Langmuir 2001, 17, 1043-1053
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Structure and Transport Properties of a Charged Spherical Microemulsion System Alex Evilevitch,*,† Vladimir Lobaskin,‡ Ulf Olsson,† Per Linse,† and Peter Schurtenberger‡ Division of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden, and Physics Department, Soft Condensed Matter Group, University of Fribourg, Perolles, CH-1700 Fribourg, Switzerland Received August 18, 2000. In Final Form: November 15, 2000
Structure and transport properties of an oil-in-water microemulsion of weakly charged spherical micelles were studied experimentally using viscosity, NMR self-diffusion, and static and dynamic light scattering as well as theoretically by Brownian dynamics and Monte Carlo simulations and the Poisson-Boltzmann equation. The micelles contain decane covered by the nonionic surfactant pentaethylene glycol dodecyl ether (C12E5) and the ionic surfactant sodium dodecyl sulfate. The system has a constant surfactant-to-oil ratio, and the total volume fraction of surfactant and oil, Φ, is varied between 0.01 e Φ e 0.46. The micelles were made weakly charged by replacing a small fraction (0.01, 0.04, and 0.06) of the nonionic surfactant with ionic surfactant, retaining the micellar size. Comparison between self-diffusion and viscosity coefficients measured as a function of concentration showed that the system obeys the generalized Stokes-Einstein relation at lower micellar concentrations. At higher micellar concentrations, a slightly modified equation can be used upon the addition of an extra frictional factor due to stronger interactions. The collective diffusion coefficient shows a maximum as a function of the volume fraction. This result is in good agreement with predictions based on a charged hard-sphere model with hydrodynamic interactions. Other static and dynamic properties such as osmotic pressure, osmotic compressibility, and self-diffusion coefficient were obtained theoretically from simulations based on a charged-sphere model. The static and dynamic properties of the charged hard-sphere model qualitatively describe the behavior of the charged microemulsion micelles. At high volume fractions, Φ > 0.1, the agreement is quantitative, but at Φ < 0.1 the effect of the charge is smaller than what is predicted from the model.
1. Introduction Microemulsions are thermodynamically stable isotropic fluid mixtures of water, oil, and surfactant. The surfactants assemble as dividing surfaces between oil and water domains. Previous studies on the structure of microemulsions have shown that the structure can vary from discrete swollen micelles in solution to disordered bicontinuous networks as a function of either temperature or composition.1 In many respects, the properties of swollen micelles resemble those of small colloidal particles. In particular, by introducing a small amount of charged surfactants such microemulsions composed by micelles are electrostatically stabilized. The stabilization of colloidal particles in general by electrostatic particle-particle repulsion has a long history.2 The use of surface charges allows for a control of the phase behavior and the rheological properties by manipulating, for example, the salt content or the pH of the solution (if titrable surface groups exist). An important factor in predicting electrostatic stabilization is the surface charge density of the particles, a property often difficult to uniquely measure experimentally.3,4 † ‡
Lund University. University of Fribourg.
(1) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain, Where Physics, Chemistry, Biology, and Technology Meet, 2nd ed.; WileyVCH: New York, 1999. (2) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989. (3) Yamanaka, J.; Yoshida, H.; Koga, T.; Ise, N.; Hashimoto, T. Langmuir 1999, 15, 4198. (4) Horn, M. F.; Richtering, W.; Bergenholtz, J.; Willenbacher, N.; Wagner, N. M. J. Colloid Interface Sci. 2000, 225, 166.
The Derjaguin-Landau-Verwey-Overbeek (DLVO)5 theory constitutes the classical theoretical foundation for describing charged stabilized colloidal suspensions. In this theory, the attractive van der Waals force promoting aggregation is counteracted by a repulsive force described on the basis of the Debye-Hu¨ckel solution of the linear Poisson-Boltzmann equation. During the last two decades, the DLVO theory has been challenged by experimental observations indicating that charged latex particles in aqueous solution attract each other and that the attraction has an electrostatic origin.3,6,7 Such observations have renewed theoretical interest in charged colloidal systems.6,8-10 In this work, we were particularly interested in the case in which the microemulsion consists of spherical oilswollen micelles dissolved in water. To obtain the desired system, the temperature for our three-component mixture was kept at the phase boundary between the single microemulsion phase (L1) and one in coexistence with excess oil (L1 + O), termed “emulsification failure”.11 At the emulsification failure phase boundary, the microemulsion spheres are of low polydispersity12 (≈ 16%) and have a concentration invariant size. Therefore, they can (5) Verwey, E. J.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (6) Arora, A. K.; Tata, B. V. R. Ordering and Phase Transitions in Charged Colloids; VCH: New York, 1996. (7) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (8) Schmitz, K. S. Langmuir 1999, 15, 4093. (9) Vlachy, V. Annu. Rev. Chem. 1999, 50, 145. (10) Hansen, J.-P.; Lo¨wen, H. Submitted for publication. (11) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113. (12) Bagger-Jo¨rgensen, H. Polymer Effects on Microemulsions and Lamellar Phases. Ph.D. Thesis, Lund University, Sweden, 1997.
