Foam Engineering: Fundamentals and Applications

By Paul Stevenson

Containing contributions from leading academic and industrial researchers, this book provides a much needed update of foam science research.  

The first section of the book presents an accessible summary of the theory and fundamentals of foams. This includes chapters on morphology, drainage, Ostwald ripening, coalescence, rheology, and pneumatic foams.

The second section demonstrates how this theory is used in a wide range of industrial applications, including foam fractionation, froth flotation and foam mitigation. It includes chapters on suprafroths, flotation of oil sands, foams in enhancing petroleum recovery, Gas-liquid Mass Transfer in foam, foams in glass manufacturing, fire-fighting foam technology and consumer product foams.

Key features:

• Foam fractionation is an exciting and emerging technology, starting to gain significant attention
• Discusses a vital topic for many industries, especially mineral processing, petroleum engineering, bioengineering, consumer products  and food sector
• Links foam science theory to industrial applications, making it accessible to an engineering science audience
• Summarizes the latest developments in this rapidly progressing area of research
• Contains contributions from leading international researchers from academia and industry

• Language: English
• Publisher: Wiley
• Release date: Jan 3, 2012
• ISBN: 9781119961093

Book preview

1

Introduction

Paul Stevenson

1.1 Gas–Liquid Foam in Products and Processes

A gas–liquid foam, such as those found on the top of one’s bath or one’s beer, is a multiphase mixture that generally exhibits several physical properties that make it amenable to be used in multifarious industrial applications:

1. High specific surface area. The amount of gas–liquid surface area per unit volume of material that is attainable in a foam is greater than that in comparable two-phase systems. This property makes gas–liquid foam particularly attractive for interphase mass transfer operations. Examples of such processes are froth flotation, in which valuable hydrophobic particles are recovered from a slurry, the recovery of oil sands, and the stripping of gases from effluent by absorption into the liquid phase.

2. Low interphase slip velocity. The slip velocity between gas and liquid phases is the absolute velocity of the liquid phase relative to the gas phase, and this is typically much smaller in a foam than in a bubbly gas–liquid mixture. This is because the large specific surface area is able to impart a relatively large amount of shear stress on the liquid phase, thereby limiting the relative slip velocity between phases. A high contact time between gas and liquid phases can be engendered, which can also enhance the amount of mass transfer from liquid to gas, gas to liquid, or liquid to interface.

3. Large expansion ratio. Because the volumetric liquid fraction of a foam can be very low, the expansion ratio (i.e. the quotient of total volume and the volume of liquid used to create that foam) can be very high. This property is harnessed in the use of the material for fighting fires and to displace hydrocarbons from reservoirs.

4. A finite yield stress. Because gas–liquid foams can support a finite shear stress before exhibiting strain, they are very effective for use in delivering active agents contained in liquids in household and personal care products (such as bathroom cleaner and shaving foam), as well as in topical pharmaceutical treatments.

Thus, the geometrical, hydrodynamical and rheological properties of gas–liquid foam can be harnessed to make it a uniquely versatile multiphase mixture for a variety of process applications and product designs. It is therefore a material that is of broad interest to chemical engineers.

However, these physical properties of gas–liquid foam are determined by the underlying physics of the material. The rheology of foam is dependent upon, inter alia, the liquid fraction in the foam, which is in turn dependent of the rate of liquid drainage. This is a function of the rate at which bubbles coalesce and how the bubble size distribution evolves because of inter-bubble gas diffusion. The performance of a froth flotation column is dependent upon the stability of the foam, but the very attachment of particles to interfaces can have a profound influence upon this stability. In fact, the underlying physical processes that dictate the performance of a foam in a process or product application are generally highly interdependent.

It is precisely because of this interdependency, and how the interdependent fundamental physical processes impact upon the applications of foam, that it is hoped that this volume will have utility, for it seems axiomatic that those motivated by applications of foam would need to know about the underlying physics, and vice versa.

1.2 Content of This Volume

This volume is split into two major sections, within which the chapters broadly:

1. Give a treatment of one or another aspect of the fundamental physical nature or behaviour of gas–liquid foam

2. Consider a process or product application of foam

The first part provides a chapter in which the topology of gas–liquid foam is described followed by expositions of how this can change through liquid drainage, inter-bubble gas diffusion and coalescence, although these processes are highly mutually interdependent. Further, there are chapters on the rheology of foam and how particles can enhance stability, since these topics are rooted in fundamental physics, but have an important impact upon applications of foam. There is a chapter on the hydrodynamics of pneumatic foam, which underpins the processes of froth flotation, foam fractionation and gas–liquid mass transfer, and one on the formation and stability of non-aqueous foams. Finally in the ‘Fundamentals’ section there is a chapter on ‘Suprafroth’, which is a novel class of magnetic froth in which coarsening is promoted by the application of a magnetic field and therefore is reversible.

In the second part, ‘Applications’, there are chapters on processes and products that exploit the properties of foam. Froth flotation, foam fractionation and foam gas absorption are unit operations for different types of separation processes that rely upon pneumatic gas–liquid foam for their operation, and each is treated in an individual chapter. In addition there is a dedicated chapter on the flotation of oil sands because the technical challenges of this process are dissimilar to those of phase froth flotation of minerals and coal and because the supply of hydrocarbon resources from this source is likely to become increasingly important over the next century. However, foams also find utility in the enhanced recovery from oil reservoirs and this is described in a chapter. Foams manifest in a variety of manufacturing processes, and there is a description of foam behaviour and control in the production of glass. One of the most common applications of foam is in firefighting, as is discussed in a dedicated chapter. There is an important chapter on the creation and application of foams in consumer products; such products are typically of high added-value and therefore this field is rich with opportunities for innovation and development. Finally, a chapter on blast-mitigation using foam is given.

