Vibration of soap films and Plateau borders, as elementary blocks
of a vibrating liquid foam
a,b,
a,1
a
a
a
a
F. Elias , S. Kosgodagan Acharige , L. Rose , C. Gay , V. Leroy, , C. Derec ,
a
Laboratoire Matière et Systèmes Complexes (MSC), Univ. Paris-Diderot, CNRS UMR 7057 - Paris, France
b
Sorbonne Universités, UPMC Université Paris 6, UFR 925 - Paris, France
Abstract
The propagation of an acoustic wave in a liquid foam results from the coupling of a pressure
wave 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 here
the transverse vibration of those constitutive elements. The measurement of the velocity and
attenuation of the transverse wave on each element isolated on a rigid frame, compared with an
analytical 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 wave
propagating on a single soap film, we show that (i) the wave velocity is set by the surface tension
and the inertial mass of the film loaded by the surrounding air, and (ii) that the damping of the
wave is mainly due to the viscous dissipation in the air. In the case of a transverse wave propagating along the junction line between three soap films (Plateau border), the dispersion relation
reveals two different scalings at low frequency and at high frequency, which are interpreted by
considering the role of the vibration of the adjacent soap films, and the role of the inertia of
the liquid inside the channel. The attenuation of the transverse wave along the liquid channel is
measured in the low frequency regime. In both investigated cases (transverse wave propagating
on a soap film as well as on a liquid channel), we show that the surrounding gas plays a dominant
role, 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 the
microstructure 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 minimize
its interfacial energy, is structured by a few geometrical rules known as Plateau’s rules: the
soap films between bubbles meet by three in liquid channel also called Plateau borders, which
Email address: [email protected] (F. Elias)
Present adress: Laboratoire de Physique (UMR CNRS 5672), ENS de Lyon, 46, allée d’Italie, F-69364 Lyon cedex
07, France
1
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 the
foam 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 of
foams, such as their complex rheological response under shear [Cohen-Addad (2013)], their
electrical 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 propagation in foams. Foams are nevertheless used for mitigating blast waves, thanks to their strong
acoustic attenuation [Clark (1984); Raspet (1987)]. At smaller acoustic amplitudes, the acoustic
velocity 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 on
the frequency and the bubbles average diameter d [Ben Salem (2013); Pierre (2014)]. Several
regimes of propagation have been identified: two non-dispersive regimes at low and high frequencies, 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)]. The
propagation of an acoustic wave in a liquid foam couples a pressure wave which propagates in
the gas and the vibration of the liquid skeleton. Due to the cellular geometry of the foam, the
compression wave generates transverse vibrating waves in the liquid phase. Hence, the relevant
length scale for describing the acoustic propagation in foams is not the acoustic wavelength in
2
the foam, which is, in general, several orders of magnitude larger than d , but the wavelength of
the 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 isolate
one by one the constitutive elements of the liquid skeleton, and we measure the dispersion relation 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 measure
the phase velocity and the attenuation of the wave. We develop a theoretical modeling to identify
the 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 channel
must be taken into account, as well as the tensile surface forces exerted by the soap films on the
Plateau border. Does the Plateau border vibrate like the free border of a liquid membrane, or does
it behave as an inertial liquid string? In section 3, we show that the answer is in between: two
regimes are identified as a function of the frequency. In section 4, we finally replace the results
in the light of the recent studies of the acoustic propagation in a liquid foam, and we discuss the
2
The typical acoustic wavelength in air is about 3 mm at 100 kHz, and about 30 cm at 1 kHz
2
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 in
order 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 rigid
frame. When the frame is vibrated vertically using an electromagnetic shaker, a wave is created at
the periphery of the soap film and travels up to its center where it is totally reflected (in the linear
limit where the amplitude is small compared to the wavelength). Then a transverse standing vibration 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 the
antisymmetric mode, as sketched in Fig. 2. To characterize the antisymmetric wave appearing on
the soap film, we develop two experimental setups in order to determine the complete dispersion
relation (real part and imaginary part): the first setup gives access to the measurement of the
wavelength as a function of the frequency for different thicknesses of the soap film; the second
setup 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 is
the amplitude of the wave, λ the wavelength, and e the thickness.
