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Langmuir 1999, 15, 5414-5421
Colloidal Dispersions in Lyotropic Lamellar Phases Golchi Salamat† and Eric W. Kaler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received July 23, 1998. In Final Form: April 22, 1999 Phase behavior studies and small-angle neutron scattering measurements of nonionic lamellar phases that contain dispersed colloidal particles show a strong interplay between semiflexible bilayers and particles. Addition of charged silica particles to lamellar phases formed with either mixtures of the nonionic surfactant n-dodecylpentaoxyethylene glycol monoether (C12E5) or C12E5 and hexanol (C6E0) in water changes the temperature of the phase transition from the single lamellar phase to its adjacent two-phase region. Close to the ternary phase boundary, particles weaken the long-range correlation between the highly undulated surfactant bilayers, as evidenced from the changes in the structure factor of the bilayers measured in the small-angle neutron scattering experiments. A self-consistent thermodynamic model is developed based on the free energy contributions of the binary interactions describing the phase behavior of the C12E5water-colloid mixture. This model shows the effect of concentration, temperature, particle size, and bilayer rigidity on the phase behavior, and agrees well with the measurements.
I. Introduction Stable microstructured solutions containing dispersed colloidal particles display interesting properties and have diverse industrial applications. Recent studies have focused on the stability and phase behavior of colloidal particles in surfactant solutions of various microstructures.1-15 Colloidal silica dispersed in weakly structured C4E1-water mixtures,1-3 or in oil-in-water microemulsions,4 and polystyrene latex particles dispersed in aqueous micellar solutions of C12E5,15 or in solutions of oppositely charged rodlike micelles,14 directly influence the phase behavior of the surfactant solutions. The addition of particles induces phase separation into two phases that may have similar or different microstructures. The macroscopic phase behavior displayed by these mixtures depends on the surfactant microstructure and the physical properties of the colloids. The particular microstructure of interest in this paper is the lamellar (LR) phase formed by nonionic surfactants or mixtures of surfactant and cosurfactant. The LR phase is made of stacks of periodic bilayers separated by a solvent (water or oil). The stability of LR phase is governed by, * To whom correspondence should be addressed. Telephone: (302) 831-3553. Fax: (302) 831-8201. Email:
[email protected]. † Current address: Oilfield Chemical Products, Schlumberger Dowell, P.O. Box 4610, Houston, TX 77210-4610. (1) Jayalakshmi, Y.; Kaler, E. W. Phys. Rev. Lett. 1997, 78, 1379. (2) Jayalakshmi, Y.; Salamat, G.; Kaler, E. W. To be submitted for publication. (3) Kline, S. R.; Kaler, E. W. Langmuir 1994, 10, 412. (4) Kline, S. R.; Kaler, E. W. J. Colloid Interface Sci. 1998, 203, 392. (5) Quilliet, C.; Fabre, P.; Cabuil, V. J. Phys. Chem. 1993, 97, 287. (6) Fabre, P.; Ober, R.; Veyssie, M.; Cabuil, V. J. Magn. Magn. Mater. 1990, 85, 77. (7) Nallet, F.; Prost, J. Europhys. Lett. 1987, 4, 307. (8) Fabre, P.; Casagrande, C.; Veyssie, M.; Cabuil, V.; Massart, R. Phys. Rev. Lett. 1990, 64, 539. (9) Dabadie, J. C.; Fabre, P.; Veyssie, M.; Cabuil, V.; Massart, R. J. Phys.: Condens. Matter 1990, 2, SA291. (10) Quilliet, C.; Fabre, P.; Veyssie, M. J. Phys. II Fr. 1993, 3, 1371. (11) Ponsinet, V.; Fabre, P.; Veyssie, M.; Auvray, L. J. Phys. II Fr. 1993, 3, 1021. (12) Ponsinet, V.; Fabre, P. J. Phys. Chem. 1996, 100, 5035. (13) Raghunathan, V. A.; Richetti, P.; Roux, D. Langmuir 1996, 12, 3789. (14) Koehler, R. D. Ph.D. thesis, University of Delaware, 1995. (15) Koehler, R. D.; Kaler, E. W. Langmuir 1997, 13, 2463.
