Vibration of soap films and Plateau borders, as elementary blocksof a vibrating liquid foam

F. Eliasa,b,, S. Kosgodagan Acharigea,1, L. Rosea, C. Gaya, V. Leroy,a, C. Dereca,

aLaboratoire Matiere et Systemes Complexes (MSC), Univ. Paris-Diderot, CNRS UMR 7057 - Paris, FrancebSorbonne Universites, UPMC Universite Paris 6, UFR 925 - Paris, France

Abstract

The propagation of an acoustic wave in a liquid foam results from the coupling of a pressurewave in the gas phase and the vibration of the liquid backbone of the foam. At the bubble scale,the foam liquid skeleton is made of soap films connected by liquid channels. We study herethe transverse vibration of those constitutive elements. The measurement of the velocity andattenuation of the transverse wave on each element isolated on a rigid frame, compared with ananalytical modeling, reveals the main sources of inertia, elastic restoring forces and dissipation,for frequencies ranging from a few tens of Hz to a few kHz. In the case of a transverse wavepropagating on a single soap film, we show that (i) the wave velocity is set by the surface tensionand the inertial mass of the film loaded by the surrounding air, and (ii) that the damping of thewave is mainly due to the viscous dissipation in the air. In the case of a transverse wave propa-gating along the junction line between three soap films (Plateau border), the dispersion relationreveals two different scalings at low frequency and at high frequency, which are interpreted byconsidering the role of the vibration of the adjacent soap films, and the role of the inertia ofthe liquid inside the channel. The attenuation of the transverse wave along the liquid channel ismeasured in the low frequency regime. In both investigated cases (transverse wave propagatingon a soap film as well as on a liquid channel), we show that the surrounding gas plays a dominantrole, whereas the role played by the interfacial rheology is negligible.

Keywords: soap film, Plateau border, transverse wave, fast dynamics, interfacial rheology,foam acoustics.

1. Introduction

Foams are typical examples of complex fluids, whose macroscopic properties depend on themicrostructure of the material. A liquid foam is a dispersion of gas bubbles in a liquid matrix,stabilized by tensioactive molecules or particles. The liquid skeleton, organized to minimizeits interfacial energy, is structured by a few geometrical rules known as Plateau’s rules: thesoap films between bubbles meet by three in liquid channel also called Plateau borders, which

Email address: florence.elias@univ-paris-diderot.fr (F. Elias)1Present adress: Laboratoire de Physique (UMR CNRS 5672), ENS de Lyon, 46, allee d’Italie, F-69364 Lyon cedex

07, France

Preprint submitted to Colloids and Surfaces A February 18, 2017

Soap film

Plateau border

Vertex

Figure 1: At the bubble scale, the vibrating skeleton of the foam is composed of coupled soap films and Plateau borders,and four plateau borders meet at a vertex. Here, the bubble size is millimetric and the volume fraction of liquid in thefoam is about 1 percent.

themselves meet fourfold at the vertices of the liquid network (see Fig. 1) [Hutzler (1999); Cantat(2013)]. Models at the scale of the bubble are needed to explain the macroscopic behaviour offoams, such as their complex rheological response under shear [Cohen-Addad (2013)], theirelectrical conductivity [Lemlich (1978)], the drainage of the liquid phase out of the foam [Stone(2003)], or the filtration of solid particles by a liquid foam [Haffner (2015)].

Amongst the foam physical behaviours, a still little explored domain is the acoustic propa-gation in foams. Foams are nevertheless used for mitigating blast waves, thanks to their strongacoustic attenuation [Clark (1984); Raspet (1987)]. At smaller acoustic amplitudes, the acousticvelocity and attenuation in the foam have been shown to depend on the liquid content [Goldfarb(1997)] and on the bubble size [Mujica (2002)].

It has been recently shown that the acoustic propagation in a foam strongly depends onthe frequency and the bubbles average diameter d [Ben Salem (2013); Pierre (2014)]. Severalregimes of propagation have been identified: two non-dispersive regimes at low and high fre-quencies, separated by a resonance, with a maximal attenuation and a negative density behavior.A model at the scale of the bubble successfully explains these three regimes [Pierre (2014)]. Thepropagation of an acoustic wave in a liquid foam couples a pressure wave which propagates inthe gas and the vibration of the liquid skeleton. Due to the cellular geometry of the foam, thecompression wave generates transverse vibrating waves in the liquid phase. Hence, the relevantlength scale for describing the acoustic propagation in foams is not the acoustic wavelength inthe foam, which is, in general, several orders of magnitude larger than d 2, but the wavelength ofthe vibration wave on the liquid interfaces, at the forcing frequency.

In this article, we investigate the vibrations of the liquid skeleton of a dry foam. We isolateone by one the constitutive elements of the liquid skeleton, and we measure the dispersion rela-tion and the attenuation of a transverse wave propagating on each of these elements. In section 2,we study an isolated soap film submitted to a transverse vibration (bending wave). We measurethe phase velocity and the attenuation of the wave. We develop a theoretical modeling to identifythe relevant parameters that contribute to the inertial and elastic response of the vibrating films,and to the dissipative effects.

