Effect of double-layer repulsion on foam film drainage

This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies areencouraged to visit:

http://www.elsevier.com/copyright

http://www.elsevier.com/copyright

Author's personal copy

Available online at www.sciencedirect.com

Colloids and Surfaces A: Physicochem. Eng. Aspects 319 (2008) 34–42

Effect of double-layer repulsion on foam film drainage

Stoyan I. Karakashev a,∗, Emil D. Manev b, Anh V. Nguyen a

a Division of Chemical Engineering, School of Engineering, The University of Queensland, Brisbane, Qld 4072, Australiab Department of Physical Chemistry, Faculty of Chemistry/Sofia University, 1 J. Bourchier Avenue, 1164 Sofia, Bulgaria

Received 19 February 2007; received in revised form 13 June 2007; accepted 15 July 2007Available online 2 August 2007

Abstract

This paper presents an experimental validation of new theoretical development for foam film drainage, which focuses on the role of surfaceforces. The drainage of microscopic foam films (with radii smaller than 100 �m) from aqueous solutions of 10−6 to 10−4 mol/L sodium dodecylsulphate (SDS) was studied by means of an improved Scheludko micro-interferometric technique which consisted of a conventional Scheludkocell, a high-speed camera system, and the software for digital analysis Optimas used for the digitisation of the interferometric images to obtain themonochromatic light intensity. The experimental technique allowed fast processing of the interferometric data for determining the transient filmthickness with high accuracy. The zeta-potential of the air–water interface was determined from the electrophoretic mobility of micro-bubbles in SDSsolutions of the same concentrations. Advanced predictions for the electrical double-layer repulsion at either constant surface potential or constantsurface charge were employed. Significant discrepancy between the theoretical prediction and the experimental data was obtained. The analysisshowed that the adsorption layer, which is located on the film surfaces, is far away from equilibrium, while the theory assumes condition close toequilibrium. In this term the interaction between the film surfaces is affected by the dynamics of the adsorption layers during the film drainage.© 2007 Elsevier B.V. All rights reserved.

Keywords: Foam film; Surface forces; Double-layer interaction

1. Introduction

The foam drainage is influenced by the pressure squeezingthe liquid out of the film separating the bubbles and by theliquid outflow along the Plateau-Gibbs channels. The film thin-ning, therefore, is important in determining the overall foambehaviour. In turn, both the surface forces and the surface rheol-ogy of the adsorbed surfactants are significant for the drainageand stability of the individual liquid films. An intensive researchin this area has been conducted over the past four decades.

The first theory for film drainage belongs to Scheludko [1],who applied the Stefan–Reynolds equation obtained from thelubrication approximation for describing the velocity of filmthinning. He assumed the immobile, plane-parallel surfaces offoam films thinning under the action of the capillary and DLVOsurface forces. Further study on film drainage aimed to elucidatethe effects of the (DLVO and non-DLVO) surface forces, thesurface mobility, and the film thickness in homogeneity on the

∗ Corresponding author. Tel.: +61 7 336 59054.E-mail address: [email protected] (S.I. Karakashev).

kinetics of film thinning [2]. Briefly, the further investigationswere focused on the three key factors influencing film drainage:(i) surface rheology, (ii) interface shape, and (iii) surface forces.All of these factors have not been equally considered in theavailable models for film drainage.

Regarding the surface rheology, Radoev et al. [2] developeda theory focusing on the mobility of film surfaces. The authorsincluded the surfactant adsorption, the Marangoni effect [2], andthe surface diffusion of the surfactant molecules [3]. The furthertheoretical development after Radoev et al. included the effectof the surface shear viscosity [4–6]. Due to the complexity ofthe solution, this effect was only approximately described. Theproblem was re-visited with the numerical solutions in the recentyears [7–9], but without any experimental validation. Recently,Karakashev and Nguyen [10] developed a new analytical modelwhich precisely accounted for the effect of the surface shearviscosity and elasticity, and the surfactant diffusions. The focusof the validation of the new complex model was on the interfacialproperties.

One of the significant, experimentally established facts wasthat foam films could only be planar at very small radii (smallerthan 100 �m) [11–14] which have to be considered in obtaining

0927-7757/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.colsurfa.2007.07.021


S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 319 (2008) 34–42 35

Nomenclature

a adsorption lengthA Hamaker–Lifshitz functionA function in Eq. (6)A0 zero frequency term of the Hamaker–Lifshitz

functionB function in Eq. (6)Bo Boussinesq numberc bulk concentration of the surfactantcel bulk concentration of the elelctrolyteC function in Eq. (6)D diffusion coefficient of surfactant in the bulk solu-

tionDs surface diffusion coefficient for surfactantE Gibbs elasticityf function in Eq. (2)f function used for calculation of the electrostatic

disjoining pressure with Eq. (5)F Faraday number or driving force for film drainageh film thicknessh dimensionless film thicknessh Plank constant divided to 2πI instantaneous intensity of the photocurrent;Imax maximal intensity of the photocurrentImin minimal intensity of the photocurrentL order of interferenceMa Marangoni numbern characteristic refractive index of water in the UV