among other things, the competition between the elastic energy of the membrane (i.e., the bilayer bending constant, κb) and the thermal energy, kBT. According to Helfrich,16 thermal undulations of nonionic membranes stabilize the LR phase because the undulations give rise to a steric repulsive force. Therefore, in the absence of any other repulsive force, such as electrostatic repulsions, extremely rigid bilayers (κb . kBT) collapse due to attractive van der Waals interactions. Lyotropic LR phases swell to up to thousands of angstroms by adding an appropriate solvent, so the bilayer separation (d) is inversely proportional to the bilayer (or surfactant) volume fraction, φs. Several experimental and theoretical attempts17-23 have been made to characterize and quantify the elastic properties of LR membranes. Light scattering,18 small-angle neutron scattering,21 and osmotic stress experiments17 have independently measured a bilayer bending constant on the order of 2 kBT for nonionic bilayers composed of ethoxylated alcohol surfactants (and alcohol). Some of the same studies have also measured values for the compressibility modulus at constant chemical potential (B h ), which is a measure of interbilayer interaction. Several studies5-11,24-26 include either particles or polymer between the bilayers and examine their effect on the bilayer stability and elastic properties. Two kinds of experimental studies have been performed in this area. One is the inclusion of hydrophobically modified iron oxide particles in an oil-dilute LR phase.5-11,24 Measurements on this system10,12,27 seem to show that particles change (16) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (17) Bagger-Jorgensen, H.; Olsson, U. Langmuir 1996, 12, 4057. (18) Freyssingeas, E.; Nallet, F.; Roux, D. Langmuir 1996, 12, 6028. (19) Nallet, F.; Roux, D.; Prost, J. J. Phys. Fr. 1989, 3147. (20) Golubovic, L.; Lubensky, T. C. Phys. Rev. B 1989, 39, 12110. (21) Strey, R.; Schomacker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc. Faraday Trans. 1990, 86, 2253. (22) Safinya, C. R.; Sirota, E. B.; Roux, D.; Smith, G. S. Phys. Rev. Lett. 1989, 62, 1134. (23) Quist, P.-O. Langmuir 1995, 11, 2201. (24) Fabre, P.; Quilliet, C.; Veyssie, M.; Nallet, F.; Roux, D.; Cabuil, V.; Massart, R. Europhys. Lett. 1992, 20, 229. (25) Ligoure, C.; Bouglet, G.; Porte, G. Phys. Rev. Lett. 1993, 71, 3600. (26) Ficheux, M.-F.; Bellocq, A.-M.; Nallet, F. J. Phys. II Fr. 1995, 5, 823. (27) Nallet, F.; Roux, D.; Quilliet, C.; Fabre, P.; Milner, S. T. J. Phys. II Fr. 1994, 4, 1477.
10.1021/la980928t CCC: $18.00 © 1999 American Chemical Society Published on Web 06/09/1999
Colloidal Dispersions in Lyotropic Lamellar Phases
Langmuir, Vol. 15, No. 16, 1999 5415
some of the elastic properties of the bilayers in stable mixtures, but under some conditions the particles aggregate or induce phase separation. Other studies25,26 incorporate water-soluble polymers in electrostatically stabilized LR phases. Nonionic PEG polymers26 are soluble in a LR phase up to high concentrations (50% in water), and PVP polymers25 induce a phase separation into two LR phases at higher salt concentrations. The main objective of this work is to link the macroscopic stability of the LR phases containing particle inclusions to their microscopic interdomain interactions and their equilibrium thermodynamic states. In previous experimental investigations1,2 we showed that model colloidal silica in an isotropic solution is a true third component in the phase behavior of these mixtures. The similarities between our findings there and others15,28 in isotropic mixtures prompt us to investigate the parallelism between the behavior of particles in isotropic solutions and the behavior in more structured anisotropic lamellar phases.
after phase separation was determined from the 1D swelling law (dφβ ) δ, where δ is bilayer thickness), and by measuring the concentration of C12E5 (by drying), and C6E0 (by gas chromatography). Small-Angle Neutron Scattering (SANS). Samples were prepared in the one-phase region and were held at 35 °C in a specially designed scattering cell. This cell allowed partial orientation of the lamellae, and consisted of a rectangular (13 mm × 11 mm) quartz cell holding 25 thin (193 µm thick) equally spaced quartz plates. To increase the scattering signal from the bilayers, the neutron scattering length density of colloidal silica and that of quartz plates were matched by the neutron scattering length density of the H2O/D2O solvent (64% D2O). All neutron scattering experiments were performed using the NG-7 30-m SANS spectrometer at the National Institute for Standards and Technology at Gaithersburg, MD. Samples were measured using 6 Å wavelength neutrons having a wavelength spread of ∆λ/λ ) 10%. The raw 2-D intensity data was anisotropic, so plots of I vs qz were made by sector averaging the reduced data in the anisotropic (z) direction.
II. Experimental Section
To calculate the equilibrium distribution of surfactants and particles between the bilayer phases, the total free energy of the mixture is approximated as the sum of the contributions of the individual binary pairs. The free energy of particles in the LR phase is estimated as that of hard spheres confined between flat bilayers, with the assumptions that particles do not penetrate into the membranes and have no specific interactions with them. The free energy of the LR phase is written by excluding the volume occupied by colloidal particles from the volume available for membrane undulations, and with neglect of any effect that particles might have on the intrinsic elastic properties of the bilayers. The free energy of particles in the L3 phase is written by excluding the volume occupied by the surfactant and alcohol to the particles, and vice versa. The Lr Phase. Helfrich repulsions stabilize undulating bilayers in the LR phase.16 The free energy density of this phase is
