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Historical perspective The science of foaming Wiebke Drenckhan a, ⁎ , Arnaud Saint-Jalmes b a Laboratoire de Physique des Solides, Université Paris-Sud, UMR 8502, Orsay, France b Institut de Physique de Rennes, CNRS UMR 6251, Université Rennes-1, Rennes, France abstract article info Available online xxxx Keywords: Foaming Liquid foam Bubble generation Gas/liquid interface Foaming techniques Bubble size distribution The generation of liquid foams is at the heart of numerous natural, technical or scientific processes. Even though the subject of foam generation has a long-standing history, many recent progresses have been made in an attempt to elucidate the fundamental processes at play. We review the subject by providing an overview of the relevant key mechanisms of bubble generation within a coherent hydrodynamic context; and we discuss dif- ferent foaming techniques which exploit these mechanisms. © 2015 Elsevier B.V. All rights reserved. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2. Fundamentals of foam formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.2. The physics of deforming interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.3. The physics of rupturing gas ligaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.4. Bubble generation requiring topological changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.4.1. Bubbling into a stationary liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.4.2. Bubbling into a flowing liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.4.3. Breakup of bubbles under shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.4.4. Gas entrainment at free surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 2.5. Bubble generation without topological changes — phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3. Techniques of foam formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.2. Mechanical foaming techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.2.1. Foaming by bubbling into a stationary liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.2.2. Foaming via co-injection of gas and liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.2.3. Gas entrainment at free surfaces and bubble break-up under shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.3. Foaming through phase transitions and by electro-chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.3.1. Nucleation and growth of bubbles in supersaturated liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.3.2. Cavitation and boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.3.3. Foaming by electrolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 3.4. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 1. Introduction Liquid foams consist of gas bubbles which are closely packed within a liquid carrier matrix [1–4]. Coalescence of the bubbles is hindered by the addition of stabilising agents, which may be low molecular weight surfactants, polymers, proteins, nano-particles, or their mixtures [5]. The resulting network of gas/liquid interfaces provides liquid foams with numerous complex and useful properties. Amongst these are thermal and acoustic properties, or the fact that they strongly scatter light. Most exploited are their rheological properties, since they have the unique feature that the same foam may behave like an elastic/plastic Advances in Colloid and Interface Science xxx (2015) xxx–xxx ⁎ Corresponding author. E-mail address: [email protected] (W. Drenckhan). CIS-01529; No of Pages 32 http://dx.doi.org/10.1016/j.cis.2015.04.001 0001-8686/© 2015 Elsevier B.V. All rights reserved. Contents lists available at ScienceDirect Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis Please cite this article as: Drenckhan W, Saint-Jalmes A, The science of foaming, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/ j.cis.2015.04.001 solid or like a viscous liquid depending on how it is manipulated. The result is that liquid foams have found their way into numerous applica- tions of very different kind, including cosmetic and laundry applica- tions, fire fighting, oil recovery, soil remidation or foam fractionation. Liquid foams are also very important as templates for the generation of solid foams. Last but not at all the least, foams may also occur as a byproduct in many natural or industrial processes. In the latter case, their presence may interfere with production in a harmful manner; paper or paint industry being the most prominent examples. The physical properties of a foam depend crucially on its structural properties — such as the gas fraction Φ (= gas volume / foam volume) or the bubble size distribution. A large number of foaming techniques has been developed in the past in order to obtain control over these pa- rameters. Depending on the technique, bubble sizes may range from micrometres to centimetres, bubbles can be extremely monodisperse or highly polydisperse, and gas fractions can be varied over the entire range. Unfortunately, to most users' despair, each individual technique typically covers only a relatively small range of these parameters. More- over, the choice of a foaming technique is not only guided by the prop- erties of the obtained foam, but also by how rapidly it can be generated. Hence, any academic or industrial foaming application needs to start with a wise choice of the appropriate foaming technique(s) — which can be a daunting task. All foaming techniques have one aspect in common: the generation of bubbles within a liquid. This implies the creation of gas/liquid inter- faces of interfacial tension γ which, in turn, implies an energy input of at least U =4γr B 2 per bubble. For typical interfacial tensions and bubble sizes, this is many orders of magnitude larger than thermal energies (kT), which means that bubble formation is not a spontaneous process and that one needs to put a lot of energy into a liquid in order to create a foam. What distinguishes the different foaming techniques is how ex- actly one chooses to put this energy into the liquid. This may be done by physical, chemical or even biological means (see Table 1). Physical means include mechanical action (gas sparging, whipping, shaking, etc.) or phase transitions (boiling, cavitation, effervescence, etc.). Chem- ical techniques create bubbles either by a gas-releasing chemical or electro-chemical reaction (electrolysis), while the most common bio- logical approach relies on gas-generating species such as yeast. These different techniques lead to the final foam in a one-step or in a two-step process. In a one-step process, the mechanism which gener- ates the bubbles leads directly to the final foam with a well-defined gas fraction. Only few foaming techniques allow for this. Most tech- niques rely on two types of two-step processes. In the first case, loose bubbles are generated in the liquid which are then compacted to give the final foam — for example through gravity or pressure-driven drain- age of the liquid. In the second case, the initial bubbling mechanism cre- ates a coarse foam containing large bubbles, which are then broken into smaller bubbles to create the final foam. In order to describe the science of foaming we have decided to take the following approach: First of all, we concentrate on the description of physical foaming techniques, leaving out considerations of chemical and biological foaming (with the exception of electro-chemical foaming). We provide first a detailed description of the different funda- mental mechanisms which lead to the creation of isolated bubbles in a liquid (Section 2). We then continue with the description of commonly used foaming techniques (Section 3). Considering the large number of available foaming techniques, we have decided to stay on the level of a general overview, providing the reader mostly with the main ideas of the different concepts or techniques, rather than with an in- depth analysis. We hope that this can help the reader to orient his general reflexions before going into detail by following the provided references. Before entering into the analysis, some comments on the physico- chemical aspects of foaming are required. Any foaming solution needs to contain stabilising agents in order to control the stability of the ob- tained foam. It is evident that the properties of the obtained foam de- pend sensitively on the fact whether bubble coalescence occurs during the foaming process. In fact, in many of the more complex foaming tech- niques, the final foam properties are given by a subtle equilibrium be- tween bubble generation and bubble coalescence. Moreover, the stabilising agents may have characteristic adsorption times or energy barriers, which need to be compared with the characteristic generation times and energy input of the foaming process. For example, adsorption times play an important role in setting the interfacial tension during bubble generation. If the bubbling process is faster than the equilibra- tion time of the interface, the surface tension will be between that of the pure solvent and the equilibrium surface tension, its precise value depending on the bubbling speed [6]. Last but not least, the stabilising agents often confer visco-elastic properties to the gas/liquid interfaces [7]. This may add important interfacial stresses during the process of bubble creation, which may alter significantly the finally obtained foam. For the sake of simplicity, we shall neglect all these different ef- fects in most of our argumentations by assuming that bubbles are indef- initely stable and that their interfaces have one constant value γ for the interfacial tension which is obtained instantaneously upon bubble creation. The formulation of the bulk liquid can also modify significantly the rheology of the liquid phase, leading eventually to non-Newtonian behaviour and very different conditions for bubble formation [8,9]. For the sake of clarity we will limit our description to the foaming of simple, i.e. Newtonian fluids. 2. Fundamentals of foam formation 2.1. Introduction In this section we shall concentrate on the description of the funda- mental mechanisms which lead to the generation of individual bubbles. We have chosen to group them into two conceptually different catego- ries. In the first category, the gas/liquid interface needs to undergo a to- pological change in order to create the final bubble. This topological change can occur in many different ways. The most common ones are sketched in Fig. 1. For example, a bubble may detach from a nozzle or from a deformed free surface; or a large bubble may break into smaller ones. What differentiates these mechanisms is which of the two phases is flowing and under which conditions. The different mechanisms are discussed in more detail in Section 2.4. Even if the initial mechanism of creating a pocket-like gas/liquid interface may be very different, the final break-up mechanism which leads to the topological change is the same: the gas/liquid interface has to be deformed into a slender fila- ment which is physically unstable and breaks to make the topological transition. The physics of this instability is described in more detail in Section 2.3. Table 1 Classification of different foaming techniques. Global mechanism Sub-mechanism Examples Physical foaming Mechanical foaming Bubbling, sparging, foam generation in porous media, wave breaking, shaking, rotor–stator mixers, kitchen blender, double syringe technique Phase transition Champagne, beer, extrusion, cream dispenser, shaving foam Chemical foaming Chemical reaction Fizzy drink tablets, baking powder, polyurethane foaming Electro-chemical reaction Microflotation Biological foaming Yeast Baking 2 W. Drenckhan, A. Saint-Jalmes / Advances in Colloid and Interface Science xxx (2015) xxx–xxx Please cite this article as: Drenckhan W, Saint-Jalmes A, The science of foaming, Adv Colloid Interface Sci (2015), http://dx.doi.org/10.1016/ j.cis.2015.04.001

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Page 1: The science of foaming - Université Paris-Sud · The science of foaming ... surfactants, polymers, proteins, nano-particles, or their mixtures [5]. ... providing the reader mostly

Historicalperspective

Thescience

offoaming

Wiebke

Drenckhan

a,⁎,Arnaud

Saint-Jalmes

b

aLaboratoire

dePhysique

desSolides,U

niversitéParis-Sud,U

MR8502,O

rsay,FrancebInstitutde

Physiquede

Rennes,CNRS

UMR6251,U

niversitéRennes-1,Rennes,France

abstract

article

in

fo

Available

onlinexxxx

Keywords:

Foaming

Liquidfoam

Bubblegeneration

Gas/liquid

interfaceFoam

ingtechniques

Bubblesize

distribution

Thegeneration

ofliquidfoam

sisatthe

heartofnumerous

natural,technicalorscientific

processes.Eventhough

thesubject

offoam

generation

has

along-standin

ghistory,m

anyrecent

progresseshave

beenmade

inan

attempt

toelucidate

thefundam

entalprocesses

atplay.W

ereview

thesubject

byprovidin

gan

overviewof

therelevantkey

mechanism

sofbubble

generationwithin

acoherenthydrodynam

iccontext;and

wediscuss

dif-ferentfoam

ingtechniques

which

exploitthesemechanism

s.©

2015Elsevier

B.V.A

llrightsreserved.

Conten

ts

1.Introduction

...............................................................

02.

Fundamentals

offoamform

ation.....................................................

02.1.

Introduction...........................................................

02.2.

Thephysics

ofdeforming

interfaces.................................................

02.3.

Thephysics

ofrupturinggas

ligaments

...............................................

02.4.

Bubblegeneration

requiringtopologicalchanges

...........................................

02.4.1.

Bubblinginto

astationary

liquid..............................................

02.4.2.

Bubblinginto

aflow

ingliquid

...............................................

02.4.3.

Breakupofbubbles

undershear

..............................................

02.4.4.

Gas

entrainment

atfreesurfaces

..............................................

02.5.

Bubblegeneration

without

topologicalchanges—

phasetransitions

..................................

03.

Techniquesoffoam

formation

.......................................................

03.1.

Introduction...........................................................

03.2.

Mechanicalfoam

ingtechniques

..................................................

03.2.1.

Foaming

bybubbling

intoastationary

liquid.........................................

03.2.2.

Foaming

viaco-injection

ofgasand

liquid..........................................

03.2.3.

Gas

entrainment

atfreesurfaces

andbubble

break-upunder

shear..............................

03.3.

Foaming

throughphase

transitionsand

byelectro-chem

istry.....................................

03.3.1.

Nucleation

andgrow

thofbubbles

insupersaturated

liquids..................................

03.3.2.

Cavitationand

boiling...................................................

03.3.3.

Foaming

byelectrolysis

..................................................

03.4.

Conclusionsand

outlook......................................................

0Acknow

ledgements

..............................................................

0References

..................................................................

0

1.Introduction

Liquidfoam

sconsistofgas

bubbleswhich

areclosely

packedwithin

aliquid

carriermatrix

[1–4].Coalescenceofthe

bubblesishindered

by

theaddition

ofstabilisingagents,w

hichmay

below

molecular

weight

surfactants,polymers,proteins,nano-particles,or

theirmixtures

[5].The

resultingnetw

orkof

gas/liquidinterfaces

providesliquid

foams

with

numerous

complex

anduseful

properties.Amongst

these

aretherm

alandacoustic

properties,orthe

factthat

theystrongly

scatterlight.M

ostexploited

aretheir

rheologicalproperties,sincethey

havethe

uniquefeature

thatthesam

efoam

may

behavelike

anelastic/plastic

Advances

inColloid

andInterface

Sciencexxx

(2015)xxx–xxx

⁎Corresponding

author.E-m

ailaddress:Wiebke.D

[email protected]

(W.D

renckhan).

CIS-01529;NoofPages

32

http://dx.doi.org/10.1016/j.cis.2015.04.0010001-8686/©

2015Elsevier

B.V.A

llrightsreserved.

Contents

listsavailable

atScienceD

irect

Advances

inColloid

andInterface

Science

journ

alhomepage:www.e

lsevie

r.com/lo

cate

/cis

Pleasecite

thisarticle

as:Drenckhan

W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

ColloidInterface

Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

solidor

likeaviscous

liquiddepending

onhow

itismanipulated.The

resultisthatliquid

foamshave

foundtheir

way

intonum

erousapplica-

tionsof

verydifferen

tkind,including

cosmetic

andlaundry

applica-tions,fire

fighting,oilrecovery,soilremidation

orfoam

fractionation.Liquid

foamsare

alsovery

important

astem

platesfor

thegeneration

ofsolidfoam

s.Lastbut

notat

alltheleast,foam

smay

alsooccur

asa

byproductin

man

ynaturalor

industrialprocesses.Inthe

lattercase,

their

presence

may

interferewith

productionin

aharm

fulmanner;

paperor

paintindustrybeing

themost

prominentexam

ples.The

physicalpropertiesofa

foamdepend

cruciallyon

itsstructural

properties—

suchas

thegas

fractionΦ

(=gas

volume/foam

volume)

orthe

bubblesize

distribution.Alarge

number

offoaming

techniqueshas

beendeveloped

inthe

pastinorder

toobtain

controloverthese

pa-ram

eters.Depen

dingon

thetechnique,bubble

sizesmay

rangefrom

microm

etresto

centimetres,bubbles

canbe

extremely

monodisperse

orhighly

polydisperse,andgas

fractionscan

bevaried

overthe

entirerange.U

nfortunately,tomost

users'despair,eachindividualtechnique

typicallycovers

onlyarelatively

smallrange

oftheseparam

eters.More-

over,thechoice

ofafoam

ingtechnique

isnotonly

guidedby

theprop-

ertiesofthe

obtainedfoam

,butalsoby

howrapidly

itcanbe

generated.Hence,any

academic

orindustrialfoam

ingapplication

needsto

startwith

awise

choiceofthe

appropriatefoam

ingtechnique(s)

—which

canbe

adaunting

task.Allfoam

ingtechniques

haveone

aspectincom

mon:the

generationofbubbles

within

aliquid.This

implies

thecreation

ofgas/liquidinter-

facesofinterfacialtension

γwhich,in

turn,implies

anenergy

inputof

atleastU=

4γrB 2

perbubble.For

typicalinterfacialtensionsand

bubblesizes,this

isman

yorders

ofmagnitude

largerthan

thermalenergies

(kT),which

means

thatbubbleform

ationisnot

aspontaneous

processand

thatoneneeds

toputa

lotofenergyinto

aliquid

inorder

tocreate

afoam

.Whatdistinguishes

thedifferentfoam

ingtechniques

ishow

ex-actly

onechooses

toputthis

energyinto

theliquid.This

may

bedone

byph

ysical,chemical

oreven

biologicalmeans

(seeTable

1).Physicalmeans

include

mechanical

action(gas

sparging,whipping,sh

aking,etc.)

orphasetransitions(boiling,cavitation,effervescence,etc.).Chem

-ical

techniquescreate

bubbleseith

erby

agas-releasing

chemicalor

electro-chemicalreaction

(electrolysis),while

themost

common

bio-logicalapproach

relieson

gas-generatingspecies

suchas

yeast.These

differenttechniqueslead

tothe

finalfoamin

aone-step

orina

two-step

process.Inaone-step

process,themechanism

which

gener-ates

thebubbles

leadsdirectly

tothefinalfoam

with

awell-defi

nedgas

fraction.O

nly

fewfoam

ingtechniques

allowfor

this.M

osttech-

niquesrely

ontw

otypes

oftwo-step

processes.Inthe

firstcase,loose

bubblesare

generatedin

theliquid

which

arethen

compacted

togive

thefinalfoam

—for

example

throughgravity

orpressure-driven

drain-age

oftheliquid.In

thesecond

case,theinitialbubbling

mechanism

cre-ates

acoarse

foamcontaining

largebubbles,w

hichare

thenbroken

intosm

allerbubbles

tocreate

thefinalfoam

.

Inorder

todescribe

thescience

offoaming

wehave

decidedto

takethe

following

approach:Firstofall,weconcentrate

onthe

descriptionof

physicalfoam

ingtechniques,leavin

gout

considerationsof

chemical

andbiological

foaming

(with

the

exceptionof

electro-chem

icalfoam

ing).Weprovide

firstadetailed

descriptionofthe

differentfunda-mentalm

echanismswhich

leadto

thecreation

ofisolatedbubbles

ina

liquid(Section

2).Wethen

continuewith

thedescription

ofcommonly

usedfoam

ingtech

niques(Section

3).Consideringthelarge

number

ofavailablefoam

ingtechniques,w

ehave

decidedto

stayon

thelevel

ofageneral

overview,providin

gthereader

mostly

with

themain

ideasofthe

differentconcepts

ortechniques,rather

thanwith

anin-

depthanalysis.W

ehope

that

this

canhelp

thereader

toorient

hisgeneralrefl

exionsbefore

goinginto

detailbyfollow

ingthe

providedreferences.

Beforeentering

intothe

analysis,somecom

ments

onthe

physico-chem

icalaspectsoffoam

ingare

required.Any

foaming

solutionneeds

tocontain

stabilisingagents

inorder

tocontrolthe

stabilityofthe

ob-tained

foam.It

isevident

thatthe

propertiesofthe

obtainedfoam

de-pend

sensitivelyon

thefactw

hetherbubble

coalescenceoccurs

duringthe

foaming

process.Infact,in

many

ofthemore

complex

foaming

tech-niques,the

finalfoamproperties

aregiven

byasubtle

equilibriumbe-

tween

bubblegeneration

andbubble

coalescence.

Moreover,

the

stabilisingagents

may

havecharacteristic

adsorptiontim

esor

energybarriers,w

hichneed

tobe

compared

with

thecharacteristic

generationtim

esand

energyinputofthe

foaming

process.Forexample,adsorption

times

playan

important

rolein

settingthe

interfacialtensionduring

bubblegeneration.Ifthe

bubblingprocess

isfaster

thanthe

equilibra-tion

timeofthe

interface,thesurface

tensionwillbe

between

thatof

thepure

solventand

theequilibrium

surfacetension,its

precisevalue

dependingon

thebubbling

speed[6].Last

butnot

least,thestabilising

agentsoften

confervisco-elastic

propertiesto

thegas/liquid

interfaces[7].This

may

addim

portantinterfacialstresses

duringthe

processof

bubblecreation

,which

may

altersign

ificantly

thefinally

obtained

foam.For

thesake

ofsimplicity,w

eshallneglect

allthesedifferent

ef-fects

inmostofourargum

entationsby

assuming

thatbubblesare

indef-initely

stableand

thattheirinterfaces

haveone

constantvalueγfor

theinterfacial

tensionwhich

isobtained

instantaneously

uponbubble

creation.The

formulation

ofthebulk

liquidcan

alsomodify

significantlythe

rheology

oftheliquid

phase,leading

eventuallyto

non-N

ewtonian

behaviourand

verydifferentconditions

forbubble

formation

[8,9].Fort he

sakeofclarity

wewilllim

itourdescriptionto

thefoam

ingofsim

ple,i.e.N

ewtonian

fluids.

2.Fundam

entals

offoamform

ation

2.1.Introduction

Inthis

sectionweshallconcentrate

onthe

descriptionofthe

funda-mentalm

echanismswhich

leadto

thegeneration

ofindividualbubbles.W

ehave

chosento

groupthem

intotw

oconceptually

differentcatego-ries.In

thefirstcategory,the

gas/liquidinterface

needsto

undergoato-

pologicalchangein

orderto

createthefinalbubble.This

topologicalchange

canoccur

inmany

differentways.The

most

common

onesare

sketchedin

Fig.1.Forexam

ple,abubble

may

detachfrom

anozzle

orfrom

adeform

edfree

surface;oralarge

bubblemay

breakinto

smaller

ones.Whatdifferentiates

thesemechanism

siswhich

ofthetw

ophases

isflow

ingand

underwhich

conditions.Thedifferent

mechanism

sare

discussedin

more

detailinSection

2.4.Evenifthe

initialmechanism

ofcreatingapocket-like

gas/liquidinterface

may

bevery

different,thefinalbreak-up

mechanism

which

leadsto

thetopologicalchange

isthe

same:

thegas/liquid

interfacehas

tobe

deformed

intoaslen

derfila-

ment

which

isphysically

unstableand

breaksto

make

thetopological

transition.Thephysics

ofthisinstability

isdescribed

inmore

detailinSection

2.3.