2.1. Wavelength
2.1.1. Setup and measurements
A 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 wavelength
λ 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 and
concentrate into bright areas (caustics) pointing out the positions of the antinodes of the standing
wave. The transverse displacement on the soap film of cylindrical symmetry is:
i(ωt+φ0 )
ζ(r, t) = A0 J0 (q f r)e
(1)
where ζ(r, t) is the vertical displacement at a distance r from the center of the film and at time
t, A0 is the amplitude at the center of the film, J0 is the Bessel function of the first kind and
′
′′
of 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 given
thickness 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 on
each 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 the
cylindrical tube, the soap film begins to drain and its thickness decreases from few microns to
around 100 nanometers within few minutes. During the drainage process, measurements are
made at different thicknesses. Each frequency sweep is fast enough (duration 2 s) to consider
3
that the thickness remains constant during the sweep (see Fig. 4) . Independently the thickness
at the center of the soap film is measured using a spectrometer, just before the beginning of each
sweep.
PSD
video
camera
semitransparent
plate
parallel
white light
laser
spectrometer
soap film
shaker
translation stage
(a)
(b)
Figure 3: Vibration of a single film: setups. (a) Measurement of the wavelength by visualisation of the antinodes after
reflection of a parallel light beam on the soap film: this setup allows a rapid measurement (faster than the draining time
of the soap film), but the vibration amplitude can not be measured. (b) To quantify the attenuation of the wave, the
amplitude of vibration as a function of the frequency is measured in one point of the film by collecting the deflection by
the film of a laser beam in a position sensitive detector (PSD). The profile of the deformation along the diameter of the
film is obtained in about 15 seconds using a translation stage.
The experiments have been performed using solutions with different concentrations of surfactants, and for each solution at different thicknesses (see Table 1). The experimental results for
′
the wavenumber q f as a function of the frequency are shown in Fig. 5. We can note that at a
′
given frequency, q f tends to increase with thickness.
2.1.2. Comparison to theoretical predictions
In 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.
3
The exposure time of each image is both much smaller than the frequency sweep duration (thus one image is associated to a single exciting frequency f ) and much larger than 1/ f (so that each image corresponds to the average enveloppe
of the soap film vibration).
4
(a)
(c)
(b)
(e)
(d)
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. The
colors are due to the interference of the light reflected by the soap film interfaces (Newton colors ), using a white light
illumination: different colors correspond to different soap film thicknesses. Note that the color pattern does not change
during the sweep, showing that the thickness profile remains the same.
c (g/l)
1.2
1.4
1.6
2.0
5.0
γ (mN/m)
37 ±2
34 ±2
35 ±2
36 ±0.5
35 ±0.5
emeas (µm)
•1.2 - ▪0.8 - ⬩0.3
•1.3 - ▪1.1 - ⬩1.1
•2.5 - ▪1.3 - ⬩0.7
•2.1 - ▪1.5 - ⬩1.1
•1.0 - ▪0.8 - ⬩0.5
Table 1: Different solutions used: bulk TTAB concentration c, measured surface tension γ. For each solution the experiments 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 incompressible fluids, of respective densities ρ and ρa . The thickness of the soap film is assumed to be
constant and equal to e (see Fig. 2). The calculations are performed in the approximation of the
long wavelength limit, i.e. λ large compared to the thickness and to the amplitude of vibration of
the soap film.
