Effect of a Nonadsorbing Polymer on the Stability of a Two-Solvent

Langmuir 2000, 16, 2581-2594

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Effect of a Nonadsorbing Polymer on the Stability of a Two-Solvent Lamellar Phase: Experimental and Theoretical Study of Critical Points of Lamellar/Lamellar Phase Separations L. Porcar,* J. Marignan, and C. Ligoure Groupe de Dynamique des Phases Condense´ es C.C.26, Unite´ Mixte de Recherche UM II / CNRS 5581, Universite´ Montpellier II, F-34095, Montpellier, Cedex 5, France

T. Gulik-Krzywicki CNRS, Centre de Ge´ ne´ tique Mole´ culaire, 91198, Gif sur Yvette, France Received September 9, 1999. In Final Form: November 23, 1999 We investigate how the elastic properties of a two-solvent lyotropic lamellar phase are modified by the addition of a nonadsorbing water-soluble homopolymer (polyvynilpyrolidone). The initial two-solvent lyotropic lamellar phase consists of a regular stack of surfactant monolayers made from a mixture of nonionic surfactants (TX100/TX35) and a cationic surfactant (cetylpyridinium chloride) which separate successive layers of oil (decane) or water. By modifying the relative proportions of the various components of the lamellar phase or the temperature, we are able to control the magnitudes of the different contributions to physical interactions between the bilayers. From small-angle X-ray measurements on this system, we have identified four different critical LR/LR phase separations. At such critical points the corresponding smectic susceptibility B h -1 diverges. In an attempt to better understand these experimental results we present a general calculation of the smectic compression modulus B h of a four-component lyotropic lamellar phase. This model is applied to our specific system. This allows us to understand the occurrence of three of the observed critical points of our system.

1. Introduction Phase diagrams of surfactant in water exhibit an interesting variety of structures.1 Among them is the lamellar LR phase, which is perhaps the best characterized lyotropic liquid crystalline phase known to date. It consists of infinite planar bilayers of one or more amphiphilic components periodically stacked in space separated by a solvent. The long-range periodicity is set by a balance between several repulsive interactions (electrostatic, hydration, and steric interactions) and the attractive van der Waals interaction.2-6 In recent years much attention has been focused on pseudo ternary lamellar systems of surfactants, water, and polymers.7-20 Previous studies * Present address: Neutron Scattering Group, Solid State Division, Building 7962 Mail Stop 6393, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831-6393. E-mail: porcarl@ ornl.gov. (1) Tanford, C. The Hydrophobic Effect 2nd ed.; Wiley: New York, 1980. (2) Roux, D.; Safinya, C. R. J. Phys. (Paris) 1988, 49, 307. (3) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: Orlando, FL, 1985. (4) Helfrich, W. Z. Naturforsch. 1978, 33A, 305. (5) Rand, R. P. Annu. Rev. Biophys. Bioeng. 1981, 10, 277. (6) Parsegian, V. A.; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777. (7) Ligoure, C.; Bouglet, G.; Porte, G. Phys. Rev. Lett. 1993, 71, 3600. (8) Ligoure, C.; Bouglet, G.; Porte, G.; Diat, O. J. Phys. II (France) 1997, 7, 473. (9) Bouglet, G.; Ligoure, C.; Bellocq, A. M.; Dufourc, E.; Mosser, G. Phys. Rev. E. 1998, 57, 834. (10) Bouglet, G.; Ligoure, C. Euro. Phys. J. B 1999, 9, 137. (11) Porcar, L.; Ligoure, C.; Marignan, J. J. Phys. II (France) 1997, 5, 493. (12) Singh, M.; Ober, R.; Kleman, M. J. Phys. Chem. 1993, 97, 11 108. (13) Warriner, H. E.; Davidson, P.; Slack, N.; Schellhorn, M.; Eiselt, P.; Idziak, S. H. J.; Schmidt, H.-W.; Safinya, C. J. Chem. Phys. 1997, 107, 9.

have shown that macromolecules which are solubilized into a lamellar phase can occupy various sites: indeed, they can be confined within the water layers,7-12 fully incorporated into the surfactant bilayers,18,19 localized in the bilayers or in the solvent15,16 or even adsorb onto the membranes by a specific group.13,14,17,20 The sites of location will depend both on the architecture of the polymer and on the interaction between the polymer and the membrane. The simplest architecture is that of water-soluble homopolymers, where all monomers of the macromolecules are identical. In this case two situations are encountered: the polymer can either adsorb onto the membrane or not. Both situations have been theoretically21 and experimentally studied.7-12,15,16 Ficheux et al.16 reported the effect of poly(ethylene oxide) on the stability of the lamellar phase made from a mixture of sodium dodecyl sulfate, octanol, and water. They showed that the polymer is partially associated to the surfactant bilayers as expected for an adsorbing polymer. The phase diagram of this system exhibits two critical phase separation points between two different lamellar phases. Ligoure and coworkers7-11 have studied the opposite case of a nonad(14) Deme´, B.; Dubois, M.; Zemb, T.; Cabane, B. J. Phys. Chem. 1996, 100, 3828. (15) Ke´kicheff, P.; Cabane, B.; Rawiso, M. J. Coll. Int. Sci. 1984, 102, 51. (16) Ficheux, M.-F.; Bellocq, A.-M.; Nallet, F. J. Phys. II (France) 1995, 5, 823. (17) Castro-Roman, F.; Porte, G.; Ligoure, C. Phys. Rev. Lett. 1999, 82(1), 109. (18) Radlinska, Z. E.; Gulik-Krzywicki, T.; Lafuma, F.; Langevin, D.; Urbach, W.; Willians, C. E.; Ober, R. Phys. Rev. Lett. 1995, 74(21), 4237. (19) Nicot, C.; Waks, M.; Ober, R.; Gulik-Krzywicki, T.; Urbach, W. Phys. Rev. Lett. 1996, 77(16), 3485. (20) Illiopoulos, I.; Olson, U. J. Phys. Chem. 1994, 98, 1500. (21) Brooks, J. T.; Cates, M. E. J. Chem. Phys. 1993, 99, 5464.

