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Dynamics of Thinning of Foam Films Stabilized by n-Dodecyl-β-maltoside Silke St€ockle, Pedro Blecua, Helmuth M€ohwald, and Rumen Krastev*,† Max-Planck Institute of Colloids and Interfaces, 14424 Golm/Potsdam, Germany. † Present address: NMI Natural and Medical Sciences Institute at the University of T€ ubingen, 72770 Reutlingen, Germany. Received September 29, 2009. Revised Manuscript Received January 12, 2010 We studied the process of thinning of thin liquid films stabilized with the nonionic surfactant n-dodecyl-β-maltoside (β-C12G2) with primary interest in interfacial diffusion processes during the thinning process dependent on surfactant concentration. The surfactant concentration in the film forming solutions was varied from 0.01 to 1.0 mM through the critical micellar concentration of 0.16 mM at constant electrolyte (NaCl) concentration, nominally 0.2 M. This assures the formation of Newton black films at the end of the thinning process. The velocity of thinning was analyzed combining previously developed theoretical approaches. From the model, which accounts for diffusion processes in the bulk of the film and in the interfaces, an analytical function was derived and fitted numerically to the experimental data. Quantitative information about the mobility of the surfactant molecules at the film surfaces could be obtained. We find that above a surfactant concentration of 0.12 mM (β-C12G2) the film surfaces behave as immobile and nondeformable which decelerates the thinning process. This follows the predictions for Reynolds flow of liquid between two nondeformable disks. Moreover, we could apply the theory on free area dependent diffusion coefficients on our results and show that it is in reasonable ranges applicable on the used surfactant system.
Introduction The interfacial and bulk properties of nanometer thick liquid layers have become the object of research during the past decades because these studies supply information about the properties of matter under confinement, when the surface interaction forces are active.1-5 The dynamic properties of thin liquid films contribute to the research area of nanofluidics, which has gained interest in the past years following the rise of microfluidics in the 1990s.6 Thin liquid film studies complement the classical investigations on the properties of fluid interfaces. Well-defined films have turned out to be suitable experimental systems because of their simple geometry and reproducible preparation.1 The practical interest in these films stems from the fact that several of their properties are determined by the same forces, which also govern the stability of colloidal dispersion systems like foams, emulsions, and sols. Foam films, the thin liquid layers separating gas bubbles, are the basic building elements of foams. Their properties are similar to those of emulsion films separating oil droplets or to liquid films between particles in concentrated dispersions. A foam film is schematically visualized in Figure 1. It consists of surfactant molecules, which are adsorbed as a monolayer at the film interfaces and separated by an aqueous core (bulk) of the solution from which the film is formed (Figure 1a). The properties of such films are determined by the properties of the interfacial adsorption layers as well as the properties of the bulk solution, which in turn depend on various factors, such as the type and concentration of solvent, electrolyte, and surfactant. The properties of foam films that have been investigated under equilibrium conditions include the film thickness h, which allows calculating the interactions between the film surfaces and the electrostatic double layer *Corresponding author. E-mail:
[email protected]. (1) Israelachvili, J. N. Intermolecular & Surface Forces; Academic Press: London, 1992. (2) Fleer, G. J.; et al. et al. Polymers at Interfaces; Chapman & Hall: 1993. (3) Asnacios, A.; et al. Phys. Rev. Lett. 1997, 78, 4974–4977. (4) Qu, D.; et al. Colloids Surf., A 2007, 303, 97–109. (5) Becker, T.; Mugele, F. Phys. Rev. Lett. 2003, 91, 1661041–1661044. (6) Eijkel, J.; Berg, A. Microfluid. Nanofluid. 2005, 1, 249–267.
