Adsorption and transport of surfactant/protein

onto a foam lamella

within a foam fractionation column with reflux

A Thesis submitted to The University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2014

By

Denny Vitasari

School of Chemical Engineering and Analytical Science

Contents

List of Figures vi

List of Tables vii

Nomenclature viii

Abstract xii

Declaration xiii

Copyright Statement xiv

Acknowledgements xv

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature review 6

2.1 Description of a foam fractionation column . . . . . . . . . . . . . . . . 7

2.2 Dynamics of adsorption of surfactant on a bubble surface . . . . . . . . . 11

2.3 Equilibrium of adsorption of surfactant on a bubble surface . . . . . . . . 17

2.4 Adsorption of mixed protein-surfactant . . . . . . . . . . . . . . . . . . . 21

2.4.1 Adsorption equilibrium of individual protein and surfactant . . . 21

2.4.2 Adsorption equilibrium of mixed protein-surfactant . . . . . . . 23

ii

2.5 Foam Drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Drainage of a foam lamella . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6.1 Reynolds model film drainage for a rigid interface . . . . . . . . 31

2.6.2 Power law film drainage model for a mobile interface . . . . . . . 32

2.6.3 Exponential film drainage model for viscous film . . . . . . . . . 33

2.7 Surface rheological forces on foam films . . . . . . . . . . . . . . . . . . 35

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Dynamics of adsorption of surfactant on a bubble surface 39

3.1 Mathematical model of the adsorption dynamics . . . . . . . . . . . . . . 40

3.2 Numerical solution of the adsorption equation . . . . . . . . . . . . . . . 43

3.3 Numerical simulation using Henry adsorption isotherm . . . . . . . . . . 46

3.4 Rescaling of parameters for Henry and Langmuir comparison at early

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Numerical simulation of Ward-Tordai equation . . . . . . . . . . . . . . 48

3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6.1 Verification of numerical solution . . . . . . . . . . . . . . . . . 50

3.6.2 Comparison of simulations using Langmuir and Henry isotherms 52

3.6.3 Rescaling of parameters and numerical simulation of Ward-Tordai

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Simulation of dynamics of adsorption of mixed protein-surfactant on a bub-

ble surface 58

5 Surfactant transport onto a foam lamella 61

6 Surfactant transport onto a foam lamella in the presence of surface viscous

stress 66

7 Conclusions 69

8 Future work 72

iii

8.1 Adsorption of mixed protein-surfactant onto a bubble surface . . . . . . . 72

8.2 Surfactant transport onto a foam lamella . . . . . . . . . . . . . . . . . . 73

8.3 Future experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.3.1 Measurement of surface rheological properties . . . . . . . . . . 75

8.3.2 Enrichment of a foam fractionation column . . . . . . . . . . . . 75

8.4 Recommendation for the design of a foam fractionation column . . . . . 76

References 78

A Analytical verification of the numerical simulation of dynamics of adsorption

of surfactant on the bubble surface 87

A.1 Derivation of Ward-Tordai equation from the Laplace transformation of

the diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B Numerical simulation of Ward-Tordai equation 94

C Equation for desorption of surfactant in the presence of protein in the bulk

solution 97

D Typical parameters for calculation of surfactant transport onto a foam lamella 98

D.1 Typical operating condition of foam fractionation column with reflux . . . 98

D.2 Radius of curvature of Plateau border and liquid fraction . . . . . . . . . 99

D.3 The residence time within a foam fractionation column . . . . . . . . . . 102

D.4 Early time evolution of film thickness . . . . . . . . . . . . . . . . . . . 103

D.5 Amount of surfactant in the film and in the Plateau border . . . . . . . . . 105

Final word count: No. of Words 70111

iv

List of Figures

2.1 Illustration of foam fractionation column using batchwise operation: (a)

simple batch column; (b) batch column with reflux. . . . . . . . . . . . . 8

2.2 Illustration of foam fractionation column using continuous operation: (a)

simple continuous column; (b) stripper continuous column; (c) enricher

continuous column; (d) combined continuous column. . . . . . . . . . . 9

2.3 Diagram of dynamics of adsorption of surfactant on a bubble surface

showing the surface, subsurface and bulk solution. The surfactant molecules

diffuse towards the subsurface and the adsorption occurs from the sub-

surface onto the surface. The thickness of the subsurface is negligible

compared to the adsorption length L. . . . . . . . . . . . . . . . . . . . . 13

2.4 Typical curve for surface tension γ versus logarithm of bulk concentration

C. The surface tension decreases with the increase of bulk concentration

up to the CMC. Above the CMC, the surface tension does not change with

the increase of bulk concentration. . . . . . . . . . . . . . . . . . . . . . 20

2.5 Typical curve for surface excess Γ versus bulk concentration C following

the Langmuir isotherm. The curve tends to level off when concentration

increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Illustration of a foam network: a. network of Plateau borders; b. a film

and two Plateau borders at each end. . . . . . . . . . . . . . . . . . . . . 27

2.7 Schematic diagram of half a foam lamella with an adjacent Plateau border 31

v

3.1 Dimensionless bulk concentration in the subsurface C ′ vs dimensionless

time t′ simulated using various methods and adsorption isotherms at very

low initial bulk concentration (Ci = 0.2 mol m−3). The analytical re-

sult overestimates the numerical results due to numerical error. The inset

shows the correlation at early time. At very early time, due to assump-

tion of infinite adsorption length, the subsurface concentration calculated

analytically using the Henry isotherm is lower than that calculated numer-

ically – where the adsorption length is finite. . . . . . . . . . . . . . . . . 51

3.2 Dimensionless bulk concentration C ′ in the subsurface vs dimensionless

time t′ simulated using the Henry and Langmuir isotherms. The simu-

lation is using Ci = 200 mol m−3 while the other parameters are as pre-

sented in Tab. 3.1. The Langmuir isotherm with Γ′ = (1+κ)C ′(0, t′)/(1+

κC ′(0, t′)) has less subsurface concentration than the Henry isotherm with

Γ′ = C ′(0, t′). The inset is the profile at early time where the value of C ′

calculated using the Henry isotherm is higher than that calculated using

the Langmuir isotherm. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Dimensionless surface concentration Γ′ vs dimensionless time t′ simu-

lated using the Henry and Langmuir isotherms. The simulation is using

Ci = 200 mol m−3 while the other parameters are as presented in Tab. 3.1.

The Langmuir model has more surface excess than the Henry model. The

inset is the profile at early time. At very early time, the surface excess

calculated using the Henry model is higher than that calculated using the

Langmuir model as the effect of higher C ′ calculated using the Henry

model at this very early time. . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Rescaled dimensionless surface concentration Γ′′ vs dimensionless time

t′′ simulated using the Langmuir and Henry isotherms compared with the

simulation result using the Ward-Tordai equation. The rescaling collapses

the data at early times. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

vi

List of Tables

3.1 The values of parameters used in the simulation of dynamics of adsorption

of surfactant onto a bubble surface taken from the data of the study by

Chang and Franses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

D.1 Typical operating condition of a foam fractionation column in the study

by Martin, et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

D.2 Surface mobility, the Henry constant and ratio of surfactant in the Plateau

border and in the film of various surfactant and/or surface active protein.

Data were taken from literatures. . . . . . . . . . . . . . . . . . . . . . . 107

vii

Nomenclature

αp intermolecular interaction parameter of protein

αs intermolecular interaction parameter of surfactant

αps intermolecular interaction parameter of protein and surfactant

τ function of the radius of the hemisphere of a bubble (s)

a activity coefficient

u mean velocity of liquid flowing inside a straight Plateau border (m3 s−1)

v mean velocity of a surfactant molecules (m s−1)

δ half film thickness (m)

∆P excess pressure within a foam lamella (Pa)

δ0 initial half of film thickness (m)

εa potential adsorption barrier (m2 kg s−2)

Γ surface concentration (mol m−2)

γ surface tension (N m−1)

Γ′ dimensionless surface concentration

Γ′′ rescaled dimensionless surface concentration

γ0 surface tension of pure solvent (N m−1)

Γe equilibrium surface concentration (mol m−2)

Γm maximum surface concentration (mol m−2)

Γp surface concentration of protein (mol m−2)

Γs surface concentration of surfactant (mol m−2)

Γfilm surfactant surface concentration on the film (mol m−2)

γPb surface tension at the Plateau border (N m−1)

viii

κ dimensionless Langmuir adsorption coefficient

λ length of Plateau borders per unit volume (m)

B Bond number (dimensionless)

H Henry constant (m)

Π disjoining pressure (Pa)

µ liquid viscosity (Pa s)

µs surface viscosity (Pa m s)

ω0 molar area of solvent (m2 mol−1)

ωp molar area of protein (m2 mol−1)

ωs molar area of surfactant (m2 mol−1)

φ liquid fraction in the foam

Π surface pressure (N m−1)

ρ density (kg m−3)

τ dummy variable (s)

τ ′ dimensionless dummy variable

τ ′′ rescaled dimensionless dummy variable

θ angle of a straight Plateau border with vertical

θp protein surface coverage

θs surfactant surface coverage

Γe rescaled equilibrium surface concentration (mol m−2)

H rescaled Henry constant (m)

h rescaled adsorption depth (m)

A cross sectional area of a Plateau border (m2)

a radius of curvature of a Plateau border (m)

Acut area of the oblique cut through a Plateau border (m2)

Afilm surface area of a film (m2)

bp adsorption coefficient of protein (m3 mol−1)

bs adsorption coefficient of surfactant (m3 mol−1)

C concentration (mol m−3)

C ′ dimensionless bulk concentration

ix

Ci initial bulk concentration (mol m−3)

cp bulk concentration of protein (mol m−3)

cs bulk concentration of surfactant (mol m−3)

D diffusion coefficient (m2 s−1)

d thickness of the subsurface (m)

dc foam fractionation column diameter (m)

F Laplace transformation of C ′

f constant for the shape of the cross section of a Plateau border

g gravity acceleration (m s−2)

H foam height (m)

h adsorption depth (m)

JF superficial liquid velocity (m s−1)

K Langmuir constant (m3 mol−1)

k Boltzmann constant (m2 kg s−2 K−1)

ka rate of adsorption

kd rate of desorption

L diffusion length/ half of lamella length/ length of the sides of a hexagonal film (m)

l dimensionless adsorption length

ls rescaled dimensionless adsorption length

Pc capillary pressure (Pa)

pg pressure in the gas (atm)

pl pressure in the liquid in a Plateau border (atm)

QPb volumetric flowrate of liquid in a Plateau border (m3 s−1)

R ideal gas constant (J K−1 mol−1)

Rb radius of a spherical cell giving the volume equal to the Kelvin’s cell (m)

rb radius of a bubble (m)

Rcap radius of the hemisphere of a bubble (m)

s Laplace variable

Sfilm amount of surfactant on the film surface (mol)

SPb amount of surfactant on the Plateau border surface (mol)

x

T temperature (K)

t time (s)

t′ dimensionless time

t′′ rescaled dimensionless time

tb time required to form a bubble (s)

tr residence time inside a foam fractionation column (s)

tmobile time for a film with a mobile interface to thin (s)

trigid time for a film with a rigid interface to thin (s)

u liquid velocity (m s−1)

uAvg average liquid velocity (m s−1)

v superficial gas velocity (m s−1)

vb interstitial gas velocity (m s−1)

Vcell volume of cell/bubble (m3)

VPb volume of one Plateau border (m3)

x distance from the bubble surface/ distance from the centre of a lamella along x

axis (m)

x′ dimensionless distance from the bubble surface/ dimensionless distance from the

centre of a lamella along x axis

x′′ rescaled dimensionless distance from the bubble surface

y vertical coordinate in a foam fractionation column (m)

z distance from the centre of a film (m)

xi

Adsorption and transport of surfactant/protein onto a foam lamellawithin a foam fractionation column with reflux

Denny Vitasari, The University of Manchester, August 28, 2014Submitted for the degree of Doctor of Philosophy

Abstract

Foam fractionation is an economical and environmentally friendly separation methodfor surface active material using a rising column of foam. The system of foam fractiona-tion column with reflux is selected since such a system can improve the enrichment of theproduct collected from the top of the column. Due to the reflux, it is assumed that there ismore surface active material (surfactant and/or protein) in the Plateau border than that inthe foam lamella, so that the Plateau border acts as a surfactant/protein reservoir. The aimof this thesis is to investigate the adsorption and transport of surface active material suchas surfactant and/or protein onto the surface of a lamella in a foam fractionation columnwith reflux using mathematical simulation.

There are two steps involved in adsorption of surface active material onto a bubblesurface within foam, which are diffusion from the bulk solution into the subsurface, alayer next to the interface, followed by adsorption of that material from the subsurfaceonto the interface. The diffusion follows the Fick’s second law, while the adsorption mayfollow the Henry, Langmuir or Frumkin isotherms, depending on the properties of thesurface active material. The adsorption of mixed protein-surfactant follows the Frumkinisotherm. When there is a competition between protein and surfactant, the protein arrivesonto the interface at a later time due to a slower diffusion rate and it displaces the surfac-tant molecules already on the surface since protein has a higher affinity for that surfacethan surfactant.

The surfactant transport from a Plateau border onto a foam lamella is determined bythe interaction of forces applied on the lamella surface, such as film drainage, due to thepressure gradient between the lamella and the Plateau border, the Marangoni effect, dueto the gradient of surface tension, and surface viscosity, as a reaction to surface motion. Inthis thesis, there are two different models of film drainage. One approach uses assumptionof a film with a mobile interface and the other model assumes a film with a rigid interface.In the absence of surface viscosity, the Marangoni effect dominates the film drainageresulting in accumulation of surfactant on the surface of the foam lamella in the case ofa lamella with a rigid interface. In the case of a film with a mobile interface, the filmdrainage dominates the Marangoni effect and surfactant is washed away from the surfaceof the lamella. When the drainage is very fast, such as that which is achieved by a filmwith a mobile interface, the film could be predicted to attain the thickness of a commonblack film, well within the residence time in a foam fractionation column, at which pointthe film stops draining and surfactant starts to accumulate on the lamella surface. Thedesirable condition in operation of a foam fractionation column however is when theMarangoni effect dominates the film drainage and surfactant accumulates on the surfaceof a foam lamella such as the one achieved by a film with a rigid interface.

In the presence of surface viscosity and the absence of film drainage, the surface viscousforces oppose the Marangoni effect and reduce the amount of surfactant transport ontothe foam lamella. A larger surface viscosity results in less surfactant transport onto thefoam lamella. In addition, the characteristic time scale required for surfactant transport isshorter with a shorter film length.

xii

Declaration

I declare that no portion of this work referred to in this thesis has been submitted in sup-port of an application for another degree or qualification of this or any other university orother institute of learning.

Denny Vitasari

xiii

Copyright Statement

i The author of this thesis (including any appendices and/or schedules to this thesis)owns any copyright in it (the “Copyright”)1 and s/he has given The University ofManchester the right to use such Copyright for any administrative, promotional,educational and/or teaching purposes.

ii Copies of this thesis, either in full or in extracts, may be made only in accordancewith the regulations of the John Rylands University Library of Manchester. Detailsof these regulations may be obtained from the Librarian. This page must form partof any such copies made.

iii The ownership of any patents, designs, trade marks and any and all other intellec-tual property rights except for the Copyright (the “Intellectual Property Rights”)and any reproductions of copyright works, for example graphs and tables (“Repro-ductions”), which may be described in this thesis, may not be owned by the authorand may be owned by third parties. Such Intellectual Property Rights and Repro-ductions cannot and must not be made available for use without the prior writtenpermission of the owner(s) of the relevant Intellectual Property Rights and/or Re-productions.

iv Further information on the conditions under which disclosure, publication and ex-ploitation of this thesis, the Copyright and any Intellectual Property Rights and/orReproductions described in it may take place is available from the Head of Schoolof Chemical Engineering and Analytical Science (or the Vice-President) and theDean of the Faculty of Life Sciences, for the Faculty of Life Sciences’ candidates.

1This excludes material already printed in academic journals, for which the copyright belongs to saidjournal and publisher. Pages for which the author does not own the copyright are numbered differently fromthe rest of the thesis.

xiv

Acknowledgements

I would like express my heartfelt gratitude and acknowledgement to my two supervisors:Dr Paul Grassia – Thank you for giving me such opportunity to carry out this interestingproject as a PhD student. I cannot thank enough for your dedication, guidance, supportand supervision through my PhD. And thank you for always encouraging me to reachbeyond my limit.

Dr Peter Martin – Thank you for all your support and for keep challenging my opin-ions that motivate me to learn even more.I have learnt a lot from both of you, not only as supervisors, but also as wonderful men-tors. Many thanks to both of you for making all achievements are possible.

Thank you to Dr Alastair Martin for invaluable discussions and feedback on my workthat enrich my knowledge.

Thank to Directorate General of Education Republic Indonesia and Universitas Muham-madiyah Surakarta for financial support to carry out my PhD project.

I also acknowledge everyone in the Multiphase Processing Group as well as everyonein Multi-scale and Multi-phase Systems Group, past and present, for all supports andvaluable discussions to improve my work. Also special thanks to my colleagues frompast C-19 for help and motivation, particularly for the lunchtime bridge that stimulatedmy brain through applied maths. All appreciation also for my colleagues at B-9 and D-39for nice time together and lots of entertainment to make my writing time far from tension.

I would like to thank to all family and friends for encouragements, comforts andprayers that made every obstacles were passable. Thank you to my sister Rima, mynephew Sakti and my niece Bintang also my best friends Anita, Zakky and Wahyu foralways listened and provided me escape whenever needed.

Finally I would like to thank to Mum and Dad who always supported and be on mysides throughout the years. And also for your patience while I am away most of the timewhen you needed me. Without you, I will not be where I am now. As a token of gratitudeand love, I would like to dedicate my thesis to both of you.

Ultimately thank to Allah Almighty for making all things possible. Everything hap-pens for a reason and I could only thank enough by fulfilling that reason.

xv

Chapter 1

Introduction

This chapter consists of two main parts. First is the motivation of this thesis and the

second is the structure of the thesis. At first the background of research on the adsorption

and transport of protein/surfactant onto a foam lamella is discussed. The importance of

the study as well as the application of the topic in industry is described. Finally the lay-out

and the content of the remainder of the thesis is outlined.

1.1 Motivation

Foam fractionation is an economical and environmentally friendly separation method for

surface active material such as protein and/or surfactant [1–5] based on bubble separation

techniques [6]. Foam fractionation can be conducted using semi batch or continuous pro-

cesses [1, 7] using a foam fractionation column. The foam fractionation process is one

of the alternatives for separation of dilute and structurally unstable material due to sensi-

tivity towards changes of e.g. temperature or pH [4]. This method is proven to be more

cost-effective for the intended purpose compared to other methods such as ion exchange,

electrophoresis, ultrafiltration and column chromatography [2, 4, 8] since it has simple

operation, high efficiency and low energy consumption [9]. The other processes require

1

solvent for the separation which is not the case for foam fractionation. For this reason,

foam fractionation is more ecological than other methods [4]. Although recently there

are significant number of application-related research on foam fractionation column fol-

lowed by vast number of publication, the commercial application of the technique is very

limited [10]. Therefore, the known existing foam fractionation columns are at laboratory

scale. One example of the geometry of a laboratory scale foam fractionation column is

described in Appendix D.

Foam in a foam fractionation column consists of air bubbles separated by thin liquid

films [11]. Due to its amphiphilic nature, the surface active material is adsorbed onto

the surface of the bubbles [12]. Since air bubbles have much lower density than the

liquid, they will be lifted up to the top of the column carrying the surface active material

with them. The enriched foamate is then collected from the top of the column. The

amount of adsorbed material determines the efficiency of a foam fractionation column

[1]. The adsorbed materials also have a role to stabilise the liquid film, preventing the

bubble coalescence and foam collapse [13–16] by reducing the mobility of the film surface

[14]. The design of a foam fractionation process needs to optimise the efficiency of the

adsorption as well as the stability of the foam.

Protein and surfactant may coexist in commercial foam separations [12] depending on

the nature of the solution and also when there is modification of the solution to improve

its separation efficiency [12, 17, 18]. Surfactant is mixed with protein to modify the ad-

sorption as well as the rheological characteristic of the adsorbed protein layer [19–21].

When adsorbed on the interface, protein may exhibit an immobile surface that stabilize

the liquid film [12, 17]. On the other hand, surfactant may form a mobile interface that

may stabilize the film later on due to the Gibbs-Marangoni effect [12]. When those two

components coexist in a foam film, the effect may be different from that which results

from the pure substance [12]. Protein and surfactant molecules may also compete to oc-

cupy the interface [20,22]. Therefore, it is important to study the adsorption behaviour of

mixed protein-surfactant in order to estimate the efficiency of a foam fractionation column

as well as the stability of the foam itself.

2

There are many studies examining the kinetics of adsorption of mixed protein-surfactant.

Those studies [19,22–25] determine the adsorption dynamics as well as adsorption isotherms

of mixed protein-surfactant on a bubble surface based on experimental data. Studies on

the mathematical modelling of adsorption behaviour of mixed protein surfactant are less

common. A predictive mathematical model for the adsorption dynamics and adsorption

isotherm of mixed protein-surfactant has a significant importance for examining the effi-

ciency of a foam fractionation column. Using mathematical model, the cost of conducting

experiments can be minimised without losing the important information on the adsorption

behaviour. Therefore, this study aims to develop a mathematical model for adsorption of

mixed protein-surfactant on a bubble surface that can be applied to determine the effi-

ciency of a foam fractionation column.

Due to low liquid content within the foam (less than 10%), the air bubbles take polyhe-

dral shapes [26], separated by thin liquid film named foam lamella [11]. Three films meet

and form an interstitial channel named Plateau border [11, 27]. Most of the liquid within

a foam is contained in the Plateau borders [27] and connected into a network within the

foam. Due to the curvature of the Plateau border, the pressure within it is lower than that

in the film [28]. As a consequence, there will be liquid drainage from the film towards the

Plateau border [28, 29].

Some foam fractionation columns employ a reflux system to improve the enrichment

of the rising liquid [6, 30, 31], by returning some part of the collapsed foamate to the

top of the column. The rising stream from the bottom of the column is enriched by the

falling stream from the top [30]. As falling stream drains through the Plateau border

network, it exchanges mass with adjacent films. This enrichment takes place within the

Plateau borders since most of the liquid within a foam is inside the Plateau border. As a

consequence, the concentration of surface active material in the Plateau border is higher

than that in the foam film.

As the surface active material is adsorbed from the bulk liquid onto the surface of the

Plateau border and the film, the gradient of concentration between the Plateau border and

3

the foam film leads to a gradient of surface tension. When there is gradient of surface

tension, the Marangoni effect takes place, causing the Marangoni flow from the region

with low surface tension to the region with higher surface tension. For systems of interest

here, the direction of the Marangoni flow is towards the centre of the film, opposite to the

direction of the film drainage. This Marangoni effect is important for transport of surface

active material from the surface of Plateau borders onto the surface of foam lamellae.

Moreover, the Marangoni effect contributes to the stabilisation of the liquid film [14, 32]

as its direction is opposite to the film drainage, therefore reduces the surface mobility.

Therefore this transport plays an important role in determination of the efficiency of a

foam fractionation column.

Studies on the film drainage and its effect on foam and foam film stabilisation have

been carried out [14,28,29,32–35]. However, those studies only consider the film thinning

and did not explore the transport of surfactant onto the foam lamella. On the other hand,

the transport of surfactant onto the surface of the foam lamella due to the interaction of

forces on it also plays an important role in determining the separation efficiency of a foam

fractionation column. Therefore, this study examines the transport of surfactant onto the

foam lamella and the phenomena involved in that transport. Those phenomena include

the film drainage, the Marangoni forces and the viscous forces that arise in response to

the resultant of other forces.

The purpose of a foam fractionation column is to separate surface active materials

from the solution by attachment of those materials onto the surface of the air bubbles.

Therefore, studying the dynamics and equilibrium of adsorption of surface active material

onto the foam as well as the transport of that surface active material onto the foam film is

fundamental for predicting the efficiency of a foam fractionation column. The knowledge

of adsorption and transport of surface active materials onto a foam film will be useful

to model the whole foam fractionation column system. In this study, foam fractionation

column with reflux is selected as it gives a better separation efficiency.

4

1.2 Thesis overview

This thesis is presented in the alternative format, as allowed by The University of Manch-

ester thesis submission guidelines. In this thesis, journal articles (submitted or accepted)

are presented in the place of standard thesis chapters. Introduction to the journal article

is given in the beginning of every chapter that consists of journal article to describe the

connection among the articles.

The thesis is organised as follows. The background theory for general operation of a

foam fractionation column, adsorption of mixed protein-surfactant, foam drainage as well

as film drainage is presented in Chapter 2. The model of dynamics of adsorption of sur-

factant on a bubble surface is discussed in Chapter 3. This model gives a basic knowledge

for a more complex model of dynamics of adsorption of mixed protein-surfactant which

is described in Chapter 4. Once the surface active material is adsorbed onto the interface,

it will be transported along the surface of the foam film due to interactions of forces on

the film surface. The model of transport of surfactant onto a foam lamella is presented in

Chapter 5. Surface viscous stress may occur as a response to the surface motion. Chap-

ter 6 deals with the transport of surfactant onto a foam lamella in the presence of surface

viscous stress. In the end, the findings are summarised in Chapter 7 and direction to the

future work is outlined in Chapter 8.

5

Chapter 2

Literature review

This thesis concerns various models of processes occurring in a foam fractionation col-

umn. A better physical understanding of foam fractionation column resulting from this

study is expected to help in the improvement of the operation and performance of a foam

fractionation column. Detailed knowledge of adsorption dynamics as well as of adsorp-

tion equilibrium of surfactant on the bubble surface is important to improve the perfor-

mance of foam fractionation columns.

This literature review is comprised of a general description of foam fractionation

which is presented in Section 2.1. Section 2.2 describes the study of kinetics of adsorp-

tion on the bubble surface which is a gas-liquid interface. The equilibrium of adsorption

of single surfactant on a bubble surface is discussed in Section 2.3 and the equilibrium

of adsorption of mixed protein-surfactant on a bubble surface is discussed in Section 2.4.

The theory of foam drainage is presented in Section 2.5, while the theory of film drainage

is discussed in Section 2.6. The surface rheological forces present on a film surface are

covered in Section 2.7. In the end, the contents of the literature review are summarised in

Section 2.8.

6

2.1 Description of a foam fractionation column

Foam fractionation is a process of enrichment of streams containing surface active mate-

rial using the principles of adsorption of the surface active substance(s) onto the surface

of bubbles in rising foam. The process is normally conducted in a foam fractionation

column. The separation in a foam fractionation column is due to the difference of sur-

face activity – the tendency of material to accumulate at surface – of the solute to be

adsorbed [6]. The air bubbles in the solution provide interfacial surface to adsorb the

surface active material. The foam created rises up carrying the solute upward [36]. The

foam is collapsed at the top of the column (e.g. using mechanical foam breaker). When

the foam is collapsed at the top of the column, the liquid collected from it is richer in

surface active material than the liquid at the bottom [31]. A foam fractionation column

can be equipped with a reflux system to enhance the separation [36]: this returns enriched

product back to the column with the aim of enriching it even further.

The foam fractionation process is applied for separation of various materials including

proteins [2, 37–40], biosurfactants [41–45], microorganisms [46, 47], and other surface

active materials [38].

Foam consists of air bubbles separated by thin films named foam lamellae. The thin

films form cells and connected by channels called Plateau borders (will be discussed

later), and three Plateau borders meet to form 120◦angle at a vertex. This work studies

the phenomena in the foam lamella and their effect on the efficiency of the separation in

a foam fractionation column.

A foam fractionation column can be operated continuously or batchwise. In batchwise

operation, product is collected as collapsed foam from the top of the column [6]. Batch-

wise operation can be run with or without reflux. The diagram of a batchwise column

and a batchwise column with reflux is presented in Fig. 2.1. The whole collapsed foam

is returned to the column during an operation of batchwise column with reflux [6]. The

batchwise operation is for a small scale, normally for high economical value product.

7

Gas Sparger

Foam breaker

(a)Gas

Reflux

Sparger

Foam breaker

(b)

Figure 2.1: Illustration of foam fractionation column using batchwise operation: (a) sim-ple batch column; (b) batch column with reflux.

When a continuous column is operated, there is a continuous feed and product taken

as collapsed foam as well as product taken from the liquid pool at the bottom [6, 7].

Continuous foam fractionation consists of two different methods, which are continuous

separation and continuous separation with reflux. Some of the collapsed foam collected

is returned through the top of the column to obtain a better separation efficiency in the

continuous column with reflux. There are four operation modes of continuous columns,

which are simple mode, stripper, enricher, and combined mode [6]. The diagram of those

four operation modes of continuous columns is presented in Fig. 2.2.

Feed is introduced to the liquid pool whereas product is taken from the bottom and

and from collapsed overflow in the simple column mode as presented in Fig. 2.2a. The

product taken from the overflow is rich in surface active material. The simple model is

good for foam with high stability. The foam at the top of the column will be drier than that

at the bottom, therefore has more tendency to collapse. In the stripping mode, feed enters

the column well above the top of the liquid pool and trickles down along the rising foam,

as shown in Fig. 2.2b. The feed would be stripped by mass transfer between the rising

stream from the liquid pool and the descending feed. The stripping mode is used when the

8

Feed

Gas Sparger

Foamate product

Bottom

Foam breaker

(a)

Feed

Gas Sparger

Foamate product

Bottom

Foam breaker

(b)

Feed

Gas

Reflux

Sparger

Foamate product

Bottom

Foam breaker

(c)

Feed

Gas

Reflux

Sparger

Foamate product

Bottom

Foam breaker

(d)

Figure 2.2: Illustration of foam fractionation column using continuous operation: (a) sim-ple continuous column; (b) stripper continuous column; (c) enricher continuous column;(d) combined continuous column.

solute is invaluable and need to be removed from the solution [48]. The stripping mode is

also appropriate for a system of dilute and low surface activity surfactant. The addition of

liquid from the top of the column will improve the foam stability [49]. Some part of the

collapsed overflow is returned to the top of the column as reflux to improve enrichment in

the enriching column presented in Fig. 2.2c. The reflux improves the separation efficiency

of the column as there will be transport of surfactant from the surface of surfactant-rich

Plateau border towards the film [50]. There is mass transfer between the falling richer

9

reflux stream and the rising liquid from the pool. As feed is introduced near the bottom

of the liquid pool, the mass transfer occurring will increase the concentration of the rising

liquid. The combined mode combines the stripping and enriching function of the column,

as the diagram is in Fig. 2.2d. Feed is introduced above the liquid pool, while there is

reflux to enrich the rising liquid [6]. So far the application of the combined mode of foam

fractionation column is difficult to establish due to the complexity of the operation [48].

The main principle of separation using a foam fractionation column is based on two

mechanisms. First is the selective adsorption of surface active component onto gas-liquid

interface of rising bubbles. The second phenomenon is formation of a foam on the top

of the liquid pool. The foam is relatively rich in adsorbed material, therefore it can be

recovered from the top of the column.

One of the important factors to determine the efficiency of a foam fractionation col-

umn is adsorption of surface active material onto the interface of a foam lamella or a

Plateau border. Therefore, it is important to understand the kinetics and equilibrium of

adsorption of the surface active substance onto the gas-liquid interface. Since the pro-

cess to be examined is separation of protein from solution (one of the main application

areas for foam fractionation nowadays [9, 41, 42, 51–53]), therefore this study considers

the mass transfer of mixed protein-surfactant on the gas-liquid interface. The interstitial

liquid flow in the foam can also play an important role in determining the efficiency of

a foam fractionation column. Therefore, it is also crucial to understand foam drainage

and/or foam film drainage. The model of mass transfer on the gas-liquid interface will

be coupled with the model of foam/film drainage to develop a better understanding of

separation in a foam fractionation column.

10

2.2 Dynamics of adsorption of surfactant on a bubble sur-

face

This study of a foam fractionation column examines the separation of surface active sub-

stances such as surfactant and surface active protein. The dynamics of adsorption of those

surface active substances on a bubble surface follow the same principles, regardless of

the type of the surface active substance. Therefore this section discusses the dynamics

of adsorption of surfactant and can also be applied to the dynamics of adsorption of any

other substances [22]. The dynamics of adsorption of surfactant on a bubble surface can

be used to predict the efficiency of a foam fractionation column. The dynamics of adsorp-

tion of the surface active material on a bubble surface are governed by a two-step process

as follows [54]:

1. Molecular transfer between the surface layer and the subsurface layer, a layer well

below the surface layer.

2. The replacement of molecules between the subsurface layer and the bulk solution.

The first step is an adsorption, while the second step is a diffusion. The subsurface

layer acts as a boundary between the region of adsorption and diffusion. The surfac-

tant adsorbs onto the interface, leaving the concentration in the subsurface lower. As a

consequence, the diffusion of surfactant in the bulk solution restores the concentration

of surfactant in the subsurface. The diagram of the dynamics of adsorption is presented

in Fig. 2.3. In reality, the interface between two bulk phases is not sharp, but there is

a region where the density and local pressure vary [55, 56]. The model considers sur-

factant below the so called CMC (critical micelle concentration) [6, 54, 57]. When the

concentration of surfactant is above CMC, the surfactant molecules will form micelles –

agglomeration of surface molecules, where the hydrophilic head arranged to be outside

the hydrophobic core of the cluster. The concentration in the actual foam fractionation

11

column can be above CMC. However, due to limitation of the model, this study only con-

sider concentration of surfactant below CMC. As the concentration of the solution is low,

the molecular diffusion can be modelled using the one-dimensional diffusion/adsorption

equation [54]. The model can also apply for a surfactant concentration above CMC, pro-

vided the overall micellar lifetime is shorter than the time taken to reach the equilibrium

surface tension [58]. When the micellar lifetime is longer than the time required to reach

the equilibrium surface tension, the monomers will stay in the micelles therefore limit the

adsorption [58].

There are two models of adsorption of surfactant onto a bubble surface. The first

model is the diffusion controlled adsorption, which occurs when there is spontaneous

adsorption once the surfactant reaches the subsurface. The second model is the mixed

kinetic-diffusion which assumes that once the surfactant reaches the subsurface, the ad-

sorption is not spontaneous, and could be a rate limiting which can be due to the existence

of an adsorption barrier [58].

When the diffusion is the rate limiting step of the mass transfer, the mass balance of

the adsorption process can be expressed using the equation of Fick’s second law [59]. The

diffusion equation is presented in the following equation:

∂C(x, t)

∂t= D

∂2C(x, t)

∂x2. (2.2.1)

The initial and boundary conditions are:

D∂C(0, t)

∂x=

dΓ(t)

dt(2.2.2)

C(x, 0) = Ci (2.2.3)

Γ(0) = 0 (2.2.4)

C(∞, t) = Ci (2.2.5)

where C is the surfactant concentration in the solution, D is the diffusion coefficient, x is

the distance from the subsurface, t is time, Γ is the surface concentration of surface active

12

adsorption

diffusion

surface

subsurface

C(x,t)

bulk solution

air

x

x=0

L

Γ(x,t)

Figure 2.3: Diagram of dynamics of adsorption of surfactant on a bubble surface showingthe surface, subsurface and bulk solution. The surfactant molecules diffuse towards thesubsurface and the adsorption occurs from the subsurface onto the surface. The thicknessof the subsurface is negligible compared to the adsorption length L.

material and Ci is the initial concentration of the solution. The boundary condition at the

interface, where x = 0, needs information about the adsorption isotherm (see Section 2.3)

to determine the surface concentration of surfactant.

Numerical simulation can be applied to the diffusion equation to obtain the solution

based on the boundary and initial conditions set. However, an analytical solution is useful

when a better understanding of the adsorption process is needed [60]. Ward and Tordai

[61] first developed an integration of the diffusion equation using Laplace transformation,

now known as the Ward-Tordai equation. The equation has often been used to determine

adsorption dynamics on the gas-liquid interface [59, 60, 62]. That analytical solution of

the diffusion equation is presented in Eq. (2.2.6).

Γ(t) = 2Ci(Dt/π)1/2 − (D/π)1/2

∫ t

0

C(0, τ)

(t− τ)1/2dτ (2.2.6)

where τ is a dummy variable [54].

The Ward-Tordai equation as proposed by Ward and Tordai [61] in Eq. (2.2.6) is only

suitable for planar interface. A study by Liu et al. [63] derived the modification of the

13

Ward-Tordai equation for a spherical interface. The equation was derived from the Fick’s

diffusion equation which was solved using Laplace transformation. This results in the

additional second term of Ward-Tordai equation as follows [63]:

Γ(t) =

√D

π

{2Ci√t−∫ t

0

C(τ)√(t− τ)

dτ

}+D

rb

{Cit−

∫ t

0

C(τ)dτ

}(2.2.7)

where rb is the radius of the bubble. This study considers the adsorption of surfactant

to the surface of a foam film (lamella) in a foam fractionation column with a flat sur-

face. Therefore, the dynamics of adsorption is modelled using the Ward-Tordai equation

presented in Eq. (2.2.6).

The net adsorption kinetics in Eq. (2.2.6), first established by Ward and Tordai, is

based on an assumption that the transfer of surfactant from the bulk solution to the in-

terface is fully controlled by the diffusion as the adsorption occurs spontaneously [61].

Another commonly used model is assuming that the diffusion mechanism is similar to

what occurs in the first model, however, the adsorption does not occur spontaneously due

to an adsorption barrier. This adsorption barrier presents as the effect of at least one of

the following: increasing surface pressure (the difference between surface tension with-

out and with adsorbed material), incorrect orientation of the monomers, unavailability of

the vacant sites on the surface or the presence of micelles the time scale for break-up of

which is longer than the time scale for adsorption [58]. This model is known as the mixed

kinetic model, also takes into account the exchange kinetics of surfactant between the sur-

face and the subsurface adjacent to it [64]. The mixed kinetic model adopts the following

physical assumptions:

1. The adsorption of the surfactant occurs only between the surface and the subsurface.

2. The thickness of the subsurface, d is only a few mean free paths of the adsorbing

molecules, hence it is negligible in the macroscopic point of view.

3. At the distance of x > d the transport mechanism of the molecules is through

diffusion with the diffusion coefficient of D.

14

4. The concentration at the distance d is C(d, t) ≡ C1(t).

5. The concentration at the surface, C(0, t) ≡ C0(t), is a function of the adsorption Γ

and is calculated using a particular adsorption kinetic equation.

In most approaches, the adsorption flux has a linear dependence on the concentration

C1 and the desorption flux depends only on adsorption, therefore the general form for the

adsorption kinetics is as follows [64, 65]:

dΓ

dt= kag(Γ)C1 − kdf(Γ) (2.2.8)

where g(Γ) and f(Γ) are functions to be specified, ka and kd are the rate of adsorption and

desorption, respectively, and the values depend on the model considered for the surface

phase. The rate of adsorption can be described as follows:

ka =v

4exp

(− εakT

)(2.2.9)

where εa is the potential adsorption barrier, k is the Boltzmann constant, T is the absolute

temperature and v is the mean velocity of the surfactant molecules. As a consequence,

Eq. (2.2.8) after dividing and multiplying the right hand side by the thickness of the sub-

surface, d can be rearranged as follows:

dΓ

dt= D∗g(Γ)

(C1 − C0

d

)(2.2.10)

where C0 can be described as follows:

C0 ≡kdka

f(Γ)

g(Γ)(2.2.11)

and D∗ is determined as follows:

D∗ =vd

4exp

(− εakT

). (2.2.12)

15

It can be found from the definition of C0 in Eq. (2.2.11) that C1 = C0 at equilibrium,

where dΓ/dt = 0. The macroscopic view of Eq. (2.2.10) would be in the following

equation:dΓ

dt= D∗g(Γ)

∂C

∂x

∣∣∣∣x=0+

. (2.2.13)

It can be noted from Eq. (2.2.13) that there is a discontinuity of the diffusion coefficient,

where it is constant in the bulk and dependent on the adsorption at the surface by D(0) =

g(Γ)D∗.

By solving the Fick’s equation using Laplace transform techniques for x > 0 with the

following boundary condition:

C(0, t) = C0(t) (2.2.14)

and the following initial condition:

C(x, 0) = Ci (2.2.15)

and by substituting C(x, t) into Eq. (2.2.13), the equation for Γ(t) can be obtained as

follows:1

g(Γ)

dΓ

dt=

√Da

π

Ci√

t+

1

2

∫ t

0

C0(τ)√(t− τ)3/2

dτ

(2.2.16)

where C0 is obtained using Eq. (2.2.11) and the apparent diffusion coefficient is intro-

duced as follows:

Da =D∗2

D= D exp

(−2

εakT

). (2.2.17)

Eq. (2.2.16) is integrated to obtain the more useful following form:

∫ Γ

0

1

g(Γ)dΓ =

√Da

π

[2Ci√t−∫ t

0

C0(τ)√(t− τ)

dτ

]. (2.2.18)

Eq. (2.2.18) represents the general equation for adsorption kinetics of various surface

models.

Eq. (2.2.16) is obtained in the case of D = vd/4 [64]. Therefore, the value of D∗ is

reduced to the actual diffusion coefficient when the adsorption barrier is zero (i.e. εa = 0).

16

In the case of no adsorption barrier and g(Γ) = 1, Eq. (2.2.18) is reduced to the Ward-

Tordai equation.

A knowledge of adsorption equilibrium is required to solve the diffusion equation via

the boundary condition on the gas-liquid interface. The model of adsorption equilibrium

depends on the adsorbed components and the concentration of solute. In this study, mod-

els for adsorption of individual surfactant and adsorption of mixed protein-surfactant will

be used to develop concentration profiles of surfactant and protein in the foam fractiona-

tion process.

2.3 Equilibrium of adsorption of surfactant on a bubble

surface

Gibbs equilibrium explains the adsorption equilibrium of the surfactant on the surface of

the bubbles as in Eq. (2.3.1) [6, 55].

dγ = −RT∑

Γid ln ai (2.3.1)

where γ is the surface tension, ai is the activity of component i, R is the gas constant, T

is the absolute temperature, and Γi is the surface excess or surface concentration.

Eq. (2.3.1) is obtained from the Gibbs-Duhem equation for an open system [55]. At

constant temperature and pressure, the equation gives the following relation:

dγ =

∑i

nSi dµSi

B(2.3.2)

where ni is the quantity of component adsorbed, µi is the chemical potential of the com-

ponent, B is the area of the interface and superscript S indicates that the system is at

17

constant entropy. If we determine the surface excess of the component Γi is the amount

of species i in unit area of the interface, or can be presented as follows:

Γi =niB

(2.3.3)

then we have the Gibbs adsorption isotherm as follows:

dγ =∑

i

Γidµi (2.3.4)

For a two-component system the equation can be simplified as follows:

dγ = Γ1dµ1 + Γ2dµ2 (2.3.5)

where the subscript 1 and 2 refer to the solvent and the solute respectively. Since the

interface between two bulk phases is not sharp, the actual amount of a species in the

whole system differs from the sum of the amounts in the two bulk phases by an excess

obtained in the surface region as presented in Eq. (2.3.5). Since the values of Γ1 and Γ2

are defined relative to an arbitrary surface region as a plane of infinitesimal thickness, it

is possible to position the surface such as Γ1 = 0 hence the surface excess of the solvent

is zero [55]. As a consequence Eq. (2.3.5) becomes the following equation [55]:

Γ2 = −(

dγ

dµ2

)

T

= − a

RT

dγ

da. (2.3.6)

For more than one solute, the equation can be be presented as Eq. (2.3.1).

The Gibbs equilibrium results in Γi, a term called surface excess or surface concen-

tration. It determines the concentration of adsorbed component i at the bubble surface

(gmol cm−2) [66]. However, the determination of the Gibbs equilibrium is difficult since

it requires the knowledge of activity coefficients. Moreover, it needs a precise determina-

tion of small change in surface tension, which is very difficult to measure.

For special cases, such as the adsorption of a single nonionic surfactant in pure solvent

at low concentration (below critical micelle concentration/CMC), the activity coefficient

18

is constant [66] and the activity, ai can be replaced by concentration C [6]. Therefore, the

equilibrium concentration can be simplified into Eq. (2.3.7)

Γ = − 1

RT

dγ

d lnC(2.3.7)

where Γ is the concentration of surfactant at the surface and C is the concentration of

surfactant at the bulk solution [66].

The surface tension decreases (as concentration increases) below the CMC and is

constant at the CMC and above. Below the CMC, if the curve of γ vs lnC forms a

straight line as presented in Fig. 2.4, this results in a constant Γ from Eq. (2.3.7). This

case is referred to as a Gibbs isotherm.

Other so called isotherms are possible. For example the surface excess well below the

CMC can often be expressed by Eq. (2.3.8)

Γ = HC (2.3.8)

which is called the Henry isotherm, H is the Henry constant, which is the ratio between

surface excess and bulk concentration in the limit of small concentration. The expression

of Eq. (2.3.8) in terms of surface tension is as follows:

γ0 − γ = RTHC = RTΓ (2.3.9)

where γ0 is the surface tension of pure solvent and γ is the surface tension of the solution.

As concentration increases, the Γ vs C curve tends to level off [7], as presented in

Fig 2.5. For non-ionic material, the entire curve of surface excess can be predicted by a

Langmuir isotherm as in Eq. (2.3.10)

Γ =HC

1 +KC(2.3.10)

19

γ

ln CCMC

Figure 2.4: Typical curve for surface tension γ versus logarithm of bulk concentrationC. The surface tension decreases with the increase of bulk concentration up to the CMC.Above the CMC, the surface tension does not change with the increase of bulk concentra-tion.

Γ

CFigure 2.5: Typical curve for surface excess Γ versus bulk concentration C following theLangmuir isotherm. The curve tends to level off when concentration increases.

where H and K are constants [66]. Eq. (2.3.10) coupled to Eq. (2.3.7) implies a surface

tension as follows:

γ0 − γ = RTHK

ln(1 +KC) = −RT HK

ln

(1− ΓK

H

). (2.3.11)

When the system consists of mixed protein-surfactant, a simple isotherm model such

as Langmuir isotherm is not applicable due to the complexity of the adsorption. Com-

monly the adsorption of mixed protein-surfactant can be modelled using a so called

20

Frumkin isotherm [25]. The following section describes the adsorption behaviour of

mixed protein-surfactant as well as single protein and single surfactant with high molec-

ular weight.

2.4 Adsorption of mixed protein-surfactant

2.4.1 Adsorption equilibrium of individual protein and surfactant

Most of the models which describe protein and mixed protein-surfactant adsorption as-

sume that protein molecules undergo multiple adsorption states at the surface [25, 67],

where the protein molecules can be folded or unfolded on the interface. During the ad-

sorption, protein molecules can exist in different states resulting in different molar areas

which vary from a maximum value ωmax, giving very low surface coverage to the mini-

mum value ωmin, at high surface coverage. Although there are infinite numbers of protein

molecule conformations, they can be approximated by a discrete and limited spectrum of

basic states. An equation of state for the surface layer can be obtained using the assump-

tion that molecules in different states are in equilibrium with each other, but behave as

independent components. Another assumption is that the molar area of the solvent, ω0,

is much smaller than the minimum molar area of the protein. The equation of state for

individual protein adsorption without any surfactant in the solution is as follows [68]:

− Πω0

RT= ln(1− θp) + θp(1− ω0/ωp) + αpθ

2p (2.4.1)

where Π = γ0 − γ is the surface pressure, γ0 is the surface tension of the solvent, R is

the gas law constant, T is the temperature, αp is the intermolecular interaction parameter,

Γp = Σni=1Γpi is the total adsorption of protein in all n states, θp = ωpΓp = Σn

i=1ωiΓpi and

ωp is the average molecular area of an adsorbed protein molecule, ωi = ω1 + (i−1)ω0 for

(1 ≤ i ≤ n) are the molar area of the i states with ω1 = ωmin and ωmax = ω1 +(n−1)ω0.

21

The equation of the adsorption isotherm for each state (i) of the protein is defined by:

bpicp =ωpΓpi

(1− θp)ωi/ωpexp[−2αp(ωi/ωp)θp] (2.4.2)

where cp is the bulk concentration of the protein and bpi are the equilibrium adsorption

constants of state i. If it is assumed that bpi are constant for any state of the protein ad-

sorption (bpi = bp, for any i), therefore the adsorption constant for the protein molecule as

a whole is Σbp = nbp. By this assumption the distribution function of various adsorption

states can be calculated from Eq. (2.4.2):

Γpi = Γp(1− θp)

ωi−ω1ωp exp[2αpθp

ωi−ω1

ωp]

∑ni=1(1− θp)

ωi−ω1ωp exp[2αpθp

ωi−ω1

ωp]. (2.4.3)

Studies found that Eqs. (2.4.1) – (2.4.3) are valid only for relatively low protein con-

centration. Under these circumstances, the increase of protein concentration in the solu-

tion results in an increase of surface pressure [68]. Higher protein concentration leads to

aggregation of molecules in the surface layers or formation of a bilayer. As a consequence

the equations become more complicated [24].

Previously we described the adsorption isotherm and equation of state for non-ionic

surfactant as following the Langmuir model as presented in Eqs. (2.3.10) and (2.3.11).

More generally however, a non-ionic surfactant can be assumed to satisfy the Frumkin

model as follows [69]:

− ΠωsRT

= ln(1− θs) + αsθ2s (2.4.4)

bscs =θs

(1− θs)exp[−2αsθs] (2.4.5)

where θs = ωsΓs, Γs is the surface concentration of surfactant, bs and αs are the adsorp-

tion equilibrium constant and the interaction constant of the surfactant, respectively. The

molar area of an adsorbed surfactant, ωs is not necessarily constant, but can be dependent

22

on surface pressure [70]:

ωs = ωs0(1− εΠθs) (2.4.6)

where ωs0 is the molar area at zero surface pressure and ε is the two-dimensional relative

surface layer compressibility coefficient, which characterises the intrinsic compressibility

of the molecules in the surface layer. This intrinsic compressibility, in this case, represents

the change of the tilt angle of the alkyl chain due to surface layer compression. The tilt

angle decreases upon compression, therefore the molar area becomes smaller and the

monolayer becomes thicker [70].

2.4.2 Adsorption equilibrium of mixed protein-surfactant

The protein molecules are much larger than the surfactant molecules. Moreover, protein

molecules are assumed to undergo conformation changes during the adsorption process.

As an effect, simple isotherm models such as Henry and Langmuir, which are based on

assumption of ideality of entropy and enthalpy cannot represent the protein adsorption.

Due to this reason, the model of adsorption of protein and surfactant mixture applies an

assumption of nonideality of entropy and enthalpy in the surface layer, and assumes that

ω0∼= ωs. Based on those simplifications the equation of state of adsorption of protein and

non-ionic surfactant mixture is as follows [68]:

− Πω0

RT= ln(1− θp − θs) + θp(1− ω0/ωp) + αpθ

2p + αsθ

2s + 2αpsθpθs (2.4.7)

where αps is a parameter that describes the interaction between the protein and surfactant

mixture. The adsorption isotherm for each state (i) of the protein is then derived as:

bpicp =ωpΓpi

(1− θp − θs)ωi/ωpexp[−2αp(ωi/ωp)θp − 2αpsθs]. (2.4.8)

The protein adsorption isotherm for the state of the minimum molar area of the protein,

23

ω1 = ωmin, is then described as:

bp1cp =ωpΓ1

(1− θp − θs)ω1/ωpexp[−2αp(ω1/ωp)θp − 2αpsθs]. (2.4.9)

The adsorption isotherm equation for the surfactant is as follows:

bscs =θs

(1− θp − θs)exp[−2αsθs − 2αpsθp] (2.4.10)

where the subscript p and s refer to characteristic parameters for the individual protein and

surfactant, respectively. The distribution of protein adsorption is given by the following

equation, noting the assumption that bpi = bp1 = bp [25]:

Γpi = Γp(1− θp − θs)

ωi−ω1ωp exp[2αpθp

ωi−ω1

ωp]

∑ni=1(1− θp − θs)

ωi−ω1ωp exp[2αpθp

ωi−ω1

ωp]. (2.4.11)

The total surface concentration of protein molecules and the total surface coverage are

calculated as follow where Γp = Σni=1Γpi is the total adsorption of protein in all n states

and θp = ωpΓp = Σni=1ωiΓpi is the total surface coverage of protein in all n states. The

theoretical description for protein-surfactant mixture can be elaborated in the following

way: from the known values of T , ω0, ωmin, ωmax, αp, bp, ωs0, αs and bs for the individual

protein and/or surfactant and αps as a single additional parameter for the mixture, the

dependent parameters of ωp, Γp, Γs, θp, θs and Π as a function of the concentration cs and

cp can be calculated [24].

The adsorption characteristic of protein-ionic surfactant mixture is essentially dif-

ferent from the adsorption of protein-non-ionic surfactant mixture given in the previous

equations. Interaction of a protein molecule with m ionized groups at a concentration cp

with ionic surfactant molecules at concentration cs will form complexes due to Coulomb

forces. These complexes are determined by the average activity of ions (cmp cs)1/(1+m) par-

ticipating in the reaction. The respective equation of state of the surface layer is similar

24

to the equation for mixed protein-non-ionic surfactant solution [71]:

− Πω0

RT= ln(1− θps − θs) + θps(1− ω0/ωps) + αpsθ

2ps + αsθ

2s + 2αspsθpsθs (2.4.12)

where θps = ωpsΓps is the coverage of the interface by adsorbed protein/surfactant com-

plexes and αsps is the parameter which describes the interaction of non-associated surfac-

tant with the protein/surfactant complexes. The corresponding adsorption isotherms for

protein-surfactant complexes in state i = 1 (a similar isotherm can be calculated for any

of the possible states i) and for the free surfactant that does not form complexes are as

follows:

bps(cmp cs)

1/(1+m) = bpscm/(1+m)p c1/(1+m)

s

=ωpsΓps1

(1− θps − θs)ω1/ωpsexp[−2αps(ω1/ωps)θps − 2αspsθs] (2.4.13)

bs(cscc)1/2 =

θs(1− θps − θs)

exp[−2αsθs − 2αspsθps] (2.4.14)

where cc is the surfactant counter-ion concentration. The subscript ps refers to the pro-

tein/surfactant complex [25].

Besides the adsorption of surface active material onto the bubble surface, the intersti-

tial liquid flow within the foam column also determines the efficiency of a foam fraction-

ation column. This study therefore also examines the fluid dynamics in a foam film and

its effect on the transport of surfactant onto a foam lamella. This fluid dynamics is also

determined by the liquid drainage within the foam fractionation column. Thus, a brief

discussion about foam drainage is presented in the following Section 2.5.

25

2.5 Foam Drainage

The mass transfer of surfactant and protein to the gas-liquid interface does not provide

any information of foam liquid fraction or the relative flow rate between draining liquid

and rising foam on the top of liquid pool [27]. Therefore, a study on foam drainage is

required to give a complete model of the foam fractionation process.

The liquid content of foam immediately after it is formed is higher than the liquid

content in hydrostatic equilibrium which is achieved a certain time after foam forma-

tion [72]. The reduction of liquid content is due to foam drainage. When foam forms,

foam drainage begins instantly as liquid flows through foam due to capillary and gravity

forces [73]. As a consequence, the liquid content of foam changes with time to reach its

hydrostatic equilibrium. This type of liquid flow is known as free drainage. The drainage

of foam contributes to foam (in)stability [74] where dry foam has a fragile structure and

thinner films that promote foam collapse [72]. The design of foam fractionation pro-

cesses and equipment depends on the knowledge of liquid volume fraction and drainage

rate. Therefore studies on foam drainage have a significant importance in the industrial

application of foam fractionation [72].

Dry foam consists of films which join to form channels with finite width, known as

Plateau borders as illustrated in Fig. 2.6. It is assumed that the most significant liquid

streams only occur in the Plateau borders. Typically, drainage occurs as mentioned above

due to gravity and capillary forces within the Plateau borders. The early experiments

of foam drainage were based on free drainage. However, this experimental method has

a drawback since it is difficult to determine the start of the experiment as liquid drains

instantly when foam is formed [75]. A controlled forced drainage was then introduced

[76] to quantify foam drainage more easily. Liquid is added constantly to the top of

the column to keep the constant liquid flow throughout the foam. The forced drainage

experiments are reminiscent of foam fractionation with reflux, the system to be examined

in this study. A simple model of foam drainage was proposed by Verbist et al. [77]. The

26

film

Plateau bo

rder

a b

Plateau bo

rder

Figure 2.6: Illustration of a foam network: a. network of Plateau borders; b. a film andtwo Plateau borders at each end.

model of foam drainage is based on relatively dry foam and a number of simplifying

assumptions apply compared to the real case. Those assumptions are as follows [77]:

1. Liquid flow in the film does not contribute to the drainage, the flow only happens

along the Plateau borders.

2. The flow along Plateau borders is Poiseuille type, therefore the velocity at the

boundaries is equal to zero.

3. The contribution to the viscous dissipation from the shearing motion of the flow

through the junction is insignificant, therefore can be ignored.

4. There is a constant surface tension and a constant liquid (bulk) viscosity in the

whole system.

The foam drainage model [77] is examined in a single Plateau border with cross sec-

tional area of A(y, t) changing with vertical coordinate y and time t. Assuming the liquid

27

is incompressible and the upward direction is positive, the equation of continuity may be

applied as follows:∂

∂tA(y, t) +

∂

∂y[A(y, t)u(y, t)] = 0 (2.5.1)

where the velocity u is averaged across the cross-section of the Plateau border.

The radius of the curvature of the Plateau border depends on the pressure difference

between the liquid in the Plateau border and the surounding gas following the Laplace-

Young law:

pg − pl =γ

a(2.5.2)

where a is the radius of the curvature of a Plateau border wall. The cross sectional area

of the Plateau border is related to the radius a as

A =(√

3− π

2

)a2 = C2a2 (2.5.3)

where C is a geometric constant equal to√√

3− π/2 [77].

For a volume element of A(y, t)dy of the Plateau border, the dissipative force due to

flow will be −fµu/A, where f ' 49, representing the shape of the cross-section of the

Plateau border, and µ is the viscosity of the liquid [11]. In this case, dissipation is balanced

by gravity, ρg and capillarity, −∂pl/∂y, resulting in equation for a vertical Plateau border

as follows:

− ρg − ∂

∂ypl −

fµu

A= 0. (2.5.4)

Substitution of pl from Eq. (2.5.2) into Eq. (2.5.4), using average over orientations of

Plateau borders results in velocity as a function of the cross-section:

u =1

3fµ

(−ρgA− Cγ

2√A

∂A

∂y

)(2.5.5)

the value of 3 coming from averaging over all possible Plateau border orientations. The

value of u can be substituted to the continuity equation to obtain the drainage equation

∂A

∂t+

∂

∂y

(− ρg

3fµA2 − Cγ

6fµ

√A∂A

∂y

)= 0 (2.5.6)

28

The foam drainage equation proposed by Verbist et al. [77] as presented in Eq. (2.5.6)

apply for a foam system without foam bursting/coalescence along the column. Neethling

et al. [78] introduced the length of Plateau border per unit volume, λ to incorporate the

foam bursting/coalescence into the drainage equation where the value of λ is allowed to

vary due to coalescence/bursting in a prescribed way in space and time. The volumetric

fraction of water then can be calculated as follows:

φ = Aλ (2.5.7)

where λ is a function of the radius of the bubble as presented in the following equation:

λ =5√

3

πψrb2(2.5.8)

where ψ = (1 +√

5)/2 ≈ 1.618. The drainage equation proposed now includes the

impact of gas velocity v, due to the bubbling up of the foam, therefore the liquid velocity

expression is as follows:

u =1

3fµ

(−ρgA− Cγ

2√A

∂A

∂y

)+ v (2.5.9)

as a consequence, the drainage equation for a system with bubble bursting/coalescence

can be presented as follows [79]:

∂(Aλ)

∂t+

∂

∂y

((− ρg

3fµA2 − Cγ

6fµ

√A∂A

∂y+ vA

)λ

)= 0. (2.5.10)

The knowledge of foam drainage provides the information of the relative flow rate

between draining and rising liquid within a foam fractionation column. Drainage theory

has been used to good effect to determine the distribution of liquid along a fractionation

column [80]. The foam drainage equation presented in this section has been developed in

a single Plateau border with assumption that the liquid flow only happened in the Plateau

border and liquid flow in the film does not contribute to the drainage. In reality, liquid is

also draining from the foam film. The drainage of liquid from the film not only contributes

29

to the liquid content within the foam but also defines the transport of surface active mate-

rial onto the film surface. Therefore, a study of surface active material onto a foam lamella

with relation to the film drainage is important for estimation of the efficiency of a foam

fractionation column. The following section discusses the drainage of a foam lamella.

2.6 Drainage of a foam lamella

Typically, the liquid within the foam lamella is in much smaller quantity than the liquid

within the Plateau border. Therefore, some models of foam drainage do not take into

account the drainage of the lamella. Although the contribution of liquid from the drainage

of lamellae to the liquid within the Plateau borders may be insignificant, the drainage

of the lamellae itself influences the transport of surface active materials onto the foam

lamella. This transport of surface active materials determines the efficiency of a foam

fractionation column, where the accumulation of surfactant on the foam films is what

causes enrichment. Therefore, it is important to study the models of drainage of a foam

lamella and apply it in the model of transport of surface active materials onto the foam

lamella.

A schematic diagram of a two-dimensional slice of half a foam lamella and a Plateau

border is presented in Fig. 2.7. One assumption taken in this study is that the interface

between the air bubble and the liquid film is symmetric [28]. The curvature at the Plateau

border causes the pressure in that region to be lower than that in the film. As a conse-

quence, there is suction of liquid from the film to the Plateau border [28], hence film

drainage occurs.

The rate of film drainage, the reduction of film half thickness δ by time, is the product

of interaction of the film with the Plateau borders around its edges and also other complex

phenomena such as dimpling and non-uniformity of film thickness. Evolution of film

thickness (∂δ/∂t) can be measured experimentally, using a device such as Scheludko

30

z

x

δ

Air

AirL

u(z,x,t)

γPb

γF

lamella Plateau border

Figure 2.7: Schematic diagram of half a foam lamella with an adjacent Plateau border

cell [81–83], therefore an empirical formula of ∂δ/∂t can be obtained. Other alternatives

are by employing theoretical estimates of the thinning rate obtained from literature. The

selection of the model of film drainage is based on the mobility of the film surface. One

of the factors affecting the mobility of the interface is the selection of the surfactant.

Some surfactants generate rigid interfaces, while others produce mobile interfaces. In this

section, there are three models of film drainage discussed, which are the Reynolds [84]

model for a rigid interface, the power law model for a mobile interface [28, 29, 33], and

the exponential model [85].

2.6.1 Reynolds model film drainage for a rigid interface

When a film has a rigid interface, the liquid flow within it is Poiseuille like. A no-slip

boundary assumption is taken for the velocity profile within the film. The film drainage for

a rigid interface developed by Reynolds [84] is based on the application of the lubrication

approximation to the Navier-Stokes equation to develop this following equation:

dδ

dt= −δ

3∆P

3µL2(2.6.1)

where ∆P is the excess pressure within the film as the driving force of the film drainage.

This excess pressure is the capillary pressure (Pc) due to the suction from the Plateau

31

border minus the disjoining pressure (Π) that occurs mainly as a result of electrostatic

and van der Waals’ contribution [86] as presented in the following equation:

∆P = Pc −Π. (2.6.2)

The capillary pressure for a dry foam, where the films form polyhedral shapes is the ratio

between the surface tension at the Plateau border γPb and the radius of curvature of the

Plateau border a [87] as can be expressed in the following equation:

Pc =γPba

(2.6.3)

where γPb is the surface tension at the Plateau border. If the film is sufficiently thick, the

disjoining pressure is negligible compared to the capillary pressure [11].

2.6.2 Power law film drainage model for a mobile interface

The assumption of plug-like flow within a film is taken when a mobile interface is as-

sumed. The equation for film drainage using the assumption of a mobile interface was

developed by Breward and Howell [28]. A similar drainage model was also developed by

Stewart and Davis [29, 33]. This later model is actually the same as the model developed

by Breward and Howell [28], but uses a different set of scalings to obtain various di-

mensionless model parameters. The equation developed by Breward and Howell uses the

initial thickness of the film to obtain the dimensionless film thickness, while the model

by Stewart and Davis determines the dimensionless film thickness based on half of the

film length. When the dimensionless equations from those two models are converted into

dimensional form, both of them result in a single dimensional equation for film drainage

as follows:dδ

dt= −3γPbδ

3/2

8µL√a. (2.6.4)

32

The rate of film drainage modelled using assumption of a mobile interface is much

faster than that modelled using the assumption of a rigid interface [50]. This can also be

seen from the power of δ in Eq. (2.6.4) which is smaller than that in Eq. (2.6.1). Since δ

falls as time evolves, we obtain dδ/dt∣∣mobile

� dδ/dt∣∣rigid

.