Materials. The silica dispersion Ludox TM (Dupont) was used as supplied. The dispersion contained discrete spheres 26 nm in diameter and was supplied as 50 wt % silica. The particles are stabilized by the dissociation of surface hydroxyl groups, so the surfaces are negative at pH 9. The surface charge is screened by sodium salts at concentrations of 0.78 g/L NaCl, and 4.45 g/L Na2SO4 (as determined by capillary electrophoresis measurements). C12E5 (Nikko Surfactants, 99+% pure) was used as supplied. C6E0 (hexanol) was purchased from Fluka (98+% pure), and was used as received. Water was distilled and deionized before pH adjustment with NaOH to match the pH 9 of the Ludox stock solution. D2O was purchased from Cambridge Isotope Laboratory (deuteration 99.9%, low paramagnetic, low conductivity). Phase Behavior. Samples were prepared by diluting pure C12E5 with water until the desired concentrations were obtained. Samples were vortex mixed for about a minute before the silica solution was added to achieve the final compositions. To prevent surfactant degradation, all solutions were deoxygenated with argon, and the headspace of the final mixture was purged with argon prior to being placed in a thermostated temperature bath ((0.1 °C). For samples containing C6E0, a stock solution of C12E5C6E0 was first made at the desired molar ratio (C6E0/C12E5, na/ns ) 1.04) and then diluted to the appropriate concentration with water. The phase behavior was mapped out as a function of temperature for several particle and bilayer concentrations. Some samples were made with mixtures of D2O/water as the solvent to determine the phase boundaries before small-angle neutron scattering experiments. Composition Determination. The partitioning of the colloidal silica between the LR phase and L3 phase was determined at a fixed temperature and particle concentration as a function of bilayer concentration. A sample of C12E5-C6E0-water-D2Ocolloidal silica mixture was placed in a closed 25 mL graduated cylinder equipped with a separating funnel at the bottom. Teflon tubing was attached to the bottom of the funnel to withdraw the bottom phase. The cylinder was mounted inside a constanttemperature bath, and the sample was allowed to phase separate after complete mixing with a magnetic stir bar. Equilibration was assumed when the position of the meniscus between the coexisting phases no longer changed with time. Small portions of the coexisting phases were extracted and cooled to 2 °C, where the surfactant microstructure is micellar (the dispersions remained stable). The partition coefficient (Kp) of the particles is proportional to the ratio of the absorbance (Lambda 2 UV/vis spectrometer at 350 nm) of the upper phase (LR) to that of the lower phase. Some samples with high concentrations of silica particles were further diluted with water prior to the measurement. The measured light absorption at 350 nm of the surfactant solution was negligible. The interbilayer spacing in the LR phase (28) Beysens, D.; Esteve, D. Phys. Rev. Lett. 1985, 54, 2123.
III. Thermodynamic Model
gLa )
3π2(kBT)2 φσ3 128kb (1 - φ )2δ3
(1)
σ
where kB is the Boltzmann constant, T is the temperature, and φs is the volume fraction of the surfactant. For dilute nonionic lamellar phases, the bending constant of the bilayers is on the order of kBT,17 and long-range interactions between membranes are absent. A logarithmic (or crumpling) correction to the dilution law is needed to account for the lack of direct correlation between the layers12,18,20. By including the correction, the average d spacing for a dilute LR phase is given by:18
d ∼ (A - B ln(φs))/φs
(2)
Theoretical estimates for the parameters A and B in eq 2 are18
[
A)δ 1+
(
)]
kBT 16κbδ3 ln 8πκb 3πkBTvs
kBT B)δ 4πκb
(3a)
(3b)
where vs is the volume of the surfactant. This correction can also be incorporated into the free energy expression of eq 1.
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The L3 Phase. The isotropic L3 phase consists of a disordered, bicontinuous multiconnected surfactant bilayer structure separating two water-rich domains. Generally, this phase occupies a narrow region in the temperature-concentration plane and is bordered by a dilute phase (L1) at high temperature and a LR phase at low temperature. There is an extensive literature concerning the form of the free energy expression of the L3 phase and the possible driving forces for the phase transition from the LR phase to L3 phase.29-33 A free energy model based on the flexible and parallel surface model and as described by Daicic et al.29 is adopted here. Within this model, the surfactant membrane is considered to be an ideally connected geometrical surface composed of two oppositely oriented monolayers that are each displaced from the mid-surface by a distance δ/2. The total free energy of this phase is dominated by the monolayer mean curvature energy, and contributions such as fluctuations and entropy are neglected. This model captures the essential features of the L3 phase that are observed experimentally, namely the sequence of phase transitions, the finite swelling of the structure, and the narrow extent of the phase in composition. The free energy density of the L3 phase is given by29
(
gL3 ) 2κb
Ho2
(δ/2)
φs +
Ho (δ/2)
3
φ 2 s
+
)
1 φ5 4(δ/2)3
(4)
where Ho is the spontaneous curvature and is a function of the swelling of the L3 phase. For phase equilibrium calculations, the term linear in φs is absorbed into the standard chemical potential of surfactant in the L3 phase.29 For bilayer phases in equilibrium (L3/LR), the standard chemical potential29 and the bending constant18 are assumed to be the same in both phases. The Dilute Phase. The dilute phase (L1) adjacent to the L3 phase in aqueous C12E5 solution21 is not well characterized. We described this phase as dispersed surfactant monomers in water with a large chemical potential compared to the chemical potential in bilayer phases.29,34 This assumption is reasonable since the L1 phase appears at very low surfactant concentrations. The chemical potentials for surfactant (µs) and water (µw) are given by
µs(L1) ) µ°s(L1) + kBT ln
( ) ( )
µw(L1) ) µ°w(L1) - kBT ln
vwφs vs
(5a)
vwφs vs
(5b)
where vw is the volume of a water molecule, and µ°x is the standard chemical potential for water or surfactant. Free Energy of Colloidal Particles. For calculations of three-component phase equilibrium only the phase behavior of the bilayer phases LR and L3 is considered. The total free energy of each bilayer phase is the free energy of that phase (eq 1 for the LR phase and eq 4 for (29) Daicic, J.; Olsson, U.; Wennerstro¨m, H.; Jerke, G.; Schurtenberger, P. J. Phys. II Fr. 1995, 5, 199. (30) Anderson, D.; Wennerstro¨m, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243. (31) Roux, D.; Cates, M. E.; Olsson, U.; Ball, R. C.; Nallet, F.; Bellocq, A. M. Europhys. Lett. 1990, 11, 229. (32) Roux, D. Physica A 1995, 213, 168. (33) Cates, M. E.; Roux, D.; Andelman, D.; Milner, S. T.; Safran, S. A. Europhysics Lett. 1988, 5, 733. (34) Tanford, C. The hydrophobic effect: Formation of micelles and biological membranes; Wiley: New York, 1980.
