Bubble Rearrangements Research Articles

Foam coarsening under a steady shear: interplay between bubble rearrangement and film thinning dynamics.

Aqueous foams are unstable and age by drainage and coarsening. Today, these effects are well described, as also their impact on foam properties. In that respect, the foam viscoelastic properties evolve in time as a consequence of coarsening which tends to increase the mean bubble size. Here, we investigate the reverse coupling, and study if and how the continuous flow of a foam can impact its dynamics of coarsening. We introduce a new protocol where brief oscillatory measurements are inserted during a constant steady shear, allowing us to monitor the relative variation of the bubble size with time (obtained from the one of the elastic modulus G') as a function of the applied shear rate. It turns out that the coarsening rate is strongly impacted by the applied shear: this rate is continuously reduced above a critical shear rate, which itself decreases with the bubble size. This coarsening-rate reduction is interpreted as the result of out-of-equilibrium and shear-dependent film thicknesses, being higher than at rest. The critical shear rate, above which films are dynamically sustained at higher thickness than at equilibrium, emerges from the competition between the rate of rearrangements and the time required to drain the thick film created during the rearrangement. We thus report here a first experimental proof and measurements of out-of-equilibrium film thicknesses within a sheared foam, and of the impact this has on coarsening.

Sub-millimetric visualization and stability measurement for supercritical CO2 foams: Effect of ionic complexation on tubular and diverging flows

Development of natural gas resources as fuel for energy transition requires sustainable recovery processes with manageable amount of water used for unconventional stimulation of wellbores to protect the environment. To manage stability issues and bubble morphological alternations within the waterless fluids such as dry foams, microstructure and physiochemical properties were correlated to ionic complexation, bubble local plastic deformation (T1 event) in vicinity of constriction and elastoplastic behavior of CO2-water interface travelling from wellbore to fracture and flow back. This work presents facile sub-millimetric observation and step by step image analysis protocol for dry scCO2 foam transport and stability evaluation across the vertical view cell (tubular flow) and after constriction (diverging flow) and highlights the effect of viscoelastic components of interface on bubble plastic deformation and interfacial stability. The scCO2 foam’s microscopic structure was stabilized with (i) Zwitterionic Surfactants (ZS) and (ii) ZS-conjugated Polyelectrolyte Complex Nanoparticles (PECNP). PECNP-Surfactant scCO2 foam structure was noticeably affected by confinement ratio between the feeding tube and view cell, thus excessive lamella breakup and rearrangement of scCO2 bubbles occurred with higher rate in widening flow, which contributed to formation of less uniform dispersion of scCO2 droplets in PECNP-Surfactant scCO2 foam. However, the PECNP-Surfactant complexes residing in CO2-water lamella stabilized the deformed microstructure for significantly longer time. Bubble shape deformity was attributed to bubble elongation due to elastoplastic properties of lamella interface originated by the presence PECNP nanoparticle verified by higher dilatational elasticity at interface. Stability of deformed bubbles in pattern of PECNP-surfactant scCO2 foam with longer resistivity to lamella breakage, coalescence, and gravity drainage (up to 160 min) were compared to sudden bubble growth and expansion (after 30 min) detected for narrower distribution of smaller scCO2 bubbles stabilized in mixtures of ZS in high salinity brine (33.3 kppm). Higher resistance to morphological change (coalescence, rupture, burst) was observed for viscous lamella of PECNP-surfactant in scCO2 foam originated from higher viscosity and dilatational elasticity of lamella interface.