Even though it was stated previously that the matter of whether a film is rigid or mobile

is sensitive to surfactant type, it can also (for a given surfactant) be sensitive to time. A

film surface that originally has uniform surfactant concentration (hence no gradient in

surface tension) may act initially as mobile, but the film drainage may set up surface

tension gradients which make the film surface appear to become more rigid over time.

This is the topic that will be explored later on in the thesis (see Chapter 5). Even though

here we are concerned primarily with foam fractionation, in which foams are produced

with surfactant, for the sake of completeness we note that a sufficiently viscous liquid can

also produce a foam in the absence of any surfactant. The film drainage model for such

foam is discussed in Subsection 2.6.3.

2.6.3 Exponential film drainage model for viscous film

A highly viscous liquid can also produce a persistent foam without the presence of any

surfactant [26,85]. This type of foam can be obtained such as in the production of metallic

foam [26] and also in a glass furnaces and during volcanic eruption [85]. The profile

of film drainage for this type of foam is different from that profile of surfactant laden

film. In the absence of surfactant, the flow in the film is plug flow. Debrégeas et al.

[85] conducted an experiment to determine the film drainage equation for highly viscous

liquid in the absence of surfactant. In the experiment, an air bubble is injected to create

a hemispherical bubble with certain radius on the surface of the liquid. The film of the

bubble thinned under the action of gravity (rather than capillary suction from the Plateau

border). It turned out that the velocity field in the film was independent of film thickness.

The thinning of the liquid on the top of the bubble could then be modelled using an

33

exponential equation as follows:

δ = δ0 exp(−t/τ) (2.6.5)

where τ is a function of the radius of the hemisphere as presented in the following equa-

tion:1

τ=ρgRcap

µ(2.6.6)

where ρ is the density of the liquid, g is the gravity acceleration, Rcap is the radius of the

hemisphere and µ is the viscosity of the liquid.

We emphasise that the model developed by Debrégeas et al. [85] applies on a hemi-

spherical film draining under gravity (rather than capillary suction). Since the present

study considers films with a flat surface, the common shape in a dry foam, the following

discussions will only consider the model of foam drainage for a film with a rigid interface

proposed by Reynolds [84] and the power law model for a film with a mobile interface as

proposed by Breward and Howell [28] and also Stewart and Davis [29, 33].

Another reason for not considering the exponential film thinning model proposed by

Debrégeas et al. [85] is that in this model, the foam film produced has a large Bond

number. The Bond number B is the ratio between the gravity to the capillary forces and

can be described as follows:

B =ρgR

γ/a. (2.6.7)

where R is the radius of the hemisphere and a is the radius of curvature of the Plateau

border. If B > 1, the surface tension forces have insignificant effect compared to gravity

[88]. In the experiment of film drainage proposed by Reynolds [84] and Breward and

Howell [28] and also Stewart and Davis [29, 33], the Bond number was B < 1. On the

other hand, due to the large radius of curvature of the bubble and the comparatively high

density liquid employed, in the experiment of film drainage by Debrégeas et al. [85] it

happens instead that B � 1. This large Bond number B indicates that the effect of surface

tension is negligible. Therefore, this model is not suitable for simulation of surfactant

34

transport onto a foam film where the effect of surface tension (and its gradient) play a

significant role.

Film drainage is one of the factors that determines the transport of surfactant onto a

foam lamella. In this study, the drainage of foam lamella is caused by capillary suction

from the Plateau border. Capillary suction is however not the only force driving motion

on the scale of foam lamella. The forces involved in the motion of liquid within and on

the surface of a lamella is discussed in the following section.

2.7 Surface rheological forces on foam films

One of the forces determining the movement of liquid on the scale of foam lamella is

the capillary suction due to the curvature of the Plateau border [28] as discussed in Sec-

tion 2.6. Besides this capillary suction, there are other forces involve in the movement of

liquid on the scale of a lamella. Those forces include Marangoni forces due to a gradient

of surface tension and surface viscous stresses. The resultant of those forces determines

the transport of surfactant onto a foam lamella.

According to Eqs. (2.3.9), (2.3.11), (2.4.1), (2.4.4), (2.4.7) or (2.4.12), surface tension

depends on the surface coverage, i.e. surface excess of surfactant. If surface coverage

is non-uniform along the film, then the surface tension is likewise non-uniform. The

gradient of surface tension along the film leads to so called Marangoni stresses on the

film [14,32,89,90]. These Marangoni stresses typically need to be balanced by the viscous

shear stress upon the liquid in the bulk of the film as presented in the following equation

[14, 34, 50]:

µ

(∂u

∂z

) ∣∣∣∣z=δ

=∂γ

∂x(2.7.1)

where µ is the liquid viscosity, u is the liquid velocity along the film length (x axis), z is

the distance from the centre of the film along the z axis (the film thickness), δ is half of

35

the film thickness, γ is the surface tension and x is the distance from the centre of the film

along the x axis.

Those stresses drive the fluid motion in the film as well as on the film surface. In

addition, there can be surface viscous stress which couples the motion of a given point on

the film surface to motion of neighbouring surface points [50, 89]. In the presence of the

surface viscous stress, the stresses on the film now obeys the following equation [50, 89]:

µ

(∂u

∂z

) ∣∣∣∣z=δ

=∂γ

∂x+ µs

∂us∂x

(2.7.2)

where µs is the surface viscosity, us is the velocity on the film surface. These effects of

the surface rheological forces (Marangoni stresses and surface viscosity) will be discussed

in more detail later: see Chapters 5 and 6.

The Marangoni effect occurs when there is gradient of concentration of surfactant or

protein on the surface of a foam lamella. In this present study, the systems of interest

are where the concentration of surfactant/protein on the surface of the Plateau border is

higher than that on the surface of the lamella. When the Marangoni effect dominates the

film drainage, surfactant/protein is transported from the Plateau border towards the centre

of the film [50]. The surface viscous effect opposes the surface motion [91]. In the case

where the Marangoni effect is dominating, the presence of surface viscous effect reduces

the amount of surfactant transport onto the surface of a lamella as the surface viscous

forces oppose the Marangoni flow towards the centre of the film. Depending on the extent

of the surface viscous effect, the reduction of the amount of surfactant transport can be

quite severe.

2.8 Conclusions

Foam fractionation is a separation by adsorption of surfactant on the bubble surface in

a rising foam in a column. The foam fractionation column can be operated batchwise

36

or continuously. An external reflux is sometimes added into the column to improve the

enrichment of the surfactant in the product collected from the top of the column.

The adsorption dynamics of surfactant on the bubble surface is comprised of two steps:

adsorption and diffusion. For a diffusion controlled mechanism, the mass balance can be

expressed by the Fick’s second law. The diffusion equation can be solved numerically by

selecting the initial and boundary conditions appropriately. An analytical solution of the

diffusion equation, known as the Ward-Tordai equation is also applied by many studies.

When the adsorption is the rate-limiting step, a mixed adsorption model can be applied to

the dynamics of adsorption of surfactant on a bubble surface. The Ward-Tordai equation

can also be modified for compatibility with this mixed adsorption model.

The equilibrium of adsorption of single surfactant follows Gibbs’ equilibrium. Based

on the assumption that the surfactant concentration is relatively low and below the CMC,

the equilibrium of adsorption can be expressed using a linear equation or more generally

using a Langmuir isotherm. For more complex surfactant or for mixed protein-surfactant,

the adsorption equilibrium can be expressed using a Frumkin isotherm. When modelling

adsorption of protein, it has to be considered that the protein exhibits multiple states due

to folding and unfolding on the interface during the adsorption. The model of adsorp-

tion of protein-surfactant includes parameters that involve a combination of protein and

surfactant properties.

The model of foam drainage using the principle of forced drainage which is similar

to the system of foam fractionation column with reflux has been described. The foam

drainage model uses the assumption that the liquid flow is only within the Plateau border.

The more comprehensive model of foam drainage by incorporating bubble bursting/coa-

lescence has also been discussed.

The drainage of liquid from the foam film gives only small contribution to the foam

drainage. However, the film drainage determines the transport of surfactant onto a film

surface. Three models of film drainage have been discussed here. Those models are the

37

Reynolds drainage model for a film with a rigid interface, the power law drainage model

for a film with a mobile interface, and the exponential drainage model for a film with

highly viscous liquid in the absence of surfactant. In the system of interest here (i.e.

foam fractionation), we focus on the respective cases of rigid interface and the power law

drainage for a mobile interface.

Besides film drainage, there are other forces driving the motion of liquid within the

film. The gradient of surface tension leads to Marangoni stresses balanced to the viscous

shear stress upon the liquid in the bulk of the film. There can be surface viscous stresses

which couple the motion between nearby points on the film surface.

38

Chapter 3

Dynamics of adsorption of surfactant on a

bubble surface

Foam consists of air bubbles separated by foam lamella. The separation in a foam frac-

tionation column occurs by transferring surfactant or other surface active materials onto

the surface of air bubbles that is lifted up to the top of the column [6]. One of the pro-

cesses of transfer of surfactant onto a bubble surface is by adsorption. There are two steps

of this adsorption. The first step is diffusion of surfactant from the bulk solution to a layer

next to the surface, named the subsurface layer. Once the surfactant molecules reach the

subsurface layer, adsorption of surfactant from the subsurface to the bubble surface takes

place [54].

This chapter reports the results of a study of the simulation of dynamics of adsorption

of surfactant onto a bubble surface. This model also applies for adsorption dynamics of

other surface active materials. The diffusion of surfactant from bulk solution to the sub-

surface follows the Fick’s Law [92]. The adsorption of surfactant from the subsurface

to the interface is very fast, therefore the diffusion process is rate limiting [54, 93]. The

equilibrium of adsorption is modelled using the Langmuir [94] and Henry [56] isotherms.

The diffusion equation results in a differential equation. On the bubble surface, the sur-

factant concentration is in equilibrium with that in the subsurface layer. Therefore, the

39

adsorption isotherm applies on the bubble surface. The differential equation is solved nu-

merically using a finite difference method [95]. An analytical solution of the differential

equation using Laplace transformation verifies the numerical models. The analytical solu-

tion only applies on simple model such as the case with the Henry isotherm. The equation

using more complicated adsorption isotherm, such as the Langmuir isotherm can only be

solved numerically. The Laplace transformation of the differential equation also results

in a Ward-Tordai equation [61], a common model for adsorption of surfactant on a bubble

surface. The numerical solution is also compared with the solution of the Ward-Tordai

equation. The simulations of adsorption dynamics are presented in dimensionless forms.

3.1 Mathematical model of the adsorption dynamics

The diffusion of the surfactant from the bulk solution to the subsurface is rate limiting

of the whole adsorption process. That diffusion follows the Fick’s law of diffusion as

presented in Section 2.2, Eq. 2.2.1 as follows:

∂C(x, t)

∂t= D

∂2C(x, t)

∂x2

where C(x, t) is the concentration of surfactant at any given point in the bulk solution

at any given time, t is the time, x is the distance from the bubble surface and D is the

diffusion coefficient of surfactant in the solution. The differential equation is solved using

the boundary and initial conditions. The boundary condition at the bubble surface (x = 0)

is as follows:

D∂C(0, t)

∂x=

dΓ(t)

dt(3.1.1)

where Γ(t) is the surface concentration of surfactant on the bubble surface. The surfactant

surface concentration Γ(t) is in equilibrium with the surfactant bulk concentration at the

subsurfaceC(0, t), therefore the adsorption isotherm applies at this boundary. Commonly,

adsorption of surfactant onto a bubble surface is modelled using the Langmuir isotherm

40

[66] as follows:

Γ(t) = ΓmKC(0, t)

1 +KC(0, t)(3.1.2)

where Γm is the maximum surface concentration, represents the maximum packing on

the interface [96] and K is the Langmuir adsorption coefficient. The other boundary

condition is taken at position away from the bubble surface, at x = L as follows:

∂C(L, t)

∂x= 0 (3.1.3)

where L is the diffusion length.

The bulk concentration of surfactant in the subsurface and the surface concentration

of surfactant is initially zero. The initial bulk concentration of surfactant outside the

subsurface is uniform and equals to the initial surfactant bulk concentration. Therefore,

the initial conditions can be described as follow:

C(x, 0) = Ci; x > 0 (3.1.4)

C(0, 0) = 0 (3.1.5)

and

Γ(0) = 0 (3.1.6)

where Ci is the initial bulk concentration of surfactant.

Dimensional analysis was carried out to determine the dimensionless forms of the

adsorption dynamics as well as the boundary conditions. The dimensionless forms of the

adsorption dynamics are as follow:

∂C ′(x′, t′)

∂t′=∂2C ′(x′, t′)

∂x′2(3.1.7)

where C ′ = C/Ci is the dimensionless bulk concentration of surfactant, t′ = (D/h2)t

is the dimensionless time, h = Γe/Ci is the adsorption depth, which is the depth de-

pleted by surfactant adsorption [96], Γe = ΓmKCi/(1 +KCi) is the equilibrium surface

41

concentration of surfactant at a given initial bulk concentration Ci [96] and x′ = x/h

is the dimensionless distance from the bubble surface. The dimensionless form of the

adsorption equilibrium at the gas-liquid interface is as follow:

Γ′(t′) =(1 + κ)C ′(0, t′)

1 + κC ′(0, t′)(3.1.8)

where Γ′ = Γ/Γe is the dimensionless surface concentration of surfactant and κ = KCi

is the dimensionless Langmuir adsorption coefficient.

The dimensionless form of the boundary conditions are presented in the following

equations:∂C ′(0, t′)

∂x′=

dΓ′(t′)

dt′(3.1.9)

at x′ = 0, and∂C ′(l, t′)

∂x′= 0 (3.1.10)

at x′ = l, where l = L/h is the dimensionless adsorption length and h is the adsorption

depth. The initial conditions are also converted into dimensionless forms as follow:

C ′(x′, 0) = 1, x > 0 (3.1.11)

C ′(0, 0) = 0 (3.1.12)

and

Γ′(0) = 0. (3.1.13)

In what follows, the differential equation and the associated boundary and initial con-

ditions are solved numerically. The result of the numerical simulation is then verified by

the result of analytical simulation using Laplace transformation.

42

3.2 Numerical solution of the adsorption equation

The partial differential equation of surfactant adsorption was solved numerically using

Crank Nicolson method [95]. The numerical simulation involves the following finite

difference approximations on space and time intervals:

(∂C ′

∂t′

)

(i,n+1/2)

=1

2

[(∂2C ′

∂x′2

)

(i,n+1)

+

(∂2C ′

∂x′2

)

(i,n)

](3.2.1)

(∂C ′

∂x′

)

(i,n)

=C ′(i+1,n) − C ′(i−1,n)

2∆x′(3.2.2)

(∂2C ′

∂x′2

)

(i,n)

=C ′(i+1,n) − 2C ′(i,n) + C ′(i−1,n)

∆x′2(3.2.3)

x′i = i∆x′; t′n = n∆t′; i = 0, 1, 2, ..., I; n = 0, 1, 2, ... (3.2.4)

where i represents the counter over spatial steps and n is the counter over time steps.

The Crank Nicolson method is based on second order central difference in space and

central difference in time [95]. Application of Eqs. (3.2.1), (3.2.2) and (3.2.3) on dis-

cretisation of Eq. (3.1.7) results in equations at internal points of the finite difference as

presented in Eq. (3.2.5).

−(

∆t′

∆x′2

)C ′(i−1,n+1) + 2

(1 +

∆t′

∆x′2

)C ′(i,n+1) −

(∆t′

∆x′2

)C ′(i+1,n+1) =

(∆t′

∆x′2

)C ′(i−1,n) + 2

(1− ∆t′

∆x′2

)C ′(i,n) +

(∆t′

∆x′2

)C ′(i+1,n). (3.2.5)

43

The series of equations are then rearranged into a system of matrices to obtain the

values of C(i+1,n+1) at each time step using the Gaussian elimination. At the boundaries,

the finite difference equations are set based on the boundary conditions equations at x′ = 0

and at x′ = l. The finite difference equation for the boundary condition at x′ = 0 is:

(dΓ′

dt′

)

(n+1/2)

=1

2

[(∂C ′

∂x′

)

(0,n+1)

+

(∂C ′

∂x′

)

(0,n)

](3.2.6)

where dΓ′/dt′ can also be expressed as the following:

dΓ′

dt′=

(∂Γ′

∂C ′

)(∂C ′

∂t′

). (3.2.7)

Therefore, Eq. (3.2.6) can also be expressed as follows:

(dΓ′

dt′

)

(n+1/2)

=

1

2∆t′

[(∂Γ′

∂C ′

)

(0,n+1)

(C ′(0,n+1) − C ′(0,n)

)+

(∂Γ′

∂C ′

)

(0,n)

(C ′(0,n+1) − C ′(0,n)

)]. (3.2.8)

As a consequence, the boundary expressed by Eq. (3.1.9) can be discretised into the fol-

lowing form:

1

2∆t′

[(∂Γ′

∂C ′

)

(0,n+1)

(C ′(0,n+1) − C ′(0,n)

)+

(∂Γ′

∂C ′

)

(0,n)

(C ′(0,n+1) − C ′(0,n)

)]

=1

2∆x′(C ′(1,n+1) − C ′(0,n+1) + C ′(1,n) − C ′(0,n)

). (3.2.9)

Eq. (3.2.9) is rearranged to bring unknown terms to the left hand side.

− ∆t′

∆x′C ′(1,n+1) +

[(∂Γ′

∂C ′

)

(0,n+1)

+

(∂Γ′

∂C ′

)

(0,n)

+∆t′

∆x′

]C ′(0,n+1) =

[(∂Γ′

∂C ′

)

(0,n+1)

+

(∂Γ′

∂C ′

)

(0,n)

− ∆t′

∆x′

]C ′(0,n) +

∆t′

∆x′C ′(1,n) (3.2.10)

where (∂Γ′

∂C ′

)

(0,n)

=1 + κ

(1 + κC ′(0,n))2

(3.2.11)

with an analogous expression for (∂Γ′/∂C ′)(0,n+1).

44

The boundary condition at x′ = l is approximated using backward finite difference as

follows:1

2∆x′(C ′(I−2,n+1) − 4C ′(I−1,n+1) + 3C ′(I,n+1)

)= 0. (3.2.12)

The profile of adsorption dynamics was obtained by solving Eq. (3.2.5) together with

Eqs. (3.2.10) and (3.2.12) using the Gaussian Elimination method. Eq. (3.2.10) is non-

linear due to the differential of adsorption isotherm in Eq. (3.2.11). Therefore, the whole

system of Eqs. (3.2.5), (3.2.10) and (3.2.12) was solved using subsequent iterations. At

each time step, the value of (∂Γ′/∂C ′)(0,n+1) was first estimated using information from

the previous time step, or in other words, the initial guess is that C ′(0,n+1) = C ′(0,n). Then

the system of linearized equations was solved iteratively until the values of C ′(0,n+1) reach

convergence.

The iteration process was accomplished using the Newton-Raphson method. Assign-

ing that z = C ′(0,n+1), the initial guess of z was used to calculate the new value using the

following equation [95]:

zm+1 = zm −f(zm)

f ′(zm)(3.2.13)

where m is a counter over Newton-Rhapson iterations and f is obtained by subtracting

the left hand side of Eq. (3.2.10) from the right hand side of that equation at given C ′(0,n),

C ′(1,n) and C ′(1,n+1). The differential f ′ is obtained by finite difference as follows :

f ′(zm) =f(zm + ∆z)− f(zm −∆z)

2∆z. (3.2.14)

where ∆z = zm/100. The iteration was carried out to reach convergence of zm+1 and zm.

Once concentration distribution of surfactant across the solution domain was deter-

mined, the surface concentration, Γ′, at each time step was then calculated using the

following equation:

Γ′n =(1 + κ)C ′(0,n)

1 + κC ′(0,n)

. (3.2.15)

45

Obtaining C ′(1,n+1) via Eq. (3.2.5) with i = 1 requires the value of C ′(0,n+1) but this is

not available until Eq. (3.2.10) is solved. However, solving Eq. (3.2.10) itself requires the

value of C ′(1,n+1) to be provided. Thus a full solution of the system needs to iterate back

and forth between Eq. (3.2.5) and Eq. (3.2.10) as well as performing Newton Raphson

iterations to solve Eq. (3.2.10) at each stage.

The analytical verification of the numerical solution can be found in Appendix A.

3.3 Numerical simulation using Henry adsorption isotherm

An additional comparison of the results from numerical and analytical solution is obtain-

able by performing a numerical simulation using the Henry adsorption isotherm. The

dimensionless form of adsorption dynamics to be solved numerically are presented in

Eqs. (A.0.3) – (A.0.9). The partial differential equation was solved numerically using the

Crank Nicolson method and applying the boundary conditions. The internal points of the

finite difference equations are the same as in Eq. (3.2.5):

Since Γ′(t′) = C ′(0, t′), therefore we have:

(∂Γ′

∂C ′

)

(0,n)

= 1. (3.3.1)

As a consequence, the finite difference equation for the boundary condition at x′ = 0 to

replace Eq. (3.2.10) becomes:

− ∆t′

∆x′C ′(1,n+1) +

(2 +

∆t′

∆x′

)C ′(0,n+1) =

(2− ∆t′

∆x′

)C ′(0,n) +

∆t′

∆x′C ′(1,n). (3.3.2)

The boundary condition at x′ = l is approximated using backward finite difference as

presented in Eq. (3.2.12).

46

3.4 Rescaling of parameters for Henry and Langmuir com-

parison at early time

The parameters in both simulations using the Langmuir and Henry isotherms are rescaled

to compare the value of C ′|x=0 and Γ′ at early times, where the values of C ′|x=0 are

very low. At this low concentration on the surface, it is expected that the Langmuir

adsorption behaves similar to the Henry adsorption. The rescaling is carried out by giving

a parameter of Γe = Γe(1 + κ) so Γe = Γmκ = HCi, H = H(1 + κ), h = H = Γe/Ci =

Γe(1 + κ)/Ci = h(1 + κ) and Γ′′ = Γ/Γe or Γ′′ = Γ/Γe(1 + κ). Note that Γ′ = Γ/Γm,

therefore the rescaled parameters become:

Γ′′ =Γm

κC(0,t)′

1+κC(0,t)′

Γmκ=

C(0, t)′

1 + κC(0, t)′=

Γ′

1 + κ

h =ΓeCi

t′′ =D

h2t =

DC2i

Γe2 t =

DC2i

Γ2mκ

2t =

t′

(1 + κ)2

x′′ =Ci

Γe=

CiΓmκ

x =x′

1 + κ.

Hence the differential equations are rescaled as follow:

∂C ′

∂x′′= (1 + κ)

∂C ′

∂x′

∂Γ′′

∂t′′= (1 + κ)

∂Γ′

∂t′

∂C ′

∂t′′= (1 + κ)2∂C

′

∂t′.

As a consequence, the new dimensionless diffusion equation is:

∂C ′

∂t′′=∂2C ′

∂x′′2(3.4.1)

with boundary conditions as follow:

∂C ′(0, t′′)

∂x′′=

dΓ′′(t′′)

dt′′(3.4.2)

47

and∂C ′(ls, t′′)

∂x′′= 0 (3.4.3)

where ls = l/(1 + κ), and the initial conditions are as follow:

C ′(x′, 0) = 1, x > 0 (3.4.4)

C ′(0, 0) = 0 (3.4.5)

and

Γ′′(0) = 0. (3.4.6)

The rescaled equations apply to diffusion using both the Langmuir and Henry isotherms,

where in the limit κ → 0 for the Henry isotherm can be recovered from the Lang-

muir isotherm. Specifically the Langmuir isotherm becomes Γ′′(t′′) = C ′(0, t′′)/(1 +

κC ′(0, t′′)), whereas the Henry isotherm corresponds to Γ′′(t′′) = C ′(0, t′′). Note that in

this rescaled system, provided C ′(0, t′′) is very small (which is always the case at suffi-

ciently early times), the Langmuir and Henry isotherms now agree, regardless the value of

κ in the Langmuir case. As a result, the early time evolution C ′(0, t′′) and Γ′′(t′′) are the

same, regardless of whether we consider Langmuir or Henry isotherm. This correspon-

dence between the two system at early times was in fact the motivation for performing a

rescaling in this fashion.

3.5 Numerical simulation of Ward-Tordai equation

The results of the numerical solution are also compared with the result of an established

equation for dynamics of adsorption on gas-liquid interface proposed by Ward and Tordai

[61]. In this chapter, only the Henry isotherm is used in the simulation using the Ward-

Tordai equation. The equation is an analytical solution for the differential equation of

adsorption dynamics as follows:

Γ(t) =

√D

π

[2Ci√t−∫ t

0

C(0, τ)√t− τ dτ

]. (3.5.1)

48

Dimensional analysis has been carried out to the Ward-Tordai equation results in the fol-

lowing equation:

Γ′′ =2√π

[√t′′ − 1

2

∫ t′′

0

C ′(0, τ ′′)√t′′ − τ ′′dτ

′′]

(3.5.2)

where Γ′′ = Γ/HCi, for a Henry isotherm we have t′′ = (D/H2)t and C ′ = C/Ci. The

numerical solution of the Ward-Tordai equation is discussed in Appendix B.

3.6 Results and discussion

This section presents the result of numerical and analytical simulations of the kinetics

of adsorption. At first, the numerical simulation using the Langmuir isotherm is verified

using the numerical simulation using the Henry isotherm. An analytical solution of the

differential equation using the Laplace transformation confirms the numerical solution

using the Henry isotherm. The dimensionless form of the adsorption dynamics equation

using the Langmuir and Henry isotherms are rescaled and the results of numerical simu-

lation based on the rescaled parameters are compared with the results of simulation using

the Ward-Tordai equation with a Henry isotherm. The values of the parameters used in

this study are taken from the data of the study by Chang and Franses [97]. The details of

values of the parameters taken is presented in Table 3.1.

Table 3.1: The values of parameters used in the simulation of dynamics of adsorption ofsurfactant onto a bubble surface taken from the data of the study by Chang and Franses[97].

Parameter Value UnitΓm 7.1× 10−6 mol m−2

K 5.5× 10−3 m3 mol−1

Ci 0.2 - 200 mol m−3

D 8.8× 10−13 m2 s−1

49

3.6.1 Verification of numerical solution

Numerical simulations of the dynamics of adsorption of surfactant on a bubble surface

was carried out using the finite difference method. There are two different approaches

to the adsorption isotherm. One simulation is using the Langmuir isotherm, while the

other simulation is using the Henry isotherm. There are 1000 spatial steps used in the

simulation, where the length of each dimensionless spatial step is 0.1 unit. The simulation

was carried out to 100 unit of dimensionless time which is discretised into 1000 time steps.

Therefore, each time step values 0.1 unit. The results of the numerical simulations are

verified using the analytical solution using the Laplace transformation. Since the Laplace

equation is too complicated to be inverted using the table of inverse Laplace transform,

the inversion of the Laplace transformation is obtained numerically using the Fourier Fast

Transformation as presented in Eqs.(A.0.24) and (A.0.25). The parameters used for the

numerical inversion of the Laplace transformation are as follows: N = 512, a = 0.1,

T = 4.5/a = 45, ∆t′ = 2T/N = 0.18. The results of the simulation are presented in

Fig. 3.1.

The analytical simulation using the Laplace transformation employs the boundary

condition where the bulk concentration at infinite distance from the bubble surface is

equal to the initial bulk concentration. This assumption does not apply to the numerical

simulation, either using the Henry or the Langmuir isotherm. For the numerical simula-

tions, one of the boundary conditions was set to 90 times of the characteristic length of

adsorption (adsorption depth h) and a zero derivative of concentration was applied there

(∂C ′/∂x′|x′=l = 0). Due to those different approaches, there is a slight deviation of the

simulation result using analytical and numerical method. Another reason for the discrep-

ancy is due to the error of the numerical inversion of the Laplace equation. The error of

the Laplace inversion can be minimised by selecting more points when performing the

Fourier Fast Transform (FFT).

The characteristic length h represents the characteristic distance over which surfactant

molecules have to diffuse to supply the interface [98]. When a gas-liquid interface is

50

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

C'

t'

Numerical LangmuirNumerical HenryAnalytical Henry

0

0.5

1

0 5 10

Figure 3.1: Dimensionless bulk concentration in the subsurface C ′ vs dimensionless timet′ simulated using various methods and adsorption isotherms at very low initial bulk con-centration (Ci = 0.2 mol m−3). The analytical result overestimates the numerical resultsdue to numerical error. The inset shows the correlation at early time. At very early time,due to assumption of infinite adsorption length, the subsurface concentration calculatedanalytically using the Henry isotherm is lower than that calculated numerically – wherethe adsorption length is finite.

formed, surfactant in the bulk immediately next to it will adsorb, resulting a depletion of

the local concentration. As a consequence, surfactant diffuses from the bulk solution to

supply this region. The distance where the diffusion starts to occur is the characteristic

length [96].

The simulation presented in Fig. 3.1 was carried out at very low initial bulk concen-

tration (Ci = 0.2 mol m−3 and K = 5.5× 10−3 m3 mol−1). At this low concentration, the

Langmuir isotherm will give a concentration profile very close to the profile given by the

Henry isotherm as can be seen in Fig. 3.1. Since the concentration at the surface is very

small, its change will be insignificant to the adsorption isotherm. As a consequence, the

Langmuir adsorption isotherm will behave similarly to the Henry isotherm.

51

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

C'

t'

LangmuirHenry

0

0.5

1

0 5 10

Figure 3.2: Dimensionless bulk concentration C ′ in the subsurface vs dimensionlesstime t′ simulated using the Henry and Langmuir isotherms. The simulation is usingCi = 200 mol m−3 while the other parameters are as presented in Tab. 3.1. The Langmuirisotherm with Γ′ = (1 + κ)C ′(0, t′)/(1 + κC ′(0, t′)) has less subsurface concentrationthan the Henry isotherm with Γ′ = C ′(0, t′). The inset is the profile at early time wherethe value of C ′ calculated using the Henry isotherm is higher than that calculated usingthe Langmuir isotherm.

3.6.2 Comparison of simulations using Langmuir and Henry isotherms

When the result of simulation using the Langmuir isotherm at higher initial bulk concen-

tration (Ci = 200 mol m−3 and K = 5.5× 10−3 m3 mol−1) is compared to the simulation

result using the Henry isotherm, it is apparent that there is a slight difference of concen-

tration at the subsurface, as can be seen in Fig. 3.2. At higher concentration, there is an

effect of the subsurface concentration on the Langmuir adsorption isotherm as presented

in Eq. (3.1.8). As a consequence, the surface concentration calculated using the Langmuir

isotherm will differ from the surface concentration calculated using the Henry isotherm

as presented in Fig. 3.3.

There is an interesting phenomenon seen in the comparison of the simulation results

using the Langmuir and Henry isotherms at high surfactant concentration. On the one

52

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

Γ'

t'

LangmuirHenry

0

0.5

1

0 5 10

Figure 3.3: Dimensionless surface concentration Γ′ vs dimensionless time t′ simulatedusing the Henry and Langmuir isotherms. The simulation is using Ci = 200 mol m−3

while the other parameters are as presented in Tab. 3.1. The Langmuir model has moresurface excess than the Henry model. The inset is the profile at early time. At very earlytime, the surface excess calculated using the Henry model is higher than that calculatedusing the Langmuir model as the effect of higher C ′ calculated using the Henry model atthis very early time.

hand, the bulk concentration at the subsurface calculated using the Langmuir isotherm

is lower than that calculated using the Henry isotherm. On the other hand, the surface

concentration calculated using the Langmuir isotherm is higher than that obtained using

the Henry isotherm as presented in Fig. 3.3. From Figs. 3.2 and 3.3 it can be seen that the

adsorption rate is greater for the Langmuir isotherm.

Since the diffusion rate is not affected by the adsorption isotherm, the greater adsorp-

tion for the Langmuir isotherm results in lower bulk concentration at the subsurface. The

greater adsorption rate using Langmuir isotherm leads to a higher surface concentration.

As stated in Eq. (3.1.8), for the Langmuir isotherm, Γ′ = (1 + κ)C ′(0, t)/(1 + κC ′(0, t)),

while for the Henry isotherm Γ′ = C ′. As a consequence, for a given C ′ the value of

Γ′ obtained using the Langmuir isotherm is bigger than that calculated using the Henry

isotherm. For a given Γ′, the value of C ′(0, t) is bigger when calculated using the Henry

isotherm than that calculated using the Langmuir isotherm.

53

3.6.3 Rescaling of parameters and numerical simulation of Ward-

Tordai equation

The dimensionless parameters for the adsorption kinetics using the Langmuir and Henry

isotherms have been rescaled to compare the values ofC ′|x′=0 and Γ′ at early times, where

the values of C ′|x′=0 are very low. The rescaling results in parameters t′′, x′′ and Γ′′ as

presented in Section 3.4. The values of Γ′′ as a function of t′′ from Langmuir and Henry

isotherms are compared with the results of simulation using the Ward-Tordai equation.

The comparison is presented in Fig. 3.4.

Fig. 3.4 shows that the rescaled surface concentration calculated using the Henry

isotherm matches the rescaled surface concentration calculated using the Langmuir isotherm

at very early time. As time becomes longer, the surface concentration of the Langmuir

isotherm differs from the surface concentration of the Henry isotherm. At longer times,

there will be increasingly large values of C(0, t′′) calculated using both isotherms. For

the Henry isotherm, it was obtained that Γ′′(t′′) = C(0, t′′), whereas for the Langmuir

isotherm Γ′′(t′′) = C(0, t′′)/(1 + κC(0, t′′)). Therefore, increase of C(0, t′′) results in

a larger denominator in the Langmuir isotherm. As a consequence, the value of Γ′′(t′′)

calculated using the Langmuir isotherm (in this new scaling) is lower than the value cal-

culated using the Henry isotherm.

The value of Γ′′ is calculated numerically, using the finite difference method and the

result is compared with the value of Γ′′ calculated using the Ward-Tordai equation. Both

simulations are using the Henry isotherm. At early time, the surface concentration of sur-

factant is equivalent with the square root of time as mentioned in Eq. (3.5.1). In Fig. 3.4,

the value of Γ′′ obtained using the Ward-Tordai equation is slightly higher than that ob-

tained using the Crank-Nicolson finite difference approximation. The small difference

is possibly due to different boundary conditions taken for those two methods, or may

be simply a result of the discretisation. The Ward-Tordai equation is obtained using a

boundary condition of infinite diffusion length, while the Crank-Nicolson method of fi-

nite difference requires a finite diffusion length.

54

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1

Γ"

t"

LangmuirHenry

Ward-Tordai

Figure 3.4: Rescaled dimensionless surface concentration Γ′′ vs dimensionless time t′′

simulated using the Langmuir and Henry isotherms compared with the simulation resultusing the Ward-Tordai equation. The rescaling collapses the data at early times.

The simulation using the Ward and Tordai equation only provides concentration of

surfactant on the surface and in the subsurface. On the other hand, simulation using

the Crank-Nicolson finite difference method is able and need to simulate a concentra-

tion profile along the diffusion length. The profile of concentration along the adsorption

length, however is not presented in this thesis Calculation of the concentration profile

using the Crank-Nicolson method using the Langmuir isotherm takes four orders of mag-

nitude longer time compared to the calculation using the Ward-Tordai equation. In this

simulation, the computations were carried out for 100 unit of dimensionless time. For

1000 time steps, the simulation using the Ward-Tordai equation needed only 6 s to run,

the numerical simulation (employing the diffusion equation) using the Henry isotherm

needed 44 s, while the numerical simulation using the Langmuir isotherm needed 15

hr. The large difference on simulation time is because there are multiple loops to move

through during the simulations using the finite difference method. One other possible

cause for the very expensive computation using the finite difference method in the case

of Langmuis isotherm is that the fact that the back and forth iteration between solving

55

Eqs. (3.2.5) and (3.2.10) only exhibits slow convergence.

Most of the experimental data on dynamics of adsorption of surfactant on a bubble

surface are presented as surface pressure vs time. The conversion of the data into surface

excess vs time needs additional information, such as the surface tension of the pure solvent

and temperature. The parameters for this simulation was taken from a study by Chang

et al., however that literature does not provide the experimental data for the particular

surfactant. Therefore, the verification of the numerical model in this study was carried

out using the analytical method.

3.7 Conclusions

From the simulation results, it can be concluded that the analytical solution using the

Laplace transformation confirms the numerical solution using the Henry isotherm. Ma-

nipulation of the Laplace transformation of the diffusion equation using the Henry isotherm,

and solution of the rearranged equation using convolution results in the Ward-Tordai equa-

tion. It is also evident that the numerical solution using the Henry isotherm confirms the

numerical solution of the Langmuir isotherm at a low surfactant concentration. Moreover,

through comparison with Laplace transformation and with Ward-Tordai equation, the nu-

merical solution for the equation of adsorption dynamics on foam surface is verified and

applicable to both the Henry and Langmuir isotherms. Using the dimensionless form of

Γ′(t′) = (1+κ)C ′(0, t′))/(1+κC ′(0, t′)) for the Langmuir isotherm and Γ′(t′) = C ′(0, t′)

for the Henry isotherm, at high bulk concentration, it is obtained that the adsorption rate

using the Langmuir isotherm is greater than the adsorption rate using the Henry isotherm.

The simulation of adsorption dynamics using rescaled parameters using the Langmuir

isotherm is verified by the simulation using the Henry isotherm at very early time. The re-

sults of the numerical simulation using the Crank-Nicolson method for a Henry isotherm

are in accordance with the results of simulation of the Ward-Tordai equation using the

Henry isotherm. At early time, the surface concentration is proportional to square root of

time, a fact which is evident from direct inspection of the Ward-Tordai equation.

56

The adsorption dynamics presented in this chapter apply only for an adsorption of

single surfactant with a simple adsorption isotherm. The model for adsorption of mixed

protein-surfactant will be more complex than the model of adsorption for a single surfac-

tant. Therefore, it is important to modify the adsorption dynamics model to be applicable

to adsorption of mixed protein-surfactant. The next chapter presents the simulation of

adsorption dynamics of mixed protein-surfactant.

57

Chapter 4Simulation of dynamics of adsorption of

mixed protein-surfactant on a bubble

surface

This chapter is a copy of an article published in Colloids and Surfaces A: Physicochemi-

cal and Engineering Aspects, 438:63–76, 2013 DOI 10.1016/j.colsurfa.2012.12.007. Au-

thors: Denny Vitasari, Paul Grassia, Peter Martin. This paper discusses modelling the

dynamics of adsorption of mixed protein and surfactant on a bubble surface using the

Ward-Tordai equation combined with the Frumkin adsorption isotherm.

Foam fractionation tends to be popular for separation of biochemical products, partic-

ularly when the target substance is very dilute. The other advantage of foam fractionation

process is its ability to handle structurally unstable products which may exhibits high sen-

sibility to temperature and/or pH [4]. In this separation method, different surface active

substances may coexist. Competition of those substances to adsorb onto the interfacial

area may occur due to their amphiphilic nature [12]. This competition can result in a

complex adsorption behaviour which is different from adsorption of a single substance.

Therefore, it is important to study the adsorption behaviour of this mixed solution in the

design of a foam fractionation process.

58

Recently, the utilisation of foam fractionation process to separate protein from aque-

ous solution has risen due to an increasing demand and market potential for protein based

active pharmaceutical ingredients [4, 99]. The separation is commonly carried out in a

foam fractionation column. During the separation of protein using a foam fractionation

column, surfactant may be added to the solution to stabilize the foam [4, 5, 12] and/or

increase the adsorption of the protein onto the interface due to complexation of the pro-

tein with surfactant [100]. The addition of surfactant to the protein solution causes a

competition between protein and surfactant molecules to adsorb onto the interface. As

a consequence the adsorption of mixed protein-surfactant solution needs to be be mod-

elled using a different approach from the one for a single surfactant. Concerning the

significance of adsorption of mixed protein-surfactant in industry, the paper in this chap-

ter discuss the development of mathematical model of dynamics of adsorption of mixed

protein-surfactant onto a bubble surface within a foam fractionation column. This model

is useful for prediction of the performance of a foam fractionation column.

The Henry and Langmuir adsorption isotherms used in the model of adsorption dy-

namics in Chapter 3 applies for adsorption of a single solute only [101]. Therefore, that

model of dynamics of adsorption of surfactant is not suitable for adsorption of mixed

protein-surfactant. Another adsorption isotherm, named the Frumkin isotherm is required

to model the equilibrium of adsorption of mixed protein-surfactant [101]. This model

is based on a surface equation of state that involves the surface and bulk concentration

of both protein and surfactant. Due to the competition on the interface, the adsorption

isotherm of surfactant is affected by the concentration of protein and vice versa. The ki-

netics of the adsorption is modelled using the Ward-Tordai equation similar to the one

for a single surfactant adsorption. Therefore, the model of dynamics of adsorption of

mixed protein-surfactant results in simultaneous mathematical equations of the adsorp-

tion isotherm as well as the Ward-Tordai equations for both protein and surfactant.

The protein molecules may undergo changes of conformational states during the ad-

sorption [18, 68, 102] resulting in different molar areas [22]. The conformation changes

may affect the adsorption as there will be changes in molar area and adsorption coefficient

59

of the protein. In this study, the changes of conformations is not taken into consideration

since the emphasis is more on the competition between protein and surfactant molecules

to adsorb onto the interface.

The study was carried out using base case parameters obtained from literature. This

base case study examines the competition between protein and surfactant molecules on the

interface. The diffusion of protein molecules is slower than that of surfactant molecules.

However, the adsorption constant of protein is higher than that of surfactant, therefore

protein is more easily adsorbed onto the interface. Once the protein reaches the interface,

it will displace the surfactant on the interface. The result of this simulation is compared

with experimental data available in the literature. A parametric study was also carried out

varying the diffusion and adsorption parameters. The parametric study was carried out

using various protein and surfactant concentrations to see the effect on the displacement

of surfactant on the interface by the protein. Simulation using various relative affinity

between protein and surfactant was also carried out to study the extent of competition be-

tween protein and surfactant molecules to adsorb onto the interface. The effect of molec-

ular size that may impact the diffusivity, surface affinity as well as surface activity was

also examined. A phase diagram showing the protein-surfactant competitive behaviour is

developed based on the parametric study.

60

Publication 1

Simulation of dynamics of adsorption of mixed protein-

surfactant on a bubble surface

61

Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

Contents lists available at ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

jo ur nal ho me page: www.elsev ier .com/ locate /co lsur fa

Simulation of dynamics of adsorption of mixed protein–surfactant on a bubblesurface

Denny Vitasari, Paul Grassia ∗, Peter MartinSchool of Chemical Engineering and Analytical Science, The Mill, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

h i g h l i g h t s

� Interfacial adsorption dynamics ofmixed protein–surfactant systemsare considered.

� Ward–Tordai equation (for sur-factant dynamics) and Frumkinisotherm are used.

� Surfactant adsorbs more quickly butcan be displaced by high surfaceaffinity protein.

� Protein–surfactant concentration,surface affinity and molecular sizeare varied.

� A phase diagram for displacementvs non-displacement of surfactant ispresented.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 August 2012Received in revised form 3 December 2012Accepted 5 December 2012Available online 14 December 2012

Keywords:Mixed protein–surfactantAdsorptionDynamics of adsorption

a b s t r a c t

The dynamics of adsorption of mixed protein–surfactant on a bubble surface is simulated mathematically.The model for the adsorption dynamics is developed based on the Ward–Tordai equation combined withthe Frumkin adsorption isotherm. The simultaneous equations are solved using the Newton methodfor iteration. Base case adsorption and diffusion parameter values for the simulation were sourced fromliterature. It was found that protein arrives on the surface at a later time than surfactant. At this later time,the protein replaces the surfactant resulting in depletion of surfactant on the surface. There is, however,less protein adsorbed in the presence of more surfactant in the bulk. In contrast, more protein stays in thesubsurface layer under these conditions. In addition to the base case simulation and a comparison to theexperimental data available in the literature, a parametric study was performed to explore the effects ofvarying adsorption and diffusion parameters. The parametric study varying the protein surface affinityrevealed that below a certain critical affinity, protein tends not to replace surfactant on the surface, eventhough the affinity of protein remains higher than that of surfactant. Therefore, protein molecules need tohave sufficiently high affinity to displace surfactant molecules from the surface. Another parametric studysetting a fixed protein surface affinity and varying relative diffusivity and surface affinity of surfactant (fora specified maximum possible surface capacity of surfactant) concluded that with high relative diffusivityand low surfactant affinity (relative to protein), the displacement of surfactant on the surface is morelikely to occur.

© 2012 Elsevier B.V. All rights reserved.

∗ Corresponding author. Tel.: +44 161 306 8851; fax: +44 161 306 9321.E-mail address: paul.grassia@manchester.ac.uk (P. Grassia).

1. Introduction

Mixtures of protein and surfactant have a practical significancein industry, for example in the stabilisation of emulsions and foams[1]. Moreover, these mixtures can be found easily in biological

0927-7757/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.colsurfa.2012.12.007

64 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

Nomenclature

˛p intermolecular interaction parameter of protein˛s intermolecular interaction parameter of surfactant˛ps parameter describing the interaction between pro-

tein and surfactant in the mixture� surface tension of the solution (N m−1)�0 surface tension of the solvent (N m−1)�m relative surface capacity of surfactant to protein�p surface concentration of protein (mol m−2)�s surface concentration of the adsorbed surfactant

(mol m−2)�pm maximum surface concentration of protein

(mol m−2)�sm maximum surface concentration of surfactant

(mol m−2)�p dimensionless surface affinity of protein�s dimensionless surface affinity of surfactantD relative diffusivity of surfactant to proteinω0 molar area of the solvent (m2 mol−1)ωp molar area of the adsorbed protein (m2 mol−1)ωs molar area of the surfactant (m2 mol−1)� surface pressure (N m−1)�′ dimensionless surface pressure� dummy integration variable (s)� ′ dimensionless dummy integration variable�p fractional surface coverage of protein�s fractional surface coverage of surfactantbp equilibrium adsorption constant of protein

(m3 mol−1)bs equilibrium adsorption constant of surfactant

(m3 mol−1)cb relative bulk concentration of surfactant to proteinCp concentration of protein in the subsurface layer

(mol m−3)C ′

p dimensionless concentration of protein in the sub-surface layer

Cs concentration of surfactant in the subsurface layer(mol m−3)

C ′s dimensionless concentration of surfactant in the

subsurface layerCpb bulk concentration of protein (mol m−3)Csb bulk concentration of surfactant (mol m−3)Dp diffusion coefficient of protein in the solvent

(m2 s−1)Ds diffusion coefficient of surfactant in the solvent

(m2 s−1)i label for each protein conformational stateL half the thickness of the film (m)N number of protein conformational statesR gas law constant (J mol−1 K−1)T temperature (K)t time (s)t′ dimensionless timex distance from the film surface (m)

systems, such as blood serum which contains human serum albu-min (HSA) and a number of compounds including low-molecularweight surface active molecules [2]. Proteins are often mixedwith low-molecular weight surfactants to improve the quality offoam produced. The ability of proteins to unfold at the interfacegenerates films with high surface elasticity and provides stericresistance to avoid coalescence of films [3]. As a consequence, stud-ies on the adsorption phenomena of mixed protein–surfactant have

significant practical as well as scientific importance. Manyworks attempt to determine the behaviour of adsorption ofprotein–surfactant mixtures. Those studies [2,4–7] describe boththe adsorption isotherms and adsorption dynamics of mixedprotein–surfactant on the liquid–gas interfaces based on experi-mental data. On the other hand, studies on mathematical modellingto predict the adsorption behaviour of mixed protein–surfactant onthe liquid–gas interfaces are less common. Nevertheless, studies ofthis nature have a significant importance for the design of a processinvolving the adsorption of mixed protein–surfactant, such as foamfractionation or emulsification.

One of the existing studies on the mathematical simulationof adsorption dynamics on the bubble surface was conducted byMulqueen et al. [8]. This study modelled the adsorption of aqueoussurfactant mixtures on the bubble surface. The study also observedcompetition between surfactants where there was a displace-ment of less surface active surfactant. Various other simulationsof adsorption dynamics of surfactants were also carried out andthe results were compared with experimental data [9,10]. Li et al.[11] developed a numerical solution for an adsorption dynamicsequation applied to the adsorption of single non-ionic surfactant.A solution consisting of mixed protein and surfactant may behavedifferently from a solution of two or more surfactants due to thepotentially very different adsorption and transport properties ofprotein and surfactant. Therefore, a study on the adsorption dynam-ics of mixed protein–surfactant examining the significance of theparameters on the adsorption behaviour is needed.

The adsorption isotherms and adsorption dynamics of pro-tein and surfactant mixtures in the previous studies [5,6,12] weredetermined based on the experimental data of equilibrium surfacetension and dynamic surface tension, respectively. Such analysis aswas performed of the equations governing adsorption behaviourwas done primarily with a view to extract parameter valuesfrom the available experimental data. However, so far a thor-ough parametric simulation study, elucidating detailed behaviourof adsorption dynamics, has not been developed in previous stud-ies. When a robust simulation of dynamics of adsorption of mixedprotein–surfactant is made available, it will save the resource asso-ciated with carrying out a multitude of laboratory experiments todetermine the dynamics of adsorption. The mathematical simula-tion can be developed using the model of adsorption dynamics andalso the adsorption isotherm of mixed protein–surfactant on theliquid–gas interfaces developed in the previous studies.

Addition of surfactant into a protein solution can modify theproperties of the adsorption layer on liquid–gas interfaces. Theinterfacial properties of the mixtures of protein–surfactant arestrongly affected by the interaction of the components of the mix-tures which becomes a significant factor in the formation andstabilisation of the foam or emulsion produced [6]. The nature andconcentration of surfactant in the bulk determines the mechanismof the protein–surfactant interactions [13]. Moreover protein, whenmixed with surfactant molecules can have different interactionsboth in the bulk and on the surface. The interaction, which can behydrophobic or an electrostatic type, has the possibility to changethe conformation of the protein, in the bulk and/or on the sur-face [14,15]. Adding different amounts of surfactant in the mixturescreates more complex interactions and adsorption layer composi-tion since surfactant molecules and protein molecules compete foraccess to the interface. At sufficiently high surfactant concentrationin the mixture, the surfactant molecules can completely displacethe protein from the surface [15], whereas protein would com-pletely displace surfactant in the case of sufficiently high proteinconcentration.

Mixed ionic surfactant and protein behaves differently from themixture of non-ionic surfactant and protein. At low surfactant con-centration, the behaviour of the adsorption layer formed by mixed

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 65

protein and non-ionic surfactant is determined mainly by com-petitive adsorption between molecules of the two components.On the other hand a Coulombic interaction between ionic surfac-tant molecules and protein molecules with oppositely charged ionswas observed at low surfactant concentration [16]. The presentstudy however considers only the interaction between protein andnon-ionic surfactant, looking into the phenomena of competitiveadsorption between molecules of the two components involved.

In summary then, this paper presents the simulation andparametric study of the dynamics of adsorption of mixedprotein–surfactant on the bubble surface. The study is laid outas follows. Section 2 describes the theory of adsorption equilib-rium of mixed protein–surfactant using the Frumkin isotherm.Section 3 explains the derivation of adsorption dynamics fromFick’s law of diffusion and the relevant boundary conditions. Thissection also mentions the application of the Ward–Tordai equa-tion for determining the adsorption dynamics. Section 4 discussesthe dimensional analysis of the equations of adsorption dynamics.Section 5 explains the simulation methods and related parametervalues. The simulation results are presented in Section 6 along withdiscussion of the phenomena observed. At the end, the summaryof the study is found in Section 7.

2. Adsorption equilibrium of mixed protein–surfactant

The adsorbed protein changes its conformation, subject to thelocal conditions. The conformations can be well approximated by adiscrete and limited number of states. The protein molar area variesconsecutively from a maximum (ωmax) to a minimum value (ωmin).The equation of state of the surface layer then can be presented asfollows [17]:

−�ω0

RT= ln(1 − �p − �s) + �p

(1 −

(ω0

ωp

))+ ˛p�2

p + ˛s�2s

+ 2˛ps�p�s (1)

where � = �0 − � is the surface pressure, �0 is the surface tension ofthe solvent, � is the surface tension of the solution, R is the gas lawconstant, T is the temperature, �p = ωp�p = N

i=1ωi�pi is the pro-tein surface coverage fraction, �p = N

i=1�pi is the total adsorptionof protein in all N conformational states assumed to be availableto it, �pi is the adsorption of protein in one particular conforma-tion state, ωp is the average molar area of the adsorbed proteinmolecules, ωi = ω1 + (i − 1)ω0 with (1 ≤ i ≤ N) are the molar areas ofeach state i with ωmin = ω1 and ωmax = ω1 + (N − 1)ω0, �s = ωs�s isthe surfactant surface coverage fraction, ωs is the molar area of theadsorbed surfactant, �s is the surface concentration of surfactant,ω0 is the molar area of the solvent, ˛p is the intermolecular inter-action parameter of protein, ˛s is the intermolecular interactionparameter of surfactant, ˛ps is a parameter describing the interac-tion between the protein and surfactant mixture. The adsorptionisotherm for each state (i) of the protein is then derived as:

bpiCp = ωp�pi

(1 − �p − �s)ωi/ωp

exp

[−2˛p

(ωi

ωp

)�p − 2˛ps�s

](2)

where Cp is the concentration of the protein in the subsurface layerand bpi are the equilibrium adsorption constants of protein in thestate i.

If it is assumed that bpi are constant for any states of the proteinadsorption (bpi = bp, for any i), therefore the adsorption constantfor the protein molecules as a whole may be approximated by∑

bp = Nbp. More strictly, retaining the assumption that bpi = bp forall states i, the distribution of protein adsorption is given by the

following equation [6]:

�pi=�p(1 − �p − �s)

((ωi−ω1)/ωp) exp[2˛p�p((ωi − ω1)/ωp)]∑Ni=1(1 − �p − �s)

((ωi−ω1)/ωp) exp[2˛p�p((ωi − ω1)/ωp)].

(3)

It then follows (using the definitions of �p, ωp and �p given earlier)

bpCp = �p(1 − �p − �s)(ω1/ωp) exp[−2˛p�p(ω1/ωp) − 2˛ps�s]∑N

i=1(1 − �p − �s)((ωi−ω1)/ωp) exp[2˛p�p((ωi − ω1)/ωp)]

.

(4)

In the special case where all conformational states i of the protein(out of N total states) have comparable molar areas it follows thatωp ≈ ω1 and thus

NbpCp ≈ �p exp[−2˛p�p − 2˛ps�s](1 − �p − �s)

, (5)

hence the approximation that the adsorption constant for ‘proteinas a whole’ is Nbp.

Thus far we have only discussed adsorption of protein. Theadsorption isotherm equation for the surfactant is analogous asfollows:

bsCs = �s

(1 − �p − �s)exp[−2˛s�s − 2˛ps�p] (6)

where Cs is the concentration of the surfactant in the subsurfacelayer and bs is the equilibrium adsorption constant of surfactant.

The theoretical description for protein–surfactant mixture canbe elaborated in the following way: from the known values of T,ω0, ωmin, ωmax, ˛p, bp, ωs, ˛s and bs for the individual protein andsurfactant and ˛ps as single additional parameter for the mixture,the dependent parameters of ωp, �p, �s, �p, �s and � as a functionof the concentration Cs and Cp can be calculated [4].

3. Adsorption dynamics of mixed protein–surfactant atgas–liquid interfaces

Adsorption dynamics of both protein and surfactant towards thegas–liquid interface follow Fick’s equation [18,19]:

∂C

∂t= D

∂2C

∂x2(7)

where t is time, x is the distance from the film surface and subjectto boundary conditions:

D∂C

∂x

∣∣∣x=0

= d�

dt(8)

∂C

∂x

∣∣∣x=L

= 0 (9)

where L is half the thickness of the film. The initial conditions are:

C(x, 0) = Cb, x > 0

C(0, 0) = 0

�(0) = 0 (10)

where Cb is the bulk concentration of protein or surfactant.The Laplace transformation of the diffusion equation results in a

general dynamics of adsorption equation on a liquid–gas interfaceas proposed by Ward and Tordai [20]. The equation describes theevolution of surface concentration due to transfer from the subsur-face. The Ward–Tordai equation applies to the case of an infinitely

66 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

thick film1 (L→ ∞) as one of the boundary conditions. The pro-file of surface concentration of both protein and surfactant overtime (�p(t) and �s(t)) can be defined using the adsorption dynamicsequation and the corresponding adsorption isotherms as presentedin Eqs. (11) and (12) which follow:

�p(t) =√

Dp

�

[2Cpb

√t −

∫ t

0

Cp(�)√t − �

d�

](11)

�s(t) =√

Ds

�

[2Csb

√t −

∫ t

0

Cs(�)√t − �

d�

](12)

where Dp and Ds are diffusion coefficients of protein and surfactantin the solvent, respectively, Cpb and Csb are the bulk concentrationsof protein and surfactant, respectively and � is a dummy integra-tion variable. The first term on the right hand side of the equationrepresents the diffusive transport to the surface. This diffusionis mitigated by a reduction in diffusive driving force as surfac-tant and/or protein on the surface builds up, which is presentedby the second term of the right hand side of the equation. TheWard–Tordai equation presented in Eqs. (11) and (12) is applicableon a planar interface. This shape of interface is selected as in com-mon applications such as foam fractionation, polyhedral bubbleswith nearly planar films occur in the system, due to low fraction ofliquid [21].

The Ward–Tordai equation solves the adsorption dynamics interms of time only. Therefore, it is cheaper to solve than thepartial differential equation in Eq. (7) which also evaluates theconcentration profile across the film thickness. Considering thisadvantage, the Ward–Tordai equation is used to model the adsorp-tion dynamics of mixed protein–surfactant in this study. Theknowledge of adsorption isotherms and adsorption dynamics ofmixed protein–surfactant as well as the individual protein or sur-factant on gas–liquid interfaces is then applied to the simulation ofadsorption dynamics of mixed protein–surfactant.

4. Dimensional analysis of the equation of adsorptiondynamics

As indicated in Section 3 the dynamics of adsorption of mixedprotein–surfactant is modelled using the Ward–Tordai equation.The adsorption isotherm of protein and surfactant on the bubblesurface is defined using the Frumkin isotherm. For a single confor-mational state of protein, Eq. (2) can be simplified as:

bpCp = �p[1 − �p − �s

] exp[−2˛p�p − 2˛ps�s

](13)

A dimensional analysis is carried out upon the equations ofdynamics of adsorption of mixed protein–surfactant which are Eqs.(6), (11)–(13). The dimensionless form of the equations governingthe dynamics of adsorption involves dimensionless groups that wecall D, �m, �p, �s and cb, precise definitions of which will be givenshortly. The resulting dimensionless equations are as follows.

Dynamics of protein adsorption:

�p(t′) = 1√�

[2√

t′ −∫ t′

0

C ′p(� ′)√t′ − � ′ d� ′

](14)

Adsorption isotherm of protein:

C ′p(t′) = �p(t′)

�p

[1 − �p(t′) − �s(t′)

] exp[−2˛p�p(t′) − 2˛ps�s(t′)] (15)

1 Infinitely thick in this context means that material which eventually adsorbsonto the surface is only a small fraction of what was initially present in the bulk.

Dynamics of surfactant adsorption:

�s(t′) = cb

√D

�m

1√�

[2√

t′ −∫ t′

0

C ′s(�

′)√t′ − � ′ d� ′

](16)

Adsorption isotherm of surfactant:

C ′s(t

′) = �s(t′)�s[1 − �p(t′) − �s(t′)]

exp[−2˛s�s(t′) − 2˛ps�p(t′)] (17)

And the equation of state:

�′=− ln(1 − �p − �s) − �p(1 − ω0/ωp) − ˛p�2p − ˛s�

2s −2˛ps�p�s

(18)

where �′ = �ω0/(RT) is the dimensionless surface pressure,�p = �p/�pm is the dimensionless surface concentration of protein(i.e. the coverage fraction), �s = �s/�sm is the dimensionless sur-face concentration (coverage fraction) of surfactant, �pm = 1/ωp

and �sm = 1/ωs are the maximum surface concentrations of proteinand surfactant respectively (both measures of surface capacity),t′ = (Dp t)/(�pm/Cpb)2 is the dimensionless time, C ′

p = Cp/Cpb is thedimensionless bulk concentration of protein at the layer next to thesurface, C ′

s = Cs/Csb is the dimensionless bulk concentration of sur-factant at the layer next to the surface, Cpb and Csb are the initial bulkconcentrations of protein and surfactant respectively, �p = bpCpband �s = bsCsb are the dimensionless adsorption equilibrium con-stants of protein and surfactant, respectively (both measures ofsurface affinity), cb = Csb/Cpb is the relative bulk concentration, D =Ds/Dp is the relative diffusivity and �m = �sm/�pm = ωp/ωs is the rel-ative capacity which is equivalent to the ratio between molar areasof protein and surfactant.

As alluded to above, the parameters �p and �s measure the affin-ity of the surface for protein and surfactant, respectively. Smallvalues of the parameters �p and �s tend to indicate that there islittle protein or surfactant on the surface (low fractional coverage).Meanwhile, large values of �p and �s tend to indicate that the sur-face is nearly saturated with protein and/or surfactant (high overallfractional coverage). The parameter D is typically bigger than unityas surfactant has a larger diffusivity than protein. Surfactant hasa lower molecular weight than protein, therefore it diffuses morereadily. The value of �m is typically far bigger than unity due tomuch larger molar area of protein than that of surfactant. The fac-tor 1/�m in Eq. (16) indicates that surfactant molecules can occupya relatively small fraction of surface of protein even if compara-tively many of them are adsorbed. The factor ω0/ωp in Eq. (18) isalso expected to be near 1/�m since the molar area of solvent ω0 isunlikely to be very different from that of surfactant, ωs.

5. Methods and parameter values

The parameters used in the simulation are obtained from a studyby Miller et al. [5] using Bovine ˇ-lactoglobulin (BLG) protein andnonionic decyl dimethyl phosphine oxide (C10DMPO) surfactant.Those parameters are listed in Table 1. The parameters from thestudy by Miller et al. were set as a base case when the simula-tion involved variation of the material parameters such as diffusionconstant of surfactant Ds (or analogously D in dimensionless form),surface capacity of surfactant �sm (or analogously dimensionless�m), adsorption coefficient of surfactant, bs (or analogously �s) andadsorption coefficient of protein, bp (or analogously �p). The dimen-sionless form of the parameters and their ranges of variation arepresented in Table 2.

A numerical algorithm was developed to solve Eqs. (14)–(17).Full details of the algorithm are given in Appendix A. The algorithmwas implemented in Matlab, and took less than a minute to run ona standard desktop PC.

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 67

Table 1The values of base case parameters used in the simulation of adsorption dynamics.

Variable name Symbol Value Unit

Characteristic time scale (equivalent to 1 unit of dimensionless time) (�pm/Cpb)2/Dp 1033 sNumber of simulation steps (see Appendix A) n 20,000Diffusivity of protein Dp 5 × 10−11 m2 s−1

Diffusivity of surfactant Ds 4 × 10−10 m2 s−1

Bulk concentration of protein Cpb 1 × 10−3 mol m−3

Bulk concentration of surfactant Csb 1 × 10−2 mol m−3

Gas constant R 8.3144621 J mol−1 K−1

Temperature T 298 KMolar area of solvent ω0 3.5 × 105 m2 mol−1

Molar area of protein ωp 4.4 × 106 m2 mol−1

Molar area of surfactant ωs 2.5 × 105 m2 mol−1

Surface capacity of protein �pm 2.27 × 10−7 mol m−2

Surface capacity of surfactant �sm 4.00 × 10−6 mol m−2

Protein interaction parameter ˛p 0.4Surfactant interaction parameter ˛s −0.25Protein–surfactant interaction parameter ˛ps 0.075Protein adsorption constant bp 1.4 × 103 m3 mol−1

Surfactant adsorption constant bs 21.9 m3 mol−1

Table 2The dimensionless form of the parameters for simulation of adsorption dynamics.

Dimensionless parameter name Symbol Definition Base case Range

Surface affinity of protein �p bpCpb 1.4 0–1.4Surface affinity of surfactant �s bsCsb 0.219 0–2.19Relative diffusivity D Ds/Dp 8 1–10Relative capacity �m �s/�p 17.6 1–17.6Relative bulk concentration cb Csb/Cpb 10 0–100

6. Results and discussion

This study examines the adsorption dynamics of mixedprotein–surfactant on a bubble surface in a situation typical of afoam fractionation process. A parametric study was also carried outto explore the effect of varying adsorption and diffusion parame-ters. The results of this study are presented in the following order.Section 6.1 discusses the competition between protein and sur-factant molecules on the surface, and also the growth of surfacepressure with the arrival of protein. This subsection also reports thecomparison between the simulation results and the experimentaldata obtained from the literature. Section 6.2 deals with the effectof variation of surfactant concentration in the bulk solution (or rel-ative bulk concentration) on the concentration of both protein andsurfactant on the surface and in the subsurface. Section 6.3 exam-ines the influence of relative affinity on the adsorption of proteinand/or surfactant onto the surface. Section 6.4 explains the effect ofmolecular size, which determines the diffusivity, adsorption capac-ity and surface affinity, on the displacement of surfactant from thesurface by protein. Section 6.5 summarises the findings in the formof a ‘phase diagram’.