the L3 phase) plus the free energy of the colloidal particles in that phase. The free energy of colloidal particles in the L3 phase is described by the Carnahan-Starling (CS) equation of state in which the particles behave as hard spheres.35,36 With the exception of the excluded volume of the surfactant, any other contributions of the L3 phase to the free energy of colloidal particles are neglected. The chemical potential of colloidal particles in the LR phase is calculated for the case in which particles are confined between two rigid plates. At low particle densities, any dependence of the total chemical potential on position in the water layer (i.e., in the direction perpendicular to the bilayers) can be neglected, so the first correction to the ideal gas limit is
µp/kBT = ln(Λ3Feff) + Feff〈Vexcluded〉
(6)
where µp is the total chemical potential, Λ is the thermal de Broglie wavelength, and Feff is the effective particle number density equal to Fdw/dw - σ)) (F ) particle number density) which reflects the fact that the center of particles of diameter σ are excluded from a region closer than σ/2 to the bilayer surface. 〈Vexcluded〉 is the average volume excluded to a particle between the bilayer sheets by the presence of a second particle and is
{
πσ3 Feff(8 - 3/D*3), D* > 1 6 〈Vexcluded〉 ) πσ3 F (6D* - D*3), D* < 1 6 eff
(7)
where D* ) (d-δ-σ) / σ. Equation 7 describes the limiting behavior at low densities (Feff e 0.05) as verified by Monte Carlo simulations in the canonical (NVT) ensemble. Phase Diagrams. The equilibrium phase behavior of the C12E5-water mixture was established by minimizing the total free energy density with respect to the surfactant concentration in each phase subject to mass balance constraints and equality of chemical potentials of each component in coexisting phases. To construct a phase diagram showing phase behavior as a function of temperature, the spontaneous curvature was related to temperature via30
-Ho )
2c2 (T - T0) 2.2δ
(8)
where T0 is defined as the temperature where Ho ) 0. The proportionality constant c varies with surfactant type. The total free energy for the ternary mixture was minimized with respect to surfactant and particle concentrations with appropriate constraints. IV. Results Phase Behavior. Figure 1 shows the experimental phase map for γ ) 5 wt % (γ ) (C12E5/C12E5 + H2O) (×100 in wt %) in water with and without C6E0. Addition of a medium-chain alcohol to binary mixtures of an ethoxylated alcohol (CiEj) and water lowers the phase behavior on the temperature scale.37 This is advantageous because the rate of degradation of C12E5 is lower at lower temperatures.38,39 At 5 wt % C12E5 in the absence of alcohol, the (35) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (36) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 53, 600. (37) Jonstromer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (38) Douglas, C. B.; Kaler, E. W. J. Chem. Soc., Faraday Trans. 1994, 90, 471.
Colloidal Dispersions in Lyotropic Lamellar Phases
Figure 1. Phase maps of C12E5-H2O (left) and C12E5-C6E0H2O (right) for 5 wt % C12E5 in water, and a molar ratio of C6E0 to C12E5 (na/ns) ) 1.04. There is a small (>0.1 °C) multiphase region below the lower LR phase boundary.
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Figure 3. Partition coefficient (Kp ) NpLa/NL3 for colloidal silica between the LR and L3 phases as a function of the reduced d spacing (dw/σ). Data are for 36 °C for a particle volume fraction equal to 0.01 in the C12E5-C6E0-D2O mixture as a function of dw/σ with γ ) 3 (filled circle), 4 (open square), 5 (open circle), 6 (closed square), and (7 triangle). na/ns is fixed at 1.04. The model results are shown by the solid line. Table 1. d Spacing in the Lr Phase at 36 °C from SANS Measurements of C12E5/C6E0 Phases Without Particles, and As Calculated from Measured Concentrations Using the Dilution Law for Lr Phases in Equilibrium with L3 Phases d spacing (Å)
Figure 2. Phase map of C12E5-C6E0-H2O-colloidal silica determined for a fixed bilayer composition of 5 wt % C12E5 in water, with na/ns ) 1.04.