1.3 A Personal View of Collaboration in Foam Research

I had been doing postdoctoral work in the UK into multiphase flow through subsea oil flowlines when, in 2002, I travelled to Newcastle, Australia, to commence research on froth flotation of coal. I confess to not knowing what flotation was, but when I was travelling to work by train on my first morning I saw a coal train pass that seemed to be at least one mile long, so I thought it must be a field worthy of engagement. I had never considered foams beyond those encountered in domestic life.

However, once in Australia, it soon became clear to me that there was nothing specific for me to do, so I was left to my own devices from the outset. I inherited a pneumatic foam column that lived in a dingy dark-room, and for six months I would go there each morning and watch foam rise up a column and collect the overflow in a bucket. When it got too hot, I went to the excellent and well-air-conditioned library to read about foam drainage. I especially remember reading articles on drainage of Denis Weaire’s (co-author of Chapter 2 herein) group from Trinity College Dublin, and the work that Stephan Koehler (author of Chapter 3) carried out at Harvard. Despite having had a relatively rigorous education in a good chemical engineering department, I felt totally out of my depth when trying to get to grips with this work. I’d come across vector notation as an undergraduate, but it still daunted me. One afternoon I read the words ‘self-similar ansatz’, and immediately retired for the day. During this time, I shared an office with Noel Lambert (joint author of Chapter 11), now Chief Process Engineer of CleanProTech, who would come into the office coated in coal dust and issue instructions down the telephone to organise the next day’s flotation plant trials. I found the mathematical approach of Denis and Stephan difficult to comprehend, but Noel’s world was completely alien to me. And yet we were all working on one or another aspect of foam.

I learnt enough from Noel to realise that flotation was an incredibly physically complicated process and that plant experience was of paramount importance when trying to improve and innovate. In this context, methods that claimed to be able to simulate the entire flotation process by numerical solutions of sets of equations based upon oversimplified physics seemed particularly contrived. Similarly, there was a plethora of dimensionally inconsistent data fits in the flotation literature that were by their very nature only relevant to the experiments from which they were developed, but upon which general predictive capability was claimed. It is not surprising that some physicists appear to view some work of engineers with caution.

However, it was a chemical engineer who, arguably, was the first researcher to make significant process in both the fundamental science of gas–liquid foam and the process applications. Among his many achievements, Robert Lemlich of the University of Cincinnati proposed what is often now known as the ‘channel-dominated foam drainage model’, and he used this to propose a preliminary mechanistic model for the process of foam fractionation. Thus, the desire for a better understanding of a process technology for the separation of surface-active molecules from aqueous solution was the driver for the development of what some regard as the ‘standard model’ of foam drainage. Robert Lemlich’s career was characterised by trying to describe and innovate process technologies that harnessed foam by building a better understanding of the underlying physics. Lemlich’s contributions, which are often not given the credit that they deserve, demonstrate the value of a combined approach of physical understanding and practical application. Lemlich, and his co-workers, were able to effect these developments within their own research group. Those of us who do not possess Lemlich’s skill and insight may not be able to make similar progress single-handedly, but can still benefit from cross-disciplinary collaboration to achieve similar goals.

As a chemical engineer working on the fundamentals of gas–liquid foam and its process applications, I have collaborated with physicists and have found that the biggest impediments to interdisciplinary research in foam are caused by semantic problems. For example, as a former student of chemical engineering, I learnt about Wallis’s models of one-dimensional two-phase flow, and I therefore frequently invoke the concept of a ‘superficial velocity’ (i.e. the volumetric flowrate of a particular phase divided by the cross-sectional area of the pipe or channel). However, I have discovered that this is not a term universally known by the scientific community, and its use by me has caused some consternation in the past. Equally, I am quite sure that I have inadvertently disregarded research studies because I have failed to understand the language and methods correctly. However, I have recently found that perseverance, an open mind and a willingness to ask and to answer what may superficially appear to be trivial questions can overcome some difficulties.

The contributors to this volume may be from differing disciplines of science and engineering, but all are leading experts in their fields and all are active in developing the science and technology of foam fundamentals and applications. It is very much hoped that, in bringing together this diverse cohort of authors into a single volume, genuine cross-disciplinary research will be stimulated that can effectively address problems in the fundamental nature of gas–liquid foam as well as innovate new processes that can harness its unique properties. In addition, it is anticipated that engineering practitioners who design products and processes that rely on gas–liquid foam will benefit from gaining an insight into the physics of the material.

Part I

Fundamentals

2

Foam Morphology

Denis Weaire, Steven T. Tobin, Aaron J. Meagher and Stefan Hutzler

2.1 Introduction

When bubbles congregate together to form a foam, they create fascinating structures that change and evolve as they age [1], are deformed [2], or lose liquid [3]. Foams are usually disordered mixtures of bubbles of many sizes, but they may also be monodisperse, in which case ordered structures may also be found. They may be relatively wet or dry, i.e. contain a greater or lesser amount of liquid.

While the familiar foams of industry and everyday life are three-dimensional, laboratory experiments create two-dimensional foams of various kinds, offering attractive possibilities of easy experiments, computer simulations and visualizations, and more elementary theory. One form of 2D foam consists of a thin sandwich of bubbles between two glass plates. Let us begin with the 3D case, recognizing its greater practical importance.

