Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 9, ICA 203 Montreal Montreal, Canada 2-7 June 203 Physical Acoustics Session 2pPA: Material Characterization 2pPA. Probing acoustics of liquid foams by optical diffusive wave spectroscopy Marion Erpelding, Reine-Marie Guillermic, Juliette Pierre, Benjamin Dollet*, Arnaud Saint-Jalmes and Jerôme Crassous *Corresponding author's address: Institut de Physique de Rennes, Rennes, 35042, Bretagne, France, We have studied the effect of an external acoustic wave on bubble displacements inside an aqueous foam. The signature of the acousticinduced bubble displacements is found using a multiple light scattering technique, and occurs as a modulation on the photon correlation curve. Measurements for various sound frequencies and amplitudes are compared to analytical and numerical simulations. These comparisons finally allow us to elucidate the non-trivial acoustic displacement profile inside the foam; in particular, we find that the acoustic wave creates a localized shear in the vicinity of the solid walls holding the foam. This study of how bubbles "dance" inside a foam as a response to sound turns out to provide new insights on both foam rheology at high frequencies, foam acoustics and sound transmission into a foam, and analysis of light scattering data in samples undergoing non-homogeneous deformations. Published by the Acoustical Society of America through the American Institute of Physics 203 Acoustical Society of America [DOI: 0.2/ ] Received 22 Jan 203; published 2 Jun 203 Proceedings of Meetings on Acoustics, Vol. 9, (203) Page

2 INTRODUCTION Inside an aqueous foam, bubbles are highly packed and compressed on each other. They are jammed, and cannot move apart by themselves. Nevertheless, bubbles are soft objects, and can easily be distorted : consequently, their shape, position and topology (numbers of neighbors) are never irreversibly frozen. Foam coarsening is a first mechanism which modifies the organization of bubbles inside the foam [, 2]. This coarsening results from gas diffusion from bubbles to others; some bubbles grow and others shrink, which induces a constant modification of the bubble shapes. To conserve the local mechanical equilibrium, some discrete bubble rearrangements must be induced as a consequence of coarsening. The evolution with time of the rate of bubble rearrangements and of the mean bubble radius have been studied by many techniques, in particular multiple light scattering like Diffusive Wave Spectroscopy (DWS) [3]. Applying a shear stress is another way to modify the bubbles shapes and organization [4]. As for coarsening, bubbles can be unjammed by shear, and techniques like DWS have also been useful to capture the links between macroscopic properties and bubble-scale effects [5, 6, 7]. Despite the large research activity in this field, foam rheology remains a complex topics, with many open issues. The full dependance with amplitude and frequency of the viscoelastic moduli is still not understood, especially at high frequencies [8]. Under continuous shear, the occurrence of shear localization or wall slip is also still pending. In order to bring new clues, finding some other ways of externally soliciting a foam is a possible approach. Unaccessible ranges of deformation or frequencies, as well as new regimes could be investigated. Another motivation of studying the effect of a sound wave on a foam is the foam acoustics itself: despite its rather unusual acoustic properties, not much has been done on this topic of sound propagating inside a foam. As a matter of fact, experimental results on acoustics of foams mostly deal with speed of sound and wave attenuation [9, 0, ], while almost no results are reported on the local motion of the bubbles and on the coupling between foam elasticity and sound wave propagation. In this article, we show that coupling an acoustic signal (as a "source") and a multiple light scattering signal (as a "detector") turns out to be fruitful for many issues on foams. These first results shed lights on wall slip, on shear viscoelasticity, on acoustics, and on shear localization and how to detect it. MATERIALS AND METHODS Our samples are made of commercial shaving foam (Gillette) poured into a U-shaped PVC cell with transparent glass walls. The inner dimensions of the cell are mm. To produce an acoustic perturbation, a loudspeaker (Jeulin) is fixed approximately 45 cm above the foam. It receives a sinusoidal input signal of controlled amplitude A mv and frequency f from a function generator. We have used frequencies f of 400 Hz, khz, 4 khz and 0 khz. For each of these values, we have measured the acoustic amplitude A a,db of the sound waves in air as a function of the input amplitude A mv using a sonometer (Lutron SL-400). The resulting acoustic displacements A a in air can be computed from the data as: A a = p 0 ρ a c a ω 0A a,db/20, () where A a,db is the measured acoustic amplitude in air in decibels, ρ a is the density of air and c a is the celerity of sound in air at ambient temperature and atmospheric pressure, ω = 2πf is the pulsation of the acoustic wave, and p 0 = 20 μpa the reference pressure from the definition of a decibel. As mentionned in the introduction, light scattering has been used to study Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 2

