Multiple light scattering as a probe of foams and emulsions

Current Opinion in Colloid & Interface Science 19 (2014) 242–252

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Current Opinion in Colloid & Interface Science journal homepage: www.elsevier.com/locate/cocis

Multiple light scattering as a probe of foams and emulsions Reinhard Höhler a,b,⁎, Sylvie Cohen-Addad a,b, Douglas J. Durian c a b c

Université Paris 6, UMR 7588 CNRS-UPMC, INSP, 4 Place Jussieu, 75252 Paris Cedex 05, France Université Paris-Est, UPEM, 5 boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France Department of Physics and Astronomy, 209 South 33rd Street, University of Pennsylvania, Philadelphia, PA 19104-6396, USA

a r t i c l e

i n f o

Article history: Received 20 December 2013 Received in revised form 10 April 2014 Accepted 10 April 2014 Available online 18 April 2014 Keywords: Foams Emulsions Multiple light scattering Diffusing-wave spectroscopy

a b s t r a c t Light propagating in foams or emulsions is strongly scattered by the gas–liquid or liquid–liquid interfaces. This feature makes it generally impossible to directly observe the structure and dynamics deep within the bulk of such materials. However, multiple light scattering can be used as the basis of non-invasive experimental techniques that probe the average bubble size, droplet size or the dispersed volume fraction. If the sample is illuminated with a laser, the transmitted or backscattered light forms a speckled interference pattern whose temporal fluctuations reveal the dynamics of internal structural changes. Such changes can be due to coarsening, flocculation, or applied strain. We briefly recall the fundamental principles of multiple light scattering and present an overview of the experimental techniques that have been developed in recent years. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Foams are familiar from everyday life as an aggregation of gas bubbles in a liquid [1–5]. The presence of surfactants, or other macromolecules or particles, can prevent bubble coalescence by stabilizing the thin liquid film between neighboring bubbles. Then the bubble packing fraction can easily range from about 2/3, when the bubbles are nearly spherical and closely packed, to about 1, when the bubbles are nearly polyhedral and space-filling. And the bubble size can range from tens of microns up to centimeters, depending on production method. For example, bubbles produced by blowing or shaking or carbonation, like in the kitchen sink or on a glass of ale, are visible to the eye. But the bubbles produced by strong turbulence or aerosols, like a shaving cream, are not. Emulsions are analogously comprised of liquid droplets packed in another immiscible liquid, and are stabilized by similar mechanisms [6, 7]. The droplet size is usually in the range 100 nm to a few microns. In this review, we focus on concentrated emulsions where droplets are closely packed. Both foams and emulsions are not truly stable. With time, the two phases tend to separate even in the absence of coalescence. On earth, this is hastened by gravity and the density mismatch between the dispersed bubbles or droplets and the continuous liquid phase. Furthermore, surface tension causes the texture of foams and emulsions to coarsen, by driving gas or liquid-transfer from smaller bubbles or droplets into larger ones. The relative importance and the speed of the various evolution mechanisms are of prime importance for applications, both where the stability of special-purpose foams or ⁎ Corresponding author at: Université Paris 6, UMR 7588 CNRS-UPMC, INSP, 4 Place Jussieu, 75252 Paris Cedex 05, France. E-mail address: [email protected] (R. Höhler).

http://dx.doi.org/10.1016/j.cocis.2014.04.005 1359-0294/© 2014 Elsevier Ltd. All rights reserved.

emulsions must be controlled and where unwanted foams must be prevented or destroyed. Fundamental issues of stability are of longstanding interest in colloid and interface science, as well as in physics and chemistry and even mathematics. In a completely different direction, there is also strong pure and applied interest in the flow and rheology of foams and emulsions [5,8,9,7]. At heart is the stability of the packing structure against external forces, and how the bubbles rearrange when it fails. In fact the archetypal model system exhibiting a jamming transition as a function of both packing fraction and applied load is a wet foam, modeled as an over-damped collection of repulsive frictionless spheres [10–12]. A prohibitive difficulty in measuring the local phenomena underlying the stability and rheology of foams and emulsions is that they tend to be opaque, more so for smaller bubbles or droplets and larger volume fractions of the continuous phase. The scattering of light from the maze of liquid–liquid or gas–liquid interfaces causes incident photons to do a random walk in the sample, and generally restricts optical imaging to the sample surface, to index matched emulsions, or extremely dry foams. However, the strong light scattering also opens up the possibility for new diagnostic tools. For example, the step-size in the random walk can be deduced from the probability that incident photons diffusely transmit through a sample, and gives information about the size and packing fraction of the bubbles or droplets in the interior of the sample [13,14]. And if the incident light is coherent, from a laser, then the multiply-scattered light forms a speckle pattern that dances around according to the motion of the bubbles or droplets — be it thermal or due to evolution mechanisms or applied forces [13,15–24]. These two techniques, based either on diffuse light transmission or on speckle patterns, are known respectively as diffuse-transmission spectroscopy (DTS) [25] and diffusing-wave spectroscopy (DWS) [26–29]. A brief overview of

