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Insertion of Small Anionic Particles in Negatively Charged Lamellar Phases I. Grillo,*,† P. Levitz,‡ and Th. Zemb§ ILL, DS/LSS, Avenue des Martyrs, 38042 Grenoble Ce´ dex 9, France, CNRS, CRMD, 1 bis rue de la Fe´ rollerie, 45071 Orle´ ans Ce´ dex 2, France, and CEA Saclay, DRECAM/SCM, 91191 Gif/Yvette Ce´ dex, France Received September 30, 1999. In Final Form: February 18, 2000
We have studied the structure of ternary systems made of anionic anisotropic Laponite particles in coexistence with negatively charged lamellar phases of AOT. First, we measured the adsorption of surfactant monomers on the mineral area. Then, after determination of the stable monophasic domain in the ternary system, we studied the occurrence of possible microstructures. SAXS measurements combined with osmotic pressure measurements show that small inorganic negatively charged particles can be forced to be included in the hydrophobic part of the surfactant bilayers when the interlayer water thickness becomes smaller than the particle thickness.
Clay particles are used industrially as fillers in complex formulations. It is therefore important to analyze the stability of mixed colloidal systems including a filler. One class of mixed colloidal systems is obtained when solid colloidal particles have to be introduced into ordered surfactant phases, such as lamellar phases. The case of spherical particles coexisting with lyotropic liquid crystals has been considered in refs 1-4. We describe here the first microstructural investigation in the single-phase region of a mixed system of anionic flat particles and anionic lamellar phases.
edge surface ( 2%, we have fitted the experimental curve and extrapolate it by the following empirical equation:
Πp(S/L) ) 9.9 × 105(S/L)1.9
(1)
Experimental Section The main prerequisite for these experiments was the use of solid mineral colloidal particles with a simple and well-defined anisotropic shape. For this reason, we have used Laponite RD, a synthetic trioctaedric hectorite manufactured by Laporte, Ltd. The general composition is Si8Mg5.45Li0.4H4O24Na0.7. The particle density is 2.65. The particle shape may be correctly described as a disk of 300 Å diameter and 10 Å thickness.5 The total specific area developed is Σp ) 800 m2/g, of which 50.3 m2/g represents the particle edge. Suspension concentrations are defined by S/L, the mass of the dried clay divided by the mass of the aqueous solution. The pH’s of solutions were adjusted to 8-9 by addition of NaOH, to avoid a congruent dissolution of particles. Nevertheless, when Laponite is in the presence of AOT, we measured the pH after a few days at 8. Under these conditions, the liquid/ soft solid transition is found around S/L ) 1.5%.5,6 Each particle bears about 1000 structural negative charges (one charge per 70 Å2), on the basal area. Only the edges are pH dependent. Amphoteric groups (MgOH, LiOH, and SiOH) are located on the * To whom correspondence should be addressed. † ILL, DS/LSS. ‡ CNRS, CRMD. § CEA Saclay, DRECAM/SCM. (1) Alexeev, V. L.; Ilekti, P.; Persello, J.; Lambard, J.; Gulik, T.; Cabane, B. Langmuir 1996, 12, 2392-2401. (2) Me´nager, C.; Belloni, L.; Cabuil, V.; Dubois, M.; Gulik-Krzywicki, T.; Zemb, Th. Langmuir 1996, 12, 3516-3522. (3) Ponsinet, V.; Fabre, P.; Veyssie, M.; Auvray, L. J. Phys. II 1993, 3, 1021-1039. (4) Ramos, L.; Fabre, P.; Dubois, E. J. Phys. Chem. 1996, 100, 45334537. (5) Mourchid, A.; Delville, A.; Lambard, J.; Le´colier, E.; Levitz, P. Langmuir 1995, 11, 1942-1950. (6) Mourchid, A.; Le´colier, E.; van Damme, H.; Levitz, P. Langmuir 1998, 14, 4718.
We used sodium bis(2-ethylhexyl)sulfosuccinate, usually called AOT, as a model anionic surfactant. The general formula is C20H37O4SO3- Na+. AOT was purchased by Fluka and used as received. The density is 1.13, and the molecular volume vm ) 650 Å3. The critical micellar concentration (cmc) at pH ) 8-9 determined by surface tension measurements is 6.3 × 10-4 mol/ L. Using the Gibbs law, we deduce an area per polar head of 86 Å2 at the air/liquid interface, by explicitly counting one surfactant plus its counterion (supposed to be 100% bound) as one molecule.9 The phase diagram of AOT in water or in brine has been studied for 25 years.10-13 With w designated as the weight percent of AOT, at pH ) 9, the maximum swelling of the lamellar phase (LR) is obtained for w ) 10%. Between w ) 1.4% and 10%, micelles are in equilibrium with the lamellar phase at its maximum swelling. Between w ) 10% and 17%, the existence of a singlephase lamellar domain is still under discussion, since the samples are birefringent but slightly turbid. Nevertheless, the smectic period follows the ideal one-dimensional swelling relation (eq 2) and no phase separation occurs, even under centrifugation or several months of decantation. Frances et al. have suggested the presence of a spherulite phase.14 Above w ) 17%, a single-phase LR exists. In the case of an ideal swelling law, the period of the (7) Cases, J.-M. Chim. Phys. 1969, 66, 1602-1611. (8) Thompson, D. W.; Butterworth, J. T. J. Colloid Interface Sci. 1992, 151, 236-243. (9) An, S. W.; Lu, J. R.; Thommas, R. K.; Penfold, J. Langmuir 1996, 12, 2446-2453. (10) Fontell, K. J. Colloid Interface Sci. 1973, 44, 156-164. (11) Fontell, K. J. Colloid Interface Sci. 1973, 44, 318-329. (12) Skouri, M.; Marignan, J.; May, R. Colloid Polym. Sci. 1991, 269, 929-937. (13) Balinov, B.; Olsson, U.; So¨derman, O. J. Phys. Chem. 1991, 95, 5931-5936. (14) Frances, E. I.; Hart, T. J. J. Colloid Interface Sci. 1983, 94, 1-13.