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sorbing polymer confined between the surfactant bilayers of the lamellar phase. Ligoure et al.7-10 have incorporated a large water-soluble polymer, PVP (polyvynilpyrolidone), in the LR phase of the ternary system CPCl (cetylpyridinium chloride)/hexanol/water. The smectic order is stabilized by long-range electrostatic interactions. They have shown that the presence of a nonadsorbing polymer in the solvent of the lamellar phase induces an effective destabilizing interaction between the bilayers which depends both on the smectic periodicity dp and on the volume fraction of the polymer in the solvent φ h . This interaction is of purely entropic origin, and similar to the depletion interaction between colloidal particles in a polymer solution.22 In the absence of added salt, this destabilizing interaction cannot compete with the strong stabilizing electrostatic interaction: this explains why PVP can be incorporated in any proportions without any modification of the structure of the lamellar phase.8 On the contrary, lamellar/lamellar phase separations are encountered when the electrostatic interaction is screened by the addition of salt. In this case the authors found a critical point of lamellar/lamellar phase separation for a particular set of component concentrations. For such a critical point, the corresponding modulus of smectic compressibility B h vanishes. A simple theoretical model was developed that predicts the smectic compression modulus of such a lamellar phase doped by a nonadsorbing polymer.8 At a critical point, the calculated polymermediated and electrostatic contributions to this modulus balance and agree well with the experimental observations. Moreover, Bouglet and Ligoure10 reported systematic measurements of this modulus by analysis of smallangle X-ray and neutron scattering spectra of lamellar samples of the same system. The measurements essentially confirm the theoretical approach. The system studied in this work consists of a doped amphiphilic lamellar phase formed from a mixture of nonionic surfactants (polyoxyethylene(10)isooctylphenyl ether or Triton X100 and polyoxyethylene(3)isooctylphenyl ether or Triton X35), cationic surfactant (cetylpyridinium chloride), two solvents, water and decane, and a watersoluble polymer: polyvynilpyrolidone. Since the ratio TX100/TX35 is kept constant in all samples, this system can be viewed as a four-component lamellar phase. It is also a case of a nonadsorbing polymer doped lamellar phase. Because of the large number of compounds constituting the mixed lamellar phase, the phase behavior is rich. In particular, small-angle X-ray scattering measurements show the occurrence of four critical points of lamellar/lamellar phase separation (in a previous paper,11 we reported the existence of one of these critical points). By decreasing the amount of cationic surfactant, one diminishes the surface charge density of the bilayers, and therefore the repulsive electrostatic interaction is decreased until it cannot compete with the attractive polymer-mediated interaction. Again, the existence of this critical point can be quantitatively explained, using the same theoretical approach, with a careful calculation of the electrostatic interaction taking into account the often wrongly neglected Donnan effect. The aim of the present paper is to give a comprehensive insight of the stability of a polymer-containing lamellar phase, by focusing on the different critical points of lamellar phase separations that we have observed in this system. The paper is organized as follows. In Section 2 we present the experimental system we have investigated: (22) Joanny, J. F.; Leibler, L.; de Gennes, P. G. J. Pol. Sci.: Pol. Phys. 1979, 17, 1073.

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the structural parameters of the lamellar phase are studied by X-ray scattering and cryofracture experiments. We will show that PVP strongly modifies the texture of the lamellar phase, but does not affect the smectic periodicity. In Section 3, we report the occurrence of four critical points of lamellar/lamellar phase separation. The first one is monitored by decreasing the surface charge density of the bilayers. The second one occurs by decreasing the thickness of the aqueous solvent at a constant mass fraction of polymer per surfactant. The third one is induced by a decrease of the oil layer thickness. Finally the last one is governed by decreasing the temperature of a lamellar sample. It is important to note that any of these critical phase separations will occur in the absence of polymer. In Section 4, we build up a general formalism in order to calculate the smectic compression modulus B h of a four-component lamellar phase as a function of the relevant potentials of interactions between membranes. We also review in this section the numerical calculation of the electrostatic interaction, taking into account the Donnan effect. In Section 5 the existence of the four critical points is discussed with the aid of the theoretical model developed in Section 4. 2. Experimental System 2.1. System Studied and Samples Preparation. Our experimental system consists of lamellar phases obtained from a ternary mixture of two nonionic commercial surfactants: Triton X100 (TX100), Triton X35 (TX35), and water.23 TX100 and TX35 were used as received from Labosi and Sigma companies. Water was doubly distilled and deionized. The weight ratio TX100/TX35 ) 55/45 g/g was kept constant in all samples. TX35 plays the role of a cosurfactant, so that for all samples we will study, the lamellar structure is preserved. In the following we will denote as TX the mixture of TX100 and TX35 with the specific weight ratio 55/45 and consider, as usual for surfactant/cosurfactant systems, this mixture as an effective single surfactant. The initial weight ratio TX/water of 50/50 g/g was chosen as it corresponds to a composition located in the center of the stability zone of the lamellar phase of aqueous solutions of TX (see Figure 1a). For this specific composition, the samples are very viscous. Moreover, the electron density contrast between TX and water is weak (see Table 1). Decane (purchased from Fluka and used as received) was incorporated in order to reduce the viscosity of the solutions (making mixing and manipulations easier) and to obtain a better electron density contrast for X-ray experiments (Table 1). The corresponding threecomponent lamellar phase is stable over a wide region of the phase diagram (see Figure 1b). This neutral lamellar phase is stabilized by Helfrich’s interactions, and we are not able to add a significant amount of polymer without inducing phase separations. To allow the incorporation of polymer in our lamellar phases, we have doped the membrane by a cationic surfactant: cetylpyridinium chloride (CPCl). CPCl was purchased from Fluka and was recrystallized twice in ethanol. Polyvynilpyrolidone (PVP) is a watersoluble neutral polymer. This is a commercial product obtained from Sigma and was used as received. The reported molecular weight is 10 000 g/mol. Samples were prepared by mixing the appropriate amounts of PVP, CPCl, and water in order to obtain a single homogeneous phase. TX and decane are then added, and the mixtures are stirred and centrifuged several times in order to obtain homogeneous mixtures. The samples are then left un(23) Rouviere, J. Information Chimie 1991, 325, 158.

Critical Points of Two-Solvent Lamellar Phase Separations

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metal deposit). The replicas were washed with organic solvents and distilled water and observed with a Philips 301 electron microscope. 2.3. Two-Solvent Lamellar Phase. We have checked that decane behaves as a solvent and inserts between the aliphatic parts of two adjacent TX monolayers, without any modification of the surfactant monolayers. Doping the membrane by a cationic surfactant CPCl allows us to change the surface charge density of the bilayers and hence control the stabilizing interaction between them. It has been shown that the presence of a low quantity of ionic surfactants in a neutral membrane does not affect the thickness of the membrane.24,25 We have checked this result for our particular system; we have obtained for a mixture of TX and CPCl with a molecular ratio CPCl/TX ) 1/5 and for a neutral TX membrane the same membrane thickness δ ) 32.5 Å from the dilution rule. Neutral polymers such as PVP are recognized to have little interaction with cationic surfactants, contrary to the case of anionic surfactants.26-28 This has indeed been observed in the mixed lamellar phase made from CPCl and hexanol in which PVP was incorporated.29 So, a four component-lamellar phase is obtained when PVP is introduced into the two-solvent lamellar phase. Figure 2 is a diagram of the situation under study, in which surfactant monolayers are separated by aqueous layers of thickness d h and oil layers of thickness do. For each sample, the relevant definitions given below are used: Figure 1. (a) Ternary phase diagram of TX35, TX100, and water at 20 °C given by ref 23 (H hexagonal phase, C cubic phase, I isotropic phase and LR lamellar phase). Dotted line represents the weight ratio TX100/TX35 ) 55/45. (b) Ternary phase diagram of water, TX, decane at 20 °C. The large lamellar phase is bounded by a two-phase region of liquid crystals and clear isotropic solution.

