Surfactant Films in Biliquid Foams - Langmuir (ACS Publications)

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Surfactant Films in Biliquid Foams† O. Sonneville-Aubrun,‡,§ V. Bergeron,‡,| T. Gulik-Krzywicki,⊥ Bo Jo¨nsson,∇ H. Wennerstro¨m,X P. Lindner,× and B. Cabane*,∞ Equipe mixte CEA-RP, Rhoˆ ne-Poulenc, F 93308 Aubervilliers, France, Centre de Ge´ ne´ tique mole´ culaire, CNRS, F 91198 Gif sur Yvette, France, Physical Chemistry 1 and 2, Center for Chemistry and Chemical Engineering, University of Lund, P.O. Box 124, S 221 00 Lund, Sweden, Institut Laue Langevin, B.P. 156, F 38042 Grenoble, France, and Laboratoire PMMH, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France Received May 18, 1999. In Final Form: October 20, 1999 Centrifugation can be used to remove the continuous aqueous phase of an oil-in-water emulsion. The cream that remains after most of the water has been removed has the structure of a biliquid foam; it can be redispersed in water. Examination of this cream through electron microscopy shows polyhedral oil cells separated by thin films. The thickness of these films has been measured through small-angle neutron scattering. The results yield a disjoining pressure isotherm, where the film thickness is solely determined by the pressure applied to extract water during centrifugation. For hexadecane-in-water biliquid foams, stabilized with sodium dodecyl sulfate (SDS), this isotherm has two states, the common black film (CBF; water thickness beyond 25 Å) and the Newton black film (NBF; (water thickness of 13 Å). At low pressures (1-50 atm), the films are in the CBF state, where the measured disjoining pressure matches the entropic pressure of the counterions, calculated from the Poisson-Boltzmann equation. At high pressures (20-300 atm), ionic correlations in the counterion layer reduce the disjoining pressure and the films jump discontinuously to the NBF. The thickness of the NBF is stabilized by hydration forces, which resist the dehydration of counterions and headgroups. The surface density of SDS molecules in these films has also been measured. As water is extracted, the concentration of counterions increases, and they screen the headgroups more efficiently; as a result, the surface density of SDS in the monolayers rises. In the NBF state, the monolayers are tightly packed, with an orientational order that exceeds that of the lamellar phase. This tighter packing of surfactant molecules may explain the surprisingly high metastability of biliquid foams when the films are in the NBF state.

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edge or at a vertex. The polyhedral cell types that meet these requirements have been examined in a beautiful paper by Lissant.1 The mechanical consequence is that these foams resist flow, because the surface tension opposes the deformation of cells. Consequently, these biliquid foams are also called “gel emulsions”.2,3 Finally, the thermodynamic consequence of surface tension is that all biliquid foams are out of equilibrium; indeed, the state of lowest free energy is a macroscopic separation of the two liquids with a minimal area of their interface. This stable state can be reached by film rupture and coalescence of the cells or by transfer of molecules from smaller cells into larger ones until the small ones vanish. Both processes are opposed by the surfactant molecules in the films. Consequently, the metastability of biliquid foams is determined by the state of the surfactant films. In regular foams, much is known about the state of the films, thanks to studies of macroscopic films drawn from ionic surfactant solutions.4-8 As water is extracted from a freely suspended film, the interference colors of reflected light show that the total film thickness decreases regularly to about 10 nm, where the film appears black because it

(1) Lissant, K. J. J. Colloid Interface Sci. 1966, 22, 462.

(2) Solans, C.; Pons, R.; Zhu, S.; Davis, H. T.; Evans, D. F.; Nakamura, K.; Kunieda, H. Langmuir 1993, 9, 1479. (3) Kunieda, H.; Fukui, Y.; Uchiyama, H.; Solans, C. Langmuir 1996, 12, 2136. (4) Jones, M. N.; Mysels, K. J.; Scholten, P. C. Trans. Faraday Soc. 1966, 62, 1336. (5) Exerowa, D.; Nikolov, A.; Zacharieva, M. J. Colloid Interface Sci. 1981, 81, 419. (6) De Feijter, J. A.; Vrij, A. J. Colloid Interface Sci. 1978, 64, 269. (7) Benattar, J. J.; Schalchli, A.; Be´lorgey, O. J. Phys. II (France) 1992, 2, 955. (8) Bergeron, V.; Radke, C. J. Langmuir 1992, 8, 3020.

Introduction Biliquid foams are arrays of cells filled with a liquid and separated by thin films containing another liquid. Their structure is in all respects similar to that of ordinary foams, except that the cells are filled with a liquid instead of a gas. A common type of biliquid foam is made by concentrating emulsions; these biliquid foams are also called “high internal phase ratio emulsions”.1 For instance, the continuous aqueous phase of an oil-in-water emulsion can be extracted until the droplets come into contact and deform to fill the available space. In this case the cells are filled with oil, and the films that separate them are made of two surfactant monolayers separated by a layer of water. The films that separate oil and water have a surface tension, and this has structural, mechanical, and thermodynamic consequences. The structural consequence is the formation of structures that tend to minimize the area of films. Indeed, the films tend to be flat (if they separate cells of comparable internal pressures) and meet according to rules that ensure the equilibrium of tensions along an This work used the neutron beams of LLB and ILL. * To whom correspondence should be addressed. ‡ Rho ˆ ne-Poulenc. § Current address: L’OREAL, Centre de Chevilly, BP 553, 94152 Chevilly la Rue, France. | Current address: Rho ˆ ne Poulenc, CRIT-C, B.P. 62, 69192 Saint Fons Cedex, France. ⊥ CNRS. ∇ Physical Chemistry 2, University of Lund. X Physical Chemistry 1, University of Lund. × Institut Laue Langevin. ∞ ESPCI.

10.1021/la990599k CCC: $19.00 © 2000 American Chemical Society Published on Web 01/08/2000

Surfactant Films in Biliquid Foams

is too thin to reflect light. This state is called the common black film (CBF). If more water is extracted, there is a discontinuous jump to another state, where the film is even thinner; this is the Newton black film (NBF).4,9,10 The NBF state can have a high metastability, presumably related to the organization of surfactant molecules in the monolayers. The transition from CBF to NBF is caused by a change in the balance of forces applied to the films and forces within the films (the disjoining pressure). Films that separate oil droplets rather than air bubbles have been comparatively less studied. Macroscopic films separating large oil domains are difficult to stabilize, because they can easily be ruptured by mechanical vibrations transmitted through the oil.11 On the other hand, microscopic films separating droplets in biliquid foams1-3 and in flocculated emulsions12,13 have longer lifetimes, thanks to their small dimensions. Flocculated emulsions have been studied by Poulin et al. With ionic surfactants, they found that the addition of salt causes a transition from a state where the opposing surfactant monolayers repel each other to a state where they attract. In the adhesive state, the film thickness is only a few nanometers, as in the NBF state of regular foams. This raises a set of interesting questions: (i) Is the NBF a general feature of all ionic surfactant films? (ii) What effect does it have on the adhesion of droplets? (iii) What effect does it have on the metastability of the films? In principle, these questions could be answered by studying biliquid foams, where the films can be studied at different thicknesses. However, the films of biliquid foams have, to our knowledge, not been studied yet. To understand the state of the surfactant films in biliquid foams, we need two sets of information. First, we need to determine the balance of forces in the film: so far we do not know what forces cause the transition from CBF to NBF and what forces then stabilize the thickness of these films. Second, we need to know the state of surfactant molecules in the monolayers: are they in a constrained liquid state, as in lamellar surfactant phases, what is their surface density and their orientational order, and what are the consequences of their organization for the strength of the monolayers? This information is directly related to the metastability of the films. For the purpose of obtaining this information, biliquid foams made by concentrating oil-in-water emulsions through centrifugation are useful systems. Indeed, the osmotic pressures that are applied to extract the aqueous phase of emulsions measure the balance of forces that work to retain water in the films.14 Consequently, centrifugation experiments can be used as a surface force apparatus to measure this balance of forces. In a first step, we used osmotic pressures calculated from centrifugation experiments, and film thicknesses measured through small-angle neutron scattering (SANS), to obtain the equation of state for the thickness of the films. In a second step, we used chemical analysis to determine the surface density of surfactant molecules in the films and nuclear magnetic resonance (NMR) to determine their orientational order. The results show that the compression of the foams (i.e., the extraction of water) produces substantial changes in the structures of the films. (9) Newton, I. Opticks; Smith and Walford: London, 1704; Book II, Part I, Obs. 17. (10) Gibbs, J. G. Trans. Connecticut Acad. 1876-8, 3, 343; Collected Works; Longmans Green: New York, 1931. (11) Bergeron, V. To be published. (12) Aronson, M. P.; Princen, H. M. Nature (London) 1980, 286, 370. (13) Poulin, P.; Nallet, F.; Cabane, B.; Bibette, J. Phys. Rev. Lett. 1996, 77, 3248. (14) Narsinham, G. Colloids Surf. 1992, 62, 41.

