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Elasticity of foam bubbles measured by profile analysis tensiometry
Colloids and Surfaces A: Physicochem. Eng. Aspects 369 (2010) 136–140
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Elasticity of foam bubbles measured by profile analysis tensiometry Stoyan I. Karakashev a,∗ , Roumen Tsekov a , Emil D. Manev a , Anh V. Nguyen b a b
Department of Physical Chemistry, University of Sofia, 1 James Bourchier Ave., 1164 Sofia, Bulgaria School of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia
a r t i c l e
i n f o
Article history: Received 23 April 2010 Received in revised form 18 July 2010 Accepted 4 August 2010 Available online 12 August 2010 Keywords: Elastic modulus Gibbs elasticity Foam films Soap bubble Adsorption frequency
a b s t r a c t Elastic modulus of foam bubbles, stabilized with tetraethylene glycol octyl ether (C8 E4 ) and 1 × 10−5 M NaCl, was determined by cyclic expansion and shrinking of foam bubbles with frequency of 0.1 Hz and volumetric amplitude of 2 mm3 . The film tension was monitored by a commercial profile analysis tensiometer (Sinterface Technologies, GmbH). The elastic moduli of foam bubbles were obtained as a function of surfactant concentration in the range of 2 × 10−3 –1 × 10−2 M. The theory of Lucassen and van den Tempel [1] for the elastic modulus of a single liquid/air interface at a given frequency was employed. In the theoretical analysis the bulk diffusion coefficient of surfactant molecules was considered as a unknown model parameter which was obtained by matching the theory with the experimental data. Hence, the dependence of the bulk diffusion coefficient of C8 E4 molecules upon the C8 E4 concentration was obtained. The diffusion coefficient reached a maximum at 5 × 10−3 M C8 E4 (D = 8.5 × 10−11 m2 /s). In the experimental surfactant concentration range (2 × 10−3 –1 × 10−2 M, CMC = 7.5 × 10−3 M) the foam bubbles were relatively dry, with visible interferometric fringes corresponding to thin films stabilized by repulsion of the electrostatic disjoining pressure. Hence, the overall dynamics of periodical expansion and shrinking of the foam bubbles occurred within the thin film state. © 2010 Elsevier B.V. All rights reserved.
1. Background The Gibbs elasticity of thin liquid films (TLF) has been defined by Gibbs in 1876 in his famous work “On the equilibrium of the heterogeneous substances” [2]: EG =
2d d ln A
(1)
where is the surface tension of each film surface with area A of the film. He assumed that the solution in the foam lamella becomes depleted of surfactant as a result of the adsorption on the surfaces, and hence the surface tension increases with the expansion of the lamella (film). The film volume and the total number of surfactant molecules were assumed constant during the film expansion. The first experimental determination of the elastic moduli of foam films was conducted by Mysels et al. [3,4] in rectangular vertical frames, adopted later by Prins et al. [5,6] as well. The method consists of measuring the increased tension of a foam lamella stretched it to a measured extent, with concurrently determining the local film thickness of the lamella from the interference pattern. These first attempts to determine the elastic moduli were critically evaluated in the literature [3,7,8]. The arguments [3] were focused on the
∗ Corresponding author. Tel.: +359 2 8161241. E-mail address: [email protected]fia.bg (S.I. Karakashev). 0927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2010.08.008
dynamic nature of the film elasticity showing that the film dynamics effects should last for milliseconds, while the time-scale of the stretching was of the order of seconds, and this was accompanied by increased values of the film tension. Hence, a different limiting factor – presumably the depletion of surfactant in the intra-lamellar liquid – had to be critical the film [3]. According to Kitchener [8] the latter effect corresponded to the exact definition of the Gibbs elasticity as defined in [2]. This should be valid for thin films in which the electrostatic disjoining pressure was significant, but in many cases the thickness of the foam lamella was of the order of microns, for which the electrostatic disjoining pressure was negligibly weak. The methodology of Mysels et al. [3,4] was limited to relatively stable foam lamellae with high level of surfactant adsorption in the surface layers. In addition, the expanding of the film surfaces was related only to dilatational (not shear) visco-elasticity. The surface shear viscosity of insoluble surfactant monolayers was studied earlier in a series of work [9–11]. In order to study the surface elastic moduli at lower adsorption levels Lucassen and van den Tempel employed the theory of capillary waves on visco-elastic surface adsorption layers [12,13] and introduced a new methodology [1,14], operating with longitudinal surface waves and stress relaxation. The elastic modulus was regarded as a complex value, however the surface viscosity was not considered. The elasticities of the monolayers in Langmuir through [15,16] have been measured directly by determining the changes in surface tension caused by small amplitude sinusoidal compression/expansion cycles. This