6.1. Competition between protein and surfactant molecules onthe surface

Fig. 1, corresponding to the base case parameter values, showsthe comparison between the dimensionless subsurface concentra-tion of protein and the dimensionless subsurface concentration ofsurfactant at base case. This simulation was carried out using thesevalues of dimensionless groups: �p = 1.4, �s = 0.219, D = 8, �m =17.6, cb = 10. Since the diffusivity coefficient of surfactant is higherthan the diffusivity coefficient of protein, surfactant is more rapidlytransferred to the subsurface. At early time, there is therefore moresurfactant adsorbed on the surface. Due to the faster diffusion (highD), and also lower surface affinity (viz. the low value of the param-eter �s), surfactant reaches its final concentration in the subsurfacefaster than protein. The subsequent arrival of additional protein

in the subsurface provides more protein molecules to adsorb tothe interface while there is limited further change of surfactantconcentration in the subsurface. As protein has a relatively highersurface affinity (measured by the values of �p vs �s), proteinmolecules compete strongly with surfactant molecules on the sur-face. Therefore, protein molecules replace surfactant molecules onthe surface resulting in lower surface concentration of surfactant aspresented in Fig. 2. This overshoot phenomenon also occurs in theadsorption of mixed surfactants as reported by Mulqueen et al. [8].

Fig. 3 shows the growth of the surface pressure with the additionof protein in the bulk solution where the values of the dimen-sionless groups are: �p = 1.4, �s = 0.219, D = 8, �m = 17.6, cb =10. The time scale of this figure is taken up to five units, longerthan the time scale of Figs. 1 and 2 which is up to one unit. Thelonger time scale in Fig. 3 has been selected to show the final surfacepressure is approached on that time scale. The surface pressure ishigher in the presence of protein and surfactant in the bulk solutioncompared to the surface pressure resulting from pure surfactant orpure protein solution. For a fixed amount of surfactant on the sur-face (dashed–dotted line), which was set to be the final surface

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C'

t'

protein

surfactant

Fig. 1. Dimensionless subsurface concentration as a function of dimensionless timeobserved at dimensionless parameters of: �p = 1.4, �s = 0.219, D = 8, �m = 17.6, cb =10.

68 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

θ

t'

θp

θs

Fig. 2. Surface coverage as a function of dimensionless time observed at dimension-less parameters of: �p = 1.4, �s = 0.219, D = 8, �m = 17.6, cb = 10.

concentration of surfactant in the presence of protein (�s = 0.0728),in general, lower surface pressure occurs compared to that obtainedfrom simulation of dynamic concentrations of protein and surfac-tant on the surface. This happens since the fixed surfactant surfaceconcentration is mostly lower than the dynamic surface concentra-tion, the only exceptions being at very early times and at the finaltime. Although difficult to resolve on the scale of the graph, at veryearly time, the surface pressure is higher in the case of fixed surfacecoverage of surfactant due to finite surfactant concentration on thesurface initially. By contrast, in the dynamic case, the surface con-centration of surfactant has to grow from zero at very early times.At final time, of course the surface pressure of those both cases willbe equal since by that time the dynamic surface concentration ofsurfactant reaches its final value that is equal to the selected fixedsurface concentration. The graph also indicates that protein con-centration on the surface is able to increase (and thereby influencesurface pressure) even with the presence of significant surfactanton the surface. Protein with its higher affinity is able to competewith surfactant to adsorb on the surface.

An experiment on adsorption of mixed protein–surfactant ona bubble surface using a solution of protein (BLG) and surfactant(C10DMPO) was carried out by Miller et al. [5] and the experimen-tal data for surface pressure are now compared to the results ofour simulation. The parameters applicable in the experiment arethe same parameters used as base case parameters in the simu-lation. The experimental data for surface pressure are convertedto dimensionless form using the known parameters. The resultsof the simulation are then compared to the dimensionless form ofthe experimental data. In the conversion of the experimental data,

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

Π'

t'

pure surfactant

dynamic θs and θp fixed θs, dynamic θp

pure protein

Fig. 3. Dimensionless surface pressure as a function of dimensionless time at variousprotein–surfactant composition using dimensionless parameters of: �p = 1.4, �s =0.219, D = 8, �m = 17.6, cb = 10 (solid line: pure surfactant; dashed line: dynamicprotein and dynamic surfactant surface concentration; dotted line: dynamic proteinand fixed surfactant surface concentration; dashed–dotted line: pure protein).

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7

Π'

t'

simulation κp = 1.4 simulation κp = 53.2

experiment

Fig. 4. Comparison of dimensionless simulation results and experimental data (con-verted into dimensionless form) obtained from a study by Miller et al. [5] usingthe following dimensionless parameters: �p = 1.4, �s = 0.219, D = 8, cb = 10, �m =17.6 (taking ωp = ωmax

p = 4.4 × 106 m3 mol−1). Simulation data are also shown fora larger �p value (�p = 53.2), representing an approximate way of accounting forprotein being spread over a total of N = 38 conformational states as Miller et al. [5]assume.

there are two different extremal values of molar area ωp of protein,which are the minimum molar area (ωmin

p = 4.4 × 106 m3 mol−1

corresponding to conformational state i = 1) and the maximummolar area (ωmax

p = 1.8 × 107 m3 mol−1 corresponding to confor-mational state i = N). The base case in our simulation was taken atωp = ωmin

p = 4.4 × 106 m3 mol−1 which gives the value of �m = 17.6.

If instead we take ωp = ωmaxp = 1.8 × 107 m3 mol−1, then the value

�m = 72 is obtained. The comparisons of the dimensionless experi-mental data and the simulation results at ωp = ωmin

p and ωp = ωmaxp

are presented in Figs. 4 and 5, respectively.The graphs shown in Figs. 4 and 5 indicate that there are discrep-

ancies between the simulation results and the experimental data.It is found that the simulation underestimates the experimentalresults, therefore the surface pressure predicted from the simu-lation is much lower than that obtained using experiment. Thosediscrepancies occur as in the simulation there is an assumptionthat there is no possibility of changing the conformational stateof the protein. This assumption was taken to simplify the mathe-matical simulation since the present study has more emphasis onthe competition between protein and surfactant molecules. Usingan assumption of a single conformational state of protein resultsin a lower value of protein surface coverage �p compared to thatcalculated using an assumption of multiple protein conformationalstates. The surface pressure �′ via Eq. (18) is then correspondinglyalso much lower.

0

0.5

1

1.5

2

2.5

3

3.5

0 20 4 0 60 80 100

Π'

t'

simulation κp = 1.4 simulation κp = 53.2

experiment

Fig. 5. Comparison of dimensionless simulation results and experimental data (con-verted into dimensionless form) obtained from study by Miller et al. [5] using thefollowing dimensionless parameters: �p = 1.4, �s = 0.219, D = 8, cb = 10, �m = 72(taking ωp = ωmax

p = 1.8 × 107 m3 mol−1). The dimensionless time range is muchbigger than the one in Fig. 4 because of the way times were non-dimensionalised inSection 4. As in Fig. 4, simulation data are also shown for a larger �p value (�p = 53.2).

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 69

Earlier (see Section 2) we stated that a way to approximate theeffect of protein in multiple conformational states was to treat theadsorption coefficient bp (and hence its dimensionless analogue �p)for ‘protein as a whole’ to be considerably larger than the bp valuefor an individual conformational state. Miller et al. [5] considereda total of N = 38 conformational states. Hence we take, in lieu ofbp = 1.4 × 103 m3 mol−1 (as given in Table 1), a value that is N timeslarger, i.e. bp = 5.32 × 104 m3 mol−1. Analogously, in lieu of base casevalue �p = 1.4 reported in Table 2, we consider the value �p = 53.2which is N times larger. The corresponding surface pressure dataare also plotted in Figs. 4 and 5. They give surface pressures which,although still disagreeing with experiment, have moved closer tothe experimental values, suggesting that multiple conformationalstates of protein were in fact accessed experimentally.

Comparing Figs. 4 and 5 we note that they cover the samedomain of t′/�2

m even though they cover different domains of t′. Fora given value of t′/�2

m, corresponding (roughly) to a given value of�s (as can be seen directly from Eq. (16), assuming the first termon the right hand side is dominant), the value of �p is larger when�m is larger (which follows from Eq. (14) because t′ itself is thenlarger at the given t′/�2

m value, again assuming the first term onthe right dominates). Moreover (for a large �p value, such as thevalue �p ≈ 53.2 considered in Figs. 4 and 5), it is terms in �p thattend to make the dominant contribution to the surface pressure �′

(Eq. (18)). Hence the fact that �p is larger at any given t′/�2m in the

case of Fig. 5 (compared to Fig. 4), explains why Fig. 5 appears toshow a sharper transition for the �′ data between the initial value(identically zero) and the final value (steady state).

When �p is large, there is a curious inflection in the �′ vs t′

simulation data (particularly noticeable in Fig. 4): this is not seen inthe experimental data. Nor would any such inflection be seen in thegraph of simulated �p vs t′ (not plotted here): it only appears for �′

vs t′ simulation data. Moreover the inflection only appears for large�p – it disappears for smaller �p (as Fig. 4 itself makes apparent).The reason for the inflection appears to be that (for the case ofa very large �p value), simulation data for �p + �s approach veryclose to unity. Surface pressure in Eq. (18) is then dominated by thelogarithmic term, the argument of which is now much smaller thanunity, making the logarithm itself very sensitive to the precise valueof the argument. Even very modest rates of change for �p + �s on theapproach to final steady state can therefore influence the surfacepressure dramatically. The experimental data in Fig. 4 do seem notto attain a final surface pressure quite so high as the simulationresult does, so presumably the experimental �p + �s values likewisedo not approach quite so closely to unity – thereby the experimentaldata seem to avoid the inflection.

Having considered the base case behaviour and the comparisonwith experimental results, we now proceed to perform a parametricstudy of the mixed protein–surfactant system as adsoprtion and/ordiffusion parameters are changed.

6.2. Effect of surfactant concentration on the adsorption

Figs. 6–11 present the profiles of subsurface and surface concen-tration of protein and surfactant. The simulations were performedat the following conditions: �p = 1.4, �s = 0 − 2.19, D = 8, �m =17.6, cb = 0 − 100. The values of �s and cb change (relative tothe base case values presented in Tables 1 and 2) due to thechange of surfactant bulk concentration, Csb, which varies from0 − 10−1 mol m−3.

It is obvious in Fig. 6 that the surface coverage of surfactantincreases with the addition of surfactant in the bulk solution. Atearly time the surface coverage of surfactant increases dramaticallyup to a maximum point. Beyond the maximum point, the surfacecoverage of surfactant decreases due to the presence of proteinarriving on the surface. Surfactant reaches its maximum surface

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1

θ s

t'

cb = 1; κs = 0.0219 cb = 10; κs = 0.2190 cb = 100; κs = 2.1900

Fig. 6. Surface coverage of surfactant as a function of dimensionless time at variousbulk concentrations of surfactant where �p = 1.4, �s = 0.0219 − 2.19, D = 8, �m =17.6, cb = 1 − 100 (solid line is the final surface concentration of surfactant atrespective bulk concentrations in the absence of protein).

coverage faster at higher surfactant bulk concentration. The max-imum surface coverage of surfactant with the presence of proteinis somewhat lower than the final surface coverage achieved by theadsorption of surfactant on its own. Eq. (17) shows that the pres-ence of protein reduces the amount of surfactant adsorbed on thesurface. That occurs because there is a smaller denominator on theright hand side of the equation due to the inclusion of protein sur-face concentration which does not appear in the equation for puresurfactant. This then makes the numerator (i.e. surfactant surfaceconcentration) likewise smaller.

In the special case of a Langmuir isotherm (the special casefor which ˛s = ˛p = ˛ps = 0) it is possible to derive a simple formulafor the extent to which presence of adsorbed protein in a mixedprotein–surfactant system will reduce the amount of adsorbed sur-factant.

If we assume that before significant protein arrives on thesurface, surfactant subsurface concentration C ′

s attains the samesurfactant concentration as in the bulk – scaled to equal unity here– and that the surfactant subsurface concentration is relativelyinsensitive to the subsequent arrival of protein, then we find fora Langmuir isotherm

�pures,final

�s= 1

(1 − �p)

where �pures,final

is the final surface coverage in a pure surfactant sys-tem, and �p and �s refer to the mixed protein–surfactant system.Here �p may evolve gradually over time, leading to a correspondingevolution in �s.

0

0.005

0.01

0.015

0.02

0 0.05 0.1 0.15 0.2 0.25

θ s

t'

displaced surfactant pure surfactant

mixed protein-surfactant avg adsorption - desorption

Fig. 7. Process of surfactant displacement where �p = 1.4.�s = 0.0219, D = 8, �m =17.6, cb = 1.

70 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

Table 3Summary of maximum surface coverage of surfactant in mixed protein–surfactant adsorption where tmax is the dimensionless time when the maximum overshoot occurs,�s, max is the maximum surface coverage at the overshoot, �s, final is the final surface coverage of mixed protein–surfactant adsorption, �pure

s, finalis the final surface coverage of

adsorption of pure surfactant, tavmax is the time required for a maximum averaged surface coverage between desorption and adsorption to occur and �av

s,max is the maximumaveraged surface coverage.

cb �s tmax �s, max �s, max/�s, final �pures, final

/�s, final tavmax �av

s, max

1 0.0219 0.0844 0.0134 1.3673 2.1633 0.0833 0.016810 0.2190 0.0620 0.1152 1.3458 1.9579 0.0623 0.1386

100 2.1900 0.0130 0.5228 1.2380 1.4603 0.0139 0.5672

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25

θ s

t'

displaced surfactant

pure surfactant mixed protein-surfactant

avg adsorption - desorption

Fig. 8. Process of surfactant displacement where �p = 1.4, �s = 2.19, D = 8, �m =17.6, cb = 100.

The change in �s (in the mixed system) per unit small incrementin �p is therefore −�pure

s,finalwhich evaluates to −�s/(1 − �s) for the

Langmuir isotherm. When �s 1 there is plenty of free space onthe surface for protein to adsorb, without affecting any surfactantwhich might already be there. When �s 1 however, the only wayfor protein to adsorb is to remove a nearly equivalent coverage ofsurfactant.

The above analysis indicates how much adsorbed surfactantmust be removed in the presence of adsorbed protein, but doesnot make any prediction regarding the time scale upon which sur-factant desorbs. The maximum of �s with respect to time in a casesuch as Fig. 6 should occur when the surfactant adsorption ratehas fallen to a level that equals the rate of displacing surfactantdue to the arrival of protein onto the surface. In order to inves-tigate this notion further, a (desorption) simulation was carriedout where at the initial time there is surfactant on the surfacewith coverage equal to the equilibrium surface coverage of sur-factant without the presence of protein in the solution (proteinthen being instantaneously added to the bulk). The curve obtainedfrom this simulation was then considered alongside the curve of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

θ p

t'

cb = 0 ; κs = 0 cb = 1 ; κs = 0.0219 cb = 10 ; κs = 0.2190 cb = 100; κs = 2.1900

Fig. 9. Surface coverage of protein as a function of dimensionless time at vari-ous bulk concentrations of surfactant where �p = 1.4, �s = 0 − 2.19, D = 8, �m =17.6, cb = 0 − 100.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Cs'

t'

cb = 1; κs = 0.0219 cb = 10; κs = 0.2190 cb = 100; κs = 2.1900

Fig. 10. Dimensionless subsurface concentration of surfactant as a function ofdimensionless time at various bulk concentrations of surfactant where �p =1.4, �s = 0.0219 − 2.19, D = 8, �m = 17.6, cb = 1 − 100.

adsorption of pure surfactant. An example of the result of thesimulation for �s = 0.0219 is presented in Fig. 7. If one takes anaverage of the desorption simulation data for �s and the pure surfac-tant adsorption �s data, qualitative agreement is obtained with thedata from the mixed protein–surfactant system. The time at whichthe above-mentioned average has a local maximum (i.e. the timeat desorption and adsorption rates are matched) however showsgood quantitative agreement with the time at which the mixedprotein–surfactant system also has its local maximum for �s. Asummary of the overshoot obtained in Fig. 6 and the comparisonobtained from the simulation demonstrated in Fig. 7 is presentedin Table 3.

For pure surfactant adsorption, it is possible to estimate roughlythe time for surfactant surface coverage to grow to saturation byretaining the leading square root term in Eq. (16) and then settingC ′

s equal to unity for saturation in Eq. (17). The resulting time scaleis only weakly sensitive to �s for small �s (assuming �s/cb is heldfixed, as is the case here). However, the time for growth to saturatewill start to decrease sharply for any given �s/cb when �s is large(compare, e.g. the pure surfactant data in Figs. 7 and 8).

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Cp'

t'

cb = 0; κs = 0 cb = 1; κs = 0.0219 cb = 10; κs = 0.2190 cb = 100; κs = 2.1900

Fig. 11. Dimensionless subsurface concentration of protein as a function of dimen-sionless time at various bulk concentrations of surfactant where �p = 1.4, �s =0 − 2.19, D = 8, �m = 17.6, cb = 0 − 100.

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 71

In the meantime for the desorption case, the initial growth of �p

will obey via Eq. (14)

�p∼√

t′

regardless of the cb and/or �s values. In the surfactant desorptionproblem therefore, the initial value of �p is infinite, whereas it turnsout that the initial �s is finite. Since �p therefore evolves much morequickly than �s, any changes in the value of C ′

s in Eq. (17) will nowbe driven by the changes in �p (the subsurface C ′

s must rise at leasttemporarily slightly above the unit concentration of surfactant inthe bulk to drive surfactant away from the surface).

At early times, for the simpler special case of a Langmuirisotherm (with the Frumkin isotherm expected to be qualitativelysimilar), it is possible to show via Eq. (17) that:

C ′s∼1 + �p

1 − (�s)initial

In the case of small �s we derive C ′s − 1∼�p∼√

t′ because(�s)initial 1. In the case of large �s, however Eq. (17) implies(again consider the simpler special case of a Langmuir isotherm,with the behaviour of a Frumkin isotherm expected to be simi-lar) we have 1 − (�s)initial∼�−1

s and thus C ′s − 1∼�s�p∼�s

√t′. Hence

the subsurface concentration of surfactant is perturbed (by proteinadsorption) to a greater extent when �s is large.

The perturbation to C ′s then drives the evolution of �s. By sub-

stituting the perturbation of C ′s into the integral in (16), the rate

of change of �s, i.e. the surfactant desorption rate, is in order ofmagnitude terms

�s∼ − cb

√D/�m

for small �s, but

�s∼ − �scb

√D/�m

for large �s (the ratio cb/�s is held fixed here, as �s is changed).Thus both the adsorption dynamics of pure surfactant, and the

desorption dynamics of initially adsorbed surfactant (upon addi-tion of protein) are pushed to earlier times as �s increases. Asa consequence, the matching point (where adsorption and des-orption rates are equal) occurs at earlier times for larger �s. Thisphenomenon is presented in Table 3, showing that the times toachieve the matching point (i.e. tav

max in the table) and the time toachieve maximum �s for the mixed protein–surfactant system (i.e.tmax in the table) are in a good agreement. For small �s the max-imum of �s with respect to time is relatively insensitive to �s atfixed �s/cb. However, with larger �s, the maximum �s is achievedat significantly shorter time as shown by the comparison betweenFigs. 7 and 8.

The more surfactant in the bulk solution results in less proteinadsorbed onto the surface as can be seen in Fig. 9. More surfac-tant in the bulk solution increases the surfactant concentration onthe surface. As a consequence, the large surface coverage of surfac-tant will reduce the surface coverage of protein as calculated usingEq. (15). As surface coverage of surfactant (�s) rises, the denomina-tor of the equation becomes smaller, the numerator then becomessmaller to compensate, hence resulting in smaller surface coverageof protein2 (�p). Nonetheless, low surfactant bulk concentration(cb = 1 and �s = 0.0219), does not have a significant impact on thesurface coverage of protein (compared to the surfactant free case).

2 When surfactant in the bulk increases, the ratio of maximum and final surfactantcoverage decreases. It can be explained by simplification in the special case of aLangmuir isotherm where �pure

s,final/�s = 1/(1 − �p). Reduced �p due to increased cb

and �s then leads to reduced �s,max/�s,final at least to the extent that �s,max approaches�pure

s,final.

0.01

0.1

1

0 1 2 3 4 5

θ p

t'

κp = 1.40 κp = 0.70 κp = 0.35

Fig. 12. Dimensionless surface concentration of protein as a function of dimension-less time at various protein affinity. The parameters used in the simulations are:�p = 0.35 − 1.4, �s = 0.219, D = 8, �m = 17.6, cb = 10.

Small value of �s does not have a significant impact on the denom-inator of the equation, therefore �p does not change substantially.

The higher concentration of surfactant in the bulk solution alsoresults in higher concentration of surfactant in the subsurface layer(at any given time) as can be seen in Fig. 10. Larger �s and cb impliesmore surfactant in the bulk. At large �s (and also cb) the surfacesaturates with surfactant, therefore surfactant accumulates in thesubsurface instead of adsorbing immediately onto the surface.

When the surfactant concentration in the subsurface reaches itssaturation point, the concentration of protein keeps growing as aresult of slower diffusion rate.3 However, in the presence of highersurfactant concentration in the bulk, as shown in Fig. 11, there ismore surfactant already adsorbed on the surface. Therefore, thereis less space for protein to adsorb onto the surface. With less pro-tein adsorbed onto the surface and continuous diffusion of proteininto the subsurface, the concentration of protein in the subsurfacewill be higher, in the presence of higher surfactant concentrationin the bulk as shown in Fig. 11. Surface concentration of surfac-tant is smaller than surface concentration of protein when the bulkconcentration of surfactant is equal or lower than the bulk concen-tration of protein. Therefore, surfactant has an insignificant effecton Eq. (15) under these conditions. However, as the surface con-centration (of surfactant) grows due to higher bulk concentration,the denominator of the right hand side of Eq. (15) becomes smallerresulting a higher subsurface concentration of protein.

6.3. Effect of relative affinity on the adsorption

A parametric study on the effect of the relative affinity on theadsorption was conducted. In this parametric study, the relativeaffinity was varied by varying the affinity of protein while theother parameters were kept fixed. The value of the parametersused in the simulation are: �p = 0 − 1.4, �s = 0.219, D = 8, �m =17.6, cb = 10.

Figs. 12–14 represent the adsorption of mixedprotein–surfactant at various protein affinities. The affinity ofsurfactant is set fixed at �s = 0.219. At a quite low affinity, protein(�p = 0.35) cannot compete with surfactant on the surface. As aresult, the surface concentration of protein reaches its saturationcomparatively soon and at low concentration. At higher affinitymore protein adsorbs on the surface as presented in Fig. 12. It alsoappears in Fig. 13 that the surface coverage of surfactant decreasesafter a certain time with the addition of protein with higher affinity

3 There is also a cooperative effect, namely comparatively high surface affinityof protein which can keep subsurface concentrations relatively low, and therebymaintain a mass transfer driving force between the subsurface and bulk.

72 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5

θ s

t'

κp = 1.40 κp = 0.70 κp = 0.35 κp = 0

Fig. 13. Dimensionless surface concentration of surfactant as a function of dimen-sionless time at various protein affinity. The parameters used in the simulations are:�p = 0 − 1.4, �s = 0.219, D = 8, �m = 17.6, cb = 10.

(�p = 0.70 and �p = 1.40) due to the displacement of surfactant byprotein. This displacement of surfactant by protein has alreadybeen described in Figs. 2 and 7. At low affinity of protein, there ishowever no depletion of surface coverage of surfactant. Havingthe affinity of protein less than that of surfactant is not a necessarycondition to suppress the overshoot in the surfactant surfacecoverage profile. When �p is considerably smaller than one, eventhough it is bigger than �s, the adsorption of protein does notnecessarily result in significant desorption of surfactant. Thedisplacement of surfactant molecules by protein molecules on thesurface depends on the (absolute as well as relative) affinities4 andthe rates of protein and surfactant to reach the surface.

The occurrence of an overshoot in Fig. 13 can be explainedusing Eq. (17). When the value of �s in that equation is rela-tively small, due to a much smaller value of �s compared to�p, then �s can be eliminated from the denominator resultingin (�s)eff = �s(1 − �p)/exp(− ˛ps�p), where (�s)eff is the effective �s

when �s is very small. Here (�s)eff is a decreasing function of �p,since referring to Table 1, ˛ps is a quite small parameter. ThereforeEq. (17) can be simplified to:

C ′s ≈ �s

�s(1 − �s)exp(−˛ps�p) = �s

(�s)eff(19)

At early time, �p is quite small and can be eliminated from Eq. (19),therefore �s is affected mainly by the parameter �s. At later time,as �p grows, it influences the equation, resulting in smaller valueof �s with the increase of �p. The gradual decrease of �s followingthe overshoot occurs when the value of �p starts to affect (�s)eff,hence starts reducing �s. The overshoot will not occur if �s is notsufficiently small compared to �p, as in that case �s has an effect onEq. (17) and it cannot be simplified to Eq. (19). However, a largerratio between �p and �s results in more displacement of surfactantby protein. Under those circumstances, not only the final, but alsothe maximum concentration of surfactant on the surface is lowerwith larger protein affinity.

Fig. 14 shows that the surface pressure is higher at a larger affin-ity of protein. Despite the displacement of surfactant by protein,and despite the fact that surface pressure is more sensitive to �s

than to �p (a result of the factor ω0/ωp in Eq. (18)) surface pressurestill manages to grow with increasing �p.

4 Whenever �s �p an overshoot is certainly expected, but if �s and �p are bothsmall, then one tends not to see an overshoot in the case where �s is only slightlyless than �p . The reason for this is clear: when �s and �p are both small, a great dealof free space is available on the surface, and protein can adsorb without displacingsurfactant. When �s and �p are both larger, even though the ratio of �s and �p ismaintained the same, an overshoot in surfactant concentration becomes more likely.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5

Π'

t'

κp = 1.40 κp = 0.70 κp = 0.35 κp = 0

Fig. 14. Dimensionless surface pressure as a function of dimensionless time at var-ious protein affinity. The parameters used in the simulations are: �p = 0 − 1.4, �s =0.219, D = 8, �m = 17.6, cb = 10.

6.4. Effect of molecular size on the adsorption

Molecular size may determine diffusivity, surface capacity andsurface affinity of solutes, which in this study are protein and sur-factant. The dependency on molecular size may however impactdifferently on diffusivity, surface capacity and surface affinity. TheStokes–Einstein relation [22] for determining diffusivity of solute ina solution shows that the diffusivity is determined by the maximumlinear dimension of molecules [23]. Therefore, the diffusivity of asolute is clearly very sensitive to the size of the solute molecule,but does not depend much on its orientational conformation. Onthe other hand, the surface capacity is conformation and orien-tation dependent. The conformation and orientation are in turngoverned by the spatial distribution of polar residues around themolecule surface. In the case of a surfactant molecule with a sin-gle polar head group and a non-polar tail, the molecule will tendto orient itself with the tail directed normal to the surface. If thetail is relatively straight therefore, the surface capacity (i.e. max-imal packing density) of surfactant only weakly depends on thelength of the (normally directed) surfactant tail. The surface affin-ity of surfactant meanwhile increases with the increase of the chainlength for homologous series of a particular surfactant type, eventhough the surface capacity may remain almost the same [24]. Thus,from the physical point of view, it is reasonable to say that whenthe molecular size significantly changes, there is a dependence ofsurface affinity on the molecular size [25].

In view of the arguments above, Fig. 15 shows a parametricstudy on surfactant adsorption by varying the relative diffusivity(D) and the affinity of surfactant (�s). The relative capacity (�m)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

θ s

t'

D = 10; κs = 0.219 D = 10; κs = 2.190 D = 1; κs = 0.219 D = 1; κs = 2.190

Fig. 15. Surface coverage of surfactant as a function of dimensionless time at variousrelative diffusivity and relative capacity using the following dimensionless parame-ters: �p = 1.4, �s = 0.219 − 2.19, D = 1 − 10, �m = 10, cb = 10. Straight line on thetop indicates the final value of �s in the absence of protein in the bulk solution.

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 73

was assumed to be fixed as it depends only weakly on the chainlength of the surfactant. The values and ranges of parameters usedin the simulation are as follows: �p = 1.4, �s = 0.219 − 2.19, D =1 − 10, �m = 10, cb = 10.

The typical case in the adsorption of mixed protein–surfactant isthat surfactant has a bigger diffusivity and bigger surface capacitythan protein, but a smaller surface affinity. In Fig. 15, the data forD = 10, �m = 10 and �s = 0.219, shown by the solid line, representthat particular case. Due to higher diffusivity, surfactant is trans-ported faster than protein to the subsurface. On the other hand, thelarger surface capacity indicates that surfactant molecules are rela-tively small compared to protein molecules. As a result, individualsurfactant molecules do not occupy much space on surface, hencetake a longer time to attain any given surface coverage fraction.The lower surface affinity also implies the surfactant saturates at alower coverage fraction.

By contrast, the case with higher surface affinity of surfactant(�s = 2.19) results in more surface coverage of surfactant (�s). It isobvious that with higher surface capacity, surfactant has more ten-dency to adsorb onto the surface. This high surfactant affinity stillseems to exhibit a (weak) overshoot for surfactant – despite thefact that the ratio �p/�s is now less than unity.

A lower relative diffusivity (D = 1), which means a lower dif-fusivity of surfactant, also tends to suppress the overshoot. Lowerdiffusivity of surfactant leads to slightly slower surfactant growthon the surface. As a consequence, the difference between the surfac-tant growth rate and the protein growth rate becomes less marked.

Indeed the sharpness of the overshoot in Fig. 15 increases withthe parameter cb

√D/�m which clearly from Eq. (16) affects the

growth rate of �s. There are three separate effects occurring fromthat parameter (only one of which is explored in Fig. 15). Large cbimplies a large driving force for mass transfer, large D implies fastermass transfer (although this is partially offset by faster growth inthe thickness of the mass transfer boundary layer associated withEq. (7) – hence the proportionality with

√D rather than D itself),

and finally large �m implies individual surfactant molecules takeup little space on the surface so that surface coverage grows onlyslowly.

6.5. Identification of region where surfactant displacement occurs

A parametric study was carried out to identify the region of sur-factant displacement. The result of the parametric study is shownin Fig. 16. This parametric study was carried out using parametersin the following ranges: �p = 0 − 1.53, �s = 0.0219 − 0.219, D =1 − 10, �m = 1 − 1260, cb = 1 − 10. The displacement of surfactant

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70

c b D

1/2

/ Γ

m

κp/κs

base case

κs = 0.2190κs = 0.0219

Fig. 16. Phase diagram showing the region of surfactant displacement byprotein. The parameters used in this simulation range as follows: �p = 0 −15.33, �s = 0.219, D = 1 − 10, �m = 1 − 1260, cb = 10 (solid line) and �p = 0 −1.53, �s = 0.0219, D = 1 − 10, �m = 1 − 200, cb = 1 (dashed line). The base casewas at �p = 1.4, �s = 0.219, D = 8, �m = 17.6, cb = 10. The displacement of surfac-tant occurs in the region above and to the right of the curve.

occurs in the region above and to the right of the curve in that fig-ure. In practical cases of adsorption of mixed protein–surfactant(including the base case in Table 2), it is most likely that there isdisplacement of surfactant by protein. As we have seen in Figs. 6, 13and 15, the displacement results in an overshoot of the curveof surface coverage of surfactant. The displacement of surfactantcan be avoided when the affinity of protein is not sufficientlylarge compared to the affinity of surfactant. Small relative diffu-sivity and large relative capacity also result in the removal of theovershoot. However very significant changes in relative diffusiv-ity and/or relative capacity about our base case values would beneeded to eliminate surfactant displacement (taking also our basecase �s = 0.219, �p = 1.4 and cb = 10 as considered here).

Fig. 16 also shows that the critical protein surface affinity forsurfactant displacement to occur is sensitive to the surface affin-ity of surfactant: data are for �s = 0.0219, �p = 0 − 1.53, D = 1 −10, �m = 1 − 200, cb = 1. With low (absolute) surfactant affinity,displacement of surfactant is less likely to happen even though thesurface affinity of protein is relatively high. Therefore, the boundarybetween surfactant displacement and no surfactant displacementin the graph depends not only on relative surface affinities, butalso depends on the (absolute) surface affinities. A small �s needs amassive �p/�s to displace surfactant, whereas larger �s will exhibitsurfactant displacement with more modest �p/�s.

Reducing cb is also an option for avoiding surfactant displace-ment, but the most straightforward experimental way of doingthat – namely reducing surfactant concentration in the mixture –might impact upon foam quality. Moreover changes in surfactantconcentration also affect the value of �s and hence the ratio �p/�s.

7. Conclusions

Simulations of dynamics of adsorption of mixed protein–surfactant have been carried out using base parameters of theadsorption of mixed BLG protein and C10DMPO non-ionic sur-factant. The simulation studies the displacement of surfactant byprotein and the effect of surfactant bulk concentration on theadsorption. This study also reports the comparison between thesimulation results and the experimental data obtained from litera-ture. Parametric studies were also conducted to examine the effectof protein affinity and the effect of molecular size on the adsorption.

The simulation results show that protein arrives on the surfaceat a later time than surfactant. At early time surfactant dominatesthe surface. However, as the protein reaches the surface, it tendsto replace the surfactant and reduces the concentration of sur-factant on the surface. Surfactant, therefore reaches a maximumsurface concentration following which it depletes in competitionwith protein. The maximum surface concentration of surfactant inthe presence of protein is somewhat lower than the final concen-tration achieved by adsorption of pure surfactant of the same bulkconcentration. There is also less protein adsorbed with more surfac-tant in the bulk solution. The smaller amount of protein adsorbedon the surface moreover results in more protein molecules inthe subsurface. Comparison of simulation results with experimen-tal data obtained from literature shows a discrepancy due to theassumption of no conformational change of the protein: multipleconformational states of protein need to be considered.

The parametric study when varying the surface affinity of pro-tein reveals that below a critical affinity protein tends not to replacesurfactant on the surface. The critical affinity of protein to preventsurfactant displacement need not necessarily be smaller than theaffinity of surfactant. The molecular size of surfactant, which hasa strong influence on its diffusivity (therefore influences relativediffusivity) and surface affinity, affects the adsorption and the dis-placement of the surfactant. The displacement of surfactant is more

74 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

likely to occur at high relative diffusivity and low surface affinity(relative to protein affinity) of surfactant. Higher surface coverageis also achieved with higher surface affinity of surfactant.

The study only investigated the adsorption dynamics of mixednon-ionic surfactant and protein, where conformational changes ofprotein do not occur. A real engineering process such as foam frac-tionation may involve protein conformation changes in additionto interaction between protein and surfactant molecules. There-fore, a further study investigating the effect of protein conformationalongside the interaction of protein and surfactant molecules andthe effect on adsorption is needed.

Appendix A. Solving the equations of dynamics ofadsorption using Newton iteration

The equations of dynamics of adsorption of mixedprotein–surfactant are solved numerically, using the techniquesdescribed below. The integral from the right hand side of Eqs. (14)and (16) is represented in numerical form using the trapezoidalrule. The numerical form of the dynamics of adsorption of proteinand surfactant, respectively, at time step i and using a timeincrement of ıt′, are:

�p(i ıt′) = 1√�

{2√

i ıt′ − 23

√ıt′

i − 1C ′

p(ıt′)

− ıt′

2

i−1∑

k=1

[C ′

p(k ıt′)√i ıt′ − k ıt′

+ C ′p((k + 1) ıt′)√

i ıt′ − (k + 1) ıt′

]

− 43

√ıt′C ′

p(i ıt′) − 23

√ıt′C ′

p((i − 1) ıt′)

}(A.1)

and

�s(i ıt′) = Cb

√D

�m

1√�

{2√

i ıt′ − 23

√ıt′

i − 1C ′

s(ıt′)

−ıt′

2

i−1∑

k=1

[C ′

s(k ıt′)√i ıt′ − k ıt′

+ C ′s((k + 1) ıt′)√

i ıt′ − (k + 1) ıt′

]

−43

√ıt′C ′

s(i ıt′) − 23

√ıt′C ′

s((i − 1) ıt′)

}(A.2)

Note that we have treated the first and last integration inter-

vals within the trapezoidal rule in a special fashion to take properaccount of weak singularities in the integrand that occur there.

There are four equations to be solved using Newton iterationwhich are Eqs. (15), (17), (A.1) and (A.2) with four unknowns(C ′

p(iıt′), C ′s(iıt′), �p(iıt′), �s(iıt′)). The roots of the equations are

then obtained using Newton method of iteration. The equations tobe solved in the Newton method of iteration are (defining functionsp, f, s, g):

0 = p(i ıt′) ≡ �p(i ıt′) − 1√�

{2√

i ıt′ − 23

√ıt′

i − 1C ′

p(ıt′)

−ıt′

2

i−1∑

k=1

[C ′

p(k ıt′)√i ıt′ − k ıt′

+ C ′p((k + 1) ıt′)√

i ıt′ − (k + 1) ıt′

]

−43

√ıt′C ′

p(i ıt′) − 23

√ıt′C ′

p((i − 1) ıt′)

}(A.3)

0 = f (i ıt′) ≡ C ′p(i ıt′) − �p(i ıt′)

�p

[1 − �p(i ıt′) − �s(i ıt′)

]

× exp[−2˛p�p(i ıt′) − 2˛ps�s(i ıt′)

](A.4)

0 = s(i ıt′) ≡ �s(i ıt′) − Cb

√D

�m

1√�

{2√

i ıt′ − 23

√ıt′

i − 1C ′

s(ıt′)

−ıt′

2

i−1∑

k=1

[C ′

s(k ıt′)√i ıt′ − k ıt′

+ C ′s((k + 1) ıt′)√

i ıt′ − (k + 1) ıt′

]

− 43

√ıt′C ′

s(i ıt′) − 23

√ıt′C ′

s((i − 1) ıt′)

}(A.5)

0 = g(i ıt′) ≡ C ′s(i ıt′) − �s(i ıt′)

�s

[1 − �p(i ıt′) − �s(i ıt′)

]

× exp[−2˛s�s(i ıt′) − 2˛ps�p(i ıt′)

](A.6)

The derivatives of the equations are presented in the followingequations where the variables are presented in short hand formas follows: C ′

p to denote C ′p(i ıt′), C ′

s to denote C ′s(i ıt′), �p to denote

�p(iıt′) and �s to denote �s(iıt′).Derivatives of the equation representing the dynamics of

adsorption of protein:

∂p

∂C ′p

= 43

1√�

√ıt′ (A.7)

∂p

∂�p= 1 (A.8)

∂p

∂C ′s

= 0 (A.9)

∂p

∂�s= 0 (A.10)

Derivatives of the equation representing the adsorption isothermof protein:

∂f

∂C ′p

= 1 (A.11)

∂f

∂�p=[

2˛p�p

�p(1 − �p − �s)− (1 − �s)

�p(1 − �p − �s)2

]exp(−2˛p�p − 2˛ps�s)

(A.12)

∂f

∂C ′s

= 0 (A.13)

∂f

∂�s=[

2˛ps�p

�p(1 − �p − �s)− �p

�p(1 − �p − �s)2

]exp

(−2˛p�p − 2˛ps�s

)

(A.14)

Derivatives of the equation representing the dynamics ofadsorption of surfactant:

∂s

∂C ′p

= 0 (A.15)

D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76 75

∂s

∂�p= 0 (A.16)

∂s

∂C ′s

= 43

Cb

√D

�m

1√�

√ıt′ (A.17)

∂s

∂�s= 1 (A.18)

Derivatives of the equation representing the adsorptionisotherm of surfactant:

∂g

∂C ′p

= 0 (A.19)

∂g

∂�p=[

2˛ps�s

�s(1 − �p − �s)− �s

�s(1 − �p − �s)2

]exp(−2˛s�s − 2˛ps�p)

(A.20)

∂g

∂C ′s

= 1 (A.21)

∂g

∂�s=[

2˛s�s

�s(1 − �p − �s)− (1 − �p)

�s(1 − �p − �s)2

]exp(−2˛s�s − 2˛ps�p)

(A.22)

The Newton method of iteration is based on the first order Tay-lor expansion of the functions about some estimated solutions ofC ′0

p , C ′0s , �0

p and �0s as presented in the following equations. The solu-

tions from the time step i − 1 provide suitable estimates to start theNewton iteration at time step i.

p(C ′p, �p, C ′

s, �s) = 0 = p(C ′0p , �0

p, C ′0s , �0

s )

+ ∂p

∂C ′p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′p − C ′0

p )

+ ∂p

∂�p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�p − �0p )

+ ∂p

∂C ′s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′s − C ′0

s )

+ ∂p

∂�s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�s − �0s ) (A.23)

f (C ′p, �p, C ′

s, �s) = 0 = f (C ′0p , �0

p, C ′0s , �0

s )

+ ∂f

∂C ′p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′p − C ′0

p )

+ ∂f

∂�p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�p − �0p )

+ ∂f

∂C ′s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′s − C ′0

s )

+ ∂f

∂�s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�s − �0s ) (A.24)

s(C ′p, �p, C ′

s, �s) = 0 = s(C ′0p , �0

p, C ′0s , �0

s )

+ ∂s

∂C ′p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′p − C ′0

p )

+ ∂s

∂�p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�p − �0p )

+ ∂s

∂C ′s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′s − C ′0

s )

+ ∂s

∂�s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�s − �0s ) (A.25)

g(C ′p, �p, C ′

s, �s) = 0 = g(C ′0p , �0

p, C ′0s , �0

s )

+ ∂g

∂C ′p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′p − C ′0

p )

+ ∂g

∂�p

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�p − �0p )

+ ∂g

∂C ′s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(C ′s − C ′0

s )

+ ∂g

∂�s

∣∣∣∣(C ′0

p ,�0p ,C ′0

s ,�0s )

(�s − �0s ) (A.26)

where C ′0p , �0

p , C ′0s , �0

s are the initial guesses of C ′p, �p, C ′

s, �s. We define

ıC ′p = C ′

p − C ′0p

ı�p = �p − �0p

ıC ′s = C ′

s − C ′0s

ı�s = �s − �0s

The matrix format of Eqs. (A.23)–(A.26) is as follows:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(∂p

∂C ′p

)0 (∂p

∂�p

)0 (∂p

∂C ′s

)0 (∂p

∂�s

)0

(∂f

∂C ′p

)0 (∂f

∂�p

)0 (∂f

∂C ′s

)0 (∂f

∂�s

)0

(∂s

∂C ′p

)0 (∂s

∂�p

)0 (∂s

∂C ′s

)0 (∂s

∂�s

)0

(∂g

∂C ′p

)0 (∂g

∂�p

)0 (∂g

∂C ′s

)0 (∂g

∂�s

)0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ıC ′p

ı�p

ıC ′s

ı�s

⎤⎥⎥⎥⎥⎥⎦

= −

⎡⎢⎢⎢⎢⎢⎣

p0

f 0

s0

g0

⎤⎥⎥⎥⎥⎥⎦

(A.27)

Superscripts ‘0’ appended to p, f, s, g from Eqs. (A.3)–(A.6) and theirderivatives denote function evaluations at C ′0

p , �0p , C ′0

s , �0s .

Eq. (A.27) can be solved using the Cramer’s rule [26] resultingin the first set of values of the correction vector. A new estimate ofthe solution can now be calculated from the previous estimate byadding the correction vector onto it, and the process iterated. Forexample:

C ′(m+1)p = C ′(m)

p + ıC ′(m)p (A.28)

76 D. Vitasari et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 438 (2013) 63– 76

where m is the number of iterations. As mentioned previously,the initial guess of the variables are obtained from the valuesdetermined in the previous step. The above procedure applies toobtained data at time iıt′ for i ≥ 2. At t′ = ıt′, the values of �p(ıt′)and �s(ıt′) are estimated via simplification of the Ward–Tordaiequation, using the assumption that the effect of subsurface con-centration on the evolution of the equation is negligible in a firstapproximation at very low concentration in the subsurface. Thesimplified equations are presented as follows5:

�p(ıt′) = 1√�

{2√

ıt′ − �

2

√ıt′C ′

p(ıt′)}

(A.29)

�s(ıt′) = cb

√D

�m

1√�

{2√

ıt′ − �

2

√ıt′C ′

s(ıt′)}

(A.30)

Those equations need to be solved simultaneously with Eqs. (15)and (17) to obtain C ′

p(ıt′), C ′s(ıt′), �p(ıt′) and �s(ıt′). The values of

C ′p(ıt′) and C ′

s(ıt′) on the right hand side of Eqs. (A.29) and (A.30) areapproximated by �p(ıt′)/�p and �s(ıt′)/�s (Henry’s law analogues ofEqs. (15) and (17), and these �p(ıt′) and �s(ıt′) terms when appear-ing on the right hand side can be replaced in turn by the leadingterm from Eqs. (A.29) and (A.30)).

In the case of non-zero initial surfactant concentration on thesurface, Eq. (A.30) is generalized into the following form:

�s(ıt′) − (�s)initial = cb

√D

�m

1√�

×{

2√

ıt′ − 23

√ıt′C ′

s(0) − 43

√ıt′C ′

s(ıt′)}

(A.31)

where (�s)initial is the initial surface concentration of surfactant. Thevalue of C ′

s(ıt) on the right hand side of Eq. (A.31) can be obtained viafirst determining leading order approximations for �p(ıt′) (retainthe first term from Eq. (A.29)) and for �s(ıt′) − (�s)initial which is

�s(ıt′) − (�s)initial ≈ cb

√D

�m

2√�

√ıt′(1 − C ′

s(0))

and then substituting the values obtained into Eq. (17).Moreover in the important special case where C ′

s(0) = 1 (whichis the case contemplated in Fig. 7) the initial evolution of �s is muchslower than that of �p, so the perturbation to C ′

s at early times isprimarily due to growth in the value of �p and not due to changesin the value of �s. This leads to an increase in C ′

s in the subsurface –making its concentration higher than the bulk – which is necessaryfor driving the surfactant away from the subsurface towards thebulk (the desorption calculations of Fig. 7).

References

[1] R. Miller, D.O. Grigorief, E.V. Aksenenko, S.A. Zholob, M.E. Leser, M. Michel, V.B.Fainerman, Thermodynamic and adsorption kinetic studies of protein +surfac-tant mixtures, in: E. Dickinson (Ed.), Food Colloids: Interaction, Microstructureand Processing, Royal Society of Chemistry, Cambridge, 2005, pp. 120–130.

[2] R. Miller, V.B. Fainerman, A.V. Makievski, J. Krägel, R. Wüstneck, Adsorptioncharacteristics of mixed monolayers of a globular protein and a non-ionic sur-factant, Colloids Surf. A: Physicochem. Eng. Aspects 161 (2000) 151–157.

[3] V.S. Alahverdjieva, K. Khristov, D. Exerowa, R. Miller, Correlation betweenadsorption isotherms, thin liquid films and foam properties of protein/

5 The factor �/2 on the right hand sides of Eqs. (A.29) and (A.30) arises from mildsingularities in both numerator and denominator of the integrands Eqs. (14) and(16)

surfactant mixtures: lysozyme/C10DMPO and lysozyme/SDS, Colloids Surf. A:Physicochem. Eng. Aspects 323 (2008) 132–138.

[4] V.B. Fainerman, S.A. Zholob, M. Leser, M. Michel, R. Miller, Competitive adsorp-tion from mixed nonionic surfactant/protein solutions, J. Colloid Interface Sci.274 (2004) 496–501.

[5] R. Miller, V.B. Fainerman, M.E. Leser, M. Michel, Kinetics of adsorption of pro-teins and surfactants, Curr. Opin. Colloid Interface Sci. 9 (2004) 350–356.

[6] C. Kotsmar, V. Pradines, V.S. Alahverdjieva, E.V. Aksenenko, V.B. Fainerman,V.I. Kovalchuk, J. Krägel, M.E. Leser, B.A. Noskov, R. Miller, Thermodynamics,adsorption kinetics and rheology of mixed protein–surfactant interfacial layers,Adv. Colloid Interface Sci. 150 (2009) 41–54.

[7] C. Kotsmar, E.V. Aksenenko, V.B. Fainerman, V. Pradines, J. Krägel, R. Miller,Equilibrium and dynamics of adsorption of mixed ˇ-casein/surfactant solutionsat the water/hexane interface, Colloids Surf. A: Physicochem. Eng. Aspects 354(2010) 210–217.

[8] M. Mulqueen, K.J. Stebe, D. Blankschtein, Dynamic interfacial adsorptionin aqueous surfactant mixtures: theoretical study, Langmuir 17 (2001)5196–5207.

[9] S.N. Moorkanikkara, D. Blankschtein, New methodology to determine the rate-limiting adsorption kinetics mechanism from experimental dynamic surfacetension data, J. Colloid Interface Sci. 302 (2006) 1–19.

[10] S.N. Moorkanikkara, D. Blankschtein, New theoretical framework for design-ing nonionic surfactant mixtures that exhibit a desired adsorption kineticsbehavior, Langmuir 26 (2010) 18728–18733.

[11] X. Li, R. Shaw, G.M. Evans, P. Stevenson, A simple numerical solution to theWard–Tordai equation for the adsorption of non-ionic surfactants, Comput.Chem. Eng. 34 (2010) 146–153.

[12] V. Pradines, V.B. Fainerman, E.V. Aksenenko, J. Krägel, R. Wüstneck, R. Miller,Adsorption of protein–surfactant complexes at the water/oil interface, Lang-muir 27 (2011) 965–971.

[13] S.F. Santos, D. Zanette, H. Fischer, R. Itri, A systematic study of bovine serumalbumin (BSA) and sodium dodecyl sulfate (SDS) interactions by surface ten-sion and small angle X-ray scattering, J. Colloid Interface Sci. 262 (2003)400–408.

[14] J. Krägel, M. O’Neill, A.V. Makievski, M. Michel, M.E. Leser, R. Miller, Dynam-ics of mixed protein–surfactant layers adsorbed at the water/air and water/oilinterface, Colloids Surf. B: Biointerfaces 31 (2003) 107–114, Food Colloids,Biopolymers and Materials Special Issue.

[15] C. Kotsmar, D.O. Grigoriev, F. Xu, E.V. Aksenenko, V.B. Fainerman, M.E. Leser, R.Miller, Equilibrium of adsorption of mixed milk protein/surfactant solutions atthe water/air interface, Langmuir 24 (2008) 13977–13984.

[16] V.B. Fainerman, S.A. Zholob, M.E. Leser, M. Michel, R. Miller, Adsorption frommixed ionic surfactant/protein solutions: analysis of ion binding, J. Phys. Chem.B 108 (2004) 16780–16785.

[17] V.B. Fainerman, E.H. Lucassen-Reynders, R. Miller, Description of the adsorp-tion behaviour of proteins at water/fluid interfaces in the framework ofa two-dimensional solution model, Adv. Colloid Interface Sci. 106 (2003)237–259.

[18] R.Z. Guzman, R.G. Carbonell, P.K. Kilpatrick, The adsorption of proteins togas–liquid interfaces, J. Colloid Interface Sci. 114 (1986) 536–547.

[19] C.H. Chang, E.I. Franses, Adsorption dynamics of surfactants at the air/waterinterface: a critical review of mathematical models, data, and mechanisms,Colloids and Surf. A: Physicochem. Eng. Aspects 100 (1995) 1–45.

[20] A.F.H. Ward, L. Tordai, Time-dependence of boundary tensions of solutions I.The role of diffusion in time-effects, J. Chem. Phys. 14 (1946) 453–461.

[21] R.A. Leonard, R. Lemlich, A study of interstitial liquid flow in foam. Part I.Theoretical model and application to foam fractionation, AIChE J. 11 (1965)18–25.

[22] A. Einstein, Investigations on the Theory of the Brownian Movement, DoverPublications, Inc., New York, 1956.

[23] S. Kim, S.J. Karrila, Microhydrodynamics: Principles and Selected Applications,Dover Civil and Mechanical Engineering Series, Dover Publications, Mineola,NY, 2005.

[24] E.D. Shchukin, A.V. Pertsov, E.A. Amelina, A.S. Zelenev, Colloid and SurfaceChemistry. Chapter II. The Adsorption Phenomena. Structure and Properties ofAdsorption Layers at the Liquid–Gas Interface, Volume 12 of Studies in InterfaceScience, Elsevier, Amsterdam, 2001, pp. 64–164.

[25] V.B. Fainerman, R. Miller, Adsorption and interfacial tension isotherms forproteins, in: D. Möbius, R. Miller (Eds.), Proteins at Liquid Interfaces, Vol-ume 7 of Studies in Interface Science, Elsevier, Amsterdam, Oxford, 1998,pp. 51–102.

[26] A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineerswith MATLAB Applications, Prentice Hall PTR, Upper Saddle River, NJ, 1999.

Chapter 5

Surfactant transport onto a foam lamella

This chapter is a copy of article published in Chemical Engineering Science, 102:405–

423, 2013 DOI 10.1016/j.ces.2013.08.041. Authors: Denny Vitasari, Paul Grassia, Peter

Martin. This paper studies the evolution of surfactant concentration on the surface of

a foam film within a foam fractionation column with reflux. The study examines the

interaction between film drainage and the Marangoni force in determining the transport

of surfactant onto the surface of the film.

The transport of surface active material onto a foam lamella is through two processes.

One process is by adsorption of surface active material from the bulk solution onto the

gas-liquid interface on the surface of the lamella as discussed in Chapter 4. The other

process is the transport of the surface active material on the surface of lamella itself due

to the interaction of forces applied on the surface.

This study models the transport of surfactant onto the surface of a foam lamella. The

model for other surface active materials as well as for the mixed surfactant and/or protein

is similar to the present simulation. In this model an assumption of insoluble surfactant,

where the surfactant molecules stay on the surface once they are adsorbed, is taken as the

insoluble surfactant which is more simple to model. Practically, this can be be achieved

61

when the thickness of the film is substantially less than the Henry constant. The justifica-

tion of this assumption is discussed in Appendix D.

In the insoluble case certainly material that moves via the Marangoni flow onto the

surface of a lamella does not subsequently leak off the surface. In the case of soluble

surfactant, the Marangoni effect not only convects surfactant rich material from Plateau

border surface along the film surface, but also convects the surfactant rich material from

the bulk of the Plateau border to the bulk of the film. In this case, there may be less mate-

rial adsorbed on the surface, but nevertheless more material overall in the film. In the case

of soluble surfactant, as the film thins under drainage, surfactant that has been collected

from Plateau border is sent back to into the Plateau border. As a consequence, the amount

of surfactant recovered might peak at certain time. On the other hand, in the insoluble

case it goes on gradually increasing all times. If the peak recovery only corresponds to a

very slight enrichment over and above the surfactant in the Plateau border, it could be de-

sirable to allow the film to thin a little more, and sacrifice some of the recovery in favour

of more enrichment.

In the operation of a foam fractionation column with reflux, surfactant-rich liquid is

introduced back to the top of the column to increase the enrichment. This liquid flows

through the Plateau borders. As a consequence, the concentration of surfactant in the

Plateau borders is higher than that in the foam lamellae, therefore the concentration of

surfactant on the surface of Plateau borders is also higher than that on the surface of

lamellae. As a consequence, there will be Marangoni flow in the direction towards the

centre of the film (see Section 2.7). Besides the Marangoni effect, film drainage also

takes place inside the film, in the direction towards the Plateau border, which is opposite

to the Marangoni flow (see Section 2.6). The interaction between the Marangoni force

and the film drainage determines the transport of surfactant onto the surface of a foam

lamella.

This study considers two extremal models for film drainage that applies to the liquid

film inside a foam system. One model is assuming that the film surface is completely

62

mobile, therefore the flow within the film is plug-like flow [28]. The other model is us-

ing an assumption that the film surface is completely rigid, therefore the liquid velocity

profile in the film is parabolic, where there is no slip condition on the film surface [84].

In reality, the mobility of the interface can be between the completely rigid and the com-

pletely mobile. In this case, the rate of film drainage can be determined empirically using

experimental data such as obtained using Scheludko cell [81–83]. The important focus

of this study is to predict the spatial distribution of surfactant and its changes over time.

This study also compares the distribution obtained using an assumption of a mobile and

a rigid interface and also examines whether that predicted distribution remains consistent

with the assigned model.

The evolution of surfactant surface concentration is simulated using the material point

method [103]. The material point method is a method in computational fluid mechan-

ics which combines the fixed Eulerian meshes and moving Lagrangian points along the

mesh [104, 105] on solving continuity equations. On the mesh, the information carried

by the material points is used to solve the continuity equation [106]. The solution of the

equation on the mesh is then used to update the material points. This technique gives

the material point method a capacity to handle a system with large deformation without

overly distorting the mesh [104]. Therefore, this method is appropriate for modelling

the fluid flow on the surface of a foam film where there is large deformation due to the

Marangoni effect as well as film drainage.

The desirable condition for operation of a foam fractionation column is when the

Marangoni effect dominates the film drainage. At this condition, surfactant will accumu-

late on the surface of the lamella. This condition is achieved when the film drainage is

not very fast, like the one modelled using a rigid interface. When the film drainage is

very fast, like the one modelled using a mobile interface, the lamella becomes very thin

and possibly reaches the thickness of a common black film, where the film stops thinning

due to the electrostatic repulsion between the interfaces [107–109], during the typical res-

idence time in a foam fractionation column with reflux. In the absence of film drainage,

when the thickness of a common black film is achieved, the film drainage stops, therefore

63

surfactant will accumulate on the surface of the lamella due to the Marangoni effect. Long

before that happen however, Marangoni forces are likely to become sufficiently strong as

to violate the assumptions underlying the mobile interface model.

64

Publication 2

Surfactant transport onto a foam lamella

65

Surfactant transport onto a foam lamella

Denny Vitasari, Paul Grassia n, Peter MartinSchool of Chemical Engineering and Analytical Science, The Mill, The University of Manchester, Oxford Road, Manchester M13 9PL, UK

H I G H L I G H T S

� In foam fractionation with reflux,films contact surfactant-rich Plateauborders.

� Marangoni forces pull surfactantonto films while drainage sweepssurfactant away.

� For surfactants giving rigid film sur-faces, Marangoni dominates.

� For surfactants giving mobile filmsurfaces, film drainage dominates.

� In the mobile case, surfactant migratesback onto very thin films once drain-age stops.

G R A P H I C A L A B S T R A C T

a r t i c l e i n f o

Article history:Received 31 May 2013Received in revised form23 July 2013Accepted 18 August 2013Available online 26 August 2013

Keywords:SurfactantInterfacial tensionFilmsMathematical modellingMarangoni forcesFilm drainage with rigid/mobile filminterface

a b s t r a c t

A modelling study of surfactant evolution on a film surface coupled to lamellar film drainage has beencarried out. This study simulates the surfactant transport onto a lamella in a foam fractionation columnwith reflux: such columns offer greater control over enrichment and recovery of the surface activematerial. Insoluble surfactant is assumed during the simulation as such surfactants potentially derivemore benefit from a reflux system. There are two terms involved in the equation for surfactant flux whichare the Marangoni flow – from the Plateau border to the centre of the lamella – and the film drainage –

which is directed from the centre of the lamella to the Plateau border. The extent of film drainage isbounded using two extremal assumptions of mobile or rigid interfaces. On a mobile interface, the filmdrainage dominates the Marangoni effect, while on a rigid interface, the Marangoni effect is dominant.The numerical simulation was carried out using a material point method followed by a bookkeepingoperation to regrid the film. Analytical solutions for the case of no drainage and for a (quasi) steady statein which Marangoni flows and drainage are balanced were used to verify the numerical simulation. Fromthe simulations, it can be concluded that the film drainage obtained using surfactant with a mobileinterface is much faster than that determined using surfactant with a rigid interface, meaning thatsurfactant tends to be washed out of the film in the mobile case. The desirable condition in a foamfractionation column is however where the Marangoni flow dominates the liquid drainage which can beachieved when using surfactant that gives a rigid interface. The (quasi) steady state solution verifies thesimulation result at later time for the case of film with a rigid interface. An asymptotic solution in thecase of film with a mobile interface gives a prediction of the surface concentration of surfactant at thecentre of the film, but also predicts growing surfactant concentration gradients and growing Marangonistresses near the film's edge, which may then invalidate model assumptions.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Liquid flow in a foam film/lamella can be significantly influ-enced by the adsorption of surfactant onto the liquid–gas

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ces

Chemical Engineering Science

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ces.2013.08.041

n Corresponding author. Tel.: þ44 161 306 8851; fax: þ44 161 306 9321.E-mail address: paul.grassia@manchester.ac.uk (P. Grassia).

Chemical Engineering Science 102 (2013) 405–423

interface. The variations of surfactant concentration on the inter-face lead to Marangoni stresses that have a significant effect on thevelocity profile of the film drainage (Yeo et al., 2001). On the otherhand, the dynamics of surfactant evolution on the interface alongthe film is affected by convective flow and hence by film drainage(Kraynik, 1983). The evolution of surfactant on the interface of thefilm determines the efficiency of separation using a foam fractio-nation column utilised e.g. for purifying/enriching that surfactant(Martin et al., 2010). Therefore, a study of the surfactant evolutionon the surface of the film coupled to lamellar film drainage isimportant to understand the performance of a foam fractionationcolumn.

In a foam fractionation column, the foam formed usually hasa small fraction of liquid which is less than 5%, known as dry foam(Koehler et al., 2000; Xie et al., 2004). With this small amount ofliquid in the foam, the bubbles are present in polyhedral shapesdue to crowding (Brush and Davis, 2005). Having such polyhedralshapes, foam consists of thin films known as foam lamellae. Thethin films meet at Plateau borders, which form the edges of thepolyhedra (Weaire and Hutzler, 1999a). Liquid in foam lamellaedrains to Plateau borders due to the difference of liquid pressurebetween those two regions, where the Plateau border has a lowerpressure (Weaire and Hutzler, 1999b).

Some foam fractionation columns employ a reflux system toenhance the enrichment of the foamate (i.e. the fractionation producttaken from the top of the column), thereby improving the separationefficiency (Lemlich, 1968; Lemlich and Lavi, 1961; Martin et al., 2010;Stevenson and Jameson, 2007). In a foam fractionation column withreflux, a portion of the collapsed foamate is returned back to the topof the column. This collapsed foamate is richer than the risinginterstitial liquid in the column. When these streams mix, itenhances the adsorption onto the bubble surface, since surfactantin the reflux can be pulled onto a bubble surface by Marangonistresses. Therefore, a higher concentration of overhead product of thecolumn is achieved (Lemlich, 1968). Considering the advantages offoam fractionation column with reflux, this study models thesurfactant transport onto and liquid drainage from a lamella in afoam fractionation column with reflux. As we shall see, this surfac-tant transport involves a complex interplay of the Marangoni stressesarising due to reflux and the aforementioned film drainage flow.

This study is laid out as follows. Section 2 deals with themathematical model of surfactant transport and film drainage. Thissection describes the assumptions taken in the study as well as thegoverning equations of the system. Section 3 reports the dimen-sional analysis upon the governing equations. Section 4 considersthe analytical solution of the governing equations (where available).These analytical solutions are used to benchmark the numericalsimulation. Section 5 studies the numerical simulation employed tosolve the governing equations as well as to improve the accuracy ofthe results. Section 6 deals with the results and discussion, whileSection 7 describes the conclusions of the study.

2. Mathematical model of surfactant transport and filmdrainage

A schematic diagram of a two-dimensional slice through alamella is shown in Fig. 1. In this study, the interfaces between theliquid film and the air bubble are assumed to be symmetric(Breward and Howell, 2002). At the Plateau border, the curvatureof the gas–liquid interface causes lower pressure in the liquid. As aconsequence, there is a suction of liquid from the lamella to thePlateau border (Breward and Howell, 2002). The surface concen-tration of surfactant on the Plateau border interface is higher thanthe surface concentration of liquid on the lamella interface due tothe enrichment of surfactant concentration from the reflux stream.

Therefore, the interface at the Plateau border has a lower surfacetension than the interface on the lamella. Because of that gradientof surface tension, there is a Marangoni flow on the surface fromthe low surface tension to the higher surface tension.

2.1. Assumptions of the model

The model of lamella drainage developed in this study is basedon the following assumptions:

� At initial time the surface concentration of surfactant along thefilm is uniform.

� Since the thickness of the film is much smaller than the length ofthe film, the associated Reynolds number, utilising a length scalebased on the film thickness, is very small (Yeo et al., 2001) and thelubrication approximation is assumed (Ivanov, 1980; Ivanov andDimitrov, 1988; Kralchevsky et al., 1997; Karakashev et al., 2010;Radoev et al., 1974; Reynolds, 1886; Traykov and Ivanov, 1977)

� The liquid is assumed to be incompressible and Newtonian.� The surface concentration of surfactant on the interface of the

Plateau border is fixed at the value set by the reflux to the foamfractionation column.

