Foam Dynamics Research Articles

Dynamics and stability of metallic foams: Network modeling

We formulate a large-scale network model for the dynamics and stability of a planar (gas–liquid) foam with low liquid fraction composed of approximately polygonal gas bubbles. We restrict attention to clean liquids (free of surfactants and colloidal particles) as a model for molten metallic foams produced using batch-processing techniques for manufacturing porous-metal solids. The liquid is primarily confined within highly curved regions around the bubble vertices, Plateau borders, interconnected by thin liquid films that form the edges of the bubbles and drain due to capillary-viscous suction. The model incorporates direct coupling between the pressure and area of the bubbles, surface-tension forces on the gas–liquid interfaces and draining and elongational flows in the films. The model also explicitly accounts for van der Waals instabilities that grow to rupture a liquid film once it becomes sufficiently thin, leading to bubble coalescence and hence coarsening of the foam. We consider foams confined wit...

Pinch-off dynamics in foams, emulsions and suspensions

The break up of a non-Newtonian yield stress fluid bridge (acrylic paint, mayonnaise, hair gel, foam, and bentonite) was investigated. The minimum neck radius, hmin, was measured as a function of time and fit to a power law with exponent n1. A rotational rheometer was used to measure the shear stress-rate of strain curve which was fit to a Herschel–Bulkley model with exponent n2. For the pure, as purchased fluids, the exponent from the time dependence of hmin (n1) and the rheology power law index (n2) were quantitatively the same. These results provide the first experimental confirmation of this relationship predicted by Suryo and Basaran. In the pure fluids, the pinch-off did not produce a satellite drop. In contrast, when the non-Newtonian fluids were diluted with a Newtonian fluid, the relationship between n1 and n2 was more complicated even though the diluted fluids were still Herschel–Bulkley, and the pinch-off produced satellite drops whose size was a continuous function of dilution.

The effect of expansion conditions on the batch foaming dynamics of St–MMA copolymer

In this study, we synthesized styrene–methyl methacrylate copolymer particles by suspension polymerization process to study the foaming dynamics via visual batch foaming apparatus. The weight ratio of styrene–methyl methacrylate was 53–47. The foaming system consisted of a self-sealing observation cell equipped with two glass windows and a stereo microscope attached with a high-speed digital camera as well as a pressure and temperature controller. The required pentane pressure was supplied by heating a small pentane flask. The synthesized copolymer particles impregnated by pentane and then the foaming dynamics were recorded after a rapid pressure release. The effect of different foaming conditions, such as temperature, impregnation time, and impregnation pressure on the expansion ratio of styrene–methyl methacrylate copolymer was investigated. It was concluded that impregnation pressure, time, and temperature have different effects on the foaming ratio at impregnation pressures lower and higher than 4 bar.

Decadal evolution of a degassing magma reservoir unravelled from fire fountains produced at Etna volcano (Italy) between 1989 and 2001

Between 1989 and 2001, five eruptions at Etna displayed a regular alternation between repose periods and episodes rich in gas, termed quasi-fire fountains and consisting of a series of Strombolian explosions sometimes leading to a fire fountain. This behaviour results from the coalescence of a foam layer trapped at the top of the reservoir which was periodically rebuilt prior to each episode (Vergniolle and Jaupart, J Geophys Res 95:2793–2809, 1990). Visual observations of fire fountains are combined with the foam dynamics to estimate the five degassing parameters characteristic of the degassing reservoir, i.e. the number of bubbles, gas volume fraction, bubble diameter, reservoir thickness and reservoir volume. The study of decadal cycles of eruptive patterns (Allard et al., Earth Sci Rev 78:85–114, 2006) suggests that the first eruption with fire fountains occurred in 1995 while the last one happened in 2001. The number of bubbles and the gas volume fraction increase smoothly from the beginning of the cycle (1995) to its end (2001). The increasing number of bubbles per cubic metre, from 0.61–20×105 to 0.1–3.4×109, results from cooling of the magma within the reservoir. The simultaneously decreasing bubble diameter, from 0.67–0.43 to 0.30–0.19 mm, is related to the decreasing amount of dissolved volatiles. Meanwhile, the thickness and the volume of the degassing reservoir diminish, from values typical of the magma reservoir to values characteristic of a very thin bubbly layer, marking the quasi-exhaustion of volatiles. The magma reservoir has a slender vertical shape, with a maximum thickness of 3,300–8,200 m and a radius of 240 m (Vergniolle 2008), making its detection from seismic studies difficult. Its volume, at most 0.58–1.4 km3, is in agreement with geochemical studies (0.5 km3) (Le Cloarec and Pennisi, J Volcanol Geotherm Res 108:141–155, 2001). The time evolution of both the total gas volume expelled per eruption, and the inter-eruptive gas flux results from the competition between the increasing number of bubbles and the decreasing bubble diameter. The smooth temporal evolution of the five degassing parameters also points towards bubbles being produced by a self-induced mechanism within the magma reservoir rather than by a magmatic reinjection prior to each eruption. The decadal cycles are therefore initiated by a magmatic reinjection, in agreement with a typical return time of 14–80 years (Albarede 1993). Hence, the 1995 eruption results from a fresh magma being newly emplaced while the magma from the following eruptions is progressively depleted in volatiles species until reaching a state of quasi-exhaustion in 2001. A magmatic reinjection of 0.13–0.6 km3 every few decades is sufficient to explain the expelled gas volume, including SO2. A scenario is also proposed for the alternation between gas-rich summit eruptions and gas-poor flank eruptions which are observed during decadal cycles. The scenario proposed for Etna could also be at work at Piton de la Fournaise and Erta ’Ale volcanoes.