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potential. The contact angle θ between film and meniscus (Figure 1b) gives insight into the interaction free energy per unit of film area.7,8 Gas permeability measurements yield information on the density of the surfactant layers stabilizing the foam film and thus on long-term foam stability.9 Studies on foam films have been intensively performed since the middle of the last century. The main part of these studies has been summarized in books and reviews.8,10-16 There are two types of black films corresponding to two equilibrium states of interaction between the film surfaces: common black films (CBF) and Newton black films (NBF). The thickness of CBF is in the range of 10-100 nm which is controlled by the balance between the van der Waals attraction and the electrostatic double-layer repulsions as described by the classical DLVO theory.17,18 The theory originally described the colloidal stability determined by a balance between repulsion among diffuse layers of ions (double-layer repulsion) and the molecular interaction between two separated phases (van der Waals attraction). It was successfully applied to describe the interaction forces between two fluid interfaces of foam films.10 The interaction forces per unit area of the film are termed disjoining pressure (Π). The thickness of a CBF decreases on addition of electrolyte into the film forming solution or by (7) Ivanov, I. B.; Toshev, B. V. Colloid Polym. Sci. 1975, 253, 593–599. (8) Exerowa, D.; Kruglyakov, P. M. Foam and Foam Films;Theory, Experiment, Application; Elsevier: Amsterdam, 1998. (9) Farajzadeh, R.; Krastev, R.; Zitha, P. L. J. Adv. Colloid Interface Sci. 2008, 137, 27–44. (10) Sheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391–464. (11) Thin Liquid Films; Ivanov, I. B., Ed.; Dekker: New York, 1988. (12) Foams: Theory, Measurements, and Applications; Prud'homme, R. K., Khan, S. A., Eds.; Dekker: New York, 1996. (13) Bergeron, V. J. Phys.: Condens. Matter 1999, 11, R212–R238. (14) von Klitzing, R.; M€uller, H. J. Curr. Opin. Colloid Interface Sci. 2002, 7, 42– 49. (15) Coons, J. E.; et al. Adv. Colloid Interface Sci. 2003, 105, 3–62. (16) Manev, E. D.; Nguyen, A. V. Int. J. Miner. Process. 2005, 77, 1–45. (17) Derjaguin, B. V. Theory of Stability of Colloids and Thin Films; Johnson, R. K., Translator; Consultants Bureau: New York, 1989. (18) Verwey, E. J. W.; Overbeek, J. T. G. The Theory of the Stability of Liophobic Colloids; Elsevier: Amsterdam, 1948.
Published on Web 02/04/2010
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Figure 2. Molecular structure of n-dodecyl-β-maltoside (β-C12G2).
Figure 1. Cross section of a free-standing liquid film prepared in the glass capillary cell according to Scheludko-Exerowa.8 (a) Surfactant molecules adsorbed at the film interfaces; electrolyte and free surfactant molecules in the core and meniscus of the film. (b) Indication of the thickness of the film h. The arrows show the forces acting in and on the film. Attractive van der Waals forces (ΠvdW) and capillary pressure (Pc) counteract repulsive electrostatic (Πel) and steric forces (Πst). θ shows the contact angle between film and meniscus.
expulsion of solution on applying an external pressure. NBFs are formed either at high salt concentration when the electrostatic double-layer repulsion is suppressed or at a high external pressure. Once a NBF is formed, its thickness cannot be reduced further. NBFs possess the smallest possible thickness, in the range of about 4-10 nm, depending on the type of surfactant molecules. These films are practically bilayers of surfactant molecules with almost no free water. They are stabilized due to the balance between steric repulsions in the film, caused by a direct contact between the hydrophilic heads of the surfactant molecules, and the van der Waals attraction. See refs 8 and 11 for a detailed summary of the properties and structure of thin foam films. Analysis of the dynamics of thinning of foam films has been proven to be a powerful measurement tool. As reviewed in ref 15, knowledge on the drainage of foam films is essential for accurate predictions of the stability and lifetime of a film. The analysis of the film’s thinning rate, the rupture thickness, or the black spot formation/expansion are essential pieces of information for characterization of the mechanical properties of a film and its interfaces as well as for flow and diffusion effects on a molecular level.19-22 Sugar-based “natural” surfactants are gaining ever-growing awareness.23,24 They are of low toxicity, biodegradable, and are widely used in protein solubilization and membrane studies.25 It is expected that the use of sugar surfactants as foaming agents will increase. This makes it necessary to accumulate information about the properties of foam films stabilized with such surfactants. The surfactant used in the present study is the sugar-based, nonionic surfactant n-dodecyl-β-maltoside (β-C12G2). Different investigations on foam films stabilized by β-C12G2 have been (19) Scheludko, A.; Platikanov, D. Kolloid Z. 1960, 175, 150–158. (20) Radoev, B.; Dimitrov, D. S.; Ivanov, I. B. Colloid Polym. Sci. 1974, 252, 50– 55. (21) Jachimska, B.; Warszynski, P.; Mazysa, K. Prog. Colloid Polym. Sci. 2000, 116, 120–128. (22) Jachimska, B.; Warszynski, P.; Malysa, K. Colloids Surf., A 2001, 192, 177– 193. (23) Holmberg, K. Curr. Opin. Colloid Interface Sci. 2001, 6, 148–159. (24) Stubenrauch, C. Curr. Opin. Colloid Interface Sci. 2001, 6, 160–170. (25) Hill, K., Rybinski, W. Stoll, G. Alkyl Polyglycosides: Technology, Properties, and Applications; Wiley-VCH Verlag GmbH: Berlin, 1997. (26) Claesson, P. M.; et al. Sugar-Based Surfactants: Fundamentals and Applications; Ruiz, C., Ed.; CRC Press: Boca raton, FL, 2008; Chapter 4. (27) Stubenrauch, C.; Schlarmann, J.; Strey, R. Phys. Chem. Chem. Phys. 2002, 4, 4504–4513.