LR phase exists from 55 to 67.5 °C, and is in equilibrium with a L3 phase at the upper and lower phase boundaries.21 C12E5 forms micelles in water at temperatures below 33 °C. At a molar ratio of C6E0 to C12E5 of 1.04, the LR phase exists from 21.8 to 39.3 °C, and the micellar phase exists below 7 °C. Addition of 0.8 vol % colloidal silica to the LR phase composed of mixtures of C12E5, C6E0, and water lowers the upper phase boundary between the LR phase and the L3 phase from 39.3 to 36.7 °C (Figure 2). The particles partition preferentially into the L3 phase as was apparent from the strong blue color of this phase. This amount of silica changes the phase boundary between the micellar phase (1φ) and its adjacent two-phase region (Figure 1) by only 0.3 °C. Adding only electrolyte equivalent to the amount present in this mixture from the silica stock solution reproduces only the 0.3 °C shift in the phase boundaries, showing that the effect of silica on delaying the formation of a particular phase as a function of temperature depends on the solution microstructure. Silica particles partition between the LR phase and L3 phase. To examine the effect of interbilayer spacing on (39) Douglas, C. B.; Kaler, E. W. Langmuir 1991, 7, 1097.
γ (wt %)
SANS
dilution law
3 4 5 6 7
1076 726 628 503 409
1025 741 581 467 391
the partitioning of particles, the bilayer concentration was varied from γ ) 3 to 7 wt % for a fixed amount of silica (1 vol %), and at a constant temperature (36 °C). Adding particles drives a phase transition from the LR phase to coexisting LR and L3 phases at this temperature. Figure 3 shows the partition coefficient (Kp) as a function of reduced length, dw/σ (dw ) d - δ). Colloidal silica preferentially partitions into the L3 phase (Kp < 1) for all bilayer separations studied. Kp decreases from 0.69 to 0.17 as the reduced length changes from 3.8 to 1.4. Interbilayer spacings after phase separation (Figure 3) were determined from the 1D swelling law (Table 1). The interbilayer spacing was also determined from small-angle neutron scattering measurements for these samples at 36 °C in the absence of particles. Table 1 shows there is a smaller d spacing in the LR phase (with the exception of γ ) 4 wt %) after the phase separation, indicating that more solvent partitions into the L3 phase in the presence of particles. Small-Angle Neutron Scattering (SANS). The smallangle neutron scattering spectra from LR samples containing particles at 35 °C are shown in Figure 4 at a concentration of γ ) 5 wt % (na/ns ) 1.04). The position of the faint but noticeable quasi-Bragg peak yields the average d spacing between the bilayers (d ) 2π/qmax ∼ 570 Å). Two important features of the scattering curves with increasing particle concentration are an increase in the low q intensity and a gradual disappearance of the quasi-Bragg peak. The theoretical description of the SANS spectra for highly undulating bilayers can be approximated by writing the averaged intensity in the direction perpendicular to
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Figure 4. Scattered intensity from samples containing partially oriented bilayers in LR phases as a function of qz (the scattering vector in the direction perpendicular to the bilayer stacking). Samples contain 5 wt % C12E5 in water, na/ns ) 1.04 and are at 35 °C. The volume fraction of particles are φp ) 0 (0), φp ) 0.004 (O), φp ) 0.006 (4). Solid lines are the model fits. In the fit C ) 1.7 × 10-25 J/Å3, B ) 8.2 × 10-27 J/Å3, B h ) 1.4 × 10-29 J/Å3, and K ˜ 1 ) 1 × 10-24 J/Å. χt-1 are 3.45 × 10-24, 3.83 × 10-24, and 5.17 × 10-24 J/Å3 for samples with φp ) 0, 0.004 and 0.006, respectively.
the bilayers as (see ref 40 for details):
I(qz,0) )
Az 1+
2
qz ξzeff2
+
ABragg (|qz/qmax| - 1)2 + R
+ b (10)
where Az and ABragg are the amplitudes of the small-angle scattering and Bragg scattering, respectively, R is related to the resolution of the spectrometer, and b is the background isotropic incoherent scattering. ζzeff is the effective correlation length for concentration fluctuations along the z-axis and is given by40
ξzeff2
(
)
2 ˜1 B 2 χt C K ) ξz + B B B h
(11)
where ξz is the correlation length, B is the compression modulus at constant layer spacing, B h is the compression modulus at constant chemical potential, χt-1 is the osmotic compressibility, C is the lowest order coupling constant between layer displacement and concentration fluctuations, and K ˜ 1 is the second-order bending constant of the LR phase. The lines through the data points in Figure 4 are the ˜ 1 are fits to eq 10. Values for C, B, B h , χt-1, and K calculated18,19,40,41 and kept constant while Az, ABragg, R, and b are fit to the data. There is a negligible change in the total osmotic compressibility due to the presence of the particles.42 Phase Behavior Model. Figure 5, a and b, shows the experimental and theoretical phase diagram of C12E5water for the LR phase and adjacent phases, respectively. The horizontal lines in the coexistence regions are tie lines. Figure 5a shows the experimental temperature-composition phase diagram21 for C12E5-water. Figure 5b is the model calculation with parameter values listed in Table 2. The model phase diagram shows the correct sequence (40) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. Fr. 1990, 51, 2333. (41) Gelbart, W. M.; Ben-Shaul, A.; Roux, D. Micelles, Membranes, Microemulsions, and Monolayers; Springer Verlag: Berlin, 1994. (42) Salamat, G. Ph.D. thesis, University of Delaware, 1998.