regionn0 refractive index of the film liquidN dimensionless diffusivity numberp pressure inside the filmPσ capillary pressureq =1.185, see Eq. (8)r Fresnel reflection coefficientR film radiusRg universal gas constantT absolute temperatureV velocity of film drainagey dimensionless surface potential at infinite thick-

ness of the film

Greek symbolsΓ Gibbs surface excess of the surfactantΔ =(I − Imin)/(Imax − Imin)ε static dielectric permittivity of the mediumε0 static dielectric constant of vacuumκ Debye constantλ wavelength of the monochromatic light or char-

acteristic wavelength of retarded van der Waalsinteraction

λk kth root of the Bessel function of the first kind andthe zero order

μ bulk viscosityμs surface shear viscosity

ξ characteristic absorption wavelength of the waterin the UV region

π =3.1415926Π disjoining pressureΠel disjoining pressure due to electrostatic double

layersΠvdW disjoining pressure due to van der Waals interac-

tionσ surface tension of gas/liquid interfaceψ surface potentialω characteristic absorption frequency of the water

in the UV region

the experimental validation of the available theories developedfor plane-parallel films. Manev et al. [11,13,14] observed thatthe increase in the film radius correlates with the enlarged cor-rugations of the film surfaces. Such films drain faster than theplanar films [11,13–15]. The influence of the thickness corruga-tion on the film drainage was modelled and validated by Manevet al. [11], Tsekov [16], and Tsekov and Evstatieva [17].

Another important aspect of the theoretical development isthe prediction of the surface force interaction between the filmsurfaces through the disjoining pressure introduced by Derjaguinsome time ago. The DLVO theory of stability of lyophobic col-loids includes essentially the van der Waals and electrostaticdouble layer interactions [18,19].

The superposition approximation by Derjaguin for the elec-trostatic disjoining pressure is based on the assumption of weakoverlapping of the diffuse electric layers between the film sur-faces. It is widely employed, but is only justified for surfacepotential smaller than 25 mV. The computation of the electro-static interactions was refined [20,21] to extend the range ofthe applicability to higher surface potential values and differentsurface charging mechanisms. The double-layer interaction inthinning films can take place under the conditions of: (i) con-stant surface potential; (ii) constant surface charge; (iii) surfacecharge regulation. This surface charging condition depends onthe state of the adsorption layers on the film surfaces.

The van der Waals disjoining pressure is described byeither the microscopic (Hamaker) or macroscopic (Lifshitz)approaches. The theory of van der Waals interactions hassignificantly been developed, allowing a better determinationof the electrolyte screening and electromagnetic retardationeffects—see Ref. [22] for further details. The theory of the non-DLVO interactions in thin liquid films has undergone significantdevelopment. Of the non-DLVO interactions, the hydrophobicattraction appears to be the most significant but least under-stood and has intensively been studied over the last two decades[23–30].

Surfactant adsorption affects both surface rheology and sur-face forces. The advance in the theory of the surface forcesallows one to account for significantly stronger interactionbetween the film surfaces under either constant surface poten-tial or constant surface charge. However, the properties of the


36 S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 319 (2008) 34–42

adsorption layer in the dynamic conditions of the film drainageare not considered by the theory. In addition, the distributionof the surface charges during the drainage is not homogeneous,which results in variable disjoining pressure upon the film sur-faces. The rheology of the film surfaces is affected by thedynamics of the adsorption layer additionally. It is, therefore,significant to examine how surface rheology and surface forcesjointly influence the foam film drainage. In a previous investi-gation [10] we have developed an advanced analytical modelwhich accounted for the effects of surface viscosities and diffu-sion, and surface forces. The focus of the previous paper was onthe effect of interfacial rheology on foam film drainage. The aimof this paper is to investigate the effect of surface force interac-tions on foam film drainage. Specifically, the experiments werecarried out with a range of surfactant concentration, from 10−6 to10−4 mol/L of sodium dodecyl sulphate. The research focus wason the double-layer forces with different surface charging mech-anisms. Two special models for the double-layer forces includethe double-layer interaction at constant surface potential and thedouble-layer interaction at constant surface charge density. Thecomparison with the data for foam film drainage shows that thedrainage cannot be described by the present theory of double-layer interaction at either constant surface potential or constantsurface charge density.

2. Theoretical background

2.1. Surfactant adsorption

The surfactant type and concentration – together with theelectrolyte – command the foam film drainage and stability. Thesurfactant adsorption provides the films with the ability to resistthe local deformations and rupture in the process of thinning.Certain specific effects, related to the individual structure of theadsorbed layers of different surfactants, are also expected.

The surfactant adsorption affects the dynamic properties ofthe liquid surfaces, which govern the resistance of the film tothe changes in the shape and size. The surface tension (whichdepends on the surfactant type and concentration) and foamdrainage rate are related through the surface elasticity and vis-cosity of the film. The elasticity, defined by Gibbs, expresses theresistance of the film to local deformations. Any increase in filmarea is accompanied by a local increase in surface tension and,consequently, a tendency to contraction. The tangential force(per unit area) toward restoring the initial, undeformed, state ofthe film is the Gibbs elasticity E:

E = − dσ

d lnΓ(1)

Here σ is the surface tension of gas/liquid interface and Γ is thesurface excess of the surfactant.