When the soap films meet at a Plateau border junction, the inertia of the liquid in the channelmust be taken into account, as well as the tensile surface forces exerted by the soap films on thePlateau border. Does the Plateau border vibrate like the free border of a liquid membrane, or doesit behave as an inertial liquid string? In section 3, we show that the answer is in between: tworegimes are identified as a function of the frequency. In section 4, we finally replace the resultsin the light of the recent studies of the acoustic propagation in a liquid foam, and we discuss the

2The typical acoustic wavelength in air is about 3 mm at 100 kHz, and about 30 cm at 1 kHz2

outlooks of this work.This article is a synthesis of results obtained in previous publications [Kosgodagan (2014)

and Derec (2015)], and complementary results. All those results are explained in this article inorder to get a complete picture of the dynamics of the studied systems.

2. Transverse vibration of a single soap film

We consider a horizontal soap film, surrounded by air, which is freely suspended on a rigidframe. When the frame is vibrated vertically using an electromagnetic shaker, a wave is created atthe periphery of the soap film and travels up to its center where it is totally reflected (in the linearlimit where the amplitude is small compared to the wavelength). Then a transverse standing vi-bration takes place on the soap film. Several vibration modes are theoretically predicted [Couder(1989)]: a so-called symmetric mode, where the interfaces of the film undulate in anti-phase,and an antisymmetric mode where they undulate in phase with each other. In our experiments,no thickness variation is associated to vibration, which means that the soap film vibrates in theantisymmetric mode, as sketched in Fig. 2. To characterize the antisymmetric wave appearing onthe soap film, we develop two experimental setups in order to determine the complete dispersionrelation (real part and imaginary part): the first setup gives access to the measurement of thewavelength as a function of the frequency for different thicknesses of the soap film; the secondsetup allows to measure the dissipation as a function of the frequency, in the case of thin films.The results are compared to the predictions of a model detailed in a previous article [Kosgodagan(2014)].

Figure 2: Antisymmetric wave on a soap film, i.e. with the interfaces of the film vibrating in phase with each other. A isthe amplitude of the wave, λ the wavelength, and e the thickness.

2.1. Wavelength

2.1.1. Setup and measurementsA horizontal soap film is formed upon a cylindrical plexiglas tube of 16 mm of diameter,

which is mounted on an electromagnetic shaker driven by a harmonic excitation. The wavelengthλ is measured by visualisation of the reflection of a parallel light beam by the soap film (see Fig.3a). When the soap film is vibrated, the light beam is deflected by the local slope of the film andconcentrate into bright areas (caustics) pointing out the positions of the antinodes of the standingwave. The transverse displacement on the soap film of cylindrical symmetry is:

ζ(r, t) = A0J0(q f r)ei(ωt+φ0) (1)

where ζ(r, t) is the vertical displacement at a distance r from the center of the film and at timet, A0 is the amplitude at the center of the film, J0 is the Bessel function of the first kind andof zero order, q f = q′f + iq′′f is the complex wavenumber (the subscript ‘ f ’ stands for f ilm),ω = 2π f is the exciting angular frequency, f is the frequency and φ0 the phase at r = 0 and t = 0.

3

The positions of the antinodes correspond to the extrema of the Bessel function J0. For a giventhickness of the soap film, a linear frequency sweep is imposed in the frequency range 100 Hz- 5 kHz. Images are recorded during the frequency sweep and the wavelength λ is measured oneach image using image analysis. The soap films are made from a solution of water with TTAB(tetradecyltrimethylammonium bromide) at different concentrations. After its creation upon thecylindrical tube, the soap film begins to drain and its thickness decreases from few microns toaround 100 nanometers within few minutes. During the drainage process, measurements aremade at different thicknesses. Each frequency sweep is fast enough (duration 2 s) to considerthat the thickness remains constant during the sweep (see Fig. 4) 3. Independently the thicknessat the center of the soap film is measured using a spectrometer, just before the beginning of eachsweep.

soap film

shaker

parallelwhite light

spectrometer

translation stage

PSD

lasersemi-

transparent plate

videocamera

(a) (b)

Figure 3: Vibration of a single film: setups. (a) Measurement of the wavelength by visualisation of the antinodes afterreflection of a parallel light beam on the soap film: this setup allows a rapid measurement (faster than the draining timeof the soap film), but the vibration amplitude can not be measured. (b) To quantify the attenuation of the wave, theamplitude of vibration as a function of the frequency is measured in one point of the film by collecting the deflection bythe film of a laser beam in a position sensitive detector (PSD). The profile of the deformation along the diameter of thefilm is obtained in about 15 seconds using a translation stage.

The experiments have been performed using solutions with different concentrations of sur-factants, and for each solution at different thicknesses (see Table 1). The experimental results forthe wavenumber q′f as a function of the frequency are shown in Fig. 5. We can note that at agiven frequency, q′f tends to increase with thickness.

2.1.2. Comparison to theoretical predictionsIn order to understand how the wavelength is modified by the film thickness and surface

tension, we now compare our experimental results to the predictions of a model (detailed in[Kosgodagan (2014)]). We just recall here the main steps of this modelling.

3The exposure time of each image is both much smaller than the frequency sweep duration (thus one image is associ-ated to a single exciting frequency f ) and much larger than 1/ f (so that each image corresponds to the average enveloppeof the soap film vibration).

