Foam Engineering: Fundamentals and Applications docx

13.5 Occurrences of Foams at the Surface and Downstream 29814.2.1 Adsorption Kinetics at Quiescent Interface 311 14.5 Foam Fractionation Devices and Process Intensification 317 14.6 Conc

Foam Engineering

Foam Engineering

Fundamentals and Applications

Edited by Paul Stevenson

Department of Chemical and Materials Engineering, Faculty of Engineering, University of Auckland, New Zealand

A John Wiley & Sons, Ltd., Publication

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Library of Congress Cataloging-in-Publication Data

Foam engineering : fundamentals and applications / [edited by] Paul Stevenson – 1st ed.

1.3 A Personal View of Collaboration in Foam Research 3

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

2.7 Structures in Transition: Instabilities and Topological Changes 21

3.7 Rigid Interfaces and Neglecting Nodes: The Original

3.10 The Network Model: Combining Nodes and Channels 48

3.12 Interpreting Forced Drainage Experiments: A Detailed Look 51

4.3.1 Inter-bubble Gas Diffusion through Thin Films 61

5.2.1 Experimental Studies Dealing with Isolated Thin Liquid Films 765.2.2 Theoretical Description of the Rupture of an Isolated

5.4 What Are the Key Parameters in the Coalescence Process? 81

Nikolai D Denkov, Slavka S Tcholakova, Reinhard Höhler

and Sylvie Cohen-Addad

6.6.1 Predominant Viscous Friction in the Foam Films 1086.6.2 Predominant Viscous Friction in the Surfactant

G Kaptay and N Babcsán

7.3 On the Thermodynamic Stability of Particle Stabilized Foams 125

7.4 On the Ability of Particles to Stabilize Foams during

8.2.3 The ‘Vertical Foam Misapprehension’ 152

8.2.6 The Influence of Humidity upon Pneumatic Foam

8.2.8 Shear Stress Imparted by the Column Wall 157

Nomenclature 164References 165

Lok Kumar Shrestha and Kenji Aramaki

9.2 Phase Behavior of Diglycerol Fatty Acid Esters in Oils 173

9.3.3 Effect of Hydrophobic Chain Length of Surfactant 181

9.3.7 Non-aqueous Foam Stabilization Mechanism 201

Acknowledgements 203References 204

Ruslan Prozorov and Paul C Canfield

Contents ix

Paul Stevenson and Noel W.A Lambert

References 247

Laurier L Schramm and Randy J Mikula

Laurier L Schramm and E Eddy Isaacs

13.2 Foam Applications for the Upstream Petroleum Industry 284

13.3.2 Well Stimulation Foams: Fracturing, Acidizing, and Unloading 288

13.4.2 Foam Applications in Primary and Secondary Oil Recovery 29213.4.3 Foam Applications in Enhanced (Tertiary) Oil Recovery 293

13.5 Occurrences of Foams at the Surface and Downstream 298

14.2.1 Adsorption Kinetics at Quiescent Interface 311

14.5 Foam Fractionation Devices and Process Intensification 317

14.6 Concluding Remarks about Industrial Practice 324Nomenclature 325References 326

15.7 Calculation of Specific Interfacial Area in Foam 342

15.9 Mass Transfer and Equilibrium Considerations 345

15.9.3 Estimation of Mass Transfer Coefficient 34615.10 Towards an Integrated Model of Foam Gas–Liquid

Contactors 347

Nomenclature 351Acknowledgements 351References 352

16.3.2 Surface Active Agents and Surface Tension of Gas/Melt Interface 36816.3.3 Drainage and Stability of a Single Molten Glass Film 369

16.6 Measures for Reducing Glass Foaming in Glass Melting Furnaces 396

16.6.5 Atmosphere Composition and Flame Luminosity 39916.6.6 Control Foaming in Reduced-pressure Refining 400

17.7 The Future 453Acknowledgements 454References 454

Peter J Martin

A Britan, H Shapiro and G Ben-Dor

19.2.5 Foam Impedance and the Barrier Thickness 488

Paul Stevenson is a senior lecturer in the Chemical and Materials Engineering Department

of the University of Auckland, New Zealand He took BA (Hons) and MEng degrees in

chemical engineering from the University of Cambridge, UK, where he completed doctoral

studies in multiphase flow in oil flowlines After post-doctoral research with the 2nd