The Navier-Stokes equations are written for the liquid and for the air, leading to the general expressions of the pressure and velocity fields in both fluids. These quantities are then linked by the
continuity conditions at the liquid-air interfaces. The continuity of the tangential stress describes
the equilibrium between the tangential viscous forces, in the liquid and in the air, and the gradient of the interfacial stress that depends on the viscoelastic interfacial modulus. The equation of
continuity of the normal stress balances the pressure jump across the liquid-air interfaces with
the 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 our
experimental conditions, this relation is simplified to the leading order, and we obtain the phase
′
velocity v linking the real part of the wavenumber q f = 2π/λ and ω = 2π f :
ω
v = λf = ′ ≃
qf
√
2γ
ρe + 2ρa /q′f
(2)
The comparison between√the experimental results and this prediction is shown in Fig. 6a, where
′
the 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) − 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 is
compared to the value of emeas , measured in the centre of the soap film, and taking into account
′
5
2
′
′
Figure 5: Experimental results: real part of the wave number q f = 2π/λ as a function of the frequency f for different
film thicknesses and solution concentrations. The legend of the symbols is detailed in Table 1.
4
the thickness gradient within the films . The thickness gradient is estimated by the observation
of the interference fringes (see photograph in Fig. 6b): from one interference order to the next
one, 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 the
adjusted average thickness ead j is compatible with the measured interval emeas , centered on the
measurement in the middle of the film using the spectrometer.
Finally, we conclude that the data collapse on the same master curve, which is well described
by the theoretical prediction, showing experimentally that the role of γ and e are well described
by Eq. 2.
2.2. Dissipation of the wave
In 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 to
the excitation (see [Stenvot (1988)]). Here, the decay length is much larger than the size of the
soap film, thus this technique is not possible. In [Kosgodagan (2014)], we have measured the
wave attenuation using the second setup (Fig. 3b). The principle was to measure the amplitude
of vibration of the soap film as a function of the frequency in order to extract the measurement
of the attenuation from the widths of the resonances. The amplitude was obtained by recording
with 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 translation
stage. After a fit with the appropriate Bessel function, we were able to extract the value of the
amplitude 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 the
frequency in about 200 minutes, which is long compared to the characteristic time of drainage
of the solution used for the previous experiment. Therefore experiments were performed with a
′
thin film (thickness of a few tens of nanometers), such as e ≪ 2ρa /(ρ q f ) (see Eq. 2), hence the
4
The thickness gradient is due to the fact that the soap films are slightly tilted from horizontal to avoid the formation
of a liquid dimple in the center.
6
f
qf0 q (1/m)
’ (1/m)
4000
3000
2000
1000
0
0
1000
2000
3000
(Hz)
f f (Hz)
3
3000
emeas (µm)
p racine / 2pi
qf0 ...q’f x/2⇡
(Hz)
4000
2
2000
1
1000
0 00
0
4000
1000
1
2000
(µm)
f (Hz)
adj
f e(Hz)
(a)
2
3000
4000
3
4000
emeas (µm)
p
qf0 ...q’f x/2⇡
(Hz)
racine / 2pi
3
3000
2000
1000
0
0
2
1
1000
2000
3000
4000
(Hz)
f f (Hz)
0
0
1
2
3
eadj (µm)
(b)
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 the
film) 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 film
diameter is 16 mm.
′
film thickness e played no role in the investigated phenomena. The soap film was made of water
added with TTAB (2.8 g/l), glycerol (10 wt%) and dodecanol (0.04 wt%). Its surface tension was
22.5 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 this
expression :
α
A0 =
(3)
∣J0 (q f R)∣
where q f = q f + iq f and α, R and q f were three fitting parameters. The parameters α and R
described 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 intrinsic
attenuation 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 simplification to the leading order in our experimental conditions, predicts the following imaginary
′
′′
′′
7
part of the dispersion relation:
′′
′2
−q f ≃ q f
δa (ω)
3 + q′f e ρ/ρa
(4)
√
where δa (ω) = ηa /(2ρa ω) 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 experimental conditions is the viscous friction in the air. As shown in [Kosgodagan (2014)], the
attenuation 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 (wavelength and dissipation) of a transverse antisymmetric wave on a soap film. The experiments
validate 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, due
to surface tension, and the inertia of the system where the inertia of the air must be taken into
account (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 source
of dissipation in the system is the viscous friction in the air.
