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Faraday wave lattice as an elastic metamaterial L. Domino,1 M. Tarpin,1 S. Patinet,1 and A. Eddi1 1Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636 ; PSL - ESPCI, 10 rue Vauquelin, 75005 Paris, France; Sorbonne Université - UPMC, Univ. Paris 06; Sorbonne Paris Cité - UDD, Univ. Paris 07 Metamaterialsenabletheemergenceofnovelphysicalpropertiesduetotheexistenceofanunder- lyingsub-wavelengthstructure. Here,weusetheFaradayinstabilitytoshapethefluid-airinterface with a regular pattern. This pattern undergoes an oscillating secondary instability and exhibits spontaneous vibrations that are analogous to transverse elastic waves. By locally forcing these waves, we fully characterize their dispersion relation and show that a Faraday pattern presents an effective shear elasticity. We propose a physical mechanism combining surface tension with the Faraday structured interface that quantitatively predicts the elastic wave phase speed, revealing that the liquid interface behaves as an elastic metamaterial. PACSnumbers: 05.45.-a,47.35.Pq,62.30.+d,81.05.Xj 6 1 An artificial material made of organized sub- hexagons, triangles... [15]). Although the pattern selec- 0 wavelength functional building blocks is called a meta- tionofthisinstabilityisquitecomplex,forasquarevessel 2 material [1, 2] when it exhibits properties that differ it is most often a square pattern that is obtained, with n greatly from that of the unit cell. These new physical its two main directions aligned with the sides of the con- a properties are intrinsic of the presence of an underlying tainer. The pattern becomes unstable upon increasing J structure. Although metamaterials are still strongly as- the driving amplitude, and leads to a chaotic state [18] 9 sociated with negative index materials in optics [3], they called “defect-mediated turbulence” (DMT) [19]. Tran- 2 also refer to structures with mechanical [4], acoustic [5] sition to chaos is achieved by a phase instability called or even thermodynamic properties [6]. By engineering the oscillatory transition phase (OTP) [20–22]. Similar ] h building blocks from micro to metric scale, several new oscillatory motions were observed and characterized in p mechanical properties emerge in metamaterials, such as 1D systems such as Taylor-Couette [23], falling liquid - s cloaking in elastic plates [7], auxetic behavior [8, 9], ul- columns [24], Rayleigh-Bénard convection rolls [25], and s tralight materials [10] or seismic wave control [11]. So viscous fingering [26]. Very few 2D systems exhibit this a l far the main challenge has been to design appropriate kind of secondary oscillatory modes: vibrated granular c unit cells to obtain efficient metamaterial constructions. materials[27], liquidcolumns[28]andbouncingdroplets . s Here, we propose a novel approach that uses stationary crystalline aggregates [29]. c i waves to produce the underlying structure of a macro- In this letter, we first show that the Faraday struc- s y scopic metamaterial. ture exhibits spontaneous in-plane transverse waves. We h Spatial patterns arising in systems driven away from studytheirpropagationinthe2Dstructureandlinktheir p equilibrium have been extensively studied over the last existence to the emergence of an effective elastic shear [ decades [12]. The Faraday instability is often used as modulus of the fluid-air interface. We propose a physi- 1 a model system in non-linear physics and the patterns cal interpretation that quantifies the appearance of this v emerging from a vertically vibrated fluid layer are well effective mechanical property revealing that a Faraday 4 documented [13–17]. This hydrodynamic instability ap- wave lattice behaves as an elastic metamaterial. 2 pears at the interface between two fluids subjected to a 0 Experimental set-up. – Our experimental set-up con- vertical oscillation. Above a certain threshold of acceler- 8 sists of a square vessel (13 cm×13 cm) filled with a 0 ationac,thesurfaceshowsastationarydeformationthat thin layer of silicone oil (viscosity η = 5 mPa.s, density 1. oscillatesathalftheexcitationfrequency. Thispatternis ρ=0.965 kg.L−1 and surface tension σ =20.9 mN.m−1) both stable in time and regular in space, with a Faraday 0 of thickness h = 3 to 5 mm. The vessel is mounted 6 wavelength λF defined by the inviscid gravity-capillary on a vibration exciter (Bruël & Kjær), driven with a 1 wave dispersion relation computer-controlled amplifier. The