The elasticity is considered as a major factor for the foam andfilm conduct. Elasticity created by the presence of surfactant isthe necessary condition for the formation of a foam column,which does not decompose instantly (see, e.g. [2]). First of all,the visco-elastic effect of the surfactant is obligatory for thedynamic stabilization of the liquid films, in order that they can

form and drain while withstanding tangential stresses due tothe outflow on their surfaces. The drainage in the presence ofsurfactants at the film surfaces may be substantially reducedby an opposing gradient of the surface tension, the so-called‘dynamic elasticity’ or ‘Marangoni effect’ [12].

2.2. Modelling of foam film drainage

Since the focus of this paper is on the effect of surface forces,the experiments were designed to simplify the theoretical anal-ysis and were carried out with very small films (with radiussmaller than 100 �m) which were plane parallel. Foam filmswith greater radius were dimpled and were not analysed. Forthe small foam films with plane-parallel surfaces, the refineddrainage theory [10] can be applied. To avoid repetition, only abrief description and the major results are given here. Followingthe conventional approach the modelling was developed usingthe lubrication approximation obtained from the continuity andStokes equations. The mass balance equation for surfactants wassolved to obtain the surfactant concentration gradient along thefilm surfaces. The full tangential stress balance applied at the filmsurfaces was solved analytically. The equation for the stress bal-ance contains three terms, including the bulk flow stress appliedat the film surfaces, the stress due to the surface tension gradientalong the film surfaces (the Marangoni effect) and the surfacestress due to surface viscosities which have been proved signif-icant for the liquid drainage in foam films and foams. The foamfilm drainage model with the full stress balance equation has onlybeen solved numerically [31]. In terms of the Marangoni (Ma),Boussinesq (Bo), and diffusivity (N) dimensionless numbers, thevelocity, V ≡ −dh/dt, for the film drainage can be described as

V = 2h3

3μR2

Pσ −Π

f(2)

where h is the film thickness, P� and Π the capillary anddisjoining pressures, respectively, R the film radius, and μ isthe liquid viscosity. The correction factor, f, in Eq. (2) accountsfor the deviation from the Stefan–Reynolds prediction [1,32]for the film drainage velocity with plane-parallel, tangentiallyimmobile (non-slip) surfaces. The factor deviates from unitydue to the interfacial properties of the adsorbed surfactants andcan be described by

f = 1 − 32∞∑k=1

6/λ4k − (h/λk)

2

6 + Boλ2kh+Ma h/[1 + (N/λk) tanh(λkh/2)]

(3)

where h ≡ h/R, Ma ≡ ER/(Dsμ), Bo ≡μs/(Rμ), N = (D/DS)(R/a), and λk is the kth root of the Bessel function of the firstkind and zero order, σ the surface tension, E = −(dσ/d lnΓ )and Γ are the Gibbs elasticity and surface excess, respectively,a = (dΓ/dc) the adsorption length, c the surfactant concentra-tion, D and Ds the bulk and surface diffusion coefficients, andμs is the surface shear viscosity.

If the surface shear viscosity is not accounted for, Bo = 0, andthe model is reduced to the well-known theory of Radoev et al.[3] since

∑∞k=1λ

−4k = 1/32 and tanh(x) = x + O(x3).



2.3. Effect of the tangential mobility, surface deformability,and thickness heterogeneity

Eqs. (2) and (3) cannot always explain the experimentallyobserved acceleration of the drainage rate, especially for foamfilms of large radii (R > 0.1 mm). In this case, the experimentswith both foam and emulsion films show a much weaker depen-dence on the radius, namely V ∝ R−0.8 while Eq. (2) requiresVRe ∝ R−2. In other words, the larger the film radius, the big-ger the deviations from Reynold’s law (drainage is much fasterthan VRe ∝ R−2). Therefore, tangential mobility alone is not suf-ficient to describe adequately the deviations from Eq. (2). Thesecond condition for the applicability of Eq. (2) requires non-deformable film surfaces. However, while small films (radiusR � 0.1 mm) may preserve their quasi-parallel profile, the thick-ness of large films is not homogeneous during the thinningprocess. The larger the film, the more pronounced the thick-ness non-homogeneity. The experimental studies indicate thatthe deformability of the fluid film interfaces dominates filmdrainage. Moreover, the experiments have shown that the ampli-tude of thickness heterogeneity increases with the film radius.

According to the theory of Manev et al. the liquid outflowbetween the corrugated surfaces corresponds to the decompo-sition of the film from a single entity into several (thicker andthinner) domains, the number of which increases with increasingfilm radius. This spontaneously formed structure – an ensembleof several smaller films of different thickness thinning together– results in a more favourable hydrodynamic situation (sincesmaller films drain faster) and thus in accelerated thinning. Therate of thinning is now described by

V =(

1

6μ

) (h12ΔP8

4σ3R4

)1/5

(4)

The most significant characteristics of Eq. (4) is that the rateof thinning must be inversely proportional to a power of 4/5of the film radius (R−0.8), which is perfectly in line with allavailable experimental data on foam and emulsion films.