2.2 Basic Rules of Foam Morphology

2.2.1 Foams, Wet and Dry

Foams may be classified as dry or wet according to liquid content, which may be represented by liquid volume fraction φ. This ranges from much less than 1% to about 30%. Engineers call the gas fraction (i.e. 1 − φ) the foam quality. Foams used in firefighting are classified by their expansion ratio, which is defined by φ −1. At each extreme (the dry and wet limits) the bubbles come together to form a structure which resembles one of the classic idealized paradigms of nature’s morphology: the division of space into cells in the dry limit and the close-packing of spheres in the wet limit (see Fig. 2.1).

Fig. 2.1 Shown are examples of 3D dry and wet foams, as obtained from experiment (a and c) and computer simulations (b and d). Typical 3D foams are polydisperse, consisting of bubbles of many different sizes. (a) Reproduced with kind permission of M. Boran. (d) Reproduced with permission from Wiley-VCH Verlag GmbH & Co. KGaA. (b) and (d) are simulations carried out by A. Kraynik [4].

Bubble size is important in determining which picture is more relevant in equilibrium under gravity. If the average bubble diameter is less than the capillary length l 0, defined as

(2.1)    

where γ is the surface tension of the liquid, g is acceleration due to gravity and Δ ρ is the density difference of the gas and liquid, a thin layer of foam consisting of small bubbles will be wet (i.e. have a liquid fraction larger than about 20%). Larger bubbles in equilibrium under gravity form a dry foam.

Fig. 2.2 Plateau’s rules of equilibrium require tetrahedral junctions for dry foams. They are prevalent for small values of liquid fraction, but wet foams can contain junctions of more then four edges (or six cells) [7].

2.2.2 The Dry Limit

In the dry limit the soap films that constitute the interface between bubbles may be idealized as infinitesimally thin curved surfaces, which are generally not simply spherical. These surfaces constitute the faces of polyhedral cells. Many varieties of polyhedra are found in equilibrated dry foams, as enumerated, for example, in the classic observations of Matzke [5] (see Fig. 2.21). But they are subject to important geometrical and topological restrictions, first stated by Plateau [6],¹ foam morphologist par excellence. His rules, illustrated in Fig. 2.2, are as follows.

Faces (films) must meet three at a time. The angles at which they meet must everywhere be 120 degrees, so that three cells are joined symmetrically at a cell edge.

Edges must meet four at a time. The angles between edges are arccos (−1/3) ≈ 109.43 degrees, the Maraldi angle, where six cells meet symmetrically at every corner.

It may seem intuitively reasonable that such rules follow somehow from local equilibrium of surface tension forces at the points in question. In part this is indeed true, but it is not obvious upon naive consideration why conjunctions of more than six cells are not possible. Plateau observed only tetrahedral junctions in the soap film configurations that he created in wire frames; in due course a colleague, Lamarle [8], supplied a very longwinded mathematical proof. We still await something more expeditious. Taylor [9] has provided a more refined and rigorous modern proof, but it is even less transparent.

Returning to the surfaces that constitute the cell faces, there is a further rule, well known as the Laplace–Young law in the general context of fluid interfaces. It expresses the balance of forces on a small element of soap film in terms of a pressure difference Δ p,

(2.2)    

Fig. 2.3 A photograph of the surface of a foam. The curvatures of the films are made visible by the reflections of light on the surface.

Fig. 2.4 Simulations of foams are usually carried out with K. Brakke’s Surface Evolver [10]. This software approximates surfaces with a triangulated mesh or tessellation. This mesh can be refined (i.e. the number of triangles used can be increased) to improve the accuracy of the approximation. (a) to (c) show the same surfaces as the refinement of the tessellation is increased. Note how the curvature of the surfaces becomes much smoother.

Here γ is surface tension and r is the mean radius of curvature. It is related to the two principal radii of curvature, R 1 and R 2, by the expression

(2.3)    

In the general case R 1 differs from R 2; for the case of a sphere R 1 = R 2.

The surface is therefore free to have a complicated form, difficult to formulate mathematically; see Fig. 2.3. It is for this reason that almost all detailed descriptions of dry foam structures are numerical in character, consisting of some sort of tessellation, as shown in Fig. 2.4. In modern times they are usually carried out with the freely available Surface Evolver software of Ken Brakke [10].²

2.2.3 The Wet Limit

In the wet limit, the bubbles are spherical (see Fig. 2.1c, d). There are some restrictions on the possibilities for such a packing of hard spheres, familiar in the idealized models used in the field of granular materials. Each sphere must be in contact with at least three others (with the rare exception of ‘rattlers’, small spheres trapped in large cages). The average number of these contacts is six in disordered packings. The latter result, from the elementary theory of mechanical constraints that was originated by James Clerk Maxwell, is not to be considered exact, but is generally valid in practice (at least approximately).

2.2.4 Between the Two Limits

A real foam must lie somewhere between these two idealized limiting cases. Let us start from the dry end, first considering the addition of an amount of liquid that is large enough that we may still neglect the liquid content of the films, but nevertheless still close to the dry limit. The liquid occupies the interstitial space associated with the cell edges. These swell to form what are called Plateau borders.

For a small enough liquid fraction, Plateau’s rules should still apply in some approximate sense. They are progressively violated as the liquid fraction is increased, and our understanding of this intermediate regime is limited. Progressing towards the wet limit we reach a regime in which the cells are slightly deformed spheres, but these are not easy to describe, other than by simulation or rather over-idealized models. For example, the bubbles are sometimes represented by overlapping spheres [4] (or circles in 2D [11]).

2.3 Two-dimensional Foams

The merits of the much simpler 2D foam may now be obvious. Its structure may be modelled using only circular arcs, with curvatures consistent with local gas and liquid pressures. It was C.S. Smith [12] who did most to promote this system as an object of study, although many before him, including Lord Kelvin, had occasional recourse to it.