� The changes of interfacial tension associated with dilationaland shear surface viscosity are negligible compared to thosedue to the Marangoni effect (Karakashev et al., 2010).

� The lamella film is taken to be flat and has a uniform thicknessalong its length (Stewart and Davis, 2012).This last condition is not always true as based on observation,most foam or emulsion films have non-uniform thickness. Indeedon purely geometric grounds, the longer a film, the greater theextent to which its thickness can vary, without incurring excessivecurvatures on the film surface. These lesser curvatures then limitthe pressure fields within the film that would otherwise tend toinduce fluid motions driving it back towards uniform thickness. Atypical film has a central part which is thicker than its periphery.This peripheral ‘dimple’ is surrounded by a thinner ‘barrier ring’(Joye et al., 1992). Moreover, Joye et al. (1994) implied that there isa hydrodynamical instability within the film that causes anasymmetrical drainage, resulting in non-uniformity of film thick-ness. This asymmetric drainage increases the rate of surfactantmass transport and film drainage of liquid. Moreover, Joye et al.(1992) described in their study that non-uniform dimples will notformwhen the radius of curvature of the Plateau border times thefilm thickness is much smaller than the square of film radius. Thisimplies that dimples are more likely on larger films. Despite theabove, modelling studies of Marangoni-induced flows in the flat,uniform thickness film limit remain useful (Grassia and Homsy,1998; Smith and Davis, 1983). They provide a convenient basecase – where we can study the main physical effects present in afractionation column with reflux, namely transport of surfactantfrom Plateau borders onto foam films due to the competitionbetween Marangoni process and film drainage – without the

γ

γ

Fig. 1. Two-dimensional slice of a lamella.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423406

geometric complexity of a non-flat, non-uniform thickness filmfor which sophisticated numerical treatments are required (Sayeand Sethian, 2013). Under certain special circumstances, i.e. filmswith extremely mobile surfaces, do actually remain flat (Howelland Stone, 2005). Alternatively, if hydrodynamic instabilities thatvary film thickness subsequently set in – then to the extent thatthey onset to enhance transport rates (Manev et al., 1997) – thenwe expect them to magnify the phenomena present in ourbase case.

2.2. Velocity profile across and along the film

The liquid drainage from a lamella to the Plateau border isgoverned by the continuity equation and the momentum balanceequation (see e.g. Gambaryan-Roisman, 2011). The governingequations are (see also Fig. 1):

∂u∂x

þ∂w∂z

¼ 0 ð1Þ

ρ∂u∂t

þu∂u∂x

þw∂u∂z

� �¼�∂p

∂xþμ

∂2u∂x2

þ∂2u∂z2

� �ð2Þ

ρ∂w∂t

þu∂w∂x

þw∂w∂z

� �¼�∂p

∂zþμ

∂2w∂x2

þ∂2w∂z2

� �ð3Þ

where u is the liquid velocity along the x-axis, w is the liquidvelocity along the z-axis, x is the distance from the centre of thelamella along the x-axis, z is the distance from the centre of thelamella along the z-axis, ρ is the liquid density, p is the liquidpressure, and μ is the viscosity of liquid.

Eq. (2) is simplified considering our assumption that the Rey-nolds number is very small, hence the flow field is obtained quasi-statically based on the instantaneous stress field. As a consequence,the left hand side of the equation is approximately zero. Anotherassumption is that ∂2u=∂x2 is negligible. This assumption is basedon the lubrication approximation arising from the thin geometry ofthe film. As an effect, ∂2u=∂x25∂2u=∂z2 making ∂2u=∂x2 negligible.Therefore, Eq. (2) can be presented as follows:

∂p∂x

¼ μ∂2u∂z2

: ð4Þ

Using the same assumption that Reynolds number is very small, Eq.(3) can also be simplified by setting the left hand side equals to zero.Due to the very thin geometry of the film, the second term on theright hand side of Eq. (3) is also negligible as w5u. Therefore, thesimplified form of the equation is

∂p∂z

¼ 0: ð5Þ

From Eq. (5), it can be seen that there is no pressure gradient acrossthe film.

The equation to determine the profile of liquid velocity acrossthe lamella is obtained by solving Eq. (4) applying the followingboundary conditions:

∂u∂z

� �����z ¼ 0

¼ 0 ð6Þ

and

μ∂u∂z

� �����z ¼ δ

¼ ∂γ∂x

ð7Þ

where δ is half of the lamella thickness and γ is the surfacetension. Using the assumption mentioned in Eq. (5), the equationof motion in Eq. (4) is solved using the boundary conditionsresulting in the following equation:

1δ∂γ∂xz¼ μ

∂u∂z: ð8Þ

Integration of Eq. (8) with respect to u and z results in:

u¼ 12μδ

∂γ∂xz2þB ð9Þ

where B is an integration constant (to be determined shortly).The thinning of the lamella can be described as (Breward and

Howell, 2002)

∂δ∂t

¼� u∂δ∂x

þδ∂u∂x

� �ð10Þ

where t is the time and u is the average liquid velocity across thelamella. Since the thickness of the lamella is assumed uniformalong the film, i.e. ∂δ=∂x¼ 0, Eq. (10) can be solved to determinethe (average) velocity of liquid in the lamella as follows:

u ¼�1δdδdt

x: ð11Þ

In the absence of surfactant, there can be no stress at the interface,therefore the velocity profile of liquid in the lamella would be plugflow. In the presence of surfactant, there are gradients of surfacetension on the interface. As a consequence, the velocity profile ofliquid in the lamella is determined by coupling Eqs. (9) and (11) toobtain the following equation:

u¼�xδdδdt

þ z2

2μδ∂γ∂x�1δ

Z δ

0

z2

2μδ∂γ∂x

dz: ð12Þ

Integration of the last part on the right hand side of Eq. (12) resultsin the equation for the velocity profile of liquid in the lamella asfollows:

u¼�xδdδdt

þ1μ

z2

2δ�δ6

� �∂γ∂x: ð13Þ

Using the equation for velocity profile, the surface velocity ofliquid can be determined where z¼ δ. The equation for surfacevelocity is as follows:

us ¼�xδdδdt

þ δ3μ

∂γ∂x

ð14Þ

where us denotes the liquid velocity at the interface. The surfacetension is influenced by the surface concentration of surfactant, themolar amount of surfactant present at any location on the surface.Therefore, the surface tension can also be expressed as a function ofthe surface concentration of surfactant. In this study, the surfacetension and the surface concentration of surfactant are assumed to betrying to attain some equilibrium value.1 Therefore, the relationbetween the surface tension and the surface concentration of surfac-tant is assumed to obey the following equation (Durand and Stone,2006):

dγd ln Γ

��G ð15Þ

where Γ is the surface concentration and G is a Gibbs parameter(which is assumed to be constant). Eq. (14) then becomes:

us ¼�xδdδdt

�Gδ3μ

∂ ln Γ∂x

: ð16Þ

In the domain sketched in Fig. 1, typically ∂ ln Γ=∂x40 since we havesurfactant rich reflux material in Plateau borders migrating onto films.Film drainage meanwhile implies dδ=dto0. Clearly (according to Eq.(16)) on the surface there is a competition between Gibbs–Marangoniflow and film drainage.

Knowledge of the film thinning rate dδ=dt is required to solveEq. (16). Note however that dδ=dt is a parameter that we must

1 Equilibrium for the purposes discussed here, can be taken to represent theassumed constant surface tension and constant surfactant surface concentrationassociated with reflux material in Plateau borders.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 407

input to this (uniform film thickness) model – it is not somethingwe can predict within the framework of our model – withoutdescribing how the film interacts with Plateau borders around itsedges (possibly considering also complex phenomena e.g. dimpling,non-uniformities of the film, as mentioned previously). Evolution offilm thickness and hence dδ=dt can however in principle bemeasured experimentally via a device such as Scheludko cell(Denkov et al., 1999; Politova et al., 2012; Scheludko, 1967) so anempirical dδ=dt formula could be obtained, even in the absence of adetailed model. Alternatively, we could employ theoretical esti-mates of the thinning rate sourced from the literature.

In what follows two approaches for estimating this thinning rateare discussed. One approach is based on the assumption of a mobileinterface, while the other approach is based on the assumption of arigid surface, using the so-called Reynolds equation. The detailedthinning rate equations are presented in the following subsections.

An important point to note is that the two approaches turn outto make completely different predictions regarding which effect(Gibbs–Marangoni vs film drainage) is the dominant contributionto the film surface velocity in the long time limit as the filmbecomes extremely thin. Thus it is conceptually useful to studyboth approaches, and to understand the different predictions thatthey make. An actual interface will be somewhere between thetwo extremes of completely mobile and completely rigid.

It is well known that (Denkov et al., 2009) certain surfactantsproduce ‘mobile’ bubble surfaces, while others produce ‘rigid’ bubblesurfaces. This does not however imply that one can universally adopt amobile film thinning equation for one type of surfactant and a rigidfilm thinning equation for another type. A nominally ‘rigid’ surfactantactually imparts rigidity to the surface by tending to set up aconcentration field of surfactant on the surface that producesMarangoni stresses which then cancel any surface flows. Such aconcentration field however requires time to set up – even anominally ‘rigid’ surfactant produces no Marangoni stresses at aparticular initial instant, if its initial concentration is perfectlyuniform. Likewise a nominally ‘mobile’ surfactant could potentiallylead to a significant reduction in some local surface velocity at somepoint on a film, if a local surfactant concentration gradient hap-pened to be large. Thus an important focus here will be upon how,over time, the predicted spatial distribution of surfactant concen-tration differs between the mobile and rigid film thinning models,and indeed whether that predicted distribution remains consistentwith those underlying models.

2.3. Thinning rate for a mobile interface

The rate of film thinning is determined based on the studies byBreward and Howell (2002) and Stewart and Davis (2012) (see alsoStewart and Davis, 2013). The dimensionless form of film thinningrate equation determined by Breward and Howell (2002) wasgiven as follows:

dhldt′

¼�QBH ¼�3ffiffiffi2

ph3=2l

16ð17Þ

where hl ¼ h=h0 is the dimensionless film thickness, h¼ 2δ is thedimensional film thickness, h0 is the initial film thickness, t′¼ ðU=LÞtis the dimensionless time, U ¼ ðγ=μÞ

ffiffiffiffiffiffiffiffiffiffih0=a

pis the velocity scale, L is

half length of the film, a is the radius of curvature of the Plateauborder, and QBH is the dimensionless liquid flux from lamella to thePlateau border determined by Breward and Howell (2002).

Stewart and Davis (2012) determined different forms of dimen-sionless thinning rate equations based on two distinct velocityscalings. We give all the scalings here – to demonstrate that theresulting formulae are consistent. Using the same velocity scalingas determined by Breward and Howell (2002), the dimensionless

thinning equation is as follows:

dhL

dt′¼�QSD0

¼� 316

2a′Ph3Lh′0

!1=2

ð18Þ

where hL ¼ h=L is an alternative dimensionless film thickness, QSD0

is the dimensionless liquid flux from the lamella to the Plateauborder determined by Stewart and Davis (2012), a′¼ a=L is thedimensionless radius of curvature of the Plateau border, P ¼ 1=a0is the difference between liquid pressure in the lamella and in thePlateau border that drives the capillary-viscous suction of liquidfrom the lamella, h′0 ¼ h0=L is the initial film thickness in dimen-sionless form. Yet another form of the thinning rate can be derivedby using the capillary velocity scale, where Uc ¼ γ=μ (in lieu of thescale U described previously), so that the dimensionless equationfor film thinning is as follows:

dhL

dt″¼�QSDc

¼� 316

ð2Ph3L Þ1=2 ð19Þ

where t″¼ ðUc=LÞt, QSDcis the dimensionless liquid flux derived

using the capillary velocity scale.Regardless which of the dimensionless forms we adopt, the

dimensionless form of the film thinning equations by Breward andHowell (2002) as well as by Stewart and Davis (2012) is thenconverted to the dimensional form using the relevant definitionsof the dimensionless variables. All three above equations result insingle dimensional equation as follows:

dhdt

¼�3ffiffiffi2

p

16γPbh

3=2

μLffiffiffia

p ð20Þ

where γPb is a representative surface tension taken here to be thatin the Plateau border. The value of γPb is chosen in Eq. (20) sincethe drainage flux of the liquid occurs between the lamella and thePlateau border.

In this study, half of the film thickness δ is used as thedependent variable instead of the full thickness. RedefiningEq. (20) results in the following equation:

dδdt

¼�38γPbδ

3=2

μLffiffiffia

p : ð21Þ

Eq. (21) is then applied to Eq. (16) for the simulation of surfactanttransport on the film surface in the case of mobile interface. Theequation for surface velocity in the case of mobile interface is thendescribed via the following equation:

us ¼ 38γPbx

ffiffiffiδ

p

μLffiffiffia

p �Gδ3μ

∂ ln Γ∂x

: ð22Þ

2.4. Thinning rate for a rigid interface

When the film has rigid interfaces, the drainage follows thetheory developed by Reynolds (1886) (see also Coons et al., 2003,2005). The thinning rate is determined by application of thelubrication approximation to the Navier–Stokes equation resultingin the Reynolds equation as follows:

dδdt

¼�δ3ΔP

3μL2ð23Þ

where ΔP is the excess pressure in the foam film as the drivingforce of the film drainage and can be expressed as ΔP ¼ Pc�Π,where Pc is the capillary pressure and Π is the disjoining pressure(Wang and Yoon, 2008). The capillary pressure for a foamwith lowliquid fraction where the films form polyhedral shapes is the ratiobetween the surface tension at the Plateau border and the radiusof curvature of the border (Coons et al., 2003) as presented in the

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423408

following equation:

Pc ¼γPba: ð24Þ

The disjoining pressure is primarily the sum of electrostatic andvan der Waals' contribution (Wang and Yoon, 2005, 2008) aspresented in the following equation:

Π ¼ΠelþΠvw ð25Þwhere Πel and Πvw are the electrostatic and van der Waals' terms,respectively. In the first instance, in this simulation, the film isassumed to be sufficiently thick, at least2 several tens of nano-metres, that the disjoining pressure is relatively very small com-pared with the capillary pressure (Weaire and Hutzler, 1999c). As aconsequence, ΔP can be determined solely from the capillarypressure: ΔP � Pc . The resulting formula for dδ=dt is as follows:

dδdt

¼�δ3γPb3μL2a

: ð26Þ

Using the assumption of a rigid film, the equation of surfacevelocity in Eq. (16) can be expressed as

us ¼xδ2γPb3μL2a

�Gδ3μ

∂ ln Γ∂x

: ð27Þ

The thinning rate predicted using the assumption of rigidinterface is much slower than the one predicted using the mobilecase. The following comparison of dδ=dt in Eqs. (21) and (26) forrigid and mobile cases, respectively, proves this explicitly

dδdt

� �rigid

dδdt

� �mobile

¼ δ3γPb3μL2a

� 83μL

ffiffiffia

p

γPbδ3=2 ¼

89δL

ffiffiffiδa

r:

,ð28Þ

Since δoa and δ 5 L, it is clear that the thinning rate of the rigidinterface case is always much slower than that predicted by themobile case.

Whether a given surfactant behaves as mobile or as rigid for thepurposes of starting to work with the model really depends onwhether, for the initial film thickness, the film drainage term inEq. (16) dominates the surface flow or whether the Marangoni termdominates. However, deciding this relies on knowing the (initial)dδ=dt value that is input to the model, which in turn depends onwhether the interface is mobile or rigid. If the film drainage stilldominates Marangoni assuming a conservative (i.e. rigid) model forthe drainage rate, then films are certainly mobile. Likewise ifMarangoni dominates drainage using a generous (i.e. mobile)estimate for drainage rate then films are certainly mobile. There ishowever an intermediate regime between these extremes, wherethe model could require empirical dδ=dt data obtained (as notedpreviously) experimentally via e.g. a Scheludko cell. Moreover, a filmdrainage model that starts off as being valid at initial time mightbecome invalid at later time if the spatial distribution of surfactantconcentration changes significantly (since we have already notedthat spatial distribution of surfactant can affect the choice of filmdrainage model).

2.5. Mass balance of surfactant on the interface

The foregoing analysis has concerned itself with determiningsimple (albeit plausible) fluid flow fields for a foam film. We nowadopt an approach similar to that of Ubal et al. (2010, 2011), i.e. we usethose flow fields within a mass balance equation. Specifically a surfac-tant mass balance using the assumption of insoluble surfactant isdeveloped. Using this assumption, the surfactant stays on the interface

once it is adsorbed.3 Therefore, there is no surfactant present withinthe bulk of film (i.e. surfactant diffusion is ignored). The surfactantmass balance for insoluble surfactant is presented as follows:

∂Γ∂t

þ ∂∂xðusΓÞ ¼ 0: ð29Þ

Implementing the formula to calculate us from Eq. (16) into thesurfactant mass balance results in the following equation:

∂Γ∂t

¼ ∂∂x

Γxδ

dδdt

þGδ3μ

∂Γ∂x

� �: ð30Þ

Rearranging Eq. (30) after differentiation of the first term on theright hand side results in:

∂Γ∂t

¼ x∂Γ∂x

þΓ� �

1δdδdt

þGδ3μ

∂2Γ∂x2

: ð31Þ

In the case of no film drainage, where the thickness of the filmδ remains constant with respect to time, Eq. (31) can be expressedas follows:

∂Γ∂t

¼ Gδ3μ

∂2Γ∂x2

: ð32Þ

Clearly Eq. (32) resembles a ‘diffusion’ equation where the ‘diffu-sion coefficient’ can be expressed as Ds ¼ Gδ=ð3μÞ.

Eq. (32) can be solved analytically to obtain the evolution ofsurfactant concentration on the interface by applying the correctinitial and boundary conditions:

Γðx;0Þ ¼ΓF0

δð0Þ ¼ δ0ΓðL; tÞ ¼ΓPb

usð0; tÞ ¼ 0 ð33Þwhere ΓF0 is the initial surface concentration of surfactant on thefilm interface, δ0 is the initial half film thickness with δ¼ δ0always in the case of Eq. (32), ΓPb is the surface concentration ofsurfactant on the Plateau border interface.4

A solution to Eqs. (30) and/or (31) can be obtained numericallyusing the material point method (Embley and Grassia, 2011). Themethod can be described as follows. First, the length of the film isdivided into a number of equal spatial intervals. Initially thesurface concentration of surfactant on the interface of the lamellais assumed to be uniform with the value of ΓF0. The surfaceconcentration of surfactant on the interface of the Plateau borderis fixed all the time with the value of ΓPb. The basic idea behind

2 This is corroborated by the fact that rigid films actually drain very slowly –

see later – and so do not have sufficient time in the fractionation column to achievea film thickness that gives double layer repulsion.

3 The case of a soluble surfactant is rather more complicated. Surfactant carriedfrom the Plateau border onto the film surface by the Marangoni surface flow cansubsequently diffuse (Miller et al., 2004; Ward and Tordai, 1946) away from thesurface into the film interior (Vitasari et al., in press) and then be convected by flowin the interior (Ubal et al., 2010). This leads to a complicated surfactant concentra-tion field in the film interior. Streamlines will tend to convect material near the filmsurface away from Plateau borders, but following such a streamline along, it willeventually penetrate deeper into the film itself and then change direction –

carrying flow outward towards the Plateau border again. Soluble surfactant willtherefore cause some of the reflux material (carried from Plateau border to film) toleak back out to the Plateau border. Soluble surfactants in fractionation therebypotentially derive less benefit from reflux than insoluble surfactants do. For thisreason we focus on the insoluble surfactant case here.

4 For simplicity, we make the assumption that the Plateau borders provide a largereservoir of surfactant, which means that ΓPb is assumed constant despite surfactantexchange with the films. This can be true even in the case of a nominally “insoluble”surfactant. “Insoluble” in this context means that on the thickness scale of the films,surfactant is predominantly present as surface excess – but in the substantially thickerPlateau borders, a significant amount of surfactant in solution could be present(provided the volume of liquid in the Plateau borders exceeds that in the films), whichcan replenish Plateau border surfaces maintaining them at surface concentration ΓPb .The case of what might be termed an ‘extremely insoluble’ surfactant is distincthowever. If the surfactant is so extremely insoluble that it is present mostly as surfaceexcess even in the Plateau borders, then reflux-enriched Plateau borders becomeexhausted very quickly, as soon as they start exchanging surfactant with films.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 409

the material point method, which conserves surfactant betweenneighbouring material points, is presented in Fig. 2. When theliquid on the surface moves, the position of xðjÞ (the jth materialpoint) changes owing to the surface velocity. Therefore, thedistance between two material points Δx also changes. The newvalue of surface concentration Γ on each spatial interval is thenevaluated using the new Δx.

3. Dimensional analysis of the equations

A dimensional analysis5 is carried out upon the equation ofsurface velocity. The half length is used as the characteristic lengthscale. The dimensionless form of the generic equation of surfacevelocity is as follows:

u′s ¼�x′

δ′dδ′dt′

� δ′3δ′0

∂ ln Γ′∂x′

ð34Þ

where u′s ¼ usμ=Gδ

′0 is the dimensionless surface velocity,

δ′0 ¼ δ0=L is the dimensionless initial half thickness of the film,δ′¼ δ=L is the dimensionless half thickness of the film, x′¼ x=L isthe dimensionless coordinate position on the film surface, L is halfof the length of the film, t′¼ ðGδ′0=LμÞt is the dimensionless time,Γ′¼Γ=ΓPb is the dimensionless surface excess of surfactant.The dimensional analysis can be carried out specifically for theequations governing both mobile and rigid surfaces. The detailsare described below.

A dimensional analysis for a mobile interface was carried outon Eq. (22) results in the following equation for surface velocity:

u′s ¼

1

δ′0

38x′

ffiffiffiffiffiδ′

p

Gffiffiffiffia′

p �δ′3∂ ln Γ′∂x′

!ð35Þ

where a′¼ a=L is the dimensionless curvature of the Plateauborder, a is the curvature of the Plateau border, G ¼ G=γPb is thedimensionless Gibbs parameter. The dimensionless form of thethinning equation is as follows:

dδ′dt′

¼�38

δ′3=2

δ′0Gffiffiffiffia′

p : ð36Þ

An analogous dimensional analysis is carried out in Eq. (27)applicable to the case of a film with rigid interfaces. The dimen-sionless equation is as follows:

u′s ¼

1

δ′0

x′δ′2

3Ga′�δ′3∂ ln Γ′∂x′

!: ð37Þ

The film thinning rate for the rigid interface is

dδ′dt′

¼� δ′3

3δ′0Ga′: ð38Þ

The dimensional analysis of the surfactant mass balance aspresented in Eq. (30) results in the following equation (applicableto either the mobile or the rigid case):

∂Γ′∂t′

¼ ∂∂x′

Γ′x′δ′

dδ′dt′

þ δ′3δ′0

∂Γ′∂x′

!: ð39Þ

Rearranging Eq. (39) results in the following equation:

∂Γ′∂t′

¼ x′∂Γ′∂x′

þΓ′� �

1δ′

dδ′dt′

þ δ′3δ′0

∂2Γ′∂x′2

: ð40Þ

The dimensionless initial and boundary conditions are thendescribed as follows:

Γ′ðx′;0Þ ¼Γ′F0

δ′ð0Þ ¼ δ′0Γ′ð1; t′Þ ¼ 1u′sð0; t′Þ ¼ 0 ð41Þ

where Γ′F0 ¼ΓF0=ΓPb is the initial dimensionless surface excess of

surfactant on the film interface.Using the boundary conditions given in Eq. (41), an analytical

solution for film thinning rate (δ′ as a function of t′) for a film witha mobile or rigid interface can be obtained by integrating Eqs. (36)and (38), respectively. The analytical solution for the thinning ratefor a mobile interface is presented in the following equation:

δ′¼ 1ffiffiffiffiffiδ′0

q þ 316

t′δ′0G

ffiffiffiffia′

p

0B@

1CA

�2

: ð42Þ

Meanwhile, the solution for thinning rate for film with a rigidinterface is as follows:

δ′¼ 1

δ′20þ23

t′δ′0Ga′

!�1=2

: ð43Þ

Therefore, the time to thin the film to a certain thickness can beestimated using either Eq. (42) or (43).

As alluded to previously, Eq. (42) implies much more rapidthinning than Eq. (43). Indeed (when converted back to dimen-sional form) Eq. (42) enables the film to thin so rapidly that thinfilm colloidal forces (i.e. disjoining forces) would become signifi-cant within the time scale that the film would reside in thefractionation column. We can model this by saying that at somecritical cut-off value of δ′¼ δ′cut�off (say with δ′cut�off orders ofmagnitude smaller than δ′0) thinning stops (Valkovska et al., 2002;Vrij, 1966), and δ′ remains constant at the value δ′cut�off . We shallreturn to this idea later on in the paper.

4. Analytical solution of the equation of surfactant transport

The equation of surfactant transport can be solved analyticallyin some special cases. Assuming that there is no film drainagefrom the lamella results in a relatively simple equation which ispossible to solve analytically. Another simplification can beobtained at (quasi) steady state conditions where the surfacevelocity vanishes ðu′

s ¼ 0Þ. The solution for u′s ¼ 0 is strictly speak-

ing a (quasi) steady solution rather than a true steady solutionsince dδ′=dt′ and δ′ are themselves functions of time. In yetanother case, in particular where the film drainage is very fast,hence the drainage dominates the Marangoni effect (exceptpossibly in a narrow boundary layer near the Plateau border),

Fig. 2. Schematic drawing for material point method on evolution of surfactant onlamella interface.

5 The scales employed in the dimensional analysis are different yet again fromthose adopted by Breward and Howell (2002) or Stewart and Davis (2012), becausewe consider a system where Marangoni effects are an essential part of the physics,which influences the choice of the scales.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423410

the surfactant transport can be modelled using an asymptoticsolution. The analytical solutions of the surfactant transportequation in those special cases are described in the followingsubsections.

4.1. Analytical solution in the case of no film drainage

In the case of no film drainage ðdδ′=dt′¼ 0Þ, the dimensionlessform of surfactant mass balance in Eq. (40) can be expressed as

∂Γ′∂t′

¼ δ′3δ′0

∂2Γ′∂x′2

¼ 13∂2Γ′∂x′2

ð44Þ

since δ′¼ δ′0 when dδ′=dt′¼ 0. Mathematically this is a simplediffusion equation. At early time, Eq. (44) can be solved analyti-cally using the following boundary conditions in lieu of those inEq. (41):

Γ′ð�1; tÞ ¼Γ′F0

Γ′ð1; t′Þ ¼ 1 ð45Þwhereas an approximation valid at early time we have shiftedone boundary condition to x′-�1. The solution for Γ′ can beobtained via standard techniques e.g. Laplace transforms, and theresulting solution is

Γ′¼Γ′F0þðΓ′

Pb�Γ′F0Þerfc

ffiffiffi3

pð1�x′Þ2ffiffiffiffit′

p" #

: ð46Þ

Due to the shift of the boundary condition assuming thatΓ′ð�1; tÞ ¼Γ′

F0, we obtain ∂Γ′=∂x′Z0 at x′¼ 0. However, inreality, symmetry demands that ∂Γ′=∂x′¼ 0 at x′¼ 0. Therefore, amethod of reflection is applied to Eq. (46) to correct the boundarycondition, resulting in this following equation:

Γ′¼Γ′F0þð1�Γ′

F0Þ erfc

ffiffiffi3

pð1�x′Þ2ffiffiffiffit′

p" #

þerfc

ffiffiffi3

pð1þx′Þ2ffiffiffiffit′

p" #( )

: ð47Þ

Eq. (47) again only applies at early times. As time progresses,Γ′ð1; tÞ would begin to exceed Γ′

Pb, as the reflection starts toviolate the boundary condition we have imposed. Additionalreflections could be added to correct this, as presented in thefollowing equation:

Γ′¼Γ′F0þð1�Γ′

F0Þ erfc

ffiffiffi3

pð1�x′Þ2ffiffiffiffit′

p" #

þerfc

ffiffiffi3

pð1þx′Þ2ffiffiffiffit′

p" #( )

�ð1�Γ′F0Þ erfc

ffiffiffi3

pð3�x′Þ2ffiffiffiffit′

p" #

þerfc

ffiffiffi3

pð3þx′Þ2ffiffiffiffit′

p" #( )

þð1�Γ′F0Þ erfc

ffiffiffi3

pð5�x′Þ2ffiffiffiffit′

p" #

þerfc

ffiffiffi3

pð5þx′Þ2ffiffiffiffit′

p" #( )

� … ð48ÞThe more terms that are included in this series, the more accurateit becomes at later times.

Another method of solving Eq. (40) analytically is using themethod of separation of variable. This method uses the assump-tion that Γ′¼ Xðx′ÞTðt′Þ, and solves the differential equations forX and T separately in terms of x′ and t′. The boundary conditionsused in this separation of variable method are as follows:

Γ′ðx;0Þ ¼Γ′F0

Γ′ð1; t′Þ ¼Γ′Pb ð49Þ

and the analytical solution is a Fourier series as presented in thefollowing equation:

Γ′¼Γ′PbþðΓ′

Pb�Γ′F0Þ ∑

1

b ¼ 1

4bπ

sinbπ2

� �exp � bπ

2ffiffiffi3

p� �2

t′

" #cos

bπ2x′

� �ð50Þ

where b¼1, 3, 5, …. Eq. (50) applies for any time. However, at earlytime, a large number of terms for the Fourier series is needed andformulae such as Eqs. (46) or (47) are more convenient. Therefore, inthis simulation, the Fourier series is applied for the profile of surfaceconcentration of surfactant on the lamella interface at late times only.

The solutions discussed above assume no film drainage.Another analytical solution can be obtained in the presence offilm drainage at the (quasi) steady state of the surfactant transportwhere u′

s ¼ 0, which implies that film drainage exactly balancesthe Marangoni effect. The (quasi) steady state solution is describedin the following subsection.

4.2. Analytical solution at (quasi) steady state

At (quasi) steady state, where u′s ¼ 0 (i.e. no surfactant flux on

the surface) the distribution of Γ′ vs x′ can be solved analytically inthe presence of film drainage. Applying the (quasi) steady state toEq. (34) results in the following differential equation:

3δ′0x′δ′2

dδ′dt′

¼�∂ ln Γ′∂x′

: ð51Þ

Integration of Eq. (51) results in the following equation:

ln Γ′¼ ð1�x′2Þ2

3δ′0δ′2

dδ′dt′

: ð52Þ

From Eq. (52) it can be seen that having a large value of dδ′=dt′,small δ′ and/or large δ′0 will give a large departure from a uniformsurfactant distribution, i.e. a large departure from Γ′¼ 1. This isphysically reasonable since large dδ′=dt′ means large film drainagerate – hence there is more surfactant swept to the direction of thePlateau border – therefore less surfactant transported onto thefilm from the Plateau border. Small δ′ and/or comparatively largeδ′0 together mean that there is a great deal of scope for Marangoniflow to weaken over time as the film thins because viscous effectsin the subphase become more significant. Likewise small δ′ alsoaccelerates film drainage velocity for a given dδ′=dt′ as flux mustescape through a narrower film. These effects present as a 1=δ′2

factor in Eq. (52).The values of Γ′ and x′ from Eq. (52) would instantaneously

give u′s ¼ 0 but would not lead to a true steady state since δ′ and

dδ′=dt′ change with time. The true steady state for Γ′ is onlyachieved if dδ′=dt′p�δ′2 as can be seen from Eq. (52). Neithermobile nor rigid interfaces satisfy that condition. With amobile interface we have dδ′=dt′p�δ′3=2 where the film drainsmore rapidly than in the “steady state” case. On the other hand,the rigid case drains more slowly with dδ′=dt′p�δ′3. It ishowever interesting to speculate whether the numerical solu-tions might approach a quasistatic limit in which Γ′ vs x′satisfies the above formulae at the instantaneous value of δ′.To achieve this, we expect that time required to attain quasi-steady state (estimated as δ′0=δ′ based on the first term and lastterm of Eq. (36)) should be less than6 the time for which δ′remains in neighbourhood of its present value δ′=ðdδ′=dt′Þ,therefore δ′0=δ′5δ′=ðdδ′=dt′Þ hence dδ′=dt′5δ′2=δ′0. It shouldbe possible to achieve this for a rigid case, but is more difficult toachieve for a mobile film. The prediction that the quasisteady stateshould be readily achievable in the case of a surfactant with a rigidinterface offers a posteriori support for the choice of a Reynoldsequation-based film drainage model (Eq. (38)), since the quasisteadystate obeys u′

s ¼ 0, which is precisely the assumption employed bythat film drainage model.

6 A somewhat analogous system has been considered by Baldessari et al.(2007) during which evolution dynamics take place on time scales less than thefilm drainage time.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 411

The (quasi) static surfactant concentration distributions for thetwo extremal cases of mobile and rigid interfaces are readilydetermined from the equation of surface velocities as follows.

The (quasi) static equation for a rigid interface is as follows:

x′δ′Ga′

¼ ∂ ln Γ′∂x′

ð53Þ

and the integral of Eq. (53) is as follows:

ln Γ′¼�ð1�x′2Þδ′2G a′

: ð54Þ

In the case of surfactant with a rigid interface, non-uniformsurfactant distribution is more likely to occur for small G (sincesharp surfactant concentration gradients are needed to compen-sate for weak Gibbs elasticity) and small a′ (since capillary suctiondriving film drainage is then stronger). Usually however the non-uniformities in surfactant concentration predicted by Eq. (54) areweak, because δ′ is necessarily smaller than δ′0 which itself is avery small parameter. Over time, these non-uniformities decayeven further as δ′ decays.

The analogous formula for a mobile case is as follows:

ln Γ′¼� 916

ð1�x′2ÞGffiffiffiffiffiffiffiffiδ′a′

p : ð55Þ

This exhibits similar trends with respect to G and a′ as the rigidcase does – but the trend with respect to δ′ is different. Non-

uniformities can start off quite large since ln Γ′ scales like 1=ffiffiffiffiffiδ′

p

which is 1=ffiffiffiffiffiδ′0

q⪢1 initially. The non-uniformities7 then grow even

more as δ′ decays. It is however questionable whether this mobilefilm quasistatic solution has any physical relevance as the timerequired to set it up will be much longer than the time for whichthe film thickness remains in the neighbourhood of any particularδ′. We will return to this point shortly.

The value of spatially averaged surface excess of surfactant,⟨Γ′⟩¼ R 1

0 Γ′ dx is an important physical quantity that representsthe amount of surfactant on the film. For a (quasi) steady conditionthis can be estimated using Eqs. (54) and (55), for a rigid interfaceand for a mobile interface respectively. The value of ⟨Γ′⟩ for filmwith a rigid interface is as follows:

⟨Γ′⟩¼

ffiffiffiffiffiffi2π

perf 1

2i2δ′Ga′

� 1=2� �

2i δ′Ga′

� 1=2exp δ′

2Ga′

� ð56Þ

and the equation for film with a mobile interface is as follows:

⟨Γ′⟩¼2ffiffiffiffiπ

perf 3

4i1

Gffiffiffiffiffiffiδ′a′

p� 1=2� �

3i 1Gffiffiffiffiffiffiδ′a′

p� 1=2

exp 916G

ffiffiffiffiffiffiδ′a′

p� ð57Þ

Since the formula for the change of δ′ with t′ is known, it ispossible to obtain a prediction of ⟨Γ′⟩ vs t′ using Eqs. (56) and (57)for films with a rigid interface and with a mobile interface,respectively.

4.3. Asymptotic solution for the case of film with a mobile interface

We have already alluded to the fact that the quasisteadyformula may be less relevant to the mobile film case, because filmthicknesses may change more quickly than the quasisteady statesurfactant distribution (corresponding to any instantaneous thick-ness) can be established. Indeed while a mobile film starts to thin,

the velocity contribution from film drainage may well dominatethat from the Marangoni effect. A solution with Marangoni termsdiscarded altogether however cannot satisfy boundary condi-tions at both the centre of the film and the Plateau border. It isnecessary to consider a boundary layer near the Plateau borderwhich matches the inner region near the film centre (where theMarangoni terms are negligible) with the outer region near thePlateau border (where Marangoni terms are retained). The narrowerthe boundary layer, the larger the Marangoni terms associated withit, so it is also important to keep track of whether Marangoni stressesmight reach levels sufficiently large as to invalidate the film drainagemodel (Eq. (36)) which was derived (Breward and Howell, 2002)ignoring such stresses altogether. The development of a boundarylayer solution is discussed in this subsection.

The equation of surfactant mass balance as presented inEq. (39) is as follows:

∂Γ′∂t′

¼ x′∂Γ′∂x′

þΓ′� �

1δ′

dδ′dt′

þ δ′3δ′0

∂2Γ′∂x′2

:

If the Marangoni term is neglected far from the Plateau border,the above equation admits a solution for which ∂Γ′=∂x′¼ 0. As aconsequence, far from the Plateau border we have the followingcondition:

∂Γ′∂t′

¼Γ′δ′

dδ′dt′

: ð58Þ

Integration of Eq. (58) from time 0 to t′ and surfactant coverageand Γ′

F0 to Γ′ results in the following solution:

Γ′innerðt′Þ ¼

δ′ðt′Þδ′0

Γ′F0 ð59Þ

where Γ′inner denotes Γ′ far from the Plateau border. From Eq. (59)

we have a good approximation to the value of Γ′inner as long as the

value of δ′ðt′Þ is known. Eq. (59) has a very simple geometricinterpretation whereby, as the film thins and stretches, thesurfactant molecules on its surface move further apart.

Eq. (59) clearly does not satisfy the boundary condition atx′¼ 1, so an adjustment must be made over a narrow boundarylayer to ensure that this condition is satisfied. Indeed the surfaceconcentration at the end of the film is fixed, therefore at x′¼ 1 wehave ∂Γ′=∂t′¼ 0. Assuming we can treat ∂Γ′=∂t′ as roughly zeroover the entire boundary layer (i.e. surfactant does not accumulatethere), as a consequence, from Eq. (31) we obtain the followingequation:

Γ′δ′

dδ′dt′

þ δ′3δ′0

∂Γ′∂x′

��C ð60Þ

where C is a constant that represents the surfactant flux from filmto Plateau border. Applying the boundary condition at the centreof film, where at x′¼ 0 we have ∂Γ′=∂x′¼ 0, the value of C can bedetermined as follows:

Γ′inner

δ′dδ′dt′

��C: ð61Þ

Therefore, the asymptotic equation can be determined as follows:

½Γ′inner�Γ′�1

δ′dδ′dt′

� δ′3δ′0

∂Γ′∂x′

: ð62Þ

As this asymptotic behaviour applies to the case of a film with amobile interface, therefore we substitute the drainage equation inEq. (36) into Eq. (62) which results in the following equation:

½Γ′�Γ′inner�

9

8Gffiffiffiffiffiffiffiffiδ′a′

p � ∂Γ′∂x′

: ð63Þ

Integration of Eq. (63) results in an asymptotic solution for thesurfactant evolution in the case of a film with a mobile interface as

7 If non-uniformities in surfactant distribution (and the stresses they generate)become sufficiently large, moreover, the assumption of a mobile film also tends tobe invalid.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423412

follows:

Γ′�Γ′innerþ½1�Γ′

inner�exp�9ð1�x′Þ8G

ffiffiffiffiffiffiffiffiδ′a′

p� �

: ð64Þ

The spatially averaged surfactant concentration on the filmsurface can be estimated using the following equation:

⟨Γ′⟩¼Γ′innerþ½1�Γ′

inner�89G

ffiffiffiffiffiffiffiffiδ′a′

p�exp � 9

8Gffiffiffiffiffiffiffiffiδ′a′

p� �

þ1� �

; ð65Þ

but we should also note that the dimensionless Marangoni stressat the end of the film is (using (Eq. 63))

�Gd ln Γ′dx′

����x′ ¼ 1

�� 9

8ffiffiffiffiffiffiffiffiδ′a′

p ½1�Γ′inner� ð66Þ

which typically will be larger than the capillary suction pressureinto the Plateau border (1=a′ in dimensionless form). This theninvalidates the assumption that Marangoni stresses can beneglected in deriving the film drainage model (which was the basisfor Eq. (36)). Physically what is happening here is that compara-tively fast film drainage is driving a tendency for the surfactantconcentration gradient to sharpen into a boundary layer. Howeverlong before it sharpens down to a boundary layer of the size (oforder G

ffiffiffiffiffiffiffiffiδ′a′

p) predicted by Eq. (64), Marangoni stresses become

significant, which will then feedback upon the film drainage rate.The exceedingly low levels of surfactant coverage away from theboundary layer predicted by Eq. (59) coupled to Eq. (42) need notthen be realised.

To date we have described analytical solutions in certain specialcases – e.g. no film drainage, no surfactant flux on the interfaceand a boundary layer solution. To consider more general cases weturn (in the next section) to a numerical scheme. Obviouslyhowever, the analytical special cases provide useful benchmarksfor the numerical scheme.

5. Numerical simulation of surfactant transport on a lamellainterface

Here we describe the numerical scheme – which is similar to ascheme already discussed in the literature (Embley and Grassia,2011), but with minor modifications to account for the boundaryconditions imposed where a film meets a surfactant rich Plateauborder in a foam fractionation system with reflux.

5.1. Material point method

The surface velocity equations are discretised into N time stepsand Jþ1 spatial points (i.e. J spatial intervals) to obtain a numericalsolution for the evolution of the surface concentration of surfac-tant on the lamella interface. Our numerical method tracks themotion of the Jþ1 spatial points treated as material points(Embley and Grassia, 2011). The accuracy of the numerical simula-tion is improved using a predictor–corrector method. In thissimulation the predictor–corrector method used is Heun's method(Loney, 2001; Press et al., 2007). This method improves uponEuler's method using the principle of Taylor expansion. Heun'smethod for a material point location x′ can be written as follows:

x′ðj;nÞ ¼ x′ðj;n�1Þ þdx′dt′

� �mΔt′

¼ x′ðj;n�1Þ þk12þk2

2

� �Δt′ ð67Þ

where Δt′ denotes the time step, the subscript m on the righthand side denotes that the derivative follows a material point, k1and k2 are the rate of change of the predictor and the corrector,

respectively, and are defined as follows:

k1 ¼dx′dt′

� �m

ðx′ðj;n�1Þ ; t′ðn�1ÞÞ ¼ u′

sðj;n�1Þ���

k2 ¼dx′dt′

� �m

ðx′ðj;n�1Þ þu′sðj;n�1ÞΔt′; t′ðn�1Þ þΔt′Þ ¼ u′

sðx′ðj;n�1Þ þu′sðj;n�1ÞΔt′; t′ðn�1Þ þΔt′Þ:

����Eq. (67) applies for a time step n and a spatial point j wherej¼ 0;1;2;…; J. Once x′ðj;nÞ is available, Γ′

ðj;nÞ follows via surfactantmass conservation

Γ′ðj;nÞ ¼

Γ′ðj;n�1Þðx′ðj;n�1Þ�x′ðj�1;n�1ÞÞ

x′ðj;nÞ�x′ðj�1;nÞ; j¼ 1;2;3;…; J ð68Þ

Velocities are now evaluated as

u′sðj;nÞ ¼

1

δ′0

38

x′ðj;nÞ

ffiffiffiffiffiffiffiffiδ′ðnÞ

qGffiffiffiffia′

p �δ′ðnÞ3

ln Γ′ðjþ1;nÞ�ln Γ′

ðj;nÞ½x′ðjþ1;nÞ�x′ðj�1;nÞ�=2

0@

1A; j¼ 1;2;3;…; J

ð69Þfor the mobile interface, and

u′sðj;nÞ ¼

1

δ′0

x′ðj;nÞδ′2ðnÞ

3Ga′�δ

′ðnÞ3

ln Γ′ðjþ1;nÞ�ln Γ′

ðj;nÞ½x′ðjþ1;nÞ�x′ðj�1;nÞ�=2

!; j¼ 1;2;3;…; J

ð70Þfor the rigid interface, where n¼ 1;2;3;…;N.

The above equations need to be solved with suitable initialconditions which can be described as follows:

x′ðj;0Þ ¼ jΔx′; j¼ 0;1;2;…; J

Γ′ðj;0Þ ¼Γ′

F0; jr J

u′sðj;0Þ ¼ 0; j¼ 1;2;3;…; J�1

u′sðJ;0Þ ¼

1

δ′0

38

ffiffiffiffiffiδ′0

qGffiffiffiffia′

p �δ′0

3ln 1�ln Γ′

F0

Δx′=2

0@

1A ðmobile interfaceÞ

u′sðJ;0Þ ¼

1

δ′0

δ′203Ga′

�δ′0

3ln 1�ln Γ′

F0

Δx′=2

!ðrigid interfaceÞ

whereΔx′ is the distance between two initial points at initial time.Meanwhile, the boundary conditions are as follows:

x′ðJ;nÞ ¼ 1

u′sð0;nÞ ¼ 0

Γ′ðJþ1;nÞ ¼ 1

where n¼ 0;1;2;…;N.The material point method is illustrated in Fig. 2. Note in

particular that the surface velocity at point J (adjacent to thePlateau border) is a physically important quantity as it controlssurfactant flux from Plateau border to the film. The space betweenmaterial intervals J and Jþ1 is taken to be Δx′=2. If a hypotheticalpoint Jþ1 in the Plateau border is added infinitesimally to theright hand side of J, then the value of Γ′ corresponding to thatinfinitesimal interval is Γ′

Pb ¼ 1: the equations are then formallythe same as Eqs. (69) and (70).

5.2. Bookkeeping operation

On every time step, the material point positions change,leading to an uneven spatial interval over time. To restore eveninterval spacing, a bookkeeping operation is carried out on everytime step. The mechanism of the bookkeeping operation is illu-strated in Fig. 3.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 413

As the interval ðΔx′Þ is restored, the value of Γ′ is corrected.The new values of x′ very simply are calculated as follows:

x′newðj;nÞ ¼ jΔx′; j¼ 0;1;2;…; J ð71Þ

The bookkeeping operation proceeds by identifying whichof the ‘old’ intervals (pre-bookkeeping) overlap with the ‘new’

intervals (post-bookkeeping) – and then takes a weighted averageover the surfactant concentration in the old intervals to obtain thatin the new intervals. There are three cases to consider. One case iswhere the material points on both ends of an interval move to theleft (away from the Plateau border) as illustrated in Fig. 3a. Thebookkeeping equation for this case is as follows:

Γ′newðj;nÞ ¼

Γ′oldðjþ1;nÞ½x′newðj;nÞ �x′oldðj;nÞ�þΓ′old

ðj;nÞ½x′oldðj;nÞ�x′newðj�1;nÞ�x′newðj;nÞ �x′newðj�1;nÞ

ð72Þ

where as alluded to above the superscripts old and new denote thevalue of Γ′ and x′ before and after the bookkeeping operationrespectively. The new value of Γ′ and x′ is then applied for thecomputation using the material point method for the nexttime step.

Another case is where the material point on the right end of aninterval moves leftward (away from the Plateau border), but thematerial point on the left end of the interval moves rightward

towards the Plateau border as illustrated in Fig. 3b. The equationfor this bookkeeping operation is as follows:

Γ′newðj;nÞ ¼

Γ′oldðjþ1;nÞ½x′newðj;nÞ �x′oldðj;nÞ�þΓ′old

ðj;nÞ½x′oldðj;nÞ�x′oldðj�1;nÞ�þΓ′oldðj�1;nÞ½x′oldðj�1;nÞ�x′newðj�1;nÞ�

x′newðj;nÞ �x′newðj�1;nÞ:

ð73Þ

The third case is where the material points on both ends of aninterval are moving rightward towards the Plateau border asillustrated in Fig. 3c. The equation for the bookkeeping operationfor this case is as follows:

Γ′newðj;nÞ ¼

Γ′oldðj;nÞ½x′newðj;nÞ �x′oldðj�1;nÞ�þΓ′old

ðj�1;nÞ½x′oldðj�1;nÞ�x′newðj�1;nÞ�x′newðj;nÞ �x′newðj�1;nÞ

: ð74Þ

The case where the right hand material point moves right(towards the Plateau border) and the left hand material pointmoves left (away from the Plateau border) is trivial – in that caseΓ′new (post-bookkeeping) inherits the value of Γ′old (pre-bookkeeping).

The interval J is a special case of the formulae given above sincethe surfactant concentration on the surface of Plateau border isfixed, therefore Γ′old

ðJþ1;nÞ ¼ 1.

Fig. 3. Illustration of the bookkeeping operation with three different cases, during a time step: (a) ‘old’ material points on both ends of an interval move to the left, (b) ‘old’material point on the right of an interval moves leftward while ‘old’material point on the left end moves rightward, (c) ‘old’material points on both ends of an interval moveto the right. The ‘new’ material points are placed in such a way as to ‘undo’ this motion.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423414

6. Results and discussion

A simulation of the evolution of surfactant surface concentra-tion on a lamella surface was carried out. The parameters for thesimulation are presented in Table 1, while dimensionless para-meters are presented in Table 2. All parameter values except forthe values of ΓPb, ΓF0, δcut�off and a were taken from the study byDurand and Stone (2006) using sodium dodecyl sulfate (SDS) assurfactant at bulk concentration of 4.8 g L�1 . The values of ΓPb

and ΓF0 in Table 1 were selected to be lower than the maximumsurface excess for SDS ðΓmax ¼ 10� 10�6 mol m�2Þ based on thestudy by Chang et al. (1992). The value of δcut�off was taken fromWeaire and Hutzler (1999c). The selected value of radius ofcurvature of the Plateau border a could be shown (based onformulae given by Weaire and Hutzler, 1999b for a typical cellshape of a so-called Kelvin cell, Weaire and Hutzler, 1999d) tocorrespond to a well defined liquid fraction of 0.17% within thefoam fractionation column, so a comparatively dry foam. Thisliquid fraction estimate does however only consider liquid con-tained in films (so the true liquid fraction of the foam would be alittle higher).

It is instructive to compare estimates of the amount of liquid infilms vs liquid in Plateau borders for the parameter values givenabove. Assuming that the shape of the film is hexagonal with sidelength L and thickness δ, the volume of the film, V film can bedetermined using the following equation:

V film ¼ 3ffiffiffi3

pL2δ: ð75Þ

The length of the Plateau border can be identified as the half-lamellar length L (as sketched in Fig. 1), since a hexagon of sidelength L can be drawn in a circumscribing circle of radius L.Meanwhile, the cross-sectional area of the Plateau border APb is asfollows:

APb ¼ ðffiffiffi3

p�π=2Þa2 ð76Þ

and the volume of the Plateau border VPb is as follows:

VPb ¼ ðffiffiffi3

p�π=2Þa2L: ð77Þ

Since each hexagonal film would have six Plateau borders, buteach Plateau border is shared between three films, therefore eachfilm can ‘claim’ two Plateau borders. As a consequence the ratio ofliquid in the films and in the Plateau borders is as follows:

V film

VPb� 3

ffiffiffi3

pL2δ

2ðffiffiffi3

p�π=2Þa2L

� 16:11Lδa2

� 6:44 ð78Þ

where the values of L, a and initial δ (i.e. δ0) have been taken fromTable 1.

According to our initial condition, the ratio of the amount of liquidin the film to that in the Plateau border is 6.44, corresponding (inrough order of magnitude terms) to comparable amounts of liquid inthe Plateau borders and in the films. Interestingly we have been ableto estimate that a film should attain a thickness comparable to thevalue of δ0 as chosen by Durand and Stone (2006) within the typicaltime that it would take to form a bubble in a fractionation column(at least for the case of a rigid film where film thinning is compara-tively slow). Thus the choice for δ0 made by Durand and Stone (2006)seems reasonable, and is employed here.

Over time however the balance of liquid will certainly shifttowards the Plateau borders. The time that a bubble film resides ina flotation column is substantially more than the time to form thebubble in the first place, giving films ample opportunity to drain,i.e. become thinner than δ0 under the action of capillary suction(Kraynik, 1983).

The film drainage process can in principle change the value ofthe Plateau border radius of curvature a, and thereby change thecapillary pressure driving film drainage. In the interests of simpli-city however we ignore this complication and treat a as a constant.Note that the value of a only scales as the square root of theliquid volume in the Plateau borders, so a given relative changein Plateau border volume implies a lesser relative change in a.

Table 1Parameters for simulation of surfactant transport onto a foam lamella taken from Durand and Stone(2006), Chang et al. (1992), and Weaire and Hutzler (1999c).

Parameters Symbol Value Unit

Characteristic ‘Marangoni’ time scale L2μ=ðGδ0Þ 3.125�10�2 s

(equivalent to one unit of dimensionless time)Characteristic thinning time scale δ0=jdδ=dtj0 1.48�10�3 s(mobile interface)Characteristic thinning time scale δ0=jdδ=dtj0 2.08 s(rigid interface)Initial half lamella thickness δ0 20�10�6 mHalf thickness of common black film δcut�off 15�10�9 mHalf lamella length L 5�10�3 mLiquid viscosity μ 1�10�3 Pa sSurfactant surface excess at the Plateau border ΓPb 2�10�6 mol m�2

Initial surfactant surface excess at the lamella ΓF0 1�10�6 mol m�2

Radius of curvature of the Plateau border a 5�10�4 mSurface tension at the Plateau border γPb 45�10�3 N m�1

Gibbs parameter G 40�10�3 N m�1

Table 2Dimensionless parameters used in the simulation of surfactant transport onto a foam lamella.

Parameters Symbol Definition Value

Dimensionless initial film thickness δ′0 δ0=L 4�10�3

Dimensionless half thickness of common black film δ′cut�off δcut�off=L 3�10�6

Dimensionless radius of curvature of PB a′ a/L 0.1Dimensionless Gibbs parameter G G=γPb 0.89

Dimensionless initial surfactant surface excess Γ′F0 ΓF0=ΓPb 0.5

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 415

Transient increases in a moreover tend to enhance foam drainage(i.e. liquid transport out of the foam through the network ofPlateau borders, Grassia et al., 2001; Neethling et al., 2000; Verbistet al., 1996; Weaire and Hutzler, 1999e) thereby restoring a backtowards its former value (prior to having received liquid from thefilms): thus substantial amounts of liquid drained from films toPlateau borders could simply escape from the foam altogether bytraversing the Plateau border network (Kraynik, 1983). An addi-tional complication is that liquid arriving from the films mightdilute the surfactant in the reflux-enriched Plateau borders.However, since (in the interests of simplicity) our model treatsthe Plateau borders as a reservoir of surfactant, such a dilutioneffect will not be captured here.

Film drainage can (in principle, given adequate time) reduce filmthickness from δ0 (given as 20 μm) down to a level δcut�off , at whichcolloidal forces (e.g. disjoining forces) prevent any further filmdrainage. This could correspond as in Table 1 to a common blackfilm with a typical half thickness of 15 nm (Weaire and Hutzler,1999c). Given that δcut�off 5δ0, it is evident that liquid will beoverwhelmingly in the Plateau borders once the films have drainedsufficiently. In fact, as soon as liquid is overwhelmingly in the borders,even if film thickness is still many times greater than δcut�off , filmdrainage no longer has any significant effect upon Plateau border radiiand/or in diluting Plateau border surfactant, and the simplifyingassumptions discussed above would become valid.

To summarise, using the parameter values in Tables 1 and 2 andemploying the simplifying assumptions detailed above, the mate-rial point method combined with the bookkeeping operation areemployed to obtain the numerical solution for the equation ofsurfactant evolution on the film surface. First of all, as a bench-mark, the numerical simulation in the case without film drainagewill be verified using the analytical solution at the same condition.

6.1. Verification of the numerical simulation using the analyticalsolution of the case without film drainage

The first simulation is based on the assumption of no filmdrainage, therefore the thickness of the lamella is fixed and hencedδ′=dt′¼ 0. The result of this simulation based on an assumptionof fixed thickness of the lamella is presented in Fig. 4. Using thisassumption, the velocity of liquid on the lamella surface isdetermined wholly by the Marangoni stress.

As one would expect, the surface concentration of surfactant onthe lamella increases with time. Since the surface concentration onthe Plateau border is higher than the surface concentration on thefilm, there is flow on the surface in the direction away from thePlateau border. As a consequence, some surfactant is transferred tothe film from the Plateau border.

The simulation result presented in Fig. 4 corresponding todimensionless spatial interval Δx′¼ 1:25� 10�2 and time stepΔt′¼ 3:125� 10�5 was verified using the analytical solution of theequation of mass transfer. At very early time ðt′¼ 0:05Þ, the resultof the numerical simulation was verified using the sum ofequations obtained via a reflection technique and presented inEq. (47). At this early time, one reflection is sufficient since thereflection will not introduce any significant error into the value ofΓ′ at x′¼ 1. The comparison between the solution obtained fromthe reflection technique and the numerical simulation result ispresented in Fig. 5. The result from the numerical simulationshows a good agreement with the analytical solution. Althoughnot shown in the graph in Fig. 5, the analytical formula using thecomplementary error function without reflection and that usingmultiple reflections show an essentially identical solution withthat obtained via the reflection technique at this very early time.Therefore, it can be concluded that the numerical simulation is

reliable to model the surfactant transport onto the foam lamella atearly times.

At later time t′¼ 2, the result of the simulation is also verifiedusing analytical solution. At this later time, the solution obtainedusing the Fourier series in Eq. (50) should be employed to verifythe simulation result. Fourier series is a convenient description atlate time, as it converges after a few terms of the series, unlike thecase of early time. By contrast, the complementary error functionsolution at later time needs many reflections which will compli-cate the description.

The results of numerical simulation at t′¼ 2 were verified usinganalytical solution obtained using the Fourier series out to 11terms. The simulations were carried out at various numbers ofspatial grid points to investigate the effect of number of spatialgrid points upon the truncation error. The effect of the number oftime steps upon the truncation error is not presented in this studysince the time step used in these tests was checked to besufficiently small as to have no influence on the error (i.e. to besmaller than the error arising from the spatial discretisation). Thenumerical simulations using various time steps at equal number ofspatial grid points thus show essentially identical results. It wasfound that more spatial grid points reduces the average error aspresented in Table 3. The analytical solution and the simulationresults using various numbers of spatial grid points are presentedin Fig. 6. The more spatial grid points utilised in the simulationdoes however normally demand finer time increments (Presset al., 2007). Therefore, in this simulation the number of time

Fig. 4. Evolution of dimensionless surfactant surface concentration Γ′ along thefilm length x′ in the absence of film drainage. The labels on the lines represent thedimensionless time t′.

Fig. 5. Verification of the result of numerical simulation at early time usingsolution of the reflection technique to the complementary error function att′¼ 0:05.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423416

steps was increased whenever a greater number of spatial gridpoints was utilised. In the simulations to follow we usedΔt′¼ 0:2ðΔx′Þ2.

The amount of surfactant transported onto the film surfaceincreases with time (in the absence of film drainage) due to theMarangoni effect. The change of the spatial average of dimension-less surfactant surface concentration on the film surface ⟨Γ′⟩ withdimensionless time t′ is presented in Fig. 7 based on our chosenΓ′

F0 value, initially ⟨Γ′⟩¼ 0:5. In that figure, the time was extendedto 5 units to show the value of ⟨Γ′⟩ near the final equilibrium. Thedimensionless time to reach ⟨Γ′⟩¼ 0:9 is t′¼ 1:66 which corre-sponds to dimensional time t¼0.05 s and to achieve ⟨Γ′⟩¼ 0:99 ittakes t′¼ 4:47 which corresponds to t¼0.14 s. Those times aremuch shorter than the typical residence time in a foam fractiona-tion column at about 12 s as reported in the study by Martin et al.(2010). Therefore, from this simulation, surfactant on the filmsurface achieves equilibrium with surfactant on the Plateau bordersurface at very short time in the absence of film drainage. In thepresence of film drainage however, the Marangoni effect has tocompete with the film drainage, therefore the surfactant transportonto the film surface is expected to slow down.

The case of surfactant transport with film drainage is toocomplicated to solve analytically. Nevertheless, considering thatthe numerical solution has been verified by the analytical solutionin the case without film drainage, the case with film drainageshould be able to be solved numerically using the same methodwith similar numerical time steps and numbers of grid elementsas reported above. The following subsection discusses the numer-ical solution of the surfactant transport involving film drainage.

6.2. Numerical simulation of the case with film drainage

In this simulation, there are, as we have stated, two approachesto predict the rate of film drainage. One uses an assumption of amobile film surface, and the other uses the assumption of a rigidfilm surface. Therefore, some notion of the degree of surface

mobility needs to be available in order to select the appropriatefilm drainage equation. The choice of surfactant determines thesurface mobility as studied by Golemanov et al. (2008), although(as alluded to at the end of Sections 2.2 and 4.3) one must alsomonitor the spatial and temporal evolution of the predictedsurfactant concentration to check that the assumed film drainagemodels continue to remain appropriate.

The predicted drainage rate of film with a mobile interface isvery fast. Without any evaporation, film thinning is limited to afinal film thickness of the order of 30 nm ðδcut�off ¼ 15 nmÞ to forma film known as a common black film (Weaire and Hutzler, 1999c).The film does not thin further due to electrostatic repulsion withinit (Valkovska et al., 2002; Vrij, 1966; Weaire and Hutzler, 1999c).In this simulation with a mobile interface, the film reaches thethickness of common black film at t′¼ 3:6, corresponding todimensional time t¼0.1 s, which is much shorter than the resi-dence time in the foam fractionation column at t′¼ 383, whichcorresponds to dimensional time about t¼12 s. After reaching thethickness of common black film, the lamella does not thin furtherdue to electrostatic repulsion between the layers (assuming ofcourse sufficient surfactant still remains to stabilise the film). Weincorporate this repulsion in an approximate fashion as a sharpcut-off of the thinning rate at a specified δ′ taken as 3�10�6 (seeTable 2), although in reality the decay of the drainage rate wouldbe spread over a range of δ′ values: one could model thatcomparatively easily via so-called double layer theory, with anelectrostatic disjoining pressure which would start to grow as soonas film thickness fell to within a few Debye lengths (Bergeron,1999) of the cut-off thickness (since the electrostatic interactionbetween surfaces or particles has an exponential behaviour with acharacteristic length equal to the Debye length, Israelachvili, 1985),and which would exactly balance the Plateau border capillarysuction pressure precisely when that cut-off thickness wasachieved. The predicted thinning of a film with a mobile interfaceis presented in Fig. 8. The time scale is taken to t′¼ 400, close tothe residence time inside a foam fractionation column.

On the other hand, a film with a rigid interface drains orders ofmagnitudes slower than one with a mobile interface. Reachingδcut�off takes dimensionless time t′¼ 5:93� 107 which correspondsto t¼1.85�106 s. Therefore, the film with a rigid interface will notreach the thickness of common black film during the residence timein the foam fractionation column. The thinning of film with a rigidinterface is presented in Fig. 9. Note that, even though the film fails toapproach anything near a common black film within the residencetime available, significant thinning does in fact take place. It is a poorapproximation to treat the film as having a fixed thickness. This

Table 3Average error of the value of Γ′ along the film withvarious numbers of spatial grid points evaluated fortime t′¼ 2.

Number of spatial gridelements

Average error(%)

20 0.5880 0.15

140 0.08

Fig. 6. Verification of the result of numerical simulation at a comparatively latetime using the Fourier series for various numerical spatial discretisations at t′¼ 2.

Fig. 7. Profile of spatially averaged surfactant surface concentration ⟨Γ′⟩ vs time t′obtained from numerical simulation in the special case of no film drainage.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 417

tallies with Table 1, which reports a characteristic thinning time scalethat is rather less than the residence time.

The choice of film thinning rate equation, impacts on the surfactantconcentration. When the assumption of a mobile interface is utilised,the film thinning rate follows the equation developed by Brewardand Howell (2002). The result of the consequent simulation for Γ′ ispresented in Fig. 10. In this simulation, drainage is driving liquidrapidly out of the film. On the surface, however there can be aMarangoni force that causes the liquid to flow in the direction awayfrom the Plateau border. If the Marangoni flow were to dominate thefilm drainage, surfactant would accumulate on the surface. However,in this simulation, the film drainage (initially) dominates theMarangoni flow. Therefore, surfactant will be washed away from thesurface. As surfactant is washed away, the surface concentration ofsurfactant on the film surface becomes lower. According to the modelpredictions, Marangoni stresses are unimportant away from the filmedges, but near the film edges they might retain importance. One wayto see that Marangoni stresses are indeed relevant is by reference tothe arguments of Section 4.3, which suggest that even though Γ′changes rapidly in the neighbourhood of the edge of the film, thesurfactant flux (which is the product u′

sΓ′) does not. Hence sharp risesinΓ′ approaching the film edge, imply sharp falls in u′

s, showing that aMarangoni contribution to the film surface velocity is opposing acontribution from the film drainage flux. Nevertheless, the argumentsof Section 4.3 also indicate that the Marangoni stresses are now so

large in this region, that the formula utilised for film drainage rate(and hence for film drainage surfactant flux) will be unreliable.