Strain manipulation of ferroelectric skyrmion bubbles in a freestanding PbTiO3 film: A phase field simulation

So far nanoscale topological polar structures have been mainly observed in ferroelectric superlattices, nanodots, and complex heterostructures, originating from the intricate interplay of built-in electric field, polarization, strain, and their gradient-related energies. However, solid substrates hinder the continuous strain manipulation of the ferroelectric topological textures. Recently, a breakthrough in fabricating freestanding films has demonstrated the possibility of continuous strain manipulation, but there has been little experimental or theoretical investigation of whether polar topological structures exist in freestanding films. Herein, by performing phase field simulation on ${(\mathrm{PbTi}{\mathrm{O}}_{3})}_{20}/{(\mathrm{SrTi}{\mathrm{O}}_{3})}_{10}$ freestanding bilayers, we observed the stabilized ferroelectric skyrmion bubbles in a ferroelectric layer. A thickness-dependent phase diagram indicates that the skyrmion bubbles exist when the ferroelectric layer is between eight and 30 unit cells. Meanwhile, we demonstrated that the uniaxial strain is an effective way to manipulate the skyrmion bubble in freestanding films, which not only induces consolidation and rearrangement of skyrmion bubbles, but also induces a phase transition from the topological skyrmion bubble state to a standard ${a}_{1}/{a}_{2}$ domain state. These results provide us guidance with a mechanical approach to control topological polar structures in freestanding films.

A computational method to simulate mono- and poly-disperse two-dimensional foams flowing in obstructed channel

A modified phase-field model is presented to numerically study the dynamics of flowing foam in an obstructed channel. The bubbles are described as smooth deformable fields interacting with one another through a repulsive potential. A strength of the model lies in its ability to simulate foams with wide range of gas fraction. The foam motion, composed of about hundred two-dimensional gas elements, was analyzed for gas fractions ranging from 0.4 to 0.99, that is below and beyond the jamming transition. Simulations are preformed near the quasi-static limit, indicating that the bubble rearrangement in the obstructed channel is primarily driven by the soft collisions and not by the hydrodynamics. Foam compression and relaxation upstream and downstream of the obstacle are reproduced and qualitatively match previous experimental and numerical observations. Striking dynamics, such as bubbles being squeezed by their neighbors in negative flow direction, are also revealed at intermediate gas fractions.

Microfluidics approach to investigate foam hysteretic behaviour

Foam stability often refers to the foam left to evolve with time in static conditions. However, in everyday life, foams are submitted to numerous deformations. A feature of foam stability is represented by the foam’s ability to resist to the deformation and to recover its initial properties after deformation. The technique developed here allows for a qualitative evaluation of the property of foam recovery after a deformation in a flow-focusing microfluidic device. The foam hysteretic behaviour was evaluated by introducing the analogous of a standard three-step test in which the recovery of viscosity is commonly studied over three deformation stages. The foam behaviour is analysed over an induced cycle of ascendant and descendant deformation at the wall, well controlled by varying the gas pressure for a constant liquid pressure. Thus, the recovery of the two-row foam pattern used as reference is studied after a high deformation phase corresponding to the bamboo pattern and the level of hysteresis is measured qualitatively. The samples investigated comprise a range of Newtonian aqueous solutions containing 5 cmc (critical micellar concentration) of sodium dodecyl sulphate (SDS). A retardation effect was observed leading to hysteresis caused by the increase in viscosity. A higher surface elasticity produced a smaller but non-negligible hysteresis due to an excess in elastic energy caused by the increase of the duration of the bubble rearrangements. The present study has gone some way towards enhancing our understanding of the mechanisms triggering or enhancing foam hysteresis in a microchannel. The findings will be of interest to many industrial processes where foams are submitted to a series of deformation steps along the process line from food industrial applications to biological systems.Graphical abstractA schematic of the three-step test consisting in an ascending and descending pressure ramps obtained by varying the gas pressure for a constant liquid pressure.

Using X-ray tomoscopy to explore the dynamics of foaming metal

The complex flow of liquid metal in evolving metallic foams is still poorly understood due to difficulties in studying hot and opaque systems. We apply X-ray tomoscopy –the continuous acquisition of tomographic (3D) images– to clarify key dynamic phenomena in liquid aluminium foam such as nucleation and growth, bubble rearrangements, liquid retraction, coalescence and the rupture of films. Each phenomenon takes place on a typical timescale which we cover by obtaining 208 full tomograms per second over a period of up to one minute. An additional data processing algorithm provides information on the 1 ms scale. Here we show that bubble coalescence is not only caused by gravity-induced drainage, as experiments under weightlessness show, and by stresses caused by foam growth, but also by local pressure peaks caused by the blowing agent. Moreover, details of foam expansion and phenomena such as rupture cascades and film thinning before rupture are quantified. These findings allow us to propose a way to obtain foams with smaller and more equally sized bubbles.