2.3.1 The Dry Limit in 2D

In the dry limit the 2D foam consists of polygonal cells, as in Fig. 2.5. Since the vertices can only be threefold (a Plateau condition), it follows easily that the average number of sides of a cell is exactly six (Euler’s theorem) [13].

Fig. 2.5 Examples of experimental and simulation images of 2D dry foam. Recently there has been renewed interest in experiments with various types of 2D foam, in particular with regard to their rheological properties [14–17].

Fig. 2.6 Examples of experimental and simulation images of a 2D wet foam. In contrast to the dry system shown in Fig. 2.5, the Plateau borders between bubbles can touch four or more bubbles. As with Fig. 2.5(b), the simulation was carried out with the PLAT [18] software, and includes periodic boundary conditions.

2.3.2 The Wet Limit in 2D

In the wet limit, the cells are touching circular disks, as shown in Fig. 2.6. Just as in the 3D case, we make contact with close-packed structures and hence with the theory of granular materials [19]. Bubbles, the epitome of soft particles, become effectively hard particles in this limit.

2.3.3 Between the Two Limits in 2D

As in 3D, it is not so obvious what happens in the intermediate regime, as Plateau’s requirement of threefold vertices is relaxed, so that stable vertices (in reality liquid-filled junctions) of higher order can appear, as shown in Figs 2.7 and 2.8.

Fig. 2.7 In a 2D foam the fraction of n-sided Plateau borders (left y-axis) varies with liquid fraction φ. In the dry case (i.e. φ = 0) all Plateau borders have three sides. As φ is increased, the fraction of Plateau borders with four sides begins to increase, and eventually five and more sided Plateau borders begin to appear. The dots represent the average number of sides of Plateau borders (right scale).

Fig. 2.8 Examples of 2D foams with varying liquid fraction φ. The average number of contact per cell is seen to vary smoothly from six (for dry foams) to four (for wet foams). These images result from early computer simulations, demonstrating the rigidity loss of the foam at φ ≈ 0.16 [20] (the structure loses mechanical stability as the bubbles come apart at this value of φ).

Fig. 2.9 Simulation of 2D hexagonal foam with liquid fraction increasing from left to right (simulations carried out with the PLAT software) [18, 21, 24]. The dry (leftmost) honeycomb is the structure that optimally partitions 2D space. Note that for the honeycomb, the average contact number remains six even as φ is varied, in contrast to the 2D foam shown in Fig. 2.8.

Fig. 2.10 Experimental packing of bubbles into the honeycomb configuration for the case of a dry foam, an intermediate foam and the wet case (the dry foam is confined between two glass plates, while the intermediate and wet cases are free-floating Bragg rafts). This progression is approximately equivalent to that shown for the simulations in Fig. 2.9. Note that in the wet (rightmost) case the bubbles appear separated due to an optical effect.

The following relation connects the average number z of sides of Plateau borders with the average number n of sides (i.e. films) of the cells.

(2.4)    

As seen in Figs 2.7 and 2.8, these quantities vary continuously over the full range of φ in the case of a typical disordered foam. Contrast this behaviour with that of the ordered honeycomb (see Figs 2.9 and 2.10) for which there is no such variation (z = 3, n = 6). For this reason, early models of the mechanical properties of foams (which were based on the honeycomb) were misleading.

For liquid fractions small enough that no such higher order vertices appear, a useful theorem is available. The Decoration Theorem [21] states that such a 2D foam has a skeleton that is a dry foam in equilibrium, whose vertices may be decorated with Plateau borders to recover exactly the original structure.

Fig. 2.11 Examples of 2D finite clusters for varying numbers of bubbles. Each cluster has minimal perimeter length (equivalent to surface area of a 3D foam). Such calculations were carried out for clusters with up to 200 bubbles [26]. The authors would like to thank S. Cox for providing the above figure.

2.4 Ordered Foams

2.4.1 Two Dimensions

2.4.1.1 The 2D Honeycomb Structure

The paragon of perfection of foam structure is surely the 2D hexagonal honeycomb (see Fig. 2.9, leftmost). It may be made by trapping monodisperse bubbles between two glass plates (see Fig. 2.10 for examples). It has been presumed for centuries that this structure minimizes line length (for given bubble size). The proof of this was a long time in coming [22, 23]; it is nevertheless elementary. Plateau borders may be added, the Decoration Theorem being entirely trivial in this case, up to the point where the bubbles form touching circles – the wet limit. See Fig. 2.9.

2.4.1.2 2D Dry Cluster

Finite 2D clusters display an interesting sequence of minimal structures (see Fig. 2.11), and have been studied both experimentally and in simulations in recent years [25, 26].

2.4.1.3 2D Confinement

Ordered 2D foams confined in narrow channels are of particular importance to what has been termed discrete microfluidics [27, 28]. Here, trains of bubbles are pushed through networks of channels, the design of which allows for a number of tightly controlled manipulations. Fig. 2.12 shows how neighbouring bubbles may be separated in a simple U-bend; other geometries allow for the controlled injection of bubbles into a moving train, or the separation of a double row of bubbles into two single rows. Dynamic simulation methods are on hand to help interpretation of such processes [27].

Fig. 2.12 Example of a 2D dry foam in a U-bend. The foam is being pushed though the tube. Note the (temporary) formation of an unstable fourfold vertex occurring in the bubbles in the U-bend. This leads to a topological T1 or neighbour-switching, which is discussed in Section 2.7.