The film drains and thins however only until it reaches thethickness of a common black film. At this thickness, there is nofurther thinning, therefore only the Marangoni effect takes placethereafter and this determines the subsequent surfactant trans-port. As a result, surfactant begins to accumulate on the filmsurface again, albeit starting from a rather low concentration level.However, since the film is very thin, the Marangoni effect is weakand the surfactant accumulation is much slower than that whichoccurs on a substantially thicker film such as in the case withoutfilm drainage presented in Fig. 4. Remember however that here weswitch off the film drainage abruptly at a certain cut-off thickness.Were we to reduce the rate of film drainage rate more gradually(by allowing a film disjoining pressure to build up with falling filmthickness) then we could potentially begin transferring surfactantback onto the film at a slightly larger thickness, where the masstransfer rate would be likewise slightly larger.

The following simulation result presented in Fig. 11 is based onthe assumption of a rigid interface. The thinning rate of the filmthen follows the Reynolds equation. Comparing Eq. (36) withEq. (38), the thinning rate determined using the assumption of arigid interface is much slower than the thinning rate using theassumption of a mobile interface. As a consequence, the Marangoniflow dominates the film drainage, therefore, surfactant accumulateson the film surface right from the beginning. At any given timehowever there is less surfactant on the surface than there wouldhave been in the absence of film drainage. Although it is difficultto distinguish on the scale of the graph, from Fig. 4, Γ′ðx′¼ 0;t′¼ 2Þ ¼ 0:87 in the case without film drainage. This value is higherthan Γ′ðx′¼ 0; t′¼ 2Þ ¼ 0:86 in the case of film drainage in a rigidfilm as presented in Fig. 11.

One of the interesting results to obtain from the simulation ofsurfactant transport onto a foam lamella is the net accumulation ofsurfactant on the film surface with time. Fig. 12 presents thespatially averaged surfactant concentration on the film surface ⟨Γ′⟩with time t′ for films without drainage, and for a draining filmwith a mobile interface and that with a rigid interface. In the caseof a film without drainage and a draining film with a rigidinterface, since the Marangoni effect dominates, the amount ofsurfactant on the film surface increases with time and reaches afinal equilibrium at a much shorter time than the residence time inthe foam column. On the other hand, the amount of surfactant onthe surface of the film with a mobile interface decreases with timeuntil a particular time and subsequently increases with time due

0.000001

0.00001

0.0001

0.001

0.01 0.1 1 10 100

δ’

t’

Fig. 8. Change of dimensionless film thickness δ′ with dimensionless time t′ in thecase of a mobile interface.

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.01 0.1 1 10 100

δ’

t’

Fig. 9. Change of dimensionless film thickness δ′ with dimensionless time t′ in thecase of a rigid interface.

Fig. 10. Evolution predicted for dimensionless surfactant surface concentration Γ′along the film length x′ with film drainage using the assumption of a mobileinterface. Labels on the lines represent the dimensionless time t′. The inset iszoomed in within the spatial range 0.95–1 spatial unit.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423418

to the absence of any further film drainage. With a mobileinterface, as long as the thickness of film is above the thicknessof a common black film, the drainage is very fast and dominatesthe Marangoni effect. As a consequence, the concentration ofsurfactant on the film surface decreases with time (although inreality the decrease is unlikely to be anywhere near as dramatic asFig. 12 suggests, because Marangoni stresses will feed back to thefilm thinning rate, reducing the rate at which surfactant can bewashed off the film). In our model, the subsequent increase of theamount of surfactant on the surface of a film with a mobileinterface only occurs when there is no further film drainage oncethe film is as thin as a common black film. Since the film is verythin, the Marangoni effect is however weak, and the accumulationof surfactant is quite slow. Indeed the surfactant coverage has noteven managed to regain its initial value by around 400 time units(which as we have said is roughly the typical residence time in thecolumn).

6.3. Approach of the surfactant transport to (quasi) steady state

One of the interesting questions for the numerical simulation ofsurfactant transport is whether it approaches the analytical solu-tion at (quasi) steady state. The equations for (quasi) steady state

for a film with a mobile and with a rigid interface in Eqs. (55) and(54) respectively are obtained assuming u′

s ¼ 0. As pointed outearlier, this is not a true steady state since δ′ and dδ′=dt′ changewith time. This subsection discusses the comparison between thesimulation results and the (quasi) static solution at various times,both for films with a rigid and with a mobile interface. Distribu-tions of surfactant surface concentration along the film werecalculated using the (quasi) static equations for both models andthe results were tested to see if they agreed with the numericalsolutions.

The (quasi) static solution and the simulation of surfactanttransport on a filmwith a rigid interface is presented in Fig. 13. The(quasi) steady state distribution of surfactant surface concentra-tion along the film surface changes only very slowly with time. Theagreement between the (quasi) static solution and the numericalsimulation only occurs at later times. At early time, the numericalsolution does not agree with the (quasi) static solution. It can beseen in Eq. (54) that the surface concentration profile along thefilm from the (quasi) static solution will only evolve with time dueto the change of δ′, as the values of G and a′ are taken as fixed. The(spatial) variation of the (quasi) static Γ′ is on the order of δ′=Ga′which is invariably a small parameter, even initially when δ′¼ δ′0.Moreover, since the film thickness does not change rapidly withtime, therefore the profile of surfactant surface concentrationpredicted from a (quasi) static solution does not change greatlywith time either.

Thus agreement between the numerical and (quasi) staticsolution is then only possible at reasonably long times: since the(quasi) steady solution has weak spatial variation in Γ′, it isnecessary to wait for the numerical solution to have similarlyweak spatial variation before attempting a comparison. The(spatially averaged) accumulation of surfactant with time obtainedfrom numerical simulation and from the (quasi) static solution ispresented in Fig. 14. The solution approaches the quasistatic onearound t′¼ 10 time units.

The simulation results for a film with a mobile interface do notagree at all well with the (quasi) steady state solution. Thevariation of Γ′ across the film is controlled in the (quasi) steadycase by the parameter ðG

ffiffiffiffiffiffiffiffiδ′a′

p�1 and this parameter is large even

at the initial film thickness. Thus the (quasi) steady approachwould predict barely any surfactant on the film. Moreover, asalluded to previously, δ′ is now changing very quickly that it variessignificantly on the time scale that would be required to establisha quasisteady profile in the first place. A potentially betterapproximation is to consider the picture discussed in Section 4.3,

Fig. 11. Evolution of dimensionless surfactant surface concentration Γ′ along thefilm length x′with film drainage using the assumption of a rigid interface. Labels onthe lines represent the dimensionless time t′.

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100

⟨Γ’⟩

t’

no drainagemobile

rigid

Fig. 12. Predictions for spatially averaged dimensionless surfactant surface con-centration ⟨Γ′⟩ vs dimensionless time t′ on the surface of films in the case of no filmdrainage as well as with drainage assuming films with a mobile and with a rigidinterface.

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Γ’

x’

analytical t’ = 0.2numerical t’ = 0.2analytical t’= 2.0numerical t’ = 2.0analytical t’ = 10.0numerical t’ = 10.0

Fig. 13. Comparison of (quasi) steady state solution and results of simulation atvarious dimensionless times for a film with a rigid interface ðt′¼ 0:2;2 and 10Þ.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 419

with an inner region of uniform surfactant concentration matchedon to a boundary layer near the Plateau border: this is discussed inthe next subsection.

6.4. Asymptotic boundary-layer solution for the case of mobileinterface

The simulation result is compared with the asymptotic solution inEq. (64). The profile of surfactant surface concentration obtainedfrom numerical simulation and from asymptotic solution is pre-sented in Fig. 15. The asymptotic solution represents the conditionwhere the film drainage tends to dominate the Marangoni effect, dueto the film thinning very rapidly. As a consequence, the asymptoticsolution has not been designed to accommodate the accumulation ofsurfactant on the film surface once the film stops thinning, so werestrict our comparison to times before this event occurs.

The shape of the concentration profile obtained from simula-tion results and from asymptotic solution is presented in Fig. 16.In the interest of simplicity, only several simulation data arecompared to the asymptotic solution. Those data are taken at‘early’, ‘intermediate’ and ‘late’ time (t′¼ 0:02;0:2 and 2 respec-tively). In the graph, the horizontal axis was taken to be 1�x′ tofocus on the neighbourhood of the Plateau border.

We used 120 numerical spatial grid intervals along the lengthof the full film, but since most of the variation in Fig. 16 is confinedto the interval 1�x′o0:05, the spatial variation across this intervaleffectively utilised only several grid points. Since the film drainagerate is very fast, surfactant is washed away towards the Plateauborder. On the other hand, the gradient of surfactant surfaceconcentration between the film surface and the surface of thePlateau border causes a boundary layer near the Plateau border.That boundary layer narrows gradually over time as the film thins.The Marangoni stresses in that increasingly narrow layer actuallyrise, and although not accounted for in the model results pre-sented below, could call into question the film thinning rateequation used which ignores them.

The asymptotic solution gives a good approximation to thevalue of Γ′ at the centre of the film x′¼ 0 as predicted by thenumerical simulation, as well as to the values of Γ′ within theboundary layer. Any small discrepancy between the value of Γ′obtained from the numerical simulation and from the asymptoticsolution at x′¼ 0 is likely to be due to truncation error of thenumerical simulation (given the comparatively small number ofgrid points over which the variation of Γ′ is realised).

It is noteworthy that the early time decrease of δ′ for the case ofa film with a mobile interface as presented in Fig. 8 is reflected in

the predictions of spatially averaged ⟨Γ′⟩ in Fig. 12. This is partlydue to the prediction in Eq. (59) that Γ′

inner is proportional to δ′.Nevertheless, the magnitude of the decrease in ⟨Γ′⟩ is lessdramatic than that of δ′ as ⟨Γ′⟩ accounts for surfactant retainedin the boundary layer in addition to any surfactant that is in theinner region as can be seen in Eq. (65).

6.5. Growth of surfactant coverage for a common black film

In the mobile case, once a common black film is attained (at apredicted time t′ around 3.6 as mentioned previously), surfactantstarts to return to the film as Fig. 15 shows.

Once the common black film is attained (as mentioned inSection 6.2), Eq. (44) describes the evolution of surfactant cover-age. The solution takes a form similar to Eq. (50) but with verylittle surfactant initially present (Γ′

F051, surfactant having beenfor the most part already been washed away) and with time t′rescaled by the factor δ′0=δ

′cut�off (which takes the value 1333

according to Table 2). The solution will then take a form similar tothe ‘no drainage’ case shown in Fig. 12, but evolving 1333 timesmore slowly. Remembering (as stated previously) that a residencetime in the fractionation column has been estimated at 383 units,it is apparent that the surfactant coverage on the film is unable toequilibrate with that on the Plateau border in the residence time

0.01

0.1

0.01 0.1 1 10

1 -⟨Γ

’⟩

t’

(quasi) steadynumerical

Fig. 14. Profile of surfactant surface concentration with time for film with a rigidinterface obtained from (quasi) static solution and numerical simulation.

0

0.1

0.2

0.3

0.4

0.5

0.01 0.1 1 10

⟨Γ’⟩

t’

asymptoticnumerical

Fig. 15. Predicted profile of surfactant surface concentration vs time on a film witha mobile interface obtained from numerical simulation and asymptotic solution.

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

Γ’

1 - x’

t’

t’

t’

t’

t’

t’

t’ = 0.02

t’ = 0.2

t’ = 2

Fig. 16. Comparison of asymptotic solution and results of numerical simulation atvarious dimensionless times ðt′¼ 0:02;0:2 and 2Þ. The numerical simulation resultsare represented by dashed lines while the asymptotic solutions are shown bysolid lines.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423420

available (as Fig. 12 makes evident). The situation could potentiallybe improved with a surfactant exhibiting a stronger electrostaticrepulsion (which might increase δ′cut�off and thereby decrease theratio δ′0=δ

′cut�off ) and/or by increasing residence times (e.g. by

changing column height).

7. Conclusions

Simulation of transport of surfactant onto a foam lamella insidea foam fractionation column with reflux has been carried out. Theparameters used in the simulation were taken, for the most partfrom a study by Durand and Stone (2006). Our simulation uses anassumption of insoluble surfactant, as such surfactants potentiallyderive more benefit from reflux. The simulation studies thetransport of surfactant onto a flat, planar foam lamella, althoughreal lamellae may curve and exhibit thickness variations: suchcomplications become more likely for bigger lamellae, and mayenhance surfactant transport over and above our predictions. Thesimulation considers two possible models of film drainage, whichare distinguished based on the mobility of the film surface. Thosetwo models of film drainage are that for a mobile interface andthat for a rigid interface. Which model is appropriate for a givensurfactant is dependent on system parameter values (e.g. thedimensionless Gibbs elasticity) and the dimensionless Plateauborder radius (itself related to liquid fraction), and may even besensitive to the evolving spatial distribution of surfactant along thefilm surface. However, definitive selection of a particular filmdrainage model is strictly speaking outside the framework pre-sented here, and experimental/empirical thinning data could beemployed if available. The simulation was also carried out for thecase without film drainage – which gives a fixed film thickness –

and the results were verified using analytical solutions which wereobtained from complementary error function and Fourier series.This study also reports the comparison of the simulation resultswith the (quasi) steady state solution for both models for rigid andmobile interfaces. Only the rigid interface results fit well with(quasi) steady solution. An alternative asymptotic boundary layermodel for surfactant transport on a film with a mobile interfacehowever verified the simulation result.

There are two terms involved in the transport of surfactant onto afoam lamella. Those two terms are film drainage, the direction ofwhich is towards the Plateau border, and Marangoni flow, thedirection of which is towards the centre of the film. The film drainagetends to wash surfactant away to the Plateau border whereas theMarangoni flow pulls the surfactant onto the film surface. The nettransport of surfactant from the surface of the Plateau border to thesurface of the foam lamella depends on the interaction betweenthose two flows. In the case of no film drainage, the surfactanttransport is wholly determined by the Marangoni effect. Therefore,surfactant accumulates on the film surface with time.

The verification of the simulation results, using analytical solu-tions in the case where film drainage does not occur, has demon-strated that the numerical simulation is properly benchmarked forestimating the transport of surfactant onto a foam lamella. It wasalso found, as expected, that a greater number of spatial grid pointsgives a better accuracy of the simulation. However, having a largenumber of spatial grid points requires a significantly larger numberof time increments. As a consequence, there is a trade off betweenthe accuracy of the numerical simulation and the cost in terms ofsimulation time.

The film drainage of a lamella with a mobile interface is muchfaster than that of a lamella with a rigid interface. As a consequence,in the case of a lamella with a mobile interface, the film drainagedominates the Marangoni flow. As a result, the liquid flow within thefilm washes away the surfactant on the film surface towards the

Plateau border. On the other hand, for a filmwith a rigid interface, thedrainage of liquid from the film is very slow, therefore the Marangonieffect dominates the film drainage. This results in accumulation ofsurfactant on the film surface. The desirable condition for a process ina foam fractionation column is where the Marangoni effect dominatesthe film drainage for a better separation effectiveness when surfactantattaches to the foam lamellae. Therefore, the selection of surfactantthat gives films with a rigid interface is prudent.

Our findings suggest that a system with a mobile interface couldonly potentially be used to good effect if the residence time in thefractionation column is sufficiently long to enable the film to draindown to a common black film at which point drainage stops. At thispoint, surfactant which initially washed off the film is pulled backonto it. Nevertheless, the time required to return surfactant back tothe film may be rather longer than the available residence time in atypical fractionation column. Although the process of surfactantremoval followed by surfactant return seems to be typical for mobilefilms, there are numerous ways in which the calculations presentedhere could be improved. According to the predictions, sharpsurfactant concentration gradients develop near the edges of films,which lead to Marangoni stresses that are likely to reduce the filmdrainage rates significantly. Therefore, surfactant will not be washedoff mobile films to anywhere near the same extent as predicted, andmay also start to be pulled back onto them even before the commonblack film thickness is achieved. Growing disjoining pressures, asthe films thin towards the common black state, also suppress filmdrainage rates, with similar consequences for mass transfer asdescribed above. The rigid system meanwhile is unlikely to reacha common black film in a reasonable residence time inside a foamfractionation column – but this is not problematic as surfactant isnever predicted to be washed off it in the first place.

Simple analytical formulae are available (quasisteady state and/or asymptotic boundary layer solution) which may be useful forestimating surfactant coverage in engineering design calculations ofa foam fractionation column. The (quasi) steady state approach toestimate surfactant coverage in the case of a film with a rigidinterface only confirms the simulation results at late time where thedrainage rate is already extremely slow and the Marangoni effect isweak due to a thinner film and weak surfactant concentrationdifferences along it: the quasisteady solution does however corre-spond to vanishing net velocities on the film surface, so conformswith the conventional notion of a ‘rigid’ system. The (quasi) steadystate solution is not useful for describing the case of a film with amobile interface, as film thickness then changes more rapidly thansurfactant can equilibrate at any given thickness, with the resultthat (quasi) steady solution merely predicts an exceedingly smallsurfactant surface concentration. The distribution of surfactantconcentration along the film surface in the case of a film with amobile interface can however be modelled using an asymptoticboundary layer equation. This asymptotic solution predicts not onlythe value of surfactant surface concentration at the centre of thefilm, but also the boundary layer near the Plateau border (althoughthe large Marangoni stresses predicted in that layer do beg thequestion of whether the film actually remains mobile).

In conclusion, it is worth recalling that the work we havepresented here goes some way to understanding the operation of afoam fractionation column, but is still not a complete picture ofthat process. We have modelled a key process in fractionation,namely the transport processes (driven by Marangoni effects and/or film drainage) occurring on a single film in contact with aPlateau border that contains surfactant-rich reflux. We are ableto predict the amount of surfactant transport onto the film as afunction of the time of contact between the film and the Plateauborder, a result which could be used in principle to estimatesurfactant coverage of foam film given bubble residence time in afractionation column.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 421

Nevertheless, there are numerous assumptions in our modelwhich could influence the validity such estimates: one strongassumption is that the film remains of uniform thickness. A realfilm, particularly one that is fairly large, could develop non-uniformities in thickness, potentially as a way of enhancingtransport processes. Another assumption here is that the Plateauborder acts as an inexhaustible surfactant reservoir (from thepoint of view of the film) and also an inexhaustible liquid sink(into which liquid can drain). In reality one must perform asurfactant mass balance between Plateau border and film, so thatborders gradually become depleted in surfactant as it transfersonto films: this aspect needs to be built into the film-leveltransport processes that we have modelled here. If there aresignificant volumes of liquid retained initially in the films (com-pared to the liquid volumes in the Plateau border) it is alsopossible that liquid draining from the films may dilute the Plateauborder surfactant yet further. Moreover, since the films betweenbubbles and the liquid in Plateau borders tend to move through afractionation column with different velocities, a bubble is not inconstant contact with just one element of Plateau border fluid:rather it sees a sequence of different fluid elements, each with apotentially different surfactant concentration.8

Where and how a bubble film collects most surfactant from thePlateau borders depends in part on how the column is operated(e.g. flux of bubble films relative to liquid flux in the Plateauborders, and whether reflux is added as a batch or continuously).In batch mode, the films at the top of the column have first accessto the surfactant-rich reflux fluid. In continuous mode however,the surfactant-lean films at the bottom of the column may benefitmost: if films low down in the column manage to achieve nearequilibrium with the borders, comparatively little mass transferneeds to take place higher up, with the result that surfactant-richPlateau border liquid penetrates deep down into the column. Thepoints raised above are all important engineering questions (as yetunanswered) needed for developing a comprehensive model ofthe design and operation of a foam fractionation column withreflux. The process that we have considered here (i.e. Marangoniand/or film-drainage driven surfactant transport onto or off anindividual foam film) nevertheless provides an essential ingredientof any such comprehensive model.

Nomenclature

a radius of curvature of a Plateau border (m)a′ dimensionless radius of curvature of a Plateau borderAPb cross-sectional area of the Plateau border (m2)G Gibbs parameter (N m�1)G dimensionless Gibbs parameterh film thickness (m)h0 initial film thickness (m)hL dimensionless film thickness based on half lamella

lengthhl dimensionless film thickness based on initial film

thickness

h′0

dimensionless initial film thickness based on halflamella length

L half lamella length (m)p liquid pressure (N m�2)ΔP excess pressure in a foam film (N m�2)P difference between dimensionless liquid pressure in

the lamella and Plateau borderPc capillary pressure (N m�2)QBH dimensionless liquid flux from lamella to the Plateau

border determined by Breward and Howell (2002)QSD0

dimensionless liquid flux from lamella to the Plateauborder determined by Stewart and Davis (2012)

QSDcdimensionless liquid flux from lamella to the Plateauborder using the capillary velocity scale determined byStewart and Davis (2012)

t time (s)t′ dimensionless timet″ dimensionless time (based on an alternative scaling)u liquid velocity along x axis (m s�1)us liquid surface velocity (m s�1)u′s dimensionless liquid surface velocity

u average liquid velocity across a lamella (m s�1)U velocity scale for dimensional analysis of the film

drainage model (m s�1)Uc capillary velocity scale (m s�1)V film volume of film (m3)VPb volume of the Plateau border (m3)w liquid velocity along the z-axis (m s�1)x distance from the centre of a lamella along the x-axis

(m)x′ dimensionless distance from the centre of a lamella

along the x-axisz distance from the centre of a lamella along the z-axis

(m)δ half of lamella thickness (m)δ′ dimensionless half lamella thicknessδ0 initial half lamella thickness (m)

δ′0 dimensionless initial half lamella thickness

δcut�off cut-off half lamella thickness at which drainage ceases(m)

δ′cut�offdimensionless cut-off half lamella thickness at whichdrainage ceases

γ surface tension (N m�1)γPb surface tension at the Plateau border (N m�1)Γ surface excess of surfactant (mol m�2)ΓF0 initial surfactant surface excess on the film (mol m�2)ΓPb surfactant surface excess on the Plateau border

(mol m�2)Γ′ dimensionless surfactant surface excess

Γ′inner

dimensionless surfactant surface excess towards thecentre of the film

Γ′F0

dimensionless initial surfactant surface excess on thefilm

Γ′Pb

dimensionless surfactant surface excess on the Plateauborder

μ liquid viscosity (Pa s)Π disjoining pressure (N m�2)Πel electrostatic force per area (N m�2)Πvw van der Waals' force per area (N m�2)ρ liquid density (kg m�3)

Acknowledgements

Denny Vitasari would like to give thanks for support providedby Directorate General of Higher Education Republic of Indonesiaand Universitas Muhammadiyah Surakarta Indonesia.

8 Relative motion between bubbles and Plateau border liquid tends to mitigatethe effect of liquid draining into Plateau borders and diluting any (reflux-enriched)surfactant there. Film drainage is most significant towards the bottom of thecolumn where the bubbles and films are freshly formed, so this is where the mostsignificant dilution effects are liable to occur. Liquid in the diluted Plateau borderswill however flow downwards and thus away from the bubbles and hence will notcontact bubbles any further.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423422

Part of this work also was carried out while Paul Grassia was aRoyal Academy of Engineering/Leverhulme Trust Senior ResearchFellow and funding from the fellowship is gratefully acknowledged.

References

Baldessari, F., Homsy, G.M., Leal, L.G., 2007. Linear stability of a draining filmsqueezed between two approaching droplets. Journal of Colloid and InterfaceScience 307, 188–202.

Bergeron, V., 1999. Forces and structure in thin liquid soap films. Journal of Physics:Condensed Matter 11, R215–R238.

Breward, C.J.W., Howell, P.D., 2002. The drainage of a foam lamella. Journal of FluidMechanics 458, 379–406.

Brush, L.N., Davis, S.H., 2005. A new law of thinning in foam dynamics. Journal ofFluid Mechanics 534, 227–236.

Chang, C.H., Wang, N.H.L., Franses, E.I., 1992. Adsorption dynamics of single andbinary surfactants at the air/water interface. Colloids and Surfaces 62, 321–332.

Coons, J.E., Halley, P.J., McGlashan, S.A., Tran-Cong, T., 2003. A review of drainageand spontaneous rupture in free standing thin films with tangentially immobileinterfaces. Advances in Colloid and Interface Science 105, 3–62.

Coons, J.E., Halley, P.J., McGlashan, S.A., Tran-Cong, T., 2005. Bounding film drainagein common thin films. Colloids and Surfaces A: Physicochemical and Engineer-ing Aspects 263, 197–204.

Denkov, N.D., Cooper, P., Martin, J.Y., 1999. Mechanisms of action of mixed solid–liquid antifoams. 1. Dynamics of foam film rupture. Langmuir 15, 8514–8529.

Denkov, N.D., Tcholakova, S., Golemanov, K., Ananthpadmanabhan, K.P., Lips, A.,2009. The role of surfactant type and bubble surface mobility in foam rheology.Soft Matter 5, 3389–3408.

Durand, M., Stone, H.A., 2006. Relaxation time of the topological T1 process in atwo-dimensional foam. Physical Review Letters 97, 226101.

Embley, B., Grassia, P., 2011. Viscous froth simulations with surfactant mass transferand Marangoni effects: deviations from Plateau's rules. Colloids and Surfaces A:Physicochemical and Engineering Aspects 382, 8–17.

Gambaryan-Roisman, T., 2011. Dynamics of free liquid films during formation ofpolymer foams. Colloids and Surfaces A: Physicochemical and EngineeringAspects 382, 113–117.

Golemanov, K., Denkov, N.D., Tcholakova, S., Vethamuthu, M., Lips, A., 2008.Surfactant mixtures for control of bubble surface mobility in foam studies.Langmuir 24, 9956–9961.

Grassia, P., Cilliers, J.J., Neethling, S.J., Ventura-Medina, E., 2001. Quasi-one-dimensional foam drainage. European Physical Journal E 6, 325–348.

Grassia, P., Homsy, G.M., 1998. Thermocapillary and buoyant flows with lowfrequency jitter. I. Jitter confined to the plane. Physics of Fluids 10, 1273–1290.

Howell, P.D., Stone, H.A., 2005. On the absence of marginal pinching in thin freefilms. European Journal of Applied Mathematics 16, 569–582.

Israelachvili, J.N., 1985. Intermolecular and Surface Forces: With Applications toColloidal and Biological Systems. Academic Press, London.

Ivanov, I.B., 1980. Effect of surface mobility on the dynamic behavior of thin liquidfilms. Pure and Applied Chemistry 52, 1241–1262.

Ivanov, I.B., Dimitrov, D.S., 1988. Thin film drainage. In: Ivanov, I.B. (Ed.), Thin LiquidFilms: Fundamentals and Applications. Marcel Dekker, Inc., New York,pp. 382–489.

Joye, J.L., Hirasaki, G.J., Miller, C.A., 1992. Dimple formation and behavior duringaxisymmetrical foam film drainage. Langmuir 8, 3083–3092.

Joye, J.L., Hirasaki, G.J., Miller, C.A., 1994. Asymmetric drainage in foam films.Langmuir 10, 3174–3179.

Karakashev, S.I., Ivanova, D.S., Angarska, Z.K., Manev, E.D., Tsekov, R.,Radoev, B., Slavchov, R., Nguyen, A.V., 2010. Comparative validation ofthe analytical models for the Marangoni effect on foam film drainage.Colloids and Surfaces A: Physicochemical and Engineering Aspects 365,122–136.

Koehler, S.A., Hilgenfeldt, S., Stone, H.A., 2000. A generalized view of foamdrainage: experiment and theory. Langmuir 16, 6327–6341.

Kralchevsky, P.A., Danov, K.D., Denkov, N.D., 1997. Chemical physics of colloidsystems and interfaces. In: Birdi, K.S. (Ed.), Handbook of Surface and ColloidChemistry, 3rd ed. CRC Press, Boca Raton, pp. 333–477.

Kraynik, A.M., 1983. Foam Drainage. Technical Report SAND83-0844, SandiaNational Labs., Albuquerque, NM, USA.

Lemlich, R., 1968. Principles of foam fractionation. Progress in Separation andPurification 1, 1–56.

Lemlich, R., Lavi, E., 1961. Foam fractionation with reflux. Science 134, 191.Loney, N.W., 2001. Applied Mathematical Methods for Chemical Engineers. CRC

Press, Boca Raton, FL.Manev, E., Tsekov, R., Radoev, B., 1997. Effect of thickness non-homogeneity on the

kinetic behaviour of microscopic foam film. Journal of Dispersion Science andTechnology 18, 769–788.

Martin, P.J., Dutton, H.M., Winterburn, J.B., Baker, S., Russell, A.B., 2010. Foamfractionation with reflux. Chemical Engineering Science 65, 3825–3835.

Miller, R., Fainerman, V.B., Leser, M.E., Michel, M., 2004. Kinetics of adsorption ofproteins and surfactants. Current Opinion in Colloid & Interface Science 9,350–356.

Neethling, S.J., Cilliers, J.J., Woodburn, E.T., 2000. Prediction of the water distribu-tion in a flowing foam. Chemical Engineering Science 55, 4021–4028.

Politova, N., Tcholakova, S., Golemanov, K., Denkov, N.D., Vethamuthu, M., Ana-nthapadmanabhan, K.P., 2012. Effect of cationic polymers on foam rheologicalproperties. Langmuir 28, 1115–1126.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 2007. NumericalRecipes: The Art of Scientific Computing, 3rd ed. Cambridge University Press,Cambridge.

Radoev, B.P., Dimitrov, D.S., Ivanov, I.B., 1974. Hydrodynamics of thin liquid films.Effect of the surfactant on the rate of thinning. Colloid and Polymer Science252, 50–55.

Reynolds, O., 1886. On the theory of lubrication and its application toMr. Beauchamp Tower's experiments, including an experimental determinationof the viscosity of olive oil. Philosophical Transactions of the Royal Society ofLondon 177, 157–234.

Saye, R.I., Sethian, J.A., 2013. Multiscale modeling of membrane rearrangement,drainage and rupture in evolving foams. Science 340, 720–724.

Scheludko, A., 1967. Thin liquid films. Advances in Colloid and Interface Science 1,391–464.

Smith, M.K., Davis, S.H., 1983. Instabilities of dynamic thermocapillary liquid layers.Part 1. Convective instabilities. Journal of Fluid Mechanics 132, 119–144.

Stevenson, P., Jameson, G.J., 2007. Modelling continuous foam fractionation withreflux. Chemical Engineering and Processing: Process Intensification 46,1286–1291.

Stewart, P.S., Davis, S.H., 2012. Dynamics and stability of metallic foams: networkmodeling. Journal of Rheology 56, 543–574.

Stewart, P.S., Davis, S.H., 2013. Self-similar coalescence of clean foams. Journal ofFluid Mechanics 722, 645–664.

Traykov, T.T., Ivanov, I.B., 1977. Hydrodynamics of thin liquid films. Effect ofsurfactants on the velocity of thinning of emulsion films. International Journalof Multiphase Flow 3, 471–483.

Ubal, S., Grassia, P., Harrison, C.H., Korchinsky, W.J., 2011. Numerical simulation ofmulti-component mass transfer in rigid or circulating drops: multi-componenteffects even in the presence of weak coupling. Colloids and Surfaces A:Physicochemical and Engineering Aspects 380, 6–15.

Ubal, S., Harrison, C.H., Grassia, P., Korchinsky, W.J., 2010. Numerical simulation ofmass transfer in circulating drops. Chemical Engineering Science 65,2934–2956.

Valkovska, D.S., Danov, K.D., Ivanov, I.B., 2002. Stability of draining plane-parallelfilms containing surfactants. Advances in Colloid and Interface Science 96,101–129.

Verbist, G., Weaire, D., Kraynik, A.M., 1996. The foam drainage equation. Journal ofPhysics: Condensed Matter 8, 3715–3731.

Vitasari, D., Grassia, P., Martin, P. Simulation of dynamics of adsorption of mixedprotein-surfactant on a bubble surface. Colloids and Surfaces A: Physicochem-ical and Engineering Aspects. Doi: http://dx.doi.org/10.1016/j.colsurfa.2012.12.007, in press.

Vrij, A., 1966. Possible mechanism for the spontaneous rupture of thin, free liquidfilms. Discussions of the Faraday Society 42, 23–33.

Wang, L., Yoon, R.-H., 2005. Hydrophobic forces in thin aqueous films and their rolein film thinning. Colloids and Surfaces A: Physicochemical and EngineeringAspects 263, 267–274.

Wang, L., Yoon, R.-H., 2008. Effects of surface forces and film elasticity on foamstability. International Journal of Mineral Processing 85, 101–110.

Ward, A.F.H., Tordai, L., 1946. Time-dependence of boundary tensions of solutions. I.The role of diffusion in time-effects. Journal of Chemical Physics 14, 453–461.

Weaire, D., Hutzler, S., 1999a. Local equilibrium rules. In: The Physics of Foams.Oxford University Press, Oxford, New York, pp. 21–27.

Weaire, D., Hutzler, S., 1999b. Quantitative description of foam structures. In:The Physics of Foams. Oxford University Press, Oxford, New York, pp. 28–46.

Weaire, D., Hutzler, S., 1999c. Foam collapse. In: The Physics of Foams. OxfordUniversity Press, Oxford, New York, pp. 144–150.

Weaire, D., Hutzler, S., 1999d. Ordered foams. In: The Physics of Foams. OxfordUniversity Press, Oxford, New York, pp. 151–174.

Weaire, D., Hutzler, S., 1999e. Drainage. In: The Physics of Foams. Oxford UniversityPress, Oxford, New York, pp. 127–143.

Xie, W., Neethling, S.J., Cilliers, J.J., 2004. A novel approach for estimating theaverage bubble size for foams flowing in vertical columns. Chemical Engineer-ing Science 59, 81–86.

Yeo, L.Y., Matar, O.K., de Ortiz, E.S.P., Hewitt, G.F., 2001. The dynamics of Marangoni-driven local film drainage between two drops. Journal of Colloid and InterfaceScience 241, 233–247.

D. Vitasari et al. / Chemical Engineering Science 102 (2013) 405–423 423

Chapter 6

Surfactant transport onto a foam lamella in

the presence of surface viscous stress

This chapter is in preparation for submission to Journal Colloids and Surfaces A: Physic-

ochemical and Engineering Aspects special issue for EUFOAM Conference. This chapter

studies the motion on the surface of a foam lamella and the transport of surfactant onto a

lamella surface due to that motion in the presence of surface viscous stress. In this present

study, the effect of film drainage upon the surface motion is neglected to simplify the real

problem to provide a benchmark for a more complicated case when the film drainage

exists.

A surface viscous effect can occur when there is a motion on the surface of a foam

lamella and it opposes the direction of the surface motion [91] as has been discussed in

Section 2.7. The surface viscous effect is quantified using the term surface viscosity – the

force per unit length of an element on a surface, per unit surface strain rate. This study

is looking at a system of foam lamellae and Plateau borders within a foam fractionation

column with reflux, where the Plateau borders have a higher surfactant concentration and

act as surfactant reservoir. On the surface of a foam lamella, the film drainage towards

the Plateau border and the Marangoni effect towards the centre of the film compete with

each other to determine the direction of the surface motion [50] as discussed in Chapter 5.

66

In the absence of the film drainage as presented in this study, the Marangoni effect deter-

mines the direction of the surface motion towards the centre of the film. This condition

is favourable for the transport of surfactant onto a foam lamella as surfactant will be ac-

cumulated on the surface of the lamella. However, the surface viscous effect opposes this

motion and reduces the amount of surfactant transport.

This study also develops a benchmark for the case of very small surface viscosity,

where the surface velocity can be readily determined using integration of a Green’s func-

tion. The Green’s function for the case of small surface viscosity is easier to develop

where the boundary conditions at the centre and the end of film can be effectively pushed

away to infinity and the parameters of the system can be solved in isolation to each other.

The Green’s function for the case of large surface viscosity is rather more complicated to

obtain, therefore it is not the interest of this study.

The profile of surface velocity for arbitrary gradient of surface concentration is ob-

tained numerically using a finite difference method [95]. The evolution of surfactant sur-

face concentration can be calculated from the surface velocity profile using the material

point method [103]. The larger the surface viscosity is, the smaller the amount of surfac-

tant transport onto the foam lamella. In the case of large surface viscosity, the total tension

along the film is nearly uniform, moreover the profile of surfactant surface concentration

can be modelled using an integro-differential equation. For a large surface viscosity, the

velocity on the surface is proportional to the distance from the centre of the film and the

distribution of surfactant coverage on the film surface may be nearly uniform except at a

boundary layer near the Plateau border. The shorter the film length is, the shorter time

scale for transport of surfactant.

67

Publication 3

Surfactant transport onto a foam lamella in the presence

of surface viscous stress

68

Surfactant transport onto a foam lamella in the presence ofsurface viscous stress

Denny Vitasari, Paul Grassia∗, Peter MartinSchool of Chemical Engineering and Analytical Science, The Mill, The University of Manchester

Oxford Road, Manchester M13 9PL, UK

Abstract

Surfactant transport onto a foam lamella in the presence of surface viscosity and in the ab-sence of film drainage has been simulated. In the case of very small surface viscosity, the modelcan be simplified and the equation for surface flow can be tackled analytically as a benchmarkfor more complicated systems. Analytical solution and Green’s function have been used to com-pute the profile of surface viscosity in special cases. It was found that the surface viscous effectslows down the surface velocity which results from balance between the Marangoni stress, bulkviscous stress and surface viscous stress. The larger the surface viscosity is, the slower thesurface flow. The effect of surface viscous stress is of larger magnitude where there is largergradient of surface concentration that also gives larger Marangoni stress. The evolutions of sur-face velocity and surfactant surface concentration are modelled using a finite difference methodcoupled with the material point method. The finite difference solution has been verified us-ing the analytical solution and the solution obtained from integration of the relevant Green’sfunction in the special case where the change of surface concentration is linear with position.It was found that the surface velocity slows down particularly near the Plateau border as theeffect of surface viscous stress. The larger the surface viscosity is, the lower the surfactantsurface concentration on the lamella at any given time. The boundary condition at the end ofthe film implies also that a drier foam (i.e. smaller radius of curvature of the Plateau border)leads to less surfactant transport onto the films. Moreover, the shorter the film length is, also theshorter the characteristic time for surfactant transport onto the surface of a lamella. For a largesurface viscosity, the surface concentration of surfactant is nearly uniform except at positionsnear the Plateau border where the velocity and surfactant concentration fields need to satisfy theboundary condition at the end of the film.

1. Introduction

Foam fractionation is an inexpensive and environmentally friendly separation process forrecovery of surface active compounds from dilute aqueous solutions [1]. The foam fractionationprocess is applied for separation of various materials including proteins [1–5], biosurfactants[6–10], microorganisms [11, 12], and other surface active materials [1]. Foam fractionation iscommonly performed in a foam fractionation column, where gas is sparged from the bottomof the column to create bubbles which form a rising foam. The principle of the separation in afoam fractionation column is by selective adsorption of solute(s) to the surface of gas bubblesmoving towards the top of the column [13]. Commonly, the liquid content within the foamcolumn is below 10% [14, 15] and this foam is known as dry foam [16]. As an effect, the liquidseparating the gas bubbles forms thin films called foam lamellae. The films form polyhedral

∗Corresponding authorEmail address: paul.grassia@manchester.ac.uk (Paul Grassia)

Preprint submitted to Elsevier August 28, 2014

cells and meet in channels at the edges of the polyhedra. The channels where the lamellae meetare called Plateau borders [16].

Some foam fractionation columns are equipped with reflux to increase separation efficiency[17–20] by enhancing the enrichment of the foamate (the fractionation product taken from thetop of the column). In a foam fractionation column with reflux, a portion of the collapsed foa-mate is returned back to the top of the column. This collapsed foamate is richer in surfactantthan the rising liquid in the channels within the foam column. Those streams mix in the channelsand enhance the adsorption of surfactant onto the surface of the foam lamella when the surfac-tant in the reflux can be pulled onto the surface of the foam lamella by so called Marangonistresses. As a consequence, a higher concentration of the foamate at the top of the column isachieved [17]. Considering that a foam fractionation column with reflux provides those advan-tages, this study models the surfactant transport onto a foam lamella in a fractionation columnwith reflux.

Surfactant is transferred onto the surface of a foam lamella through two mechanisms. Oncebubbles are introduced into a surfactant solution, adsorption of surfactant onto the surface of thebubble occurs and foam is created. Due to reflux, the concentration of surfactant on the surfaceof Plateau borders is higher than that on the surface of films. Therefore, there is extra transfer ofsurfactant from the surface of the Plateau border onto the surface of the film due to the resultingMarangoni stresses. Therefore, there are two processes involved in the surfactant transfer. Oneis from the bulk solution to the lamella surface [21, 22], and the other is transport of surfactantalong the lamella surface [23]. This study, like [23] considers the transport of surfactant on thelamella surface. An assumption of insoluble surfactant is taken, where the surfactant will stayon the surface once it is adsorbed.

The transport of surfactant on the surface of the lamella is governed by the film drainagetowards the Plateau border, the Marangoni effect in the direction towards the centre of the filmand possibly also the surface viscosity. Due to forces applied on it, the film surface will bemoving towards the Plateau border and/or towards the centre of the film. As an effect, there arechanges in either (or both) shape and area of the film surface elements. Due to these changes,surface viscosity takes effect [24]: shear surface viscosity against the change of shape of thesurface element and the dilational surface viscosity opposing the change in area of the surfaceelement [25]. The dilational surface viscosity is potentially much smaller than the shear surfaceviscosity [26]. However, the dilational changes in particular establish surface pressure gradients[27] that lead to surface elastic (Marangoni) effects which may be even larger than any surfaceviscous ones. The surface viscosity of adsorbed monolayers plays a crucial role in the technicalproblems such as the formation, stability and rheology of emulsions or foams as well as inthe process of film coating, spraying, flotation and oil recovery [26]. Leonard and Lemlich[28] identified how surface mobility determines the velocity profile within a Plateau border ina foam. This surface mobility was defined as the ratio of the bulk viscosity times the radiusof curvature of the Plateau border to the surface viscosity. When surface mobility is low, theflow pattern within the Plateau border is mainly determined by the bulk viscosity with no slipon the walls of the border. However when surface mobility is high, significant motion on thewalls takes place. Earlier, Scriven [29, 30] also recognised the effect of surface viscosity onthe surface motion, although did not explicitly determine the surface mobility. Saffman andDelbrück [31] developed a model of protein moving through a viscous lipid cell membraneseparating much bigger domains of less viscous water and found that there is a critical size ofprotein where the drag coefficient due to the membrane is comparable to the drag coefficientdue to the fluid within the membrane. The size of protein in this case enters a surface mobilityparameter analogous to that of Leonard and Lemlich [28]. When the protein is small the effectof the outer liquid upon the velocity of liquid within the membrane can be ignored and thiscondition is analogous to the immobile surface proposed by Leonard and Lemlich. On the other

2

hand, a large protein size increases traction on the surface of the membrane by the outer liquid,which results in a mobile surface of the membrane [31, 32].

Our previous study of foam fractionation with reflux [23] incorporated Marangoni effectsupon surfactant transport along foam lamellae, but ignored the role that surface viscosity mightplay. Accordingly, in this work, we shall consider how surface viscosity modifies this surfactanttransport process. Our main finding is that surface viscosity reduces (possibly quite substan-tially) the amount of surfactant transferred onto a lamella, thereby reducing the efficiency of thereflux process during fractionation.

This study examines the surfactant transport onto a foam lamella in the presence of surfaceviscosity. This work is laid out as follows. Section 2 deals with the mathematical model of sur-face velocity profile in the presence of surface viscosity, including the assumptions used and theboundary conditions assigned. Section 3 discusses the various methods to measure surface vis-cosity and the variability of the measurement results. Section 4 presents the parameters for thesimulation. Section 5 deals with the case where the surface viscosity is very small and discussesthe special case where the gradient of surfactant surface concentration is constant along the film.Solutions for special cases where the surface concentration jumps at one point along the filmsurface are obtained using Green’s function and the derivation of the solution is discussed inSection 6. The general solution technique for an arbitrary profile of surface concentration ispresented in Section 7. The profile of surface velocity where the surface viscosity is very largeis discussed in Section 8, and the profile of surfactant surface concentration for the case of largesurface viscosity is explored in Section 9. The results of the simulation presenting the evolu-tion of surface velocity and surfactant surface concentration are discussed in Section 10 and thesummary of the study is outlined in Section 11.

2. Mathematical model of the surface velocity

Under dynamic conditions, deformation of the surface of a foam film occurs and this leadsto surface stresses. Those surface stresses can be an elastic stress or viscous stress. The vis-cous stress can originate from shear deformation and dilational deformation, while the elasticstress originates from dilation and is characterised by a Gibbs elasticity parameter. The viscousstresses (per unit strain rate) are characterized by a constant called surface shear viscosity andsurface dilational viscosity, respectively.

A mathematical model of the surface velocity taking into account the surface viscosity isdeveloped. In this study, a 2-D bulk flow field will be considered (Fig. 1): changes in thelocation of surface elements are then always accompanied by changes in area. Accordingly, thevalue of surface viscosity assigned is the combination of both the shear surface viscosity andthe dilational surface viscosity, but is possibly dominated by the former [26].

2.1. Assumptions of the modelThe mathematical model of surface velocity in this study is developed based on the follow-

ing assumptions:

• At initial time, the surface concentration of surfactant along the film is uniform.

• The surface concentration of surfactant on the Plateau border is fixed all the time to thevalue set by the reflux to the foam fractionation column.

• The film is taken to be flat and has uniform thickness along its length [23, 33, 34].

• The thickness of the film is much smaller than its length, consequently the associatedReynolds number, with length scale based on the film thickness, is very small [35] and alubrication approximation is assumed [36–42].

3

us(film)us(Pb)

symmetry point

x

z

δ

Lfilm LPb

Film Plateau border

μμs

us(Pb)

γPb

γfilm

Figure 1: Two-dimensional slice of half of a foam lamella and an adjacent Plateau border. Thefilm half length is Lfilm and the half film thickness is δ, the Plateau border length is LPb,surface tensions are ΓPb and Γfilm on the surface of Plateau border and on the surface of filmrespectively, surface and bulk viscosity are µ and µs respectively, surface velocities are us(Pb)and us(film) on the surface of Plateau border and on the surface of film respectively.

• The liquid is incompressible and Newtonian.

2.2. Surface velocity profileThe equation for the profile of surface velocity taking into account the surface viscosity is

developed based on the lubrication theory. The profile of the surface velocity can be expressedby the following equation:

us = −xδ

dδ

dt+

δ

3µ

[∂γ

∂x+

∂

∂x

(µs∂us∂x

)](1)

where us = dx/dt is the surface velocity, x is the distance from the centre of the lamella alongthe x axis, t is time, δ is the half thickness of the lamella, µ is the liquid viscosity, γ is thesurface tension, µs is the surface viscosity. Applying the Gibbs equilibrium [43] to the surfacetension to convert it into surface excess of surfactant results in the following equation:

3µ

δ

(us +

x

δ

dδ

dt

)=

∂

∂x

(−G ln

Γ

ΓPb

)+

∂

∂x

(µs∂us∂x

)(2)

where G is the Gibbs parameter, Γ is the surface excess of surfactant on the surface of the filmand ΓPb is the surface excess of surfactant on the surface of the Plateau border. This is generallylarger than the surface excess on the film (Γ or ΓF0 initially at time t = 0).

A dimensional analysis was carried out upon Eq. (2) results in the following equation:

3

δ′

(u′s +

x′

δ′dδ′

dt′

)= − 1

δ′0

∂ ln Γ′

∂x′+

∂

∂x′

(µs∂u′s∂x′

)(3)

where u′s = usµ/Gδ′0 is the dimensionless surface velocity, δ′0 = δ0/L is the dimensionless

initial half film thickness, δ0 is the initial half film thickness, L is half of the film length, δ′ =δ/L is the dimensionless half film thickness, x′ = x/L is the dimensionless distance fromthe centre of the lamella along the x axis, t′ = tGδ′0/Lµ is the dimensionless time, Γ′ =Γ/ΓPb is the dimensionless surface excess of surfactant and µs = µs/µL is the dimensionlesssurface viscosity. The dimensionless surface viscosity is the reciprocal of the surface mobilityparameter identified by Leonard and Lemlich in their study [28, 44].

4

2.3. Case without film drainageThe previous study [23] which ignored surface viscosity looked at the interplay between

Marangoni and film drainage effects (but ignored surface viscosity). However it demonstratedthat for a typical fractionation process, the time scale to drain films down to exceedingly thinvalues (down to the level of a so called ‘common black film’ of thickness δcb) can be muchlonger than typical residence time in the fractionation column. Hence, in this study, we lookat the case without film drainage so that dδ′/dt′ = 0 and δ′ = δ′0. Therefore, Eq. (3) can besimplified into the following equation:

3u′sδ′0

= − 1

δ′0

∂ ln Γ′

∂x′+

∂

∂x′

(µs∂u′s∂x′

). (4)

As a result, the equation for surface velocity can be described as follows:

u′s = −1

3

∂ ln Γ′

∂x′+δ′0µs

3

∂2u′s∂x′2

. (5)

The parameter δ′0µs is important since it determines the effect of surface viscosity upon thesystem (an estimated value for this parameter will be provided shortly). If δ′0µs is small, theeffect of surface viscosity is weak. On the other hand, if δ′0µs is very large, there is balancebetween the Marangoni effect and surface viscosity, therefore we obtain u′s ≈ 0 (or at leastu′s � 1). As a result, Eq. (5) can be simplified into this following equation:

µs∂2u′s∂x′2

=1

δ′0

∂ ln Γ′

∂x′. (6)

Eq. (6) indicates that the total film tension which consists of the Gibbs-Marangoni and surfaceviscosity contributions is spatially uniform when δ′0µs is very large. In the opposite limit, whenthe effect of surface viscosity is very weak, the equation of surface velocity profile can besimplified as follows:

u′s = −1

3

∂ ln Γ′

∂x′(7)

which indicates that film surface velocity is driven by Marangoni effects [23].Note the mathematical difference between the expression of surface velocity u′s in Eq. (7)

and Eqs. (5) – (6), the latter being differential equations for u′s that can only be solved usingsuitable boundary conditions. In the presence of surface viscosity, the boundary condition at theend of the film is obtained by extrapolating the velocity field onto the Plateau border itself. Thisvelocity field contributes to the surface velocity even in the absence of the Marangoni effectson the surface of Plateau border. The determination of the boundary condition is presented inthe following subsection.

2.4. Boundary conditionsThe surface velocity u′s must vanish at the film centre due to symmetry of the film as pre-

sented in Fig. 2(a). Due to symmetry at the Plateau border, the value of u′s must also fall to zeroat the ‘symmetry point’ of a tricuspid Plateau border (see Fig. 2). The ‘symmetry point’ is themidpoint of the curved section of each tricuspid arc. Since each of the tricuspid curves subtendsan arc π/3 , the angle subtended between the cusp and the symmetry point is π/6, making thedistance between the cusp and the midpoint be L′Pb = a′π/6, where a′ = a/L is the dimen-sionless radius of curvature of the Plateau border and a is the radius of curvature of the Plateauborder. Therefore, we can take the distance from the centre of the film to the ‘symmetry point’of a tricuspid Plateau border as 1 + a′π/6. A diagrammatic illustration of a film and a Plateau

5

π/3

π/6

a'

Plateau border

film

L' = a'π/6

symmetry point

film

film

L' = 1

film

Pb(a)

(b)L' = 1

(c)

L' = a'π/6Pb

x'u' s

u' s(Pb)

u' s(film)

0

u' s(film)

u' s(Pb)

u' s(Pb)

u' s(Pb)u' s(film)

centre of the film

Figure 2: Illustration of half of a film and an adjacent Plateau border. (a) Cross section of halflength of a film with a Plateau border at the end of the film. (b) Cross section of a Plateauborder. (c) Velocity profile on the surface of a film and a Plateau border next to it for the case ofno drainage. The total distance along the film to the midpoint of the Plateau border is 1 + a′π/6where for the purpose of computing, we assume that the curved shape of the Plateau bordersurface can be ‘uncurled’ onto a straight line.

border is presented in Fig. 2. The illustration in Fig. 2 shows that on symmetry grounds, the ve-locity component along the surface must accelerate away from the point where it vanishes at thesymmetry point (or equivalently decelerates towards the symmetry point). The role of surfaceviscosity is to couple the motion of nearby points to one another: given the comparatively smallsize of the Plateau border (compared to the length of the film), points towards the end of thefilm may well be influenced by the fact that the symmetry point of the Plateau border remainsstationary. This is a phenomenon that we shall investigate in this subsection. Therefore, if thesurface viscosity is large, the fact that the velocity component must be zero at the symmetrypoint will couple back to the film keeping the surface flow very small at the end of the film.

The surface of the tricuspid Plateau border is of course a complicated 2-D shape. As asimplifying approximation, we unfold it into a 1-D line. By ‘uncurling’ the surface of thetricuspid Plateau border, we keep the geometry as well as the mathematics simple but this stillenables us to capture the essential physics of the flow accelerating away from the symmetrypoint (or equivalently decelerating towards the symmetry point).

As mentioned previously, the equation for surface velocity presented in Eq. (5) is a secondorder differential equation for u′s. There are two boundary conditions required to solve thatdifferential equation. One boundary condition is at x′ = 0 we have u′s|x′=0 = 0. The otherboundary condition is taken at x′ = 1 by imposing the constraint at the symmetry point ofa tricuspid Plateau border supposing that u′s is nearly linear in x′ over the domain 1 < x′ <1 + a′π/6 (i.e. the surface velocity decelerates uniformly between the end of the film and thesymmetry point of the Plateau border). Therefore, the boundary point at the end of the film nearthe Plateau border is as follows:

∂u′s∂x′

∣∣∣x′=1

=0− u′s|x′=1

a′π/6(8)

6

or in other form:u′s|x′=1 +

a′π

6

∂u′s∂x′

∣∣∣x′=1

= 0. (9)

The velocity boundary condition at the end of film in Eqs. (8) and (9) is shifted to x′ = 1instead of at the symmetry point itself at x′ = 1 + a′π/6. The reason for shifting the boundarycondition in this fashion is that the fluid flow and the surfactant mass transfer model that weanalyse considers only the domain of the film (i.e. 0 < x′ < 1) without any detailed model forthe Plateau border (1 < x′ < 1 + a′π/6). The detailed flow field in the Plateau border itself isvery complex – with flow changing direction from either within the border or along its axis toalong the film (i.e. normal to the border axis) – and it is beyond the scope of this study.

Having a boundary condition at x′ = 1 is also convenient when solving the mass conser-vation equation for surfactant (since one only needs to solve for Γ′ in the domain 0 < x′ < 1)and one can simply assume that surfactant coverage on the Plateau border is fixed. This followsbecause typically in a foam, the bulk of the liquid is in the Plateau borders (meaning that thePlateau border surfaces can continually be replenished by surfactant arriving from the Plateauborder interiors). This does not contradict the notion of the surfactant being nominally ‘insol-uble’ which only requires that surfactant be present primarily as surface excess on the lengthscale of the (exceedingly thin) films.

This completes our model of surfactant transport on a foam lamella in the presence of sur-face viscosity. However in order to compute with the model we require values of model param-eters, in particular values for the surface rheological material properties. These are not alwaysreadily obtained however (as we explain in the next section).

3. Variability of measurements of surface viscosity

Measuring the viscoelastic properties of a surface is very challenging since those propertiesare affected by environmental contamination [45]. Moreover, it is not always easy to distinguishbetween surface elasticity and surface viscosity [45], and, as a possible consequence it hasbeen observed that different experimental methods to determine the surface viscosity result indifferent measured values of surface viscosity [27, 45, 46]. In the discussion that follows, wetake the determination of surface viscosity of SDS as an example case since that property ofSDS is widely examined in various studies. It was found that the surface viscosity of SDSmeasured using different techniques varies up to several orders of magnitude [27, 46], eventhough there is only a narrow distribution of SDS concentration in the solution measured [46].

Some studies [26, 46, 47] reported values of surface viscosity of SDS solution orders ofmagnitude smaller than the value of surface viscosity reported in other studies such as by Du-rand and Stone [43] and Liu and Duncan [48]: see Appendix A for an indication of just howmuch reported values vary. The variation of measured surface viscosity of SDS can howeverbe an artifact of various measurement methods [45, 46]. The fact that SDS is a soluble surfac-tant also contributes to the inconsistency where the three dimensional layer neighbouring thesurface also governs the resistance to shear deformation at the surface [46]. It is also suggestedthat the variation of the surface viscosity measurement results is due to the different purities ofthe test samples since surface viscosity and indeed surface tension itself are very sensitive tocontamination [45, 49].

Tian et al. [45] mentioned that the rate of surface contraction or expansion determines thesurface viscoelasticity. It was found that different experimental methods applying different sur-face contraction or expansion rates result in different reported values of surface viscosity. Thereason for this is potentially because the attempts to measure surface viscosity may have beenactually observing Gibbs elasticity. When a surface is expanded, additional tension occurs dueto surface viscosity. That surface expansion also results in the reduction of surfactant surface

7

coverage, thereby increasing the surface tension even more. As a result (as mentioned previ-ously), it is not easy to distinguish between surface viscosity and Gibbs elasticity. These similareffects are seen in bulk rheology of stretched polymer solutions as has been observed by Grassiaand Hinch [50] and by Doyle and Shaqfeh [51]. In particular, Doyle and Shaqfeh [51], in theirstudy of polymer stretch and relaxation, also observed that for a very rapid stretch the polymerviscous stresses dominate the elastic ones. However, it is much more common that the elasticforces dominate the viscous ones. Therefore returning to the case of a film, having differentrates of surface expansion or contraction will result in different surface stresses dominating,which will affect the measurement of surface viscosity, as it is not easy to distinguish betweensurface viscosity and Gibbs elasticity.

Interestingly, the variation of surface viscosity obtained using different measurement meth-ods is somewhat analogous to the various results obtained from the measurement of extensionalviscosity of the so called M1 fluid [52] using various measurement methods in polymer rhe-ology. The M1 polymeric fluid (polyisobutylene in polybutene) was prepared and distributedamong researchers in different laboratories to measure its extensional viscosity using variousmeasurement methods. Despite the same liquid and hence the same composition being mea-sured, there are large variations of measured extensional viscosity up to seven orders of magni-tude (for comparison, the surface rheology data in Appendix A vary over five or six orders ofmagnitude). The difference between the measurement results is also due to the different extentsof deformation that occurred before the fluid reached the measuring zone, as well as being dueto the difference of the measurement instruments themselves [52].

Zell et al. [27] developed equipment to measure surface viscosity using the most precisetechnique to date. The method uses interfacial micro-rheometry technique with ferromagneticmicrobuttons. The surface shear rheology is determined by measuring the rotational displace-ment (strain) of the microbutton held at the liquid interface within the surfactant monolayer.This strain occurs as a response to an externally-applied torque (stress). This technique hasadvantages among other methods of measurement of interfacial rheology as it measures thesurface viscosity unambiguously, while other methods combine multiple distinct rheologicalproperties into one measurement [27]. This study found that the surface shear viscosity of SDSis < 0.01µPa m s, which is smaller than the values of surface viscosity of SDS reported in theprevious studies which ranged up to 5 × 103 µPa m s. This finding demonstrates that the val-ues of surface viscosity from previous studies are presumably due to contribution of other (e.g.elastic) properties (although there may also be geometric factors to consider – see AppendixA). Since the surface viscosity of SDS was found to be very small, hence insignificant to thesurface velocity profile, in this study the value of surface viscosity of bovine serum albumin(BSA) together with a cosurfactant propylene glycol alginate (PGA) which is much larger thanthat of SDS will be used [43]. Moreover, fractionation is commonly used for enriching proteins– so applying our study to the case of protein such as BSA is extremely appropriate.

4. Parameters for simulation

The present work performs a simulation study of the effect of surface viscosity on the trans-port of bovine serum albumin (BSA) together with a cosurfactant propylene glycol alginate(PGA) onto a foam film during a process of foam fractionation. The parameters used in thissimulation are presented in Tab. 1 while the relevant dimensionless parameters are presented inTab. 2. All parameters except the values of ΓPb, ΓF0, δcb, δ′0 and a were taken from the studyby Durand and Stone [43] using protein BSA together with a cosurfactant PGA, both at con-centration of 4.0 g L−1. The value of a was estimated as presented in the work by Vitasari et al.[23], while the value of δcb was taken from Weaire and Hutzler [16]. The values of ΓPb, ΓF0 areestimates, taken as lower than the maximum surface excess Γmax of BSA which was reported

8

Table 1: Parameters for simulation of surfactant transport onto a foam lamella incorporatingsurface viscosity taken from a study by Durand and Stone [43]. The value of surface viscosity µsas well as the error bar is the one observed by Durand and Stone [43]. Some other experimentssuggested much smaller values of µs than that reported by Durand and Stone [43].

Parameters Symbol Value UnitCharacteristic ‘Marangoni’ time scale L2µ/(Gδ0) 3.125× 10−2 s(equivalent to one unit of dimensionless time)Initial half lamella thickness δ0 20× 10−6 mHalf thickness of common black film δcb 15× 10−9 mHalf lamella length L 5× 10−3 mLiquid viscosity µ 7× 10−3 Pa sSurface viscosity µs 31± 12× 10−3 Pa m sSurfactant surface excess on the Plateau border ΓPb 3× 10−8 mol m−2

Initial surfactant surface excess on the lamella ΓF0 1.5× 10−8 mol m−2

Radius of curvature of the Plateau border a 5× 10−4 mSurface tension on the Plateau border γPb 55× 10−3 N m−1

Gibbs parameter G 65± 12× 10−3 N m−1

Table 2: Base case dimensionless parameters used in the simulation of surfactant onto a foamlamella incorporating surface viscosity. The values were taken from a study by Durand andStone [43].

Parameters Symbol Definition ValueDimensionless initial film thickness δ′0 δ0/L 4× 10−3

Dimensionless half thickness of common black film δ′cb δcb/L 3× 10−6

Dimensionless radius of curvature of PB a′ a/L 0.1Dimensionless surface viscosity µs µs/µL 8.8± 3.4× 102

Dimensionless initial surfactant surface excess Γ′0 Γ0/ΓPb 0.5

in a study by Fainerman et al. which is Γmax = 5× 10−8 [53]. Those parameters are base-caseparameters and variations about the base-case will be applied in the simulations.

The parameter δ′0µs is the key parameter to determine the effect of surface viscosity uponthe system. Using the base-case parameters presented in Tab. 1, the value of that key parameteris δ′0µs = 3.5± 1.4. However, there is a lot of uncertainty associated with this parameter.

The value of µs = 65 ± 12 × 10−3 Pa m s used was taken from a study by Durand andStone [43] using BSA together with a cosurfactant propylene glycol alginate (PGA). However,this was obtained using the same technique Durand and Stone used to obtain a value of surfaceviscosity of SDS, which tends to be higher than most of the other surface viscosity valuesreported elsewhere in the literature (see Appendix A). Hence while it is likely that the surfaceviscosity of BSA will be substantially larger than that of SDS, it still might not be quite so largeas the value used in this study. This will affect the parameter value for δ′0µs.

There is also uncertainty on the film thickness. We have supposed that films have an (arbi-trary) initial thickness estimated by Durand and Stone. However, we could in principle considerthat mass transfer takes place on much thinner films: in other words we could assume any δ0

value down as low as δcb. Again, this will affect the parameter value for δ′0µs.Hence we have a base case value for δ′0µs of 3.5 ± 1.4 but with the possibility of much

smaller values. Given the possible wide range of variation for δ′0µs, it is useful to understand

9

Table 3: Range of the dimensionless parameters used in the simulation of surfactant transportonto a foam lamella incorporating surface viscosity specific to the case of a small viscosityparameter.

Parameters Symbol RangeDimensionless initial film thickness δ′0 3.0× 10−6 − 1.5× 10−4

Dimensionless surface viscosity µs 8.8− 8.8× 102

Dimensionless viscosity parameter δ′0µs 2.7× 10−4 − 5.4× 10−2

Dimensionless radius of curvature of PB a′ 0.10− 0.25

how the system behaves in the limiting cases of large and small values of this parameter. Thisis achieved in the following sections.

In the simulations with δ′0µs � 1 it is possible to choose a film thickness δ′0 much smallerthan the base case value, whilst retaining the base case µs. Yet another way of ensuringδ′0µs � 1 is to choose a µs much smaller than the base case. The ranges of the parametersused specifically in the simulation with δ′0µs � 1 are presented in Tab. 3.

5. Case where δ′0µs is very small

In the case of very small δ′0µs, the second term of the right hand side of Eq. (5) can beignored except possibly for a boundary layer near the Plateau border. Therefore, Eq. (5) can besimplified into this following equation:

u′s = −1

3

∂ ln Γ′

∂x′. (10)

To model the boundary layer, Eq. (5) is solved as non-homogeneous linear differential equationby rearranging it into the following equation:

δ′0µs3

∂2u′s∂x′2

− u′s =1

3

∂ ln Γ′

∂x′. (11)

Although this equation can be solved very readily by Green’s function techniques, it isuseful to consider a simple special case of the equation first (to obtain intuition about how thesolutions behave). We consider this in the next subsection.

