Mosaics and Two-Dimensional Foams of Freely Suspended Soap Films J. Mathe´,* J.-M. di Meglio,† and B. Tinland Figure 1. Semi-developed formula of a fluorinated betaine. Institut Charles Sadron (CNRS UPR 022), 6 rue Boussingault, 67083 Strasbourg Cedex, France Received September 15, 2000. In Final Form: July 16, 2001

1. Introduction Common soap films exhibit colors during their thinning when observed in white light.1-3 These colors originate from the interferences between light rays reflected by the two walls of the film. Upon drainage, when the film thickness approaches its final thermodynamic value, one can sometimes observe an interesting phenomenon of stratification due to the structuration of micelles:4,5 during the last stage of drainage, layers of micelles are ejected one by one, creating steps of thickness.2,3,6 This stepwise thinning of the film is revealed by the nucleation of very thin circular domains in expansion surrounded by a thicker rim6 and sometimes by very thick satellite pockets of liquid. Thin film stratification has been observed in a variety of different systems. Langevin et al.7 reported this stepwise thinning on different kinds of surfactant solutions and notably with zwitterionic surfactants at low concentration; the stratification is then attributed to swollen lamellar phases. Langevin observed these steps on the thinning front (i.e., the region of thickness abrupt change). Stratification is also observed with confined particles as colloidal spheres: the stratification is then explained as a layerby-layer expulsion out of the colloidal crystal formed by the spherical particles.8 The addition of polyelectrolytes9,10 in a soap film can influence not only the stability of the film but also its thinning. In the semidilute regime of polymer concentration, stratifications also appear, whose step heights correspond to the correlation length of the polymer solution. Similar phenomena have been also observed on thermotropic smectic films; observations of two-dimensional foams11 and mosaics12 have been performed on smectic films at the water-air interface. Some observations have been carried out with freely suspended films:13,14 stratifications are visible only at the meniscus of the film or around a defect, and the measured step heights were equal to the thickness of a single smectic layer. †

Universite´ Louis Pasteur and Institut Universitaire de France.

(1) Newton, I. Opticks; Dover: New York, 1952. (2) Johnnott, E. S. Philos. Mag. 1906, 70, 1339. (3) Perrin, J. Ann. Phys. 1918, 10, 160. (4) Nikolov, A. D.; Wasan, D. T. J. Colloid Interface Sci. 1989, 133, 1. (5) Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B.; Wasan, D. T. J. Colloid Interface Sci. 1989, 133, 13. (6) Bergeron, V.; Jimenez-Laguna, A. I.; Radke, C. J. Langmuir 1992, 8, 3027. (7) Langevin, D.; Sonin, A. A. Adv. Colloid Interface Sci. 1994, 51, 1. (8) Wasan, D. T.; Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B. Colloids Surf. 1992, 67, 139. (9) Klitzing, R. V.; Espert, A.; Asnacios, A.; Hellweg, T.; Colin, A.; Langevin, D. Colloids Surf., A 1999, 149, 131. (10) Bergeron, V.; Langevin, D.; Asnacios, A. Langmuir 1996, 12, 1550. (11) Friedenberg, M. C.; Fuller, G. C.; Franck, C. W.; Robertson, C. R. Langmuir 1994, 10, 1251. (12) de Mul, M. N. G.; Adin Mann, J., Jr. Langmuir 1998, 14, 2455. (13) Ge´minard, J.-C.; Holyst, R.; Oswald, P. Phys. Rev. Lett. 1997, 78, 1924.

In this note, we report an original stratification with freely suspended lyotropic films (soap films made out of fluorinated surfactant aqueous solution): the stratification generates two-dimensional foams or regions with different thicknesses that gather to form mosaic structures. We compare these multilayers to copolymer lamellar phases incorporating edge dislocations. 2. Experimental Section We have used a fluorinated betaine (Figure 1) (a zwitterionic molecule) as surfactant, kindly provided by Elf-Atochem, in solution in Milli-Q water; the critical micellar concentration is 0.01% w/w and a lamellar phase appears above 0.4% w/w.15 All the results presented in this note have been obtained with a 0.55% w/w solution where lamellae are then expected to form; the viscosity of the solution does not depart significantly from the viscosity of pure water. In these conditions, stratification phenomena are always observed. We have used the so-called porous plate method first introduced by Mysels16 and later developed by Exerowa et al.17,18 The soap film is formed on a conical hole (5 mm in diameter) drilled in a sintered-glass disk that acts as a reservoir for the soapy solution. We did not use in the reported experiments the opportunity to monitor the pressure inside the film; the pressure of the film is then determined by the Laplace pressure difference due to the pores of the glass disk. The film is made by stretching a liquid film over the hole using a laboratory spatula. The experiment is done in an open air cell without control of the humidity. The observation of the film is performed with an optical microscope under reflection of white light. The measurement of the thickness is done by an interferometric method, as originally described by Scheludko.19 It consists of the study of the interferences between light rays reflected by the two interfaces of the thin liquid film. The relation between reflected intensity I and thickness h for monochromatic light of wavelength λ is

