Dynamic Surface Phenomena 9789067643009, 9780429070921, 9067643009

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Dynamic Surface Phenomena

Dynamic Surface Phenom ena Paul Joosf

Universitaire Instelling Anwerpen, Wilrijk, Belgium Dedicated to my grandchildren Delphine, Audric and Justine

Editors:

Valentin B. Fainerman - Institute o f Technical Ecology, Donetsk, Ukraine Giuseppe Loglio - University o f Florence, Italy Emmi H. Lucassen-Reynders - Unilever, Vlaardingen, The Netherlands Reinhard Miller - MPI Kolloid- und Grenzflächenforschung, Berlin, Germany Peter Petrov - Procter & Gamble, Brussels, Belgium (f died on January 26, 1997 before completing the manuscript)

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First published 1999 by VSP BV Published 2021 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1999 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works ISBN-13: 978-90-6764-300-9 (hbk) DOI: 10.1201/9780429070921 This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organiza-tion that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http ://www.taylorandfrancis.com and the CRC Press Web site at http ://www.crcpress.com

CONTENTS Foreword

ix

1.

The definition of surface tension 1.1. The molecular definition of the surface tension 1.2. The mechanical definition of the surface tension 1.2.1. The equation of Kelvin 1.2.2. The equation of Thomson 1.3. The thermodynamic definition of the surface tension

1 1 3 5 7 9

2.

Surface thermodynamics 2.1. The Gibbs equation 2.2. The position of the Gibbs dividing surface 2.3. The application of the Gibbs equation 2.4. The chemical potential at the surface 2.5. Adsorption isotherms 2.6. The adsorption derivation for a surfactant mixture 2.7. Kinetic derivation of the Langmuir isotherm 2.8. Surface activity coefficients 2.9. Traube’s rule, the hydrophobic effect and inter-chain penetration 2.10. The principle of Le Chatelier - Braun

14 14 17 22 25 27 35 42 44

3.

Diffusion 3.1. Diffusion equations 3.2. The diffusion penetration depth 3.3. The boundary condition for the diffusion equation

58 58 69 71

4.

Methods for measuring the dynamicsurface tension 4.1. Different experimental methods 4.2. Classification of experimental methods 4.3. Introduction to hydrodynamics 4.3.1. The continuity equation 4.3.2. The Stokes equation

75 75 77 79 79 80

48 52

contents

4.4.

4.5.

4.6.

4.7.

4.3.3. The Navier-Stokes equation 4.3.4. Boundary conditions Stress relaxation methods 4.4.1. The vibration or oscillating jet method 4.4.2. The inclined plate method 4.4.3. The pulsed drop technique 4.4.4. Axisymmetric drop shape analysis 4.4.5. The drop volume method 4.4.6. Stepwise surface deformation in a Langmuir trough Dilation or 0 methods 4.5.1. The experiments of Van Voorst Vader and Van den Tempel 4.5.2. The stripe method 4.5.3. The dynamic capillary method Methods with constant surface deformation 4.6.1. Experiments in a Langmuir trough 4.6.2. Epansion of a drop 4.6.2.1. Expansion of a drop with initially clean surface 4.6.2.2. The maximum bubble pressure method 4.6.2.3. The drop method of Mac Leod and Radke Small amplitude periodic surface deformation

Diffusion controlled adsorption kinetics 5.1. The equation of Ward and Tordai 5.2. The Sutherland equation 5.3. The short and long time approximation of the Ward and Tordai equation 5.4. The approximation making use of the diffusion penetration depth 5.5. The rate of adsorption for diffusion controlled kinetics 5.6. The Ward and Tordai equation for a surfactant mixture 5.7. The rate of adsorption for diffusion to a drop surface Diffusion controlled adsorption kinetics to a continuously deformed surface 6.1. Diffusion to an expanding surface 6.1.1. Surface expansion with constant rate dfi/dt 6.1.2. Expansion with constant dilation rate 6.1.3. Expanding drop with a constant volume flow rate 6.1.4. The drop volume technique

82 82 85 85 95 98 99 99 99 100 100 100 101 108 108 109 109 112 115 115 119 119 124 126 134 137 138 141 144 144 153 154 158 160

contents

vii

The diffusion penetration depth Expansion at constant surface tension The long time approximation Integration of the convective diffusion equation 6.1.8.1. Stress relaxation experiments 6.1.8.2. Continuously expanding surface 6.1.8.3. Continuously compressed surface Diffusion to a compressed surface 6.2.1. Sutherland equation for surface compression 6.2.2. Compression with a constant speed dQ/dt 6.2.3. Compression with a constant dilation rate 0 6.2.4. The diffusion penetration depth 6.2.5. Compressed surface at a constant surface pressure

164 165 168 169 173 174 175 177 181 184 185 188 189

6.1.5. 6.1.6. 6.1.7. 6.1.8.

6.2.

7.

The interfacial tension during mass transfer across an interface 7.1. The transfer of an alkanol between two immiscible phases 7.2. The transfer of a fatty acid between two immiscible phases 7.3. Mass transfer in partly miscible systems Appendix

196 196 203 209 221

8.

Periodic surface deformation 8.1. Periodic deformation on a flat surface 8.2. Surface disturbances for other geometries 8.2.1. Thin layers (foam films) 8.2.2. Elasticity of a drop 8.2.2.1. Diffusion outside the drop 8.2.2.2. Diffusion inside the drop 8.2.3. Elasticity of a liquid cylinder 8.2.3.1. Diffusion outside the cylinder 8.2.3.2. Diffusion inside the cylinder 8.3. The elasticity for a surfactant mixture 8.4. Relation between complex elasticity and dynamic surface tensions

275 223 232 232 236 236 237 239 239 241 243

Transfer controlled adsorption kinetics 9.1. Departure in electrode kinetics 9.2. The kinetic equation 9.3. Experimental confirmation 9.4. Small amplitude periodic deformation 9.5. Experiments with constant dilations

258 259 262 269 278 280

9.

247

viii

contents

10. Micellar solutions 10.1. Local equilibrium between the monomers and micelles 10.2. Periodic small amplitude oscillations 10.3. Approximate solutions for micellar systems 10.4. The diffusion penetration depth for micellar systems 10.5. Use of the diffusion-micellisation penetration depth to describe the dynamic surface tension for micellar solutions

285 288 294 299 307

11. Reorientation at the surface 11.1. Surface reorientation with diffusion equilibrium 11.2. Surface reorientation in an insoluble monolayer 11.3. Surface reaction with diffusion 11.4. Surface expansion with a constant dilation rate for diffusion equilibrium between the bulk and the subsurface 11.5. Small amplitude periodic area deformations 11.6. Small amplitude surface deformation with diffusional exchange 11.7. Unsolved problems 11.8. The dynamic interfacial tension during the mass transfer of decanoic acid from hexane to water containing 0.1 NaOH

314 316 318 320

339

List of used PhD Theses Symbol index Subject index

352 354 357

309

324 327 334 336

IX

Foreword

Paul Joos was bom on May 26, 1936 in Ghent, Belgium. He studied chemistry at the University of Ghent, obtaining his PhD in 1964 on foamability of surfactant and in particular of saponine solutions. This research topic was of both fundamental and applied interest at that time, and retained its relevance for him throughout his entire career. In 1970, after having finished the PhD, Paul Joos joined the Physical Chemistry group of Max van den Tempel at the Vlaardingen laboratory. The original idea was a three-month stay, for him to familiarise himself with the longitudinal wave technique, but he remained there to late 1973. Here he also worked on interfacial instability and on the equation of state for adsorbed proteins. In fact, he was the first to use a 2-D solution approach for protein adsorption. After this he worked in the Pharmacy and Cell Biology Departments of the Ghent University until 1974, before he moved to the Universitaire Instelling Antwerpen (UIA), where he became Professor in the Biochemistry Department where his continued interest in proteins became very fruitful. In Gent Paul Joos was involved in studies on biological substances such as lipids, cholesterol and proteins. After starting his work at the UIA in Wilrijk he changed the focus of research principally to the study of surfactants at liquid interfaces. In his scientific work he went into many topics of surface science, such as thermodynamics and dynamics of adsorbed and spread layers, mechanical properties of interfacial layers, transport phenomena in bulk and across interfaces, the effect of micelles and o f molecular interaction on adsorption layer behaviour. The first widely recognised achievement of his work is the description of surfactant surface layers in equilibrium by thermodynamic models. His early paper in Bull. Soc. Chim. Beiges (76,1967, 591) is still very frequently cited. It describes a mixed adsorption layer comprised of molecules with different molar interfacial areas. This topic is of high actuality and he returned to this problem very recently with new approaches. The use of the "Principle of Braun-Le Chatelier at Surfaces" (with G. Serrien, J. Colloid Interface Science, 145(1991)291) represents a logical continuation o f this work on interfacial models and is a completely new theoretical

X

basis for a further improvement of the quantitative understanding of adsorption layers. The application of this physical principle to interfacial problems generated a series of new models founded on this generic idea. Paul Joos was strongly involved in studies of interfacial dynamics since the early 70s. Under his supervision existing techniques were modified and new machines for measuring various surface and inter facial characteristics were invented, such as dynamic surface and interfacial tensions, surface potentials, and shear and dilational surface rheology. New procedures have been developed, for example with M. van Uffelen the peak tensiometry (Colloids and Surfaces A, 85(1994)119). This phenomenon is based on the competitive effects of surface area expansion and simultaneous adsorption of surfactant from the adjacent bulk. While the surface expansion leads to an increase in surface tension, the adsorption flux results in a decrease. When both effects are counterbalanced a maximum (peak) appears. The localisation and height of the peak are characteristic for the surfactant under study and its adsorption mechanism. All instruments Paul Joos used at Antwerp were made in his laboratory. As an example for the complexity of the instruments, the apparatus for simultaneous measuring dynamic surface tensions and potentials developed with G. Geeraerts and F. Ville should be noted here as being of particular importance (Colloids & Surfaces A, 95(1995)281). A modified oscillating jet apparatus was used to measure the dynamics surface tension and surface potential of a surfactant system simultaneously. The maximum bubble pressure method was also remarkably improved by Paul Joos in a successful international collaboration with A.V. Makievski and V.B. Fainerman (J. Colloid Interface Sci. 166(1994)6). In these experiments the complicated growing bubble system had to be described. The concept of an effective surface age was established which made this method as the most reliable dynamic surface tension measurement for very short adsorption times. Paul Joos directed his attention to various surfactant systems, and quite often his work initiated research in new scientific domains. This is the case for his studies on surfactant mixtures with R. van den Bogaert (J. Phys. Chem. 84(1980)190), on micellar surfactant solutions with E. Rillaerts (J. Phys. Chem. 86(1982)3471), on the adsorption kinetics of surfactants at the

XI

water/oil interface and the mass transfer across water/oil interfaces with J. van Hunsel and G. Bleys (J. Colloid Interface Sci. 114(1986)432, Langmuir (1987)1069). Also the work on the dynamics of protein layer formation at liquid interfaces with G. Serrien and G. Geeraert (Colloids Surfaces 68(1992)219) was a significant new contribution. Further activities have been the studies of adsorption/desorption processes of surfactants at deforming surfaces such as growing drops and bubbles with M. van Uffelen (J. Colloid Interface Sci. 171(1995)297), or in the compressed/expanded layers on a Langmuir trough with P. Petrov (J. Colloid Interface Sci. 181(1996)530). This type of experiments became a domain of his school and led to the development of many new theoretical models. The list of outstanding contributions from the school of Paul Joos in Antwerp could be easily extended with many more of his 148 publications. Additionally, the impact of his work in conference contributions and personal discussions is hard to assess properly so soon after his untimely death. The present textbook provides a comprehensive introduction into the fast developing research field of dynamic processes at liquid/gas and liquid/liquid interfaces to graduated students, scientists and engineers interested in the fundamentals of non-equilibrium interfacial properties. It also addresses to some extent application fields, such as foams and emulsions. Theory and experiments on dynamic adsorption layers are considered systematically and discussed with respect to processes at interfaces. The original references given for each topic is a subjective selection of the large variety of literature, done by the author and completed by the editors as best they could. It is both an introduction for beginners in the present field as well as a systematic preparation of a vast range of the current scientific investigations generalised together with accumulated knowledge for those being already insider. It is the first extensive review available on the subject of dynamics of adsorption and gives a general summary of the current state of adsorption kinetics theory and experiments. The book also reviews recent progress in newdesigned set-ups and improved and generalised known methods for studying interfacial relaxations. So far insufficiently described processes like adsorption from micellar solutions or

XU adsorption with transfer across the interface are included. Also the very recent developments of interfacial molecular processes on the dynamics of adsorption layers, such as molecular reorientation and aggregation, are described in detail. The book also describes present theories of the effect of dynamic adsorption layers on mobile surfaces, such as deformed drops and bubbles, based on both diffusion and kinetic controlled adsorption models and introduces efficient approximate analytical methods to solve the mathematical problem of coupling between surfactant transport and hydrodynamics. Unfortunately, it was not given to him to finish this important work. On January 26, 1997 Paul Joos, Professor at the Universitaire Instelling Antwerpen, died after a short serious disease. The loss to the scientific community of this excellent scientist whose continuous creative ideas have opened up so many new approaches in the field of surface science is immeasurable. The range of his interests was wide: it included both theory and experimental techniques, both statics and dynamics, both simple small molecules and complex large ones. He had a real gift for simplifying complex theory and finding the right experimental conditions to test it. It was the aim of the editors to keep the style and stay true to the Joos way of interpretation, adding and completing only those parts which Paul Joos was unable to do himself. The editors are very grateful to S. Siegmund, Dr. A.V. Makievski and L. Makievska for essential support in completing the material and designing the artwork. They also want to express their gratitude to the Universitaire Instelling Antwerpen for financial support in completing the book.

The Editors Valentin B. Fainerman, Donetsk, Ukraine Giuseppe Loglio, Florence, Italy Emmi H. Lucassen-Reynders, Oegstgeest, The Netherlands Peter Petrov, Brussels, Belgium Reinhard Miller, Berlin, Germany

March 1999

Xlll Acknowledgement The family of Prof. Dr. P. Joos is most grateful to all those who made the possible realisation and publishing of this book. More specifically, the author’s family would like to express their thanks to the editors Valentin B. Fainerman, Giuseppe Loglio, Emmi H. Lucassen-Reynders, Peter Petrov and Reinhard Miller. Without their help, their inspiration and long distance collaboration which is required for a book of this nature, Professor Dr. P. Joos’ work would never have seen the light of day. A special mention and token of gratitude should be extended to Dr. P. Petrov, one of Prof. Dr. P Joos’ former PhD students, who in addition to sorting all the graphs, meticulously went over and validated every mathematical equation contained in this work. In addition to those we have cited in the text for their assistance, a special thanks should go to the Universitaire Instelling Antwerpen (UIA) for its financial support. A great big thank you also goes out to Prof Dr. P. Joos’ friends and colleagues, both in Belgium and abroad, without whom the colloquium on May 26, 1997 at the UIA honouring Professor Joos wouldn’t have happened. This colloquium was organised by a close and dear friend of P. Joos, Prof. Dr. R. Vochten, without whom the colloquium would not have been the success it was, contributing further to the publication of this book. All those cited not only contributed to the publication of this book, but they also made it possible for the lifework of P. Joos to be accomplished and made known to the world at large. This was also the wish of his wife, Beatrice Van Cauwenberghe, who unfortunately passed away before she could see the realisation of this dream. It was she who, years ago first inspired her husband to write this book. Lastly, we wish to express our gratitude to all those who stood by his side during his last months among us, especially Jacqueline Aerens who was a big support to him.

His children, Sven Joos and Sabine Christ.

Chapter 1 The Definition of the Surface Tension

Before we start to discuss surface tension we should define its physical. Surface tension can be defined on a molecular level by the theory of Bakker, mechanically by the Laplace equation, and thermodynamically as proposed by Gibbs.

1.1.

The mo lecular

de finit ion o f t h e s u r f a c e te ns ion

Let us consider two immiscible bulk phases, e.g. water and air, brought in contact with each other. In some books the surface tension is defined in the following way (see fig. 1.1a). In the bulk a water molecule is surrounded with many others which attract this molecule. These attraction forces Fa are symmetrical around the water molecule so that the net force Fn on it is zero. Near to the surface these attraction forces are no longer symmetrical because the contribution of the molecules in the bulk is larger than that from the molecules near the surface. As a result there should be a net force direction inside to the bulk. This picture is wrong. Indeed if a net force acts on a molecule, it should be accelerated into the bulk in view of Newton’ s law. Therefore the molecules near a flat surface will distribute themselves in such a way that the net force acting on them, in a direction normal to the surface (see fig. 1.1b) is zero. This means that the pressure gradient in the direction normal to the surface is zero,

2 ( 1. 1)

p being the pressure and z the space co-ordinate normal (or perpendicular) to the surface. In the bulk the forces acting on a molecule are symmetrical. This means that the pressure is isotropic and is a scalar. The pressure in the normal direction, p N is equal to that in the direction parallel to the surface, p T

Pn

~

Pt

Fig. 1.1.

(1*2)

Distribution of the attraction forces Fa around a molecule in liquid medium a) non-symmetrical distribution (the net force Fn * 0) b) symmetrical distribution (Fn = 0)

As required by eq.(l.l) p N is a constant not depending on z. p \ is the tangential pressure far away from the surface. Near to the surface, the forces acting on a molecule in directions normal

3 and tangential to the surface are no longer equal. This means that the pressure is anisotropic and a tensor. The tangential pressure p T is a function of z, or

(1.3) At some distance z, we have a net tangential pressure p°T - p T(z) * 0 which contracts the surface, and this results in a tension per unit length, called the surface (air/liquid) or interfacial (liquid/liquid) tension a. According to Bakker [1] this tension is defined as

o

=

j[ p ° T -

(1.4)

Pr(z)]dz = j [ / \ - PT(z)]dz

in view of eq. (1.2). The dimension of the surface tension is dyn cm'1 (CGS units) or N m'1 (SI units). From these arguments it is evident that the phase boundary between two immiscible phases is not a mathematical plane (with zero thickness) but a more or less sharp transition zone between theses phases. The mechanical properties of such a surface are characterized by a hypothetical stretched membrane. Eq (1.4) relates the surface tension to the locally varying pressure p, but it does not provide us with a method for measurement of a because the local pressure in the surface region is not experimentally accessible.

1.2.