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The aim of this paper is to report these changes, find out how they make sense according to the known forces in the films, and examine whether they give a clue to the unusual metastability of the films. Methods Emulsification. Coarse emulsions were made by mixing hexadecane with aqueous solutions of sodium dodecyl sulfate (SDS) in a laboratory mixer. These coarse emulsions were made finer by repeated passages through a laboratory homogenizer (Microfluidics MT110). The highly fragmented droplets subsequently recoalesced in the outlet pipe of the device, until the amount of surfactant per droplet was enough to prevent further recoalescence.15 The stationary state where fragmentation and recoalescence balance each other was reached after 10 cycles through the homogenizer. At this stage, a homogeneous distribution of droplet sizes was obtained, with a width of about 10% according to quasi-elastic light scattering and to neutron scattering. The average droplet diameter in the homogenizer was controlled by changing the surfactant-to-oil ratio.15 It could be set in the range from 80 nm to 3 µm by choosing the ratio of SDS to hexadecane between 10-1 and 10-4. In all cases, the oil/water interfaces were not saturated with SDS molecules. However, to achieve a high metastability of the emulsion, it was necessary to produce saturated interfaces. This was done after the emulsification was completed, by diluting the emulsion with an aqueous solution of SDS. The surfaces were considered as saturated when the concentration of SDS in the aqueous phase reached the critical micelle concentration (cmc ) 2.3 × 10-3 g/g). According to the adsorption isotherm,16 this equilibrium corresponds to a surface coverage of 2 molecules/nm2. Thus, the amount of SDS needed was calculated according to this surface coverage plus the amount needed to reach the cmc in the aqueous phase. In one case, we verified that the concentration of SDS in the aqueous phase was actually at the cmc. Centrifugation. The aqueous phase of the emulsions was extracted by centrifugation at speeds of 5000-50 000 rpm, using various rotors with swinging holders. The compression of emulsions occurs in two stages. In the first stage (in our case a few minutes), the emulsion droplets (hexadecane density 773 kg/m3) migrate to the top of the tube where they form a white “cream” layer. In a second stage (in our case 10 h), water is drained through the system of films and plateau borders. During this second stage, the cream turns into a biliquid foam with an oil volume fraction in excess of 0.9 and can become transparent and elastic (gellike). Once centrifugation equilibrium is established, flat plane-parallel films are formed and the centrifugal force is balanced by repulsive forces within the films that oppose any further extraction of water; therefore, at all locations in the foam, the centrifugal force is equal to the gradient of the osmotic pressure (also called the disjoining pressure of the films).14 Consider a slice of biliquid foam in the centrifugation tube at a distance z from the axis of rotation (Figure 1). The volume fraction of oil in this slice is φ(z), the difference in density between hexadecane and water is ∆F, and the rotation speed is ω. The equilibrium between the centrifugal force and the gradient of osmotic pressure (15) Taisne, L.; Walstra, P.; Cabane, B. J. Colloid Interface Sci. 1996, 184, 378. (16) Rehfeld, S. J. Phys. Chem. 1967, 71, 768.

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Figure 1. Sketch of the centrifugation experiment and photograph of the slices cut out of the biliquid foam that formed at the top of the centrifugation tube.

exerted on this slice by the neighboring slices can then be written as follows:

dΠ/dz ) -∆Fφ(z) ω2z

(1)

When the cream is highly compressed, the volume fraction φ(z) is close to unity throughout the cream. The differential equation can then be integrated between the bottom of the cream, located at a distance z0, where the osmotic pressure Π(z0) vanishes and any location in the biliquid foam located at a distance z:

Π(z) ) (1/2)∆Fω2 (z02 - z2) + Π(z0)

(2)

In a typical experiment, z ) 50 mm at the top of the cream and z0 - z ) 35 mm. For such conditions the maximum pressure is Π ) 53 atm at a rotation speed of 30 000 rpm. We have verified that this equation correctly predicts the osmotic pressure by using different combinations of centrifugation speeds ω and heights (z0 - z) in the biliquid foam. Indeed, different combinations that gave the same predicted pressure according to eq 2 were found

to yield the same state of the foam, as measured by the amount of water that remained in the films. Recovery of Biliquid Foam Samples. Because the biliquid foam is elastic, it could be pulled out as a whole from the centrifugation tube. To facilitate this operation, the centrifugation tube was lined, before use, with a film of poly(ethylene terephthalate) (PET). Immediately after centrifugation, the PET film with the biliquid foam in it was pulled out, and the aqueous subphase was left in the tube. The biliquid foam was then cut into slices of equal thickness; typically, 10 slices were obtained from a sample that produced a total foam height of 35 mm. The slices closest to the aqueous subphase were soft and white. These slices were discarded because there was a small uncertainty in their water content (in these slices the gradient in the water content was high, and at the end of centrifugation there was some diffusion of water from the subphase). All other slices were fairly rigid and transparent, indicating that their water content was quite low. The osmotic pressure in each slice could then be calculated according to eq 2. These slices were then stored in hermetically closed Eppendorf tubes (volume 1.5 mL). They could be kept in this way for long times (1 month) with no change in structure (i.e., no coalescence) as long as water evaporation was prevented. Dehydration through Vapor Exchange. Some slices of biliquid foams were submitted to a stronger dehydration, through equilibrium at a set relative humidity. The equilibrium was reached by placing them in closed tubes that contained a saturated salt solution. The osmotic pressure of the biliquid foam was then calculated from the known vapor pressure of the salt solution. Chemical Analysis. In some biliquid foams made through centrifugation, all slices were analyzed to determine the amounts of water and surfactant that formed the films; the experimental protocols are detailed in Appendix A. The amount of water was compared with the total extension of the films, to obtain the average film thickness. Indeed, in slices that were submitted to a high osmotic pressure, the plateau borders were collapsed, and most of the water was in the films. In this case the film thickness h could be calculated from the volume fraction ∆φ of water and from the droplet diameter according to

∆φ/φ ) 3h/D

(3)

Thicknesses calculated in this way were identical with those measured through neutron scattering (see below) when the applied pressures were at least 10 times the Laplace pressure of the undeformed droplets. The amount of surfactant was also compared with the total extension of the films, calculated from the droplet diameter D, to obtain the surface density of SDS in the monolayers, or the area per SDS molecule at the oil/water interface. In the case of the original emulsion, we found that the area per SDS molecule was 52 Å2, which is in excellent agreement with the value determined by Rehfeld at a macroscopic interface between heptadecane and an aqueous SDS solution at the cmc16 and with that determined by Bergeron in the case of hexadecane.11 Freeze Fracture Microscopy. Small pieces of biliquid foam, about 20-50 µm thick, were deposited on a thin copper holder. Either the holder was rapidly quenched in liquid propane or the sample was first squeezed between the holder and a thin copper plate before being quenched in liquid propane. Both types of preparations were fractured in vacuo (about 10-7 Torr) with the liquidnitrogen knife inside a Balzers 301 freeze etching unit.


Figure 2. Profile of the density of scattering length ∆F(z) across a film of deuterated SDS and water separating two oil cells in the biliquid foam. h is the thickness of the water layer and δ that of the SDS monolayers.