What Is the Mechanism of Soap Film Entrainment?

Classical Frankel's law describes the formation of soap films and their evolution upon pulling, a model situation of film dynamics in foams (formation, rheology, and destabilization). With the purpose of relating film pulling to foam dynamics, we have built a new setup able to give an instantaneous measurement of film thickness, thus allowing us to determine film thickness profile during pulling. We found that only the lower part of the film is of uniform thickness and follows Frankel's law, provided the entrainment velocity is small. We show that this is due to confinement effects: there is not enough surfactant in the bulk to fully cover the newly created surfaces which results in immobile film surfaces. At large velocities, surfaces become mobile and then Frankel's law breaks down, leading to a faster drainage and thus to a nonstationary thickness at the bottom of the film. These findings should help in understanding the large dispersion of previous experimental data reported during the last 40 years and clarifying the pulling phenomenon of thin liquid films.

A level set method to study foam processing: a validation study

SUMMARYWe have developed a production‐level foam processing computational model suitable for predicting the self‐expansion of foam in complex geometries. The model is based on a finite element representation of the equations of motion, with the movement of the free surface represented using the level set method. An empirically based time‐dependent and temperature‐dependent density model is used to encapsulate the complex physics of foam nucleation and growth in a numerically tractable manner. The evolving density drives the dynamics of foam self‐expansion. This continuum‐level model uses a homogenized description of foam, which does not include the gas explicitly, but allows varying local fields, such as temperature and gas volume fraction, and material models. In addition, material models vary with the location of the level set interface, taking properties of the displaced air phase in the negative level set region and the foam in the positive region. The level set zero describes the location of the interface, where surface forces are applied using the continuous surface force treatment. The variation from foam to gas properties is handled with a diffuse interface method using a smooth Heaviside function and equation averaging. Material model development was guided and populated by careful experiments. Results from the model are compared with temperature‐instrumented flow visualization experiments giving the location of the foam front as a function of time for a physically blown, epoxy foam. Good qualitative agreement is seen between simulations and experiments, although some of the subtleties of the filling process are lost to the model. Published 2011. This article is a US Government work and is in the public domain in the USA.

The Euclidean three-point function in loop and perturbative gravity

We compute the leading order of the three-point function in loop quantum gravity, using the vertex expansion of the Euclidean version of the new spin foam dynamics, in the region of γ < 1. We find results consistent with Regge calculus in the limit γ → 0, j → ∞. We also compute the tree-level three-point function of perturbative quantum general relativity in position space and discuss the possibility of directly comparing the two results.

Roles of transient and local equilibrium foam behavior in porous media: Traveling wave

In foam EOR, complex dynamics of bubble creation and destruction controls foam properties. Recently, there has been consensus that local equilibrium between bubble creation and destruction adequately describes foam displacements. We assume that local equilibrium applies throughout a foam displacement on the field scale, with the exception of an entrance region and at shock fronts, where saturations and texture (bubble size) change abruptly. We find a range of conditions in which the local-equilibrium condition applies even within the shock front. In a waterflood, the width of a shock transition zone is determined by capillary-pressure gradients. For foam, this equation is joined by one for evolving foam texture. One expects that slow foam dynamics widens the traveling wave at the shock considerably. Theory and simulations show that the width of and mobility inside a shock front can affect foam sweep. If there is no gas ahead of the foam, as is common in published simulations, we prove that foam texture is everywhere at local equilibrium within the shock, regardless of the foam model, as previously observed for one dynamic foam model. If there is gas initially in the formation, slow foam generation and coalescence processes can actually narrow the shock from that assuming local equilibrium. In other cases, the dynamics of the traveling wave leads to oscillations near the shock; these are not numerical artifacts, but reflections of the models. Multiple steady states seen in experiment for some injection rates can be predicted by certain foam models. The approach of solving for the traveling wave can rule out some of these states for certain displacements.