mW mPVP + FW FPVP φS ) mCPCl mTX35 mTX100 mdec mW mPVP + + + + + FCPCl FTX35 FTX100 Fdec FW FPVP

Table 1. Characteristic Data of the Chemical Products TX100 TX35 water CPCl decane PVP

M (gr/mol)

d (gr/cm3)

F (eÅ3)

646 338 18 339 142 10 000

1.06 1.02 1 0.98 0.73 1.07

0.35 0.338 0.334 0.298 0.253 0.345

disturbed at 20 °C for several days. Characterization of the lamellar structure was made from observations with a polarizing microscope and small-angle X-ray scattering experiments. 2.2. Experimental Techniques. Small-Angle X-ray Scattering (SAXS). Small-angle X-ray scattering experiments have been performed using a low resolution instrument having an incident wavelength equal to 1.54 Å (λ Cu KR1). The data are collected using a position sensitive detector and a slit collimation system. These experimental parameters allow studies in a range of wave vectors q from 0.03 to 0.2 Å-1. If not mentioned, all the SAXS experiments are carried out at 20 °C. Freeze Fracture Experiments. A thin layer of each sample (20-30 µm) was placed on a thin copper holder and then rapidly quenched in liquid propane. The frozen samples were then fractured at -125 °C in a vacuum (pressure lower than 10-6 Torr) with a liquid nitrogen cooled knife in a Blazers 301 freeze-etching unit. The replicas were made using unidirectional shadowing at an angle of 35° with platinum carbon (1-1.5 mm of mean

φo )

mdec Fdec mCPCl mTX35 mTX100 mdec mW mPVP + + + + + FCPCl FTX35 FTX100 Fdec FW FPVP

mCPCl mTX35 mTX100 + + FCPCl FTX35 FTX100 φsurf ) mCPCl mTX35 mTX100 mdec mW mPVP + + + + + FCPCl FTX35 FTX100 Fdec FW FPVP and

mPVP FPVP φ h PVP ) mPVP mW + FPVP FW where mi and Fi are the weight in grams and the density of material i (Table 1). (24) Schomacker, R.; Strey, R. J. Phys. Chem. 1994, 98, 3908. (25) Jonstromer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (26) Hayakawa, K.; Kwat, J. T. C. Cationic Surfactants; Rubingh, D. N., Holland, P. M., Eds.; Marcel Dekker: New York, 1991. (27) Garcia-Mateos, I.; Perez, S.; Velazquez, M. M. J. Coll. Inter. Sci. 1997, 194, 356. (28) Anthony, O.; Zana, R. Langmuir 1994, 10(11), 4048. (29) Bouglet, G. Ph.D. Thesis, Universite´ Montpellier II, 1997.

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Figure 2. Schematic drawing for a four-component lyotropic smectic A phase. Surfactant monolayers are separated by aqueous and oil layers of thickness respectively d h and do.

The measured lamellar period dp of this two-solvent lamellar phase is then given by:

dp )

2δsurf + do 2δsurf + do 2δsurf + d h ) ) φsurf + φo 1 - φS φsurf + φS

h are the surfactant monolayer where δsurf, do, and d thickness, the decane layer thickness, and solvent (water + PVP) layer thickness, while φsurf, φo, and φS are the surfactant, the decane, and the solvent volume fraction, respectively. Whatever the CPCl/TX ratio, the solvent, and decane concentrations, 2δsurf is always equal to 32.5 Å. 2.4. Effect of Polymer on the Two-Solvent Lamellar Phase. We investigated a series of lamellar phases of constant solvent, oil, and surfactant volume fractions φS ) 0.44, φo ) 0.17, φsurf ) 0.39 (CPCl/TX ) 1/5) in an aqueous solution of PVP. The volume fraction of polymer in the solvent φ h PVP was varied from 0 to 19%. All the samples remained homogeneous, transparent, and birefringent. Figure 3 shows the X-ray scattering patterns of this series. The first Bragg peak position qB remains remarkably constant whatever the polymer concentration as inferred from the inset of Figure 3, where the lamellar periodicity is plotted as a function of polymer concentration in the solvent (φ h PVP ) 5.7% denotes the crossover between the 2D semidilute regime and the 3D semidilute regime8,10). Then the first Bragg peak position is in agreement with the picture where the polymer is distributed into water only and leaves the bilayer (surfactant + oil) thickness unchanged; our experimental system therefore pictures solutions of nonadsorbing polymer confined to infinite soft slits made of fluid membranes. We note however in Figure 3 a softening of the Bragg singularities by increasing the amount of polymer. In terms of the current theory of scattering by smectic media,30 one expects I ≈ (q - qo)-x, with x ) 1 - η for an isotropic powder sample. The dimensionless parameter η which characterizes the profiles of the smectic Bragg singularities is defined in terms of the smectic elastic constants:

η)

kBTqB2 8π(KB h )1/2

where K is the smectic curvature modulus (related to the

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Figure 3. Intensity versus q (Å-1) curves for the series of samples at constant solvent, oil, and surfactant volume fractions φS ) 0.44, φ0 ) 0.17, φsurf ) 0.39 (CPCl/TX ) 1/5) but with several volume fraction of polymer in water φ h PVP. Inset: Lamellar periodicity versus polymer volume fraction in water φ h PVP. The lamellar periodicity is independent of polymer / concentration. The line is a guide for the eye. φ h PVP is the crossover between the 2D semidilute regime and the 3D semidilute regime the overlap volume fraction.8,10

