Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content

Simply solvable model capturing the approach to statistical self-similarity for the diffusive coarsening of bubbles, droplets, and grains

Aqueous foams and a wide range of related systems are believed to coarsen by diffusion between neighboring domains into a statistically self-similar scaling state, after the decay of initial transients, such that dimensionless domain size and shape distributions become time independent and the average grows as a power law. Partial integrodifferential equations for the time evolution of the size distribution for such phase separating systems can be formulated for arbitrary initial conditions, but these are cumbersome for analyzing data on nonscaling state preparations. Here we show that essential features of the approach to the scaling state are captured by an exactly-solvable ordinary differential equation for the evolution of the average bubble size. The key ingredient is to characterize the bubble size distribution approximately, using the average size of all bubbles and the average size of the critical bubbles, which instantaneously neither grow nor shrink. The difference between these two averages serves as a proxy for the width of the size distribution. Solution of our model shows that behavior is controlled by a signed length δ that is proportional to the width of the initial distribution relative to that in the scaling state. In particular, δ is negative if the initial preparation is too monodisperse, and is positive if it is too polydisperse. To test our approach, we compare with data for quasi-two dimensional dry foams created with three different initial amounts of polydispersity. This allows us to readily identify the critical radius from the average area of six-sided bubbles, whose growth rate is zero by the von Neumann law. The growth of the average and critical radii agree quite well with exact solution, though the most monodisperse sample crosses over to the scaling state faster than expected. A simpler approximate solution of our model performs equally well. Our approach is applicable to 3d foams, which we demonstrate by re-analyzing prior data, as well as to froths of dilute droplets and to phase separation kinetics for more general systems such as emulsions, binary mixtures, and alloys.

Dynamic Foam Characteristics during Cultivation of Arthrospira platensis

This study is aimed at understanding the serious foaming problems during microalgal cultivation in industrial raceway ponds by studying the dynamic foam properties in Arthrospira platensis cultivation. A. platensis was cultivated in a 4 L bowl bioreactor for 4 days, during which the foam height above the algal solution increased from 0 to 30 mm with a bubble diameter of 1.8 mm, and biomass yield reached 1.5 g/L. The algal solution surface tension decreased from 55 to 45 mN/m, which favored the adsorption of microalgae on the bubble to generate more stable foams. This resulted in increased foam stability (FS) from 1 to 10 s, foam capacity (FC) from 0.3 to 1.2, foam expansion (FE) from 15 to 43, and foam maximum density (FMD) from 0.02 to 0.07. These results show a decrease in CO₂ flow rate and operation temperature when using the Foamscan instrument, which minimized the foaming phenomenon in algal solutions to a significantly lower and acceptable level.

Unraveling hidden rules behind the wet-to-dry transition of bubble array by glass-box physics rule learner

A liquid–gas foam, here called bubble array, is a ubiquitous phenomenon widely observed in daily lives, food, pharmaceutical and cosmetic products, and even bio- and nano-technologies. This intriguing phenomenon has been often studied in a well-controlled environment in laboratories, computations, or analytical models. Still, real-world bubble undergoes complex nonlinear transitions from wet to dry conditions, which are hard to describe by unified rules as a whole. Here, we show that a few early-phase snapshots of bubble array can be learned by a glass-box physics rule learner (GPRL) leading to prediction rules of future bubble array. Unlike the black-box machine learning approach, the glass-box approach seeks to unravel expressive rules of the phenomenon that can evolve. Without known principles, GPRL identifies plausible rules of bubble prediction with an elongated bubble array data that transitions from wet to dry states. Then, the best-so-far GPRL-identified rule is applied to an independent circular bubble array, demonstrating the potential generality of the rule. We explain how GPRL uses the spatio-temporal convolved information of early bubbles to mimic the scientist’s perception of bubble sides, shapes, and inter-bubble influences. This research will help combine foam physics and machine learning to better understand and control bubbles.

