Hierarchical bubble size distributions in coarsening wet liquid foams.

Coarsening of two-phase systems is crucial for the stability of dense particle packings such as alloys, foams, emulsions, or supersaturated solutions. Mean field theories predict an asymptotic scaling state with a broad particle size distribution. Aqueous foams are good model systems for investigations of coarsening-induced structures, because the continuous liquid as well as the dispersed gas phases are uniform and isotropic. We present coarsening experiments on wet foams, with liquid fractions up to their unjamming point and beyond, that are performed under microgravity to avoid gravitational drainage. As time elapses, a self-similar regime is reached where the normalized bubble size distribution is invariant. Unexpectedly, the distribution features an excess of small roaming bubbles, mobile within the network of jammed larger bubbles. These roaming bubbles are reminiscent of rattlers in granular materials (grains not subjected to contact forces). We identify a critical liquid fraction [Formula: see text], above which the bubble assembly unjams and the two bubble populations merge into a single narrow distribution of bubbly liquids. Unexpectedly, [Formula: see text] is larger than the random close packing fraction of the foam [Formula: see text]. This is because, between [Formula: see text] and [Formula: see text], the large bubbles remain connected due to a weak adhesion between bubbles. We present models that identify the physical mechanisms explaining our observations. We propose a new comprehensive view of the coarsening phenomenon in wet foams. Our results should be applicable to other phase-separating systems and they may also help to control the elaboration of solid foams with hierarchical structures.

Coarsening transitions of wet liquid foams under microgravity conditions.

We report foam coarsening studies which were performed in the International Space Station (ISS) to suppress drainage due to gravity. Foams and bubbly liquids with controlled liquid fractions ϕ between 15 and 50% were investigated to study the transition between bubble growth laws previously reported near the dry limit ϕ → 0 and the dilute limit ϕ → 1 (Ostwald ripening). We determined the coarsening rates for the driest foams and the bubbly liquids, they are in close agreement with theoretical predictions. We observe a sharp cross-over between the respective laws at a critical value ϕ*. At liquid fractions beyond this transition, neighboring bubbles are no longer all in contact, like at a jamming transition. Remarkably ϕ* is significantly larger than the random close packing volume fraction of the bubbles ϕrcp which was determined independently. We attribute the differences between ϕ* and ϕrcp to a weakly adhesive bubble interaction that we have studied in complementary ground-based experiments.

Elastic behavior of confined soap froth.

The elastic response of ordered 3D soap froth, in which N layers of cells are confined between two rigid walls, is analyzed. Surface Evolver simulations are used to compute the equilibrium structure, which consists of a layer of Fejes-Toth cells at each wall and N - 2 core layers of Kelvin cells. The reference state corresponds to the plate spacing ho that achieves isotropic stress; and the foam confinement is varied by changing h. The foam is sheared in two orthogonal directions to determine elastic behavior up to the elastic limit or yield strain, which corresponds to the onset of topological transitions. The shear moduli in both directions decrease as N increases and slowly converge to the values of bulk Kelvin foam; the dependence on N is well described by a three-layer model consisting of Fejes-Toth cells and a core of Kelvin cells that deform like bulk foam. The influence of foam confinement on the elastic limit is studied. The topological transitions are compared to those in bulk Kelvin foam.

Many-body interactions in soft jammed materials.

In jammed packings of soft frictionless particles such as foams or emulsions, stress is transmitted via a network of mechanical contacts between neighbors. In generic simplified models of such materials, particle interaction energies are assumed to be pairwise additive. We report ab initio simulations of foam microstructures, showing that in general, this fundamental assumption is not justified: the conservation of bubble volumes introduces a many-body coupling between all the contacts of a given particle. It strongly modifies the relation between forces and displacements at individual contacts, in a way that cannot be captured by an effective two-body interaction. We report the impact of this effect on the linear and nonlinear elastic response of ordered bubble packings with coordination numbers ranging from 6 to 12, used as simple model systems, and we present an analytical model without free parameters which is valid as long as bubbles have an approximately spherical shape. It predicts the many-body coupling of particle contact forces, as well as the macroscopic mechanical response. For packing fractions approaching the jamming transition where contact forces go to zero, we derive an asymptotic two-body interaction law. It contains a logarithmic term, yielding a critical scaling that cannot be approximated by a power law.

Stabilization of foams by the combined effects of an insoluble gas species and gelation.

Liquid foams are unstable due to aging processes such as drainage, coalescence or coarsening. Since these processes modify the foam structure, they can be a severe limitation to the elaboration of solid foams with controlled structures inherited from their liquid precursors. Such applications call for a thorough understanding of foam stabilization. Here we study how coarsening can be inhibited by the combined effects of a mixture of gas containing a species insoluble in the foaming solution and of gelation of the foaming solution. We present experiments with model ordered liquid foams and hydrogel foams. They allow us to identify the underlying physical mechanisms of stabilization and their governing parameters, namely the bubble radius Ro, the foam shear modulus G and the number ηo of insoluble trapped gas molecules per bubble. We propose a scaling model that predicts the stability diagram of an ideal monodisperse perfectly ordered foam as a function of Ro, G and ηo, in qualitative agreement with our data. We show that the domain of stable foams is governed by a characteristic elasto-capillary radius set by the ratio of surface tension to storage modulus.