Th e mechanical

defin ition

of

t he

su r f a c e

tension

(l a w o f

Lap lac e ) Let us consider an infinitesimal rectangular curved surface between two immiscible liquids. The lengths of this surface element are dlx and dl2 (see fig. 1.2). At the point A, the surface tension a, stretches the surface over a distance dl2, giving rise to a force dF{ = a i//,. The same holds for point B. At point A and B we draw lines normal to these forces meeting each other in point 0. The angle between these lines with the line PO is a, P being the centre of the

4 rectangular surface elements. These forces are now decomposed into two normal forces dFXN = Gdlxsin a = g dlxa and dF2N = g dlxsin a = g dlxa (since the angle a is infinitesimal and sina « a) and two tangential forces dFXT = Gdlxcosa and dF2T = Gdlxc o s a . The two tangential forces cancel each other, and the resulting total normal force dF'N =2Gdixa is

and dQ = dl\dh is the area of the surface element we obtain for the normal force directed inside

force dFÿ = — — . R\ is the radius of curvature of dl\. The total normal force dFN

dF1=adl]

Fig. 1.2.

dR-adl*

Schematic of an infinitesimal rectangular element of a curved surface

acting over the whole surface element dQ is dFN = dF’N +dFH = ai/Qi — + — I . This \R \ F i) normal force is balanced by the pressure difference p t - p Q, p t is the pressure inside and Pq the pressure outside the curved surface. It is obvious that p t > p Q. (In a balloon the pressure inside is larger than this outside.) This pressure difference gives rise to a force dFp equal to dFN, dFp = (p '~ p 0)dQ = dFN whence

5 \

(1.5)

1 — + — is the curvature of the surface element and Rx and R2 the mean radii of curvature, R\ R2 being complicated functions of the co-ordinates ( 1. 6)

If the considered surface element in a spherical cap with radius r, R{ = R2 = r and eq. (1.5) reduces to

Pi ~Po

2a

(1.7)

= A/? = —

Eq (1.5) is known as the law of Laplace [2]. Each method for measuring the surface tension relies on the Laplace equation. This equation is also a hydrodynamic boundary condition for normal stress. Moreover the Laplace equation has consequences for the thermodynamic properties of the system. To illustrate this we will consider the equations of Kelvin and Thompson. 1.2.1. Th e

equa t io n o f

Ke l

v in

Let us consider a liquid phase in equilibrium with its vapour. For a flat surface (r - » oo), the pressure inside and outside the liquid is equal (the gravitational contribution is neglected). The equilibrium state requires that the chemical potentials in the vapour phase, p r , and in the liquid phase, \xL , are equal. At a certain temperature T0 the pressure on the flat surface is p 0. Hence ^ r(P o -7o) = l-t £(/’0>7o)

( 1. 8)

6 If the liquid phase is a drop with radius r, the pressure inside and outside the drop is increased correspondingly by dpL and dpv . At constant temperature, T0, the equilibrium condition, expressed by eq. (1.8) must be replaced by \iy (p0 + dpvo ,Ta) = \ i L{pa + dpL0, T0)

(1.9)

By expanding in a Taylor series around the reference pressure p Q, we have

( 1. 10)

From thermodynamics we know that fW dp

dp ) T

T

= VL

( 1.11)

where vv and vL are the volumes per mole in the vapour and liquid phase. For water under standard conditions they are vv = 22.4 dm3 and vL = 18cm3. From eqs. (1.8), (1.10) and (1.11) we obtain ( 1. 12)

Vv dPv = vLdpL According to the law of Laplace (1.7) dp l = dp + d\

2a

(1.13)

where we have replaced dp for dpy . Substitution of eq. (1.13) in eq. (1.12) gives

(vr ~ vL)dP = v L

2a d

(1.14)

7 Since \ v » vL in the 1. h. s. of eq. (1.14) the molar volume of the liquid phase can be omitted. By using the gas law =

RT

(1.15)

P

the equation becomes (1.16)

Integration of this equation, between the limits p Q and p , and keeping in mind that at p = p 0, r -> oo , we obtain ^ = In— rRT Pq

(1.17)

This equation is due to Kelvin and indicates that for a drop the vapour pressure is higher than for a flat surface. 1.2.2. Th e

e q u a t io n o f

Th o m s o n

For obtaining the equation of Thomson, we consider the same water-vapour system at constant pressure outside the drop as in the previous section. In order to keep the equilibrium state, the temperature inside and outside the drop must be different. (We should remark that for obtaining the Kelvin equation, the temperature in the entire system was kept constant, but doing this the pressure inside and outside the drop must be different at equilibrium.) This means that eq. (1.9) is replaced by V -i\p o Jo +dT) = M / ’o +dPL’1o +dT)

(1-18)

Again expanding in a Taylor series around the reference temperature T0 eq. (1.18) becomes

8 (1.19)

From the thermodynamics we know

'd\iL dT

( 1.20) P

where sL and ¿y are the entropies per mole in the liquid and vapour phases. Substituting the derivatives in eq (1.19) by using eqs. (1.11) and (1.20) and remembering the relation (1.8) we obtain - sy dT = - s L dT + vL dpL

( 1. 21 )

According to the Laplace equation in this case we have dpL =d\

2a

( 1. 22)

Hence, eq. (1.21) results in

-(■V

- ^ ) d T = V L

(1.23)

The entropy per mole in the vapour phase is always larger than this in the liquid phase and the difference between both is related to the molar heat of vaporisation, H SV Si // = -

(1.24)

In this way we obtain for eq. (1.23)

vA

/ 2 a >l dT - 7 ) - h Y

(1.25)

9 Integration of this equation between the limits T and T0, remembering the condition that at T =T0, r -» oo, gives the equation of Thomson. 2v l o rH

(1.26)

This equation indicates that the temperature of the liquid phase in a drop which is in equilibrium with its vapour is lower than the temperature of the gas phase. The smaller the drop the bigger the temperature drop is (super-cooling of a saturated vapour). A combination of the equations of Kelvin and Thomson gives the equation of Clapeyron. Indeed from eqs. (1.16) and (1.25) we obtain d ln p H ~dTr = ^ l f I

(1.27)

which is called the equation of Clapeyron. As a consequence for a drop with radius r the vapour/liquid equilibrium line is shifted if the radius changes, but remains equidistant to it for a flat surface. By these two examples we have given a general idea for the implications of the law of Laplace on the thermodynamic properties of the system. More detailed considerations are beyond the scope of this book. For the interested readers, reference is made to the book of Defay and Prigogine [3].

1.3.

Th e thermo

dyn ami c definition

o f t h e s u r f a c e t ens io n

The first law of thermodynamics state that if we supply some heat d Q , to the system, we can obtain some work dW from the system. The difference between both is the increase of internal energy dU of the system,

10 dQ - dW = dU

(1.28)

dU is a function of state, but dQ and dW are not. Note that in the convention adopted here dQ> 0 if heat is supplied to the system, and dW > 0 if work is done by the system. As a consequence, for an exothermic reaction, the reaction heat is negative. For reversible processes, and if only expansion work is considered eq. (1.28) is expressed as dU = T d S - pdV

(1.29)

T is the absolute temperature, S is the entropy, p is the pressure and V the volume of the system. If some surface work is done too, e.g. by expanding the surface, this work term must be included in eq. (1.29) dU = T d S - p d V + g dQ

(1.30)

with Q the area of the surface. The work due to expansion of the surface is +cjdQ which becomes clear from the following argument. Let us consider a cylinder containing a gas phase with volume V0 under a pressure p Q and a drop with radius r, a volume Vl and an internal pressure pt (fig. 1.3). If the piston moves up the change of the volume is dV = dV0 +dVi

(1.31)

The work done by the expansion is dW = - p 0dV = - p 0dV0 - p 0dVl Since the drop is spherical, we have for its volume

(1.32)

11 4nr* F = — — -> ' 3

3t dVt = 4nr2dr

(1.33a)

and for the area

Q = 4nr2->dQ = Snrdr

(1.33b)

From this we obtain (1.34)

Fig. 1.3.

Two phase system expanding in a cylinder

In view of the Laplace law 2a

Po=P,- —

(1.35)

The substitution of eqs. (1.34) and (1.35) in eq. (1.32) results in dW = - p 0dV0 - p,dV\ + tjdO.

(1.36)

12

The terms p 0dV0 and

are the usual works of volume expansion. On the other hand it is

clear that the term +adQ is the work due to change of the area, hence eq. (1.30) is obtained. Using this equation we have for the other functions of state, the enthalpy H, the Helmholtz free energy F, and the Gibbs free energy G [4]: dU = T d S - p d V + o dQ

(1.37a)

d H = T d S + Vdp + g dQ

(1.37b)

dF = -S d T - p d V + c dQ

(1.37c)

dG = -S d T + Vdp + o dQ

(1.37d)

Note that these functions of state apply only for a closed system. For open systems a work term , due to the exchange of material must be added to the r.h.s. of eqs. (1.37). Here p, is the chemical potential and nl the number of moles of component i. From eqs. (1.37) we obtain a =

ÔU)

_ fdH)

_ (dF)

en)sy ~Uq JS/, ~\ ôqJt

ÔG ÔQ

(1.38) T,P

From these relations we can define that the surface tension is the increase of internal energy if we expand the surface keeping the entropy and volume constant, and so on. Hence a has the dimension of a surface energy per unit surface area. The mechanical definition is more general than the thermodynamic one because the latter applies only to reversible changes of area, i.e., to equilibrium conditions, whereas the mechanical interpretation in terms of Laplace's law does not require thermodynamic equilibrium [5, 6].

13 Re f

erences

1. G. Bakker, Handbuch der Experimentalphysik, Akademische Verlagsgesellschaft, Leipzig, Vol. VI, 1928, p. 18,412 2. P.S. de Laplace, Supplément au Lièvre X du Traité de Méchanique Céleste Sur l’Action Capillaire, Gauthier-Villars, Paris, 1806, p. 349 3. R. Defay and I. Prigogine “Surface tension and adsorption” (with collaboration by A. Bellemans), Longmans Green, London (1966). 4. J.W. Gibbs, “The Scientific Papers“, Vol. 1, Longmans, Green & Co., New York, 1906, reprinted from Transactions of the Connecticut Academy, Vol. Ill, (1976) p. 108, and (1978) p. 343 5. A.W. Adamson, Physical Chemistry of Surfaces, John Wiley & Sons. Inc., New York, Chichester, Brisbane, Toronto, Singapore, 1990 6. J. Lyklema, Fundamental of Interface and Colloid Science, Vol. 1 - Fundamentals, London, Academic Press, 1991

Chapter 2 Surface Thermodynamics at Equilibrium

Before we can start the description of surfaces out of equilibrium, we must know the behaviour of the system at equilibrium. Therefore, this chapter is concerned with capillary systems at equilibrium. We have seen that for a closed capillary system, the internal energy is given by eq. (1.30). For an open system the exchange of matter with the surrounding is allowed, hence this equation is written as

dU = TdS - pdV + odCl + YJHA

(2.1)

rii being the number of moles and //, the chemical potential of species /. This equation is sufficient to obtain all the relations we need.

2.1.

The Gibbs e q u a t io n

We consider a system of two immiscible phases a and P (Fig. 2.1). The total internal energy of such a system is given by eq. (2.1). The internal energy for each phase separately is dU a = TdSa - p d V a + X u A , ° dU &= TdSp - p d V p + X X dn?

(2.2)

where the superscripts a and P refer to the phases a and /?, respectively. Assuming a flat surface between the phases a and /?, from the condition of equilibrium it is required that the

15 pressure p, the temperature T and the chemical potentials

are identical over the whole

system. PHASE P P.

pZ,

V*

SURFACE p. Z p.

V‘ PHASE a

Fig. 2.1.

The border between two phases: equivalent to the real situation (filled range transition zone) and Gibbs’ dividing surface Q

The internal energy of the surface dU s is dU s = d U - d U a-d U *

(2.3)

and with eqs. (2.1) and (2.2), eq. (2.3) becomes dU s = TdSs + adQ + £ p,ifrif

(2-4)

where the entropy of the surface is defined as dSs = d S - d S a - d S &

(2.5)

and n f , the number of moles of species i in the surface, is given by dnf = dn, -d n * -dn®

(2.6)

We have discussed that in reality the surface is a transition region between the two phases a and p. Following the formalism of Gibbs [1], we define the surface as a mathematical plane

16 between these phases. By definition a mathematical plane has no thickness, hence no volume. Consequently V = Va +

and the pressure term in the expression for internal energy cancels.

The Gibbs' equation is now obtained by using Euler's theorem for homogeneous functions. In this way from eq. (2.4) the Gibbs' equation results. It is perhaps more clear, and certainly recommended to do this more explicitly. If u* and sf are the internal energy and entropy per mole for species / at the surface it follows u s = X u fn f -> dU s = X u fd n f

(2.7a)

5 s = Z s fn f -> dSs = Y .* ? dn?

(2.7b)

If the area per mole of species / is 0)h the total surface area is £2 = X W,n f -> dQ = X ©¡dnf

(2.8)

This equation is the same as V - nivi , where V is the total volume and v, the molar volume of species /. To make eq. (2.8) clearer, let us consider a table with area i2, that we cover with coins of different size. Each species of coins i has an area eoh We can cover this table with coins in different ways, but eq. (2.8) always holds and cot is constant for each /. In this way eq. (2.4) becomes X u fd n f = r]T sfd n f +

©¡dnf + £ \i,dnf

(2.9)

Now we integrate eq. (2.9) with the restriction that the composition of the surface remains constant. Hence we obtain = TY j sf nf + g Z ° W

r j

(2.27)

whence dp =

2a

Vn -V:

(2.28)

21 and since vo » v, substitution of eq. (2.28) in eq. (2.24) gives

(2.29) giving the dependence of the surface tension as a function of the radius. For a flat surface

oo,

a -» a 0 and integration of eq. (2.29) gives £o

a

2rv,

(2.30)

= 1+ -

This equation is due to Defay and indicates that the surface tension of a drop is smaller than that of a flat surface. If we could make surface tension measurements as a function of the drop size, (at this moment such measurements are nearly impossible), we could obtain the adsorption of the pure component. In the book of Defay and Prigogine [2] one experiment with water is reported

cto =

75.2 dyn cm'1 (r -»

oo)

and a = 72.5 dyn cm'1 for r = 1.72x1 O'7 cm (!). By

eq. (2.30) (with v* = 18 cm3 m o l1) we obtain T = 1.77xlO‘10 mol cm 2, being the adsorption of the pure solvent. This value is not unexpected, although the accuracy of this experiment is an open question. For a surface with a density equal to the bulk we expect T = 1.67xl0‘9 mol cm'2 (as was calculated above). In fact a much lower value is obtained, and this agrees with the assumption that the density in the surface must be less than this in the bulk. Strictly we have an adsorption of air also. In a similar way the adsorption of air could be obtained by measuring the surface tension of a bubble in water as a function of the bubble size. An equation similar as to eq. (2.30) is obtained

(2.31) In principle the adsorptions of the solvents (if we are doing experiments at a hexane/water interface) can be found by doing surface tension measurements as a function of the concentration at a flat interface, and by doing experiments at constant concentration as a function of the drop size, once for a hexane drop in water, and once for a water drop in hexane. Hence the thermodynamics provide a way to obtain both adsorption of the solvent and the

22 surfactant by using surface tension data on a flat surface, and as a function of the drop size. However the possibilities of doing such experiments are nearly impossible and is not a question of thermodynamics. (The technical difficulties are immense.) This argument may be seen as fussy, but it has consequences for the boundary condition for the conservation of mass in dynamic systems. Indeed this boundary condition reads /

(2.32)

V where t is the time, D the diffusion coefficient of the surfactant, and

d cs is the concentration d z)0

gradient at the surface. This concentration gradient does not depend on the position of the dividing surface. If the value of T in eq. (2.32) should do, this equation has no meaning. In practice however, this is of no importance since x \/ xq «

2.3.

1

(cf. eq. (2.19)).

The app lica tion o f t h e Gib bs e qu ati on

For a system with only one surfactant, the adsorption is obtained from eq. (2.19) from surface tension measurements as a function of the concentration, considering that Toxi/xo is neglected. For a system containing two surfactants the Gibbs equation is written as

-i/a = /?r[r,i/lnc,+r2i/lnc2]

(2.33)

hence for the adsorptions we have da RT ,dln c )

r, =-

1 da • r : ■ R T \d In

(2.34)

To obtain the adsorptions T, and T2 at a given composition of the bulk, equilibrium surface tension measurements have to be performed as a function of cj, keeping ci constant and vice versa. We can also do experiments, by keeping c\/ci constant, this means that the system is diluted, and we have

23

( r . + r , ) - - ^RT

.¿/Inc \J

(2.34) Cl /Cl

For an ionic surfactant, for example SDS, we have the adsorption of two kinds of ions, the Na+ and the DS\ The Gibbs equation is now written as - d a = RT^T+d \n c + +T_i/lnc_]

(2 .3 5 )

T+ is the adsorption of the Na+ and T_ that of the DS', analogously c+ and c. are the bulk concentrations of both kinds of ions. If no indifferent electrolyte (e.g. NaCl) is added to the system, c+= c= c and since the surface must be electrically neutral T+ = T. = T eq. (2 .3 5 ) reduces to da 2RT d in e 1

(2.36)

If an excess amount of indifferent electrolyte (NaCl) is added to the system, then

C+ - C NaCl

+ CSDS

- C NaCl 1 +

L SPS

(2.37)

: NaCl .

If the indifferent electrolyte is in such an excess that cSDS/c NaCI = 0, c+ can be considered as constant and the Gibbs equation reduces to da RT d \n c SDS 1

In general, (if we note the concentration of SDS with

(2.38)

cq

and the concentration of NaCl with c)

since the surface is electrically neutral (T+ = T. = T), the Gibbs equation reads - d a = ^ r r |c / l n c 0 +d\n(cQ+c)J and after rearrangement

(2.39)

24

- d c = R TT 1+

c + cn

4 co + c) dine* dcn

(2.40)

If we keep the concentration of NaCl constant, dc = 0 and eq. (2.40) reduces to

- do = RTT 1 +

-

i/lncn

It is seen that for c » Co eq. (2.38) is obtained and for c q -

(2.41)

c

eq. (2.40) reduces to (2.36).

If we are doing experiments with a fatty acid HZ in NaOH the Gibbs equation reads - d o = R T [ r HZd \n c HZ+r ^ d \ n c Na. + Tz .d \n c z _\

(2.42)

and if cNa+ is in excess (e.g. we have done experiments with NaOH and NaCl in such a way that cNa+ = 0.1M ), eq. (2.42) becomes

- des = RT]THZd In cm + Yr d In cr J

(2.43)

The dissociation of the fatty acid HZ dln Y/ =

(2.56)

0

This equation is the analogue of the Gibbs-Duhem equation in bulk

= 0 . Equation

(2.56) relates the activity coefficients with each other. In other words if we are dealing with a one surfactant system and we have an expression for the surface activity coefficient of the surfactant our choice for the surface activity coefficient is not free. The feature that the expression for the chemical potential is consistent with the Gibbs equation, gives us support for the correctness of this expression. Merely, as a matter of convenience, we will write eq. (2.51) as a function of the surface pressure /Z If 1 . This equation applies for a system with only one surfactant as well as for a surfactant mixture. Since the sum of the mole fractions in the surface is equal to unity *o + X xj =

1

(/ * 0 ) we have for a one surfactant

system

n = - - ^ l n [ y 0( l - * f ) ] G>0 1 J

(2.61)

and for a mixture

n = - f ; lnM 1 _ ^

s )]

(2-62)

For a one surfactant system it is indicated to adopt the convention of Lucassen-Reynders and van den Tempel (eq. (2.21)) and we have (if we drop the subscript 1)

fl = -R T r " In Yo

(2.63)

giving the relation between surface pressure and adsorption, provided the activity coefficient of the solvent, y 0, is known. This activity coefficient depends on the composition of the surface. If we have a second surfactant 2, it may happen that T™ = r 1Q0. For such a situation, the convention of Lucassen-Reynders - van den Tempel still applies and eq. (2.62) becomes

29 n = _ R T T * in

1-

Yo

m ii) r® j

(2.64)

with r ” = r " = T” . If the saturation adsorptions are not equal, a choice has to be made for the molar area of the solvent

coq

(or r 0°°), e.g. ro" = — = 16.7-1 O'10 mol cm 1. Making use of ©o

eq. (2 .2 2 ), eq. (2.62) result in

RT fl = - — Inyo ©n

1-

© 1+ E

r /(© o-® y).

(2.65)

The standard state for the solvent is defined for the pure substance (xq -> 1,jcq ->1), the standard states for the solutes are defined at infinite dilution [x1] -» 0 ,x f

oj, and eq. (2 .5 9 )

is written as ( 013 _ r O S ' ) rj Sy = — exp RT j expr Yj

r t

)

( 2 . 66)

For dilute solutions, the mole fraction of the solutes (but not that of the solvent) is proportional to its concentration. Indeed, if «o is the number of moles of the solvent, rij that of the surfactant of species j, and for diluted solutions (no » Znj), we have

xf = 1 " o + Z nj

nj

nj v0

“ n0

n'!)+ R T ln x f -aco,

(2.139)

"j , T. P

and for a non-ideal systems dG^ dnf j s 1

= p®5(p ,r ) + R T ln x f + R T \n yi - aw,

(2.140)

Hj, T, P

hence we have

RT lny, =

f SAG K dnf

rij,Ttp

dnf

H (nf + ns \ n f ) a(ns f 5 \a+3 «0 + n -j

(2.141) ij.T.P

Doing this derivation we obtain the activity coefficients for the solvent yo and for the surfactant J m n y 0 = //[afx * )“" ' ^ ) * - ( a + p - l X ^ ) a (xs )P

RTlny = //[p(x*)a (xs )M - ( a + p - 0 ( x £ ) a (xi )li

(2.142)

Substitution of these equations in eq. (2.73) gives the generalized Langmuir isotherm

47 c a

r exp* r ° ° -r

p - ( a + p) —

(2.143a)

with h

H_

RT

(2.143b)

Substitution of eqs. (2.142a) and (2.143b) in eq. (2.63) in view of eq. (2.73) gives the generalized Frumkin equation

n = -RTr°° Ini 1- p r I - RTT*h

(2.144)

Eqs (2.143) and (2.144) can be used to predict the surface pressure as a function of concentrations. An analytical explicit expression like the von Szyszkowski equation cannot been obtained. Restricting ourselves to a regular surface behavior a = J3 = 1. Eq. (2.143a) simplifies to: c a

(2.145)

and eq. (2.144) to

n = -RTr°° ln(^l - p ^ ) -

(2.146)

and of course for h = 0 the common equations (Langmuir, von Szyszkowski and Frumkin) are obtained. It is also found that eq. (2.343a) meets the Gibbs-Duhem equation (2.56). If we should like to consider regular surface behaviour for a mixed surfactant systems, the heat of mixing should be written as [18] AH = («o + n? + n%)(//1x0s x f + H2x s0 x% + Hl2x?x$ )

(2.147)

48 //,, H2 and Hx2 are parameters for the heat of mixing for the single component system, and for mixing of the surfactants 1 and 2 in the surface, respectively.