5.1. Special case where ∂ ln Γ′/∂x′ = βx′

One special case for varying Γ′ can be assumed for which the solution for Eq. (11) can beobtained particularly simply. This special case results by assigning βx′ = ∂ ln Γ′/∂x′, where βis a constant. Eq. (11) can be rearranged as follows:

∂2u′s∂x′2

− u′sδ′0µs/3

=βx′

δ′0µs. (12)

This is a non-homogeneous equation to solve for u′s. The solution is presented in the followingequation, but readers interested on the details may refer to Appendix B.

u′s = −βx′

3+

β (1 + a′π/6)

3(

1 + a′π/(6√δ′0µs/3)

) exp

(x′ − 1√δ′0µs/3

). (13)

For illustration, in this computation, various values of δ′0µs are applied and the results arepresented in Fig. 3. From the figure can be seen that the variation of δ′0µs for a particular value

10

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1

us’

x’

δ’0

-µ

s = 2.7 × 10

-2

δ’0

-µ

s = 2.7 × 10

-3

δ’0

-µ

s = 2.7 × 10

-4

Figure 3: Dimensionless surface velocity u′s on the film surface for a special case where∂Γ′/∂x′ = βx′ at various δ′0µs � 1. This computation was carried out using a′ = 0.1 andβ = 0.5. The value of δ′0µs = 2.7 × 10−2 is obtained using δ′0 = 3 × 10−5 and µs = 886;the value of δ′0µs = 2.7 × 10−3 is obtained using δ′0 = 3 × 10−6 and µs = 886; the value ofδ′0µs = 2.7×10−5 is obtained using δ′0 = 3×10−5 and µs = 8.86. The value of µs = 886 is theone reported by Durand and Stone [43], while the value of µs = 8.86 was taken arbitrarily, tobe smaller than that reported by Durand and Stone [43]. The value of δ′0 = 3× 10−5 was takenarbitrarily to satisfy δ′0µs � 1, this value is being larger than the thickness of a common blackfilm, while δ′0 = 3× 10−6 corresponds to the thickness of a common black film. All three casesshow a linear region over much of the domain where u′s ≈ −β/3, and also show a deviationfrom that linear region is grater and occurs sooner for larger values of δ′0µs

of β does not affect the surface velocity at positions far from the Plateau border. It can bereadily seen that the second term of the right hand side of in Eq. (B.11) is much smaller than βfor a small value of δ′0µs. The role of this second term is to make a local adjustment in the slope∂u′s/∂x

′ so as to satisfy the boundary condition at x′ = 1. Therefore, near the Plateau border,the speed of the surface slows down due to the effect of the surface viscosity. The larger thevalue of δ′0µs, the slower is the movement of the surface in the direction towards the centre ofthe film. It can be seen in Fig. 3 that the magnitude of u′s

∣∣x′=1

decreases with the larger valueof δ′0µs. A graph that summarises the values of u′s

∣∣x′=1

over various δ′0µs � 1 is presented inFig. 4.

The analysis which has been discussed in this subsection is only applicable for a specialcase where ∂ ln Γ′/∂x′ = βx′ where β is a constant. In reality, the profile of ∂Γ′/∂x′ is not ingeneral linear in x′. However, a general solution to solve the profile of surface velocity along thefilm can be obtained via the so called Green’s function technique. The solution using Green’sfunction technique is discussed in the following Section 6.

6. Green’s function solution for the case of very small δ′0µs

The Green’s function solution for u′s vs x′ is obtained when ∂ ln Γ′/∂x′ is concentrated asa Dirac delta function at a particular location x′0. Then the general solution is obtained by

11

-0.2

-0.1

0

0 0.01 0.02 0.03

us’

| x’

=1

δ0’-µs

Figure 4: Surface velocity at the edge of the film u′s∣∣x′=1

for various values of δ′0µs � 1 for aspecial case where ∂Γ′/∂x′ = βx′ where β = 0.5, a′ = 0.1 and the limiting value for u′s

∣∣x′=1

is−β/3 which is achieved at δ′0µs = 0.

integrating the Green’s function over all possible x′0 locations. In what follows we consider thenature of Green’s function for various different situations. The first situation considers a jumpin Γ′ at an arbitrary but given location along the film: this is significant because any Γ′ vs x′

function can be idealised as a sequence of small jumps at a multitude of different locations.The second situation considers a jump in Γ′ that occurs very close to the centre of the film: itturns out that symmetry constrains limit the effectiveness of such a jump in producing surfacemotion. The third situation considers a jump in Γ′ at the end of the film: again this is significantbecause it corresponds to the initial distribution of surfactant.

6.1. Value of Γ′ jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0The system is illustrated in Fig. 5. There is a jump at the point of x′ = x′0 where Γ′ = Γ′0

(for some specified value Γ0) at x′ ≤ x′0 and Γ′ = 1 at x′ ≥ x′0. Due to the jump in the valueof Γ′, the value of ∂Γ′/∂x′ is infinite at the jump point and equals zero everywhere else. As aconsequence ∂ ln Γ′/∂x′ can be defined using a Dirac delta function as follows:

∂ ln Γ′

∂x′= ln

(1

Γ′0

)δ (x′ − x′0) (14)

where δ(x′ − x′0) is a Dirac delta function at x′ = x′0. The Dirac delta function vanishes at allpoints except at x′ = x′0, therefore, only complementary solutions are required to solve Eq. (5).However, these solutions need to be properly matched at x′ = x′0. Again a detailed derivationfollows: some readers may prefer to skip straight to the results in Subsection 6.1.1.

Due to the step change in Γ′ leading to Eq. (14), Eq. (5) can be solved using a Green’s

12

Γ'

x'

Γ'0

0x'

1

1

Figure 5: Illustration of the system solved using Green’s equation where there is a jump of thevalue of Γ′ = Γ′0 at x′ ≤ x′0 to Γ′ = 1 at x′ ≥ x′0. The illustration shows the surface excess Γ′

vs position on the film surface x′.

function G as follows:

u′s = ln

(1

Γ′0

)G =

d1 exp

(x′√δ′0µs/3

)+ d2 exp

(− x′√

δ′0µs/3

)if x′ < x′0

d3 exp

(x′√δ′0µs/3

)+ d4 exp

(− x′√

δ′0µs/3

)if x′ > x′0.

(15)

Within the range 0 ≤ x′ ≤ 1 the corresponding equations for u′s for the positions away fromthe point x′0 can be determined as follows:

u′s = d1 exp

(−x′0 + x′√δ′0µs/3

)+ d2 exp

(−x′√δ′0µs/3

)(16)

for x′ < x′0 and

u′s = d3 exp

(x′0 − 2 + x′√

δ′0µs/3

)+ d4 exp

(x′0 − x′√δ′0µs/3

)(17)

for x′ > x′0, where the parameters d1, d3 and d4 are defined by d1 = d1 exp(x′0/√δ′0µs/3),

d3 = d3 exp((−x′0 + 2)/√δ′0µs/3) and d4 = d4 exp(−x′0/

√δ′0µs/3).

The coefficients d1, d2, d1 and d4 are solved as follows. Readers interested on the details ofthe derivation may refer to Appendix C. The value of d1 = d4 can be determined as follows:

d1 = d4 = − ln(1/Γ′0)

6√δ′0µs/3

. (18)

While the value of d2 can be determined as follows:

d2 = −d1 exp

(−x′0√δ′0µs/3

). (19)

13

And we have the value of d3 as follows:

d3 = −d4

(1− a′π/6√

δ′0µs/3

)

1 + a′π/6√δ′0µs/3

. (20)

There is a critical value of a′ = a′crit that gives d3 = 0. For a value of a′ different from a′critthe sign of d3 will vary depends on whether a′ < a′crit or a′ > a′crit. The value of a′crit can becalculated using this following formula:

a′crit =6

π

√δ′0µs

3. (21)

As a consequence Eq. (C.17) can also be presented in the following equation:

d3 = −d4

(1− a′

a′crit

)

1 + a′a′crit

. (22)

Remember here that δ′0µs is taken to be a small parameter, and hence a′crit is small, but a′ is alsoa small parameter (scaling with the liquid fraction of the foam). Thus the ratio a′/a′crit (whichdetermines the sign of d3) is a ratio between two small parameters. One can therefore obtaindifferent signs of d3 by varying either δ′0 or µs or a′.

In the limit of δ′0µs → 0, we also have a′crit → 0. As a consequence, d3 equates to d4.However, u′s at boundary x′ = 1 which is multiplied by a factor exp((x′0 − 1)/

√δ′0µs/3) =

exp((x′0−1)/(a′critπ/6) in Eq. (C.3) vanishes at the boundary x′0 = 1 as a′crit → 0. In that case,and in the special case where x′0 = 0.5, the Green’s function obtained is perfectly symmetricabout x′ = 0.5. However selecting a′crit non-zero or shifting x′0 away from 0.5 (or both) tendsto break that symmetry.

6.1.1. Computed velocity fieldThe velocity field computed for the case where there is a jump at x′0 = 0.5 is presented

in Fig. 6. The result in Fig. 6 is from a computation where a′ is fixed at 0.1 and the valueof δ′0µs varies. The surface velocity is negative indicating surfactant is being pulled onto thefilm by the lower surface tension there. Moreover, the surface velocity is largest in magnitudeat the jump point due to the sharp gradient of surface concentration of surfactant at that pointthat drives the Marangoni effect. In general, there are three balanced forces involved in thetransport of surfactant on the film surface. The Marangoni effect occurs due to the gradient ofsurface concentration and is balanced by the bulk viscous shear stress immediately below thesurface and also by the surface viscosity on the surface. In the special case considered here,the Marangoni stress only occurs at the jump point (i.e. the point where surface concnetrationundergoes jump), since that is the only point with a surfactant concentration gradient. TheMarangoni stress is balanced by a surface viscous stress at the jump point. Everywhere else,the Marangoni force is zero, and there is a balance between the surface viscosity and the bulkviscous shear stress immediately below the surface. That balance requires the velocity to decayover a finite distance. The surface strain rate ∂u′/∂x′ either side of the jump point can be shownto scale inversely with the surface viscosity. The velocity scales as the product of the strainrate and the decay distance. Even though the decay distance (scaling as square root of surfaceviscosity) decreases with decreasing surface viscosity, that decrease turns out to be less rapidthan the increase in the strain rate at the jump point. The net result is that the velocity at thejump point grows.

14

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0 0.2 0.4 0.6 0.8 1

u s’

x’

δ’0-µs = 5.4 × 10-2

δ’0-µs = 2.7 × 10-2

δ’0-µs = 5.4 × 10-3

δ’0-µs = 2.7 × 10-3

δ’0-µs = 5.4 × 10-4

-0.1

-0.05

0

0.8 0.9 1x’

u s’

Figure 6: Dimensionless surface velocity u′s on the film surface for a special case where Γ′

jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0 where x′0 = 0.5 and Γ′0 = 0.5. The valueof a′ used in this computation is 0.1. The values of δ′0µs = 5.4× 10−4, δ′0µs = 2.7× 10−3 andδ′0µs = 5.4× 10−3 give a′ > a′crit while the values of δ′0µs = 2.7× 10−2 and δ′0µs = 5.4× 10−2

give a′ < a′crit. Small δ′0µs gives sharply peaked velocity profiles. The inset is zoomed in withinthe spatial range of 0.8 to 1 spatial unit.

15

-0.02

-0.015

-0.01

-0.005

0

0 0.02 0.04 0.06 0.08 0.1

us’

| x’

=1

δ0’-µs

Figure 7: Surface velocity at the edge of the film u′s∣∣x′=1

over a range of values of δ′0µs for aspecial case where Γ′ jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0 where x′0 = 0.5and Γ′0 = 0.5. The value of a′ used in this computation is 0.1. The values of δ′0µs vary fromδ′0µs = 1.1× 10−3 to δ′0µs = 1.1× 10−1. Clearly increasing δ′0µs decreases u′s|x′=1.

The inset in Fig. 6 zooms in near x′ = 1 to show the small (but non-zero) values of u′s thatapply there. The value of u′s < 0 at x′ = 1 actually shows that there is surfactant flux from thePlateau border to the film. The value of u′s at x′ = 1 is a physically important quantity, sincein the presence of surface viscosity, there is surfactant flux at x′ = 1 even though there is noMarangoni force at that point (Marangoni being concentrated instead at x′ = x′0 in the specialcase considered here). As δ′0µs becomes larger, u′s

∣∣x′=1

becomes bigger in magnitude, meaningthat there is more surfactant flux from the Plateau border to the centre of the film. When δ′0µsbecomes smaller, the magnitude of u′s

∣∣x′=1

becomes exponentially small. The change of u′s∣∣x′=1

with the variation of δ′0µs for the case where x′0 = 0.5 is presented in Fig. 7. The change ofu′s∣∣x′=1

with the variation of δ′0µs also shows the same trend for different choices of x′0.It is stated in Eq. (14) that ∂Γ′/∂ lnx′ is effectively ln(1/Γ′0)δ(x′−x′0) whereas the calcula-

tion result also shows that u′s is also approaching a Dirac delta function in the limit as δ′0µs → 0.If u′s is integrated in the range of x′ = 0 to x′ = 1, the integral can also be rearranged as follows:

∫ 1

0

u′sdx′ =

∫ x′0

0

u′sdx′ +

∫ 1

x′0

u′sdx′ (23)

=δ′0µs

3

∫ x′0

0

∂2u′s∂x′2

dx′ +δ′0µs

3

∫ 1

x′0

∂2u′s∂x′2

dx′dx′ (24)

=δ′0µs

3

[∂u′s∂x′

∣∣∣∣x′0

0

+∂u′s∂x′

∣∣∣∣1

x′0

]. (25)

Using the fact that ∂u′s/∂x′ is the largest near x′ = x′0 and much smaller near x′ = 0 and x′ = 1

16

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.85 0.9 0.95 1 1.05 1.1 1.15

us’

x’

a’ = 0.10

a’ = a’crit

= 0.18

a’ = 0.30

Figure 8: Dimensionless surface velocity u′s on the film surface in the neighbourhood of x′ = 1for a special case where Γ′ jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0 wherex′0 = 0.5 and Γ′0 = 0.5. The value of δ′0µs is fixed at 2.7 × 10−2 to give a′crit = 0.18. The datais plotted in the domain 1 < x′ < 1 + a′π/6 are the straight line extrapolations of the functionu′s at x′ = 1. Smaller a′ gives smaller (magnitude) of u′s|x′=1 .

(provided δ′0µs is very small), we can express the equation in the following form:∫ 1

0

u′sdx′ =

δ′0µs3

[∂u′s∂x′

(x′−0)

+∂u′s∂x′

(x′+0)]. (26)

The solution of Eq. (26) can be presented as follows:∫ 1

0

u′sdx′ = −1

3ln

(1

Γ′0

). (27)

Eq. (27) is consistent with u′s becoming −(1/3) ln(1/Γ′0)δ(x′ − x′0) in the limit of δ′0µs ap-proaching zero. In this case, the system is behaving as if the effect of surface viscosity werenegligible and the Marangoni effect were balanced by the bulk viscous shear stress immediatelybelow the surface.

6.1.2. Effect of varying a′

A computation where the value of δ′0µs is fixed and a′ varies was also carried out. In thiscomputation the value of δ′0µs = 2.7 × 10−2 is set at δ′0 = 3 × 10−5 and µs = 886, while thevalue of a′ ranges between 0.1 and 0.3. Those values of a′ are taken at below and above a′critfor the specified value of δ′0µs. The result of the calculation is presented in Fig. 8. The figureonly presents the profile of surface velocity in the neighbourhood of x′ = 1 to show the effectof the change of a′ upon the velocity profile at the end of the film.

In the neighbourhood of x′ = 1 when a′ < a′crit, the profile of surface velocity is perturbedupward from the profile where a′ = a′crit. On the other hand the profile of surface velocity isperturbed downward in the neighbourhood of x′ = 1 when a′ > a′crit. This effect occurs sincethe value of d3 has a negative sign when a′ > a′crit quantity. Therefore, the surface velocity at

17

the end of film is larger in magnitude than that obtained at a′ < a′crit where the value of d3 hasa positive sign. The larger radius of curvature of Plateau border a′ reduces the extent to whichsurface viscosity slows down the surface velocity at the end of a film (see also Eq. (30) to bederived shortly).

6.1.3. Effect of varying x′0When x′0 is taken at different positions on the film surface, the effect is that the curve is

shifted along according to the value of x′0 taken as presented in Fig. 9. The surface velocityis largest in magnitude at the jump point for any jump point set. If x′0 is shifted closer to thePlateau border, there is more surfactant flux from the Plateau border to the film surface. WhenEq. (17) is applied to x′ = 1, it results in the following equation:

u′s|x′=1 = d3 exp

(x′0 − 1√δ′0µs/3

)+ d4 exp

(x′0 − 1√δ′0µs/3

). (28)

Using the definition of d3 Eq. (28) can be rearranged into the following format:

u′s|x′=1 =

−d4

(1− a′

a′crit

)

1 + a′a′crit

+ d4

exp

(x′0 − 1√δ′0µs/3

). (29)

Eq. (29) can be simplified into the following form:

u′s|x′=1 = d4

(2a′

a′crit + a′

)exp

(x′0 − 1√δ′0µs/3

). (30)

As x′0 moves closer to unity, the magnitude of the exponential in Eq. (28) is larger and the valueof u′s|x′=1 is also bigger in magnitude. This indicates that there is more surfactant flux from thePlateau border to the film when the jump point (x′0) is shifted closer to the Plateau border asalso appears in Fig. 9.

6.2. Value of Γ′ jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0 where x′0 �√δ′0µs

The analysis of Subsection 6.1 presupposed that√δ′0µs � 1 whereas x′0 was order of unity.

If however the value of x′0 is very small so that x′0 �√δ′0µs, the gradient of surface velocity at

0 ≤ x′ ≤ x′0 is potentially very steep and Eq. (16) can be simplified into the following equation:

u′s = d1x′

x′0(31)

and the derivative of Eq. (32) is as follows:

du′sdx′

=d1

x′0(32)

Considering that d1 = d4, we can obtain the following equation (see Eq. (C.8) in Appendix C):

1

3ln

(1

Γ′0

)=δ′0µs

3

[− d1√

δ′0µs/3exp

(x′0 − x′+0√δ′0µs/3

)− d1

x′0

]. (33)

18

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.2 0.4 0.6 0.8 1

us’

x’

x0’ = 0.25

x0’ = 0.50

x0’ = 0.75

Figure 9: Dimensionless surface velocity u′s on the film surface for a special case where Γ′

jumps from Γ′ = Γ′0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′0. The position of x′0 varies between 0.25and 0.75 and Γ′0 = 0.5. The value of a′ used in this computation is 0.1 and the value of δ′0µsis fixed at 2.7 × 10−2. The curves have broadly similar shapes, but the location of the peak isshifted.

We can take that (x′0−x′+0 )/√δ′0µs/3 ≈ 0, therefore Eq. (33) can be simplified into the follow-

ing equation:1

3ln

(1

Γ′0

)=δ′0µs

3

[− d1√

δ′0µs/3− d1

x′0

]. (34)

The value of d1/√δ′0µs/3 is much smaller than the value of d1/x

′0 since x′0 �

√δ′0µs/3,

therefore the first term on the right hand side of Eq. (34) can be ignored to find the value of d1

as follows:

d1 = −x′0 ln(1/Γ′0)

δ′0µs(35)

The result of computation for the case of a jump of the value Γ′ from Γ′ = Γ′0 for x′ ≤ x′0to Γ′ = 1 for x′ ≥ x′0 where x′0 �

√δ′0µs is presented in Fig. 10. In the neighbourhood of

x′ = x′0 in this case, the value of u′s is determined by d1 – as presented in Eq. (32) – which isproportional to x′0 as presented in Eq. (35). As a consequence, the selected value of x′0 �

√δ′0µ

affects the value of highest magnitude of the surface velocity u′s which occurs at the jump point.A smaller x′0 gives a smaller magnitude of u′s.

The value of d1 obtained using Eq. (35) is much smaller than that obtained using Eq. (18).As a result, the step change of surfactant concentration very close to the centre of the film leadsto very little overall surface motion as the surface viscosity gives a strong coupling to the fixedpoint at the centre of the film ( u′s = 0 at x′ = 0).

6.3. Value of Γ′ jumps from Γ′ = Γ′0 for x′ ≤ 1 to Γ′ = 1 for x′ ≥ 1

Now consider the case x′0 → 1. The illustration in Fig. 11 represents the system in this case.That represents the initial surface concentration of surfactant on a foam lamella. Therefore, thecomputation carried out in this subsection is in fact computing the initial velocity profile, hence

19

-0.3

-0.2

-0.1

0

0 0.2 0.4 0.6 0.8 1

us’

x’

x0’ = 0.001

x0’ = 0.005

x0’ = 0.010

-0.3

-0.2

-0.1

0

0 0.05 0.1

Figure 10: Illustration of the system solved using Green’s function where there is a jump of thevalue of Γ′ = Γ′0 at x′ ≤ x′0 to Γ′ = 1 at x′ ≥ x′0 where x′0 �

√δ′0µs. The value of a′ used in

this computation is 0.1. The value of δ′0µs is fixed at 2.7× 10−2. The closer x′0 moves to 0, thelower the surface speeds. The iset is the profile of surface velocity near x′ = 0.

its importance. Again a detailed derivation of the velocity formula is given, although somereaders may prefer to jump straight to the results Subsection 6.3.1.

The surface velocity for x′ ≤ 1 can be described by the following equation:

u′s = d5 exp

(−1 + x′√δ′0µs/3

)+ d6 exp

(−x′√δ′0µs/3

). (36)

Near x′ = 1, for δ′0µs � 1, the second exponential on the right hand side is very small andnegligible, therefore d5 can be determined as follows (for detailed determination see AppendixD):

d5 = − ln(1/Γ′0)

3√δ′0µs/3

(a′crita′ + 1

) . (37)

If a′ � a′crit the value of d5 will be of similar order of magnitude as d4 given by Eq. (18)although it is twice as big as d4. On the other hand if a′ � a′crit, the value of d5 will be smallerin magnitude than d4: the foam is now sufficiently dry, i.e. the value of a′ is sufficiently small,that the effect of a jump in Γ′ at the end of the film feels the limitation of the zero velocityconstraint at the symmetry point of the Plateau border.

The value of d6 is determined a posteriori using the boundary condition at x′ = 0 whereu′s = 0 to obtain this following equation:

d6 = −d5 exp

(−1√δ′0µs/3

). (38)

Eq. (38) has the same form as Eq. (C.14) when x′0 is shifted to x′0 = 1. For a small√δ′0µs/3

20

Γ'

x'

Γ'0

1

1

Figure 11: Illustration of the system solved using Green’s function where there is a jump of thevalue of Γ′ = Γ′0 at x′ ≤ 1 to Γ′ = 1 at x′ ≥ 1. The illustration shows the surface excess Γ′ vsposition on the film surface x′.

the value of d6 is much smaller than d5 thus negligible as anticipated above.

6.3.1. Computed velocity fieldThe computation result is presented in Fig. 12. The gradient of surface velocity is larger for

a smaller value of δ′0µs. There are three different stresses that affect the surface velocity. TheMarangoni stress occurs only at the jump point where there is gradient of surface concentration.At the jump point, the surface viscous stress balances the Marangoni stress. The bulk viscousstress occurs everywhere, and away from the jump point is balanced by surface viscous stress.The surface shear rate scales inversely with the surface viscosity. Meanwhile, the surface veloc-ity scales as the shear rate multiplied by the decay distance for the shear. This decay distancescales only like the square root of surface viscosity, the net result being that surface velocitygrows as surface viscosity falls.

Another computation for the case where there is a jump in the value of Γ′ from Γ′ = Γ′0 forx′ ≤ 1 to Γ′ = 1 for x′ ≥ 1 is carried out using fixed value of δ′0µs = 2.7 × 10−2 that givesa′crit = 0.18. In this computation, the value of a′ varies between 0.1 and 0.3 in the range of a′

smaller than a′crit to a′ larger than a′crit. The result of this computation is presented in Fig. 13.It can be seen that the surface velocity for the cases of a′ > a′crit and a′ < a′crit differs

from the surface velocity obtained from a′ = a′crit even though those three computations wereusing the identical value of δ′0µs = 2.3 × 10−2. As stated in Eq. (D.2), the surface velocity inthe neighbourhood of x′ = 1 depends only on the value of d5 which depends on the value ofa′crit/a

′ as mentioned in Eq. (D.8). When a′ < a′crit the surface velocity at the end of the film isperturbed upward from that obtained using a′ = a′crit. On the other hand, when a′ > a′crit thesurface velocity perturbed downward from that obtained using a′ = a′crit. For a smaller a′ theend of the film is closer to the symmetry point of the Plateau border, therefore the requirementthat the surface velocity is zero at the symmetry point also constrains the motion at the end ofthe film.

7. General solution for arbitrary ∂ ln Γ′/∂x′

The solutions discussed in Sections 5 and 6 are for special cases of the profile of ∂ ln Γ′/∂x′.The general solution for arbitrary ∂ ln Γ′/∂x′ is required since the real cases do not always fol-low those special cases. In this section, two methods are discussed. One method uses the

21

-14

-12

-10

-8

-6

-4

-2

0

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

us’

x’

δ’0

-µ

s = 5.4 × 10

-2

δ’0

-µ

s = 2.7 × 10

-2

δ’0

-µ

s = 5.4 × 10

-3

δ’0

-µ

s = 2.7 × 10

-3

δ’0

-µ

s = 5.4 × 10

-4

Figure 12: Dimensionless surface velocity u′s on the film surface in the neighbourhood of x′ = 1for a special case where Γ′ jumps from Γ′ = Γ′0 for x′ ≤ 1 to Γ′ = 1 for x′ ≥ 1. The valueof Γ′0 is 0.5. The value of a′ used in this computation is 0.1. The values of δ′0µs = 5.4× 10−4,δ′0µs = 2.7× 10−3 and δ′0µs = 5.4× 10−3 give a′ > a′crit while the values of δ′0µs = 2.7× 10−2

and δ′0µs = 5.4 × 10−2 give a′ < a′crit. When a′ < a′crit the magnitude of u′s|x′=1 is muchsmaller than when a′ > a′crit

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.5 0.6 0.7 0.8 0.9 1 1.1

us’

x’

a’ = 0.10

a’ = a’crit

= 0.18

a’ = 0.30

Figure 13: Dimensionless surface velocity u′s on the film surface in the neighbourhood of x′ = 1for a special case where Γ′ jumps from Γ′ = Γ′0 for x′ ≤ 1 to Γ′ = 1 for x′ ≥ 1. A jump in∂u′s/∂x

′ is seen at the end of film. The value of δ′0µs is fixed at 2.7× 10−2 to give a′crit = 0.18.

22

integration of the Green’s function while the other method uses the finite difference approxima-tion for the differential equation. The solution using the integration of the Green’s function tobe discussed in this study is only for the case where δ′0µs � 1, as solution for this case is moresimple than the case where δ′0µs is large. When δ′0µs is very small, the boundary conditions atx′ = 0 and x′ = 1 can be pushed to infinity, therefore some coefficients of the exponential termscan be computed in isolation from others (as was done in Section 6).

7.1. Integration of the Green’s functionA general solution for arbitrary function of ∂ ln Γ′/∂x′ can be derived by integrating the

Green’s function presented in Eqs. (C.2) and (C.3). The differential equation for the profile ofsurface velocity for arbitrary ∂ ln Γ′/∂x′ is as in Eq. (5) as follows:

δ′0µs3

∂2u′s∂x′2

− u′s =1

3

∂ ln Γ′

∂x′.

The associated Green’s function for Eq. (5) is modification of Eqs. (C.2) and (C.3) as follows:

G =

D1 exp

(−x′0+x′√δ′0µs/3

)+D2 exp

(−x′0−x′√δ′0µs/3

)if x′ < x′0

D3 exp

(x′0−2+x′√δ′0µs/3

)+D4 exp

(x′0−x′√δ′0µs/3

)if x′ > x′0

(39)

where the parameters D1, D2, D3 and D4 are defined by1:

D1 = D4 = − 1

6√δ′0µs/3

(40)

D2 =1

6√δ′0µs/3

(41)

D3 =1

6√δ′0µs/3

[1− a′π/(6

√δ′0µs/3)

1 + a′π/(6√δ′0µs/3)

](42)

Note that D1, D2, D3 and D4 given by Eqs. (40) - (42) are based on assumption that δ0µs � 1.The velocity profile for arbitrary ∂ ln Γ′/∂x′ is calculated using integration of the associated

Green’s function as presented in the following equation:

u′s(x′) =

∫ 1

0

G(x′|x′0)∂ ln Γ′(x′0)

∂x′0dx′0. (43)

The integration needs only to be done for 0 ≤ x′0 ≤ 1 since ∂ ln Γ′/∂x′ vanishes for 1 ≤ x′0 ≤1 + a′π/6. Using the approximation that δ′0µs � 1, Eq. (43) can also be presented in the form:

u′s(x′) =

∫ x′

0

[D3 exp

(x′0 − 2 + x′√

δ′0µs/3

)+D4 exp

(x′0 − x′√δ′0µs/3

)]∂ ln Γ′(x′0)

∂x′0dx′0

+

∫ 1

x′

[D1 exp

(−x′0 + x′√δ′0µs/3

)+D2 exp

(−x′0 − x′√δ′0µs/3

)]∂ ln Γ′(x′0)

∂x′0dx′0. (44)

1The values of D1, D3 and D4 are obtained from d1, d3 and d4, respectively, divided by factor of ln(1/Γ′0),where D2 = d2 exp(x′0/

√δ′0µs/3)/ ln(1/Γ′0).

23

This solution does not correctly account for contributions from x′0 values very close to eitherboundary where the form of the Green’s function changes under those circumstances. Giventhat ∂ ln Γ′/∂x′ can be an arbitrarily complicated function, the integral in Eq. (44) cannot ingeneral be solved analytically. Therefore, a numerical scheme, such as the trapezoidal rule, isrequired to solve the integral. The numerical integration for arbitrary ∂ ln Γ′/∂x′ will not bediscussed in this study. Instead, the Green’s function method in the case of ∂ ln Γ′/∂x′ = βx′

will be illustrated. In this special case, the equation is simpler and the integral can be solvedanalytically.

7.2. Integration of the Green’s function for special case where ∂ ln Γ′/∂x′ = βx′

For the special case, already solved in Subsection 6.1, where ∂ ln Γ′/∂x′ = βx′ the integralin Eq. (44) can be presented as follows:

u′s(x′) =

∫ x′

0

[D3 exp

(x′0 − 2 + x′√

δ′0µs/3

)+D4 exp

(x′0 − x′√δ′0µs/3

)]βx′0dx′0

+

∫ 1

x′

[D1 exp

(−x′0 + x′√δ′0µs/3

)+D2 exp

(−x′0 − x′√δ′0µs/3

)]βx′0dx′0. (45)

The solution of Eq. (45) is obtained using integration by parts. The equation obtained is simpli-fied into the following form:

u′s(x′) = β

{x′√δ′0µs

3

[D4 +D3 exp

(−2 + 2x′√δ′0µs/3

)+D2 exp

(−2x′√δ′0µs/3

)+D1

]

+δ′0µs

3D4 exp

(−x′√δ′0µs/3

)+δ′0µs

3D3

[− exp

(−2 + 2x′√δ′0µs/3

)+ exp

(−2 + x′√δ′0µs/3

)]

+δ′0µs

3D2

[− exp

(−1− x′√δ′0µs/3

)+ exp

(−2x′√δ′0µs/3

)]− δ′0µs

3D1 exp

(−1 + x′√δ′0µs/3

)

+

√δ′0µs

3

[−D2 exp

(−1− x′√δ′0µs/3

)−D1 exp

(−1 + x′√δ′0µs/3

)]}. (46)

The terms D1βδ′0µs/3 and −D4βδ

′0µs/3 cancel each other since D1 = D4.

It can be seen that for x′ �√δ′0µs and 1− x′ �

√δ′0µs the exponential terms in Eq. (46)

result in very small values. Therefore Eq. (46) in that double limit can be simplified into thefollowing equation:

u′s(x′) ≈ βx′

√δ′0µs

3(D4 +D1)

≈ −βx′

3. (47)

The definition of D1 = D4 is given by Eq. (40). In the limit of x′ → 0, Eq. (46) can besimplified into the following equation:

u′s|x′→0 ≈δ′0µs

3(D4 +D2) (48)

with exponentially small corrections. By the definition of D4 in Eq. (40) and D2 in Eq. (41),D4 +D2 vanishes, leaving u′s being exponentially small (compatible with our boundary condi-

24

tion at x′ = 0). It is remarkable that the integration of the Green’s function managed to lead tothis exponentially small results, since Subsection 6.2 suggested a modification to the Green’sfunction is needed in the limit of x′ → 0.

When x′ → 1 we can simplify Eq. (46) into the following equation:

u′s|x′→1 ≈ β

[√δ′0µs

3(D4 +D1) +

√δ′0µs

3

(1−

√δ′0µs

3

)D3 exp

(−2 + 2x′√δ′0µs/3

)

−√δ′0µs

3

(1 +

√δ′0µs

3

)D1 exp

(−1 + x′√δ′0µs/3

)]. (49)

Eq. (49) contains exponential terms which grow near the boundary. Concerning that x′ → 1and

√δ′0µs/3� 1, Eq. (49) can be simplified into the following form:

u′s|x′→1 ≈ −β

3+β

3

1

(1 + a′/a′crit)(50)

Eq. (50) relies on the definition ofD1 –D4 described in Eqs. (40) – (42). The form of Eq. (50) issimilar to the one presented in Eq. (B.11) in Section 5. In the case of x′ → 1, it can be assumedthat exp((x′−1)/

√δ′0µs/3) ≈ 1 and the value of a′π/6 is very small in Eq. (B.11) and therefore

it can be neglected. The small discrepancy between Eqs. (B.11) and (50) as x′ → 1 can beattributed to the Green’s function coefficient described in Eqs. (40) – (42) needing modificationas x′ → 0 in Eq. (39). It is interesting also that the arguments of the exponential terms differbetween Eqs. (B.11) and (50).

The example discussed above for the case where ∂ ln Γ′/∂x′ = βx′ solved using the inte-gration of the relevant Green’s function gives a benchmark for other cases applying arbitrarygradient of surface concentration. From the solution for this special case obtained using theGreen’s function approach, it can be seen that for most of the domain along the film surface(away from the boundaries) the solution follows u′s ≈ −(1/3)∂ ln Γ′/∂x′. This is actually thesame velocity field as occurs in the absence of surface viscosity [23] and is consistent with thefact that the small δ′0µs case is a small perturbation away from the case without surface viscosity.

So far in the present study we have only treated velocity fields and not how those fieldscouple to mass transfer (a topic to be considered later in Section 10). However, given thevelocity fields with weak surface viscosity are very close to those with no surface viscosity, itfollows that the mass transfer process in the weak surface viscosity case will be similar to themass transfer already considered in [23].

7.3. Finite difference methodThe solutions of the velocity profile obtained using Green’s function in Section 6 are only

applicable for specific cases of ∂ ln Γ′/∂x′ where there is a jump of Γ′ at one point along theaxis of x′. A general solution for arbitrary function of ∂ ln Γ′/∂x′, can be solved by a Green’sfunction approach (Subsection 7.1), but this generally involves some numerics (i.e. a numericalquadrature involving a Green’s function). An alternative approach can be obtained simply bynumerical solution of the differential equation in Eq. (5). In this case a finite difference methodis selected. For reference, Eq. (5) is presented here as follows:

δ′0µs3

∂2u′s∂x′2

− u′s =1

3

∂ ln Γ′

∂x′.

where the boundary conditions are determined at the centre of the lamella and at the lamellaedge, the boundary with the Plateau border. At the centre of the lamella where x′ = 0 the

25

surface velocity is u′s = 0, while at the lamella edge where x′ = 1 the condition is as presentedin Eq. (9) as follows:

u′s|x′=1 +a′π

6

∂u′s∂x′

∣∣∣x′=1

= 0.

The solution of Eq. (5) is determined using the following approximations. For a spatial step∆x′ the derivative of ∂2u′s/∂x

′2 is approximated as follows:

∂2u′s∂x′2

∣∣∣∣(i)

≈u′s(i+1) − 2u′s(i) + u′s(i−1)

∆x′2(51)

and the derivative of ∂ ln Γ′/∂x′ is approximated as follows:

∂ ln Γ′

∂x′

∣∣∣∣(i)

≈ ln Γ(i+1) − ln Γ(i)

∆x′. (52)

The numerical form of Eq. (9) then can be derived as follows:

δ′0µs3

(u′s(i+1) − 2u′s(i) + u′s(i−1)

∆x′2

)− u′s(i) =

1

3

(ln Γ(i+1) − ln Γ(i)

∆x′

)(53)

where i = 1, 2, 3, ..., I − 1 and I is the number of spatial increments. The boundary conditionsare applied at i = 0 and at i = I and the numerical equations for the boundary conditions areas follow:

u′s(0) = 0 (54)

and

u′s(I) +a′π

6

(u′s(I) − u′s(I−1)

∆x′

)= 0. (55)

Recall that this latter boundary condition arises from matching surface plus viscous tensionsat the end of the film and in the Plateau border, and assuming a uniform velocity decelerationacross the Plateau border to attain zero velocity at the border’s symmetry point.

Eq. (53) is rearranged to obtain equations for u′s(i) for each i = 1, 2, 3, ..., I − 1 as follows:

δ′0µs3∆x′2

u′s(i−1) −(

1 +2δ′0µs3∆x′2

)u′s(i) +

δ′0µs3∆x′2

u′s(i+1) =1

3

(ln Γ(i+1) − ln Γ(i)

∆x′

). (56)

The linear equations obtained are simultaneous and cannot be solved individually. However,this system of simultaneous equations has a very convenient structure. When the equations arewritten in a matrix form, they provide the interior of a tridiagonal matrix that can be solvedusing Gaussian elimination. The exterior of the matrix is derived by rearranging the equationsfor the boundary conditions. At x′(0) = 0 we have the following equation:

u′s(0) = 0 (57)

and at x′(I) = 1 the equation for the boundary condition in Eq. (55) is rearranged as follows:

− a′π

6∆xu′s(I−1) +

(1 +

a′π

6∆x

)u′s(I) = 0. (58)

26

-0.8

-0.6

-0.4

-0.2

0

0.5 0.6 0.7 0.8 0.9 1

us’

x’

Green’s function

Finite Difference

Figure 14: Comparison between the finite difference approximation for u′s and the solution ofGreen’s function for the special case where there is a jump of the value of Γ′ from Γ′ = Γ′0 atx′ ≤ 1 to Γ′ = 1 at x′ ≥ 1. The parameters are as follows: Γ′0 = 0.5, δ′0µs = 2.7 × 10−2 anda′ = 0.1. A good agreement between the Green’s function and finite difference techniques isobserved.

7.3.1. Benchmarking the finite difference solutionWe can benchmark the finite difference solution as follows. The finite difference approx-

imation for u′s in the special case where there is a jump of the value of Γ′ from Γ′ = Γ′0 atx′ ≤ 1 to Γ′ = 1 at x′ ≥ 1 is verified using the associated Green’s function solution. Thecomparison between the finite difference approximation and the Green’s function solution forthat special case is presented in Fig. 14. The computation was carried out using parameters ofδ′0µs = 2.7× 10−2 and a′ = 0.1. The graphs in Fig. 14 show that the Green’s function solutionconfirms the finite difference approximation for u′s in the special case where there is a jump ofthe value of Γ′ from Γ′ = Γ′0 at x′ ≤ 1 to Γ′ = 1 at x′ ≥ 1.

The result of computation using the finite difference method for the special case where∂ ln Γ′/∂x′ = βx′ is compared with the result from analytical solution of the differential equa-tion as well as the result from integration of the Green’s function of the same special case.Those results are obtained from simulation using δ′0µs � 1. Recall that the solution for theGreen’s function equation is easier to find with very small δ′0µs where the boundary conditionsat x′ = 0 and x′ = 1 can be pushed away to infinity. The Green’s function solution for the caseof larger δ′0µs is much more complicated to solve. In this particular simulation, the value used isδ′0µs = 5.4× 10−3. The comparison of the results in presented in Fig. 15. It is shown in Fig. 15that the graph obtained using finite difference method is confirmed by the results of integrationof the Green’s function as well as the results of the analytical solution.

8. Case where δ′0µs is very large

Through these benchmarks we have demonstrated that our finite difference technique forsolving the velocity field is reliable. We will use this finite difference later in mass transfercalculations. Before doing that, however we consider yet another limit of interest – namely the

27

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1

us’

x’

Green’s functionAnalytical

Finite difference

Figure 15: Comparison between the integration of Green’s function solution, analytical solutionand finite difference approximation for the special case where ∂ ln Γ′/∂x′ = βx′. The parame-ters are as follows: δ′0µs = 5.4 × 10−3 (obtained from δ′0 = 6 × 10−6 and µs = 886), a′ = 0.1and Γ′F0 = 0.5. Results are virtually indistinguishable between all three cases.

limit of large δ′0µs. The case where δ′0µs is very large corresponds to the situation where the totalfilm tension (incorporating the Marangoni effect and surface viscosity) is spatially uniform. Aformula for the surface velocity is then available as follows (detailed derivation is presented inAppendix E):

u′s = − 1

δ′0µs

[∫ x′

0

ln

(1

Γ′

)dx′ − x′

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]. (59)

Evaluating Eq. (E.7) for x′ = 1 we obtain this following equation:

u′s|x′=1 = − 1

δ′0µs

[1− 1

1 + a′π/6

] ∫ 1

0

ln

(1

Γ′

)dx′. (60)

Surfactant rich reflux is added to the top of the column and flows down the Plateau borderwhere it contacts and flows onto the films. Since Γ′ = Γ/ΓPb, it is generally less than unity andhence ln Γ′ < 0. As a consequence, the value of surface velocity at x′ = 1 as calculated usingEq. (60) is a negative quantity which indicates that the surface at the end of the film is movingin the direction away from the Plateau border. As a consequence, the film tension (includingsurface viscosity effect) is a little larger than the equilibrium tension (defined here as the tensionthat would have resulted at the Plateau border in the absence of any surface viscous effects).

We can verify as follows that Eq. (E.7) is the velocity field that we seek.

∂u′s∂x′

= − 1

δ′0µs

[ln

(1

Γ′

)− 1

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]. (61)

28

Since Γ′ = 1 at x′ = 1, we can also obtain the following equation:

∂u′s∂x′

∣∣∣∣x′=1

=1

δ′0µs

(1

1 + a′π/6

)∫ 1

0

ln

(1

Γ′

)dx′ = −u

′s|x′=1

a′π/6(62)

as the boundary condition at x′ = 1 demands. Moreover, from Eqs. (61) and (62) we also havethe following equation:

∂u′s∂x′

+1

δ′0µsln

(1

Γ′

)=∂u′s∂x′

∣∣∣∣x′=1

(63)

which says that the surface viscous plus surface elastic tension in the film balance the surfaceviscous tensions in the Plateau border exactly as we expect. At the Plateau border, there is nosurface elastic tension as there is no gradient of surface concentration and in fact Γ′Pb = 1 andhence ln(1/Γ′P b) = 0 all over the surface of the Plateau border.

8.1. Profile of surface velocity: Special case of uniform surfactant concentration away fromx′ = 1

Having derived a formula for surface velocity in Eq. (E.7) we now analyse the formula inthis case of large δ′0µs. We have already stated that the total film tension is spatially uniform.At early times, we anticipate a profile of surfactant concentration that is nearly uniform exceptin a thin region near x′ = 1 (the end of the film in contact with the Plateau border). It turns out(as we shall demonstrate later) that such a profile can be maintained spatially near uniform,even though the uniform concentration value starts to evolve over time. This special case is suf-ficiently important, that it is worthwhile to compute the velocity field that it produces. Supposethat Γ′ ≈ Γ′0 except at the thin region near x′ = 1. The value of Γ′0 is assumed to be significantlyless than one. Eq. (E.7) can be simplified into the following equation:

u′s ≈ −1

δ′0µs

x′a′π/6

1 + a′π/6ln

(1

Γ′0

). (64)

Therefore the derivative of Eq. (64) can be presented as follows:

∂u′s∂x′≈ − 1

δ′0µs

a′π/6

1 + a′π/6ln

(1

Γ′0

). (65)

This ∂u′s/∂x′ applies for the overwhelming majority of the length of the film. However in the

thin region near x′ = 1 the value of ∂u′s/∂x′ is rather different. In fact at x′ = 1 Eq. (62) now

results in the following equation:

∂u′s∂x′

∣∣∣∣x′=1

≈ 1

δ′0µs

1

1 + a′π/6ln

(1

Γ′0

). (66)

It is shown that outside the thin region near x′ = 1, ∂u′s/∂x′ is a factor of a′π/6 smaller than

(and of opposite sign to) ∂u′s/∂x′ at x′ = 1. Therefore, there is an obvious asymmetry in the

graph of u′s vs x′ with a gradual slope outside the thin region and a sharper slope inside thethin region. Note also that u′s is an order a′/(δ′0µs) quantity, instead of just order 1/(δ′0µs).Therefore, the surface velocity can be decreased either by increasing δ′0µs or by decreasing a′.In other words, the drier the foam (i.e. smaller a′), the lower the velocity at the edge of the film,and the less mass transfer from Plateau border to film.

Another direct approach to determining the surface velocity at the end of the film uses thebalance of surface tension and surface viscous term on the film surface to the surface viscousterm on the Plateau border. The dimensionless elastic tension is −(1/δ′0) ln Γ′0, while the di-mensionless strain rate in the film (assumed nearly spatially uniform) is u′s|x′=1 and the dimen-

29

-0.02

-0.015

-0.01

-0.005

0

0 0.2 0.4 0.6 0.8 1

us’

x’

a’ = 0.02

a’ = 0.10

a’ = 0.20

Figure 16: Dimensionless surface velocity u′s on the film surface for a special case whereln(1/Γ′) = β(1 − x′2). The parameters are a′ = 0.02 − 0.2 and the value of δ′0µs is fixedat 5.4. The surface velocity changes its slope around x′ = 0.6 instead of near the end of thefilm.

sionless strain rate in the Plateau border is u′s|x′=1/(a′π/6), the dimensionless viscous tensions

being µs times the strain rates. Balancing tensions and viscous terms leads to the followingequation:

− 1

δ′0ln Γ′0 + µsu

′s|x′=1 =

−µsu′s|x′=1

a′π/6(67)

which upon rearrangement is identical to Eq. (64).

8.2. Profile of surface velocity: Special case of ln(1/Γ′) = β(1− x′2)

The surface velocity equation for the case of presented in Eq. (E.7) shows that (for nearuniform concentration field) the surface velocity is an order a′/(δ′0µs) quantity rather than anorder of 1/(δ′0µs) quantity (one of the main findings of Subsection 8.1). For a non-uniformconcentration field however we see order 1/(δ′0µs) velocity values in the interior of the filmas we now demonstrate. However these fall back to order a′/(δ′0µs) as x′ → 1 (which is alsodemonstrated below).

We now explain the reasons for this behaviour. Substitution of the value of ln(1/Γ′) intoEq. (E.7) results in the following equation:

u′s = − 1

δ′0µs

[∫ x′

0

β(1− x′2)dx′ − x′

1 + a′π/6

∫ 1

0

β(1− x′2)dx′]

= − β

δ′0µs

[(x′ − x′3

3

)− 2x′

3(1 + a′π/6)

]. (68)

Fig. 16 shows the result of calculation using the above equation where β = 0.5 and is fixedat that value. Likewise δ′0µs is fixed at the value 5.4. In the absence of film drainage, the surface

30

velocity is in the direction towards the centre of the film due to the Marangoni effect. Thereforethe surface velocity is negative. The surface velocity changes its slope around x′ = 0.6. Theexact position of the turnaround of u′s can be determined using the derivative of Eq. (68) withrespect to x′ as follows:

∂u′s∂x′

= − 1

δ′0µs

[β(1− x′2

)− 2β

3 (1 + a′π/6)

]. (69)

The minimum surface velocity u′s (i.e. maximum surface speed, given the velocity is negative)is obtained while ∂u′s/∂x

′ = 0, therefore the value of x′ that gives the minimum u′s can becalculated as a function of a′ as follows:

x′min =

√1− 2

3 (1 + a′π/6)(70)

where x′min is the position where u′s is minimum. For a range of a′ between 0.02 and 0.2, x′min is0.58 – 0.63 resulting in minimum surface velocity is between u′s = −0.0062 and u′s = −0.0078.The magnitude of surface velocity is very small compared to the case of δ′0µs � 1 (as presentedin Fig. 3). Moreover, in the case of δ′0µs � 1, the slope of the surface velocity profile onlychanges near the end of the film. In the case of δ′0µs � 1, the velocity field is of order β/(δ′0µs).However, the boundary conditions require that there is no motion at x′ = 0 and x′ = 1 + a′π/6.With a large surface viscosity, the effect of zero velocity at the boundaries tends to influence asignificant portion of the film. Since x′ = 1 is near to x′ = 1 + a′π/6 (the value of a′ beingsmall), the zero velocity at x′ = 1 + a′π/6 implies a low velocity at x′ = 1. The velocity atx′ = 1 can be determined as follows:

u′s|x′=1 = − 2β

3δ′0µs

(a′π/6

1 + a′π/6

). (71)

Therefore, at x′ = 1 the surface velocity is of order a′β/(δ′0µs). This is a contrast with theprofile of surface velocity discussed in Subsection 8.1. In this case, ln(1/Γ′) jumps from afinite value to zero. If we indentify the value ln(1/Γ′0) from Subsection 8.1 with the value βhere, the velocity field is of order a′β/(δ′0µs) along the entire film. This results in the surfacevelocity having a constant slope over most the film length, apart from an arbitrarily thin regionnear the Plateau border. The surface velocity at x′ = 1 can be described as follows:

u′s|x′=1 = − β

δ′0µs

(a′π/6

1 + a′π/6

)(72)

which is very similar to Eq. (71) apart from a factor 2/3.The graphs in Fig. 16 show that the smaller a′ is, the smaller the magnitude of u′s|x′=1.

This demonstrates that the value of a′ limits the amount of surface velocity at the end of thefilm, hence the surfactant transport from the Plateau border onto the film. In the case of largesurface viscosity, the boundary condition at x′ = 1 + a′π/6 has a big effect on slowing downthe surfactant transfer onto the film at x′ = 1. Even though the more spread out concentrationfield gives a much higher velocity part way along the film than a less spread out concentrationfield would, at the end of the film the velocity field is always small.

8.3. Magnitude of surface velocity fieldThe surface velocity equation for the case of δ′0µs � 1 presented above shows that the

surface velocity at the end of the film is in an order of a′/(δ′0µs) quantity rather than an order of1/(δ′0µs) quantity (a fact we already noted back in Subsection 8.1). For a′ � 1, we can make

31

simplification as follows:1

1 + a′π/6≈ 1− a′π/6. (73)

Therefore, Eq. (E.7) can also be presented as follows:

u′s ≈ −1

δ′0µs

[∫ x′

0

ln

(1

Γ′

)dx′ + x′

∫ 1

0

ln

(1

Γ′

)dx′ − x′a

′π

6

∫ 1

0

ln

(1

Γ′

)]dx′. (74)

Moreover in the limit as x′ → 1

u′s|x′=1 ≈ −1

δ′0µs

[−a′π

6

∫ 1

0

ln

(1

Γ′

)dx′]. (75)

In the case of nearly uniform value of Γ′ the values of the first and second terms on the righthand side of Eq. (74) almost cancel and u′s is in order of a′/(δ′0µs) everywhere (including atx′ = 1).

In the case of non-uniform values of Γ′, Eq. (75) shows that u′′s |x′=1 is in order a′/(δ′0µs).Moreover, it is only necessary to vary Γ′ within a distance O(a′) of the Plateau border to makesignificant changes to the velocity field.

9. Profile of surfactant surface concentration Γ′ in the case of δ′0µs � 1

So far the main emphasis of this work has been on obtaining surfactant velocity fields, ratherthan on surfactant mass transfer. This has followed the view that the main difference betweenthis work (including surface viscosity) and previous work [23] (excluding surface viscosity) isthe determination of the surface velocity field. Once the surface velocity is known, the methodsfor determining surfactant mass transfer are similar regardless of whether surface viscosity isincluded or excluded.

We have already mentioned in Subsection 7.2 that in the case of small δ′0µs, the evolution ofthe surfactant concentration field is just a small perturbation away from the case without surfaceviscosity (already considered by [23]). In what follows we consider the opposite limit, i.e. thecase of large δ′0µs, for which the surfactant concentration field should be very different fromthat in [23].

The general equation governing the evolution of Γ′ is as presented in the following equation:

∂Γ′

∂t′+

∂

∂x′(u′sΓ

′) = 0. (76)

Substitution of Eqs. (E.7) and (61) into Eq. (76) results in the following equation:

∂Γ′

∂t′− 1

δ′0µs

[ln

(1

Γ′

)− 1

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]

Γ′

− 1

δ′0µs

[∫ x′

0

ln

(1

Γ′

)dx′ − x′

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]∂Γ′

∂x′= 0. (77)

Eq. (77) is an integro-differential equation that can be solved numerically. The value of Γ′ atany point on the film surface is known and can be discretised into a set of known values at givenx′ positions. The evolution of Γ′ at a later time can be calculated using Heun’s method. In thissimulation, the second order of Heun’s method is applied for the integration of Eq. (77). Thespatial derivatives in the equation are approximated using centred differences for spatial deriva-tives (there being I spatial interval in total), while the integrals are solved using the trapezoidal

32

rule. The spatially discretised Eq. (77) is as follows:

(∂Γ′

∂t′

)

(i,n)

=1

δ′0µs

[ln

(1

Γ′(i,n)

)− 1

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]

Γ′(i,n)

+1

δ′0µs

[∫ i∆x′

0

ln

(1

Γ′

)dx′ − i∆x′

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]

Γ′(i+1,n) − Γ′(i−1,n)

2∆x′(78)

where the integrals can be presented in the following numerical trapezoidal rule formulae:

∫ 1

0

ln

(1

Γ′

)dx′ ≈ 1

2

[ln

(1

Γ′(0,n)

)+ 2

I−1∑

j=1

ln

(1

Γ′(j,n)

)]∆x′ (79)

as Γ′(I,n) = 1 always, so ln(1/Γ′(I,n)) = 0, therefore it is omitted from Eq. (79), and

∫ i∆x′

0

ln

(1

Γ′

)dx′ ≈ 1

2

[ln

(1

Γ′(0,n)

)+ ln

(1

Γ′(i,n)

)+ 2

i−1∑

j=1

ln

(1

Γ′(j,n)

)]∆x′ (80)

where the subscripts i = 0, 1, 2, ..., I and j = 1, 2, 3, ..., I − 1 denote the number of spatialsteps while subscript n = 0, 1, 2, ...N denotes the number of time steps, ∆x′ is the spatial stepand ∆t′ is the time step. Eq. (78) also applies for the case i = 0, however in that special caseit is necessary to impose an extra condition that Γ′(−1,n) (appearing on the right hand side ofEq. (78)) is identical to Γ′(1,n) (which follows from the x′ = 0 boundary condition).

The Heun’s method to calculate Γ′ can be presented as follows:

Γ′(i,n) = Γ′(i,n−1) +1

2

[(dΓ′

dt′

)

(i,n−1)

+

(dΓ′

dt′

)

(i,n)

]∆t′

= Γ′(i,n−1) +

(k1(i)

2+k2(i)

2

)∆t′ (81)

where ∆t′ denotes the time step, k1 and k2 are the rate change of the predictor and the corrector,respectively, and are defined as follows:

k1(i) =

(dΓ′

dt′

) ∣∣∣∣(i∆x′,(n−1)∆t′)

for the set of Γ′ values {Γ′(i,n−1)}, whilst

k2(i) =

(dΓ′

dt′

) ∣∣∣∣(i∆x′,n∆t′)

for the set of Γ′ values {Γ′(i,n−1) + k1(i)∆t′}.

We will study these discretised systems numerically in due course. First however we derivea number of analytical results.

9.1. Specific case where Γ′ is close to some uniform value Γ′0In the specific case where Γ′ is close to Γ′0 (apart possibly for a very thin region near x′ = 1),

the fact that Eqs. (64) – (65) suggest a near uniform strain rate in the film has important im-plications for mass transfer. Specifically the near uniform strain rate drives a near uniform rateof change of surfactant concentration, which in turns ensures that the surfactant concentration

33

field (whilst changing in time) remains close to spatially uniform (being therefore consistentwith the assumptions under which Eqs. (64) – (65) were derived).

An ordinary differential equation for the time evolution of Γ′0 can be derived as follows:

∂Γ′0∂t′≈ 1

δ′0µs

a′π/6

1 + a′π/6ln

(1

Γ′0

)Γ′0. (82)

Integration of Eq. (82) from t′ = 0 to t′ results in the following equation:

ln

(ln

(1

Γ′0

))≈ ln

(ln

(1

Γ′00

))− 1

δ′0µs

t′a′π/6

1 + a′π/6(83)

where Γ′00 is the initial value of Γ′0. Eq. (83) can also be expressed as follows:

ln

(1

Γ′0

)≈ ln

(1

Γ′00

)exp

(− 1

δ′0µs

t′a′π/6

1 + a′π/6

)(84)

and we can deduce as follows:

Γ′0 ≈ exp

(− ln

(1

Γ′00

)exp

(− 1

δ′0µs

t′a′π/6

1 + a′π/6

)). (85)

Eq. (85) applies only for positions outside the thin region near x′ = 1. From the equation,it is clear that Γ′0 → Γ′00 if t′ � δ′0µs/(a

′π/6), however Γ′0 → 1 if t′ � δ′0µs/(a′π/6). This

prediction for Γ′0 relies of course on the assumption that the surfactant surface coverage is nearlyspatially uniform. This prediction will be compared with the numerical solution obtained fromthe integro-differential equation in Eq. (77) as well as the solution obtained using the finitedifference method.

The value of Γ′0 calculated using Eq. (85) is for positions outside the thin region near x′ = 1.Inside the thin region near x′ = 1, Γ′ varies significantly over a small spatial distance: indeedat the initial time, Γ′ must exhibit a step from Γ′00 to 1 at x′ = 1. In the case of large surfaceviscosity, although Γ′ exhibits this step function behaviour, value of u′s and ∂u′s/∂x

′ given byEqs. (E.7) and (61) remain finite. This is in contrast with the case of small surface viscosity orin the absence of surface viscosity, where u′s ∼ −1

3∂ ln Γ′/∂x′, becoming infinite in the limit

where Γ′ is a step function.Given the finite value of u′s and ∂u′s/∂x

′ in the limit of large surface viscosity, but the infiniteinitial value of ∂Γ′/∂x′, the approximation of the mass balance in Eq. (76) in the region nearx′ = 1 can be presented as follows:

∂Γ′

∂t′+ u′s|x′=1

∂Γ′

∂x′≈ 0. (86)

As Γ′ varies only within a thin region near x′ = 1 as predicted by Eq. (74) the solution ofEq. (86) can be expressed in the following equation:

Γ ≈ Γs(x′ − u′s|x′=1t

′ − 1) (87)

where Γs(x − u′s|x′=1t′) is a step function jumping from Γ′0 to 1 as its argument reaches zero.

The step occurs at x′s = 1 + u′s|x′=1t′, where x′s denotes the position where the step occurs.

Eq. (87) restricts consideration to values of t′ sufficiently small that the step has not displacedany significant distance away from x′ = 1 (hence u′s at the step is still well represented by u′s atx′ = 1, which can moreover be estimated via Eq. (64)).

It is apparent that an approximation in which Γ′ is near uniform (with a value of Γ′0) except

34

in a thin region near x′ = 1 is a valid one, but the region where Γ′ ≈ Γ′0 is continually shrinkingwith time. There is now a pertinent question of whether the shrinkage of this region affects thetime evolution of Γ′0. This is addressed in the next section where to simplify the analysis werestrict consideration to the case where Γ′0 is close to unity.

9.2. Approximation for the case where Γ′0 is close to unityConcerning that the surfactant surface concentration is nearly uniform except in the thin

region near x′ = 1, the value of Γ′ can be modelled at least approximately using the followingequation:

Γ′ ≈ Γ′0 + exp

(−1− x′

αa′

)(1− Γ′0) (88)

where α is a constant that determines the size of the region (αa′) over which surfactant con-centration is non-uniform. We choose this to be proportional to a′ to ensure that the size of thenon-uniform region is comparable with the size of Plateau border. Recall from Subsection 8.3that this permits significant velocity change across the non-uniform concentration region.

The equation of surface velocity for this case is obtained as follows (for a detailed derivation,readers may refer to Appendix F):

u′s =1

δ′0µs

[αa′ exp

(−1− x′

αa′

)(1− Γ′0)−

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)x′(1− Γ′0)

]. (89)

In what follows we subject the formula Eq. (89) to a careful analysis.

9.2.1. Velocity at and near the end of the film x′ = 1

At the end of the film where x′ = 1 we obtain the following equation for surface velocity:

u′s|x′=1 =1

δ′0µs

[αa′ (1− Γ′0)−

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)(1− Γ′0)

]

=1

δ′0µs

(1

1 + a′π/6

)(αa′2π/6− a′π/6

)(1− Γ′0)

=1

δ′0µs(−a′π/6)

(1− αa′

1 + a′π/6

)(1− Γ′0) . (90)

For u′s|αa′�(1−x′)�1, the equation of the surface velocity can be determined as follows:

u′s|αa′�(1−x′)�1 = − 1

δ′0µs

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)(1− Γ′0) (91)

When the value of α is non-zero, it gives a larger magnitude of u′s in the region where Γ′ isuniform, but paradoxically a very slightly smaller magnitude of u′s at the end of the film x′ = 1.The function of u′s contains a uniform gradient contribution (that becomes more negative withincreasing α) and a positive exponential contribution (confined near the end of the film). Asα changes, the change in the negative contribution almost offsets the positive one, with only avery weak residual effect at x′ = 1.

This result can be verified from a consideration of velocity gradients. By differentiatingEq. (F.11), the velocity gradient along the film can be determined as follows:

∂u′s∂x′

=1

δ′0µs

[exp

(−1− x′

αa′

)(1− Γ′0)−

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)(1− Γ′0)

]. (92)

35

0.9

0.92

0.94

0.96

0.98

1

0 0.2 0.4 0.6 0.8 1

Γ’

x’

α = 0.5, a’ = 0.1

α = 0.7, a’ = 0.1

α = 0.5, a’ = 0.2

α = 0.7, a’ = 0.2

Figure 17: Effect of the value of αa′ upon the surfactant surface concentration in the case ofΓ′0 is close to unity. The calculation was carried out using a′ = 0.1 − 0.2 and Γ′0 = 0.9 andα = 0.5 − 0.7. The smaller αa′, the more abrupt deviation from the linear behaviour near theend of the film over a region of size αa′.

Therefore, at x′ = 1 the velocity gradient can be derived as follows:

∂u′s∂x′

∣∣∣∣x′=1

=1

δ′0µs

[(1− Γ′0)−

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)(1− Γ′0)

]. (93)

Eq. (93) can be simplified as follows:

∂u′s∂x′

∣∣∣∣x′=1

=1

δ′0µs

(1− αa′

1 + a′π/6

)(1− Γ′0). (94)

The gradient of surface velocity is here perturbed away from the case of α → 0 with only asmall change due to the non-zero value of α. Moreover, at x′ = 1, ∂u′s/∂x

′ differs by a factorof −a′π/6 from u′s, which is actually the requirement of the boundary condition at that point.

9.2.2. Effect of α on surfactant concentration profilesThe effect upon the value of αa′ to the surfactant surface concentration in the case of Γ′0

is close to unity is presented in Fig. 17. This computation calculates the surfactant surfaceconcentration at a particular time where the surfactant surface concentration away from x′ = 1is Γ′0 = 0.9. Various values of α between 0.5 and 0.7 and also various values of a′ between0.1 and 0.2 are applied. The values are selected to obtain αa′ on either sides of a′π/6, which isthe distance from end of film to the symmetry point of the Plateau border. Larger values of αa′

give a larger surfactant surface concentration, that results in a larger departure of the surfactantsurface concentration from Γ′0.

The effect of the value of αa′ upon the surface velocity in the case of Γ′0 is close to unity ispresented in Fig. 18.

