HAL Id: hal-01313968https://hal-univ-rennes1.archives-ouvertes.fr/hal-01313968
Submitted on 14 Jun 2016
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Synchronized diffusive-wave spectroscopy: Principle andapplication to sound propagation in aqueous foamsJérôme Crassous, Patrick Chasle, Juliette Pierre, Arnaud Saint-Jalmes,
Benjamin Dollet
To cite this version:Jérôme Crassous, Patrick Chasle, Juliette Pierre, Arnaud Saint-Jalmes, Benjamin Dollet. Synchro-nized diffusive-wave spectroscopy: Principle and application to sound propagation in aqueous foams.Physical Review E , American Physical Society (APS), 2016, 93 (3), pp.032611. �10.1103/Phys-RevE.93.032611�. �hal-01313968�
ACCEPTED MANUSCRIP
T
Synchronized Diffusive Wave Spectroscopy: Principle and Application to SoundPropagation in Aqueous Foams
Jerome Crassous,∗ Patrick Chasle, Juliette Pierre, Arnaud Saint-Jalmes, and Benjamin DolletInstitut de Physique de Rennes, UMR CNRS 6251, Universite de Rennes 1,
Campus de Beaulieu, F-35042 RENNES Cedex, France(Dated: January 18, 2016)
We present an experimental method to measure oscillatory strains in turbid material. The materialis illuminated with a laser, and the speckle patterns are recorded. The analysis of the deformationsof the optical pathlength shows that the speckle patterns are modulated at the strain frequency.By recording those patterns synchronously with the strain source, we are able to measure theamplitude and the phase of the strain. This method is tested on the specific case of an aqueous foamwhere an acoustic wave propagates. The effects of material internal dynamics and of heterogeneousdeformations are also discussed.
PACS numbers: 42.25.Dd Wave propagation in random media, 78.35.+c Brillouin and Rayleigh scattering;other light scattering, 62.20.F- Deformation and plasticity 83.80.Iz Emulsions and foams
I. INTRODUCTION
Soft materials are commonly used in the food, cosmet-ics and pharmaceutical industries; this is in part due tothe wide range of texture and rheology which can be en-countered when dealing with gels, pastes, concentratedcolloidal solutions, foams or emulsions. Together withthese mechanical aspects, the ability to encapsulate anddeliver chemicals provides some increased interest in suchmaterials. As well, the non-trivial and often dispersivenature of wave propagation in such materials is also an-other specificity which can be optimized to get originalthermal or acoustical macroscopic properties. The draw-backs of such a high potentiality in applications are thestructural and dynamical complexity of such soft mate-rials; they are disordered, often opaque, and generallydriven in a out-of-equilibrium state as they are formed.This results in time evolution (aging) and complex intrin-sic dynamical mechanisms, with possible intermittencyand heterogeneity in space and time. Understandingthese evolutions inside the materials remains a genericand important issue. It is for instance important to de-termine the microscopic origins of the intrinsic rearrange-ments, and how they can be coupled and superimposedonto rearrangements induced by external forcing. Aswell, transport properties and wave propagation in suchdisordered and amorphous materials need to be investi-gated, both on fundamental and practical standpoints.Numerous techniques to scan these materials have beendeveloped, especially light scattering methods, taking ad-vantage of being non-intrusive. When dealing with turbidmedia, diffusive wave spectroscopy (DWS) is well suited,as it can monitor the internal dynamics in highly diffu-sive samples [1, 2]. This technique has been improved indifferent ways to better resolve the space and time fluc-tuations, to reduce the measurement timescales, and to
get more information on the homogeneity of the dynam-ics [3–9]. Despite these improvements, the optimizationof existing light scattering methods or the design of newones remains an active axis of research.
As a typical example of soft, opaque and aging materi-als, aqueous foams — dispersions of gas into a liquid —have been used as model systems to test and demonstratethe efficiency of DWS [10, 11]. With time, DWS actuallybecame a major tool for investigating foams and has pro-vided important insights on foam properties at the scaleof the bubbles, especially when it was coupled to macro-scopic rheology [12, 13]. However, and despite their wideuse in industries and active academic studies [14, 15], var-ious issues on aqueous foams remain open. These pend-ing problems mostly concern (i) the time destabilizationof a foam and how this can be controlled by the physico-chemical parameters, and (ii) the rheology of foams andits specificities when compared to other soft glassy ma-terials. In that respect, new progress requires techniquesto follow how the bubble diameter D and the foam liq-uid fraction φl evolve in time; in fact, as a consequence ofdrainage and coarsening, these two crucial quantities canhardly be kept constant, as the bubble diameter increaseswith time and the initial liquid content of the foam de-creases with time. Optical and electrical methods can beused to monitor these aging effects, but the whole evo-lution of D and φl can still not be simply inferred withthese techniques. More recently, the acoustic propertiesof aqueous foams have also been widely investigated, withthe aim to design original methods to probe foam prop-erties [16–21]. The propagation of sound in foams turnsout to be quite complex, with non-trivial effects, likethe occurrence of negative density [20]. Though promis-ing, the experimental approach based on transducers addalso some experimental limitations due to the mismatchof acoustic impedance between the transducer and thefoam; hence, the injected acoustic wave depends on thefoam properties itself. As a consequence, measuring allthe sound features into a foam remains tricky. As analternative, the direct measurement of the acoustic de-
ACCEPTED MANUSCRIP
T
2
formation (amplitude and phase) at any given locationis another approach to extract the sound velocity and itsattenuation. Along this direction, we previously showedthat the acoustic deformation can be detected by DWS[22]. This was a first attempt to use light scattering tech-niques to investigate the sound propagation into a foam.