Figure 5. (a, top) Measured partial phase diagram of C12E5H2O mixtures showing the LR phase, the L3 phase, and its adjacent phases as a function of temperature over a range of C12E5 concentration (ref 21). (b, bottom) Calculated partial phase diagram of C12E5-H2O mixtures over the same composition and temperature range shown in (a), with κb ) 2kBT, and c ) 0.11 (see Table 2 for other parameter values). Table 2. Physical Parameters Used in the Thermodynamic Model volume of C12E5 (vs) (Å3) volume of H2O (vw) (Å3) bilayer thickness (δ) (Å) bilayer bending constant (kb) T0 (°C) ∆µstandard ) µ dilute - µ bilayer phase
706 30 30 (ref 21) 2kBT (refs 17, 21) 64.5 (ref 30) 15.5kBT (ref 34)
of phase transitions, but otherwise provides only a qualitative description of the experimental diagram. We chose a value of 2 kBT for the bending constant, which represents an average value of the existing theoretical and experimental measurements. A larger value of the bending constant stabilizes the LR phase at lower surfactant concentrations and expands the two-phase region (LR - L3) in the temperature-composition plane, while a smaller value shrinks the two-phase region and shifts the LR region to much higher surfactant concentrations. Model calculations show that adding colloidal particles dramatically changes the phase behavior displayed in Figure 5b. Figure 6 is a calculated phase map of colloidal particle concentration as a function of C12E5 concentration at constant temperature (Ho ) - 5 × 10-5), κb (2 kBT), and σ (50 Å). Figure 6 shows the effect of increasing particle concentration on the position of the phase boundaries between the LR phase and L3 phase (corresponding to a horizontal line in Figure 5b). The phase boundary at φp
Colloidal Dispersions in Lyotropic Lamellar Phases
Langmuir, Vol. 15, No. 16, 1999 5419
Figure 6. A calculated phase map for C12E5-H2O-colloidal particles at constant temperature. The volume fraction of colloidal particles is shown as a function of surfactant volume fraction with Ho ) -5 × 10-5, σ ) 50 Å, and κb ) 2kBT. The solid lines in the two phase region are tielines.
) 0 in Figure 6 corresponds to the phase boundary of the C12E5-water mixture. Addition of particles expands the region of coexistence of L3 and LR phases. The tie lines show the partitioning of colloidal particles and C12E5 between the bilayer phases. Colloidal particles partition favorably into the L3 phase while the surfactant partitions favorably into the LR phase. At high C12E5 concentrations, where dw ∼ σ, the model predicts a 3φ region. These calculations are done for particles with sizes smaller than those used in the experiments. This is because we wish to compare the experimental results for LR/L3 partitioning observed for lamellar phases containing hexanol, which form at lower surfactant concentrations and so have larger values of the water layer spacing dw. In order to compare with experiment we have used a value of the particle diameter that allows for matching of the experimental and calculated values of the particle volume fraction at a given number density; in that case the reduced length dw/σ is set. This is the key parameter setting the particle free energy (eqs 6 and 7). Figure 3 compares the experimental partition coefficient (Kp) to the model. The solid curve shows the partitioning of particles between rigid bilayers and the bulk phase for the same reduced lengths and particle concentration as the experiment. It is interesting to note that the model predicts a somewhat higher Kp than that measured. The phase boundaries between bilayer phases in the ternary system shift dramatically as a function of temperature. Figure 7 shows the model phase behavior at two temperatures separated by 0.8 °C and projected onto the φp-φs plane. The shaded area shows the coexisting phases at the higher temperature, and the area enclosed by solid lines shows the coexisting phases at the lower temperature (redrawn from Figure 6). κb and σ are kept constant at 2kBT and 50 Å, respectively. Noticeable changes in the phase behavior with increasing temperature are the formation of L3 phase in place of the L3-LR coexistence phases at low φp, and the overall shrinkage of both bilayer phases (Figure 7 inset compares the LR phases) and their coexisting two-phase region. In addition to the temperature, varying elastic properties of the bilayers, particle size, or the interbilayer interactions (eq 2) affect the phase behavior. Incorporating a crumpling correction to the free energy expression of the LR phase increases the stability of this phase both in the binary and ternary mixtures. In the C12E5-water
Figure 7. Two calculated phase maps for C12E5-H2O-colloidal particles at two temperatures projected on the same diagram. The volume fraction of colloidal particles is shown as a function of surfactant volume fraction with σ ) 50 Å and κb ) 2kBT. Open circles are calculations at Ho ) -5 × 10-4, and solid lines (with no symbols) are calculations for Ho ) -5 × 10-5. Lower panel is the enlargement of the LR region.
mixture, the crumpling correction relocates the phase boundaries shown in Figure 5b between the LR and L3 phases, and moves the position of the LR phase to lower C12E5 concentrations (not shown). In the ternary mixture, this correction expands the LR phase on a constant temperature plane. Figure 8 is a phase map of particle concentration as a function of C12E5 concentration and shows the extension of the LR phase to higher particle concentration with the crumpling correction (open squares) when compared to the previous results shown in Figure 6 (filled circles in Figure 8) under the same conditions. Here the particle size is set at 61 Å in the model with the crumpling correction to match approximately the reduced length at φp ) 0 to the calculations with no correction. An increase in the bilayer rigidity lowers the free energy of the lamellar phase (eq 1). Model calculations show that increasing the bilayer rigidity from 2 kBT to 5 kBT stabilizes the LR phase at lower surfactant concentrations in the Tφs plane. In the ternary mixture, increasing kb from 2kBT to 5kBT expands the LR phase to higher particle concentrations, as shown in Figure 8. Incorporation of larger particles into the LR phase destabilizes this phase at high surfactant concentrations. Figure 9 shows the model phase behavior for two different particle sizes, and so dw/σ, at constant κb and Ho (50 Å (open circles) and 28 Å (closed circles)). For clarity, only the LR region and its adjacent two phase are shown. The LR phase extends to higher
5420 Langmuir, Vol. 15, No. 16, 1999
Figure 8. A calculated phase map for C12E5-H2O-colloidal particles. The volume fraction of colloidal particles is shown as a function of surfactant volume fraction with Ho ) -5 × 10-5. Filled circles are calculations with κb ) 2kBT and σ ) 50 Å; open squares are calculations including the crumpling (logarithmic) correction with κb ) 2kBT and σ ) 61 Å; and filled triangles are calculations with no logarithmic correction with κb ) 5kBT and σ ) 50 Å.
