Effect of sodium dodecyl sulphate and dodecanol mixtures on foam film drainage: Examining influence of surface rheology and intermolecular forces

Colloids and Surfaces A: Physicochem. Eng. Aspects 293 (2007) 229–240

Effect of sodium dodecyl sulphate and dodecanol mixtures on foam film drainage: Examining influence of surface rheology and intermolecular forces Stoyan I. Karakashev, Anh V. Nguyen ∗ Discipline of Chemical Engineering, School of Engineering, The University of Newcastle, Callaghan, New South Wales 2308, Australia Received 21 December 2005; received in revised form 17 July 2006; accepted 24 July 2006 Available online 3 August 2006

Abstract Intermolecular forces and surface rheology of adsorbed surfactants can jointly influence the drainage and stability of foam films significantly, but have been separately considered in the literature. Here we use mixtures of sodium dodecyl sulphate and dodecanol to produce different surface shear viscosities and to elucidate the role of intermolecular forces and surface rheology in foam film drainage. The thickness of plane parallel foam films with very small radius versus time was obtained with an improved Scheludko cell set-up used in conjunction with a high-speed video camera system operating at a frequency of 1000 frames per second. The surface tension and surface potential were measured by pendant bubble tensiometry and micro-electrophoresis, respectively. The surface shear viscosity was determined by a deep channel surface shear viscometer. The modelling considered surface tension gradient, surface shear viscosity and diffusion, and DLVO and non-DLVO intermolecular forces. Comparison between experimental data and theoretical models for foam film drainage shows that the film drainage is influenced by the Marangoni effect due to surface tension gradient, producing foam films with immobile interfaces. Intermolecular forces, in particular, non-DLVO hydrophobic attractive forces, have the most significant effect on the film drainage. © 2006 Elsevier B.V. All rights reserved. Keywords: Foam film drainage; Surface rheology; DLVO forces; Hydrophobic forces; Marangoni effect; Shear viscosity; SDS and dodecanol

1. Introduction The liquid drainage in foam films has been intensively investigated during the last four decades. One of the pioneers in this area was Scheludko [1] who applied the Stefan–Reynolds equation [2] to describe the film drainage. He considered plane parallel rigid surfaces of the film due to the Marangoni effect produced by the surface tension gradient. This assumption has been validated [3–5] for foam films with very small radii (smaller than 0.1 mm). Radoev et al. [6] developed the theory further by considering mobile film surfaces. They obtained the exact analytical solution of the equation for the surfactant mass balance inside the film. According to this model, the velocity on the film surfaces is a result of the combined effects of the liquid outflow, Marangoni effect, and the surfactant adsorption.

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0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.07.047

Later, Radoev et al. [7] included the effect of the surface diffusion on the interfacial velocity. However, for the sake of simplicity they replaced the exact analytical solution of the film mass balance equation by an approximate solution. This approximation has been well accepted in the further theoretical models. The additional development of the theory [8–10] included the effect of the surface shear viscosity on the film drainage. However, the authors were not able to obtain the exact analytical solution of the boundary equation for surface stress and approximate models for the surface viscosity effect were obtained. Manev and Nguyen [11] observed experimentally that the increase in the film radius correlates with the film thickness fluctuations, which accelerate the film drainage. Analytical solutions for the limiting cases of small and large deformation of the bubble caps were also obtained [12]. This model works with an interpolated equation for the shape of the film surfaces and neglects the effect of surface viscosity. The influence of the thickness fluctuations on the film drainage has been modelled by Manev et al. [3] and Tsekov [5,13].

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Nomenclature ak a˜ A A0 b cel C C0 D Ds E f F h hs h ¯ I Imin Imax J0 J1 l n n0 p Pσ q r r˜ Rg R T u U v V VRe z

Fourier–Bessel coefficients adsorption length, a˜ = (∂Γ/∂C)0 Hamaker–Lifshitz function zero frequency term of the Hamaker–Lifshitz function bulk diffusivity number in Eq. (4) electrolyte concentration surfactant concentration in the film solution surfactant concentration in the bulk solution diffusion coefficient of surfactant in the bulk solution surface diffusion coefficient for surfactant Gibbs elasticity, E = −(∂σ/∂ ln Γ )0 function used for calculation of the electrostatic disjoining pressure with Eq. (8) Faraday number or driving force for film drainage film thickness characteristic surface diffusion length Planck constant divided to 2␲ instantaneous intensity of the photocurrent; minimal intensity of the photocurrent maximal intensity of the photocurrent Bessel function of the first kind and the zero order Bessel function of the first kind and the first order order of interference fringes characteristic refractive index of water in the UV region refractive index of the film liquid pressure inside the film capillary pressure = 1.185, see Eq. (7) radial coordinate Fresnel reflection coefficient universal gas constant film radius absolute temperature radial liquid velocity surface velocity axial liquid velocity or speed of light velocity of film drainage Stefan–Reynolds velocity of film drainage, Eq. (2) axial coordinate

Greek symbols αk defined by Eq. (17) defined by Eq. (23) βk Δ = (I − Imin )/(Imax − Imin ) Γ surfactant adsorption excess (density) Γ0 surfactant adsorption excess at equilibrium κ Debye constant λ wavelength of the monochromatic light or characteristic wavelength of retarded van der waals interaction

