J. Phys. Chem. 1996, 100, 4533-4537
4533
Compatibility between Solid Particles and a Lamellar Phase: A Crucial Role of the Membrane Interactions L. Ramos* and P. Fabre Laboratoire de Physique de la Matie` re Condense´ e, Colle` ge de France, CNRS, URA 792, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France
E. Dubois Laboratoire de Physico-Chimie Inorganique, UniVersite´ P et M. Curie, CNRS, URA 1662, 4 place Jussieu, 75232 Paris Cedex 05, France ReceiVed: July 11, 1995; In Final Form: October 30, 1995X
From previous studies, we know that Helfrich-stabilized smectic phases can incorporate solid particles of typical size 10 nm between their membranes; the incorporation of magnetic particles yields a “ferrosmectic” phase. We present here a study of the compatibility between such solid particles and a lamellar phase versus the interactions between membranes. The type of stabilizing forces, from purely electrostatic to purely entropic, is varied via the concentration of salt (NaCl) in a swollen lamellar phase. The experimental results reported in this paper show that the Helfrich phases can incorporate particles within their lamellar matrix, whereas particles are excluded from the electrostatic phases. A semiquantitative interpretation based on considerations about the interactions between particles and membranes is given.
Introduction Lyotropic lamellar phases, in which amphiphilic molecules in solution self-assemble to form layers that stack with longrange periodicity, have long been subjects of intense interest. In particular, much attention has been focused on the understanding of the interactions between membranes.1 The stability of these lyotropic lamellar phases results from opposing forces between membranes: the attractive long-range van der Waals force and the repulsive forces. In the case of charged membranes separated by sufficiently large layers of water (for interlamellar spacings larger than 20 Å, the repulsive hydration forces become negligible), the dominant repulsive force is electrostatic and originates from the mixing entropy of the counterions. When the membranes are neutral or separated by brine at sufficiently high salt content, this repulsive force becomes short range and can no longer explain the stability. In that case, Helfrich (2) showed that a steric long-range repulsion may occur, which originates from thermally induced out-ofplane fluctuations of the membranes: these are entropic phases. From previous studies,3,4 we know that Helfrich-stabilized oil-swollen lamellar phases are able to incorporate solid particles of typical size 10 nm within their oil layers. These hybrid phases have been called ferrosmectics and have been extensively studied.5 The incorporation of particles considerably modifies the nature and intensity of the repulsion between membranes, whose behavior cannot be interpreted in the framework of a pure Helfrich model.5 In this respect, it was tempting to explore the generality of such hybrid phases versus various types of stabilizing interactions in the system. An approach to the problem is to investigate a system where both the electrostatic and the steric interactions are present and to explore the different regimes. The study presented here deals with a lamellar phase that can be swollen as well with water as with brine, the crossover between regimes being driven by screening of the electrostatic interactions. X
Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-4533$12.00/0
The first part of this paper consists of a description of the experimental methods. We then present our experimental results concerning the stability of the doped systems and conclude with a quantitative interpretation of the behavior of the phases. Preparation of the Doped Phases and General Experimental Methods The doping objects are magnetic particles dispersed in water (i.e., “ferrofluid”).6 The ferrofluid we used was prepared through the method described in ref 7. The particles have a magnetic core of maghemite (γ-Fe2O3) and are coated by a shell of adsorbed sodium citrate. They are thus negatively charged, with a surface charge density of about 5 elementary charges per nm2. By successive partial precipitations and solubilizations, a size selection was performed,8 which gave a suspension of very small particles. The diameter a of the solid core of the particles follows a log-normal size distribution,9 the characteristic parameters of which were determined by small-angle X-ray scattering (SAXS): a0 ) 30 Å (ln a0 ) 〈ln a〉) and σ ) 0.4 (standard deviation). The citrate layer can be roughly estimated to be about 10 Å, which gives an effective diameter of 50 Å for the particles. Their negative surface charges ensure the stability of the colloidal solution through electrostatic repulsion, counterbalancing the van der Waals and magnetic attractions. This stability may however be endangered when the electrostatic repulsion is screened with added salt. For this reason, we chose a ferrofluid with these very small particles, so as to ensure colloidal stability up to high salt content (>0.5 M), for volume fraction of particles Φ varying from 0.1% to 2.8%. The undoped lamellar phase was the ternary (pseudoternary) mixture sodium dodecyl sulfate (SDS)/pentanol/water (brine). The membranes, constituted of SDS and pentanol, are negatively charged. Their high flexibility enables the Helfrich interaction to compete with the van der Waals attraction and electrostatic repulsion. This explains why these lamellar phases can be swollen as well with pure water as with brine. To obtain the doped lamellar phase, we first prepared the ferrofluid with a given volume fraction in particles Φ, and a © 1996 American Chemical Society
4534 J. Phys. Chem., Vol. 100, No. 11, 1996
Figure 1. Schematic phase diagrams (weight percent) of the ternary systems SDS/pentanol/aqueous solvent. Only the lamellar zones are represented for three solvents: pure water, 0.1 M brine, and 0.5 M brine. The data points correspond to the samples studied.