R. Höhler et al. / Current Opinion in Colloid & Interface Science 19 (2014) 242–252

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the principal variants of these techniques is provided in Appendix A. They offer advantages over confocal microscopy or MRI or x-ray tomography in giving good ensemble averaging over a large sample volume as well as in the time and length scales probed by DWS (as small as 10−8 s and 1 Å). For analysis of DTS and DWS data a strong requirement is that the structure and dynamics of the systems be spatially homogeneous and stationary in time, like in an equilibrium Brownian suspension. In a densely packed foam subject to coarsening or slow shear, the bubble rearrangements are intermittent — local clusters suddenly undergo avalanche-like neighbor changes while the surrounding bubbles remain in place. This is an example of a spatiotemporal dynamic heterogeneity, thought to be generic to any nearly jammed system [30]. Usual DWS still applies, if the sampled volume is large, and it gives the time between rearrangements [13]. But it cannot probe the dynamics of individual avalanches or rearrangements. For this, time-resolved versions of DWS based on multi-speckle detection with a digital video camera have recently been developed. In speckle-visibility spectroscopy (SVS), the dynamics within the sample are deduced from a single frame by the degradation of the speckle visibility due to its dance during the exposure [31,32]. In time-resolved correlation (TRC), the same information is deduced instead from the correlation of the speckle patterns obtained by two very short frames separated in time by an amount comparable to the exposure duration used in SVS [21,33–40]. In both methods, time evolution is then resolved by repetition. Below we review these developments, starting with the basic physics of photon diffusion and leading to DWS and the time-resolved versions SVS and TRC. While these are applicable to any opaque sample, such as dense colloids and granular media, the emphasis will be on foams and emulsions. Before concluding, we discuss prospects for extension to the resolution of dynamics in space as well as in time [41, 42]. This is a challenge not just due to bubble size, but also due to the basic physics of photon diffusion and unavoidable sampling of the detected photons over a range of different path lengths and scattering sites [43]. 2. Light propagation in foams and emulsions Light propagating in a foam or an emulsion frequently encounters gas–liquid or liquid–liquid interfaces. We first consider a bubble or droplet dispersion not only sufficiently concentrated to be turbid, but also so diluted that the bubbles or droplets can be considered as independent scatterers. If their diameter d is so small that k d b 10 where k is the light wave number, the scattering cross section σ can be calculated from Mie theory. If drops or bubbles are large so that their interfaces are flat at the length scale of the wavelength, the light is reflected and refracted following the laws of geometrical optics and the propagation can be simulated by ray-tracing. The scattering mean free path ℓ is given by 1 / ρσ, where ρ is the scatterer concentration [45]. Photons propagating inside a sample change directions each time they encounter a scatterer and the distance ℓ⁎ = ℓ/〈1 − cosθ〉 over which the “memory” of the initial propagation direction is lost is called the transport mean free path. Here θ is the scattering angle and the average is taken over the ensemble of scattering events. When bubbles or droplets are closely packed, complex optical patterns are observed at the surface of the sample, depending on the packing structure (cf. Fig. 1) [44]. In this regime, the bubbles or droplets can no longer be considered as independent scatterers. However, ray tracing simulations of light propagation in random sphere packings with different refractive indices in the dispersed and continuous phases indicate that if each encounter of a ray with an interface is considered as a scattering event, the relation ℓ⁎ = ℓ/〈1 − cosθ〉 still holds to a good approximation (Fig. 2). Such random sphere packings are models of foams, emulsions or granular materials near the random close packing fraction. Moreover, we recall that for independent point scatterers, ℓ⁎ scales as the inverse of the scatterer concentration. By analogy, one would expect

Fig. 1. Ray tracing simulations of the pattern seen at the surface of ordered bubble packings. The structures are a) hexagonal close packed, b) face centered cubic (111) and c) face centered cubic (100). The pictures on the left are obtained for spherical bubbles, while on the right, the bubbles are assumed to be squeezed by 50% perpendicular to the imaging direction, as illustrated at the top of the figure. The refractive index of the continuous phase is supposed to be that of water. The liquid between neighboring bubbles has the shape of a concave lens and transmits a distorted image of the bubbles below. Reprinted from Ref. [44] with permission from Elsevier.

that ℓ⁎ scales as the inverse of the amount of interfacial area per unit volume, in agreement with experiments [46,47]. Recent experiments and simulations have evidenced effects beyond the scope of the independent scatterer approximation. Light propagation in foam is indeed strongly constrained by the shape of the interfaces of thin films at the contacts between neighboring bubbles, and of liquid channels called Plateau borders at the junctions between films [4]. If the refractive index inside the Plateau borders is larger than outside, the light is guided, like by an optical fiber [49,50]. As a consequence, the light rays are more likely to travel through Plateau borders than through other parts of the structure. If the continuous phase is absorbing, this means that the penetration of the diffusing light into the sample will be significantly reduced due to this effect. Moreover, ray-tracing studies where scattering by Plateau borders was neglected have predicted a strong variation of ℓ⁎ with film thickness or light wavelength, due to the interference of light undergoing multiple reflections in the films [51]. However, this regime has so far not been evidenced experimentally (cf. Fig. 3), and it is not clear below what liquid fraction the Plateau border contribution to the scattering is negligible. A

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Fig. 2. Ratio between the transport mean free path ℓ⁎ and the radius R of spherical particles that are randomly close packed, versus the ratio of refractive indices inside and outside the particles. The continuous curve shows the mean distance between scattering events ℓ/R divided by 1 − 〈cosθ〉 where θ is the scattering angle. The data are obtained using a ray tracing simulation [48]. Reprinted with kind permission of The European Physical Journal (EPJ).

quantitative theory predicting how ℓ⁎ is related to the structure and composition of real foams and emulsions is an important objective for further theoretical work. To determine ℓ⁎ experimentally, the propagation of light on length scales much larger than ℓ⁎ is studied. Under these conditions, photon trajectories can be described as persistent random walks, and the light propagation is as a diffusion process governed by Fick's laws [45]. The light diffusion constant is given by ℓ⁎c / 3 where c is the propagation velocity in the continuous phase. In the framework of the diffusion approximation, the entry of light into the multiply scattering sample is modeled as a source of diffuse photons, either located at the depth of the order of ℓ⁎ below the illuminated sample surface, or distributed in space with an intensity that decreases exponentially with penetration depth [45]. For anisotropic scatterers, it is more accurate to model the source as exponential in ℓ, with a discontinuity at the source points in proportion to 〈 cosθ 〉 [52]. Fick's laws can be used to deduce ℓ⁎ from a diffuse light transmission measurement: a slab of material with thickness L is illuminated on one side and the transmitted light intensity is detected on the other side. The diffusion model predicts in this case that a fraction T = (1 + ze)/(L/ℓ⁎ + 2ze) of the incident intensity is diffusely transmitted. Simulations have shown that this result is accurate to within 1%, down to a sample thickness of 10 ℓ⁎; full treatment of the source gives a more complex expression, similarly accurate even for much thinner samples [52]. Here ze is a parameter depending on the optical boundary conditions [53,14]. For dry foams bounded by glass plates held in air, ze is close to the value 0.88. In this case, most of the exiting

