LETTER pubs.acs.org/Langmuir
What Is the Mechanism of Soap Film Entrainment? Laurie Saulnier,*,† Frederic Restagno,† Jer^ome Delacotte,‡ Dominique Langevin,† and Emmanuelle Rio*,† † ‡
Laboratoire de physique des solides, CNRS & Universite Paris-Sud 11, 91405 Orsay cedex, France Laboratoire de physique statistique, CNRS & Ecole Normale Superieure, 75005 Paris, France ABSTRACT: Classical Frankel’s law describes the formation of soap films and their evolution upon pulling, a model situation of film dynamics in foams (formation, rheology, and destabilization). With the purpose of relating film pulling to foam dynamics, we have built a new setup able to give an instantaneous measurement of film thickness, thus allowing us to determine film thickness profile during pulling. We found that only the lower part of the film is of uniform thickness and follows Frankel’s law, provided the entrainment velocity is small. We show that this is due to confinement effects: there is not enough surfactant in the bulk to fully cover the newly created surfaces which results in immobile film surfaces. At large velocities, surfaces become mobile and then Frankel’s law breaks down, leading to a faster drainage and thus to a nonstationary thickness at the bottom of the film. These findings should help in understanding the large dispersion of previous experimental data reported during the last 40 years and clarifying the pulling phenomenon of thin liquid films.
F
oams are dispersions, in a liquid matrix, of gas bubbles stabilized by surface active agents. They are presently the object of much interest, due to their interesting structural and dynamical properties as well as their numerous practical applications.1 One of the problems posed by foams is the control of their stability which is directly linked to the stability of the thin liquid films formed between air bubbles. The dynamics of these films during topological rearrangements is a key to understand foam rheology and foam stability, but only a few recent works began to address these issues.2,3 Model liquid films can be formed on frames using soapy water and have been extensively studied over the years:4 their beautiful colors and patterns are well-known, but the origin of their shape and rupture is surprisingly not yet elucidated. Even the common expectation that film rupture is a stochastic process has been questioned very recently with experiments on pure water films.5 The prediction of the ability of a solution to generate stable foams remains therefore an empirical question. A simple experiment allowing the study of film pulling was proposed already in the 1960s by Frankel, in which a soap film is pulled out of a solution at a controlled speed V. A number of such experiments have been performed.610 Gibbs addressed the present problem of a film pulled from a soapy solution and proposed that the profile results from a competition between the film weight and the surface concentration gradients.11,12 However, it is usually assumed that the film thickness hF follows the so-called Frankel’s law:4 hF ¼ 1:89k1 Ca2=3
ð1Þ 1
where the capillary length k is (γ/Fg) with γ being the liquid surface tension, F the liquid density, and g the gravitational constant. The capillary number Ca = ηV/γ is the plate velocity normalized by the liquid viscosity η and surface tension γ. Equation 1 has been established by Frankel by analogy with the LandauLevichDerjaguin model for a similar experiment, the entrainment of a liquid by a moving plate (LLD experiment).13 1/2
r 2011 American Chemical Society
Frankel assumed that a film of constant thickness is pulled out of a liquid bath, with the film thickness being obtained from a balance between viscous forces and capillary suction in a region called the dynamic meniscus. Note that, in the original paper, Frankel did not mean to describe a vertical film but a horizontal one, in order to avoid to take into account the liquid weight effect. The motors of liquid entrainment are the two liquidair interfaces which are assumed to behave as solid plates. The film surfaces are of course not solid but may remain immobile due to surfactant concentration gradients at the surfaces, giving rise to surface tension gradients and therefore to Marangoni forces. The surface concentration gradient can be quantified by the surface elastic modulus E.11,14 So, whereas free surfaces of pure liquids are fully mobile, the presence of surfactants leads to a reduction in surface mobility. The elastic moduli actually vary with the nature of the surfactants: for example, sodium dodecyl sulfate (SDS) leads to rather mobile surfaces, whereas mixtures of SDS and dodecanol lead to fully immobile surfaces.15 The maximum observable thickness is given by Frankel’s law, whereas smaller thicknesses could be expected for surfactants with small elastic moduli. In a recent paper,16 Van Nierop et al. compiled the available data h(V) obtained during the last 40 years. They showed that even if the data are all in rather good agreement with Frankel’s law, they are relatively scattered. Aiming at a better understanding of foam stability, we started a study of film rupture under pulling and adopted the Frankel experimental configuration. Thanks to an improved setup, we could perform a detailed study of the film behavior before rupture, and we found a number of unsuspected deviations between theory and the actual film thickness variation with pulling velocity. This study is described below. Soap films are Received: June 14, 2011 Revised: October 13, 2011 Published: October 17, 2011 13406
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Figure 1. (a) Schematic diagram of experimental setup. A stainless steel frame (4) gridded with three fishing lines (6) is immersed in a glass cell (5) filled with a surfactant solution. This cell is translated downward thanks to a motion controlled stage (1) at a chosen speed, creating a soap film on the frame. A force sensor (2) allows detection of film rupture. An optical fiber (3) is used to measure film thickness. (b) Observation of a film during pulling. Note the presence of interference fringes, indicating nonuniform film thickness.