10.1021/la991288c CCC: $19.00 © 2000 American Chemical Society Published on Web 05/05/2000
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Figure 1. Swelling law of lamellar phases of AOT in H2O. The smectic periodicity d* is plotted against the inverse of the surfactant volume fraction Φ: ([) experimental points; (full line) one-dimensional ideal swelling law. lamellar phases follows
d* )
δ Φ
(2)
where Φ is the volume fraction of surfactant and δ is the bilayer thickness. The water layer thickness D is always given by d* δ. We have measured by SAXS the smectic period d* for surfactant concentrations between 20% and 60%. The experimental dilution law, with the evolution of d* as a function of the inverse of the volume fraction Φ, is shown in Figure 1. Using the linear swelling law (eq 2), we obtain a membrane thickness δ ) 19.5 Å, in good agreement with previous studies.11,12 From the bilayer thickness, we deduce an area per headgroup of σ ) vm/δ ) 67 Å2, close to the area found at the air/liquid interface by surface tension determinations, and thus a specific area Σs ) 900 m2/g.
Microstructure Anisotropic colloids such as platelets offer new possible insertion mechanisms in surfactant lamellar phases. We consider three microstructures, two single-phase ones and a two-phase one, schematically represented in Figure 2. In the single-phase domain of the ternary system, the particles may either be dispersed in the water layers, between the surfactant membranes, according to the dispersion model (Figure 2a), or enter the bilayers according to the insertion model (Figure 2b). In the first case, the particles add volume; in the second one, they add surface. In the two-phase domain (Figure 2c), the sample is microphase-separated into a Laponite-rich phase and a surfactant-rich phase, a situation already encountered and described by Morvan15 for larger clay particles (montmorillonite) in coexistence with polyelectrolytes.
Figure 2. Different possible microstructures for the ternary system in the lamellar domain of the surfactant: (a) dispersion model with particles in the water, between the surfactant bilayers; (b) insertion model where particles enter the surfactant bilayers; (c) exclusion model where particles are ejected from the lamellar phase.
Laponite suspensions. A mother solution was prepared by addition of 1.5 × 10-2 mol/L of AOT in a clay solution (S/L ) 0.2%, 0.5%, or 0.7%). Samples were then obtained by dilution of the mother solution by the clay suspension and were kept for equilibration for 1 day. As the particles were 2.65 times denser than the surfactant monomers, 12 h of centrifugation at 15 000 rpm allowed us to separate particles with adsorbed monomers from the unbound surfactant molecules, in equilibrium in the bulk. The equilibrium surfactant concentration, ceq, was measured by a TOC (Total Organic Carbon) method, in the supernatant. A standard TOC analyzer DC-180 from Dorhmann, Rosemont, was used. Adsorption in moles/gram is the amount of the adsorbed surfactant, that is, the difference between the total initial surfactant concentration, cin, and the equilibrium concentration, ceq, divided by the mass of dried clay in the sample:
Determination of the Adsorption Isotherm Inserting colloidal particles into a surfactant solution may induce adsorption of a part of the available surfactant, which is consequently no longer included in the bulk phase.16 Therefore, we have first to determine the adsorption isotherm of AOT on Laponite particles. We considered first the dilute part of the phase diagram, around the cmc of the surfactant and below the liquid/solid transition of (15) Morvan, M.; Espinat, D.; Lambard, J.; Zemb, Th. Colloids Areas A 1994, 82, 193-203. (16) Grillo, I.; Levitz, P.; Zemb, Th. Eur. J. B 1999, 10, 29-34. (17) Brahimi, B.; Labbe´, P.; Reverdy, G. Langmuir 1992, 8, 19081915. (18) Grillo, I. The`se de l′universite´ Paris XI, 1998.