The method described in [22] allows only to obtain theamplitude of the acoustic waves. The phase of the wavemay be obtained by synchronizing the acquisition of scat-tered light with the acoustic signal. This has been doneby Wintzenrieth et al [23]. These authors used a methodbased on the visibility of the speckle pattern [9]. Thevariations of the phase difference between the beginningof time integration and the harmonic excitation producesa variation of the visibility. The relative phase betweenthe excitation and material deformation may then be ob-tained.
Following those previous works, we propose here an-other DWS experimental scheme where the deformationamplitude and phase can be inferred from synchronizedmeasurements and the combination of four partial corre-lation functions. The system chosen to make this proofof concept consists in an aqueous foam, where the sheararises from an acoustic forcing. This method is testedon the acoustics propagation into a liquid foam and com-pared to previous measurements.
In this article, we first describe this DWS scheme,named ”Synchronized DWS”, and present the mathemat-ical formalism — valid for any diffusive system under asinusoidal shear deformation — showing that the shearamplitude and phase of this deformation can be derivedby measuring four different correlation functions, mon-itored synchronously with the deformation. Then, weexplain how this can be tested by doing experiments onfoams under an acoustic forcing. Various results are pre-sented, either at a given location for different imposedamplitude and phase, or while scanning the foam as afunction of the distance from the source. Further exper-imental tests, discussions and comparisons are given inthe last section.
II. SYNCHRONIZED DIFFUSING WAVESPECTROSCOPY
A. Principle of the method
Diffusing wave spectroscopy is a experimental toolwhich has been successfully used for concentrated col-loidal suspensions, emulsions, foams or granular materi-als. The principle of DWS is to record multiply scat-tered coherent light, and to monitor the time evolutionof this scattered light, in terms of correlation functions.Shortly speaking, the correlation of the scattered inten-sity arises from interference between many photon paths.If a system is at rest, the interference pattern is frozen,and the scattered intensity does not show temporal evo-lution. If scatterers move inside the sample, interference
FIG. 1. (a) A sinusoidal shear of displacement γ. Time t0is such that γ(t0) = 0. (b) Snapshots of a portion of opticalpath (full line) on some scatterers (black circle). The zone isperiodically sheared and the path periodically deformed. (c)Variation of a segment joining two points r and r + l∗e. A isthe displacement field of the scatterers.
pattern fluctuates, and the intensity correlation functiondecreases with time. We consider here the case of a phys-ical system of scatterers which is periodically sheared atfrequency T . We plot in Fig. 1a a sinusoidal shear defor-mation. Fig. 1b shows an optical path joining some scat-terers. Since they are displaced into the sheared zone,the path is also periodically deformed. Let t0 be a timeat which the strain vanishes. The paths at times t0+T/4and t0 + 3T/4 are symmetrically sheared. This shear in-duce a phase shift along this ray, thus a loss of correlationof the scattered light between these two times. The am-plitude of the shear may then be related to this loss ofcorrelation. Because γ(t0) = γ(t0 + T/2), the paths attimes t0 and t0 + T/2 are identical. The system appearsfrozen, and the intensity of the scattered line is perfectlycorrelated. Hence, the time t0 may be deduced from themaximum of correlation of speckle patterns acquired attime t and t + T/2. The phase difference between theshear at a given point of the material and the source ofthe shear may then be estimated. Therefore, in principle,by a synchronized monitoring at times t, t+T/4, t+T/2and t+3T/4, one can derive the amplitude and the phaseof the deformation. The derivation of the relevant math-ematical combinations allowing us to get these quantitiesfrom the partial correlations is given below. The princi-ple of parallel lock-in detection that we use is not newand has been used in detection of ultrasonic modulationsuch as described in [24, 25], but to our knowledge neverused with DWS.
ACCEPTED MANUSCRIP
T
3
B. Link between the intensity correlation functionand material deformation
In the following, we note l∗ the transport mean freepath into the material, f = 1/T the frequency of thestrain modulation and ω = 2πf . The length of a photonpath into the sample is noted s. We consider here a par-allel detection of the scattered light. For this a camera isused as sensor, and the intensity of many pixels are mea-sured simultaneously. We note I(t) the array of scatteredintensity acquired at a time t. We compute the normal-ized correlation function of the scattered intensity:
gI(t, t+ τ) =〈I(t)I(t+ τ)〉 − 〈I〉2
〈I2〉 − 〈I〉2, (1)
where 〈·〉 is here an average on the pixels of the cam-era. The electronic noise of the camera is removed fol-lowing a procedure explained in [26]. The normaliza-tion of the correlation function ensured that gI(t, t) = 1whatever the contrast of the image. From the Siegertrelation gI(t, t + τ) = |gE(t, t + τ)|2 we obtain the nor-malized correlation function of the scattered electric fieldE : gE(t, t + τ) = (1/I0)〈E∗(t) · E(t + τ)〉. The normal-ization constant I0 is the mean scattered intensity.