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performed.26 Quantitative values of the disjoining pressure Π, the thickness h,27 the gas permeability,28 the pH dependence,29 and the contact angle θ are available in the literature. Moreover, temperature, surfactant, and electrolyte concentration dependent equilibrium thickness and disjoining pressure measurements were reported.30 The present paper presents the properties of foam films from β-C12G2 in the nonequilibrium state during the process of film formation (thinning). The experiments were conducted with focus on the effect of the surfactant concentration on the thinning velocity. A sufficient amount of ions suppresses the electrostatic double-layer repulsion between the film interfaces, so that we can follow the thinning of the films until the formation of the very thin NBF. The thinning behavior was analyzed with a combination of existing theoretical models and analytical and numerical methods. We obtained quantitative information on the mobility of the surfactant molecules in the air-liquid interfaces and could apply the free area model to explain the concentration dependence of the surface diffusion coefficients.
Experimental Section Substances. The nonionic surfactant n-dodecyl-β-malto-
side (β-C12G2), with a purity of g99% obtained from Glycon, Luckenwalde, Germany, was used without further purification. The hydrophilic group of the maltoside consists of two sugar rings connected via an ether bond as illustrated in Figure 2. Sodium chloride (NaCl for analysis; Merck, Germany) was purified by heat treatment at 600 C for at least 5 h to remove surface-active contaminations. A water purification system (Elga Labwater, Germany) was used to purify the water. The specific resistance of the water used was 18.2 MΩ cm, the pH was 5.5, and the total organic carbon (TOC) value was less than 10 ppb. All solutions for the experiments were prepared by dilution and mixing of concentrated stock solutions made from β-C12G2 and NaCl. An electrolyte concentration of 0.2 M was chosen for all the experimental solutions. The solutions were prepared and stored in glassware at room temperature for at least 12 h before usage. Surface Tension. The surface tension (σ) of the solutions was measured with a Kr€ uss tensiometer K11 with an accuracy of (0.1 mN/m using the Wilhelmy plate mode. During measurements, the solutions were kept in polytetrafluoroethylene (PTFE) beakers. The experiments, performed at 25 C, complement the already published data30 of the surface tension of β-C12G2 solutions containing 0.2 M NaCl and concentrations of surfactant between 0 M and 4 10-2 mM. The experimentally obtained isotherm was fitted with the software IsoFit31 using the Langmuir-Szyskowski model according to eq 1,32,33 where σ represents the surface tension of a solution which depends on the surfactant concentration c. The value of σ0 is the surface tension of the surfactant free solvent containing (28) Muruganathan, R. M.; et al. Langmuir 2003, 19, 3062–3065. (29) Stubenrauch, C.; Cohen, R.; Exerowa, D. Langmuir 2006, 23, 1684–1693. (30) Muruganathan, R. M.; et al. Langmuir 2004, 20, 6352–6358. (31) M€obius, D.; Miller, R.; Fainerman, V. B. Surfactants: Chemistry, Interfacial Properties, Applications; Elsevier: Amsterdam, 2001. (32) Langmuir, I. J. Am. Chem. Soc. 1916, 38, 2221–2295. (33) Szyskowski, B. V. Z. Phys. Chem. (Munich) 1908, 64, 385–414.
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Figure 3. Surface tension of aqueous solution containing 0.2 M NaCl and β-C12G: solid symbols, data measured in the present work; open symbols, data from ref 30. The solid line indicates the best fit to the data using the Langmuir isotherm. 0.2 M NaCl and determined to be σ0 = 72.2 mN/m. Γ σ -σ0 ¼ -RTΓ¥ ln 1 Γ¥ Γ a ¼ 1þ Γ¥ c
ð1Þ
The two fitting parameters in eq 1 are Γ¥, which is the maximum possible concentration of adsorbed surfactant molecules at the interface, and a, which is the Langmuir adsorption constant and describes the affinity of molecules to be adsorbed at the surface. We obtain the following values for the fitting parameters: Γ¥ = 4.09 10-6 mol/m2 and a = 3.97 10-3 mM. Γ¥ corresponds to a molecular area of 0.41 nm2 which is about twice the cross section of an aliphatic chain. Hence, these chains are disordered and the density is determined by the sugar moiety. The critical micelle concentration (cmc) of the surfactant was determined from the kink point of the curve to be 0.16 mM. The values for a, Γ¥, and the cmc are similar to those found in the literature34-36 and given by the manufacturers. The results using more complicated adsorption isotherms (e.g., Frumkin adsorption model,37 which takes into account the interaction between the adsorbed molecules at the interface) are not presented here. These isotherms give very similar fits to that obtained using the Langmuir isotherm, and the differences in the parameters a and Γ¥ are smaller than the error of the used fitting routine.