given salt (NaCl) concentration C0, and then added SDS, whose amount is directly related to the lamellar periodicity. These two steps enabled us to check the stability of the ferrofluid with salt. Finally, we added the required quantity of pentanol. The phases were prepared in glass tubes and maintained at room temperature. They were then held in sealed glass capillaries of rectangular cross section (100 µm × 1 mm) and observed under a polarizing microscope. The lamellar phase was unambiguously identified through its typical texture of oily streaks between crossed polarizers. Moreover, the uniform red color due to the iron oxide particles we observed indicated, on one hand, that particles were incorporated between the membranes and, on the other hand, that there were no aggregates. We monitored in this way the stability of the doped system. We also used SAXS for determining the lamellar periodicity, as well as deriving some qualitative information about the interactions. Experimental Results In this part, we present our experimental results concerning the initial system and the doped lamellar phases. In the first part, we discuss the characterization of the undoped phases when the repulsive interactions between membranes change progressively from purely electrostatic to purely entropic, and subsequently, we address the stability of the hybrid phases obtained by incorporation of particles between membranes. The Initial System. The two ternary systems SDS/pentanol/ water and SDS/pentanol/brine (NaCl concentration C0 ) 0.5 M) have been previously studied by Roux:10 purely electrostatic (entropic) repulsions stabilize the lamellar phases swollen with water (brine). In order to investigate “intermediate” phases, where both repulsive forces could compete, we also prepared lamellar phases with other salt concentrations (namely, 0.01, 0.1, and 0.25 M). A schematic phase diagram is given in Figure 1, where the compositions of the lamellar phases investigated, swollen with pure water and brine (0.1 and 0.5 M), are shown with data points. A rapid comparison of the localization of the different lamellar zones underscores two major effects of the addition of salt. On one hand, salt stabilizes the lamellar phase over a much larger dilution range: the lamellar phase containing pure water is stable to a maximum dilution of 71 vol % water, which corresponds to a maximum periodicity of 90 Å, while phases can be swollen with brine (C0 g 0.1 M) up to periodicities larger than 200 Å. On the other hand, less pentanol is required for a phase swollen with brine than for a phase swollen with pure
Ramos et al. water: the variation of alcohol content is largely correlated with the variation of the area per polar head of the ionic surfactant SDS due to a screening of the electrostatic repulsions. We controlled the smectic nature of the phases and measured their periodicity by SAXS. The features of a spectrum also provide information about the interactions between membranes: the spectrum of an electrostatic lamellar phase displays sharp Bragg peaks and no scattering when q tends to 0, and the transition from such an electrostatic phase to an entropic one induces strong scattering when q tends to 0, broadening of the Bragg peaks, and possible removal of higher order Bragg singularities.11 These effects are enhanced as the phases are swollen. As an illustration, we give in Figure 2 the spectra of four phases with equal periodicity swollen with different solvents, namely, water and brine with C0 ) 0.01, 0.1, and 0.5 M. These spectra exhibit a continuous variation from the typical features of electrostatic phases to the typical features of entropic phases, which agrees with the continuous evolution from one type of interaction to the other. The Doped Phases. We now examine phases in which particles were incorporated. The parameters we explored for the doped systems were C, the salt (NaCl) concentration in the ferrofluid, Φ, the volume fraction of particles, and d, the lamellar periodicity, as they appear to be the relevant parameters for the existence and stability of the lamellar phases. At a given value Φ of 0.35%, the results obtained for the different combinations of C0 and d explored and the stability of the corresponding phases are reported in Figure 3. The graph clearly indicates two diagonally separated regions: in the upper zone (higher salt content and periodicity), the doped phases are stable; the particles are incorporated within the lamellar structure. Particles are, however, excluded from the lamellar phases of the lower zone where they slowly aggregate and phase separate. Notice that the indicated periodicities are those of the initial phases and not those of the doped ones, but since few particles are introduced within the layers, it is reasonable to assume that the periodicity does not change. We have to make this assumption because SAXS does not allow to measure the periodicity of the doped phases: the scattering from the magnetic