Quantifying the long-range structure of foams and other cellular patterns with hyperuniformity disorder length spectroscopy

We investigate the local and long-range structure of several space-filling cellular patterns: bubbles in a quasi-two-dimensional foam, and Voronoi constructions made around points that are uncorrelated (Poisson patterns), low discrepancy (Halton patterns), and displaced from a lattice by Gaussian noise (Einstein patterns). We study local structure with distributions of quantities including cell areas and side numbers. The former is the widest for the bubbles making foams the most locally disordered, while the latter show no major differences between the cellular patterns. To study long-range structure, we begin by representing the cellular systems as patterns of points, both unweighted and weighted by cell area. For this, foams are represented by their bubble centroids and the Voronoi constructions are represented by the centroids as well as the points from which they are created. Long-range structure is then quantified in two ways: by the spectral density, and by a real-space analog where the variance of density fluctuations for a set of measuring windows of diameter D is made more intuitive by conversion to the distance h(D) from the window boundary where these fluctuations effectively occur. The unweighted bubble centroids have h(D) that collapses for the different ages of the foam with random Poissonian fluctuations at long distances. The area-weighted bubble centroids and area-weighted Voronoi points all have constant h(D)=h_{e} for large D; the bubble centroids have the smallest value h_{e}=0.084sqrt[〈a〉], meaning they are the most uniform. Area-weighted Voronoi centroids exhibit collapse of h(D) to the same constant h_{e}=0.084sqrt[〈a〉] as for the bubble centroids. A similar analysis is performed on the edges of the cells and the spectra of h(D) for the foam edges show h(D)∼D^{1-ε} where ε=0.30±0.15.

Experimentally testing a generalized coarsening model for individual bubbles in quasi-two-dimensional wet foams

We present high-precision data for the time evolution of bubble area A(t) and circularity shape parameter C(t) for several bubbles in a quasi-two-dimensional foams consisting of bubbles squashed between parallel plates. In order to fully compare with earlier predictions, foam wetness is systematically varied by controlling the height of the sample above a liquid reservoir which in turn controls the radius r of the inflation of the Plateau borders. For very dry foams, where the borders are very small, classic von Neumann behavior is observed where a bubble's growth rate depends only on its number n of sides. For wet foams, the inflated borders impede gas exchange and cause deviations from von Neumann's law that are found to be in accord with the generalized coarsening equation. In particular, the overall growth rate varies linearly with the film height, which decrease as surface Plateau borders inflate. More interestingly, the deviation from dA/dt∝(n-6) von Neumann behavior grows in proportion to nCr/sqrt[A]. This is highlighted definitively by data for six-sided bubbles, which are forbidden to grow or shrink except for the existence of this term. It is tested quantitatively by variation of all four relevant quantities: n, C, r, and A.

Scaling law for the kinetics of water imbibition in polydisperse foams

We investigated the kinetics of water imbibition in polydisperse foams. We used a Hele-Shaw cell, and horizontal imbibition was observed for a timescale of up to 10³ s in which the gravity effect was negligible. While several papers have reported kinetics for imbibition in foams, imbibition kinetics in polydisperse foams and its variations in longer timescales are not well understood. The tip position of imbibition was proportional to the square root of time in the initial stage of imbibition, but it showed plateauing in the late stage of imbibition. We evaluated the proportional constant A in the initial stage of imbibition as a kinetic constant for the time-dependent increase in the tip position, which showed a clear dependency on the initial and final water volume fractions in the foams. Conversely, the mean initial radius of the curvature and the channel length in the Plateau borders did not show any clear correlations with A, although both valuables are frequently used in modeling for liquid imbibition in foams. On the basis of the t^1/2 dependence, the correlation of A with the water volume fraction and the increase in the water volume fraction during imbibition, we proposed a simple equation to describe the tip position over the entire period of imbibition. We used them to scale all of the experimental data, which showed good agreement with the theoretical line. This clearly showed that the water volume fraction in the foams during imbibition was the key factor to quantitatively describe the rate of water imbibition. Features in the kinetics of imbibition were discussed.

Horizontal imbibition of water in foams is scaled well by a simple mathematical expression that considers t^1/2 dependence and changes in volume fraction of water in foams.

Different strategies of foam stabilization in the use of foam as a fracturing fluid

An attractive alternative to mitigate the adverse effects of conventional water-based fluids on the efficiency of hydraulic fracturing is to inject foam-based fracking fluids into reservoirs. The efficiency of foaming fluids in subsurface applications largely depends on the stability and transportation of foam bubbles in harsh environments with high temperature, pressure and salinity, all of which inevitably lead to poor foam properties and thus limit fracturing efficiency. The aim of this paper is to elaborate popular strategies of foam stabilization under reservoir conditions. Specifically, this review first discusses three major mechanisms governing foam decay and summarizes recent progress in research on these phenomena. Since surfactants, polymers, nanoparticles and their composites are popular options for foam stabilization, their stabilizing effects, especially the synergies in composites, are also reviewed. In addition to reporting experimental results, the paper also reports recent advances in interfacial properties via molecular dynamical simulation, which provide new insights into gas/liquid interfacial properties under the influence of surfactants at molecular scale. The results of both experiments and simulations indicate that foam additives play an essential role in foam stability and the synergic effects of surfactants and nanoparticles exhibit more favorable performance.