Table1

Classificationofdifferent

foaming

techniques.

Global

mechanism

Sub-mechanism

Examples

Physicalfoam

ingMechanical

foaming

Bubbling,sparging,foamgeneration

inporous

media,w

avebreaking,shaking,rotor–stator

mixers,kitchen

blender,doublesyringe

techniquePhase

transitionCham

pagne,beer,extrusion,cream

dispenser,shavingfoam

Chemical

foaming

Chemical

reactionFizzy

drinktablets,baking

powder,polyurethane

foaming

Electro-chemical

reactionMicroflotation

Biologicalfoam

ingYeast

Baking

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esA,The

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ing,Adv

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Inthe

secondcategory,no

topologicalchangeofthe

gas/liquidinter-

faceisrequired,as

afreely

floatingbubble

isdirectly

createdwithin

theliquid

byaph

asetransition

orby

achem

icalreaction

(Table1and

Fig.2).Thedifferent

physicalmechanism

softhis

kindofbubble

crea-tion

arediscussed

inmore

detailin

Section2.5.Som

emechanism

smay

belongto

bothclasses.For

example,phase

transitionsare

oftenfavoured

bythe

presenceofw

allsor

impurities

fromwhich

thebubbles

needto

detachvia

atopologicalchange.

Inboth

categories,anum

berofdifferentstresses

areinvolved

inthe

processofbubble

creation.Webriefly

reviewthese

inSection

2.2and

discusshow

theymay

begrouped

intonon-dim

ensionalnumbers

foracoherentdescription

ofdifferentbubblingprocesses.

2.2.Thephysics

ofdeforming

interfaces

Any

processofbubble

formation

involvesthe

creationand

deforma-

tionofa

gas/liquidinterface.Even

ifthefundam

entalmechanism

may

benon-m

echanicalinnature

(likechem

icalreactions),theactualcreation

processinvolves

anum

berofm

echanicalstresses(=

forceper

area)which

needto

beconsidered.O

neofthe

most

importantcontributions

isthat

ofthesurface

tensionγ,w

hichleads

toanorm

alstressσγand

thereforeto

apressure

dropΔpacross

acurved

surfaceofm

eancurva-

tureκdescribed

bythe

Young–Laplacelaw

[15]

σγ ¼

Δp¼

2γκ:

ð1Þ

Ofsim

ilarim

portanceisthe

influenceofa

hydrostaticpressure

dif-ference,w

hichleads

tothe

well-know

nbuoyancy

forceFBon

abubble

ofvolumeVB ,w

hichplays

amajor

rolein

bubbledetachm

entandcom

-paction.The

buoyancyforce

isgiven

by

FB ¼

ΔρgV

B :ð2Þ

Surfacetension

andgravitationalstresses

are“static”

stressessince

they

donot

requirefluid

motion

.In

adynam

iccon

textim

portant“dynam

icstresses”

may

arise.Thefirstone

isatangentialviscous

stressσV ,w

hichresults

fromaviscous

shearflow

attheinterface.Itdepends

onthefluid

viscosityηLand

thedeform

ationrate

dU/dLof

thefluid

andiscom

monly

approximated

as

σV ¼

ηL UL

;ð3Þ

where

Uisthe

characteristicflow

velocityand

Lacharacteristic

lengthscale

oftheflow

.Generally

onehas

twoviscous

contributions,on

efrom

thegas

andone

fromthe

liquidphase.V

iscousflow

canalso

leadto

theoccurrence

ofadynam

icpressure,hen

ceexerting

additional

Fig.1.Differentroutes

ofbubblegeneration

which

requireatopologicalchange

to“pinch-off”

oneor

more

bubblesfrom

amain

gasbody

(bubblingfrom

nozzlefrom

[10],effervescencefrom

[11],co-flowfrom

PhDthesis

Jan-PaulRaven,cross-flowfrom

[12],air-entrainmentinspired

by[13],breakup

undershear

from[14]).“A

ctive”or

“Passive”refers

towhether

ornot

thecorresponding

phaseisflow

ingactively

inthe

process.

Fig.2.Sketchofthe

twomain

routesofbubble

generationby

physicalmeansw

hichdo

notrequire

atopologicalchange

oftheinterface.The

firstcreatesbubbles

viacavitation,boil-

ingor

effervescence.Inthe

second,thedrops

ofanem

ulsionturn

intoagas,generally

uponpressure

release.

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normalstresses

onan

interface.Gases

orliquids

inmotion

alsoexert

normalin

ertialstressesσI on

aninterface

dueto

themom

entum

ofthe

flow.These

stressesmay

beapproxim

atedas

σI ¼

ρU2:

ð4Þ

Asdiscussed

inSection

2.1,thepresen

ceof

stabilisingagents

cangive

riseto

important

interfacialstresses[16]

which

caninterfere

with

bubblegeneration

innon

-negligible

manner

[17,18].Other

stressesmay

actin

specificallydesigned

systemswhich

make

useofacoustic,

electricorm

agneticforces.Forsim

plicity,wewillnotconsiderexplicitly

anyofthese

latterstresses

here.Generally,allthe

above-mentioned

stressesactsim

ultaneouslyduring

theprocess

ofbubblegeneration.H

owever,in

most

cases,someofthe

contributionsdom

inatewhile

otherscan

beneglected.O

necom

monly

distinguishesbetw

eena“quasi-static”

anda“dynam

ic”regim

e.Inthe

quasi-staticregim

e,theflow

velocitiesare

sufficientlysm

allsothatthe

systemcan

beassum

edto

progressvia

asequence

ofstaticstates.Such

processesare

thereforeindependentofthe

flowvelocities.In

thedynam

icregim

e,viscousand/orinertialforcesneedto

betaken

intoaccount.In

thisregim

ethe

bubbleform

ationisoften

dominated

byone

ofthetw

ocontri-

butions,which

iswhy

onecom

monly

distinguishesbetw

eena“viscosity-

dominated”

andan

“inertia-dominated”

regime.

One

way

tonavigate

thecom

plexityof

thedifferent

stressesis

towork

with

non-dim

ensionalnumbers

which

allowsto

measure

ina

(somew

hat)more

quantitativeway

therelative

importance

ofeachof

theacting

stresses.Table

2sum

marises

themost

important

non-

dimensionalnum

berswhich

areused

inthe

descriptionofbubble

for-mation

.Theuse

ofthese

numbers

canbe

extremely

powerful

sincethey

providean

important

degreeofabstraction

bygrouping

togetherdifferent

parameters

usingsim

ple“scaling

arguments”.M

ostofthese

numbersare

usedthroughoutthisarticle

inorderto

classifydifferentre-

gimes

offoaming

techniques.The

definitionofthe

dimensionless

numbers

involvesthe

following

parameters:surface

tensionγ,density

ρ,viscosityηand

thegravitation-

alaccelerationg.M

oreover,itrelies

onacharacteristic

velocityUand

acharacteristic

lengthscale

L.Thelatter

twoparam

etersneed

tobe

cho-sen

with

care,asmost

flowproblem

sinvolve

complex

flowfields

andflow

geometries,hen

ceawide

rangeof

velocitiesand

lengthscales.

Moreover,

theproblem

sdeal

with

two-ph

aseflow

s;on

etherefore

needsto

choosewisely,ifthe

stressesofone

ofthephases

dominate

thebehaviour,or

ifbothneed

tobe

takeninto

account.Good

choicesofthese

parameters

arediscussed

inmore

detailthroughoutthisarticle.

Another

frequentlyused

parameter

isthe

capillarylength

lc ¼ffiffiffiffiffiffiffiffiffiffiffiγΔρg

r;

ð12Þ

which

isadim

ensionalexpressionofthe

Bondnum

ber.Itprovidesachar-

acteristiclength

beyondwhich

gravitationaleffectsneed

tobe

takeninto

accountin

comparison

tosurface

tensioneffects.Last

butnot

least,theviscosity,density

andflow

rateratio

ofthegas

(“G”)

andthe

liquid(“L”)

phase

λV ¼

ηG

ηL;λD ¼

ρG

ρL;and

λQ¼

QG

QL;

ð13Þ

areoften

takeninto

accountinthe

descriptionofbubble

formation.

Num

erousothernon-dim

ensionalnumbers

exist,forexam

ple,ifthevisco-elasticity

ofthe

interfaceswas

tobe

takeninto

account.The

interfacialmobility

may

beused

toas

ameasure

oftheinterfacialviscos-

ity,while

theMarangoninum

bercaptures

theim

portanceofan

interfa-cialelasticity

[3,7].

2.3.Thephysics

ofrupturinggas

ligaments

Here

weshalldiscuss

indetailthe

physicsofhow

thebubble

undercreation

undergoesthe

topologicalchangeto

detachfrom

anobject,a

freesurface

ora“m

other”bubble.M

anydifferent

configuration

scan

leadto

this

final

stageand

aselection

isdiscussed

indetail

inSection

2.4.How

ever,inallcases,the

physicsofthe

finalstepleading

tothe

topologicalchangeisthe

same(Fig.3):the

bubblehas

tobe

de-form

edinto

anelon

gatedcylinder

(or“gas

thread”,or

“ligament”),

which

becomes

physicallyunstable

andbreaks.Ifthe

cylinderis

rela-tively

shortwith

respectto

itswidth,it

breaksin

onepoint

toliberate

onebubble

(Fig.3a).Ifitis

verylong,it

canbreak

intomany

bubbles(Fig.3b).

Tounderstand

thisbreak-up

letusconsider

thephysics

ofalong

cy-lindricalligam

entofonefluid

within

another.Thiscan

beagas

ligament

inaliquid,a

liquidligam

entin

agas,or

aliquid

ligament

inasecond

immiscible

liquid.The

thinner

suchaligam

entis,

thehigher

isits

Table2

Summary

ofthemostim

portantdimensionless

numbers

usedto

classifydifferentregim

esofbubble

generationand

theircross-relations.

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surface-to-volumeratio.

Itwould

therefore

energeticallybe

more

favourableto

turnthe

cylinderinto

anum

berofsphericalobjects.But

thekey

questionis:how

doesthe

systemgetthere?

Thistransition

hap-pens

viaan

instabilitywhich

linksthe

ligamentcurvature

andpressure

viathe

Young–Laplaceequation

(Eq.(1)).Thisissue

hasbeen

widely

studied,andthe

understandingof

thekey

mechanism

sgoes

backto

the19th

centuryand

thepioneering

works

ofSavart,Plateauand

Ray-leigh

who

were

tryingto

understandthe

destabilisationofa

water

jetas

itfallsoutofa

faucet.Implem

entingthe

ideasofSavart[21]and

Pla-teau

[22],Rayleighdeveloped

afirst

basicmodel[23],w

hich

isnow

knownas

the“Rayleigh–Plateau

instability”.Rayleighconsidered

acy-

lindrical,infinite

thread

ofan

inviscid

fluid

inair,w

ithoutexternal

flow.Such

afluid

columnexperiences

spontaneousthickness

undula-tions

(called“varicosity”,Fig.4a)

which

aregenerally

damped.H

owev-

er,Rayleigh

couldshow

thatunder

certaincon

ditions,som

eof

the

undulationscan

actuallygrow

with

timerendering

acertain

rangeof

“varicose”modes

unstable.Thiscannot

beunderstood

ifoneconsiders

onlythe

pressuregradientsarising

frommodulations

ofthecross-section

(ofradiusr)

oftheligam

ent.Infact,as

sketchedin

Fig.4a,theYoung–La-

placelaw

includesalso

theradius

r′,inthe

planeorthogonalto

thecross-

section,andassociated

with

thedeform

ationmode

(andwith

thedefor-

mation

wavelength

λ).Inparticular,w

henthe

cross-sectiongets

small,

thereis

asaddle

pointand

thetw

ocurvatures

havedifferent

signs.Thus,depending

onwhich

radius(r

orr′)

isthe

biggest,adifferent

behaviourwillbe

found.Qualitatively,m

odesassociated

with

larger′

(wavelength

longerthan

theligam

entsection)

willbe

unstable,andvice

versa.Performing

thedetailed

mathem

aticalanalysis

assuming

thatinertiaisbalancing

surfacetension,one

obtainsthatthe

modes

hav-ing

apositive

growth

rate(sketched

inFig.4b)

arethose

whose

wave

number

k(k

=2π

/λ)fulfils

krb1:

ð14Þ

Themaxim

umin

Fi g.4bcorresponds

tothe

most“dangerous”

mode

which

iscalled

the“Rayleigh

mode”.It

isgiven

by

kr¼0:697

andλ¼

9:02r:

ð15Þ

Anim

portantpointisthatalltheunstable

modesare

finiteand

oftheorderofthe

threadradius(butsom

ewhatlargerthan

thethread

radius).Since

thefinalbubble

ordropletsizeisofthe

orderofthewavelength

ofthe

mostunstable

mode,itis

generallyalinearfunction

ofthethread

ra-dius

atwhich

theinstability

occurs.Another

interestingoutput

isthe

capillarytim

escaleassociated

with

thesedeform

ations[24]

τcap ¼

ffiffiffiffiffiffiffiffir 3ργ

s;

ð16Þ

Fig.4.TheRayleigh–Plateau

instability:(a)Schem

eshow

ingthe

differentradiiofcurvature.(b)Grow

thrate

asafunction

oftheundulation

wave

number

k=

2π/λ

multiplied

bythe

threadradius

r.(c,d)Tim

eevolution

ofunstableligam

entsof(c)

twopolym

ericfluids

[27],ofa(d)

gasligam

entin

aviscous

liquidconfined

inatube

(from[28]).

Fig.3.Theinstability

ofthingas

ligaments

leadsto

thetopologicalchange

ofthegas/liquid

interfacerequired

tocreate

oneorm

oreseparate

bubbles.(a)Bubble

pinch-offinwater/glycerol

mixtures

ofincreasingliquid

viscosityηL (from

[19]).(b)Pinch-offofa

gasbubble

containinginsoluble

SF6forη

L =0.03

Pas,show

ingthe

generationofm

anytiny

satellitebubbles

duringthe

detachment

ofthemain

bubble(tim

ebetw

eenim

ages:5μs.Scale

bar50

μm)(from

[20]).

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which

providesan

approximation

asto

howrapidly

athread

willbe-

comeunstable.

Rayleigh'ssim

plemodelw

asexpanded

inmany

differentways

inorder

todescribe

more

realisticscenarios.W

eber[25]

was

oneofthe

firsttotake

intoaccountthe

viscosityofthe

fluidofthe

cylinder.Hisre-

sultscan

berationalised

usingthe

Ohnesorge

number

Oh(Eq.(10)

inTable

2)which

canalso

beconsidered

asthe

ratioofthe

viscoustim

e-scale

overthe

capillarytim

escale

([24,26]and

references

therein).

Thestability

ofthefluid

cylinder

decreaseswith

decreasingOh,and

thegrow

thrate

andthe

wavelength

ofthemost

unstablemode

canbe

calculatedusin

gthis

“viscous”Rayleigh–Plateau

instability([24,26]

andreferences

therein).Rem

ainingin

thesituation

ofstationaryliquids

(noflow

insideor

outsidethe

cylinder),theratio

λVof

theviscosities

ofthe

innerand

outerfluidcan

betaken

intoaccount[29].The

mostunstable

wavelength

dependson

theviscosity

ratio,beingmaxim

umwhen

λV≈

1.Consider-ing

thecase

ofagas

cylinderinsidealiquid

(λV≈

1/55)

themostunsta-

blewavelength

ispredictedto

beλ=

12.5r—which

remains

closeto

thebasic

prediction(Eq.(15))

ofRayleigh.Mostsituations

which

leadto

theform

ationoflong

cylindricalliga-ments

requirethat

thefluid

insideand/or

outsidethe

ligamen

tis

flowing

(Fig.1).Inthis

case,the“tem

poralinstability”discussed

upto

nowisoften

notsufficient.One

mustalso

considerthe

“spatialinstabil-ity”,m

eaningthat

aninitially

localiseddeform

ationpropagates

alsoalong

theligam

entaxis.Researchers

havetherefore

identifiedtw

odif-

ferentscenariosforthe

instabilityto

occur:the“absolute”

andthe

“con-vective”

instability([30]and

referencestherein).

Anabsolute

instabilitycorresponds

toadeform

ationwhich

grows

toofastin

timeand

spaceto

beconvected

bythe

main

flow,itthusprop-

agatesboth

downstream

andupstream

,invadingthe

whole

ligament,

independentlyofthe

flow.A

tafixed

pointinspace,the

growth

ofanun-

stablemode

isnever

limited.By

contrast,theconvective

instabilityim

-plies

perturbationswhich

areconvected

downstream

while

theygrow

.Atagiven

pointin

space,theam

plitudeof

thedeform

ationrem

ains

finite,asthe

flowisfastenough

tosw

eepthe

deformation

away

beforeitbecom

esexponentially

large.Theconvective

regimeallow

sfor

acon-

tinuousand

longerfluidligam

enttosurvive,w

hencom

paredto

theab-

soluteregim

e.Within

thesam

ebubbling

conditionsone

cantypically

switch

between

bothscen

ariosby

appropriatelyadjusting

theflow

ratesofboth

phases.Theconsequences

ofthesedifferent

behavioursare

particularlywellevidenced

anddiscussed

inmore

detailwhen

bub-bles

areblow

nfrom

nozzles(Sections

2.4.1and

2.4.2.)Let

usfinish

with

twoim

portantrem

arks.A

ligament

doesnot

necessarilybreak

into

equal-volumebubbles.Sm

allbubbles

(called“satellite

bubbles”)can

resultfrom

thefinalstep

ofrupture

(Figs.3band

4c).Thisis

discussedin

more

detailinSection

2.4.3.Lastbut

notleast,it

isim

portantto

keepin

mind

that

geometrical

confinem

enthas

adram

aticeffecton

ligamentcurvatures

andtherefore

onits

stabil-ity.In

fact,ifaligam

entisconfinedin

onedirection

(forexample

inaflat

channel)itrem

ainsstableuntilthe

confine mentis

released[27,31].This

canprovide

anim

portantdegreeofcontroloverbubble

formation

andis

oftenused

inmicrofluidic

contexts.

2.4.Bubblegeneration

requiringtopologicalchanges

Many

differentmechanism

sexist

which

allowto

detachabubble

fromagas

pocket.Oneof

themost

common

approachesuses

the

gravity-drivendetachm

entof

abubble

which

isblow

nat

constant

pressureor

flow

rateinto

astation

arysolution

viaan

orifice

(Fig.1and

Section2.4.1).Since

gravityisnotvery

efficientwhen

smalllength

scalesare

involved(sm

allBondnum

bersBo

—Table

2),many

bubblingmechanism

sinvolve

aco-flow

oftheliquid

inorder

tomake

useofvis-

cousor

inertialforcesto

influencethe

break-upprocess.This

co-flowmay

occurinunconfined

conditions—

forexample

onecan

move

anoz-

zlethrough

thefoam

ingsolution.Butin

many

casesitismore

suitableto

work

inconfined

geometries

inwhich

dynamicstresses

canbe

imposed

more

efficientlyand

inabetter

controlledmanner.W

ediscuss

herein

particularthecase

ofconfi

nedco-fl

owand

cross-flow

(Fig.1

andSection

2.4.2).Another

important

mechanism

includesthe

entrainment

ofairat

freesurfaces

(Fig.1and

Section2.4.1)

andthe

breakupoflarge

bubblesinto

smaller

onesunder

shearflow

(Fig.1and

Section2).

Forsim

plicity,weshalldiscus

thevarious

flowproblem

sin

termsof

“flowrates

Q”and

“flowvelocities

U”.This

means

thatthepressures

ad-just

toensure

aconstantflow

rate.Inmany

experimentalscenarios

itis

more

practicaltocontrolthe

pressuresandmeasure

theflow

rates,forex-am

pleto

avoidlong

equilibrationtim

es.Thismeans,how

ever,thatflowrates

arenot

constantand

thephysicaldescription

issom

etimes

more

awkw

ard.Itcanalso

changenon-negligibly

thebehaviour

ofadevice.

2.4.1.Bubblinginto

astationary

liquidDue

toth

etech

nical

importan

ceof

the

process

ofbu

bbling

throu

ghorifi

ce s,thesu

bjecthas

beeninvestigated

inten

sivelyin

thepast.W

hileslow

bubblingcan

bedescribed

inafairly

straightfor-

ward

man

ner

(Section2.4.1.1)

thedescription

ofthedyn

amic

casehas

provento

behigh

lycom

plex(Section

2.4.1.2).This

complexity

isdueto

differen

tfeed

backmech

anism

swith

inth

esystem

,th

emost

importan

ton

ebein

gth

eintricate

couplin

gbetw

eenth

edy-

namic

forcesand

thebubble

shape.Moreover,the

physicsofth

edy-

nam

icsof

thecon

tactan

glebetw

eenth

einterface

andth

eorifi

ce,an

dof

thefinalbreak-u

pmech

anism

(s)are

only

beingelu

cidated

now.The

resultisthatdespite

theavailability

ofsomereview

articles[32,33],th

esu

bjectrem

ainsqu

itescattered

with

differen

tau

thors

presen

ting

differen

tparam

eterran

ges(su

rfaceten

sion,orifi

cedim

ension

san

dsh

ape,wettin

gcon

ditions,liquid

viscosity,gasan

dliq

uid

den

sity,etc.).