(a)

photons strike the boundary from inside gas bubbles pressed up against the walls. For wet foams confined in the same way, by contrast, ze is close to the value 1.77. In this case the exiting photons strike the boundary almost exclusively from inside the liquid residing between bubbles. To within approximately 10%, ze increases linearly with liquid fraction between these two extremes. These values are modified if the foaming liquid has a refractive index that differs significantly from that of water, or if an emulsion is considered. General expressions depending on the refractive indices of the constituents of the dispersion and its environment are given in the literature cited above. Recently, it has been shown that ℓ⁎ can also be inferred from the contrast of backscattered speckle images, observed at the surface of a sample illuminated by a laser, via a rotating multiply scattering disk. This speckle contrast has been shown to be a function of the ratio between the size b of the incident speckles created by the disk and ℓ⁎. Measuring the contrast as a function of b thus yields ℓ⁎ [54]. Contrary to a transmission experiment, this technique probes the sample structure in the vicinity of the illuminated surface. Diffuse light transmission experiments with foams of a wide range of liquid fractions and bubble sizes have established the empirical relationship ℓ⁎ = d(1.5 + 0.14/ε) shown in Fig. 3, relating ℓ⁎ to the average bubble diameter d and the continuous phase volume fraction ε. These results hardly depend on light wavelength, and for moderately wet foams, it has also been shown that ℓ⁎ is not modified if a shear flow is applied [16,55]. Therefore, measurements of ℓ⁎ are often used as a fast, noninvasive probe of the microstructure or of the dispersed volume fraction of foams or emulsions. Recently, diffuse light transmission measurements have been used to identify destabilization mechanisms in emulsions [56] or to study foam coarsening [57] and drainage [58] under microgravity conditions. 3. Diffusing-wave spectroscopy We briefly review the principle of a DWS experiment and the formalism used to interpret it [28,26]. Coherent light of wavelength λ incident on a sample is multiply scattered, such that photons propagate along random walk-like paths characterized by the transport mean free path ℓ⁎ (cf. Section 2). In the absence of absorption, the light is finally transmitted or backscattered. The interference of light from many different paths produces a speckle pattern in the far field. If the scatterers move, the photon phase evolution along the paths is modified and the speckle intensities fluctuate. A photodetector measures this fluctuation as a function of time over an area comparable to the speckle size λL / D, where D is the diameter of the region where diffuse light emerges from the sample and L is the sample-detector distance. The total scattered electric field at the detector, at time t, Eðt Þ ¼ ∑p Ep eiϕp ðt Þ , is the sum of the field contribution with amplitudes Ep due to all paths p. The temporal field fluctuations are characterized by the autocorrelation

(b)

Fig. 3. (a) The ratio of the scattering mean free path ℓ⁎ and bubble diameter d is plotted as a function of the light wavelength λ for a range of liquid volume fractions ε. The samples are aqueous foams. Reprinted from Ref. [14]. (b) The ratio between transport mean free path and the bubble diameter is plotted versus liquid fraction, for aqueous foams where ε was modified by compression or expansion. The green dashed line illustrates the phenomenological law mentioned in the text. Reprinted with permission from Ref. [14].

R. Höhler et al. / Current Opinion in Colloid & Interface Science 19 (2014) 242–252

with γ ≈ 2 [28]. In both cases, we use the notation:

function g1(t;τ) that depends on t and the delay time τ:  hEðt þ τÞÞE ðt Þi
 g 1 ðt; τ Þ ≡ : jEðt Þj2

ð1Þ

The brackets denote an average over all possible realizations of the light random walks. It can be achieved by temporal averaging for systems where the scatterers move due to Brownian motion, coarseningor shear-induced dynamics. Alternatively, the average can be performed over an ensemble of different speckles. This variant, called multispeckle DWS, is suitable for studying transient or non-ergodic dynamics and will be discussed in more detail at the end of this section. The temporal fluctuations of the speckle intensity I(t;τ) are measured by a photodetector and they are characterized by the autocorrelation function g2(t;τ): g 2 ðt; τ Þ ≡

hIðt ÞI ðt þ τÞi

ð2Þ

2

hIi :

If the electric field statistics are Gaussian, then it is related to g1(t;τ) by the Siegert relation: 2

g 2 ðt; τ Þ ¼ 1 þ βjg 1 ðt; τÞj

ð3Þ

The parameter β depends on the collection optics and on the polarization of the detected light. It is of the order of one if the number of detected speckles is close to one [59]. The function g2(t;τ) measured in a DWS experiment provides information about the scatterer dynamics, if the link between this motion and the phase shifts that it induces is established. Between two instants t and t + τ the phase of the light reaching the detector along a path p varies by an amount Δϕp(t; τ) = ϕp(t + τ) − ϕp(t). The field correlation function can thus be written as the sum: g 1 ðt; τ Þ ¼

D E X I D p hI i

p

h iE exp −iΔϕp ðt; τÞ

ð4Þ

where 〈 Ip 〉 is the average intensity of the light propagating via path p. Eq. (4) holds for single as well as multiply scattering experiments. Δϕp(t; τ) is further decomposed into a sum over the phase shift variations Δϕ(t; τ), occurring between two successive scattering events on a path, between the instants t and t + τ. These variations are modeled as Gaussian random variables. Moreover, paths are classified according to their curvilinear length s, corresponding to s/ℓ⁎ scattering events. In terms of the path length distribution function P(s) and in the limit of a large number of scattering events, g1(t; τ) is given by: Z g 1 ðt; τ Þ ¼

∞ s¼0

D E  h i 2  P ðsÞ exp −s Δϕ ðt; τÞ = 2 ℓ ds:

ð5Þ

The distribution P(s) is set by the scattering geometry and the optical boundary conditions. As an example, we consider a slab of thickness L ≫ ℓ⁎, with uniform illumination on one side. The field correlation function of the transmitted light is then [28,60]: g 1T ðt; τÞ ¼

pffiffiffiffi ⁎ X L=ℓ pffiffiffiffi : sinh X L=ℓ⁎

ð6Þ

The field correlation of light backscattered by a uniformly illuminated slab of infinite thickness is:  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 1B ðt; τÞ ¼ exp −γ X ðt; τÞ

245

ð7Þ

D E. 2 X ðt; τ Þ ¼ 3 Δϕ ðt; τÞ 2:

ð8Þ

In the following, we discuss how X(t;τ) is linked to different kinds of sample dynamics and we review the insight about foams and emulsions that has been gained by DWS experiments. In multiply scattering emulsions sufficiently diluted so that the liquid droplets undergo independent Brownian motion, X(t;τ) is determined from the mean-squared displacement of a scatterer b Δr2(t; τ) N over a time interval τ: X(t; τ) = k2 b Δr2(t; τ) N with k the wave number in the medium. The mean-squared displacement increases linearly with time and it is governed by the particle diffusion coefficient D: b Δr2(t; τ) N = 6Dτ. We obtain in this case X(t; τ) = 6τ/τo with τo = 1/k2D representing the characteristic time over which the distance between successively encountered scatterers (typically ℓ⁎) changes approximately by λ. Since D is related to the droplet radius by the Stokes–Einstein relation, measuring the characteristic decay time τo of the intensity correlation function g2(τ) yields the droplet size (typically in the range 50 nm–1 μm) without requiring any dilution down to the single scattering regime where standard particle sizing techniques can be used. Such DWS experiments have been for instance used to determine the particle size distribution upon the in vitro digestion of emulsions [20,61]. Note that the resolution limit of small intensity correlations can be improved by more than two orders of magnitude if the usual homodyne light detection is replaced by an optimized heterodyne detection scheme, allowing scatterer dynamics to be studied on extended time scales [62]. DWS is also used to study emulsion stability, in particular in food science [23]. Food emulsions often contain molecules that promote droplet adhesion, depending on the pH and the addition of salt. In the course of flocculation, the mobility of the droplets decreases as they form aggregates. This effect increases the characteristic decay time τo of the light intensity autocorrelation function and can therefore be detected with DWS [63]. If droplets or bubbles are highly concentrated, they form a jammed, solid-like packing. In the case of foams, it evolves due to diffusive gas exchange between neighboring bubbles which is driven by the Laplace pressure and which leads to intermittent local rearrangements. In contrast to Brownian motion, these dynamics are not statistically homogeneous. Upon an event, the scatterer displacements are so large that the phases of photons going through the affected regions are randomized, leading to a decay of the autocorrelation function of the form: X(τ) = 6τ/τc, where τc represents the average time between coarseninginduced bubble rearrangements at a given location [13]. This time increases as the foam ages [64]. Foams laden with colloidal particles are model system of some aerated food products. A recent DWS study of their coarsening behavior shows that the intensity correlation function exhibits two decoupled decay times [65]: A short one due to the Brownian particle motion and, a second longer one due to foam coarsening. When DWS is combined with rheological experiments, it provides information about the local dynamics that accompany a macroscopic flow or deformation. If the scatterers undergo random ballistic motion, their mean-squared displacement increases quadratically with time and their mean-squared ballistic speed is denoted b Δv2 N. Here, the argument of the correlation function (Eq. (8)) is: X(τ) = k 2 b Δv 2 N τ2 = 6(τ/τs) 2 . Like τ o, τs represents the characteristic time over which the distance between successively encountered scatterers becomes significant, compared to the light wavelength. If the material undergoes an affine deformation described by the infinitesimal strain tensor ε, the mean-squared phase shift writes [66–69]: D

2

Δϕðt; τ Þ

E

   2 n 2 2 ¼ kℓ ½TrðΔεðt; τ ÞÞ þ 2Tr Δεðt; τÞ Þg=15:

ð9Þ

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Δε(t;τ) denotes the change of the strain tensor between the instants t and t + τ. For incompressible laminar shear flow with shear rate ε,˙ Eq. (9) yields: X ðτÞ ¼

 2  3 k ℓ ε˙ τ : 15

ð10Þ

It varies quadratically with τ as for ballistic scatterer motion, and can be written as a function the characteristic decay time τs: X(τ) ≡ pffiffiffiffiffiffi  of  6(τ/τs) 2 with τs ¼ 30= kℓ⁎ε˙ . This result was first used to probe velocity gradients in concentrated Brownian suspensions [66]. This is the first example where DWS was used to detect the coexistence of two different scatterer dynamics, respectively due to Brownian motion and shear flow. A DWS study of foams subjected to shear flow has evidenced different local flow mechanisms at the bubble scale, depending on shear rate [70]. At very low rates, the flow is driven by intermittent coarseninginduced bubble rearrangements (further explored by the multispeckle experiments described below), at larger rates additional bubble rearrangements are triggered by the applied shear and plastic flow sets in. At even higher shear rates, larger than the yield strain divided by a typical rearrangement duration, bubbles undergo approximately laminar convective motion. This predominantly viscous rheological behavior at high strain rates has been evidenced independently by measurements of stress relaxation following a step-strain superposed on steady shear [71] and by observations of fingering instabilities upon air injection into foam confined between two plates [72]. DWS has also been used to detect slip flow as foam slides along a solid wall [73] or deformation of a foam in the vicinity of a solid wall as an acoustic compression wave propagates through the sample [68]. If DWS is used to probe foams or emulsions subjected to oscillatory shear strain, the intensity correlation function exhibits periodic echoes at delay times that are multiples of half of the strain period [18,19]. As long as the applied strain amplitude is well below the yield strain, the echo heights are equal to the autocorrelation measured in the absence of shear at the corresponding delay times. This indicates that the deformation is reversible, as expected for linear mechanical response [18]. If coarsening-induced rearrangements are significant on a time scale comparable to a few periods, the height of the echoes decreases accordingly and their envelope is given by the function g2 measured in the absence of strain. When the applied shear strain amplitude approaches the yield strain, the echo height decreases due to strain induced bubble rearrangements that indicate the onset of flow. DWS echoes have been used to detect the yield strain of concentrated emulsions and to measure as a function of strain amplitude the volume fraction of the sample that undergoes rearrangements during an oscillation period. By comparing the heights of successive echoes, it was shown that shearinduced rearrangements occur persistently in the same locations, suggesting a heterogeneous mechanical response where weak liquid-like regions are dispersed in an elastic solid-like matrix [19]. Moreover, DWS echo experiments have provided insight about fracture in bridging-flocculated emulsions [74]. DWS is also used for passive microrheology experiments which probe the linear viscoelastic response of turbid soft matter. In these experiments, tracer particles are embedded in the sample and their mean-square displacement 〈Δr2〉 due to thermal fluctuations is measured. In emulsions, the droplets themselves can be used as tracer particles, and in foams thermal fluctuations of the structure have also been evidenced [17]. The complex shear modulus of the material is deduced from 〈Δr 2 〉 using a generalized Stokes–Einstein relation [75]. This approach has been used to probe the linear viscoelastic relaxations of concentrated emulsions in an extended range of frequencies, typically between 0.1 and 107 Hz [76,77]. A comparison between such microrheology data and rheometry measurements obtained at low frequency shows a good agreement between the two techniques in the cases of repulsive emulsions [76] as well as

depletion-flocculated emulsions [74]. For bridging-flocculated emulsions, DWS measurements strongly overestimate the storage modulus, suggesting that the generalized Stokes Einstein relation cannot directly be applied in all cases [74]. Moreover, it has been shown that at frequencies above about 103 Hz, a contribution due to the inertia of the suspending fluid must be taken into account, otherwise passive microrheology measurements underestimate the loss modulus [78]. This technique is particularly useful in applications where the available sample volumes are small and where contact-free measurements are required. Typical examples are the emulsions encountered for biology or pharmacy [79]. If temporal averaging is not sufficient to obtain a well resolved correlation function g2 over a given time interval, an average over an ensemble of speckles can be performed. In such multispeckle detection (first introduced in single scattering experiments [33,34]) the correlation function g2 is determined from a pixel-to-pixel correlation between speckle images taken by a camera at instants t and t + τ [80,37]: g 2 ðt; τ Þ ¼

hI ðt ÞIðt þ τÞip hIðt Þip hIðt þ τÞip

:

ð11Þ

The index p is used in this expression to indicate that the averages are performed at fixed times t, over an ensemble of speckles. g1(t;τ) is deduced from g2(t;τ) using the Siegert relation in Eq. (3). An optimized signal to noise ratio is obtained if the autocorrelation function is determined as follows [21]: 2 g ðt; τ Þ−1 hIðt ÞI ðt þ τ Þip−hIðt Þip2
2 ¼ : g 1 ðt; τÞ ¼ 2 g 2 ðt; 0Þ−1 I ðt Þ p−hIðt Þip2

ð12Þ

Combining Eqs. (3), (11) and (12) yields an expression of the parameter β in terms of the moments of the speckle intensity distribution measured at a given time t: D β¼

2

Iðt Þ

E p 2

hIðt Þip

−1:

ð13Þ

If an infinite number of speckles could be detected, these moments would be well defined constants (assuming a fixed incident light intensity). However, since in an experiment, the number N of detected speckles is finite, the calculated moments and correlations fluctuate depending on the disorder in the sample, with a variance that decreases as 1/N [81,82]. In Eq. (12) this fluctuation is taken into account so that the normalization condition of the autocorrelation function |g1(t; 0)|2 = 1 holds exactly even for a finite number of detected speckles. In contrast, if |g1(t; τ)|2 is deduced from Eqs. (3) and (11) using a fixed, independently determined value of β, the condition |g1(t; 0)|2 = 1 is not accurately satisfied and further data analysis is required to disentangle the temporal variations of |g1(t; τ)|2 either due to sample dynamics or to speckle statistics [81,83]. Furthermore, detector noise can induce artifacts that may be significant, depending on experimental conditions. If the noise on successively recorded speckle images is uncorrelated, its effect can be eliminated by adjusting the normalization of |g1(t; τ)|2 (Eq. (12)), such that it extrapolates to 1 in the limit of small delay times τ. The impact of noise and speckle statistics on light intensity autocorrelation measurements has been discussed in detail in the recent literature [82,81]. In practice, successive images separated by a time interval τ are recorded. Then the speckle pattern is sampled at times ti and ti + τ, and the first and second moments of the intensity in Eq. (12) are calculated: 〈I〉p = (〈I(ti)〉p + 〈I(ti + τ)〉p)/2 and 〈I〉2p = (〈I(ti)2〉p + 〈I(ti + τ)2〉p)/2, as well as the correlation 〈I(ti) I(ti + τ)〉p. This procedure based on the simultaneous measurement of a large number of speckles using a camera constitutes a fast and accurate way to obtain and analyze the intensity autocorrelation [21]. If averaging is performed only over speckles and

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not over time, the experimental technique is called time resolved correlation (TRC) spectroscopy, as discussed in detail in Section 5. In cases where the sample dynamics are stationary, autocorrelation functions obtained either with temporal averaging or with speckle averaging are expected to be in good agreement. Such behavior has been evidenced experimentally for coarsening foams [21]. Multispeckle DWS experiments with coarsening foams have shown that in response to a transient shear of an amplitude close to or larger than the yield strain, the rate of bubble rearrangements slows down [21]. It then progressively recovers the value observed in the absence of such a mechanical perturbation. This slowing down of the dynamics differs from the overaging in colloidal glasses which is observed only if the applied strain is below the yield strain [84]. Moreover, the behavior of foams is in contrast to the shear induced acceleration of dynamics observed in colloidal glasses which is called shear rejuvenation [37]. Multispeckle DWS has also been used to measure in situ the frequency of bubble rearrangements in a foam placed in a rheometer and subjected to a constant stress, chosen well below the yield stress, too small to induce bubble rearrangements [22]. Under these conditions, foam undergoes a steady creep flow, and the measured deformation increases linearly with time. Direct comparison of the optical and mechanical data revealed that the foam viscosity scales as the inverse of the rate of coarsening induced rearrangements. This result elucidates the origin of the slow relaxation observed in foams around 1 mHz and its coupling to aging. Averaging over multiple speckles can be implemented using a photomultiplier as a photodetector, instead of a camera if the following two-cell technique is used [85]. Between the sample cell and the photodetector a reference cell is introduced which contains a diffusing medium with ergodic dynamics (a concentrated Brownian suspension for instance). The scattering by this second cell performs a speckle averaging. Under appropriate optical conditions, the field autocorrelation function measured by the detector is simply given by the product of the sample field autocorrelation function g2,S with the reference field autocorrelation function g2,R. This technique has been used to investigate non-ergodic dynamics in colloidal gels [85]. Finally, another variant of multispeckle DWS consists in placing between the laser source and the sample cell a rapidly rotating diffusing disk (typically at frequency fr = 50 Hz), and detecting light scattered by the sample with a photomultiplier. The rotating disk generates a correlation function g2,R which exhibits periodic echoes. This function is multiplied with the correlation function due to the sample g2,S and thus, the total intensity correlation function measured at the detector g2 = g2,S g2,R exhibits echoes whose envelop yields g2,S. A comparison between data for a colloidal suspension obtained either with this detection scheme or with temporal averaging shows a very good agreement [86]. 4. Time resolved diffusing-wave spectroscopy as a probe of dynamical heterogeneities If the internal dynamics of a sample change with time, then obviously the usual DWS intensity autocorrelation should only be averaged over times that are short compared to the evolution. This makes it impossible to capture the act of start–stop intermittent motion common in a variety of important situations. This includes the dynamics of localized bubble rearrangements in a foam, the intermittent avalanches on the surface of a slowly tilted pile of sand, or the cycle of grain dynamics in a periodically vibrated sample. But if the intermittency is stationary in time and uniform throughout the sampled volume, then a partial remedy is to employ higher-order intensity correlations, g n ðτ1 ; τ2 ; …; τn−1 Þ ¼