Note that the calculations have been performed considering the properties of the liquid-air
interface, by taking into account the complex interfacial viscoelasticity. We have shown that it
plays a negligible role here and can be neglected in Eqs. 2 and 4. This is very different from what
is observed for wave propagation on a surfactant monolayer, where the interfacial viscoelasticity
is the dominant cause of dissipation [Stenvot (1988)]. The respective roles of the interfacial
viscoelasticity 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 the
Plateau border (PB). In a dry foam, a Plateau border contains the liquid in a channel that separates
three soap films. Let R be the width of the channel. How does such a channel behave when the
foam liquid skeleton is vibrated: like a liquid string with its own tension and inertia or like the
passive 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 the
dispersion relation (Eq. 2) and the attenuation (Eq. 4). However, the mass of the PB is concentrated at one edge of the film. Hence, the vibrating PB consists in a coupled system of a liquid
2
channel of mass per unit length µ = ρS PB = 0.161ρR (where S PB is the PB cross-section),
and of soap membranes loaded by the air. The surface tension of the liquid interfaces induces
a restoring force not only on the liquid membranes to bring them back flat, but also on the PB
to 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 simplified
by considering that the (zOx) plane is a plane of symmetry: only the films in the (yOz) plane
′
and in the (y Oz) plane are transversally deformed, and the deformation of the PB occurs in the
plane of symmetry (xOz). The computation is detailed in a previous article [Derec (2015)] and
we summarize here the main results. In this model, the attenuation of the wave along the PB is
8
y’
2π/3
y
ζ
y
z
z
y’
x
O
x
O
y’
R
x
u
x
y
ζ
y
(a)
(b)
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 soap
film: u(z, t) is the displacement of the liquid channel and ζ(y, z, t) is the transversal displacement of one of the soap
films.
i(ωt−qz)
neglected. We consider a harmonic wave u(t, z) = u0 e
propagating along the PB (with q a
real number). In the case of a finite PB radius R and at a finite frequency, a plane harmonic wave
i(ωt−qy y−qz)
in the soap film ζ(y, z, t) = ζ0 e
is solution of the coupled equations of motion, with:
2
qy = −i
µω
3γ
(5)
and the dispersion relation:
2
q =
′
2
qf
2
2
µω
+(
)
3γ
(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 relax
exponentially in the direction y perpendicular to the PB. In the limit of thin liquid films and large
′
wavelength, 2ρa /q f ≫ ρe, Eq. 6 becomes:
2
2
ρa 2/3 4/3 ρe 2
µω
q =(γ) ω +
ω +(
)
3γ
3γ
2
(7)
The first term of the right hand side of Eq. 7 contains the inertia of the air loading the vibrating
4/3
films. It scales like ω and is thus dominant at low frequency. The second term corresponds to
the role played by the liquid inside the film. Finally, the role of the inertia of the PB is represented
2 4
in the last term; this term scales like µ ω , 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 measurements
The 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 soap
solution (Fig. 8a). The soap solution (of surface tension 30 mN/m) is made of distilled water, a
commercial dishwashing liquid (Fairy liquid 1%vol.) and glycerol(2%vol.). The PB is vibrated
transversally by plunging a thin capillary, which can be displaced along the x direction using
an 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 in
the PB at a constant flow rate Q through the capillary, so the PB radius R(Q) can be adjusted by
5
choosing the appropriate value of the flow rate as described in a previous article [Elias (2014)].