The autocorrelation function of the scattered electricfield may be decomposed as a sum on paths of lengths [1, 2]:
gE(t, t+ τ) =
∫s
P (s)〈exp(∆φs(t, t+ τ)
)〉ds. (2)
In (2), P (s) is the normalized path length distributionwith
∫sP (s)ds = 1. The complex exponential function
is averaged on all paths of length s, and ∆φs(t, t+ τ) isthe phase shift for a path of length s. We drop in thefollowing the two times t and t + τ in equations. Suchpaths may be decomposed in s/l∗ independent segments
of length l∗. It then follows that ∆φs =∑s/l∗
j=1 ∆φj ,with ∆φj the phase variation of the segment number jbetween times t and t+ τ . Since ∆φs is the sum of manyindependent variables, it is a Gaussian random variable,and then:
〈exp(∆φs)〉 = exp
[−1
2〈∆φ2s〉
]
= exp
−1
2
s/l∗∑j=1
〈∆φ2j 〉
. (3)
In (3), 〈·〉 is an average on all orientations of the seg-ments, and we also assume that 〈∆φj〉 = 0, i.e. there isno dilatation of compression of the medium. If the defor-mation is homogeneous, 〈∆φ2j 〉 does not depend on the
number of the segment, and∑s/l∗
j=1 〈∆φ2j 〉 = s〈∆φ2〉/l∗.For small enough phase shifts, exp
[s〈∆φ2〉/2l∗
]' 1 −
s〈∆φ2〉/2l∗ and (2) becomes: gE(t, t + τ) '∫sP (s)[1 −
s〈∆φ2〉/2l∗]ds. After integration we obtain:
gE(t, t+ τ) ' 1−A〈∆φ2(t, t+ τ)〉, (4)
with A a constant depending on P (s). Following the ap-proach detailed in numerous studies [27? –29] we nowrelate 〈∆φ2〉 to the deformation of the material. We con-sider a segment joining the points r and r + l∗e withe the unit vector joining the two points as shown onFig. 1c. The displacements at time t are A(r, t) andA(r+l∗e, t), and the new distance between the two pointsis ' l∗
(1+[(e ·∇)A(r, t)
]·e). The variation of the phase
between the two times t and t+ τ is then k times the dif-ference of length:
∆φ = kl∗[(e · ∇)
(A(r, t+ τ)−A(r, t)
)]· e. (5)
For a harmonic displacement with a polarization alongthe unit vector a, we can write A(r, t) = [α(r) cos(ωt) +β(r) sin(ωt)]a. The phase variation is then:
∆φ ' 2kl∗(e · a) sin(ωτ/2)[−(e · ∇α(r)
)sin(ωt+ ωτ/2)
+(e · ∇β(r)
)cos(ωt+ ωτ/2)
]. (6)
For simplicity, we suppose that ∇α(r) and ∇β(r) areparallel to the same unit vector n. We will justify thisassumption for our experiments in Sec. IV C. We maythen write:
∇α(r) = γ(r) cos(Ψ0(r)
)n (7a)
∇β(r) = −γ(r) sin(Ψ0(r)
)n, (7b)
with γ(r) the amplitude and Ψ0(r) the phase of the de-formation. Performing average on segment orientations:
〈∆φ2〉 ' 4k2l∗2η(a,n)γ2(r) sin2(ωτ/2)
× sin2(ωt+ ωτ/2 + Ψ0(r)) (8)
with η(a,n) = 〈(e · a)2(e · n)2〉e. For transverse wavesη(a,n) = 2/15. Combining with (7), we finally obtain:
γ2(r) sin2(ωτ/2) × sin2(ωt+ ωτ/2 + Ψ0(r))
' C × [1− gE(t, t+ τ)], (9)
with C a constant depending on geometry and of opticalconstants.
C. Synchronized acquisition
The correlation functions are now measured at fourdifferent combinations of times t(mod T ) and t+ τ(modT ):
g(1) = gE(0, T/2) (10a)
g(2) = gE(T/4, 3T/4) (10b)
g(3) = gE(0, T/4) (10c)
g(4) = gE(T/4, T/2). (10d)
With (9), we have:
ACCEPTED MANUSCRIP
T
4
C × [1− g(1)] = γ2(r) sin2[Ψ0(r) + π/2] (11a)
C × [1− g(2)] = γ2(r) sin2[Ψ0(r) + π] (11b)
C × [1− g(3)] =1
2γ2(r) sin2[Ψ0(r) + π/4] (11c)
C × [1− g(4)] =1
2γ2(r) sin2[Ψ0(r) + 3π/4], (11d)
and then:
γ2(r) = C ×[2− g(1) − g(2)
], (12a)
cot 2Ψ0(r) =[ g(1) − g(2)
2− g(1) − g(2)]/[ g(3) − g(4)
2− g(3) − g(4)]. (12b)
The amplitude and the phase of the shear may thenbe simply obtained with the above combinations of cor-relation functions, all measured synchronously with theharmonic excitation.
III. EXPERIMENTS
A. Experimental Setup
FIG. 2. Schematic drawing of the setup. A cylindrical glasscell is filled with a liquid foam. A loudspeaker generates anacoustic wave. A laser illuminates the foam and the scatteredlight is detected by a camera. A lens is placed to expand thebeam. The camera and the laser may be translated along thez direction.
The preceding formalism is in principle applicable toevery scattering medium which is periodically sheared.In order to test this formalism, we performed experimenton aqueous foams, in which we apply a sinusoidal defor-mation by an external acoustic forcing. In our geometry,the strain occurs because there is a displacement fieldgenerated by the acoustic wave into the bulk, and noacoustic displacement at the glass walls [22]. This cre-ates a periodically sheared layer near the lateral glassboundaries. The structure of this sheared layer will bediscussed in Sec. IV C.
The experimental setup is drawn on Fig. 2. A cylin-drical glass container (diameter 19 cm, height 50 cm)
is filled with a liquid foam. A loudspeaker is placedabove the cylinder, and generates a acoustic wave at a fre-quency f . The acoustic wave emitted by the loudspeakeris partially reflected by the foam surface, and partiallytransmitted within the foam. The sheared layer is illu-minated with the beam of a continuous laser (633 nm,15 mW). The beam is slightly expanded to a diameterof a few millimeters and illuminates the foam. The scat-tered light is recorded with a CSMOS camera (Photon-Focus MV1-D1312-160-CL) operating at a frequency 4f :sequences of images at 4f are acquired in a burst mode.The beginning of the acquisition sequence is triggeredsynchronously to the loudspeaker excitation. A delaybetween the trigger and the first acquired image may be
added in order to imposed a given phase shift ψ(imp)0 . All
the experiments presented here have been performed atfrequency f = 800 Hz and the exposure time of everyimage is 11 µs. The number of pixels of each image isNp = 13120.