2.9.

Tr a u b e 's rule , t h e h yd rop hob ic ef fect

a nd inte rch ain

PENETRATION For obtaining the adsorption isotherms, (see section 2.5.) we referred to the standard state of the solvent to the pure component, and for this of the surfactant to infinite dilution. Now we define the standard state of the surfactant also with respect to the pure component. For the surfactants we have in mind liquids such as methanol, ethanol and propanol which are miscible with water. As the solutions are not diluted we will not use concentrations but mole fractions. We start with eq. (2.51). Equilibrium requires the equality of the chemical potential in the bulk and the surface li-B{p,T) + R n n x ? = p™ ( p j ) + RT In x? -trco,

(2.148)

The difference of the standard chemical potentials is defined by making reference to the pure system for which x f = 1 and x f = 1, hence from eq. (2.148) H?s (p,7’) = n,05( p ,r ) - a ,c o /

(2.149)

a, is the surface tension of the pure system. In this way eq. (2.148) becomes R T \n x f = R T ln x B - ( a , -c)co,

(2.150)

Since the sum of the mole fractions in the surface is unity it follows from eq. (2.150) written for water (subscript 0), and the surfactant (subscript 1), and defining the Gibbs dividing surface as co0 = coj = ( f 00) 1 that f o = - R T F * In Xq exp

V

f

\

r t t

00;

+ x x exp

\

,n?

(2.158)

53 where p,° is the standard chemical potential and the summation in Eq. 2.158 is taken positive for the reaction products and negative for the reactants. In fact the law of Guldberg and Waage (Eq. 2.157) is also a result of this principle of Braun-Lé Chatelier, but here we are interested in how the constant K depends on the intensive variables. For the variation of temperature, pressure (gas reactions) or electrical potential, E, (redox processes) we have the well-known equations: d \n K AH dT ~ RT2 d \n K dp d ln K dE

Van't Hoff

AV AT nF

(Nemst)

~ RT

Here AH is the heat of reaction, AV = ^ v, v, (vi the volume per molecule), F is the Faraday constant. In view of this, it is expected that for a reaction occurring at the surface the equilibrium constant should depend on the surface tension. An example of such a reaction is the folding-unfolding equilibrium for a protein at the surface. The Gibbs free energy at the surface and at p and T constant is dGs =

(T ,x f,a )d n f

(2.159)

It is understood that the summation is positive for the reaction products and negative for the reactants, n f are the number of molecules of species / at the surface, and x f are the mole fractions. By introducing the advancement of the reaction dq dGs = ' E v iV-î{x?

(2.160)

(For the reactants v, < 0, for the reaction products v, > 0.) At equilibrium we have dGs dç

='Zviyif(x?,o) =0

and it follows

(2.161)

54

2> ,

a ^ gp? dc

s )

¿/lnxf d o + Z v< Id ln x f )

(2.162)

/

Since

dni SoJ

gpf ' = RT and remembering Eq. (2.49) written as d V ln x f j c

( x s\

fa

gp,

gnJ

(2.163)

X ^ i / l n x f = d ln K

(2.164)

S

and with

Eq. (2.162) results in d ln K _ dn

Xv,
oo

while for z —»00 one should rather expect vz —>0. To solve this problem exactly, the hydrodynamic (Navier - Stokes) equations must be integrated and this is not an easy task. However the diffusion process operates in a sublayer, where the rate can be given by eq. (3.22). For larger distances eq. (3.22) does not apply, but there the concentration can be assumed as constant. Hence dc/dz -» 0 and the convective term cancels. In terms of boundary layer theory one expects that from a certain distance on, characterized by the hydrodynamic penetration depth

« p .t [3] (p is the liquid density) liquid motion disappears. Also in a certain depth VP

larger than the diffusion penetration depth 6D« V 5 7 , the diffusion process stops. As a comparison, inserting relevant data D ~ 10'6 cm2 s’1,

7

- 10‘2 g cm'1 s'1 and p ~ 1 g cm'2 one

67 finds 8d =10 25h . Hence the diffusion process occurs in a layer when the hydrodynamic process is still fully operating and therefore eq. (3.22) is a sufficient approximation. This means that the Prandtl number Pr = -2 - = 104 is much bigger than unity. In liquids this condition is pD always fulfilled (in gases Pr ~ 1). If surface deformation occurs only in the x direction eq. (3.16) can be written as: dc dt

dc dz

dc dx

( d2c V5x2

d2c' dz2

(3.23)

but since the concentration along the x direction changes much less than along the z coordinate dc

dc d2c d2c « — and — j « —r-. Thus with eq. (3.22) instead of eq. (3.23) we get dx dz dx1 dz dc dc d2c — - 0z— = D — y at 0 > 0 dt dz dz2

(3.24)

This convective diffusion equation holds for an expanding surface, while for a compressed surface (0 < 0) it states dc

dc

d2c

_ ^ 40)

This adsorption isotherm is a thermodynamic relation. If at equilibrium this Langmuir relation does not apply, we have to consider more complicated adsorption isotherms. However if the adsorption T is close to the equilibrium value Te, this part of the adsorption isotherm may be linearised:

r - r , = f ( c - co)

(3.41)

dr

where — is a thermodynamic parameter at T = Te. For some cases, to be discussed later, this dc condition of local equilibrium between surface and subsurface does not hold, and the Gibbs adsorption equation does not apply. For such cases the adsorption isotherm must be replaced by

74 a kinetic equation. We do not know anything about kinetic equations a priori, but in general we have dT dt

(3.41)

with the requirement that at t - » co, eq. (3.41) has to result in the adsorption isotherm. If we have ideal surface behavior, eq. (3.41) can be written as

(3.42) which is identical to eq. (2.123) when we replace the bulk concentration by the subsurface concentration. If regular surface behaviour applies, the aspect for the kinetic equation is not clear at a first sight. References 1. P. Petrov, PhD thesis, Antwerp 1997 2. M. van Uffelen, PhD thesis, Antwerp 1994 3. R.E. Bird, W.E. Stewart and E.N. Lightfoot, in “Transport Phenomena” Wiley, New York (1960), Chapter 21. 4.

J. Cranck, “The Mathematics of Diffusion”, Oxford University Press, London (1956).

Chapter 4 Methods for Measuring the Dynamic Surface Tension

4.1.

Ge n e r

a l r emar k s

It is assumed that the current methods for measuring equilibrium surface and interfacial tensions are known to the reader. Any surface or interfacial tension corresponding to an interface not in equilibrium is a dynamic interfacial or surface tension. Hence speaking about a dynamic surface tension is meaningless, without a specification how the surface is deformed to bring it out of equilibrium and the time elapsed from the initial start of deformation. Experimentally measurement of the dynamic surface tension at the air/water interface is easier than at the oil/water interface, therefore the dynamic surface tension of the air/water interface will be considered first. For a diffusion controlled adsorption process the relaxation to equilibrium is characterized by the diffusion relaxation time xD, defined as

(4.1) For most surfactant systems, the diffusion coefficient D does not change much and is of the order of 10*6 - 10’5 cm2 s'1. Hence in a broad time range this diffusion relaxation time for different kinds of surfactants, is not ruled by the diffusion coefficient but by the thermodynamic parameter dT/dc. Assuming a Langmuir adsorption isotherm we can calculate dT/dc and therefore the diffusion relaxation time is given by

76

-18

1

D

l«J

1

2 1 ^J->00^ exp 4nU "j /?rrj l*J

HP

(4.2)

From this it follows that for a given surfactant system, the diffusion relaxation time strongly depends on the surfactant concentration or on the surface pressure. To compare different surfactant systems, we define the diffusion relaxation time at concentration

0, and we

obtain

T0D

~

(4.3)

D\ a )

For different surfactant systems the saturation adsorption V o does not change over a broad range say between 2xlO'10 and 7xlO'10 mol cm'2, and cannot explain the broad time range of the diffusion relaxation time for different surfactant systems. The only parameter left is the Langmuir - von Szyszkowski distribution constant a, which indeed changes considerably from one surfactant system to another. For example, for decanol with D = 5x1 O'6 cm2 s'1 (which is a reasonable estimate for the diffusion coefficient), T00 = 6x1 O'10 mol cm'2 and a = 1.39x1 O'8 mol cm'3 we obtain for the diffusion relaxation time at c -» 0 x°D=373 s and for propanol (with the parameters D = 5x1o"6 cm2 s'1,

r00=

6xlO'10 mol cm'2, a = IJSxlO ’4 mol cm'3)

= 2.4xl0’6s. The practical

conclusion is that the experimental methods must be available to allow measurements of the surface tension over a very broad range of time. Fortunately for low surface active surfactants as propanol (having a high value of a) the adsorption process is much slower as predicted by diffusion, and the time domain required for measuring the dynamic surface tension is somewhat narrowed but still extend from a millisecond to several minutes (hours) as for protein solutions. It is also clear that the dynamic surface tension must be measured in a time window, comparable with the characteristic time of the relaxation process. It is meaningless to measure the dynamic surface tension of a decanol solution with the oscillating jet method (operating in a

77 time window of say 2 to 30 milliseconds), since the time of observation is much smaller than the characteristic time for the relaxation process. If we measure the dynamic surface tension of a decanol solution with the oscillating jet method we obtain the same as for pure water (solvent), indicating that the equilibration process did not start yet. On the other side if we would measure the dynamic surface tension of an aqueous propanol solution by a slow method, say by the drop pendent method, the equilibrium surface tension is obtained, indicating that in this time domain the equilibration process has been finished, or that the observation time is longer than the characteristic time. As a conclusion, the time of observation must be comparable with the characteristic time of the equilibration process. Up to here we have confined our attention to a diffusion controlled adsorption process, where the equilibrium adsorption isotherm is given by the Langmuir equation, however the made conclusions are general. Hence we must have methods allowing measurements over a broad time window. A particular experimental method can be suited for a given surfactant system, another not. A further requirement is that the experimental data obtained by different methods having connected time domains should match.

4.2.

Cla ssification

o f t h e e x pe r im e n t a l met hod s

The experimental methods can be divided in three or four classes. (/) Stress relaxation methods: In the stress relaxation methods the surface is deformed (expanded or compressed) quickly and the relaxation to its equilibrium is measured. Theoretically the surface deformation should be a step function of the time. In practice the surface deformation needs some time and it is required that during this deformation time the relaxation process does not start. (Time of surface deformation is much shorter than the relaxation time.) A classical example of a stress relaxation experiment is that of a Langmuir trough. A surfactant solution is poured into a Langmuir trough, after surface equilibration the surface is suddenly expanded or compressed with a barrier, and afterwards the surface tension is measured as a function of time using a Wilhelmy plate.

78 In some methods (oscillating jet, inclined plate) the surface is expanded in such a way that initially a bare surface with no adsorbed surfactant is obtained, and the decay of the surface tension with time is measured. (//) Dilatation or 9 methods: Originally this method is due to Van Voorst Vader and Van den Tempel. The surface of a surfactant solution, initially at equilibrium, is expanded with a constant dilatation rate 0 = ¿/InQ! d t , after a transient period a steady state surface tension is obtained. From such experiments one obtains the dynamic surface tension at the steady state as a function of 9. If the dilatation rate 9 is high enough, the surface tension of the pure solvent is obtained. (///) 9 - t methods: With those methods, the surface is expanded with a time dependent dilatation rate, and the surface tension is given as a function of time. If the initial dilatation rate is high enough, a bare surface is obtained, but in contrast to a stress relaxation experiment, during surface equilibration the surface is still deformed. Here the dynamic surface tension depends on time and on the dilatation at this time. As it will be shown theoretically both parameters can be combined in a single one - the adsorption time or effective time. An example of this is a growing drop with a constant flow rate. A drop is formed at the tip of a capillary and is growing with a dilatation rate corresponding to a constant liquid flow rate. The surface tension is obtained when the drop detaches from the capillary. The surface tension depends on the drop formation time and the flow rate. (iv) Small amplitude periodic surface deformations: This method is due to Lucassen and Van den Tempel. A surfactant surface initially in equilibrium with the bulk, is subjected to small amplitude surface deformations with a frequency co. The relative change in area is AQ ¿/InQ = — - « 1 Q0

(4.4)

where D$ is the initial area and AD the amplitude of the surface deformations. As a result the surface tension also shows periodic changes with the same frequency and the amplitude Aa. One measures the ratio of the two amplitudes giving the modulus of elasticity (s), defined as:

79 (4.5) In general when a relaxation process is going on there is a phase difference between the oscillations.

4.3.

In t

r o d u c t io n t o h y d r o d y n a m ic s

Before we can analyze the different experimental techniques one should have some knowledge about hydrodynamics. The hydrodynamic equations are rather simple, but their integration is merely a matter for mathematicians and theoretical physicists. There are two hydrodynamic equations: the continuity equation being an expression for the conservation of mass and the Navier - Stokes equation being the expression for the conservation of momentum. 4.3.1. Th e

c o n t in u it y equa t io n

If a liquid flows with a velocity v, the amount of material transported through a surface of unit area perpendicular to this flow direction per unit of time is the flux J. It is given by J = pv

(4.6)

where p is the density of the liquid. Note that this is an equation similar to that for convective diffusion. The amount of material transport into or out of a volume element dV surrounded by the area dQ is fpvJQ and gives rise to a change in density in this volume element f~ ^ d V . a v dt Hence the conservation of mass requires

(4.7)

We want to replace the surface integral by a volume integral and this is done with the equation of Gauss -Ostrogradski as expressed by eq. (3.14), in this way eq. (4.7) is rewritten as

80

J^rf/vpv +

(4.8)

=0

and since this must be taken independently of the magnitude of the volume element, it follows

+ ¿/'v(pv) = 0

(4.9)

If now as in liquids, the density is constant, this equation reduces to rf/vv = 0

4.3.2. Th e S to

(4.10)

k e s e q u a t io n

Let us consider the stream lines surrounded by an imaginary envelope, and two cross sections separated by a distance dx (see Fig. 4.1).

F

X

Q Fig. 4.1.

Schematic for a force Fx acting along the x-axis on the cross section Q

On the right side the pressure is p' and on the left side p. The force Fx acting along the x

direction on the cross sections Q is

81 (4.11)

Fx = -{ p ’- p ) n = - ^ d x Q

We remark that Qdx is the volume element dV between these cross sections and the force per unit of volume,^, is

Jp

dV

(4.12)

dx

In this volume other body forces may be present e.g. gravity, electrical, magnetic forces noted as f e, which are conservative forces and can be expressed as a gradient of the potential Ee

Je

(4.13)

dx

Hence the total force per unit volume f is

(4.14) According to the law of Newton, a force acting on a body gives rise to an acceleration: dy_x_ f t =p- dt

(4.15)

Now attention must be paid to this acceleration. It may happen that at a certain distance x the velocity changes with time

* 0, but even then when this local acceleration is zero

(steady state) the acceleration can still depend on the distance, hence dvx dt

fd Q V dt J

dvx dx dx dt

Since dx/dt = vx , we have

(4.16)

82

dt

dt

(4.17)

x dx

where we have dropped the subscript x. This is an example of a hydrodynamic time differentiation. In general relations of type of (4.17) are written as d• — =-

+« W

(4.18)

Usually the symbol — is written as — . We shall not use this notation in order to avoid dt Dt confusion with the diffusion coefficient D. Finally from eqs. (4.12) - (4.17) the balance for conservation of momentum reads ^x

dP

f

(4.19)

If dv j d t = 0 , eq. (4.19) is integrated and results in the equation of Bernoulli

pv2 + p + Ee = const.

(4.20)

It can be shown, that eq. (4.19) can be generalized and written in vector notation giving the Stokes equation

8\

p— ^pv.gradv = -grad/7-

(4.21)

ot

4.3.3. Th e N a v ie r - S t o k e s e qua

t io n

The Stokes equation only applies if there are no friction forces, e.g. when the viscosity can be neglected. If friction forces cannot be neglected an additional term must be added to eq. (4.21). It can be shown using tensor calculus that this friction force/„ is given by

83

fv = t|V2v

(4.22)

The Stokes equation taking into account friction forces becomes the Navier - Stokes equation which states UV

p—

2

+ pv.gradv = -gradp + r |V v - f e

(4.23)

4.3.4. B o u n d a r y c o n d it io n s The integration of the Stokes and the Navier - Stokes equations requires boundary conditions at a solid wall and at a free surface. The Stokes equation is a first order differential equation and requires one boundary condition at the solid surface and one at the free surface. If the solid surface lies in the plane xy of the spatial coordinates x andy, it is clear that at the solid wall this boundary condition is (4.23) At the free surface we have the boundary condition of normal stress. This is the equation of normal stress, i.e. the equation of Laplace as discussed before. If frictional forces are present the boundary conditions at the solid wall are v2 = 0,

= 0,

vy = 0. These are the non-slip conditions. At the free surface we have the boundary condition for tangential stress. A free surface without surfactants cannot sustain a tangential stress, hence

(4.24)

but for a surface containing adsorbed material eq. (4.24) is not valid. Indeed, it may happen that the surface tension over the whole surface is not uniform, this means that a gradient of surface tension, d a /d x, is present due to local differences in the adsorption. The system will try to return to a uniform surface, therefore surface elements having a higher adsorption (or lower

84 surface tension) will expand and surface elements with higher surface tension will contract. This can be seen as a “surface wind“. Since the surface is not an autonomous phase motion in the surface will generate a motion in the adjacent liquid layer, therefore

do dx

dvx ^ dz

(4.25)

This phenomenon is known as the Marangoni effect. Some particular properties of surfactant solutions, such as foaming and emulsification, are related to this boundary condition. To be complete we must mention that a surface tension gradient can be generated by a temperature gradient along the surface as well. Then eq. (4.25) becomes

da dx

do dT dT dx

dvx ^ dz

(4.25a)

whereas for a surfactant solution without temperature gradient we have

do dx

do dT dT dx

dvx ^ dz

(4.25b)

For the boundary condition of normal stress, a viscous term must be included in the Laplace equation:

Pn = - P o

+ 2 'V

dz

(4.26)

with p N the external normal pressure and p a the capillary pressure according to Laplace [1].

85 4.4.

St r es s r e la x a ti o n metho ds

4.4.1. Th e

v ib r a t in g or o s c il l a t in g j e t m e t h o d

In this technique a horizontal liquid jet is issued from the orifice of an elliptic capillary with a small eccentricity (Fig. 4.2) [2 - 4]. In this way small amplitude standing waves are generated on the surface of the jet. Let us consider a cross section of the elliptical jet (see Fig. 4.3a). At point 1 the curvature is higher than at point 2, and in view of the law of Laplace the pressure at point 1, p\, is higher than at point 2, p 2. As a result there is liquid flow inside the jet from point 1 towards point 2. For low viscous liquids, such as water, a circular cross section (Fig. 4.3b) is obtained, with an overshoot in the opposite direction (Fig. 4.3c). In this way standing waves are generated due to the pressure difference pi - P2, hence due to the surface tension. It is understandable that the wave length of the standing waves is a function of the surface tension. The theory for this hydrodynamic problem was given by Bohr [2]. In order to illuminate the physical aspects we will present here a somewhat simplified theory for a jet with zero viscosity. This means we have to deal with the equation of continuity and Stokes’ equation, which are written in cylindrical coordinates. The continuity equation eq. (4.10) is written as

(4.27) where vr is the radial velocity,

vq

the angular velocity and vz the velocity component along the z

coordinate. Since the undisturbed velocity along the z coordinate is constant for a horizontal jet, we assume that disturbances along the other coordinates are more important and we approximate d v jd z = 0. In this way eq. (4.27) becomes

(4.28)

86

Fig. 4.2.

Schematic of a horizontal liquid jet issued from an elliptic orifice, parameters described in the text, according to G. Geeraets, PhD thesis, Antwerp 1994

P-P

R-R

Pi > P2

P i= P :

Pi < P2

b)

c)

a) Fig. 4.3.