Γ(mol/g) )
cin - ceq S/L
(3)
The adsorption isotherm obtained at 22 °C is shown in Figure 3. The adsorption is independent of the initial clay concentration. It reaches a plateau slightly above the cmc at 0.28 mmol/g, a value around 10 times lower than that found for adsorption of nonionic16 or cationic17,18 surfactants on Laponite. Indeed, nonionic and cationic surfactants are adsorbed on the particle faces while AOT remains on the edges. Adsorption of anionic molecules on globally anionic clay is possible, as the pH ) 8-9. Thus, positive charges are
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Grillo et al. Table 1. SAXS Parameters for the Lamellar Phases of AOT with the Laponite Fitting Procedure, Using the Models Described in the Appendix AOT monomer volume (Å3) δT (Å) δH (Å) FT (cm-2) (SAXS) FH (cm-2) (SAXS) Favg (cm-2) (SAXS)
650 8.1 1.9 6.70 × 1010 2.78 × 1011 1.03 × 1011
Laponite Vp (Å3) FP (cm-2) (SAXS)
7.1 × 105 2.23 × 1010
H2O vw (Å3) FH2O (cm-2) (SAXS)
Figure 3. Adsorption isotherm of AOT on Laponite particles. Adsorption in millimoles per gram is plotted against the equilibrium surfactant concentration ceq. Experiments are carried out at 22 °C, for three clay concentrations: (9) S/L ) 0.2%; ([) S/L ) 0.5%; (2) S/L ) 0.7%. The vertical arrow indicates the cmc of AOT.
present on the edge of the particles, where anionic monomers are expected to be strongly adsorbed due to electrostatic attraction. We introduce the coverage parameter c to describe geometrically the adsorption layer.16 c is the ratio between the total area developed by the surfactant and the lateral area of the particles:
c)
NaΓσ Σlateral clay
(4)
Na is Avogadro’s number. In a schematic picture where the monomers would be perpendicular to the solid area, c ) 1 would correspond to a complete monolayer around the particle, and c ) 2 to a bilayer. For this description, we chose as reference state a molecular area of σ ) 67 Å2 in the lamellar phase.16 As ) 50.3 m2/g. We found at the seen previously, Σlateral clay plateau of adsorption c ) 2.1: enough surfactant is adsorbed to form a complete bilayer on each particle edge. Identification of the One-Phase Domain in the Phase Diagram We focus now on the lamellar domain of AOT in water, for w > 20%, well above the cmc. Samples were systematically prepared for w ) 20% to 60% and S/L ) 0.01% to 4%. Stable one-phase domains were identified by visual inspection. Sample birefringence was checked between crossed polarizers, and turbidity characterized a demixing and a two-phase sample. The resulting phase diagram is presented in Figure 4. The continuous line separates the single-phase domain from the two-phase one. The experimenal result is an increase of S/L while the surfactant concentration increases. For w < 20%, particles are ejected from the lamellar phase. Above 20%, the maximal clay concentration S/L increases with w, from 0.01% (w ) 20%) up to 0.7% (w ) 60%). Above w ) 60%, the insertion limit is prolonged, following the dotted line. As the AOT samples in water are themselves slightly turbid, it is not obvious whether the turbidity is really due to the demixing of the clay particles in this ternary system. The crosses and the thin dotted lines represent the composition of the phases in osmotic equilibrium in two two-phase samples (w ) 30%, S/L ) 5% and 10%). The phases have been separated by centrifugation (12 h at
30 9.33 × 1010
25 000 rpm). AOT and Laponite concentrations have been measured by COT and atomic adsorption of Si, respectively. The structure of the phases has been determined by SAXS. The sample is microphase-separated into a Laponite-rich phase and a surfactant-rich phase, according to Figure 2c. Structural Characterization of the Samples by SAXS The microstructure and smectic period of the samples have been investigated by SAXS, on the Huxley-Holmes and Guinier cameras available in the laboratory.19 Figure 5 shows spectra of lamellar phases of AOT obtained either for w ) 20% and S/L ) 0.5%, in the two-phase domain of the ternary system, or for w ) 65% and S/L ) 0, 0.2, and 0.5%, in the one-phase domain. Points correspond to the experimental curve, and full and dotted lines are the best fits obtained, using the two possible single-phase models described in the appendix and the parameters given in Table 1. In the fitting procedure, the only variables are the scaling factor A, the background bg, and Caille´’s parameter η. The Bragg peak positions are measured on the scattering curves after the radial averaging. These peaks, sharp even when plotted on the log scale, emerged out of a q-2 behavior, the signature of a Laponite particle. The same peak positions are found before and after division by the particle form factor. Due to the two camera resolutions and the q-range of the lamellar periods studied here, the accuracy of the position is better than 1%. We focus now on the most informative lines in the phase diagram; similar results with other surfactant and clay concentrations are presented in ref 18. Two-Phase Domain of the Ternary System. At w ) 20% (Figure 5a), only one Bragg peak is present, and we clearly see, at 0.15 Å-1 < q < 0.4 Å-1, the form factor of the surfactant bilayer. This oscillation and the q-value where the form factor equals zero allow the ratio ∆FT/∆FH in eq 23 to be adjusted. We find that the form factor goes to zero for q ) 0.1120 Å-1. The fitting parameters are summarized in Table 2. The increase of the intensity at low q is due to the platelet particles. Nevertheless, in this two-phase sample, we lose the q-2 slope, a signature of a dilute suspension of randomly oriented cylinders. Clusters of excluded particles are in osmotic equilibrium with a lamellar phase. From the Bragg peak position, at q0 ) 0.0575 Å-1, we deduce the surfactant concentration in the lamellar phase with eq 2, Φ ) 18.2% (w ) 20.4%). From eq 17, we calculate that this phase imposes an osmotic pressure of 1.1 × 105 (19) Rapport d′activite´ 1996-1997 du Service de Chimie Mole´culaire; DRECAM, CEA Saclay; http://www-drecam.cea.fr/scm/science9697/ fthem.htm.