Experimental Instruments and Measurement Methods. The thinning process of foam films was monitored with an imageenhanced optoelectronic device based on an Olympus IX50 microscope. The films were prepared in a Scheludko-Exerowa glass ring cell as schematically shown in Figure 4.8 The ring cell (Figure 4a) is placed in a closed vessel (Figure 4b), which is kept at constant temperature in the device support (Figure 4c) connected to a thermostat. The atmosphere in the vessel is saturated with vapor of the respective solution. Two hours is allowed for reaching of equilibrium before starting measurements. The cell is loaded from the top (Figure 4d) with the film forming solution, which fills up the connected glass ring (Figure 4e and t1). With a piston (f) the liquid can be removed from the capillary. The drop forms a horizontal, biconcave, very thick film (Figure 4, t2) and starts thinning spontaneously, driven by the difference in the pressure in the film and its meniscus immediately after formation. The pressure in the meniscus is lower with the value of the capillary pressure Pc = 2σ/Rc due to (34) Stubenrauch, C.; Miller, R. J. Phys. Chem. B 2004, 108, 6412–6421. (35) Santini, E.; et al. Colloids Surf., A 2007, 298, 12–21. (36) Angarska, J. K.; Stubenrauch, C.; Manev, E. Colloids Surf., A 2007, 309, 189–197. (37) Frumkin, A. Z. Phys. Chem. (Munich) 1925, 116.
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the curvature in the meniscus. The additional forces are interfacial forces (molecular interactions) between the two surfaces of the film. They come into play when the film thickness decreases to a certain value and result in the disjoining pressure (Π). These forces expel the liquid from the film until equilibrium is reached. At equilibrium, the capillary pressure is balanced by the action of the positive electrostatic disjoining pressure: Pc = Π (Figure 4, t3). As illustrated in Figure 4g, a small area of the film is illuminated and the light reflects back from both film interfaces. The reflected light intensity I(t) filtered at 545.6 nm (Figure 4h) is recorded over time. The last maximum (k = 0 in eq 2) of light intensity of the periodically repeating positive and negative interference of reflected light is the maximum light intensity Imax and used to calibrate the measurement. The interference of the reflected light stands in direct relation to the distance between the film interfaces, i.e., to the film thickness. The thickness of the film is calculated from the intensity of the reflected light according to eq 2. The resulting equivalent film thickness hw depends on the thickness of the bulk of the film (h2 in Figure 1b) and the thickness of the hydrophobic part of the adsorbed surfactant molecules (h1 in Figure 1) with respect to the respective refractive indices n1 and n2. The difference of hw to the absolute thickness h = h2 þ 2h1 is in our case below the statistical error of the measurement method, especially since we consider films thicker than 30 nm. The accuracy of thickness measurements is (0.2 nm.38 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 λ @ IðtÞ=Imax A kπ ( arcsin hw ¼ 2πn 1 þ ½ðn2 -1Þ=2n2 ð1 -IðtÞ=Imax Þ ð2Þ
hw ¼ h2 þ 2h1
n1 2 -1 n2 2 -1
ð3Þ
Only films with plane parallel surfaces are suitable to be used for observation of the thinning behavior based on the interference of reflected light. It was shown39 that foam films with a radius below 100 μm should fulfill this criterion. We choose the radii of the films in the range of 70 ( 3.5 μm. In all the experiments, the temperature was kept constant at 23 C and controlled with an accuracy of (0.1 C. Additionally, the room temperature was adjusted to be equal to the temperature of the measurement and was kept constant with an accuracy of (0.5 C. An experimentally chosen equilibration time of ∼10 min was allowed between each measurement. For surfactant concentrations above 0.02 mM, the I(t) curves show considerable deviations when allowing less time for reaching equilibrium. Such curves tend to become very wavy, which indicates irregularities in thickness of the film interfaces. Concerning films from solutions with very low surfactant concentrations (e0.02 mM), we found no substantial difference in the smoothness of the I(t) curves independent of equilibration time. In order to be able to compare different measured data sets, we have to introduce to the data one common origin of time. We choose the occurrence of Imax as the reference point of time where we set t = 0. In this way, we obtain a distribution of measured data points as shown exemplary for one of the concentrations of surfactant in Figure 5. The deviation between single h(t) curves (see Figure 5) can be attributed to the fact that the setup does not allow to determine the radius of the film more precisely than (7 μm, which leads to small changes in the thinning velocity. Local surface corrugations (38) Toca-Herrera, J. L. Wechselwirkungskr€ afte und Struktur in PhospholipidSchaumfilmen; GCA-Verlag: Berlin, 2000. (39) Sedev, R.; Kolarov, T.; Exerowa, D. Colloid Polym. Sci. 1995, 273, 906–911.