formed on a 6 cm2 rectangular stainless steel frame gridded with 100 μm fishing lines (purchased from Mitchell), one horizontal at the top of the frame, two vertical separated by 1.3 cm, as shown in Figure 1. To form a film, the frame is immersed in a 15 mL glass cell, filled with surfactant solution, supported by a motorized linear stage (Newport UTS 150CC) coupled with a motion controller (Newport SMC100CC) which allows one to vary the velocity from 0.2 to 20 mm 3 s1 with an accuracy of 0.1 μm 3 s1. In order to avoid the presence of electrostatic surface potential and subsequent adsorption barriers, we have used a nonionic surfactant, hexaoxyethylene dodecyl ether (C12E6). The surfactant was purchased from Sigma-Aldrich (purity g 98.0%) and used as received. Its surface properties have been previously characterized: surface tension has been measured using the classical Wilhelmy plate technique, and the surface elastic modulus using an oscillating bubble instrument (Teclis).17 All solutions were prepared with ultrapure water (purified with a Millipore system). All materials used for solution preparation and storage, and which may have been in contact with the surfactant solutions studied (frame and wire) were cleaned with Deconex 22 HPF (purchased from Borer chemie, diluted 20 times in Millipore water before use) and rinsed with Millipore water. For C12E6 in water at 22 °C, the critical micellar concentration (cmc) is 7 105 M. Surface tension above the cmc is constant and equal to 34.5 mN 3 m1. The solution viscosity was measured with a low shear rheometer and found to be equal to water viscosity. A polychromatic light, emitted by a tungsten halogen lamp, is transmitted by an optical fiber, at normal incidence relative to the soap film. A spectrometer (USB 400 Ocean Optics), with a bandwidth from 350 to 1000 nm, measures the reflected light spectrum. This spectrum can be fitted by adjusting the parameters A, signal amplitude, j, signal phase, B, background, and h, thickness of the film in the analytical expression of reflectivity: R = A sin((4πnh/λ) + j) + B, where R is the light reflectivity, λ is the wavelength, and n is the refractive index of the solution. Spectrum acquisition frequency can be chosen up to 250 Hz to allow almost instantaneous measurement of the thickness of the liquid film during generation, whatever the entrainment velocity is, as long as film
Figure 2. Vertical thickness profile of a C12E6 (concentration = 3 cmc) film during entrainment at a speed V = 2 mm 3 s1. z denotes the vertical position along the film, with z = 0 cm being the position of the wire (see Figure 1). h corresponds to the entrained film thickness. The film breaks just after 12 s. The inset figure shows the time evolution of the length L of the upper part of the film.
lifetime is reasonable. All the thickness measurements were made in the horizontal middle of the frame in order to avoid any edge effects. Moreover, an accuracy better than 5% was found regarding the experiment reproducibility, which corresponds to error bars smaller than the symbols used in the figures. 13407
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Figure 3. Normalized entrained film thickness, hk, versus capillary number, Ca, for solutions of C12E6: 1 cmc (b), 3 cmc (O), 5 cmc (red solid diamond), 10 cmc (red open diamond), and Frankel’s law (—). The inset figure shows the values of the parameter htr at the beginning of the transition (b) versus the normalized concentration of C12E6 and theoretical value (—) calculated with σ = 1.