Q'Q

Schematic of the cross section of an elliptical jet, explanations see text, according to G. Geeraets, PhD thesis, Antwerp 1994

Since the jet is stationary, dv / dt = 0, and for a horizontal jet with zero viscosity, the Stokes equation is approximated by

87 dv dp pv^ = - ¥

dvQ

1

pv* l k = ~ 7

dp

ae

(4.29a)

(4.29b)

Because we have neglected the viscosity, the flow is conservative and can be expressed as a gradient of a potential cp

1

(4.30a)

Substitution of eqs. (4.30) into the continuity eq. (4.28) gives d2cp dr2

1 dtp r dr

1 d29 r 2 d02

(4.31)

This equation is integrated by the method of separation of variables cp(e,r) = f ( r ) y ( e )

(4.32)

and eq. (4.31) reduces to r 2 f"(r) + r f '( r ) - n 2 f(r) = 0

(4.33)

vj/"(0 ) - « 2cp(0 ) = 0

(4.34)

where n2 is a constant for the separation of variables. For an ellipse n -= 2. Eq. (4.33) is of the Euler type with the integral

88 f(r)= Kxr n +K 2r n

(4.35)

but since at r = 0,f(r) must remain finite the integration constant Ki = 0. Thus f(r) = Kxr n

(4.36)

The integral of (4.34) is

VO) =

(4.37)

hence eq. (4.32) becomes q>(r,0,r) = K r neM eikz

(4.38)

(p depends on the coordinate z because standing waves are generated with a wave number k. The integration constant, K, is now obtained from the Laplace law

p =a

J _ Ji VA, R , + R,

(4.39)

with R\ the radius of curvature normal to the axis of the jet, and R2 the radius of curvature along this axis. Since small amplitude waves are generated R\«Ri and Eq. (4.39) approximates to

P =T

(4-40)

First we calculate the pressure, and substitute eq. (4.30) into eq. (4.29) d2cp dp pV-’ dr dz ~~dz

(4.41a)

89 52cp dp pVz 56 dz = 50

(4.41b)

After integration both equations result in

(4.42)

p = pv2—

Hence the boundary condition for normal stress, expressed by eq. (4.40) at the surface of the jet (for r = R) gives

P VZ (

d< p"|

a \d z )R

=1

(4.43)

R\

The radius of curvature R\ in polar coordinates is

1 X\

R l + 2 R 'l - R R ” ( r 2 + r ,2Y

1

1+ 2

R f

1+

Ru

JT

R2 ~ R

^,2^%

-

(4.44)

R 'J

For an ellipse with small eccentricity R '/R « 1 and eq. (4.44) reduces to

JL

Rt ~ I r ~ R 2

(4.45)

We consider now that the jet oscillates around the radius a of the undisturbed cylindrical jet R = a + Z(Q,z ) with a»Z(0, z). In this way eq. (4.45) becomes

(4.46)

90 1

1

R{

l

d2Z

(4.47)

(Z + a)2 d02

Z +a

and since the amplitude of the waves is small, we expand it in a series

Rx

aV

a)

a2 \

(4.48)

a) dQ2

retaining terms up to a 2 only 1 Ri

1

Z

a

a2

1 d2Z a2 502

(4.49)

If we now consider that R « a, the boundary condition, eq. (4.43), becomes p v .p q A a

1

Z

1 d2Z

a

a2

a2 502

(4.50)

v> is the hydrodynamic derivative of R, (cf. eq. (4.18)) so that we have ,

,

facp'l

dR

dR

dR

(4.51)

Because the waves are stationary dR/St = 0 eq. (4.51) reads ( Sep')

dZ

(4.52)

l * J / v' & =0

The function Z can now be eliminated from eqs. (4.50) and (4.52). For doing this we take the derivation of eq. (4.50) along the coordinate z

pvz ( g2q>']

a \ d z 2) a

a 2 dz

a 2 dzdO2

(4.53)

91 and substitution of eq. (4.52) into eq. (4.53) gives the final expression for the boundary condition of normal stress

3 q> U r æ 2J

2 2 i P V2 a

>

(

a

\

dV dz2.

+

(4.54)

a

Substitution of eq. (4.38) into eq. (4.54) yields the dispersion equation

a 3pyz

(r? -n ) = k 2

(4.55)

We remark that the flow rate F is given by F = vzna2

(4.56)

and with

(4.57)

* -T n = 2 for an ellipse eq. (4.55) finally reads 4pF 2 cr = ■ 6dk2

(4.58)

For obtaining this equation we have neglected dv2/ dz in eq. (4.28), if we would not do this we obtain 4pF2 6aX2

1 5 n 2a 2 1+ ; t t

which is a rather important correction.

(4.59)

Bohr even took into account the effect of viscosity and the finite wave amplitude, yielding finally to the more complex relationship

(4.60)

with b the amplitude of the wave. These correction factors are usually small and in practice eq. (4.59) can be used. If we do measurements for pure liquids with a constant surface tension, we observe that the surface tension depends on the distance from the orifice, and is higher than the static value. Since a dynamic surface tension for pure liquids does not make sense, this must be an artifact. At longer distances from the orifice the surface tension becomes equal to the static ones. The reason for this is that in the capillary we have a velocity profile according to Poiseuille, while in the jet there is a plug flow. Hence near to the orifice we have a transition region from Poiseuille flow to plug flow, and this transition is not accounted for in the theory. Therefore in practice the first waves have to be disregarded. The extent of this transition region can be obtained numerically for instance by the finite element method. For a liquid film falling through a slit (the curtain coating technique) such calculations are performed [3, 4] and this transition region is indeed very narrow. For an aqueous surfactant solution, at the orifice of the jet a new surface is created. By travelling along the jet surfactant is adsorbed giving rise to a decrease in surface tension. The age of the surface, f, corresponding to the surface tension at a distance z from the orifice is

t=

z v.

(4.61)

with vz the mean velocity in the jet. From eq. (4.56) it follows that the age o f the jet is

93 Hence to obtain the dynamic surface tension with time one must measure the flow rate F, (this is easily done), the distance from the orifice z, the mean radius of the capillary and the wave length. A water (solution) head is mounted (eventually with a pump) to give a constant flow. The mean radius of the orifice of the capillary is measured with a microscope, and the distance z by a traveling microscope (Fig. 4.4).

Fig. 4.4.

Schematic of an oscillating jet method, description see text, according to G. Geeraets, PhD thesis, Antwerp 1994

The wave length of the jet is obtained by a parallel light beam coming through a horizontal slot. The waves of the jet act as lenses, focusing the light beam on a mat glass plate (Fig. 4.2). This procedure is due to Stokes. The main difficulty is however to obtain a capillary giving a stable jet. For this purpose we have used capillaries produced by the coworkers of the late

94 Prof. Defay. They were manufactured in the following way: a capillary was heated and immediately afterwards subjected to a mechanical pressure. Most of these capillaries do not give a stable jet and are rejected, but a few of them produced stable standing waves of the jet. Along the jet the surface tension decreases so that a surface tension gradient (da/dz

2

with a « 0.02 cm, F « 1 cm s’ and — ^ -2 dyne cm' this convection term for the Marangoni flow is

t ici^

da

= “ 3 *10”3 and hence negligible. This conclusion is rather general. For a fast

flowing liquid the Marangoni effect is unimportant. The oscillating jet method allows surface tension measurements, the first waves being rejected, in a time window between 3-30 ms. A typical velocity of the jet is vx = 400 cm s'1. Hence 1 cm along the jet is travelled by the liquid for 2.5x10’3 s. If we assume the surface tension drop for this distance is ^ - = 2 0 dyn cm l (which is a relatively high value) the Marangoni effect generated by this gradient can be 6 da da assumed as — = rj—— or vM = — — where 8 = r| dx dx P* \ 1/2 , t | da Marangoni flow. Thus vM =[ — J

1/2

is the penetration depth of the

10 cm s . This is only 2.5% of the jet velocity,

hence the Marangoni effect along the jet can safely be neglected. 4.4.2.

Th

e

in c l in e d

p l a t e m e t h o d

In this method [7, 8], the surfactant solution is pumped over an inclined plate with an angle of inclination a. At the inlet a new surface is created and when the liquid film flows in a canal of width b (about 2 cm) over the plate the surface ages and the surface tension decreases. The surface tension is measured by a Wilhelmy plate (Fig. 4.5). The age of the surface is given by eq. (4.61)

,=v

(4.68)

with x the distance from the inlet to the place where the surface tension is measured. There is no pressure gradient along the plate so that the non linear term vgrad v in the Navier Stokes equation can be neglected, This term would be important only at the inlet, where it is assumed that the dilatation 0 or dvx /dx is high enough for creating a fresh surface.

96

pump Fig. 4.5.

Sketch of the apparatus of the inclined plate method, the transducer is equipped with a Wilhelmy plate, x - direction along the inclined plate; according to [8]

Considering that the flow is stationary (dvx/dt = o) this equation becomes

= -p g sin a

(4.69)

v* is the velocity along the canal, z the coordinate normal to the surface and pgsina the

buoyancy term. Integration of eq. (4.69) gives:

97

TI

O, and therefore the dynamic surface tension relaxes to its equilibrium value a c corresponding to a height he. The time for equilibration can

107 be rather long (hours) because the surfactant has to be supplied not only to the expanded meniscus, but also to the expanding liquid film adhering to the wall of the capillary. Hence

h*

(4.87)

R A pg

hence R A p g (h -h e)

Aa = a - a e

(4.88)

h -h e

Strictly there is also a difference Ahv due to the viscous dissipation in the capillary given as

A/jv =

8rjlv

(4.89)

R2Apg

which in most cases can be neglected. L is the length of the liquid in the capillary. The main experimental difficulty in this method is that the capillary must be thoroughly cleaned especially for measurements at the oil/water interface. A correction in all of these capillary methods is required as we have assumed a Poiseuille flow inside

the

capillary.

This

is

strictly

not

correct.

A

Poiseuille

flow

requires

vz ~ ( r 2 - r 2) whereas in the meniscus vz does not depend on r. Hence there is a transition region where dvz / dr * 0 . This can be accounted for by introducing a small distance Ac for which Ap R 2 Ax

8rpd 1+

x

(4.90)

108 This correction disappears for x-»oo. An experimental proof is that the withdrawal of a liquid in a capillary (receding meniscus) is excellently described by the Washbum-Rideal-Lucas equation.

4.6.

Met

h o d s w it h c o n s t a n t s u r f a c e d e f o r m a t io n

For these kind of experiments, a surface initially in equilibrium is deformed with a dilatation for a particularly defined time length. Since we have a continuous surface deformation a convective current occurs, which must be accounted for in the diffusion equation as argued in Chapter 3. Another feature is that during the formation of a new surface (expansion), surface elements are formed at different times. Experiments are also possible where a clean surface is created at time t = 0, but contrary to stress relaxation experiments, the surface still expands with time. 4.6.1.

E

x p e r im e n t s in a

L

a n g m u ir

t r o u g h

The surface in a Langmuir trough, initially in equilibrium with its bulk is continuously deformed, and the jump in surface tension monitored as a function of time. The most simple experiments are when the surface is linearly expanded or compressed. If Dq is the initial surface area, the area at any time t is

Q = Q 0 +~~^t = f20(l + QLt)

where

(4.91)

= const, (linear surface deformation). For an expanding surface dQ/dt > 0 and for a

compressed one dQ/dt < 0. The parameter a is defined as _}_dQ Q 0 dt

(4.92)

and is consequently positive or negative for an expanded or compressed surface, respectively. From this equation it follows that the dilatation 0 is given by

109 dlnQ dt

e = -

a 1+ at

(4.93)

a is the initial dilatation at t->0. Hence for an expanding surface the dilatation decreases with 0 1 time, but for an proceeding compression the surface — = — —- increases with time (since a-l, which means for Q-*0.

4.6.2.

E

x p a n s io n o f a

4.6.2.1. Ex p a n s io n

d r o p

o f a d r o p w it h in it ia l l y c l e a n s u r f a c e

A clean surface is formed at the tip of a capillary, growing further with a constant volume flow rate. A spherical shape of the drop is assumed. At a time td the drop detaches from the tip of the capillary and its volume F is measured. However this volume cannot be related directly to the surface tension at the moment of detachment as one does for a static drop. The reason is hydrodynamics. Because of the constant flow into the drop two counteractive effects occur [34]: (i) During the drop detachment the neck of the drop becomes thinner. Simultaneously it is blown up by the liquid flown from the tip which will increase the measured volume F above the volume Fo, corresponding to the surface tension a at the moment the detachment started, (ii) An opposite effect is due to the extra impulse of flowing liquid. During the detachment of a static drop surface tension forces are equal to gravity forces. In a dynamic drop, a circulation current exists caused by the stream of the falling drop, which creates an extra force enhancing the detachment. This decreases the volume F of a detaching drop below the volume Fo. As a result even for pure liquids, for which the surface tension is constant during drop growth, the drop volume F depends on the dropping time td (being the time between two consecutive drops) or on the flow rate. According to Brady and Brown [35] McGee has obtained the following empirical relation between the volume F and the dropping time V = V0 + S td-01i

(4.94)

where S is a parameter depending on the surface tension, called also the slope of McGee.

110

Fig. 4.13.

Dependence of drop volume V [cm3] on dropping time td [s] at Vo= 0.2381 cm3, r=0.4025cm, S=4.70xl0'2cm3 s3/4, yielding the interfacial tension cr=50.3 mN/m, system water/hexane without surfactant [37]

Fig. 4.14.

Dependence of the extrapolated drop volume Vo [cm3] at t-*oo as a function of the slope S [cm3s3/4] (r=0.4025cm): ( • ) system water/hexane; (A) hexane/aqueous solution of PFAC (perfluoroammonium caprylate); (♦ ) hexane/aqueous solution of Triton X-100, according to [37]

Ill

55 i

30 -

♦♦♦

25 20 "-----------r-

0 Fig. 4.15.

10

20

30 t[s]

40

50

60

Dynamic interfacial tension a and the boundary condition far away from the surface (we are considering semi infinity medium) c -> c0 at z -> oo

(5.3)

co is the bulk concentration. The initial conditions are c —Cq and T = T0 at t = 0

(5.4)

Ward and Tordai integrated the diffusion equation using Green's functions [1], but this can be done also, as shown by Hansen [2] by the method of Laplace transformations. This method will be followed here. We take the Laplace transform with respect to the variable t of eq. (5.1), remembering the initial condition (5.4)

SC - C Q

D

dz

(5.5)

c is the Laplace transform of concentration c 00

c(z) = fe~s‘c(z,t)dt = l {c (z ,Ì)) 0

(5.6)

For integrating eq. (5.5) we first consider the reduced equation

sc p —D

d 1cR dz2

with the integral

(5.7)

121

cR - A exp

f \

+ Z?exp

(5.8)

Since for z -> oo the concentration c and hence the image function cR must remain finite the integration constant B = 0. The integral of eq. (5.5) is equal to the integral of the reduced equation (5.8) (whith B = 0) and a function of the form of the second member, which is a constant, hence ( c - A exp

V

+K

(5.9)

Substitution of eq. (5.9) in eq. (5.5) gives K, and the result is / c - A exp

V

(5.10)

At z = 0, the image function of the subsurface concentration (this is the concentration at z = 0) is cs, whence from eq. (5.9) we have

+^

(5.11)

The integration constant A is obtained from the boundary condition of the conservation of mass at the surface eq. (5.2). The Laplace transform of this boundary condition taking into account again the initial condition eq. (5.4) is

*r-r0=

(5.12)

T is the image function of the adsorption T. Substituting the derivative in (5.12) taken from eq. (5.10) gives

122 s r - r 0 = -y fD s a

(5.13)

and the integration constant A is obtained from eq. (5.11) yielding

r=

(5.14)

We have now to take the inverse Laplace transform to obtain the result. The important inverse transforms needed are:

To obtain the inverse transform of -f= we make use of the theorem of Borel also called the y/S

convolution or Faltung theorem:

L~x

2 77 (*->■)- _ f c,(X) ■dk = —?= fcs(t - k ) d ^ ■Jnk l yln(t - X)

In this way the inverse transform of eq. (5.15) results in the famous equation of Ward and Tordai

r = r0 + 2

1/2

(5.15)

In fact we have transformed our diffusion equation into an integral equation. The convolution integral (internal product o f Volterra) is called the back diffusion integral. This comes from the

following. If the surface should be a sink, the subsurface concentration should always be zero, and in eq. (5.15) the convolution integral should become zero. (This is in fact the short time

123 approximation of the adsorption process considered later). In reality the surface is not a sink but has some finite capacity and the convolution integral accounts for this. The equation of Ward and Tordai is unsuitable to describe the experimental results since it contains two unknown functions T and cs. If the adsorption process is diffusion controlled there is local equilibrium between the surface adsorption and the subsurface concentration. The relation between both is given by an adsorption isotherm, e.g. the Langmuir isotherm

r =r

cs(t)

(5.16)

a + cs(t)

or a more complicated isotherm like one for regular surface behavior cs(t) a

_r_

exph 1 - 2 J->00

(5.17)

With the Ward and Tordai equation and the adsorption isotherm the adsorption as a function of time can be obtained numerically [3,4]. Nowadays computer programs are available. If the adsorption process is not diffusion controlled, the equation of Ward and Tordai still applies, but the condition of local equilibrium between surface and subsurface is to be replaced by a kinetic equation. As was discussed in Chapter 2 when the Langmuir equation applies, the respective kinetic equation can be written as

— = kxc , ( t ( \ ~ — dt 1' r*)

r°

(5.18)

If the surface expansion is high, e.g. in an oscillating jet experiment the initial adsorption r 0 = 0 and the Ward and Tordai equation becomes

T=2

V7

c0V 7- jc s(t - \ ) d j k o

(5 .1 9 )

124 If we consider the desorption from a slightly soluble spread monolayer Co = 0 and the Ward and Tordai equation becomes

r = r 0 - 2^—J /

5.2.

1/2

Th e Su t

V7

\c s{t - \ ) d j k

(5.20)

h e r l a n d e q u a t io n

The Sutherland equation is a special expression of the Ward and Tordai equation for the case of small deviations from equilibrium. Eq. (5.14) can be rewritten as re

s

r 0- r s

D ie »

Vs \ s

-c t

(5.21)

where Te is the equilibrium adsorption. If we adopt the notations r - T , = AT

(5.22a)

and

r0-r,= A r e

(5 .2 2 b)

their Laplace transforms are

__ _ r

¿{AT} = Ar = r - — s

(

\

a Fa

r0re

and also cs - c 0 = A cs with the Laplace transform

(5.23)

125 ¿{Ac,} = A

S

With this eq. (5.21) becomes

— Ac,s

(5.24)

We assume now that the deviations from equilibrium are small and we may linearize the adsorption isotherm

(It should be noted that in eq. (5.25) the condition for local equilibrium between surface and subsurface is accounted for). Taking the Laplace transform of eq. (5.25), substituting the results into eq. (5.24) and defining the diffusion relaxation time x D or diffusion relaxation frequency

(5.26) we obtain

(5.27) The inverse Laplace transform of this equation gives

(5.28) Since we have considered small deviations from equilibrium we can linearize the jump in adsorption Ar with the jump in surface tension A a , and eq. (5.28) gives the Sutherland equation

126

Act = Acr0 exp

e rfc j—

(5.29)

with Act = ct -

(5.50)

co

oo,

Dynamic surface tension of Triton X-405 solutions; c0=2.54 10'8 mol/cm3 (• ) , co=7.63 10'8 mol/cm3 (A),c0=1.53TO‘7 mol/cm3 (♦),c0=2.5410'7 mol/cm3 (■), solid lines - long time approximation of eq. (5.42), according to [9]

—

I

- » oo

but since T must remain finite (T

lim Co /— ► oo Therefore Hansen assumes

Te) it follows that

(5.51)

132

s(/sin20 ) - c o - v - “ ne

(5.52)

where a is a constant to be evaluated. Substitution of eq. (5.52) in eq. (5.50) gives

T = 2|

D tV 12 [

*}(

a

\

(5.53)

and after integration f = a Jn D

(5.54)

Substitution of a from eq. (5.54) into eq. (5.52) gives

C0 -C 5 =

sinGVnDt

(5.55)

V7iDt

since for large times sin0 -»1. Finally from the Gibbs equation, eq. (5.55) gives

a =

+-

2 ( 1 N'1/2 c0 \nD t

r t t

(5.56)

This is the long time approximation of Hansen which is also obtained from the Sutherland equation using the asymptotic expansion for

exp{x)erfc4x » -7 = for* -> yJKX

00

If A cs° = cs{t = 0 ) - c 0 = ~Cq (since cs (/=0) « 0) than eq. (5.28) we have

(5.57) AT ^ Acs ^ cs - c 0 . Hence from AT0 A c5° c0

133

& Cs -

f~ ~

yjnt

(5.58)

- Cs

and with the definition of the diffusion relaxation time

Ac, =

c0 dT _ yfnDt dc

c0

td

given by eq. (5.26) we finally obtain

Te

(5.59)

JnD t c0

dr

r

In the linearized domain — = — and eq. (5.59) is identical to eq. (5.55) dc cn Summarizing, we have two expressions for the long time approximation, one given by eq. (5.42) and the other by eq. (5.56). We remark that both equations are only different by a numerical factor RTT2

cr = a e + - ^ - V P

(5-6°)

According to Hansen p=0.318 and according to eq. (5.42) p=0.785. The question is which value is more correct? In order to answer this we compute from eq. (5.15) (with T0 = 0 ) together with eq. (5.16) the dynamic surface tension with time for a hypothetical surfactant system with the following parameters cq = 6.85x1 O’6 mol cm'3t D = 5x1 O'6 cm2 s '\ r 00= 5.9xlO'10 mol cm 2t a = 1.5X10-6 mol cm'3. From the computer results we select data in the long time domain and plot them as a function of t ~^2. The results a are shown in Fig. 5.4a as a function of t and in Fig. 5.4b as a function of f m. Through the points of Fig. 5.4b we can draw a straight line from which we obtain a slope giving P = 0.75, a value closer to that predicted by eq. (5.42). Nevertheless the data for t~{/2 -» 0 obtained by this equation does not extrapolate through the equilibrium surface tension, but to a somewhat lower value. For very long times the equation of Hansen is more accurate, but there the dynamic surface tension is very close to the equilibrium value. In this very long time domain, the Hansen approximation applies, but requires extremely precise dynamic surface tension measurements which are hard to obtain experimentally. For moderate

134 and more practical data, the approximation given by eq. (5.42) seems better. Hence the approximation of Hansen is applicable in the suitable time domain, but it is of minor practical importance. This point was considered recently in more detail by Miller et al. [8].