The liquid foam is made using the method describedin [30]: it provides large quantities of homogeneous foam,with initial bubble size of the order of 100 microns in di-ameter. The production rate is 0.1 L/s, so that our 50-cmhigh cell is filled in 1 min. The gas used is C2F6, to slowdown the coarsening [31]. Experiments are performed af-ter waiting times between 30 mn to 5 hours, correspond-ing to bubble diameter of 200 to 700 microns. For theexperimental results shown here, we used sodium dodecylsulfate (SDS) as surfactant, with a concentration equalto 10 times the surfactant critical micelle concentration(cmc). The initial liquid fraction is φl ' 5%. Comple-mentary tests were made using a more complex chemicalformulation, providing a high interfacial viscoelasticity[32].
B. Computations of correlation functions
The raw data are a numbered succession of images ofNp = 13120 pixels acquired at intervals 1/4f = 312.5 µs.Because the light intensity levels are low, we need to care-fully remove the electronic noise of the sensor. We definethe normalized intensity correlation function between twoimages n and m as:
gn,mI =〈ImIn〉 − 〈Im〉〈In〉 − aelec − belecδnm
〈I2n〉 − 〈In〉2 − aelec − belec(13)
We noted In the intensity vector of Np pixels of theimage n, and 〈·〉 designs here an average on all the pix-els. We checked that the electronic noise in is such that〈inim〉 = aelec + belecδnm. The values of aelec and belecare determined by an independent measurement, andwe checked that for a well aged foam at rest, we getgn,m�nI = 0 and gn,n+1
I = 1.
For a cycle of oscillation n, we compute the quantities
ACCEPTED MANUSCRIP
T
5
g(i)I (n) as:
g(0)I (n) =
1
4
p=3∑p=0
g4n+p,4n+4+pI (14a)
g(1)I (n) =
1
2(g4n,4n+2I + g4n+2,4n+4
I ) (14b)
g(2)I (n) =
1
2(g4n+1,4n+3I + g4n+3,4n+5
I ) (14c)
g(3)I (n) =
1
2(g4n,4n+1I + g4n+1,4n+4
I ) (14d)
g(4)I (n) =
1
2(g4n+1,4n+2I + g4n+2,4n+5
I ). (14e)
If we suppose a scattering medium where the dynam-ics are only due to the acoustic wave (i.e. no internaldynamics), we have for i 6= 0:
g(i) =
√g(i)I (4n), (15)
where the g(i) are the quantities defined in (10). The
interest of computing g(0)I (n) and of the somewhat tricky
definitions of g(i6=0)I (n) is to allow some corrections of the
scattering medium dynamic. This will be explained inSec. IV A.
FIG. 3. Correlations functions of the scattered intensitiesg(i)I (n) as a function the number n of the oscillation cycle.
Colors are i = 0: purple, 1: black, 2: red, 3: green and 4:blue.
Fig. 3 shows the evolution of the function g(i)I (n) as a
function of the cycle of oscillation n. In this experimentthe value of the acoustic excitation is kept constant, with
a fixed phase. g(0)I is close to 1, the fluctuations being
due to the intrinsic dynamics of the foam (coarsening
events). The values of g(i6=0)I fluctuate around their mean
values, and the values depend on the partial correlationsfunctions number i.
In the following, we will average g(i) on 250 cycles ofoscillations, i.e. on sequences of acquisition 0.31 s.
C. Amplitude and phase measurements
In this Section, we demonstrate that we can determinethe imposed variation of amplitude and phase — at agiven location — from our model of combinations of cor-relations functions.
The amplitude of the strain is obtained from (12a) as:γ2(r) = C ×
[2 − g(1) − g(1)
]. We perform experiments
where the amplitude of the strain is varied. For this, wechange the level of excitation of the loudspeaker. In ourexperiments, we remain always below 90 dB of acousticlevel in air, hence the pressure amplitude in air is below1 Pa, and it is even lower in foam because only a fractionof acoustic energy is transmitted to the foam. Hence, weremain in the regime of linear acoustics. Indeed, sincefoam is mostly made of gas, the order of magnitude ofits bulk modulus is given by the atmospheric pressure,105 Pa, and deviations from linear acoustics are expectedonly when the pressure amplitude of the acoustic wavebecomes a significant fraction of the bulk modulus.
FIG. 4. Symbols: 1 − (g(1) + g(2))/2 as a function of theamplitude of excitation of the acoustic wave. The straightline is of slope 2.
Fig. 4 shows the evolution of 1 − (g(1) + g(2))/2 asthe function of the excitation amplitude. Every symbolcorrespond to one measurement of duration 0.31 s. Wesee that the decorrelation increases with the amplitudeof the acoustic excitation. The straight line is line ofslope 2, showing that 1 − (g(1) + g(2))/2 varies as γ2 atleast at small amplitudes. This is in agreement with (4).It should be remained that (4) has been obtained in thelimit of small phase shifts, and deviations to the γ2 areexpected at large strain amplitudes.
The phase of the strain is obtained from (12b). Inorder to test this relation, we proceed as follow: we fix theamplitude of the acoustic wave, and we begin to acquire
images with a given phase shift ψ(imp)0 (see Sec. III A).