The squeezed preparations were fractured by removing the upper plate with the cold knife. A replica of the sample was made using unidirectional shadowing with platinumcarbon at an angle of 35°. The mean thickness of the metal deposit was 1-1.5 nm. The replicas were subsequently washed with THF, ethanol, and distilled water and then observed in a Philips EM410 microscope. Contrast in the resulting images is related to the depth fluctuations of the metal deposit. To check that there were no artifacts caused by the crystallization of water, we also made biliquid foams where the aqueous phase was a mixture of 1/3 glycerol and 2/3 water. These samples gave images with the same features as the biliquid foams made with pure water. SANS. Some slices of biliquid foams were examined through SANS. In a biliquid foam, small-angle scattering is produced by differences in the scattering length density between the aqueous films and the cell interiors which comprise the dispersed oil phase. There is a range of scattering vectors q where the interference patterns reflect the profile of scattering length density F across a bilayer. We used this scattering to determine the thickness of the bilayers, as was previously done by Poulin et al. in the case of flocculated emulsions.13 The instruments used were PACE at Laboratoire Le´on Brillouin (LLB) and D11 at Institut Laue Langevin (ILL). Poulin et al. used emulsions made of C12D26, SDS, and D2O, in which case the SDS monolayers (low F) stand out in a background made of C12D26 and D2O (high F). We used the opposite contrast, with C16H34, deuterated SDS, and H2O, where the SDS monolayers (high F) stand out in a background made of C12H26 and H2O (low F). A schematic of the scattering length density profile is shown in Figure 2. The scattering length densities are F ) -0.43 for C16H34, F ) 7.13 for the deuterated chains of SDS, and F ) -0.43 for the water in the films (a H2O + D2O mixture chosen to match the scattering length density of the oil, hereafter called H2O because it contained only 2% D2O). Scattering curves (intensity I vs magnitude q of the scattering vector) were obtained by radial averaging of interference patterns. At high q, these scattering curves show an oscillation that reflects the film profile: the period of the oscillation corresponds to the spacing δ + h between the middle planes of the SDS monolayers. We calculated the scattering for a set of bilayers that had random orientations and similar profiles, as was previously done by Poulin et al.13 (Appendix B). This calculated scattering reproduced the oscillation observed in the experimental scattering curves. From the fit we obtained three parameters that characterize the profile of the bilayers: the thickness of SDS monolayers, δ, the thickness of water layers, h, and the average fluctuation in thickness, σ.

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NMR. The same biliquid foams, made with deuterated SDS, were examined through deuterium NMR. For each CD2 (or CD3) group of the SDS chain, the NMR spectrum is split by the residual quadrupolar interaction left by the anisotropy of chain motions in the bilayer.17-23 For a set of bilayers with random orientations, a powder spectrum (“Pake doublet”) is obtained, which is a superposition of the spectra with splittings corresponding to all orientations. The experimental spectra were submitted to a “depakeing” procedure, which restitutes the spectrum from bilayers oriented with their normal along the magnetic field. These spectra show a set of discrete doublets, each corresponding to the line of a particular type of CD bond along the chain. The frequency splitting of each doublet in the “depaked” spectrum, ∆νQ, is related to the orientational order parameter SCD of the CD bond and to the frequency νQ ) 167 kHz of the quadrupolar interaction by

∆νQ ) (3/2)νQSCD

(4)

The plot of order parameter Sj vs the position j of the carbon along the SDS chain constitutes the order parameter profile of the chain in the bilayer; it reflects the average anisotropy of chain motions imposed by steric constraints from neighboring chains. In a highly ordered bilayer, with all chains stretched along the normal, the magnitude of the order parameter of the chain would be S ) 1 and that of the CD bond would be 1/2. Results Centrifugation Equilibrium. For each experiment, an emulsion was centrifuged until centrifugation equilibrium was reached; i.e., the centrifugal force was equilibrated by the osmotic pressure gradient. Then the centrifugation was stopped, and the sample was sliced to provide a set of biliquid foams with different osmotic pressures and water contents. In this way the ultracentrifuge was used as a surface force instrument. In a first step, we verified that the measured water contents were an equilibrium property of a biliquid foam submitted to the osmotic pressure calculated as in eq 2. Indeed, we found that the profile of water content in the foam was stable after a few hours: measurements made after 13 h and after 23 h of centrifugation gave identical profiles for the water content. Conversely, samples that were left at rest in equilibrium with the supernatant swelled up with water in less than 1 h. This is in agreement with the known rates of water self-diffusion in bilayers. Accordingly, it was assumed that the water equilibrium was reached after times shorter than 13 h. Another question was whether the distribution of surfactant in the samples was also an equilibrium property of the biliquid foam submitted to these pressures. Measurements of the surfactant concentration in various slices gave values that matched the equilibrium surface density of SDS in monolayers (at low pressures) or in bilayers (at high pressures) (see below). However, the diffusion rates of surfactant molecules in bilayers may not be fast enough (17) Charvolin, J.; Manneville, P.; Deloche, B. Chem. Phys. Lett. 1973, 23, 345. (18) de Gennes, P. G. Phys. Lett. 1974, A47, 123. (19) Davis, J. H. Biochim. Biophys. Acta 1983, 737, 117. (20) Ke´kicheff, P.; Cabane, B.; Rawiso, M. J. Colloid Interface Sci. 1984, 102, 51. (21) Charvolin, J.; Hendrikx, Y. In Nuclear Magnetic Resonance of Liquid Crystals; Emsley, J. W., Ed.; D. Reidel: Dordrecht, The Netherlands, 1985. (22) So¨derman, O.; Walderhaug, H.; Henriksson, U.; Stilbs, P. J. Phys. Chem. 1985, 89, 3693. (23) Samulski, E. Exxon Symposium.

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to provide the SDS needed to reach these values. Therefore, there must be another mechanism that is responsible for the adjustment of the surface density of SDS to its equilibrium value. We presume that this mechanism is a limited coalescence of the emulsion during the initial stage of compression. The question of resistance to coalescence is also quite interesting. Coalescence of the biliquid foam was easily noticed, as it produced an oil layer on top of the sample, where the osmotic pressure is highest. We did observe coalescence in biliquid foams where the monolayers were not saturated with SDS: there was an oil layer that grew regularly with centrifugation time. We did not find an oil layer in biliquid foams where the monolayers were saturated with SDS, unless the samples were kept for extremely long times (1 month) at the highest pressures (water content set by centrifugation at the highest speeds or by vapor exchange). By “saturated with SDS”, we mean the following: All biliquid foams described in this work were made from emulsions with an aqueous phase that contained SDS at the cmc. The surface density of SDS in the monolayers of these emulsions is that set by the concentration of SDS in the aqueous phase, i.e., 52 Å2/ molecule. During centrifugation, this surface density may change, because of a limited number of coalescence events that occur as the droplets are compressed into a biliquid foam. This produces a biliquid foam that has extremely high metastability. This metastability was checked by reswelling them with water: metastable biliquid foams gave emulsions with the same average droplet size as the original emulsion. We now describe observations made on these highly metastable biliquid foams, starting with observations at a mesoscopic scale, which give the overall organization of the biliquid foam (cell sizes and shapes and packing of cells). Then we go on to observations at a higher level of resolution (thickness of films that separate cells). We finish with observations at the local scale (state of the surfactant molecules in the films). Cell Sizes and Shapes. Images of the foam were obtained through freeze fracture electron microscopy. They show the trace of a fracture across the biliquid foam. The contrast depends on how the fracture was deviated by structures in the foam. This was found to depend on the nature of the oil. With hexadecane, the oil crystallized during the cooling stage of the freeze fracture procedure, and the fracture followed the boundaries of crystallites. Because crystal growth stopped at cell boundaries, these images reflect the overall shapes of cells. On the other hand, the growth of crystals made it impossible to observe the original state of the films. With mineral oil or squalane, the oil was quenched in an amorphous state, and the fracture propagated straight across the cells. In this case the films that separated cells were visualized because they produced small steps in the fracture as it crossed from one cell to the next. Biliquid Foams Made of Hexadecane/SDS/Water. Images from the most compressed slices of a biliquid foam which had been centrifuged at 20 atm are shown in Figure 3. They show an array of polyhedral cells. The cell sizes are fairly homogeneous, and they match the diameters of the original emulsion droplets (1 µm). The cells are polyhedral, indicating that the pressure applied to this slice far exceeded the Laplace pressure of undeformed droplets (PL ) 0.4 atm). The films that separate cells are not visible in this slice nor are the Plateau borders that join them. Images from the least compressed slices (not shown here) show rounded cells separated by thick aqueous films.


Figure 3. Freeze fracture image of the most compressed slice in a hexadecane-SDS-water biliquid foam. Scale bar 1 µm. The droplet size of the original emulsion was 1 µm, and the SDS concentration in the aqueous phase was the cmc. The osmotic pressure exerted by centrifugation was 20 atm.