Linear and non-linear wall friction of wet foams.

We study the wall slip of aqueous foams with a high liquid content. We use a set-up where, driven by buoyancy, a foam creeps along an inclined smooth solid wall which is immersed in the foaming solution. This configuration allows the force driving the bubble motion and the bubble confinement in the vicinity of the wall to be tuned independently. First, we consider bubble monolayers with small Bond number Bo < 1 and measure the relation between the friction force F and the bubble velocity V. For bubbles which are so small that they are almost spherical, the friction law F ∝ V is Stokes-like. The analysis shows that the minimal thickness of the lubricating contact between the bubble and the wall is governed by DLVO long-range forces. Our results are the first evidence of this predicted linear friction regime for creeping bubbles. Due to buoyancy, large bubbles flatten against the wall. In this case, dissipation arises because of viscous flow in the dynamic meniscus between the contact film and the spherical part of the bubble. It leads to a non-linear Bretherton-like friction law F ∝ V(2/3), as expected for slipping bubbles with mobile liquid-gas interfaces. The Stokes-like friction dominates for capillary numbers Ca larger than the crossover value Ca* ∼ Bo(3/2). The overall friction force can be expressed as the sum of these two contributions. On this basis, we then study 3D foams close to the jamming transition with osmotic pressures Π small compared to the capillary pressure Pc. We measure the wall shear stress τ as a function of the capillary number, and we evidence two friction regimes that are consistent with those found for the monolayer. Similarly to this latter case, the total shear stress can be expressed as the sum of the Stokes-like friction term τ ∝ Ca and the Bretherton-like one τ ∝ Ca(2/3). However, for a 3D foam, the crossover at a capillary number Ca** between both regimes is governed by the ratio of the osmotic pressure to the capillary pressure, such that Ca** ∼ (Π/Pc)(3/2).

Closed-cell crystalline foams: self-assembling, resonant metamaterials.

Internal degrees of freedom and periodic structure are critical requirements in the design of acoustic/elastic metamaterials since they can give rise to extraordinary properties like negative effective mass and stiffness. However, they are challenging to realize in three dimensions. Closed-cell, crystalline foams are a particularly advantageous basis to develop metamaterials as they intrinsically have a complex microstructure, exhibiting internal resonances. Recently self-assembly techniques have been implemented to produce such foams: a Kelvin (body centered cubic) foam, a face centered cubic foam, and a Weaire-Phelan structure. Numerical models are employed to demonstrate that such foams are superanisotropic, selectively behaving as a fluid or a solid, pentamode solids as a result of fluid-structure interaction, in addition to having regimes characterized by film resonances and high density of states. Microstructural deformations obtained from numerical models allow the derivation of equivalent mechanical models.

Bubble rearrangement duration in foams near the jamming point.

We investigate the dynamics of bubble rearrangements in coarsening foams, using a time-resolved multiple light scattering technique. We measure the average duration of such events as a function of the foam confinement pressure. Rearrangements slow down as the pressure is decreased toward the jamming point. Our results are explained by a scaling law based on the balance of pressure and Darcy flow, highlighting an analogy between wet foams with mobile interfaces and suspensions of hard grains.

Strain-induced yielding in bubble clusters.

We study how shearing clusters of two or four bubbles induces bubble separation or topological rearrangement. The critical deformation at which this yielding occurs is measured as a function of shear rate, liquid composition, and liquid content in the cluster. We establish a geometrical yield criterion in the quasistatic case on the basis of these experimental data as well as simulations. In the dynamic regime, the deformation where the cluster yields increases with the strain rate, and we derive a scaling law describing this phenomenon based on the dynamical inertial rupture of the liquid meniscus linking the two bubbles. Our experiments show that the same scaling law applies to two- and four-bubble clusters.

Jamming and flow of random-close-packed spherical bubbles: an analogy with granular materials.

Wet foams are random-close-packed assemblies of approximately spherical gas bubbles in a liquid. We report rheological experiments with this material, showing that even though the stiffness and frictional interactions of bubbles strongly distinguish them from solid spherical grains, jamming and flow of wet foams and granular materials are governed by closely analogous laws.

Dynamics of yielding observed in a three-dimensional aqueous dry foam.