Salamat and Kaler
Figure 10. Experimental and theoretical phase maps for C12E5-H2O-colloidal particles. The relocation of the transition temperature between the LR and LR-L3 phases with the addition of colloidal particles to C12E5-H2O mixtures is shown as ∆Ttrans., defined as the difference in transition temperature in the presence or absence of particles, over a range of particle concentrations. The filled circles are measured values for C12E5H2O-colloidal silica with dw/σ ∼ 1.8 ( 0.2. The solid line is the model calculation with the crumpling (logarithmic) correction with dw/σ ) 1.6, c ) 0.09, and κb ) 2kBT. The dashed line is the model calculation without the logarithmic correction with dw/σ ) 1.6, c ) 0.11, and κb ) 2kBT. Table 3. Experimental and Model Parameters Used To Determine the Phase Boundary between the Lr and Lr-L3 Phases in Figure 10a dw/σ EXP (b) LC (s) N/LC (- - -)
1.8 ( 1.6 1.6
0.2b
κb/kBT 1.3 ( 2 2
c
vs (Å3)
δ (Å)
0.09 0.11e
706 706 706
30d 30 30
1c
a LC, logarithmic correction; N/LC, no logarithmic correction. From SANS and eq 6 in ref 21. The ( is calculated from the average d spacing and Ludox polydispersity. c References 21, 18, and 17. d Reference 21 e Reference 30.
b
V. Discussion Figure 9. A calculated phase map for C12E5-H2O-colloidal particles. The volume fraction of colloidal particles is shown as a function of surfactant volume fraction with κb ) 2kBT, and Ho ) -5 × 10-4. Open circles are calculations for σ ) 50 Å, and filled circles are for σ ) 28 Å.
particle and surfactant concentrations when the size of dispersed particles is reduced. In general the transition temperature, Ttrans, between adjacent phases is lowered as the colloidal particle concentration increases.1,43 Figure 10 (measured at γ ) 6.43 wt %) shows the change in the transition temperature, ∆Ttrans (defined as the difference in transition temperature in the presence or absence of particles) as a function of colloidal particle concentration. Experimental measurements are shown by filled circles, and model predictions with and without the crumpling correction are shown by lines. The model parameters are scaled to the experimental parameters as shown in Table 3. Both model predictions agree well with the measured values. If the crumpling correction is used (solid line), the proportionality constant, c, is slightly lower than the value determined by Anderson et al.30 If no correction is made (dashed line), the concentration of particles in the LR phase is about 0.54 vol %, short of the experimental results (i.e., ∼0.8 vol %). (43) Poulin, P.; Raghunathan, V. A.; Richetti, P.; Roux, D. J. Phys. II Fr. 1994, 4, 1557.
Colloidal silica induces phase separation in mixtures of C12E5-C6E0-water and C12E5-water near the LR-(LRL3) phase boundary (Figures 2 and 10) as a result of an unfavorable interaction between colloidal silica and surfactant bilayers in the LR phase. Two dominant factors contributing to phase separation are the large confinement free energy of the particles between undulating bilayer sheets, and a decrease in the stabilizing steric repulsive force between LR bilayers due to the dispersed particles. Inclusion of spherical particles between the bilayers creates a distortion in the anisotropic medium. Particles trigger the formation of an isotropic medium, which is not favored by the surfactant mixture at that concentration and temperature. Similar phase behavior is reported for iron oxide particles in a charged oil-dilute LR phase,6 latex particles in a nematic phase,13,43 or water-soluble polymers in a charged LR phase.25,26 Even though some of these studies did not report a phase boundary between the bilayer phase and its adjacent two-phase region, they all report a particle (or polymer)-induced phase separation of the bilayer phases. An increase in the small angle scattering (SAS) and a broadening of the quasi-Bragg peak in the SANS spectra following the addition of the silica is due to an increase in the incoherent thermal fluctuations and a decrease in the long-range correlation between the bilayers. The h -1, the bilayer intensity as qz f 0 is proportional to B
Colloidal Dispersions in Lyotropic Lamellar Phases
compressibility modulus, so an increase in SAS stems from suppresion of the Helfrich repulsions by the particles; this eventually will cause phase separation at a higher colloidal particle concentration. Previous studies26 of water-soluble polymers in a charged LR phase also showed a broadening of the Bragg peak and an increase in the SAS with polymer addition. In contrast, another study11 of an oil-dilute LR phase showed an increase in the order of the bilayers and a constant or decreasing SAS with addition of iron oxide spheres. The reason for the difference between the two observations could be related to the different types of LR phases (i.e., oil-dilute vs water-dilute, or charged vs uncharged bilayers), and to the different types of particles in the mixtures. Silica particles are charged and are stabilized by steric and electrostatic forces, while the iron oxide particles are stabilized by the adsorption of a longtail surfactant.5 Such adsorbed surfactant could introduce additional interparticle or particle-bilayer interactions. Figure 3 shows the depletion of particles from the LR phase. Particles always partition preferentially into the isotropic phase (Kp < 1) regardless of the interbilayer separation, and the partition coefficients show a systematic dependence on the reduced length. Although, the model calculation (solid curve) closely follows the experimental points, it predicts higher values of Kp for all bilayer spacings. This is probably because the effect of bilayer undulations