S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 369 (2010) 136–140
approach was first applied to insoluble protein monolayers [14] and later to slightly soluble surfactants [17]. The authors derived the following expression for the elastic modulus of the surface monolayer: ε=
1+2
ε0
(2)
ω0 /ω + 2ω0 /ω
where ε0 = − d/d ln , is the surface tension and is the surfactant molar surface concentration, ω is the cyclic frequency of compression/expansion, while ω0 is called the adsorption frequency of the surfactant and is expressed by: ω0 =
D 2(d /dc)
2
(3)
In Eq. (3) D is the bulk diffusion coefficient, while d /dc is the socalled adsorption length. It can be determined from the surfactant surface tension isotherm, processed with an appropriate adsorption model (e.g. the Szyskowski–Langmuir model). Eq. (2) shows that the compression and expansion of the surface layer compete with adsorption and desorption of the surfactant molecules. Hence, the elastic modulus depends on the frequency of the compression and expansion. Loglio et al. [18] used the approach of Lucassen and van den Tempel [1] and determined the visco-elastic modulus as a complex value consisting of the real (elastic) and imaginary (loss) components. In addition, they introduced the Fourier transform operator for relating the time domain behavior of ε(, A) to the frequency domain ε(ω). They advanced later their theory [19–21] by devising a computational procedure allowing accurate evaluation of the spectra of the visco-elastic modulus from the observed transient surface tension decay. The numerical simulations [19,20] showed a maximum of the loss modulus at a given frequency, while the elastic modulus reached a plateau but unfortunately this numerical procedure was very complicated. The methodology of Lucassen and van den Tempel was adopted in the literature as a successful tool to study the elasticity of surfactant monolayers and their adsorption–desorption kinetics [22–41]. Meanwhile, new ways for determining the visco-elastic modulus were sought and developed [42–44]. For example, Hard and Lofgren [42] exploited laser light scattering techniques to determine the surface elastic modulus and surface viscosity. Hühnerfuss et al. [43] developed further an experimental approach for measuring the surface potential variations over propagating water waves covered with monolayers. Vogel and Mobius [44] determined the surface dilatational elasticity by a resonance of longitudinal and transverse surface waves. Hajiloo and Slattery [45] used the approach of Lucassen and van den Tempel [1,46] and preformed a first order perturbation analysis for longitudinal and transverse waves between air and aqueous surfactant solution. They exploited the linear model of Boussinesq [47] to describe the surface adsorption layer (called the Maxwell model) and hence the shear and dilatational viscosities were considered theoretically. Unfortunately, their model was very complicated and difficult to use. All these works were focused on studying the viscoelasticity of adsorption layers on flat liquid/air interface (in the Langmuir trough). Concurrently with the approach of Lucassen and van den Tempel for determining the visco-elastic moduli in Langmuir trough, Lunkenheimer et al. [48–54] developed independently the oscillating bubble method for the determination of the elasticity (called by them the Marangoni elastic modulus) of adsorption layers. The two approaches agreed theoretically. A good review on the basic methods for studying the rheology of adsorption layers can be found in Ref. [55]. The oscillating bubble method a small bubble fixed at the tip of a capillary. The bubble is submerged in the surfactant solution and connected to a gas compartment, which is
137
excited to harmonic oscillations by a piezo-drive vibrating with a given frequency and a small amplitude. The pressure in the gas compartment, the cross-sectional area and the amplitude of the bubble (the variation of bubble length) are measured as a function of the excitation frequency. Hence, the phase shift between the excitation signal and the bubble response is measured at different frequencies. The elastic and loss moduli can be determined by this procedure. The theory and the experiment of the oscillation bubble method have been advanced by the others [56–62]. Wantke and Fruhner applied this method to determine the dilatational viscosity of the surfactant adsorption layer [63]. Kovalchuk et al. [64] performed detailed analysis and showed that the frequency–amplitude relationship depends on the bubble volume and the surfactant concentration up to frequency of 400 Hz. Stubenrauch and Miller [65] applied profile analysis tensiometry (PAT) to study the rheology of adsorption layers at low frequencies (up to 0.1 Hz). A good analysis of the Gibbs elasticity and the visco-elastic modulus can be found in Ref. [66]. Leser et al. [67] reported that the frequency limit of PAT for determining the visco-elastic modulus is 1 Hz. A good review on the both methods can be found in Ref. [68]. Wantke and Fruhner [63] and Ortegren et al. [69] exploited the oscillation bubble method at high frequencies, at which the adsorption–desorption mechanism had no contribution to the elastic modulus, and showed