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Figure 4. Phase diagram of Laponite particles in lamellar phases of AOT. The full line prolonged by the dotted line corresponds to the maximal clay concentration in the surfactant lamellar phase.
Figure 5. SAXS of lamellar phases of AOT with Laponite particles for different surfactant and clay concentrations, in the twophase domain [(a) w ) 20%, S/L ) 0.5%] and the single-phase doamin [(b) w ) 65%, S/L ) 0%; (c) w ) 62%, S/L ) 0.2%; (d) w ) 66%, S/L ) 0.5%. (]) experimental points; (s) dispersion model; (- - -) inclusion model. For w ) 20% and S/L ) 0.5%, the sample is two phase. In the single-phase domain, for w ) 62%, S/L ) 0.2 and w ) 66%, S/L ) 0.5%, the water layer thickness is larger than the particle thickness and the best agreement between experiment and fitting is obtained with the dispersion model (eq 24).
Pa. From eq 1, a Laponite solution with an equivalent osmotic pressure corresponds to S/L ) 36%. At this concentration, the particle positions in the dense particle
phase are correlated and it would be necessary to introduce a specific structure factor to simulate the scattered intensity.
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Table 2. Parameters Corresponding to the Best Fit of the Experimental Scattering Curves Presented in Figure 5, Obtained with the Dispersion Model wt % S/L % A bg (cm-1) η
65 0 0.16 0.005 0.15
62 0.2 0.2 0.05 0.15
66 0.5 0.16 0.05 0.17
Table 3. Relative Humidity p/p0 and Corresponding Osmotic Pressures for the Six Saturated Salt Solutions at 22 °C, Used for the Experiment salt
p/p0
Π (Pa)
salt
p/p0
Π (Pa)
K2Cr2O7 K2SO4 BaCl2 KCl
0.98 0.95 0.90 0.854
2.74 × 106 6.96 × 106 1.44 × 107 2.14 × 107
NaCl Mg(NO3) MgCl2
0.753 0.529 0.328
3.85 × 107 8.64 × 107 1.51 × 108
Single-Phase Domain of the Ternary System. Three X-ray spectra of single-phase samples are shown in Figure 5a-c. In the single-phase domain, the characteristic q-2 slope from platelets is present in the samples with Laponite. The best fits are obtained using eq 24 derived from the dispersion model. The corresponding η values are given in Table 2. Addition of these small particle concentrations does not modify the compression parameter η introduced by Caille´21. Since η is not influenced by the presence of the clay particles, the bending constant kc of the bilayer should not be affected either. X-ray Grazing Incidence under Controlled Osmotic Pressure The experimental setup previously described in ref 20 was used. Five samples were deposited on quartz plates and equilibrated above a removable reservoir containing a saturated salt solution, in a closed, hermetic, and thermostated box. Around 12 h were necessary to reach the osmotic equilibrium. Temperature and pressure were followed with a Novasina captor. This setup was installed on a rotating table in order to orient the quartz lamellae in grazing incidence (θ, 2θ), in the direct focused X-ray beam of a Huxley-Holmes camera.19 The chemical potential of water in the sample was imposed by a saturated salt solution. A relative humidity of p/p0 set a pressure equal to
()
kT p Π)ln νw p0
(5)
vw is the molecular volume of water, k is the Boltzmann constant, and T is the temperature. The values of relative humidity are known for many salts,21 and six different values were used to cover a pressure range from 2 × 106 to 2 × 108 Pa. The osmotic pressures calculated at 22 °C with eq 5 are given in Table 3. The maximal uncertainty in the relative humidity estimated, at (0.05 for p/p0 ) 0.98, yields an error of (1.5 × 105 Pa in the osmotic pressure values. The water content changes with the osmotic pressure. The relevant parameter to characterize the samples is RS, the ratio between the surfactant bilayer area and that of the particles, which remains constant.
Rs )
Σsms Σs w(1 + S/L) ) Σpmp Σp S/L(1 - w)
(20) Dubois, M.; Zemb, Th. J. Phys. IV 1998, 8, 55-62. (21) O′Brien, F. E. M. J. Sci. Instrum. 1948, 21, 73-76.
(6)
Figure 6. Raw SAXS data obtained on the two-dimensional multiwire gas detector17 with the ternary sample RS ) 120. On the right-hand side is shown the shadow of the beam stop; on the left is shown the partial ring of the oriented lamellar phase(s): (a) average relative humidity p/p0 ) 0.930, Π ) 9.85 × 106 Pa; (b) average relative humidity p/p0 ) 0.918, Π ) 1.20 × 107 Pa; (c) average relative humidity p/p0 ) 0.869, Π ) 1.90 × 107 Pa.