〈 IðtÞIðt þ τ1 ÞIðt þ τ2 Þ⋯Iðt þ τn−1 Þ 〉 = 〈 I 〉

n

ð14Þ

where n ≥ 2, I(t) is the detected intensity at time t, collected at a single detector with area comparable to speckle size, and 〈 ⋅ ⋅ ⋅ 〉 denotes an average over t. These can be measured in real-time with a standard digital

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correlator and a custom-built digital delay line [87], or afterwards from a recorded sequence of speckle images [83], and they contain extra information beyond g2(τ) whenever the electric field statistics are nonGaussian [87]. If the dynamics intermittently change between static (“off”) and flowing (“on”), such as for avalanches in sand or bubble rearrangements in foam, the new information includes the probabilities for switching or not between the on and off states after a given time interval [88]. This can be seen strikingly in the case of periodic switching, where g1(t;τ) decays monotonically but g4(τ, τ + T) ∝ 〈I(0)I(T) ⋅ I(τ)I(τ + T) 〉 is periodic in τ [89]. This particular slice of g4 is similar to what is measured in the frequency domain by the so-called second spectrum [90]. For bubble rearrangements in foams, the field statistics are Gaussian if the volume sampled by the photons is large compared to the volume of a rearrangement event [91]. Note that the time averaging over t must be long enough to sample both speckle and rearrangement statistics. To measure the evolution of dynamics within individual rearrangement or avalanche events, and to use such information to accumulate event statistics, it is necessary to perform the ensemble average over speckle statistics essentially instantly with the use of a digital camera [33–38]. The ensemble average over multiple speckles means that the Siegert relation may be invoked, no matter whether or not the sample dynamics are ergodic. In time-resolved correlation (TRC) [39,40], the intensity autocorrelation g2(t;τ) is found as a function of time t by multiplying together the two speckle images taken at times t and t + τ. The fluctuations of g2(t;τ) are minimized if Eq. (12) is used for this purpose. In a TRC experiment, the exposure duration must be chosen so short that there is negligible speckle motion during each exposure. In speckle-visibility spectroscopy [31,32], by contrast, the exposure duration is purposely chosen long enough that the speckle motion somewhat blurs out the contrast. In both methods, the values returned by pixels i = 1, 2, … N for a single exposure of duration T, taken over times t to t + T, represent the integrated intensity during the exposure, Si,T(t) = ∫ tt + T Ii(t′)dt′/T. For SVS the quantity of primary interest is the n variance v2(t; T) = 〈 I2 〉 T/〈 I 〉 2 − 1 where 〈 In 〉 T = ∑ N i = 1 (Si,T) /N. This can be computed off-line for a sequence of images, but in principle it could be found in real time – or even on-board a camera – so that experimental runs are not limited by capacity to store video images. To relate the variance to motion within the sample, the Siegert relation and the assumption that the electric field autocorrelation g1(t;τ) is even, are combined to give the fundamental equations of SVS as Z v2 ðt; T Þ ¼ 2β

T 0

dτ 2 ð1−τ=T Þjg 1 ðt; τÞj : T

ð15Þ

Thus the variance is a special average of |g1(t; τ)|2 over the duration of the exposure. For short exposures compared to speckle motion v2(t; T) → β, while for long exposures v2(t; T) → 0; thus a normalized variance may be defined as V2(t; T) = v2(t; T)/β, which decays from 1 to 0 with increasing exposure duration. For optimal signal to noise, the speckle to pixel size should be arranged to give β ≈ 1/2 and the average detected intensity should be about 50 for an 8-bit camera [32]. For the common case of quasi elastically scattered light with Lorentzian spectrum of line width Γ, the field autocorrelation is | g1(t; τ)| = exp(−Γτ) and the normalized variance is   2 V 2 ðt; T Þ ¼ ½expð−2xÞ−1 þ 2x= 2x

ð16Þ

where x = ΓT. This is appropriate for diffusive motion and both singlyscattered and multiply-scattered light in transmission, as well as for ballistic motion light in backscattering. For the latqffiffiffiffiffiffiffiffiffiffiffiffiffi
and multiply-scattered  ter, Γ ¼ 4π Δv2 =λ. See Table I of Ref. [32] for V2(t;T) for other spectra. As an experimental check that the correct form is used, one may compare the consistency of results either for different T or for higher moments than just the variance [32].

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100

50

0 1

0.8

m=2 m=4 m=8 m=16

0.6

0.4 10

FOAM

8

(spot-in / spot-out)

6 4 2 0 39

40

41

42

43

44

Fig. 4. Example rearrangement event in a coarsening foam captured by SVS in terms of pixel gray-scale levels, variance ratios, and Lorentzian linewidth versus time t [32]. Here the foam age is t0 = 6 h and the exposure duration is T = 10 ms. Reprinted with permission from Ref. [32] Copyright 2005 American Institute of Physics.

SVS and Eq. (16) were first used to characterize evolving grain dynamics for oscillatory shaking [31] and for avalanche flows on a heap [92]. They were also used to study rearrangement events for a coarsening foam [32,93], as illustrated for one event in Fig. 4. The top panel shows a space time plot for 100 pixels, with an exposure duration of T = 10 ms that is short compared to the fastest speckle motion. At early and late times in the plot the speckle is nearly static, with the bubbles in the scattering volume being essentially fixed in place. But from times 41–42 s the speckle is in motion, indicating a rearrangement event. Since T is short, V2(t;T) is nearly one even during rearrangements. The middle plot thus shows the variance for longer exposures, V2(t;mT) where m = 2,4,8 and 16. As the duration increases, and the speckle becomes less visible during rearrangements, the variance drops. But to within uncertainty almost the same line width Γ is recovered by the use of Eq. (16), as demonstrated in the bottom panel of Fig. 4. From plots of the linewidth or of the time resolved correlation versus time, it is straightforward to accumulate statistics on the durations τd of different rearrangement events, estimated as the full width at half maximum of the peaks. Alternatively, τd can be deduced from the temporal autocorrelation of g2(t; τ) data, expressed by the function g4(τ, τ + T) introduced above. This latter correlation is significant for delay times T smaller than τd and it vanishes for larger delay times. Both approaches have been shown to yield consistent results for bubble rearrangements in coarsening foams [83]. Moreover, the decay of g4(τ, τ + T) with T for T N τd provides information about the correlation between successive rearrangements in the scattering volume. The time scale τd is important for foam rheology, since if the shear rate is fast compared to 1/τd then successive rearrangements merge together and the material flows more like a viscous liquid with no capacity for elastic energy storage [22,72,70,71]. The distribution of rearrangement durations has been studied by SVS [93] as well as by TRC for different bubble packing fractions and interfacial properties [24,83]. The results collected in Fig. 5 agree well for all samples, and show a pronounced peak with an asymmetric tail extending to long durations. For systems with low interfacial rigidity, the data in Fig. 6 show that the average rearrangement duration strongly increases with decreasing