4

(b)(a) (c) (d) (e)

Figure 4: Images of the vibrating soap film during the frequency sweep. (a) f = 0.41 kHz, (b) f = 0.77 kHz, (c) f =

1.2 kHz, (d) f = 2.5 kHz, (e) f = 5.4 kHz. The soap film diameter is 16 mm and the total sweep duration is 2s. Thecolors are due to the interference of the light reflected by the soap film interfaces (Newton colors ), using a white lightillumination: different colors correspond to different soap film thicknesses. Note that the color pattern does not changeduring the sweep, showing that the thickness profile remains the same.

c (g/l) γ (mN/m) emeas (µm)1.2 37 ±2 •1.2 - ▪0.8 - ⬩0.31.4 34 ±2 •1.3 - ▪1.1 - ⬩1.11.6 35 ±2 •2.5 - ▪1.3 - ⬩0.72.0 36 ±0.5 •2.1 - ▪1.5 - ⬩1.15.0 35 ±0.5 •1.0 - ▪0.8 - ⬩0.5

Table 1: Different solutions used: bulk TTAB concentration c, measured surface tension γ. For each solution the ex-periments have been performed at three different times during the drainage process: the corresponding thickness emeas,measured at the center of the film with the spectrometer, is indicated.

The liquid of the soap film and the surrounding gas are both considered as viscous and incom-pressible fluids, of respective densities ρ and ρa. The thickness of the soap film is assumed to beconstant and equal to e (see Fig. 2). The calculations are performed in the approximation of thelong wavelength limit, i.e. λ large compared to the thickness and to the amplitude of vibration ofthe soap film.The Navier-Stokes equations are written for the liquid and for the air, leading to the general ex-pressions of the pressure and velocity fields in both fluids. These quantities are then linked by thecontinuity conditions at the liquid-air interfaces. The continuity of the tangential stress describesthe equilibrium between the tangential viscous forces, in the liquid and in the air, and the gradi-ent of the interfacial stress that depends on the viscoelastic interfacial modulus. The equation ofcontinuity of the normal stress balances the pressure jump across the liquid-air interfaces withthe normal force due to the interfacial curvature and the viscous normal forces.The calculations give a whole expression of the complex wavenumber q f = q′f + iq′′f . In ourexperimental conditions, this relation is simplified to the leading order, and we obtain the phasevelocity v linking the real part of the wavenumber q′f = 2π/λ and ω = 2π f :

v = λ f =ω

q′f≃

√2γ

ρe + 2ρa/q′f(2)

The comparison between the experimental results and this prediction is shown in Fig. 6a, wherethe product (q′f/2π) ×

√2γ/(ρe + 2ρa/q′f ) is plotted as a function of f . For each experiment,

the thickness has been set to ead j = ⟨(2γ/ρ)(q′f/ω)2 − 2ρa/(ρq′f )⟩ given by Eq. 2, where⟨...⟩ stands for the average over all the data in the same frequency sweep. In Fig. 6, ead j iscompared to the value of emeas, measured in the centre of the soap film, and taking into account

5

Figure 5: Experimental results: real part of the wave number q′f = 2π/λ as a function of the frequency f for differentfilm thicknesses and solution concentrations. The legend of the symbols is detailed in Table 1.

the thickness gradient within the films 4. The thickness gradient is estimated by the observationof the interference fringes (see photograph in Fig. 6b): from one interference order to the nextone, the optical path varies by δ ≃ 550 nm, therefore the thickness varies by δ/(2n) ≃ 200 nm,where n = 1.38 is the measured refractive index of the soap solution. Fig. 6b shows that theadjusted average thickness ead j is compatible with the measured interval emeas, centered on themeasurement in the middle of the film using the spectrometer.

Finally, we conclude that the data collapse on the same master curve, which is well describedby the theoretical prediction, showing experimentally that the role of γ and e are well describedby Eq. 2.

2.2. Dissipation of the waveIn the case of a monolayer at the surface of a liquid, the attenuation of a transverse wave

is determined by measuring the decay of the wave amplitude as a function of the distance tothe excitation (see [Stenvot (1988)]). Here, the decay length is much larger than the size of thesoap film, thus this technique is not possible. In [Kosgodagan (2014)], we have measured thewave attenuation using the second setup (Fig. 3b). The principle was to measure the amplitudeof vibration of the soap film as a function of the frequency in order to extract the measurementof the attenuation from the widths of the resonances. The amplitude was obtained by recordingwith a position-sensitive detector (PSD) the deflection of a laser beam (of waist around 100 µm)reflected by the soap film. The deflection was measured along a film diameter using a translationstage. After a fit with the appropriate Bessel function, we were able to extract the value of theamplitude of deformation A0 at the center of the film (see Eq. 1). For each scan along a diameter,the frequency was changed. We were thus able to obtain the variation of A0 as a function of thefrequency in about 200 minutes, which is long compared to the characteristic time of drainageof the solution used for the previous experiment. Therefore experiments were performed with athin film (thickness of a few tens of nanometers), such as e ≪ 2ρa/(ρ q′f ) (see Eq. 2), hence the

4The thickness gradient is due to the fact that the soap films are slightly tilted from horizontal to avoid the formationof a liquid dimple in the center.