Sometim

esth

eresu

ltsseem

contrad

ictory.How

ever,inmost

casesth

esecon

tradiction

sresu

ltfrom

thefact

that

experimen

tsare

donein

differentregim

eswhere

differentpa-

rameters

may

have

avery

differen

t—

evenop

posite

—influen

ce.The

samechallen

geholds

formodelling

attempts

which

necessarilyneed

tosim

plifyth

eproblem

.Theappropriate

approximation

sde-

pen

don

ceagain

onth

ebu

bblingregim

e,which

has

ledto

avast

number

oftheoreticaland(sem

i-)empiricalm

odels,eachdescribing

generally

acertain

parameter

range

only.A

nexten

siveoverview

ofexisting

models

isgiven

inTable

3in

reference

[33].Wesh

allpresenthere

someofthe

keyingredients.

2.4.1.1.Thequasi-static

regime.A

tthe

outset,letus

discussthe

simple

scenario

ofbubblin

gat

constantgas

flow

rateQGinto

astationary

foaming

solutionusing

averticalorifice

ofcircularcross-sectionand

ra-dius

RO(see

Figs.1and

5).Letusassum

ethatthe

bubblingoccurs

suffi-ciently

slowly

tobe

inthe

quasi-staticregim

e(Fig.5),i.e.Ca

≪1and

We≪

1(Table

2),andthatthe

generationprocess

istherefore

dominat-

edby

surfacetension

andgravitationalforces

(buoyancy).Ph

otographsand

simulations

ofa

typicalquasi-static

blowing

processare

shown

inFig.

6a.During

this

processthebubble

goesthrough

awell-defined

seriesofpressure

stateswhich

aresketched

inFig.6b.The

pressureinitially

increases

(which

somerefer

toas

the

“nucleationstage”)

beforereaching

amaxim

umwhen

thebubble

isa

hemisphere,i.e.w

henthe

bubbleradius

RBisequalto

theorifice

radiusRO .A

tthis

point

ΔPmax ¼

2γRO;

ð17Þ

which

followsnaturally

fromthe

Young–Laplace

law(Eq.(1)).This

pressureis

important,as

anybubblin

gapplication

needsto

providepressures

abovethis

value.Due

tothe

inversedependence

oftheblow

-ing

pressureon

theorifice

radius,thepressures

forvery

smallorifices

may

becomeofthe

orderofa

fewbar,w

hichcan

beunfeasible

forcer-

tainapplications.

6W.D

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as:Drenckhan

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esA,The

scienceoffoam

ing,Adv

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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Acertain

reductionofthe

maxim

umpressure

may

beobtained

usingorifices

with

more

complex

geometricalcross-sections

[159].Ifthe

appliedpressure

ishigher

thanthe

maxim

umpressure,the

bubblegrow

sbeyond

thehem

isphericalshape(leading

toapressure

decrease)and

detachesvia

theinstability

mechanism

sdiscussed

inSection

2.3when

thesurface

tensionforce

Fγ(w

hichkeeps

thebubble

“stuck”to

theorifi

ce)is

oftheorder

ofthebuoyancy

forceFG .If

the

orificesize

ROismuch

smaller

thanthe

capillarylength

(Eq.(10)),i.e.ifR

O≪

lc ,onemay

approximate

theshape

ofthedetaching

bubbleby

asphere

ofvolume4/3π

RB 3.This

leadsto

abuoyancy

forceof

FG=

4/

3ΔρgR

B 3,while

thesurface

tensionforce

canbe

written

asFγ=

2πγRO .

RBmay

thereforebe

approximated

as

RB �

γΔρg

��

2=3R

1=3

O¼

l 2=3

cR

1=3

O¼

ROBo

−1=

3for

RO ≪

lcð

Þ;ð18Þ

where

thelen

gthscale

Lin

thedefi

nition

ofth

eBon

dnumber

Bo(Eq.(5))

has

beench

osento

betheradius

oftheorifi

ceRO .In

order

Fig.5.Variation

ofbubblingregim

ewith

increasingflow

velocities.Theblack

arrowindicates

where

break-upoccurs.

Fig.6.Blowing

bubblesfrom

anorifice

intoastationary

solution.(a)Bubble

evolutionas

afunction

oftimefor

twodifferentorifice

materials

(wetting

andnon-w

etting)com

paringeach

timephotographs

froman

experiment(top

row)and

computer

simulations

(bottomrow

)(adapted

from[36]).(b)

Schematic

pressureevolution

duringthe

generation.(c)Finalbubble

volumeas

afunction

oforificewetting

properties(expressed

incontactangle)

fortw

odifferent

orificediam

eters(adapted

from[37]).

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forthe

bubbleto

detach,Botherefore

needsto

beatleastofthe

orderofunity.

Ifth

eorifi

cerad

iusis

ofth

eord

erof

thecap

illarylen

gth,i.e.

RO≥lc ,on

eneeds

totake

into

accountth

egravity-driven

deforma-

tionof

thebu

bblebefore

detachmen

t.Research

ersfindth

atth

isis

wellcap

tured

bysligh

tlymod

ifyingth

eexp

onen

tin

Eq.(18)su

chthat

RB �

RO Bo

−α=

3for

RO≥lc

ðÞ;

ð19Þ

with

α≈

1.06[34].In

bothcases,it

isclear

thatthebubble

sizeisset

bytheorifi

cedim

ensionsand

bythe

capillarylength

lcwith

smaller

orifices

leadingto

smaller

thebubbles.Sin

cethis

isnot

astrong

de-pen

denceitisoften

more

appropriateto

work

with

co-flow

scenari-

os(Section

2.4.2),especially

ifsm

allbubbles

andhigh

prod

uction

ratesare

needed.Th

epinch

-offprocess

(seealso

Section2.3)

isgen

erallymuch

fasterthan

thebubble

blowing

process,which

iswhy

itsduration

iscom

mon

lyneglected

inmodels.A

fterthepinch

-offthegas

thread

re-tracts

fullyback

to(often

eveninto)

theorifi

ceand

theperiodic

pro-cess

restarts.Due

tothe

periodicityofthis

process,foamsgen

eratedvia

bubblin

gin

thequ

asi-staticregim

eten

dto

behigh

lymon

odis-

perse.

Typical

bubble

sizeswhich

canbe

prod

uced

with

this

ap-

proach

arebetw

eena

fewhundred

microm

etresan

da

fewmillim

etres.In

somecases,on

emay

belookin

gto

prod

uce

largebu

bbles.Alook

atEq.(18)

suggests

theuse

ofalarge

orifice

forth

ispurp

ose.How

ever,when

thesize

ofth

eorifi

ceis

largerth

anth

ecap

illarylength

lcone

has

todealw

ithgravity-driven

instabilitiesofthe

gas/liquid

interface

leadingto

aneffect

called“w

eaping”

(liquiden

terstheorifi

ce).Anelegan

tapproach

isto

“tune”

gravityinstead

bylet-

tingthe

bubblegrow

underneath

aninclined

plane(Fig.7)

[35].The

pre-factorsin

Eqs.(18)an

d(19)

areofthe

orderofunity,but

dependsensitively

onthe

geometry

oftheorifi

ceand

onits

wettin

gprop

ertieswith

respect

toth

efoam

ingliqu

id.Th

emore

theliqu

idwets

theorifi

ce,themore

thegas

thread

remain

sconfi

ned

with

inthe

orifice

boundaries,leadingto

smaller

bubblesand

toaclearer

de-pen

denceofthe

bubblesize

onthe

orifice

dimensions.This

issh

own

intheim

agesequences

inFig.6a

(adaptedfrom

[36])and

alsoin

thefinalbubble

sizemeasurem

entsofFig.6c

(adaptedfrom

[37]),which

show

show

thebu

bblesize

dep

endson

thewettin

gcon

dition

s(expressed

bythe

contactangle

oftheliquid

onthe

nozzlematerial).

Italso

show

sth

atth

eorifi

cesize

plays

aless

importan

trole

inth

ecase

ofanon-w

ettingorifi

ce.Eq.(18)

allowsus

toapproxim

atein

which

rangeoffl

owrates

we

canapproxim

atethebubbling

processas

quasi-static.Since

viscousforces

tendto

play

amore

importan

trole

atlow

ervelocities

than

inertialforces,w

emay

requirethat

Ca≪

1,hence

QG ¼

UπR2O ≪

γηπR2O:

ð20Þ

Using

Eq.(18),wecan

calculateacorresponding

bubblingfrequency

fBfB ¼

QG

VB≪

Δρg

ηRO:

ð21Þ

Thisgives

fB≪

100fora

nozzlewith

RO=

100μm

.Bubblingfrequen-

ciesin

thetruly

quasi-staticregim

eare

therefore

rarelyabove

afew

bubblesper

second,which

istoo

slowfor

most

applicationswhich

thereforetend

tobe

runin

adynam

icregim

e(Section

2.4.1.2).

2.4.1.2.Thedynam

icregim

e.Upon

moving

away

fromquasi-static

bub-bling

conditions,differentdynamic

forcesneed

tobe

consideredin

thebubbling

processwhich

aresketched

inFig.8a.For

example,viscous

dragexerted

bythe

liquidon

thebubble

slowsdow

nthe

detachment

andtherefore

leadsto

largerbubbles.Bubbledetachm

entisalso

resistedby

inertialforcesofthe

liquid,while

theinertialforces

ofthegas

helpin

pushingthe

bubbleaw

ayfrom

theorifice.

Images

ofthebubble

generationprocess

with

increasinggas

flowrate

aresketched

inFig.5

andsom

ecorresponding

graphsin

Fig.8b.Ifsurface

tensionand

inertialforcescan

beneglected,researchers

find

[38]thatthe

resultingbubble

sizemay

beapproxim

atedby

RB �

ηL Q

G

Δρg

��

1=4�

RO

CaBo

��

1=4;

ð22Þ

where

theconstant

ofproportionalityisoforder

1and

dependson

theorifi

cecon

ditions(geom

etry,wetting).

Theim

portantobservation

hereisthat

unlikein

thequasi-static

regime,the

bubblesize

nowde-

pendson

theviscosity

ηLofthe

liquidand

thegas

flowrate

QG .M

orespecifically,the

bubblesize

increaseswith

ηLand

QG .

Upon

furtherincrease

ofQG ,inertial

forcesbegin

toplay

anon

-negligible

role(W

eGN1).W

ithgrow

ingW

ebernum

berWeGof

the

gas,theform

ingbubble

becomes

more

elongated,sinceinertialforces

becomeincreasingly

important

with

respectto

therestoring

surfacetension

forces.Thepinch-offpointofthe

gasthread

moves

increasinglyaw

ayfrom

theorifice,leading

eventuallyto

theform

ationofa

gas-jetwhich

breaksup

intobubbles

atits

tip.Thisjet

widens

away

fromthe

nozzle,asthe

gasisdecelerated

dueto

viscousfriction.For

sufficientlyhigh

WeG ,bubble

detachmentisentirely

drivenby

thedynam

icallygen-

eratedbubble

shape.Theinfluence

ofgravitybecom

esthereforenegligi-

ble,which

hasalso

beenconfirm

edin

micro-gravity

experiments

[39].One

cantherefore

thinkoftw

oextrem

escenarios,one

inwhich

bub-ble

pinch-offisdriven

entirelyby

gravity(Bo

N1)

versusone

where

itisdriven

byinertialforces.W

henthe

bubbleform

ationisdom

inatedby

gravityand

viscousforces,one

generallyspeaks

ofthe“dripping”

re-gim

e(in

analogyto

thedripping

tapproblem

).When

inertialforcesdom

inate,onespeaks

ofthe“jetting

regime”.A

number

oftheoreticaland

experimen

talstudies

havesh

ownthat

inertial

forcesdom

inatethe

pinch-offwhen

WeGN1–10

[33,39–41].Inmost

situations,gravityand

inertia

acttogether,

which

issh

own

intheph

asediagram

inFig.

8b,which

usestheBo

and

theWeGnum

berto

predictunder

Fig.7.Blowing

largebubbles

viatuning

theeffective

gravityacting

onthe

bubble.(a)Schem

aticrepresentation.(b)

Experimentalrealisation

from[35].

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esA,The

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which

flowconditions

bubbledetachm

entoccurs

(unstableregion).A

number

ofstudieshave

investigatedthe

transitionfrom

thedripping

tothe

jettingregim

e—

andwewilldiscuss

thephysics

ofthesetransi-

tionsin

more

detailwhen

discussingbubbling

underco-flowconditions

inSection

2.4.2.What

isim

portantto

noteat

thispoint

isthat

atthe

transition,importantnon-lineareffects

canplay

arole,leading

tomulti-

plebubble

periodsand

evenchaotic

bubbling.Thisleads

tofoam

swith

bubblesofm

ultipleorpolydisperse

sizesand

isdiscussed

inmore

detailin

Section3.2.1.1.

Ifsurfacetension

andviscousforces

canbe

neglected,thebubble

sizein

theinertialregim

emay

beapproxim

atedby

[32,38]

RB �

Q2=

5G

g1=

5 �ROFrG

2=5¼

RO

WeG

Bo

��

1=5;

ð23Þ

where

thefactor

ofproportionalityisoforder

one,itsvalue

dependingon

themodelused.In

thisregim

epolydispersity

ofthebubbles

tendsto

increasesignificantly.

Inmostpracticalcases,gravity,surface

tension,viscosityand

inertiaact

simultaneously.In

thiscase,the

modelneeds

totake

intoaccount

thevarious

contributions,renderingitquite

complex.O

neofthe

fre-quently

usedmodels

isthe

oneproposed

byJam

ialahmadietal.[42]

RB ¼

RO

5:0

Bo1:08 þ

9:261

FrG0:36

GaG0:39

þ2:14

FrG0:51

"#

1=3:

ð24Þ

Notice

theim

portanceofthe

Froudenum

berFr,w

hichcaptures

theratio

ofinertial

andgravitational

stressesand

which

canalso

beexpressed

asFr

2=

We/Bo

inorder

tomake

thelin

kwith

previousdiscussions.

2.4.2.Bubblinginto

aflow

ingliquid

Scientistshave

developedalarge

number

ofbubbling

techniques

which

involvethe

activeflow

oftheliquid

phase.Thiscreates

additionalviscous

andinertial

forceswhich

canplay

avery

important

rolein

bubbledetachm

ent.Thefirst

groupoftechniques

involvesthe

co-flow(Fig.9a,b)

orthe

cross-flow(Fig.9c)

inunconfined

conditions.Thelat-

termay

beachieved

bydragging

theorifice

throughalarge

liquidpool,

anapproach

originallyused

byBragg

[43]toblow

smallbubbles.The

in-fluence

oftheflow

ingliquid

phasecan

beincreased

ifco-or

cross-flowtake

placeunder

confinedconditions

(Fig.9d–g),sincehigh

shearrates

andhigh

flowvelocities

canbe

obtainedin

theconfined

flowregions.

Theconfining

geometry

thereforeplays

anim

portantrole

inthe

bub-bling

process,asitfixes

theflow

fieldand

thereforethe

dynamicforces

actingon

thebubble.Sm

ithwas

oneofthe

firstscientiststo

usethis

ap-proach

systematically

inorderto

blowsm

all(er)bubbles

forBragg

[44].Tow

ardsthe

endofthe

20thcentury

thesubjecthas

foundan

explosiverebirth

with

emerging

microfl

uidictechniques

[45–49].Amore

sub-stantialbody

ofwork

hasbeen

doneon

droplet,ratherthanbubble

gen-eration,

butwith

many

physical

mechanism

sbeing

thesam

e,the

interestedreader

willfind

important

inspirationin

thedroplet

litera-ture

[50–52].Inmost

oftheconfined

co/cross-flowdevices,the

influ-en

ceof

gravitycan

beneglected

becauseof

the

small

devicedim

ensions(i.e.

Bo≪

1)or

theim

portanceof

thedynam

icforces

(Ca≫

1,and/orW

e≫

1, etc.).Thisiswhy

wewillnotdiscuss

theinflu-

enceofgravity

inthe

following.

2.4.2.1.Thequasi-static

regime.W

henboth,the

gasand

theliquid

flowrate,are

smallenough

sothatthe

dynamicforces

canbe

neglected,un-confined

flowgives

thesam

eresults

asbubble

blowing

intoastationary

liquid(Section

2.4.1).Inthe

absenceofgravity

thereisno

detachment

force,hencebubbles

cannotbe

generated.Thisis

verydifferent

when

thebubble

isgenerated

under

appropriatelyconfi

nedconditions.In

this

case,thebubble

canblock

theen

tirechannel

orconstriction

inwhich

itisgenerated.The

liquidflow

thenleads

toafilling

ofthechan-

nelortheconstriction,w

hichpinches

offthebubble

(Fig.1).Thebubble

iscarried

away

bythe

slowflow

andthe

periodicprocess

restarts.Thisregim

ehas

beencalled

the“squeezing”

regime[48]

sinceitisentirely

controlledby

thenorm

alpressuresacting

onthe

interface.Sincethe

bubblecontinues

tobe

blownatconstantflow

rateQGwhile

theliquid

fillsthe

constrictionofvolum

eVcat

constantflow

rateQL ,one

findsa

simple

relationshipwhich

relatesthe

finalbubblevolum

eto

thechan-

nelgeometry

andthe

flowrates

toagood

approximation

[53,54]

VB

�VCQ

G

QL:

ð25Þ

Theprefactor

dependson

theprecise

geometry.The

volumeofthe

constrictionmay

bereplaced

bythe

3rdpow

erofa

characteristicchan-

neldimension,ifthe

geometry

doesnotcontain

awell-defined

constric-tion.W

iththis

approach,onecan

thereforegenerate

foamsover

awide

rangeofbubble

sizes(10–1000

μm)and

gasfractions

Φ,since

Φ¼

QG

QG þ

QL:

ð26Þ

Thesqueezing

regimeleads

tohighly

monodisperse

foams(PI

b2%).

Someexam

plesofobtained

foamsare

shownin

Fig.10.

2.4.2.2.Thedynam

icregim

e.Thesim

plestcasein

thedynam

icregim

eis

when

theviscous

dragwhich

isexerted

onthe

bubbleby

thefluid

over-com

esthecapillary

forcewhich

attachesthebubble

tothe

orifice

(Fig.8a).Anexam

plefrom

anexperim

entisshow

nin

Fig.11b.Thevis-

cousStokes

force,which

may

beapproxim

atedas

FV=

6πRB η

L UL ,plays

thesam

erole

asgravity

inSection

2.4.1.Equatingthe

viscousand

thecapillary

force(2πγ

RO )

leadsto

asim

pleexpression

forthe

obtainedbubble

size

RB �

γηl U

RO ¼

Ca−1

LRO:

ð27Þ

Fig.8.(a)Sum

mary

ofthedifferentforces

actingon

abubble

duringcreation.(b)

Sketchof

variationofbubble

radiusRBand

polydispersityPIw

ithgas

flowrate

QG .Corresponding

images

areshow

nin

Fig.5.(c)Stability

diagramofthe

bubblingprocess.Bubbles

detachfrom

theorifice

ifeithertheBond

number

Boorthe

Webernum

berWeGofthe

gasare

suf-ficiently

high.Inspired

from[40].

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Hence,the

obtainedbubble

sizeisinversely

proportionaltothe

cap-illary

numberCa

L oftheliquid

anddepends

linearlyon

theorifice

radiusRO .In

thecase

ofconfinedgeom

etriesone

observesawell-defined

tran-sition

fromthe

“squeezing”(Section

2.4.2.1)to

the“dripping”

regime

when

viscousforces

startto

win

oversurface

tensionforces,i.e.above

acriticalCa

L[57,58].