Fig. 5a shows the evolution of g(i) as a function of the
imposed phase shift ψ(imp)0 . We see that the correlation
functions evolves with the imposed phase shift. As itmay be seen from (9) and (10), advancing the imposedphase of π/2 has the same effect of the permutations ofg(1) with g(2) and of g(3) with g(4), reflecting the ωt+ψ0
ACCEPTED MANUSCRIP
T
6
FIG. 5. (a) Variations of g(i) as a function of an imposedphase shift. i = 1: black, 2: red, 3: green and 4: blue.Symbols are results from measurements and continuous curvesare sinusoids of period π for eyes guidelines. (b) Computedphase shift as a function of the imposed one. Symbols aremeasurements and the line is the linear fit with a slope 1.
dependence of (9). The phase is obtained from (12b) as:
Ψ0 =1
2arccot
[( g(1) − g(2)
2− g(1) − g(2))/( g(3) − g(4)
2− g(3) − g(4))](16)
The unwrapped phase Ψ0 calculated from the data ofFig. 5a is plotted on Fig. 5b. As it may be seen, thephase of the acoustic wave is measured with high pre-cision. This shows that the phase of the strain may beextracted from DWS synchronized with the strain sourcewith great accuracy.
D. Application to the measurement of phasevelocity in a foam
Building on the previous experimental proofs of con-cept at a single position, we investigated how the phaseevolve inside a foam as a function of the distance of prop-agation into our cylindrical cell. This is the relevant ex-periment in terms of foam acoustics, as it should provideinformation which can be compared to previous ones. Inpractice, the laser and the camera are therefore trans-lated of a distance ∆x = 2 mm between successive mea-surements. The ageing time of the foam is here 45 mn.This is a short enough time to avoid drainage and to pre-vent liquid fraction variations in the scanned part of the
foam [33].
FIG. 6. Sound phase as a function of the distance of propaga-tion: comparison between the measured phase (symbol) andthe model of Eq. (17).
Fig. 6 shows the evolution of the phase as a func-tion of the depth. The phase shift is roughly 4π, cor-responding to two acoustic wavelengths. In this ex-periment, the bottom “wall” is the top surface of thedrained liquid in the bottom of the glass cylinder. It islocated at a the distance H = 202 mm. We see that,far from the bottom of the foam, the phase evolves lin-early with the distance; however, small oscillations ofthe phase occurs as the surface of the drained liquidis approached. Rather than only focusing on the linearregime away from the bottom, we can take into accountall these features by including interferences between thepropagating and reflected wave into the cylinder. Letui(x) = u0 exp[−(jka + βa)x] be the incident acousticwave, with u0 the amplitude, ka the wave vector of theacoustic wave and βa the attenuation coefficient. The re-flected wave is ur(x) = Ru0 exp (jka + βa)x with R thecomplex reflection coefficient. We set that, at the bot-tom, the drained liquid is a perfectly reflecting surface,which yields the boundary condition: ur(H)+ui(H) = 0.This assumption comes from the fact that the acous-tic impedance Z = ρc of a foam is orders of magni-tude lower than that of the liquid, because both ρ andc are much lower in foam than liquid. It follows thatR = e−2(jka+βa)H , and that the argument of u = ur + uiis:
arg u = −jkax− arctane−2βa(H−x) sin 2ka(H − x)
1 + e−2βa(H−z) cos 2ka(H − x).
(17)The plain line of Fig. 6 shows the evolution of arg(u)
as the function of the distance as predicted by (17). Thefree parameters are ka, βa and a constant phase which isadded to arg(u). The shown adjustment is obtained withka = 0.09 mm−1 and βa = 0.009 mm−1. The value of thevelocity is thus v = ω/ka = 56 m/s. These values fullyagree with the previous studies. For the velocity, it isexpected to have a value following the Wood’s law, valid
ACCEPTED MANUSCRIP
T
7
for these bubble sizes and frequencies [18, 19]. Indeed,the Wood’s law predicts a velocity of 55 m/s for φl=0.05.Although we have not monitored the variation of liquidfraction, the fact that this value is so close to the experi-mental value also suggests that drainage has not modifiedmuch the profile of liquid fraction over the height of thefoam probed in Fig. 6. Also, Pierre et al. [19] have shownby acoustic measurements in an impedance tube that at800 Hz, the attenuation coefficient equals 8 ± 1 m−1 forSDS foams, a value insensitive (at this frequency) to theliquid fraction, which is in excellent agreement with ourmeasurement.
IV. REMARKS ON THE METHODS AND ONTHE EXPERIMENTAL TESTS
Following the experimental tests demonstrating thefeasibility of measuring the amplitude and the phase ofa acoustic wave propagating into a foam, we want topoint out a few other remarks on this synchronized DWSmethod.
A. Corrections of internal dynamics
We want here to give some information about the sen-sitivity of the determination of the correlation functions.The first point to address is the effect of the internaldynamics of the scattering medium on the correlationfunctions. Fig. 3 shows the evolutions of the correlation
functions g(i)I with time. The fluctuations are due to the
internal dynamics of the system. Indeed, it is well knownthat reorganizations occurs in foam which create varia-tions on path lengths [10, 11]. When those fluctuationsof correlations are small compared to the decorrelationsdue to the acoustic wave, their effects may be safely ne-glected. However, in the case of very small strains, theireffects must be taken into account. A simple way consistsin considering that the internal and the acoustic wavesdynamics are uncorrelated. Defining ∆φint(t, t + τ) thevariation of phase due to the internal dynamic, (4) be-comes:
gE(t, t+ τ) ' 1−A[〈∆φ2(t, t+ τ)〉+ 〈∆φ2int(t, t+ τ)〉
],
(18)with ∆φ(t, t + τ) the phase shift due to the acousticwave. Since ∆φ(t, t + T ) = 0, we have gE(t, t + T ) '1 − A〈∆φ2int(t, t + T )〉. If the internal dynamic is dueto many independent rearrangements which occurs ata constant rate between t and t + T , we expect that〈∆φ2int(t, t+ τ)〉 = (τ/T )〈∆φ2int(t, t+ T )〉, and then:
1−A〈∆φ2(t, t+ τ)〉 ' (19)
gE(t, t+ τ)+τ
T
[1− gE(t, t+ T ))
].