6

(a)

0 1000 2000 3000 40000

1000

2000

3000

4000

f (Hz)

q’f x

raci

ne /

2pi

q0 fp

.../2⇡

(Hz)

f (Hz)

(b)

0 1000 2000 3000 40000

1000

2000

3000

4000

f (Hz)

q’f x

raci

ne /

2pi

0 1000 2000 3000 40000

1000

2000

3000

4000

f (Hz)

q f’ (1/

m)

q0 fp

.../2⇡

(Hz)

q0 f(1

/m

)

f (Hz)

f (Hz)0 1 2 3

0

1

2

3

eadj (µm)

e mea

s (µm

)

0 1 2 30

1

2

3

eadj

(µm)

e mea

s (µm

)

Figure 6: Comparison between the experimental results presented in Fig. 5 and the prediction from Eq. 2. (a) (q′f /2π)×√2γ/(ρe + 2ρa/q′f ) is plotted as a function of the frequency f for different film thicknesses and solution concentrations.

The data are aligned along the first bisector (line) as predicted by Eq. 2. The legend of the symbols is detailed in Table 1.For each soap film, the thickness has been adjusted to the best plot (see text). (b) Measured thickness (at the center of thefilm) versus adjusted thickness. The error bars are given by the thickness variation within each soap film. Photograph:the interference pattern allows to estimate the thickness variation within the film (estimated here to 1 µm). The soap filmdiameter is 16 mm.

film thickness e played no role in the investigated phenomena. The soap film was made of wateradded with TTAB (2.8 g/l), glycerol (10 wt%) and dodecanol (0.04 wt%). Its surface tension was22.5 mN/m.

The experimental measurements of the maximum amplitude A0 as a function of the frequency(in the range 300 Hz - 1600 Hz) exhibit four resonances, each of which was fitted using thisexpression :

A0 =α

∣J0(q f R)∣(3)

where q f = q′f + iq′′f and α, R and q′′f were three fitting parameters. The parameters α and Rdescribed the effective forcing applied to the soap film by the vibrating cell. The characterisa-tion of the width of each resonance curve was given by q′′f , which described then the intrinsicattenuation of the wave on the soap film at the corresponding frequency.

The model mentioned in Sec.2.1.2 and described in detail in [Kosgodagan (2014)], after sim-plification to the leading order in our experimental conditions, predicts the following imaginary

7

part of the dispersion relation:

−q′′f ≃ q′ 2fδa(ω)

3 + q′f e ρ/ρa(4)

where δa(ω) =√ηa/(2ρaω) is the thickness of the viscous boundary layer in the air. This

relation predicts that the dominant source of dissipation of the antisymmetric wave in our ex-perimental conditions is the viscous friction in the air. As shown in [Kosgodagan (2014)], theattenuation measured experimentally is very well described by this theoretical prediction.

2.3. Conclusion

We have developed two setups in order to measure the complex dispersion relation (wave-length and dissipation) of a transverse antisymmetric wave on a soap film. The experimentsvalidate the predictions of a model simplified in our experimental conditions. Namely, equation(2) shows that the phase velocity is given by the ratio between the elastic restoring force, dueto surface tension, and the inertia of the system where the inertia of the air must be taken intoaccount (as already observed, see [Joosten (1984); Couder (1989); Vega (1998); Afenchenko(1998)]). The imaginary part of the dispersion relation (Eq. 4) shows that the dominant sourceof dissipation in the system is the viscous friction in the air.

Note that the calculations have been performed considering the properties of the liquid-airinterface, by taking into account the complex interfacial viscoelasticity. We have shown that itplays a negligible role here and can be neglected in Eqs. 2 and 4. This is very different from whatis observed for wave propagation on a surfactant monolayer, where the interfacial viscoelasticityis the dominant cause of dissipation [Stenvot (1988)]. The respective roles of the interfacialviscoelasticity in the case of a monolayer and in the case of a film are discussed in Appendix A.

3. Transverse vibration of a Plateau border

Let us now go one step further into the complexity of foam vibrations by considering thePlateau border (PB). In a dry foam, a Plateau border contains the liquid in a channel that separatesthree soap films. Let R be the width of the channel. How does such a channel behave when thefoam liquid skeleton is vibrated: like a liquid string with its own tension and inertia or like thepassive geometrical boundary of a vibrating liquid membrane?

3.1. Model: coupling soap films and a liquid channel

The PB is at the junction between three soap films. Each soap film vibrates following thedispersion relation (Eq. 2) and the attenuation (Eq. 4). However, the mass of the PB is concen-trated at one edge of the film. Hence, the vibrating PB consists in a coupled system of a liquidchannel of mass per unit length µ = ρS PB = 0.161ρR2 (where S PB is the PB cross-section),and of soap membranes loaded by the air. The surface tension of the liquid interfaces inducesa restoring force not only on the liquid membranes to bring them back flat, but also on the PBto bring it back straight via the tension forces exerted by the soap films onto the liquid channel[Elias (2014)]. In order to derive the equations of motion of the system, the problem is simplifiedby considering that the (zOx) plane is a plane of symmetry: only the films in the (yOz) planeand in the (y′Oz) plane are transversally deformed, and the deformation of the PB occurs in theplane of symmetry (xOz). The computation is detailed in a previous article [Derec (2015)] andwe summarize here the main results. In this model, the attenuation of the wave along the PB is