We measured the thickness of a C12E6 (concentration = 3 cmc) film as a function of the position along the z axis (with z = 0 cm being the position of the fishing line) at different times during entrainment at V = 2 mm 3 s1 (Ca = 5.8 105). Although it is possible to measure time evolution of thickness at a given position during one film generation, it is necessary to generate a new film when changing the position of the optical fiber. The reconstruction of the time evolution of the film profile during entrainment is shown in Figure 2 from the top left to the bottom right. First of all, none of these profiles are of uniform thickness, which is evidenced as well by the interference fringes in Figure 1b. Two different parts of the film can be observed: (i) the bottom of the film has a well-defined thickness, constant along a height of few millimeters and stationary during the experiment. It corresponds to a freshly formed film element. (ii) The upper part of the film is not of uniform thickness, with the film being always thinner close to the upper boundary. Let us first discuss the upper region of the film and assume that thinning is due to gravity drainage. Mysels and Shinoda modeled such a drainage in the case of a pure liquid film, with rigid interfaces, pulled out of a solution.4 The time evolution of the length L of the upper part of the film can be expressed as follows: LðtÞ ¼
Fg 2 ht 4η
ð2Þ
The line of separation between the two parts of the film moves down with a velocity v such that v ∼ Ca1/3V. When Ca1/3 , 1, then v , V and the gravity drainage is slower than the pulling velocity. The inset of Figure 2 shows that L is linear in time as predicted by eq 2, with v ∼ 1.5 mm 3 s1. Using F = 1.0 g 3 cm3, g = 9.81 m 3 s2, η = 1 mPa 3 s, and h = 4.5 μm, eq 2 gives v = 0.05 mm 3 s1. The experimental value of the thinning velocity is 30 times higher than the value calculated with eq 2, which leads to the conclusion that the surfaces of the film may not be totally rigid, leading to a faster drainage. However, this model seems to catch the main draining mechanism since the scaling behavior is well described and the theoretical and experimental values are comparable. Let us now concentrate on the “Frankel’s film” area. Due to non-negligible size of the upper part of the film, if one wants to
compare the thickness of a soap film to Frankel’s law, the thickness measurement has to be done with care: (i) a thickness profile should be established to verify the existence of a “Frankel’s film” area; (ii) the thickness should be measured in the “Frankel’s film” area. It is not clear, in the previous studies,610 that the combination of position of the monochromatic light and extrapolation time (end of the film generation) gives a measurement of the thickness of a freshly formed film element. Without such precautions, the measured thickness could be underestimated and it may be one partial explanation for the deviation from one experiment to another, observed among all existing data collected by Van Nierop et al.16 We did the film entrainment experiment at different velocities and various concentrations (1, 3, 5, and 10 cmc) of C12E6 solutions. Results are shown in Figure 3 where the thickness normalized by the capillary length k1 (here k1 = 1.76 mm) is plotted versus the capillary number Ca. At low capillary number, data are consistent with Frankel’s law, for all concentrations. For high capillary numbers, the entrained thickness is lower than the prediction of Frankel’s law. Note that we did not measure thicknesses above Frankel’s values, as other researchers.16 The saturation at high capillary numbers was already observed and attributed to the finite wire thickness. However, in our case, the wire diameter is here far above all the measured thicknesses (Figure 3). In order to investigate further the role of the wire thickness, we have used two wire radii (namely 10 and 100 μm) and found that the entrained thickness was the same (data not shown). The observed saturation is therefore intrinsic to the studied solution. Another experimental observation is that the observed deviation from the Frankel’s law increases with the surfactant concentration. Let us note that a similar effect of surfactant concentration has already been observed in a LLD experimental configuration: when a solid plate is pulled out of a liquid bath, a thin film of thickness h is also entrained. Theory leads to an expression13 similar to Frankel’s law: h ¼ 2:38βk1 Ca2=3
ð3Þ
where β is a prefactor taking into account the effect of the surfactant: β = 1 in the case of rigid interfaces and β = 0.39 for pure liquids. Addition of surfactants thus thickens the film until the free surface becomes fully immobile. β has been shown to decrease at high concentration and high pull-off velocities which has been attributed to “reservoir” effects.1820 The transition occurs when the parameter σ = Γ/ch has an order of magnitude of 1 (in the case of nonionic surfactants) with Γ and c being the surface and bulk concentrations, respectively. If σ is larger than 1, the film does not contain enough surfactants to refill the surface during the pulling, leading to large surface concentration gradients and thus to large surface elastic moduli, helping to maintain the thickness. The thickness at the transition htr was determined for each concentration by the intersection between the “plateau” observed at large Ca and the theoretical curve shown in Figure 3. The inset of Figure 3 shows the values of h at the beginning of the transition as a function of the concentration normalized by the cmc. The solid line in the inset shows the calculated values of htr for σ = 1 (Γ = 2 molecules 3 nm2). The transition is therefore consistent with the assumption σ = 1 at high concentrations. However, the transition at c = cmc is not understood and will be discussed in the following. Note that no transition at c = cmc was observed in the LLD configuration by Delacotte et al.20 To our knowledge, this is the first time that this reservoir 13408
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the film surfaces cannot be considered as rigid. This prevents the use of the experiment as a tool to measure surface rheology as proposed earlier.16 The experiment will now be used to investigate film rupture. Preliminary results showed us that the film rupture occurs at a rather well-defined film thickness, hence the importance of a good understanding of what controls film thickness before rupture, which was achieved in the work reported in this paper.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected] (L.S.); emmanuelle.rio@ u-psud.fr (E.R.). Figure 4. Vertical thickness profile of a C12E6 (concentration = 10 cmc) film during entrainment at a speed V = 15 mm 3 s1. The film does not break during entrainment.