that the model of Lucassen and van den Tempel [1] produced values of the elastic modulus deviating substantially from the data for d/d ln as determined using the experimental data for surface tension and the surface excess predicted by the surface tension isotherm. This would be true especially at high levels of surface coverage. Kovalchuk et al. [70] reported that this deviation would be due to the existence of twodimensional intrinsic compressibility of the surfactant layer, which should be correctly accounted for by an appropriate adsorption model. In addition, Wantke et al. [71] showed the that subsurface adsorption layer contributed to the surface elastic modulus as well. Yeung and Zhang [72] performed numerical simulations indicating that when the surfactant molecules entangled, the rheology of the expanding/shrinking surface could be described successfully by the Maxwell model of visco-elasticity, although they acknowledged this result as unphysical. The experimental technique based on the bubble/drop oscillating method underwent significant advance in the last decade [60,73]. For example, the second harmonic generation spectroscopy was combined with the oscillating bubble method [74]. Meanwhile Bianco and Marmur [75] developed a new experimental approach for measuring the Gibbs elasticity of oscillating soap bubble. This methodology was based on the determination of the film tension upon the expansion and shrinking of the soap bubble surface at a given low frequency. The methodology was advanced by Kovalchuk et al. [76]. The inhomogeneous distribution of the soap bubble liquid due to gravity was accounted for. In contrast to the numerous works on the visco-elastic moduli of single gas/liquid surface layers, the literature data on elasticity of soap bubbles has remained scarce. In addition, the questions on the origin of the Gibbs elasticity of foam films remained unanswered. For example, it is not clear why the tension of thick foam films (∼2–3 m) varies upon the change of the film surface area with speed much lower than the speed of relaxation of the adsorption layer. Obviously, the electrostatic disjoining pressure should not be relevant for those thick films. Hence, it should not be any depletion of surfactant molecules in such kind of foam films. Consequently, the approach of Lucassen and van den Tempel could be applied. The present work is devoted to studying the elastic modulus of soap bubbles in a wide range of surfactant concentrations. The applicability of Lucassen and van den Tempel approach is critically evaluated.
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S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 369 (2010) 136–140
Fig. 1. Sketch of the profile analysis tensiometer system for studying elastic modulus of soap bubble (not to scale).
2. Experimental The elasticity of soap bubbles produced by aqueous solutions of C8 E4 (Sigma–Aldrich Ltd.) in the concentration range 2 × 10−3 –1 × 10−2 M and in the presence 1 × 10−5 M NaCl (Sigma–Aldrich Ltd.), was measured. The temperature was kept constant during the experiments at 25 ◦ C. The surface tension isotherm was determined by the Harkins–Brown method [77]. All the measurements were obtained using a commercially available profile analysis tensiometer (PAT 1 D module, Sinterface Technologies, Ltd., Germany), with frequency of 0.1 Hz and amplitude of 2 mm3 . The tensiometer consisted of (1) a mechanical unit for creating and controlling the test fluid–liquid interface in a 2 cm × 2 cm × 2 cm cuvette made from optical grade silica, (2) an optical unit for monitoring the evolution of the interface profile and (3) a computer with the Sinterface software, PAT-1D and a data acquisition system for operating the instrument, storing the raw data for the interface profiles, and processing the data off-line. The mechanical unit had a water bath for controlling the temperature. The soap bubble was produced (see Fig. 1) by a dual tube – a narrower internal tube placed in a wider external tube. The surfactant solution flowing in the external tube was controlled by two syringe, while air flows through the internal tube, as controlled by another syringe. The two syringes were mounted on the panel of a motorized pumps, controlled by the computer. Once formed, the soap bubble was illuminated, equilibrated and its image was captured by the CCD video camera, stored, and processed by the software. The edge (the interface profile) of the bubble was digitally identified with sub-pixel resolution and fitted with the numerical solution of the Young–Laplace equation, allowing the determination of the film tension, volume and area of the bubble. The cyclic dependence of film tension on time was determined by changing the bubble volume as a sinusoidal function of time. Only bubbles with low amount of water were formed. Otherwise, the gravity deformed the soap bubble thus giving redundant data due to the fact that the software provided by Sinterface did not consider the effect of gravity. Hence the experimental data were obtained within these limitations. The average error of thus obtained values of the elastic moduli was about 5%.
Fig. 2. Surface tension isotherm of C8 E4 : experimental points and fit by the Szyskowski–Langmuir model.