Upon

increasingthe

gasorthe

liquidflow

rate,oneobserves

thatthepointofbubble

detachmentm

ovesaw

ayfrom

theorifice,leading

tothe

creationof

ajet

whose

lengthdepen

dson

theflow

velocities.Thischange

ofbehaviourmarks

anim

portanttransitionfrom

the“dripping”

(or“bubbling”)to

the“jetting”

regime[51,59,60].This

transitionisgen-

erallyrationalised

usingthe

conceptofabsoluteand

convectiveinstabil-

ities,which

wediscussed

attheend

ofSection2.3.In

thesqueezing

anddripping

regimes,the

break-upofthe

gasligamentis

causedby

anabso-

luteinstability,w

hichinvades

theentire

systemand

thereforeoccurs

al-ways

attheorifice.W

iththe

presenceofnon-negligible

viscousand/or

inertialforces,theevolution

oftheinstability

canbe

slowed

downand

hencebe

“convected”with

theflow

.Thesystem

cantherefore

createa

jetwhich

breaksat

thepointw

herethe

jetis“old

enough”so

thatthe

capillaryinstability

hadenough

timeto

develop.Twodifferentextrem

esofjetting

scenariosexist.In

thefirstcase,the

jettingisdom

inatedby

theexternalflow

(Fig.11c).Thisleads

tothe

formation

ofajetw

hichthins

away

fromthe

orificeas

theinner

fluidisbeing

accelerated,leadingto

theform

ationofbubbles

which

aresm

allerthan

theorifi

cediam

eter(since

thebubble

diameter

isof

theorder

ofthe

jetdiam

eteras

discussedin

Section2.3).In

thesecond

case,thejetting

isdom

inatedby

inertialcontributionsofthe

innerfluid,leading

toajet

ofincreasingthickness

asthe

innerfluid

isdecelerated

bythe

outerfluid

(Fig.11d),and

tobubble

sizeswhich

arelarger

than

theorifi

cediam

eter.Thiscase

was

alreadydiscussed

inSection

2.4.1.2.In

thefirstcase,researchers

havebeen

ableto

showthatthe

transi-tion

between

drippingand

jettingiswelldefined

bythe

Weou

t andRe

out

numbers

ofthe

outerfluid

[61].Thisis

show

nin

aphase

diagramin

Fig.11e(redraw

nfrom

[61]).Thetransition

dependsstrongly

onthe

densityratio

λDand

onthe

viscosityratio

λVofboth

fluids.Interestingly,one

cansee

that

Weout and

Reout are

proportionalatlow

Renum

ber,meanin

gthat

the

transitionoccurs

ata

fixed

Canum

bersince

Ca=

We/Re.This

observationwas

confirmed

bydifferentexperim

ents

andawide

rangeoffluid

couplesand

flowconditions

[51,61].Fig.11fshow

san

example

ofatypicalphase

diagramobtained

forthegeom

etryshow

nin

Fig.11a(from

[61]).What

thisphase

diagramalso

showsis

thatthe

capillarynum

berofthe

outerflow

definesthe

transitiononly

ifinertial

forcesof

theinner

fluid

canbe

neglected.In

thelim

itof

smallCa

out ,the

transitionseem

sto

bedefined

byacriticalW

ebernum

-ber

Weinofthe

innerfluid.Forinterm

ediatecases

(andifReynold

num-

bersare

sufficientlylow

)one

observesthatthe

transitionoccurs

when

Wein+

Weou

t ≈O(1)

[61].The

discussionbetw

eendripping

andjetting

may

seemsom

ewhat

academic.H

owever,it

isvery

usefulevenfor

thepractitioner.The

firstreason

isthat

asone

entersthe

jettingregim

e,monodispersity

com-

monly

drops.Thisisdue

tothe

factthatthemaxim

umofthe

curvegiv-

ingthe

unstablewavelengths

(Fig.4b)is

quitesm

ooth,meaning

thatthe

systemmay

selectawider

rangeofbubble

sizes.Moreover,w

henthe

systemisin

thejetting

state,thebubbles

donot

needto

be“nucle-

ated”with

theassociated

problemsofthe

initialentrypressure,helping

with

pressurereduction

indevices

andfor

parallelisation.Lastbut

notleast,w

ithbubble

sizesbeing

oftheorder

ofthejet

diameter

atbreak-up

(Section2.3),creating

thinjets

isan

efficientway

tocreate

alarge

number

ofsmallbubbles,dow

nto

afew

microm

etres[62,63].

Theobtained

bubblesizesdepend

onthe

operatingregim

eofthe

de-vice.Bubble

sizesin

thedripping

regimetend

tobe

welldescribed

byEq.(27).Bubble

sizesin

thejettin

gregim

edom

inatedby

theouter

fluidphase

arefound

tobe

wellcaptured

bysim

plescalings

ofthetype

RB

RO �

QG

QL

��

β;

ð28Þ

with

thepre-factor

andβdepending

onthe

flowgeom

etryand

onthe

viscosityratio

λV .Typically,β

hasbeen

foundto

liebetw

een0.4

and0.5.For

example,researchers

foundexperim

entallyand

theoreticallythatβ

=0.5

ifQG/QL→

0,andβ=

2/12

ifλV=

ηG/η

L→

0[26].For

highReynolds

andW

ebernum

bersitwas

foundthatβ

=2/5[64].

Upto

nowwehave

only

discussedperiodic

bubblingregim

es.Itneeds

tobe

saidthat

justas

inthecase

ofbubblin

gfrom

anorifi

ceinto

astation

aryliquid

(Section3.2.1)

co-flow

canlead

toperiod-

Fig.9.Selectionofthe

most

commonly

usedgeom

etriesto

generatebubbles

inconditions

where

both,thegas

andthe

liquidare

flowing.The

flowmay

eitherbe

unconfined(a–c)

orconfined

(d–g).Flowpatterns

aretypically

thatofco-flow

ofbothphases

(a,b,d–f)or

ofcross-flow(c,g).

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doublingand

chaoticbubbling

[65](Section

3.2.2).Suchphenom

enaoccur

oftenatthe

transitionbetw

eenregim

es.

2.4.3.Breakupofbubbles

undershear

Inthis

sectionweconsider

thebubbles

aspassive

objects,which

arealready

inthe

liquid.Weask

ourselveshow

thesecan

bebroken

intosm

allerbubbles

bythestresses

imposed

byan

activeextern

alshearflow

(Fig.1).Thisprocess

hasbeen

widely

studied.Eventhough

itis

conceptuallysim

ple,aproperphysicaldescription

turnsouttobe

highlycom

plex.How

ever,thegeneralphenom

enologyis

welldocum

ented,and

canbe

summarised.

Sincemost

oftheexperim

ental

work

hasbeen

doneon

dropsrather

thanon

bubbles,andsince

theconcepts

arevery

similar,w

eshallstay

onarather

generallevelhereby

simply

consideringtw

oim

miscible

fluidsLet

uscon

sideradrop

ofafluid

(gasor

liquid)of

viscosityηD

suspendedin

afluid

ofviscosityηCexposed

toashear

flow(Fig.12a).

Thebubble/drop

isfirst

deformed

into

anelongated,

steadysh

apewhose

geometry

dependson

thecapillary

number

Caofthe

imposed

shearflow

[66–70].Examples

ofresultingbubble

shapesare

shownin

Fig.12b(adapted

from[70]).W

ithincreasing

Caand

ηCthe

bubblesbe-

comeincreasingly

elongated—

butstill

reachasteady

equilibriumsh

ape.Note

howstrongly

suchabubble

canbe

steadilydeform

ed

without“breaking”.The

deformation

iscom

monly

measured

bythe

as-pect

ratioD=

(A−

B)/(A+

B)(Fig.12a).

Upon

furtherincrease

ofthecapillary

number,a

criticalcapillarynum

berCa*

(andhence

acriticaldeform

ationD*)

isreached

beyond

which

thebubble

canno

longersustain

asteady

shape.A

saconse-

quence,Dincreases

andeventually

reachesapoint

where

capillaryin-

stabilities(Section

2.3)break

theelongated

objectinto

pieces.Thus,theinstability

ofthe

aspectratio

isdirectly

linkedto

thebreakup

ofthe

bubble—

eventhough

theyare

notsim

ultaneous.W

hatsets

thecriticalcapillary

number

Ca*?Anim

portantpiece

ofwork

was

doneby

H.P.G

race[71]

forthe

caseofdroplets.H

eshow

edthat

Ca*depends

stronglyon

theviscosity

ratioλV ,w

hichisshow

nin

thegraph

inFig.13a

andby

correspondingph

otographsof

bubblesand

dropstaken

justbefore

Ca*isreached

inFig.13b.It

isstraightfor-

ward

tosee

theasym

metry

ofthecurve:

with

twogiven

fluids

ofdifferentviscosities,results

aredifferentdepending

onwhich

fluidisin-

sideor

outsidethe

drop.When

theinnerfluid

hasamuch

lower

viscos-ity

(asfora

bubbleinside

water,w

hereλV~10

−2),the

resultsshow

thatthe

bubblecan

supportreasonably

highCa*

≈3before

beingirrevers-

iblyextended

andruptured.

Once

theelongated

mother-bubble

(ormother-drop)

hasruptured

justaboveCa*,the

resultingsize

ofthecreated

objectsisdirectly

linked

Fig.10.Examples

ofmonodisperse

foamsgenerated

viaco-flow

ofgasand

foaming

solutionthrough

aconstriction.(a)

Examples

offoamsofdifferentbubble

sizeand

liquidfraction

ina

channeljustafterthegeneration

(from[55]).(b)

Ordered

foamswith

differentliquidfraction

(from[56]).(c)

Monodisperse

foamundergravity

(courtesyA.van

derNet)

(allbubblesizes

200–500μm

).

Fig.11.Thedripping

tojetting

transitionin

co-flowconditions.(a)

Schemeofa

commonly

usedsetup.(b)

Dripping

regime—

thebubble/drop

iscreated

atthetip

oftheorifice.(c)

Jettingregim

edom

inatedby

theexternalflow

.(d)Jetting

regimedom

inatedby

theinternalflow

.(a–dfrom

[59])(e)

Transitionbetw

eendripping

andjetting

fordifferentdensity

andviscosity

ratioswhen

thebehaviour

isdom

inatedby

theflow

oftheexternalfluid

(redrawnfrom

[61]).(f)Transition

between

drippingand

jettingwhen

atlowReynolds

number

andwhen

both,the

internalandthe

externalfloware

important

(from[59]).

11W.D

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tothe

criticalelongationD*:the

ligamentradius

atCa*provides

anesti-

mation

ofthediam

eterofthe

newly

createdentities

(seeSection

2.3).Since

boththe

criticalcapillarynum

berCa*and

theassociated

deforma-

tionD*increase

with

decreasingλVforλ

Vb1(Fig.13),this

implies

thatthe

sizeofthe

resultingobjects

decreasesfor

decreasingλV .

Theabove

descriptionscorrespond

todeform

ationsgenerated

bya

pureshear

flow.In

fact,itwas

rapidlydiscovered

thatthe

deformation

andstability

ofbubblesand

dropsin

aflow

fielddepend

stronglyon

thetype

ofimposed

flow[68,72,73].Elongational,rotationalor

simple

shearflowdoes

notdeformabubble

inthe

sameway,and

consequent-ly,the

criticalcapillarynum

berCa*

dependson

thetype

offlow.A

vastbody

oftheoreticalandnum

ericalresearchdeals

with

suchissues,con-

sideringgradually

varyingtypes

offlow,w

hichare

shownin

Fig.14a.Such

flowfields

canbe

expressedvia

thevelocity

gradient

∇u¼

12G

1þα

1−α

0−1þ

α−1−

α0

00

0

2435;

ð29Þ

where

Gfixes

theam

plitudeofflow

field.Thebalance

between

shearand

vorticityof

aflow

canthen

besim

plydescribed

bythefactor

α,

which

givespurely

extensionalflow

inthecase

ofα=

1and

simple

shear

flow

forα=

0(see

flow

patternsin

Fig.14a).Experimen

tally,such

kindof

flow

scan

begenerated

byfour

independen

tlyrotating

cylinders.The

capillarynum

berof

these

flow

sis

thentypically

expressedas

Ca¼

GηR

B

γ;

ð30Þ

where

RBisthe

radiusofthe

undeformed

bubble/drop.Asshow

nin

Fig.14b,thedependence

ofCa*on

theviscosity

ratioλV

dependsnon-negligibly

onα.In

thelim

itofbubbles(sm

allλV ),the

crit-icalcapillary

number

Ca*decreases

asone

goesfrom

α=

0(pure

shearflow

)to

α=

1(purely

extensionalflow

).For

curiosity,note

the

differenceswith

theregim

eat

highλV :

with

onlysim

pleshear

(α=

0)one

cannotbreak

veryviscous

drops,while

with

extensionalshearitispossible

(αapproaching

1).Besides

theRayleigh–Plateau

instability(Section

2.3)which

breaksthe

elongatedligam

ents,thereare

twoother

important

mechanism

swhich

canalso

leadto

thecreation

ofsmaller

bubbles/drops.Thefirst

phenomenon

iscalled

“tipstream

ing”and

isshow

nin

Fig.15.Itcreatesathin

threadwhich

startsatthe

tipofan

elongatedbubble/drop,w

hereithas

anon

-trivialshape[74].This

thread

breaksinto

tinybubbles/

drops.This

isvery

differentfrom

therupture

ofthe

main

bubble(drop)

[75].Asshow

nin

thegraph

ofFig.13a,thisphenom

enonoccurs

wellbelow

thecriticalcapillary

number

Ca*for

breakup.Thepresence

ofsurfactantsplays

anim

portantrolein

thisprocess

andnum

erousim

-portantadvances

havebeen

made

recentlyin

orderto

elucidatethe

keymechanism

satplay

inthis

process[[]].

Anotherm

odeofrupture

oftheelongated

bubble/dropwhich

occurswellbefore

theonset

ofthecapillary

instabilitytakes

placewhen

theflow

issuddenly

stopped,orifthe

bubble/dropisconvected

towards

aregion

oflow

ercapillary

number

[14,26,69].Assh

owin

Fig.16a,inthe

absenceof

deforming

stresses,thebubble/drop

tendsto

goback

toits

minim

alsurfaceshape,w

hichisasphere.In

thisprocess

ofcon-traction

,aneffect

called“en

d-pinching”can

occur:abulbous

headform

sat

eachen

dof

thefilam

ent.Thefilam

entbecom

esunstable

inthe

vicinityofthe

head,creatingtw

osm

allerheads

attherem

ainingfil-

amentand

soforth.This

canbe

nicelyseen

inthe

rightimage

sequenceofFig.16a.W

hetherend-pinching

occursdepends

onanum

berofpa-

rameters.Recently,a

phasediagram

was

proposedby

Driessen

etal.

[26](Fig.

16b),show

ing

underwhich

condition

therelaxation

ofstretched

dropsofdifferent

viscositiesin

airare

stableor

not.Itseem

sthat

asin

thecase

ofinfinitelylong

fluidcylinders

(Section2.3),one

ofthe

controlparameters

isthe

Ohnesorge

number

Oh(Table

2).Thesec-

ondcontrolparam

eteris

thedeform

ationofthe

dropat

themom

entwhen

thedeform

ingsh

earflow

isstopped.O

nceagain

,most

ofthe

Fig.12.(a)Schem

eofa

bubblebeing

deformed

inashear

flow.(b)

Photographsofthe

equilibriumshape

ofasheared

bubbleatdifferentCapillary

numbers

Caand

atdifferentviscosityratios

λV(adapted

from[70]).D

isameasure

ofthedeform

ationgiven

byD=

(A−

B)/(A+

B).

Fig.13.(a)The

“Grace

plot”:Criticalcapillarynum

berCa*

beyondwhich

thedrop/bubble

becomes

unstablein

apure

shearflowas

afunction

oftheviscosity

ratioλVofthe

dispersedand

continuousphase(adapted

from[71]).(b)Photographsofdrop/bubble

shapesatthecriticalpoint,generated

inan

extensionalflowfora

wide

rangeofviscosity

ratiosλV(adapted

from[68]).

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experiments

havebeen

performed

with

drops,ratherthan

with

bub-bles,but

thephysicalph

enom

enacon

cernedhere

arevalid

forboth

systems.

2.4.4.Gas

entrainmentatfree

surfacesIfone

considersthe

flowofliquid

with

afree

surface,many

sourcesof

perturbationsof

theinterface

canlead

tothe

entrain

men

tof

the

surrounding,passiveair.This

phenomenon

isgenerally

relatedto

thepresence

ofageom

etricalcon

straintor

anabrupt

changeof

cross-section

orspeed

between

neighbouringfluid

zones[13].W

ell-known

examples

includethe

plungingjet

(Fig.17a)[80],in

which

theregion

ofgas

incorporation

opens

likea

cone,

orthe

hydraulicjum

p(Fig.17c)

(Section3.2.3),w

here

airis

entrained

when

theaverage

flow

speedof

aliquid

slowsdow

nsuddenly.Sim

ilarsituation

sarise

when

theflow

isaccelerated

temporarily

asitpasses

overobstacles

orwhen

thefree

interfacewith

airvanishes,forexample

when

flowdisap-

pearsunder

aninclined

plane(Fig.17d).

Theplunging

jetisthe

most

documented

situationin

foam-related

work

[80].Due

toits

practicalrelevance(for

example

inthe

kitchensink

andthe

bathtub),itisused

asastandardised

foamability

technique(Ross–M

ilestest,Section

3.2.3).Moreover,itcorresponds

toaprototype

(orparadigm

)experim

ent,with

which

onecan

probethe

basicmecha-

nismsat

work

inmore

complex

hydrodynamic

processes[80].O

neof

thekey

parameters

inthis

processis

theim

pactvelocity

ofthejet

U,

i.e.itscapillary

number

Ca=

ηL U

/γ.Below

athreshold

value,asteady

deformation

oftheinterface

isobtained

without

theincorporation

of

gas(insetofFig.18).A

nim

portantfeatureisthatthe

radiusofcurvature

ratthe

edgeofthe

gas“cusp”

vanishesexponentially

with

thecapillary

numberCa,i.e.r~

exp(−Ca)

[80].Thissteady

interfacialdeformation

be-com

esunstablebeyond

acriticalcapillary

numberCa

c (andan

associatedcriticalcusp

radiusrc ):a

viscouspum

pingofthe

entrainedfluid

intothe

narrowing

nosegenerates

alubrication

pressure,which

finallyprevents

anysteady

forcebalance

[80].Asin

thecase

ofbubblebreak-up

undershear,above

thisCa

c thecusp

isincreasingly

stretcheduntiliteventually

breaksinto

smaller

bubblesby

themechanism

sdiscussed

inSection

2.3.The

entireprocess

issketched

inFig.1.It

canproduce

quitemonodis-

persefoam

swith

bubble-sizesand

monodispersity

dependingon

thejet

speed—

asthe

regularbath

tuberor

dishwasher

may

confirm.

Thephysics

ofplungingjets

hasbeen

investigatedin

detailforhigh-

viscosityliquidsin

air,asbothexperim

entsand

modelisation

turnoutto

beeasier

toperform

thanfor

low-viscosity

fluids.Different

investiga-tion

slead

totheprediction

that,as

longas

inertia

canbe

neglected(sm

allRe),thecriticalcapillary

number

Cac scales

logarithmically

with

theviscosity

ratioλV[80,81]

Cac �

−ln

ηG

ηL

��

¼ln

λV

ðÞ:

ð31Þ

Thisis

shownin

Fig.18(adapted

from[80]).Lorenceau

etal.[81]

haveshow

nthat

thisrelationship

canbe

extendedover

fiveorders

ofmagnitude

tosm

allerviscosity

ratioswhen

working

with

twoliquids.

Fig.14.(a)Differenttypesofflow

fieldsinwhich

thebubbles/dropsare

deformed,ranging

frompurely

extensional(α=

1)to

pureshearflow

(α=

1)(adaptedfrom

[68]).(b)Effectofthe

flowtype

onthe

criticalcapillarynum

berCa*

beyondwhich

thebubble/drop

shapebecom

esunstable

(adaptedfrom

[68]).

Fig.15.Tip-streaming:production

oftinybubbles

atthetip

ofbubblessheared

inaCouette

cell(aand

bfrom

[79])and

atthetip

ofadrop

(cfrom

[24]—Leonhard,H

.,1996,“Experiments

ontipstream

ing,”unpublished).

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Anothernon-trivialresultconcerns

therate

ofgasentrained.Itturns

outthat

itdepends

onlyon

thegas

viscosity,and,surprisingly,noton

thefluid

viscosity[80].This

isdue

tothe

factthatthegas

filmdecouples

thejet

fromthe

restofthefluid.

Incontrast

tothe

previouslydescribed

visco-capillaryentrainm

entmodes,

thephysics

oflow

-viscosityplunging

jetsis

commonly

characterisedby

highReynolds

numbers

(ReN100)

andlow

capillarynum

bers(Ca

b1).U

ndersuch

conditions,differentinertialinstabilities

ofthegas/liquid

interface

candevelop

alreadybefore

theim

pact,sothatthe

cross-sectionofthe

jetisno

longerconstant.One

oftheseinsta-

bilitiesisthe

Kelvin–H

elmholtz

instability[82].It

ariseswhen

twoim

-miscible

fluids

with

differentdensities

flow

sideby

sideat

verydifferent

velocities(Fig.19a).Beyond

acriticalvelocity

differencecer-

tainwave

lengthsofnaturalinterfacialfl

uctuationsbecom

eunstable

andare

amplified

bythe

flow[82].A

nexam

pleofthis

instabilityon

asim

plewater

jetin

airisvisible

onthe

leftside

ofthejet

inFig.19b.If

thisinstability

developsbefore

theviscous

pumping

effectstarts

tobe

important

theymodify

inanon-trivialm

annerthe

entrainment

andbreak-up

mechanism

.Differentscenarios

arisewhich

areoften

studiedin

separateregim

esdepending

ontheinitialam

ountof

disturbancerelative

tothe

speedofthe

jet[80].

TheKelvin–H

elmholtz

instability

canlead

toanother

typeof

airentrainm

entwhen

itisdriven

tosufficiently

highW

ebernum

bersWe

(i.e.bothRe

andCa

needto

behigh

sinceWe=

ReCa).Inthis

casethe

destabilisationofthe

interfaceisgreatly

amplified,leading

tocom

plexflow

structureswhich

tendto

leadto

theform

ationofsm

allerligaments

which

thenagain

undergothe

typeofcapillary

instabilitiesdiscussed

inSection

2.3.This

kindof

fluid

break-upis

called“atom

isation”or

“fragmentation”

(rightside

ofthejet

inFig.19b

andwave

breakingin

Fig.19c)[30,83–86].U

nderthe

rightflowconditions,it

canlead

tothe

entrainment

ofasignificant

amount

ofairand

henceto

foamgenera-

tion,as

shown

bythe

example

ofbreaking

waves

inFig.