Since all the correlation functions g(i)I with i 6= 0 are
measured on a the same mean delay τ = T/2, we may re-
move the internal dynamics by computing a “corrected”value:
g(i6=0)I;corr =
√g(i6=0)I (4n) +
1
2
[1−
√g(0)I (4n)
]. (20)
FIG. 7. Symbols: uncorrected (top) and corrected (bottom)
values of g(i)I as a function of an imposed phase shift. Curves
are sinusoidal guidelines. Color codes are the same as forFig. 3
The effect of such a correction is striking on a smallamplitude measurement. Fig. 7 shows a measurementwhere the phase is controlled and the amplitude of theacoustic wave is very small. If the internal dynamicsis not corrected, one can hardly detect the evolutionsof g(i) with the imposed phase. However as far as theinternal dynamics is corrected using (20), the variationsare clearly visible. This shows that acoustic propagationmay be probed even if the internal dynamic dominatesthe variations of path lengths.
B. Sensitivity and comparison with thestroboscopic method
We want here to quantify the uncertainties on the de-termination of the amplitude and of the phase. We focuson the noise on measurements during 250 oscillations cy-cles. Due to the multispeckle scheme used here, the mea-surement of correlation functions are averaged on manycoherence areas. We record images of 13120 pixels, andthe coherence area of ≈ 4 pixels. We then have ≈ 3000coherence areas on every image. Because the scatteringmedium used here has a internal dynamics, the specklepattern evolves slowly. The typical timescale of this evo-
ACCEPTED MANUSCRIP
T
8
lution may be measured in a experiment where no oscilla-tions take place, and we found an evolution time ≈ 0.1 s,which is 1/3 of the experiment duration. With both thetime and the multispeckle averages, one phase and ampli-tude measurement is averaged on ≈ 3×3000 realizations.
The noise for the phase determination is estimated asthe difference between the measured and the imposedphase in a experiment where the phase is imposed. Wefound a typical noise of 0.02 rad, which does not seemto depend strongly on the amplitude of oscillation. Thestandard deviations on the values of 1 − (g(1) + g(2))/2are typically of order 10−3 when signals are uncorrectedfrom internal dynamics and of 2× 10−4 when the signalsare corrected. We may then expect to measure 1−gE ontypically 3 orders of magnitude.
An alternative method to characterize mechanicalwave in light diffusing medium has been proposed re-cently by Wintzenrieth et al [23]. When a scatteringmedium is submitted to a strain modulation, the contrastof the time-averaged speckle pattern is lowered comparedto contrast of the same static medium. This has beenused to detect ultrasonic waves in optical tomographymeasurements [34]. By synchronizing the beginning ofthe light acquisition with the acoustic wave, and by mea-suring the visibility of the speckle pattern, the phase ofthe strain modulation may be obtained [23]. A quanti-tative comparison between this method and the methodreported in this study is not straightforward. Such acomparison requires quantitative models for the differentsources of noises, for instance noises due to internal dy-namics of the scattering medium, or noise due to sensors,and this analysis is far from being trivial. Nevertheless,we may notice that the visibility methods correspondto measurements of correlation functions integrated overtime. If gE(t) is the electric field correlation function
the visibility is V (T ) = 1T
∫ T0
2(1− tτ )|gE(t)|2dt [9]. It is
then easy to show that gE(t) is related to the second time
derivative of the visibility: |gE(t)|2 = 12d2[V (t)t2]
dt2 . Gen-erally speaking, a time derivative is a source of noise.Hence, we may expect that synchronized acquisitions,which avoid such derivatives, allow to obtain correla-tion functions with a better signal-to-noise ratio. Thisis in agreement with the fact that we are able to mea-sure 1 − gE on typically 3 orders of magnitude, whileWintzenrieth et al. [23] obtains a dynamical range for1− gE of one order of magnitude. This is also confirmedby numerical simulations where we compared correlationsfunctions obtain directly by correlation of signal, andby correlation functions obtained by time-derivative ofvisibility. However, only an experimental comparison ofthe two methods on different experimental systems mayquantitatively confirm these differences.