Biliquid Foams Made with Squalane or Mineral Oil. Replacing hexadecane by squalane or mineral oil suppresses oil crystallization problems. The most compressed slices of the biliquid foam show a smooth fracture plane with thin steps caused by the films that separate cells (Figures 4 and 5). Some films do not show up at all; consequently, the foam looks more polydisperse than it really is. The films are extremely thin, at the limit of resolution of the picture (below 5 nm). The junctions of films (indicated by arrows in Figures 4 and 5) are not rounded, indicating that the Plateau borders are completely collapsed. This was expected, because the pressure applied to these slices (P ) 30 atm) far exceeds the Laplace pressure of the droplets. Two minor features can also be noticed on the micrographs. At various locations in the films, some nanometric droplets appear to be trapped (an arrow in Figure 4 indicates a place where the fracture went along a film with trapped droplets in it). It may be that, upon cooling, the films collapsed and expelled some water into these droplets. Therefore, it is important to measure the film thickness of the original foam, without freezing, as we did using SANS. Also, in rare places, the fracture went through regions that contained threadlike objects. A likely explanation for this feature is that a few coalescence events occurred during the preparation of the biliquid foam. The released surfactant formed small pieces of mesophases that remained trapped between droplets. The images from the less compressed slices show a similar array of very thin films, with more nanometric


Figure 4. Freeze fracture image of the most compressed slice in a squalane-SDS-water biliquid foam. Scale bar 1 µm. The droplet size of the original emulsion was 1 µm, and the SDS concentration in the aqueous phase was the cmc. The osmotic pressure exerted by centrifugation was 20 atm.

droplets trapped at various places in the films. The droplets are larger than those in Figures 4 and 5, which makes sense because these less compressed slices contained more water. It is likely that water contained in the films and Plateau borders was expelled to droplets during the cooling procedure, because the interactions of monolayers that form the films become attractive at low temperatures.24 At this stage, we have a picture of the biliquid foam at a mesoscopic scale: a homogeneous array of polyhedral cells, densely packed, and separated by very thin films. In the following, we present a study of these films, which control the stability and most of the properties of the biliquid foam. Thickness of Films. The structures of the films are reflected in the shapes of the SANS curves. Figure 6 presents the scattering curves of two biliquid foams, made from hexadecane, deuterated SDS, and water, submitted to pressures of either 20 or 57 atm. The low q part of the scattering curves reflects the overall shapes and orientations of the films. In this range, all curves from biliquid foams show a q-2 decay, which is the law expected for randomly oriented flat films. The high q part of the curves reflects the average profile of scattering density across the films. In this range, all curves have an oscillation, which is expected if the profile is that indicated in Figure 2, and the film thickness is nearly uniform.13 Thus, the general shape of scattering curves confirms that the (24) Poulin, P.; Essafi, W.; Bibette, J. J. Phys. Chem. B 1999, 103, 5157.

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Figure 5. Freeze fracture image of the least compressed slice in a squalane-SDS-water biliquid foam. Scale bar 1 µm. The droplet size of the original emulsion was 1 µm, and the SDS concentration in the aqueous phase was the cmc. The osmotic pressure exerted by centrifugation was 2 atm.

Figure 6. Scattering curves from slices of a biliquid foam made of deuterated SDS, C16H34, and H2O. Average cell diameter 0.9 µm. Open triangles: slice located at an applied pressure Π ) 20 atm; fit parameters δ ) 8 Å, h ) 27 Å, and σ ) 8 Å. Filled diamonds: slice located at an applied pressure Π ) 57 atm; fit parameters δ ) 8 Å, h ) 15 Å, and σ ) 5 Å.

biliquid foam can be described as a homogeneous array of flat films with a uniform thickness. Three parameters of the film profile were determined from a fit of the scattering curves (Appendix B): the average thickness of SDS monolayers, δ, the average thickness of water layers, h, and the average fluctuation in thickness, σ. The value of δ was 8 Å in all fits, which is consistent with the thickness of SDS monolayers in

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Figure 7. Equilibrium thicknesses of SDS films in slices of biliquid foams made with cell sizes 1 µm. Vertical scale: osmotic pressures calculated from the locations of the slices in a centrifugation tube, or set by the vapor pressure of a salt solution in equilibrium with the biliquid foam. Horizontal scale: water thickness of the films, from a fit of the oscillation in the neutron scattering curves. Filled diamonds: data from centrifugation equilibria. Open triangles: data from vapor pressure equilibria.

lamellar phases.20 The values found for h were in the range 10-80 Å. At low water content they were consistent with the total amount of water in the slice, determined through chemical analysis, but the precision of the scattering experiment was better. At high water content the values calculated through chemical analysis were higher, indicating that some water was also localized in the Plateau borders. The values of σ indicated a larger fluctuation in water thickness for thick films (σ ) 8 Å) than for thin ones (σ ) 5 Å). The shift of the oscillation (Figure 6) indicates that the films became thinner with increasing pressure. The value of the film thickness was measured in each slice of a foam that had been centrifuged to give an osmotic pressure gradient. The experiment was repeated at different speeds to extend the range of osmotic pressures. In this way the equilibrium water thickness was measured for osmotic pressures ranging from 1 to 60 atm. In addition, the water thickness was also measured for slices that had been dehydrated through vapor exchange to reach osmotic pressures above 30 atm. All of these results were collected to give a disjoining pressure isotherm for the system. Disjoining Pressure Isotherm. Figure 7 presents the plot of equilibrium water thickness versus imposed osmotic pressure, measured for biliquid foams with cell sizes of 1 µm. This shows a regular decrease of the water thickness, from 60 to 25 Å, when the osmotic pressure is raised from 2 to 40 atm. These data result from many experiments at different centrifugation speeds and foam heights. The fact that they all fall on the same curve demonstrates that the measurement of pressures according to the height of the slice in the biliquid foam column, as described above, is accurate. A further increase in pressure resulted in a discontinuous jump in the thickness, to another group of data points at a thickness of about 15 Å. In this branch of the equation of state, the data points obtained through vapor equilibrium show that the equilibrium thickness is practically independent of applied pressure in the range


20-200 atm. The application of very high pressures (>100 atm) for a long time caused some coalescence (formation of macroscopic oil domains), but no further decrease in the thickness. This constant thickness of 15 Å corresponds to a hydration of 8 water molecules/SDS molecule, according to the area per SDS molecule in the monolayers; this is just enough water to hydrate the SO3- and Na+ ions. The thicker branch of the isotherm, at 25 Å, corresponds to 20 water molecules/SDS molecule, which provides some free water besides the hydration shell of the ions. This transition between two states of the films, a relatively thick one and a collapsed one, has already been observed in soap films separating air bubbles:4 it is referred to as the transition from the common black film CBF to the NBF. The thickness measured here for the NBF of the biliquid foam is in very good agreement with that measured by Benattar et al. in soap films separating air bubbles.7 In emulsions, Poulin et al.13 found that salt addition caused a similar transition between a repulsive state and an attractive state of the films. In the present case (biliquid foams), the transition was obtained solely by the application of an osmotic pressure. CBF to NBF Transition. The observation of two branches in the equation of state (the CBF and NBF) raises questions about the thermodynamic nature of these branches: (a) Are they really distinct thermodynamic states? (b) Is the NBF the equilibrium state at the given osmotic pressure? One way to answer these questions is to nucleate NBF from the CBF and, conversely, to reswell the NBF to reach the CBF. This nucleation problem was studied early on by Jones et al. in the case of soap films in air.4 They found that the NBF could be nucleated from the CBF by application of very high osmotic pressures (evaporation) or by addition of salt and application of very low pressures (gravity). We have made similar observations for the films in biliquid foams. In the absence of salt, the NBF state was nucleated at very high osmotic pressures and long equilibration times (equilibrium with the vapor of saturated salt solutions). The addition of salt (0.15 M NaCl) caused biliquid foams to nucleate the NBF at lower pressures in a centrifugation experiment (above 20 atm, all slices were in the NBF state, whereas in the absence of salt, the CBF state persisted up to 50 atm). The coexistence of the NBF with the CBF state was observed through SANS in biliquid foams made of very fine droplets (0.15 µm). In this case, the scattering curves of the most compressed slices (pressures 52 and 46 atm) have one oscillation at the thickness of the NBF (14 Å); the scattering curves of the middle slices (pressures 39 and 32 atm) have two successive oscillations, and the scattering curves of the least compressed slices (pressures 25 and 17 atm) have one oscillation at the thickness of the CBF (26 Å) (Figure 8). Remarkably, the pressure/thickness relation is the same for this very fine biliquid foam as for the coarser one with micron-sized cells (Figure 9). Finally, we have induced the reverse transition, from the NBF to the CBF, in two ways. Some slices of biliquid foams in the NBF state were extracted and immersed in water. Immediately, they turned from transparent to white, and later they redispersed entirely to give the original emulsion. Other biliquid foams were centrifuged at high speed (30 000 rpm for 15 h), and the analysis of one sample confirmed that the four most compressed slices had crossed into the NBF state (thickness 13-16 Å). Then the remaining samples were centrifuged at a lower speed