3 coarsening-induced or shear-induced effects. Here we have used DWS to investigate the deformation of a foam by an acoustic wave. Light scattering techniques are classical tools to investigate the structure and dynamics of random media. They consist of illuminating a diffusing sample with coherent light, and analyzing the intensity fluctuations of the transmitted or backscattered light. These fluctuations arise from the modification of the optical paths followed by photons propagating inside the sample, due to its internal dynamics or to an externally imposed deformation. Light propagation in random media can be described as a random walk in the diffusion approximation. The intensity fluctuations of the output light can be related to deformations of the sample [2, 3, 4]. Here, the sample is illuminated with a unpolarized HeNe laser beam of wavelength λ = 633 nm and power 5 mw. The backscattered light is collected after a linear polarizer and propagates through a monomode optical fiber into a photomultiplier tube. The output signal is then analyzed by a linear correlator (correlator.com Flex02-2D/C). Figure shows a sketch of the experimental setup. FIGURE : Sketch of the experimental setup. The laser beam (λ = 633 nm) illuminates the center of the foam sample. A monomode optical fiber fixed on a fiber mount (FM) carries the backscattered light into a photomultiplier tube (PM). The polarizer (P) ensures us that only one direction of the depolarized backscattered light is collected. The signal is then analyzed by a linear correlator. Acoustic waves are generated by the loudspeaker fixed above the foam that receives a sinusoidal input signal from a function generator (FG). Experiments are organized as follows: after pouring the foam into the cell, we systematically wait for approximately one hour before starting the measurements [3]. Then, for a given frequency and amplitude of the acoustic signal, we record the correlation function g 2 (τ) ofthe backscattered light intensity between times t and t + τ: I(t)I(t + τ) g 2 (τ) = I(t) I(t + τ), (2) where represents a temporal average over the duration of the measurement. In our case, measurements always last 2 minutes. Successive measurements on a given sample are all performed within half an hour. This ensures us that the foam does not age significantly during the experiment. In the absence of acoustic perturbation, g 2 (τ) decreases monotonically with τ with a characteristic time associated to the coarsening of the foam, as shown on the inset of Fig. 2. When we add the acoustic perturbation, we observe a modulation of g 2 (τ), i.e., a succession of partial decorrelations and recorrelations that adds up to the decorrelation observed in the absence of acoustic perturbation. This shows that in addition to aging, the foam is cyclicly deformed by the acoustic wave (see Fig. 2). We observe that for all frequencies, the amplitude of the modulation increases with the amplitude of the acoustic wave. As an example, we plot on Fig. 2 the typical measured g 2 (τ) ata fixed frequency f = 400 Hz for two different acoustic amplitudes A a, db = 90.8 db and 80.9 db. Moreover, in the range τ < μs, the amplitude of the modulation is roughly constant Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 3

4 g 2 (τ) τ (µs) 5 20x0 3 FIGURE 2: Examples of measured g 2 (τ) when the foam is submitted to acoustic waves of frequency 400 Hz and amplitudes 90.8 db (upper curve) and 80.9 db (lower curve). Inset: log-lin plot of a typical g 2 (τ) measured in the absence of acoustic wave. with τ, signing reversibility of the displacement. Indeed, although irreversible bubble rearrangements can be induced by the acoustic wave, yielding a decrease of the modulation of g 2 (τ) with time, we are in an experimental situation in which the foam has been submitted to a large number acoustic oscillations before we perform the measurement. As a consequence, these effects, arising shortly after the acoustic perturbation is turned on, are not relevant here. Furthermore, the acoustic displacements are always small compared to bubbles sizes, and plastic effects are then