C. Structure of the sheared layer
The method explained in this paper permits to mea-sure the amplitude and the phase of a sheared zone. We
discuss in this subsection the origin and the structure ofthe shear created by an acoustic wave near a solid sur-face which is perpendicular to the propagation directionof the acoustic wave. In a previous paper [22], we haveconsidered the experimental situation of an acoustic wavepropagating into a foam. The wave propagates parallelto the wall. Because of the non-slip boundary conditionat the boundary, the acoustic displacement cancels at thewall. The variation of the displacement with the distancefrom the wall occurs in a viscoelastic boundary layer,which depends on the rheology of the material. We treatthe foam as a continuous visco-elastic fluid, with ρ thedensity and G = G0 exp j∆ the visco-elastic modulus ofthe material. Let u(z, t) = Re[u(z)eiωt] be the acousticdisplacement, and u0 = limz→∞ u(z) the displacementfar from the wall. The force balance writes:
ρ∂2U(z, t)
∂t2= G
∂2U(z, t)
∂z2, (21)
with U(z, t) = u(z, t)− u0. The solution of (21) is:
U(z, t) = U0 exp (jωt) exp (κz), (22)
with U0 + u0 the amplitude of displacement at z = 0and κ = κ0 exp jϕ. We have κ0 = ω
√ρ/G0 and
ϕ = (π − ∆)/2. The strain is γ = ∂U(z, t)/∂z =U0κ exp (jωt) exp (κz), and we obtain:
|γ(z)| = κ0 exp [−κ0 cos(ϕ)z] (23a)
Ψ0(z) = ϕ+ κ0 sin(ϕ)z. (23b)
In this model, the amplitude and the phase of thestrain depend on z. As explained in [22], it is this shearedlayer that is probed with light scattering. In Sec. II, wemade the hypothesis that the deformation is homoge-neous into the volume probed by the light. Using this
hypothesis, every term of the sum∑s/l∗
j=1 〈∆φ2j 〉 of phase
shift used in (3) is the same: 〈∆φ2〉. The deformationbeing in fact heterogenous, we must consider average val-ues of 〈∆φ2〉 on volumes Vs probed by paths of length s:∑s/l∗
j=1 〈∆φ2j 〉 = (s/l∗)〈∆φ2〉Vs . Because long paths probelarger volumes than short paths, the volume Vs dependson s.
The fact that the phase of the strain is heterogeneoushas an interesting consequence. Indeed, let us consider asegment of the random walk located at a distance z fromthe wall. The variance of the phase shift (8) depends onz:
〈∆φ2(z)〉 ∝ γ2(z) sin2(ωτ/2) sin2[ωt+ ωτ/2 + Ψ0(z)].(24)
Since Ψ0 depends on z, 〈∆φ2(z)〉 cannot be canceled si-multaneously at every z. It follows that the correlationfunction g(i) with i 6= 0 cannot be equal to one. This iswhat is observed on the Fig. 5a: the maximum values ofthe correlation functions are not 1. Complementary ex-periments with foam of high interfacial viscoelasticity [32]exhibit higher values of correlation recovery. This prop-erty may be used as a useful non-destructive method to
ACCEPTED MANUSCRIP
T
9
estimate the value of ϕ and will be discussed in a forth-coming paper.
Finally, we can justify the assumption on the displace-ment field made to derive (7) from (6). At the fre-quency of 800 Hz, the thickness ξ = 1/κ0 cos(ϕ) of thesheared layer is about 2 mm [22], whereas the acous-tic wavelength in the foam is λa = 2π/ka = 6 cm (seeSec. III D). Hence, ξ/λa � 1, and to a good approxima-tion, the displacement field in the sheared layer writes :A(r, t) = [α(z) cos(ωt)+β(z) sin(ωt)]ex, with ex the unitvector in the direction of propagation. Hence, ∇α and∇β are both along ez.
V. CONCLUSION
We have investigated the situation of a turbid mediaundergoing a periodical shear, and wondered if and howthe amplitude and the phase of this deformation could belocally measured by light scattering. We have proposeda new approach — named here “synchronized DWS” —which requires to acquire four different correlations func-tions. Then by combining these correlations, the ampli-tude and the phase can be derived, at any locations. Thisapproach has been successfully tested by experiments onfoams, where a sinusoidal shear is obtained as an acousticwave is propagating. In a first set of tests, we have foundan excellent agreement between the imposed forcing andthe measurements. These first results fully validated theprinciples and our formalism. Further experiments al-lowed us to scan a foam as a function of the distanceof propagation of the sound. It turns out that we havebeen able to monitor the phase of the deformation alonga few wavelengths. The full analysis — including in-
terference effects at the bottom of the vessel — allowedus to quantify the phase velocity and the sound atten-uation. The obtained values are in excellent agreementwith previous measurements, and definitively confirmsthat this approach based on the measurement by DWS ofthe acoustic deformation is relevant. Moreover, we haveshown that possible internal dynamics and heterogeneityin space do not prevent these types of measurements, asthey can also be taken into account.
In the future, these results can be pursued along differ-ent routes. For instance, this method may be used to getinformation about the shear layer in a direction perpen-dicular to the displacement. We saw in Sec. IV C that thestructure of this sheared layer depends on the rheologicalproperties of the material. Correlation data should thenpermit to obtain informations about high frequency rhe-ological properties with λa . l∗. A deeper understandingwill be required to clarify the links between the correla-tion data and the viscoelasticity of the material. As well,measurements can be done not only in the direction ofpropagation, but also in the orthogonal one. It is thuspossible to create a 2D mapping of the amplitude and ofthe phase; this might be especially relevant in the caseswhere sound could be diffracted by obstacles or slits.
ACKNOWLEDGMENTS
We acknowledge funding support from Agence Na-tionale de la Recherche in the frame of the SAMOUSSEproject (ANR-11-BS09-001, coordinated by FlorenceElias). We thank all the participants of this project forfruitful discussions. We also thank the GdR “Mousseset Emulsions” for providing another network for discus-sions.
[1] Pine, D.J., Weitz, D.A., Zhu, J.X., and Herbolzheimer,E. Diffusing-wave spectroscopy: dynamic light scatter-ing in the multiple scattering limit. J. Phys. France,51(18):2101–2127, 1990.
[2] In W. Brown, editor, Dynamic Light Scattering: TheMethod and Some Applications. Oxford University Press,1993.
[3] S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, andH. Sillescu. J. Chem. Phys., 104:1758, 1996.
[4] L. Cipelletti and D. Weitz. Rev. Sci. Instrum., 70:3214,1999.