10.1021/la0011883 CCC: $20.00 © 2001 American Chemical Society Published on Web 01/19/2001
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be compared with a colloidal dispersion of sterically stabilized colloid particles.13 Olsson and Schurtenberger have previously shown that such systems behave to a good approximation as a dispersion of hard spheres when they are stabilized by nonionic surfactants.14 Properties such as collective and self-diffusion, low-shear viscosity, and osmotic compressibility agreed quite well with existing theoretical models and with experimental results from model hard-sphere systems. These studies are here further extended by introducing a charge to the stabilizing surfactant film through the addition of a small amount of ionic surfactant sodium dodecyl sulfate (SDS). By this procedure, we obtained a system of charged micelles interacting not only by a hard-sphere potential but also by a long-range screened Coulomb potential. Phase diagrams for the different charge densities can be found in refs 15 and 16. The aim of this work is to describe and characterize a charged micellar microemulsion system by studying its static and dynamic properties. In section 2, we describe the preparation and general features of the charged microemulsion systems. Sections 3 and 4 contain a description of experimental techniques used and theoretical models. In section 5, we present experimental data on collective and self-diffusion, osmotic compressibility, and shear viscosity of the microemulsion systems together with results of a charged hard-sphere model. Finally, in sections 6 and 7 we discuss the presented data and provide our conclusions. 2. Experimental System The microemulsion micelles can be pictured as hydrocarbon spheres, containing oil (decane) and the alkyl chains of the surfactant (pentaethylene glycol dodecyl ether, C12E5), coated with a layer of “end-grafted” pentaethylene oxide chains. In the present work, the surfactant, Φs, to oil, Φo, volume fraction ratio was kept constant at Φs/Φo ) 0.815. The alkyl chains make up approximately half of the surfactant volume. The hydrocarbon volume fraction Φhc is thus given by Φhc ) Φo + 0.5Φs ) 0.775Φ, where Φ is Φ ) Φo + Φs. The radius of the hydrocarbon core, Rhc, is 75 Å at the L1/(L1 + O) phase boundary.17 In the uncharged case, the equivalent hardsphere radius RHS has been determined to 86 Å. This relates hard-sphere volume fraction to the total volume fraction Φ by ΦHS ) 1.14Φ.14 The introduction of charges at the micellar surface takes place by dissolving the dodecyl chain of SDS in the hydrocarbon core of the micelle. We have studied systems with β ) 0.01, 0.04, and 0.06, where β is the mole fraction of SDS in the surfactant mixture, β ) nSDS /(nC12E5 + nSDS). The micelles contain approximately 1460 surfactant molecules and 3000 oil molecules. Small-angle neutron scattering data18 indicate that the micelles remain spherical and do not change their radius within the present range of charge. This holds as long as we are near the lower temperature phase boundary of the microemulsion phase.16 Materials. The nonionic surfactant pentaethylene glycol dodecyl ether (C12E5) was obtained from Nikko (13) Safran, S. A. Statistical thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley: Reading, MA, 1994; Vol. 90. (14) Olsson, U.; Schurtenberger, P. Langmuir 1993, 9, 3394. (15) Fukuda, K.; Olsson, U.; Wu¨rz, U. Langmuir 1994, 10, 3222. (16) Rajagopalan, V.; Bagger-Jo¨rgensen, H.; Fukuda, K.; Olsson, U.; Jo¨nsson, B. Langmuir 1996, 12, 2939. (17) Olsson, U.; Schurtenberger, P. Prog. Colloid Polym. Sci. 1997, 104, 1157. (18) Pedersen, J. S.; Mortensen, K.; Rajagopalan, V.; Olsson, U.; Schurtenberger, P. To be published.