expected to be negligible. We now examine the effect of the cell walls on the response of the foam to the acoustic perturbation. To eliminate the effect of the lateral walls, the foam is now poured directly onto a horizontal PVC plate, instead of into the U-shaped cell. Furthermore, one has to be aware that in DWS experiments on foams, the volume over which the deformation is probed increases with the age of the foam. Indeed, since the size of the foam bubbles increases when the foam ages, it becomes more and more transparent so that light propagates inside over an average distance that increases with the age of the foam before being backscattered. For this reason, these experiments dealing with the effects of the walls have to be performed shortly after preparing the foam, which is then opaque enough in order not to probe the deformation in the vicinity of the bottom boundary. We fix the frequency and amplitude of the acoustic signal. First, we observe a significant decrease of g 2 (τ) with τ, due to the fact that the foam sample is "young" so that it ages faster than in the situation of Fig. 2. Second, almost no modulation is observed on g 2 (τ) (bottom curve on Fig. 3). Then, for the same acoustic perturbation, we add a glass wall against the illuminated side of the foam sample and observe a clear modulation of g 2 (τ) (top curve of Fig. 3) g 2 (τ) τ (µs) 5 20x0 3 FIGURE 3: Correlation functions g 2 (τ) measured for the same acoustic perturbation in the presence (top curve) and in the absence (bottom curve) of wall on the illuminated side of the foam sample. The frequency and amplitude of the acoustic wave in both cases are 400 Hz and 82.3 db, respectively. Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 4

5 LIGHT SCATTERING MODEL Our experiments show that the foam is deformed by the acoustic wave. In this section, we recall a theoretical framework describing the effect of an acoustic wave on the backscattered light correlation functions, for a foam constrained between solid walls [5]. The effect of an acoustic perturbation on the coherence of light multiply scattered by a medium has been the subject of some theoretical and experimental studies. The basic mechanism has been identified in [6] by considering the light scattered by a colloidal suspension in which ultrasonic waves propagate. This propagation induces a deformation of the sample, modifying the optical paths inside. In [6], the propagation of a high-frequency (2 MHz) ultrasonic wave is considered. In this case, the deformation of the material arises from its small compression and dilation along the direction of propagation of the acoustic wave and then scales with the acoustic frequency. In our experiment, we analyze backscattered light, thus we measure the acoustic deformation of the foam mostly near its boundary. Let A f = A f (z)sin(ωt) e x by the acoustically-induced displacement in the foam (x: direction parallel to the boundary, z: direction orthogonal to the boundary). In the following, we make the assumption that the deformation caused by the propagation of an acoustic wave has a vanishing amplitude at the solid wall (no-slip boundary condition, as will be justified later. Let then ξ be the extent of the deformed zone in the z direction. We consider a simplified model for the propagation of a photon inside the foam: it consists of a random walk made of a succession of independent steps separated by a constant scattering mean free path l. We also consider that the directions of the different steps are uncorrelated. We assume that the distance between scattering events is large compared to the optical wavelength λ and that there is no correlation between different paths (weak scattering approximation). Then the autocorrelation function g (τ) = E(t) E (t + τ) t of the scattered electric field E(t) can be shown to be equal to [5]: g (τ) γ 2 π k2 2 l A f ξ ( cos(ωτ)) if ka f cos(ωτ) < (3) γ l ξ ka 0 f ξ cos(ωτ) if < kaf cos(ωτ) < l, (4) where k is the light wavevector. Equations (3) and (4) deserve some comments. Let us first discuss the limit ξ : this corresponds to a constant shear Γ = A f /ξ in all the material. In this limit, the situation described by (3) does not occur, and g varies linearly with Γ in the limit klγ <<, as predicted by (4). When the extent of the sheared zone ξ is finite, the situation is somewhat different. The decorrelation g at early delay times is dominated by the contribution of long photon paths in the medium [7]. However, these long paths being essentially in undeformed parts of the material, the decorrelation is smaller than in the case of an homogeneous shear. This is the situation described by (3). DISCUSSION Quadratic dependence of Δg with the acoustic amplitude To compare our experimental results with the theory presented above, we first compute from the data the amplitude Δg (τ) of the modulation of the correlation function of the scattered intensity. The correlation functions of the scattered electric field and intensity are indeed linked by the Siegert relation [8] : g 2 (τ) = + β g (τ) 2, (5) Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 5