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 continuous
forcing (Fig. 8b); the second set-up is designed to measure the velocity of a transverse pulse
along the liquid channel (Fig. 8c).
flow rate
shaker
capillary
grid
f
x
x
x
h
y
y
zoom
frame
δ
camera
high-speed
camera
(a)
(b)
(c)
Figure 8: Sketch of the setup. (a) Side view : a vertical Plateau border is formed on a prismatic frame; it is transversally
deformed 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 the
image of a square grid reflected on the soap film; in (c), the displacement of the Plateau border is measured at a fixed
height 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: observations
The 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 square
when the soap film is flat, and undulates when the film undulates transversally (Fig. 9). The light
is 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 by
the rigid edges of the frame. The transverse displacement of the soap film is a superposition of
waves travelling in different directions as shown in Fig. 9. Those images show that when the
5
The liquid flow rate within the PB is chosen small enough so that the vertical flow velocity inside the PB remains
negligible with respect to the transverse velocity of the PB during the vibration.
10
capillary
Plateau border
(a)
(b)
(c)
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 estimated
by measuring the mean distance between neighbouring antinodes: we found λexp ∼ 14.5± 1.5 mm, 6.7± 0.3 mm and
′
5.1± 0.5 mm (from left to right). Theses values can be compared to the data computed from Eq. 2 (with λ = 2π/q f and
′
for 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 deformed
grid, 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 the
wavelength of the antisymmetric wave on an infinite soap film, remains relevant even when the
soap 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 close
to the PB and strongly attenuated in the direction perpendicular to the PB. This is consistent with
the 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, using
the setup described in Fig. 8c. A transverse pulse was emitted at the tip of the capillary, and
the pulse propagation along the PB was followed using a high-speed camera fitted with a zoom
objective. The dimensions of the frame was δ = 20 cm and h = 23 cm, large enough to postpone
sufficiently the arrival of the reflected pulse coming from the soap film boundaries or from the
other 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 from
a Fourier analysis. Soap solution could be added at a constant flow rate in the PB through the
capillary to tune the radius R. Two regimes were identified on the dispersion relation q( f ): at
2/3
low frequency, q ∝ f
whatever the R, whereas the dispersion relation tends asymptotically
2
towards a power law q ∝ f at high frequency. The larger the PB radius, the smaller the
frequency at which the transition between both regimes is observed. The low-frequency regime
′
corresponds to Eq. 2 in the limit q f e ≪ ρa /ρ (which is the case in the experiment): the PB
vibrates at low frequency like the passive border of the soap film, its own inertia having no effect
on the wave propagation velocity. The high-frequency regime corresponds to a balance between
the PB inertia µü and the restoring forces exerted by the soap films pulling on the PB. In other
words, the dispersion relation of the transverse wave propagating along a PB is equal to that of a
′
liquid membrane at low frequency (q ≃ q f in Eq. 6), and is equal to that of a liquid string at high
2
frequency (q ≃ µω /(3γ)).
3.5. Dissipation
Although 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 measurement
consists in studying the relaxation of the vibration of the PB after the excitation is stopped. A
loudspeaker, placed in front of the PB, emits a plane wave propagating in the x direction. A
monochromatic continuous sound causes the vibration of the PB and the adjacent soap films at
the forcing frequency f0 , as long as t < 0. The forcing is stopped at t = 0 and the relaxation
u(z, t) is measured using a high speed camera. At this stage, u(t) at a fixed height is wiggly and
the data must be treated in order to extract an attenuation time. We analyse the Fourier transform
û( f ) around the forcing frequency: û( f ) presents a maximum at f ≃ f0 . Assuming that, around
f0 , u(t) is a damped sinusoid at the forcing frequency, the measurement of the width ∆ f at the
half of the maximum height (see insert in Fig. 10) gives the attenuation time at f = f0 :
√
3
.