ΣS and ΣP are the specific areas developed by the surfactant bilayers and the clay particle faces, respectively, and mS and mP, are their masses in the sample. For this experiment, RS ) 58, 120, 250, and infinity (AOT only). Samples were prepared in the single-phase domain of the ternary system, around w ) 50%, to have a homogeneous particle dispersion. In Figure 6, raw SAXS data obtained on the two-dimensional multiwire gas detector15 with the ternary sample RS ) 120 are shown for three relative humidities p/p0. On the right is shown the shadow of the beam stop; on the left is shown the partial ring of the oriented lamellar phase(s). The osmotic pressure as a function of the lamellar period d* is plotted in Figure 7. The smectic period decreases as the osmotic pressure increases, and three domains may be distinguished.
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Figure 7. Evolution of the osmotic pressure as a function of the lamellar period for four surface ratios between surfactant bilayers and clay particles: ([) RS ) 58; (2) RS ) 120; (9) RS ) 250; (b) RS ) infinite.
(a) For Π < 1.7 × 107 Pa, an exponential relation, as in eq 7, is found. (b) In the intermediate regime, a reversible equilibrium between two lamellar phases is found in the three samples containing AOT and Laponite, at around 1.4 × 107 Pa, the pressure set by BaCl2. The water layer thickness is then close to the particle thickness. This equilibrium between two lamellar phases is not obtained with pure AOT. This equilibrium domain exists in a narrow pressure range, and no calibrated salt solution is available to explore such a small pressure range. Samples were equilibrated with KCl (Π ) 2.14 × 107 Pa), and the salt solution is changed for one of K2SO4 (Π ) 6.96 × 106 Pa). The transition was followed in real time on the gas detector, with 5-min acquisitions every 10 min, for the sample RS ) 120. Relative humidity and temperature were measured before and after the experiment to calculate the average pressure during the exposure. When the equilibrium (at 6.96 × 106 Pa) was reached, K2SO4 was replaced by KCl and the experiment was repeated. The same equilibrium periods were found: the transition is reversible, since we systematically observe the SAXS spectra (a-c) shown in Figure 6 (or in the reverse order) while hydrating or dehydrating the sample. We find that the equilibrium between two lamellar phases is present for 1.2 × 107 Pa < Π < 1.7 × 107 Pa. The variations of the periods with the osmotic pressure are presented in Figure 8. These real-time measurements are possible due to the high flux of the Huxley-Holmes X-ray camera. As the sample is out of equilibrium, we have to find a compromise between the pressure variation and the statistics obtained in the Bragg peak. During a 5-mn acquisition, the pressure variation is around 2 × 105 Pa, corresponding to an uncertainty of 2% in the pressure and (0.2 Å on the Bragg peak position. (c) For Π > 1.7 × 107 Pa, the interlamellar distance decreases from 8 to 2 Å. The hydration pressure dominates the total pressure and decays exponentially as in eq 7.
Figure 8. Periods and osmotic pressures of the coexisting lamellar phases during the transition from a single-phase sample with particles between the bilayers to a single-phase sample with particles in the bilayers. RS ) 120. This figure is a magnification of Figure 7, in the narrow pressure range from 106 to 4 × 107 Pa. Table 4. Amplitude and Decay Length of the Hydration Pressure, Obtained by Linear Regression of the Experimental Points in Figure 7a
a
RS
Π0 (Pa)
λ (Å)
∞ 250 120 58
2.81 × 2.84 × 108 2.63 × 108 2.84 × 108
2.57 2.53 2.59 2.53
108
The reference of the spacing scale is 19.5 Å.
The Π0 and λ values obtained after linear regression are given in Table 4. The origin of the spacing scale dhyd is taken at 19.5 Å. The presence of Laponite does not change Π0 and λ. λ is comparable to the average size of a water molecule as already measured by McIntosh and Parsegian
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on lipid bilayers.22,23 λ depends on the nature of the interlamellar space. Thus, its invariance proves that the interlamellar space remains the same with and without Laponite. In all the measurements performed, the area developed by the surfactant is at least 50 times greater than that due to the clay particles. As long as the ratio RS is >1, the physical properties of the bilayer are not significantly modified and Π0 remains constant. Discussion: Determination of the Microstructure of Clay Particles in the Lamellar Phase Thanks to the adsorption isotherm, we know that the amount of surfactant adsorbed is 0.28 mmol/g or 0.12 g/g. From the phase diagram, we know that < 0.7% of the Laponite enters the lamellar phase. As a consequence, the amount of surfactant adsorbed is 10 Å.24,25 Hydration pressure decreases exponentially as
(
Πhyd ) Π0 exp -
)
d* - dhyd λ
(7)
where the decay length λ is of the order of a few angstroms, close to the size of a water molecule.22,23 The amplitude Π0 is around 108-109 Pa, choosing as the reference of the spacing scale dhyd ≈ δ, the bilayer thickness deduced from the swelling relation. We were working at low ionic strength. The following expression (eq 8), neglecting residual salt, is numerically equivalent to the complete case developed in ref 26. It is appropriate to estimate the repulsive electrostatic component of the osmotic pressure, which is a function of the lamellar period d*, the membrane thickness δ, and the area per polar head σ.26
πkT (1 - 2/F) Πelec(d*,δ,σ) ) 2LB(d* - δ)2
(8)
where LB is the Bjerrum length, equal to 7.2 Å in water, and F is a numerical parameter given by F ) πLBD/2σ, with D the bilayer separation. For the surfactant concentrations used in the following, the steric interaction is at least 100 times smaller than the electrostatic one and will therefore be neglected. The long-range attractive van der Waals force can be written as27,28 (22) McIntosh, T. J.; Simon, S. A. Biochemistry 1986, 25, 40584066. (23) Parsegian, V. A.; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777-4782. (24) Parsegian, V. A.; Rand, R. P. Langmuir 1991, 7, 1299-1301. (25) McIntosh, T. J.; Simon, S. A. Biochemistry 1986, 25, 40584066. (26) Dubois, M.; Zemb, Th.; Belloni, L.; Delville, A.; Levitz, P.; Setton, R. J. J. Chem. Phys. 1992, 96, 2278. (27) Ninham, B. W.; Parsegian, V. A. Biophys. J. 1970, 10, 646-674. (28) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976.