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Figure 4. Experimental setup for observation of thin foam films and interferometric measurement of their thickness. A Scheludko-Exerowa glass ring cell (a) sitting in a cell holder (b) surrounded by a thermostated support (c). The inner void of the cell (d) is filled with the film forming solution, which fills the glass ring (e and t1). With a piston (f) the liquid is removed from the capillary to transform the drop (t1) into a biconcave film (t2). Spontaneous thinning leads to the formation of a flat, ultrathin film (t3). The change in thickness is obtained in situ with a microinterferometer measurement setup (g). A small area of the film is illuminated, and the light reflected at both film interfaces shows thickness-dependent interference patterns. The resulting light intensity is detected and recorded digitally (h).
Figure 5. Dynamics of thinning of foam films from aqueous solution containing 0.02 mM β-C12G2 and 0.2 M NaCl. The experimentally obtained data from eight measurements are shown as squares. in small areas of the film can alter the thinning behavior so that the h(t) curves are wavy.
Results and Discussion We recorded the dynamics of thinning of foam films stabilized by different amounts of the nonionic surfactant β-C12G2 in the presence of 0.2 M NaCl. In Figure 6 we show experimental data sets of the change of the film thickness over the time during the film formation process. Each data set is an average of multiple measurements. We use a combination of previously developed theoretical approaches for the quantitative analysis of the dynamics of film thinning. We follow refs 11, 20, and 40 and refer the reader to them, as well as to references therein, for a more detailed derivation of the theory. Here, we will present a qualitative (40) Ivanov, I. B. Pure Appl. Chem. 1980, 52, 1241–1262.
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understanding of the main assumptions and necessary steps in order to obtain a physical fitting function. The drainage of thin liquid films has been intensively investigated during the past decades. A pioneer work in this area10 was based on the lubrication approximation by Stefan and Reynolds,41,42 a simplification of the Navier-Stokes equations to describe the outflow of a fluid from in between two rigid disks. The theory is valid (among other criteria15) for a Newtonian incompressible liquid with constant viscosity in between plane parallel and rigid surfaces. The approximation is limited to zero flow velocity at the interfaces and to a constant force applied on the disks. When this theory was applied to foam films, it was found that they drain faster than predicted. This phenomenon could be explained with additional forces acting on the film interfaces due to their submicrometer distance. This is formulated in the modified Stefan-Reynolds equation for film thinning10 2 Pc -ΠðhÞ VRe ¼ h3 3 μRf 2
ð4Þ
where VRe is the thinning velocity, h is the average film thickness, μ is the dynamic viscosity of water, and Rf is the film radius. The additional forces on the interfaces, i.e., the disjoining pressure, adds to Pc and influences the velocity of thinning of the film. The two main components of the disjoining pressure according to the classical DLVO theory are the attractive van der Waals interaction (ΠvdW) and the repulsive electrostatic force (Πel). The other components of the disjoining pressure included in the extended DLVO theory, which consider also the action of different short-range interactions in the films, are summarized (41) Stefan, J. Sitzungsber. Kais. Akad. Wiss. Math. Naturwiss. Cl. 1874, II, 713– 735. (42) Reynolds, O. Philos. Trans. R. Soc. London 1886, 177, 157–234.
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Figure 6. Experimentally recorded thinning dynamics for foam films from solution with 0.2 M NaCl and β-C12G2 from 0.01 to 1 mM. Each of the curves is an average result of multiple measurements. The error bars indicate the standard deviation of the average. The theoretical fits are shown as solid lines.