8

O x

z

y

O

z

yζ

y’

x

y

y’R

(a) (b)

x

x

y

y’

ζ

2π/3

u

Figure 7: Sketch of a Plateau border at the junction between three soap films: (a) side view (top) and top view (bottom):R is both the width and the radius of curvature of the PB. (b) Transversally vibrated Plateau border and adjacent soapfilm: u(z, t) is the displacement of the liquid channel and ζ(y, z, t) is the transversal displacement of one of the soapfilms.

neglected. We consider a harmonic wave u(t, z) = u0 ei(ωt−qz) propagating along the PB (with q areal number). In the case of a finite PB radius R and at a finite frequency, a plane harmonic wavein the soap film ζ(y, z, t) = ζ0 ei(ωt−qyy−qz) is solution of the coupled equations of motion, with:

qy = −iµω

2

3γ (5)

and the dispersion relation:

q2= q

′2f + (µω

2

3γ )2

(6)

where q′f is the wavenumber on the soap film, given by Eq. 2 [Derec (2015)]. According to Eq.5, which implies that iqy is a positive real number, the deformation in the soap film must relaxexponentially in the direction y perpendicular to the PB. In the limit of thin liquid films and largewavelength, 2ρa/q′f ≫ ρe, Eq. 6 becomes:

q2= (ρa

γ )2/3

ω4/3+ρe3γ ω

2+ (µω

2

3γ )2

(7)

The first term of the right hand side of Eq. 7 contains the inertia of the air loading the vibratingfilms. It scales like ω4/3 and is thus dominant at low frequency. The second term corresponds tothe role played by the liquid inside the film. Finally, the role of the inertia of the PB is representedin the last term; this term scales like µ2

ω4, hence the inertia of the PB plays a relevant role in the

dispersion relation at high frequency or for a thick PB.

9

3.2. Setup and measurementsThe transverse deformation of the soap films and of the PB can be visualized using the setup

presented in Fig. 8. A vertical PB is isolated by pulling a rigid prismatic frame out of a soapsolution (Fig. 8a). The soap solution (of surface tension 30 mN/m) is made of distilled water, acommercial dishwashing liquid (Fairy liquid 1%vol.) and glycerol(2%vol.). The PB is vibratedtransversally by plunging a thin capillary, which can be displaced along the x direction usingan electromagnetic shaker. As a result, a vibrating transverse wave propagates along the films(yOz) and (y′Oz) (see notations on Fig. 7), and along the PB. Soap solution can be injected inthe PB at a constant flow rate Q through the capillary, so the PB radius R(Q) can be adjusted bychoosing the appropriate value of the flow rate as described in a previous article [Elias (2014)].5

To observe the deformation of the PB under a transverse vibration, two setups are developed.The first setup allows to visualize the deformation of the adjacent soap films under continuousforcing (Fig. 8b); the second set-up is designed to measure the velocity of a transverse pulsealong the liquid channel (Fig. 8c).

camera

gridf

flow rate

frame

capillary

(a) (b)

xx

y

δ

h

high-speed camera

x

y

(c)

shaker

zoom

Figure 8: Sketch of the setup. (a) Side view : a vertical Plateau border is formed on a prismatic frame; it is transversallydeformed by vibrating a glass capillary inserted in the top vertex; the capillary is connected to an electromagnetic shaker.The PB radius can be varied by injecting soap solution at a constant flow rate within the PB trough the capillary. (b) and(c) Top views: in (b), the transverse deformation of one soap film under harmonic forcing is visualized by recording theimage of a square grid reflected on the soap film; in (c), the displacement of the Plateau border is measured at a fixedheight using a high-speed camera fitted with a zoom objective. Set-up (b) allows a global visualisation, and set-up (c)permits a quantitative measurement of the propagation of the transverse wave (velocity and attenuation).

3.3. Harmonic forcing: observationsThe transverse deformation of the film in the plane (yOz) is visualized by recording the

image of a square grid reflected by the soap film (Fig. 8b): the image of the grid remains squarewhen the soap film is flat, and undulates when the film undulates transversally (Fig. 9). The lightis strobed at the forcing frequency to allow the capture of a stable image at each frequency.

The transverse harmonic wave propagates from the tip of the capillary and is reflected bythe rigid edges of the frame. The transverse displacement of the soap film is a superposition ofwaves travelling in different directions as shown in Fig. 9. Those images show that when the

5The liquid flow rate within the PB is chosen small enough so that the vertical flow velocity inside the PB remainsnegligible with respect to the transverse velocity of the PB during the vibration.

10

(b)(a) (c)

Plateau bordercapillary

Figure 9: Image of the square grid projected on the plane of the deformed film (Fig. 8b), at different frequencies f =

250 Hz (a), f = 650 Hz (b) and f = 1000 Hz (c). The white bar in (a) represents 1 cm. The dimensions of the frame(defined in Fig. 8a) are δ = 5 cm and h = 12 cm for this experiment. On each image the wavelength has been estimatedby measuring the mean distance between neighbouring antinodes: we found λexp ∼ 14.5± 1.5 mm, 6.7± 0.3 mm and5.1± 0.5 mm (from left to right). Theses values can be compared to the data computed from Eq. 2 (with λ = 2π/q′f andfor a thin film q′f e ≪ ρa/ρ): λ=13 mm, 6.4 mm and 4.7 mm respectively for the corresponding frequencies.

forcing frequency increases, the wave on the soap film is damped far from the tip of the capillary.The wavelength on the soap film can be directly measured from the images of the deformedgrid, and the values are compared to the predictions given by Eq. 2 (see caption of Fig. 9):the comparison is good, which suggests that the model presented in section 2, describing thewavelength of the antisymmetric wave on an infinite soap film, remains relevant even when thesoap film is connected to a Plateau border. This will be attested in the next section.