Fig. 5.4.

Dynamic surface tensions a(t) calculated from the Ward and Tordai eqs. (5.15), (5.16); roo=5.910'10mol/cm2, a = 1.5'4 O'6 mol/cm3, co=6.85 10‘6 mol/cm3, D = 5 10"6 cm2/s, T=300K; a) a as function of t, b) a as function of t‘1/2, dotted line - linear plot, solid line - eq. (5.56)

5.4.

a ppr o x im a t io n m a k in g u s e o f t h e d if f u s io n pe n e t r a t io n

Th e

DEPTH In the Ward and Tordai equation the convolution integral complicates the analytical evaluation of the adsorption as a function of time. In order to avoid this we make use of the concept of the diffusion penetration depth. In this way, the integration of the diffusion equation is avoided. This procedure is even more useful in more complicated situations where the integration of the diffusion equation and the boundary condition become much more difficult. By the concept of the diffusion penetration depth, the concentration gradient is approximated by

135

(5.61)

-shzDt is the diffusion penetration depth given by eq. (3.29). In this way the mass balance at the surface becomes

If we assume now that the concentration differences are small, and we may linearize (see eq. (5.25)), it follows from eq. (5.62)

¿ ( r - r .)

(5.63)

dt

where we have introduced the diffusion relaxation time xD (see eq. (5.26)). Integration of this equation with the initial condition: at / = 0 T = r 0 and remembering that Ar = T - Te and Ar0 = r 0 - r e gives f Ar = Ar0exp ■

(5.64)

V

In view of the linearization

Aa = Aa 0 exp

Act

Aa0

----- we obtain A ry

/ \

(5.65)

As eq. (5.47) this equation is an approximation of the Sutherland equation. In another way we can use eq. (5.62) in combination with the adsorption isotherm, which is assumed to be the Langmuir equation (eq. (5.16)). This results in

136 &

i ( p V

av \ f d V/2 i - r / r e v ° _ r * - r v ~ I ti/J i - r / r “ c°

n i

dt ~ \ n t )

(5.66)

Integration of this equation with the initial conditions t = 0, T = 0 results in

2 cq

P l 1/2 n.

H

r*

r+ (rM- r e)in

re

(5.67)

re- r

in which the Langmuir equation at equilibrium is used. In Fig. 5.5 we compare the adsorption with time using eq. (5.67) and the Ward and Tordai equation with the Langmuir adsorption isotherm using the parameters Co = 3x1 O'7 mol cm '\ a = 2.9x1 O'8 mol cm'3, D = 5x1 O'6 cm2 s '\ f® - 4.12x1 O’10 mol cm'2.

4 3,5 1 ¥

o°S

3

/* •

‘

I 2,5 ~2 1,5

0,5 0 0

0,2

0,6

0,4

0,8

t[s] Fig. 5.5.

Adsorption T [mole cm'2], as a function of time, t; ro=3.75 10'10 mol/cm2, D=2.7 10'6 cm2s'1, a=2.9 10'8 mol/cm3, c=3 10'7 mol/cm3, r°°=4.12 10*10 mol/cm2. (♦ ) small times eq. (5.34), ( • ) large times eq. (5.42), (■) penetration theory eq. (5.65), (A) Ward and Tordai equation eq. (5.15), (□) first-term of the Ward and Tordai eq. (5.15), according to [10]

It seems that the eq. (5.67) overestimates somewhat the exact data obtained by the Ward and Tordai equation. Nevertheless this equation seems a good approximation of the Ward and Tordai equation if the Langmuir adsorption isotherm is used.

137

5.5.

The r a t e o f ad sorpti on for diffusion

contro

lle d kinet

ics

Let us consider the short time approximation eq. (5.30) or eq. (5.62) with cs = 0. For all surfactant

systems

the

diffusion

coefficient

is

not

very

different,

say

\0'e cm2s'x< D < 10'5 cm2s']. Hence the rate of adsorption depends little on the D-value (a factor of 3) but depends strongly on the bulk concentration. If we want to do an experiment, the equilibrium surface tension must be substantially lower than that of the pure solvent, say at least several dyn cm'1. This means that if we restrict ourselves to a Langmuir isotherm cq/ci may not be too low. Since the Langmuir - von Szyszkowski constant a, for a homologous series of surfactants strongly depends on the chain length, this means that for the same value of cq/ ci, the bulk concentration will be higher for short chain length surfactants and lower for a long chain length. This means if the adsorption process is diffusion controlled a short chain length surfactant adsorbs rather quickly and the long chain length slowly. The rate of adsorption depends merely on the concentration (or chain length) and not on the diffusion coefficient. If te is the time needed to reach the equilibrium adsorption we can have a rough idea about this time applying eq. (5.30) r I* ta =4 D VCq

TE

f

poo ^ 2

4D \ a )

f\ i + -a

n 7pooA2 -211 exp 4D RT r 00

(5.68)

where the final term results from the von Szyszkowski equation (2.79). Since the equilibrium co surface tension (surface pressure) is determined by 1+ — , (assuming a Langmuir isotherm) it a becomes clear that this characteristic time te, depends on the saturation adsorption and on the parameter a. For different surfactant systems, the saturation adsorption does not differ strongly, say 2x1 O'10 mol cm'2 < r00 < 7x1 O'10 mol cm'2. Hence the factor determining the rate of adsorption is again mainly the surface activity. The more active the surfactant, the slower the adsorption rate. The same conclusion is obtained from the diffusion relaxation time, which for a surfactant system, following a Langmuir isotherm is:

138 f

/'r x 'l Z° ~ D \ d c )

~ D

\ a

a J

exp

411

V RTT°

(5.69)

i ^ r the lowest value of Co is obviously zero, and with this yields for

0

TD

lin V d

{

(5.70)

>

from where we see once again that the parameter a rules the rate of adsorption. For instance for propanol (using D = 5x1 O'6 cm2 s'1) x°D = 2.4x1 O'6 s and for decanol x ^= 370 si Hence we have two surfactant systems with very different adsorption rates. It is incorrect to intuitively attribute this to a different value of the diffusion coefficient, the very reason is rather that the slowest is more surface active than the fastest one. This of course holds true only if we speak here about surfactants at one and the same bulk concentration.

5.6.

Th e W a r

d and

To

r d a i e q u a t io n f o r a s u r f a c t a n t m ix t u r e

If we have a mixture of two surfactants, we have to consider two diffusion equations dcx d2c{ —L = A — t dt dz2

(5.71a)

foz n dt ~ Dl dz2

(5.71b)

with two boundary conditions for conservation of mass at the surface dY\ dt

(5.72)

139

o If the initial conditions are

= 0 a n d r2 = 0 at t = 0 the integration of these equations gives

two equations of the Ward and Tordai type

(5.73a)

(5.73b)

This result is general and applies also to other experimental situations: adsorption to a continuously deformed surface or adsorption to a periodically deformed surface. The diffusion equations with the boundary conditions result in two equations containing the subsurface concentrations. These two subsurface concentrations are now related to an adsorption isotherm. If for both components, the saturation adsorptions are equal T® = r 2c0 = T 00, and we have ideal surface behaviour, these adsorption isotherms are given by the generalized Langmuir equations

(5.74a)

(5.74b)

In this way the Ward and Tordai equations can be solved numerically.

140

3 n 2.5 [ U1D/I0UI

♦

2

♦

01]

J

1.5 " ■ ♦ ♦ 1 ♦ 0,5 4 o *0

100

200

300

400

500

600

t [m sec] ^

Adsorption as a function of time (calculated) for a sodium myristate-sodium laurate mixture; concentration are co=3 10‘7 mol/cm3 for sodium myristate (♦ ) and co=1.52 10‘6mol/cm3 for sodium laurate (■), D=2.8 10-6 cm2/s, according to [11]

[u i/ mu i] O Fig. 5.7.

Dynamic Surface pressure for a mixture of sodium myristate (1.52 10’3mol/l)sodium laurate (3 1O'4 mol/1); different runs (□, ■, • , A , ♦), solid line calculated by the generalized von Szyszkowski equation (2.82), according to [11]

141 An interesting aspect is that if one of the surfactants (say surfactant 1) adsorbs faster than the other one (surfactant 2), the adsorption of the last one (minor component) at small times is equal to zero and eq. (5.74) reduces to

(5.75)

As a result the adsorption of surfactant 1 does not depend on that of the minor component (at least in the small time domain). Later when also the minor component adsorbs, eq. (5.74) has to be accounted for ( r 2 * 0) and the adsorption of surfactant 1 as a function of time passes through a maximum (see Fig. 5.6). In Fig. 5.7 we have presented another example and it is seen that the results are well described by the theory.

5.7.

Th e W a r d

and

To

r d a i e q u a t io n f o r d if f u s io n t o a d r o p s u r f a c e

Until now we have considered the diffusion to/from a flat surface between two semi infinite media (air and water). Here we will consider the diffusion from an infinite bulk phase to the surface of a drop. The diffusion equation is now written in spherical coordinates

(5.76)

r is the radial coordinate. The image function c is given by (at / = 0; c = cq)

(5.77)

The integral of eq. (5.77) is

(5.78) The boundary condition after Laplace transformation becomes (at t = 0; f = 0)

142

sr

-AfX

(5.79)

and with eq. (5.78), eq. (5.79) becomes /

DA j r = - —5R2

i—

\

R + \ exp

(5.80)

is the radius of the drop. From eq. (5.78) we have for the image function of the subsurface concentration cs at r = R

c .^ e x p

(

I— \ »0

/

(5.81)

s

Elimination of the integration constant A between eqs. (5.80) and (5.81) gives Ca

i—

s^s

c.

Cn D

c. D

r*

S2 R

s R

(5.82)

Taking the inverse Laplace transform, (remembering the theorem of Borel) gives /

I r \.

r = 2-)j~co - 2[~J

V2 "7/

l cj(t -

ry

t

+— C0t - jcs(t -

\)d k

(5.83)

One can see that for a large drop this equation reduces to that of Ward and Tordai. From eq. (5.83) we have for the short time approximation 1/2

■ - £ )

1+

« o)

2R ,

The effect of a curved interface depends on the ratio between the diffusion penetration depth JnD t and the radius of the drop R. If JnD t « 2 R this equation reduces to the short time

143 approximation of Ward and Tordai (5.30). For the long time approximation we factorize cs out of the integral in (5.83), and using the Gibbs equation we obtain RTY2 ( n \ ' 2 1 a = G* + c0 U Dt) i + V ^ 2R For *JnDt « 2 R this again results into the long time approximation, eq. (4.42) of Ward and Tordai. In practice the radius of the drop is R « 0.1cm and with D = 5x10‘6 cm2 s'1 it follows that the long time approximation of Ward and Tordai remains applicable for a time less than 30 minutes. R

e f e r e n c e s

1. A.F.H. Ward and L. Tordai, J. Chem. Phys., 14 (1946) 453. 2. R.S. Hansen, J. Colloid Sci., 16 (1961) 549. 3. V.G. Levich, B.I. Khaikin and E.D. Belokonos, Electrochim., 1 (1965) 1273. 4. R. Miller, Colloid Polymer Sci., 258 (1980) 179. 5. V.B. Fainerman, A.V. Makievski and P. Joos, Colloids Surfaces A, 90(1994)213 6. E. Rillaerts and P. Joos, J, Phys. Chem., 86 (1982) 3471. 7. R.S. Hansen, J. Phys Chem., 64 (1960) 637. 8. A.V. Makievski, V.B. Fainerman, R. Miller, M. Bree, L. Liggieri and F. Ravera, Colloids Surfaces A, 122 (1997) 269. 9. J. Van Hunsel, G. Bleys and P. Joos, J. Colloid Interface Sci., 114 (1986) 432 10. R. Van den Bogaert and P. Joos, J. Phys. Chem. 83 (1979) 2244 11. R. Van den Bogaert and P. Joos, J. Phys. Chem. 84 (1980) 190

Chapter 6 Diffusion Controlled Adsorption Kinetics to a Continuously Deformed Surface In Chapter 5, we have considered diffusion controlled adsorption kinetics to a surface which undergoes stepwise deformation. Here we consider the adsorption to a surface which is continuously deformed. As already stated a continuous surface deformation gives rise to a convection current which has to be accounted for in the diffusion equation. As was shown in Chapter 3 the convection term in this equation is somewhat different for surface expansion and surface compression, that is why both situations will be considered separately here.

6.1.

Diff usi on t o a n e x pa n d in g s u r f a c e

For a continuously expanding surface, the convective diffusion equation should be written as dc Ht

( 6. 1)

and with the mass balance as boundary condition

In order to integrate eq. (6.1) we reduce it to a common diffusion equation (eq. (5.1)). For doing this we follow the method given by Levich [1], by introducing a new variable u u = zi{t)

(6.3)

145 with f(/) a function of time to be specified later. We have to express the convective diffusion equation in this new variable u. Note that in eq. (6.1) c = c ( z ,/), but with eq. (6.3) c - c[t, u(t, z)]. Therefore from (6.3) we have dc(z,t) de de du dc dc „ ---“---- = — + ----— = — + — z f ' M dt dt du dt dt du

(6.4)

where for convenience we only wrote c instead of c[t9u (t, z)]. We also have dc(z9t)

de du

dc , .

dz

du dz

du

(6.5)

and d2c(z,t) dz

du

du ( f ) ôz

d2c du2

f 2W

( 6 .6)

In this way eq. (6.1) transforms into

| +|4fW -efW ] =D0r=W

(6.7)

Now we choose the new function f(/) in such a way that the convective term cancels d f(t) — 0f(i) = 0 dt Since 0 =

( 6 . 8)

inteSraiinS (6-8) with the additional condition f(r) = 1 at t = 0 we obtain

fW = £

(6-9)

Q 0 is the area of the surface before the expansion is started. It is also essential that at t = 0 the surface is in equilibrium with the bulk so that T = Te (the equilibrium adsorption). Having eliminated the convective term, the convective diffusion equation becomes

146 de ~dt

-dBìh,ì

( 6. 10)

which is further transformed by introducing another time variable t ( 6. 11)

defined in such a way that

£ - ( ■ ( ,)

We require that at t = 0;

] ( £ ) ,»

t

(6.12)

= 0 (in this way the initial condition remains unchanged). Hence

eq. (6.11) becomes dc dz

d2c du2

(6.13)

In this way the convective diffusion equation is transformed into the common diffusion equation. Before we can integrate this equation, the boundary condition eq. (6.2) must be expressed in the new variables u and x as well. Hence dT dx d In f (/) dx dx dt + dx dt

( dcA du_ \d u ) 0 dz

(6.14)

A d In f(i) since 0 = ---- ----- according to eq. (6.8). With eqs. (6.3) and (6.12) we get dt

dx

du) r

(6.15)

In this way our problem is reduced to that of Ward and Tordai, with the result (since at t = 0; r = Te and f(0 = 1)

147

(6.16)

This equation may now be evaluated numerically using the adsorption isotherm relating cs with T. In order to obtain an analytical solution eq. (6.16) may also be approximated by assuming that over the whole time domain cs(t - k) is more or less constant with respect to the change in t

[2, 3]. This means that cs may be factorized out of the convolution integral. The same

procedure was followed for obtaining the long time approximation for the Ward and Tordai equation (section 5.3). The result is (6.17) We should remark that although this approximation seems quite crude, in fact it does not influence the results so much. As an example, if we specify that the surface is expanded with constant dilatation 0 the substitution of f(f) into (6.16) and (6.17) with the correct expression for this kind of deformation gives (see later section 6.1.2. eq. (6.41))

(6.18a)

(6.18b) This equation can be evaluated numerically. As an example some numerical results are shown in Fig. 6.1. If only moderate deviations from equilibrium are considered, we can make the following linearization (6.19)

148

0

1

2

3

4

5

t[s] Fig. 6.1.

Comparison of dynamic adsorption predicted by the exact eq. (6.41) (solid line) and by the approximation of eq. (6.18) (dotted line) assuming a linear adsorption isotherm, parameters used are rQ O =3.60 10'10 mol/cm2, D = 2.5 1O'6 cm2/s, 0 = 0.5 s'1, dc/dr =1541 cm'1 and Tss = rt-»ao = 2.64 1O'10 mol/cm2, equilibrium adsorption r c (x—x), according to [2]

In fact eq. (6.19) involves the adsorption isotherm (boundary condition for local equilibrium between the surface and the composition in the subsurface). Substitution of eq. (6.19) in eq. (6.17) results in Ar

f(r)

1+

4, y /2

( 6 .20)

™ D)

where we have introduced

Ar

=r - r

( 6.21)

the diffusion relaxation time x Ddefined by eq. (2.85) and the effective time

( 6.22)

149 In fact we are not measuring the jump in adsorption A r (which is negative since T < Te due to the ex p a n sio n o f the surface) but m erely the ju m p in surface ten sio n A ct (p o sitiv e for exp an d in g su rfaces), w h ic h is obtained by linearization (w e are co n sid erin g sm all ju m p s)

da Acj = — Ar dT

(6.23)

resulting in

A ct = s n

Q -Q n Q

1+

(At

v /2

(6.24)

\TIX D)

where the Gibbs elasticity So is defined as do r

(6.25)

For a stepwise expansion and teff - t eq. (6.24) reduces to

Aa = Act

•

1+

(6.26)

4,

d)

which is an approximation of the Sutherland equation (cf. eq. (5.47)). On the other hand we can define the surface elasticity s as Aa 8 = ——Q AQ

(6.27)

Hence, eq. (6.24) gives us the elasticity of the surface as a function of the effective time

1+

Ateff

1/2

(6.28)

150 For different kinds of surface expansions we can describe experiments with eqs. (6.24) or (6.28), with so and

as adjustable parameters [4]. These parameters are related to the

td

adsorption T, hence in this way the adsorption value can be found by dynamic experiments. Indeed do r e dT

so

RTF?