Multiscale Simulation of the mechanical properties and the structural dynamics of a copper foam

AbstractIn this contribution, a simulation chain for the prediction of the structural dynamics behavior of metal foam is proposed: Starting with investigations of the mesoscopic structure and topology, the linear elastic material properties and their scatter are computed with the help of mesoscopic non‐representative volume elements. Also their probability functions are estimated. In order to be able to describe the mechanical properties as random fields, the autocorrelation function and the power spectral density are determined.The next step in the proposed simulation chain is to generate realisations of the random fields with the help of the Karhunen‐Loeve expansion or the Spectral Representation. Afterwards, these realisations are used in Monte‐Carlo‐Simulations on the macroscale in order to predict the eigenfrequencies of beams and their scatter.With this multiscale approach, the mechanical properties of an open‐celled Copper foam are determined by simulations and the eigenfrequencies are compared to experimental results. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Numerical Analysis of Foam Motion in Porous Media Using a New Stochastic Bubble Population Model

We present a numerical analysis of the stochastic population balance (SPB) theory for foam motion in porous media. The theory condenses into a set of non-linear partial differential equations in the saturation, pressure, and bubble density. We solve the equations using the IMPES method and perform sensitivity and parametric analysis. Finally, we compare the saturation profiles obtained numerically with those obtained from CT scan foam experiments. The agreement between the theory and experiments confirms that the stochastic population balance model describes adequately foam dynamics in porous media.

Growth of Aqueous Foam on Flexible Membranes

I consider the coarsening dynamics of two-dimensional dry foam sandwiched by deformable membranes. The time-varying deformation of the confining membranes gives rise to a significant alteration in the evolution of polygonal cells of bubbles when compared to the case of rigid membranes. This alteration is attributed to the correlation between the rate of inter-cell gas transfer and temporal fluctuation in surface curvature within a cell domain. Existing material constants are referred to show that the correlation can yield sizeable consequences in actual coarsening experiments.

Emergence of foams from the breakdown of the phase field crystal model

The phase field crystal (PFC) model captures the elastic and topological properties of crystals with a single scalar field at small undercooling. At large undercooling, new foamlike behavior emerges. We characterize this foam phase of the PFC equation and propose a modified PFC equation that may be used for the simulation of foam dynamics. This minimal model reproduces von Neumann's rule for two-dimensional dry foams and Lifshitz-Slyozov coarsening for wet foams. We also measure the coordination number distribution and find that its second moment is larger than previously reported experimental and theoretical studies of soap froths, a finding that we attribute to the wetness of the foam increasing with time.

Numerical simulations of two-dimensional foam by the immersed boundary method

In this paper, we present an immersed boundary (IB) method to simulate a dry foam, i.e., a foam in which most of the volume is attributed to its gas phase. Dry foam dynamics involves the interaction between a gas and a collection of thin liquid-film internal boundaries that partition the gas into discrete cells or bubbles. The liquid-film boundaries are flexible, contract under the influence of surface tension, and are permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. Such problems are conventionally studied by assuming that the pressure is uniform within each bubble. Here, we introduce instead an IB method that takes into account the non-equilibrium fluid mechanics of the gas. To model gas diffusion across the internal liquid-film boundaries, we allow normal slip between the boundary and the gas at a velocity proportional to the (normal) force generated by the boundary surface tension. We implement this method in the two-dimensional case, and test it by verifying the von Neumann relation, which governs the coarsening of a two-dimensional dry foam. The method is further validated by a convergence study, which confirms its first-order accuracy.

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.

Mechanical response and fracture dynamics of polymeric foams

Heterogeneous materials usually break through a process of microcracking that eventually leads to final rupture after accumulation and coalescence of many microcracks. The statistical properties of microcracking rupture have been known to resemble critical point statistics, with many of the physical quantities obeying power law distributions. However, there is no clear understanding of the origin of these distributions and of the specific values observed for the power law exponents. In this paper, we review the special case of polymeric foams that have the advantage of containing a single material component, the polymer, as opposed to usual heterogeneous materials such as composites. First, we briefly review the typical features of the polymeric foam mechanical response up to rupture that have been widely studied previously. Then, we focus on a less well-known aspect: the rupture dynamics of polymeric foams. We not only show that polymeric foams behave like other heterogeneous materials, i.e. they display power law statistics, but we are also able to test the effect on the power laws of the following properties: the foam heterogeneity by changing its density, the foam mechanical response by changing its temperature and the mechanical history by comparing creep tests and tensile tests.

