eq.(1) and represented by arrows do correspond neither to the minima in threshold, nor to the maxima. However, these frequencies multiplied by 1.15 give the maxima positions, at f /f c = 1.05, 2.05 and 3.15 for ℓ =2, 3 and 4 respectively. This numerical factor depends on the geometry of the excitation mode (bouncing in this case) : e.g. another (smaller) factor is obtained when the droplet is stuck on a vibrated solid surface [11,12].
As shown in [8], the bouncing ability of droplets is due to the cooperation of (i) the droplet deformation, that stores potential energy, and (ii) the vertical force resulting from the squeezing of the intervening air layer between the droplet and the bath. The forced motion of the bath provides some energy to the droplet, a part of which helps the droplet to bounce (translational energy) while the other part increases internal motions inside the droplet, which are eventually dissipated by viscosity. The proportion of energy supplied to the translational / internal motion varies with the forcing frequency (i.e. when the energy is provided in the oscillation cycle). Minima (resp. maxima) in the threshold curve correspond to a maximum (resp. minimum) of the translational to internal energy ratio. Maxima may be related to the cut-off frequency recently observed and theoretically explained by Gilet et al. [8]. This fact is confirmed experimentally, since the maxima in threshold correspond to the boundaries between modes. At those points, corresponding to a forcing frequency of 1.15 f R , the most energy is spent in internal motions : the droplet resonates and absorbs energy.
The droplet behaviour has been observed in the vicinity of the first maximum in the threshold curve. At 115 Hz, the droplet moves along a linear trajectory analogously as the walkers observed in [2]. In this latter case, 50 cSt droplets were produced on a 50 cSt silicon oil bath. The mechanism for the walking motion is the interaction between the droplet and the bath surface wave generated by the bouncing. Such a mechanism does not hold in the case of a 1000 cSt silicon oil bath. Indeed, the generated waves are rapidly damped and cannot be responsible for the motion of the droplet. A movie of the moving droplet has been recorded using a high speed camera (Fig. 5). The images are separated by 1 ms. The deformation is not axi-symmetrical as in the resonant minima of the threshold acceleration curve. On the other hand, the mode is related to ℓ = 2 and m = 1. This mode is characterized by two lines of nodes that are orthogonal. It results the existence of two fixed points located on the equator of the droplet. As the the line of nodes does not follow the axi-symmetrical geometry, those fixed points move. More precisely, they turn giving the droplet a straight direction motion. As far as Rayleigh frequencies are concerned, we observe a break of degeneracy for the m parameter! Some tracers have been placed in the droplet. The reflection of the light on the particles allows to follow the inner fluid motion, which clearly reveals that the droplet rolls over the bath surface. The initial speed v of the droplet has been measured in respect to the forcing parameters (amplitude and frequency). A scaling is found when considering that the phenomenon can occur only above a cut-off frequency f 0 ≈ 103 Hz and the amplitude threshold where α is a constant. The initial speed is represented versus the characteristic speed v * = (A − A th (f ))(f − f 0 ) in Fig.5. The proportionality between the initial speed of the roller and v * is remarkable. Three characteristic frequencies are involved in the rolling process : (i) the excitation time 100 Hz, (ii) the rotation of the liquid in-side the droplet about 10 Hz and (iii) the translational motion about 0.2 Hz (the droplet travels a distance corresponding to its circumference once per 5 seconds). The three processes are interrelated, i.e. the deformation induces the internal motion which induces the rolling and the translational motion. The deformation of low viscosity bouncing droplets is emphasized on a high viscosity bath : the droplet oscillations are much less damped than the bath oscillations. Depending on the forcing frequency, droplets need a different amount of supplied energy to achieve a sustained periodic bouncing. When the forcing frequency corresponds to a multiple of the eigenfrequency of the bouncing droplet (f ∼ (ℓ − 1)f c ), the supplied energy is mainly lost into internal motions, so the threshold acceleration Γ th is maximum. On the other hand, Γ th is minimum when f /f c = ℓ − 3/2.
A new self-propelled mode has been discovered for forcing frequencies between the ℓ = 2 and ℓ = 3 modes (between roughly 100 and 150 Hz) for a sufficiently low droplet viscosity. This excited mode is a non-axisymmetrical mode Y 1 2 characterized by an internal rotation of the fluid inside the droplet. The droplet rolls over the vibrated bath ! This droplet displacement technique is more adapted to large low-viscosity droplets; it is therefore of considerable interest for microfluidic applications: manipulating aqueous mixtures without touching them.
SD/TG would like to thank FNRS/FRIA for financial support. The Authors want to thank also COST P21 Action 'Physics of droplets' (ESF) for financial help. J.P. Lecomte (Dow Corning, Seneffe, Belgium) is thank for silicon oils. A. Kudrolli (Clark University, MA, USA), J.Y. Raty (University of Liège), P. Brunet (University of Lille, France) and R. D'Hulst are acknowledged for fruitful discussions.