6 where β is an experimental parameter fixed by the optical setup [8]. We first determine β as the limit g 2 (τ 0), and then we compute g (τ) from (5). We define the amplitude Δg (τ) of the modulation of g (τ) as follows: Δg = gmax,2 + g max,3 g min, (6) 2 where g max,2 and g max,3 are the second and third maxima of g, and g min is the minimum in-between (see Fig. 2). This definition of the amplitude modulation corrects the modulation of g from a slow linear decrease of g due to the foam aging. Δ g (τ) Hz khz 4kHz 0kHz k opt A FIGURE 4: Plot of Δg as a function of ka a, where k is the optical wavevector and A a is the acoustic displacement in air. Symbols are experimental data and lines are quadratic fits. Fig. 4 is a log-log plot of Δg as a function of ka a for different values of the acoustic frequency. For each frequency, Δg is found experimentally to vary quadratically with ka a.as shown before, this cannot be interpreted without supposing that the deformation of the foam must be localized in the vicinity of the wall, over a distance ξ shorter than the acoustic wavelength. We also need to check if we are in the required condition for Eq. (3) to be valid., i.e. ka f <. For our experiment, the values of the acoustic displacement of the foam are then expected to be smaller than t f A a, with t f < 0.2 the transmission coefficient from air to foam [5]. In our case, the acoustic amplitude is such that 0.02 < ka a < 0, and then < t f ka a < 2. Since we expect A f to be less than or of the order of A a, we may expect ka f to be less than or of the order of unity. Thus, Eq. (3) should hold, and we expect: Δg = g (0) g (π/ω) = 0.56(kA f ) 2 l /ξ. (7) This quadratic dependence of Δg with the acoustic amplitude is the observed experimental behavior (Fig. 4). Shear length and no-slip boundary condition The model of deformation used to compute theoretically and numerically the modulation of intensity correlation functions assumes that the foam deformation is localized in a thin layer near the walls. The presence of such a sheared zone in our experiment suggests that the foam is not free to move at the wall. This might appear surprising, since a glass boundary is usually considered as smooth, and hence slip is expected to occur on it. However, one shall be aware that the acoustic displacement is not quasi-static, so that viscous effects on the boundary must be taken in account. We have proven that due to the combination of high frequencies and tiny displacements specific to an acoustic forcing, the foam indeed displays an effective no-slip boundary condition, and predicted the length ξ over which the foam close to the wall is significantly sheared [5]. For this, we take the following scaling of the foam shear modulus: Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 6

7 G = G 0 [ + (iω/ω c ) /2 ], with for Gillette foam: G 0 = N m 2 and ω c = rad s [9], at frequencies above 5 rad s,uptoafew0 2 rad s. In our experiments, ω ω c, hence G G 0 (iω/ω c ) /2. Then: G0 ξ ω /4 c ω 3/4 ρ f sin(π/8), (8) where ρ f = 0.06 g cm 3 is the foam density. Computed values of ξ are 2.8,.4, 0.5 and 0.2 mm for f = 0.4,, 4 and 0 khz, respectively: ξ is of the order of mm, which means that the foam is sheared over typically ten layers of bubbles. The corresponding values of the acoustic wavelength λ ac = c/f, with c 50 m/s [9], are 2.5, 5,.25 and 0.5 cm: they are one to two orders of magnitude higher than ξ, which justifies our main approximation λ ac ξ. It is worth mentioning that 3D foam shear experiments usually do not exhibit shear localization. Actually, the mechanism of localization in our experiments is rather specific, since the occurrence of no-slip and the importance of inertia are crucially related to the combination of high frequencies and small displacements which are classical in acoustics, but very unusual in classical shear rheometry, where much lower frequencies and higher displacements are applied. Anyway, this theoretical analysis supports the experimental data, where the modulation of g 2 shows the occurrence of shear close to the wall. Comparison between theory and experiments The previous modeling has allowed us to extract theoretical values of the localization length ξ. We can now try to compare their orders of magnitude to our experimental data. We measure from the data (Fig. 