particles hinders the Bragg peaks; a direct verification would require a study by small-angle neutron scattering with a contrast variation technique. As for the effect of Φ, our preliminary results do not indicate a major influence of Φ on the stability: one of the doped phases we tested remained stable over a wide range of volume fractionsfrom 0.1% to 1.4%swithout any noticeable changes. Since the lamellar phase swollen with pure water is unable to incorporate particles, it appeared interesting to study the isotropic phases adjacent to this lamellar phase. In that respect, we progressively added pentanol to an alcohol-free mixture of SDS and ferrofluid (9.1 wt % SDS), so as to explore the whole lamellar domain: particles were stable within the isotropic zones neighboring the lamellar one, which is easily understood since these phases are microemulsions, whereas they aggregated immediately in the whole lamellar zone. The same protocol performed for different volume fractions of particles in the ferrofluid (from 0.1% to 1.4%) gave the same results. Moreover, we noticed that the aggregation of particles outside the lamellar matrix is a reversible process, since particles were again uniformly dispersed when pentanol was added. These experiments show, on one hand, that colloidal particles are excluded from the ternary lamellar phase SDS/water/pentanol with a periodicity of 88 Å, from Φ ) 0.1% to Φ ) 1.4%, and aggregate and, on the other hand, that it is possible to incorporate these colloidal particles in lamellar phases swollen with brine,
Solid Particles and a Lamellar Phase
J. Phys. Chem., Vol. 100, No. 11, 1996 4535
Figure 2. SAXS spectra of lamellar phases, with equal periodicity (90 Å), swollen with various solvents: (a) pure water, (b) 0.01 M brine, (c) 0.1 M brine, (d) 0.5 M brine. Notice the continuous variation from a typical spectra (a) of an electrostatic phase (first- and second-order sharp Bragg peaks with no scattering at q ) 0) to a typical spectrum (d) of an entropic phase (broadening of the Bragg singularity, strong scattering at q ) 0). Notice also the change in scale of the y axis.
Figure 3. Stability of the doped phases investigated versus two relevant parameters: the lamellar periodicity and the salt (NaCl) concentration of the swelling solvent. The curve is the plot of dmin versus C0 calculated from expression 1.
provided appropriate values of the two relevant parameters, the lamellar periodicity d and the salt concentration C0, are used. These results thus indicate that there is a combined effect of the screening of the charges and of the size of the particles compared to the available space. We will analyze these contributions in the next section. Discussion In the first section, we evaluate the evolution of the interaction potentials of the initial lamellar phases when screening the electrostatic repulsion. With this approach, we interpret, in the second section, our experimental results concerning the stability of the doped phases.
Interactions in the System. In order to quantify the relative importance of the Helfrich and electrostatic repulsions in the different phases studied, we evaluate the ratio VHelf/Velec, where VHelf and Velec are respectively the Helfrich and electrostatic repulsive potentials per unit area. The two potentials are calculated knowing the characteristics of the membrane, that is, its composition, surface charges, and bending modulus, which are explained in the following. Properties of the Membrane. The membrane is constituted of a mixture of SDS and pentanol. We assumed that all surfactant and cosurfactant molecules are incorporated in the membrane, as the solubilities of SDS and pentanol in pure water and brine are much smaller than the concentration we used. We also assumed that the membrane thickness δ did not largely vary in the regions of the phase diagrams we explored and took an experimental value of δ ) 20 Å (10) for the calculations developed below. We first evaluate kc, the bending modulus of the membrane, and especially its variations when charges are added in the surrounding solvent. For charged surfactant membranes in water, electrostatic interactions induce a positive contribution, δkc|elec, to the bare bending constant;12 this leads to a renormalization of this constant, which can be written as kc ) ki + δkc|elec, where ki is the intrinsic modulus, which originates from the molecular packing and the loss of conformational entropy of the hydrophobic chains. Both theoretical calculations13 and experiments14 show that ki is very small as soon as a short-chained alcohol is incorporated in the membrane. Different measurements on the present system15 and analogous systems16 give values for ki around kT (in the range 0.2-2kT). The evaluation of δkc|elec is more delicate, since it may be expressed by different analytical expressions, depending on three relevant length scales:17 the lamellar periodicity, d, the Gouy-
4536 J. Phys. Chem., Vol. 100, No. 11, 1996
Ramos et al.