2.4. Surface forces

The disjoining pressure,Π, on the right-hand side of Eq. (2)is critical to the modelling of thin film drainage. Every kind ofmolecular interactions between the film surfaces correspondsto a certain type of disjoining pressure. Generally, they can bedivided into the DLVO (electrostatic and van der Waals) andnon-DLVO pressures which include the steric, hydration andhydrophobic components [10]. Among the non-DLVO forces,only the hydrophobic one is long-ranged and might play animportant role in the foam film drainage. Generally, only theelectrostatic and van der Waals interactions are expected to besignificant for the film drainage giving Π =Πel +ΠvdW. Thepressure components are briefly described below.

2.4.1. DLVO forces2.4.1.1. Double-layer disjoining pressure. The electrostaticdisjoining pressure, Πel, arises in thin films from the over-

lap of the diffuse electric layers at the two film surfacesand can be determined from the Poisson–Boltzmann equa-tion which is highly non-linear. Consequently, approximateexpressions for Πel are frequently used. The Debye–Huckellinearization is limited by the condition of low surface poten-tials used in the linearization. For small surface potentials, ψ,the superposition approximation is often used, giving Πel =64celRgT tanh2(y/4) exp(−κh), where Rg is the universal gasconstant and cel is the molar concentration of electrolytes inthe solution. The normalised surface potential, y, is defined asy = zFψ/(RgT ), where F is the Faraday constant. The Debyeconstant, κ, for binary z:z electrolyte solutions is defined as

κ = {2celF

2z2/(εε0RgT )}1/2

, where ε and ε0 are the permit-tivity of vacuum and the dielectric constant of the solution (∼80 for water). The superposition approximation is applicable forthe cases of weak overlap of the diffuse layers. For the strongoverlap of the diffuse layers at a small film thickness, the cal-culation of Πel depends on the charging mechanism at the filmsurfaces. Under the condition of constant surface potential, theexact numerical solution to the non-linear Poisson–Boltzmannequation can be semi-analytically described as [20]:

Πel = 32celRgT tanh2(y

4

)

×{

1

1 + cosh κh+ f (y) sinh2 y

4exp[−f (y)kh]

}(5)

where f is defined as f (y) = 2 cosh(0.332 |y| − 0.779) for|y| ≤ 7. Eq. (5) is the most general solution for the double-layerinteraction under the condition of constant surface potential.

Under the condition of constant surface charge density, theexact numerical solution to the non-linear Poisson–Boltzmannequation can be approximately described as [20]:

Πel(h) = 2celRgT A

[cosh(κhC) − 1]√

1 + B2 coth(κhC/2)(6)

The actual potential at the film surfaces at the constant surfacecharge interaction changes with the film thickness and can-not be used in the prediction. In place of the actual potential,the prediction described by Eq. (6) uses the surface potentialof the film surfaces at an infinitely large separation distance,where the diffuse layers do not overlap. The surface poten-tial belongs to isolated gas–liquid interface (as used to modelthe interaction at constant surface potential). The constants inEq. (6) are described as A = BC sinh(1.854 |y| − 0.585|y|2 +0.1127|y|3 − 0.00815|y|4), B = 0.571|y| exp(−0.095|y|1.857)and C = 1 − 0.00848|y|. These approximations are valid for|y| ≤ 5 or |ψ| ≤ 126 mV. For foam (symmetrical) films, Πel isalways positive corresponding to the repulsion between the filmsurfaces and strongly depends on the ionic strength of the solu-tion. Πel at low electrolyte concentration may oppose the filmdrainage, but further increase in the electrolyte concentrationsuppresses the effect.

However, Eqs. (5) and (6) are originally derived for baresolid surfaces [20] with homogeneously distributed potentialand charges. In addition, the equation of Poisson–Boltzmann



does not consider the nature of the surface. The fluid gas/liquidinterfaces of the film with the adsorption layers located on themare floatable and deformable; the adsorption of the charges hasvariable distribution on the surfaces thus affecting the interactionbetween them.

2.4.1.2. van der Waals disjoining pressure. The van der Waalsinteraction between surfaces is well described in many books,including [33–35]. The interaction can be determined usingeither the Hamaker (microscopic) and/or Lifshitz (macroscopic)approach. For the van der Waals disjoining pressure, ΠvdW, asa function of the film thickness, combining the two approachesgives

ΠvdW = −A(h, κ)

6πh3 + 1

12πh2

dA(h, κ)

dh(7)

where A(h,κ) is the Hamaker–Lifshitz function, which is a weakfunction of the film thickness, h, and the Debye constant, κ, dueto the electromagnetic retardation and electrolyte screening. TheHamaker–Lifshitz function can be decomposed into the zero andnon-zero frequency terms which are described for foam films as[28]:

A(h, κ) = 1 + 2κh

exp(−2κh)

3kBT

4

∞∑j=1

1

j3

(1 − ε

1 + ε

)2j

+ 3hω

16√

2

(1 − n2)2

(1 + n2)3/2

{1 +

(h

ξ

)q}−1/q

(8)

where kB is the Boltzmann constant, T the absolute temperature,h = 1.055 × 10−34 J s/rad the Planck constant (divided by 2π),ω the absorption frequency in the UV region—typically around2.068 × 1016 rad/s for water, n2 = 1.887 the square of the charac-teristic refractive index of the film (water) in the UV region[36],and q = 1.185. The characteristic wavelength, ξ, is described byξ ≡ v/π2ω

√2/(n2 + n4) = 5.59 nm, where v is the speed of

light and ω has unit of 1 s−1. For aqueous foam films, ε= 80,and the infinite sum in Eq. (5) is equal to 1.444.