( 6 .2 9 )

d r -J3 d c ~ c 0-Jd

where use has been made of the Gibbs adsorption equation RTY

(6.30)

- d a = -------- - d c

co

The value of T is to be evaluated from the respective equation of state. It may happen that for certain experiments (cf. sections 6.1.1. and 6.1.2.) 1 «

(A t

\

1/2

^ ^Z)>

then

eq. (6.24) reduces to

Aa = 6

i

fW-i 0

fit)

f

4t

^

= e0(f(i)-l)7r^ |

1/2 4t

(6.31)

Vk t £>f2W. Equation (6.31) or (6.24) may now be applied to different surface expansions. The jump in surface tension and the elasticity are given by approximate eqs. (6.24) and (6.28). However for small jumps we can obtain exact solutions [5]. To do this we start from the transformed diffusion equation, (6.13) and transformed boundary condition eq. (6.15) written in a new variable cp

Taking the Laplace transform of eq. (6.13) we obtain

151 _ l'O . c = — + A exp s

V

(6.33)

D

and by introducing the subsurface concentration at z - 0

c_= t +

■7 + e x p ( -

(6.34)

The Laplace transform of the boundary condition (6.32) yields $(p

(6.35)

Ac is related to A(p via — dc — Ac = — Acp 5 dy

(6.36)

with

dc

dc

(6.37)

Hence we have

— dc x , U= 0 and hence limzU(z) = 0. In this way eq. (6.113) becomes:

i

z-»0

_ — (—

(6.118)

Cq \ dzJ 0

In this boundary layer theory a model for the concentration profile has to be assumed. According to Kralchevsky, the following concentration profile is suitable

cm0.z) = c1(i) + (c0 - c i )sin

cm(t,z) = c0 at z > 8(/)

TCZ

L2S(r)J

at z < 5(/)

(6.119)

( 6. 120)

In fact the exact concentration profile is replaced by a model function. Instead of this we assume the following concentration profile (it is clear that other suitable function can be used)

171

cm = c,(t) + [c0 - i’j W j e r f a t z < 8(r)

( 6 . 121)

where 5(t) is some distance to be evaluated later. It is required that the amount of surfactant in this model concentration profile cm(t,z) is the same as in the real system

h 0

- c(z’‘)]dz = J[co - cm{ t,z )\k = [c0 - W

(7.21)

We can write eq. (7.21) in a somewhat different way by saying that if the whole system is in equilibrium with the oil bulk phase the concentration c2 should correspond to the bulk concentration in the aqueous phase K c ,°, (to avoid any confusion we should elucidate that cf is not the concentration in the aqueous phase, because it is zero, but it is just a concentration in

202 the aqueous phase which would be in equilibrium with the oil phase at concentration c®) whence eq. (7.21) becomes

n = n„

R T F ’0 In 1 +

J d I + k J d I a ,,

for H-»W

(7.22)

For a system at equilibrium, the bulk concentrations are Cj0(e) and c^e ) in water and in the organic phase, respectively, hence c®(e) = Kc^{e) . The equilibrium interfacial tension is given then by

TL = J?7T°°ln 1+

-

(7.23)

By comparing the steady state with the equilibrium values given correspondingly by eqs. (7.16) and (7.23) one sees that the steady state interfacial tensions are shifted to higher values with respect to the equilibrium value [1]. If the surfactant solved in water is nearly insoluble in the organic phase (as ethanol in the system water/hexane) the distribution constant K->0 and eq. (7.16) reduces to the equilibrium one (n -> n ,). If on the other hand, the surfactant is very soluble in the organic phase K-*oo, and from eq. (7.16) we obtain Il-»0. * A particular situation is that of SDS with a minor contamination of dodecanol solved in water and brought into contact with pure hexane. The dodecanol is immediately extracted from the

* In the discussion no consideration of the total volumes of the two phases, water and oil, have been made. In particular cases the volume ratio can lead to peculiar behaviours. For a certain distribution constant K and volume ratio the surface pressure passes through a minimum. This phenomenon is caused by the transfer of surfactant across the interface from a phase with a small volume into the other phase having a much larger volume and higher solubility of the surfactant.

203 aqueous subsurface to the hexane phase and the interfacial tensions for the pure and contaminated SDS samples becomes very similar [2]. In Fig. 7.2 we have compared the steady state and the equilibrium interfacial tensions, for hexanol diffusing from water to hexane and vice versa and we see that a nice agreement between theory and experiment is achieved. At last it should be said that steady state interfacial tensions are dynamic interfacial tensions as well.

60 50 ■ *A

40 -

A

A K

1 30 * o 20 -

■


cos(co/ + cp) At the beginning of the area deformation the amplitude P is a function of time, but very soon it becomes independent of time. As a result of the surface concentration oscillations, the bulk concentration also changes c = c0 + f(z)ei(ùt

(8.5)

The concentration oscillations depend on the distance from the surface, accounted for in eq. (8.5) by the function f(z). The concentration oscillations must fulfill the diffusion equation, and since the area is changing, a convective term must also be accounted for. As argued above (section 3.1) this convective term is

225

Q z*

3z

= z ^

È

dt

i

dz

s z A Ì M

dz

. e2

**

( 8 .6 )

However if the amplitudes are small, higher harmonic terms as e2'“ can be neglected, as we have a product of two small amplitudes. As a result the convective term can be neglected and the concentration must fulfill the simple diffusion equation 3c dt

(8.7)

Substitution of eq. (8.5) in eq. (8.7) defines f(z)

m f(z) = D

d 2 f(z )

( 8. 8)

dz1

The integration of this equation with the constrains that at z - » oo, f(z) = He”*

f(o o )

-» 0, is (8.9)

with

( 8. 10)

H is an integration constant. Hence eq. (8.5) becomes c = Cq + He

e

( 8. 11)

The boundary condition for conservation of mass at the surface is

( 8. 12)

Substitution of eqs. (8.3), (8.4) and (8.11) in eq. (8.12) gives

i(£>Pe'm + H(ùAeial(re + Pem!) = -D nH emt

(8.13)

Neglecting higher harmonics, as just argued, we obtain mP + icoAr e = -D n H

(8.13a)

Eq. (8.13) applies whether or not the adsorption process is diffusion controlled. For a diffusion controlled process there is local equilibrium between the surface and the subsurface concentration. Since the fluctuations are small we may linearize the adsorption isotherm. c5 - c 0 _ H

r-r#

p

dc dr

(8.14)

Here dddT is a constant which can be obtained from the adsorption isotherm for c = c0 . The integration constant //, being the amplitude for the subsurface concentration oscillations is eliminated between eqs (8.13) and (8.14) yielding the amplitude of the adsorption oscillations as a function of the amplitude of the area deformation.

P=

- mATe dc /co + Dn — dT

A re

(8.15)

Dn dc 1+ “ /co d l

Let us have a look at the parameter

Dn dc

/CD dT ’

Bearing in mind the definition of n given by

eq. (8.10) we can wTite Dn d c _ ( (ù A 1//2 /CD dT \ /CD/ Here co0 is the relaxation frequency defined as

(8.16)

227 It is related to the diffusion relaxation time xD by (8.18)

=1 In this way eq. (8.15) becomes -A T .

P=

(8.19)

1/2

> ♦V (Ko- )J In fact we are merely interested in the amplitude of the variations in surface tensions, which are related to the adsorption variations (still considered to be small)

A

ct

da da = — dT = ----- F

^

‘*r < 1+[ V ,/2

( 8.20)

da The parameter - — T is the limiting elasticity, eo, sometimes called the Gibbs elasticity dT do

( 8.21)

From eq. (8.20) we see that the ratio of the two amplitudes is the elasticity s(/co)

e(/cd) =

da dlnCl

Aa A

( 8.22)

Hence finally eq. (8.20) becomes

e(/co) =

1+

1/2

(\ m- )J

(8.23)

228 Previously we have defined the elasticity as a function of the effective time z[te^ j (see Chapter 6, eq (6.28)), here another elasticity as a function of the frequency is introduced s(/co), the complex elasticity. As a notation we define

and after some arrangements eq. (8.23) becomes

e (/c o )

=

We multiply with the complex conjugate (l + £ +

e(/co) =

(8.25)

i + ç -/; and obtain

s„(l + Ç + «;)

1+ 2Ç + 2Ç2

(8.26)

In general if we have a complex parameter we can measure the real part s r , and the imaginary part e , . The modulus then is given by I ( 2

2 \1/2

l=(er2 +c;J

(8.27)

and the phase angle reads

tancp = —

(8.28)

For the present situation we can only measure the modulus of elasticity |s| and the phase angle. For the real and the imaginary part in eq. (8.26) we have

229 e _

Eo(^+ (^)

_ E _____ £0^ Er ~ 1+2C + 2c2 ’ z ‘ ~ \ + 2C,+2t f

(8.29)

Hence eqs. (8.27) and (8.28) become

2 V/ 2 ( i + 2 ; + 2^2)

; tancp =

(8.30)

1+?

We see that at high frequencies, where the area oscillations are very fast, there is no diffusion exchange with the bulk (at co -» oo, £ -> 0, s r -> 6 0,

e,

-> 0, |e| -> s 0, and tancp = 0). This

means that the surface behaves as purely elastic. At lower frequencies a diffusion exchange occurs and the monolayer becomes visco-elastic, i.e. for co —> 0 , £ -» co, |s| = -?=-, tancp = 1 , n n hence cp = —. A phase angle of — is typical for a process where the diffusion exchange is fully T

i

operating. This is also observed in electrode kinetics and known as Warburg impedance. At low frequencies we have £o

(8.31)

=& and with the definitions of

|s| = -p== Vco = -7 = — Vco

Vco0

' J D cq

e 0 and co0 and the Gibbs adsorption equation (5.41) we find

(8.32)

If we have measured the modulus of elasticity over the whole frequency domain we are able to obtain the limiting (Gibbs) elasticity So and the relaxation frequency o 0. It also follows from eqs. (8.30) that e = |s|cos(p and e ( =|s|sincp

(8.33)

230 The imaginary part of the elasticity is related to the dilational surface viscosity r\d (not to be mixed up with the surface shear viscosity), by x]d = cos,

(8.34)

Eqs. (8.33) and (8.34) are very general and allow us to obtain the elasticity er and the dilational surface viscosity r\d . In Fig. 8.1 an example for the modulus of elasticity is given.

- 2 ,5

-2

- 1 ,5

-1

- 0 ,5

0

log© [s'1] Fig. 8.1.

|s| as a function of log co (method of Lucassen) for a mixture CTAB/PFAC with = 1 .2 4 '1 0 ’8 mol/cm3 and cpy=2.68 1 0 '9 mol/cm3 ( c r / c y = 5 ) , solid line calculated from eq. ( 8 .3 0 ) with cqo = 4 . 4 1 0 '2 s'1 and S o=191 mN/m, according to J. Van Hunsel, PhD thesis, Antwerp 1 9 8 8 cr x

The dynamic surface properties are also obtained in principle at high frequencies (co > 1 Hz) by the damping of capillary waves, i.e. transverse surface waves, under certain conditions. For this we refer to the review article of Lucassen-Reynders et al. [3]. In electrode kinetics one plots the imaginary part of the impedance as a function of the real part. The resulting curve is the indicatrix and the plot is called Cole-Cole plot. The imaginary

231 and the real parts of the elasticity are given by the eqs. (8.29), which are the parameter expressions for the indicatrix. We can define 1+C i + 2;+2 oo and tanh(nh)

(8.44)

1, and remembering eq. (8.16), eq. (8.44)

reduces to (8.23). For a thin layer nh « 1, i.e. h « (2D/co)'/2, and tanh(«/z) « nh, and using eq. (8.10), eq. (8.44) approximates to (8.45)

Such a thin film, having two interfaces, has an elasticity given by 2sf , exactly equal to the elasticity defined by Gibbs [8] for a soap-stabilized film in a long-time experiment. Such experiments have been reported by Mysels et al. [9], Prins et al. [10] and Krotov et al. [11] for small non-periodic area extensions. These workers introduced the term “Gibbs elasticity“ for the film elasticity defined by Gibbs; their results have confirmed the film thickness dependence

of this elasticity.

234 Eq. (8.45) indicates that the concentration in the film is continuous and can also be obtained by considering the film as a closed system. The elasticity is given by du d InQ

(8.46)

Since we consider the film as a closed system, upon deformation the volume of half the film (h < z < 0) must remain constant. We call this constant K\, whence V = K x = -AQ

(8.47)

from where we get i/lnQ = -d \n h

(8.48)

Secondly the amount of surfactant in the film is constant too, we call this constant K2, hence (8.49)

K2 =hQc + m and in view of eq. (8.47)

(8.50)

Whence k2/hQ is constant too. Differentiating eq. (8.50) we obtain h d T -T d h dc +-------;----h2

=

0

(8.51)

from where we obtain (8.52)

235

By substituting eqs (8.48) and (8.52) into eq. (8.46) and remembering the definition of the Gibbs elasticity we obtain eq. (8.45). This equation can also be expressed in a different form. From the definition of the Gibbs elasticity and the equation of Gibbs we have

(8.54)

dc j

In an actual foam, a film is connected to the volume phase by a Plateau border (Fig. 8.3). The modulus of elasticity for a surface adjoining a semi infinite medium is given by eq. (8.30), and the elasticity of the film by eq. (8.44), hence it follows that e F > |e|

(8.55)

This is true if the surfactant is present in the film. If however the surfactant is not present in the film but in the outer phase, instead of eq. (8.39), the eqs. (8.5) and (8.9) apply and this results directly in eq. (8.30). Here the elasticity in the film is equal to that of the surface adjoining semi infinite medium and eq. (8.55) does not hold. A consequence for it is B ancrofts rule [12]. For a

236

system where the surfactant is present in the oil phase a stable water in oil emulsion is obtained. The reverse is true for an oil in water emulsion. For obtaining a stable emulsion, the surfactant must be present in the continuous phase. Only in this case does eq. (8.55) apply, which means that the resistance of the film against area extension is greater than that of the (thicker) Plateau border, hence the overall area disturbance will be largely taken up by the surface adjoining the semi infinite phases, and not so much in the film. This idea is due to Van den Tempel [13]. 8.2.2.

E

l a s t ic it y o f a

8.2.2.1.

d r o p

D if f u s io n o u t s id e the d r o p

For this we have to consider the diffusion equation in spherical coordinates under the condition that the concentration variations are periodic, i.e. c = c0 + f(r)ei(0t

(8.56)

and the concentration is given by eq. (5.76). Substitution of eq. (8.56) in eq. (5.76) gives d 2 f(r) dr2

2 d i(r) 2 + -----f(r) = 0 r dr

(8.57)

where n2 is given by eq. (8.10). The integral of eq. (8.57) fulfilling the boundary condition that at z -* oo, f(r) —> 0 is

f(r) = — ex p (-w ) r

(8.58)

hence eq. (8.56) becomes H

The boundary condition for conservation of mass at the interface is

(8.59)

237 ¿In Q dt

dr dt

R

(8.60)

and with eqs. (8.3) and (8.4) we obtain

mP + mAYe = - —j-e nR(i + nR) R

(8.61)

The condition for local equilibrium between surface and subsurface at r = R, (see eq. (8.14)) is ± = J L e-nR dT PR

(8.62)

whence in the usual way we obtain

e(;cc the concentration variation,/f^, becomes zero, therefore in eq. (8.69) H, = 0. In this way eq. (8.75) becomes c = c0 + H K 0(nr)eia'

(8.76)

240 where we have dropped the subscript 1. The boundary condition for the conservation of mass at r = R gives

mP + i(oFeA = -D nH

¿K o W d(nR)

(8.77)

and since

d K 0(nR) d{nR)

-K » M

(8.78)

eq. (8.77) becomes koP + DnH K {(nR) + /coreA - 0

(8.79)

Finally the boundary condition for local equilibrium at the surface for r = R is dc H K 0(nR) dT~ P

(8.80)

Hence we obtain for the elasticity

s(/c o ) =

Eq Dn dc K {{nR) 1+ /CO d r K 0(nR)

For nR » 1

(8.81)

K ,M ) K0M

^

and eq. (8.81) again simplifies to the equation of Lucassen and van den Tempel [1, 2].

241 8 . 2 . 3.2.

D if f u s io n in s id e th e c y l in d e r

For this problem we require that for r -» 0, f (r) remains finite, whence eq. (8.75) simplifies to (8.82)

c = c0 + H l0(nR)ei"‘ In this geometry inside the cylinder the mass balance reads i/lnQ dt + r e dt

dT_

(8.83)

d lA n R ) ( and since ^ =I

'

with the boundary condition of local equilibrium we obtain in an

analogous way

(8.84)

Dn dc I {(nR) + /co dT l 0{nR) again for nR -» oo,

For nR -> 0,

11(nR) i 0M

i,M ) io M -"

.

and Lucassen’s equation results.

nR = — and we obtain an expression similar to that for thin layers or small

drops

8= ■

1+

dc_R dT 2

(8.85)

This equation can also be obtained when assuming a uniform concentration in the cylinder, with the conditions that the volume and the amount of matter is constant, as outlined for thin

242

films. Indeed, if the volume V = n r21(/ is the length of the cylinder) and the area Q = 2nrl, elimination of / gives ( 8. 86)

- T Since V is constant it follows ¿/InQ + d ln r = 0

(8.87)

hence the elasticity is da d InQ

da d ln r

( 8. 88)

Since the amount of material is constant we have cV + Q r = constant

(8.89)

or since V is constant and together with eq. (8.86) we obtain IT c + — = constant r

(8.90)

Differentiation of this equation results in dT , d ln r = ■

R dc + 2 dr

and substitution of eq. (8.91) into eq. (8.88) finally gives eq. (8.85).

(8.91)

243

8.3.

Th e elas ti cit y f or a sur fa c t a n t mix tu re

We have already considered the equation of Ward and Tordai (Chapter 5, section 5.6) and dynamic surface tensions (Chapter 2, section 2.6) for a surfactant mixture. We will do now the same for the elasticities, being somewhat simpler because we are allowed to linearize. The method is always the same and applies as well for other surface deformations where the convective term cannot be neglected (see [14]). Here we consider periodic deformations on a flat surface. Considering only two surfactants, the principal requirement is that the adsorption of surfactant 1 depends on the subsurface concentration of both surfactants r . = r ( c 1s,c21)

(892a)

(8.92b) or by inverse functions

=*(rIfr2)

(8.93a)

c2 “ c(f"l»^2)

(8.93b)

If the deviations from equilibrium are small we can linearize, otherwise we must consider the whole adsorption isotherm. We have now two adsorption variations

r, =rr + Re**

(8.94a)

(8.94b) and two concentration variations c, = c,° + H{e-n'zei(ùt

(8.95a)

244 (8.95b)

c2 =c°2 +H2e - ^ e ia‘

The concentrations are given by the diffusion equations (5.71), and n\ and «2 are defined as ,

;co

1(0

(8.96)

■«(a

For both components we have a balance for the conservation of mass at the surface „d ln£i dt

dT, dt

+ rf

dT-> dt

„ d InQ

( dcdz

•+ r,*—r~ =A dt

(8.97a)

(8.97b)

Substitution of eqs. (8.94), (8.95) and (8.3) into eq. (8.97) yields i(oP{ + /corf A =

(8.98a)

mP2 + mV l A = - D 2n2H2

(8.98b)

The boundary condition for local equilibrium are now obtained from eqs. (8.93)

dc\ = H { =

dcx

f ^C \

1 {arj

a r j_

( d c 2> dc2 = H2 = ---—

Ur,J

f p, +

f2

^

In this way eq. (8.86) becomes

(8.99a)

(8.99b)

245 O'! 1/2" 02\ 1/2 0)1 ®1I p2 = - t ; a 1+ P{ + l *© J l K*> J

CO 21

y /2

1(0 y

(8.100a)

" / 0 > 1/2 -|l/2 Px + 1 + f — p 2 = - r 2*A V /© J

(8.100b)

where co?,, oo^, co^, co22 are abbreviations for: dc. V00?! “ V a W

^ dc, ^

J CO ?2 = V a

'

w

(8.1101a)

r,

dc2

7 CO

^

(8.101b)

r,

From eq. (8.88) we obtain for Pi and P 2

~ 1+ -

= -r7

( 1 +

=-n

0

^22

+

1 1

j

*©

0

\

r 2 co12 r r l /CO J

, /CO J CO 2 2

I

r*^

y / 2"l

( O ' ) 1/2

0 >1 V 2

^ L /CO J

r

/c o

(8.102a)

^(O0n (O022 —VC0?2C0

1/2 / o y / 2" I l K n 1+ — ] r 2e l /co J V/COJ

/ co„ 0 > 1/2 /y o "\ i “ 22 1+ + — l /CO J \ m J

± [ f77o~o~

(8.102b) /7To~o~

” /(0 [ V0311°° 22 ” VC0120)21

Since the elasticity is given by Aa £=-

f

da

^

Px

( d UrJ

II

1 -1« —

o

CO

2.6,

r2 •

e02 = -r2[— J (8.104)

we obtain

U rJ J f

=*01 i +

e(zco) = ■

o

©22

1+

l

1/2'

( o ' )

\ '/2 '

+ *02

o > 1/2

V to J

1+

+

©11 K0

J

l V /©

-e 80lVCO?2 +

802t]®2\

o' ©22 1/2 11 / 0 o / 0 0 — VC011C022 ” V0) I2C02I /CO l to ) (8.105)

This expression can now be split into a real and imaginary part and from this the modulus (e) is obtained to be compared with experimental data. In Fig. 8.4 experimental results are shown for dodecyl tri(ethylene glycol) C 12E3 and dodecyl hexa(ethylene glycol) C 12E6 [15].

2 n

6 *

i

t

i

b) 0,5 0 -i------------ .------------ .------------ .------------ 1------------ . -2,5 Fig. 8.4.