4) the ratio S = Δg /(ka a ) 2. Applying Eq. (7), we obtain (A f /A a ) l /ξ = 2.5 S. Measured values of 2.5 S are 0.07, 0.3, 0.83 and 0.6 at f = 0.4,, 4 and 0 khz. These values are in qualitative agreement with the behavior expected from Eq. (8), namely that ξ decreases when the acoustic frequency increases. CONCLUSION We have shown that it is possible to monitor - by a multiple light scattering technique like DWS - how an external acoustic signal induces bubble displacements inside a foam. We have determined the origins of the modulation seen in light scattering data, and in particular understood its amplitude and the non-trivial quadratic dependance with the acoustic amplitude. We have shown that such a dependance is the signature of a specific shear profile near the walls holding the foam. Quantitatively, the experimental results are also consistent with the theoretical estimations; this especially concerns the amplitude of the deformation inside the foam and how it is related to the one in air. In agreement with the predictions, we can extract from the data that the typical displacement in the foam is of the order of a few nm, spread over a bubble diameter, thus corresponding to deformation of the order of 0 4. Note that on the general issue of the coupling between the acoustic wave outside and inside the foam, our experiments have evidenced an important feature: the transmission coefficient turns out to strongly depend on the frequency, thus on the wavelength. This is most likely due to a finite size effect, which remains to be fully understood. Another interesting result is illustrated by the set of equations (3) and (4): we have derived analytical results showing significant differences, in back scattering, between a sample undergoing an homogeneous constant shear and a sample in which sheared and non-sheared parts are present. On this specific question, performing DWS in back scattering and testing Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 7

8 whether a modulation occurs linearly or quadratically with the amplitude of deformation (whatever its source) is then a possible way to investigate the spacial homogeneity of the deformation. More generally, this study can be considered as a first landmark, already bringing some insights on some of the questions raised in the introduction : in particular, we have shown that an acoustic wave is actually an interesting way for applying controlled shear deformations on a foam, in this range of high frequencies which remains poorly understood. As well, on the foam acoustic issues, we have shown that DWS is a useful tool for extracting information at the scale of the bubble, probably crucial for explaining macroscopic sound attenuation and speed in foams. Experiments in this direction are under way. REFERENCES [] S. Jurine, S. J. Cox, and F. Graner, Dry three-dimensional bubbles: growth-rate, scaling state and correlations, Colloids Surf. A 263, 8 26 (2005). [2] S. Hilgenfeldt, S. A. Koehler, and H. A. Stone, Dynamics of coarsening foams: accelerated and self-limiting drainage, Phys. Rev. Lett. 86, (200). [3] D. J. Durian, D. A. Waitz, and D. J. Pine, Scaling behavior in shaving cream, Phys. Rev. A 44, R7902 R7905 (99). [4] H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology (Elsevier) (989). [5] A. D. Gopal and D. J. Durian, Shear-induced melting" of an aqueous foam, J. Colloid Interface Sci. 23, (999). [6] R. Höhler and S. Cohen-Addad, Rheology of liquid foam, J. Phys. Condens. Matter 7, R04 R069 (2005). [7] S. Marze, D. Langevin, and A. Saint-Jalmes, Aqueous foam slip and shear regimes determined by rheometry and multiple light scattering, J. Rheol. 52, 09 (2008). [8] A. J. Liu, S. Ramaswamy, T. G. Mason, H. Gang, and D. A. Weitz, Anomalous viscous loss in emulsions, Phys. Rev. Lett. 76, (996). [9] N. Mujica and S. Fauve, Sound velocity and absorption in a coarsening foam, Phys. Rev. E 66, (2002). [0] I. Ben Salem, R. M. Guillermic, C. Sample, V. Leroy, A. Saint-Jalmes, and B. Dollet, Propagation of ultrasound in aqueous foams: bubble size dependence and resonance effects, Soft Matter 9, (203). [] J. Pierre, F. Elias, and V. Leroy, A technique for measuring velocity and attenuation of ultrasound in liquid foams, Ultrasonics 53, (203). [2] D. Bicout, E. Akkermans, and R. Maynard, Dynamical correlations for multiple light scattering in laminar flow, J. Phys. I, (99). [3] D. Bicout and R. Maynard, Diffusive wave spectroscopy in inhomogeneous flows, Physica A 99, (993). [4] M. Erpelding, A. Amon, and J. Crassous, Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid, Phys. Rev. E 78, (2008). Proceedings of Meetings on Acoustics, Vol. 9, (203) Page 8

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