Pattern formation in pulsed gas-solid fluidized beds – The role of granular solid mechanics

Under certain conditions, gas-solid fluidized beds are known to develop a structured flow of bubbles when exposed to periodically pulsating air flows. In quasi-two-dimensional beds, periodically rising bubbles form a triangular tessellation in the vertical plane. Bubble nucleation sites at the distributor plate alternate during each cycle. This pattern sets an excellent benchmark for fundamental studies of fluidization. Notably, most common Eulerian descriptions of granular flow do not yet capture this interplay between solid mechanics and fluid-solid momentum exchange, which we show to be instrumental to the dynamic rearrangement of bubbles in a pulsed bed. We report the first successful CFD simulations of structured bubble flows in a deep, quasi-2D geometry using a Eulerian-Lagrangian CFD-DEM framework. Numerical results are in quantitative agreement with experiments. The simulated dynamics reveal that the patterns emerge from the transition of the granular collective behavior between solid-like and fluid-like, which is an outcome of dynamical coupling between gas and particles. The simulated results point out the essential role of solid frictional stresses on inducing and maintaining the formation of bubble patterns. This underscores the value of investigating pulsation-induced patterns as a prime manifestation of the mesoscopic physics underpinning fluidization, and highlights the direction for improving current practices.

Close relationship between a dry-wet transition and a bubble rearrangement in two-dimensional foam.

Liquid foams are classified into a dry foam and a wet foam, empirically judging from the liquid fraction or the shape of the gas bubbles. It is known that physical properties such as elasticity and diffusion are different between the dry foam and the wet foam. Nevertheless, definitions of those states have been vague and the dry-wet transition of foams has not been clarified yet. Here we show that the dry-wet transition is closely related to rearrangement of the gas bubbles, by simultaneously analysing the shape change of the bubbles and that of the entire foam in two dimensional foam. In addition, we also find a new state in quite low liquid fraction, which is named “superdry foam”. Whereas the shape change of the bubbles strongly depends on the change of the liquid fraction in the superdry foam, the shape of the bubbles does not change with changing the liquid fraction in the dry foam. Our results elucidate the relationship between the transitions and the macroscopic mechanical properties.

Multiscale modelling of evolving foams

We present a set of multi-scale interlinked algorithms to model the dynamics of evolving foams. These algorithms couple the key effects of macroscopic bubble rearrangement, thin film drainage, and membrane rupture. For each of the mechanisms, we construct consistent and accurate algorithms, and couple them together to work across the wide range of space and time scales that occur in foam dynamics. These algorithms include second order finite difference projection methods for computing incompressible fluid flow on the macroscale, second order finite element methods to solve thin film drainage equations in the lamellae and Plateau borders, multiphase Voronoi Implicit Interface Methods to track interconnected membrane boundaries and capture topological changes, and Lagrangian particle methods for conservative liquid redistribution during rearrangement and rupture. We derive a full set of numerical approximations that are coupled via interface jump conditions and flux boundary conditions, and show convergence for the individual mechanisms. We demonstrate our approach by computing a variety of foam dynamics, including coupled evolution of three-dimensional bubble clusters attached to an anchored membrane and collapse of a foam cluster.

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.

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.