Fig. 2.13 Experimental and simulated examples of a 3D crystalline dry foam. The simulation image is two cells of a bulk Kelvin foam (Lord Kelvin’s conjectured space-partitioning structure). The experimental structure contains this bulk Kelvin structure, but the bubbles in contact with the walls are deformed.

2.4.2 Three Dimensions

2.4.2.1 3D Dry Foam

What is the counterpart of the honeycomb in 3D; that is, how can we partition space into cells of equal volume and minimum area? This question was first asked by Lord Kelvin in 1887 [29]. His conjectured answer consisted of identical cells in a body-centred cubic arrangement, as shown in Fig. 2.13. After a hundred years of consideration of this proposition, Weaire and Phelan computed a structure of lower surface area [30, 31]. This structure is shown in Fig. 2.14.

This remains the undisputed, but rigorously unconfirmed, champion. It has proven practically impossible to create experimentally, while Kelvin’s simpler structure can be made in various ways.

Fig. 2.14 A simulation of the Weaire–Phelan structure. The structure consists of two different (but equal-volume) bubble types: an irregular dodecahedron with pentagonal faces, and a tetrakaidecahedron with two hexagonal faces and twelve pentagonal faces (All pentagonal faces are slightly curved). The structure achieves a surface area 0.3% less than the Kelvin structure. Although the Weaire–Phelan structure has not been mathematically proven to be optimal, no better structure has yet been found.

Fig. 2.15 Comparison between an experimental crystalline wet foam (left) and a simulation (right). The simulation involved the application of ray-tracing to the fcc arrangement of glass spheres [33].

2.4.2.2 3D Wet Foam

In the wet limit a 3D monodisperse foam should form a close packing of spherical bubbles, with no obvious discrimination between fcc and other possibilities. It was first observed by Bragg and Nye [32] that wet foams of small bubbles do in fact readily crystallize, perhaps too readily for our understanding. Experiments [33] suggest that the fcc structure predominates; see Fig. 2.15. Recent X-ray tomography experiments [34] have shown that ordering might be restricted to the bubble layers close to confining boundaries of the sample (Fig. 2.16), but further experiments are required to settle the issue.

Fig. 2.16 X-ray tomographic image of an ordered microfoam showing the ABC arrangement of bubbles associated with fcc crystallization. The image shows the foam as it has ordered between a flat surface (top) and a liquid interface (bottom). On increasing the number of layers in such a sample, it is found that fcc crystallization no longer extends through the bulk [34] (see Section 2.5).

Fig. 2.17 A comparison between experimental imagery and simulation for a dry structure, confined in a tube with square cross-section. The structure has six bubbles in its unit cell. The simulation image on the right is rotated 90 degrees about the vertical compared to the photograph on the right.

Again, the scenario is more complicated and largely unexplored between the two extremes of wet and dry. The ‘phase diagram’ of monodisperse foam is still unknown.

2.4.2.3 Ordered Columnar Foams

We have seen that 2D confinement induces ordering. The same is true of confinement in a narrow column or channel. The structural variations observed as the column width is changed are fascinating [35–38], and have taken on some practical importance in microfluidics and other contexts. Examples are shown in Figs 2.17, 2.18 and 2.19.

Fig. 2.18 Example of experimental and simulation images for an ordered wet foam confined in a cylinder. Bubbles of size 0.5 mm are seen to order in the same configuration as predicted by simulations of the packing of hard spheres in a similar cylindrical confinement [39].

Fig. 2.19 A progression from a wet to a dry foam (left to right), demonstrated for a simple ordered foam structure with only two bubbles in the periodic cell. The tube diameter is roughly 1 mm.

2.5 Disordered Foams

Only specially prepared laboratory foams are monodisperse, and hence perhaps ordered; see Figs 2.15–2.19. However, monodispersity does not guarantee order. X-ray tomography of the interior of large foam samples (20,000 bubbles) show that the bubbles are random-closed-packed, featuring the characteristic radial distribution function of the Bernal packing of hard spheres, first investigated by Bernal in his study of the structure of liquids, see Fig. 2.20 [40].

Fig. 2.20 Radial distribution function g(r) – a measure of local arrangements within a sample – calculated for the bulk of a monodisperse foam composed of 20,000 bubbles of diameter 800 μm ± 40 [34]. The distribution exhibits a split second peak at values of r/r0 = √2 and 2 (shown with dashed vertical lines), but g(r) quickly approaches one, corresponding to the absence of long-range order [40].

Generally, foams made by ordinary methods (e.g. shaking, sparging or gas evolution) consist of a wide range of bubbles sizes as shown in Fig. 2.1, and are inevitably disordered. Their morphology is necessarily a matter of statistics with a number of interesting regularities emerging [13].

2.6 Statistics of 3D Foams

The description of disordered foams is framed in terms of averages and distributions. They may firstly be characterized by the distribution p(V) or p(A) of cell sizes (in 3D, the cell volume V, in 2D the cell area A).

A second characteristic is the distribution p(n) of the number of faces belonging to each cell, or of sides (edges) in two dimensions. This can vary according to the preparation and treatment of the sample, even though the size distribution is unaltered. In 2D its mean (for an infinite sample) is exactly six for dry foam, by Euler’s theorem, and its second moment μ 2 is a traditional measure of (topological) disorder. One may also define a second moment for p(V) or p(A), which may be used as the measure of polydispersity. Often μ 2 is of order unity, but its value depends on how the foam was prepared and its subsequent history.

Fig. 2.21 Some exemplary polyhedral cells with 12, 13, and 16 faces, as identified by Matzke in 1946 in a disordered foam [5]. Matzke’s findings have recently been reproduced in a study by Kraynik et al. [43], which identified all 36 of his reported polyhedra in a monodisperse foam sample that was produced using computer simulations. Images courtesy of R. Gabbrielli, created with 3dt software.