Consortium on Transient Multiphase Flow at Cambridge, he took a position at the University

of Newcastle, Australia, to investigate froth flotation and foam fractionation He has worked

as a process chemist for Allied Colloids and as a chemical engineer for Croda Hydrocarbons

and British Steel Technical, all in the UK In addition he has spent periods as a Japanese

convertible bond dealer for the US investment bank D.E Shaw Securities International and

a racecourse bookmaker for his family’s business

About the Editor

Kenji Aramaki

Graduate School of Environment and

Information Sciences, Yokohama National

University, Yokohama, Japan

N Babcsán

BAY-ZOLTAN Foundation for Applied

Research, Miskolc-Tapolca, Igloói, Hungary

G Ben-Dor

Protective Technologies R&D Center,

Faculty of Engineering Sciences,

Ben-Gurion University of the Negev,

Beer Sheva, Israel

A Britan

Protective Technologies R&D Center,

Faculty of Engineering Sciences,

Ben-Gurion University of the Negev,

Beer Sheva, Israel

Paul C Canfield

Department of Physics and Astronomy,

Ames Laboratory, Iowa State University,

Ames, IA, USA

Sylvie Cohen-Addad

Université Paris 6, Paris, France, and

Université Paris-Est, Marne-la-Vallée, France

Annie Colin

Université Bordeaux, Pessac, France

Nikolai D Denkov

Department of Chemical Engineering,

Faculty of Chemistry, Sofia University,

Sofia, Bulgaria

Reinhard Höhler

Université Paris 6, Paris, France, and Université Paris-Est, Marne-la-Vallée, France

G Kaptay

BAY-ZOLTAN Foundation for Applied Research, Miskolc-Tapolca, Iglói, Hungary, and University of Miskolc, Egyetemváros, Miskolc, Hungary

Stephan A Koehler

Physics Department, Worcester Polytechnic Institute, Worcester, MA, USA

Noel W.A Lambert

Clean Process Technologies Pty Ltd, Lower Belford, NSW, Australia

Xueliang Li

Centre for Advanced Particle Processing, University of Newcastle, Callaghan, Australia

Peter J Martin

School of Chemical Engineering and Analytical Science, The University of Manchester, Manchester, UK

List of Contributors

Henry Samueli School of Engineering and

Applied Science, Mechanical and

Aerospace Engineering Department,

University of California Los Angeles,

Los Angeles, CA, USA

Olivier Pitois

Université Paris-Est, Laboratoire Navier,

IFSTTAR, France

Ruslan Prozorov

Department of Physics and Astronomy,

Ames Laboratory, Iowa State University,

Ames, IA, USA

Lok Kumar Shrestha

International Center for Young Scientists, WPI Center for Materials

Nanoarchitectonics, National Institute for Materials Science, Tsukuba Ibaraki, Japan

Paul Stevenson

Department of Chemical and Materials Engineering, Faculty of Engineering, University of Auckland, New Zealand

Slavka S Tcholakova

Department of Chemical Engineering, Faculty of Chemistry, Sofia University, Sofia, Bulgaria

I am enormously grateful to all of authors who have contributed to this volume on gas–

liquid foam One of the great pleasures of working with such accomplished scientists and

engineers from industry and academia is that everybody has known the level at which to

pitch their contributions

Special thanks are due to Laurie Schramm who, along with co-authors, has contributed

two chapters on foams in enhanced oil recovery and flotation of oil sands, and to Thomas

Martin who endured initial confusion upon my part to produce a first class chapter on

fire-fighting foams I’d like to express my gratitude to Stephan Koehler for giving me sound

advice about the selection of authors for various chapters, and to Denis Weaire who advised

upon how to engender more coherence in the volume Cat Chimney gave sterling assistance

in formatting referencing styles of a number of chapters My former colleague Noel

Lambert delivered excellent copy at tremendously short notice despite being in enormous

demand by others Gratitude is due to Sven Schröter (of Schroeter Imagery) for his excellent

photography and production of the image on the cover of this volume I could not have

completed this project without the constant and faithful assistance of my doctoral student

and friend Xueliang (Bruce) Li Bruce co-wrote two chapters with me, reviewed other

chapters and developed ideas for cover art He is truly a gentleman and scholar

Thanks should also go to the staff at Wiley (Chichester), in particular Rebecca Stubbs,