19cand

discussedbriefly

inSection

3.2.3.Gas

entrain

men

tdoes

not

necessarily

require

high

velocityflow

:trappingpockets

ofgasin

afluid

canoccu

rat

lowflow

veloc-ities

inth

epresen

ceof

confined

geometries

andwhen

flow

condi-

tionsvary

fasten

ough

soth

atth

efree

interface

getslocally

suffi

ciently

deform

edto

entrain

agas

pocket.

Wash

ing

your

han

dswith

soapis

probably

enou

ghof

anexp

erimen

tto

convin

ceyou

rselfof

this

effect.High

lyviscou

sfluidsmay

alsosim

ply

befold

edin

away

totrap

air.

2.5.Bubblegeneration

without

topologicalchanges—

phasetransitions

One

ofthemain

techniquesto

obtainbubbles

withoutthe

necessityof

atopological

transitionof

theinterfaces

istheir

generationvia

a

liquid→

gasphase

transitionwhich

occurslocally

inthe

fluidto

createbubbles.There

aretw

odifferent

classesofphase

transitionswhich

arecom

monly

usedfor

bubblegeneration

[88–90].Thefirst

classoccurs

inpure

fluids,where

theliquid

phaseturns

intovapour.A

sindicated

inthephase

diagramof

Fig.20a,

this

caneither

betriggered

bya

pressuredrop

(“cavitation”)or

bya

temperature

rise(“boiling”).

Cavitationis

commonly

createdby

rapidflow

swhich

createlocalised

pressuredrops

throughconstrictions

orrotating

propellersor

byhigh

energy

pressurewaves,such

asen

counteredin

ultrasound.

Bubbleand

foamgeneration

byboiling

isknow

nto

most

ofusfor

thecase

ofwater

ormilk

inthe

kitchen.The

secondclass

ofliquid→

gastransitions

occursin

supersaturatedliquids

with

inwhich

agas

isdissolved

which

comes

outof

solutionupon

apressure

drop(Fig.20b)

oratem

peratureincrease.By

farthe

mostcom

monly

usedmechanism

inthe

caseoffoam

generationisthe

oneencountered

uponapressure

drop,which

iscalled

“effervescence”and

which

isparticularly

wellknow

nto

loversoffizzy

drinks,beeror

champagne.

Thesedifferentw

aysofgenerating

bubblesin

anoriginally

homoge-

nousfluid

haveacom

mon

physicalmechanism

:thesystem

needsto

bedriven

intoasupercriticalstate,in

which

smallbubbles

nucleatespon-

taneouslyand

thengrow

(Fig.2).Understanding

thenucleation

ofbub-bles

inaliquid

hasalong-standing

history[88–91],w

ithanum

berof

openquestion

sthat

remain

tobe

answered.The

complexity

ofthis

Fig.16.(a)Abubble

breaksasitrelaxesfroman

elongatedshape

when

theapplied

shearisstopped(from

[68]).Theway

itrupturesdependsonitsinitialaspectratio

andthe

viscosityratio

ofthetw

ofluids.This

isshow

nin

aphase

diagramin

(b)which

showsastable

andan

unstablezone

fordeform

eddroplets

ofdifferentviscosities

inair

usingthe

aspectratio

andthe

Ohnesorge

number

Ohas

keyparam

eter(from

[26]).Here

thefullcircles

correspondto

dropletswhich

contractwithout

breaking.Theem

ptycircles

areobtained

formore

stretchedand

viscousdrop,w

hichfinally

breakas

theyrelax.

Fig.17.Differentconfigurations

leadingto

gasentrainm

entatfreesurfaces.

Adapted

from[13].

14W.D

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es/A

dvancesin

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xxx(2015)

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Pleasecite

thisarticle

as:Drenckhan

W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

ColloidInterface

Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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subjectresultsfrom

thefactthat

bubblenucleation

needsto

startwith

thecreation

ofasingularity

within

thefluid,w

hichcalls

uponamolec-

ulardescription

oftheprocess

(challengefor

theory)and

which

isex-

tremely

sensitiveto

thepresence

ofim

purities(challen

gefor

experiments).

Ifoneneglects

thevery

initialstageofthe

holeopening,one

may

as-sum

ethatthe

createdbubble

islarge

enoughso

thatonecan

applythe

conceptofa

surfacetension

andabulk

descriptionofthe

gasand

theliquid

phase.Inthis

caseone

canconsider

thetw

okey

contributions

tothe

freeenergy

ofthebubble

formation.O

nthe

onehand,once

thesystem

isin

asupercriticalstate,it

willreduce

itsfree

energyby

thephase

transitionwith

anassociated

changein

thefree

energydensity

ΔGV

ΔGbulk ¼

−VB Δ

GV ¼

−43πR3B Δ

GV:

ð32Þ

Atthe

sametim

e,thecreation

ofabubble

requiresthe

creationofa

vapour/liquidor

gas/liquidinterface,w

hichleads

toan

increasein

freeenergy

ΔGsurf ¼

AB Δ

GS ¼

4πR2B γ

:ð33Þ

Hence,the

totalchangein

freeenergy

may

beapproxim

atedby

ΔGtot ¼

ΔGsurf þ

ΔGbulk ¼

πR2B γ

−43πR3B Δ

GV:

ð34Þ

Makin

gth

eapproxim

ationthat

thesurface

tension

andthevol-

umefree

energy

areindep

enden

tof

thebu

bblesize,on

eobtain

sa

freeen

ergyfunction

which

ischaracterised

byawell-defi

ned

maxi-

mum

,assketch

edin

Fig.21.Thismaxim

umsets

acriticaln

ucleationbarrier

ΔGCwhich

isassociated

with

acriticalbu

bblerad

iusRc .A

spontan

eouslygen

eratedbubble

which

issm

allerth

anth

iscritical

sizewillsh

rinkan

ddisappear,w

hile

abubble

which

isbigger

than

this

criticalsizewillgrow

.Thenucleation

ratenof

bubblesmay

beapproxim

atedby

n�exp

−ΔGC

kT

��:

ð35Þ

Thegrow

thrate

ofthesebubbles

dependson

thesaturation

condi-tions

andcan

beapproxim

ated[88–92]—

butthis

isbeyond

thescope

ofthearticle.

Amore

mechanicalw

ayto

approximate

thecriticalradius

directlyis

touse

theLaplace

law

RC ¼

2γΔP ¼

2γσPf

ð36Þ

where

σisthe

“supersaturation”ofthe

liquid(σ

=α−

1,with

αbeing

the“saturation”)

andPf is

thefinalpressure.The

definitionofthis

super-saturation

dependson

theconsidered

scenarioand

canbe

obtainedfrom

thephysicalparam

etersofthe

system.In

thepresence

ofonephase

only(cavitation

andboiling,index

“1Ph”)one

needsto

takeinto

accountthe

vapourpressure

PVofthe

liquid.One

may

thereforedefine

ΔP¼

PV −

Pf ¼

σ1Ph P

fð37Þ

σ1Ph ¼

α−1¼

PV

Pf −

1:

ð38Þ

Inthe

caseoftw

ophases

(index“2Ph”),i.e.a

liquidbeing

supersat-urated

byadissolved

gas(effervescence),one

needsto

takeinto

ac-count

thevapour

pressureof

thesolven

tand

thepartialpressure

ofthe

dissolvedgas.The

latterbeing

generallymuch

higher,onecan

ne-glectthe

vapourpressureand

expressthe

supersaturationusing

Henry's

law

ΔP¼

Pi −

Pf ¼

HXi −

Xf

ðÞ¼

σPi

ð39Þ

σ2Ph ¼

α−1¼

Xi

Xf −

1ð40Þ

where

H=

Pf /X

f isHenry's

constantandXi and

Xf are

themole

fractionsofthe

dissolvedgas

inthe

initialandfinalstate,respectively.

Due

tothe

nucleationbarrierΔ

GC ,one

generallyfinds

thatsupersat-urations

ofafew

1000are

requiredin

verypure

liquids.Thereason

why

wesee

nucleationand

growth

atmuch

lowersupersaturationsin

every-day

lifeis

dueto

thepresence

ofim

puritiesor

pre-formed

bubbleswhich

actas

nucleationsites

(Fig.22)and

which

may

dramatically

de-crease

thenucleation

barrier.Assketched

inFig.22

(inspiredby

[90]),scientists

groupthe

differentnucleation

scenariosin

twomain

groupsof

nucleation.“H

omogen

eousnucleation

”(Typ

eI)

occurswith

outthe

presenceofany

nucleationagents

andtherefore

atveryhigh

super-saturations.“H

eterogeneousnucleation”

occursin

thepresence

ofnu-cleation

agents.Inthis

caseone

differentiatesbetw

eenthree

types,follow

ingthe

kindof

agentspresent

andthe

specific

physicalconditions

[90]:

Fig.18.Criticalcapillarynum

berCac asa

functionofthe

viscosityratio

λVbeyond

which

airisentrained

byan

impacting

liquidjet.Inset:sketch

oftheinitialentrainm

entsteps.

Adapted

from[80].

Fig.19.(a)Destabilisation

ofagas/liquid

interfacedue

tothe

Kelvin–H

elmholtz

instability.(b)

Kelvin–H

elmholtz

instabilityon

aliquid

jet.Thegrow

thofthe

instabilityatsufficiently

highWeleads

tothe

formation

ofcomplex

patternswhich

canlead

tothe

creationofsm

alldrops

(atomisation,from

)or

theentrainm

entofairbubbles,as

shownby

breakingwaves

in(c)

(from[87]).

15W.D

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es/A

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xxx–xxx

Pleasecite

thisarticle

as:Drenckhan

W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

Heterogen

eousnucleation

—Type

II(classical):Nucleation

occursin

thepresence

ofsolidim

puritieswhich

caneither

besm

allparti-cles

dispersedin

theliquid

orthesolid

wallofthe

container.Reason-ably

highsupersaturation

sof

order100

arerequired

forbubble

generationin

thiscase.

Heterogen

eousnucleation

—Type

III(classical):Nucleation

occursin

thepresence

ofpre-existinggas

cavities(on

thecontainersurface,

onsuspended

particlesor

freelyfloating

inthe

liquid)ofradius

ofcurvature

smaller

thanthe

criticalradiusRC .H

enceone

stillneedsto

nucleate,butsupersaturations

aresignificantly

lower

thanin

thecase

oftypeIand

IInucleationsince

the“holes”

inthe

liquidare

al-ready

made.

Heterogen

eousnucleation

—Type

IV(non-classical):Bubbles

aregenerated

frompre-existing

gascavities

(onthe

containersurface,

onsuspended

particlesor

freelyfloating

inthe

liquid)ofradius

ofcurvature

largerthan

thecriticalradius

RC .Since

thereisno

energybarrier

toovercom

eitis

referredto

as“non-classical”

nucleation.These

cavitiesgrow

assoon

asasm

all[93]levelofsupersaturationisreached.Type

IVnucleation

may

followtype

IIInucleation,forex-

ample

ifbubblesnucleate

incavities

onthe

wallleaving

behinda

bubblelarge

thanthe

criticalradius.

Inthe

caseofheterogeneous

nucleation,thebubble

generationata

walloratan

impurity

becomes

onewhich

requiresatopologicalchange

todetach

thebubble

fromthe

nucleatingagent.This

providesthe

same

conditionsasdiscussed

inSection

2.4.1with

thedifference

thatthebub-

bleisblow

nby

aphase

transition.

3.Techniqu

esof

foamform

ation

3.1.Introduction

Inthis

sectionwediscuss

someofthe

mostcom

monly

usedfoam

ingtechniques.These

techniquesmake

useof

thefundam

entalmecha-

nismsofbubble

generationdiscussed

inSection

2.How

ever,tothe

de-spair

ofthose

who

want

tomodel

theirbeh

aviour,they

commonly

combine

differentmechanism

s.Moreover,they

arecom

monly

runin

away

thatthegeneration

ofonebubble

cannotbeconsidered

asisolated

fromthegeneration

ofthe

others(as

wedid

throughout

most

ofSection

2).Hen

ce,eventhough

themain

mechanism

sof

most

tech-niques

may

beiden

tified,predicting

theproperties

oftheobtain

edfoam

asafunction

ofthe

differentdevice

parameters

isnot

simply

a

questionofextrapolating

ourdiscussion

fromthe

previoussection

andtherefore

remains

amajorchallenge.M

oreover,onlyfew

ofthecom

plexfoam

ingtechniques

havebeen

investigatedsystem

aticallyin

thepast.

Itisforthis

reasonthatthe

descriptionofthe

differentfoaming

tech-niques

remains

hereon

asom

ewhat

qualitativelevel.W

ehave

triedto

groupthe

differenttechniquesby

theirmain

mechanism

s,goingevery

timefrom

thesim

plestconfiguration—

where

someofthe

actingphys-

icsisunderstood

—to

themostcom

plexone.In

ordertoguide

theread-

erwho

may

consultthis

articlein

the

searchfor

anappropriate

techniquefor

hisapplication,w

ehave

compiled

someofthe

keyfea-

turesofcom

monly

usedtechniques

inTable

3.

3.2.Mechanicalfoam

ingtechniques

3.2.1.Foaming

bybubbling

intoastationary

liquidThe

processof

bubblinginto

astationary

liquidis

ofgreatim

por-tance

inanum

berofindustrialprocesses,the

most

common

examples

beingbubble

columnreactors

[94,95]or

foamflotation/fractionation

tanks[96,97].In

most

applications,bubblingoccurs

throughmany

ori-fices

inparallel,th

roughaperforated

plate(regular

distributionof

equal-sizedholes)

orthrough

aporous

disc(irregular

arrangement

oforifi

cesof

irregularcross-section,but

typicallywith

ach

aracteristicsize).Researchers

findthat

eventhough

—in

principle—

thebubbling

processon

eachindividualorifice

canbe

describedas

inthe

caseofan

Fig.20.Differentpossibilities

tonucleate

bubblesin

asolution.(a)

Liquid→

gastransition

ofonephase

viapressure

drop(“cavitation”)

oratem

peraturerise

(“boiling”).(b)Liquid

→gas

transitionupon

changeofsolubility

ofagas

dissolvedin

aliquid,here

byapressure

drop(effervescence).

Fig.21.Changeoffree

energyduring

bubblegeneration

fromasupercriticalliquid.

16W.D

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es/A

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Pleasecite

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as:Drenckhan

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esA,The

scienceoffoam

ing,Adv

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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isolatedorifice

inSection

2.4.1,anum

berofcom

plexinteractions

occurbetw

eenthe

closelyspaced

orificeson

theplate

orthe

discwhich

needto

betaken

intoaccount

[33,94].Theseinteractions

may

arisefrom

acoupling

throughthe

reservoiror

throughthe

liquidby

hydrodynamic

flowsor

directbubblecontact.In

anattem

pttoelucidate

thesecom

plexinteractions,researchers

havestarted

tolook

atmore

simplified

systems

ofafew

orificesofcontrolled

spacingand

dimensions

(Section3.2.1.2).

Characteristic

bubblesizes

with

these

kinds

oftechniques

rangefrom

afew

hundredmicrom

etresto

onecentim

etre.Thefoam

canbe

verymonodisperse,ifidenticalorifices

aredriven

undertherightcondi-

tions(Section

3.2.1.2),orpolydisperse,w

ithpolydispersities

beingtyp-

icallyup

to50%

.Foam

productionrates

dependon

thenum

berof

orifices

used.With

porousdiscs

onecan

easilygenerate

1L/m

in/cm2

ofporesurface.Since

thebubbles

needto

begenerated

inaliquid,the

foamjust

aftergeneration

hasgenerally

ahigh

liquidfraction.A

sbub-

blesrise

away

fromthe

generationsite

theliquid

isdrained

bygravity

tocreate

foamswith

lowliquid

fractions.

3.2.1.1.Complex

bubblingregim

esfor

oneorifice.In

Section2.4.1

wecon-

sideredperiodic

bubblingregim

esonly.H

owever,even

asingle

orificecan

bedriven

inhighly

non-linearbubbling

modes,depending

onthe

bubblingconditions.A

sshow

nin

Fig.23aand

bat

intermediate

gasflow

ratesthe

bubblingperiods

T(tim

ebetw

eentw

obubble

detach-ments)

bifurcate,leadingto

twoor

more

bubblingperiods.In

certainranges

afully

chaoticbubbling

isobserved

with

arandom

distributionofthe

bubblingperiods

[93,98–102].Multiple

bubblingperiods

canlead

tobubble

sizedistributions

with

severalpeaks(bi-

ortri-disperse,etc.)

while

chaoticbubbling

may

generatepolydisperse

foams.W

hichone

ofthe

differentregimes

isselected,depends

stronglyon

theliquid

proper-ties,the

orificeconditions

andthe

gasflowrate.A

ttheoutset,researchers

believedthatthis

complex

behaviourresulted

fromintricate

interactionsbetw

eenthe

detachedbubble(s)and

theone

which

isbeing

generated,orfrom

pressurefluctuations

inthe

reservoir.Theinterpretation

which

seemsto

beem

ergingnow

isthatitis

ratherthe

dynamics

ofthegas/liq-

uidmeniscus

within

andatthe

orificewhich

controlsthis

interestingbe-

haviour[93,100–102].Forexam

ple,Fig.23bshow

sthatthe

complexity

islostas

soonas

theorifice

radiusro

islarge

enough.

3.2.1.2.Bubblingfrom

severalorifices/perforatedplates.If

onebubbles

throughseveralorifices

inparallel,the

resultingbehaviour

dependson

whetherthe

bubblesareblow

nfrom

thesam

egasreservoir.A

ttheoutset

letus

considertw

oorifices

which

arefed

byaseparate

gasinjection.A

slong

asthey

aresufficiently

faraw

ayfrom

eachother

within

thefluid,

Table3

Overview

ofthekey

featuresofsom

eofthe

mostcom

monly

usedfoam

ingtechniques.

PrincipleTechnique

Bubblesize

(mm)

Polydispersity(%)

Initialgasfraction

φProduction

rate(L/m

in)Section

Bubblinginto

astationary

liquid(active

gasphase,passive

fluid

phase)Individualorifice

0.1–102–40

b0.60

b1

2.4.1and

3.2.1.1Many

orifices/perforatedplate

0.1–102–40

b0.60

b5(per

cm2)

3.2.1.2Porous

disc0.1–10

10–50b0.70

b5(per

cm2)

3.2.1.3Co-flow

oofgas

andliquid

(activegas

phase,activefluid

phase)Individualorifice

0.01–12–30

b0.99

b0.1

2.4.2and

3.2.2.1Porous

media

0.01–1N30

b0.90

b1

3.2.2.2Static

mixer

0.01–1N30

b0.95

b15

3.2.2.3Straight

tube0.01–1

N30

b0.97

b10

3.2.2.4Airentrainm

entandbreak-up

undershear

(passivegas

phase,activefluid

phase)Ross–M

iles(plunging

jet)0.1–5

10–40b0.60

b1

3.2.3.2Venturihose

0.01–0.1b30

b0.90

b5

3.2.3.2Propeller

N1

b30

b0.97

b1000

3.2.3.2Kitchen

blender0.01–1

b40

b0.97

b1

3.2.3.3Rotor–stator-m

ixer0.01–0.1

b30

b0.90

b10

3.2.3.3Foam

ingby

shaking0.1–10

b70

b0.80

Depends

3.2.3.4Phase

transitionNucleation

0.001–0.0110–40

b0.60

b1

3.3.1Aerosolcans

N0.01

N30

b0.99

b1

3.3.2Cavitation

≈0.001

N20

b0.60

b0.001

3.3.2Electro-chem

icalElectrolytic

cell0.001–0.01

10–30b0.60

b0.001

(percm

2)3.3.3

Fig.22.Differenttypes

ofnucleationand

growth

processesencountered

inbubble

formation

(inspiredfrom

[90]).

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bothorifices

functionindependently

following

thebehaviour

discussedin

Section2.4.1.H

owever,below

acriticalspacing

(which

dependsonpa-

rameters

likethe

bubblingvelocity),both

orificesbegin

tocom

municate

viahydrodynam

icinteractions

[103,104]throughthe

liquid.Theresultis

thatbothorifices

cansynchronise

suchthatthe

bubblesare

generatedin

analternating

fashionas

isshow

nin

Fig.24awith

correspondingflow

fieldsin

Fig.24b(from

[103]).Thissynchronisation

issensitive

tothe

bubblingspeed

andisgenerally

lostat

verylow

bubblingvelocity

andatvery

highbubbling

speed,when

theorifice

isin

thejetting

mode.