There is no corresponding exact result for the mean number of faces Ν in the 3D case, although it is commonly found to lie between 13 and 14. An interesting hypothetical design for the ideal 3D cell has flat faces and obeys Plateau’s rules: it would comprise 13.39 faces [42] so it cannot, of course, be realized. Nevertheless this mathematical chimera has played a role in thinking about 3D foam cells, and the Kelvin problem in particular.

In an early experimental study of monodisperse disordered foam comprising 600 bulk bubbles [5] (see Fig. 2.21), Matzke obtained Ν ≈ 13.70, which is very close to the hypothetical value above. Matzke’s result was confirmed by computer simulations involving up to 1000 bubbles [43].

2.7 Structures in Transition: Instabilities and Topological Changes

The topological structure of a foam can be characterized precisely in terms of the construction of its cells in terms of discrete elements (faces, edges, etc.), and this will usually not be changed by a small perturbation. However, when it is varied (e.g. by an imposed strain), it may be brought to a configuration in which there is a violation of Plateau’s rules by the introduction of a forbidden vertex. This dissociates rapidly and a new structure is formed.

In 2D, the possibilities are rather simple: the so-called T1 process eliminates a fourfold vertex and forms two threefold ones, as shown in Fig. 2.22.

In 3D, the most elementary possibility involves the disappearance of a triangular face or the inverse process in which a line (Plateau border) is reduced to zero length, see Fig. 2.23. But in reality, the disappearance of one triangle generally causes a neighbouring triangle to vanish too. Indeed, topological changes often come in cascades, particularly for wet foams [44, 45]. The details of their dynamics are still under investigation [46, 47].

For both dry and wet foams the processes of phase change in ordered foams (for example, from bcc to fcc) are largely unexplored. This may be of little direct importance, but the close analogy with some metallurgical phase transformations should add interest.

Fig. 2.22 Rearrangement in 2D foam. If conditions are varied in such a way that the length of one of the cell sides goes to zero resulting in a fourfold vertex, we necessarily encounter an instability and the system jumps to a different configuration that is in accord with Plateau’s rules. This is called the ‘T1 process’ or neighbour-swapping event.

Fig. 2.23 A T1 event in 3D involves the shrinkage and disappearance of a triangular face, followed by the formation of a Plateau border (or the reverse process).

2.8 Other Types of Foams

2.8.1 Emulsions

While microemulsions may bring into play additional forces, emulsions which have droplets with diameter on the order of 100 μm or more are closely analogous to foams (see the example shown in Fig. 2.24). All of the above applies, except that close matching of the densities of the two constituent liquids is possible. It follows that the emulsion is likely to be ‘wet’ if there is an excess of the continuous phase lying below or above it. That is, the droplets will be nearly spherical. By ensuring that there is less of the continuous phase, a dry emulsion of polyhedral cells may be prepared.

2.8.2 Biological Cells

Ever since the microscope was first applied to biological tissue, its foam-like cellular nature has been generally evident. As part of his eloquent case for the introduction of mathematics into biological morphology and morphogenesis, D’Arcy Wentworth Thompson envisaged a theory that was based on surface tension [50, 51]. Succeeding generations of biologists were at first intrigued by this notion, and sought to find Kelvin cells in particular. They are prevalent only in epithelial layers, so scepticism prevailed [5, 51].

Fig. 2.24 An example of a monodisperse ordered emulsion of silicone oil in water [48]. Note that this structure is the same as that displayed in Fig. 2.19 for monodisperse foam.

Today Thompson’s vision is enjoying a revival, as attempts are made to frame models of cell arrangements that are based largely on surface tension [52].

Is this a real (i.e. physical) surface tension, or do more biological principles somehow mimic its effects? Remarkably, evidence is at last appearing for the role of the physical force. Of course there will be complications, but it seems that the insights gained from foam physics will extend into biological science and medicine at this level.

2.8.3 Solid Foams

Solid foams generally have a solidifying liquid foam as a precursor. The solidification may occur due to a change in temperature, as is the case for metal foams, which may be formed by foaming a liquid melt, followed by rapid cooling [53, 54].

An interesting recent development is the formation of threads of hydrogel polymer with a crystalline cellular structure [55]. In this case, air and two different chemical solutions (one containing monomers, cross-linker, an accelerator, a surfactant, and water; the other only the initiator, a surfactant, and water) are brought together in a flow-focusing device. As soon as the ordered liquid foam emerges, it begins to solidify due to chemical reactions between its liquid components (see Fig. 2.25 for examples).

Fig. 2.25 Examples of ordered polymerized foam threads, in both swollen and dried states. The different ordered structures are created by varying the cross-section of the tube used to confine the initially liquid foam. Drying and swelling of the structures is completely reversible [55].

2.9 Conclusions

We have seen that the morphology of foams presents a variety of geometrical patterns in 2D and 3D, which we can today simulate and analyse in great detail (in particular, by using the Surface Evolver).

We have not pursued the observational side of the subject. In the case of 3D foams it presents some challenges; the multiple light scattering that gives foam its white appearance obscures its interior from view.

Nevertheless, Matzke [5] was able to use stereoscopic microscopy to record the details of thousands of large 3D bubbles, half a century ago, so perhaps we have made too many excuses in that regard, at least for dry foams. But today new techniques promise much more powerful and efficient probes of morphology, including X-ray tomography and MRI. Increasingly they can even observe changing structures, opening up the role of morphology in dynamics.