In the region where Γ′ does not vary spatially, the magnitude of u′s increases. This leads tomore rapid time evolution of Γ′0 compared to the case where α = 0 i.e. compared to Eq. (85) –

36

-0.004

-0.003

-0.002

-0.001

0

0 0.2 0.4 0.6 0.8 1

u’ s

x’

α = 0 , a’ = 0.1

α = 0.5, a’ = 0.1

α = 0.7, a’ = 0.1

α = 0 , a’ = 0.2

α = 0.5, a’ = 0.2

α = 0.7, a’ = 0.2

Figure 18: Effect of the value of αa′ upon the surface velocity in the case of Γ′0 is close tounity. The calculation was carried out using δ′0µs = 5.4 (obtained from δ′0 = 6 × 10−5 andµs = 8.86 × 104), a′ = 0.1 − 0.2 and Γ′0 = 0.9 and α = 0.5 − 0.7. The graphs show u′s beinglinear in x′ over most of the domain, but with an abrupt deviation from the linear behaviour nearthe end of the film over a region of size αa′.

which for small 1− Γ′00 becomes:

Γ′0 ≈ 1−(

1− ln

(1

Γ′00

))exp

(− 1

δ′0µs

t′a′π/6

1 + a′π/6

)(95)

which applies in the α→ 0 limit.The departure of gradient of surface velocity from the case where α→ 0 is not only affected

by the value of αa′ but also affected by the value of a′ itself. In Fig. 18, it can be seen forinstance that when a′ varies from 0.1 to 0.2, while keeping the same value of α that the changeof magnitude of surface velocity is larger than that obtaining by changing the value of α from0.5 to 0.7 while keeping the value of a′.

The value of α represents the thickness of the layer over which spatial changes in Γ′ arerealised (and hence α changing i.e. growing with time) is important. Not only does non-zero αspeed up the evolution of Γ′0, the growth over time in α enhances that speed up. Nevertheless,computing the actual evolution of Γ′0 is difficult: it is possible to write down an equation for theevolution of Γ′0 analogous to Eq. (82) incorporating the factor of α, but unless one knew how αwas changing with time, it would not be possible to integrate that equation.

9.3. Rescaling dimensionless time and surface velocity for the case of δ′0µs � 1

The velocity and time have been non-dimensionalised in Subsection 2.2 on scales thatdepend on bulk viscosity of fluid. In the large δ′0µs limit, it is more convenient to rescale,so that these parameters are independent of bulk viscosity. The new scaling is as follows:u′′s = δ′0µsu

′s = µsus/(LG) is the rescaled dimensionless surface velocity and t′′ = t′/(δ′0µs) =

37

Gt/µs is the rescaled dimensionless time. Using the new scaling, Eq. (5) is presented as follows:

u′′s =δ′0µs

3

(−∂ ln Γ′

∂x′+∂2u′′s∂x′2

)(96)

and the rescaled equation for the evolution of surfactant surface concentration Γ′ is as follows:

∂Γ′

∂t′′+

∂

∂x′(u′′sΓ

′) = 0. (97)

The important point from Eq. (96) is that (whenever δ′0µs is large), the two terms inside thebracket on the right hand side ∂ ln Γ′/∂x′ and ∂2u′′s/∂x

′2 must sum to almost zero. Whereasu′s is typically order 1/(δ′0µs) in the large δ′0µs limit, i.e. velocity decays as surface viscosityincreases, that effect is scaled out when expressed in terms of u′′s . Likewise evolution timesformerly scaled as δ′0µs (i.e. slow evolution in the large surface viscosity limit), but that effectis again scaled out when expressed in terms of t′′.

For finite δ′0µs, the format of Eq. (96) is similar to the format of Eq. (5), therefore thesolution of Eq. (96) can be obtained using a finite difference method which is analogous to thesolution of Eq. (5). Once the evolution of surface velocity u′′s is determined, the evolution ofsurfactant surface concentration Γ′′ can be calculated using Eq. (97) using the material pointmethod. In what follows we are interested in both small and large values of δ′0µs and so adopteither the original scaling or these new scalings depending on the problem of interest.

If the same set of rescalings are applied to the integro-differential equation Eq. (77), theresult is as follows:

∂Γ′

∂t′′−[ln

(1

Γ′

)− 1

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]

Γ′

−[∫ x′

0

ln

(1

Γ′

)dx′ − x′

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]∂Γ′

∂x′= 0. (98)

The discretion of this rescaled form is obvious (and entirely analogous to obtaining Eq. (78)from Eq. (77)). It is however slightly more convenient to work with Eq. (77) rather than Eq. (78)as the effect of δ′0µs has now been scaled out.

10. Evolution of surface velocity and surfactant surface concentration

Sections 8 and 9 consider some calculations of velocity and surfactant surface concentrationfields, but focussed on an idealised situation where Γ′ is uniform over position over most of thefilm – a situation that cannot be sustained indefinitely in the case of large δ′0µs. In more generalsituations, a different approach is required.

Considering that the finite difference method outlined in Subsection 7.3 has been provedreliable (see e.g. Fig 15) to solve the profile of surface velocity for specified ∂ ln Γ′/∂x′, themethod will be coupled with a material point method to model the evolution of surface ve-locity as well as the evolution of surface concentration on a foam lamella at any given time.The material point method is similar to a scheme already discussed in the literature [54]. Mi-nor modifications are applied to account for the boundary conditions imposed where a filmmeets a surfactant rich Plateau border in a foam fractionation column with reflux, similar tothe numerical scheme presented in a previous study [23]. In this numerical scheme, the surfacevelocity equation is discretised into N time steps and I + 1 spatial points treated as materialpoints [54]. The temporal accuracy of the numerical simulation is improved using a predictor-corrector method which is Heun’s method [55, 56]. The details of the material point method

38

are similar to the one presented in another study [23] except the equation for surface velocity,since in this case the surface viscosity is taken into account. In this study, the profile of surfacevelocity is solved using the finite difference method as discussed in Subsection 7.3. The surfacevelocity is simulated using Eqs. (56) – (58) applied to any given time.

The material point positions change on every time step, leading to uneven spatial intervalover time. To keep the spatial interval even, a bookkeeping operation is applied on every timestep to restore equal width intervals. The mechanism of the bookkeeping operation is similar tothe one reported in the previous study [23].

In what follows numerical results are presented for velocity fields and mass transfer. Bearingin mind the different behaviours expected between the cases δ′0µs � 1 and δ′0µs � 1, we treatthese two cases in turn.

10.1. Case where δ′0µs � 1

First, the study considers the case where δ′0µs � 1. The evolution of surface velocityin the presence of surface viscosity for this case is presented in Fig. 19. The simulation wascarried out using δ′0µs = 5.4 × 10−2 and a′ = 0.1. At initial time, the surface concentration ofsurfactant along the film is set as Γ′F0 = 0.5 and the surface concentration of surfactant at thePlateau border is Γ′Pb = 1. Therefore, at this initial time, the gradient of surface concentrationis localised at the Plateau border (x′ = 1). For this case of δ′0µs � 1 the computation wascarried out using 10000 time steps for dimensionless time t′ = 2, resulting in time interval∆t′ = 2 × 10−4. The dimensionless film length was divided by 250 spatial steps, resulting inspatial interval ∆x′ = 4× 10−3.

At initial time, the surface velocity achieves its largest magnitude at the Plateau borderdue to the highest gradient of surface concentration being at that point. The gradient of sur-factant surface concentration leads to the Marangoni effect, and results in movement on theinterface in the direction towards the centre of the film. Surface viscous forces arise balancingthe Marangoni force in the direction opposite to the Marangoni flow. As an effect, the flowneighbouring the Plateau border tends to be slower, the larger the surface viscosity. Howeverthe profile of surface velocity near the Plateau border becomes less sharp at a later time sincethe Marangoni force becomes weaker and more spread out. The weaker Marangoni force ata later time is a result of transport of surfactant to the lamella surface, therefore reducing thegradient of surface concentration at the Plateau border (x′ = 1).

As the evolution of surface velocity is determined at each time step, the evolution of surfaceconcentration of surfactant can be computed using the material point method and the simulationresult is presented in Fig. 20. Another simulation of evolution of surfactant surface concentra-tion in the absence of surface viscosity was also carried out using similar parameters as the onesapplied on the case with surface viscosity, and the result is presented in Fig. 21. Comparingthe surface concentrations calculated in the presence of surface viscosity and in its absence, itappears that there is less surfactant transported onto the foam lamella in the case with surfaceviscosity. The lower surfactant surface concentration in the presence of surface viscosity is theresult of slower surface flow when the surface viscous stresses balance the Marangoni force.

At early time, the surfactant surface concentration obtained using the model with surfaceviscosity is much lower than that obtained using the model without surface viscosity. The masstransfer of surfactant in fact grows like square root of time in the absence of surface viscosity[23], but only linearly in time in the presence of surface viscosity (a fact that follows fromSubsection 6.3). At later time, the relative difference of surface concentration obtained usingthose two models becomes smaller. At early time, there is significant gradient of surfactantsurface concentration on the Plateau border resulting in significant Marangoni stresses. Asa consequence, there may be significant surface viscous stresses to offset those Marangonistresses. As time progresses, the gradient of surface concentration becomes less significant,

39

Figure 19: Evolution of surface velocity in the absence of film drainage computed using δ′0µs =5.4 × 10−2 (obtained from δ′0 = 6 × 10−5 and µs = 886), a′ = 0.1 and Γ′F0 = 0.5. Thecomputation was carried out using the finite difference method. Labels on curves correspond tothe time t′. At early times, the only significant surface velocities are seen in the region near theend of the film. At later times, the velocity profiles are more spread out, but the magnitude ofthe peak velocity is less.

Figure 20: Evolution of surfactant surface concentration in the absence of film drainage com-puted using δ′0µs = 5.4 × 10−2 (obtained from δ′0 = 6 × 10−5 and µs = 886), a′ = 0.1 andΓ′F0 = 0.5. The computation was carried out using the finite difference method. Labels oncurves correspond to the time t′. Surfactant accumulates on the film surface over time.

40

Figure 21: Evolution of surfactant surface concentration in the absence of film drainage andabsence of surface viscosity simulated using δ′0 = 6 × 10−5, µs = 0, a′ = 0.1 and Γ′F0 = 0.5.Labels on curves correspond to the time t′. Surfactant accumulates on the film surface over timemore slowly in Fig. 20 than it does here, owing to the surface viscosity affecting the results inFig. 20.

hence there are less significant Marangoni as well as surface viscous stresses.The average surfactant surface concentration over time in the case with surface viscosity is

compared with that in the case without surface viscosity and the result is presented in Fig. 22.In the case without surface viscosity, at early time, the rate of accumulation of surfactant on thesurface of lamella is faster than that in the case with surface viscosity. This is shown by thesteep increase of surface concentration at early time for the case without surface viscosity. Thesteep increase is due to the large gradient of surface tension at early time that results in strongerMarangoni effect that transports the surfactant onto the lamella surface. In the case with surfaceviscosity, this strong Marangoni effect is offset by the surface viscous effect. As a consequencethe surface velocity slows down and this reduces the rate of accumulation of surfactant ontothe lamella surface (i.e. surfactant concentration grows linearly in time, not as square root). Astime progresses the surfactant surface concentration spreads more evenly on the lamella surface,results in a weaker Marangoni effect. Consequently, at later time, the rate of accumulation ofsurfactant on the surface of lamella is slower than that at early time.

The computation for the case of small surface viscosity is expensive. The smaller the sur-face viscosity, the finer the spatial step needed to resolve the velocity field properly. The finerspatial steps requires the much finer time steps. The computation for this case where t′ = 2used 250 spatial steps and 10000 time steps. This calculation takes 30 minutes to run. Thelonger time scale requires more time steps, therefore requires finer spatial steps. In comparison,the computation of the case without surface viscosity does not require very fine spatial step.This saves computational time. For example a computation for t′ = 2 can be done using 30spatial steps and 10000 time steps. This computation requires 30 seconds to run. Moreover,in the absence of surface viscosity, not having to do ‘extra work’ to compute the velocity fieldpresumably saves computational time. In particular, in the presence of surface viscosity, the

41

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

⟨Γ’

⟩

t’

δ’0

-µ

s = 0

δ’0

-µ

s = 5.4 × 10

-2

Figure 22: Comparison of the average surfactant surface concentration over time calculatedusing assumption of the absence of surface viscosity (δ′0µs = 0, δ′0 = 6 × 10−5, µs = 0) andassumption of the presence of surface viscosity (δ′0µs = 5.4× 10−2, δ′0 = 6× 10−5, µs = 886).The computation used a′ = 0.1 and Γ′F0 = 0.5. The computation was carried out using thefinite difference method. Surface viscosity slows down the accumulation of surfactant.

velocity field must be obtained via a matrix inversion, a step that is not required when surfaceviscosity is absent. The inversion of the matrix in the solution using the finite difference methodused the standard MATLAB inversion routine which presumably slows down the computation(a more efficient tridiagonal matrix solver should be somewhat faster).

10.2. Case where δ′0µs � 1

In the case where δ′0µs � 1, it is more convenient to rescale the dimensionless time asdescribed in Subsection 9.3. The rescaled dimensionless time is independent of the film thick-ness and the bulk viscosity. With a large surface viscosity, the mass transfer on the surface isindependent of the viscosity in the bulk solution. The rescaled dimensionless equation for thiscase is as presented in Eq. (96). The equation is solved using three different methods: the finitedifference method, an integro-differential equation and an approximate analytic solution. Thedevelopment of the solution using the methods is based on the rescaled form that permits a finitelimiting value for the velocity field in the event that bulk viscosity is neglected.

The computation for the case of δ′0µs � 1 used the following parameters: δ′0µs = 5.4(obtained from δ′0 = 6× 10−5 and µs = 8.86× 104), a′ = 0.1 and Γ′F0 = 0.5. The computationcarried out for rescaled dimensionless time t′′ = 20. The dimensionless time was divided into10000 time steps, resulting in time interval ∆t′′ = 2 × 10−3. There are 200 spatial steps,resulting in spatial interval ∆x′ = 5× 10−3.

10.2.1. Solution using finite difference method for the case of δ′0µs � 1

This subsection presents the results of computation for the case where δ′0µs � 1 using theequation of surface velocity in Eq. (96). The equation is solved using a finite difference methodto obtain the profile of surface velocity at any particular time. The profile of surface velocityobtained from simulation for this case is presented in Fig. 23.

42

Figure 23: Evolution of surface velocity in the absence of film drainage calculated usingδ′0µs = 5.4 (obtained from δ′0 = 6 × 10−5 and µs = 8.86 × 104), a′ = 0.1 and Γ′F0 = 0.5.The computation was carried out using the finite difference method. Labels on curves are therescaled dimensionless time t′′. The inset is zoomed in around x′ = 1. The velocity profiles(like those in Fig. 18) show a linear variation with x′ over much of the domain, but an abruptdeviation from linearity near x′ = 1.

At early time, the magnitude of u′s (away from x′ = 1) grows. As the width of the regionnear x′ = 1 over which the system deviates from uniform Γ′ grows, the magnitude of ∂u′′s/∂x

′

outside this region also grows. This effect has also been predicted in Subsection 9.2. At latertimes however, the magnitude of ∂u′′s/∂x

′ decays due to the gradients of surface concentrationeventually decaying as more and more surfactant accumulates on the film surface. The mag-nitude of surface velocity at the end of the film is a decreasing function of time as shown bythe inset in Fig. 23. The early time behaviour has been predicted in Subsection 9.2 as before,the subsequent decay in the velocity is due to the gradient of surface concentration at x′ = 1reducing with time due to transport of surfactant onto the lamella.

The profile of surfactant surface concentration is calculated using the material point methodbased on the calculated surface velocity presented in Fig. 23. The profile of surfactant surfaceconcentration is presented in Fig. 24.

At positions far from the Plateau border, the surface velocity is proportional to the distancefrom the centre of the film. Near the Plateau border, the surface velocity profile turns aroundto satisfy the boundary condition at x′ = 1 and the symmetry point on the Plateau border.Moreover, as discussed in Subsection 9.1, the surface concentration of surfactant is uniformlydistributed along the film except at positions near the Plateau border.

10.2.2. Calculation of distribution of surfactant surface concentration using integro-differentialequation

The result of the calculation using integro-differential equation Eq. (77) (or in rescaled form,Eq. (98)) suitably discretised is presented in Fig. 25.

The average surfactant surface concentration is compared between the result from calcula-tion using the finite difference method and that using the integro-differential equation and thecomparison is presented in Fig. 26. It is shown in the figure that the finite difference method

43

Figure 24: Evolution of surfactant surface concentration in the absence of film drainage calcu-lated using parameters δ′0µs = 5.4 (obtained from δ′0 = 6×10−5 and µs = 8.86×104), a′ = 0.1and Γ′F0 = 0.5. The computation was carried out using the finite difference method. Labelson curves are the rescaled dimensionless time t′′. The profiles show a uniform concentrationregion over much of the domain with an abrupt change in concentration at the end of the domain(similar to what was shown in Fig. 17).

Figure 25: Evolution of surfactant surface concentration in the absence of film drainage calcu-lated using integro-differential equation (applicable for δ′0µs � 1), using parameters a′ = 0.1and Γ′F0 = 0.5. Labels on curves are the rescaled dimensionless time t′′. Close agreement isseen between the integro-differential equation results here (δ′0µs →∞) and the finite differenceresults (Fig. 24 with δ′0µs = 5.4).

44

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0 5 10 15 20

⟨Γ’

⟩

t"

finite differenceintegro-differential

Figure 26: Comparison of the average surfactant surface concentration over time calculatedusing the finite difference method (δ′0µs = 5.4, µs = 8.86 × 104) and the integro-differentialequation (δ′0µs � 1). The computation used a′ = 0.1 and Γ′F0 = 0.5. Close agreement is seenbetween the results.

gives similar result to that obtained using the integro-differential equation. The computation us-ing the integro-differential equation is much faster than that using the finite difference method.The finite difference method using 200 spatial steps and 10000 time steps requires 30 minutesto compute while the integro-differential equation solved using the same numbers of spatialand time steps, respectively, requires only 90 seconds computation time. One of the reasons isthat unlike the finite difference method, the numerical solution of the integro-differential equa-tion does not need a matrix inversion. As already stated previously, our use of the standardMATLAB matrix inversion route in the finite difference method is likely to be slowing downthat computation. Remember also an important restriction here: the integro-differential equa-tion only applies to the cases where δ′0µs � 1 due to the assumption of spatially uniform filmtension to develop the equation.

10.2.3. Approximate analytic solutionUsing the assumption of uniform distribution of surfactant surface coverage in the case

of large surface viscosity, the surfactant surface concentration at position away from the thinregion near x′ = 1 can be predicted using Eq. (85). The predicted value of Γ′ at x′ = 0 atvarious times calculated using Eq. (85) is compared with the ones calculated using the finitedifference method and the integro-differential equation. To match the result of the calculationusing the finite difference method and the integro-differential equation, the dimensionless timein Eq. (85) also needs to be rescaled to t′′ = t′/(δ′0µs), and results in the following equation:

Γ′0 ≈ exp

(− ln

(1

Γ′00

)exp

(− t′′a′π/6

1 + a′π/6

)). (99)

The comparison of the result calculated using Eq. (99) with the result of the calculation usingthe finite difference method and the integro-differential equation is presented in Fig. 27. In this

45

0.5

0.51

0.52

0.53

0.54

0.55

0 0.5 1 1.5 2

Γ’

x’

= 0

t"

finite differenceintegro-differential

uniform distribution

Figure 27: Comparison of the surfactant surface concentration away from x′ = 1 over timepredicted using Eq. (85) (for a near uniform surfactant distribution) with those calculated usingthe finite difference method (δ′0µs = 5.4, µs = 8.86× 104) and the integro-differential equation(δ′0µs � 1). The computation used a′ = 0.1 and Γ′F0 = 0.5. The predictions of Eq. (85) capturethe qualitative behaviour.

figure, the data is only presented up to two units of time to show the agreement of the resultsof the three methods for calculation of Γ′ at x′ = 0. At later time, the results obtained usingEq. (85) deviates from the results obtained using the two other methods. At later time, thesurfactant surface concentration at the thin region near x′ = 1 increases gradually instead ofhaving a sharp jump from Γ′ = Γ′0 to Γ′ = 1. However, Eq. (99) (and Eq. (85)) are based on theassumption that there is a jump of surface concentration over a thin region near x′ = 1. Thesedifferent approaches to the profile of surfactant surface concentration gives discrepancy betweenthe results obtained using Eq. (99) and the two other methods. It is noted that the growth rate ofsurfactant surface concentration is predicted by Subsection 9.2 to increase as the profile of Γ′ vsx′ departs from uniformity (as demonstrated by Fig. 27) – although the predictions discussed inSubsection 9.2 were actually derived only for the limit of 1− Γ′0 � 1.

10.2.4. Small variation of Γ′ in time for the case of δ′0µs � 1

In the case of large surface viscosity, it is clear from e.g. Fig 24 or 25 that the value of Γ′

does not vary rapidly in time away from the neighbourhood of Plateau border. The plot of u′′sΓ′

vs. x′ presented in Fig. 28 is instructive here: it shows that the spatial gradients of the profile ofu′′sΓ

′ (and hence the value of ∂Γ′/∂t′) are very small except at positions near the Plateau borderat early times. This small spatial gradient of u′′sΓ

′ demonstrates in fact that Γ′ does not changevery rapidly in time. The gradient in u′′sΓ

′ occurs near the Plateau border (near x′ = 1) where∂Γ′/∂x′ is large. Fig. 29 demonstrates that both for Γ′ and u′′s the most significant gradientsoccur only near x′ = 1 to satisfy the boundary condition at that point.

10.2.5. Comparison of large and small viscosity casesThe profile of surface velocity and surfactant surface concentration in Figs. 23 and 24 are de-

veloped using rescaled dimensionless time t′ = δ′0µst′′. Therefore in terms of t′, the time scales

46

Figure 28: The profile of u′′sΓ′ vs. x′ in the absence of film drainage simulated using δ′0µs = 5.4

(obtained from δ′0 = 6 × 10−5 and µs = 8.86 × 104), a′ = 0.1 and Γ′F0 = 0.5. The com-putation was carried out using the finite difference method. Labels on curves are the rescaleddimensionless time t′′. Despite abrupt spatial variations in u′s and Γ′ near the end of the film(Figs. 17 – 25), the product u′sΓ

′ doees not display abrupt spatial variation at late times.

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

Γ’

us"

, u

s"Γ’

x’

Γ’

us"

us"Γ’

Figure 29: The profile of Γ′, u′′s and u′′sΓ′ vs. x′ in the absence of film drainage simulated using

δ′0µs = 5.4 (obtained from δ′0 = 6× 10−5 and µs = 8.86× 104), a′ = 0.1 and Γ′F0 = 0.5. Thisdata was taken at t′′ = 1. The computation was carried out using the finite difference method.Abrupt spatial variations occur only near x′ = 1.

47

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

⟨Γ’

⟩

t’

δ’0

-µ

s = 0

δ’0

-µ

s = 5.4 × 10

-2

δ’0

-µ

s = 5.4

Figure 30: Comparison of the average surfactant surface concentration over time calculatedusing assumption of the absence of surface viscosity and assumption of the presence of surfaceviscosity (δ′0µs = 0 − 5.4, µs = 0 − 8.86 × 104). The simulation was using δ′0 = 6 × 10−5,a′ = 0.1 and Γ′F0 = 0.5. The computation was carried out using the finite difference method.High surface viscosity severely limits the rate at which surfactant accumulates on the film.

in those figures are much longer than the time scale used in the simulation with smaller surfaceviscosity parameter presented in Figs. 19 and 20. The average surfactant surface concentrationat any given time for the case of large surface viscosity has been calculated and the result isrescaled to match the case of small surface viscosity. The result is presented in Fig. 30. Therate of surfactant transport onto the lamella surface is much slower with large surface viscosityparameter compared with that for a small surface viscosity parameter and that without surfaceviscosity. The larger surface viscosity results in larger surface viscous effect that opposes theMarangoni effect. As the effect, the magnitude of surface velocity is much smaller, result inless surfactant transported onto the surface of the lamella.

10.3. Variation of film lengthThe variation of film length affects the characteristic time scale for moving surfactant onto

the film. A shorter film results in the shorter characteristic time. This is also demonstrated inthe definition of the characteristic time, given by L2µ/(Gδ0).

In the limit of δ′0µs � 1 (where the effects of surface viscosity are comparatively weak,and results for mass transfer should be close to those in the absence of surface viscosity) thechange of characteristic time scale is the only effect can be seen upon the change of film lengthL. A decrease in L should lead to a very substantial decrease in (dimensional) time to achievemass transfer. Not only the Marangoni forces become stronger, but also the distance over whichsurfactant needs to travel is less.

In the limit of δ′0µs � 1 although the Marangoni time scale can be reduced significantly byreducing the film length, this time scale is already so short [23] compared to other time scalesof interest, e.g. residence time in the column. Therefore, there is unlikely to be any require-ment to reduce it further. Reducing bubble size may have other beneficial effects: increasing

48

specific surface area, and thereby increasing the total amount of adsorbed surfactant present inthe column.

In the limit of δ′0µs � 1, the characteristic time is no longer L2µ/(Gδ0) but rather the valueof L2µ/(Gδ0) × δ′0µs – which evaluates to µs/G. This is independent of the film length (sothat changing film length has only a very weak effect on characteristic time scales) in the limitδ′0µs � 1.

In other words, for a decrease in film length L, surface viscosity placing more limitationson surface motion on short film, is offset by the shorter distance over which surfactant musttravel. The only residual effect is then from a′ increasing as L decreases (and this should give avery slight speed up – constraints on surfactant motion imposed by the Plateau border are lessrelevant as the film shrinks relative to the Plateau border).

Indeed whereas characteristic time scale for surfactant transfer with large surface viscosityis nominally just increased by a factor δ′0µs (the dimensionless surface viscosity parameteridentified here) compared to the low surface viscosity case, in practice it is increased by afactor δ′0µs/a

′ (a′ is the ratio between the Plateau border size a and film size L, with a′ � 1).This additional increase in time scale arises from the zero velocity constraint imposed on thesymmetry point of the Plateau border, which is relatively close to the end of the film. With thelow surface viscosity, the Marangoni time scale is estimated as 3.125× 10−2 s, and δ′0µs = 5.4and a′ = 0.1, we obtain a time scale 1.68 s within an order of magnitude of the typical residencetime in a foam fractionation column [23].

In this study the result of calculation using shorter film (L = 2 × 10−3 m, correspond toa′ = 0.25) is compared with that using longer film (L = 5 × 10−3 m, correspond to a′ = 0.1).For the purpose of this comparison, the rescaled dimensionless equation is used where t′′ =Gt/µs is independent of film length L, the being example taken is for the case where δ′0µs � 1.The profile of surfactant surface concentration for a film with L = 2 × 10−3 m is presentedin Fig. 31. This profile was calculated using the integro-differential equation, i.e. for the casewhere δ′0µs has ceased to play any role whatsoever, and the only effect of changing film lengthL is the residual effect of changing a′.

It is shown that at any time t′′, the surfactant surface concentration obtained in the calcu-lation using shorter film in Fig. 31 is larger than that obtained using longer film as presentedin Fig. 24. For further comparison, the average surfactant surface concentration for short film(L = 2 × 10−3 m) at any given time is calculated and the result compared with the result ob-tained using longer film (L = 5 × 10−3 m). The result of calculation of surfactant surfaceconcentration is presented in Fig. 32. It is apparent that for a shorter film the rate of surfactantaccumulation is faster than that for a longer film.

11. Conclusions

Simulation of surfactant transport onto a foam lamella in the presence of surface viscousstress has been carried out. The base case parameter set for the simulation was taken fromthe experimental data reported in the literature using protein bovine serum albumin (BSA) to-gether with cosurfactant propylene glycol alginate (PGA). The simulation studies the surfacevelocity in the presence of surface viscous stress. In order to compute the surface velocities,an approximation was used, whereby the Plateau border was ‘uncurled’ onto a straight line,and the surface velocity on the border itself was assumed to vary linearly with distance fromthe symmetry point of the border, but with non-linear variation permitted on the adjacent film.The surface velocities in some special cases were determined using Green’s functions, and inother cases via finite differences. The evolution of surface concentration is modelled using finitedifference approximation coupled with the material point method.

One special case considered the effect of surface viscosity upon the surface velocity wheresurface viscosity is comparatively weak, i.e. δ0µs � 1. An analytical solution is also derived

49

Figure 31: Evolution of surfactant surface concentration in the absence of film drainage simu-lated using the integro-differential equation (applicable in the limit δ′0µs � 1) Eq. (98) usingparameters: a′ = 0.25 and Γ′F0 = 0.5. Labels on curves are the rescaled dimensionless time t′′.Evolution is more rapid here than in Fig. 25, owing to the larger a′ values used here.

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

⟨Γ’

⟩

t"

L = 2 × 10-3

m

L = 5 × 10-3

m

Figure 32: Comparison of the average surfactant surface concentration over time calculatedusing two different film length (i.e. two different a′). The computation was using L = 2 ×10−3m, corresponding to a′ = 0.25 and L = 5 × 10−3m, corresponding to a′ = 0.1 alsoΓ′F0 = 0.5. The computational method was using the integro-differential equation (applicablefor δ′0µs � 1). Faster surfactant accumulation is seen with smaller L, i.e. larger a′

50

for the surface velocity in the special case where ∂ ln Γ′/∂x′ = βx′. In this case, the surface vis-cosity only affects the surface velocity near the Plateau border. The larger the surface viscosityis, the slower the surface flow near the Plateau border.

The Green’s function solution solves the case where there is a jump of surfactant surfaceconcentration at certain positions along the lamella length. In this case, there is a critical radiusof curvature of Plateau border a′crit (scaling as the square root of surface viscosity) that deter-mines (relative to pure exponential profile) the perturbation of the surface velocity profile nearthe Plateau border, in the neighbourhood of x′ = 1. When a′ < a′crit, the profile of surfacevelocity is perturbed upward (corresponding to smaller magnitude of velocity) from the profilewhen a′ = a′crit. On the other hand, there is downward perturbation (corresponding to largermagnitude of velocity) when a′ < a′crit. The largest magnitude of surface velocity is obtainedat the jump point where the gradient of surface concentration is localised.

The surface velocity at the symmetry point of the Plateau border is zero. The surfactantflux onto the film however is governed by the surface velocity at the end of the film, and thisis typically non-zero. The larger the surface viscosity is, the smaller is the magnitude of thesurface velocity due to the greater effect of surface viscous stress. Moreover, drier foams (i.e.small radius of curvature of Plateau border a′) lead to less surfactant transport onto the foamfilms.

Computation of surface velocity for arbitrary ∂ ln Γ′/∂x′ is also carried out using the finitedifference method. The finite difference solution is verified using the analytical solution in thespecial case of ∂ ln Γ′/∂x′ = βx′ and the computation using the Green’s function where thereis a jump of the value of Γ′ from Γ′ = Γ′0 at x′ ≤ 1 to Γ′ = 1 at x′ ≥ 1. The simulation resultsshow that the both the analytical and the Green’s function solutions confirm the finite differenceone.

The finite difference approximation is coupled with the material point method to simulatethe evolution of surface velocity and surfactant surface concentration. The surface velocity isslowed down near the Plateau border due to the effect of surface viscous stress, with moreovera very significant slow down when a′ � a′crit. At a later time, the profile of surface velocityis less sharp since the surfactant surface concentration spreads out more evenly on the filmsurface, which resulting in weaker Marangoni stress. At any given time, the surfactant surfaceconcentration in the presence of surface viscosity is lower than that without surface viscosity.The rate of surfactant transport is slower with a larger surface viscosity parameter.

For a large surface viscosity, δ′0µs � 1, and in the special case where surfactant concentra-tion is nearly uniform along the film, the surface velocity profile is proportional to the distancefrom the centre of the film except at positions near the Plateau border, where the velocity fieldneeds to adjust to satisfy the boundary condition at the end of film. As a consequence, the sur-factant surface concentration remains distributed almost evenly at positions far from the Plateauborder. This uniform distribution of surface coverage also results in the uniform distributionof surface tension. The profile of surfactant surface concentration can be simulated using anintegro-differential equation. The results of calculation using the integro-differential equationagree with the results of computation using the finite difference method. It is found that thevelocity scale is greatly slowed down in the high surface viscosity limit (scaling now inverselywith the surface viscosity). Moreover a small curvature for the Plateau border a′ likewise slowsdown the surface velocity at the end of the film (and hence the surfactant mass transfer onto thefilm).

Finally when a shorter film length is applied in the calculation, the characteristic time scalefor surfactant transport is also shorter.

We would expect the trends identified above to be reflected in the rate of enrichment seenin a fractionation column operated with reflux, e.g. slower enrichment with higher surfaceviscosity, slower enrichment in a very dry foams, but faster enrichment with smaller bubbles.

51

Knowing the rate at which reflux-induced Marangoni flow enriches the film surfaces is expectedto be most relevant in the case of high surface viscosity, since for lower surface viscosity, thecharacteristic time scale for Marangoni processes (see Table 1) is really very short compared toexpected film residence times in a fractionation column.

References

[1] D. Linke, H. Zorn, B. Gerken, H. Parlar, R. G. Berger, Laccase isolation by foamfractionation–new prospects of an old process, Enzyme and Microbial Technology 40(2007) 273–277.

[2] L. Du, V. Loha, R. Tanner, Modeling a protein foam fractionation process, AppliedBiochemistry and Biotechnology 84–86 (2000) 1087–1099.

[3] D. Linke, R. Berger, Foaming of proteins: New prospects for enzyme purification pro-cesses, Journal of Biotechnology 152 (2011) 125–131.

[4] B. M. Gerken, A. Nicolai, D. Linke, H. Zorn, R. G. Berger, H. Parlar, Effective enrichmentand recovery of laccase C using continuous foam fractionation, Separation and Purifica-tion Technology 49 (2006) 291–294.

[5] L. Brown, G. Narsimhan, P. C. Wankat, Foam fractionation of globular proteins, Biotech-nology and Bioengineering 36 (1990) 947–959.

[6] J. B. Winterburn, A. B. Russell, P. J. Martin, Characterisation of HFBII biosurfactantproduction and foam fractionation with and without antifoaming agents, Applied Micro-biology and Biotechnology 90 (2011) 911–920.

[7] J. B. Winterburn, A. B. Russell, P. J. Martin, Integrated recirculating foam fractionationfor the continuous recovery of biosurfactant from fermenters, Biochemical EngineeringJournal 54 (2011) 132–139.

[8] D. A. Davis, H. C. Lynch, J. Varley, The application of foaming for the recovery ofsurfactin from B. subtilis {ATCC} 21332 cultures, Enzyme and Microbial Technology 28(2001) 346–354.

[9] C.-Y. Chen, S. C. Baker, R. C. Darton, Continuous production of biosurfactant with foamfractionation, Journal of Chemical Technology & Biotechnology 81 (2006) 1915–1922.

[10] C.-Y. Chen, S. C. Baker, R. C. Darton, Batch production of biosurfactant with foamfractionation, Journal of Chemical Technology & Biotechnology 81 (2006) 1923–1931.

[11] K. Schügerl, Recovery of proteins and microorganisms from cultivation media by foamflotation, in: New Products and New Areas of Bioprocess Engineering, Springer, 2000,2000, pp. 191–233.

[12] K. H. Bahr, K. Schügerl, Recovery of yeast from cultivation medium by continuousflotation and its dependence on cultivation conditions, Chemical Engineering Science47 (1992) 11–20.

[13] R. Lemlich, Adsorptive bubble separation methods: Foam fractionation and allied tech-niques, Industrial and Engineering Chemistry 60 (1968) 16–29.

[14] S. A. Koehler, S. Hilgenfeldt, H. A. Stone, A generalized view of foam drainage: Experi-ment and theory, Langmuir 16 (2000) 6327–6341.

52

[15] W. Xie, S. J. Neethling, J. J. Cilliers, A novel approach for estimating the average bubblesize for foams flowing in vertical columns, Chemical Engineering Science 59 (2004)81–86.

[16] D. Weaire, S. Hutzler, The Physics of Foams, Oxford University Press, New York, 1999.

[17] R. Lemlich, Progress in Separation and Purification, Interscience, New York, 1968, pp.1–56.

[18] R. Lemlich, E. Lavi, Foam fractionation with reflux, Science 134 (1961) 191.

[19] P. J. Martin, H. M. Dutton, J. B. Winterburn, S. Baker, A. B. Russell, Foam fractionationwith reflux, Chemical Engineering Science 65 (2010) 3825–3835.

[20] P. Stevenson, G. J. Jameson, Modelling continuous foam fractionation with reflux, Chem-ical Engineering and Processing: Process Intensification 46 (2007) 1286–291.

[21] D. Vitasari, P. Grassia, P. Martin, Simulation of dynamics of adsorption of mixed protein-surfactant on a bubble surface, Colloids and Surfaces A: Physicochemical and EngineeringAspects 438 (2013) 63–76.

[22] R. Miller, V. B. Fainerman, M. E. Leser, M. Michel, Kinetics of adsorption of proteinsand surfactants, Current Opinion in Colloid & Interface Science 9 (2004) 350–356.

[23] D. Vitasari, P. Grassia, P. Martin, Surfactant transport onto a foam lamella, ChemicalEngineering Science 102 (2013) 405–423.

[24] I. B. Ivanov, K. D. Danov, K. P. Ananthapadmanabhan, A. Lips, Interfacial rheology ofadsorbed layers with surface reaction: On the origin of the dilatational surface viscosity,Advances in Colloid and Interface Science 114–115 (2005) 61–92.

[25] A. D. Barber, S. Hartland, The effects of surface viscosity on the axisymmetric drainageof planar liquid films, Canadian Journal of Chemical Engineering 54 (1976) 279–284.

[26] J. T. Petkov, K. D. Danov, N. D. Denkov, R. Aust, F. Durst, Precise method for measuringthe shear surface viscosity of surfactant monolayers, Langmuir 12 (1996) 2650–2653.

[27] Z. A. Zell, A. Nowbahar, V. Mansard, L. G. Leal, S. S. Deshmukh, J. M. Mecca, C. J.Tucker, T. M. Squires, Surface shear inviscidity of soluble surfactants, Proceedings of theNational Academy of Sciences 111 (2014) 3677–3682.

[28] R. A. Leonard, R. Lemlich, A study of interstitial liquid flow in foam. Part I. Theoreticalmodel and application to foam fractionation, AIChE J. 11 (1965) 18–25.

[29] L. Scriven, Dynamics of a fluid interface equation of motion for newtonian surface fluids,Chemical Engineering Science 12 (1960) 98–108.

[30] L. E. Scriven, C. V. Sternling, On cellular convection driven by surface-tension gradients:effects of mean surface tension and surface viscosity, Journal of Fluid Mechanics 19(1964) 321–340.

[31] P. G. Saffman, M. Delbrück, Brownian motion in biological membranes, Proceedings ofthe National Academy of Sciences 72 (1975) 3111–3113.

[32] P. G. Saffman, Brownian motion in thin sheets of viscous fluid, Journal of Fluid Mechanics73 (1976) 593–602.

53

[33] P. S. Stewart, S. H. Davis, Dynamics and stability of metallic foams: Network modeling,Journal of Rheology 56 (2012) 543–574.

[34] P. S. Stewart, S. H. Davis, Self-similar coalescence of clean foams, Journal of FluidMechanics 722 (2013) 645–664.

[35] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz, G. F. Hewitt, The dynamics of Marangoni-driven local film drainage between two drops, Journal of Colloid and Interface Science241 (2001) 233–247.

[36] I. B. Ivanov, Effect of surface mobility on the dynamic behavior of thin liquid films, Pureand Applied Chemistry 52 (1980) 1241–1262.

[37] I. B. Ivanov, D. S. Dimitrov, Thin Liquid Films: Fundamentals and Applications, MarcelDekker, Incorporated, 1988, pp. 382–489.

[38] P. A. Kralchevsky, K. D. Danov, N. D. Denkov, Handbook of Surface and Colloid Chem-istry, Third Edition, Taylor & Francis, 1997, pp. 333–477.

[39] S. I. Karakashev, D. S. Ivanova, Z. K. Angarska, E. D. Manev, R. Tsekov, B. Radoev,R. Slavchov, A. V. Nguyen, Comparative validation of the analytical models for theMarangoni effect on foam film drainage, Colloids and Surfaces A: Physicochemical andEngineering Aspects 365 (2010) 122–136.

[40] B. P. Radoëv, D. S. Dimitrov, I. B. Ivanov, Hydrodynamics of thin liquid films. Effect ofthe surfactant on the rate of thinning, Colloid and Polymer Science 252 (1974) 50–55.

[41] O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’sexperiments, including an experimental determination of the viscosity of olive oil, Philo-sophical Transactions of the Royal Society of London 177 (1886) 157–234.

[42] T. T. Traykov, I. B. Ivanov, Hydrodynamics of thin liquid films. Effect of surfactants onthe velocity of thinning of emulsion films, International Journal of Multiphase Flow 3(1977) 471–483.

[43] M. Durand, H. A. Stone, Relaxation time of the topological T1 process in a two-dimensional foam, Phys. Rev. Lett. 97 (2006) 226101.

[44] A. M. Kraynik, Foam drainage, Technical Report SAND83-0844, Sandia National Labo-ratories, Albuquerque, New Mexico, USA, 1983.

[45] Y. Tian, R. G. Holt, R. E. Apfel, Investigation of liquid surface rheology of surfactantsolutions by droplet shape oscillations: Experiments, Journal of Colloid and InterfaceScience 187 (1997) 1–10.

[46] P. Stevenson, Remarks on the shear viscosity of surfaces stabilised with soluble surfac-tants, Journal of Colloid and Interface Science 290 (2005) 603–606.

[47] S. A. Koehler, S. Hilgenfeldt, E. R. Weeks, H. A. Stone, Drainage of single Plateauborders: Direct observation of rigid and mobile interfaces, Phys. Rev. E 66 (2002) 040601.

[48] X. Liu, J. H. Duncan, The effects of surfactants on spilling breaking waves, Nature 421(2003) 520–523.

54

[49] R. L. Kao, D. A. Edwards, D. Wasan, E. Chen, Measurement of interfacial dilatationalviscosity at high rates of interface expansion using the maximum bubble pressure method.I. Gas-liquid surface, Journal of Colloid and Interface Science 148 (1992) 247–256.

[50] P. Grassia, E. J. Hinch, Computer simulations of polymer chain relaxation via Brownianmotion, Journal of Fluid Mechanics 308 (1996) 255–288.

[51] P. S. Doyle, E. S. G. Shaqfeh, Dynamic simulation of freely-draining, flexible bead-rodchains: Start-up of extensional and shear flow, Journal of non-Newtonian Fluid Mechanics76 (1998) 43–78.

[52] T. Sridhar, An overview of the project M1, Journal of non-Newtonian Fluid Mechanics35 (1990) 85–92.

[53] V. B. Fainerman, E. H. Lucassen-Reynders, R. Miller, Description of the adsorption be-haviour of proteins at water/fluid interfaces in the framework of a two-dimensional solu-tion model, Advances in Colloid and Interface Science 106 (2003) 237 – 259.

[54] B. Embley, P. Grassia, Viscous froth simulations with surfactant mass transfer andMarangoni effects: Deviations from Plateau’s rules, Colloids and Surfaces A: Physic-ochemical and Engineering Aspects 382 (2011) 8–17.

[55] N. Loney, Applied mathematical methods for chemical engineers, CRC Press, 2001.

[56] W. Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, CambridgeUniversity Press, 2007.

[57] C. Barentin, C. Ybert, Surface shear viscosity of Gibbs and Langmuir monolayers, Journalof Fluid Mechanics 397 (1999) 331–349.

[58] A. M. Poskanzer, F. C. Goodrich, Surface viscosity of sodium dodecyl sulfate solutionswith and without added dodecanol, Journal of Physical Chemistry 79 (1975) 2122–2126.

[59] P. A. Harvey, A. V. Nguyen, G. J. Jameson, G. M. Evans, Influence of sodium dodecylsulphate and dowfroth frothers on froth stability, Minerals Engineering 18 (2005) 311–315.

[60] A. Patist, T. Axelberd, D. O. Shah, Effect of long chain alcohols on micellar relaxationtime and foaming properties of sodium dodecyl sulfate solutions, Journal of Colloid andInterface Science 208 (1998) 259–265.

[61] O. Pitois, C. Fritz, M. Vignes-Adler, Liquid drainage through aqueous foam: study of theflow on the bubble scale, Journal of Colloid and Interface Science 282 (2005) 458–465.

[62] A. Saint-Jalmes, Y. Zhang, D. Langevin, Quantitative description of foam drainage: Tran-sitions with surface mobility, European Physical Journal E 15 (2004) 53–60.

55

Appendix A. Variation of surface viscosity from various measurement methods

It has been found that different methods to measure surface viscosity result in differentmeasured values. Most of the existing rheometers have geometries that are specifically designedto excite shear deformation, therefore they purport to measure the shear rheology. On the otherhand, in the measurement of surface viscosity additional stresses may exist. Translating themeasurement probes may deform the surface via compression, dilation and extension in additionto shear. These deformations will produce stresses that also affect the results of the measurement[27]. The various values of surface viscosity as well as the methods utilised as have beenreported in the previous studies are summarised in Tab. 4. There are two different classesof methods to obtain the surface viscosity: by direct measurement and deduction from otherrheological measurements.

It is important to note that there is not just one surface viscosity but rather two: a surfaceshear viscosity and a surface dilational viscosity (quite apart from any surface elastic effects thatdilation might produce). Depending on the geometry of the surface deformation, either one orthe other or a combination of both surface viscosities might be relevant. There is no guaranteethat the surface shear viscosity and surface dilational viscosity will be the same: indeed theycould be very different in magnitude. In that case, the surface viscosity measured would bevery dependent on the geometry of the surface motion, which may explain part of the variationobserved in Tab. 4.

For the particular geometry that we consider (as in that considered by Durand and Stone[43]) it turns out that what matters is the sum of surface shear and dilational viscosities, andthis sum is what we refer to simply as ‘surface viscosity’ in the main text. Note that the data ofDurand and Stone [43] (shear plus dilation) and Zell et al. [27] (pure shear) would be compatibleif the surface dilational viscosity happened to be much larger than the surface shear viscosity,but Petkov et al. [26] predict the opposite.

Appendix B. Solution for special case where ∂ ln Γ′/∂x′ = βx′

The equation of surface viscosity for the special case where ∂ ln Γ′/∂x′ = βx′ is a non-homogeneous differential equation Eq. (12) as follows:

∂2u′s∂x′2

− u′sδ′0µs/3

=βx′

δ′0µs.

Each non-homogeneous equation has a corresponding homogeneous equation. The corre-sponding homogeneous form of Eq. (12) is as follows:

∂2u′s∂x′2

− u′sδ′0µs/3

= 0. (B.1)

Therefore, the complementary solution for the differential equation in Eq. (12) is as follows:

u′s(c) = C1 exp

(x′√δ′0µs/3

)+ C2 exp

(− x′√

δ′0µs/3

)(B.2)

where C1 and C2 are constants to be defined later. For this spacial case, the particular solutionfor the differential equation in Eq. (12) can be solved as follows:

u′s(p) = −βx′

3. (B.3)

This particular solution is for the special case where ∂Γ′/∂x′ = βx′. This particular solution

56

Table 4: Previously reported surface viscosity of SDS solution. The values were taken from asummary developed by Stevenson [46] and some other studies. In this table the value of surfaceviscosity is presented in µPa m s .

Investigator Shear/dilationalviscosity

Concentration(g/l)

µs (µPa m s) Method

Direct measurementZell et al. [27] shear 2.3 < 0.01 Drag coefficient of a

micro buttonPetkov et al. [26] shear 12.2 1.45 Drag coefficient of a

sphereBarentin et al. [57] shear 10 2.6 Drag coefficient of a

diskPoskanzer andGoodrich [58]

shear 1.5 10 Rotating wall knife-edge viscometer

Harvey et al. [59] shear 0.25 2 Deep-channel surfaceviscometer

Patist et al. [60] shear 7.5 5× 103 Deep-channel surfaceviscometer

Indirect measure-mentPitois et al. [61] shear 3.0 0.02 Deduced from drainage

rate of isolated Plateauborder

Koehler et al. [47] shear 1.0 0.036 Deduced from veloc-ity profile in a drainingPlateau border

Saint-Jalmes et al.[62]

shear 10 0.08 Deduced from forceddrainage experiments

Liu and Duncan [48] no information 1.73 1× 103 Deduced from the prop-agation characteristicsof longitudinal surfac-tant waves

Durand and Stone[43]

mixed 4.8 (1.3 ± 0.7) ×103

Deduced from T1 dy-namics experiments

57

u′s(p) corresponds to the solution outside the boundary layer −13∂Γ′/∂x′. The solution of the

differential equation in Eq. (12) can then be presented as follows:

u′s = C1 exp

(x′√δ′0µs/3

)+ C2 exp

(− x′√

δ′0µs/3

)− βx′

3. (B.4)

Eq. (B.4) can be rewritten into this following equation:

u′s = C1 exp

(x′ − 1√δ′0µs/3

)+ C2 exp

(− x′√

δ′0µs/3

)− βx′

3(B.5)

where C1 = C1 exp(1/√δ′0µs/3). For a very small value of δ′0µs, the first term of the right hand

side of Eq. (B.5) is negligible near x′ = 0. Therefore, the boundary condition at x′ = 0 givesC2 = 0 as the value of u′s(p) is already zero at that boundary. As a result, we have the equationfor surface velocity as follows:

u′s = −βx′

3+ C1 exp

(x′ − 1√δ′0µs/3

). (B.6)

where C1 is a constant that satisfies the boundary condition at x′ = 1. Substituting Eq. (B.6)into Eq. (9) at the boundary where x′ = 1 results in the following equation:

−β3

+ C1 +a′π

6

∂u′s∂x′

∣∣∣∣x′=1

= 0. (B.7)

The value of ∂u′s/∂x′∣∣x′=1

can then be determined from Eq. (B.6) as follows:

∂u′s∂x′

∣∣∣∣x′=1

= −β3

+C1√δ′0µs/3

. (B.8)

Substitution of Eq. (B.8) to Eq. (B.7) results in the following equation:

β

3

(1 +

a′π

6

)= C1

(1 +

a′π

6√δ′0µs/3

). (B.9)

Therefore, the value of C1 can be solved as follows:

C1 =β (1 + a′π/6)

3(

1 + a′π/(6√δ′0µs/3)

) . (B.10)

Substitution of Eq. (B.10 ) into Eq. (B.6) results in the following equation:

u′s = −βx′

3+

β (1 + a′π/6)

3(

1 + a′π/(6√δ′0µs/3)

) exp

(x′ − 1√δ′0µs/3

). (B.11)

Eq. (B.11) automatically satisfies the boundary condition at x′ = 0 provided δ′0µs is very small.

58

Appendix C. Solution using the Green’s function for the case where the value of Γ′ jumpsfrom Γ′ = Γ′

0 for x′ ≤ x′0 to Γ′ = 1 for x′ ≥ x′

0

The complementary solution for Eq. (5) is as presented in Eq. (B.2) as follows:

u′s(c) = C1 exp

(x′√δ′0µs/3

)+ C2 exp

(− x′√

δ′0µs/3

)

Due to the step change in Γ′ leading to Eq. (14), Eq. (5) can be solved using a Green’sfunction G as follows:

u′s = ln

(1

Γ′0

)G =

d1 exp

(x′√δ′0µs/3

)+ d2 exp

(− x′√

δ′0µs/3

)if x′ < x′0

d3 exp

(x′√δ′0µs/3

)+ d4 exp

(− x′√

δ′0µs/3

)if x′ > x′0.

(C.1)

Within the range 0 ≤ x′ ≤ 1 the corresponding equations for u′s for the positions away fromthe point x′0 can be determined as follows:

u′s = d1 exp

(−x′0 + x′√δ′0µs/3

)+ d2 exp

(−x′√δ′0µs/3

)(C.2)

for x′ < x′0 and

u′s = d3 exp

(x′0 − 2 + x′√

δ′0µs/3

)+ d4 exp

(x′0 − x′√δ′0µs/3

)(C.3)

for x′ > x′0, where the parameters d1, d3 and d4 are defined by d1 = d1 exp(x′0/√δ′0µs/3),

d3 = d3 exp((−x′0 + 2)/√δ′0µs/3) and d4 = d4 exp(−x′0/

√δ′0µs/3).

If δ′0µs � 1 then in the neighbourhood of x′ = x′0 (with x′0 being further than order√δ′0µs

from either boundary) this will be dominated by:

u′s = d1 exp

(−x′0 + x′√δ′0µs/3

)(C.4)

for x′ < x′0 and

u′s = d4 exp

(x′0 − x′√δ′0µs/3

)(C.5)

for x′ > x′0. If u′s is continuous at x′ = x′0, it is required that d1 = d4.To determine the value of d1 = d4, Eq. (5) is integrated around x′ = x′0 to obtain the

following equation: ∫ x′+0

x′−0

u′sdx′ = −1

3ln Γ′

∣∣∣∣x′+0

x′−0

+δ′0µs

3

∂u′s∂x′

∣∣∣∣x′+0

x′−0

. (C.6)

The integral of u′s over this infinitesimal interval is zero, therefore we have the following equa-tion:

1

3ln Γ′

∣∣∣∣x′+0

x′−0

=δ′0µs

3

∂u′s∂x′

∣∣∣∣x′+0

x′−0

. (C.7)

59

The solution of Eq. (C.7) results in the following equation:

1

3ln

(1

Γ′0

)= −d1

√δ′0µs

3

[exp

(x′0 − x′+0√δ′0µs/3

)+ exp

(−x′0 + x′−0√δ′0µs/3

)]. (C.8)

We can take that:x′0 − x′+0√δ′0µs/3

≈ 0 (C.9)

and−x′0 + x′−0√δ′0µs/3

≈ 0 (C.10)

therefore Eq. (C.8) can be simplified into the following form:

1

3ln

(1

Γ′0

)= −2d1

√δ′0µs

3. (C.11)

Therefore, the value of d1 = d4 can be determined as follows:

d1 = d4 = − ln(1/Γ′0)

6√δ′0µs/3

. (C.12)

The boundary condition at x′ = 0 is u′s = 0, therefore we can solve the value of d2 usingEq. (C.2) as follows:

0 = d1 exp

(−x′0√δ′0µs/3

)+ d2. (C.13)

Therefore the value of d2 can be determined as follows:

d2 = −d1 exp

(−x′0√δ′0µs/3

). (C.14)

In this case, we have δ′0µs � 1 and x′0 is typically order unity. Therefore, for almost all x′0 wehave x′0 �

√δ′0µs/3 that gives the value of d2 being negligibly small. The value of d3 is solved

using the boundary condition at x′ = 1 as presented in Eq. (9) to obtain the following equation:

d3 exp

(x′0 − 1√δ′0µs/3

)+ d4 exp

(x′0 − 1√δ′0µs/3

)+

a′π/6√δ′0µs/3

[d3 exp

(x′0 − 1√δ′0µs/3

)− d4 exp

(x′0 − 1√δ′0µs/3

)]= 0. (C.15)

Eq. (C.15) can be simplified into the following equation:

d3

(1 +

a′π/6√δ′0µs/3

)+ d4

(1− a′π/6√

δ′0µs/3

)= 0. (C.16)

60

Therefore, we have the value of d3 as follows:

d3 = −d4

(1− a′π/6√

δ′0µs/3

)

1 + a′π/6√δ′0µs/3

. (C.17)

Appendix D. Solution using the Green’s function for the case where the value of Γ′ jumpsfrom Γ′ = Γ′

0 for x′ ≤ 1 to Γ′ = 1 for x′ ≥ 1

The surface velocity for x′ ≤ 1 can be described by the following equation:

u′s = d5 exp

(−1 + x′√δ′0µs/3

)+ d6 exp

(−x′√δ′0µs/3

). (D.1)

Near x′ = 1, for δ′0µs � 1, the second exponential on the right hand side is very small andnegligible, therefore u′s can be determined in this region as follows:

u′s ≈ d5 exp

(−1 + x′√δ′0µs/3

). (D.2)

In fact, Eq. (D.2) is the same form as Eq. (C.4) when x′0 is shifted to x′0 = 1. The value of d5 isdetermined by integrating Eq. (5) around x′ = 1 to obtain this following equation:

1

3ln Γ′

∣∣∣∣1

1−=δ′0µs

3

∂u′s∂x′

∣∣∣∣1

1−. (D.3)

Therefore we have the following equation:

1

3ln

(1

Γ′0

)=δ′0µs

3

(∂u′s∂x′

∣∣∣∣x′=1

− d5√δ′0µs/3

). (D.4)

At x′ = 1 we have u′s = d5. Applying this value of u′s into the boundary condition at x′ = 1 aspresented in Eq. (9) one obtains the following equation:

d5 +a′π

6

∂u′s∂x′

∣∣∣∣x′=1

= 0. (D.5)

Substituting Eq. (D.5) into Eq. (D.4) we obtain this following equation:

1

3ln

(1

Γ′0

)= d5

δ′0µs3

(− 6

a′π− 1√

δ′0µs/3

). (D.6)

Eq. (D.6) can also be expressed in the term of a′crit in Eq. (21) as presented in the followingequation:

1

3ln

(1

Γ′0

)= −d5

√δ′0µs

3

(a′crita′

+ 1

). (D.7)

Therefore d5 can be determined as follows:

d5 = − ln(1/Γ′0)

3√δ′0µs/3

(a′crita′ + 1

) . (D.8)

61

If a′ � a′crit the value of d5 will be of similar order of magnitude as d4 given by Eq. (C.12)although it is twice as big as d4. On the other hand if a′ � a′crit, the value of d5 will be smallerin magnitude than d4: the foam is now sufficiently dry, i.e. the value of a′ is sufficiently small,that the effect of a jump in Γ′ at the end of the film feels the limitation of the zero velocityconstraint at the symmetry point of the Plateau border.

The value of d6 is determined a posteriori using the boundary condition at x′ = 0 whereu′s = 0 to obtain this following equation:

d6 = −d5 exp

(−1√δ′0µs/3

). (D.9)

Eq. (D.9) has the same form as Eq. (C.14) when x′0 is shifted to x′0 = 1. For a small√δ′0µs/3

the value of d6 is much smaller than d5 thus negligible as anticipated above.

Appendix E. Equation of surface velocity in the case where δ′0µs is very large

The derivation of the surface velocity equation in the case where δ′0µs is very large is asfollows:

µs∂u′s∂x′

=1

δ′0ln Γ′ + C (E.1)

where C is an integration constant. Using the boundary condition where u′s = 0 at x′ = 0,integration of Eq. (E.1) results in the following equation:

µsu′s =

1

δ′0

∫ x′

0

ln Γ′dx′ + Cx′. (E.2)

Applying the boundary condition at x′ = 1, the value of C then can be solved as follows:

C = −µsu′s|x′=1

a′π/6. (E.3)

As a consequence, Eq. (E.1) can be solved giving the following equation:

µs∂u′s∂x′− 1

δ′0ln Γ′ = −µs

u′s|x′=1

a′π/6. (E.4)

The left hand side of Eq. (E.4) represents the (uniform) extra (viscous) tension on the film dueto a combination of surface viscosity and Gibbs elasticity, which is equal to the extra tension onthe Plateau border as presented in the right hand side of the equation.

Integration of Eq. (E.4) from x′ = 0 to x′ = 1 results in the following equation:

µsu′s|x′=1

a′π/6=

1δ′0

∫ 1

0ln Γ′dx′

1 + a′π/6. (E.5)

Substituting Eq. (E.5) into Eq. (E.4), we obtain this following equation:

µs∂u′s∂x′− 1

δ′0ln Γ′ = −

1δ′0

∫ 1

0ln Γ′dx′

1 + a′π/6. (E.6)

62

Integration of Eq. (E.6) from x′ = 0 to x′ = 1 results in the equation for u′s as follows:

u′s = − 1

δ′0µs

[∫ x′

0

ln

(1

Γ′

)dx′ − x′

1 + a′π/6

∫ 1

0

ln

(1

Γ′

)dx′]. (E.7)

Appendix F. Case where δ′0µs is very large: derivation of approximation for the casewhere Γ′

0 is close to unity

For the special case when Γ′0 is close to unity we can make the following simplification ofEq. (88):

ln Γ′ ≈ −(1− Γ′0) + exp

(−1− x′

αa′

)(1− Γ′0) . (F.1)

Therefore, the spatial derivative of ln Γ′ can be derived as follows:

∂ ln Γ′

∂x′=

1

αa′exp

(−1− x′

αa′

)(1− Γ′0) . (F.2)

In the case of high surface viscosity, there is uniform tension on the film surface, therefore theequation for surface velocity is expressed as follows:

∂ ln Γ′

∂x′= δ′0µs

∂2u′s∂x′2

. (F.3)

Substitution of Eq. (88) into Eq. (F.3) results in the following equation:

1

αa′exp

(−1− x′

αa′

)(1− Γ′0) = δ′0µs

∂2u′s∂x′2

. (F.4)

The solution of u′s is given in Eq. (F.6) below, but a detailed derivation now follows.Integration of Eq. (F.4) followed by further integration results in the following equation:

u′s =1

δ′0µs

[αa′ exp

(−1− x′

αa′

)(1− Γ′0) + C1x

′ + C2

](F.5)

where C1 and C2 are integration constants. Using the boundary condition where u′s = 0 atx′ = 0 it can be calculated that C2 = 0 as the exponential term on the right hand side ofEq. (F.6) has very a small value at that boundary, a′ being small. Therefore, Eq. (F.6) can alsobe presented using the following equation:

u′s =1

δ′0µs

[αa′ exp

(−1− x′

αa′

)(1− Γ′0)− Cx′(1− Γ′0)

](F.6)

where C = −C1/(1− Γ′0). The derivative of u′s can be derived as follows:

∂u′s∂x′

=1

δ′0µs

[exp

(−1− x′

αa′

)(1− Γ′0)− C(1− Γ′0)

]. (F.7)

The value of C can be solved by substitution of Eqs. (F.6) and (F.7) into Eq. (E.4) whichuses the assumption of uniformity of the surface plus viscous tension on the film. Eq. (E.4) isas follows:

∂u′s∂x′

+1

δ′0µsln

(1

Γ′

)= −u

′s|x′=1

a′π/6.

63

The value of ln(1/Γ′) is substituted from Eq. (88) after rearrangement resulting in the followingequation:

ln

(1

Γ′

)≈ (1− Γ′0)− exp

(−1− x′

αa′

)(1− Γ′0) .

Therefore we can obtain the following equation:

1

δ′0µs

[exp

(−1− x′

αa′

)(1− Γ′0)− C(1− Γ′0)

]

+1

δ′0µs

[(1− Γ′0)− exp

(−1− x′

αa′

)(1− Γ′0)

]

= − 1

δ′0µs

1

a′π/6[αa′ (1− Γ′0)− C(1− Γ′0)] . (F.8)

Eq. (F.8) can be simplified into the following form:

−1 + C =(−αa′ + C)

a′π/6. (F.9)

Therefore, the constant C can be defined as follows:

C =

(1 +

α

π/6

)(a′π/6

1 + a′π/6

). (F.10)

As a result the equation for surface velocity for the special case where Γ′0 is close to 1 can bepresented as follows:

u′s =1

δ′0µs

[αa′ exp

(−1− x′

αa′

)(1− Γ′0)−

(1 +

α

π/6

)(a′π/6

1 + a′π/6

)x′(1− Γ′0)

]. (F.11)

64

Chapter 7

Conclusions

Foam fractionation is a method of separation of surface active materials from solution

using the principle of adsorption of the surface active substance onto the interface of air

bubbles within a rising foam. The foam fractionation operation is normally carried out

within a foam fractionation column, where the enriched product is collected from the

top of the column. To improve the separation efficiency, some columns employ a reflux

system where some of fractionation product is returned to the top of the column.

There are two steps of adsorption. The first step is diffusion of surface active material

from the bulk solution to the layer next to the surface named the subsurface. The diffusion

follows the Fick’s second law of diffusion. The next step is adsorption of surface active

material from the subsurface onto the surface of air bubbles. For a single surfactant,

the adsorption equilibrium can follow the Henry or the Langmuir isotherm. The rate

limiting step is the diffusion, therefore the surface is assumed to be in equilibrium with

the subsurface. The differential equation of the diffusion can be solved using the finite

difference method as presented in Chapter 3. This process can also be modelled using the

Ward-Tordai equation [61], which only evaluates the concentration in the subsurface and

on the surface (Chapter 3). The computation using the Ward-Tordai equation is orders of

magnitude faster than the one using the finite difference method.

69

Separation of protein from aqueous solutions using foam fractionation columns be-

comes more widely used in this recent time. In some cases, surfactant may added to the

solution to stabilize the foam. The adsorption behaviour of mixed protein-surfactant is

described in Chapter 4. The adsorption behaviour of mixed protein-surfactant is different

from that of single surfactant, where there is competition between protein and surfactant

molecules to occupy the interface. The Frumkin isotherm represents the equilibrium of

adsorption on the bubble surface. The diffusion of protein is slower than that of surfac-

tant. However, the protein will displace surfactant once it reaches the interface due to

the higher surface affinity of protein. However, there is a critical affinity of protein for

displacement of surfactant to occur. This critical affinity need not necessarily be higher

than the affinity of surfactant. However, the displacement is more likely to occur when

the affinity of protein is much higher and the diffusivity of protein is lower than those of

surfactant.