Figure 8. SANS scattering curves of different slices in a biliquid foam with an average cell size 0.15 µm, obtained after centrifugation equilibrium. Open squares: slice 6, pressure 17 atm, thickness 26 Å, CBF state. Filled triangles: slice 4, pressure 32 atm; the two minima mark the coexistence of the CBF and NBF states. Open circles: slice 2, pressure 46 atm, thickness 14 Å, NBF state.

Figure 9. Disjoining pressure isotherm of SDS films in slices of biliquid foams made with two different cell sizes. Dots: cell size 1 µm. Open squares: cell size 0.15 µm.

(10 000 rpm for 8.5 h). Their analysis indicated that all slices but the upper one had reswelled back into the CBF state (thicknesses 13, 30, 37, 38, and 55 Å respectively). A longer equilibration time at this lower pressure would have been necessary to allow water to diffuse back into the first slice. At this stage we may summarize all of the results on the thickness of films in biliquid foams made from oilin-water emulsions stabilized by SDS. (1) NBF and CBF are distinct thermodynamic states of the films in biliquid foams. (2) These states may coexist at a certain pressure, which depends on the ionic strength of the aqueous phase. (3) At lower pressures, the films remain in or return entirely to the CBF state. (4) At higher pressures, the films cross entirely to the NBF state. These observations also raise important questions concerning possible differences in the state of the monolayers in these two types of films.

Langmuir, Vol. 16, No. 4, 2000 1573

Figure 10. Variation of the surface density of SDS films with their state of hydration. Vertical scale: area per SDS molecule in the monolayers, from chemical analysis. Horizontal scale: inverse film thickness, from neutron scattering experiments. The gap between 1/h ) 0.04 and 0.06 Å-1 corresponds to the CBF to NBF transition. Symbols: dots, biliquid foam; open circles, lamellar mesophase. Full line: calculations according to the PB theory (eq 24).

State of the Monolayers. The state of surfactant molecules in the films was determined in three steps. First, SANS experiments were performed to find out whether the surfactant was only located in the films or possibly also in domains of mesophases that would have originated from coalescence events. For this purpose, the contrast was between water (D2O) and the other components (“protonated” oil and SDS). With this contrast, the scattering curves of biliquid foams give monotonic q-2 decays, and mesophases give diffraction peaks. With some unstable emulsions, diffraction peaks were indeed observed. With the highly metastable biliquid foams, however, only the monotonic q-2 decay was observed, indicating that most of the surfactant was contained in the films. Then the surface density of surfactant molecules in the films was measured, and their orientational order was also determined. Surface Density. The total amount of surfactant was measured (Methods section) and compared with the surface area of the films to yield the surface density of surfactant in the monolayers. The results are expressed as an area per molecule in the monolayers that form the films. For the biliquid foams prepared at low pressures, the area per SDS molecule was found to be 52 Å2. This is exactly the area calculated for the original emulsion, and it matches the area per SDS molecule at a macroscopic interface between heptadecane and an aqueous SDS solution at the cmc.16 In both cases, the good agreement results from the fact that the original emulsions were prepared with an aqueous phase that contained SDS at the cmc. For biliquid foams prepared at higher pressures, the area per SDS molecule was found to be lower (Figure 10). The data show a steady decrease of the area per molecule when the film thickness is reduced, from large thickness (CBF, left-hand side of the figure) to low thickness (NBF, right-hand side of the figure). This raises two interesting questions: Why does a decrease of water thickness cause an increase in the surface density of SDS in the monolayers? What is the source of the additional SDS that comes to increase this surface density? One possible source would be the aqueous phase that is in equilibrium with the biliquid foam during centrifugation; however, it is unlikely that SDS ions could diffuse back fast enough through the

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Figure 11. Order parameter profiles of SDS molecules in the monolayers, at various states of hydration. Vertical scale: lefthand side, frequency splitting of the doublets in the 2D NMR spectra; right-hand side, corresponding orientational order parameters. Horizontal scale: location of the deuterium nuclei along the SDS chain, starting from the head and ending with the methyl group. Upper set of data (filled diamonds): biliquid foam in the NBF state, 22 °C. Middle set (circles): biliquid foam in the CBF state, 22 °C. Lower set (triangles): lamellar phase of SDS-water at 70 °C, data from ref 21.

films. A more likely source would come from a limited coalescence in the CBF state, which would reduce the total surface area by about 20%; the system would then form stronger films in the NBF state. For biliquid foams that were prepared at very high pressures, the area per SDS molecule reached 38 Å2. This is exactly the area that has been found in lamellar phases of the SDS/water system at the same water thickness (Figure 10). This agreement suggests that the surface density reached by NBF films is an equilibrium value. Orientational Order of the SDS Molecules. The organization of surfactant molecules in the monolayers is also characterized by their alignment along the normal to the monolayer. In surfactant mesophases, it has been shown that this orientational order arises from steric constraints imposed on each chain by neighboring chains; it becomes stronger when the chains are more tightly packed.17-23 For instance, in the lamellar phase of potassium dodecanoate, the CD2 groups at positions 2-7 have order parameters SCD between 0.2 and 0.15, and these values rise to 0.25-0.20 when the area per molecule is reduced from 41 to 31 Å2. Figure 11 presents the order parameters of CD2 groups, calculated from deuterium NMR spectra (shown in Appendix C). Also shown, for comparison, is the order parameter profile of SDS molecules in the lamellar phase of the SDS/water system at 65 °C.20 All of these order parameters are in the range 0.1-0.2. Accordingly, the SDS chains of the monolayers in biliquid foams are in a constrained liquid state, just as they are in a lamellar phase. In CBF, the order parameters of CD2 groups are higher than those of the lamellar phase, particularly near the headgroup. This difference is not caused by a denser packing of the chains (the area per molecule is larger in CBF), but it could originate from a stronger anchoring of the headgroups, because the biliquid foam was examined at 22 °C and the lamellar phase at 70 °C. In NBF, the order parameters are even higher, all along the chain; this higher order likely results from a denser packing of SDS molecules. In addition, the NMR spectra of NBF have sharper peaks than those of CBF (Appendix C). This reduced width reflects a decrease in fluctuations of the order parameter, such as surface density fluctua-