TABLE 1: Characteristic Parameters of the Phases Studieda κ-1 (Å)
solvent water 0.01 M brine
30
0.1 M brine
9.6
0.25 M brine
6.1
0.5 M brine
4.3
d (Å)
σ (C/m2)
88 83 90 91 114 170 199 110 118 142 72 88 122 168 186
0.11 0.11 0.11 0.18 0.18 0.16 0.14 0.24 0.26 0.20 0.32 0.33 0.35 0.32 0.32
φ0 (mV)
Velec (J/m2)
VHelf (J/m2)
E
151 151 117 117 112 105 109 113 100 106 108 110 106 106
5.2 × 10-4 4.7 × 10-4 3.8 × 10-4 6.2 × 10-6 5.6 × 10-7 1.6 × 10-9 7.2 × 10-11 7.4 × 10-9 2.0 × 10-9 4.0 × 10-11 1.1 × 10-7 2.6 × 10-9 9.5 × 10-13 2.3 × 10-17 3.5 × 10-19
4.1 × 10-6 1.0 × 10-5 8.1 × 10-6 1.3 × 10-5 7.7 × 10-6 3.0 × 10-6 2.1 × 10-6 9.0 × 10-6 7.6 × 10-6 4.9 × 10-6 2.9 × 10-5 1.7 × 10-5 7.6 × 10-6 3.6 × 10-6 2.9 × 10-6
26kT 8.2kT 0.4kT 0.09kT 2.8kT 1.4kT 0.2kT 8.3kT 0.2kT 7 × 10-4kT 9 × 10-5 kT
a κ-1 is the Debye length, and d is the lamellar periodicity: σ, φ , V 0 Helf, and Velec are the surface charge density, surface charge potential of the membrane, and the Helfrich and electrostatic potentials, respectively. E is the interaction energy between a particle and the membranes. All quantities were calculated as explained in the text.
Chapman length λ ) e/2πlσ, and the Debye-Hu¨ckel screening length κ-1 ) (8πn∞l)-1/2; in the last two relationships, n∞ is the bulk electrolyte concentration, l ) e2/4π0kT is the Bjerrum length, which is around 7 Å for a solution of dielectric constant ) 80, and σ is the surface charge density of the membrane. As for σ, we assumed that all charges are dissociated, we made the approximation of a constant area per polar head for pentanol (10 Å2), and we derived the variation of this area for SDS with salt content from a capacitor model:18 we obtained 60 Å2 for pure water, and 34 Å2 (respectively 24 and 23) for salt concentration 0.1 M (respectively 0.25 and 0.5). The resulting numerical values for σ are reported in Table 1. In our system, the phases swollen with brine belong to the so-called intermediate regime (κd > 1 and κλ < 1), where the electrostatic contribution reads δkc|elec ) kT/πlκ;17,19 it increases from 0.2kT to 1.4kT with decreasing salt concentration (from 0.5 to 0.01 M). The phases swollen with pure water belong to the GouyChapman regime (λ < d < κ-1); in this region, although δkc|elec has not been calculated precisely, scaling and continuity arguments yield δkc|elec ) kTd/πl.20 This gives, for the phase considered, an electrostatic contribution equal to 4kT. If we compare this value with experimental results,10,21 we tend to think that it is probably slightly overevaluated. In the calculations developed below, we have added the electrostatic contributions given above to a constant intrinsic part taken equal to kT. RepulsiVe Potentials. (i). The Helfrich potential may be written as2
VHelf )
()
kT 3π2 kT 128 kc (d - δ)2
where d - δ is the intermembrane spacing. For the phases investigated, the numerical values were calculated using the kc values given in the preceding section and are reported in Table 1. (ii) The electrostatic interaction potential is strongly dependent on the swelling solvent. In the case of pure water, the exact solution is known;22 its expansion at large separations reads10
Velec )
[
(
) ]
2 e πkT e 1+ ... + 4l(d - δ) σl(d - δ) σl(d - δ)
On the contrary, the phases swollen with brine belong to a regime where the weak oVerlap approximation is valid.1 The electrostatic repulsion potential is expressed as Velec ) (64kTγ2κ-1[NaCl])e-κ(d-δ) with γ ) tanh(eφ0/4kT). The surface
potential φ0 is calculated using the Grahame equation, once the surface charge density σ is known: σ ) x80kT sinh(eφ0/ 2kT)x[NaCl]. The numerical values are reported in Table 1. We also evaluated errors made in calculating the potentials: the errors appear to be essentially due to inaccuracy in the bending modulus of the membrane and lead to an uncertainty on the order of a factor 10 for the ratio VHelf/Velec. The large range, from 10-2 to 1013, shows that, for the phases we investigated, the interactions between membranes continuously change from purely electrostatic repulsions (VHelf/Velec , 1) to purely entropic ones (VHelf/Velec . 