A fitting procedure with two matching parameters, ∞ and k, was conducted via “Solver” function of Microsoft Excel. The theoretical curve is plotted in Fig. 2 and the obtained values of the fitting parameters are ∞ = 3.4 × 10−6 mol/m2 ± 5 × 10−8 mol/m2 and k = 25 m3 /mol ± 2.5 m3 /mol. Eq. (4) is equivalent to the following equation of state:
= 0 + RT∞ ln 1 −
∞
(5)
where ∞ is the maximal surfactant molar surface concentration. The relation between and k can be presented via Langmuir adsorption isotherm: = ∞
kc 1 + kc
(6)
The Gibbs elasticity of the soap bubble can be calculated by the expression: EG = 2ε0 =
−2d 2RT∞ = d ∞ −
(7)
and Eq. (2) can be transformed in the following expression: 2ε = EG
1+2
1
(8)
ω0 /ω + 2ω0 /ω
The values of the Gibbs elasticity of a single air/liquid interface and the adsorption length, calculated from the surface tension isotherm using the Szyskowski–Langmuir model are presented in Fig. 3. These values are utilized further to calculate 2ε/EG and ω0 using Eqs. (2) and (3). In the expression for ω0 , the diffusion coefficient D in Eq. (3) for C8 E4 molecules in the soap bubble is a unknown parameter. Hence the ratio 2ε/EG from Eq. (7) and the experimental data Eexp /EG can be adjusted by fitting the diffusion coefficient.
3. Results and discussion The surface tension isotherm of C8 E4 as measured by Harkins–Brown (average error 0.5 mN/m) method is presented in Fig. 2. These experimental data were processed with the Szyskowski–Langmuir equation of state given as: = 0 − RT∞ ln (1 + kc)
(4)
Fig. 3. The Gibbs elasticity EG /2 of single air/water interfaces and the adsorption length, d /dc, as a function of the surfactant concentration.
S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 369 (2010) 136–140
Fig. 4. The experimental points Eexp /EG and theoretical line 2ε/EG (obtained by adjusted diffusion coefficient D) versus C8 E4 concentration (error bars correspond to 5% experimental error).
139
frequency of bubble oscillation and the Gibbs elasticity, calculated from the surfactant surface tension isotherm are not the same. In addition, the literature data on soap bubble surface rheology are very scarce. Our study showed that the film tension values of soap bubbles prepared from C8 E4 aqueous solutions were larger than the expected values of twice surface tension. The elastic moduli values were significantly lower than the values of the Gibbs elasticity, calculated by the surface tension isotherm using the measured surface tension data. Our fit of the ratio 2ε/EG , calculated with the Lucassen and van den Tempel model (Eqs. (2) and (3)) to the experimental data Eexp /EG gave the dependence of the bulk diffusion coefficient of C8 E4 molecules upon C8 E4 concentration which was significantly lower than expected for a single C8 E4 molecule. All this indicates that there should be an exchange of surfactant molecules between the film surfaces and the film solution (2ε EG ), but this exchange could be impeded by some unknown factor. Furthermore, the increased viscous dissipation of the film liquid would be very much possible during the soap bubble oscillation, as compared to the case of the surface of a semi-infinite bulk phase (e.g. single bubbles or drop). Acknowledgment Stoyan Karakashev thanks the EC/‘Marie Curie Actions’ for the financial support of the work through the DEFFED – Project No. 230626/2009. References
Fig. 5. Bulk diffusion coefficient of C8 E4 molecules in the foam film as a function of the C8 E4 concentration.
The experiments on expanding soap bubbles were conducted in the concentration range of 2 × 10−3 –1 × 10−2 M C8 E4 since at lower surfactant concentrations the soap bubbles were not stable. The experimental values for Eexp /EG and the theoretical ratio 2ε/EG versus the surfactant concentration are presented in Fig. 4. The fitting procedure was performed by the variation of the values of the bulk diffusion coefficient at each surfactant concentration. Fig. 5 presents the diffusion coefficient upon the surfactant concentration. One can see a maximum for the diffusion coefficient at 5 × 10−3 M C8 E4 . The obtained values are in the range of 3.5 × 10−11 m2 /s and 8.5 × 10−11 m2 /s, which is about an ordered of magnitude lower than what is expected for single molecules of the surfactant in the bulk. This reflects a substantial combined effect of molecules, which is expected since the considered concentrations are near CMC. Fig. 4 shows that the values of the elastic moduli are substantially lower than the values of the Gibbs elasticity obtained from the surface tension isotherm. In addition, the ratio Eexp /EG decreases upon the increase of the C8 E4 concentration until reaching a plateau before the CMC. This result is in accord with Refs. [63,69], reporting deviations of the values of elastic moduli from the Gibbs elasticity, when the model of Lucassen and van den Tempel is applied to bubbles oscillating with high frequency. 4. Conclusions This paper is dedicated to studying the elasticity of soap bubbles. Although numerous works have been devoted to investigating the rheology of adsorption surface layers, there still are obscure aspects in their behavior. For example, it is not clear, what is the origin of the surfactant depletion in foam films of large thickness (of the order of several microns), causing the variation of the film tension at slow expansion or shrinking rates of the film surface. In addition, according to the literature data [69] the elastic modulus at high