(8)
τ( f0 ) ≃
π∆f
The values of τ obtained as a function of the exciting frequency are shown in Fig. 10, for
6
different R. 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 a
′′
soap film, τ f . This time can be computed using Eqs 2 and 4, and using τ f = 1/(vq f ) where
′
′
v = ω/q f is the phase velocity. In the approximation q f ≪ 2ρa /(ρe), we get:
γ 1/3 2ρa 1/2
1
τf ( f ) ≃ 3 (ρ ) ( η )
.
a
a
(2π f )7/6
(9)
In Fig. 10, the data τ( f ) are adjusted for comparison by a power law τ = B f
where
−1/6
the prefactor B is a fitting parameter. The best fit gives B ∼ 7.4 s
whereas Eq. 9 predicts a
−7/6
6
In this setup, the capillary (see Fig. 8a) has been removed to avoid any obstacle to the PB oscillation. R is still tuned
by injecting soap solution at a constant flow rate in the PB, but the liquid is injected via the three top vertices of the
frame.
12
prefactor of 35.6, about 5 times larger: the attenuation time of the wave on the PB is significantly
smaller than the attenuation time of the wave on a single infinite soap film, i.e. the dissipation
is stronger. This additional dissipation does not seem to come from a dissipation in the liquid
channel (e.g. viscous dissipation), since the data in Fig. 10 corresponding to different PB radii
collapse on the same plot (within error bars). Therefore, the additional source of dissipation is
likely to reside in the flow of the air around the PB. The geometry of the soap films, which meet in
the PB, confining the surrounding air in an edge, is in fact somewhat different from the geometry
of an infinite soap film. Moreover, the finite size of the soap film has not been considered in
the theory. This is likely to enhance the dissipation prefactor. Furthermore, another source of
attenuation could emerge from the finite size of the PB itself whose vertices are also set into
vibration by the loudspeaker.
FT [u(t)]
80
60
∆f
τ (ms)
0
400
f
0
500
f (Hz)
40
20
0
0
100
200
300
400
500
f0 (Hz)
Figure 10: Characteristic attenuation times τ of a transverse wave propagating along a Plateau border, after the forcing
at frequency f0 has been stopped. The different symbols correspond to different flow rates injected in the PB and thus
different radii: ▲ R < 0.1 mm, ● R ≃ 0.2 mm , ▶ R ≃ 0.3 mm and R ≃ 0.35 mm. The dashed line represents the
7/6
best fit of the data using a power law τ = B/ f
(see text). Insert: Fourier transform of the signal around the forcing
frequency f0 = 430 Hz: the width ∆ f of the curve is linked to τ using Eq. 8.
3.6. Conclusion
In 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 membrane
loaded 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 low
frequency regime of the dispersion relation. In this regime, the attenuation varies with frequency
as expected for a transverse wave along a single infinite soap film, although the prefactor is about
five times stronger. This is probably the result of the air flow structure in the 120 degree sectors
between the films which is more confined than that in the single flat film geometry assumed in
the theory.
In the high-frequency regime of the dispersion relation, the transverse wave was strongly
attenuated and no measurement was possible. However, the image presented Fig. 9c shows that
13
at 1000 Hz, a wave propagates along the Plateau border although it is strongly attenuated in
the soap films. Our efforts concentrate now on measuring this attenuation in the high-frequency
regime, 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 suspended
on a frame and of a single vertical PB isolated on a prismatic frame. In the first case, we have
shown that the phase velocity of the bending wave on the film is fixed by the balance between
the surface tension and the inertial mass of the film loaded by the surrounding air. The inertial
mass of the air, which depends on the frequency, can even be dominant at low frequency. We
have 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 regimes
in the wave velocity: a low-frequency regime, dominated by the vibration of the adjacent soap
films weighted by the air, and a high-frequency regime, where the inertia of the PB plays a
role. Those two regimes are very well described by a model that couples the motions of the
soap film and of the Plateau border. We have measured the characteristic damping time of the
transverse wave along the PB in the low-frequency regime. The results are compatible with a
power-law behaviour as a function of the frequency given by the viscous dissipation in the air
set in motion by a vibrating soap film, but numerical difference between the adjusted and the
estimated prefactor indicates that the model underestimates the source of dissipation. The same
observation was reported in similar measurements, conducted on the transverse vibration of a
circular PB attached to a frame by soap films [Seiwert (2016)]. In this reference, the damping
rate of the wave was measured in the range 20 Hz - 200 Hz and compared to an estimated
dissipation by viscous friction in the air: the same five times discrepancy as in our case was
reported. Therefore, the sources of dissipations of the vibrating transverse wave along a single
PB remain to be modeled. Our experiments as well as ref. [Seiwert (2016)] indicate that neither
the PB width nor the bulk viscosity seem to play a role in those dissipative effects in the low
frequency regime. Furthermore, in the high frequency regime, the attenuation was so strong that
no measurement was possible using our experimental setup.