2.4.2. Non-DLVO forcesGenerally the non-DLVO forces act in these special kinds of

interaction, which do not have electrostatic or van der Waals ori-gin. Some of them are: (i) steric interactions [36]—they exist incases when polymeric molecules are adsorbed on both surfacesof the film and at enough small thickness their chains overlapeach other and stop the further thinning of the film; (ii) structural(solvatation) interactions [37,38]—at film thickness ca. severalmolecular spaces often is observed repulsion. This kind of forceis called structural (in water-hydration) repulsion and it is causedby the corpuscular structure of the solvent; (iii) fluctuation struc-tural interactions [39]—they originate from the fluctuation of theliquid density inside the film and they can become significant inshort distances between the two surfaces, resulting in attractionand repulsion; (iv) depletion interactions [40]—they act usu-ally between solid particles at very short distances, i.e. whenthe liquid layer between the particles is squeezed out result-ing in attraction force, caused by the surrounding medium; (v)

hydrophobic interactions—according to the literature sources[23–26,41,42], in contrast to other non-DLVO forces, whichare usually manifested under special conditions at very shortdistances, this kind of interactions are supposed to act as long-ranged attraction between hydrophobic surfaces in aqueousmedium. If present, they could play significant role in the proper-ties of regular foams, emulsions and suspensions of hydrophobicparticles in aqueous media. The authors of the present workthink that the nature of the hydrophobic interactions as a sourceof an attractive (non-DLVO) force is not yet theoretically wellgrounded and their existence needs direct experimental verifi-cation.

2.5. Numerical computation

In order to apply the above theory, Eq. (2) was converted intothe first-order differential equation in terms of the film thickness,h, versus time, t. The differential equation was then numeri-cally integrated using a fourth-order Runge–Kutta method. Amacro was written using the VBA (Visual Basic for Appli-cation) programming language available in Microsoft Excel.Eqs. (2) and (3) for the electrostatic disjoining pressure in athinning film under the conditions of either constant surfacepotential or constant surface charge were used in conjunctionwith the van der Waals pressure components to determine thetotal disjoining pressure. The computation procedure was as fol-lows. The experimental data for the zeta-potential were usedfor the surface potential in the drainage equation. Then Eq.(2) was integrated to obtain the film thickness versus time.The computed thickness was compared with the experimentaldata.

3. Experimental

The drainage of small foam films was measured by the Sche-ludko micro-interferometric method [43] with a computerisedmicro-interferometric experimental set-up. Shown in Fig. 1 is theschematic of the improved experimental set-up which consistsof the following major units:

Fig. 1. Experimental micro-interferometric set-up with the Scheludko cell, ametallurgical inverted microscope, and a high-speed video camera system.



• The ‘Scheludko’ cell for producing the foam films.• A metallurgical inverted microscope (Epiphot 300, Nikon,

Japan) for illuminating and observing the film, and the adja-cent interference fringes (Newton rings) in the reflected light.

• A high-speed video camera system (Phantom 4, Photo-SonicsInc., USA) for registering transient interferometric images.

• A computer for controlling the high-speed video camerasystem, recording the transient interferometric images, andstoring the transient images for further off-line processingand analysis.

A droplet of the investigated solution was formed inside thefilm holder with an inner radius of ∼4 mm. The film holderwas connected through the capillary tube with a gastight micro-syringe for regulating the amount of liquid inside the drop. Themicroscopic film was formed between the surfaces of the doubleconcave meniscus by pumping out the liquid from the drop (seeFig. 1). The radius of the film was dependent on the amount ofthe withdrawn liquid, but it was smaller than 100 �m, render-ing the film surfaces plane parallel. The film was illuminated bymonochromatic light with wavelength λ= 546 nm. As a result ofthe interference of the light reflected from the two film surfaces,a set of bright and dark fringes (Newton rings) was observedand recorded in the computer using the high-speed video cam-era system. The rings appeared as a result of the interferometricextrema (minima and maxima), corresponding to the film thick-ness values multiple of λ/(4n0), where n0 is the refractive indexof the film liquid.

The transient film images were usually recorded within 14 mstime intervals. The interferometric images were digitalized usingthe Optimas program (Optimas 6.1, Optimas Inc., USA). Anarrow strip passing through the centre of the interferomet-ric fringes was selected and the digital signals were convertedinto photocurrent versus radial distance using a special macrodeveloped in Optimas 6.1 by the authors. The photocurrentbetween the interferometric extrema was quite smooth and with-out significant variation, which corresponded to planar films.The thickness of the film was calculated using the Scheludkointerferometric equation, which accounts for the light interfer-ence by multiple reflections from both surfaces [1,43]:

h = λ

2πn0

⎡⎣Lπ ± arcsin

√Δ(1 + r2)

(1 − r2) + 4r2Δ

⎤⎦ (10)

where r = (n0 − 1)2/(n0 + 1)2 is the Fresnel reflection coef-ficient for the air–solution interface, L = 0, 1, 2, 3. . . is theorder of interference, Δ = (I − Imin)/(Imax − Imin), I is theinstantaneous intensity of the photocurrent, Imin and Imax areits minimum and maximum.