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Elasticity of foam bubbles measured by profile analysis tensiometry Stoyan I. Karakashev a,∗ , Roumen Tsekov a , Emil D. Manev a , Anh V. Nguyen b a b
Department of Physical Chemistry, University of Sofia, 1 James Bourchier Ave., 1164 Sofia, Bulgaria School of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia
a r t i c l e
i n f o
Article history: Received 23 April 2010 Received in revised form 18 July 2010 Accepted 4 August 2010 Available online 12 August 2010 Keywords: Elastic modulus Gibbs elasticity Foam films Soap bubble Adsorption frequency
a b s t r a c t Elastic modulus of foam bubbles, stabilized with tetraethylene glycol octyl ether (C8 E4 ) and 1 × 10−5 M NaCl, was determined by cyclic expansion and shrinking of foam bubbles with frequency of 0.1 Hz and volumetric amplitude of 2 mm3 . The film tension was monitored by a commercial profile analysis tensiometer (Sinterface Technologies, GmbH). The elastic moduli of foam bubbles were obtained as a function of surfactant concentration in the range of 2 × 10−3 –1 × 10−2 M. The theory of Lucassen and van den Tempel [1] for the elastic modulus of a single liquid/air interface at a given frequency was employed. In the theoretical analysis the bulk diffusion coefficient of surfactant molecules was considered as a unknown model parameter which was obtained by matching the theory with the experimental data. Hence, the dependence of the bulk diffusion coefficient of C8 E4 molecules upon the C8 E4 concentration was obtained. The diffusion coefficient reached a maximum at 5 × 10−3 M C8 E4 (D = 8.5 × 10−11 m2 /s). In the experimental surfactant concentration range (2 × 10−3 –1 × 10−2 M, CMC = 7.5 × 10−3 M) the foam bubbles were relatively dry, with visible interferometric fringes corresponding to thin films stabilized by repulsion of the electrostatic disjoining pressure. Hence, the overall dynamics of periodical expansion and shrinking of the foam bubbles occurred within the thin film state. © 2010 Elsevier B.V. All rights reserved.
1. Background The Gibbs elasticity of thin liquid films (TLF) has been defined by Gibbs in 1876 in his famous work “On the equilibrium of the heterogeneous substances” [2]: EG =
2d d ln A
(1)
where is the surface tension of each film surface with area A of the film. He assumed that the solution in the foam lamella becomes depleted of surfactant as a result of the adsorption on the surfaces, and hence the surface tension increases with the expansion of the lamella (film). The film volume and the total number of surfactant molecules were assumed constant during the film expansion. The first experimental determination of the elastic moduli of foam films was conducted by Mysels et al. [3,4] in rectangular vertical frames, adopted later by Prins et al. [5,6] as well. The method consists of measuring the increased tension of a foam lamella stretched it to a measured extent, with concurrently determining the local film thickness of the lamella from the interference pattern. These first attempts to determine the elastic moduli were critically evaluated in the literature [3,7,8]. The arguments [3] were focused on the
∗ Corresponding author. Tel.: +359 2 8161241. E-mail address: [email protected]fia.bg (S.I. Karakashev). 0927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2010.08.008
dynamic nature of the film elasticity showing that the film dynamics effects should last for milliseconds, while the time-scale of the stretching was of the order of seconds, and this was accompanied by increased values of the film tension. Hence, a different limiting factor – presumably the depletion of surfactant in the intra-lamellar liquid – had to be critical the film [3]. According to Kitchener [8] the latter effect corresponded to the exact definition of the Gibbs elasticity as defined in [2]. This should be valid for thin films in which the electrostatic disjoining pressure was significant, but in many cases the thickness of the foam lamella was of the order of microns, for which the electrostatic disjoining pressure was negligibly weak. The methodology of Mysels et al. [3,4] was limited to relatively stable foam lamellae with high level of surfactant adsorption in the surface layers. In addition, the expanding of the film surfaces was related only to dilatational (not shear) visco-elasticity. The surface shear viscosity of insoluble surfactant monolayers was studied earlier in a series of work [9–11]. In order to study the surface elastic moduli at lower adsorption levels Lucassen and van den Tempel employed the theory of capillary waves on visco-elastic surface adsorption layers [12,13] and introduced a new methodology [1,14], operating with longitudinal surface waves and stress relaxation. The elastic modulus was regarded as a complex value, however the surface viscosity was not considered. The elasticities of the monolayers in Langmuir through [15,16] have been measured directly by determining the changes in surface tension caused by small amplitude sinusoidal compression/expansion cycles. This