That is where the cutting edge of the subject is to be found at present. The truly complex processes that underlie common phenomena of foam physics need to be understood at the local level. Modern probes and modern simulations may soon make this feasible.

Acknowledgements

This publication has emanated from research conducted with the financial support of Science Foundation Ireland (08/RFP/MTR1083). Research supported by the European Space Agency (MAP AO-99-108:C14914/02/NL/SH and AO-99-075:C14308/00/NL/SH), the Irish Research Council for Science, Engineering & Technology (IRCSET), and the SFI SURE Summer Undergraduate Research Experience programme. D. Weaire would like to thank the University of Hyderabad for graciously hosting him while this work was being completed.

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¹ Both the original text and an English translation of the work may be found at http://www.susqu.edu/brakke/PlateauBook/PlateauBook.html

² http://www.susqu.edu/brakke/

3

Foam Drainage

Stephan A. Koehler

3.1 Introduction

The term foam drainage originally described the process by which fluid flows out of a foam, such as liquid draining out of a soap froth [1, 2], or the draining head on freshly poured beer (see Fig. 3.1a). Since then many technological applications have been developed for foams, which include cleansing, water purification, and minerals extraction as well as production of cushions, food stuffs, and ultra-lightweight structural materials [3–7]. Consequently they have received much attention by the scientific community, and foam drainage has taken on a broader meaning of just fluid flow between compressed bubbles. Fig. 3.1(b) shows a close-up of a soap foam in a forced-drainage experiment, where the fluid is fluorescent and the drainage front traveling between the bubbles is clearly visible. As instrumentation and techniques have advanced, more detailed microscopic studies of fluid flow in foams have been performed on the level of individual bubbles and smaller, which are also considered to be foam drainage. For example, Fig. 3.1(c) shows a 3D image of the continuous phase of a liquid – liquid foam (oil in water emulsion) obtained by confocal microscopy. Only the channel-like network is seen because the films separating bubbles (oil droplets) are too thin to be resolved by the microscope. Using confocal microscopy with greater magnification and seeding the flow with micron-sized particles makes it possible to determine the flow fields. But as with most materials studies, an increased level of understanding leads to an increased number of unanswered questions. Thus at first glance foam drainage may appear to be a relatively straightforward fluids problem dealing with the flow between bubbles, but instead is a multifaceted process with length scales ranging from nanometers for surfactant molecules, to micrometers for films, to millimeters for bubbles, to centimeters for bulk foams.

Fig. 3.1 (a) The draining head on a freshly poured beer. (b) A forced drainage experiment, where continuous perfusion of a dry foam column from above results in a downwards-flowing drainage front, velocity V f , followed by a uniformly wet region. (c) 3D confocal image of an emulsion with average bubble size 200 μm and continuous fraction ε ≈ 0.005.

Foams are metastable dispersions of gas in liquid that are evolving in time, which complicates precise measurements and obfuscates experimental trends. Despite their having a very simple composition (merely gas bubbles, highly concentrated in a fluid with small amounts of surfactants), achieving the current level of understanding has taken over a century of considerable effort by the scientific community, which includes chemical engineers, food scientists, surfactant chemists, and soft matter physicists. To date there are only semi-empirical models. Several key questions remain unanswered, such as the role of surfactants on interfacial stresses, which were posed by chemical engineers as early as the 1940s [8].

There are two main dynamic processes, which are (i) the redistribution of liquid, also known as foam drainage or syneresis [8, 9], and (ii) the redistribution of gas between bubbles, also known as coarsening or Ostwald ripening [10, 11]. Unless appropriate precautions are taken, it is often difficult to disentangle these dynamic processes and perform systematic studies [12, 13]. For example, the shrinking head on a beer freshly poured into a glass is a complicated process that simultaneously involves the drainage of fluid as well as bubble rupture and inter-bubble gas diffusion. For certain surfactants another undesirable process is chemical degradation, such as the conversion of SDS (sodium dodecyl sulfate) to DOH (dodecanol) by hydrolysis, which can significantly impact foam properties [14]. A more tractable drainage scenario is the forced drainage experiment where a continuous perfusion from above creates a uniformly wet region that drains downwards at constant velocity (see Fig. 3.1b) [15]. Foam stability is optimized by using a surfactant with minimal chemical degradation and film rupture, as well as gas with a low diffusion rate.

Models for foam drainage deal with the flow and liquid distribution from a mean-field perspective, where the resolving length scale is over many bubbles. These theories are quite similar in spirit to fluid flow through porous solids, such as sands, soils, and packed beds, which are well described by Darcy’s law

(3.1)    

where K is the porous medium’s permeability (in units of area), and Q is the fluid’s volumetric flux (in units of length/time), ρ the (volumetric) density, μ the viscosity, p the pressure and g gravitational acceleration. The key property is the medium’s permeability, which depends on the pore structure and generally is determined experimentally. The main difference between foams and porous media is that for foams the pores are elastic and vary with flow rate. This is illustrated by the forced drainage experiment, where a constant fluid flux is poured onto the foam, which results in the drainage wave shown in Fig. 3.1(b). The foam can accommodate a large range of perfusion fluxes because the pores self-adjust such that the permeability is proportional to the flux K = (μQ) = (ρg). In contrast, performing a similar experiment by replacing the foam with a plug of porous material would result in accumulation of a puddle above if the flux exceeds q > Kρg = μ. Another important difference between porous materials and foams is the boundary condition of the flow: for solids the no-slip boundary condition holds at the pores’ walls, whereas for foams the walls are surfactant-laden liquid – gas interfaces, which generally are also flowing. Therefore foam drainage theories must take into account how the structure is modified by flow as well as flow of the liquid – gas interfaces, which contributes to a rich phenomenology of behaviors.