Measurement of film permeability in 2D foams

The coarsening of quasi-2D wet foams is well described theoretically by the model of Schimming and Durian, that takes into account the diffusion through the Plateau borders and the vertices in a rigorous manner. In this article, we describe an experimental study of coarsening in which the foam film permeability is measured in such quasi-2D wet foams. We first performed a full characterization of the structure of the studied foams. Then we measured the coarsening rates. It appears that, in these foams, the film thicknesses are still too small for the Plateau borders and the vertices to contribute, but the surface Plateau borders lead to a smaller coarsening rate compared to dry foams. This rate increases with capillary pressure and follows well the prediction of the model. We demonstrate the importance of working in controlled pressure conditions during permeability measurements. Indeed, permeability depends on film thickness itself depending on capillary pressure.

Evaluation of foam performance at different temperature for enhanced oil recovery process

Purpose

Application of foam in enhanced oil recovery requires a production of foam that is strong and stable enough to withstand a long period. There are numerous factors that may affect the performance of foam, among which is temperature. Therefore, this study aims to observe the foam performance at different temperature by evaluating the foamability and the stability of the foam.

Design/methodology/approach

In this study, bulk foam test using FoamScan was conducted to examine the effect of temperature on foam in the presence of crude oil. Nitrogen gas was sparged through the mixture of crude oil, an in-house developed surfactant, and sodium chloride solution as the brine at different temperatures to produce foam at a certain height. The crude oil was extracted from an oilfield in East Malaysia and the in-house developed surfactant was a mixture of amphoteric and anionic surfactants. A camera continuously recorded the height of foam during the generation and the collapse of the foam. The foamability and foam stability properties of each sample were taken as the indicators for foam performance. Furthermore, the entering, spreading and bridging analysis was run to observe the effect of the presence of crude oil on foam performance.

Findings

In general, the higher the temperature, the less stable the foam is. As the stability of foam is associated with the rate of liquid drainage, it was observed that as temperature increases, the rate of liquid drainage also increases. On the other hand, the entering, spreading and bridging analysis shows that there is entering of oil droplet happening on the interface of foam film that may promote the rupture of the foam film even more.

Originality/value

It was found that the temperature has a small impact on foamability, whereas the foam stability was significantly affected by the temperature. Therefore, it can be concluded that foamability is not necessarily interrelated to foam stability, contradicting to the findings of few authors.

Coalescence in Two-Dimensional Foams: A Purely Statistical Process Dependent on Film Area

While coalescence is ultimately the most drastic destabilization process in foams, its underlying processes are still unclear. To better understand them, we track individual coalescence events in two-dimensional foams at controlled capillary pressure. We obtain statistical information revealing the influence of the different parameters which have been previously proposed to explain coalescence. Our main conclusion is that coalescence probability is simply proportional to the area of the thin film separating two bubbles, suggesting that coalescence is mostly stochastic.

Border-crossing model for the diffusive coarsening of two-dimensional and quasi-two-dimensional wet foams

For dry foams, the transport of gas from small high-pressure bubbles to large low-pressure bubbles is dominated by diffusion across the thin soap films separating neighboring bubbles. For wetter foams, the film areas become smaller as the Plateau borders and vertices inflate with liquid. So-called "border-blocking" models can explain some features of wet-foam coarsening based on the presumption that the inflated borders totally block the gas flux; however, this approximation dramatically fails in the wet or unjamming limit where the bubbles become close-packed spheres and coarsening proceeds even though there are no films. Here, we account for the ever-present border-crossing flux by a new length scale defined by the average gradient of gas concentration inside the borders. We compute that it is proportional to the geometric average of film and border thicknesses, and we verify this scaling by numerical solution of the diffusion equation. We similarly consider transport across inflated vertices and surface Plateau borders in quasi-two-dimensional foams. And we show how the dA/dt=K_{0}(n-6) von Neumann law is modified by the appearance of terms that depend on bubble size and shape as well as the concentration gradient length scales. Finally, we use the modified von Neumann law to compute the growth rate of the average bubble area, which is not constant.

