2D Foam Research Articles

Geometry and Topology of Two-Dimensional Dry Foams: Computer Simulation and Experimental Characterization.

Pseudo-two-dimensional (2D) foams are commonly used in foam studies as it is experimentally easier to measure the bubble size distribution and other geometric and topological properties of these foams than it is for a 3D foam. Despite the widespread use of 2D foams in both simulation and experimental studies, many important geometric and topological relationships are still not well understood. Film size, for example, is a key parameter in the stability of bubbles and the overall structure of foams. The relationship between the size distribution of the films in a foam and that of the bubbles themselves is thus a key relationship in the modeling and simulation of unstable foams. This work uses structural simulation from Surface Evolver to statistically analyze this relationship and to ultimately formulate a relationship for the film size in 2D foams that is shown to be valid across a wide range of different bubble polydispersities. These results and other topological features are then validated using digital image analysis of experimental pseudo-2D foams produced in a vertical Hele-Shaw cell, which contains a monolayer of bubbles between two plates. From both the experimental and computational results, it is shown that there is a distribution of sizes that a film can adopt and that this distribution is very strongly dependent on the sizes of the two bubbles to which the film is attached, especially the smaller one, but that it is virtually independent of the underlying polydispersity of the foam.

Lamella Division in a Foam Flowing through a Two-Dimensional Porous Medium: A Model Fragmentation Process

We flow a 2D foam through a model 2D porous medium and study experimentally and numerically how the bubble size distribution evolves along the medium. The dominant mechanism of bubble creation is a fragmentation process occurring when bubbles pinched against obstacles are split in two smaller bubbles. We infer the statistics of these individual and local fragmentation events from the experimental data and propose a fragmentation equation to relate that statistics to the evolution of the global size distribution. The predicted evolution shows very good agreement with direct experimental measurements of the bubble size distribution.

Image-based correlation between the meso-scale structure and deformation of closed-cell foam

In the correlation between structural parameters and compressive behaviour of cellular materials, previous studies have mostly focused on averaged structural parameters and bulk material properties for different samples. This study focuses on the meso-scale correlation between structure and deformation in a 2D foam sample generated from a computed tomography slice of Alporas™ foam, for which quasi-static compression was simulated using 2D image-based finite element modelling. First, a comprehensive meso-scale structural characterisation of the 2D foam was carried out to determine the size, aspect ratio, orientation and anisotropy of individual cells, as well as the length, straightness, inclination and thickness of individual cell walls. Measurements were then conducted to obtain the axial distributions of local structural parameters averaged laterally to compression axis. Second, the meso-scale deformation was characterised by cell-wall strain, cell area ratio, digital image correlation strain and local compressive engineering strain. According to the results, the through-width sub-regions over an axial length between the average (lower bound) and the maximum (upper bound) of cell size should be used to characterise the meso-scale heterogeneity of the cell structure and deformation. It was found that the first crush band forms in a sub-region where the ratio of cell-wall thickness to cell-wall length is a minimum, in which the collapse deformation is dominated by the plastic bending and buckling of cell walls. Other morphological parameters have secondary effect on the initiation of crush band in the 2D foam. The finding of this study suggests that the measurement of local structural properties is crucial for the identification of the “weakest” region which determines the initiation of collapse and hence the corresponding collapse load of a heterogeneous cellular material.

Self-assembly behavior of a rod-like polypeptide at the air-water interface

We present a systematic study of morphological changes in monolayers of a rod-like polypeptide (poly-γ-benzyl-L-glutamate (PBLG)) on a Langmuir trough as a function of increasing surface density. Through the combination of atomic force and Brewster angle microscopy, we accessed structural information on the nano- and micro-scale. Already during the spreading process, aggregates formed, leading to a coexistence of randomly organized domains, consisting of preferentially oriented fibers, embedded in a less ordered matrix of lower density. The resulting morphology after spreading resembles a structure of a 2D foam as observed for small amphiphilic molecules. With increasing surface density, we observed that the domains became larger with fibers showing a strong tendency to coarsen. Due to this coarsening process, the plateau region of the surface pressure isotherm exhibited a continuous increase in width of the fibers. In the region of highest surface density, solid-like films composed of compacted wide fibers were observed. These fibers, which were most likely in a glassy state, were fully oriented orthogonal to the compression direction.