1). Interpretation of the Stability of the Doped Phases. Effect of the Lamellar Periodicity. The existence of a minimum value for d, for a doped phase to be stable, is clear from the data in Figure 3; however, this minimum value, dmin, was found to increase as the salt content decreases: from 90 Å for brine at 0.5 M it went up to around 140 Å when the NaCl concentration was lowered to 0.1 M. Notice that it cannot be evaluated for water and brine with C0 ) 10-2 M swollen phases, as these phases have a maximum periodicity of around 90 Å and do not incorporate particles within their structure. These results led us to the two following conclusions: (i) There must be an absolute minimum value for d, below which the volume available between membranes is too small to host a particle. We only know, however, from our results that this absolute minimum is inferior or equal to 90 Å. The incorporation of particles in such a narrow swollen phase is worth emphasizing since it corresponds to an intermembrane distance (70 Å) comparable to the size of the particles (50 Å). (ii) Since dmin varies with the salt content, an explanation based on steric arguments cannot alone explain our experimental results. This is in agreement with other experiments (23) on similar systems (lamellar phases with positively charged and rigid membranes), which have shown that, even for large interlamellar spacings (up to 800 Å), these electrostatically stabilized phases cannot incorporate particles with a volume fraction larger than 0.05%. Effects of the Interactions. The incorporation of a negatively charged particle between two negative walls (the membranes) separated by counterions in water has an energetic cost for the particles all the higher as the electrostatic interactions are stronger. In the following, we calculate this cost in energy versus the salt content and the distance between membranes and use it to interpret our experimental data. We consider a system sufficiently dilute so as to neglect the interactions between particles (given the volume fraction of
Solid Particles and a Lamellar Phase particles used experimentally, the average distance between particles is about 110 Å, which validates the approximation) and calculate the increase of energy of a particle; the particle is modelized as a sphere, of radius R (25 Å) and constant surface charge density (evaluated at 0.3 C m-2) located in the midplane of the two plane and infinite membranes whose surface charge density varies with salt content (the values are given in Table 1). Knowing the energy per unit area of plane parallel halfspaces, one can evaluate the interaction energy of a sphere with two infinite planes. When the particle is immersed in brine, the electrostatic potential decreases sufficiently rapidly with distance to use a Derjaguin approximation.24 The interaction energy reads then E(H) ) (32R/l)kTγγpe-kH, where H ) (d - δ)/2 - R is the distance of closest approach of the sphere and the membrane and where γ (γp) is related to the surface potential of the membranes (particles). The numerical values of E for the doped phases for which this expression is valid (C0 g 0.1 M), that is, when the weak oVerlap approximation remains available (i.e., H > K-1), are reported in Table 1. Notice that the other condition required to use the Derjaguin approximation, i.e., H , R, is not fulfilled anymore for the most swollen phases. However, in this case the values obtained for E should be even smaller than above. When solvent is pure water, the electrostatic potential is longrange and the calculation is more delicate since the Derjaguin approximation is not valid anymore. In a first approximation, by replacing the sphere by a thin platelet of area πR2 located in the midplane of the two membranes, one can use the potential given in the preceding section and get an interaction energy of 13kT. The important point, as explained in the next paragraph, is that this value is very large compared to kT. A crude criterium for the stability of the doped phases is to compare the interaction energy of a particle with two membranes with the average value of energy for a colloı¨dal particle in solution, of the order of kT. The curve delimiting the stability domain in the (C0,d) plane is thus derived: d > dmin with dmin ) δ + 2R + 2κ-1 ln[(32R/l)γγp],1 where γp varies with the salt concentration