tions, thickness fluctuations, or other defects, including Plateau borders. This is in line with the results from neutron scattering, which give a narrower spread of water thickness for NBF than for CBF (Figure 6). Accordingly, NBF appear to be more regular than CBF. Summary of Results. We have produced biliquid foams, with very dehydrated films (h < 80 Å), in two distinct states: the CBF and NBF states. We found that both states are metastable and that they correspond to an equilibrium of the films with respect to the water content and surfactant density. The equilibrium of water is described by a disjoining pressure isotherm. In the CBF, the isotherm was followed in both directions. The evidence is provided by experiments performed at different centrifugation speeds, experiments made with biliquid foams having different cell sizes, and reswelling experiments where the centrifugation speed was changed to reach a new local equilibrium. From the CBF to the NBF, the transition was also crossed in both directions, with longer equilibration times. The equilibrium of surfactant is described by an isotherm for the surface density of SDS molecules in the films, and it is also characterized by their orientational order. The evidence for equilibrium is provided by the fact that the surface density reached by films in the NBF state has exactly the same value as that in a lamellar phase of equivalent water thickness. The orientational order parameters are stronger than those of the lamellar phase, but this may be due to the lower temperature of the biliquid foam. These results raise some interesting questions concerning the forces that lead to the transition from CBF to NBF, the reason for the denser packing of SDS molecules in the NBF, and the consequences of this denser packing for the metastability of the films. Discussion The purpose of this discussion is to rationalize the evolution of the biliquid foams during the extraction of the continuous aqueous phase. This evolution has been observed at the mesoscopic scale (packing of cells and cell sizes and shapes) and at the microscopic scale (state of the films that separate the cells). At the mesoscopic scale, we have observed the transformation of an emulsion into a biliquid foam. The droplets pack into a regular (homogeneous and uniform) array, provided that they are sufficiently monodisperse. Then they deform into polyhedra, the films that separate them become thinner, and the Plateau borders that join these films collapse under the effect of the applied osmotic pressure. Our observations of cell sizes, shapes, and organization are in agreement with the previous work of Lissant,1 Princen,25 and Mason et al.26 At the microscopic scale, we have observed changes in the state of films that separate the cells. At that scale, the state of films is defined by two parameters: the water thickness (or hydration) and the surface density of surfactant molecules in the monolayers. The variations of these parameters according to the applied pressure are the equations of state of the films. We shall now try to rationalize these equations of state according to the interactions within the films. Film Thickness. The water thickness of the films is determined by the equilibrium of the applied pressure Πext, which tries to push water out of the films, with the disjoining pressure Π, which tends to retain water in the (25) Princen, H. M. J. Colloid Interface Sci. 1979, 71, 55. (26) Mason, T.; Bibette, J.; Weitz, D. Phys. Rev. Lett. 1995, 75, 10.


Langmuir, Vol. 16, No. 4, 2000 1575

films (also referred to as disjoining pressure of the films or the osmotic pressure of the foam). The main components of Π are due to electrical interactions, Πelec, to hydration forces, Πhydr, and to van der Waals attractions, ΠVdW:

Πext ) Π ) Πelec + Πhydr - ΠVdW

(5)

The relation of Π to thickness h is an equation of state of the films. Having determined experimentally this equation of state, we can compare it with theoretical predictions for the three components of Π and assess the validity of models that describe structure and interactions in the films. Mean-Field Theory. In this model, known as the DLVO theory, the electrical forces originate from the overlap of counterion clouds associated with each SDS monolayer, and the van der Waals attractions, from differences in polarizability between polar and apolar media. Both forces may be calculated, provided that the geometry and the composition of the film are known. The main difficulty is in locating the planes that separate the polar and apolar media and the planes that limit the volume accessible to counterions. For the calculation of van der Waals attractions, the film was described as a polar layer, of uniform polarizability, sandwiched between two apolar layers, also of uniform polarizability. The polar layer contained the hydrated polar headgroups, counterions, and free water, with the same polarizability as free water. The apolar layers contained the hydrocarbon chains of the surfactant, with the same polarizability as liquid hydrocarbon. Differences in polarizability are described by the Hamaker constant H; for a layer of water separating two hydrocarbon media, H ) 1.25kT.27 The strength of van der Waals attractions was then calculated from the separation h of the apolar layers:

ΠVdW ) -(H/6π)h-3

(6)

With the values of h measured through neutron scattering, the calculated van der Waals attractions were always very small compared to measured pressures; for instance, with h ) 13 Å, ΠVdW ) 1.2 atm. In fact, there is no realistic thickness at which the contribution from the van der Waals force to the measured pressures would be substantial. In the following, this contribution is neglected. For the calculation of electrostatic repulsions, the counterions were distributed between planes of uniform charge density Σ according to the Poisson-Boltzmann (PB) equation.28 There is an element of choice in deciding what is the volume accessible to the counterions. At one extreme (single layer model), the counterions may come all the way to the polar/apolar interface (Figure 12). In this case, the thickness of the layer that contains the centers of all counterions is

hPB ) h - Dions

(7)

At the other extreme (triple layer model), the counterions may be excluded from the vicinity of the polar/apolar interface, beneath the headgroups. In this case the thickness of the water layer accessible to counterions is

h′PB ) h - 2Dheads - Dions

(8)

(27) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3. (28) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; Wiley: New York, 1999.

Figure 12. Top: sketch of the locations of surfactant molecules, counterions, and co-ions in the film. Bottom: definition of the thickness of the diffuse layer containing the counterions, with the counterions distributed in the whole water layer (hPB), as expressed in eq 7, or the counterions excluded from the vicinity of the polar/apolar interface (h′PB), as expressed in eq 8.

For SDS monolayers, the size of hydrated headgroups, Dhead, and that of hydrated counterions, Dions, may both be taken equal to 4 Å. Then hPB may be calculated from the value of h obtained from the neutron scattering experiments. The density Σ of surface charges is the density of SDS molecules in the monolayers (2.5 molecules/nm2), or a lower density in cases where some ion pair formation takes place. Note the difference between ion pair formation, which involves a dehydration of the ions and is not described by the PB equation, and ion condensation, which is a normal feature of the counterion profile and is described by the PB equation. In the general case where the water layer is in equilibrium with a reservoir containing passive salt, the profiles of counterions C+(z) and coions C-(z) are determined by the boundary conditions (surface charge density Σ and thickness hPB) and by the equilibrium with the reservoir (this was taken to be an aqueous subphase, where the concentration of SDS is the cmc). They were calculated by solving numerically the PB equation. The electrostatic pressure of the diffuse counterions was then calculated from the gas pressure of ions in the midplane (note that here hPB is the total thickness accessible to the ions, whereas this is 2h in ref 28).

Πelec ) kT[ΣiCi(hPB/2) - 2Cs]

(9)

Disjoining pressure isotherms calculated in this way are presented in Figure 13. Taking the whole water layer to be accessible to the counterions, as in eq 7, gives a pressure that rises slower than the data in the range of intermediate distances (coming down from 40 to 25 Å), regardless of surface charge (Figure 13). Yet, in this range, the electrostatic pressure of counterions is the only significant force, and corrections to the mean-field theory would not give a faster rise of the pressure, so this deviation truly reveals a bad choice for the distribution of counterions. On the other hand, a good fit to the pressures measured in the CBF is obtained if the counterions are excluded from the vicinity of the polar/apolar interface, as in eq 8, and Σ is 2.5 charges/nm2 (Figure 13). This value of Σ implies that all sulfate groups count as surface charges, in agreement with the well-known fact that sulfate groups do not form ion pairs with Na+ counterions. This result could have been obtained in a more simple way. Indeed, in the limit where the monolayers are highly charged and the water layers that separate them are thin,

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Figure 13. Calculated disjoining pressure isotherms of the surfactant film according to PB models. (a) Lower line: the counterions are distributed in the whole water layer, as expressed in eq 7, and the area per charge is 40 Å2. Upper line: the counterions are excluded from the vicinity of the polar/ apolar interface, as expressed in eq 8, and the area per charge is 100 Å2. Dots: measured values of Π and h, from neutron scattering (as in Figure 7).

passive salt and excess surfactant are excluded from the films because the counterion concentration is already extremely high. In this limit the PB equation has a simple analytical solution from which the pressure may be calculated directly. At distances that are not extremely short (in our case beyond hPB ) 4.4 Å), the pressure becomes independent of the surface charge density:28 -1

Πelec ) πkTLB

hPB-2

(10)

where LB is the Bjerrum length, which gives the range of ionic interactions in water (LB ) e2/4π0rkT ) 7 Å in water at room temperature). This equation of state is accurate in the limit of high surface charge densities; it gives a good fit to the CBF data (Figure 13). In summary, these mean-field models do reproduce the pressures measured in the CBF, provided that the counterions are excluded from the vicinity of the polar/ apolar interface, beneath the headgroups, and that the monolayers are highly charged. None of them, however, can reproduce the transition to the NBF, unless the parameters of the van der Waals interactions are assumed to take totally unreasonable values (either H would have to be more than 10 times larger than usual or the polar headgroups of the SDS molecules would have to be fully dehydrated). Yet, it is clear that the electrostatic interactions must become attractive in the limit of very strong interactions (very high charge density or low dielectric constant) or low temperature. In this limit, the PB treatment is not adequate because it neglects correlations between counterions and therefore predicts a counterion distribution that is more disordered than the real one. In the following section, we present a numerical simulation where these correlation effects are taken into account. Monte Carlo Simulation of Ionic Distributions. In this section we use a different, but formally exact, statistical mechanical method to solve the electrostatic problem. The model system still is a continuum model where the water structure is not taken into account, and the charged monolayers are represented as planes with a uniform charge density. For a salt-free double layer we