λk μ μs

el

vdW

h π σ ε0 ω ψ

root of the Bessel function of the first kind and the zero order bulk viscosity surface viscosity disjoining pressure disjoining pressure due to electrostatic double layers Disjoining pressure due to van der waals interaction disjoining pressure due to hydrophobic effect or interaction = 3.1415926 surface tension static dielectric constant of the film (water) characteristic absorption frequency of the water in the UV region surface potential

The film surfaces are assumed to be fully immobile in these models. The effect of the surface shear viscosity on the film drainage has been numerically modelled in recent years [14–16]. However, these models have not been experimentally tested. It should be noted that, all the drainage models, excluding Radoev et al.’s model [6], for the effect of the surfactant adsorption on the surface velocity are derived using the approximate solution of the equation for surfactant mass balance inside the film. No comparison between the approximate and exact models obtained from of the mass balance equation has been made. Recently, the drainage theory is refined by precise computation of the intermolecular Derjaguin–Landau–Vervey–Overbeek (DLVO) interaction forces between the film surfaces [11,17,18]. In addition, the theory on non-DLVO interactions forces in thin liquid films underwent significant developments. Of the nonDLVO interactions, the hydrophobic attraction appears to be the most significant but least understood and has been intensively investigated [19–24] over the last two decades. Recently, the role of the hydrophobic disjoining pressure in equilibrium and dynamic (thinning) foam films has attracted the attention of many researchers [25–27]. However, these studies focus on the hypotheses and mechanisms of the hydrophobic attraction. The phenomenological thermodynamics approach may be convenient for describing the hydrophobic interaction [24,28]. No precise quantitative theory for hydrophobic attractive forces has been available. In summary, there are two independent research directions in foam film drainage, dealing with either the precise role of surface rheology or the nature of intermolecular/surface forces. The surface forces have not been examined in detail in the surface rheology research, in which only the DLVO forces are usually considered. The effects of surface rheology, such as surface viscosities, have been neglected in the surface force research by considering, for example, fully immobile film surfaces. The non-DLVO forces, in particular, the hydrophobic disjoining

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pressure, are often inferred from the film drainage data, which can be influenced by the surface rheological properties. It is, therefore, our desire to investigate how surface rheology and surface forces jointly influence foam film drainage. Aqueous solutions with mixtures of sodium dodecyl sulphate (SDS) and dodecanol are known to produce different surface viscosities as a function of the mass ratio and composition of the two components, and are well examined theoretically and experimentally [29–37]. In this study, mixtures of sodium dodecyl sulphate and dodecanol at 100:1 mass ratio were used to change the surface shear viscosity on the film surfaces by varying the total bulk concentration at the given mass ratio, while keeping minimum changes in surface tension. The foam films were produced with radii smaller than 0.1 mm, rendering the plane parallel film surfaces. The analytical drainage models for plane parallel films with surface rheology and accurately predicted DLVO surface forces have been used to describe the experimental data. The hydrophobic disjoining pressure [28] is accounted for in the refined drainage model. Better agreement between the theory and experiment has been achieved, indicating the important role of hydrophobic interaction forces in foam film drainage. 2. Experimental The thinning of foam films with radii R = 0.075 mm was recorded using a computerized micro-interferometric system. The foam film surfaces were plane parallel. The experimental set-up is schematically shown in Fig. 1 and consists of the following major parts: 1. A Scheludko cell made from glass for producing the foam films [1]. 2. A metallurgical inverted microscope (Epihot 200, Nikon, Japan) for illuminating and observing the film with the reflected light and interference fringes (the Newton rings). 3. A high-speed video camera (Phantom 4, Photo-Sonics Inc., USA) for capturing the transient interferometric images.

Fig. 1. Schematic of the micro-interferometric apparatus with an Scheludko cell, a metallurgical inverted microscope, and a high-speed video camera system.

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4. A personal computer for controlling the high-speed video camera system and recording the transient interferometric images. A double concave drop of the investigated solution was formed inside and attached to the vertical glass tube (the film holder), the inner radius of which was 4 mm. The film holder was connected to the capillary tube with a gastight micro-syringe for regulating the amount of liquid inside the concave drop. When the liquid inside the drop was pumped out, a microscopic film between the surfaces of the concave drop was formed (see Fig. 1). The radius of the film was dependent on the quantity of the liquid withdrawn. The film was illuminated by monochromatic light with the wavelength λ = 546 nm. As a result of the interference of the light reflected by the film surfaces, a set of black and white fringes (the Newton rings) was observed and recorded in the computer using the high-speed video camera system. The rings appeared as a result of the interferometric minima and maximums, corresponding to the film thickness λ/4n0 , where n0 is the refractive index of the film liquid. The transient interferometric images were usually recorded within 10 ms or 1 ms time intervals. The interferometric images were digitalized using the Optimas program (Optimas 6.1, Optimas Inc., USA). A narrow strip passing through the centre of the interferometric fringes was chosen and the digital signals were converted into photocurrent versus radial distance using a special macro developed by us in Optimas 6.1. The photocurrent between the interferometric maxima was quite smooth and without significant variation, which corresponded to the plane parallel films. The thickness of the film was calculated using the Scheludko interferometric equation, which accounts for the light interference by multiple reflections by both surfaces [1,24] and is described as ⎡ ⎤  2 λ ⎣ Δ(1 + r˜ ) ⎦ h= lπ ± asin (1) 2πn0 (1 − r˜ 2 ) + 4˜r 2 Δ In this equation r˜ = (n0 − 1)2 /(n0 + 1)2 is the Fresnel reflection coefficient for the air-solution interface, l = 0, 1, 2, 3 . . . is the order of the interference, Δ = (I − Imin )/(Imax − Imin ), I is the instantaneous intensity of the photocurrent, Imin and Imax are its minimal and maximal values. The true film thickness can be smaller than the thickness determined by Eq. (1) which is referred to as the equivalent film thickness. The difference is due to the surfactant adsorption layers at the film surfaces. However, for the range of film thickness studied in this paper, the difference was not significant and no correction to Eq. (1) has been made. The foam films were produced from aqueous solutions of SDS (Fluka Inc.) and dodecanol (Fluka Inc.) mixtures with 100:1 mass ratio. SDS was purified by four times re-crystallization with ethanol. The purity of SDS was tested by measuring the surface tension isotherm by the pendant bubble method, which showed no minimum and was compared with the literature data [38]. Three different mixtures with 3.5 mM SDS plus 10 mg/L dodecanol, 7 mM SDS plus 20 mg/L dodecanol, and 8.75 mM