Finally, the work presented in this article shows that the vibrating behaviour of soap films
does depend on the chemical composition of the soap solution only via the surface tension, but
7
it does not depend on the interfacial visco-elasticity nor on the bulk viscosity of the liquid
phase. Our experiments suggest that the same results apply to the transverse vibration of a
Plateau border. On the other hand, the physical properties of the gas (density, viscosity) are
essential. These results are consistent with recent observations reported at the foam scale, where
the acoustic propagation in foams of different compositions mainly depends on the nature on the
gas and does virtually not depend on the interfacial and bulk rheological properties of the liquid
phase [Pierre (2015)].
What is the link between the vibrating behaviour of an isolated soap film or BP and the
acoustic properties of foams? In [Pierre (2014)], different regimes of acoustic propagation in a
liquid foam were identified, depending on the frequency. A model at the bubble scale showed
7
The solution with TTAB, glycerol and dodecanol used for experiments described in Sec. 2.2 leads to soap films
with remarkable interfacial properties, with rigid interfaces due to the presence of dodecanol. Still the behaviour of these
vibrating soap films is well described by our model which predicts that the interfacial properties can be neglected.
14
that the key parameter separating those regimes was the product qa, where q is the wavenumber
of the capillary wave on the soap membranes and a the typical radius of the membrane. In the
experiments presented in section 3, f varies from 50 Hz to 1 kHz, which corresponds to q ∼ 100
−1
to 1400 m and a ∼ 20 cm, therefore qa ∼ 20 to 280, i.e. qa ≫ 1. Thus, our experiments
performed on an isolated PB correspond to the high-frequency regime of the acoustic propagation in a liquid foam. In this regime, the PB are immobile in the foam due to their inertia and
the acoustic moving mass is dominated by the soap films. The results presented here, where
the PB motion is forced by the external perturbation, cannot be directly exported to model the
acoustic propagation in a foam. However, in this frequency range, the dynamics of the system
is dominated by the soap films, which confirms the model presented in [Pierre (2014)]. Understanding the dynamics of those model systems opens the perpective to use them to investigate the
dissipation at the bubble scale, which is the still unknown ingredient to understand the acoustic
properties of liquid foams.
Appendix A: comparison with the monolayers
As stated in Section 2.3, interfacial rheology plays a negligible role in the dispersion relation
of a transverse wave on a soap film whereas it is an important or even dominant source of dissipation 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 tangential 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 velocity
Aω) of the air above the interface antinodes generate comparable horizontal displacement A (and
horizontal velocity Aω) within a distance from the interface that is comparable to the wavelength
λ. Such a tangential motion, which is maximum in the node region, shears
√ the air located near
the interface within some viscous boundary layer (whose thickness δa = ηa /(ρa ω) 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 interface creates similar displacement and velocity fields in the liquid. In particular the tangential
displacement 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 coming
from 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 of
the liquid phase. Furthermore, with typical surface tension and surface elasticity values, the tangential force exerted by the gas phase can be neglected: the tangential force exerted by the liquid
phase 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 turn
results 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 between
a 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 very
small. This is always the case for an antisymmetric wave on a soap film in the long wavelength
limit (i.e. wavelength λ much larger than the thickness e of the film and than the amplitude A of
the 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 thickness
e generates horizontal displacements of order Ae/λ of the film interfaces. (ii) As compared to
its projected length, the actual length of a film tilted with a slope of order A/λ is increased by a
2
2
2
factor of order A /λ , which generates horizontal displacements of order A /λ. In both cases, the
resulting horizontal displacements within the interface are very small compared to the vertical
3
displacement A (typically 10 times smaller in our experiments). Consequently, the liquid in
the 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 the
dynamics. In particular, most of the dissipation occurs in the gas phase, within the viscous
boundary 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 3.4
in the case of a pulse traveling along the PB: the attenuation can be determined after comparing
the amplitude of the Fourier transform of the signal at different heights. We find that the attenu−1
ation rate is a = (17 ± 7) m for 30 Hz < f < 450 Hz and for R = 0.2 mm. Note that the large
error bars are due to the fact that the attenuation length 1/a ∼ 6 cm is larger than the distance
between the points of measurement.