Sarah Tilley and Amie Marshall, who first envisaged this project and gave excellent support

as the volume developed

Last but not least I’d like to thank my second daughter, Charlotte, for being born at a

perfect time to enable me to claim my nine weeks paternity leave from the University of

Auckland, during which I worked upon this volume, as well as my wife Tracey for giving

birth to her and my first daughter Emily for being cute

Preface

Plate 1: Fig 12.10 Confocal micrograph of a bitumen froth (deaerated) from an easily

processed ore.The mineral is colored red and the bitumen is green Most of the mineral is

associated with the water phase that has been trapped as the bitumen bubbles collapsed

at the top of the separation cell Photomicrograph by V Muñoz.

Plate 2: Fig 12.12 Confocal micrograph of a bitumen froth (deaerated) from a difficult to

process ore The mineral is colored red and the bitumen is green Most of the mineral in this

case is closely associated with the bitumen (oil wet solids) and a very complex emulsified

froth structure is observed due to the high surfactant concentration in the bitumen The

bitumen component is imaged in a fluorescent mode and the decreased brightness in this

image relative to Fig 12.11 is indicative of a different bitumen chemistry in the bitumen

component Photomicrograph by V Muñoz.

(right) The use of sodium hydroxide in the extraction process changes the nature of the

bitumen, resulting in a froth that has significantly less mineral and water In these confocal

microscope images, the red colors are mineral components, the green are hydrocarbon, and

the dark areas are water Photomicrographs by V Muñoz.

Plate 4: Fig 12.17 A detail of a similar membranous sac at higher magnification showing

both mineral (red) and organic phases (green) The stronger the association between the

mineral and organic components, the greater is the proportion of yellow in this image

Photomicrograph by V Muñoz.

Foam Engineering: Fundamentals and Applications, First Edition Edited by Paul Stevenson.

© 2012 John Wiley & Sons, Ltd Published 2012 by John Wiley & Sons, Ltd.

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

Introduction 3

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

Foam Engineering: Fundamentals and Applications, First Edition Edited by Paul Stevenson.

© 2012 John Wiley & Sons, Ltd Published 2012 by John Wiley & Sons, Ltd.

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 f This ranges from much less than 1% to about 30% Engineers

call the gas fraction (i.e 1 − f) the foam quality Foams used in firefighting are classified

by their expansion ratio, which is defined by f−1 At each extreme (the dry and wet limits)

8 Foam Engineering

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)

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 l0, defined as

=Δ

where g is the surface tension of the liquid, g is acceleration due to gravity and Δr 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

equilib-rium under gravity form a dry foam

(a)

(c)

(b)

(d)

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].

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

equilib-rium 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

configura-tions 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 Δp,

border

(a) (b)

(c) (d)

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].

10 Foam Engineering

Here g is surface tension and r is the mean radius of curvature It is related to the two

prin-cipal radii of curvature, R1 and R2, by the expression

In the general case R1 differs from R2; for the case of a sphere R1= R2

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

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.

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]

2 http://www.susqu.edu/brakke/

12 Foam Engineering

(a)

(b)

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

mate-rials [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

3.5 0.5

3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

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).

14 Foam Engineering

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.

22

z n

z

=

As seen in Figs 2.7 and 2.8, these quantities vary continuously over the full range of f 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 2.9 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.

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]

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]

N = 13

N = 6

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.

16 Foam Engineering

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

proposi-tion, Weaire and Phelan computed a structure of lower surface area [30, 31] This structure

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.

2.4.2.2 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 2.14 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].

18 Foam Engineering

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

A

B C A

B C A B

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.

(b)

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 tomo graphy

of  the interior of large foam samples (20,000 bubbles) show that the bubbles are

20 Foam Engineering

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]

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

m2 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 m2 is of order

unity, but its value depends on how the foam was prepared and its subsequent history

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

r/r0

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 mm ± 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].

There is no corresponding exact result for the mean number of faces N 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 N ≈ 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

construc-tion 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

exam-ple, 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 2.21 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.

22 Foam Engineering

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.

2 3

3

5

5 6

6

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