-2

-1,5 -1 -0,5 0 log cois'*] Frequency dependence 01 the dynamic dilational modulus |s| of nonionic surfactant mixtures. 10’8 mol/cm3 pure C 12E3 (A), 7.5 10‘9 mol/cm3 C 12E3 + 2.5 10‘9 mol/cm5 C 12E6 (• ), 5 10 9 mol/cm5 C | 2E3 + 5 1O'9 mol/cm5 CI2E6 (▲), 2.5 10‘9 mol/cm5 C i 2E3 + 7.5 10'9 mol/cm5 C i 2E6 (♦), 10'8 mol/cm5 pure C i 2E6 (■), according to [15]

247

8.4.

Re l

a t io n b e t w e e n c o m pl e x e l a s t ic it y a n d d y n a m ic s u r f a c e

TENSIONS This equation comes from systems theory and has been introduced into surface science by Loglio [16-18]. It says that if we have a system (for instance an electronic device or whatsoever), being seen as a black box, the signal S(t) is transmitted to this system (the input) and we obtain a response (output) R(t). The input and output can be time dependent. The relation between both is given by the relation [19-21]

g (m ) =

where

F{S(t)}

(8.106)

} is the Fourier operator and g(/co) the transfer function. This relation (8.106) is very

general and applies if the following conditions are fulfilled: (i) Before applying the signal the system is in equilibrium or at rest. (ii) The system behaves linearly. This second condition is perhaps the most serious requirement but not always met in practice. With linearity we mean that changes in input and output are small and the system itself keeps close to equilibrium, therefore we can assume that the response is proportional to the input. Thus, linearity as meant here, must not to be mixed up with the linearity used above. Indeed we have said that the concentration variation in the subsurface, or the jump in adsorption, or the jump in surface tension are proportional to each other (we linearize the adsorption isotherm). It should be realized that the jump in subsurface concentration, adsorption, and jump in surface tension are the outputs, and the input is the relative jump in area deformation. For obtaining eq. (8.106) it is required that the area deformation (the input) and the output are small. In these cases they are proportional to each other. (iii) The obvious principle of causality The response at some time /, depends on the input signal during the time /, but not on the signal at a longer time. (iv) Time invariance.

248 This means that for the transfer of the input during t - r seconds, the impulse function holds g(t - x) = g(x - /). The impulse function depends only on the time difference. The Fourier transform of the impulse function is the transfer function g(i co). AQ In our situation the input is the relative change in area -^ -(/)a n d the output the variation in surface tension. For this eq. (8.106) becomes

e(/co) =

'is«}' ^{Aa(Q}

1

dt

J

(8.107)

This equation is the one Loglio derived as a general relationship for surface deformations. For a diffusion controlled adsorption process from a semi infinite medium to a flat surface we have seen that the elasticity is given by the Lucassen equation £o_ COq 1+ /co

e(/co) = —

(8.108)

1/2

If we consider a stepwise expansion or compression, (stress relaxation experiment) [18] for a surface in equilibrium with the bulk, the area deformation is given by AQl Q J

AQ /coQ

(8.109)

From eqs (8.107) - (8.109) we have

Areduces to the kinetic equation of the Langmuir type dr = k,cA 1 dt

r u

r

(9.39)

There is evidence that this equation is consistent with experimental results for many surfactant systems [6, 7]. If the diffusion step is of some importance c0 * cs and eq. (9.39) becomes

X dt - V 1, 5

_r_ -*2 T 00

(9.40)

Eq. (9.40), or more general the eq. (9.38) (with a = 1) replaces the adsorption isotherm [3, 8]. Hence for obtaining the adsorption as a function of time, we have to consider the equation of Ward and Tordai with eq. (9.40) as a second boundary condition at the surface. In particular when the diffusion step is fast, cs = c0 and eq (9.40) transforms in (9.39). The integration of eq. (9.39) gives r = re[ i- e x P(-fo)]

(9.41)

269 with the initial condition T = 0 at t = 0 and k=

kxc0 + k1

(9.42)

r*

Here use has been made of the Langmuir equation where the Langmuir-von Szyszkowski constant is defined by (9.43)

When we assumed ideal surface behaviour, the equation (2.78) applies and substituting it into (9.41) we obtain the surface pressure as a function of time (9.44)

9.3.

Ex pe r

im e n t a l c o n f ir m a t io n

Transfer controlled adsorption kinetics can only be observed in a very short time domain, and therefore fast methods for measuring dynamic surface tensions must be used. Such a method is the oscillating jet method. We have argued that data obtained by this method match the data of the inclined plate [9] having a somewhat larger time domain. Also both methods give data in agreement with the maximum bubble pressure method, operating in a much wider time window [10, 11]. There is still some question raised about the data obtained by the oscillating jet method. However we have checked this method measuring the dynamic surface tension of aqueous octanol solutions where a diffusion controlled adsorption mechanism has been proved. The experimental data are compared with the curve predicted by the equation of Ward and Tordai together with the Langmuir isotherm. Hence this calculated curve concerns diffusion controlled adsorption process. It is seen from Fig. 9.1 that nice agreement between the experimental data and the theoretical prediction for a diffusion controlled adsorption process is obtained. In view of this we think that the oscillating jet method gives reliable results [6].

270

Fig. 9.1.

Dynamic surface tension a as a function of time t for 1-octanol: (1), c=5 10'7 mol/cm3; (2), c=1.34 10'6 mol/cm3. Solid lines - calculated for reversible adsorption, oscillating jet experiments, according to [6]

If however experiments are performed with 1,8-octane diol, which is not as surface active as octanol and hence requires higher concentrations to give a comparable equilibrium surface tension, we see in Fig. 9.2 that the experimental data are much higher than the curve predicted for a diffusion controlled adsorption mechanism. For this system dynamic surface tensions follow a transfer controlled adsorption mechanism as described by eq. (9.44) [6]. From such experimental data we obtain the rate constant k, defined by eq. (9.42). From this equation it is seen that this rate constant depends linearly on the concentration. Hence by plotting k as a function of the concentration we obtain the adsorption rate constant k\ = 6.65x1 O'3 cm s'1and the desorption rate constant kj = 1.75xl0’8 mol cm s'1. From eq. (9.43) we obtain the Langmuir - von Szyszkowski constant a using the dynamic data (we note it by aD) as aD= 2.6x10‘6 mol cm'3. This value must be compared with the Langmuir - von Szyszkowski constant a, obtained from equilibrium surface tension measurements (we note by cir ) as aT = 3.5xl0'6 mol cm'3. The agreement is satisfactory. For a lot of other systems (lower chain alkanols like propanol, butanol, lower a,codicarbonic acids like 1,6-hexanoic diacid, 1,7-heptanoic diacid, 1,8-octanoic diacid, and a,codiols like 1,8-octane diol) similar results are obtained (see Table 9.1 and [6]). It follows that the

271 adsorption rate constant k x depends on the chain length as the Langmuir - von Szyszkowski constant aT does, but that the desorption rate constant

is nearly independent of the chain

length.

55 -i---------- .---------- .---------- .---------- i---------- :---------- . 0 2 4 6 8 10 12 t [ms] Fig. 9.2.

Dynamic surface tension a as a function of time t for 1,8-octandiol, c=3.42 10*6 mol/cm3 (♦), c=2.05T0"5 mol/cm3 (■), solid lines - calculated for irreversible adsorption, dashed lines - calculated for reversible adsorption, according to [6]

Table 9.1

Adsorption (k\) and desorption {k{) rate constants and Langmuir - von Szyszkowski constants from dynamic data ( aD) and from equilibrium data ( aT) h

ao

aT

cm sA

mol cm sA

mol cm‘3

mol cm'3

1,6-hexanoic diacid

7.5x1a4

l .l xl O’8

o U)

3.7x10’5

1,7-heptanoic diacid

2.0x1 O'3

1.9x1 O’8

l .l x l O’5

1.4x10'5

1,8-octanoic diacid

2.5x1 O'3

l .l x l O'8

4.5x1 O'6

4.2x1 O'6

propanol

5 .3 X 1 0 -4

5.9x1 O'8

1.1x10"*

1.8x10"*

butanol

1.09x1 O'3

5.2xl0’8

4.7x10"5

4.7x10'5

pentanol

5.5x1 O'3

6.7x1 O'8

1.25x1 O'5

1.25xl0‘5

X

k\

L*

surfactant

272 From the experimental fact that the desorption rate constant ki does not depend much on the chain length of the surfactant and the adsorption rate constant k\, changes in a similar manner as the Langmuir - von Szyszkowski constant, two conclusions can be deducted. First the transfer process is fast and hence fast techniques must be used to measure the dynamic surface tension. Indeed, by rearrangement of eq. (9.42) it follows

k=

n R T T ”)

( = 80exp

Vr

n t

\

r œy

(9.45)

(where ki = 5xl0'8 mol cm s"\ T00= 6xlO'10 mol cm'2) and that the transfer controlled processes can only be detected in a time scale of 1/80 s = 1.25xl0*2 s (for n = 0) or shorter. Fainerman and Lylyk used the dynamic maximum bubble pressure method and their data [12] are in complete agreement with those of the oscillating jet [7]. Secondly, the mechanism of the transfer from the subsurface to the surface is due to hydrophobic effects (cf. section 2.9), and the adsorption process is driven by gain of entropy (cf. [13]), reflected in the Langmuir - von Szyszkowski constant which depends on the chain length but does not depend much on the temperature [6, 7]. The chemical reaction for the transfer from the subsurface to the surface is the transition to an activated complex before the surfactant is adsorbed. As said in the bulk, the apolar tail of the surfactant is surrounded by structured water. Before this surfactant molecule can be adsorbed, the structured water must be stripped off, this is the activated complex. It is expected, the larger the chain length, the larger is the gain in entropy, and since this is the driving force, the larger is the rate constant k\ obtained. As a further step we performed dynamic surface tension measurements for a homologous series of alkanols. In Fig. 9.3 the results for heptanol are given. It is seen that the results are neither described by a diffusion controlled nor by a transfer controlled adsorption process. In this case both processes are operating and one has to integrate the Ward and Tordai equation with the boundary condition given by eq. (9.40), with the subsurface concentration being less than the bulk concentration 5< c

c

q

.

The experimental data are well described by this

and we have mixed adsorption kinetics. Hence we have the following conclusion: the adsorption kinetics for the alkanols with short chains are transfer controlled (up to pentanol) the alkanols with intermediate chains (hexanol, heptanol) show mixed adsorption kinetics, and the

273

larger ones (from octanol, nonanol) are diffusion controlled [6, 7, 12]. We have a transition between both adsorption kinetics by increasing the chain length. An experimental fact which substanciates this conclusion is that 1,12-dodecanoic diacid also shows diffusion controlled adsorption kinetics [6, 14]. The sodium salt, 1,12-dodecanoate being much less surface active, and requiring a much larger concentration shows transfer controlled kinetics [6, 15].

Fig. 9.3.

Dynamic surface tensions of heptanol solutions, solid lines - mixed adsorption kinetics, c o = 2 .5 8 1 0 ‘6 mol/cm3 (■), co= 5 .1 6 T O '6 mol/cm3 (♦), co = 8 .6 1 T O '6 mol/cm3 (A), according to [7]

Transfer controlled adsorption kinetics are sometimes called barrier controlled kinetics. We think this terminology is confusing. The term barrier controlled comes from the following. From the experimental data using the Ward and Tordai equation together with the adsorption isotherm, apparent diffusion coefficients are obtained. If these diffusion coefficients are of the same order of magnitude as the bulk diffusion coefficient, it is correctly concluded that the adsorption is diffusion controlled. For mixed and transfer controlled processes the diffusion coefficient is orders of magnitude lower than that in the bulk. Again it is concluded correctly that here the adsorption process is not diffusion controlled. The apparent diffusion coefficient being lower than the real one is due to the transfer reaction, but not to a layer near the surface where this diffusion coefficient should be less. This is explained historically by Defay and

274 Hommelen [16, 17], at a time when computers were not available. Therefore the term "barrier" controlled adsorption kinetics seems misleading. Recently Fainerman et al. [18] tried to explain the results from a transfer controlled process by a diffusion controlled process by saying that the surface tension a is not only a function of the adsorption but also depends on the concentration gradient a = c ^ T , ^ j . The model of a non-equilibrium surface layer is an alternative to kinetic-controlled adsorption models. In the framework of the purely diffusion-controlled adsorption kinetics the proper consideration of a non-equilibrium diffusion layer leads to a satisfactory agreement between theory and experimental data for various studied systems, in particular for short chain fatty acids and alcohols. The non-equilibrium model has been successfully applied to the concentration range of 10‘6 to 1O’5 mol/cm3 at different values of the Langmuir constant a (cf. eq. (2.80)). For a < 1O“6 mol/cm3 a correction for the non-equilibrium layer effects can be neglected. For a > 10'5 mol/cm3 and large surfactant concentration, the Aa values calculated from the proposed theory do not compensate for the discrepancy with the experimental data. An empirical formula proposed for the estimation of the non-equilibrium surface layer thickness, a key parameter for the definition of the non-equilibrium layer, leads to a better agreement with experimental results in a wide range of surfactant concentrations. However the ad hoc character of this expression restricts the validity of the non-equilibrium surface layer model as an alternative to non-diffusional kinetics at the surface of concentrated surfactant solutions. Finally, attention should be paid to a singularity of the Ward and Tordai equation with the adsorption isotherm. For obtaining the Ward and Tordai equation we assumed the initial conditions t = 0, T = 0, c = c0 . The conditions T = 0, c = c0 are in contradiction with the adsorption isotherm. Hence at t = 0 the condition for local equilibrium does not hold and eq. (9.39) must be used. Consequently at t = 0, the adsorption process is transfer controlled. The question arises if we can see this from experiments since we are limited in the time scale and can only do measurements at / > 10"35 (at best), and at this time the condition for local equilibrium can apply. In fact the two cases for transfer (cs=Co at t = 0) and for diffusion

275 controlled kinetics (cs=0 at / = 0) are only two extreme situations. In Fig. 9.4 we have sketched cs as a function of time, together with the two extreme situations. CS

Fig. 9.4.

;

i

n

1

Change of subsurface concentration cs as a function of time, 1 - purely transfer controlled adsorption kinetics cs=co, 2 - purely diffusion controlled adsorption kinetics T = f(cs), 3 - mixed transfer controlled (interval I) and diffusion controlled (interval II) kinetics

Even for transfer controlled kinetics, for very long times the adsorption process becomes diffusion controlled. This can be seen as follows. We start from eq. (9.40). Presenting the dynamic adsorption and subsurface concentration as

r = re-A r

(9.46a)

Cs = c0 - A c ,

(9.46b)

(the situation considered here assumes expansion, A r > 0 and Ac, > 0 ) and substituting these equations in eq. (9.40) we obtain dAT dt

(9.47)

276 Remembering that at equilibrium

k\co{1~ p^r] “

^2

(9.48)

p j ~o

and neglecting the second order term k { ^

f since we restrict ourselves to the situation

where both A r and A cs are small (very long times), we obtain dAT + &AT = k ’ Acs dt

(9.49)

where k is given by eq. (9.42) and (9.50) The long time approximation of the Ward and Tordai equation is

(9.51) Keeping in mind that we are close to equilibrium T = Te, the substitution of eq. (9.51) into eq. (9.49) gives dAT dt

(9.52)

+ k&T = k' Te

The integral of this linear differential equation is

AT = e -kt

/

-

r- U

\ 1/2 /

kt

h *

At / = 0, f = 0 and AT = f e whence the integration constant K =

(9.53)

, hence eq. (9.53) becomes

277

at

= r0

/ _ \ 1/2 2' k , + k \ i )

-ft

e~k‘

(9.54)

We can transform the integral to V7

«

yfkt

je^dyft = -j= J e ^ V x n “v k which is the Dawson integral

e~b f exdj~x. 0

having the asymptotic expansion for large t of Æ e-h \ e xd y f x = — r -

Î

(9.55)

^

In this way eq. (9.54) becomes >

at

=t

Since

1/2

k W DtJ

(9.56)

» 0 the first term on the r.h.s. of (9.56) can be neglected. Thus we have

AT = r — f — 1

12

(9.57)

e k U Dt)

and the dynamic surface tension is da

k* ( n \

U âJ

12

(9.58)

278 In this equation we recognize the Gibbs elasticity e0 = - T — and together with the aT eqs. (9.42), (9.50) and (9.43) we obtain k' a r 00 dT k = (a + c0)2 = Jc

(9.59)

if the Langmuir adsorption isotherm is used. In this way eq. (9.58) results in

(9.60)

c = ae + se

being the long time approximation for a diffusion controlled process. If we are very close to equilibrium, it is perhaps better to use the Hansen approximation (see section 5.3. eq. (5.55)) instead of eq. (9.51), which results in the long time approximation of Hansen instead of eq. (9.60)

a = a,

(9.61)

but the conclusion remains. In view of this it is not quite correct to say that a given surfactant system shows transfer or diffusion controlled kinetics. For simplicity however we still retain this term depending on the aspect of the surface tension decrease with time over a broad intermediate time domain, because accurate experimental data at very small times and very large times are not always available.

9.4.

Smal l amplitu

de periodic

d e fo r ma ti o n

The conclusion obtained at the end of the last section, saying that the adsorption mechanism can change from a transfer control to a mixed or diffusion control, can be obtained, perhaps

more simply, by considering periodic surface deformation. The boundary condition for the conservation of mass at the interface, given by eq. (8.13) (see section 8.1) remains

279 iaP + i®reA = -D n H

(9.62)

However, the boundary condition for local equilibrium must be replaced by eq. (9.40) and accounting for the deformation of surface this equation reads dT dt

d InQ dt

+ r.

(9.63)

We substitute the eqs. (8.3), (8.4) and the relation cs = c0 + Hem (which is eq. (8.5) for z = 0) into (9.63) and considering eq. (9.48) and neglecting higher harmonic terms we obtain /coP + m Y eA = k'H - kP

(9.64)

k is given by eq. (9.42) and k' by (9.50). Eq. (9.64), because of the linearization for small deviations from equilibrium, is valid for any adsorption isotherm. From eqs. (9.62) and (9.64) we obtain for the amplitude of the fluctuations of the adsorption P = - m Y eA

k' + Dn /cop + (/co + k)Dn

(9.65)

The parameter k' is related to dY/dc, saying that at equilibrium the l.h.s. of eq. (9.63) is zero, and eq. (9.59) holds. Following the same procedure as given by eqs. (8.20) - (8.23) we obtain for the surface elasticity

/. x

da P

!+ — — k \/co

12

k V /coV © 1+ 1+ /coA k A /co

1+ 12

,

zcoco(

12

7,2

^V/GMOr

1+ | 1+d

12

l-^

where the diffusion relaxation frequency co0 is defined by eq. (8.17). co For large frequencies — » 1 (this means short times) eq. (9.66) reduces to K

(9.66)

280

e(ico) =

1+

(9.67) ---

/CD

Considering the relation of Loglio (cf. section 8.4., eq. (8.84)), we obtain for a stress relaxation experiment

A a = F~l

80

1

zco ,

1+

AQ ^

—

Q

/CD

[

Q

\ /cd + &

= Acr0e

(9.68)

which is the linearization of eq. (9.44). Hence at high frequencies (short times) the transfer process dominates. For low frequencies

CD

— « 1 (this means long times) from Eq. (9.65) the Lucassen equation results, indicating that K diffusion dominates.

9.5.

E xper imen ts w it h c o n st a n t d il a t io n s

The dynamic capillary method (cf. section 4.5.3.) allows us to obtain the dynamic surface tensions in the steady state at large values for the dilatation rate 0. The highest value for 0 is somewhat less than 10s"1 corresponding to an adsorption time / = ~ - being somewhat higher 20

than 50 ms. Keeping in mind what we have said about the time window for a transfer controlled adsorption process, we expect to observe only a diffusion controlled adsorption process. Experiments were performed with Triton X-405 solutions at the air/water interface [19] and as expected the dynamic surface tensions follow the relationship

g

r t f

= a* +*

:

710

2D.

12

(9.69)

as confirmed by Fig. 9.5. Results for the same surfactant but at the hexane/water interface cannot be fitted by a diffusion controlled adsorption mechanism [19, 20] as seen in Fig. 9.6.

281

Fig. 9.5.

Dynamic surface tension a(t) at the air/water interface of Triton X-405 solutions, co=2.54 10'7 (■) c0=1.53 10'7 (♦ ) c0=7.6310'8 (A) c0=2.5410'8 mol/cm3 (• ), inclined plate method, solid lines - calculated from Eq. (9.69) (diffusioncontrolled kinetics), according to [19]

For the theoretical approach we assume that the deviations from equilibrium are small and linearization of the adsorption isotherm is allowed. Hence in the steady state we have an equation similar to eq. (9.63)

e r = *

f k

- ‘ o ) - * ( r - r CM C, dTl/dlncr = 0, indicating that micelles are not adsorbed.