in the term Πst. The disjoining pressure is given as Π ¼ ΠvdW þ Πel þ Πst
ð5Þ
Our system contains a sufficient amount of NaCl so that the electrostatic repulsion between the surfaces is screened, and therefore the term Πel can be neglected. The steric forces are also not considered, as they have a very short-range of action (up to few nanometers), and this is much below the thickness of the films we work with. Under these conditions, eq 5 reduces to1 Π ¼ ΠvdW ¼ -
AH 6πh3
ð6Þ
with AH the Hamaker constant of a gas-liquid-gas system. It was further found that the conditions under which eq 4 was obtained are not met due to the fluidity of the film interfaces.20 The interfaces, a fluid mosaic of adsorbed surfactant molecules, neither are necessarily nondeformable nor are the surfactant molecules at the interfaces immobile. Here, we neglect deformability according to ref 40 and explain briefly how the fluidity of the interfaces and resulting fluxes influence the thinning velocity of foam films (Figure 7). First, it is assumed that the amount of surfactant at the surface remains constant during the thinning process.40 This dynamic equilibrium between surfactant adsorption to and desorption from the film interfaces is assumed to be diffusion controlled according to the bulk diffusion flux jb with jb = -Dbrc and Db the diffusive constant of molecules in the bulk (see Figure 7a). Second, as shown in Figure 7b, it is assumed that there are opposing interfacial fluxes: the convective flux jc and the surface flux js. The convective flux jc is a result of the pressure driven hydrodynamic Langmuir 2010, 26(7), 4865–4872
Figure 7. Fluxes, which determine the effective velocity of thinning of a thin liquid film. (a) Diffusion-controlled adsorption process: the bulk diffusion flux jb determines the surfactant mass balance between bulk and film surface. (b) The convective flux jc, is a result of the hydrodynamic thinning of the film. It drags surfactant molecules at the interfaces toward the periphery of the film. The gradient in the surface tension is reduced by surface diffusion processes js and adsorption processes (a).
outflow of liquid from the film. At the interfaces, jc = vsΓ drags surfactant molecules with the velocity vs toward the periphery of the film if the adsorption density Γ of molecules is low enough and the spacing in between the molecules is high enough. This situation of a nonzero velocity at the interfaces accelerates the thinning process of the film (similar to slip of liquid at solid interfaces). The random diffusion of molecules at the film interfaces; here surface diffusion flux js = -DsrΓ, with Ds the diffusive constant of molecules in the surface, compensates for the interfacial outflow of molecules together with the adsorption and desorption processes of surfactant molecules between bulk and interfaces. DOI: 10.1021/la9036748
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If there were no interfacial or bulk diffusion (js = jb = 0), a surface tension gradient between film center and periphery would be created and this would be compensated by a (Marangoni) back-flow. Dependent on the gradient, this could reduce the interfacial velocity to zero and the thinning process should follow the Reynolds model for solid interfaces. Instead, because of adsorption/desorption and diffusion processes, the concentration of surfactant at the interface during the thinning process is assumed to be constant. An increase in the effective thinning velocity of a foam film is linked to the interfacial mobility of surfactant molecules and can be directly related to the surface and bulk diffusion coefficients. Hence, the thinning of thin parallel axisymmetric films with fluid interfaces is noted according to ref 20 as the ratio of the experimentally obtained thinning velocity V and the theoretical thinning velocity VRe for the system with rigid interfaces. In the case that V > VRe (V/VRe > 1), the contributions from bulk diffusion and surface diffusion are expressed by the parameters b and hs according to eq 7. V hs ¼ 1þbþ VRe h
ð7Þ
These parameters are directly connected to the diffusion coefficients of the molecules in the bulkDb and at the film interfaces Ds b ¼
3μDb ðaþcÞ2 ; Γ¥ 2 kB Tc
hs ¼
6μDs a Γ¥ kB Tc
ð8Þ
with known values for the dynamic viscosity of water μ, the temperature T, the Boltzmann’s constant kB, and the concentration of surfactant under equilibrium conditions c. The Langmuir adsorption coefficient a and the maximum adsorption density Γ¥ can be deduced from the Langmuir adsorption isotherm. That makes b and hs useful fitting parameters for the analysis of experimental thinning data with eq 7 as fitting function. Following ref 40, eq 7 was rewritten with the required input and fitting parameters and with numerical and analytical software tools. We were further able to solve this expression analytically for t. Thus, we end up with an analytical formula of the symbolic form t = T(h, AH, Pc, Rf, b, hs) (see Supporting Information for more details) with h, AH, Pc, and Rf as input parameters and b and hs as fitting parameters. An average value of Rf was given as input parameter because of minor deviations between the single experiments. A first estimate for b and hs/h in eq 7 shows that b is expected to be by a factor of 103 smaller than the value of hs/h over the complete range of surfactant concentrations. Hence, we neglect the contribution of b, and we do not expect reliable information on the bulk diffusion coefficient for our system from this method. According to the above-described method, we fitted the experimental data of foam films stabilized by β-C12G2 in the presence of 0.2 M NaCl with respect to