Fig. 5. Distribution of the duration τd of rearrangement events in coarsening foams, as measured by SVS [93] (small black dots) and TRC [83] (all other symbols). To within scatter, the shape of the distribution can be represented either by a sum of two Gaussians or a log-normal (solid black curve). The average of the distribution, bτdN, depends on bubble packing fraction and surface interfacial properties [83].

osmotic (confinement) pressure, as the jamming transition is approached. This increase may be explained by a balance of the forces acting on the bubbles as they move upon a rearrangement by a distance of the order of their size. They are pushed by contact forces that scale as the osmotic pressure and hindered by the viscous resistance opposed to the liquid redistribution, depending on the liquid viscosity η. This leads to the observed behavior of the rearrangement time which scales roughly as η/Π. If the interfacial rigidity is high, additional dissipation mechanisms dominate, and the rearrangement duration becomes independent of osmotic pressure [24,83]. 5. Resolving dynamics in space and time: a challenging perspective Many open questions concerning the behavior of foams and emulsions are closely linked to temporal as well as spatial correlations of bubble or droplet dynamics. An applied stress beyond the yield stress

Fig. 6. Average rearrangement duration τd vs osmotic (confinement) pressure Π for foams with an average bubble diameter d = 135 ± 5 µm and different foaming solutions: (●) SLES-CAPB solution (surface tension σ = 30 mN/m and low interfacial rigidity), (○) SLES-CAPB-MAc solution (σ = 22 mN/m and high interfacial rigidity). High osmotic pressures (Πd/σ ≫ 1) correspond to dry foams whereas in the limit where Π goes to zero, the foam becomes so wet that the bubble packing is not jammed any more. Reprinted from Ref. [83].

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induces plastic flow in these materials which is driven by local intermittent rearrangements of the bubble or droplet packing [5,4]. Their duration sets an elementary rheological relaxation time scale [12], and their spatial extent may influence the coupling between neighboring events. This coupling is a key feature in a recent generic model of soft dispersion rheology [94]. Bubble rearrangements can not only be induced by applying stress, they also accompany the coarsening process [5,13, 21,22,93]. The duration and the sample volume in which elastic energy is released upon such coarsening induced events govern the creep flow of foams [22]. The coalescence of bubbles is another example of collective dynamics, observed when unstable foams progressively collapse [95,4]. Yet another domain where spatially correlated bubble dynamics matter is the propagation of acoustic waves. Experiments show that they are strongly attenuated [96], and the underlying dissipation mechanisms are not yet fully understood. In all these fields, a noninvasive fast probe able to resolve the dynamics of structural changes in space and time is of great interest. Confocal microscopy has emerged as a powerful tool [97], but it is ill adapted to foams where the light scattering by gas–liquid interfaces is very strong. The 3D structure of index matched emulsions can be studied using this technique, but a trade off between acquisition speed and the sample volume limits its ability to capture fast highly collective dynamics. Multiple scattering of coherent light is a fast, non-invasive probe, but the spatial resolution of this method is limited due to the diffusive propagation of light in the sample. As an illustrative example, we consider an experimental setup where coherent light shines on the surface of a coarsening foam sample, confined between two parallel transparent plates, held at a distance L. The photons penetrating into the foam via the first plate follow random paths and are either backscattered or transmitted through the sample via the second plate. Let us assume that the bubble packing structure abruptly changes in a region of radius ξ (with ξ ≪ L) and that during this event the rest of the sample remains quiescent. Under these conditions, only the phase of the photons going through the part of the sample where the structure changes is perturbed. If it is located very close to the second plate, the transmitted speckle intensity observed here will fluctuate in a region of radius close to ξ. However, if a rearrangement occurs deep inside the sample, speckle fluctuations are observed in a more extended region of the second plate, and they are weaker. These two effects occur because photons that have visited the dynamic region undergo lateral random displacements on their way to the sample surface and because they are detected together with other photons which have followed paths going through the quiescent part of the sample. Therefore, in a diffuse transmission experiment, the extent of regions undergoing rearrangements is difficult to establish quantitatively. In addition, fluctuations due to simultaneous dynamics in different parts of the sample can add up, so that the measured data are hard to interpret. If instead of the transmitted light, the backscattered light is observed, the size of the region where speckles fluctuate also increases with the distance between the rearrangement and the observed sample surface, as investigated in detail by Zakharov et al. [98]. However, in such an experiment, the detected photons typically penetrate only a few scattering mean free paths into the sample [43]. Therefore, speckle intensity fluctuations are in this case due to bubble dynamics in a much more restricted and well defined part of the sample, compared to a transmission experiment. In many applications, it would be of great interest to go beyond the limitations of the transmission and backscattering experiments discussed above, and to resolve the spatial distribution of dynamic activity in a sample in 3D. This is possible using tomographic DWS techniques such as diffuse optical correlation tomography where light is injected and observed at several different positions at the sample surface [99,100]. To analyze such data, the transport of the temporal electric field correlation through the turbid material is modeled quantitatively using a diffusion equation [41,99]. However, these approaches that have been developed for medical applications have not yet been used for studying foams and emulsions.