Fig. 10.1.

Two stages of micelle break-down: 1. Releasing few monomers (ps range), 2. total micelle disintegration (ms range)

From this it becomes clear we must consider the diffusion of the monomers and the micelles, taking into account the micellar reaction [3, 4]. For this we consider a first order reaction, and not as a polymerisation reaction [5, 6]. A polymerisation reaction is perhaps more suited for the fast process in the microsecond range. Hence, for a continuously expanding surface one can write

- k xcx + k2nc

(10.1a)

(10.1b) where C\ and ci are the concentration of the monomers and the micelles, D\ and Di the corresponding diffusion coefficients, k\ the rate constant for micellisation and ki that for demicellisation, and n is the aggregation number of the micelles. The use of a first order chemical reaction as expressed by eq. (10.1) and not a polymerisation reaction is corroborated by experiments [6]. It is clear that from the dynamic surface tension of micellar solutions, the rate constants can be obtained. It is however not our purpose to obtain

287 them from these kind of experiments, since more appropriated techniques are available (p and T jumps, stopped flow, etc). It is our purpose to investigate how the micellisation affects the dynamic surface tension. The diffusion coefficient of the micelles is approximately given by [7] D

( 10.2)

2 = «-|/3

D,

This can be seen as follows. For a spherical particle with radius a the diffusion coefficient is given by the Stokes - Einstein equation

D=

RT

(10.3)

N A6wc\a

N A is the Avogadro number and r\ is the medium viscosity. The volume per gram v , is related to the radius by 471 3 M _ — a = ---- v 3 NA

(10.4)

where M is the molecular weight. Hence one has for the diffusion coefficient of monomers

1

(10.5)

6m\NA U A /v! )

and for that of the micelles

A

RT 4*NA 6nr\NA 3a M v 2j

1/3

( 10.6)

hence

Ei A

\I3 Kv2n )

= r

(10.7)

288 and if v2 =

eq. (10.2) results. This ratio is denoted by y 2. A typical value for y 2 = 0.25 (for

n = 64) [4]. In general, even without a convection term, the diffusion equation (10.1) is difficult to integrate analytically. That is why first we will consider the more simple case of diffusion without convection.

10.1. Local

equili

bri um bet wee n t h e mo no mer s a n d mice lles

In the following considerations we will drop the subscript 1 for the diffusion coefficient of the monomers: D= Dx, and the diffusion coefficient for the micelles is D2 = y 2D . Without the convection term the diffusion eqs. (10.1) are

dc, dt

d2c, dz

— - = D — j - - k xcx+nk2c2

dc2 ~dt

Dy

2 d 2c 2 ^1^*1 I 7 T - + — -*2*2 dz n

(10.8a)

(10.8b)

The reaction terms in eqs. (10.8) can be eliminated giving ô(c, +nc2) _ Ô2 L , = D ^ r ( ct +tï ï 2Ci) dt ÔZ'

(10.9)

To obtain c\ and ci we need a second relation being the micellisation reaction. Here we will consider the case where the micellisation - demicellisation process is so fast that locally there is equilibrium between the micelles and the monomers. This means that the kinetic equation for micellisation is replaced by an equilibrium relation k\c \ - k2nc2

This relation holds also for the system at equilibrium with

( 10. 10)

289

ct , C M C ^ c , . (‘ ‘ - CMC) n

( 10. 11)

c® is the total concentration at equilibrium ( 10. 12)

c°T =CMC + nc°2 hence the ratio of the rate constants noted by the constant p reads k\ _ nci _ k2 cx

ct

~ CMC =P CMC

(10.13)

Out of equilibrium, the mass balance given by eq. (10.12) is (10.14) ct

is the local total concentration. From eqs (10.13) and (10.14) one obtains

CT

PCt

C\ = -— - ; nc7 = — —

1 i+p

2 i+p

(10.15)

Substitution of eq. (10.15) in eq. (10.9) yields dcT

"aT=

£*(i + Py 2) d2cT

1+p

8?~

(10.16)

This equation must be integrated with the boundary conditions c2 -> c®, C| -> c®, and cT -> c° at z —> oo

(10.17)

and the boundary condition for conservation of mass at the surface. For the monomers we have as usual

290 dT

dCi

dt

° \ dz

(10.18)

and since the micelles are not surface active the flux of micelles to the surface is zero

From eqs (10.18) and (10.19) we have

^

=D^

c' +y2nc^

(10-20)

and finally with eq.(10.15) we obtain

dr _ fl(i+v2pVac/\ dt (i +p) l &/0

( 10.21)

In this way, we have transformed the problem to this of Ward and Tordai. At / = 0 we assume T = 0, hence

T=2

^ ( i + y 2p ) (1 + P)t i

r,

J[c? -c£(t-

( 10.22)

It should be recalled that c? and csT are the total bulk and subsurface concentrations. Since the adsorption T depends on the subsurface concentration of the monomers cf, and not on that of the micelles c\ , we express Cj and csT as a function of c,° and cf by mean of the eq. (10.15)

cr ~

(r - X) - (l + p)[ci° - Ci (t - A.)j

hence finally we obtain from eq. (10.22)

(10.23)

291

(10.24)

We can define an apparent diffusion coefficient D* [3,6]

D* = Z)(l + p)(l + py2)

(10.25)

In eq. (10.24), excluding a transfer process, the subsurface concentration is related to the adsorption isotherm. Hence formally our adsorption process has the aspect of a diffusion controlled process with the restriction that the apparent diffusion coefficient is much larger than that of the monomers. Especially for the long time approximations (cf. section 5.3, eq. (5.42)) we have 1/2

(10.26)

while we should recall that c? is the CMC. If instead of the present situation we do expansion experiments with a constant dilatation rate 0, the equation of Van Voorst Vader (subsection 6.1.2, eq. (6.38)) still applies with the only restriction that D must be replaced by D*:

(10.27)

If we use a polymerisation reaction instead of a first order kinetic process, the following apparent diffusion coefficient D*p should be

Z);=Z)(l + «2p)(l + «2y2p)

(10.28)

292

which is quite different from eq. (10.25) for w= 100. Eq. (10.28) does not describe the experiments in contrast to eq. (10.25). There are only a few surfactant systems where the condition of local equilibrium between micelles and monomers are met. At the CMC, the time scale for surface equilibration must be much larger than that for the attainment of micelle-monomer equilibration (time scale say more than several milliseconds), this means that the CMC must be comparatively low. This condition is met for the Brij-58 surfactant. In Fig. 10.2 we give some experimental results. The apparent diffusion coefficient as a function of P is given in Fig. 10.3. It is seen that the experimental results are described by an apparent diffusion coefficient given by eq. (10.25).

Fig. 10.2.

Dynamic surface tension a of a micellar Brij-58 solution (co=80 CMC) plotted versus t’1/2, inclined plate method (♦), dynamic drop-volume method (■), according to [6]

293

Fig. 10.3.

Ratio of the apparent diffusion coefficient D* and the monomer diffusion coefficient D, air/water interface, drop-volume method (A), inclined plate method (), solid lines: 1 - calculated from eq. (10.25) with y2=0.25, n =100, 2 calculated from eq. (10.28) with n =50, 3 - calculated from eq. (10.28) with n = 5, according to [6]

8 7

6 ¥ 1

5 4

j j i < 3 A• S* | 2 • X ■” 1 0 0

100

200

300

400

500

600

t[s] Fig. 10.4.

Compression of Brij-58 adsorbed monolayer at co=10'8 mol/cm3 (CMC) and different compression rates a= dlnQ/dt: a=-20.9 10’3 s"1 (A); a=-5.65 10‘3 s'1( • ); a=-2.92 10'3 s'1 (♦); a=-1.48 10‘3 s'1 (■), dotted lines are calculated according to Eq. (10.27) with so=41.4 mN m '1 and t d =7.58 s , according to [8]

294 Until now we considered an expanding surface, where the micelles disintegrate. For a compressed surface, there is a region of over-saturation [8] where the effect of micellisation is not apparent (see Fig. 10.4). This behaviour, probably due to a nucléation process for the micelle formation is not yet understood.

10.2. P e r

io d ic s m a l l a m p l it u d e o s c il l a t io n s

For small amplitude oscillations a rigorous expression for the elasticity can be obtained. The area deformations and the adsorption oscillations are given by eqs. (8.3) and (8.4), the oscillations of the monomer and micelle concentrations are periodic too and depend on the spatial co-ordinate z, i.e. (10.29a)

c2 = 4

+ f 2( r ) e "

(10.29b)

The oscillations of the concentrations fulfil the diffusion equations plus reaction terms given by eqs. (10.8). Again we assume the micellisation reaction as a first order chemical reaction and not as a polymerisation reaction as done by Lucassen [3,4]. Since at equilibrium kxc,° = k2nc\ , with the notation p from eq. (10.13), and after substitution of eqs. (10.29) and (10.25) in eq. (10.8) we obtain

(10.30a)

to f 1( z ) = Dy

d 2 f 2( z ) dz

k,p f i ( z ) ~ k 2 f i ( z) n

(10.30b)

These equations can be rewritten as

(10.31a)

295

f, (z) + r f2(z) + j

d 2 f2(z) , 2... = 0 dz

(10.31b)

where p, q, r and s are constants D

*" ■ l +p^’

D____ r !Q)(l + P^) ’

n(l + k ) . . , : DY2n . k - k2 ’ /CO^P to

(10.32)

With the notation

(10.33) eqs. (10.31) are written as (qD2 + l)f,(z ) + /?f2(z) = 0

(10.34a)

f,(z) + ( 5Z)2 + r ) f 2(z) = 0

(10.34b)

yielding the characteristic equation (qD2 + l \ s D 2 + r ) - p = 0

(10.35)

or

D4 +

qr + s Z)2 + r - P = 0 qs qs .

(10.36)

This biquadratic equation has 4 roots. Two of them are being rejected because their real part is positive and do not vanish for z -> oo. (We require that fi(°o) = 0 and f2(°o) = 0 at z -» oo.) The two remaining roots are n] and n\ . Hence we have

296 f i{z) = Ee-n'z + Fe~"lZ

(10.37a)

f 2(z)= Me'"'-’ + Ne-"*Z

(10.37b)

with Re(«j ) > 0 ; Re(«2) > 0. Because we have integrated the eqs. (10.31) using the characteristic equation we do not have 4 independent integration constants, but only 2. Therefore we substitute eqs. (10.37) into the differential eqs. (10.31), say into the first one, and we have Ee~n'z + Fe~”2Z + pMe~n': + pNe~niZ -\-qn^Ee~n[Z + qnlFe~niZ = 0

(10.38)

Since this condition must be fulfilled for any z, E(\ + qn[) + p M = 0

(10.39a)

F(\ + qn2 2) + p N = 0

(10.39b)

and only two integration constant are left

M =_ h ^ L E , N ^ i± 3 H L F p p

(10.40)

We could have equally well substituted eqs. (10.37) into the second differential equation of (10.31) and we obtained similarly E F M = - ------- N = - ----------- T r + sn j r + sn2

(10.41)

From eqs. (10.40) and (10.41) we have 1+ qn^ p

1 r + sn{

l+ p

1 r + sn2

----------- — ---------- - a n d ----------- = -----------T

297 or (l + qrii^r + sni) = p and (\ + q n \^ r + sn\} = p

(10.43)

and due to Vieta’s formulae 2 2 = -----r ~ P and1 nf2 +ni2 = ------- 0(s+*)|'- re\ s

s

s

\

sJ

(10.67)

with co0 the relaxation frequency as defined earlier, eq. (8.17). Denoting

s

s

s

( 10.68)

s

we obtain

AT =

Arn

Ar0

s + y](D0{s + k)

(s +k) + y[(o0(s + k) - k

(10.69)

To obtain the inverse transform we make use of the relation L-'{f(s + k)} = e -klL-'{f(s)} hence

(10.70)

302

Us + k) + y]a0{s + k) - k

We split

s + ^J(o0s - k

s + yfoy^s-k

= e~ktL-1

1 S

+ J CO -

(10.71)

k\

into partial fractions

(10.72)

- y[s~i\yfs -

with the roots y[s[ and

of the equation

(10.73)

s + y](o0s - k = 0 giving

fi\ =

-y i^ O - y l ® 0 +4k i— _ V®0 + \A°0 +4k O >\ S2 ~ o

(10.74)

Hence - to

{(5+*)+VtooC$+*)-*l V®o+4kL [Vi-V^r

V»o + 4*

[•/sT exp(s2/)erfc ^ ¡ 7 - ^

exp(s,i)erfc , / v ]

Js-fil (10.75)

Since we consider small jumps Ar

Ac

A r0

Ac0

hence finally from eq. (10.69) we obtain

(1 0 .7 6 )

303

Act =

Aane

-kt

Vtoo+4*

exp(i1/ ') e r f c ^ 7j

^/i7exp(i2i)erfc

(10.77)

This equation is due to Dushkin et ai. [9] and is a generalization of the Sutherland equation. In general, when the jump is not small and the linearization expressed by eq. (10.65) is not allowed, we must use eq. (10.64) and we restrict ourselves to T0 = 0 at t = 0. Then eq. (10.64) becomes -

r=

,jD(s + k ) ^D js + k) _ Ccs-2 c0

(10.78)

and we have to obtain the inverse Laplace transform. Keeping in mind eq. (10.70), we obtain for the transform of the first term on the r.h.s. of eq. (10.78) ■Js + k | _ j^._1f Vj + *

=

t

f

i

dt =

o \

|(s+ * )-* Jr /(

-kt

= V k L~x\ ^ K \dt

0

+ yfk cxp(kt)erf J k tjd t = Jj -j== + J k erf Jkt dt J 0' \ J n t

(10.79)

The integrals are obtained by integration by parts, or can be found in standard tables [11], hence i [ 4s +k Jk

- + k t ) erf Jkt +

i!« -‘

(10.80)

The other transform is

r 'j —

5

L e-* - L + v/*erfVfc yjnt

and with the convolution theorem

(10.81)

304 r e~b - [ = jc,(t - X ) \ j = i + VÂerf J k i dX

L~l \ c. —

(10.82)

Hence finally we obtain from (10.78) -kt

\ + kt) erf f k t + J ^ -e kt - jc (f - X) - yfk erf yfh d k \ 2 ) \ 7i { yfnt

(10.83)

This is the generalised equation of Ward and Tordai accounting additionally for the micellisation process. The short time approximation of eq. (10.83) for cs « 0 is

2 + k t \er]f Æ

r ~ n c°

+1| V “

(10.84)

For the long time approximation we write eq. (10.78) as

r - j s J —

S

(10.85)

ia -c

Vs

and by inverse transformation accounting for the convolution integral we have

r = V 5 j[c 0 - Cj( / - * ) ]

-a y/nX

+ f k erf yfkk dX

( 10. 86)

being equivalent to eq. (10.83). As argued before, for the long time approximation we factorize cQ- cs out of the integral, i.e.

r = VD(c0 - c 1)J

-a

0L yfnX

+ 4k erf fkX dX

Doing the integration in the same way as before [see eq. (10.79)] we obtain

(10.87)

305

( 10. 88)

and for k i » 1 r = V M (c0 - c i )i

(10.89)

If we linearize in this long time domain and making use of the Gibbs equation, eq. (10.89) finally results in

a = ae

RTT2

1

c0

-JDk t

(10.90)

with c0= CMC. Hence by plotting a as a function of t~x a linear relation must be obtained. For a diffusion controlled process cr is a linear function of f 12. Eq. (10.90) can be used to analyse the experimental results [6]. (Experimental data are easier accessible in the long time domain.) If we do experiments at or just below the CMC the adsorption process is diffusion controlled, hence (cf. eq. (5.40))

(10.91)

If we do experiments above the CMC, according to eq. (10.90) we have da \ ■dt 1 c>c mc

RTT2 c0VDk

(10.92)

Since at concentration above the CMC the adsorption is constant, it follows from eqs. (10.91) and (10.92)

306

Fig. 10.5.

Dynamic surface tension of Triton X-100 at a concentration twice the CMC as a function of the dilatation rate 1/0 (a) and l/Vo (b), inclined plate method, according to [5]

In Fig. 10.5a we have plotted the dynamic surface tension for a micellar solution of Triton X-100 as a function of t~l (more precisely as a function of 0'1) and in agreement with eq. (10.92) a linear relation is obtained. We have also plotted the results as a function of f m and it is seen that for large times there is a linear relation, indicating that here the condition for local equilibrium between micelles and monomers is fulfilled (Fig. 10.5b). In Table 10.1 the rate constants obtained in this way are summarised.

307 Table 10.1.

Demicellisation constants for Triton X-100, according to [5] cj/CMC

kF l

2

4.5

3

30.5

5

74.4

7.5

298

10

608

It is expected that the rate constant depends on the concentration of micelles by the equation of Kreshek [12]

k = k0

ct - CMC CMC ,

(10.94)

Here c, is the total concentration. From Table 1 it follows that for Triton X-100 solutions the equation of Kreshek does not apply [5]. Of course the question remains if the diffusion of the micelles can be neglected. 1 0 .4 .

Th

e d if f u s io n pe n e t r a t io n d e pt h f o r a m ic e l l a r s y s t e m s

We start from eq. (10.63) and integrate this equation under the condition that the subsurface concentration is constant. The image function is given by eq. (10.62), but because cs is constant the equation is written as c

Cs

f

s +k ~ D )

The image function for the concentration gradient at z = 0 is

(10.95)

308 (co ~ cs) y/s + k

de) dz)

0

(10.96)

V ï)

The inverse transform gives the concentration gradient at the surface -kt

l -

—, = + yfk erf yfkt

Jd

yJUt

(10.97)

This concentration gradient is also approximated by the introduction of a penetration depth 5dm (diffusion and micellisation) (cf. section 5.4, eq. (5.61)) d c ] _____ o dzJn 5

(10.98)

and from eqs. (10.97) and (10.98) it follows for the diffusion micellisation-penetration depth yJnDt

§ DM

- yjnkt erf Jk t + e kt

yfnkt erf yfkt + e kt

(10.99)

This result is due to Danckwerts [13] (5 D = yJnDt is the diffusion penetration depth as given before). It is seen that for k t » 0 eq. (10.99) reduces to

( 10. 100)

SDU~ J j e and for kt -»

oo;

d DM -» ô D as expected.

309

10.5.

Use

o f t h e d i f f u s i o n -m i c e l l i s a t i o n p e n e t r a t i o n d e p t h t o

DESCRIBE THE DYNAMIC SURFACE TENSION FOR MICELLAR SOLUTIONS

By the concept of the penetration depth, the integration of the diffusion equation including micellar reaction is avoided. The boundary condition for conservation of mass at the surface is approximated by

( 10. 101)

If we consider small jumps, hence we are allowed to linearize and eq. (10.101) becomes dr

D dc{

dt

dT

r - r e) &

S 5 r-re t

D

( 10. 102)

5

where we have introduced the diffusion relaxation time as defined before, eq. (2.85). Substitution of eq. (10.99) into eq. (10.102) gives [14]

dt

r ~re

(10.103)

0 ” £>')12

where x K = k 1 is the relaxation time for micellisation. For k - 0, this means no micellisation reaction, eq. (10.103) reduces to eq. (5.63) and after intégration to eq. (5.64). Intégration of eq. (10.103), with the initial condition that T = T0 at / = 0 gives

r - r , = ( r - r 0)exP -

12 7

, \ 1,2 /

t exp ——

+

1,2

f V2

x K)

\x K

(10.104)

Since we assumed small jumps there is a linearity between the surface tension and the adsorption variations:

310 \ o = a e + Aa0 exp

J

\ 1 \ ( t. \) ( 1 1 t t I exp " --- + erf — K2 x KJ Kx k )

2 l

(10.105)

This equation is an approximation for the Dushkin equation and can be used to analyse experimental results. From eq. (10.105) we obtain

A a 0

ln-

f T K1

i ' 1 U J

1,2 exp

r

r

+

1/2

(\ _

r

\2

x Kj

erf

(10.106)

*k )

Let us look now at the function 1/2

f(x) = ( - J

(10.107)

e~x + y - + x ) e r f 4 x

If we plot this function versus x (see Fig. 10.6) we observe that for x > 1 this function is approximated by (10.108)

fW = | j + »

t Aa0 hence for — >1 eq. (10.106) can be approximated by In—— xY Aa

^ V /2f i +^

V'f D'

0) interfacial tensions of cholesterol solutions in heptane (dotted line), according to [1]

316 The extrapolated interfacial tension at t -» a>, agrees with the situation where diffusion and transfer equilibrium are established, but the reorientation process has not started yet. At the surface the following reorientation reaction takes place:

r,

(11.11)

f ) Jr V

(&.'