bilayer bending modulus κ of the membrane and the smectic periodicity dp by K ) κ/dp. The softening of the profile corresponds to an increase of η and consequently to a decrease of the product of the two elastic constants K and B h . However it has been argued theoretically and confirmed experimentally by NMR experiments that the presence of a nonadsorbing polymer does not change the bending rigidity κ of a fluid membrane.9,31 We conclude that the broadening of the Bragg singularities reveals the softening contribution to B h due to the polymer confinement. Although PVP does not modify the structural properties of the lamellar phases, in as far as the bilayers are stabilized by sufficiently strong electrostatic interactions, it has a large impact on the large-scale organization of the lamellar samples. We have performed freeze fracture electron microscopy experiments on three different lamellar samples with the same surfactant composition (molecular ratio CPCl/TX equal to 1/5) and the same volume fraction of the surfactant, oil, and solvent (φsurf ) 0.49, φo ) 0.21 and φS ) 0.3). One is polymer free whereas for the others, water was replaced by an aqueous solution of PVP h PVP ) 10%. The with volume fractions of φ h PVP ) 5% and φ corresponding cryo-TEM micrographs are shown in Figures 4 and 5a,b. Figure 4 shows the polymer free sample: the image is typical of a lamellar phase with a perfect stacking of flat bilayers. The interlayer distance (about 65 Å) can be estimated from the part of the replica where the fracture surface is perpendicular to the bilayer: this value is in very good agreement with the value measured from SAXS experiments (67 Å). The sample with φ h PVP ) 5% exhibits a modified texture (Figure 5a), there are many defects where the lamellae are bent. Finally, the texture of the sample with φ h PVP ) 10% consists of dense packing of multilamellar vesicles (spherulites). It is clear then that the presence of PVP in the solvent induces a spontaneous bending of the bilayers. Observation of this spontaneous formation of spherulites indicates a large negative contribution of the nonadsorbing polymer to the Gaussian (30) Caille, A. C. R. Acad. Sc. Paris 1972, 274, Serie B, 891. (31) Brooks, J. T. Ph.D. thesis, University of Cambridge, 1993.


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Figure 4. Freeze fracture micrograph of the lamellar phase of the system TX/CPCl/water/decane with φsurf ) 0.49, φo ) 0.21, φS ) 0.3 and a surface per charged head Σ ) 260 Å2. We observe a lamellar phase with a perfect stacking of bilayers. Bar: 460 nm.

bending modulus of the bilayer32 in contradiction with the theoretical predictions of Brooks.31 Experimentally this result seems to be quite general (the same phenomenon was observed in a similar system described by Bouglet et al.9) and is presently not understood. 3. Critical Lamellar/Lamellar Phase Separations In this section we report several critical LR/LR phase separations which occur in our mixed lamellar system, in the presence of polymer only, by varying in a controlled way a single physicochemical parameter in each case. All these critical points were characterized by SAXS measurements. Far below a critical point of lamellar/lamellar phase separation the scattering pattern of a monophasic lamellar phase exhibits the first (and eventually higher) harmonic of the structure factor of a smectic. Approaching a critical point modifies the scattering patterns in the following way: (i) increasing strong small angle scattering, (ii) broadening of the Bragg peak, (iii) at the critical point the first Bragg peak is replaced by a rounded maximum, and (iv) beyond the critical point, the Bragg maximum is split in two and reveals the coexistence of two lamellar phases (one polymer rich and one surfactant rich lamellar phase) of different periodicities which continue to separate and progressively sharpen. Even after a very long equilibration time (more than three years), the lamellar phases do not separate macroscopically which nullifies the possibility of determining their composition. For a second-order phase separation, the corresponding susceptibility (the smectic compressibility B h -1) diverges. Therefore η increases and becomes larger than 1, so that the exponent -x ) -(1 - η) becomes positive and the Bragg singularity vanishes, replaced by a rounded maxi(32) Boltenhagen, P.; Kleman, M.; Lavrentovitch, O. J. Phys. II France 1994, 4, 1439.

mum. The intensity scattered at low q (q , qB) is also related to the smectic compressibility: I(0) ≈ kBT(B h -1) 33-35 Then the following exand increases dramatically. perimental observations occur: (i) rounded maximum, (ii) increase of the intensity at low q, and (iii) continuous splitting of the Bragg peak at the transition; these are characteristic of the occurrence of a second order LR/LR phase separation. 3.1. Effect of the Surface Area per Charged Headgroup. This effect was reported in a previous paper.11 We investigated a series of samples of lamellar phases which differed only by their surface charge densities. The volume fraction of solvent and oil were fixed in all samples (φS ) 0.43, φo ) 0.17), as well as the volume fraction of PVP in water φ h PVP ) 14%. The total volume fraction of surfactants (TX + CPCl) is also fixed φsurf ) 0.4, but we increase the surface area per charged headgroup Σ by changing the ratio of ionic (CPCl) to nonionic (TX) surfactant (recall that the weight ratio TX100/TX35 ) 55/45 g/g is preserved). The molecular ratio CPCl/TX was been varied from 1/2.6 to 1/12, which corresponds to an increase of the surface area per charged headgroup from Σ ) 160 ( 5 Å2 to S ) 600 ( 5 Å2. The surface area per charged headgroup is determined by:

Σ)

(

2MCPCl 1 mTX35 mTX100 + + + δNA FCPCl mCPClFTX35 mCPClFTX100 mdec mCPClFdec

)

where MCPCl is the CPCl molecular weight, NA the Avogadro’s number, and δ the bilayer (surfactant + oil) (33) Porte, G.; Marignan, J.; Bassereau, P.; May, R. Europhys. Lett. 1988, 7(8), 713.

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Figure 5. Freeze fracture micrograph of the lamellar phase of the system TX/CPCl/water/decane/PVP with φsurf ) 0.49, φo ) 0.21, φS ) 0.3 and a surface per charged head Σ ) 260 Å2. (a) For φ h PVP ) 5% note the presence of many defects where the lamellae are bent. (b) φ h PVP ) 10%. Note the presence of spherulites. Bar: 460 nm.

thickness. In Figure 6 we show the X-ray scattering patterns of some samples of this series. In the range 150 (34) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. France 1990, 51, 2333.

Å2 < Σ < 265 Å2, the samples are monophasic and (35) Zemb, T.; Gazeau, D.; Dubois, M.; Gulik-Krzywicki, T. Europhys. Lett. 1993, 21(7), 759.


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Figure 7. Variation of the smectic periodicity dp as a function of the surface area per charge headgroup Σ at constant volume fraction of surfactant, oil and solvent φsurf ) 0.4, φo ) 0.17, φS ) 0.4 and constant volume fraction of polymer in the solvent φ h PVP ) 0.14.