[5] V. Viasnoff, F. Lequeux, and D.J. Pine. Rev. Sci. In-strum., 73:2336, 2002.
[6] L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, andS. Mazoyer. J. Phys.: Condens. Matter, 15:S257, 2003.
[7] P.K. Dixon and D.J. Durian. Phys. Rev. Lett., 90:184302,2003.
[8] K.N. Pham, S.U. Egelhaaf, A. Moussaid, and P.N; Pusey.Rev. Sci. Instrum., 75:2419, 2004.
[9] R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon,and D. J. Durian. Speckle-visibility spectroscopy: A toolto study time-varying dynamics. Review of Scientific In-
struments, 76(9):–, 2005.[10] D. J. Durian, D. A. Weitz, and D. J. Pine. Scaling be-
havior in shaving cream. Phys. Rev. A, 44:R7902–R7905,Dec 1991.
[11] D. J. Durian, D. A. Weitz, and D. J. Pine. Multiplelight-scattering probes of foam structure and dynamics.Science, 252(5006):686–688, 1991.
[12] Reinhard Hohler, Sylvie Cohen-Addad, and Douglas J.Durian. Multiple light scattering as a probe of foamsand emulsions. current opinion in colloid and interfacescience, 2014.
[13] R. Hohler and S. Cohen-Addad. J. Phys. Condens. Mat-ter, 17:R1041, 2005.
[14] R. K. Prud’homme and S.A. Khan. Foams: Theory, Mea-surements, and Applications. OMarcel Dekker Inc., NewYork, 1997.
[15] I. Cantat, S. Cohen-Addad, F. Elias, F. Graner,R. Hohler, O. Pitois, F. Rouyer, and A. Saint-Jalmes.Foams - Structure and Dynamics. Oxford UniversityPress, 1 edition, 2013.
[16] K. B. Kann. Sound waves in foams. Colloids Surf. A,263:315–319, 2005.
ACCEPTED MANUSCRIP
T
10
[17] I. Ben Salem, R.-M. Guillermic, C. Sample, V. Leroy,A. Saint-Jalmes, and B. Dollet. Propagation of ultra-sound in aqueous foams: bubble size dependence andresonance effects. Soft Matter, 9:1194–1202, 2013.
[18] J. Pierre, F. Elias, and V. Leroy. A technique for mea-suring velocity and attenuation of ultrasound in liquidfoams. Ultrasonics, 53:622–629, 2013.
[19] J. Pierre, R.-M. Guillermic, F. Elias, W. Drenckhan, andV. Leroy. Acoustic characterization of liquid foams withan impedance tube. Eur. Phys. J. E, 36:113, 2013.
[20] J. Pierre, B. Dollet, and V. Leroy. Resonant acousticpropagation and negative density in liquid foams. Phys-ical Review Letters, 112:148307, 2014.
[21] J. Pierre, B. Giraudet, P. Chasle, B. Dollet, and A. Saint-Jalmes. Sound propagation in liquid foams: Unravelingthe balance between physical and chemical parameters.Phys. Rev. E, 91:042311, 2015.
[22] M. Erpelding, R. M. Guillermic, B. Dollet, A. Saint-Jalmes, and J. Crassous. Investigating acoustic-induceddeformations in a foam using multiple light scattering.Phys. Rev. E, 82:021409, 2010.
[23] Frederic Wintzenrieth, Sylvie Cohen-Addad, MarieLe Merrer, and Reinhard Hohler. Laser-speckle-visibilityacoustic spectroscopy in soft turbid media. Phys. Rev.E, 89:012308, Jan 2014.
[24] S. Leveque, A. C. Boccara, M. Lebec, and H. Saint-Jalmes. Ultrasonic tagging of photon paths in scatter-ing media:?parallel speckle modulation processing. Opt.Lett., 24(3):181–183, Feb 1999.
[25] Gang Yao and Lihong V. Wang. Theoretical and exper-
imental studies of ultrasound-modulated optical tomog-raphy in biological tissue. Appl. Opt., 39(4):659–664, Feb2000.
[26] Linda Djaoui and Jerome Crassous. Probing creep mo-tion in granular materials with light scattering. GranularMatter, 7(4):185–190, 2005.
[27] D Bicout, E Akkermans, and R Maynard. Dynamiccorrelations for multiple light-scattering in laminar-flow.JOURNAL DE PHYSIQUE I, 1(4):471–491, APR 1991.
[28] D Bicout and R Maynard. Diffusing Wave Spectroscopyin inhomogeneous flows. PHYSICA A, 199(3-4):387–411,NOV 1 1993.
[29] W Leutz and G Maret. Ultrasonic Modulation if multiplyscattered-light. PHYSICA B, 204(1-4):14–19, JAN 1995.
[30] A. Saint-Jalmes, M.U. Vera, and D. J. Durian. Eur. Phys.J. B, 12:67, 1999.
[31] A. Saint-Jalmes. Soft Matter, 2:836, 2006.[32] K. Golemanov, N. D. Denkov, S. Tcholakova,
M. Vethamuthu, and A. Lips. Surfactant mixturesfor control of bubble surface mobility in foam studies.Langmuir, 24(18):9956–9961, 2008. PMID: 18698860.
[33] Arnaud Saint-Jalmes and Dominique Langevin. Timeevolution of aqueous foams: drainage and coarsening.Journal of Physics: Condensed Matter, 14(40):9397,2002.
[34] Roger Zemp, Sava Sakadzic, and Lihong V. Wang.Stochastic explanation of speckle contrast detection inultrasound-modulated optical tomography. Phys. Rev.E, 73:061920, Jun 2006.