Scaling of critical velocity for bubble raft fracture under tension

The behavior of materials under tension is a rich area of both fluid and solid mechanics. For simple fluids, the breakup of a liquid as it is pulled apart generally exhibits an instability driven, pinch-off type behavior. In contrast, solid materials typically exhibit various forms of fracture under tension. The interaction of these two distinct failure modes is of particular interest for complex fluids, such as foams, pastes, slurries, etc. The rheological properties of complex fluids are well-known to combine features of solid and fluid behaviors, and it is unclear how this translates to their failure under tension. In this paper, we present experimental results for a model complex fluid, a bubble raft. As expected, the system exhibits both pinch-off and fracture when subjected to elongation under constant velocity. We report on the critical velocity vc below which pinch-off occurs and above which fracture occurs as a function of initial system width W, length L, bubble size R, and fluid viscosity for both monodisperse and polydisperse systems. Though both exhibit a transition from pinch-off to fracture, the behavior as a function of L/W is qualitatively different for the two systems. For the polydisperse systems, the results for the critical velocity are consistent with a simple scaling law vcτ/R∼L/W, where the fluid viscosity sets the typical time for bubble rearrangements τ. We show that this scaling can be understood in terms of the dynamics of local bubble rearrangements (T1 events). For the monodisperse systems, we observe a critical value for L/W below which the system only exhibits fracture.

Foam invasion through a single pore

We investigate experimentally the behavior of liquid foams pumped at a given flow rate through a single pore, in the situation where the pore diameter is smaller than the bubble diameter. Results reveal that foam invasion can be observed only within a restricted range of values for the dimensionless flow rate and the foam liquid fraction. Within this foam invasion regime, the liquid content of invading foams is measured to be three times higher than the initial liquid content. Outside this regime, both gas alone and liquid alone invasion regimes can be observed. The gas invasion regime results from the rupture of foam films during local T1, during bubble rearrangements events induced by foam flow, whereas the liquid invasion regime is allowed by the formation of a stable cluster of jammed bubbles at the pore's opening.

Unexpected spatial distribution of bubble rearrangements in coarsening foams

Foams are ideal model systems to study stress-driven dynamics, as stress-imbalances within the system are continuously generated by the coarsening process, which unlike thermal fluctuations, can be conveniently quantified by optical means. However, the high turbidity of foams generally hinders the detailed study of the temporal and spatial distribution of rearrangement events, such that definite assessments regarding their contribution to the overall dynamics could not be made so far. In this paper, we use novel light scattering techniques to measure the frequency and position of events within a large sample volume. As recently reported (A. S. Gittings and D. J. Durian, Phys. Rev. E, 2008, 78, 066313), we find that the foam dynamics is determined by two distinct processes: intermittent bubble rearrangements of finite duration and a spatially homogeneous quasicontinuous process. Our experiments show that the convolution of these two processes determines the age-dependence of the mean dynamics, such that relations between intermittent rearrangements and coarsening process can not be established by considering means. By contrast the use of the recently introduced photon correlation imaging technique (A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, Phys. Rev. Lett., 2009, 102, 085702) enables us to assess that the event frequency is directly determined by the strain-rate imposed by the coarsening process. Surprisingly, we also find that, although the distribution of successive events in time is consistent with a random process, the spatial distribution of successive events is not random: rearrangements are more likely to occur within a recently rearranged zone. This implies that a topological rearrangement is likely to lead to an unstable configuration, such that a small amount of coarsening-induced strain is sufficient to trigger another event.

Comparison of low-amplitude oscillatory shear in experimental and computational studies of model foams

A fundamental difference between fluids and solids is their response to applied shear. Solids possess static shear moduli, while fluids do not. Complex fluids such as foams display an intermediate response to shear with nontrivial frequency-dependent shear moduli. In this paper, we conduct coordinated experiments and numerical simulations of model foams subjected to boundary-driven oscillatory planar shear. Our studies are performed on bubble rafts (experiments) and the bubble model (simulations) in two dimensions. We focus on the low-amplitude flow regime in which T1 events, i.e., bubble rearrangement events where originally touching bubbles switch nearest neighbors, do not occur, yet the system transitions from solid- to liquidlike behavior as the driving frequency is increased. In both simulations and experiments, we observe two distinct flow regimes. At low frequencies omega, the velocity profile of the bubbles increases linearly with distance from the stationary wall, and there is a nonzero total phase shift between the moving boundary and interior bubbles. In this frequency regime, the total phase shift scales as a power law Delta approximately omegan with n approximately 3. In contrast, for frequencies above a crossover frequency omega>omegap, the total phase shift Delta scales linearly with the driving frequency. At even higher frequencies above a characteristic frequency omeganl>omegap, the velocity profile changes from linear to nonlinear. We fully characterize this transition from solid- to liquidlike flow behavior in both the simulations and experiments and find qualitative and quantitative agreements for the characteristic frequencies.