Iftheorifices

arefed

fromthe

samereservoir,as

isthe

casefor

per-forated

plates(Fig.24c–e),on

eadds

anim

portantcouplin

gthrough

fluctuationsin

thepressure

andflow

fieldofthe

gaswithin

thereser-

voir.Letus

imagine

thatat

thebeginning

allorificesare

closed.Asthe

pressureincreases

inthe

reservoir,thegas/liquid

interfacesat

theori-

fices

startto

curveoutw

ardsuntilthe

maxim

umpressure

isreached

when

theinterface

isahem

isphere(Fig.6b).Beyond

thispoint,the

bub-ble

growsatincreasing

speed(since

theresisting

Laplacepressure

oftheinterface

decreaseswith

increasingbubble

volume).The

resultingrapid

inflowinto

thebubble

thereforeleads

toapressure

dropofa

certainex-

tensionin

theunderlying

reservoir.Thispressure

dropcan

beso

strongand

long-range,thatacertain

number

oforificesaround

theone

where

thebubble

isgenerated

willnotbe

ableto

reachthe

maxim

umpressure

forbubble

formation.The

resultisthatatlow

flowrates

onperforated

platesonly

acertain

number

ofholesare

activein

thebubbling

process.Atypicalexam

pleisshow

nin

Fig.24d.Thecharacteristic

distancebe-

tween

theactive

holesdepends

onthe

influencelength

within

theres-

ervoir,which

inturn

dependson

theflow

rate.Which

holesare

activedepends

onwhich

onemanages

tofire

first,which

generallydepends

onsm

allfluctuationsin

thehole

size,with

largerholes

firing

first.As

theflow

rateisincreased,m

oreand

more

holesstart

tobe

involvedin

thebubbling

process.Fig.24esh

owsan

example

oftheactive

hole

fractionas

afunction

oftheinjected

gasvelocity

(from[105]).This

in-crease

isdue

totw

oreasons.The

increasein

gasflow

velocityleads

toan

increasingdynam

icpressure

dropat

eachhole

[106]. Once

thedy-

namicpressure

dropisofthe

orderofthe

maxim

umcapillary

pressure,allholes

shouldbe

ableto

generatedbubbles.A

secondreason

isthatfor

sufficientlyhigh

gasflow

velocities,i.e.beyondacharacteristic

Weber

number(see

Section2.4.1.2),inertialforces

playan

importantrole

lead-ing

tothe

formation

ofagas

jet.Thisim

pliesthatthe

holesrem

ainopen

permanen

tlyand

themaxim

umcapillary

pressureloses

itsmeaning

oncethe

holeisfiring.This

canbe

seenin

Fig.24e,where

allholesare

activewhen

theflow

velocityis

about12

m/s

which

correspondsto

We≈

2and

thereforeto

thejetting

regime(Section

2.4.1.2).Sincethe

jetshave

acertain

cross-section,thefinal

number

ofactive

orifices

may

besm

allerthan

100%ifthe

orificespacing

issm

allerthan

atypical

jetdiameter

[107].Thebubble

sizeateach

orificecan

beapproxim

atedusing

thegas

flowrate

peractive

orificeand

therelations

providedin

Section2.4.1.H

owever,one

needsto

keepin

mind

thatthe

couplingcan

leadto

flowrate

fluctuationsbetw

eenthe

orificesand

thereforeto

thegeneration

ofbubblesofdifferentsizes.

Formany

purposesitis

desirableto

drivethe

porousplate

inaslow

bubblingmode

tomaintain

themonodispersity

ofthebubbles.A

nele-

ganttrick

toavoid

thepressure-coupling

oftheorifi

cesis

toincrease

theflow

resistance

between

them

inthereservoir.This

canbe

doneby

increasingthe

flowresistance

ofeachorifice

(longslender

orifices)or

byreplacing

thepart

ofthereservoir

underneath

theorifi

cesby

afine

porousmedium

[106].Apartfrom

thequestion

ofthenum

berofac-tive

orifices,couplingofthe

bubblebreak-up

canoccur

forsufficiently

closeorifices.Researchers

find[108–110]thatbubbling

issynchronised

atlow

andahigh

flowrates

(jettingregim

e),while

itisasynchronous

forinterm

ediateflow

rates.Thesynchronisation

dependson

theorifice

spacingand

organisation.

Fig.23.Perioddoubling

andchaotic

bubblingbehaviourupon

gasinjection

intoastationary

liquid(a,b)

andin

co-flowconditions(c).(a)

Left:Bubblingperiods

Tas

afunction

ofreservoirpressure.Right:Corresponding

images

forperiod-1,-2and

-4behaviour(adapted

from[99]).(b)

Bubblingperiods

asafunction

ofgasflow

rateQGfortw

odifferentorifice

sizes.Thecom

-plexity

disappearsfor

thelarger

orifice(adapted

from[102]).(c)

A:Bubbling

devicefor

thecreation

ofco-flowconditions.B:Bifurcation

diagramshow

ingthe

bubblediam

eterdbas

afunction

oftheliquid

flowrate

Q(atconstant

gaspressure)

andcorresponding

photographsofthe

bubblingprocess

(adaptedfrom

[65]).

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3.2.1.3.Spargingfrom

porousdiscsandmore

exoticspargergeom

etries.Themost

commonly

usedspargers

forfoam

generationare

probablyofpo-

roustype,as

shownin

Fig.24f.Themost

popularspargers

aresintered

glassdiscs,sincethey

arehydrophilic,easy

toclean

andsince

theyprovide

awide

rangeand

agood

controloverthe

characteristicpore

dimensions.

Inporous

spargers,allofthepreviously

discussedmechanism

sinteract

simultaneously

togive

thefinalfoam

properties.Thefinalbubble

sizedistribution

dependssensitively

onthe

pore-sizedistribution,the

wetting

propertiesand

theflow

conditions.Thereare

twoim

portantdifferencesbetw

eenporous

spargersand

perforatedplates.The

firstliesin

thewide

distributionofpore

sizes(asshownin

Fig.24f).Sincethe

initialmaxim

umcapillary

pressureto

beovercom

eisinversely

proportionaltopore

radius(Eq.(17))the

bubblingwilloccurthrough

thelargestpores.Thisisw

hyat

lowflow

rates,porousdiscs

canproduce

foamsofhigh

monodispersity

(Figure24g

from[111]).W

ithincreasing

gasflow

rate,thepressures

atthe

smaller

holesincrease

sothatan

increasingnum

berofsm

allerpores

areactive,m

akingthe

foamincreasingly

polydisperse.Asin

thecase

ofasingle

orifice,asarule

ofthumb,the

averagebubble

sizeofa

spargerisproportionalto

itsaverage

poresize.The

seconddifference

liesin

thefactthatthe

differentporesofa

spargerareless

coupledby

pressurefluc-

tuationsthan

theorifices

ofaperforated

plate,sincethe

gasencounters

ahigh

flowresistance

asitpassesthroughthe

complex

porenetw

ork.How

-ever,this

simplification

regainscom

plexityby

thefactthatdepending

onthe

dynamicconditions

thegas

may

choosedifferentpreferred

passagesthrough

theporous

material.D

ependingon

theflow

conditions,bubblesize

distributionsfrom

porousspargers

havebeen

characterisedby

thepresence

ofseveralpeaks[112]orbyasm

oothW

eibulldistribution[113].

Thegas

fractionofthe

foamgenerated

justabovethe

spargertends

tobe

verylow

,thefinalgas

fractionofthe

foambeing

createdby

acom

-petition

between

foamrise

andliquid

drainage[114].Care

needsto

be

takenwhen

thegas

influxisso

highthat

afoam

iscreated

rightat

theoutlet

ofthesparger.In

thiscase,bubble

generationmechanism

smay

changedram

atically,leadingto

adependence

ofthebubble

sizewith

flow

ratewhich

may

becom

pletelydifferent

(eventheinverse)

towhat

hasbeen

discussedabove.

Aselection

ofmore

exoticspargers

hasbeen

developedin

thepast,

generallywith

theaim

toobtain

abetter

controloverthe

bubblesize

distributionwhile

increasingthe

foamproduction

rate.Amon

gstthe

most

elegantare

techniqueswhich

employ

theinstability

ofgas/liquidinterfaces

inparticulargeom

etricconditions.Exam

plesinclude

periodicbubble

generationat

anarrow

gap[116,117]

orat

astep

[118,160],which

canproduce

highlymonodisperse

bubbles,asshow

nin

Fig.25.Sim

ilarto

thecase

oforificebubbling,the

bubblesizes

may

bevaried

viathe

gap/stepwidth

andthe

gasflow

rate.

3.2.2.Foaming

viaco-injection

ofgasand

liquidAsalready

discussedin

Sec tion2.4.2,the

bubblingtechniques

oftheprevious

sectionare

oftenim

provedby

imposing

aflow

ofthefoam

ingsolution.This

helpsin

carryingthe

bubblesaw

ayfrom

thebubbling

siteand

insetting

upadditionalstresses

which

provideim

provedcontrol

overthe

bubblesizes

andthe

obtainedgas

fraction.Individualco-flowdevices

areoften

parallelisedto

increase

theproduction

rates(Section

3.2.2.1).One

common

approachof

parallelisingthe

co-flowmechanism

isto

letgas

andfoam

ingsolution

flow

through

complex

poregeom

etries.Inthis

case,bothphases

travelthroughan

intercon-nected

network

ofrandomor

purpose-shapedpores

(Sections3.2.2.2

and3.2.2.3),being

exposedto

arange

ofdifferentmechanism

swhich

turnone

largegas

pocketsinto

many

smallbubbles.O

neofthe

keydif-

ferencesbetw

eenco-fl

owin

individual

pores(as

discussedin

Section2.4.2)

andco-flow

inmore

complex

pore-networks,is

thatthe

Fig.24.Bubblingwith

increasingcom

plexity.(a)Synchronisation

ofbubblingfrom

neighbouringorifices

viainteraction

throughthe

liquid(from

[103]).(b)Sim

ulationofthe

flow-field

aroundgenerated

bubblesat

twoorifices

fortw

odifferentorifice

spacing(from

[103]).(c)Exam

pleofbubble

generationon

aperforated

plate(adapted

from[115]).(d)

Bubblingata

perforatedTeflon

plateshow

ingthat

bubblingoccurs

onlyvia

alim

itednum

berofholes

[106].(e)Variation

ofactivehole

fractionat

which

bubblesare

generatedwith

gasflow

ratefor

differentliquidheights

(from[105]).(f–g)

Image

ofaporous

discand

ofbubblesgenerated

fromitfor

increasinggas

flowrate

(from[111]).

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systemcan

choosebetw

eenmany

differentflowpaths,adding

yetan-

otherlevelofcom

plexity.Different

typesofpore

geometries

andflow

conditionscan

beused

forefficient

foamgeneration.W

ediscuss

some

ofthemhere.

3.2.2.1.Complex

bubblegeneration

andparallelisation

ofco-flowdevices.

Asin

thecase

ofbubblin

gfrom

anorifi

ceinto

astation

aryliquid

(Section3.2.1.1),even

individualco-flowdevicesm

ayexhibita

complex

behaviourat

intermediate

bubblingregim

es.Anexam

pleis

shownin

Fig.23cforthe

co-flowthrough

aconstriction,w

hichshow

speriod

dou-bling

andchaotic

bubblingfor

certainranges

ofliquidflow

ratesatcon-

stantgas

pressure[65].

Evenifco-fl

owdevices

cangenerate

upto

106bubbles/m

in,this

tendsto

correspondto

smallfoam

productionrates

sincebubble

sizesare

small.Efforts

havetherefore

beenmade

toparallelise

theindividual

devices[62,119].In

thesqueezing

ordripping

regime(Section

2.4.2)com

plexcoupling

isobserved

between

the

differentorifi

ces(Figure

26afrom

[119]),which

islargely

dueto

thecom

pressibilityof

theair

andthe

dynamics

ofthemeniscus.A

sin

thecase

oftheporous

disc(Section

3.2.1)this

couplingcan

bereduced

byincreasing

the

pressures/flowrates

andtherefore

moving

towards

thejetting

regime,

where

most

ofthephysics

happensat

thetip

ofthejet

andnot

atthe

outletof

theorifi

ce.An

extremeexam

pleof

parallelisationof

the

break-upofthin

gasjets

isshow

nin

Fig.26b[62].These

gasjets

canbe-

comevery

thinunder

extremeco-flow

conditions,andbubble

sizescan

thereforebe

oftheorder

ofamicrom

etre.

3.2.2.2.Foamgeneration

inporous

media.In

thelim

itofvery

smallpore

sizesand

lowflow

velocitiesone

findsthe

applicationoffoam

genera-tion

inporous

media

(oftensandstone)

foren

hancedoil

recovery[120–124].In

this

field,typical

poresizes

areof

theorder

ofsom

emicrom

etres,leadingto

stronglylam

inarflow

which

isdom

inatedby

surfacetension

andviscous

forces(i.e.sm

allReand

Wenum

bers).

Inaporous

medium

,atsufficiently

lowflow

rates,bothfluids

may

followseparate

paths(“no-foam

regime”

inFig.27a

andc),leading

toalow

flowresistance

andhence

asm

allpressuredrop

acrossthe

porousmaterial.U

ponincreasing

theflow

rates,oneobserves

anincrease

inthe

pressuredrop,being

associatedwith

thecreation

ofacoarse

(“weak”)

foam(Fig.27a

andc).W

henacriticalpressure

gradient∇P*

isreached

inthe

porousmedium

uponfurther

increaseofthe

flowrates,the

pres-sure

dropjum

psabruptly

toamuch

highervalueas

aresultofthe

onsetofefficient

foaming

mechanism

staking

placewithin

thepores.Three

differentfoam

ingmechanism

shave

beenidentified

[120,121]which

areshow

nin

Fig.27d.“Lamellea

division”divides

analready

existingfoam

lamella,w

hile“Leave-behind”

createsone

dueto

theflow

aroundobstacles.The

thirdmechanism

is“Snap-off”,w

hichoccurs

inthe

porethroats

(narrowsection

between

twopores).Sim

ilartothe

phenomena

discussedin

Section2.4.2,

pressureand

flow

condition

slead

toa

destabilisationofthe

gasthread

inthe

pore,which

breaksup

togener-

atesm

allbubbles.Atthis

stage,about10

differentmechanism

softhis

snap-off

havebeen

identifi

ed[125],

themost

common

onebein

gprobably

“Roof-snap-off”,which

occurswhen

gasinvades

aliquid-

filled

pore,sim

ilarto

the

caseof

co-flow

through

acon

striction(Section

2.4.2).The

differencebein

ghere

that

one

dealswith

aparallelisation

ofmany

co-flowdevices

ofdifferentdimension,i.e.a

sit-uation

ofevenhighercom

plexitythan

theone

shownin

Fig.26afor

theparallelisation

oftwoidenticaldevices.A

livelydebate

istaking

placein

thecom

munity,w

hether“Snap-off”

or“Lam

elleadivision”

isthe

main

foaming

mechanism

[125].Since

foaming

andfoam

flowin

aporous

medium

aredom

inatedby

what

happensin

thepore

throats

(snap-off

andlam

ellaneed

tobe

pushedthrough

thepore

throats)the

criticalpressuregradient

∇P*

atwhich

foaming

occursisrelated

tothe

dimensions

ofthepore

throatsRP[124]

∇P ��

RP−2;

ð41Þ

Fig.25.(a)Sparging

fromanarrow

gap(gap

width:75

μm(top

image)

&100

μm(bottom

image))

viaan

instabilityofthe

air/water

interface(adapted

from[116]).(b)

Generation

ofequal-volum

ebubbles

onastep

usingthe

samemechanism

asin

(a)(adapted

from[118]).

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esA,The

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anddepends

onthe

geometry

ofthepores.This

means

thatfor

small

porethroats,required

pressuregradients

forreliable

foamproduction

canbe

extremely

high.Above

thiscriticalpressure

gradient(andthe

as-sociated

criticalflowrates),foam

isgenerated

efficientlywith

bubblesizes

beingofthe

orderofthe

poresizes,i.e.m

icrometric.Bubble

sizesand

gasfraction

canbe

tunedby

tuning

theoveralland

therelative

flowrates,w

iththe

bubblesize

beingroughly

inverselyproportional

tothe

totalflowrate

Q=

QG+

QLin

theporous

medium

[126]

RB

�Q

−1:

ð42Þ

Anexam

pleofthis

isshow

nin

Fig.28.Unfortunately,little

system-

aticwork

hasbeen

doneatthis

stageto

establishaquantitative

linkbe-

tween

theporous

material,

theflow

conditionsand

thefinal

foamproperties.In

realporousmedia

with

microscopic

poresizes,pressures

tendto

becomeso

highthatthey

leadto

frequentfilmrupture.H

ence,an

understandingofthe

globalmechanism

sneeds

totake

intoaccount

coalescence.Aninteresting

(andfor

thepractitioner

important)

findingis

thatthere

isawide

rangeofpressure

gradientswhich

leadto

anunstable,

oftenoscillatory

flow

,andhen

ceoscillating

foamproperties.This

issh

ownby

thedotted

linein

Fig.27a.Researchersassociate

this

with

theoccurrence

ofintricatefeedback

mechanism

swithin

theporous

ma-

terial[122,124].Fig.27bshow

show

thetransition

between

theweak

andthestrong

foamregim

edepends

onthegas

andtheliquid

flow

rate.Anybody

lookingto

generateahom

ogeneousfoam

shouldthere-

forestrive

forsufficiently

highflow

ratesin

orderto

bein

the“strong

foam”regim

e.Most

considerationsoffoam

generationin

porousmedia

takeinto

accountindividual

bubbleson

ly.Asin

thecase

offoam

generationun

dersh

ear(Section

3.2.3),researchers

haverecen

tlybeen

ableto

showthatcollective

bubbleeffects

may

playanon-negligible

rolein

en-couraging

thebreak-up

oflargerbubbles

intosm

allerones.A

nillustra-

tiveexam

ple(from

[127])isshow

nin

Fig.29,where

onecan

seehow

thebubble

which

traversesthe

bottomofthe

constrictionbreaks

thebubble

which

passesatthe

top,which

thenbreaks

thebubble

atthebot-

tom.A

nindividualbubble

ofthesam

esize

which

traversesthisconstric-tion

atthesam

evelocity

doesnot

break.

3.2.2.3.High-speed

foaming

throughfibre

matrices

andstatic

mixers.In

orderto

obtainhigh

erfoam

ingrates,

scientists

usea

rangeof

purpose-designedporous-type

media

throughwhich

liquidand

gasare

co-injectedat

highvelocities

(upto

severalmetres

persecond).In

thiscase,bubble

generationprofits

alsofrom

thepresence

ofstrongvis-

cousand

inertialforces.One

classofdevices

isprovided

bystatic

mixers

(Fig.30a–c)[129–131]

which

arepurpose-designed

suchthat

arapid

flowofthe

gasand

theliquid

throughthe

interconnectednetw

orkof

poresleads

toan

optimised

shearingaction

onthe

bubbles,andhence

tobubble

break-up(Section

2.3).Thesemixers

canbe

drivenin

thelam

-inar

regime,leading

typicallyto

bubblesizes

oftheorder

of100

μm

(Fig.30d)with

awide

rangeofgas

fractions.Orthey

canbe

drivenat

highflow

ratesin

theturbulen

tregim

e,leading

tosm

allerbubble

sizesand

lower

gasfractions.In

[130,131], onecan

findaguideline

forsuch

staticmixers,linking

thetype

ofmixer

tobe

usedas

afunction

ofthe

fluidproperties

andapplications

foreseen.Other

typesofstatic

mixers

may

includethe

passagethrough

finegrids

ofdifferent

mesh

sizesor

through

systemslike

compact

steelwool

[132](or

similar

typesof

fibrous

materials).The

advantageof

suchsystem

sisthatthey

haveareasonably

highperm

eability,despitehaving

asm

allcharacteristiclength

scalewhich

fixesthe

obtainedbub-

blesize.These

“mixing

units”may

bepositioned

attheentry

ofa“foam

homogeniser”

which

may

simply

consistofavery

longtube

which

actson

thepre-foam

edsystem

sas

discussedin

Section3.2.2.4.Such

kindof

combinations

areoften

usedin

“compressed

airfoam

systems”,typical-

lyknow

forthegeneration

offirefighting

foams[132,133].The

obtainedfoam

scan

coverawide

rangeofbubble

sizes(dow

nto

someten

sof

microm

etres)and

canhave

awide

rangeofgas

fractions.