Due to low liquid fraction (< 5%) in the foam, the bubbles form polyhedral shapes

separated by liquid films named foam lamellae. Three films meet at a channel named

Plateau border. The adsorption of surface active materials occurs from the bulk liquid

within the lamellae as well as within the Plateau borders. Due to the reflux system, the

concentration of surface active material in the Plateau border is higher than that within the

lamella. When the adsorption takes place, the concentration of surface active material on

the surface of Plateau border is higher than that on the surface of lamella. This gradient of

surface concentration, hence gradient of surface tension causes the so called Marangoni

flow, from a region with low surface tension to a region with higher surface tension.

Within the lamella, liquid drainage also takes place due to the curvature of the Plateau

border that results in a lower pressure within the Plateau border. The interaction between

the Marangoni force and the film drainage determines the transport of surfactant or other

surface active materials onto the foam lamella. Chapter 5 studies the model of surfactant

transport onto a foam lamella. This model also applies for other surface active materials.

In this study there are two different models of film drainage used for calculation of

evolution of surfactant surface concentration on the interface of a foam lamella. One

70

model is assuming that the lamella surface is completely mobile, therefore the the liquid

forms a plug-like flow profile [28]. The other model is using assumption that the surface

of the lamella is completely rigid and the flow profile is parabolic [84]. The film drainage

modelled using a mobile interface is much faster than that modelled using assumption

of a rigid interface. The film drainage dominates the Marangoni flow in the case of a

lamella with a mobile interface, therefore surfactant is washed away from the surface of

the foam lamella. Having a mobile interface, a film possibly achieves the thickness of

a common black film, when the drainage stops to occur, during the residence time in a

foam fractionation column with reflux. In the absence of film drainage, at the thickness

of common black film, surfactant will accumulate on the surface of the lamella. The

desirable condition for operation of a foam fractionation column is when the Marangoni

flow dominates the film drainage which can be achieved by employing surfactant that

gives a rigid interface.

Surface viscosity takes effect when there is motion on the surface of a foam lamella.

The surface viscous force generated opposes the direction of the surface motion. This

present study models the surfactant transport onto a foam lamella in the presence of sur-

face viscosity when there is no film drainage. In the absence of film drainage, the surface

motion, hence the transport of surfactant, is a result of the Marangoni effect due to the

gradient of surface concentration. Therefore, surfactant accumulates on the lamella sur-

face due to surface motion towards the centre of the lamella. The surface viscous effect

reduces the amount of surfactant transport and a larger the surface viscosity results in

less surfactant transport onto a foam lamella. For a large surface viscosity, not only the

surfactant transport is slow, but also the profile of surface velocity is proportional to the

distance from the centre of a lamella, resulting in nearly uniform distribution of surfac-

tant surface coverage, except within a boundary layer near the Plateau border. In addition,

the characteristic time scale required for surfactant transport is shorter with a shorter film

length.

71

Chapter 8

Future work

The possible direction for further studies in this field is enormous, and a number of pos-

sible ideas can be discussed. More details can be put into the model of adsorption of

mixed protein-surfactant on the film surface by considering the nature of protein and sur-

factant. The model for surfactant transport onto a foam lamella can also be improved and

make it more available for general cases. Moreover, the knowledge of adsorption and

and transport of protein and/or surfactant onto a foam lamella will give a basic theory

for the protein/surfactant mass transfer in a foam network. In this section, a few possible

directions to the future research based on the outcome of this thesis are discussed.

8.1 Adsorption of mixed protein-surfactant onto a bub-

ble surface

Protein molecules may undergo changes of conformational states when adsorbed onto

the surface of an air bubble [24, 110]. These conformational changes not only affect the

surface rheology [111], but also affect the adsorption dynamics [22, 112]. The current

proposed model does not consider the conformational change of protein in the adsorption

72

of mixed protein-surfactant. As a consequence, the simulation using assumption of a

single protein conformational state either underestimates or overestimates the real event

when there is variation of protein conformational states. Therefore, a more comprehensive

model for dynamics of adsorption of mixed protein-surfactant on a bubble surface which

counts the changes of protein states is required.

Some proteins have ionised groups. When this type of protein is mixed with ionic

surfactants, the ionised groups experience Coulombic interactions with the oppositely

charged surfactants and form complexes [71]. The surface activity of these complexes is

different from pure protein or surfactant, hence the adsorption behaviour is also different.

Mixture of protein and ionic surfactant are commonly found in industry as well as in foam

system. Therefore, it is also important to study the adsorption behaviour of mixed protein

and ionic surfactant.

8.2 Surfactant transport onto a foam lamella

Desorption of surfactant from the surface of foam lamella may occur in a system with

soluble surfactant. Convection will occur in the interior of the film as the effect of des-

orption of surfactant [113]. Near the film surface, there will be convection of materials

in the direction away from the Plateau border. However, near the centre of the film this

stream will penetrate deeper into the film interior and will change direction towards the

Plateau border. This will cause some of the surfactant to leak back to the Plateau border

and reduce the benefit of reflux in the foam column. A study of mass transfer of this sol-

uble surfactant is therefore necessary for determination of separation efficiency in a foam

fractionation column.

A foam film is not always flat and the film thickness is not always uniform. Hydro-

dynamical instability may occur within the film. This hydrodynamical instability results

in asymmetrical film drainage [114]. Moreover, a peripheral dimple, where the central

73

part of the film is thicker than its periphery, commonly occurs, particularly for a film with

large radius [115]. The non-uniformity of film thickness affects the transport of surfac-

tant as well as the film drainage. A model of surfactant transport onto a foam lamella with

asymmetric drainage can be developed for a more accurate description of the performance

of a foam fractionation column.

The model considered here assumes a foam film in contact with a Plateau border of

fixed surfactant concentration. In reality as a foam film moves through a fractionation

column, it sees different surfactant concentrations in different Plateau borders depending

on where it currently is in the column [30], whilst the Plateau borders themselves change

their surfactant concentration (i.e. become depleted as the liquid draining through the

loses surfactant to neighbouring borders). Moreover, if the column is operated in a batch

wise (rather than continuous) fashion, surfactant concentrations vary with time as well as

space. All these effects need to be incorporated into a full model of foam fractionation

i.e. the surfactant transport onto films needs to be coupled to the mass balance in Plateau

borders and to Plateau border liquid drainage.

For the case of surfactant transport onto a foam lamella incorporating the surface

viscous effect in this present study, it has been assumed that there has been a very simple

1-D flow field on the film surface, with the Plateau border adjacent to the film being

‘uncurled’ into a flat surface and with zero flow assumed on the symmetry line on the

surface of the Plateau border. In reality, there could well be flow along the symmetry line

of the border, with surfactant transport switching direction from along the border (on the

symmetry line) to away from the border (where the border joins the film). If this case

occurs in the system, the flow field on the surface of the film and the Plateau border will

be no longer a one-dimensional model. The model of surfactant transport onto a foam

lamella in the presence of surface viscous effect in the present study does not incorporate

film drainage. In reality, there is suction of liquid from the film towards the Plateau border

due to lower pressure within the Plateau border [28]. If the film drainage is very fast, like

the one experienced by a film with a mobile interface, surfactant will not accumulate on

the surface of the film but instead will be washed away towards the Plateau border. The

74

surface viscous effect reduces the mobility of the film surface and may limit the removal

of surfactant from the surface of the foam lamella. Therefore, the film drainage along

with the other forces (Marangoni forces and surface viscous forces) determine the extent

of transport of surfactant onto a foam lamella. As a consequence, the present model needs

to be improved to incorporate the film drainage.

8.3 Future experimental studies

8.3.1 Measurement of surface rheological properties

It is discussed in Chapter 6 that most of the methods on measuring surface rheological

properties are not very accurate due to environmental contamination as well as interfer-

ence of surface elasticity on the measurement of surface viscosity. Results from different

methods show significant variation of surface viscosity, even though there is only narrow

distribution of surfactant concentration. Therefore, an accurate method for measurement

of surface rheological properties is required. The method needs to be able to distinguish

surface elasticity from surface viscosity.

8.3.2 Enrichment of a foam fractionation column

A PhD project combining modelling and experimental work on determining the perfor-

mance of a foam fractionation column can be developed. This model combines the knowl-

edge on adsorption of surface active material(s) onto a bubble surface and the transport of

surfactant onto a lamella surface with the knowledge on liquid drainage in a foam frac-

tionation column. A model for dynamics of adsorption of surfactant and/or protein on a

bubble surface as well as a model of surfactant transport onto a foam lamella has been

developed. For the next step, an experimental study can be designed using this model

75

to examine the amount of enrichment as a function of adsorption dynamics, surface rhe-

ology, liquid fraction and bubble size. The experiment on adsorption dynamics can be

carried on on a single bubble to measure the dynamic surface tension of a particular sur-

factant and/or protein solution. The knowledge of adsorption dynamics gives information

of the amount of surfactant or protein is adsorbed on the film or Plateau border surface if

the bulk concentration is known. Therefore an experimental method to measure the bulk

concentration at different height of the column is required. Combining the dynamics of

adsorption with the transport of surfactant and/or protein onto a foam lamella will give a

prediction of the enrichment of the column. There are factors affecting the enrichment,

such as the superficial gas velocity and interstitial liquid velocity, bubble size and liquid

content in the foam. The liquid content determines the radius of curvature of the Plateau

border which determines the transport of surfactant onto a foam lamella.

8.4 Recommendation for the design of a foam fractiona-

tion column

The results of the simulation give information on the time scales of the separation process

in a foam fractionation. For instance, using the parameters for dynamics of adsorption in

Chapter 4, it was found that the characteristic time scale for the protein to adsorb fully to

a surface is 1033 s [112]. Meanwhile, Chapter 5 gives a bubble residence time in a typical

column of around 12 s [50]. In spite of this large difference in estimated time scales, we

still expect to see significant adsorption taking place in a fractionation column. The full

adsorption case relies on material moving over a distance of around 2.27 × 10−4 m (the

ratio between the parameters Γ′pm and Cpb in Chapter 4). The assumed film thicknesses

in Chapter 5 however are an order of magnitude smaller (2 × 10−5 m initial thickness)

speeding up the adsorption time by two orders of magnitude. Moreover, the film thins

due to drainage whilst it is in the fractionation column (in Chapter 5, characteristic film

drainage time 2.08 s for the initial film thickness mentioned previously and for a charac-

76

teristic bubble size L 5 × 10−3 m), and this ongoing film drainage will reduce cross-film

adsorption distances, and hence cross-film adsorption times even more.

Film drainage is actually a central importance to the fractionation process, since if

all we were to achieve was to move surface active material from the interior of the films

to foam film surfaces we would not actually be enriching the surface active material.

Enrichment relies not only on adsorbing surface active material to the film surface (to the

extent that most of it is now present on the surface and not in the interior of the film), but

also removing liquid from the interior (by film drainage).

One way to increase the amount of surface active material on the surface is via reflux.

Owing to Marangoni effects, additional surfactant is then pulled from the now surfactant

rich Plateau borders to the film surfaces. We now have surfactant transport in the direction

along the film, not just across its thickness. This is a rapid process (characteristic time

scale in Chapter 5 given as 3.125×10−2 s) significantly faster than either adsorption across

the film thickness or film drainage.

Film drainage across reduces the efficacy of the Marangoni mass transfer process, but

for realistic film drainage models (e.g. the Reynolds drainage model) the reduction is very

slight. This is more than offset by the fact that drainage (by removing liquid but leaving

behind surfactant) is essential to achieve enrichment. The Marangoni effect can however

be adversely affected by systems exhibiting high surface viscosity.

77

References

[1] A. P. Shea, C. L. Crofcheck, F. A. Payne, and Y. L. Xiong, “Foam fractionation ofα-lactalbumin and β-lactoglobulin from a whey solution,” Asia-Pacific Journal ofChemical Engineering, vol. 4, no. 2, pp. 191–203, 2009.

[2] B. M. Gerken, A. Nicolai, D. Linke, H. Zorn, R. G. Berger, and H. Parlar, “Effec-tive enrichment and recovery of laccase C using continuous foam fractionation,”Separation and Purification Technology, vol. 49, no. 3, pp. 291–294, 2006.

[3] V. Burapatana, A. Prokop, and R. D. Tanner, “Enhancing cellulase foam frac-tionation with addition of surfactant,” Applied Biochemistry and Biotechnology,vol. 122, no. 1-3, pp. 541–552, 2005.

[4] B. Burghoff, “Foam fractionation applications,” Journal of Biotechnology,vol. 161, no. 2, pp. 126–137, 2012.

[5] V. Burapatana, E. A. Booth, I. M. Snyder, A. Prokop, and R. D. Tanner, “A pro-posed mechanism for detergent-assisted foam fractionation of lysozyme and cel-lulase restored with β-cyclodextrin,” in Applied Biochemistry and Biotecnology(J. R. Mielenz, K. T. Klasson, W. S. Adney, and J. D. McMillan, eds.), ABABSymposium, pp. 777–791, Humana Press, 2007.

[6] R. Lemlich, “Adsorptive bubble separation methods: Foam fractionation and alliedtechniques,” Industrial and Engineering Chemistry, vol. 60, no. 10, pp. 16–29,1968.

[7] R. Lemlich, “Principles of foam fractionation,” in Progress in Separation and Pu-rification (E. S. Perry, ed.), vol. 1, pp. 1–56, New York: Interscience, 1968.

[8] F. Uraizee and G. Narsimhan, “Foam fractionation of proteins and enzymes: I.Applications,” Enzyme and Microbial Technology, vol. 12, no. 3, pp. 232 – 233,1990.

[9] F. Zhang, Z. Wu, H. Yin, and J. Bai, “Effect of ionic strength on the foam frac-tionation of BSA with existence of antifoaming agent,” Chemical Engineering andProcessing: Process Intensification, vol. 49, no. 10, pp. 1084–1088, 2010.

[10] P. Stevenson, Foam Engineering: Fundamentals and Applications. Wiley, 2012.

[11] D. Weaire and S. Hutzler, The Physics of Foams. Oxford University Press, NewYork, 1999.

78

[12] J. Maldonado-Valderrama and D. Langevin, “On the difference between foams sta-bilized by surfactants and whole casein or β-casein. comparison of foams, foamfilms, and liquid surfaces studies,” The Journal of Physical Chemistry B, vol. 112,no. 13, pp. 3989–3996, 2008.

[13] S. Boonyasuwat, S. Chavadej, P. Malakul, and J. F. Scamehorn, “Anionic andcationic surfactant recovery from water using a multistage foam fractionator,”Chemical Engineering Journal, vol. 93, no. 3, pp. 241–252, 2003.

[14] L. Y. Yeo, O. K. Matar, E. P. de Ortiz, and G. F. Hewitt, “Film drainage betweentwo surfactant-coated drops colliding at constant approach velocity,” Journal ofColloid and Interface Science, vol. 257, pp. 93–107, 2003.

[15] B. P. Radoëv, D. S. Dimitrov, and I. B. Ivanov, “Hydrodynamics of thin liquidfilms effect of the surfactant on the rate of thinning,” Colloid and Polymer Science,vol. 252, no. 1, pp. 50–55, 1974.

[16] T. T. Traykov and I. B. Ivanov, “Hydrodynamics of thin liquid films. effect of sur-factants on the velocity of thinning of emulsion films,” International Journal ofMultiphase Flow, vol. 3, no. 5, pp. 471–483, 1977.

[17] A. Mackie and P. Wilde, “The role of interactions in defining the structure of mixedprotein-surfactant interfaces,” Advances in Colloid and Interface Science, vol. 117,pp. 3–13, 2005.

[18] J. Maldonado-Valderrama and J. M. R. Patino, “Interfacial rheology of protein-surfactant mixtures,” Current Opinion in Colloid & Interface Science, vol. 15,no. 4, pp. 271–282, 2010.

[19] R. Miller, V. Fainerman, A. Makievski, J. Krägel, and R. Wüstneck, “Adsorp-tion characteristics of mixed monolayers of a globular protein and a non-ionicsurfactant,” Colloids and Surfaces A: Physicochemical and Engineering Aspects,vol. 161, no. 1, pp. 151 – 157, 2000.

[20] E. Dickinson, “Adsorbed protein layers at fluid interfaces: Interactions, structureand surface rheology,” Colloids and Surfaces B: Biointerfaces, vol. 15, no. 2,pp. 161–176, 1999.

[21] J. Krägel, R. Wüstneck, F. Husband, P. J. Wilde, A. V. Makievski, D. O. Grigoriev,and J. B. Li, “Properties of mixed protein/surfactant adsorption layers,” Colloidsand Surfaces B: Biointerfaces, vol. 12, no. 3-6, pp. 399–407, 1999.

[22] R. Miller, V. B. Fainerman, M. E. Leser, and M. Michel, “Kinetics of adsorption ofproteins and surfactants,” Current Opinion in Colloid & Interface Science, vol. 9,no. 5, pp. 350 – 356, 2004.

[23] A. A. Mikhailovskaya, B. A. Noskov, S.-Y. Lin, G. Loglio, and R. Miller, “Forma-tion of protein/surfactant adsorption layer at the air/water interface as studied bydilational surface rheology,” The Journal of Physical Chemistry B, vol. 115, no. 33,pp. 9971–9979, 2011.

[24] V. B. Fainerman, S. A. Zholob, M. Leser, M. Michel, and R. Miller, “Competitiveadsorption from mixed nonionic surfactant/protein solutions,” Journal of Colloidand Interface Science, vol. 274, no. 2, pp. 496–501, 2004.

79

[25] C. Kotsmar, V. Pradines, V. S. Alahverdjieva, E. V. Aksenenko, V. B. Fainerman,V. I. Kovalchuk, J. Krägel, M. E. Leser, B. A. Noskov, and R. Miller, “Thermo-dynamics, adsorption kinetics and rheology of mixed protein-surfactant interfaciallayers,” Advances in Colloid and Interface Science, vol. 150, pp. 41–54, 2009.

[26] L. N. Brush and S. H. Davis, “A new law of thinning in foam dynamics,” Journalof Fluid Mechanics, vol. 534, pp. 227–236, 2005.

[27] R. A. Leonard and R. Lemlich, “A study of interstitial liquid flow in foam. Part I.Theoretical model and application to foam fractionation,” AIChE J., vol. 11, no. 1,pp. 18–25, 1965.

[28] C. J. W. Breward and P. D. Howell, “The drainage of a foam lamella,” Journal ofFluid Mechanics, vol. 458, pp. 379–406, 2002.

[29] P. S. Stewart and S. H. Davis, “Dynamics and stability of metallic foams: Networkmodeling,” Journal of Rheology, vol. 56, no. 3, pp. 543–574, 2012.

[30] P. J. Martin, H. M. Dutton, J. B. Winterburn, S. Baker, and A. B. Russell, “Foamfractionation with reflux,” Chemical Engineering Science, vol. 65, no. 12, pp. 3825– 3835, 2010.

[31] P. Stevenson and G. J. Jameson, “Modelling continuous foam fractionation withreflux,” Chemical Engineering and Processing: Process Intensification, vol. 46,no. 12, pp. 1286–1291, 2007.

[32] S. I. Karakashev and A. V. Nguyen, “Effect of sodium dodecyl sulphate and do-decanol mixtures on foam film drainage: Examining influence of surface rheologyand intermolecular forces,” Colloids and Surfaces A: Physicochemical and Engi-neering Aspects, vol. 293, pp. 229–240, 2007.

[33] P. S. Stewart and S. H. Davis, “Self-similar coalescence of clean foams,” Journalof Fluid Mechanics, vol. 722, pp. 645–664, 2013.

[34] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz, and G. F. Hewitt, “The dynamics ofMarangoni-driven local film drainage between two drops,” Journal of Colloid andInterface Science, vol. 241, no. 1, pp. 233–247, 2001.

[35] T. Gambaryan-Roisman, “Dynamics of free liquid films during formation of poly-mer foams,” Colloids and Surfaces A: Physicochemical and Engineering Aspects,vol. 382, pp. 113–117, 2011.

[36] C. A. Brunner and R. Lemlich, “Foam fractionation. Standard separator and reflux-ing columns,” Industrial & Engineering Chemistry Fundamentals, vol. 2, no. 4,pp. 297–300, 1963.

[37] L. Du, V. Loha, and R. Tanner, “Modeling a protein foam fractionation process,”Applied Biochemistry and Biotechnology, vol. 84–86, pp. 1087–1099, 2000.

[38] D. Linke, H. Zorn, B. Gerken, H. Parlar, and R. G. Berger, “Laccase isolationby foam fractionation - new prospects of an old process,” Enzyme and MicrobialTechnology, vol. 40, no. 2, pp. 273 – 277, 2007.

[39] D. Linke and R. G. Berger, “Foaming of proteins: New prospects for enzyme pu-rification processes,” Journal of Biotechnology, vol. 152, no. 4, pp. 125–131, 2011.

80

[40] L. Brown, G. Narsimhan, and P. C. Wankat, “Foam fractionation of globular pro-teins,” Biotechnology and Bioengineering, vol. 36, no. 9, pp. 947–959, 1990.

[41] J. B. Winterburn, A. B. Russell, and P. J. Martin, “Characterisation of HFBII bio-surfactant production and foam fractionation with and without antifoaming agents,”Applied Microbiology and Biotechnology, vol. 90, no. 3, pp. 911–920, 2011.

[42] J. B. Winterburn, A. B. Russell, and P. J. Martin, “Integrated recirculating foamfractionation for the continuous recovery of biosurfactant from fermenters,” Bio-chemical Engineering Journal, vol. 54, no. 2, pp. 132–139, 2011.

[43] D. A. Davis, H. C. Lynch, and J. Varley, “The application of foaming for the recov-ery of surfactin from B. subtilis {ATCC} 21332 cultures,” Enzyme and MicrobialTechnology, vol. 28, pp. 346–354, 2001.

[44] C.-Y. Chen, S. C. Baker, and R. C. Darton, “Continuous production of biosurfac-tant with foam fractionation,” Journal of Chemical Technology & Biotechnology,vol. 81, no. 12, pp. 1915–1922, 2006.

[45] C.-Y. Chen, S. C. Baker, and R. C. Darton, “Batch production of biosurfactant withfoam fractionation,” Journal of Chemical Technology & Biotechnology, vol. 81,no. 12, pp. 1923–1931, 2006.

[46] K. Schügerl, “Recovery of proteins and microorganisms from cultivation mediaby foam flotation,” in New Products and New Areas of Bioprocess Engineering,pp. 191–233, Springer, 2000.

[47] K. H. Bahr and K. Schügerl, “Recovery of yeast from cultivation medium by con-tinuous flotation and its dependence on cultivation conditions,” Chemical Engi-neering Science, vol. 47, no. 1, pp. 11–20, 1992.

[48] P. Stevenson and X. Li, Foam Fractionation: Principles and Process Design. Tay-lor & Francis, 2014.

[49] J. Merz, G. Schembecker, S. Riemer, M. Nimtz, and H. Zorn, “Purification andidentification of a novel cutinase from coprinopsis cinerea by adsorptive bubbleseparation,” Separation and Purification Technology, vol. 69, no. 1, pp. 57 – 62,2009.

[50] D. Vitasari, P. Grassia, and P. Martin, “Surfactant transport onto a foam lamella,”Chemical Engineering Science, vol. 102, pp. 405–423, 2013.

[51] V. Rangarajan and R. Sen, “An inexpensive strategy for facilitated recovery of met-als and fermentation products by foam fractionation process,” Colloids and Sur-faces B: Biointerfaces, vol. 104, no. 0, pp. 99–106, 2013.

[52] T. Nakabayashi, Y. Takakusagi, K. Iwabata, and K. Sakaguchi, “Foam fractiona-tion of protein: Correlation of protein adsorption onto bubbles with a pH-inducedconformational transition,” Analytical Biochemistry, vol. 419, no. 2, pp. 173–179,2011.

[53] Z. Xu, Z. Wu, and Y. Zhao, “Foam fractionation of protein with the presence ofantifoam agent,” Separation Science and Technology, vol. 45, no. 16, pp. 2481–2488, 2010.

81

[54] C. H. Chang and E. I. Franses, “Adsorption dynamics of surfactants at the air/waterinterface: A critical review of mathematical models, data, and mechanisms,” Col-loids and Surfaces A: Physicochemical and Engineering Aspects, vol. 100, pp. 1–45, 1995.

[55] K. S. Birdi, ed., Handbook of Surface and Colloid Chemistry, Third Edition. Taylor& Francis, 1997.

[56] A. W. Adamson and A. P. Gast, Physical chemistry of surfaces. Wiley, 1997.

[57] K. J. Mysels, Soap Films: Studies of Their Thinning and a Bibliography. PergamonPress, 1959.

[58] J. Eastoe and J. S. Dalton, “Dynamic surface tension and adsorption mechanisms ofsurfactants at the air-water interface,” Advances in Colloid and Interface Science,vol. 85, pp. 103–144, 2000.

[59] J. Liu and U. Messow, “Diffusion-controlled adsorption kinetics at the air/solutioninterface,” Colloid & Polymer Science, vol. 278, pp. 124–129, 2000.

[60] X. Li, R. Shaw, G. M. Evans, and P. Stevenson, “A simple numerical solution tothe Ward-Tordai equation for the adsorption of non-ionic surfactants,” Computers& Chemical Engineering, vol. 34, no. 2, pp. 146–153, 2010.

[61] A. F. H. Ward and L. Tordai, “Time-dependence of boundary tensions of solutionsI. The role of diffusion in time-effects,” Journal of Chemical Physics, vol. 14, no. 7,pp. 453–461, 1946.

[62] S. N. Moorkanikkara and D. Blankschtein, “New methodology to determine therate-limiting adsorption kinetics mechanism from experimental dynamic surfacetension data,” Journal of Colloid and Interface Science, vol. 302, no. 1, pp. 1–19,2006.

[63] J. Liu, C. Wang, and U. Messow, “Adsorption kinetics at air/solution interface stud-ied by maximum bubble pressure method,” Colloid and Polymer Science, vol. 283,pp. 139–144, 2004.

[64] L. Liggieri, F. Ravera, and A. Passerone, “A diffusion-based approach to mixedadsorption kinetics,” Colloids and Surfaces A: Physicochemical and EngineeringAspects, vol. 114, pp. 351–359, 1996.

[65] Z. Adamczyk, “Nonequilibrium surface tension for mixed adsorption kinetics,”Journal of Colloid and Interface Science, vol. 120, no. 2, pp. 477–485, 1987.

[66] R. Lemlich, Progress in Separation and Purification, ch. Principles of Foam Frac-tionation, pp. 1–56. Interscience, New York, 1968.

[67] R. Miller, V. B. Fainerman, E. V. Aksenenko, M. E. Leser, and M. Michel, “Dy-namic surface tension and adsorption kinetics of beta-casein at the solution/air in-terface,” Langmuir, vol. 20, no. 3, pp. 771–777, 2004.

[68] V. B. Fainerman, E. H. Lucassen-Reynders, and R. Miller, “Description of theadsorption behaviour of proteins at water/fluid interfaces in the framework of a two-dimensional solution model,” Advances in Colloid and Interface Science, vol. 106,no. 1-3, pp. 237–259, 2003.

82

[69] V. B. Fainerman and E. H. Lucassen-Reynders, “Adsorption of single and mixedionic surfactants at fluid interfaces,” Advances in Colloid and Interface Science,vol. 96, no. 1–3, pp. 295–323, 2002.

[70] V. B. Fainerman, R. Miller, and V. I. Kovalchuk, “Influence of the two-dimensionalcompressibility on the surface pressure isotherm and dilational elasticity of do-decyldimethylphosphine oxide,” The Journal of Physical Chemistry B, vol. 107,no. 25, pp. 6119–6121, 2003.

[71] V. B. Fainerman, S. A. Zholob, M. E. Leser, M. Michel, and R. Miller, “Adsorp-tion from mixed ionic surfactant/protein solutions: Analysis of ion binding,” TheJournal of Physical Chemistry B, vol. 108, no. 43, pp. 16780–16785, 2004.

[72] P. M. Kruglyakov, S. I. Karakashev, A. V. Nguyen, and N. G. Vilkova, “Foamdrainage,” Current Opinion in Colloid & Interface Science, vol. 13, no. 3, pp. 163–170, 2008.

[73] S. A. Koehler, S. Hilgenfeldt, and H. A. Stone, “A generalized view of foamdrainage: Experiment and theory,” Langmuir, vol. 16, no. 15, pp. 6327–6341, 2000.

[74] A. M. Kraynik, “Foam drainage,” Tech. Rep. SAND83-0844, Sandia National Lab-oratories, Albuquerque, New Mexico, USA, 1983.

[75] S. Hutzler, D. Weaire, A. Saugey, S. Cox, and N. Peron, “The physics of foamdrainage,” in MIT European Detergents Conference, pp. 191–206, October 2005.

[76] D. Weaire, N. Pittet, S. Hutzler, and D. Pardal, “Steady-state drainage of an aque-ous foam,” Phys. Rev. Lett., vol. 71, pp. 2670–2673, Oct 1993.

[77] G. Verbist, D. Weaire, and A. M. Kraynik, “The foam drainage equation,” Journalof Physics: Condensed Matter, vol. 8, no. 21, pp. 3715–3731, 1996.

[78] S. J. Neethling, J. J. Cilliers, and E. T. Woodburn, “Prediction of the water distribu-tion in a flowing foam,” Chemical Engineering Science, vol. 55, no. 19, pp. 4021–4028, 2000.

[79] S. J. Neethling, H. T. Lee, and P. Grassia, “The growth, drainage and breakdownof foams,” Colloids and Surfaces A: Physicochemical and Engineering Aspects,vol. 263, no. 1–3, pp. 184–196, 2005. A collection of papers presented at the 5thEuropean Conference on Foams, Emulsions, and Applications, EUFOAM 2004,University of Marne-la-Vallee, Champs sur Marne (France), 5–8 July, 2004.

[80] S. Hutzler, S. T. Tobin, A. J. Meagher, A. Marguerite, and D. Weaire, “A modelsystem for foam fractionation,” Proceedings of the Royal Society A: Mathematical,Physical and Engineering Science, vol. 469, 2013.

[81] N. D. Denkov, P. Cooper, and J. Y. Martin, “Mechanisms of action of mixed solid-liquid antifoams. 1. Dynamics of foam film rupture,” Langmuir, vol. 15, no. 24,pp. 8514–8529, 1999.

[82] N. Politova, S. Tcholakova, K. Golemanov, N. D. Denkov, M. Vethamuthu, andK. P. Ananthapadmanabhan, “Effect of cationic polymers on foam rheologicalproperties,” Langmuir, vol. 28, no. 2, pp. 1115–1126, 2012.

83

[83] A. Sheludko, “Thin liquid films,” Advances in Colloid and Interface Science, vol. 1,no. 4, pp. 391–464, 1967.

[84] O. Reynolds, “On the theory of lubrication and its application to Mr. BeauchampTower’s experiments, including an experimental determination of the viscosity ofolive oil,” Philosophical Transactions of the Royal Society of London, vol. 177,pp. 157–234, 1886.

[85] G. Debrégeas, P.-G. de Gennes, and F. Brochard-Wyart, “The life and death of"bare" viscous bubbles,” Science, vol. 279, no. 5357, pp. 1704–1707, 1998.

[86] L. Wang and R.-H. Yoon, “Effects of surface forces and film elasticity on foamstability,” International Journal of Mineral Processing, vol. 85, no. 4, pp. 101–110, 2008.

[87] J. E. Coons, P. J. Halley, S. A. McGlashan, and T. Tran-Cong, “A review of drainageand spontaneous rupture in free standing thin films with tangentially immobile in-terfaces,” Advances in Colloid and Interface Science, vol. 105, pp. 3–62, 2003.

[88] T. C. Ransohoff and C. J. Radke, “Mechanisms of foam generation in glass-beadpacks,” SPE Reservoir Engineering, vol. 3, no. 2, pp. 573–585, 1988.

[89] C. V. Sternling and L. E. Scriven, “Interfacial turbulence: Hydrodynamic instabilityand the marangoni effect,” AIChE Journal, vol. 5, no. 4, pp. 514–523, 1959.

[90] K. D. Danov, D. S. Valkovska, and I. B. Ivanov, “Effect of surfactants on the filmdrainage,” Journal of Colloid and Interface Science, vol. 211, pp. 291–303, 1999.

[91] I. B. Ivanov, K. D. Danov, K. P. Ananthapadmanabhan, and A. Lips, “Interfacialrheology of adsorbed layers with surface reaction: On the origin of the dilatationalsurface viscosity,” Advances in Colloid and Interface Science, vol. 114–115, no. 0,pp. 61 – 92, 2005.

[92] A. Fick, “On liquid diffusion,” The London, Edinburgh, and Dublin PhilosophicalMagazine and Journal of Science, vol. 10, no. 63, pp. 30–39, 1855.

[93] R. P. Borwankar and D. T. Wasan, “The kinetics of adsorption of surface activeagents at gas-liquid surfaces,” Chemical Engineering Science, vol. 38, no. 10,pp. 1637–1649, 1983.

[94] I. Langmuir, “The constitution and fundamental properties of solids and liq-uids. Part I. Solids,” Journal of the American Chemical Society, vol. 38, no. 11,pp. 2221–2295, 1916.

[95] A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineerswith MATLAB Applications. Upper Saddle River, NJ: Prentice Hall PTR, 1999.

[96] J. K. Ferri and K. J. Stebe, “Which surfactants reduce surface tension faster? Ascaling argument for diffusion-controlled adsorption,” Advances in Colloid andInterface Science, vol. 85, no. 1, pp. 61 – 97, 2000.

[97] C. H. Chang, N. H. L. Wang, and E. I. Franses, “Adsorption dynamics of single andbinary surfactants at the air/water interface,” Colloids and Surfaces, vol. 62, no. 4,pp. 321–332, 1992.

84

[98] F. Jin, R. Balasubramaniam, and K. J. Stebe, “Surfactant adsorption to sphericalparticles: The intrinsic length scale governing the shift from diffusion to kinetic-controlled mass transfer,” The Journal of Adhesion, vol. 80, no. 9, pp. 773–796,2004.

[99] C. E. Lockwood, P. M. Bummer, and M. Jay, “Purification of proteins using foamfractionation,” Pharmaceutical Research, vol. 14, pp. 1511–1515, 1997.

[100] G. Mukhopadhyay, J. Khanam, and A. Nanda, “Protein removal from whey wasteby foam fractionation in a batch process,” Separation Science and Technology,vol. 45, no. 9, pp. 1331–1339, 2010.

[101] P. A. Kralchevsky, K. D. Danov, and N. D. Denkov, Handbook of Surface andColloid Chemistry, Third Edition, ch. Chemical physics of colloid systems andinterfaces, pp. 333–477. Taylor & Francis, 1997.

[102] B. S. Murray, “Rheological properties of protein films,” Current Opinion in Colloid& Interface Science, vol. 16, no. 1, pp. 27–35, 2011.

[103] B. Embley and P. Grassia, “Viscous froth simulations with surfactant mass transferand Marangoni effects: Deviations from Plateau’s rules,” Colloids and Surfaces A:Physicochemical and Engineering Aspects, vol. 382, pp. 8–17, 2011.

[104] D. Z. Zhang, Q. Zou, W. B. VanderHeyden, and X. Ma, “Material point method ap-plied to multiphase flows,” Journal of Computational Physics, vol. 227, pp. 3159–3173, 2008.

[105] S. G. Bardenhagen and E. M. Kober, “The generalized interpolation material pointmethod,” CMES: Computer Modeling in Engineering & Sciences, vol. 5, no. 6,pp. 477–496, 2004.

[106] S. G. Bardenhagen, J. U. Brackbill, and D. Sulsky, “The material-point method forgranular materials,” Computer Methods in Applied Mechanics and Engineering,vol. 187, pp. 529–541, 2000.

[107] D. S. Valkovska, K. D. Danov, and I. B. Ivanov, “Stability of draining plane-parallelfilms containing surfactants,” Advances in Colloid and Interface Science, vol. 96,pp. 101–129, 2002.

[108] A. Vrij, “Possible mechanism for the spontaneous rupture of thin, free liquid films,”Discussions of the Faraday Society, vol. 42, pp. 23–33, 1966.

[109] D. Weaire and S. Hutzler, The Physics of Foams, ch. Foam Collapse, pp. 144–150.Oxford University Press, New York, 1999.

[110] V. B. Fainerman and R. Miller, “Adsorption and interfacial tension isotherms forproteins,” in Proteins at Liquid Interfaces (D. Möbius and R. Miller, eds.), vol. 7of Studies in Interface Science, pp. 51 – 102, Elsevier, 1998.

[111] E. M. Freer, K. S. Yim, G. G. Fuller, and C. J. Radke, “Interfacial rheology ofglobular and flexible proteins at the hexadecane/water interface: Comparison ofshear and dilatation deformation,” The Journal of Physical Chemistry B, vol. 108,no. 12, pp. 3835–3844, 2004.

85

[112] D. Vitasari, P. Grassia, and P. Martin, “Simulation of dynamics of adsorption ofmixed protein-surfactant on a bubble surface,” Colloids and Surfaces A: Physico-chemical and Engineering Aspects, vol. 438, pp. 63–76, 2013.

[113] S. Ubal, C. H. Harrison, P. Grassia, and W. J. Korchinsky, “Numerical simulation ofmass transfer in circulating drops,” Chemical Engineering Science, vol. 65, no. 10,pp. 2934–2956, 2010.

[114] J.-L. Joye, G. J. Hirasaki, and C. A. Miller, “Asymmetric drainage in foam films,”Langmuir, vol. 10, no. 9, pp. 3174–3179, 1994.

[115] J. L. Joye, G. J. Hirasaki, and C. A. Miller, “Dimple formation and behavior duringaxisymmetrical foam film drainage,” Langmuir, vol. 8, no. 12, pp. 3083–3092,1992.

[116] J. T. Hsu and J. S. Dranoff, “Numerical inversion of certain Laplace transformsby the direct application of fast Fourier transform (FFT) algorithm,” Computers &Chemical Engineering, vol. 11, no. 2, pp. 101–110, 1987.

[117] J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of com-plex Fourier series,” Mathematics of Computation, vol. 19, no. 90, pp. pp. 297–301,1965.

[118] H. Hassanzadeh and M. Pooladi-Darvish, “Comparison of different numericalLaplace inversion methods for engineering applications,” Applied Mathematics andComputation, vol. 189, no. 2, pp. 1966 – 1981, 2007.

[119] P. Grassia, J. J. Cilliers, S. J. Neethling, and E. Ventura-Medina, “Quasi-one-dimensional foam drainage,” European Physical Journal E, vol. 6, pp. 325–348,2001.

[120] D. Weaire and S. Hutzler, The Physics of Foams, ch. Drainage, pp. 127–143. Ox-ford University Press, New York, 1999.

[121] D. Weaire and S. Hutzler, The Physics of Foams, ch. Quantitative Description ofFoam Structures, pp. 28–46. Oxford, New York: Oxford University Press, 1999.

[122] V. Pradines, V. B. Fainerman, E. V. Aksenenko, J. Krägel, R. Wüstneck, andR. Miller, “Adsorption of protein-surfactant complexes at the water/oil interface,”Langmuir, vol. 27, pp. 965–971, 2011.

86

Appendix A

Analytical verification of the numerical

simulation of dynamics of adsorption of

surfactant on the bubble surface

This appendix supports discussion of dynamics of adsorption of surfactant on a bubble

surface in Chapter 3.

An analytical solution of the partial differential equation of the adsorption of surfac-

tant on gas-liquid interfaces was developed to verify the numerical model formulated. The

partial differential equation was solved analytically using the Laplace transformations.

The Laplace transformation of the adsorption model in the case of Langmuir isotherm is

complicated by the nonlinearity of the equation. Therefore, the simple Henry isotherm

was applied to the adsorption equation to be solved using the Laplace transformation. The

equation of Henry adsorption isotherm is as follows:

Γ(t) = HC(0, t) (A.0.1)

whereH is Henry isotherm constant.

87

Another simplification of the analytical solution is the selection of the boundary con-

dition away from the bubble surface (x = L), where the concentration is equal to its bulk

value. This condition is achieved at infinite adsorption layer thickness (L = ∞). As a

consequence, the equation for this boundary becomes:

C(∞, t) = Ci. (A.0.2)

The dimensionless forms of the adsorption dynamics is then developed based on the as-

sumption of adsorption isotherm and boundary conditions as follows:

∂C ′(x′, t′)

∂t′=∂2C ′(x′, t′)

∂x′2. (A.0.3)

The adsorption equilibrium at the gas-liquid interface is described as follows:

Γ′(t′) = C ′(0, t′). (A.0.4)

Where the boundary conditions are as follow:

∂C ′(0, t′)

∂x′=

dΓ′(t′)

dt′(A.0.5)

at x′ = 0, and

C ′(∞, t′) = 1 (A.0.6)

at x′ =∞. The initial conditions are as follow:

C ′(x′, 0) = 1, x′ > 0 (A.0.7)

C ′(0, 0) = 0 (A.0.8)

Γ′(0) = 0 (A.0.9)

where C ′ = C/Ci, Γ′ = Γ/Γe, x′ = x/h, t′ = (D/h2)t, h = Γe/Ci, Γe = HCi. In

order to compare with the dimensionless parameters set for adsorption dynamics using

88

the Langmuir isotherm, the Henry isotherm constant is set asH = (ΓmK)/(1 +KCi).

The Laplace transformation of the partial differential equation is as follows:

L{∂C ′(x′, t′)

∂t′

}= L

{∂2C ′(x′, t′)

∂x′2

}. (A.0.10)

The Laplace transformation of the derivatives (with s being the Laplace variable, and F

being the transform of C ′) are as follow:

L{∂C ′(x′, t′)

∂t′

}= sF (x′, s)− C ′(x′, 0) (A.0.11)

and

L{∂2C ′(x′, t)

∂x′2

}=

∂2

∂x′2F (x′, s). (A.0.12)

Therefore the Laplace transform in Eq. (A.0.10) takes the following form:

∂2

∂x′2F (x′, s)− sF (x′, s) = −1 (A.0.13)

Eq. (A.0.13) is a non-homogeneous second order partial differential equation, therefore

the solution can be obtained as sum of the complementary solution in the following equa-

tion:

Fc = A1ex′√s + A2e

−x′√s (A.0.14)

and the particular solution as follows:

Fp =1

s(A.0.15)

As a consequence, the overall solution of Eq. (A.0.13) can be determined as follows:

F = A1ex′√s + A2e

−x′√s +1

s. (A.0.16)

89

Given the boundary condition away from the bubble surface at x′ =∞ is as follows:

C ′(∞,t) = 1

F(∞,s) =1

s

we can deduce as follows:1

s= A1e

∞ + A2e−∞ +

1

s.

Therefore, we can conclude that A1 = 0, and this results in the following equation:

F = A2e−x′√s +

1

s. (A.0.17)

The boundary condition at x′ = 0 is as follows:

∂C ′(0, t′)

∂x′=

∂Γ′(t′)

∂C ′(0, t′)

∂C ′(0, t′)

∂t′=∂C ′(0, t′)

∂t′(A.0.18)

since ∂Γ′(t′)/∂C ′(0, t′) = 1. Eq. (A.0.18) subjected to a Laplace transformation is as

follows:

L{∂C ′(0, t′)

∂x′

}= L

{∂C ′(0, t′)

∂t′

}(A.0.19)

and results in the following equation:

∂F (0, s)

∂x′= [sF (0, s)− C ′(0, 0)] . (A.0.20)

Since C ′(0, 0) = 0, Eq. (A.0.20) can also be expressed as follows:

∂F (0, s)

∂x′= sF (0, s). (A.0.21)

From the definition of F in Eq. (A.0.17), the derivative in Eq. (A.0.21) can be described

as follows:∂F (x′, s)

∂x′= −A2

√se−x

′√s.

90

Therefore, Eq. (A.0.21) can also be presented as follows:

[−A2

√se−x

′√s = s

(A2e

−x′√s +1

s

)]

x′=0

.

As a result, the value of A2 can be defined as follows:

A2 = − 1

(s+√s). (A.0.22)

Using the definition of the constants A1 and A2, the general solution for F (x′, s) can be

determined as follows:

F (x′, s) = − e−x′√s

(s+√s)

+1

s(A.0.23)

It is not possible to invert this form of Laplace equation using a table of inverse

Laplace transforms owing to its complexity. As a consequence, the Laplace inversion

was solved numerically using the Fast Fourier Transform (FFT) algorithm given by the

following equation [116]:

C ′(x′, j∆t′) =exp(aj∆t′)

2T

N−1∑

k=0

[A(x′, k) exp

(2πkj

Ni

)](A.0.24)

where:

A(x′, k) =

N/2−1∑

n=−N/2F[x′, a+

π

T(k + nN)i

](A.0.25)

where i denotes the imaginary number, N∆t′ = 2T and j = 0, 1, 2, ...N − 1, so we

are interested in (dimensionless) times up to 2T . The value of N has to be a power of

two [117]. The parameter a must be chosen, such that it is greater than the real part of

any singularities of F (s). The suggested value of aT is between 4 and 5 [118]. Previous

studies have found that Laplace inversion using Fourier transform gave the most accurate

results [116, 118]. The FFT algorithm enables reduction of calculation time as well as

the error and also enlarging the time interval by a factor of two. Moreover, many chem-

ical engineering problems tranformed into Laplace form can be solved by inverting the

transform function using FFT algorithm without any difficulty [116].

91

A.1 Derivation of Ward-Tordai equation from the Laplace

transformation of the diffusion equation

The Laplace transformation of the diffusion equation using the Henry isotherm can be ma-

nipulated to obtain the Ward-Tordai equation [61]. The big advantage of teh Ward-Tordai

equation is that it is an ordinary differential equation (or strictly speaking an integro-

differential equation) for the surface concentration, therefore it is less expensive to solve

than approaches involving partial differential equation of the coupled surface plus bulk

concentration. The Laplace transformation of the boundary condition of the diffusion

equation at x′ = 0 is presented in Eq. (A.0.19).

L{∂C ′(0, t′)

∂x′

}= L

{∂C ′(0, t′)

∂t′

}

where using the definition of F in Eq. (A.0.17) we can obtain the following equations:

L{∂C ′(x′, t′)

∂x′

}= −A2

√se−x

′√s

= −√s[L{C ′} − 1

s

] (A.1.1)

and

L{∂C ′(x′, t′)

∂t′

}= s

(A2

√se−x

′√s +1

s

)

= sL{C ′}.(A.1.2)

Therefore, we can have the following equation:

1√s−√sL{C ′(0, t′)} = sL{C ′(0, t′)}. (A.1.3)

Dividing through by s, the equation of Laplace transform can be solved using con-

volution to the second part of the left hand side of Eq. (A.1.3) results in the following

equation:

2

√t′

π−∫ t′

0

1

π

C ′(0, τ ′)√t′ − τ ′ dτ

′ = C ′(0, t′) (A.1.4)

92

where τ ′ is a dummy variable. In the Henry isotherm, the value of Γ′(t′) = C ′(0, t′).

Replacing C ′(0, t′) with Γ′(t′) and rearranging the equation results in the Ward-Tordai

equation as follows:

Γ′ =2

π

[√t′ − 1

2

∫ t′

0

C ′(0, τ ′)√t− τ ′ dτ

′]. (A.1.5)

93

Appendix B

Numerical simulation of Ward-Tordai

equation

This appendix supports discussion of dynamics of adsorption of surfactant on the bubble

surface in Chapter 3.

The dimensionless Ward-Tordai equation is as follows:

Γ′′ =2√π

[√t′′ − 1

2

∫ t′′

0

C ′(0, τ ′′)√t′′ − τ ′′dτ

′′]

To make it possible to do numerical simulation of Eq. (3.5.2), it is manipulated to the

following new form using a time step δt′′ and using the integer i to count the steps:

Γ′′(iδt′′) =2√π

√iδt′′

− 1√π

[∫ (i−1)δt′′

0

C ′0(τ ′′)√iδt′′ − τ ′′

dτ ′′ +

∫ iδt′′

(i−1)δt′′

C ′0(τ ′′)√iδt′′ − τ ′′

dτ ′′]. (B.0.1)

For a Henry isotherm we can take C ′(0, t′′) = Γ′′(t′′). As a consequence Eq. (B.0.1) can

94

be written as:

Γ′′(iδt′′) =2√π

√iδt′′

− 1√π

[∫ (i−1)δt′′

0

Γ′′(τ ′′)√iδt′′ − τ ′′

dτ ′′ +

∫ iδt′′

(i−1)δt′′

Γ′′(τ ′′)√iδt′′ − τ ′′

dτ ′′]

(B.0.2)

Therefore, the algorithm of numerical solution of Ward-Tordai equation is as follows:

At first it is known that Γ′′(0) = 0. For a first approximation, it is taken that Γ′′(δt′′) ≈

2/√π(√δt′′), then the right hand side of Eq. (B.0.2) is approximated analytically to obtain

the formula Γ′′(2δt′′) ≈ 2/√π(√

2δt′′−√π/2(2δt′′)). The values of Γ′′ for the next steps

are then calculated numerically using the following algorithm:

for i = 3 to n

initialization

z(0) = 0

I0 = 0

for j = 1 to i− 1

z(j) =Γ′′(jδt′′)√iδt′′ − jδt′′

evaluate the integrand

I =δt′′

2[z(j−1) + z(j)]

trapezoidal rule

I0 = I + I0

evaluate the integral up to time(i− 1)δt

next j

In(i) = I0

Γ′′(iδt′′) =

2√π

√iδt′′ − 1√

π

[In(i) + 2

3

√δt′′Γ′′((i− 1)δt′′)

]

1 + 43√π

√δt′′

next i

95

The last part of the right hand side of the equation (as well as the denominator) to calculate

Γ(iδt′′) in the algorithm is obtained using the following details:

∫ iδt′′

(i−1)δt′′

Γ′′(τ ′′)√iδt′′ − τ ′′

dτ ′′ ≈∫ iδt′′

(i−1)δt′′

Γ′′(iδt′′) + [Γ′′((i− 1)δt′′)− Γ′′(iδt′′)] iδt′′−τ ′′δt′′√

iδt′′ − τ ′′dτ ′′

≈∫ iδt′′

(i−1)δt′′

Γ′′(iδt′′)√iδt′′ − τ ′′

dτ ′′

+

∫ iδt′′

(i−1)δt′′(Γ′′((i− 1)δt′′)− Γ′′(iδt′′))

√iδt′′ − τ ′′δt′′

dτ ′′

≈ Γ(iδt′′)[−2√iδt′′ − τ ′′]

∣∣∣∣iδt′′

(i−1)δt′′

+ [Γ′′((i− 1)δt′′)− Γ′′(iδt′′)][−2

3(iδt′′ − τ ′′)3/2]

∣∣∣∣iδt′′

(i−1)δt′′

≈ 2√δt′′Γ′′(iδt′′) +

2

3

√δt′′[Γ′′((i− 1)δt′′)− Γ′′(iδt′′)]

≈ 4

3

√δt′′Γ′′(iδt′′) +

2

3

√δt′′Γ′′((i− 1)δt′′)

This algorithm, whilst more complicated than a conventional quadrature, allows us to

account properly for the fact that the integrand in Eq. (B.0.2) is divergent, even though

the integral itself is not.

The values of Γ′′ obtained from simulation using the Ward-Tordai equation are com-

pared to the values obtained from numerical simulations of adsorption dynamics using

the Langmuir and Henry isotherm.

96

Appendix C

Equation for desorption of surfactant in

the presence of protein in the bulk solution

This appendix supports discussion of dynamics of adsorption of mixwed protein-surfactant

on a bubble surface in Chapter 3.

The Langmuir isotherm for the case when the surfactant molecules are displaced in

the presence of protein in the bulk solution is derived based on Eq. (17) in Chapter 4. In

this case, a particular amount of surfactant present on the surface when protein is suddenly

added in the bulk solution causing desorption of surfactant molecules from the surface.

In this case, the initial value of θp is infinite, while the initial value of θs is finite. As

a consequence, θp changes much more rapidly than θs. Hence, the Taylor expansion in

θp is keeping θs constant at (θs)initial. As there is desorption C ′s needs to be higher than

one (i.e. to drive surfactant away from the subsurface towards the bulk). This is what is

showed in the following equation:

C ′s ∼ 1 +θp

1− (θs)initial

97

Appendix D

Typical parameters for calculation of

surfactant transport onto a foam lamella

This appendix supports discussion of surfactant transport onto a foam lamella in Chap-

ter 5.

D.1 Typical operating condition of foam fractionation col-

umn with reflux

Some parameters such as half of film thickness (δ), radius of curvature of Plateau bor-

der (a) and residence time (tR) inside a foam fractionation column are derived using the

typical operating conditions as described in a study by Martin et al. [30]. In that study,

there are several different operating conditions, therefore, in this present study, just one

of those various conditions is taken. The parameters of δ, a and tR are derived using the

operating conditions presented in Tab. D.1.

98

Table D.1: Typical operating condition of a foam fractionation column in the study byMartin, et al. [30].

Parameter Symbol Value UnitColumn diameter dc 0.052 mColumn height Hc 0.265 mFoam height H 0.123 mSuperficial liquid velocity JF 22.85× 10−6 m s−1

Superficial gas velocity v 10.26× 10−3 m s−1

Liquid density ρ 1000 kg m−3

Gravity acceleration g 9.8 m s−2

D.2 Radius of curvature of Plateau border and liquid frac-

tion

The volumetric flowrate of liquid in the Plateau border (averaged over orientation) QPb

is equal to the cross sectional area of the Plateau border times the average liquid velocity

within that Plateau border as presented in the following equation [119]:

QPb = 〈Acut〉 × uAvg (D.2.1)

where Acut is the area of the oblique cut through the Plateau border and uAvg is the

average velocity of liquid within the Plateau border. Based on the study by Grassia et

al. [119] the value of Acut can be deduced as follows:

〈Acut〉 =

∫ π/2

0

A

cos θ2 cos θ sin θdθ = 2A (D.2.2)

where A is the cross sectional area of Plateau border can be calculated as follows [77]:

A =(√

3− π/2)a2 (D.2.3)

99

The mean velocity u of liquid flowing inside a straight Plateau border making an angle

of θ with the vertical is calculated using the following formula [120]:

u(θ) =1

49

ρg

µA cos θ (D.2.4)

where ρ is the liquid density, g is the gravitational acceleration and µ is the liquid viscos-

ity. For Plateau borders with various angle between 0 and π/2. The vertical component

of that velocity is as follows:

u =1

49

ρg

µA cos2 θ (D.2.5)

and the average of that velocity over orientations is as follows:

uavg =1

49

ρgA

µ

∫ π/20

cos2 θ sin θ dθ∫ π/2

0sin θ dθ

(D.2.6)

where sin θ comes from 3D geometry as there is more area near the equator than near the

poles. The ratio of the two integrals in Eq. (D.2.6) evaluates to 1/3, therefore that equation

is simplified to the following equation:

uavg =1

147

ρgA

µ(D.2.7)

The liquid flux is derived as follows:

φuavg = QPbuavg = λAuavg (D.2.8)

where φ is the liquid fraction and λ is the length of Plateau borders per unit volume (the

detailed definition of λ is discussed later). The liquid flux is downward relative to gas,

therefore we have the following equation:

100

(vb −

1

147

ρgA

µ

)Aλ = JF (D.2.9)

where JF is the superficial liquid velocity, vb = v/(1 − φ) is the interstitial gas velocity

and v is the superficial gas velocity. If the foam is quite dry (φ� 1) and provided (1/3×

49)(ρgA/µ) is not exceedingly close to v, therefore we obtain the following equation:

(v − 1

147

ρgA

µ

)Aλ = JF . (D.2.10)

Assuming that the bubbles are monodispersed Kelvin’s cells, the value of λ can be

derived using this following formula [121]:

λ =5.35

V2/3cell

(D.2.11)

where Vcell is the volume of a cell/bubble. The Kelvin’s cell consists of a regular octahe-

dron with the eight corners sawn off to form a 14 sided cell with equal length sides. If we

have a hexagonal film with the length of the sides is L, the radius of the circumscribing

circle is also L. Therefore, we assume that half of the film length L equals to the length

of the sides of the Kelvin’s cell. As a consequence, the volume of a Kelvin cell can be be

determined using the following formula [120]:

Vcell = 8√

2L3. (D.2.12)

Knowing the volume of a single cell, the radius of a spherical cell giving the volume equal

to the Kelvin’s cell can be calculated as follows:

Rb =

(3

4

Vcellπ

)1/3

. (D.2.13)

101

The liquid fraction within the foam φ can be calculated using this following equation

[78]:

φ = Aλ. (D.2.14)

Using Eq. (D.2.3) which relates a to A and Eqs. (D.2.11) – (D.2.12) which relate λ to

L we provide a way to relate a/L to φ. However, this approximation of liquid fraction

within the column only considers the liquid within the Plateau border. Since there are

comparable amounts of liquid within the films and within the Plateau borders, the liquid

fraction obtained underestimates the actual value.

The volume of the Kelvin’s cell obtained using Eq. (D.2.12) is 1.41 × 10−6 m3 as-

suming that L = 5 × 10−3 m. The radius of the spherical cell is Rb = 6.96 × 10−3 m.

Therefore, λ can be calculated using Eq. (D.2.11) and the result is 42, 463 m−2. The cross

sectional area of the Plateau border calculated using Eq. (D.2.10), data in Tab. D.1 as

well as in Tab. 1, and the result is 1.95× 10−7 m2. The radius of curvature of the Plateau

border is calculated from the cross sectional area using Eq. (D.2.3) and obtained to be

very roughly 1 × 10−3 m. Therefore, the radius of curvature of the Plateau border of

a = 5 × 10−4 m which is taken in the simulation is also sensible. Based on this finding,

the radius of curvature of the Plateau border used in future calculation is a = 5× 10−4 m.

The cross sectional area of the Plateau border obtained using the value of a employed in

the simulation is 4.03 × 10−8 m2. The liquid fraction obtained from Eq. (D.2.14) using

that value of a is then φ = 1.71× 10−3.

D.3 The residence time within a foam fractionation col-

umn

The residence time of a bubble within a foam fractionation column tR equals the foam

height divided by the superficial bubble velocity as described in the following equation:

102

tR =H

vb(D.3.1)

where H is the foam height and vb is the interstitial gas velocity which can be calculated

from the superficial gas velocity using the following equation [31]:

vb =v

1− φ (D.3.2)

Using the liquid fraction of φ = 1.71 × 10−3 and the data available in Tab. D.1, the

interstitial bubble velocity is vb = 10.28 × 10−3 m s−1 and the bubble residence time

calculated using this liquid fraction is 12 s.

D.4 Early time evolution of film thickness

It has been assumed in our calculations that bubbles already have δ � a by the time

they have finished forming in a fractionation column (so our film drainage calculations

for films begin with a nominal ‘initial’ film thickness δ0 which is much less than a). It is

desired to check whether this assumption is of δ0 much less than a is correct, and indeed

whether our selected δ0 value (quoted in Tab. 1) is a sensible one.

The time for film to thin from the thickness of a to the nominal ‘initial’ thickness of δ0

is compared to the time to form a bubble as it is assumed that bubbles already have δ � a

by the time they have finished forming. The time required to form a bubble is as follows:

tb =2Rb

vb. (D.4.1)

The time for film to thin depends on the mobility of the film surface. For film with a

mobile surface, the equation is as follows:

103

tmobile =16

3

µL√a

γPb

(1√δ0

− 1√a

)(D.4.2)

while the equation for a film with a rigid interface is as follows:

trigid =3

2

µL2a

γPb

(1

δ20

− 1

a2

). (D.4.3)

The time to form a bubble is calculated to be 0.68 s. The time for film to thin is 0.0024

s for a film with a mobile interface and 1.04 s for a film with a rigid interface. Therefore,

the initial film thickness taken in the simulation is reasonable i.e. comparable with the

bubble formation time (at least for one type of interface).

In addition of Eq. (D.4.2) – (D.4.3), there are some approximate formulae based on

the fact that a� δ0 therefore the terms of 1/√a and/or 1/a2 is negligible. The equations

can be simplifiend into the following equation for a film with a mobile interface:

tmobile =16

3

µL√a

γPb√δ0

(D.4.4)

as wel as for a film with a rigid interface:

trigid =3

2

µL2a

γPbδ20

. (D.4.5)

Using these approximate formulae the time for film to thin is 0.0030 s for a film with a

mobile interface and 1.04 s for a film with a rigid interface.

104

D.5 Amount of surfactant in the film and in the Plateau

border

It has been assumed that (as far as the films are concerned) surfactant is ‘insoluble’. Si-

multaneously it has been assumed that Plateau borders act as reservoirs of surfactant for

the films (and can supply surfactant to the films without themselves becoming depleted).

As the surface area of the Plateau borders is (on geometric grounds) much less than that

of the films, our assumptions require that there must be some dissolved surfactant in the

Plateau borders (which can continually replenish the surfactant on the film surfaces).

Whether or not these assumptions are reasonable depends on the Henry constant of

the surfactant and its relation to the length scales of the film and Plateau border geometry.

The Henry constant of SDS is 1.10× 10−6 m [97] which is smaller than the assumed

film thickness (at least initially if we assume initial thickness δ0 = 20 × 10−6 m). SDS

however is likely to behave as a mobile surfactant so that the characteristic thinning time

of the film is very short (1.48 × 10−3 s). The film will very rapidly thin down to the cut-

off length δcut−off (15 × 10−9 m in Tab. 1), which is now much smaller than the Henry

constant. Assuming that most film surfactant is present as surface excess is then quite

reasonable.

Meanwhile the curvature radius of the Plateau border is a = 5 × 10−4 m. This is

now larger than the Henry constant, meaning that most of the Plateau border surfactant

is present in solution. It is necessary to check the ratio between the amount of dissolved

surfactant in the Plateau borders to the amount of surface surfactant on the films. A simple

check is performed assuming that each film ‘claims’ two Plateau borders (this assumes

that a typical film is of hexagonal shape, but that Plateau borders attach to three films).

The ratio is as follows:SPbSfilm

=2ΓPbVPb/HΓfilmAfilm

(D.5.1)

105

where SPb and Sfilm are the amount of surfactant on the Plateau border surface and

on the film surface, respectively, VPb is the volume of one Plateau border and Afilm is

the surface area of a film. We assume that ΓPb exceeds Γfilm, but that ΓPb and Γfilm are

nonetheless similar order of magnitude, so we treat ΓPb/Γfilm as O(1) quantitiy. There-

fore, the volume of a Plateau border with curvature radius a and length L is calculated

using this following equation:

VPb = (√

3− π/2)a2L. (D.5.2)

Treating the film as a hexagon with side length of L, and having two faces, the surface

area of a film is as follows:

Afilm = 3L2√

3 (D.5.3)

The volume of one Plateau border is VPb = 2.0× 10−10 m3 and the surface area of a film

is Afilm = 1.3× 10−4 m2. The ratio of surfactant on the Plateau borders and on the films

is 2.82, therefore most of the surfactant is in the Plateau borders although not perhaps in

such an overwhelming amount that we could truly consider the Plateau borders to act as

surfactant reservoirs (at least for the liquid fraction, φ = 0.17% that we currently assume).

The surfactant SDS assumed in this simulation gives a mobile interface. Surfactant

or surface-active protein that gives a rigid interface tends to have a larger Henry constant,

i.e. less soluble. Examples of some surfactants and protein with various surface mobility

is presented in Tab. D.2. As a consequence, there is less protein in the solution within the

Plateau border. One example of ratio of surface active substance on the film surface and

dissolved in the Plateau border is calculated using protein BLG with Henry constant of

3.18 × 10−4 m [22] that gives the ratio of protein in the Plateau borders and on the films

of 9.75× 10−3 which is much smaller than the ratio obtained from a surfactant that gives

a mobile film.

There are however ways in which the balance between the total amount of surfactant

in the Plateau borders and on the films could be altered.

106

Table D.2: Surface mobility, the Henry constant and ratio of surfactant in the Plateauborder and in the film of various surfactant and/or surface active protein. Data were takenfrom literatures [97, 122].

Surfactant/protein Surface mobility H (m) SPb/SfilmSDS mobile 1.10× 10−6 2.82C10DMPO mobile 4.97× 10−6 0.62n-Propanol mobile 4.2× 10−7 7.38C10E4 rigid 3.93× 10−3 7.89× 10−4

BLG rigid 3.18× 10−4 9.75× 10−4

HSA rigid 1.00× 10−5 0.31

One of these would simply be to operate at a substantially higher liquid fraction (re-

member that our calculation assumes a very tiny liquid fraction of 0.17% so there is scope

for increasing this). Yet another case where the balance of surfactant shifts towards the

Plateau borders would be the case of higher concentrations (higher C and higher Γ, closer

to Γinfty). Here, instead of

Γ ∼ CH

we have for e.g. a Langmuir isotherm:

Γ =C

H/(1 + CH/Γinfty)

meaning that:

C =Γ

H

(1 +

CHΓinfty

)

and henceSPbSfilm

=2ΓPbVPbΓfilmH

(1 +

CHΓinfty

)

and so is increased significantly whenever CH/Γinfty is bigger than one.

107