ΠvdW(d*,δ) )
(
)
A 1 2 1 + 6π D3 d*3 (D + 2δ)3
(9)
where A is the Hamaker constant, derived from the summations over the dielectric susceptibilities of water and surfactant. In the case of surfactant bilayers separated by water, A is of the order of kT.29 The validity of the decomposition into three terms has been checked by Ricoul et al.30 Finally, the osmotic pressure of a lamellar phase is equal to
Πs ) Πelec(d/s ,δ,σ) + ΠvdW(d/s ,δ) + Πhyd(d/s ) (10) For the ternary system, we have described at the beginning of the paper the different possible microstructures (Figure 2). We now search the expressions for the swelling law and the osmotic pressure as a function of the surfactant and clay concentrations for each possible organization. In the dispersion regime (Figure 2a), due to their density, which is 2.65 times higher than that of water, particles modify the water layer volume. Assuming the conservation of volume of the components, the smectic period d/disp of this ternary system in this state should follow the swelling law, as in eq 2, where the surfactant volume fraction is
Φ)
Vs ) Vs + Vp + Vwater dsw(1 + S/L) 1 + w(ds - 1) + S/L(w(ds - dp) + dp)
(11)
The osmotic pressure is estimated using eqs 1 and 10 and according to the hypothesis
Πdisp ) Πs(d/disp,δ,σ) + ΠLaponite(S/L)
(12)
In the insertion mode (Figure 2b), particles in the bilayer increase the surface-to-volume ratio, since they participate in the overall mixed bilayer area. Due to the volume conservation, the membrane thickness is, on average, expected to be as follows:
δinser )
Ss Sp δ+ e Ss + Sp Ss + Sp
(13)
SS and SP are the surfactant and clay areas developed in the sample. Thus, the lamellar swelling law is
d/inser ) Φmembr )
δinser Φmembr
Vs + V p ) Vs + Vp + Vwater dsw + S/L(w(ds - 1) + dp) 1 + w(ds - 1) + S/L(w(ds - dp) + dp)
(14)
The electrostatic pressure is calculated using eq 10, with the parameters of the mixed bilayer:
Πinser ) Πs(d/inser,δinser,σinser)
(15)
σinser is the average surface area per charge: (29) Parsegian, V. A. Langmuir 1996, 12, 4057-4059. (30) Ricoul, F.; Dubois, M.; Belloni, L.; Zemb, Th.; Rico-Lattes, I. Langmuir 1998, 14, 2645-2655.
Negatively Charged Lamellar Phases
σinser )
Ss + S p Ss/σs + Sp/σp
Langmuir, Vol. 16, No. 11, 2000 4837
(16)
We consider now the third possible microstructure (Figure 2c): in this two-phase regime, period and osmotic calculations are indissociable. The main problem is to know the partition of water in the surfactant phase and in the particle phase, satisfying the equality of the osmotic pressures in the two phases:
Πs(d/excl,δ,σ) ) Πp((S/L)excl)
(17)
This equality is solved with an iterative method, where we increase regularly the volume of water in the surfactant phase and we calculate at each step the electrostatic pressure of both phases. If the volume fraction of water in the surfactant phase increases, the lamellar pressure decreases. In parallel, the volume fraction of water in the clay suspension decreases and the osmotic pressure increases. The intersection of the two osmotic curves gives the equilibrium-phase concentrations and the osmotic equilibrium. As the curves decrease or increase monotonically, the solution is unique. In Figure 9, the osmotic pressure calculated with eqs 12, 15, and 17 is plotted against the lamellar period d*, the “pure” lamellar phase (AOT in water), and the two possible single-phase models for the ternary lamellar system. At a given pressure, the period of an insertion model is smaller than the period of a binary lamellar phase, which is itself smaller than the period of a dispersion model.
d/inser e d/pure e d/disp
Figure 9. Osmotic pressure of mixed and pure lamellar phases as a function of the period d*, calculated with eqs 12, 15, and 17, for S/L ) 20%. The chosen clay concentration does not correspond to a real concentration in the single-phase ternary domain but allows here a good separation between the different curves: (thick line) insertion model; (dotted line) dispersion model; (thin line) pure lamellar phase. At a given osmotic / / pressure, dinser < d/pure < ddisp .