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approach was first applied to insoluble protein monolayers [14] and later to slightly soluble surfactants [17]. The authors derived the following expression for the elastic modulus of the surface monolayer: ε=
1+2
ε0
(2)
ω0 /ω + 2ω0 /ω
where ε0 = − d/d ln , is the surface tension and is the surfactant molar surface concentration, ω is the cyclic frequency of compression/expansion, while ω0 is called the adsorption frequency of the surfactant and is expressed by: ω0 =
D 2(d /dc)
2
(3)
In Eq. (3) D is the bulk diffusion coefficient, while d /dc is the socalled adsorption length. It can be determined from the surfactant surface tension isotherm, processed with an appropriate adsorption model (e.g. the Szyskowski–Langmuir model). Eq. (2) shows that the compression and expansion of the surface layer compete with adsorption and desorption of the surfactant molecules. Hence, the elastic modulus depends on the frequency of the compression and expansion. Loglio et al. [18] used the approach of Lucassen and van den Tempel [1] and determined the visco-elastic modulus as a complex value consisting of the real (elastic) and imaginary (loss) components. In addition, they introduced the Fourier transform operator for relating the time domain behavior of ε(, A) to the frequency domain ε(ω). They advanced later their theory [19–21] by devising a computational procedure allowing accurate evaluation of the spectra of the visco-elastic modulus from the observed transient surface tension decay. The numerical simulations [19,20] showed a maximum of the loss modulus at a given frequency, while the elastic modulus reached a plateau but unfortunately this numerical procedure was very complicated. The methodology of Lucassen and van den Tempel was adopted in the literature as a successful tool to study the elasticity of surfactant monolayers and their adsorption–desorption kinetics [22–41]. Meanwhile, new ways for determining the visco-elastic modulus were sought and developed [42–44]. For example, Hard and Lofgren [42] exploited laser light scattering techniques to determine the surface elastic modulus and surface viscosity. Hühnerfuss et al. [43] developed further an experimental approach for measuring the surface potential variations over propagating water waves covered with monolayers. Vogel and Mobius [44] determined the surface dilatational elasticity by a resonance of longitudinal and transverse surface waves. Hajiloo and Slattery [45] used the approach of Lucassen and van den Tempel [1,46] and preformed a first order perturbation analysis for longitudinal and transverse waves between air and aqueous surfactant solution. They exploited the linear model of Boussinesq [47] to describe the surface adsorption layer (called the Maxwell model) and hence the shear and dilatational viscosities were considered theoretically. Unfortunately, their model was very complicated and difficult to use. All these works were focused on studying the viscoelasticity of adsorption layers on flat liquid/air interface (in the Langmuir trough). Concurrently with the approach of Lucassen and van den Tempel for determining the visco-elastic moduli in Langmuir trough, Lunkenheimer et al. [48–54] developed independently the oscillating bubble method for the determination of the elasticity (called by them the Marangoni elastic modulus) of adsorption layers. The two approaches agreed theoretically. A good review on the basic methods for studying the rheology of adsorption layers can be found in Ref. [55]. The oscillating bubble method a small bubble fixed at the tip of a capillary. The bubble is submerged in the surfactant solution and connected to a gas compartment, which is
137
excited to harmonic oscillations by a piezo-drive vibrating with a given frequency and a small amplitude. The pressure in the gas compartment, the cross-sectional area and the amplitude of the bubble (the variation of bubble length) are measured as a function of the excitation frequency. Hence, the phase shift between the excitation signal and the bubble response is measured at different frequencies. The elastic and loss moduli can be determined by this procedure. The theory and the experiment of the oscillation bubble method have been advanced by the others [56–62]. Wantke and Fruhner applied this method to determine the dilatational viscosity of the surfactant adsorption layer [63]. Kovalchuk et al. [64] performed detailed analysis and showed that the frequency–amplitude relationship depends on the bubble volume and the surfactant concentration up to frequency of 400 Hz. Stubenrauch and Miller [65] applied profile analysis tensiometry (PAT) to study the rheology of adsorption layers at low frequencies (up to 0.1 Hz). A good analysis of the Gibbs elasticity and the visco-elastic modulus can be found in Ref. [66]. Leser et al. [67] reported that the frequency limit of PAT for determining the visco-elastic modulus is 1 Hz. A good review on the both methods can be found in Ref. [68]. Wantke and Fruhner [63] and Ortegren et al. [69] exploited the oscillation bubble method at high frequencies, at which the adsorption–desorption