acceleration deliv- : v ered by the vibration exciter is monitored using a cali- Xi ω2 =(gk + σk3)tanh(k h), (1) brated accelerometer. The bath acceleration acos2πf0t r F F ρ F F is sinusoidal, with frequency f0 ranging from 72 Hz to a 120Hz. Aboveagiventhresholdaccelerationa , theliq- c wherek =2π/λ istheFaradaywavenumber, g =9.81 uid interface spontaneously destabilizes and presents a F F m.s−2 is the acceleration of gravity, σ is the surface ten- regular square pattern of standing waves (see Fig. 1(a)) sion of the fluid, h the fluid depth and ρ its density. For with Faraday frequency f = f /2. The size of the pat- F 0 specificexperimentalconditions,onecanachievethefor- tern is about 25×25 Faraday wavelengths. We define mation of well structured and stable patterns (squares, the normalized control parameter ε=(a−a )/a . c c 2 Fourier transforms. In Fig. 2(b) we show a typical spatial 2D spectrum obtained at 1.57±0.1 Hz. The Fourier peakscorrespondingtothestableFaradaypattern(open diamonds) are split into two symmetric sub-peaks indi- cating the standing nature of the pattern oscillations. These sub-peaks are located at a distance k /4 from the f original one, confirming the wavelength observed in Fig. 1(d). The peak in the k direction (resp. k ) is split x y in the k direction (resp. k ) revealing that the spon- y x taneous oscillations correspond to the propagation of a standing transverse wave in the initial Faraday square CCD lattice. Thesespontaneousoscillationsofthepatterncor- respondtoaHopfbifurcationthattakesplaceclosetothe threshold of transition to chaos. This is confirmed when measuring their amplitude at the vibration frequency as a function of the control parameter ε, where we observe the supercritical nature of this bifurcation [Fig. 2(c)]. FIG. 1. (a) Side view of the standing Faraday instability This figure also shows that the amplitude of the vibra- wave pattern obtained for ε > 0. The length of the white tions is the same for both directions. At ε (cid:39) 1, the segment represents the Faraday wavelength (here λ = 5.1 pattern becomes unstable and we observe the formation F mm). (b) Sketch of the experimental setup. (c) Top view of of defects. We note that this threshold value is different the stable square pattern. The white segment has a length from what was found elsewhere [18, 20] as it depends on equaltoλ =5.1mm. (d)TopviewoftheoscillatingFaraday F pattern. Thewhitesegmenthasalengthequalto4λ =20.4 F mm, which is the wavelength of the spontaneous oscillations. The set-up and its imaging system is schematically shown in Fig. 1(b). Diffused white light is shone on the container with a uniform square LED light and a beam- splitterinclinedat45◦enablesustoimagethevesselfrom the top, using a 2048×2048 pixels CCD camera. An ex- ample of the stable pattern obtained is shown in Fig. 1(c),whereonlyafewwavelengthsarerepresented. This image is obtained by strobing the motion at an appropriate frequency, i.e. 18 Hz when the forcing frequency f is 72 Hz (section Spontaneous secondary instability) 0 and 30 Hz when the forcing frequency f is 120 Hz (sec- 0 tion Forced vibrations). Each white dot corresponds to anhorizontalslopeofthefluidinterface,whetheramax- imum, a minimum or a saddle point [16]. There are 4 white spots per Faraday unit cell [Fig. 1(c)]. Spontaneous secondary instability. – Upon increasing the driving amplitude to about twice the threshold value, spontaneous oscillations of the square lattice ap- pear (Fig. 1(d) and Supplementary Movies 1 and 2). These oscillations are in-plane modulations of the pattern along its two main directions. They exhibit a spa- FIG. 2. (a) Typical Fourier spectrum x˜ (ω) of the peak mn tialperiodicityλ=4λ , correspondingtothewhiteseg- (18,18) in the center of the pattern. (b) Modulus of spatial F ment presented in this figure. We label each bright spot Fourier2Dspectrum|yˆ(kx,ky,f)|forε=0.976,averagedfor with indexes (m,n) and we detect their in-plane posi- f =1.57±0.1 Hz. Open diamonds show the position of the Fourier peaks for a stable pattern [Fig. 1(c)]. (c) Amplitude tion (x (t),y (t)) using a standard custom Matlab mn mn of the spontaneous vibration as a function of the normalized algorithm. A typical spectrum corresponding to the pa- controlparameter,averagedoveralltheantinodesofthelat- rameters of Fig. 1(d) is presented in Fig. 2(a). The tice. The up (red) and down (blue) triangles correspond to measured frequency is f = 1.52 Hz with an amplitude |y˜|and|x˜|,respectively. Dashedlineisasquarerootfit. Grey of 0.07 λ . To analyse in more detail the spatial struc- shadesdenote(fromlefttoright): stablepattern,spontaneous F tureofthelatticedynamics, weperformspatio-temporal vibrations of the lattice and chaotic behavior. 