Determination of the constitutive relation and critical condition for the shock compression of cellular solids

This study aims at understanding the constitutive relation and critical condition for the shock compression of cellular solids. A 2D virtual foam is constructed from the cross-section of a closed-cell aluminium foam imaged by micro X-ray computed tomography, which enables the realistic consideration of meso-scale structural effect in numerical modelling. Quasi-static and shock compressions of the 2D foam are simulated. A series of Hugoniot relations between shock speed (and other mechanical quantities) and impact speed are determined from the finite element (FE) simulations. It is found that the shock speed increases approximately linearly with impact speed, similar to that observed for condensed solids, but the related material constants for cellular solids have different physical implications, whereas the shock strain, stress and energy increase with impact speed nonlinearly, due to shock-enhanced cell compaction and cell-wall plastic deformation. Based on conservation laws in continuum mechanics, other Hugoniot relations are derived from the basic linear one, which agree well with those obtained from the FE simulations. It is thus demonstrated that the unique linear Hugoniot relation can be used to characterise the shock constitutive behaviour which is distinct from the quasi-static one. Furthermore, a new analytical method based on the linear Hugoniot relation is proposed to estimate the critical impact speed for shock initiation, which has reasonable agreement with the present FE simulations and previous experimental and numerical results, and outperforms the existing methods.

Three-dimensional foam flow resolved by fast X-ray tomographic microscopy

Adapting fast tomographic microscopy, we managed to capture the evolution of the local structure of the bubble network of a 3D foam flowing around a sphere. As for the 2D foam flow around a circular obstacle, we observed an axisymmetric velocity field with a recirculation zone, and indications of a negative wake downstream the obstacle. The bubble deformations, quantified by a shape tensor, are smaller than in 2D, due to a purely 3D feature: the azimuthal bubble shape variation. Moreover, we were able to detect plastic rearrangements, characterized by the neighbor swapping of four bubbles. Their spatial structure suggests that rearrangements are triggered when films faces get smaller than a characteristic area.

Using Surface Evolver to measure pressures and energies of real 2D foams submitted to quasi-static deformations

Static 2D foams have the interesting property that their energy is measurable by summing up the length of their films, so that a simple optical picture of a 2D foam should enable measurement of its energy and other quantities such as its bubbles’ pressures. This operation is of course unrealizable in most experiments since the optical resolution limits the accuracy of length measurements. Here we show that, using image analysis tools alongside an iterative procedure based on the Surface Evolver (Brakke, 1992) to analyze optical images of a 2D foam, we are able to measure accurately its energy and its bubbles’ pressures up to a single multiplying factor. We determine this factor, and validate this procedure, by comparing experimental measurements of the pressure and the work done on a 2D foam experiencing a quasi-static localized deformation with the energy and pressures computed using our procedure.

Bubble migration in two-dimensional foam sheared in a wide-gap Couette device: Effects of non-Newtonian rheology

We report experiments on the migration of a large bubble in an otherwise monodisperse two-dimensional (2D) foam sheared in a wide-gap Couette device. The bubble migrates away from the walls toward an equilibrium position between the center of the gap and the inner cylinder. This differs from the situation in a narrow-gap Couette device, where the equilibrium position is at the center of the gap [Mohammadigoushki and Feng, Phys. Rev. Lett. 109, 084502 (2012)]. The shift in equilibrium position is attributed to the non-Newtonian rheology of the foam, which is brought out by the nonhomogeneous shearing in a wide-gap geometry. Two aspects of the rheology, shear-thinning and the first normal stress difference, are examined separately by comparing with bubble migration in a xanthan gum solution and a Boger fluid. Shear-thinning shifts the equilibrium position inward while the normal stress does the opposite. Bubble migration in the 2D foam is the outcome of the competition between the two effects.