effect is observed in soap films and compared to the LLD observations. It shows that (i) concentration effects can be observed far above the cmc, (ii) increasing the amount of surfactant can lead to thinner films, (iii) the same reservoir effect is observed in Frankel’s geometry as in LLD experiment, and (iv) σ seems to be a good parameter to identify the thickness deviation resulting from confinement effects. When entrained at high velocity, the film has a higher thickness and the confinement effects disappear. This leads to a replenishment of the surface and a decrease of the surface elastic modulus E. When the surface elasticity decreases, it has been shown in LLD geometry that liquid entrainment decreases.21,22 We have studied in more detail this transition by measuring the profile of a thicker film. The evolution of the film profile against time during entrainment of a C12E6 film for a high concentration (c = 10 cmc) and a Ca higher than Ca at the transition (V = 15 mm 3 s1, Ca = 4.35 104) is shown in Figure 4. First of all, the two different areas of the film observed and described previously are not as clearly separated after the transition as before (Figure 2); second, the thickness at the bottom of the film, although homogeneous along a few millimeters, is not stationary anymore. We then think that the disappearance of confinement effects leads to films with mobile interfaces, which is expected to account for both a nonstationary and smaller thickness (relative to the value predicted by Frankel’s law) and a faster drainage. Moreover, note that the profile of a film at c = cmc after the transition still clearly exhibits a thinning part at the top and an area of homogeneous and stationary thickness at the bottom (data not shown) even if a transition appears (Figure 3). The behavior at c = cmc is then intermediate. Faster drainage, nonstationary profile, and smaller thickness then seem interrelated in a way which should be modeled in more detail in order to explain better our results. To summarize, a soap film entrained on a frame is made of two parts, a “Frankel’s film” at the bottom and a film that drains in the upper part. Thanks to an original setup allowing an instantaneous measurement of the film thickness with a good spatial resolution (200 μm), we were able to measure the spatial and temporal evolution of film thickness during film formation and to enlighten for the first time the existence of two zones. Consequently, checking the Frankel prediction for thickness is not straightforward, probably explaining the large scatter of previous experimental data. Furthermore, Frankel predictions appear to fail once
’ ACKNOWLEDGMENT L.S. benefitted from a CNES/CNRS PhD grant. F.R. thanks David Quere to have introduced him in the beautiful world of soap films. We are gratefull to D. Brunello for experiment interfacing. ’ REFERENCES (1) Weaire, D.; Hutzler, S. The physics of foams; Clarendon Press: Oxford, 1999. (2) Durand, M.; Stone, H. A. Phys. Rev. Lett. 2006, 97, 226101. (3) Biance, A. L.; Delbos, A.; Pitois, O. Phys. Rev. Lett. 2011, 106, 068301. (4) J. Mysels, S. F.; Shinoda, K. Soap Films: studies of their thinning; Pergamon: New York, 1959. (5) Vakarelski, I. U.; Manica, R.; Tang, X. S.; O’Shea, S. J.; Stevens, G. W.; Grieser, F.; Dagastine, R. R.; Chan, D. Y. C. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 11177–11182. (6) Shen, A. Q.; Gleason, B.; McKinley, G. H.; Stone, H. A. Phys. Fluids 2002, 14, 4055–4068. (7) Lionti-Addad, S.; di Meglio, J. M. Langmuir 1992, 8, 324–327. (8) Cohen-Addad, S.; di Meglio, J.-M. Langmuir 1994, 10, 773–778. (9) Adelizzi, E. A.; Troian, S. M. Langmuir 2004, 20, 7482–7492. (10) Berg, S.; Adelizzi, E. A.; Troian, S. M. Langmuir 2005, 21, 3867–3876. (11) Gibbs, J. W. The Scientific Papers of J. W. Gibbs; Dover: New York, 1961; Vol. 1. (12) de Gennes, P.-G. Langmuir 2001, 17, 2416–2419. (13) Landau, L.; Levich, B. Acta Physicochim. URSS 1942, 17, 42–54. (14) Lucassen-Reynders, E. H. Anionic Surfactants-Physical Chemistry of Surfactant Action; Marcel Dekker: New York, 1981. (15) Durand, M.; Langevin, D. Eur. Phys. J. E 2002, 7, 35–44. (16) Van Nierop, E. A.; Scheid, B.; Stone, H. A. J. Fluid Mech. 2008, 602, 119. (17) Georgieva, D.; Cagna, A.; Langevin, D. Soft Matter 2009, 5, 2063–2071. (18) Quere, D.; Ramdame, O. Langmuir 1997, 13, 2911. (19) Quere, D.; de Ryck, A.; Ramdane, O. O. Europhys. Lett. 1997, 37, 305–310. (20) Delacotte, J.; Montel, L.; Restagno, F.; Scheid, B.; Dollet, B.; Stone, H. A.; Langevin, D.; Rio, E. arXiv:1106.1972v1. (21) Park, C. W. J. Colloid Interface Sci. 1991, 146, 382–394. (22) Ratulowski, J.; Chang, H. C. J. Fluid Mech. 1990, 210, 303–328.
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