3 the depth of the liquid layer, the fluid viscosity and the forcing frequency. Here, we want to point out that the spontaneous oscillations occur at a frequency f much lower than the Faraday frequency f whereas their spatial wavelength F λ (cid:39) 2.0 cm is 4 times larger than λ . In our experi- F mental conditions and at this frequency f, the gravito- capillarydispersionrelation(eq. 1)givesawavelengthof λ =23.26cmmuchlargerthanλ. Thismeansthatthe gc transverse standing wave responsible for the pattern oscillations is governed by a different physical mechanism. Forced vibrations. – We now investigate the charac- teristics of these oscillating modes of the Faraday wave pattern by forcing the vibrations of stable square patterns. We set the Faraday vertical forcing frequency to f =120Hz(resultinginλ =3.5mm),theliquiddepth 0 F to h = 3 mm and the forcing acceleration to ε = 0.81 in order to get a stable and larger initial Faraday square pattern(itssizeisnow35×35Faradaywavelengths). We addtothevesselacustom-madeforcingdeviceconsisting of a comb dipping into the liquid to a small depth [Fig. FIG.3. (a)DeviceusedtoforcethevibrationsoftheFaraday 3(a)]. It is mounted so that it is aligned with one side pattern. The black arrow shows the vertical motion of the of the container, and it vibrates vertically along with it. wholevessel,thedottedwhitearrowshowsthedirectionofthe The comb is set in motion by a second vibration exciter combvibration. Thewidthofthevesselis12cm. (b)Mapof (Bruël & Kjær) to oscillate horizontally in the reference therealpartoftheFFTpeakRe(y˜)foraforcingfrequencyof frameofthecontaineratfrequenciesrangingfrom0.5Hz 3.7Hz. Theforcingdeviceisontheleft,eachpixelrepresents a bright point of our images. (c) Dispersion relation f(k ) to 10 Hz. The distance between the comb teeth is set to T and f(k ). Blue down triangles: transversal waves in the 2λ , and the amplitude of the forcing sinusoidal motion L F y direction. Red up triangles: transversal waves in the x is set to half the Faraday wavelength. This allows us to direction. Opencircles: longitudinalwavesintheydirection. generate a sinusoidal oscillation of the line of Faraday Solid black line: linear fit. Dashed line: gravity-capillary peaks located below the forcing comb. wave dispersion relation. Dotted line and grey background: We observe a transversal wave that propagates away prediction from eq. 4 and its associated uncertainty. from the forcing device at the forcing frequency f. We detect the position (x (t),y (t)) of each bright spot mn mn and perform a temporal Fourier transform to obtain presents the dispersion relations f(kT) (blue and red tri- (x˜mn(f),y˜mn(f)). Fig. 3(b) displays Re(y˜(f)) for the angles) and f(kL) (open circles) that we obtain for f excitation frequency f = 3.7 Hz. We observe a periodic ranging from 0.5 to 10 Hz. We first notice that f(kL) patternthatdecaysalongthexdirection(indexedasm) obeysthestandardsurfacewavesdispersionrelationpre- away from the forcing device. This corresponds to the dictedbyeq. 1(dashedline). Thismeansthattheforcing propagation of a transverse wave in the x direction at device induces gravito-capillary waves. Their dispersion the forcing frequency f (the motion is along y). We de- relationappearsquitelinearinFig. 3(c)sincetheshallow fine the spatial phase φy of the pattern vibration along water approximation applies (kFh (cid:28) 1). On the other y as φy(x,t)=φ0exp[(−α+ikT)x+i2πft] where 1/α is hand, the dispersion relation for transverse waves f(kT) the decay length of the oscillation and k its wavenum- is markedly different. We observe a linear increase of f T ber. FromtheexperimentaldataweextractαandkT for with kT with a much lower slope. A linear fit gives the each value of f. The decay length 1/α does not depend phase speed of the transverse waves cT =4.60 cm.s−1. significantly on f and its typical value is 1/α (cid:39) 11.5λ , Physical interpretation. – Theseexperimentalresults F whereasthevalueofk dependsonf. Duetotheimper- show that there exists a new type of wave propagating T fectionsoftheforcingdevice, wealsonoticethepresence at the fluid-air interface. They are transverse waves as- of a periodicity in the y direction (indexed as