3.2.2.4.High

rateco-injection

instraighttubes.Efficientfoam

ingdoes

notnecessarily

requirecom

plexfoam

inggeom

etries.Under

certaincondi-

tions,a

simple

tubeis

sufficient.Co-injection

ofgas

andliquid

intoa

straighttube

atsufficiently

highflow

velocitiesleads

tocom

plexpat-

ternsoftw

ophase

flow(Fig.31a)

[134],many

ofwhich

generatebub-

blesand

thereforefoam

s,especiallyin

thepresence

ofsurfactants.Thephysics

oftheseprocesses

ishighly

complex,m

ixingdifferent

typesof

airentrain

men

t(Section

2.4.4)and

bubblebreak-up.U

ntilrecen

tly,this

hasbeen

studiedmostly

forengineering

purposeswhere

bubblegeneration

needsto

beavoided

oratleastunderstood

tooptim

isepres-

suredrops

andheatflow

problems.Researchers

havenow

realisedthat

someofthese

regimes

areextrem

elyefficientfoam

generators(Fig.31b

andc),providin

gfoam

swith

smallbubbles

andwell-controlled

gasfractions

[87,132,133,135].One

example

consistsofthe

co-injectionofgas

andfoam

ingsolution

atconstant

pressure(or

flowrate)

intoastraight

tubeof

typicallymillim

etricdim

ensions[87,135]

(Fig.31b).Thetube

needsto

bemuch

longerthan

itsdiam

eter(typically

afew

hundredtim

eslonger

thanthick)

inorder

forthe

mechanism

towork

efficiently.Moreover,the

Weber

numbers

Weof the

flowneed

tobe

sufficientlyhigh

sothatiner-

tialforcesplayarole

inthe

foaming

process.Upon

choosingthe

rightflowconditions,foam

scan

beobtained

with

verysm

allbubblesizes

(orderof

10μm

)(Fig.31d),at

veryhigh

productionrates

(litresper

second)and

overnearly

theentire

rangeofgas

fractions.Sm

allerquantities

offoams,butw

ithavery

precisecontrolover

thegas

fraction,can

beobtained

usingthe“D

ouble-Syringetechnique”

(Fig.31c)[87,161].This

usesasim

ilarmechanism

asin

theprevious

case,buthere

twosyringes

areconnected

toeither

sideofa

shorttubein

orderto

pushthetw

o-phase

mixture

backand

forththrough

the

tube.Atthe

outset,onesyringe

containsthe

gas,while

theother

onecontains

thefoam

ingliquid.Both

mix

efficientlyafter

afew

cycles,giv-ingrise

tofoam

swith

verysm

allbubbles(around

10μm

)and

with

Fig.26.Parallelisationofco-flow

devices.(a)Com

plexcoupling

ofparallelbubblingdevices

(from[119]).The

leftsideshow

sphotographs

andthe

rightsidethe

bubblingactivity

ofeachhole.From

topto

bottom,the

gaspressure

isincreased

atconstantliquidflow

rate.Thenotation

is[P

G(psi),Q

L (mL/h)].(b)

Parallelisationofm

icro-bubblegeneration

inthe

jettingregim

e(from

[62]).One

cansee

thingas

threadsform

edby

rapidgas/liquid

co-flowcom

ingoutofcapillaries

inatip-stream

ingmanner

(Section2.3).These

gas-threadsbreak

intomicrom

etricbubbles

furtherdow

nstream.

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ing,Adv

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extremely

well-controlled

gasfractions

(Fig.31e).Obtained

bubblesize

distributionsare

similar

tothose

obtainedin

thecontinuous

flow.M

ostinterestingly,ascan

beseen

inFig.31e,they

areindependentofthe

finalliquid

fractionofthe

foam[87].

3.2.3.Gas

entrainmentatfree

surfacesand

bubblebreak-up

undershear

3.2.3.1.Breakingwavesand

hydraulicjum

ps.Asdiscussed

inSection

2.4.4,gas

entrainm

entis

mostly

associatedwith

highfluid

velocityand/or

underabrupt

variationsofthe

velocityfield

oftheflow

.Thisoccurs

inmany

naturalprocesses,awell-know

nexam

plebeing

thatofcascades.

Itisalso

atthe

originoffoam

ingat

theshore

breakon

thesea

[13,80]or

athydraulic

jumps.In

theseprocesses,m

anyofthe

basicsituations

describedin

Section2.4.4

may

simultaneously

occur,andany

under-standing

ofsuchcom

plexflow

srelies

onhow

faritcan

bedescribed

bycom

biningthe

elementary

mechanism

s.Researchershavedeveloped

anum

beroflab-scaleprototypes

inan

attemptto

simplify

thecom

plex-ity

oftheprocesses

[80,136].In

thecase

offoamgeneration

bywave

breaking,asshow

nin

Fig.32b,afulldescription

ofhowexactly

thewaves

trapthe

surroundingair

hasnot

yetbeen

obtained.Nevertheless,the

main

time-sequences

oftheph

enom

enologyare

knownand

summarised

inFig.32a

[80].First,amajor

partof

airen

trainment

occursas

theplunging

wave

hitsits

frontface(Fig.32a

—im

ageA).This

isthe

moststudied

andunderstood

mode

ofentrainment

(Section2.4.4),as

onemight

expectto

linkitto

thephysics

ofaplunging

jetconfiguration

(Section3.2.3.2).H

owever,

itsign

ificantly

differsfrom

theprototypal

scenario

discussedin

Section2.4.4

sincethe

impacting

jetisnot

continuous,itsgeom

etryis

much

more

complex

(with

anglesofim

pactchanging

with

time)

andthe

liquidreceiving

theim

pactundergoesitselfa

complex

flow(parallel

tothe

surfaceand

moving

upslope).With

suchcom

plexity,simulations

havebeen

apow

erfulsourceto

providesom

eprelim

inaryinsights

ofhow

much

aircan

beentrainm

entbythis

firstim

pactofthe

wave

(seereferences

in[80]).Then,there

areseveralpost-jetim

pactentrainment

modes,

allmostly

linkedto

thesplash

ofthefirst

impact.

Thisis

schematised

inFig.32a

—im

agesBand

Cwhere

subsequentim

pactsof

theforw

ardand

backward

splashesen

trappedsom

eair.

Lastly,som

eturbulen

ten

trainmen

tsoccur

allover

thevarious

subsequent

splashes,andat

theleading

edgeof

theturbulent

region(Fig.32a

—im

ageD).G

oingmuch

beyondaphenom

enologicaldescriptionofthis

processisrendered

challengingalso

becausethe

breakingwave

situa-tion

correspondsto

flowsof

low-viscosity

fluids,which

addsfurther

complexity

(disturbances,instabilities)

asdiscussed

beforefor

the

basicplunging

jetconfiguration

(seeSection

2.4.4).Let

usmen

tionth

atwhile

aplun

gingwave

isareason

ablyeffi

-cien

tmean

toincorp

oratelarge

amou

nts

ofair

into

water

(as

Fig.27.Foamgeneration

inporousm

ediain

thelow

velocitylim

it.(a)Sketchofthe

variationofthe

pressuregradientacrossthe

porousmaterialasa

functionofthe

totalflowrate

(inspiredby

[124])show

ingthe

differentfoaming

regimes.(b)

3D-illustration

of(a)considering

thegas

andliquid

flowrate

independently(from

[124]).(c)Differentflow

scenariosencountered

inthe

porousmaterial(from

[122]).(d)The

threemain

mechanism

sresponsible

forfoam

generation(from

[127],adaptedfrom

[120]).

0,00,4

0,81,2

1,60

100

200

300

400

500

6000,025 L/m

in0,057 L/m

in0,089 L/m

in0,121 L/m

in0,153 L/m

in

Averag bubble radius <RB> ( m)

Total flow rate Q

G + QL (L/m

in)

Liquid flow rate Q

L

µ

Fig.28.Dependence

ofaveragebubble

radiusoffoam

generatedby

co-flowofgas

andliq-

uidthrough

aporous

medium

generatedby

closelypacking

2mm

glassspheres.

Data

courtesyofR.-M

.Guillerm

ic,pre-work

to[128].

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everyonecan

experien

cewhile

lookingat

thesh

orebreak

ona

beach),it

has

not

(yet)been

transform

edinto

aded

icatedfoam

ing

device.Gas

entrainments

within

hydraulicjum

ps(Fig.17c)

belongto

the

samefam

ilyof

self-aeratedsituations,w

hereair

isincorporated

asa

consequenceofrapidly

flowing

jetsassociated

tostrong

free-surfacede-

formations.It

isclear

thatnot

allhydraulicjum

psprovide

airentrain-

ment.

Usual

jumps

inasink

giveno

signsof

foamingor

bubbling.

How

ever,ifon

eincreases

thevelocity

oftheim

pactingjet,one

canstart

tosee

thefirst

bubblesentrained

inthe

fluid,originatingat

thejum

pwhere

thefluid

thickness

abruptlyvaries,as

shownin

Fig.32c.In

fact,itis

onlyfor

highenough

Froudenum

berFr

(typicallyFr

N3)

that

airis

entrained

[137],andthis

canbecom

ean

efficien

tfoam

ingmechanism

,asseen

inFig.32d.Experim

entsshow

thatinorder

toob-

tainsignificantairentrainm

ent,alarge

turbulentshearregion—

usuallycalled

a‘roller’—

needsto

beform

ed.Thisroller

islocated

atthetop

ofthe

thickfluid

region,asshow

nin

theschem

eofFig.32c

[137].Thegas

isthen

entrainedatthe

toeofthis

highshearregion,characterised

byin-

tensiveturbulen

ce.Thelength

ofthis

rolleris

directlyrelated

tothe

Froudenum

ber(see

referencesin

[137]).Similarly,the

amount

ofgasentrained

alsoincreases

with

theFroude

number.H

owever,there

arestillno

clearmodels

unambiguously

describingthiseffect,and

onlyphe-

nomenologicallaw

sare

proposedwhich

lacksatisfying

agreementw

ithexperim

entaldata.Infact,the

geometry

oftheexperim

entalsetupplays

arole,as

wellas

theam

ountofpre-entrainedgas

inthe

impacting

jet.

Eventhough

theentrainm

entconditionsby

bothmechanism

s,wave

breakingand

hydraulicjum

ps,havebeen

investigatedintensively

inthe

past,whatis

lackingalm

ostentirelyatthis

stageare

thoroughinvestiga-

tionsinto

theobtained

bubblesize

distributions,which

would

beim

-portant

forfoam

ingapplications.

3.2.3.2.Plungingjets

andfoam

inghoses.O

nefoam

ingtechnique

which

makes

systematic

useofair

entrainment

andbubble

break-upunder

shear

byaplun

gingjet

(Section2.4.4)

inacontrolled

manner

hasbeen

turned

intoastandardised

testmethod:

theRoss–M

ilestest

[138].Thehistoric

set-upisshow

nin

Fig.33awith

sometypicalexam

-ple

curves(Fig.33b)

andim

agesofthe

entrainment

process(Fig.33c).

Thisfoam

ingtest

belongsto

theASTM

databankof

testmethods

(identifi

edun

derthenum

berASTM

D1173-53(2001)).

Inpractice,

50mlofthe

solutionisplaced

inacylinder.U

singapipette

andafunnel

fixedata

heightof90cm

,200mlofthe

samesolution

isallow

edto

fallinto

thecylinder.A

ssoon

asallof200

mlsolution

hasfallen,the

foamheight

isrecorded.N

otethat

ifthe

impact

isinitially

ontheliquid

layer,thisisno

longerthe

caseonce

thefew

firstlayers

ofbubblesare

created.During

theprocess,bubbles

might

alsobe

destroyed(or

not)by

thefallin

gliquid,th

usmixin

gfoam

ingand

stabilityissues

atthe

s ametim

e.Foamsproduced

with

thistechnique

canbe

highlymonodis-

perseor

highlypolydisperse,w

ithpolydispersity

increasingwith

im-

pactvelocity

ofthejet.A

veragebubble

sizestend

tobe

oftheorder

ofafew

hundredmicrom

etresallthe

way

upto

afew

millim

etres.Foamproduction

ratesare

low,ofthe

orderof1

L/min.The

initiallyobtained

gasfraction

sare

verylow

andgenerally

increaseby

gravity-drivendrainage

ofthefluid

asthe

foamaccum

ulatesat

theliquid

surface.Themost

popularuse

ofthisfoam

ingtechnique

isprobably

thatoffilling

ourbathtub

with

foam.

Other

techniqueshave

beendeveloped

touse

airentrainm

entin

amore

systematic

andcontrolled

manner

forfoam

generation.Thisair

entrainmentm

ayhappen

attheoutletofnozzles

(Fig.34a)or

atspecif-ically

designedcross-sectionsofV

enturi-typenozzles,in

which

thefluid

flowcreates

anunderpressure

andtherefore

entrainssurrounding

air(Fig.34b).W

iththis

approach,lowto

intermediate

gasfractions

canbe

obtained.Inorder

toincrease

thegas

fractionofthe

foam,air

needsto

beinjected

actively.This

canbe

doneusing

compressed

air(Fig.34d)

orapropeller

system(Fig.34d–g).In

thelatter

case,anair/

water

mixture

isprojected

athighspeed

ontoafine

mesh.This

gener-ates

amixture

ofbubble

blowing/air

entrainment

throughthe

mesh

andbubble

break-upby

shear.Foamproduction

ratescan

reachmore

than

acubic

metre

perminute.

How

ever,bubble

sizestend

tobe

millim

etricin

thiscase,giving

riseto

areasonably

coarsefoam

.

3.2.3.3.Breakupunder

activeshear.In

orderto

obtainsm

allerbubbles,a

wide

rangeoffoam

ingdevices

hasbeen

developedby

engineerswhich

usemore

activelythe

fundamental

mechanism

sof

bubblebreak-up

Fig.29.Bubble–bubbleinteractionsm

ayplay

anim

portantrolein

bubblebreak-up.H

ereit

canbe

seenthatthe

bubblepassing

throughthe

bottomofthe

constrictionbreaks

theone

passingon

thetop,w

hichthen

breaksthe

oneon

thebottom

.From

[127].

Fig.30.High

speedco-flow

ofgasand

liquidthrough

purpose-designedstatic

mixers

generatesefficiently

foams.(a–c)

Examples

ofstaticmixer

geometries.(d)

Typicalbubblesize

distributionsobtained

inastatic

mixer

fordifferenttotalflow

ratesatconstantgas

fraction(Φ

=0.92)

(from[130]).

23W.D

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

undershear

describedin

Section2.4.3.In

thesedevices,the

foaming

fluidand

thegas

aredrastically

andheterogeneously

shearedin

orderto

effi-ciently

createafoam

with

sub-millim

etricbubbles.The

main

issuein

de-signing

andoptim

isingsuch

devicesisnotonly

toobtain

thedesired

foamin

termsofbubble

sizeand

gasfraction,butalso

toensure

thatthedevice

isefficient(rapid

foaming

atlowpow

erconsum

ption).One

highlypopular

foaming

devicewhich

isbased

onair

entrain-ment

andsystem

aticbubble-breakup

undershear

iscertainly

known

tomost

ofthe

readers:thekitchen

blender.

Inthis

device,air

isen

trainedat

thefree

surfaceof

theblended

liquid(Section

2.4.4),which

createslarge

bubbles,which

arethen

brokenunder

thecontinu-

ousshearin

gaction

ofthe

blender.The

interactionbetw

eenair

entrainm

entand

bubblebreak-up

leadsto

agradualin

creaseof

the

gasfraction

(asshow

nin

Fig.35a)and

adecrease

oftheaverage

bubblesize

overtime—

thisisnoticed

bythe

userasthefoam

becomes

increas-ingly

white

andsolid-like.This

processhappens

untilanequilibrium

stateisreached,w

hosecharacteristic

gasfraction

andbubble

sizede-

pendon

therheologicalproperties

oftheliquid

andthe

rotatingvelocity

oftheblender.A

sarule

ofthumb,the

higherthe

viscosityofthe

liquid,the

smaller

thebubbles

andthe

lower

thegas

fractionwhich

canbe

ob-tained

(Fig.35b).Depending

onthe

beatingtim

eand

thebeating

speed,obtained

bubblesizes

canrange

between

sometens

ofmicrom

etresand

afew

millim

etres.Extremeshearing

conditionsand

unusualfoamfor-

mulations

may

leadto

sub-micron

bubblesizes

[139].Ascan

beseen

Fig.31.(a)Typicaltw

o-phaseflow

patternsin

straighttubeswith

increasingratio

ofgasto

liquidflow

rate.Someofthese

regimes

canbe

usedto

generatefoam

sathigh

productionrates

andwith

smallbubble

sizes.One

example

isthe

continuousflow

inasim

plecapillary

(orlargertube)show

nin

(b),asecond

example

pushesthe

gas/liquidmixtures

repeatedlyback

andforth

throughatube

usingadouble

syringeshow

nin

(c).Bothtechniquesprovide

accesstoawide

rangeofgasfractionsand

toextrem

elysm

allbubblesizes.Exam

plesofsizedistributions

areshow

nin

(d)for

thestraighttube

andin

(e)for

thedouble

syringe.Panels

b–efrom

.[87]

Fig.32.(a)Schem

e(from

[80])and

(b)photograph

ofairentrainm

entbywave

breaking.(c)Schem

eand

(d)photograph

ofairentrainm

entat

ahydraulic

jump(both

from[13]).

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esA,The

scienceoffoam

ing,Adv

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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inFig.35a,the

initialfoaming

ratein

thecase

oflow-viscosity

liquidscan

bequite

rapid(about

1L/m

in),butthe

finalbubblebreak-up

may

takemuch

longer.Themain

issuewith

thekitchen

blenderfor

largerscale

applicationsisthe

factthatit

isanon-continuous

process.In

orderto

obtaincontinuity

andamore

explicitcontroloverthe

airinput(rather

thanrelying

onpassive

airentrainm

ent)and

theshearing

process,anum

berofadvanced

blendershave

beendeveloped

inthe

past.Afirst

classofdevices

arerotor–stator

mixers

(Fig.36a–d)which

consistofa

setofnarrow

ly-spacedstatic

“stators”and

rapidlyrotating

“rotors”through

which

thefoam

ingsolution

andthe

gasare

pressedsi-

multaneously.A

largebody

ofliteraturedeals

with

theoptim

aldesignof

therotor/stator

asafunction

ofthefluids

tobe

sheared[141–143].In

thiscontext,the

New

tonnum

ber(N

e)(defined

asthe

ratiobetw

eenthe

mechanicaldriving

forceand

theinertialforce)

isoften

usedto

de-scribe

thedevice

behaviour[141–143].

Inthese

devicesone

findsthat

themechanicalenergy

inputis

themostinfluentialparam

eterwhich

controlsthe

propertiesofthe

obtain-ed

foam.It

isafunction

oftherotor

speed,thefluid

viscosityand

thetim

ewhich

thegas/liquid

mixture

spendsin

thedevice

[142–145].The

widely-used

“Ultra-Turrax®

”(Fig.36e)

belongsto

thiscategory

ofrotor–stator

mixers,w

iththe

specialtythat

ittypically

containsonly

onerotor–stator

pair.Bubblesizes

obtained

inthese

kindof

devicescan

bevery

small,dow

nto

theorderofa

fewmicrom

etres,with

reason-ably

highfoam

productionrates.

Another

classofdevices

isbased

onthe

“Narrow

Annular

Gap

Unit”

shownin

Fig.37aand

b.Inthese

devices,gasand

liquidare

generallypre-m

ixedusing

aporous

plate(Section

3.2.1.3).Thispre-foam

isthen

injectedbetw

eenthe

impellers.Souidiet

al.[79]couldshow

thatoneof

themain

mechanism

sofbubble

generationin

thisdevice

appearsto

be“tip

streaming”

(asdiscussed

inSection

2.3and

shownin

Fig.15)which

leadsto

verysm

allbubblesizes

ofthe

orderof

10μm

(Fi g.37c).Themain

advantageisthat

thisdevice

requiresmuch

lower

rotationspeeds

(lower

energyinput)

thanthe

more

classicrotor–stator

devices[79,146].

Besideen

gineeringissues

concerningthedesign

ofsuch

kindof

mixers,one

hasto

beaw

areofthe

effectofbubble“cooperativity”,as

al-ready

mentioned

inthe

introduction.Asthe

bubblesbegin

tobe

closelypacked

theycannotbe

consideredany

more

asisolated

objects(as

donein

Section2).The

effectofthegas

volumefraction

hasbeen

investigatedin

controlledmodelexperim

entsusing

ashear

rheometer

[17,147,148](Fig.38a).The

authorsevidence

differentbreak-upregim

es.Inparticu-

lar,as

inthecase

ofcooperative

bubblebreakup

inconstrictions

(Fig.29),theyrealised

thatbubble–bubbleinteraction

becomes

impor-

tantandleads

tocriticalcapillary

numbersw

ellbelowthose

ofanisolat-

edbubble/drop

(Fig.38b).Theauthors

explainthis

enhancedinstability

bya“m

icrostructure-induced”instability

which

leadsto

apinch-offof

anelongated

bubbleby

itsneighbours

(Fig.38c)and

toacriticalstress

which

needsto

beobtained

inthe

foam(rather

thanacriticalcapillary

number)

[149].

Fig.33.Foamgeneration

byaplunging

jet.(a)The

historicRoss–M

ilestestw

hichlets

afoam

ingsolution

ofdefinedvolum

edrop

intothe

samesolution

fromawell-defined

heightandmeasures

theobtained

foamheight(from

[138]).(b)Exam

plesofthe

evolutionoffoam

heightwith

heightoverwhich

thesolutions

fall(from[138]).(c)

Images

ofplungingjets

andthe

accompanying

bubblegeneration

with

increasingim

pactvelocity(from

[80]).

25W.D

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as:Drenckhan

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esA,The

scienceoffoam

ing,Adv

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3.2.3.4.Foaming

byshaking.O

neofthe

simplesttechniques

tofoam

aso-

lutionconsists

ofshakingaclosed

containerwhich

ispartially

filledwith

thesolution.Ifthis

isdone

inacontrolled

manner,the

containerisshak-

enat

specificfrequency,am

plitudeand

foracontrolled

duration.Thefoam

abilityofthe

solutionisthen

determined

bythe

amount

offoamcreated,and

thebubble

sizecan

alsobe

monitored

asafunction

ofthesh

akingparam

eters.Thisis

oftenreferred

toas

the“Bartsch

test”.Inthe

“Bartschtest”

theshaking

isperform

edby

acontrolled

flippingof

thecellat

controlled(and

variable)frequency.N

otethat

thisshaking

canalso

bedone

byhand,and

thisusually

providesagood

andcheap

es-tim

ationofthe

foamabilities

ofvarioussolutions.