For comparison, this measurement can be converted into a time of attenuation τ2 by setting
τ2 = 1/(v a) where v = ω/q is the phase velocity. The value of a gives 9 ms < τ2 < 55 ms in
the investigated frequency range. This value is compatible with the τ measured independently in
Sec. 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. During
periods 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 the
project SAMOUSSE (ANR-11-BS09-001).
Bibliography
Afenchenko V.O., Ezersky A.B., Kiyashko S.V., Rabinovich M.I. and Weidman P.D., Phys.Fluids, 1998, 10, 390.
Ben Salem I. et.al., Soft Matter, 2013, 9, 1194.
Cantat I., Cohen-Addad S., Elias F., Graner F., Hohler R., Pitois O. and Saint-Jalmes A., ’Les mousses, Structure et Dynamique’ (Belin) 2010. English translation: ‘Foams: Structure and Dynamics’, edited by S. J. Cox (Oxford University
Press) 2013.
Clark C. J. and Bennet E. M., ‘Method for explosive blast control using expanded foam’, U. S. Patent number 4,589,341,
1986.
Cohen-Addad S., Hohler R. and Pitois O., Annu. Rev. Fluid Mech., 2013, 45, 241.
Couder Y., Chomaz J.-M. and Rabaud M., Phys.D, 1989, 37, 384.
Derec C., Leroy V., Kaurin D., Arbogast L., Gay C. and Elias F., EPL 112, 2015, 34004.
Elias F., Janiaud E., Bacri J.-C. and Andreotti B., Phys. Fluids 26, 2014, 037101.
Goldfarb I., Orenbakh Z., Schreiber I. and Vafina F., Shock Waves, 1997, 7, 77.
16
Haffner B., Khidas Y. and Pitois O., J. Colloid Interface Sci., 2015, 458, 200.
Joosten J.G.H., J.Chem.Phys., 1984, 80, 2383.
Kosgodagan Acharige S., Elias F. and Derec C., Soft Matter, 2014, 10, 8341.
Lemlich R., J. Colloid Interface Science, 1978, 64, 107.
Mujica N. and Fauve S., Phys. Rev. E, 2002, 66, 021404.
Pierre J., Dollet B. and Leroy V., Phys. Rev. Lett., 2014, 112, 148307.
Pierre J., Giraudet B., Chasle P., Dollet B. and Saint-Jalmes A. Phys. Rev. E, 2015, 91, 042311.
Raspet R., Butler P. B. and Jahani F., Applied Acoustics, 1987, 22, 35.
Seiwert J., Pierre J. and Dollet B., J. Fluid Mech., 2016, 788, 183.
C. Stenvot and D. Langevin, Langmuir, 1988, 4, 1179.
Stone H. A., Koehler S. A., Hilgenfeldt S. and Durand M., J. Phys.: Condens. Matter, 2003, 15, S283
Vega J. M., Higuera F. J. and Weidman P. D., J. Fluid Mech., 1998, 372, 213-230
Weaire D. and Hutzler S., ‘The physics of foams’, Clarendon Press (Oxford) 1999.
17