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SDS plus 25 mg/L dodecanol were selected, which gave three distinctive values of 0.18, 0.09 and 0.03 surface poise (s.p.) for surface shear viscosity, as determined by a deep-channel surface viscometer [34,39]. The room temperature was kept constant above the melting temperature for dodecanol (∼22 ◦ C). The surface tension of the surfactant solutions versus concentration was measured by the pendant bubble method (to minimise evaporation taking place in the pendent drop method) using the optical contact measuring device (DataPhysics, Germany). The properties of the mixed adsorption layers were determined by the available modelling [36]. The surface potential of the airsolution interface was measured with a Micro-Electrophoresis Apparatus MK II (Brookhaven, UK) using micro bubbles with ∼20 ␮m in diameter. 3. Theoretical models for foam film drainage For plane parallel immobile (non-slip) surfaces, the foam film drainage theory can be described by the Stefan–Reynolds equation [1,2] as VRe =

2h3 (Pσ − Π) 3μR2

(2)

where VRe = −dh/dt described the velocity of drainage predicted by the Stefan–Reynolds lubrication theory, h is the film thickness, μ is the liquid viscosity in the bulk, R is the film radius, Pσ and are the capillary and disjoining pressures, respectively. If one of the film surfaces is fully mobile and the other surface is fully immobile, the Stefan–Reynolds lubrication approximation with slip boundary conditions can be applied, giving [24] dh −8h3 = (Pσ − Π) dt 3μR2

(3)

is the Gibbs elasticity defined as E −(∂σ/∂ln Γ )0 , where σ and Γ with the subscript 0 describe the surface tension and the surface excess of adsorbed surfactant at equilibrium, respectively. Eq. (4) was also obtained using series expansions [9]. The second term on the left hand side of Eq. (4) describes the surface mobility of foam films which depends on, through the Gibbs elasticity and the adsorption length, the surfactant adsorption mechanisms. Eq. (4) explicitly shows that, due to surface diffusion, the surface mobility of foam films must depend on the film thickness. The dependence has been validated by experiments and was recently reviewed [11]. Surface viscosities are important in the surface rheology of adsorbed surfactants and have been shown to be significant in the liquid drainage in the plateau borders of foams [40–44]. To fully account for the effect of surface shear viscosity on foam film drainage, a new drainage equation was developed in the Appendix A. The mass balance and tangential stress boundary equations are analytically solved, giving Eq. (30) for the velocity of film drainage, which can be rewritten in terms of the Marangoni (Ma), Boussinesq (Bo), and diffusivity (N) dimensionless numbers as ∞

˜ k) 6/λ4k − (h/λ VRe = 1 − 32 2 V 6 + Boλ h˜ + Ma k=1

2

k

h˜ ˜ 1+(N/λk ) tanh (λk h/2)

(5)

where h˜ ≡ h/R, Ma ER/(Ds μ), Bo μs /(Rμ), N = (D/DS )(R/˜a), and λk is the kth root of the Bessel function first kind and zero order. It is noted that since

∞ of the −4 = 1/32 and tanh(x) = x + O(x3 ), Eq. (5) simplifies λ k k=1 ˜ k  1. Eq. (5) presents to Eq. (4) in the limit as Bo = 0 and h/λ the general drainage equation for foam films which accounts for the Marangoni effect, and the effects of surface shear viscosity and diffusion. 4. Intermolecular interaction and surface forces

The velocity of wetting film drainage with fully mobile (plane parallel) surfaces is therefore four times higher than the Reynolds velocity described by Eq. (2) for fully immobile surfaces. The velocity of foam films with fully mobile surfaces is (at least theoretically) infinitely large. The drainage velocity of real foam films has a finite magnitude between the velocities for the two special cases of fully immobile and mobile foam film surfaces. In practice, foam film surfaces can be partially mobile due to a number of surface rheological factors which include surface tension gradient, surface diffusion, and surface viscosities. The effects of surface tension gradient and surfactant diffusions inside the film and at the film surfaces were considered by Radoev et al. [7] who obtained the following prediction for the film drainage velocity:   hs −1 V = VRe 1 + b + (4) h