h)

λ (2πn ) arcsinx1 + 4R(1 -∆∆)/(1 - R)

2

(1)

where ∆ ) (I - Imin)/(Imax - Imin) and R ) (n - 1)2/(n + 1)2 with n = 1.33 being the refractive index of the surfactant solution. Imax and Imin are the maximum and minimum of the reflected intensity. The determination of the film thickness thus depends on the light wavelength λ, excepted for small thickness (h , λ) where the interference patterns are mixed which allows the use of white light. We have used a mean wavelength value λ ) 550 nm to calculate the thickness.

3. Results 3.1. Observations. Just after the formation of the film, a thinning associated with the drainage is observed: the soapy solution is sucked inside the porous plate. This thinning is very fast because of the high capillary suction of the pores (between 5 and 10 µm in diameter). We observe (14) Pieranski, P.; Be´liard, L.; Tournellec, J.-P.; Leoncini, X.; Furtlehner, C.; Dumoulin, H.; Riou, E.; Jouvin, B.; Fe´ne´rol, J.-P.; Palaric, P.; Heuving, J.; Cartier, B.; Kraus, I. Physica A 1993, 194, 364. (15) Pabon, M. Private communication. (16) Mysels, K. J.; Jones, M. N. Discuss. Faraday Soc. 1966, 42, 42. (17) Exerowa, D.; Scheludko, A. C. R. Acad. Bulg. Sci. 1971, 24, 47. (18) Exerowa, D.; Kolarov, T.; Khristov, K. H. R. Colloids Surf. 1987, 22, 171. (19) Scheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391.

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Figure 2. Left: two-dimensional foam. Right: stratification in mosaics of a thin liquid film.

the thinning front seen by Langevin,7 with discrete steps of thickness; the front then disappears in the porous plate but leaves behind very stable structures, either two-dimensional foam (Figure 2, left) or some jointed regions with different thicknesses (Figure 2, right) gathered in a mosaic. The boundaries of these regions of the film are full circles or circle arcs. These mosaics are very stable; they can last for several hours. Nevertheless, they undergo two dynamic processes: (i) the constitutive tiles are attracted and sucked by the meniscus and eventually incorporate the border of the film and (ii) during their motion toward the border of the film, tiles of similar thickness merge. Eventually, the most stable state of the film is a black film formed by a bilayer of surfactant or a broken film. The formation of mosaics is very fast and thus very puzzling. They appear after the passage of the thinning front, when the film is thick. The measurement of the light reflected by the different tiles of the mosaic shows that the thickness of each tile is an integer multiple of an elementary thickness (=4.5 nm) which is consistent with the thickness of one bilayer of surfactant (Figure 3 and Table 1). Two-dimensional foams are observed only for the thinnest films. They are formed by the encounter of holes of black film in expansion described by Bergeron.20,21 3.2. Measurements of Line Tensions. A noticeable feature of Figure 2 is the presence of triple points (analogous to triple lines in wetting) where three regions of different thicknesses meet and define three contact angles (Figure 4); these angles are not equal to 120° as in usual two-dimensional foams (e.g., a monolayer of soap bubbles squeezed between two plates). Writing the mechanical equilibrium at a vertex, we can deduce a relative value of the three line tensions (τij, τjk, τik) from the three contact angles (Rij, Rjk, Rik) by the following relationship

τij + τik cos Rjk + τjk cos Rik ) 0

(2)

(τij’s and Rij’s are defined on Figure 4, right). We can get (20) Bergeron, V.; Radke, C. J. Langmuir 1992, 8, 3020. (21) Bergeron, V. Langmuir 1996, 12, 5751.