We study the onset of yielding in stable three-dimensional dry foams following the start up of steady shear flow. By means of a charge-coupled device camera equipped with a small depth-of-field objective, we visualize the Plateau border network in the bulk of the foam. The onset of yielding is identified with the deformation gamma(c) for which shear induced rearrangements start occurring. We show that gamma(c) is independent of shear rate gamma; in a quasistatic regime whereas at high strain rates, a rapid increase of gamma(c) with gamma; is observed, in qualitative agreement with theoretical models. Moreover, spatiotemporal image analyses are used to determine the velocity profile in the gap. We find that this profile remains linear up to strains far beyond gamma(c). Moreover, we have studied the strain history dependence of gamma(c).

Laser-speckle-visibility acoustic spectroscopy in soft turbid media.

We image the evolution in space and time of an acoustic wave propagating along the surface of turbid soft matter by shining coherent light on the sample. The wave locally modulates the speckle interference pattern of the backscattered light, which is recorded using a camera. We show both experimentally and theoretically how the temporal and spatial correlations in this pattern can be analyzed to obtain the acoustic wavelength and attenuation length. The technique is validated using shear waves propagating in aqueous foam. It may be applied to other kinds of acoustic waves in different forms of turbid soft matter such as biological tissues, pastes, or concentrated emulsions.

Duration of bubble rearrangements in a coarsening foam probed by time-resolved diffusing-wave spectroscopy: impact of interfacial rigidity.

In aqueous foams, the diffusive gas transfer among neighboring bubbles drives a coarsening process which is accompanied by intermittent rearrangements of the structure. Using time-resolved diffusing-wave spectroscopy, we probe the dynamics of these events as a function of the rigidity of the gas-liquid interfaces, liquid viscosity, bubble size, and confinement pressure. We present in detail two independent techniques for analyzing the light scattering data, from which we extract the rearrangement duration. Our results show that interfacial rheology has a major impact on this duration. In the case of low interfacial rigidity, the rearrangements strongly slow down as the pressure is decreased close to the value zero where the bubble packing unjams. In contrast, if the interfaces are rigid, rearrangement durations are independent of the confinement pressure in the same investigated range. Using scaling arguments, we discuss dissipation mechanisms that may explain the observed dependency of the rearrangement dynamics on foam structure, pressure, and physicochemical solution properties.

Fast relaxations in foam.

Aqueous foams present an anomalous macroscopic viscoelastic response at high frequency, previously shown to arise from collective relaxations in the disordered bubble packing. We demonstrate experimentally how these mesoscopic dynamics are in turn tuned by physico-chemical processes on the scale of the gas-liquid interfaces. Two specific local dissipation processes are identified, and we show how the rigidity of the interfaces selects the dominant one, depending on the choice of the surfactant.

Dissipation in a sheared foam: from bubble adhesion to foam rheology.

The link between the rheology of 3D aqueous foam and the adhesion of neighboring bubbles is tested by confronting experiments at two different length scales. On the one hand, the dynamics of adhesion are probed by measuring how the shape of two bubbles in contact changes as their center-to-center distance is modulated. On the other hand, the linear viscoelastic behavior of 3D foam prepared with the same soapy solution is characterized by its complex shear modulus. To connect the two sets of data, we present a model of foam viscoelasticity taking into account bubble adhesion.

Osmotic pressure and structures of monodisperse ordered foam.

We present an experimental and numerical study of the osmotic pressure in monodisperse ordered foams as a function of the liquid fraction. The data are compared to previous results obtained for disordered monodisperse and polydisperse concentrated emulsions. Moreover, we report a quantitative investigation of the transition from a bubble close packing to a bcc structure as a function of the liquid volume fraction. These findings are discussed in the context of theoretical models that have been proposed in the literature.

Rigidity percolation in particle-laden foams.

We study the viscoelastic behavior of aqueous foam mixed with solid noncolloidal particles. We show that adding a tiny amount of grains can enhance the elastic and loss shear moduli by more than 1 order of magnitude. The scaling of these moduli with solid volume fraction is in qualitative agreement with that predicted by an effective-medium rigidity percolation model. We present a simple model, based on capillary attraction, to explain the particle-size dependence of the threshold.

High-resolution diffusing-wave spectroscopy using optimized heterodyne detection.

We show experimentally and theoretically that the use of optimized heterodyne detection in a diffusing-wave spectroscopy experiment leads to the detection of much smaller intensity autocorrelations than with conventional (either homodyne or heterodyne) setups. This enhanced resolution may be useful for the study of longtime dynamics of multiple-scattering disordered systems.

Origin of the slow linear viscoelastic response of aqueous foams.

We have studied the slow linear viscoelastic response of wet aqueous foams by macroscopic creep compliance measurements, combined to a diffusing-wave spectroscopy investigation of the local dynamics. The data strongly suggest that this rheological response arises from two distinct relaxation mechanisms: The first is due to the coarsening induced bubble rearrangements and governs the steady-state creep; the second results from the interplay between surface tension and surface viscosity of the gas-liquid interfaces and gives rise to a transient relaxation.