3.2 Geometric Considerations

It turns out that dry foams with liquid fraction ε ≤ 0.02 have distinctive geometric features that greatly facilitate understanding and analysis. In particular, the continuous network is dominated by straight, slender channels that are easy to understand (see Fig. 3.1c). However, as the liquid fraction increases the channels swell and are difficult to distinguish from other geometric elements. But for the sake of tractability the analysis and approximations made for dry foams in many situations can be extended to monodisperse wet foams without sacrificing too much rigor.

Although real foams often have polydisperse disordered bubbles, it is conceptually useful to simplify the structure to an idealized monodisperse foam. A convenient idealization for dry foams is a bcc packing of regular octahedra whose six corners have been truncated to create square faces. This results in a total number of fourteen faces. Moreover, every edge has the same length, L, and hence this polygon is a special type of tetrakaidecahedron that tessellates space. A commonly used name for this shape is the Kelvin bubble in honor of Lord Kelvin, who considered minimal surface area tilings of 3D space [16]. Figure 3.2(a) shows a Kelvin bubble decorated by the neighboring channel-like network. This can be classified into three geometric components that are: (i) films, which separate two compressed bubbles (Fig. 3.2b); (ii) channels, which are regions between three compressed bubbles and the intersections of three films (Fig. 3.2c); and (iii) nodes, which are regions between four compressed bubbles and the intersections of four channels (Fig. 3.2d). Alternative designations for films are lamellae (taken from the biological term for thin layers or membranes), channels are Plateau borders (named after the famous nineteenth-century Belgian physicist), and nodes are junctions (because they serve as flow junctions between channels). The bubble from Fig. 3.2(a) is decorated by channels that are shared with two adjacent bubbles and nodes that are shared with three adjacent bubbles. The tetrakaidecahedron contains a rounded bubble that is surrounded by thirds of channels and quarters of nodes. Fig. 3.3(a) is an illustration of a portion of the channel-like network that consists of a channel and the two adjoining nodes. Note that the separation between nodes also is the edge length.

Fig. 3.2 Foam geometry: (a) Kelvin bubble, ε = 0.01, where the channels and nodes from other bubbles are included (courtesy of Andrew Kraynik); (b) film of thickness w; (c) idealized channel with transverse radius of curvature r c = r; (d) node. At the surface’s midpoint the principle radii of curvature are equal and r n = 2r.

Fig. 3.3 (a) A portion of the channel-like network, with node-to-node separation L. (b) A rhombic dodecahedral bubble decorated by interstitial fluid at ε = 0.05. The arrow indicates an eight-way node (courtesy of Andrew Kraynik). (c) The network unit for flow, which consists of a channel and two quarter nodes.

As the liquid fraction increases the six square faces of the Kelvin bubble, which resulted from truncation of the octahedron, shrink and disappear at ε * ≈ 0.1 [17]. The four nodes of each of the square faces merge into a single node, which becomes the junction between eight channels. This foam is no longer stable to shearing, and rearranges to a fcc packing where the unit cell is a rhombic dodecahedron consisting of twelve faces – see Fig. 3.3(b). The arrow shows a node with an eightfold coordinated node, i.e., the junction of eight channels. (Plateau’s rule for fourfold nodes only applies to dry foams.) The number of nodes has dropped from 24 to only 14, of which six are eightfold coordinated and the remaining eight nodes are fourfold coordinated. With increasing liquid fraction the bubbles become more spherical and at ε fcc = 1 − π/√18 ≈ 0.26 the foam is a fcc hexagonal close pack of spheres. In most situations the foam is unconfined, and thus will expand as liquid is introduced. Under normal conditions the compressibility of gas is negligible because the overpressure caused by the foam’s weight is small compared with atmospheric pressure. Therefore as liquid is introduced the gas volume, V g, stays constant and the extra liquid causes the unit cell to grow. The volume of the unit cell is the sum of the liquid and gas, V t = V g + V l, and is proportional to the edge length L ³. Denoting the edge length of a completely dry foam L 0, and realizing the liquid volume is εV t, the increasing liquid fraction causes the edge length to grow L = L 0 = (1 − ε)¹/³. The volume of the tetrakaidecahedron is 2⁷/² L ³, and that of the dodecahedron is 2⁴3−3/2 L ³. Denoting D as the equivalent diameter of the gas bubble, V g = πD ³/6, the edge length dependence on liquid fraction is

(3.2)    

Fig. 3.4(a) shows how the edge length increases with liquid fraction, and is discontinuous at ε* where structural rearrangement occurs. The decrease in the number of edges is from 36 for bcc to 24 for fcc, which equals the increase in the length of the fcc edges compared with the bcc edges.

Fig. 3.4 The liquid fraction dependence of: (a) edge length, L/D; and the channel’s aspect ratios in terms of (b) r/L and (c) r/D.

The relationship between the interfacial curvature and the gas pressure of the bubbles, pb, and the liquid pressure, p, is given by Young – Laplace’s law. The interfacial curvature can be parameterized by the inverse radius of the (total) curvature, 1/r. Recall a surface has two principle radii of curvature which are in orthogonal planes and r −1 = r 1 −1 + r 2 −1. Young – Laplace’s law states

(3.3)    

where γ is the surface tension. (Note that the liquid pressure inside films is an exception to this rule because films are so thin that short-range disjoining forces between opposing interfaces come into play, which is discussed later.) Films are essentially flat, which means the local variations in the gas pressure between nearby bubbles are small. Consequently Young – Laplace’s law results in a