Figure 14. Disjoining pressure isotherm of the surfactant film according to a Monte Carlo simulation of ionic correlations, for various values of the dielectric constant of water in the films. Upper line: PB calculation, area per charge ) 40 Å2. Shortdashed line: MC simulation, dielectric constant r ) 78. Thick full line: MC simulation, dielectric constant r ) 65.

can apply the Metropolis Monte Carlo method29 in the canonical ensemble and handle the long-range part of the electrostatic interaction as described in previous reports.30 The equation for the osmotic pressure now includes an additional term describing an attractive interaction between the two half-cells:

Πelec ) kTΣiCi(hPB/2) + Πcor

(11)

where the first term is repulsive and originates from the counterion entropy and the second one is attractive and is caused by ion-ion correlations.31,32 The attractive interactions come naturally from a free-energy perturbation approach to two interacting charge distributions.33 It can be seen as a classical analogue to the quantum mechanical dispersion force.34 The leading term from the classical perturbation expansion will be an induced dipole-induced dipole term, which decays as the sixth power of separation between the two neutral charge distributions. However, these correlations have an effect on both terms: they reduce the entropic term, because more ions are accumulated close to the charged surfaces, and they produce the electrostatic attraction Πcor. The magnitude of these effects increases when the surface charge density rises or when the separation hPB decreases and also when the temperature or the dielectric constant of the solvent decreases. The results of selected simulations are presented in Figure 14. With the same assumptions for the space available to counterions (hPB ) h - 12 Å) and the same dielectric constant for water (r ) 78), the effect of correlations reduces the calculated pressures but does not produce attractions that would overcome the entropic repulsions, as in the CBF to NBF transition. However, at short separations, the dielectric constant of water in the films (29) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (30) Jo¨nsson, B.; Wennerstro¨m, H.; Halle, B. J. Chem. Phys. 1980, 84, 2179. (31) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (32) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1984, 112, 19. (33) Woodward, C. E.; Jo¨nsson, B.; Akesson, T. J. Chem. Phys. 1988, 89, 5145. (34) Wennerstro¨m, H.; Daicic, J.; Ninham, B. Phys. Rev. A 1999, in press.


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must become lower than that of bulk water. Indeed, concentrated ionic solutions have lower dielectric constants than pure water, because the water molecules that are in the hydration shells of the ions do not have the same correlations as those in bulk water.35 We made a similar choice for the dielectric constant of water in the films (r ) 65). With this choice, the pressure calculated through the Monte Carlo simulation starts to decrease when the separation hPB is pushed below 8 Å. This decrease results from enhanced ionic correlations at very short separations. Indeed, for sufficiently high charge densities, the entropic component of the pressure decreases and the attractive ion-ion correlation becomes relatively more important. This results in a nonmonotonic repulsive interaction that eventually turns attractive.36 This is all that is needed to explain the CBF to NBF transition. At very short separations one needs a more explicitly molecular picture of the system in order to obtain quantitatively reasonable values for the pressure. In eq 5 we introduced the term Πhydr to account for deviations from the continuum picture. We have performed additional fits to our data with a short-range hydration force and different choices for the radii of the ions in eq 8. These fits are not better than those where the hydration force is simply a repulsive wall, and they require that the electrostatic interaction become still more attractive at very short separations. Thus, the inclusion of a hydration force that would extend beyond the repulsive hydration wall at h ) 13 Å does not change our conclusion that the CBF to NBF transition is caused by the electrostatic interactions that become less repulsive and then attractive in the limit of high charge densities and very short separations. State of the Monolayers. The surface density of SDS molecules in the monolayers is determined by a balance between the electrostatic repulsions of charged sulfate groups, which strive to increase it, and a hydrophobic attraction between SDS molecules, which favors small areas of interface per SDS molecule. Thus, we write the total free energy of the bilayer as

G(a,h) ) Gel + Gnonel

(12)

where a is the area of the interface per SDS molecule and h is some measure of the bilayer separation. At equilibrium there is an optimal area a0 per surfactant molecule, determined by the condition that G be minimal with respect to area:

(∂G/∂a)h ) 0

(13)

The balance of forces mentioned above is thus expressed as

(∂Gel/∂a)h + (∂Gnonel/∂a)h ) 0

(14)

The analysis of these forces was done previously in the case of lamellar phases by Jo¨nsson and Wennerstro¨m.37 We follow this analysis and apply it to biliquid foams. The nonelectrostatic force (∂Gnonel/∂a)h results from local interactions (hydrophobic and hydration) between neighboring SDS molecules within a monolayer. We assume that this component is independent of the separation h of the opposing monolayers and also constant over the small (35) Smith, C. P. In Dielectric Behavior and Structure; Hammet, L. P., Ed.; McGraw-Hill: New York, 1955; Chapter 3. (36) Wennerstro¨m, H. Langmuir 1990, 6, 834. (37) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482.

range of a0 values. Consequently, it should have the same value, γ′, as that in an isolated monolayer:

(∂Gnonel/∂a)h ) γ′

(15)

The electrostatic force (∂Gel/∂a)h results from long-range interactions, and it does vary with the separation h of the layers. It can be calculated according to the PB description of the electrostatic interactions in the bilayer. It has been demonstrated37,38 that it is related to the total electrostatic energy Uel originating from ion-ion interactions:

(∂Gel/∂a)h ) -2Uel/a

(16)

Single Monolayer. In a dilute emulsion, the surfactant monolayers are independent and exposed to a surfactant solution; the surfactant concentration is set at the cmc. In this case Uel can be evaluated using the Gouy-Chapman theory (eq 3.8.5 of ref 28):

2Uel/a ) (2kT/a)s-1[(s2 + 1)1/2 - 1]

(17)

where the dimensionless quantity s equals (eq 3.8.6 of ref 28)

s ) Σ(8kTC0*r0)-1/2 ) 29.1

(18)

where Σ is the surface density of charges in the monolayer, calculated using an area per headgroup of 52 Å2 as found for the free monolayer and C0* the concentration of free molecules expressed in molecules per cubic meter. From this we can calculate

2Uel/a ) 18 mN/m

(19)

According to the equilibrium condition (14), this electrostatic force is set to balance the nonelectrostatic force:

γ′ ) (∂Gnonel/∂a) ) -(∂Gel/∂a) ) 18 mN/m

(20)

This value is in good agreement with previously determined values;37 it differs from the oil/water interfacial tension because it contains contributions from the hydration of headgroups and from alignment of alkyl chains. Bilayer. As the monolayers are brought closer together, there is an electrostatic interaction between them that will also affect lateral packing. Experimentally we see a significant decrease of the area at separations h lower than 30 Å. In this regime of separations the Debye screening length exceeds the separation, and one can to a good approximation neglect the role of free electrolyte in the aqueous film, which mainly contains the neutralizing counterions. For the electrolyte-free case the PB equation has an analytical solution (eq 5.1.5 of ref 28) and the expression for the ion-ion energy is

2Uel/a ) (2kT/a)[1 - s/tan s]

(21)

where the dimensionless parameter s is related to the bilayer separation hPB of the PB treatment by (eq 5.1.16 of ref 28)

s tan s ) hPBe2/(4akTr0)

(22)

With this expression for Uel, the equilibrium condition (14) can be used again. We still assume that (∂Gnon el/∂a) is a local force between neighboring SDS molecules within (38) Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338.