Table 1. Thickness of Tiles of Figure 3 region no. thickness (nm)

1

2

3

4

5

6

4.3

8.8

13.0

18.4

23.1

29.6

only relative values of the line tensions since the contact angles are not independent (Rij + Rjk + Rik ) 2π). 3.3. Discussion. The relative values of line tension measured using eq 2 are represented on Figure 5. The uncertainty is very important at large j because of bad statistics: dislocations of large b are rare. Note that there is no need to take into account the possible dependence of the surface tension of the different tiles with thickness to determine the line tension values since changing the shape of a single tile with a constant surface area does not change the surface energy. Smectic multilayers and their edge dislocations have been originally studied theoretically by de Gennes22 and later by Kle´man and Williams.23 Edge dislocations have been studied by Turner and co-workers more specifically for spin-coated copolymer lamellar films;24 they derived the energy (per unit length) or line tension of an edge dislocation in a copolymer lamellar film which reads as

(

F = b2xKB

)

qc Γ + + Cb π κh

(3)

where b is the height of the dislocation (Burgers vector); K and B are the splay constant and the compression modulus, respectively; κ ) xK/B is the penetration length; Γ ) γ/xKB is a dimensionless surface tension (γ is the surface tension of the film); h is the thickness of the sample without dislocation; and qc ∼ b-1 is a cutoff needed to avoid the very distorted region of the core of the dislocation. The last term is the core energy: the theoretical description of the core is difficult since it is curved (22) de Gennes, P.-G. J. Phys. (Paris) C4 1969, 30, C4. (23) Kle´man, M.; Williams, C. E. J. Phys., Lett. (Paris) 1974, 35, L-49. (24) Turner, M. S.; Maaloum, M.; Ausserre´, D.; Joanny, J.-F.; Kunz, M. J. Phys. (Paris) II 1994, 4, 689.

Notes

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Figure 3. (a) Section of a transition from one (region 1) to two bilayers (region 2). (b) Section of a transition from two (region 2) to three bilayers (region 3). (c) Numbers represent the numbers of bilayers of each tile. (d) Three-dimensional representation of (c).

Figure 4. Left: junction of three mosaic tiles. Right: definition of contact angles and line tensions.

Figure 5. Mean line tension values (in units of τ12) between a tile of one (left) or of two (right) bilayer thickness and a tile with j bilayers. Dotted lines are guides for the eye.

at a molecular scale and this term may represent about 50% of the energy on copolymer systems. In the case of charged systems, we are not aware of a detailed description of the dislocation core. The influence of surface tension on the stratification stability has been considered by Lejcek and Oswald:25 they showed that the equilibrium position of the dislocation is governed by the parameter Γ. If Γ >

1, a dislocation is stabilized within the film; otherwise it is attracted to the film surfaces. This parameter Γ is not easy to evaluate in our case and would require a comprehensive study of the hydrodynamic modes of the film using quasi-elastic light scattering; moreover, it would (25) Lejcek, L.; Oswald, P. J. Phys. (Paris) II 1991, 1, 931.

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be experimentally difficult to locate precisely the position of the dislocation within the film. Introducing h0 as the thickness of a surfactant bilayer, the line tension between a tile made of i bilayers and a tile made of j (j > i) bilayers (b ) (j - i)h0) reads (h ) ih0)

(

)

1 τij ∼ (j - i)2 c1 + c2 + c3(j - i) xi

(4)

where the ci’s are constants with units of energy per unit length. Obviously, this prediction cannot fit our experimental values that show a nonmonotonic behavior. We believe that the origin of this discrepancy has to be searched in the very detailed structure of the core of the dislocation with large Burgers vectors b; the theory seems anyway compatible with experiments for small b. 4. Conclusion We have studied soap films made of zwitterionic surfactant in their lamellar phase. We have observed, for

the first time on freely suspended soap films, stable mosaics with the coexistence of tiles of different discrete thicknesses. We have been able to show that the line tension of the tiles strongly depends on the Burgers vector value; a theory for this type of dislocation clearly remains to be developed. The formation of mosaics might have an important input on the exceptional stability of fluorinated soap films: for instance, the lifetime of a black soap film made with the surfactant used in this study may exceed several days without extended control of humidity to be compared to a duration of a few minutes for films made with SDS (sodium dodecyl sulfate) in the same conditions; this observation corroborates that lyotropic liquid crystals enhance foam stability.26 LA0013244 (26) Friberg, S.; Ahmad, S. I. J. Colloid Interface Sci. 1971, 35, 175.