The true film thickness can be smaller than that determinedby Eq. (10), and referred to as the equivalent film thickness[43]. The difference is due to the higher refractive index of theadsorption layers at the film surfaces. However, for the rangeof film thickness and surfactant concentrations studied in thispaper, the difference was insignificant and no corrections to Eq.(10) were made.

The foam films were produced from aqueous solutions ofsodium dodecyl sulphate (SDS). The surfactant was purchasedfrom Fluka and was purified by four times re-crystallization fromethanol. The purity of SDS was tested by the surface tensionisotherm, which exhibited no minimum. Surface shear viscositywas determined with a deep-channel surface viscometer [44,45]and was reported in a previous paper [45].

Six different concentrations of SDS were chosen for produc-ing the foam films, including 10−6, 5 × 10−6, 8 × 10−6, 10−5,5 × 10−5 and 10−4 M. The temperature was kept constant at∼22 ◦C. The surface tension of the surfactant solutions versusconcentration was measured by sessile bubble tensiometry witha fully computer-controlled apparatus (System OCA 20, DataPhysics, Germany). The zeta-potential of the air–solution inter-face was measured with a Micro-Electrophoresis Apparatus MKII (Rank Brothers Ltd., UK) using micro-bubbles with ∼10 �min diameter [46].

4. Results and discussion

The experimental results for the surface tension for the SDSsolutions are shown in Fig. 2. The surface tension has no mini-mum in the vicinity of the critical micelle concentration (CMC),which is ca. 8 mmol/L, indicating that the purified surfactantcould be free from surface contaminants.

The parameters of the surfactant adsorption can be obtainedfrom the experimental data by employing appropriate adsorp-tion models. The adsorption of SDS was determined followingthe approach available from the literature [47]. Briefly, thesurface tension was calculated using the Gibbs adsorption equa-tion. The relationship between the surface excess of the co-and counter-ions was modelled with two Frumkin adsorptionisotherms. The difference between the experimental data andthe model for surface tension was minimised using the non-linear regression analysis by changing the dielectric permittivityof water in the adsorption layer. The agreement between theexperimental data (points) and the model prediction (solidline) for surface tension is shown in Fig. 2. The Gibbs elas-ticity and the adsorption length, as determined by the modeland the experimental data for surface tension, are shown inFigs. 3 and 4.

Fig. 2. Experimental (points) and theoretical (line) results for surface tensionvs. sodium dodecyl sulphate (SDS) concentration.



Fig. 3. Gibbs elasticity,E = −∂σ/∂ lnΓ , of SDS solutions as determined usingthe data shown in Fig. 2.

Fig. 4. Adsorption length, a = dΓ/dC for SDS solutions as determined usingthe data shown in Fig. 2.

The experimental results for the zeta-potential of micro-airbubbles in the aqueous solutions of SDS are shown in Fig. 5.The zeta-potential has a minimum at a concentration close tothe point of adsorption saturation. It is assumed that the zeta-potential values do not differ significantly from those of thesurface potential of the adsorption layers and can be used forcomparison with the values of the surface potential obtainedwith the drainage model.

Fig. 5. Experimental data for the zeta-potential of micro-air bubbles in aqueoussolutions of SDS. The maximum standard deviation is ca. 6 mV.

Fig. 6. Experimental data (filled circles) and theoretical results (lines) forfilm thickness vs. time for a 10−6 mol/L SDS film with the film radiusR = 63 �m. The double-layer repulsion at the constant surface potentialand constant surface charge was considered. The other parameters includeσ = 72.6 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 surfacepoise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.744 mN/m, and a = 0.275�m. Thezeta-potential is ca. −62.3 ± 6 mV, as shown in Fig. 5.

Comparison between the experimental data and the modelwas carried out with the double-layer disjoining pressure at con-stant surface potential and constant surface charge. The modelwith the double-layer disjoining pressure at constant surfacepotential and constant surface charge has failed to predict thefilm drainage correctly for all the concentrations of SDS (seeFigs. 6–11). Figs. 6–11 present the experimental and theoreti-cal curve of film thinning. The data for the value of the surfacepotential are taken from the experiment on the zeta-potential ofnitrogen micro-bubbles in aqueous solution of SDS within thesame concentration range. The experimental error of determina-tion of the zeta-potential was found to be ±6 mV. Therefore thisexperimental error was taken into account in terms of relativetheoretical error in form of error bars on the theoretical curves(see Figs. 6–11).

The experimental error in the determination of the film thick-ness is ca. ±2 nm.

Fig. 7. Experimental data (filled circles) and theoretical results (lines) forfilm thickness vs. time for a 5 × 10−6 mol/L SDS film with the filmradius R = 51 �m. The double-layer repulsion at the constant surface poten-tial and constant surface charge was considered. The other parametersinclude σ = 72.5 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 sur-face poise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.744 mN/m, and a = 0.275 �m.The zeta-potential is ca. – 56.1 ± 6 mV, as shown in Fig. 5.



Fig. 8. Experimental data (filled circles) and theoretical results (lines) forfilm thickness vs. time for a 8 × 10−6 mol/L SDS film with the film radiusR = 47 �m. The double-layer repulsion at the constant surface potentialand constant surface charge was considered. The other parameters includeσ = 72.47 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 surfacepoise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.744 mN/m, and a = 0.275 �m. Thezeta-potential is ca. −52.66 ± 6 mV, as shown in Fig. 5.