Effect of foam quality on effectiveness of hydraulic fracturing in shales

The goal of this work is to study the effect of foam quality on the gas productivity of fractured wells in shales. A numerical model of fracture propagation is developed which incorporates a 2D fracture model, proppant transport, and foam rheology. The fracture conductivity distribution is computed from the local proppant concentration, which is incorporated in a reservoir simulation model to predict the gas production by depressurization. Given the rock mechanical properties, the fracturing fluid, proppants, and rock permeability control the shape and conductivity of the fractures created during hydraulic fracturing. The shape and conductivity of the fracture, in turn, control the gas productivity of such a well. Foam quality, proppant size, proppant concentration, and rock permeability are varied. The results show that, in most cases, the gas production is optimum at an intermediate foam quality (of about 65%) because it creates the largest propped fracture area and minimizes leak-off. Only for the case of 0.1 micro-Darcy rock, low proppant concentration and large proppant size, the gas production increases monotonically with foam quality because high quality foam can place a partial monolayer of proppants. Fracturing with water as the initial pad and a high quality (65–85%) foam as the proppant carrier is more effective than using the same foam for both pad and slurry.

The effects of micro-defects and crack on the mechanical properties of metal fiber sintered sheets

The effects of three types of defect (i.e., two micro defects—broken fibers and separation of fiber joints and one macro defect—crack) on the mechanical properties of porous metal fiber sintered sheets (MFSSs) are investigated by a combination of numerical simulation, analytical modeling, and experimental test. All simulations are based upon the previously developed micromechanics random beam model (Jin et al., 2013). Broken fibers are realized by removing cell edges (i.e., fibers between two joints) in an otherwise perfect model. Their induced decreases in the elastic moduli and strengths are found to be much lower than those of two dimensional (2D) foams and Kagome grids. For the defect in the form of separation of fiber joints, both analytical and numerical models are developed. The predicted linear decreases in the moduli and strengths (except for the compressive strength) with increasing number of separated fiber joints indicate that MFSSs be insensitive to the defect of joint separation. To explore the effect of crack, fracture toughness of MFSSs is measured and is found to be significantly higher than that of metal foams of the same relative density (i.e., volume fraction of the constituent solid material). The underlying ductile mechanism of MFSSs is further investigated by numerical simulations, showing that plastic deformation spreads all over the fibers in ligament rather than concentrates around crack tip. This study shows that MFSSs are superior in view of their resistance to the considered micro-defects and crack.

Effects of statistics of cell’s size and shape irregularity on mechanical properties of 2D and 3D Voronoi foams

In the study of the mechanical properties of metallic foam, the relative density (or porosity) and the average size of cells are two key parameters of the meso-geometry, but it is also well known experimentally that the two parameters alone are not enough since the mechanical properties of metallic foams are different even with the same initial relative density and average cell size. In this paper, we have classified the irregularity of cells into two types to describe polygonal and polyhedral cells in 2D and 3D metallic foams, which are called size irregularity and shape irregularity, respectively. The former reflects the deviation of the size of a cell from the average cell size in the foam and the latter reflects the deviation of the shape of a cell from a circle with the same area (2D) or a globe with the same volume (3D). With the two kinds of definitions, effects of the irregularity of cells in aluminum foam on mechanical properties are investigated using the Voronoi tessellation technique and the finite element method. The well- designed 2D and 3D Voronoi models are constructed, of which the statistic distributions of size and shape irregularity are presented. The compression simulations of Voronoi-based models indicate that the yield stress of metallic foam is seldom affected by the size irregularity, but significantly affected by the shape irregularity. The more regular the foams, the higher will be their yield plateau at constant overall relative density and average cell size.