Another critical issue in modelling thin film drainage is the theoretical description of the disjoining pressure. Approximations are used very often. For example, the van der waals disjoining pressure is determined for the electrodynamic non-retarded interaction only [45]. The screening effect due to electrolytes is also often neglected. These forces are now determined using advanced theories. The van der waals interaction between surfaces is well described in many books, including [46,47]. The interaction can be determined using either the Hamaker approach or the Lifshitz approach. For the van der Waals disjoining pressure, Π vdW , between the foam film surfaces as a function of the film thickness, h, combining the two approaches gives  A(h, κ) A(h, κ) d 1 dA(h, κ) =− − + ΠvdW = − 2 3 dh 12πh 6πh 12πh2 dh (6)

where VRe is described by Eq. (2), b ≡ 3μD/(˜aE) is the bulk diffusivity number, hs 6μDs /E is the surface diffusion length. Here, D and Ds are the coefficients of the bulk and surface diffusion, respectively, a˜ ≡ (dΓ/dC)0 is the adsorption length, and E

where A(h, κ) is the Hamaker–Lifshitz function, which is in fact a weak function of temperature T and the Debye constant, κ, as well as a weak function of the film thickness, h, due to the “electromagnetic retardation effect” that follows from the

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limited velocity of propagation of the dispersion interactions. The term in the brackets in Eq. (6) is the van der waals interaction energy per unit area of the film surface. The influence of electrolytes on the van der waals interaction is mainly via the screening effect of the electrolytes on the zero-frequency term of the Hamaker–Lifshitz constant, A0 , which is essentially due to an electrostatic interaction, resulting in the decrease of A0 by a factor of (1 + 2κh)exp(−2κh). Consequently, the Hamaker–Lifshitz function for the van der waals interaction in foam films is described by [24] 2

3¯hω (1 − n2 ) A0 + √ A(h, κ) = (1 + 2κh)e 16 2 (1 + n2 )3/2   q −1/q h × 1+ λ −2κh

(7)

where A0 is a function of the Boltzmann constant, kB , absolute temperature T, and

dielectric constants of the film, ε0 , described 2j 3 as A0 = (3kB T/4) ∞ j=1 (1 − ε0 /1 + ε0 ) /j . For foam films, ε0 = 80, and the infinite sum in the equation for A0 is equal to 1.444, giving A0 = 1.083kB T. The second term in Eq. (7) describes the non-zero frequency term of the Hamaker–Lifshitz function, where h ¯ = 1.055 × 10−34 Js/rad is the Planck constant (divided by 2π), ω is the absorption frequency in the UV region–typically around 2.068 × 1016 rad/s for water, n2 = 1.887 is the square of the characteristic refractive index of the film (water) in the UV region [48], and q = 1.185. The characteristic wavelength, λ, is measured in the units of length and described by λ ≡ (v/π2 ω) 2/(n2 + n4 ) = 5.59 nm, where v is the speed of light and ω is converted to the unit of 1/s by dividing by 2π. Eq. (6) covers both the electromagnetic non-retarded and retarded van der waals interactions in the limits as h → 0 and h → ∞, respectively. For symmetrical foam films, vdW is negative, which corresponds to attraction. Electrostatic disjoining pressure, el , arises in thin films from dilute electrolyte solutions and is due to the overlapping of the diffuse electric layers on the two film surfaces at small separation distances and can be determined using the Poisson–Boltzmann equation which is highly non-linear. Consequently, approximate expressions for el are frequently used. The Debye–H¨ucker linearisation is limited by the low surface potentials (lower then 50 mV). For high surface potentials, ψ, the superposition approximation is often used, giving

el (h) = 64cel Rg T tanh2 (y0 /4)exp(−κh) [24], where Rg is the universal gas constant, and cel is the molar concentration of electrolytes in the solution. The normalised surface potential, y0 , is defined as y0 ≡ zFψ/(Rg T), where F is the Faraday constant. The Debye constant, κ, which for binary electrolyte with valence z is defined as κ ≡ {2cel F2 z2 /(εε0 Rg T)}1/2 , where ε is the permittivity of vacuum. The superposition approximation is applied for the moderate overlapping of the diffuse double layers, i.e. κh > 2 [47]. For the strong overlapping of the diffuse double layers at small film thickness the calculation of el depends on the surface charging mechanism occurring during the interaction. For the double layer interaction under the condition of constant surface potentials, the exact numerical solution to the non-linear

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Poisson–Boltzmann equation can be semi-analytically described as [18] y  0 Πel (h) = 32cel Rg T tanh2 4  1 y0 × + f (y0 )sinh2 exp[−f (y0 )kh] 1 + coshκh 4 (8) where function f(y0 ) is defined, for |y0 | ≤ 7, as f(y0 ) = 2 cosh(0.332|y0 | − 0.779). Eq. (8) is the most general solution for the double-layer interaction under the condition of constant surface potential, which reduces to a number of predictions in the limit as y0  1. For the double-layer interaction at constant surface charges, the exact numerical solution to the non-linear Poisson–Boltzmann equation gives [18] Πel (h) =