Fig. 9. Experimental data (filled circles) and theoretical results (lines) forfilm thickness vs. time for a 10−5 mol/L SDS film with the film radiusR = 77 �m. The double-layer repulsion at the constant surface potentialand constant surface charge was considered. The other parameters includeσ = 72.45 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 surfacepoise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.741 mN/m, and a = 0.275 �m. Thezeta-potential is ca. −49.23 ± 6 mV, as shown in Fig. 5.

Fig. 10. Experimental data (filled circles) and theoretical results (lines)for film thickness vs. time for a 5 × 10−5 mol/L SDS film with the filmradius R = 58 �m. The double-layer repulsion at the constant surface poten-tial and constant surface charge was considered. The other parametersinclude σ = 72.4 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 sur-face poise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.743 mN/m, and a = 0.274 �m.The zeta-potential is ca. −48.63 ± 6 mV, as shown in Fig. 5.

Fig. 11. Experimental data (filled circles) and theoretical results (lines) forfilm thickness vs. time for a 10−4 mol/L SDS film with the film radiusR = 60 �m. The double-layer repulsion at the constant surface potentialand constant surface charge was considered. The other parameters includeσ = 71.83 mN/m, D = 4 × 10−9 m2/s, Ds = 1 × 10−8 m2/s, μs = 0.001 surfacepoise (1 × 10−6 kg/s), μ= 0.001 Pa s, E = 3.735 mN/m, and a = 0.274 �m. Thezeta-potential is ca. −48.04 ± 6 mV, as shown in Fig. 5.

One may raise the question—why such advanced kinetic the-ory fails in predicting the drainage of the foam films? The answercan be sought in the assumptions of the theory and how muchthese assumptions overlap with the reality. The kinetic model offilm drainage has two major sides—surface rheology and sur-face force. These two sides are considered independent fromeach other. In reality they are related to the state of the surfac-tant adsorption layer. It is assumed as well that the adsorptionlayer is in state very close to equilibrium. However, this dependson the velocity of film thinning. When the film is thinning veryfast, this drives far away the adsorption layer from equilibrium.This will bring the time as a variable in the mass balance equa-tions and will cause non-linearity in the surface distribution ofthe charges. The latter will create non-uniform distribution of therepulsion force over the film surfaces. In addition, the adsorptiondepends on the film thickness.

All these effects are not accounted for in the present state ofthe kinetic theory.

5. Conclusions

The drainage of foam films of diluted (10−6 to 10−4 mol/L)aqueous solution of sodium dodecyl sulphate was investigated todetermine the role of the double-layer interactions in foam filmdrainage. The transient thickness of planar foam films of verysmall radii versus time was obtained with an improved Sche-ludko cell set-up used in conjunction with a high-speed videocamera system operating at the frequency of 1000 frames/s. Thesurface tension and zeta-potential values were measured by thesessile bubble tensiometry and micro-electrophoresis, respec-tively. The surface shear viscosity was determined by a deepchannel surface shear viscometer.

The modelling considered surface tension gradient, surfaceshear viscosity and diffusion, and DLVO intermolecular forces.Comparison between experimental data and theoretical predic-tions for foam film drainage shows significant discrepancy.



An alternative cause of the discrepancy could be the non-homogeneity in the distribution of the surface charges during thefilm thinning. The surface-active ions located on the film sur-faces are driven by the liquid outflow toward to periphery of thefilm. This effect is opposed by (i) the back surface diffusion and(ii) the additional adsorption of surface charges on the film sur-faces. The actual distribution of the charges on the film surface isresult from the balance of these forces. This creates non-uniformdistribution of the electrostatic repulsive force on the film sur-faces. In addition, the surfactant adsorption depends on filmthickness. The adsorption layer is not close to equilibrium dur-ing overall film drainage. This could bring time dependence inthe mass balance equations and non-linear effects on the filmsurfaces.

Acknowledgements

The authors gratefully acknowledge the Australian ResearchCouncil for financial support through a Discovery grant.Emil Manev is grateful also for the support by the EU’sMarie Curie Research Training Network “Self-organizationunder confinement” (SOCON, contract No. MCRTN-CT-2004-512331512331). Stoyan Karakashev devotes this work to thememory of his father Ivan Karakashev with love.

References

[1] A. Scheludko, Adv. Colloid Interf. Sci. 1 (4) (1967) 391–464.[2] B.P. Radoev, E.D. Manev, I.B. Ivanov, Godishnik na Sofiiskiya Universitet

Sv. Kliment Okhridski, Khimicheski Fakultet. 60 (1968) 59–72.[3] B.P. Radoev, D.S. Dimitrov, I.B. Ivanov, Colloid Polym. Sci. 252 (1) (1974)

50–55.[4] A.D. Barber, S. Hartland, Can. J. Chem. Eng. 54 (4) (1976) 279–284.[5] T.T. Traykov, I.B. Ivanov, Int. J. Multiphase Flow 3 (5) (1977) 471–483.[6] I.B. Ivanov, D.S. Dimitrov, Colloid Polym. Sci. 252 (11) (1974) 982–990.[7] D.S. Valkovska, K.D. Danov, J. Colloid Interf. Sci. 241 (2) (2001) 400–412.[8] D.S. Valkovska, K.D. Danov, I.B. Ivanov, Colloids Surf. A 175 (1–2) (2000)