mechanism had no contribution to the elastic modulus, and showed that the model of Lucassen and van den Tempel [1] produced values of the elastic modulus deviating substantially from the data for d/d ln as determined using the experimental data for surface tension and the surface excess predicted by the surface tension isotherm. This would be true especially at high levels of surface coverage. Kovalchuk et al. [70] reported that this deviation would be due to the existence of twodimensional intrinsic compressibility of the surfactant layer, which should be correctly accounted for by an appropriate adsorption model. In addition, Wantke et al. [71] showed the that subsurface adsorption layer contributed to the surface elastic modulus as well. Yeung and Zhang [72] performed numerical simulations indicating that when the surfactant molecules entangled, the rheology of the expanding/shrinking surface could be described successfully by the Maxwell model of visco-elasticity, although they acknowledged this result as unphysical. The experimental technique based on the bubble/drop oscillating method underwent significant advance in the last decade [60,73]. For example, the second harmonic generation spectroscopy was combined with the oscillating bubble method [74]. Meanwhile Bianco and Marmur [75] developed a new experimental approach for measuring the Gibbs elasticity of oscillating soap bubble. This methodology was based on the determination of the film tension upon the expansion and shrinking of the soap bubble surface at a given low frequency. The methodology was advanced by Kovalchuk et al. [76]. The inhomogeneous distribution of the soap bubble liquid due to gravity was accounted for. In contrast to the numerous works on the visco-elastic moduli of single gas/liquid surface layers, the literature data on elasticity of soap bubbles has remained scarce. In addition, the questions on the origin of the Gibbs elasticity of foam films remained unanswered. For example, it is not clear why the tension of thick foam films (∼2–3 m) varies upon the change of the film surface area with speed much lower than the speed of relaxation of the adsorption layer. Obviously, the electrostatic disjoining pressure should not be relevant for those thick films. Hence, it should not be any depletion of surfactant molecules in such kind of foam films. Consequently, the approach of Lucassen and van den Tempel could be applied. The present work is devoted to studying the elastic modulus of soap bubbles in a wide range of surfactant concentrations. The applicability of Lucassen and van den Tempel approach is critically evaluated.
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Fig. 1. Sketch of the profile analysis tensiometer system for studying elastic modulus of soap bubble (not to scale).
2. Experimental The elasticity of soap bubbles produced by aqueous solutions of C8 E4 (Sigma–Aldrich Ltd.) in the concentration range 2 × 10−3 –1 × 10−2 M and in the presence 1 × 10−5 M NaCl (Sigma–Aldrich Ltd.), was measured. The temperature was kept constant during the experiments at 25 ◦ C. The surface tension isotherm was determined by the Harkins–Brown method [77]. All the measurements were obtained using a commercially available profile analysis tensiometer (PAT 1 D module, Sinterface Technologies, Ltd., Germany), with frequency of 0.1 Hz and amplitude of 2 mm3 . The tensiometer consisted of (1) a mechanical unit for creating and controlling the test fluid–liquid interface in a 2 cm × 2 cm × 2 cm cuvette made from optical grade silica, (2) an optical unit for monitoring the evolution of the interface profile and (3) a computer with the Sinterface software, PAT-1D and a data acquisition system for operating the instrument, storing the raw data for the interface profiles, and processing the data off-line. The mechanical unit had a water bath for controlling the temperature. The soap bubble was produced (see Fig. 1) by a dual tube – a narrower internal tube placed in a wider external tube. The surfactant solution flowing in the external tube was controlled by two syringe, while air flows through the internal tube, as controlled by another syringe. The two syringes were mounted on the panel of a motorized pumps, controlled by the computer. Once formed, the soap bubble was illuminated, equilibrated and its image was captured by the CCD video camera, stored, and processed by the software. The edge (the interface profile) of the bubble was digitally identified with sub-pixel resolution and fitted with the numerical solution of the Young–Laplace equation, allowing the determination of the film tension, volume and area of the bubble. The cyclic dependence of film tension on time was determined by changing the bubble volume as a sinusoidal function of time. Only bubbles with low amount of water were formed. Otherwise, the gravity deformed the soap bubble thus giving redundant data due to the fact that the software provided by Sinterface did not consider the effect of gravity. Hence the experimental data were obtained within these limitations. The average error of thus obtained values of the elastic moduli was about 5%.
Fig. 2. Surface tension isotherm of C8 E4 : experimental points and fit by the Szyskowski–Langmuir model.