2cel Rg TA

sinh (κhC) 1 + B2 coth(κhC) 2

(9)

where A, B and C are a function of the surface potential, y0 , at infinite separation (i.e. isolated surfaces) described as A = BC(1.610|y0 | + 0.0857|y0 |2 + 0.0310|y0 |3 + 0.00837|y0 |4 ), B = 764.146|y0 |/1253.603 + |y0 | exp (4.468|y0 |0.320 ), C = 0.5 − 0.00808|y0 |. These approximations were developed for |y0 | ≤ 7 and are slightly different from those given in [18]. For foam (symmetrical) films el is always positive (corresponding to repulsion between the surfaces), but strongly depends on the ionic strength of the medium. el present at low electrolyte concentration may oppose the film thinning, while the further addition of electrolyte suppresses the effect. The van der waals and double-layer disjoining pressures presents the intermolecular interactions of the classical DLVO theory of stability of lyophobic colloids [49,50]. Importantly, a number of non-DLVO disjoining pressures have been identified and can have significant effects on film drainage. For example, the hydrophobic interaction can accelerate considerably the film drainage. The nature of this force is obscure at present although it has been intensively investigated in the last decades. Many explanations are available in the literature [22,27,28,51–55]. Numerous researchers have used the surface force apparatus, e.g. [56] or the atomic force microscope, e.g. [55,57–59] to measure the surface forces between hydrophobic surfaces. The hydrophobic force has been determined by subtracting the DLVO (van der waals and double-layer) forces from the measured force. The empirical models by exponential decays are often employed to describe the hydrophobic force data. Similar empirical equation for hydrophobic disjoining pressure is used in this paper and can be described as Πh = K exp(−h/λ)

(10)

where λ is the decay length and K is the (negative) pressure constant. The experimental data for the empirical constants are reviewed in [24]. Sometimes, the double exponential function with two force constants and two decay lengths are used to describe the long-ranged hydrophobic interaction. Alternatively, the measured hydrophobic interaction energy can be described

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by the power function, 1/h3 , similar to those for the non-retarded van der waals interaction [25,26,60–62]. The sum of the van der waals, double-layer and hydrophobic disjoining pressures gives a good approximation for the total disjoining pressure, Π, between the foam film surfaces in the equations for foam film drainage. The van der waals and double-layer disjoining pressures can also be summed to give the DLVO disjoining pressure, which will also be used to predict the transient thickness of foam films and to demonstrate the role of the non-DLVO hydrophobic interaction in foam film drainage. 5. Results and discussion The experimental results for surface tension of the mixtures of sodium dodecyl sulphate and dodecanol at 100:1 mass ratio are shown in Fig. 2. The surface tension has a minimum in the vicinity of the critical micelle concentration (CMC), which is at about 8 mmol/L (mM) for SDS and is characteristic for the mixed SDS and dodecanol adsorption layers. The parameters of the surfactant adsorption can be obtained from the experimental data by employing appropriate adsorption models. The adsorption of the mixtures was modelled following the available approach [36]. Briefly, the surface tension was calculated using the Gibbs adsorption equation. The correlations between the surfactant surface excess and the bulk concentration were predicted using the van der waals isotherms. The counter ion binding was accounted for by employing the Stern layer model. The distributions of the SDS surfactant and counter ions were determined from the Poisson–Boltzmann equation. The difference between the experimental data and model for surface tension was minimised using the non-linear regression analysis by changing the model parameters which included the adsorption constants, adsorption energies, and molecular interaction constants. The agreement between the experimental data and the model prediction (solid line) for surface tension is shown in Fig. 2. The Gibbs elasticity and the adsorption length determined by the model and the experimental data for surface tension are shown in Figs. 3 and 4.

Fig. 3. Gibbs elasticity, E = −(∂σ/∂ln Γ ), for mixtures of SDS and dodecanol at 100:1 mass ratio, as predicted using the data shown in Fig. 2.

Fig. 2. Experimental (points) and theoretical (line) results for surface tension versus sodium dodecyl sulphate (SDS) concentration for mixtures of SDS and dodecanol at 100:1 mass ratio.

Fig. 5. Experimental data (points) for the zeta potential of micro air bubbles in mixtures of SDS and dodecanol at 100:1 mass ratio. The maximum standard deviation is about 12 mV. The solid line presents a polynomial regression.

Fig. 4. Adsorption length, a˜ = dΓ/dC, for mixtures of SDS and dodecanol at 100:1 mass ratio, as predicted using the data shown in Fig. 2.

The experimental results for the zeta potential of micro air bubbles in the mixtures of SDS and dodecanol are shown in Fig. 5. The zeta potential has a minimum at the concentration (∼5.5 mmol/L) corresponding to the point of the saturation adsorption density. It is assumed that the zeta potentials are not

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Fig. 6. Comparison between experiment (points) and theory (lines) for transient thickness of a foam film containing 3.5 mM SDS + 10 mg/L dodecanol (μs = 0.18 surface poise and R = 0.075 mm). The line shows the drainage models described by Eqs. (2), (4) and (5). The DLVO disjoining pressures described by Eqs. (6) and (8) are used in the model predictions.

significantly different from the surface potentials of the adsorption layers and can be used to replace the surface potentials in calculating the double-layer disjoining pressure. Shown in Fig. 6 is a comparison between the experimental results and theoretical predictions for the film thickness versus time, obtained for a foam film containing 3.5 mM SDS + 10 mg/L dodecanol. The surface shear viscos-

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Fig. 7. Surface shear viscosity of mixtures of SDS and dodecanol at 100:1 mass ratio as determined by a deep channel surface shear viscometer [34,39].

ity obtained by a deep channel viscometer is about 0.18 s.p. (0.18 × 10−3 kg/s) (Fig. 7). Only the DLVO disjoining pressures as described by Eqs. (6) and (8) are used in the model predictions shown in Fig. 6. The drainage equations are described by the differential equations of the first-order for thickness, h, versus time, t. The differential equations were numerically solved using the fourth-step Runge–Kutta algorithm. A macro was written using the Visual Basic for Application (VBA) programming language available in Microsoft Excel. The inputs for the numerical integration included the capillary pressure, Pσ , and the Marangoni

Fig. 8. Transient images of a small foam film confirming the uniform film thickness during drainage (3.5 mM SDS + 10 mg/L dodecanol, R = 0.075 mm, and times: 0, 10, 20 and 30 ms).