the parameter for surface diffusion hs (Figure 6). The resulting ratio of V/VRe indicates, as shown in Figure 8, an up to 75% increase of the thinning velocity for foam films from solutions with a surfactant concentration of 0.06 mM or lower due to the mobility of the interfaces. For higher concentrations of surfactant, we can fit the data with values for hs of 10-4 nm, but we reach the resolution of the method at hs ≈ 0.5 nm and therefore we are not able to make quantitative predictions for very low surface diffusion coefficients. However, we were able to fit the data of 0.12-1 mM with a fitting parameter hs = Ds = 0, which means that the model assuming rigid interfaces (eq 4) suffices to describe the data. When applying eq 4 to fit the data sets for 0.01-0.06 mM concentration of β-C12G2 (hs = Ds = 0) the fits assuming rigid 4870 DOI: 10.1021/la9036748
interfaces deviate substantially from the measured data points, whereas the fits which account for interfacial diffusion processes pass accurately through the average data points (compare as an example Figure 9 for 0.02 mM β-C12G2). The statistical difference between the curves using an error analysis based on the reduced χ2 criteria is shown. We obtain for the assumption of mobile surfaces and molecular movement in the bulk (hs 6¼ 0) the value for χ2 of 0.057 and for the fit assuming immobile surfaces (hs = 0) the value for χ2 of 1.62. The value of χ2 = 0.057 appears too good (overfitting) which we attribute to the rather high standard deviation of the average data points. Nevertheless, a fit which gives a χ2 value of 1.62 can be accepted as statistically different to the experimental data. Using the probability distribution, we found that the fit which accounts for immobile surfaces represents the data points with a probability better than 5%. Hence, we can assume that the interfaces of the films are mobile (hs > 0) below a surfactant concentration of 0.12 mM, which leads to a faster drainage of the film. For a concentration of β-C12G2 larger than 0.12 mM the interfaces seem “blocked”. Nevertheless, the interfacial mobility concentrations of 0.12 and 0.15 mM, both below the cmc, could be either below the resolution of the method or those films have blocked interfaces and no accelerated outflow. The surface diffusion coefficients calculated according to eq 8 from the fitting parameter hs show a dependence on the concentration of β-C12G2 in the film forming solution as presented in Figure 10, top. There are two major regimes of Ds. For high concentrations of surfactant (g0.12 mM) we can predict that Ds is close to zero. We assume, for that regime, that the film surfaces are rigidlike with tightly packed surfactant molecules. With decreasing concentration of surfactant in the bulk and therefore decreasing amount of surfactant at the interface, Ds increases systematically, and we obtain the highest value for Ds of 8.6 10-8 m2/s at the lowest measured concentration of surfactant, 0.01 mM β-C12G2. The value for Ds in the range of 10-8 is analogous to that presented in the literature.21,22,40 Even though, it seems rather high, since the self-diffusion of water molecules in the bulk is 2.3 10-9m2/s.44 Generally, the diffusion coefficients at the surfaces are by a factor of 10 lower. Therefore, we accept this value as an apparent surface diffusion coefficient, which might include effects of drag or convection forces, which accelerate the thinning process and reflect the measured value of the surface diffusion coefficient. Studies on phospholipid monolayers spread at interfaces show that the surface diffusion coefficient of a monolayer of amphiphilic molecules depends on the free area per molecule af = a - a0 with a the average molecular area for a certain surfactant concentration and a0 the minimum required area per surfactant molecule. The surface coverage, as shown in Figure 10, bottom, is related to the mean free area a molecule has at the interfaces for a certain surfactant concentration. We obtained a molecular area at saturation of 0.41 nm2, which is reasonable as one expects this value to be above a0. Reported literature values range between 0.30 and 0.50 nm2 but seem higher than the minimum space a β-C12G2 molecule would need with a single hydrocarbon chain (∼0.20 nm2) and two sugar rings (maximum length of 0.12 nm). Experiments with deposition of β-C12G2 on solid substrates45 showed a minimum free area necessary for the molecules of 0.30 nm2 which could indicate that in solution the sugar rings (43) Peters, R.; Beck, K. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 7183–7187. (44) Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; Oxford University Press: Oxford, 2002. (45) Zhang, L.; Somasundaran, P.; Maltesh, C. J. Colloid Interface Sci. 1997, 191, 202–208.
Langmuir 2010, 26(7), 4865–4872
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Figure 8. Experimentally obtained velocity of thinning (V) compared to Reynolds/Scheludko’s thinning velocity (VRe). (left) Plotted versus h-1. The slope represents the fitting parameter hs; the intercepts of the lines with the ordinate are the fitting parameter (1 þ b). (right) Ratio of V/VRe for films from different concentrations of surfactant, at constant thickness of h = 40 nm. Table 1. Values for Surface Diffusion Coefficients Ds Obtained via the Thinning Dynamics of Foam Films from Solutions of Different Concentrations of β-C12G2 in the Presence of 0.2 M NaCla β-C12G2 conc (mM)
Figure 9. Experimental thinning data of a foam film from solution containing 0.02 mM β-C12G2 and 0.2 M NaCl (gray squares) fitted with (hs > 0) and without (hs = 0) contribution of the surface diffusion parameter hs (black lines).