Bran-induced effects on the evolution of bubbles and rheological properties in bread dough

Evolution of bubbles is the key to volume development in bread dough. The influence of wheat bran on bubble growth in bread dough through the mixing, fermentation, and proofing stages is described as a function of its level of addition. Confocal laser scanning microscopy in combination with image processing tools was used to obtain the bubble size and shape parameters. The relationship between bubble behavior and dough rheology was mapped using biaxial extension and dynamic oscillatory rheometry studies. With increase in level of bran addition, mean bubble size decreased corresponding to each stage and showed an inverse relationship with dough overpressure and elastic modulus. Addition of wheat bran was observed to suppress the bubble coarsening phenomenon in dough. Experimental observations indicated the plausibility of coalescence-mediated bubble growth in bread dough during the latter stages of fermentation and proofing, which was hindered in the presence of bran particles.

PRACTICAL APPLICATION

The results of this work provide an insight to the underlying mechanism by which wheat bran addition impacts the volume development in bread dough. The inferences presented in this research work can be used as a basis to study bubble dynamics in an opaque food system such as bread dough. This information would be of interest to industrial researchers working on the new product development of aerated bakery products with functional fibrous ingredients.

Curvature-driven foam coarsening on a sphere: A computer simulation

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state.

Foam drainage in the presence of solid particles

We conducted forced drainage experiments to study the liquid flow within the foams stabilized by a cationic surfactant (CTAB) in the presence of partially hydrophobic silica particles. The results show that the presence of solid particles, even when present in small amounts (0.0932 g L(-1) foam), can significantly decrease the foam permeability. The scaling behaviour (power law) between the drainage velocity and the imposed flow rate indicates that the presence of solid particles in the foams triggers a transition of the foam drainage regime from a node-dominated regime to a Plateau border-dominated regime. We applied two foam drainage equations for aqueous foams to simulate the experimental data and interpret the transition. The simulation results show that the presence of solid particles in the foams increases the rigidity of the interfaces and the viscous losses in the channels (the Plateau borders) of the foams, and decreases the foam permeability. We also generalize the theory for the effects of unattached hydrophilic particles on foam drainage by considering the effects of hydrophobicity and concentration of solid particles on the confinement of foam networks. This study explores liquid drainage in three-phase foams and is relevant to the field of hydrophobic particle separation by froth flotation, in which the wash water is commonly applied to the froth layer to improve the product grade.

A critical review of the growth, drainage and collapse of foams

This review focuses on the current knowledge regarding (i) the mechanisms governing foamability and foam stability, and (ii) models for the foam column kinetics. Although different length scales of foam structure, such as air-water interface and liquid film, have been studied to elucidate the mechanisms that control the foamability and foam stability, many questions remain unanswered. It is due to the collective effects of different mechanisms involved and the complicated structures of foam sub-structures such as foam films, Plateau borders and nodes, and foam networks like soft porous materials. The current knowledge of the effects of solid particles on liquid film stability and foam drainage is also discussed to highlight gaps in our present level of understanding foam systems with solid particles. We also critically review and summarize the models that describe macroscopic foam behaviors, such as equilibrium foam height, foam growth and collapse, within the context of the mechanisms involved.

Cooperativity flows and shear-bandings: a statistical field theory approach

Cooperativity effects have been proposed to explain the non-local rheology in the dynamics of soft jammed systems. Based on the analysis of the free-energy model proposed by L. Bocquet, A. Colin and A. Ajdari, Phys. Rev. Lett., 2009, 103, 036001, we show that cooperativity effects resulting from the non-local nature of the fluidity (inverse viscosity) are intimately related to the emergence of shear-banding configurations. This connection materializes through the onset of inhomogeneous compact solutions (compactons), wherein the fluidity is confined to finite-support subregions of the flow and strictly zero elsewhere. The compacton coexistence with regions of zero fluidity ("non-flowing vacuum") is shown to be stabilized by the presence of mechanical noise, which ultimately shapes up the equilibrium distribution of the fluidity field, the latter acting as an order parameter for the flow-noflow transitions occurring in the material.

Electrical conductivity of quasi-two-dimensional foams

Quasi-two-dimensional (quasi-2D) foams consist of monolayers of bubbles squeezed between two narrowly spaced plates. These simplified foams have served successfully in the past to shed light on numerous issues in foam physics. Here we consider the electrical conductivity of such model foams. We compare experiments to a model which we propose, and which successfully relates the structural and the conductive properties of the foam over the full range of the investigated liquid content. We show in particular that in the case of quasi-2D foams the liquid in the nodes needs to be taken into account even at low liquid content. We think that these results may provide different approaches for the characterization of foam properties and for the in situ characterization of the liquid content of foams in confining geometries, such as microfluidics.