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number (Ma), Boussinesq number (Bo) and diffusivity number (N). These dimensionless numbers were calculated using the parameters of the surfactant adsorption shown in Figs. 2–4. The surface potential used in the calculation is shown in Fig. 5. The disjoining pressure at the constant surface charge, as described by Eq. (9), was also considered in the numerical integration of the drainage equations. However, the theoretical predictions under the condition of constant surface charge are not significantly different from those calculated under the condition of constant surface potential shown in Fig. 6. The coincidence is due to the fact that the drainage models predict very thick film thickness for which the two charging mechanisms at the film surfaces produce similar net forces for the double-layer interactions. Difference between the two models can be expected for thin films. For the diffusion coefficients, the values of 4 × 10−10 m2 /s [63,64] and 4 × 10−9 m2 /s [11,65] were estimated for D and Ds , respectively. These numerical values were used in all calculations if not described otherwise. The capillary pressure was obtained from the solution of the Young–Laplace equation under the condition of constant pressure [66], giving Pσ = 2σRc /(R2c − R2 cos θ), where Rc = 2 mm is the radius of the glass cell (the film holder) and θ is the “contact” angle of the inclination of the free meniscus at the film periphery. Since the film radius, R, is significantly smaller than Rc , the effect of the contact angle and R on the capillary pressure is very small and can be neglected. The small films with radii smaller than 100 microns are plane parallel (Fig. 8). Fig. 6 shows that the DLVO theory for intermolecular interactions between foam films surfaces cannot adequately describe the film drainage. Similar results obtained with foam films containing 7 mM SDS + 20 mg/L dodecanol or 8.75 mM SDS + 25 mg/L dodecanol are shown in Fig. 9. Similar observations have also been reported for the SDS foam films [25,26]. The extended DLVO theory by incorporating hydrophobic interactions between the foam film surfaces has been applied to the modelling using Eq. (10). The parameters for the hydrophobic disjoining pressure are not available and were obtained in this paper by a non-linear regression analysis, which fitted the theoretical predictions with the experimental data by minimising the sum of the squares of the differences between the model and experimental data. The minimisation changed the parameters K and λ in Eq. (10). The final fitting results are presented in Fig. 10. The regression analysis was carried out by considering the double-layer interaction at constant surface potential. The fitting with the extended DLVO theory was also carried out using the double-layer interaction at constant surface charge which is described by Eq. (9). The fitting results are summarised in Table 1. The extended DLVO theory significantly improves the Table 1 Parameters for hydrophobic interaction obtained by best fit using Eqs. (5) and (10)

3.5 mM SDS + 10 mg dodecanol 7 mM SDS + 20 mg dodecanol 8.75 mM SDS + 25 mg dodecanol

Constant potential

Constant charge

λ (nm)

λ (nm)

K

(N/m2 )

15.644 −11244.6 17.645 −6727.7 18.352 −15244.2

K (N/m2 )

14.887 −14200.5 17.288 −7298.5 18.429 −13715.0

Fig. 9. Comparison between experiment (points) and predictions (lines) by DLVO theory for transient thickness of a foam film containing 7 mM SDS + 20 mg/L dodecanol (μs = 0.09 s.p. and R = 0.075 mm) and 8.75 mM SDS + 25 mg/L dodecanol (μs = 0.03 s.p. and R = 0.075 mm). The lines show the drainage models described by Eqs. (2), (4) and (5). The DLVO disjoining pressures are described by Eqs. (6) and (8) or (9).

theoretical prediction for the foam film drainage of the surfactant mixtures. It is noted that dodecanol could be a good candidate to generate hydrophobic interactions and experiments with dodecanolfree SDS solutions were repeated. The film drainage data obtained with the SDS solutions were similar to the results obtained with the surfactant mixtures: the hydrophobic attraction was needed to describe the drainage data and the hydrophobic attraction was observed to decrease with increasing SDS concentration. At relatively high SDS concentration but still below the CMC, the hydrophobic attraction was not needed to describe the drainage data and agreement with the classical DLVO theory was obtained (Fig. 11). A similar observation was reported by Wang and Yoon [25]. The effect of surface shear viscosity has been neglected in foam film studies. It is noted that surface shear viscosity is one of many parameters influencing the mobility of foam film surfaces, including the surface tension gradient, the diffusion in the solution phase and at the surface, and the adsorption excess gradient. These parameters can be grouped to produce three important dimensionless numbers, namely, the Marangoni number, the Boussinesq number, and the diffusivity number, which

S.I. Karakashev, A.V. Nguyen / Colloids and Surfaces A: Physicochem. Eng. Aspects 293 (2007) 229–240

237

Table 2 Adsorption parameters and dimensionless numbers obtained for the surfactant mixtures (assuming D = 4 × 10−10 m2 /s and Ds = 4 × 10−9 m2 /s) μs (s.p.) 3.5 mM SDS + 10 mg dodecanol 7 mM SDS + 20 mg dodecanol 8.75 mM SDS + 25 mg dodecanol