Foamgeneration

byshaking

mixes

aboutallthedifferentm

echani-calfoam

ingeffects

which

wehave

discusseduntilnow

andistherefore

difficultto

describequantitatively.To

disentanglethe

differenteffects,

more

simple

experiments

havebeen

developed.Flatcells

were

usedby

Capset

al.[150,151],inassociation

with

arotation

setupproviding

consecutiveupside-dow

nflips

ofthecell.A

sshow

nin

Fig.39a,the

cellsare

soflatthatthey

generateaquasi-2D

foam,i.e.only

monolayers

ofbubbles.Thisconfiguration

allowsus

tovisualise

allthebubbles,as

afunction

ofthenum

berofflips

(Fig.39b).Asequence

ofimages

oftheobtained

foamswith

increasingnum

berofflips

nf is

shownin

Fig.39a,show

inghow

thebubble

size(and

hencethe

number

ofbubblesN)de-

creasedsystem

aticallywith

thenum

berofflips.These

experiments

andfurtheranalysis

ofthissetup

[150–152]finallydem

onstratethat,thanks

tothese

seriesofsuccessive

flipsofthe

flatcell,aspatially

homogeneous

foamcan

beproduced.M

oreover,thebubble

sizedistribution

eventual-ly

saturateswith

aGam

madistribution.The

obtainedaverage

bubble

sizedepends

onthe

initialamount

ofliquidand

canbe

linkedto

thecapillary

lengthofthe

foaming

solution.

3.3.Foaming

throughphase

transitionsand

byelectro-chem

istry

3.3.1.Nucleation

andgrow

thofbubbles

insupersaturated

liquidsThe

greatclassicswhich

make

useofbubble

generationthrough

nu-cleation

andgrow

thofbubbles

inasupersaturated

liquidare

knownto

mostofus

fromthe

beverageindustry:cola,cham

pagneor

beerfoam

s.In

mostcases,the

supersaturatedgas

isCO

2due

toits

highsolubility

inwater.The

supersaturationiseitherachieved

byexplicitly

applyinghigh

pressureswhen

fillingand

closingthe

bottles(fizzy

drinks),oritarises

naturallyas

aresult

oftheferm

entationprocess

(champagne,beer).In

most

ofthesecases,researchers

havebeen

ableto

showthat

bubblesare

generatedby

heterogeneous

nucleation(types

IIIand

IV)at

the

wallor

atimpurities

(Section2.5)

[11,153].Asisobvious

tothe

carefulobserver,

continuousbubble

streams

nucleatefrom

individual

nucleationsites,as

isshow

nin

Fig.40a.Thanksto

thiseffect,supersat-

urationin

these

beveragescan

bekept

lowand

istypically

ofthe

orderof5–10

only.Acham

pagnebottle,for

example,is

typicallyat

apressure

of5–10

barand

releasesaround

5Lof

CO2to

enhance

ourpleasure.

Similar

phenomena

areused

incosm

eticor

otherfood

products.Aclassic

example

isthewhipped

creamdispenser

which

decoratessom

ekitchens

(Fig.40c).Inthis

case,itis

the‘laugh

inggas’(nitrous

oxide)which

isdissolved

underhigh

pressureand

thenreleased

tocre-

atethe

bubblesin

thecream

.

Fig.34.Controlledair

entrainmentfor

foamgeneration:A

ircan

beentrained

bythe

liquidflow

(Ventouri-type

effect)(b,c),leading

tolow

andinterm

ediategas

fractions.(c–g)Gas

frac-tions

canbe

increasedsignificantly

byactively

injectingair,for

example

throughthe

useofa

propellersystem

.Theair/w

atermixture

isthen

propelledonto

agrid,w

herestrong

shearforces

turnthe

water

dropletsinto

airbubbles.(e–f)

Jet-Xfrom

ASU

L.(g)ALPH

AHF+

deLEA

DER

inaction

with

upto

1000Loffoam

/min.

Fig.35.Foaming

with

akitchen

blender(alldata

from[140]).(a)

Changeofair

volumefraction

with

timefor

differentfoaming

mixtures

containingparticles.The

variationofthe

ratiobetw

eenthe

surfactantconcentrationCS

andthe

particleconcentration

CPleads

essentiallyto

avariation

inthe

bulkviscosity

oftheliquid.(b)

Lineardecreaseofthe

maxim

umobtained

airvolum

efraction

with

theviscosity

ofthefoam

edliquid.

26W.D

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as:Drenckhan

W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

ColloidInterface

Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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Inorder

tofoam

highlyviscous

liquidsand

largequantities,on

eoften

usesclassic

extruderscrew

s(Fig.40b)

toexert

highpressures

onthefoam

ingliquid.This

isalso

themethod

ofchoice

inorder

tofoam

thermosensitive

polymers,w

hichare

mixed

with

thegas

athigh

temperatures

underhighpressures,to

givearapid

foaming

(andsim

ul-taneous

solidification)as

theliquid

isreleased

toam

bientpressure.

3.3.2.Cavitationand

boilingAsfaras

weknow

,foaming

throughboiling

iscurrently

littleused

inasystem

aticmanner.Itis

ratherknow

nto

most

ofusas

anunpleasant

surprisewhen

thesoup

orthe

milk

areleft

unwatched

alittle

toolong

onthestove.

Theon

lysystem

aticuse

known

tous

isthat

ofgels

which

foamwhen

incon

tactwith

theskin.These

gelscon

tainsub-

stances(forexample

isopentane)which

areliquid

atroomtem

perature,but

gaseousat

bodytem

perature.Hence,th

econtact

with

theskin

(shavinggels,tooth

pastes,etc.)makes

thesegels

foam.Shearing

the

gelonthe

skinhelps

tobreak

themicro-structure

andto

getthenucle-

ationprocess

going.Asim

ilarfoam

ingsystem

isfound

inthree-phase

aerosolcans,themost

classicbeing

shavingfoam

cans.Inthis

casethe

phasetransition

isdriven

byapressure

change,ratherthan

atem

peraturechange.The

overallsystemconsists

oftwoim

miscible

liquidscreating

anem

ulsionunder

pressure(see

bottomofFigs.2

and40d,B).The

dispersedphase

isaliquid

oflowvapour

pressure(m

ostcommonly

butan/propanmix-

tures)which

remains

liquidunder

thepressure

conditionsofthe

con-tainin

gvessel.

Once

thepressure

isreleased,

thedispersed

liquidchanges

phaseand

theentire

emulsion

dropletturnsinto

agas

bubble.The

finalbubblesize

istherefore

determined

bythe

initialdropletvol-

ume[154].This

iswhy

itisrecom

mended

toshake

thecan

beforeuse

inorder

tore-em

ulsifythe

mixture.Bubble

sizesin

theseapplications

aretypically

around50

μm.

An

increasingly

popularfoam

ingtechnique

employs

cavitation(Section

2.5).Thisis

ofparticular

interestwhen

foamswith

micron-

Fig.36.Aclassic

amongstthe

foaming

devices:therotor–stator

mixer.(a)

Overallsketch

ofthedevice

containingthe

rotating“rotor”

andthe

static“stator”

(from[141])

(b)Side-view

(from[141])

and(c)

topview

(from[70])

ofanelem

entaryunit

ofarotor–stator

mixer.(d)

Photographsofa

typicalstatorand

rotor(from

[70]).(e)The

“Ultra-Turrax®

”—

awidely

usedrotor–stator

devicecontaining

onlyone

rotor/statorpair.

Fig.37.TheNarrow

Annular

Gap

Unit(N

AGU).(a)

Overalldevice

configuration(from

[146]).(b)Exam

plesofpropellers

usedfor

theshearing

action(from

[79]).(c)Exam

plesofbubble

sizedistributions

obtainedfor

differentpropeller

geometries

(rotationspeed:1600

rpm,Q

G=

10mL/m

in,GL=

30mL/m

in)(from

[79]).

27W.D

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as:Drenckhan

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esA,The

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ing,Adv

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Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

sizedbubble

aredesired.H

istorically,suchkinds

ofbubbleswere

gener-ated

usingacoustic

pressurefields

(sonication).How

ever,sincethe

pro-duction

ratesare

toolow

,hydrodynamic

cavitationis

making

itsway

intofoam

ingdevices

[155,156].Different

devicesexist

touse

arapid

flowofthe

foaming

liquidin

amanner

thatasudden,strong

pressuredrop

occurswhich

leadsto

thegeneration

ofcavitationbubbles.The

most

popularscenario

consistsof

rapidflow

through

acon

striction,

leadingto

apressure

dropjust

behind

theconstriction.Specialcare

needsto

betaken

torender

these

bubblesstable

asthey

aresw

eptaw

aywith

theflow

.Acertain

degreeofsupersaturation

with

anothergas

helps.Themost

importantingredients

arestabilising

agents[156].

3.3.3.Foaming

byelectrolysis

Anelectro-chem

icalway

ofproducinggas

bubblesis

providedby

electrolysis.Forthis

purpose,avoltage

isapplied

acrosstw

oelectrodes

placedwithin

thesam

ewater

reservoir(Fig.41a),turning

oneofthe

electrodesinto

acathode

(negativelycharged)

andone

intoan

anode(positively

charged).Sincewater

molecules

decayand

recombine

per-manently

(2H2 O

↔2H++

2OH−),the

resultingH+protons

willcom

-bine

with

electronsfrom

thecathode

toform

hydrogengas

(H2 ),w

hilethe

OH−ions

willliberate

electronsatthe

anodeto

formwater

andox-

ygen:

Cathode:

4Hþþ

4e−↔2H

2Anode

:4O

H−↔2H

2 OþO

2 þ4e

−:

ð43Þ

Hence,hydrogen

bubblesare

createdat

thecathode,w

hileoxygen

bubblesare

createdat

theanode.In

bothcases,the

bubblesare

blown

atthesurface

oftheelectrode,m

eaningthateithergravity

oranexternal

flowneeds

todetach

themfrom

thesite

where

theygrow

[162].Thebubble

sizeistherefore

controlledby

thesam

emechanism

sdiscussed

inSection

s2.4.1

and2.4.2,th

ekey

differencebein

gthat

thegas

isblow

nby

anelectro-chem

icalreaction.Asecond

keydifference

isthat

theblow

ingsites

occurgenerally

onsm

alldefectson

theelectrode

sur-face.The

bubblesare

thereforeonly

weekly

attachedto

thesurface

(asif

theywere

blownfrom

atiny

orifice)and

theresulting

bubblesizes

aretherefore

ofat

leastan

orderof

magnitude

smaller

than

bubblesblow

nfrom

orifices(typically

sometens

ofmicrom

etres,asshow

nin

Fig.41b[157]).Since

thenum

berofbubbling

sitescan

bevery

highon

anelectrode,this

isan

efficienttechniqueto

producereasonable

quan-tities

offoamswith

tinybubbles.For

thisreason,one

ofthemain

appli-cation

sof

thisfoam

ingtechnique

iselectro-fl

otation.Flotation

efficiencyincreases

with

decreasingbubble

sizesince

thetotalinterfa-

cialareaisinversely

proportionaltothe

bubblesize.A

typicalflotationcellis

shownin

Fig.41c[157].A

common

useis

that

ofwaste

water

treatment

ormineralflotation

(onsm

allscales).Thenum

berofsites

andthe

bubblesize

canbe

variedby

tuning

the

currentdensity

(Fig.41b),which

isconvenient

forapplications.

Particularcontroland

eventhe

generationofhighly

monodisperse

bubblescan

beobtained

byworking

with

stronglypointed

electrodeswhich

canproduce

micron-sized

bubbles(Figure

41cfrom

[158]).As

usual,thisgain

incontrolneeds

tobe

paidby

astrongly

reducedproduc-

tionrate.

Thekey

constraintforapplications

ofbubbleand

foamform

ationby

electrolysisisthe

restrictedchoice

ofgaseswhich

canbe

produced.Thisis

commonly

oxygenor

hydrogen,which

restrictsquite

significantly

theirfield

ofuse.

3.4.Conclusionsand

outlook

Wehope

thatthisreview

showsthatthere

aremany

ways

ofincor-porating

bubblesinto

aliquid

phase.W

ehave

concentratedhere

on

Fig.38.(a)Break-up

ofbubblesunder

shearin

afoam

with

closelypacked

bubbles(from

[149]),which

occursatcapillary

numbers

wellbelow

thoseofisolated

bubbles(b)

from[149].

(b)Com

parisonbetw

eenthe

criticalcapillarynum

berrequired

forbreak-up

ofindividualdroplets(“G

racecurve”

Section2)

andassociated

data(circles)

andbetw

eendensely

packedbubbles

(“Foams”)

anddroplets

(“Emulsions”)

(c)Proposed

reasoningfor

theearlier

break-upofbubbles

inthe

presenceofneighbouring

bubbles:amicro-structure

inducedinstability

(from[149]).

Fig.39.Foaming

byflipping

ashallow

cell(adaptedfrom

[150]).(a)Consecutiveim

agesofthefoam

inthe

cellafternf=1,2,6,12,18,and

24flips.(b)Evolution

ofnumberofbubblesw

ithnum

berofflips

fordifferent

contentsofliquid

(φ)in

thecell.

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esA,The

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physicaleffects,butthereader

shouldbe

aware

thatawealth

ofchem-

icalandbiologicalfoam

ingtechniques

isalso

available.Itneeds

tobe

addedthatthere

existsalarge

number

of“naturalfoaming

techniques”,where

foamsoccur

naturallyin

nature(on

thebeach,in

waterfalls,etc.)

orin

industrialprocesses.Inboth

cases,thisnaturalfoam

productionmay

poseserious

problems.U

nderstandingthe

mechanism

sattheir

or-igin

cantherefore

bevery

helpfultoavoid

thefoam

formation.

Itis

importan

tto

keepin

mind

thatfor

agiven

fluid

anyfoam

ingtechnique

leadsto

acharacteristic

bubblesize

distribution,gasfraction

andfoam

productionrate

(Table3).The

generationof

foamswith

awide

rangeofthese

parameters

generallyrequires

theapplication

ofdifferentfoam

ingtechniques.

Awealth

ofliteratureisavailable

tocapture

certainkey

propertiesof

differentfoam

ingtechniques.H

owever,at

this

stageon

lythemost

simple

configuration

sare

beginning

tobe

explainedin

asystem

aticmanner

—for

example

bubblingfrom

individual

nozzlesor

bubblebreakup

undershear.A

ndeven

inthese

simple

cases,many

important

questionsrem

ainopen.

Much

more

systematic

work

within

acoherent

physicalframew

orkneeds

tobe

donein

thefuture

toshed

more

lighton

themore

complex

foaming

mechanism

s.Suchwork

needsto

establishthe

roleofthe

differ-ent

dynamicforces.A

swehope

tohave

shownin

thischapter,a

highlyappropriate

framew

orkfor

takinginto

accountthese

differentforces

isprovided

byworking

with

dimensionless

hydrodynamic

numbers,as

listedin

Table2.The

useofthese

numbers

allowsus

togroup

awide

rangeofobservations

bythe

main

physicalmechanism

sofa

process.The

physicalmechanism

softhe

bubblingprocess

aregenerally

well

reflectedby

theobtained

bubblesizes

andsize

distribution.Due

tothe

difficultyofm

easuringbubble

sizedistributions

infoam

s,much

ofthisinform

ationis

lackingin

thepast

literature.Modern

characterisationtechniques

provideamore

straightforward

accessto

thisinform

ation,which

shouldbe

exploitedmore

heavilyin

thefuture.These

techniquesinclude

lightscattering,X-ray

andNMRtom

ographyor

highspeed

im-

aging.Theavailability

ofthesize

distributionsshould

helpus

tograsp

more

profoundlythe

physicalmechanism

swhich

controlthebubble

formation.H

owever,great

careneeds

tobe

takenin

thisexercise:ob-

tainedbubble

sizesand

sizedistributions

may

dependmore

onthe

foamstability

thanon

thefoam

ingtechnique

used!Coalescen

ceand

gasexchange

[1,3]betw

eenbubbles

may

completely

changethe

sizedistribution

within

afew

secondsofthe

lifetim

eofthe

foam.M

uchof

thepast

literaturehas

thereforemixed

measures

offoaming

andfoam

stability—

which

capturevery

differentphysicalphenom

ena.Anum

berofkey

challengesrem

ainto

betackled

inthis

subject.Thefirst

challengerequires

toestablish

adescription

offoaming

which

al-low

sus

tomove

away

fromthe

dilutelim

itof

individualbubbles

inorder

totake

properlyinto

accountthe

many

typesofbubble

interac-tions

which

occurinmostcom

monly

usedfoam

ingtechniques.The

sec-ond

challengeisto

considerthe

influenceofthe

complex

visco-elasticproperties

ofgas/liquid

interfaces

which

contain

stabilisingagents.

Thepresence

oftheseagents

leadsto

anum

berofinterfacialstresses

which

needto

betaken

intoaccount

inaproper

dynamic

description.First

stepsinto

combining

thephysics

ofbubble

break-upin

densefoam

sand

theinfl

uenceof

theinterfacial

viscoelasticityhave

beentaken

byevidencing

theim

portanceofa

criticalstressinthe

foam,rather

thanacriticalcapillary

number

[149].Afinalkey

challengewillbe

thedescription

ofthe

foaming

ofliquids

with

non-N

ewtonian

flow

Fig.40.Examples

offoamgeneration

viabubble

nucleationand

growth.(a)

Continuousbubble

generationand

riseatnucleation

sitesatthe

wallofa

champagne

glass(im

agefrom

[11]).(b)

Awhipped

creamdispenser

usesnitrous

oxide.(c)Afoam

extruderis

typicallyused

tofoam

highlyviscous

liquidsbringing

thegas

(oftenCO

2 )into

solutionat

highpressure.

(d)Aerosolcans

aretypically

usedfor

cosmetic

applications.Thegas

caneither

bein

solution(case

A)or

itcanbe

chosensuch

thatitisan

insolubleliquid

underthe

pressurecondition

inthe

bottle.Inboth

cases,uponrelease

oftheliquid

intoam

bientpressure,the

gascom

eseither

outofsolution

orevaporates

(from[154]).

29W.D

renckhan,A.Saint-Jalm

es/A

dvancesin

Colloidand

InterfaceScience

xxx(2015)

xxx–xxx

Pleasecite

thisarticle

as:Drenckhan

W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

ColloidInterface

Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

properties,which

changetheir

flowproperties

upondeform

ationand

with

deformation

rate.Infact,m

anyprocesses

requirethe

successfulfoam

ingofcom

plexfluids

(polymer

melts

andsolutions,particle

sus-pensions,em

ulsions,etc.).Ofgreat

importance

inthe

futurewillbe

todevelop

foaming

tech-niques

which

pushbeyond

currentlimits.The

foaming

ofhighlyviscous

liquids,forexam

ple,remains

agreatchallenge

dueto

theim

portantin-fluence

ofthefluid

viscosityon

thedifferentflow

instabilities.Another

boundaryto

bepushed

willalso

bethatofthe

obtainedbubble

sizes—

especiallytow

ardssm

allbubbles

andlow

gasfraction

.Lowdensity

foamswith

bubblesizes

wellbelow

1μm

havegiant

surfaceto

volume

ratios(1

cm2of“nano-foam

”can

containthe

surfaceofan

entirefoot-

ballfield!)which

isofinterestforcatalytic

reactions.Moreover,therm

alconduction

isreduced

byorders

ofmagnitude,w

henthe

bubblesize

issm

allerthan

themean

freepath

ofthegas

molecules

(about0.1

μmfor

airatroom

temperature

andam

bientpressure).To

sumup,decades

ofconsiderableengineering

work

haveprovided

uswith

awide

rangeofreliable

foaming

techniques.Thecurrent

andfuture

challengeis

tobe

ableto

arriveat

asatisfying

descriptionand

predictionof

thecom

plexfoam

ingprocesses.O

ntheon

ehand,th

isshould

helpto

optimise

currentfoaming

processes,onthe

otherhand,

this

shouldprovide

uswith

thepossibility

togenerate

controlled

foamswith

awiderrange

ofproperties,stimulating

newresearch

direc-tions

andleading

tothe

emergence

ofnewapplications

offoams.

Ackn

owledgem

ents

Theauthors

would

liketo

thankReinhard

Miller

formotivating

thewriting

ofthisarticle

andfor

hisim

portantcontributions

toresearch

onbubbles.W

ealso

thankBillRossen,Stubenrauch,Em

manuelle

Rio,Dom

iniqueLangevin,N

ikolaiDenkov

andJean-Eric

Poirierfor

veryuse-

fulcomments

onthis

manuscript.W

.Drenckhan

acknowledges

fundingfrom

anERC

StartingGrant(agreem

ent307280-POMCA

PS).

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dvancesin

Colloidand

InterfaceScience

xxx(2015)

xxx–xxx

Pleasecite

thisarticle

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W,Saint-Jalm

esA,The

scienceoffoam

ing,Adv

ColloidInterface

Sci(2015),http://dx.doi.org/10.1016/j.cis.2015.04.001

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