In addition, Fig. 9c shows the propagation of a transverse wave along the PB, localized closeto the PB and strongly attenuated in the direction perpendicular to the PB. This is consistent withthe prediction of Eq. 5.

3.4. Experimental determination of the dispersion relation

In [Derec (2015)], we have measured the dispersion relation of the wave along the PB, usingthe setup described in Fig. 8c. A transverse pulse was emitted at the tip of the capillary, andthe pulse propagation along the PB was followed using a high-speed camera fitted with a zoomobjective. The dimensions of the frame was δ = 20 cm and h = 23 cm, large enough to postponesufficiently the arrival of the reflected pulse coming from the soap film boundaries or from theother end of the PB (bottom vertex). The displacement u(z, t) was obtained from the image

11

analysis. The wave velocity and attenuation as a function of the frequency were deduced froma Fourier analysis. Soap solution could be added at a constant flow rate in the PB through thecapillary to tune the radius R. Two regimes were identified on the dispersion relation q( f ): atlow frequency, q ∝ f 2/3 whatever the R, whereas the dispersion relation tends asymptoticallytowards a power law q ∝ f 2 at high frequency. The larger the PB radius, the smaller thefrequency at which the transition between both regimes is observed. The low-frequency regimecorresponds to Eq. 2 in the limit q′f e ≪ ρa/ρ (which is the case in the experiment): the PBvibrates at low frequency like the passive border of the soap film, its own inertia having no effecton the wave propagation velocity. The high-frequency regime corresponds to a balance betweenthe PB inertia µu and the restoring forces exerted by the soap films pulling on the PB. In otherwords, the dispersion relation of the transverse wave propagating along a PB is equal to that of aliquid membrane at low frequency (q ≃ q′f in Eq. 6), and is equal to that of a liquid string at highfrequency (q ≃ µω

2/(3γ)).

3.5. DissipationAlthough the attenuation of the transverse wave along the Plateau border was, in a first step,

neglected in the presented model, we have measured it at low frequency. The measurementconsists in studying the relaxation of the vibration of the PB after the excitation is stopped. Aloudspeaker, placed in front of the PB, emits a plane wave propagating in the x direction. Amonochromatic continuous sound causes the vibration of the PB and the adjacent soap films atthe forcing frequency f0, as long as t < 0. The forcing is stopped at t = 0 and the relaxationu(z, t) is measured using a high speed camera. At this stage, u(t) at a fixed height is wiggly andthe data must be treated in order to extract an attenuation time. We analyse the Fourier transformu( f ) around the forcing frequency: u( f ) presents a maximum at f ≃ f0. Assuming that, aroundf0, u(t) is a damped sinusoid at the forcing frequency, the measurement of the width ∆ f at thehalf of the maximum height (see insert in Fig. 10) gives the attenuation time at f = f0:

τ( f0) ≃√

3π∆ f

. (8)

The values of τ obtained as a function of the exciting frequency are shown in Fig. 10, fordifferent R.6 Note that these values are compatible with the one obtained by another technique,as detailed in Appendix B.

Those measurements can be compared to the attenuation time of a transverse wave along asoap film, τ f . This time can be computed using Eqs 2 and 4, and using τ f = 1/(vq′′f ) wherev = ω/q′f is the phase velocity. In the approximation q′f ≪ 2ρa/(ρe), we get:

τ f ( f ) ≃ 3 ( γρa)

1/3(2ρaηa

)1/2 1

(2π f )7/6. (9)

In Fig. 10, the data τ( f ) are adjusted for comparison by a power law τ = B f−7/6 wherethe prefactor B is a fitting parameter. The best fit gives B ∼ 7.4 s−1/6 whereas Eq. 9 predicts a

6In this setup, the capillary (see Fig. 8a) has been removed to avoid any obstacle to the PB oscillation. R is still tunedby injecting soap solution at a constant flow rate in the PB, but the liquid is injected via the three top vertices of theframe.

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prefactor of 35.6, about 5 times larger: the attenuation time of the wave on the PB is significantlysmaller than the attenuation time of the wave on a single infinite soap film, i.e. the dissipationis stronger. This additional dissipation does not seem to come from a dissipation in the liquidchannel (e.g. viscous dissipation), since the data in Fig. 10 corresponding to different PB radiicollapse on the same plot (within error bars). Therefore, the additional source of dissipation islikely to reside in the flow of the air around the PB. The geometry of the soap films, which meet inthe PB, confining the surrounding air in an edge, is in fact somewhat different from the geometryof an infinite soap film. Moreover, the finite size of the soap film has not been considered inthe theory. This is likely to enhance the dissipation prefactor. Furthermore, another source ofattenuation could emerge from the finite size of the PB itself whose vertices are also set intovibration by the loudspeaker.