0.18 0.09 0.03

E (N/m)

a˜ (m)

0.0521 0.0816 0.0396

9.52 × 10−8 3.32 × 10−8 2.00 × 10−8

Bo

Ma

N

2400 1200 400

9.77 × 105

79 226 375

1.53 × 106 7.43 × 105

Fig. 11. Agreement between the experimental data (points) for transient thickness of a SDS film and prediction (line) with the classical DLVO theory (R = 0.077 mm).

are incorporated into the new drainage theory described by Eq. (5). The magnitudes of the dimensionless numbers obtained for the foam films produced using the mixtures of SDS and dodecanol are shown in Table 2. At present, the assessment of the effect of the parameters on foam film drainage is difficult because the coefficient of the surface diffusion is not precisely known and is often indirectly inferred from the film drainage experiments [11,65]. Some available data for the coefficient of surface diffusion have been obtained for common and Newton black films [67–69], where the surface mobility of adsorbed molecules is decreased due to high adsorption density. Indeed, it has been suggested that the coefficient of surface diffusion depends on the surface excess of adsorbed surfactants. The coefficient of surface diffusion can be higher than 4 × 10−9 m2 /s chosen in the modelling shown in Figs. 6–9. If Ds = 4 × 10−8 m2 /s is chosen

Fig. 10. Comparison between experimental data (points) for transient thickness of the foam films and predictions (lines) by the extended DLVO theory with the inclusion of the hydrophobic disjoining pressure described by Eq. (10). The numerical values for K and λ were obtained by the best fit using the drainage Eq. (5) and Eqs. (6), (8), and (10).

Fig. 12. Effect of surface diffusion, Ds , on the drainage mobility, V/VRe , of a foam film containing 8.75 mM and 25 mg dodecanol (μs = 0.03 s.p.). Thick and thin lines describe Eqs. (5) and (4), respectively. For Ds = 4 × 10−9 m2 /s, Ma = 7.43 × 105 and N = 375. For Ds = 4 × 10−8 m2 /s, Ma = 7.43 × 104 and N = 38. For both cases, Bo = 400.

238

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Fig. 14. Schematic of foam film in the Scheludko cell.

Fig. 13. Experimental and modelled results for transient thickness for a foam film containing 8.75 mM and 25 mg dodecanol. The model prediction is based on the same equations and parameters used to produce Fig. 10, except for the coefficient of surface diffusion.

[11], a significant increase in the foam film mobility is obtained as shown in Fig. 12. A similar dependence was obtained for nitrobenzene foam films stabilised by various concentrations of dodecanol [11]. It is noted that the Boussinesq number is independent of the coefficient of surface diffusion and remains the same for the two cases shown in Fig. 12. The effect of surface diffusion on the film drainage is shown in Fig. 13. The higher the coefficient of surface diffusion, the more mobile the film surface and the faster the film thins. 6. Conclusions Drainage of foam films of aqueous mixtures of sodium dodecyl sulphate and dodecanol at 100:1 mass ratio has been investigated to determine the role of intermolecular forces and surface rheology in foam film drainage. The transient thickness of plane parallel foam films with very small radius versus time was obtained with an improved Scheludko cell set-up used in conjunction with a high-speed video camera system operating at the frequency of 1000 frames per second. The surface tension and surface potential were measured by pendant bubble tensiometry and micro-electrophoresis, respectively. The surface shear viscosity was determined by a deep channel surface shear viscometer. The modelling considered surface tension gradient, surface shear viscosity and diffusion, and DLVO and non-DLVO intermolecular forces. Comparison between experimental data and theoretical models for foam film drainage shows that the film drainage is influenced by the Marangoni effect due to surface tension gradient, producing foam films with immobile interfaces. If the coefficient of surface diffusion for surfactant molecules is of the order of 1 × 10−8 m2 /s, the film surfaces become mobile, leading to faster drainage. Intermolecular forces, in particular, non-DLVO hydrophobic attractive force, have the most significant effect on the film drainage. Acknowledgement

Appendix A. Drainage model with surface shear viscosity A.1. Velocities of the liquid flow The equations governing the film drainage can be described using a cylindrical co-ordinate system (r, ϕ, z) with the origin located in the middle plane of the film and the z-axis aligned with the axis of the rotational symmetry of the film (Fig. 14). The lubrication approximation of the continuity and Navier–Stokes equations gives the following predictions for the radial, u, and, axial, v, liquid velocities inside the film   1 h2 ∂p u(z, r) = U(r) + z2 − (11) 2μ 4 ∂r  2    h z z3 1 ∂ ∂p z ∂ v(r, z) = − r − (rU) (12) 8 6 μr ∂r ∂r r ∂r where p is the pressure inside the film, ( the bulk liquid viscosity, U the radial velocity of the film surface, and h is the thickness of the (plane parallel) film, which can be determined from the equations for surfactant mass balance inside the film and at the film surfaces. A.2. Solution for surfactant concentration distribution The mass balance inside the film is described by the diffusion equation:   1 ∂ ∂C ∂2 C r + 2 = 0, r ∂r ∂r ∂z which can be solved for the surfactant concentration, C, to give C(r, z) = C0 +

∞

ak J0 (mk r) cosh (mk z)

(13)

k=1

where C0 is the surfactant concentration in the bulk solution, ak are the coefficients, J0 is the Bessel function of first kind and zero order, mk = λk /R, (k is the kth root of J0 ((k ) = 0, and R is the film radius. Eq. (13) can be used to determine the radial gradient of the surfactant surface excess, giving ∞

The authors gratefully acknowledge the Australian Research Council for financial support through a Discovery grant.