n), cor- sociated to the presence of a pre-existing Faraday wave responding to a longitudinal wave propagating in the y pattern and reminiscent of 2D shear waves that propa- direction with wavenumber k that we extract form Fig. gate in elastic media. Here we present a quasi-2D model L 3(b). We perform the same analysis on Re(x˜), for which inwhichweidentifytheFaradaycellularpatterntoa2D we have similar maps as Fig. 3(b). Altogether, we re- metamaterial with solid-like properties. Indeed, trans- port the existence of transverse waves along both the x verse waves in an elastic material propagate with con- and y direction, as well as longitudinal waves. Fig. 3(c) stant phase velocity c that only depends on the elastic T 4 Due to surface tension there is an energy cost that depends on the applied shear deformation ∆E(γ) = σ∆S(γ). We then define the effective elastic energy density per unit area W = σ∆S/λ2 (in J.m−2) and in- S F troduce the effective shear modulus µ of the Faraday S wave pattern: W = 2µ (cid:15)2 where (cid:15) = 1γ. Following S S xy xy 2 standardelasticitytheory[30]thetransverseelasticwave phase velocity c in a 2D elastic medium is written T (cid:115) (cid:114)µ σS c = S = γγ , (4) T ρ ρ λ2 S S F with ρ the density per unit area, defined as ρ =ρA. S S Usingequations2and3,thevelocityweobtainisc = FIG. 4. Open circles: evolution of S(γ) with γ computed T numerically for λ = 3.5mm and A /λ = 13.5%. Dark 4.84±0.63 cm.s−1, which is in excellent agreement with F 0 F line: theoreticalpredictionfromeq. 3. Inset(left): Reference the experimental result of 4.60 cm.s−1. We represent in surface. Inset (right): Sheared surface, with tanθ=γ. Fig. 3(c) the estimated dispersion relation (dotted line), the grey background representing the uncertainty. Conclusion. – We have characterized a new sec- shear modulus µ. We use the structure of the Faraday ondary instability that arises in 2D Faraday patterns wave lattice and the fluid properties to derive an effec- close to the transition towards chaos. This instability tive elastic shear modulus and quantitatively predict the leads to vibrations of the Faraday pattern similar to a transverse waves properties. 2D transverse elastic wave. We established the disper- We consider a reference state for the interface defined sion relation for these waves and showed that it differs as markedly from the standard gravito-capillary waves that (cid:18) (cid:19) (cid:18) (cid:19) x+y x−y propagateattheliquid-airinterface. Weproposeaphys- z (x,y,t)=A(t)cos π cos π , (2) 0 λF λF ical mechanism that combines the surface tension with the pre-existing Faraday wave structure at the interface. with A(t) = A cos(2πf t) the amplitude of the sta- 0 F We are able to derive an effective shear modulus µ for tionary wave. Measurements (e.g. on Fig. 1(a)) give S theFaradaywavepatternthatquantitativelyagreeswith A /λ = 13.5%±3.5%. This 2D function gives a suc- 0 F the experimental observations. cession of peaks and crests arranged in a square pattern tilted at 45◦, as represented in Fig. 4 (inset on the left). In this work, we observe the emergence of a new physical property, namely an effective 2D elasticity, at the We apply a shear strain γ =tanθ to this elementary cell liquid-air interface. Our interpretation reveals that it is [Fig. 4 (inset on the right)] and calculate analytically its intimately related to the existence of a periodic pattern surface area S(γ) imprinted on the liquid interface. From this perspective, the Faraday wave pattern creates a mechanical metama- (cid:90)λF λF(cid:90)+γy (cid:34) (cid:18)∂z (cid:19)2 (cid:18)∂z ∂z (cid:19)2(cid:35)1/2 terial at macroscopic scale. In the future, we would like S(γ)= dy dx 1+ 0 + 0 −γ 0 to investigate in more detail the limit k/k = 1/2 cor- ∂x ∂y ∂x F 0 γy responding to the edge of the first Brillouin zone in a (3) crystalline material. Another line of future research is As f (cid:28) fF we average S(γ) in time which corresponds to understand if there exists a second elastic constant to replacing A(t) with A= A02/π. Fig. 4 shows the for the medium as in usual elastic solids. More gener- numerical evaluation of S(γ) for A0/λF = 13.5%. As ally, wave-based metamaterials offer unique possibilities S(γ) is an even function (γ and −γ give the same area), as wavelengths and patterns can be dynamically tuned. (cid:12) ∂S(cid:12) = 0. For a non-zero amplitude of the Faraday The authors would like to thanks E. 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