(18)
Thus, the comparison between the binary and ternary system periods at a given pressure allows one to determine which microstructural model is relevant. What is the origin of the phase separation revealed by X-ray grazing incidence under controlled osmotic pressure, shown in Figures 6-8? For each experimental pressure, we compare periods obtained in the ternary samples with the smectic periods of AOT in water, by plotting in Figure 10 the osmotic pressure as a function of the difference d between the period of the sample containing clay and AOT and that of the sample with AOT only: d ) d*(RS) - d*(RS)∞). Below the transition pressure ΠT ) 1.4 × 107 Pa, d > 0; thus, d*(RS)∞) < d*(RS). From eq 18, we deduce that particles are dispersed in the water, between the bilayers, as shown in Figure 2a. The transition occurs when the interlamellar thickness D equals 8 Å, a value close to the particle thickness. For high pressures, above ΠT, d < 0 and d*(RS) < d*(RS)∞): particles enter the surfactant bilayers, as in the insertion model presented in Figure 2b. Thus, at sufficiently high osmotic pressures corresponding to a water layer thinner than a dressed Laponite particle, we have shown that anionic particles may be forced to penetrate surfactant bilayers. We have summarized in Figure 4 the phase diagram of Laponite particles in lamellar phases of AOT with the three different microstructures encountered. In the singlephase domain of the ternary system particles are dispersed between the bilayers, in the water, as long as the interlamellar space is larger than the particle thickness. We have shown that these flat particles are forced to enter the bilayers when the osmotic pressure increases and the water thickness is smaller than the particle thickness. In the two-phase exclusion regime, a lamellar surfactant
Figure 10. Difference between the smectic periods of the ternary sample and of the “pure” lamellar phase, for each osmotic pressure fixed by the different saturated salt solutions: ([) RS ) 58; (2) RS ) 120; (b) RS ) 250; (a) Π < ΠT and d*(RS) > d*(RS)∞), particles are dispersed in the water layer (dispersion regime); (b) narrow pressure range with two lamellar phases in equilibrium; (c) Π > ΠT and d*(RS) < d*(RS)∞), particles are forced to enter the bilayers (insertion regime).
phase is in osmotic equilibrium with dense Laponite aggregates with concentration S/L > 36%. Acknowledgment. The authors wish to thank Fre´de´ric Ne´ (CEA Saclay) and Olivier Tache´ (CEA Saclay) for their kind help during the X-ray scattering measurements in the laboratory. They acknowledge P. Timmins (ILL) for careful reading of the manuscript. Appendix: SAXS Intensities Produced by Lamellar Phases of a Ternary System Made of Water, Surfactant, and Platelet Particles For a dilute suspension of randomly oriented cylinders of thickness 2 and radius r, the scattering intensity in inverse centimeters is31 (31) Guinier, A.; Fournet, G. Small Angle Scattering of X-rays; Wiley: New York; 1955.
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Langmuir, Vol. 16, No. 11, 2000
Grillo et al.
Ip(q) (cm-1) ) KcP(q) with Kc ) ∆F2ΦVp
(19)
Kc is the contrast constant and depends on ∆F, the contrast between the particle and the solvent, Φ, the clay volume fraction, and VP, the volume of one particle. P(q), the form factor, is given by
P(q) )
π/2sin
∫0
2 (q cos R) 4J1 (qr sin R)
2
(q cos R)2
(qr sin R)2
sin R dR (20)
R is the angle between the normal to the particle and the wave vector q. J1 is the first-order Bessel function. For Laponite particles, the scattering intensity reaches a plateau for q < 10-2 Å-1. At higher q, a q-2 decrease is characteristic for two-dimensional objects. For the lamellar phase, we use the general scattering model developed first by Blaurock32 and modified by Nallet et al. to introduce explicitly bending modulus constants.33 This model combines the geometry of the bilayers and the membrane displacement due to thermally induced lattice vibrations. The only adjustable parameter is η, Caille´’s parameter,34 which is inversely proportional to the square root of the elastic and bending bilayer constants (B h and Kc) and describes the interactions between the membranes.