Internal dynamics and activated processes in soft-glassy materials

Plastic rearrangements play a crucial role in the characterization of soft-glassy materials, such as emulsions and foams. Based on numerical simulations of soft-glassy systems, we study the dynamics of plastic rearrangements at the hydrodynamic scales where thermal fluctuations can be neglected. Plastic rearrangements require an energy input, which can be either provided by external sources, or made available through time evolution in the coarsening dynamics, in which the total interfacial area decreases as a consequence of the slow evolution of the dispersed phase from smaller to large droplets/bubbles. We first demonstrate that our hydrodynamic model can quantitatively reproduce such coarsening dynamics. Then, considering periodically oscillating strains, we characterize the number of plastic rearrangements as a function of the external energy-supply, and show that they can be regarded as activated processes induced by a suitable "noise" effect. Here we use the word noise in a broad sense, referring to the internal non-equilibrium dynamics triggered by spatial random heterogeneities and coarsening. Finally, by exploring the interplay between the internal characteristic time-scale of the coarsening dynamics and the external time-scale associated with the imposed oscillating strain, we show that the system exhibits the phenomenon of stochastic resonance, thereby providing further credit to the mechanical activation scenario.

Statistical mechanics of two-dimensional shuffled foams: geometry-topology correlation in small or large disorder limits

Bubble monolayers are model systems for experiments and simulations of two-dimensional packing problems of deformable objects. We explore the relation between the distributions of the number of bubble sides (topology) and the bubble areas (geometry) in the low liquid fraction limit. We use a statistical model [M. Durand, Europhys. Lett. 90, 60002 (2010)] which takes into account Plateau laws. We predict the correlation between geometrical disorder (bubble size dispersity) and topological disorder (width of bubble side number distribution) over an extended range of bubble size dispersities. Extensive data sets arising from shuffled foam experiments, surface evolver simulations, and cellular Potts model simulations all collapse surprisingly well and coincide with the model predictions, even at extremely high size dispersity. At moderate size dispersity, we recover our earlier approximate predictions [M. Durand, J. Kafer, C. Quilliet, S. Cox, S. A. Talebi, and F. Graner, Phys. Rev. Lett. 107, 168304 (2011)]. At extremely low dispersity, when approaching the perfectly regular honeycomb pattern, we study how both geometrical and topological disorders vanish. We identify a crystallization mechanism and explore it quantitatively in the case of bidisperse foams. Due to the deformability of the bubbles, foams can crystallize over a larger range of size dispersities than hard disks. The model predicts that the crystallization transition occurs when the ratio of largest to smallest bubble radii is 1.4.

Structure and coarsening at the surface of a dry three-dimensional aqueous foam

We utilize total-internal reflection to isolate the two-dimensional surface foam formed at the planar boundary of a three-dimensional sample. The resulting images of surface Plateau borders are consistent with Plateau's laws for a truly two-dimensional foam. Samples are allowed to coarsen into a self-similar scaling state where statistical distributions appear independent of time, except for an overall scale factor. There we find that statistical measures of side number distributions, size-topology correlations, and bubble shapes are all very similar to those for two-dimensional foams. However, the size number distribution is slightly broader, and the shapes are slightly more elongated. A more obvious difference is that T2 processes now include the creation of surface bubbles, due to rearrangement in the bulk, and von Neumann's law is dramatically violated for individual bubbles. But nevertheless, our most striking finding is that von Neumann's law appears to holds on average, namely, the average rate of area change for surface bubbles appears to be proportional to the number of sides minus six, but with individual bubbles showing a wide distribution of deviations from this average behavior.

Coarsening dynamics of three-dimensional levitated foams: From wet to dry

We study diamagnetically levitated foams with widely different liquid fractions. Due to the levitation, drainage is effectively suppressed and the dynamics is driven by the coarsening of the foam bubbles. For dry foams, the bubble size is found to increases as the square root of foam age, as expected from a generalized von Neumann law. At higher liquid content the behavior changes to that of Ostwald ripening where the bubbles grow with the 1/3 power of the age. Using Diffusing Wave Spectroscopy we study the local dynamics in the different regimes and find diffusive behavior for dry foams and kinetic behavior for wet foams.