is not included in the calculated excess chemical potential of the particles. The effect of undulations would be to reduce the free volume available to the particles, so that for a given d spacing the same amount of particles would be more tightly confined if they were between undulating bilayers than between flat plates. The comparison between the model and experiments suggests that on the average, fluctuations reduce the volume available by less than 20%. The enlargement of the miscibility gap between the LR and L3 phases in Figure 6 as particles are added reflects changes in the free energy contributions of the species to each phase as concentration is varied. This is an indication of ternary phase behavior. The model phase behavior captures the essential features in the phase behavior of surfactant-water-colloidal silica mixture shown in Figures 2 and 10, namely the sequence of phases and the expansion of the two-phase region. Since the calculated binary phase behavior (Figure 5b) does not show formation of LR phase at low surfactant concentrations (∼5 wt %) where measurements were made, a more direct and absolute comparison between the model calculations and the experiments is not possible. The maximum amount of colloidal silica in the LR phase is about 1 vol % (both from experiments and theory) which is at least 10 times smaller than the amount stabilized in weakly structured isotropic solutions of C4E1.1-3 The large difference in the solubility of silica emphasizes the unfavorable effect of an anisotropic medium, and also the adverse effect of bilayer undulations on colloidal stability. The model also shows that only a limited amount of particles can be dispersed in the L3 phase, leading to the conclusion that colloidal particles are most stable in surfactant solutions which exhibit spherical microstructure.1-4 A change in the phase behavior with temperature is one of the characteristics of ternary phase behavior. Increasing temperature leads not only to change in the solubilities, but also to an increase in bilayer thermal undulations. Model calculations in Figure 7 do not include the effect that bilayer fluctuations would have on the volume available to the particles, but they do show redistribution of the components in equilibrium phases
Langmuir, Vol. 15, No. 16, 1999 5421
as indicated by the tilting of the tie lines. For a fixed concentration marked by X in Figure 7, the L3 phase forms at the higher temperature in place of two phases at the lower temperature, and a mixture in the LR phase at the lower temperature phase separates at the higher temperature (marked by X in Figure 7 inset). These predictions are consistent with experimental results shown in Figure 2. In ref 2, the tilting of the tie lines for C4E1-watercolloidal silica mixtures were measured as a function of temperature. This common characteristic between the model calculations for C12E5-water-colloidal silica mixtures and experimental measurements for C4E1-watercolloidal silica mixtures suggests that there is a universal behavior in systems containing colloidal particles, and the observed phase behavior should be viewed as ternary phase behavior. Comparing Figures 2 and 10 shows that colloidal particles delay the formation of a more structured (or more concentrated) microstructure as a function of temperature. In the present model, temperature is included explicitly to compare the phase transition driven by the particles to the phase transition driven by changes in bilayer curvature. The excellent agreement between the model predictions and experiments in Figure 10 suggests that the ternary phase separation is predominantly a consequence of the interplay of the phase behavior of the constituent binary pairs. Other factors that are not considered in the model, such as the indirect interaction of non-neighboring membranes, or effect of particles on intrinsic bilayer characteristics, are only secondary when determining particle stability and phase behavior in these mixtures. The role of defect density in the lamellar phase has not been examined, but as suggested by a referee defects could play a role in stabilizing the lamellar phase if particles occupy such sites.44,45 The model calculations show that smaller particles are stable up to higher particle and surfactant concentrations than larger particles (Figure 9); however, an increase in particle number density may trigger attractive interparticle interactions and so could lead to aggregation. Similarly, the present model may be a poor representation for large particles or when dw/σ ∼ 1, because large particles are expected to affect the intrinsic properties of the bilayer. Whether much larger particles or small particles at larger volume fractions can be stable in the LR phase is thus not yet known. VI. Conclusions Colloidal silica destabilizes LR phases composed of C12E5-C6E0-water and C12E5-water. Both experimental results and model calculations show a ternary phase separation, where colloidal silica partitions favorably into the L3 phase and surfactant partitions favorably into the LR phase. A thermodynamic model predicts changes in the phase transition temperature as a function of particle concentration that agree with the experimental results. Acknowledgment. We thank H. Ashbaugh for invaluable discussions, and acknowledge the support of the National Institute of Standards and Technology for Neutron Research, U.S. Department of Commerce, in providing the facilities used in SANS experiments. This work was supported by E. I. duPont de Nemours & Co. LA980928T (44) Turner M. S.; Sens P. Phys. Rev. E 1997, 55, 1275. (45) Ramos L.; Fabre P.; Dubois E. J. Phys. Chem. 1996, 100, 4353.