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the monolayer, independent of the separation h between layers; therefore, it has the same value that was determined in the dilute case:

γ′ ) (∂Gnonel/∂a) ) 18 mN/m

(23)

Consequently, the equilibrium condition yields

2Uel/a ) γ′ ) 18 mN/m

(24)

This equilibrium condition, where Uel is calculated according to eqs 21 and 22, can be solved for the area per molecule, a. It yields an optimal area, a0, as a function of the counterion layer thickness, hPB. According to the measured disjoining pressures, hPB is obtained from the total water layer thickness, h, by subtracting 12 Å. With this choice, a theoretical curve for a0 vs h is obtained, which is presented in Figure 10. This predicts that the area per molecule should decrease as the water thickness is reduced. This can be interpreted as an electrostatic effect: the increased density of counterions screens more efficiently the repulsions between charged headgroups within a monolayer, allowing the nonelectrostatic interactions to bring the SDS molecules closer together. This represents a specific model for a general effect predicted on purely thermodynamic grounds by Hall.39 The predicted variation is faster than that observed; i.e., the screening of headgroups by counterions is predicted to become more efficient than it really does. A slower variation would be obtained with a different choice for the distribution of counterions (eq 7 instead of eq 8), but this would be inconsistent with the analysis made for the disjoining pressures. A more likely explanation could be an effect of ionic correlations within the counterion layer or an effect of the discrete nature of the surface charges, which has not been taken into account.

the concentration of counterions is quite high, the headgroups are efficiently screened, and the area per headgroup is small, comparable to that in the lamellar mesophase of the SDS-water system. This tighter packing of the SDS molecules forces them to have a higher orientational order; it also causes the NBF films to be more homogeneous than CBF films. The coexistence of two branches in the equation of state, the CBF and NBF, shows that there is a short-range attraction between opposing monolayers. van der Waals attractions are much too weak to contribute to this attraction. On the other hand, at high charge densities, the correlations between fluctuations of the ionic concentrations can produce this attraction, provided that the dielectric constant of water in the films is slightly lower than that of bulk water. On the practical side, the transition to the NBF state may have important consequences for the metastability of the films. Indeed, during the course of this work, we have found that biliquid foams made with SDS films in the NBF state have a high metastability. This may be due to the tighter packing of the surfactant molecules, which forces them to keep a higher degree of orientational order. Consequently, some fluctuations that could cause the rupture of the films (vacancies in the monolayers and defects where opposing monolayers recombine) will be inhibited. This suggests a general route for improving the metastability of biliquid foams, which is simply to improve the packing of the surfactant molecules in the monolayers. Acknowledgment. We thank M. Airiau for help and advice with the centrifugation experiments, L. Belloni for the programs that calculate the electrostatic pressure by solving the PB equation, J. M. Neumann and M. Roux for performing the NMR experiments, and the staff of LLB and ILL for support and advice with the neutron scattering experiments.

Conclusions Oil-in-water emulsions can be compressed to give biliquid foams where the oil droplets have become polyhedral cells separated by thin films. As in regular foams, these films can be in two states: the CBF state and the NBF state. The CBF state is obtained with SDS films that are well hydrated: at least 20 water molecules per surfactant molecule, which is more than the hydration shells of the headgroups and counterions; the water thickness is beyond 25 Å. The distribution of the counterions in these water layers is well described by the PB equation, with the following boundary conditions: the counterions are confined in the space between the planes where the headgroups are located (they are excluded from the vicinity of the water/hydrocarbon interfaces) and all of these headgroups are fully ionized (no ion pair formation). As water is extracted from the films, the concentration of counterions in the films increases, resulting in a more efficient screening of the headgroups; as a result, the surface density of SDS in the monolayers rises. The films resist this dehydration, and their osmotic pressure is well described by the entropic pressure of the counterions, calculated from the PB equation. The NBF state is obtained with SDS films that are much less hydrated: only 8 water molecules/surfactant molecule, which is barely enough to hydrate the headgroups and their counterions; the corresponding water thickness is 13 Å. This thickness is stabilized by hydration forces that prevent the extraction of these water molecules. Because (39) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1972, 68, 2169.

Appendix A: Chemical Analysis Methods The amount of water in each slice was determined by dissolving it in a Karl Fischer reagent. Because the water content of each sample was low, it was necessary to dilute the reagent with anhydrous methanol, so that the total volume of reagent used for each sample was at least 1 mL. The uncertainty on the result of each measurement was calculated by titrating an alcohol containing a known amount of water; it was less than 1% for a volume of reagent equal to 5 mL. This gave a satisfactory precision for biliquid foam samples with a mass of at least 100 mg and a water content of at least 0.1%. The amount of SDS in each slice was determined by dissolving the biliquid foam in methanol and then titrating the surfactant ions DS- with a cationic indicator B+ (benzethenium chloride).40 This was a two-phase anioniccationic titration, where one phase was initially chloroform and the other was an aqueous phase containing an anionic dye A- (disulfine blue) and a cationic dye C+ (diimidium chloride). The SDS solution was added to the aqueous phase, where the DS- ions combined with the dye C+. The complex salt CDS precipitated out of water and transferred in the chloroform phase, giving it a pink color. Then the B+ indicator was added to displace the cationic dye C+ from the CDS salt. At the equivalent point, the chloroform solution contained only B+ and DS- in equal molar amounts, and it became colorless. Beyond the equivalent point, the excess B+ combined with the anionic dye A- in (40) Schmitt, T. In Analysis of surfactants; Schick, M., Ed.; Surfactant Science Series 40; M. Dekker: New York, 1992; Chapter 11.


Langmuir, Vol. 16, No. 4, 2000 1579

water to form another complex salt which also precipitated and transferred into the chloroform solution, giving it a blue color. Appendix B: Theoretical Scattering Curves For bilayers that can be taken as planar and infinitely long in both directions, measurable scattering is obtained only with scattering vectors q that are along the normal to the bilayers.41 At small scattering angles the scattering vector is normal to the neutron beam, therefore scattering is obtained only from bilayers that are parallel to the beam. Consider one of these bilayers and let z be the axis normal to the bilayer and to the neutron beam; then the intensity scattered along z has a slow decay in this direction according to the magnitude qz of the scattering vector q: -h/2 δ+h/2 eiq z dz + ∫h/2 eiq ∫-δ-h/2

P(qz) ) ∆F2|

z

z

dz|2 (25)

The result of the integration is

P(qz) ) (8∆F2/qz2) sin2(qzδ/2){1 + cos[qz(h + δ)]} (26) This form factor has a strong oscillation with a period π/(δ+h), which can be used to determine the bilayer thickness. In real bilayers, this oscillation is attenuated by fluctuations in water thickness. For a Gaussian distribution of thicknesses, centered on a thickness h and with a width σ, there will be an exponential attenuation of the form factor

P(qz) ) (8∆F2/qz2) sin2(qzδ/2){1 + e-qz σ cos[qz(h + δ)]} (27) 2 2

For finite bilayers with random orientations, the intensity has the same slow decay in all directions around the beam. An approximate expression for the intensity is obtained as follows:13 (i) replace qz with the modulus q of the scattering vector; (ii) use a Lorentz factor 2π/q2 to take into account the fraction of bilayers that are oriented sufficiently close to the beam; (iii) multiply the form factor by the total concentration of films Σ:

I(q) ) Σ(2π/q2)(8∆F2/q2) sin2(qδ/2){1 + e-q σ cos[q(h + δ)]} (28) 2 2

Scattering curves of this type start at low q with a q-2 decay, then show an oscillation near q ) π/(δ + h), and (41) Glatter, O.; Kratky, O. Small Angle X-rays Scattering; Academic Press: New York, 1982.

Figure 15. 2D NMR spectra of biliquid foams made with deuterated SDS, at different states of hydration. Bottom: water content of the biliquid foam 0.5%, films in the NBF state, water thickness 15 Å according to SANS. Middle: water content of the biliquid foam 1.1%, films in the CBF state, water thickness 34 Å. Top: water content of the biliquid foam 6.5%, films in the CBF state, water thickness below 220 Å.

end at high q with a q-4 decay. Indeed, the experimental spectra show an oscillation near q ) 0.1 Å-1, which corresponds to δ + h ≈ 30 Å (Figures 6 and 8). For a precise comparison with the calculated scattering curves, a background spectrum must be subtracted. Because the amount of biliquid foam in the scattering cell was not known with sufficient precision, this operation was performed in such a way that the subtracted spectrum decayed as q-4 at high q. The calculated spectrum was then scaled in intensity to match the experimental spectrum. Finally the fit was made with three parameters: δ, h, and σ. Appendix C: NMR Spectra NMR experiments were performed on emulsions of deuterated SDS, C16H34, and H2O (depleted in D2O), and the spectra were recorded through the quadrupolar echo. A foam with an average cell size of 1 µm was chosen to avoid orientational averaging of the quadrupolar interactions by the translational diffusion of the SDS molecules around the cells. Consequently, the amount of surfactant in one sample was low (3 × 10-5 µg/cm3), which required long accumulation times (24 h). A set of spectra for biliquid foams in different states of hydration is shown in Figure 15. LA990599K

Surfactant Films in Biliquid Foams - Langmuir (ACS Publications)

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