∂Γ ∂Γ ∂C = = a˜ ak J0 (mk r) cosh (mk h/2) ∂r ∂C ∂r k=1

(14)

S.I. Karakashev, A.V. Nguyen / Colloids and Surfaces A: Physicochem. Eng. Aspects 293 (2007) 229–240

whereJ0 (mk r) = dJ0 (mk r)/dr. Assuming small deviations from the equilibrium surfactant surface excess, Γ 0 , the adsorption length in Eq. (14) is determined by a˜ = (∂Γ/∂C)0 .

where βk is defined as  2    2h ∂σ mk h 2h2 2 cosh − 4hαk − μ s αk m k βk = ak 3μ ∂C 0 2 3μ

A.3. Radial surface velocity The mass balance at the film surface gives     1 ∂ 1 ∂ ∂Γ ∂C (rΓU) = DS r ±D r ∂r r ∂r ∂r ∂z z=±h/2

(23) Eq. (22) can now be solved for βk employing the orthogonality properties of the Bessel function, leading to (15)

where ( is the surface excess of adsorbed surfactant at film surfaces, D and Ds are the bulk and surface diffusion coefficients of surfactant, respectively. Inserting Eqs. (13) and into Eq. (14) and integrating from the film centre gives U=

∞

αk ak J0 (mk r)

(16)

βk = −

8VR2

(24)

λ3k J1 (λk )

Substituting Eq. (24) into Eq. (23) gives ak =

−4VR2 /{λ3k J1 (λk )} (h2 /3μ)(∂σ/∂C)0 cosh (mk h/2)−2hαk −(h2 /3μ)μs αk m2k (25)

k=1

where αk is described by       Ds ∂Γ λk h λk h RD cosh sinh + αk = Γ0 ∂C 0 2R Γ 0 λk 2R

(17)

Small deviation from the equilibrium adsorption of surfactant is considered in Eq. (17). A.4. Axial surface velocity The stress balance at the film surfaces gives   ∂u ∂ 1 ∂rU ∂σ ±μ = + μs ∂z z=±h/2 ∂r ∂r r ∂r

(19)

The surface tension gradient in Eq. (20) controls the Marangoni effect and can be described as a function of r using Eq. (13), which gives   ∞ ∂σ ∂σ ∂C ∼ ∂σ = ak J0 (mk r) cosh (mk h/2) (21) = ∂r ∂C ∂r ∂C 0 k=1

Eqs. (21) and (16) can be inserted into Eq. (20) for determining V. Integrating the final equation twice yields

k=1

βk J0 (mk r)

(26)

where p∞ is the pressure at the film periphery (in the bulk solution). The velocity derivative in Eq. (26) can be calculated using Eq. (12). Integration of Eq. (26) yields R F = 4πμRU(R) − π

r2 0

A.5. Marangoni effect

∞

The force, F, acting on the film surface can be determined by integrating the normal component of the pressure tensor over the film surface area, giving

0

(18)

The drainage velocity, V, can be obtained by applying Eq. (12) at the film surface, which in conjunction with Eq. (19) gives   h2 ∂ ∂ ∂rU ∂σ h ∂rU V = r + rμs − (20) 6μr ∂r ∂r ∂r r∂r r ∂r

V (r 2 − R2 ) =

A.6. Driving force for film drainage

 R  ∂v F = 2π p (r) − p∞ − 2μ rdr ∂z z=±h/2

where σ is the surface tension and μs is the surface shear viscosity. The pressure gradient obtained from Eqs. (11) and (18) is described as ∂p 2 ∂σ 2μs ∂ 1 ∂rU = + ∂r h ∂r h ∂r r ∂r

239

(22)

∂p dr ∂r

(27)

The pressure gradient and the surface velocity in Eq. (27) can be obtained from Eqs. (19) and (20). One obtains 6πμ F = 4πμRU(R) − 3 h

R (Vr 3 + 2hUr 2 )dr

(28)

0

The surface velocity given by Eq. (16) can be substituted in Eq. (28), leading to ∞

F =−

3πμVR4 − 4πμR αk mk ak J1 (λk ) 3 2h k=1

+

∞ 3

24πμR h2

αk m k a k J1 (λk ) λk 2 k=1

(29)

The driving force for film drainage can also be determined from the capillary pressure, Pσ , and the disjoining pressure, Π, and is described as F = πR2 (Pσ − Π). Substituting Eq. (25) into

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S.I. Karakashev, A.V. Nguyen / Colloids and Surfaces A: Physicochem. Eng. Aspects 293 (2007) 229–240

Eq. (29) and solving for V gives VRe 32 =1+ V 3

∞

αk 6 − (λk h/R)2 λ 4 [h/(3μ)](∂σ/∂C)0 cosh (λk h/2R) k=1 k −2αk − [hμs /(3R2 μ)]αk λ2k

(30)

where VRe is the drainage velocity predicted by the Stefan–Reynolds equation. The second term on the right hand side of Eq. (30) describes deviation from the Stefan–Reynolds theory due to the surface tension gradient (the Marangoni effect), surfactant diffusions and surface shear stress. It is noted that the surface velocity gradient in Eq. (26) is often absent in a number of predictions for the driving force, which are clearly incomplete. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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