Figure 6. Small angle X-ray scattering patterns of a series of samples at constant volume fraction of surfactant, oil, and solvent φsurf ) 0.4, φo ) 0.17, φS ) 0.43 and constant volume fraction of polymer in the solvent φ h PVP ) 0.14. The only variable parameter is the surface area per charge headgroup Σ. A critical lamellar/lamellar phase separation occurs for Σ = 320 Å.

birefringent. At Σ ) 320 Å2 the samples become cloudy but are still birefringent. The successive patterns along the series of increasing Σ have the characteristic features of a critical lamellar/lamellar phase separation as described in the preceding paragraph. In particular one can identify the value Σ ) 320 ( 5 Å2 as the critical point of the lamellar/lamellar phase separation. Qualitatively, the critical phase separation occurs by reducing the repulsive electrostatic interaction between the bilayers in the presence of polymer only, which indicates that the polymermediated interaction is attractive. Figure 7 shows the smectic periodicity (measured from the Bragg peak positions of the samples) versus the area per charged head of the bilayer, the symmetric split of the Bragg singularity in two above Σ ) 320 ( 5 Å2 confirms the critical phenomenon. 3.2. Effect of Polymer Confinement. In a second series of samples we studied the effect of confinement of the polymer solution in the lamellar phase on its stability in the following way. In all samples of this series, the composition of the surfactants (Σ ) 260 Å2), the mass fraction of polymer per surfactant (0.153 g/g of surfactant) and the mass fraction of decane per surfactant (0.3 g/g of surfactant) are fixed. However, the volume fraction of solvent is progressively decreased by decreasing the amount of water, which means that the smectic periodicity decreases. In doing so, we obviously increase the electrostatic repulsion, and the van der Waals attractive interaction between the bilayers (surfactant + oil). However, the polymer-mediated interaction is modified as well in a nontrivial way as explained in ref 8 and reviewed in the following section. As the amount of water is decreased, the volume fraction of polymer φ h PVP increases and the interlayer distance decreases. Figure 8 shows the X-ray patterns of the samples along such a concentration.

At large distances, the samples are birefringent, transparent, and monophasic. At a critical distance dp = 78 Å (φs ) 40%) they phase separate. Again, the succession of the patterns demonstrates the existence of a critical point (dp ) 78 Å) of phase separation between two different lamellar phases. 3.3. Effect of the Oil Layer Thickness. As shown in the first section, decane also behaves as a solvent, which separates the hydrophobic tails (aliphatic chains) of two successive monolayers of the lamellar phase: so that we can change the total bilayer thickness by varying the amount of oil in the system. In all samples of this series the volume ratio of aqueous solution of PVP on surfactant h PVP ) 14.6% as well as the was fixed VS/Vsurf ) 1.05 and φ surfactant composition (Σ ) 260 Å2). By simply decreasing the amount of decane φo from one sample to the next of the series, we decrease the lamellar spacing and we increase the van der Waals interactions between water layers across the oil layers. In Figure 9 the change of the aspects of the X-ray patterns of the lamellar samples by decreasing the oil amount, again shows a critical lamellar/ lamellar phase separation which occurs for the decane volume fraction φo ) 14.5%. In Figure 10, we plot the lamellar periodicity of two series of samples with the same aqueous layer thickness (the first one consisting of pure water, and the second one a PVP solution with φ h PVP ) 14.6%) along a decane dilution line. Without polymer one observes an ideal swelling of the lamellar phase by oil over the full range of decane volume fraction. For the samples containing polymer the same ideal swelling behavior with the same inverted bilayer thickness as the free polymer series occurs for an oil volume fraction φo > 14.5%. Below this critical amount, two lamellar phases of different pitches coexist. These results confirm that PVP is entirely dissolved in water layers, and that the presence of polymer is necessary in order to observe the phase separation. 3.4. Effect of Temperature. In this part we focus on the effect of increasing the temperature of a biphasic mixed lamellar sample of composition: φS ) 0.44, φo ) 0.134, h PVP ) 14%. Initially at 20 φsurf ) 0.426, Σ ) 260 Å2, and φ °C the mixture has the cloudy appearance of a birefringent emulsion without macroscopic phase separation. However, the X-ray spectrum reveals the coexistence of two lamellar phases of smectic periodicities dp ) 82 ( 1 Å and dp ) 69 ( 1 Å (see Figure 11 at 20 °C). We then increased the temperature step by step: To ensure that the equilibrium

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Figure 8. Small angle X-ray scattering patterns of a series of samples at constant surfactant composition (Σ ) 260 Å2), and fixed mass fraction of polymer per surfactant (0.153 g/g of surfactant) and mass fraction of decane per surfactant (0.3 g/g of surfactant) as a function of the solvent volume fraction φS.

is reached we performed the X-ray experiments twice at each temperature. The second measurement was made after an interval of 1 day (the same pattern was obtained). The series of patterns is shown in Figure 11. Again we observe a critical phase transition from a two-phase sample to a single lamellar phase. The critical temperature is around 30 °C. The fact that the system is temperature sensitive is not surprising, since all interactions between bilayers are essentially of entropic origin (even for the electrostatic interaction, the principal contribution is from the translational entropy of the counterions), and their magnitudes increase with temperature. We will discuss the temperature dependence of the relevant interactions in the Discussion Section. We have performed the same experiment with samples of various compositions: in all cases one passes from a two phase system to a single lamellar phase by increasing the temperature. However, in general the transition is first order and not second order, except for samples revealing a very small difference between equilibrated lamellar phase smectic periodicities at 20 °C.

4. Smectic Layer Compression Modulus of a Four-Component Lamellar Phase To understand the critical lamellar phase separation described in the previous section, we need to calculate the layer compression modulus of a four-component lamellar phase. The thermodynamic description of a threecomponent lamellar phase has been made by Nallet and co-workers.36 Oda et al.37 used a simplified approach in order to describe a two-solvent lamellar phase, whereas Ligoure et al.8 using a different approach reported the calculation of the smectic compression modulus of a doped lamellar phase and apply their result to the particular case in which the host component is a nonadsorbing polymer. Our experimental system shares both these characteristics: this is a two-solvent lamellar phase, in which a nonadsorbing polymer host component is incorporated. (36) Nallet, F.; Roux, D.; Quilliet, C.; Fabre, P.; Milner, S. T. J. Phys. II France 1994, 4, 1477. (37) Oda, R.; Lister, J. D. J. Phys. II France 1997, 7, 815.


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Figure 9. Intensity versus q (Å-1) curves for a series of samples at constant volume ratio of aqueous solution of PVP on surfactant Vs/Vsurf and φPVP ) 14.6% and constant surfactant composition (Σ ) 260 Å2) as a function of the decane volume fraction φo.

To give a theoretical description of this system, we generalize the approach of Nallet and co-workers.36 The free energy of the system is as a sum of two interactions, one between monolayers with a polymer solution layer between, and the other between monolayers with an oil layer between. We will write these two interaction energies per area as Vs and Vo respectively, then the free energy density is:

f(φS, φo, σ) )

1 [V (d φ , σ) + Vo(dpφo, σ)] dp S p S

(1)

Here, φo and φS are oil and aqueous solvent volume fraction, dpφo and dpφS are the oil and aqueous layers thickness respectively, and σ is the mean surface fraction of the guest component (polymer) per unit area of monolayer. σ is related to the volume fraction of guest component in water φ h as σ ) φ h d/2a, where d h ) dpφS is the aqueous solvent layer thickness and a is the monomer length of the guest component. We assume as is usual, the monolayers to be incompressible. The free energy density of the system then, is a function of φo, φS, and σ only. The fluctuations around equilibrium can be expressed by the expansion of f up to

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Porcar et al.