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To resolve the sample dynamics in space, photon correlation imaging techniques have been introduced [101,42,102,103]. This approach has been used to study bubble dynamics in a thin slab of foam [42]. Gillette foam is injected in a sample cell bounded by parallel transparent plates, held at a distance L. One plate is illuminated with an expanded laser beam, and the backscattered light intensity is recorded by a CCD camera. The authors divide the sample surface into square regions of interest (100 μm × 100 μm) and use an objective to map the speckles emitted from these regions onto corresponding regions of the CCD camera sensor. For each region, the time resolved intensity autocorrelation function is calculated and recorded as a function of time. These data are assembled into “dynamic activity maps” where a color scale represents the level of autocorrelation in each respective region, for a given delay time (Fig. 7). The same approach is used to record and analyze the transmitted light on the opposite surface of the sample. The authors compare the activity maps based on the backscattered and the transmitted light and determine the largest sample thickness where the activity seen on both maps is correlated in space and time. This thickness is used throughout their experiments, in the aim to capture all the events in the sample. The most striking result obtained in this pioneering work is that if a rearrangement is detected at a given position, the probability of detecting another rearrangement occurring at the same place in the near future is enhanced. Therefore, coarsening induced rearrangement dynamics cannot simply be understood in terms of the build-up of elastic energy due to diffusive gas transfer among neighboring bubbles and the release of this energy upon the events. In another recent experiment, spatially resolved speckle visibility spectroscopy was used to measure the velocity and attenuation length of transverse sound waves propagating parallel to the surface of aqueous foams [69]. Such measurements yield the complex shear modulus of the sample in a frequency range that extends to 1 kHz, more than an order of magnitude beyond the current limit of research grade rheometers, typically below 100 Hz. The setup is illustrated in Fig. 8a: a vibrating rough plate inserted into the foam excites acoustic shear waves which are attenuated so strongly that they do not reach the sample boundary. A laser shines on the free sample surface, and a camera equipped with a lens measures the visibility of backscattered speckles. The shutter time is chosen large enough so that the relative motion of neighboring bubbles on this time scale modulates the speckle visibility, and small enough so that this modulation is not averaged out. Under these conditions, the recorded speckle visibility is low in regions where the local bubble displacement due to the acoustic wave goes through zero (large bubble velocity gradients), and high wherever the acoustic bubble displacement goes through an extremum (minimal

(s)

Fig. 7. The intensity autocorrelation cI = (g2(t; τ) − 1)/β of the backscattered light is calculated as a function of position and represented by colors in a “dynamic activity map” [42]. The one shown here reveals a local rearrangement event; the length of the scale bar is 1 mm. Image courtesy of D. Sessoms.

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Fig. 8. a) Setup of a Laser Speckle Visibility Acoustic Spectroscopy experiment, probing shear wave propagation in aqueous foam contained in a trough. The wave is emitted by a vibrating rough plate inserted into the sample and a laser illuminates the region where it propagates. The wave velocity and attenuation are deduced from the speckle intensity fluctuations that a camera records as a function of position and time. b) In this diagram, the gray level indicates the level of speckle visibility as a function of propagation distance x1 and time t. The bright or dark regions correspond to small or large velocity gradients induced by the acoustic wave, as explained in the text. The phase evolutions in space and time of the propagating acoustic wave yield an oscillation of the visibility. It forms parallel dark and bright stripes whose slope yields the phase velocity of the acoustic wave. Reprinted from Ref. [69].

velocity gradients). Therefore, the phase of the acoustic wave can be deduced from the speckle visibility, depending on propagation distance and time, as illustrated in the space time plot, cf. Fig. 8b. To measure the acoustic attenuation length, the shutter time is chosen much longer than the acoustic wave period. Under these conditions, the increase of visibility with propagation distance away from the emitter yields the amplitude decay of the acoustic wave. These measurements of wave speed and attenuation do not require independent measurements of any optical characteristics of the sample. The only requirements are that ℓ⁎ must be much smaller than the acoustic wavelength and that the mechanical response of the sample must be linear. This new technique is not only able to detect shear waves, but also longitudinal ones, or even multiply scattered diffuse sound waves which cannot be captured with ordinary transducers. In addition, it provides an image of the acoustic wave propagation throughout the sample and it can therefore detect heterogeneity of its mechanical properties in the vicinity of the surface. 6. Conclusion Multiple scattering of light is the basis of many fast, non-invasive probes that reveal the structure and internal dynamics of foams, emulsions, and other forms of soft matter. Using diffuse transmission experiments the mean bubble or droplet size can be studied, provided that the volume fraction of the dispersed phase is known or vice versa. Diffusing-wave spectroscopy gives access to stationary dynamics such as the average rate of bubble or droplet rearrangements, due to aging or flow. The linear viscoelastic properties of emulsions may be inferred from DWS measurement of thermal fluctuations. The introduction of multispeckle detection techniques has widened the scope since transient or intermittent dynamics can now be resolved in time. This opens the way for light scattering studies of coalescence or flocculation dynamics, in addition to rearrangement events. In recent years, multiple light scattering techniques have emerged that provide not only temporal, but also spatial resolution. Applications range from the detection of acoustic waves traveling through a foam to spatial correlation studies of bubble rearrangements that reveal their collective dynamics. These experimental probes may help to identify the local dynamics that govern the macroscopic rheological behavior of foams or emulsions and to answer similar challenging questions raised by other soft dispersions or granular materials.

Acknowledgments This work was supported in part by NASA Microgravity Fluid Physics Grant NNX07AP20G (DJD), as well as by the European Space Agency (Contract No. MAP AO 99–108) and the French Space Agency (agreement CNES/CNRS No. 127233) (RH & SCA).

Appendix A The following light scattering techniques are presented in this review: • Diffuse transmission spectroscopy (DTS). This technique probes the bubble size, droplet size or the dispersed volume fraction, averaged over the sample volume, see Section 2. • Speckle contrast imaging. This technique probes the bubble size, droplet size or the dispersed volume fraction, averaged over a region close to the sample surface, see Section 2. • Diffusing wave spectroscopy (DWS) probes stationary bubble or droplet dynamics, averaged over the entire sample (transmission experiment) or close to the sample surface (backscattering experiment). It is also used for microrheology experiments that probe the linear viscoelastic response, see Section 3. • Time resolved correlation (TRC) spectroscopy is a time resolved variant of DWS, based on speckle intensity correlations. It can be used to study transient or intermittent dynamics, such as the duration of structural rearrangement events, see Section 4. • Speckle visibility spectroscopy (SVS) is a time resolved variant of DWS, based on speckle visibility. It can be used to study transient or intermittent dynamics, such as the duration of structural rearrangement events, see Section 4. • Laser speckle visibility acoustic spectroscopy (LSVAS) is a variant of SVS. It resolves as a function of time and position the amplitude and phase of acoustic waves propagating along the sample surface. It can be used to probe acoustic wave propagation and viscoelastic response, see Section 5. • Photon correlation imaging is a variant of TRC that provides 2D images of the dynamics in 3D samples, see Section 5. • Diffuse optical correlation tomography is a variant of TRC that resolves in 3D the dynamics in 3D samples, see Section 5.

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