Statistics of bubble rearrangement dynamics in a coarsening foam

We use speckle-visibility spectroscopy to measure the time dependence of bubble rearrangement events that are driven by coarsening in an aqueous foam. This technique gives the time trace for the average scattering site speed within a prescribed volume of the sample. Results are analyzed in terms of distributions of event times, event speeds, and event displacements. The distribution of rest times between successive events is also measured; comparison with diffusing-wave spectroscopy results shows that the spatial structure of a typical event consists of a core of only a few bubbles which undergo topology change plus a surrounding shell of bubbles which shift by an amount that decays to one wavelength at four to five bubbles away. No correlations are found between the durations, speeds, and rest times between successive events.

Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam

The plastic flow of a foam results from bubble rearrangements. We study their occurrence in experiments where a foam is forced to flow in 2D: around an obstacle; through a narrow hole; or sheared between rotating disks. We describe their orientation and frequency using a topological matrix defined in the companion paper (F. Graner, B. Dollet, C. Raufaste, and P. Marmottant, this issue, 25 (2008) DOI 10.1140/epje/i2007-10298-8), which links them with continuous plasticity at large scale. We then suggest a phenomenological equation to predict the plastic strain rate: its orientation is determined from the foam's local elastic strain; and its rate is determined from the foam's local elongation rate. We obtain a good agreement with statistical measurements. This enables us to describe the foam as a continuous medium with fluid, elastic and plastic properties. We derive its constitutive equation, then test several of its terms and predictions.

Statistics of microscopic yielding in sheared aqueous foams

We detail the statistical distribution of bubble rearrangements in a sheared two-dimensional foam. Such rearrangements, known as T1 events, are vital to mechanisms resulting in flow through microscopic mechanical yielding. We find that at a constant rate of shear, the rate of occurrence of T1 events shows only small fluctuations. In addition, we detail the spatial and orientational distribution of T1 events and relate them to the distribution of stresses in the bulk of the material. Some insights into the asymmetry of the dynamics of T1 events are also discussed.

Stokes experiment in a liquid foam

The paper reports on the quasistatic steady flow of a dry liquid foam around a fixed spherical bead, a few times larger than the typical bubble size. The force exerted on the bead is recorded with a precision and a time resolution large enough to show the succession of elastic loading of the foam, separated by sudden force drops. The foam structure is observed by direct light transmission, synchronized with the force measurement, thus allowing us to correlate the plastic events with the force variations. Scaling laws for the force signal as a function of the bubble size are detailed and interpreted with a simple elasto-plastic model. The spatial distribution of the plasticity is strongly localized in the first bubble layers around the bead and the average size of the bubble rearrangements increases with the corresponding force jump amplitude.

The Apollonian decay of beer foam bubble size distribution and the lattices of young diagrams and their correlated mixing functions

We present different methods to characterise the decay of beer foam by measuring the foam heights and recording foam images as a function of time. It turns out that the foam decay does not follow a simple exponential law but a higher‐order equation V(t) = a − bt − ct2.5, which can be explained as a superposition of two processes, that is, drainage and bubble rearrangement. The reorganisation of bubbles leads to the structure of an Apollonian gasket with a fractal dimension of D ≈ 1.3058. Starting from foam images, we study the temporal development of bubble size distributions and give a model for the evolution towards the equilibrium state based upon the idea of Ernst Ruch to describe irreversible processes by lattices of Young diagrams. These lattices generally involve a partial order, but one can force a total order by mapping the diagrams onto the interval [0, 1] using ordering functions such as the Shannon entropy. Several entropy‐like and nonentropy‐like mixing functions are discussed in comparison with the Young order, each of them giving a special prejudice for understanding the process of structure formation during beer foam decay.