Now we are in position to calculate the relevant elastic constant, i.e., the smectic layer compression modulus at constant chemical potential of guest component B h . We obtain it from eq 5 of ref 36:

B h )B-

2 -1 C2σ χ-1 c + Cc χσ - 2Cσ,cCcCσ -1 2 χ-1 c χσ - Cσ,c

(5)

where the elastic constants in the right-hand-side are given by the set of eqs 4. The full expression of B h from eqs 4 and 5 is quite complex. To verify this derivation, we have checked that we obtain the previous simpler calculated expressions for B h in the two following cases: (i) two-solvent lamellar phase without host component (σ ) 0); from eqs 4-5 we obtain: Figure 10. Evolution of the smectic periodicity dp at a constant volume ratio of aqueous solution of PVP on surfactant VS/Vsurf ) 1.05 and φPVP ) 14.6% and constant surfactant composition (Σ ) 260 Å2) as a function of the decane volume fraction φo. 2: lamellar phase without polymer (φ h PVP ) 0%). O: lamellar phase with φ h PVP ) 14.6%. The dashed line represents an ideal swelling of the lamellar phase by decane with 67 Å membrane thickness.

second order in these variables as described in refs 36-38 by:

f ) feq + 2

[

1 ∂2f ∂2f ∂2f 2 2 2 δφ + δφ + δσ + o S 2 ∂φ2 ∂φ2S ∂σ2 o

]

∂2f ∂2f ∂2f δφoδσ + 2 δφ δσ δφoδφS + 2 ∂φo∂φS ∂φo∂σ ∂φS∂σ S

(2) 36-38

as The free energy density f can also be derived function of the displacement vector u and the concentration variables δc and δσ with all necessary elastic constants:

() (

)

1 B ∂u 2 K ∂2u ∂2u 2 + + + δσ2 + 2 ∂z 2 ∂x2 ∂y2 2χσ ∂u ∂u 1 + 2Ccδc + 2Cc,σδcδσ (3) δc2 + 2Cσδσ 2χc ∂z ∂z

f ) feq +

Defining δc ) δφo - δφS and using ∂u/∂z ) δdp/dp ) (δφo + δφS)/φsurf, where φsurf is the surfactant volume fraction, we obtain:

B)

dp [V′′o (1 - φS + φo)2 + V′′S (1 - φo + φS)2] 4 χc-1 )

(4a)

dp (V′′o + V′′S) 4

(4b)

2 1 ∂ VS dp ∂σ2

(4c)

χσ-1 )

dp [V′′o (1 - φS + φo)2 - V′′S (1 - φo + φS)2] (4d) Cc ) 4 Cσ )

∂2VS 1 (1 - φo + φS) 2 ∂d h ∂σ Cc,σ ) -

2 1 ∂ VS 2 ∂d h ∂σ

h 2. where V′′o ) ∂2Vo/∂do2, V′′S ) ∂2VS/∂d

(4e)

(4f)

B h ) dp

V′′S V′′o V′′o + V′′S

(6)

This expression was first derived by Oda et al.37 (ii) one-solvent (φo ) 0) lamellar phase with a guest component; from eqs 4-5 we obtain:

[

B h ) dp V′′S -

]

(∂2VS/∂σ∂d h )2 ∂2VS/∂φ h2

(7)

This expression was first derived by Ligoure et al.8 Equation 5 is the key expression of this section, from which we will calculate the layer compression modulus in all experimental situations, where a critical point of lamellar/lamellar phase separation occurs. In fact B h is expressed as a function of the second derivatives of the interaction potentials, the volume fractions of the components of the mixed lamellar phase and the smectic periodicity. The volume fractions and the smectic periodicity are experimental parameters. We still have to describe the different contributions to the interaction potentials Vo and VS. Following Brooks and Cates,21 we assume that all contributions are additive. Such an assumption is certainly crude. However, Bouglet and Ligoure10 have experimentally shown that electrostatic and polymer-mediated interactions are additive, whereas charge effect corrections to the Helfrich’s interactions will be taken into account. Finally we have not attempted to include the complicated coupling between van der Waals and Helfrich interaction leading to a transition between bound and unbound surfactant bilayers39 (however that effect may be relatively small compared to the polymer-mediated and electrostatic interaction). Under the additivity assumption VS and Vo are given by:

VS ) VSHelf + Velec + VSVdW + Vpol

(9)

Vo ) VoHelf + VoVdW

(10)

Velec is the electrostatic interaction between two adjacent charged monolayers. In a previous paper11 we have shown that the analytical expressions of this interaction given in the literature for the two asymptotic limits (in the absence of added salt or a large added salt concentration) are valid under very restrictive conditions only, which are not encountered in our experimental system. This is why we have to compute it numerically, within the (38) Nallet, F.; Roux, D.; Prost, J. J. Phys. France 1989, 50, 3147.


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Figure 11. SAXS spectra showing the effect of the temperature on a specific LR/LR lamellar phase from room temperature to 40 °C. Sample composition: φS ) 0.44, φo ) 0.134, φsurf ) 0.426, Σ ) 260 Å2, and φ h PVP ) 14%.

framework of the Poisson-Boltzmann approach, as described by Dubois and co-workers:40

Velec ) -

1 2

∫∞dh πelec δdh

(11)

VSVdW ) -

where πelec is the ionic osmotic pressure:

πelec ) 4kBTc′s sinh[φ(d)/2]

VSVdW is the attractive van der Waals interaction between oil layers with an aqueous layer in between. It is given by a simple expression if retardation effects are neglected:3

(12)

φ is the reduced electrostatic potential calculated at the mid plane (d ) d h /2) and c′s is the salt concentration of a fictitious reservoir in chemical equilibrium with the lamellar phase. φ and c′s are calculated numerically from the set of equations (1, 2a-c) of ref 40. They obviously depend on the surface charge density and the salt concentration cs in the solution (note that even in pure water this concentration is at least greater than 10-7 mol/L because of the ionic dissociation of water). The mean salt concentration cs in the lamellar phase is lower than c′s. This salt exclusion is the well-known Donnan effect.40 The numerical expression of Velec is valid under all conditions (i.e., the absence or presence of a large amount of added salt and a high or a low surface charge density) and recovers the asymptotic limits for which analytical expressions are known.3 (39) Milner, S. T.; Roux, D., J. Phys. I France 1992, 2, 1741. (40) Dubois, M.; Zemb, T.; Belloni, L.; Delville, A.; Levitz, P.; Setton, R. J. Chem. Phys. 1992, 96, 2278.