and γ is fixed since the surface potentials for the membranes are nearly constant in the range of C0 concerned (from 0.1 to 0.5 M) as seen in Table 1. The curve dmin versus C0, plotted in Figure 3, shows a good agreement with our experimental results. The error on dmin originates mainly from the polydispersity of the size of the particles, which we have taken equal to 10 Å. Let us note that the model does not take into account the reduction of entropy of the particles due to their confinment between membranes, but the entropic effect is negligible compared to the electrostatic one. In conclusion, it is very satisfactory that the coupled influence of the two paramaters, d and C0, on the stability of doped phases be well accounted for by a simple model of electrostatic interaction between a charged particle and membranes. Conclusion The experimental study we have achieved on lamellar lyotropic phases doped with solid nanoparticles has shown that,
J. Phys. Chem., Vol. 100, No. 11, 1996 4537 contrary to a naive idea, the stability of these hybrid systems is not ensured as soon as a matching of the size has been realized, i.e., when the smectic periodicity is larger than the diameter of the particles. It appears in fact that this condition is necessary but not sufficient. The preeminent parameter is the nature of the interactions between the membranes of the lamellar system: when electrostatic interactions prevail, the solid particles are ejected from the lamellae; stability occurs when interactions are dominated by the fluctuations of the membranes as described in the Helfrich model. This has been proved by studying waterswollen phases where the addition of salt results in a continuous evolution of the ratio between the electrostatic potential and the Helfrich potential. The calculated values of those potentials are in good agreement with experiments. We have shown that the stability of the doped phases versus the interactions between membranes is related to the excess of electrostatic energy of a particle when the screening is varied. Acknowledgment. We have benefited from fruitful discussions with M. Veyssie´, M. Aubouy, and C. Gay, whom we warmly thank. We also thank V. Cabuil and the Laboratoire de Physico-Chimie Inorganique, Universite´ P. et M. Curie, Paris, for having provided us with the ferrofluid and D. Roux for fruitful comments. References and Notes (1) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1991. (2) Helfrich, W. Z. Naturforsch. 1978, 33A, 305. (3) Fabre, P.; Casagrande, C.; Veyssie M.; Cabuil V.; Massard, R. Phys. ReV. Lett. 1990, 64, 539. (4) Ponsinet, V.; Fabre, P.; Veyssie, M.; Auvray, L. J. Phys. II 1993, 3, 1021. (5) Ponsinet, V.; Fabre, P. Manuscript in preparation. (6) Rosensweig, R. E. Ferrohydrodynamics; Cambridge University Press: Cambridge, 1985. (7) Bee, A.; Massart, R.; Neveu, S. J. Magn. Magn. Mater., in press. (8) Massart, R.; Dubois, E.; Cabuil, V.; Hasmonay, E. J. Magn. Magn. Mater., in press. (9) Fabre, P.; Ober R.; Veyssie, M. J. Magn. Magn. Mater. 1990, 85, 77. (10) Roux, R.; Safinya, C. R. J. Phys. Fr. 1988, 49, 307. (11) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. 1990, 51, 2333. (12) Winterhalter, N.; Helfrich, W. J. Phys. Chem. 1988, 92, 6865. (13) Szleifer, I.; Kramer, D.; Ben-Shaul, A.; Roux, D.; Gelbart, W. M. Phys. ReV. Lett. 1989, 19, 1966. (14) Di Meglio, J. M.; Dvolaitzky, M.; Taupin, C. J. Phys. Chem. 1985, 89, 871. (15) Nallet, F.; Roux, D.; Prost, J. Phys. ReV. Lett. 1989, 62, 276. (16) Roux, D.; Nallet, F.; Freyssingeas, E.; Porte, G.; Bassereau, P.; Skouri, M.; Marignan, J. Europhys. Lett. 1992, 17 (7), 575a. (17) Pincus, P.; Joanny, J. F.; Andelman, D. Europhys. Lett. 1990, 11 (8), 763. (18) See for instance Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601. (19) Lekkerkerker, H. N. W. Physica A 1989, 159, 319. (20) Higgs, P. G.; Joanny, J. F. J. Phys. Fr. 1990, 51, 2307. (21) Safinya, C. R.; Sirota, E. B.; Roux, D.; Smith, G. S. Phys. ReV. Lett. 1989, 62, 1134. (22) Parsegian, A.; Fuller, N.; Rand, R. P. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 2750. (23) Me´nager, C. Thesis, Paris VI, 1995. (24) Derjaguin, B. V. Kolloid Z. 1934, 69, 155.
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