179–192.[9] K.D. Danov, D.S. Valkovska, I.B. Ivanov, J. Colloid Interf. Sci. 211 (2)

(1999) 291–303.[10] S.I. Karakashev, A.V. Nguyen, Colloid Surf. A 293 (2007) 229–240.[11] E. Manev, R. Tsekov, B. Radoev, J. Dispersion Sci. Technol. 18 (6/7) (1997)

769–788.[12] E. Manev, A. Sheludko, D. Ekserova, Colloid Polym. Sci. 252 (7/8) (1974)

586–593.[13] D. Manev Emil, Ann. Univ. Sofia, Fac. Chem. 75 (1985) 174.[14] B. Radoev, A. Scheludko, E. Manev, J. Colloid Interf. Sci. 95 (1) (1983)

254–265.

[15] E.D. Manev, S.V. Sazdanova, D.T. Wasan, J. Colloid Interf. Sci. 97 (2)(1984) 591–594.

[16] R. Tsekov, Colloids Surf. A 141 (2) (1998) 161–164.[17] R. Tsekov, E. Evstatieva, Prog. Colloid Polym. Sci. 126 (2004) 93–96.[18] B. Deryagin, L. Landau, Acta Physicochim. U.R.S.S. 14 (1941) 633–

662.[19] E.J.W. Verwey, J.T.G. Overbeek, Theory of the Stability of Lyophobic

Colloids, Elsevier, Amsterdam, 1948, p. 218.[20] A.V. Nguyen, G.M. Evans, G.J. Jameson, Approximate calculations of

electrical double-layer interaction between spheres, in: A.T. Hubbard (Ed.),Encyclopedia of Surface and Colloid Science, Marcel Dekker, New York,2002.

[21] A.V. Nguyen, G.M. Evans, G.J. Jameson, J. Colloid Interf. Sci. 230 (1)(2000) 205–209.

[22] A.V. Nguyen, H.J. Schulze, Colloidal Science of Flotation, Marcel Dekker,New York, 2004, p. 840.

[23] J.N. Israelachvili, et al., Colloids Surf. 2 (3) (1981) 287–291.[24] J. Israelachvili, R. Pashley, Nature (London). 300 (5890) (1982) 341.[25] Y.I. Rabinovich, B.V. Deryagin, Colloids Surf. 30 (3–4) (1988) 243–

251.[26] K. Fa, A.V. Nguyen, J.D. Miller, J. Phys. Chem. B 109 (27) (2005)

13112–13118.[27] J.C. Eriksson, S. Ljunggren, P.M. Claesson, J. Chem. Soc., Faraday Trans.

2. 85 (3) (1989) 163–176.[28] B.W. Ninham, K. Kurihara, O.I. Vinogradova, Colloid Surf. A 123–124

(1997) 7–12.[29] H.K. Christenson, P.M. Claesson, Adv. Colloid Interf. Sci. 91 (3) (2001)

391–436.[30] S. Tchaliovska, et al., J. Colloid Interf. Sci. 168 (1) (1994) 190–197.[31] D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes

and Rheology, Butterworth-Heinemann, Boston, 1991, p. 558.[32] A.V. Nguyen, J. Colloid Interf. Sci. 231 (1) (2000) 195.[33] J.H. Mahanty, B.W. Ninham, Colloid Science: Dispersion Forces, Aca-

demic Press, London, 1977, p. 236.[34] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press,

London, 1992, p. 291.[35] A.V. Nguyen, J. Colloid Interf. Sci. 229 (2) (2000) 648–651.[36] P.G. De Gennes, Adv. Colloid Interf. Sci. 27 (3/4) (1987) 189–209.[37] R.M. Pashley, J. Colloid Interf. Sci. 80 (1) (1981) 153–162.[38] J. Israelachvili, H. Wennerstrom, Nature 379 (6562) (1996) 219–225.[39] K. Katsov, J.D. Weeks, Phys. Rev. Lett. 86 (3) (2001) 440–443.[40] J. Dzubiella, H. Lowen, C.N. Likos, Phys. Rev. Lett. 91 (24) (2003)

248301/1–1248301/4.[41] A.V. Nguyen, et al., Int. J. Miner. Process. 72 (2003) 215–225.[42] R. Tsekov, H.J. Schulze, Langmuir 13 (21) (1997) 5674–5677.[43] D. Exerowa, P.M. Kruglyakov, Foam and Foam Films: Theory, Experiment,

Application, Marcel Dekker, New York, 1997, p. 796.[44] D.O. Shah, N.F. Djabbarah, D.T. Wasan, Colloid Polym. Sci. 256 (10)

(1978) 1002–1008.[45] A. Harvey Paul, et al., Miner. Eng. 18 (2005) 311.[46] K. Kubota, G.J. Jameson, J. Chem. Eng. Jpn. 26 (1) (1993) 7–12.[47] S. Karakashev, R. Tsekov, E. Manev, Langmuir 17 (17) (2001) 5403–

5405.

Documents

Effect of double-layer repulsion on foam film drainage