[

]

AH 1 1 2 + 2 2 12π d h (d h + 2do) (d h + do)2

(13)

where AH is the oil in water Hamaker constant.3,41 Similarly, VoVdW is the attractive van der Waals interaction between aqueous layers with an oil layer between

VoVdW ) -

[

]

AH 1 1 2 + 2 12π d 2 (2d h + d ) (d h + do)2 o o

(14)

VSHelf and VoHelf are the long-range undulation’s interactions between monolayers:

VSHelf )

3π2(kBT)2 1 128κmono d h2

(15a)

VoHelf )

3π2(kBT)2 1 128κmono d 2

(15b)

o

In these expressions we assume that κmono ) κbilayer/2.37 (41) Hough, D. B.; White, L. R. Adv. Coll. Inter. Sci. 1980, 14, 3.

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In this section we will compare the predictions of the model we have developed and the experiments presented in Section 3. Indeed, a critical point of lamellar/lamellar phase separation is characterized by a cancellation of the smectic compression modulus B h . For each series of

samples, we numerically calculated the variations of B h using eq 5. The theoretical critical point will be defined by B h ) 0. The interaction potentials and their derivatives (eqs 9-10) are defined by the set of equations (11, 1316), which allows us to calculate all the elastic constants (eqs 4a-f) entering in the definition of B h . The volume fractions of each component as well as the structural parameters (bilayer thickness and smectic periodicity) are experimental data, which enter directly in the calculation of B h. For the van der Waals potentials (eqs 13, 14) we used the following value for the Hamaker constant: AH ) 0.6 kBT.3 For the electrostatic interaction (equation 12) we use a finite value of cs for the salinity of the lamellar phase. This is due to the presence of ionic impurities in TX batches (about 5%), which are commercial products used as received, and was obtained from standard conductivity measurements. Moreover, it has been shown that CPCl is entirely dissociated in water,45 so that PoissonBoltzmann theory can be applied without introducing any effective charge due to the binding of Cl- counterions onto the bilayer surfaces.40 We also always include the (weak) correction to the dielectric permittivity which depends on the PVP concentration. For the Helfrich interactions (eqs 15a-b) we take κmono ) (κbilayer + δκelec)/2 = 2.5 kBT, where κbilayer = 3.2 kBT is the bending modulus of a TX bilayer measured by the logarithmic deviation to the dilution law of the lamellar phase,46,47 and δκelec ) 1.7 kBT as estimated in the previous section. Note however, that the value of κbilayer should be carefully considered: this more an estimation than a measurement, since the range of dilution is not very broad and the deviation is weak.48 Finally, for the polymer-mediated interaction, we take a ) 4 Å for the persistence length of the polymer. Since we do not know exactly this value, this reasonable choice is somewhat arbitrary, and the value at which B h cancels strongly depends on it. However, the key point is that we use the same value for all the critical points under study, which lends credit to our interpretation. Effect of the Surface Area per Charged Headgroup. In Figure 12 a) we plot the calculated smectic compression modulus for the samples investigated in Section 3.1, as a function of the surface area per charged headgroup Σ. The experimental parameter values used in h PVP ) 0.14 cS the calculations are: d h ) 33.5 Å φo ) 0.17, φ ) 7 × 10-3 mol/L. We find that B h vanishes for Σ ) 330 Å2; this theoretical value is very close to the observed critical point at Σcrit ≈ 320 Å. In a previous paper,11 we interpret this critical point as a balance between the polymermediated interaction on one hand and the electrostatic interaction on the other hand. This was done by calculating B h from eq 7 with VS ) Vpol + Velec. The corresponding calculated value for the critical point was Σ ) 300 Å2, still in fairly good agreement with the experimental observation. This indicates that the electrostatic and the polymermediated interactions are the relevant interactions whose balance induces the critical separation, whereas van der Waals and Helfrich interactions play a minor role here. Effect of Polymer Confinement. In the series of samples investigated in Section 3.2, we progressively confine a macromolecular solution between charged lamellae, by decreasing the amount of water. Doing so we

(42) Zaslavsky, B. Y.; Mihevva, L. M.; Rodnikova, M. N.; Spivak, G.; Harki, V. S.; Mahmudov, A. U. Chem. Soc. Faraday Trans. 1 1989, 85, 2857. (43) Harden, J. L.; Marques, C.; Joanny, J. F.; Andelman, D. Langmuir 1992, 8(4), 1170. (44) Andelman, D. Handbook of Physics of Biological Systems; Elsevier Science B. V. 1989; Vol. 1, Chapter 12.

(45) Porte, G.; Appell, J. Surfactants in Solutions; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1983; Vol. 2, p 805. (46) Freyssingeas, E.; Roux, D.; Nallet, F. J. Phys. Condens. Matter 1996, 8, 2801. (47) Roux, D.; Nallet, F.; Freyssingeas, E.; Porte, G.; Bassereau, P.; Skouri, M.; Marignan, J. Europhys. Lett. 1992, 17(7), 575. (48) Porcar, L. Ph.D. Thesis, Universite´ Montpellier II, 1997.

For the range of polymer and surfactant concentrations used in our experiments, the polymer chains are entangled (semidilute solution), and the corresponding correlation length ξ (mesh size) is smaller than the aqueous layer thickness. The corresponding regime of confinement of the polymer solution is called the three-dimensional semidilute regime. In this regime the polymer-mediated interaction is a depletion interaction. The expression of Vpol in this regime has been calculated by Brooks and Cates: 21

Vpol ) β

kBT a

3

φ h 9/4d h + 2F

kBT a2

(16)

where β = 1.97 and F ) 0.5β = 0.985 are universal prefactors. We will briefly discuss some refinements to the assumption of additivity of all interactions. First of all, adding polymer to water modifies the dielectric permittivity of the aqueous medium, and consequently the electrostatic interaction via the Bjerrum length. Zaslavsky et al.42 have measured the variation of the dielectric permittivity of PVP solutions as a function of PVP concentration. They found a simple linear relationship: ) w - σCp, where w = 80 is the dielectric permittivity of water, Cp is the weight fraction of polymer in water, and σ ) 0.96 ( 0.03. We have taken into account this polymer concentration dependence for the calculation of the electrostatic interaction. We have found that this modification is small. Similarly, the Hamaker constant in the van der Waals interactions should also be polymer concentration dependent; however this effect is relatively small and we simply neglect it. Third, it is known that the electrostatic interaction combines with the Helfrich interactions in a nontrivial way43,44 since it modifies not only the bending rigidity of the membranes, but also the nature of out-of-plane fluctuations. Several authors have studied the electrostatic contribution to the bending rigidity of a membrane in various regimes.43,44 All samples we have investigated belong to the so-called intermediate or Gouy-Chapman regimes. In this case the electrostatic contribution to the bending modulus has been estimated:43,44

δκelec =

kBTλD ≈ 1.7 kBT πlB

where λD and lB are the Debye and Bjerrum length, respectively. This contribution does not depend on the surface charge density. We have taken it into account in our calculation. However this correction is also weak, since the electrostatic repulsion dominates the Helfrich interaction. 5. Comparison between Experiments and Theory


Langmuir, Vol. 16, No. 6, 2000 2593

Figure 13. B h /T variation as a function of temperature: φS ) 0.43, φo ) 0.17, φsurf ) 0.44, and φ h PVP ) 14 % Σ ) 340 Å2, a ) 4 Å, AH ) 0.6 kBT, κmono ) 2.5 kBT. At 20 °C, B h /T < 0 corresponds to a LR/LR phase reported from Figure 12a. An increased temperature to 50 °C made B h > 0 which attests that the lamellar phase is stable at high temperature in accordance with experimental results.