Numerical simulation of creaming and foam formation in aerated liquid with population balance modeling

A new approach for simulating the formation of foam in aerated liquid is proposed. Foam is considered as a separate phase which is comprised of a mixture of air and liquid. A computational fluid dynamic (CFD) model has been developed for the simulation of creaming and formation of foam in aerated liquid using the proposed approach. The model accounts for the formation of foam due to transformation of both air and liquid into foam and it is destruction due to liquid drainage and bursting of bubbles. The coalescence model of Prince and Blanch (1990. Bubble coalescence and break-up in air-sparged bubble columns. AIChE J. 36, 1485–1499.) was considered for air-liquid dispersion whereas for the foam layer, the bubble coalescence due to film rupture was modeled using the model developed by Tong et al. (2011. Drainage and stability of 2D foams: foam behavior in vertical Hele-Shaw cells. Colloid Surf. A 382, 42–49). A population balance method was used to track the number density of different bubble class and the fixed pivot method was used to discretize the population balance equation. The source term for coalescence in the scalar transport equation was updated using a rectified model of Hagesaether et al. (2002. A model for turbulent binary breakup of dispersed fluid particles. Chem. Eng. Sci. 57, 3251–3267). The results obtained are in reasonable agreement with the predictions of Narsimhan (2010. Analysis of creaming and formation of foam layer in aerated liquid. J. Colloid Interface Sci. 345, 566–572.). The discrepancies were mainly attributed to assumption of uniform size bubble in Narsimhan (2010).

Visualizing the dynamics of two-dimensional foams with FoamVis

We describe an interactive computer program FoamVis that provides techniques for visualization, exploration and analysis of time-dependent 2D foam simulation data generated by the Surface Evolver. The program takes a sequence of Surface Evolver files and animates them to show velocity vectors, local strain, the position of topological changes, and the forces on objects immersed in a foam flow, and enables the user to simultaneously compare different simulations. We use FoamVis to show the correlation between different attributes, such as bubble velocity and T1 density, in a foam flowing through a constriction, the relationship between orientation and foam-induced forces acting on an elliptical object falling through a foam under gravity, and the changes in bubble size that occur during diffusion-driven coarsening of a foam containing a mixture of gases.

The jamming perspective on wet foams

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can {\em jam} into a rigid, disordered state where they withstand finite shear stresses before yielding. The jamming transition has been studied extensively, in particular in computer simulations of frictionless, soft, purely repulsive spheres. Foams and emulsions are the closest realizations of this model, and in foams, the (un)jamming point corresponds to the wet limit, where the bubbles become spherical and just form contacts. Here we sketch the relevance of the jamming perspective for the geometry and flow of foams --- and also discuss the impact that foams studies may have on theoretical studies on jamming. We first briefly review insights into the crucial role of disorder in these systems, culminating in the breakdown of the affine assumption that underlies the rich mechanics near jamming. Second, we discuss how crucial theoretical predictions, such as the square root scaling of contact number with packing fraction, and the nontrivial role of disorder and fluctuations for flow have been observed in experiments on 2D foams. Third, we discuss a scaling model for the rheology of disordered media that appears to capture the key features of the flow of foams, emulsions and soft colloidal suspensions. Finally, we discuss how best to confront predictions of this model with experimental data.

Experimental studies and numerical model validation of overflowing 2D foam to test flotation cell crowder designs

A computational fluid dynamics model of froth motion has been developed to assess different flotation cell designs. This work presents an implementation of the model in a 2D case, to compare the simulated bubble velocity distribution and streamlines to an experimental foaming system. The model uses finite elements to solve Laplace's equation for a potential function from which the foam velocity can be obtained. It requires the air recovery, or the amount of air that overflows a flotation cell as unburst bubbles, as an input parameter to calculate the foam velocity distribution and bubble streamlines. The air recovery was obtained by image analysis from a vertical, overflowing monolayer of foam (2D) created in a Hele-Shaw column, which mimicked important flowing properties of flotation froths such as coalescence. Inserts were included in the foam column to represent potential crowder designs for industrial flotation cells. Three different designs were chosen to compare the effect of insert depth and shape, including rectangles and a triangle. The effect of the insert design on the overflowing foam is obvious from visual assessment of the bubble streamlines and velocity distribution, which were closely agreed by both the experiment and model.