The viscous froth model: steady states and the high-velocity limit

The steady-state solutions of the viscous froth model for foam dynamics are analysed and shown to be of finite extent or to asymptote to straight lines. In the high-velocity limit, the solutions consist of straight lines with isolated points of infinite curvature. This analysis is helpful in the interpretation of observations of anomalous features of mobile two-dimensional foams in channels. Further physical effects need to be adduced in order to fully account for these.

Surfactant Mixtures for Control of Bubble Surface Mobility in Foam Studies

A new class of surfactant mixtures is described, which is particularly suitable for studies related to foam dynamics, such as studies of foam rheology, liquid drainage from foams and foam films, and bubble coarsening and rearrangement. These mixtures contain an anionic surfactant, a zwitterionic surfactant, and fatty acids (e.g., myristic or lauric) of low concentration. Solutions of these surfactant mixtures exhibit Newtonian behavior, and their viscosity could be varied by using glycerol. Most importantly, the dynamic surface properties of these solutions, such as their surface dilatational modulus, strongly depend on the presence and on the chain-length of fatty acid(s). Illustrative results are shown to demonstrate the dependence of solution properties on the composition of the surfactant mixture, and the resulting effects on foam rheological properties, foam film drainage, and bubble Ostwald ripening. The observed high surface modulus in the presence of fatty acids is explained with the formation of a surface condensed phase of fatty acid molecules in the surfactant adsorption layer.

LIQUID FOAM DRAINAGE: AN OVERVIEW

Liquid foams are concentrated dispersions of gas bubbles in a small amount of surfactant solution, which are perpetually out of equilibrium systems. The process of liquid draining through networks of Plateau borders in a fresh foam is so-called foam drainage, as a result of both gravitational and capillary forces, which has great effect on the stability of foams. From the view of foam physics and dynamics, this paper briefly introduces foam structure and major lifetime limiting factors of foam. The substantial progress on the theory of drainage, measuring techniques for liquid fractions, drainage in both one dimension and two dimensions, and drainage in microgravity circumstances are overviewed throughout. Remaining tasks are discussed and a multiscale methodology for foam drainage is proposed for future investigations.

From sound waves to bubbling within a magma reservoir: comparison between eruptions at Etna (2001, Italy) and Kilauea (Hawaii)

Abstract The 2001 eruption of Etna, prior to the flank eruption, has shown an alternation between episodes rich in gas, composed with a series of Strombolian explosions sometimes leading to a fire fountain, and repose periods. The regular alternation results from the coalescence of a foam trapped in the reservoir and periodically rebuilt prior to each episode. The degassing of a magma reservoir depends on bubble diameter, gas volume fraction, surface area and height of the reservoir. These four parameters are deduced from the measured gas flux, the timescale over which the gas flux decreases and the foam dynamics. The dimensionless foam thickness, 0.76 for a purely Strombolian episode, increases to 0.89 for an episode leading to a fire fountain, indicating a more efficient foam coalescence. At Etna, the bubble diameter, gas volume fraction, surface area and height of the reservoir are estimated at 0.50–0.59 mm, 0.25–0.39%, &gt;0.20 km 2 and 97–220 m, respectively. At Kilauea, the excellent agreement between the prediction of the foam model, 0.94–1.2 km 2 and that resulting from deformations, 1 km 2 , reinforces the validity of the foam model qualitatively and quantitatively. The thickness of the degassing reservoir, 16–12 m in 1959, is now 1.3–1.1 km.

A simple continuum model for the dynamics of a quasi-two dimensional foam

Nominally two-dimensional foams, created by confining a single layer of bubbles between two plates, or one plate and a liquid surface, or floating on the liquid, have provided convenient model systems for foam research. We show that, in the case of rheology, viscous drag forces due to the confining plates have large effects. They can give rise to shear bands, similar to those observed by Debrégeas et al. [G. Debrégeas, H. Tabuteau, J.-M. di Meglio, Deformation and flow of a two-dimensional foam under continuous shear, Phys. Rev. Lett. 87, (2001) 17–8305]. A continuum equation that incorporates elastic, plastic and viscous drag terms is used to analyse the rheological behaviour. We also discuss the meaning of the term quasistatic in this context.