0 100 200 300 400 5000

20

40

60

80

f0 (Hz)

τ (

ms)

400 5000

f (Hz)

FT

[u

(t)]

∆f

f0

Figure 10: Characteristic attenuation times τ of a transverse wave propagating along a Plateau border, after the forcingat frequency f0 has been stopped. The different symbols correspond to different flow rates injected in the PB and thusdifferent radii: ▲ R < 0.1 mm, ● R ≃ 0.2 mm , ▶ R ≃ 0.3 mm and � R ≃ 0.35 mm. The dashed line represents thebest fit of the data using a power law τ = B/ f 7/6 (see text). Insert: Fourier transform of the signal around the forcingfrequency f0 = 430 Hz: the width ∆ f of the curve is linked to τ using Eq. 8.

3.6. ConclusionIn this section, two regimes have been evidenced in the dispersion relation of the transverse

wave on a PB: at low frequency, the dispersion relation is similar to that of a soap membraneloaded by the air, whereas at high frequency, the inertia of the PB has to be taken into account.

The wave attenuation has been measured in a frequency range corresponding to the lowfrequency regime of the dispersion relation. In this regime, the attenuation varies with frequencyas expected for a transverse wave along a single infinite soap film, although the prefactor is aboutfive times stronger. This is probably the result of the air flow structure in the 120 degree sectorsbetween the films which is more confined than that in the single flat film geometry assumed inthe theory.

In the high-frequency regime of the dispersion relation, the transverse wave was stronglyattenuated and no measurement was possible. However, the image presented Fig. 9c shows that

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at 1000 Hz, a wave propagates along the Plateau border although it is strongly attenuated inthe soap films. Our efforts concentrate now on measuring this attenuation in the high-frequencyregime, and in modeling this attenuation in the whole frequency range [10 Hz, 1000 Hz].

4. Discussion

In summary, we have studied the vibrations of a single horizontal soap film freely suspendedon a frame and of a single vertical PB isolated on a prismatic frame. In the first case, we haveshown that the phase velocity of the bending wave on the film is fixed by the balance betweenthe surface tension and the inertial mass of the film loaded by the surrounding air. The inertialmass of the air, which depends on the frequency, can even be dominant at low frequency. Wehave also demonstrated that the wave attenuation is due to the viscous friction in the air.

In the case of a bending wave propagating along the PB, we have identified two regimesin the wave velocity: a low-frequency regime, dominated by the vibration of the adjacent soapfilms weighted by the air, and a high-frequency regime, where the inertia of the PB plays arole. Those two regimes are very well described by a model that couples the motions of thesoap film and of the Plateau border. We have measured the characteristic damping time of thetransverse wave along the PB in the low-frequency regime. The results are compatible with apower-law behaviour as a function of the frequency given by the viscous dissipation in the airset in motion by a vibrating soap film, but numerical difference between the adjusted and theestimated prefactor indicates that the model underestimates the source of dissipation. The sameobservation was reported in similar measurements, conducted on the transverse vibration of acircular PB attached to a frame by soap films [Seiwert (2016)]. In this reference, the dampingrate of the wave was measured in the range 20 Hz - 200 Hz and compared to an estimateddissipation by viscous friction in the air: the same five times discrepancy as in our case wasreported. Therefore, the sources of dissipations of the vibrating transverse wave along a singlePB remain to be modeled. Our experiments as well as ref. [Seiwert (2016)] indicate that neitherthe PB width nor the bulk viscosity seem to play a role in those dissipative effects in the lowfrequency regime. Furthermore, in the high frequency regime, the attenuation was so strong thatno measurement was possible using our experimental setup.

Finally, the work presented in this article shows that the vibrating behaviour of soap filmsdoes depend on the chemical composition of the soap solution only via the surface tension, butit does not depend on the interfacial visco-elasticity 7 nor on the bulk viscosity of the liquidphase. Our experiments suggest that the same results apply to the transverse vibration of aPlateau border. On the other hand, the physical properties of the gas (density, viscosity) areessential. These results are consistent with recent observations reported at the foam scale, wherethe acoustic propagation in foams of different compositions mainly depends on the nature on thegas and does virtually not depend on the interfacial and bulk rheological properties of the liquidphase [Pierre (2015)].

What is the link between the vibrating behaviour of an isolated soap film or BP and theacoustic properties of foams? In [Pierre (2014)], different regimes of acoustic propagation in aliquid foam were identified, depending on the frequency. A model at the bubble scale showed

7The solution with TTAB, glycerol and dodecanol used for experiments described in Sec. 2.2 leads to soap filmswith remarkable interfacial properties, with rigid interfaces due to the presence of dodecanol. Still the behaviour of thesevibrating soap films is well described by our model which predicts that the interfacial properties can be neglected.