η)
q20kT
the fitting model for lamellar phases has systematically an intensity 2-5 times higher than the experimental one, on an absolute scale), and bg is a constant necessary to take into account background and solvent scattering, if their contributions are not subtracted during data reduction. IS(q) is the intensity scattered by the lamellar phase with the period d*1. IP(q) is the intensity scattered by a clay suspension of volume fraction
Φp )
Ps/p(q) Ss/p(q) -1 I(1) S/P(q) (cm ) ) 2π d1/ q2
N/2
SS/P(q) ) 1 + 2
2
(22)
d*q2
where the form factor P(q) and the structure factor S(q) are
PX-ray(q) )
[
∆F2H
4
2
q
sin q(δH + δT) - sin(qδT) +
[
exp N-1
S(q) ) 1 + 2
∑1
2q2β(n) + ∆q2z2
]
2(1 + 2∆q2β(n))
x1 + 2∆q β(n) 2
(
cos
]
∆FT sin(qδT) ∆FH
1 + ∆q2β(n)
)
[
q2β(n) + ∆q2z2
2(1 + ∆q2β(n))
×
(27)
with
z ) (nd*1 - d*1/2) PS/P(q) ) 4Φp
[
∆FH∆FP sin q(δH + δT) - sin(qδT) + q ∆FT sin(qe) sin(qδT) ∆FH qe
][
]
(23) Vhole ) πr22(δH + δT)
∆FT and ∆FH are the contrasts between the hydrophobic chain and the polar head, respectively, of the surfactant and the solvent. β(n) is the correlation function describing the layer displacement fluctuation.33 If the dispersion model (Figure 2a) is assumed, the ternary system is composed of two types of scatterers: the lamellar phase and the clay particles, in the midplane between the bilayers. The scattered intensity is the sum of three terms: (1) (q) ) A(IS(q) + I(1) IX-ray S/P(q)) + IP(q) + bg
) ]
qz
For the insertion model, three objects are involved in the observed scattered intensity: (a) the bilayers, of thickness δ and period d2, (b) the clay particles, in the midplane of the bilayers, of elementary volume VP and total volume fraction ΦP (eq 25), and (c) the “holes” in the bilayers, due to the difference between the membrane and the platelet thicknesses. Their elementary volume is
× qz
(
cos ∑1 2 2 x1 + ∆q β(n) 1 + ∆q β(n) exp -
P(q) S(q)
(26)
where the structure factor SS/P and the form factor PS/P are given by
The scattered intensity on an absolute scale is
Is(q) (cm-1) ) 2π
(25)
I(1) S/P(q) is a cross term that takes into account the presence of clay particles between the bilayers.
(21)
8πxKcB h
Vp Vs + Vp + Vwater
(24)
(28)
and their volume fraction equals
ΦH )
NpVH Vtot
(29)
where Np is the clay particle number
Np )
Vtot p Vp
A is a scaling factor (necessary, as we have noticed that
Finally, the scattered intensity is a sum of six terms:
(32) Blaurock, A. E. Biochim. Biophys. Acta 1982, 650, 67. (33) Nallet, F.; Laversanne, R.; Roux, D. J. Phys. II 1993, 3, 487502. (34) Caille´, A. C. R. Acad. Sci. Paris B 1972, 891-893.
(2) (2) (2) I(2)(q) ) A(I(2) S (q) + IS/P(q) + IS/H(q)) + (IP (q) + (2) I(2) H/H(q) + IP/H(q)) + bg (30)
Negatively Charged Lamellar Phases
Langmuir, Vol. 16, No. 11, 2000 4839
(a) I(2) S (q) is the scattered intensity of a lamellar phase of period d2, given by eq 23. (b) I(2) S/P(q) represents the presence of clay particles in the bilayer. It has the same expression as I(1) S/P(q) (eq 26) with z ) nd2. (c) Scattering heterogeneity between membrane and hole yields S/H I(2) S/H ) 2πKc
PS/H(q)SS/H(q) q
VH ) ΦP∆FP∆FH KS/H c VP
)]
(
) ΦHVH ∆FHd∆Fp KP/H c PH/H ) 4
(33)
with for the contrast constant and form factor
) ΦHVH ∆F2H KH/H c PH/H ) 4
[
]
J1(qr sin R) 2 × qr sin R sin q(δH + δT) cos R + q(δH + δT) cos R sin qδT cos R δT∆FT/∆FH -1 qδT cos R δH + δT
∫0π/2sin R dR
[
(
)]
[
][
]
J1(qr sin R) 2 sin(q cos R) × qr sin R q cos R sin q(δH + δT) cos R + (δH + δT) cos R sin (qδT cos R) δT∆FT/∆FHd - 1 (36) qδT cos R δH + δT
∫0π/2sin R dR
[
(
)]
In the two-phase domain, the scattered intensity is the sum of the intensities of both phases, weighted by their volume fractions xS and xP:
(d) I(2) P (q) is the intensity scattered by a clay suspension of volume fraction ΦP, as in eq 19. (e) The scattered intensity due to holes has the following expression: H/H I(2) H/H(q) ) Kc PH/H(q)
(35)
where the contrast constant and the form factor are
]
∆FT sin(qδT) ‚ ∆FH sin q(δH + δT) sin(qδT) ∆FT δT (32) qδT ∆FH δH + δT q(δH + δT)
PS/H ) 4 sin q(δH + δT) - sin(qδT) +
[
P/H I(2) P/H(q) ) Kc PP/H(q)
(31)
2
The structure factor SS/H is obtained with eq 23. The contrast constant and the form factor are
[
(f) The last cross term is equal to
I(3)(q) ) AxSIS(q) + xPIP(q) + bg
(37)
Nevertheless, the lamellar phases stabilized by electrostatic repulsion induce high osmotic pressures, from 106 to 108 Pa. Thus, the Laponite concentrations (S/L) of suspensions in osmotic equilibrium with these lamellar phases vary between 10% and 500%! Such a suspension cannot be considered as dilute, and particle positions are correlated with each other. It is therefore essential to include a structure factor term to adequately describe the experimental data.
2
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