A fitting procedure with two matching parameters, ∞ and k, was conducted via “Solver” function of Microsoft Excel. The theoretical curve is plotted in Fig. 2 and the obtained values of the fitting parameters are ∞ = 3.4 × 10−6 mol/m2 ± 5 × 10−8 mol/m2 and k = 25 m3 /mol ± 2.5 m3 /mol. Eq. (4) is equivalent to the following equation of state:
= 0 + RT∞ ln 1 −
∞
(5)
where ∞ is the maximal surfactant molar surface concentration. The relation between and k can be presented via Langmuir adsorption isotherm: = ∞
kc 1 + kc
(6)
The Gibbs elasticity of the soap bubble can be calculated by the expression: EG = 2ε0 =
−2d 2RT∞ = d ∞ −
(7)
and Eq. (2) can be transformed in the following expression: 2ε = EG
1+2
1
(8)
ω0 /ω + 2ω0 /ω
The values of the Gibbs elasticity of a single air/liquid interface and the adsorption length, calculated from the surface tension isotherm using the Szyskowski–Langmuir model are presented in Fig. 3. These values are utilized further to calculate 2ε/EG and ω0 using Eqs. (2) and (3). In the expression for ω0 , the diffusion coefficient D in Eq. (3) for C8 E4 molecules in the soap bubble is a unknown parameter. Hence the ratio 2ε/EG from Eq. (7) and the experimental data Eexp /EG can be adjusted by fitting the diffusion coefficient.
3. Results and discussion The surface tension isotherm of C8 E4 as measured by Harkins–Brown (average error 0.5 mN/m) method is presented in Fig. 2. These experimental data were processed with the Szyskowski–Langmuir equation of state given as: = 0 − RT∞ ln (1 + kc)
(4)
Fig. 3. The Gibbs elasticity EG /2 of single air/water interfaces and the adsorption length, d /dc, as a function of the surfactant concentration.
S.I. Karakashev et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 369 (2010) 136–140
Fig. 4. The experimental points Eexp /EG and theoretical line 2ε/EG (obtained by adjusted diffusion coefficient D) versus C8 E4 concentration (error bars correspond to 5% experimental error).
139
frequency of bubble oscillation and the Gibbs elasticity, calculated from the surfactant surface tension isotherm are not the same. In addition, the literature data on soap bubble surface rheology are very scarce. Our study showed that the film tension values of soap bubbles prepared from C8 E4 aqueous solutions were larger than the expected values of twice surface tension. The elastic moduli values were significantly lower than the values of the Gibbs elasticity, calculated by the surface tension isotherm using the measured surface tension data. Our fit of the ratio 2ε/EG , calculated with the Lucassen and van den Tempel model (Eqs. (2) and (3)) to the experimental data Eexp /EG gave the dependence of the bulk diffusion coefficient of C8 E4 molecules upon C8 E4 concentration which was significantly lower than expected for a single C8 E4 molecule. All this indicates that there should be an exchange of surfactant molecules between the film surfaces and the film solution (2ε EG ), but this exchange could be impeded by some unknown factor. Furthermore, the increased viscous dissipation of the film liquid would be very much possible during the soap bubble oscillation, as compared to the case of the surface of a semi-infinite bulk phase (e.g. single bubbles or drop). Acknowledgment Stoyan Karakashev thanks the EC/‘Marie Curie Actions’ for the financial support of the work through the DEFFED – Project No. 230626/2009. References
Fig. 5. Bulk diffusion coefficient of C8 E4 molecules in the foam film as a function of the C8 E4 concentration.
The experiments on expanding soap bubbles were conducted in the concentration range of 2 × 10−3 –1 × 10−2 M C8 E4 since at lower surfactant concentrations the soap bubbles were not stable. The experimental values for Eexp /EG and the theoretical ratio 2ε/EG versus the surfactant concentration are presented in Fig. 4. The fitting procedure was performed by the variation of the values of the bulk diffusion coefficient at each surfactant concentration. Fig. 5 presents the diffusion coefficient upon the surfactant concentration. One can see a maximum for the diffusion coefficient at 5 × 10−3 M C8 E4 . The obtained values are in the range of 3.5 × 10−11 m2 /s and 8.5 × 10−11 m2 /s, which is about an ordered of magnitude lower than what is expected for single molecules of the surfactant in the bulk. This reflects a substantial combined effect of molecules, which is expected since the considered concentrations are near CMC. Fig. 4 shows that the values of the elastic moduli are substantially lower than the values of the Gibbs elasticity obtained from the surface tension isotherm. In addition, the ratio Eexp /EG decreases upon the increase of the C8 E4 concentration until reaching a plateau before the CMC. This result is in accord with Refs. [63,69], reporting deviations of the values of elastic moduli from the Gibbs elasticity, when the model of Lucassen and van den Tempel is applied to bubbles oscillating with high frequency. 4. Conclusions This paper is dedicated to studying the elasticity of soap bubbles. Although numerous works have been devoted to investigating the rheology of adsorption surface layers, there still are obscure aspects in their behavior. For example, it is not clear, what is the origin of the surfactant depletion in foam films of large thickness (of the order of several microns), causing the variation of the film tension at slow expansion or shrinking rates of the film surface. In addition, according to the literature data [69] the elastic modulus at high
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