Figure 12. Variation of the theoretical four-component smectic layer compression modulus B h (a) as a function of the surface area per charged head group Σ. φsurf ) 0.4, φo ) 0.17, φS ) 0.43, φ h PVP ) 0.14, a ) 4Å, AH ) 0.6 kBT, κmono ) 2.5 kBT, cS ) 7 × 10-3 mol‚L-1. We find that B h vanishes for Σ ) 330 Å2 (observed critical point at Σcrit ≈ 320 Å2). (b) as a function of the solvent volume fraction φS with the following set of experimental parameters: φo ) 0.17, Σ ) 260 Å2. We have taken into account the variation of d h, φ h PVP, , and cs with respect to φS for the calculation.

increase both the polymer-mediated and van der Waals attractions and the electrostatic and Helfrich repulsive interactions. In Figure 12b, we plot the variation of B h versus the aqueous solvent volume fraction φS with the following set of experimental data: (φo ) 0.17, Σ ) 260 h, Å2). Note that we take into account the variations of d φ h PVP, , and cs with respect to φS for this calculation. We find that B h vanishes for φS ) 39%, in very good agreement with the observed critical point at φS ≈ 39.5%. Effect of the Oil Layer Thickness. The series of samples investigated in Section 3.3 show the existence of a critical phase separation of a polymer containing lamellar phase when the thickness of the oil layers is decreased. h PVP The set of experimental parameters is: Σ ) 260 Å2, φ ) 14%, d h ) 34 Å. In this experiment we expect a variation of only the van der Waals interactions between samples. Surprisingly, our model predicts that B h increases as φo decreases, which means that the lamellar phase should be stable for low amounts of decane. This exactly the opposite of the experimental observations. We suspected that the existence of specific interactions between the oil and the fat chains of the surfactant monolayers (which were not taken into account in our model) could be responsible for this unexpected behavior. We therefore prepared a homologous series of samples, changing decane by either octane or dodecane, since the penetration of oil into the aliphatic sublayer of a surfactant monolayer

should depend on the length of alkane.49 However, this is not the case, since we observe no effect of the nature of oil upon the phase transition. However, we note that this transition does not exist for lamellar samples without polymer. In our model, we have not taking into account the difficult problem of the coupling between van der Waals and Helfrich interactions, so this effect would be nonnegligible in this case. Effect of Temperature. In Section 3.4, we found a critical temperature Tc ≈ 30 °C for a particular composition of the mixed system: φS ) 0.44, φo ) 13.4%, φsurf ) 0.426, Σ ) 260 Å2, and φPVP ) 14% separating a one single lamellar phase domain at above Tc from a two lamellar phase coexistence domain below Tc. The present observation is not restricted to a single composition; similar behavior has been observed in several biphasic LR/LR compositions very close to the critical compositions described above (Section 3.1-3). An increase in temperature of the two lamellar phases with close but different periodicities yields a transition toward a monophasic lamellar phase showing the same characteristics as described in Section 3.4. Our model can qualitatively explain this behavior. At 20 °C, the coexistence of two lamellar phases can be explained by a negative value of B h . As shown previously in Section 4, B h < 0 means that the hypothetical lamellar phase is unstable against phase separation. On the basis of the theoretical model presented above, in Figure 13, we plot the calculated variations of B h /T versus the temperature. The increase of B h /T with T demonstrates clearly the stabilizing effect of the temperature upon the lamellar phase. Moreover, note that the increment of B h as a function of the temperature is steeper than a linear variation. The theoretical critical temperature is on the same order as those observed experimentally (around 30 °C). A simple interpretation can be proposed if we suppose all interaction potentials, except the electrostatic one, undergo a roughly linear variation with the temperature as they are essentially from entropic origin. This is only a rough approximation. For instance, densities of the components are temperature dependent, and the polymer/ water interaction is usually temperature-dependent too, which implies that the effective persistence length a in Vpol (eq 16) should also depend on temperature. The (49) Chen, S. J.; Evans, D. F.; Ninham, B. W.; Mitchell, D. J.; Blum, F. D.; Pickup, S. J. Phys. Chem. 1986, 90, 842.

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electrostatic interaction, in fact, undergoes a more rapid variation with temperature. This can be shown from the following approximate analytical expression of Velec (ref 3)

Velec )

( [

)]) ( )

(

4kBT lBλ′D 1 tanh argsinh 2π πλ′DlB 2 Σ

2

exp -

d h λ′D (17)

where

xkBT exc′s

λ′D ) is the Debye length and

lB )

e2 4πkBT

is the Bjerrum length

(

cs ≈ c′s 1 -

)

4λ′D d h

Consequently, the electrostatic repulsion increases more rapidly than the attractive polymer-mediated interaction as the temperature is increased. This explains why an increase in temperature should stabilize the lamellar phase, as observed experimentally. However, our calcula-

tions are from a relatively crude approximation which neglects some indirect variations of the interactions with temperature, and we obtain only a qualitative agreement between the predicted theoretical critical points and the experimental ones. 6. Summary In this article, we have investigated how the incorporation of a water-soluble nonadsorbing homopolymer into a two-solvent lamellar phase modifies its phase behavior. We have focused on the description and the interpretation of several critical points of LR/LR phase separation which occur in this mixed system. In doing so, we have proposed a simple model, which allows us to predict the smectic compression modulus of such an effective four-component lamellar phase. Despite the fact that one of these critical points, which occurs upon decreasing the thickness of oil layers, cannot be understood by this approach, our model successfully explains the occurrence of two of these critical points with remarkably good quantitative agreements. The fourth critical point, induced by a variation of the temperature, is in good qualitative agreement with the predictions of the model. Acknowledgment. The authors are indebted to G. Porte for countless helpful discussions. One of them L.P. would thank W. A. Hamilton for a careful reading of the manuscript. LA991193A

Effect of a Nonadsorbing Polymer on the Stability of a Two-Solvent

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