FoamVis: Visualization of 2D Foam Simulation Data

Research in the field of complex fluids such as polymer solutions, particulate suspensions and foams studies how the flow of fluids with different material parameters changes as a result of various constraints. Surface Evolver, the standard solver software used to generate foam simulations, provides large, complex, time-dependent data sets with hundreds or thousands of individual bubbles and thousands of time steps. However this software has limited visualization capabilities, and no foam specific visualization software exists. We describe the foam research application area where, we believe, visualization has an important role to play. We present a novel application that provides various techniques for visualization, exploration and analysis of time-dependent 2D foam simulation data. We show new features in foam simulation data and new insights into foam behavior discovered using our application.

Nonlocal effects in the continuum theory of shear localisation in 2d foams

Goyon et al. [J. Goyon, A. Colin, G. Ovarlez, A. Ajdari and L. Bocquet, Nature 454 (2008) p. 84] have shown that nonlocal effects in the rheology of foams may be accounted for by a modification of the standard (Herschel–Bulkley) model. Here we consider the effects of this modification on the continuum theory of 2d shear localisation. We compute results for various examples, showing that the localisation length is increased, and explore the limiting cases of zero and infinite nonlocality length ξ. Velocity profiles are shown to take an exponential form in the case where ξ is large. As the formulation of the nonlocal continuum model presented in this article is general, it may also be directly applicable to other complex fluids.

Drainage and stability of 2D foams: Foam behaviour in vertical Hele-Shaw cells

An integrated numerical model was developed to predict the spatial and temporal evolution of the bubble size distribution within foams. In order to validate the modelling, experiments in a vertical pseudo-2D Hele-Shaw cell were carried out since the bubble size distribution in this system can be readily measured. The model combines a population balance sub-model for predicting the bubble size with a liquid drainage sub-model. The liquid drainage model is a version of the standard foam drainage equation that has been modified to account for a pseudo-2D geometry. It has also been modified to account for the confined shape of the bubble and the size of the gap between the vessel walls, as well as the different Plateau border shapes and orientations found in this system. The population balance model is used to predict the change of bubble size as films fail and bubbles coalesce. The population balance and liquid drainage models are fully coupled, with the bubble size distribution influencing the drainage and the capillary pressure exerted by the Plateau borders influencing coalescence. In order to validate this integrated numerical model, the size of bubbles in a rising pseudo-2D foam was measured with image analysis. The numerical model reasonably predicted the evolution of the mean bubble size and was in strong agreement with the bubble size distribution over the depth of the pseudo-2D foam. The predictions are very close to the measured experimental data.

The mechanics of liquid foams: History and new developments

The story of some recent developments in foam rheology is related at key points to the remarkable early contributions of Henry Princen, who pioneered the rigorous modern development of the subject. His general approach of introducing tractable geometrical models for foam structure and linking them to physicochemical properties at a smaller scale is still valid. It has however been amended to include the effects of disorder which are essential to the understanding of some properties, such as the variation of elastic modulus and yield stress with liquid fraction. Recent interest in foam rheology concerned the effect of shear localization in 2d foams, as was definitively demonstrated in 2001. This remains a challenge to theory, although continuum models have proved to be tractable and transparent, and have enjoyed some success.

Numerical simulation of heat transfer and fluid flow of molten metal in MMA–St copolymer lost foam casting process

In this study, a computational fluid dynamics (CFD) code was developed to calculate the filling pattern using volume of fluid (VOF) algorithm with donor–acceptor method for free surface simulation. This algorithm has been modified to include the pressure of the gas produced from foam degradation. For this purpose a heat transfer model and 2D foam degradation model were developed. In heat transfer model, radiation and conduction between foam and molten metal; and convection between gas and molten metal were considered. In order to evaluate the results of simulation, a bench scale casting apparatus was assembled and the casting was conducted in a transparent mold. The effect of several parameters such as coating thickness, foam density and vacuum level on the gap temperature, gap pressure and filling speed was studied with the developed software. It was found that the simulated results are in good agreement with experimental results.