Γ(10-6 mol/m2)
tend to bind water, which prevents the formation of tightly packed interfacial layers of β-C12G2. Langmuir 2010, 26(7), 4865–4872
Ds (m2/s)
0.01 2.92 20.7 8.62 10-8 0.02 3.41 9.2 7.66 10-8 0.04 3.72 2.2 3.67 10-8 0.06 3.83 1.3 3.25 10-8 0.12 3.94 0 0 0.15 3.98 0 0 0.4 4.09 0 0 1.0 4.09 0 0 a The amount of surface coverage at each concentration is also provided.
The free volume model43 assumes a linear dependence between ln Ds (here the apparent surface diffusion coefficient) and the inverse of af. ln Ds ¼ A -
Figure 10. Top: concentration-dependent coefficient of surface diffusion Ds of foam films during the thinning process calculated from fitting parameters hs as black squares; the line indicates the decrease of Ds following the free volume model.43 Bottom: surface coverage Γ calculated from surface tension of the bulk solution. Theoretical Γ¥ indicated as dashed line.
hs (nm)
B af
ð9Þ
The coefficient A yields to an upper value of Ds for 1/af f 0 and B stands for the minimal free area ac necessary for the displacement of a molecule. This value should be in realistic relation to the area a0. With the free volume model, we are able to approximate our data with a value of ac = 0.29 nm2 and obtain a decay of Ds for increasing surface coverage according to our experimental results (compare Figure 10, top solid line). The reasonable fit of the experimental points and the reasonable value obtained for ac suggest that even though the values we obtained for Ds are larger than expected, the additional driving force for molecular diffusion at the film surfaces is independent of the surfactant concentration. This force is equal for all concentrations which allow the data to be properly presented using the free volume model. We should add that the free volume model is applicable although the diffusion coefficients are orders of magnitude higher than expected. This is probably due to the inverse proportionality of diffusion and viscosity. Therefore, also any convective motion should obey the free volume model given by eq 9.
Conclusion The dynamics of thinning of foam films prepared from solutions of the nonionic surfactant β-C12G2 with different concentration have been investigated. The electrolyte concentration was kept constant at 0.2 M NaCl. Thin liquid films are formed from a drop of a liquid which contains the sufficient concentration of DOI: 10.1021/la9036748
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surfactant to stabilize the film. Under the actions of the capillary pressure and, below a certain thickness, also the attractive van der Waals disjoining pressure, the liquid from the drop is expelled, and a liquid film is formed. Studies on the velocity of film thinning (change of the film thickness with time) supply important information about the attraction between the film surfaces (negative disjoining pressure) but also about the dynamics of the surfactant molecules on the film surfaces. We used a classical theoretical model36 to describe the thinning process and were able to distinguish between different regimes of thinning. The thinning can be described according to the outflow of liquid from between two rigid disks (Reynold’s thinning) at concentrations higher than 0.12 mM. The results show that the “blocking” of interfaces seems to happen at lower concentrations than the spontaneous formation of micelles at the cmc of 0.16 mM. For concentrations of surfactant below 0.12 mM, we observed an accelerated thinning due to nonzero velocity at the film interfaces. The outflow velocity increased with decreasing amount of surfactant and with decreasing film thickness. The mobility of interfaces plays an important role for films thinner than 100 nm. The surface diffusion coefficients we obtained follow the free volume model
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which suggests an increase of Ds with increasing mean area per molecule. Foam films could be used as experimental models to understand the behavior of liquids in nanosized confinement. This is an important issue when micro- and nanofluidic devices are constructed. This makes the foam films a unique “device” available to obtain information about fluidic systems in nanometer dimensions. Foam films can be used to model the rigidity and mobility of fluid-fluid interfaces and can deliver important information analyzing the slip and no-slip conditions of liquid flow in nanometer distances from channel walls. Acknowledgment. We thank R. Emrich and C. Janiszewski for technical support and preparation of the experimental equipment. T. Krastev is acknowledged for the development of the algorithm for film thickness measurements and A. Valeriani for the discussion on the statistical treatment of the data. Supporting Information Available: Details of fitting routine. This material is available free of charge via the Internet at http://pubs.acs.org.
Langmuir 2010, 26(7), 4865–4872