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that the key parameter separating those regimes was the product qa, where q is the wavenumberof the capillary wave on the soap membranes and a the typical radius of the membrane. In theexperiments presented in section 3, f varies from 50 Hz to 1 kHz, which corresponds to q ∼ 100to 1400 m−1 and a ∼ 20 cm, therefore qa ∼ 20 to 280, i.e. qa ≫ 1. Thus, our experimentsperformed on an isolated PB correspond to the high-frequency regime of the acoustic propaga-tion in a liquid foam. In this regime, the PB are immobile in the foam due to their inertia andthe acoustic moving mass is dominated by the soap films. The results presented here, wherethe PB motion is forced by the external perturbation, cannot be directly exported to model theacoustic propagation in a foam. However, in this frequency range, the dynamics of the systemis dominated by the soap films, which confirms the model presented in [Pierre (2014)]. Under-standing the dynamics of those model systems opens the perpective to use them to investigate thedissipation at the bubble scale, which is the still unknown ingredient to understand the acousticproperties of liquid foams.

Appendix A: comparison with the monolayers

As stated in Section 2.3, interfacial rheology plays a negligible role in the dispersion relationof a transverse wave on a soap film whereas it is an important or even dominant source of dissipa-tion when a transverse wave propagates on a single surfactant monolayer at a liquid/gas interface.In the present Appendix, we argue that this difference results from the magnitude of the tangen-tial entrainment of the interface. Let us consider standing waves for simplicity. In both situations,above a monolayer and above a film interface, the vertical displacement A (and vertical velocityAω) of the air above the interface antinodes generate comparable horizontal displacement A (andhorizontal velocity Aω) within a distance from the interface that is comparable to the wavelengthλ. Such a tangential motion, which is maximum in the node region, shears the air located nearthe interface within some viscous boundary layer (whose thickness δa =

√ηa/(ρaω) is about 50

µm in the kHz frequency range) and transmits tangential forces to the interface.In the case of a surfactant monolayer between a gas and a liquid, the undulation of the in-

terface creates similar displacement and velocity fields in the liquid. In particular the tangentialdisplacement in the liquid is of the same order of magnitude than the vertical displacement.Hence the liquid is sheared within the viscous boundary layer δ =

√η/(ρω) ≃ 10 µm and the

interface thus also undergoes tangential forces from the liquid. The net tangential force comingfrom both viscous boundary layers results in a nonzero tangential displacement of the interface.Actually in the case of air and water, it turns out that the interface tends to follow the motion ofthe liquid phase. Furthermore, with typical surface tension and surface elasticity values, the tan-gential force exerted by the gas phase can be neglected: the tangential force exerted by the liquidphase is mainly balanced by the surface tension gradient within the interface (Marangoni stress).The higher tension in the valleys of the liquid phase and the lower tension in the crests in turnresults from the surface rheology of the interface and the amount of deformation it undergoes.As a result of this coupling, the dissipation associated with the wave at such an interface betweena liquid and a gas is sensitive to the surface rheology of the surfactant monolayer, as observed[Stenvot (1988)].

By contrast, in the case of a soap film, the tangential displacement within the interface is verysmall. This is always the case for an antisymmetric wave on a soap film in the long wavelengthlimit (i.e. wavelength λ much larger than the thickness e of the film and than the amplitude A ofthe vibration). In fact, the tangential displacement appears through two mechanisms linked to the

15

film undulation itself. (i) The film slopes are of order A/λ. Such tilting of the film of thicknesse generates horizontal displacements of order Ae/λ of the film interfaces. (ii) As compared toits projected length, the actual length of a film tilted with a slope of order A/λ is increased by afactor of order A2/λ2, which generates horizontal displacements of order A2/λ. In both cases, theresulting horizontal displacements within the interface are very small compared to the verticaldisplacement A (typically 103 times smaller in our experiments). Consequently, the liquid inthe film is sheared very weakly, and the viscous shear stress in the liquid play a negligible role.Moreover, the Marangoni stress is proportional to the gradient of the deformation of the interface,which is very small, and thus the viscoelasticity of the film interfaces has no incidence on thedynamics. In particular, most of the dissipation occurs in the gas phase, within the viscousboundary layers near the film.

Appendix B: Dissipation of a wave along a PB

We have also estimated the attenuation obtained with the experiment described in section 3.4in the case of a pulse traveling along the PB: the attenuation can be determined after comparingthe amplitude of the Fourier transform of the signal at different heights. We find that the attenu-ation rate is a = (17 ± 7) m−1 for 30 Hz < f < 450 Hz and for R = 0.2 mm. Note that the largeerror bars are due to the fact that the attenuation length 1/a ∼ 6 cm is larger than the distancebetween the points of measurement.

For comparison, this measurement can be converted into a time of attenuation τ2 by settingτ2 = 1/(v a) where v = ω/q is the phase velocity. The value of a gives 9 ms < τ2 < 55 ms inthe investigated frequency range. This value is compatible with the τ measured independently inSec. 3.5, which suggests that this estimation of the relaxation time is robust.

Acknowledgements

We are greatly indebted in J. Servais for his help in setting the experimental setup. Duringperiods of probation, several students were associated with the experiments: J.-P. Testaud, S.Massart, J.-B. Caussin, A. Moller, L. Arbogast, D. Kaurin, M. Kint. We thank W. Drenckhan,S. Hutzler, M. Saadatfar, C. Stubenrauch, A. Saint-Jalmes and B. Dollet for fruitfull discussions.This work was supported by the French Agence Nationale de la Recherche (ANR) through theproject SAMOUSSE (ANR-11-BS09-001).

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