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Chemicals Co. Ltd. (Tokyo), n-decane (99.9%) was purchased from Sigma, D2O (99.8% isotopic purity) was obtained from Dr. Glaser, AG Basel (Switzerland), and SDS was purchased from BDH (England). The SDS was recrystallized three times in methanol. All other chemicals were used as received. Sample Preparation. Samples for the measurements were prepared from a stock solution containing C12E5, decane, and the appropriate amount of SDS. The samples were prepared by weighing the desired amounts into screw-capped test tubes, which were sealed thereafter. The samples were initially mixed using a vortex mixer. As the microemulsion-oil equilibrium kinetics are very slow, the samples were then homogenized by heating the samples under a hot water tap to the lamellar phase. Then, the samples were cooled and kept in the microemulsion phase in a thermostated water bath prior to the experiments. The volume fractions of the samples were calculated using the following densities (g/cm3): 0.967 (C12E5), 1.105 (D2O),16 and 0.73 (decane). The fraction of ionic surfactant is small, and we neglect its effect on the average surfactant density. 3. Experimental Techniques Light Scattering. Static and dynamic light scattering measurements were performed at temperatures between 22.1 and 39.8 °C. Approximately 1 mL of solution was transferred into the cylindrical scattering cell (10 mm inner diameter). The scattering cell was then stoppered and centrifuged for 30-60 min at approximately 5000g at the respective temperature in order to remove dust particles from the scattering volume. Measurements were made with an ALV goniometer instrument (ALV/DLS/ SLS-5000F monomode fiber compact goniometer system with ALV-5000 fast correlator), equipped with an argon ion laser (Coherent, Inova 300-8, λ0 ) 488 nm). The instrument has a modified high-temperature cell housing that allows for measurements in a wide range of temperatures between -6 and 220 °C and provides enhanced temperature stability of (0.01 °C at the temperatures used in this study. Static light scattering (SLS) experiments were performed at 28 different angles (15° e θ e 150°) thus yielding the scattering intensity 〈Is(θ)〉. The contribution from background stray light and solvent 〈Ib(θ)〉 was measured using the same cell containing the filtered solvent only. The background was then subtracted from 〈Is(θ)〉 yielding ∆〈 Is(θ)〉. The data were then converted into absolute scattering intensities, that is, “excess Rayleigh ratios” using
∆R(θ) )
( )
∆〈IS(θ)〉 n R (θ) nref 〈Iref(θ)〉 ref
2
(3.1)
where Rref(θ) ) 39.6 × 10-4 m-1 is the Rayleigh ratio of the reference solvent toluene, n and nref ) 1.499 are the index of refraction of the solution and the reference solvent, respectively, and 〈Iref(θ)〉 is the average scattering intensity from pure reference solvent. Dynamic light scattering (DLS) experiments were conducted at scattering angles θ ) 15°, 30°, 60°, 90°, and 120°. From the DLS experiments, a collective diffusion coefficient Dc ) 〈Γ〉/Q2 was obtained by means of a secondorder cumulant analysis of the intensity autocorrelation function. Here, 〈Γ〉 is the initial decay rate.19 (19) Schurtenberger, P.; Newman, M. E. Characterization of Enviromental Particles; Buffle, J. L. H., Ed.; Lewis Publishers: Boca Raton, FL, 1993; Vol. 2.
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NMR. The proton self-diffusion studies were performed on a Bruker 200 MHz NMR spectrometer using the Fourier transform pulsed gradient spin-echo (FT-PGSE) technique. The temperature was varied between 21 and 31 °C and controlled by a thermostated air-flow. The temperature control unit was calibrated by a copper-constantan thermocouple. The sample solutions (≈400 µL) were transferred to 5 mm NMR tubes. The measured collective and self-diffusion coefficients were normalized with respect to the self-diffusion coefficient at infinite dilution for uncharged micelles D0. D0 was obtained by extrapolation to Φ ) 0 at t ) 23.5 °C. The obtained D023.5 ) 2.1 × 10-11 m2 s-1 was then used to recalculate D0 to the actual temperature according to
D0(T) )
(
)
23.5
(T/°C + 273.15) η0 D 23.5 (23.5 + 273.15) η(T) 0
(3.2)
where η023.5 ) 1.172 cP is the viscosity of the solvent (D2O) at 23.5 °C and η(T) is the solvent viscosity measured at the actual temperature. Low-Shear Viscosity. Measurements of the low-shear viscosity were carried out on a Carri-Med CSL rheometer at temperatures between 20 and 50 °C. Approximately 1 mL of solution was used per run. The rheometer was equipped with a Peltier Pt 100 temperature control system, which can be used in the temperature range from -15 to 100 °C with an accuracy and stability of (0.1 °C. The measurements were carried out using a cone and plate measuring system with the dimensions of the measuring system chosen to optimize the accuracy of the results. In all the experiments, results have been obtained from a 6 cm diameter 1° cone operating with a 23 µm gap between the center of the measuring geometry and the plate. In the temperature-dependence experiments, a solvent trap is used to prevent solvent evaporation during the course of the experiment. The relative viscosities were obtained by measuring the water viscosities at the same temperature. 4. Theoretical Approach The charged microemulsion system was theoretically treated by using several models and statistical methods. Different models and methods provide complementary information. The Primitive Model. Within the primitive asymmetric electrolyte model, charged species are described as charged hard spheres interacting through repulsive hard-core and Coulombic forces, the latter attenuated by the dielectric permittivity of the solvent. The microemulsion micelles (also referred to as macroions in the model) are represented by spheres of the radius Rhc ) 75 Å carrying the charge ZM ) -14.5, -58, and -87, corresponding to β ) 0.01, 0.04, and 0.06, respectively, whereas the counterions are represented by spheres of the radius RI ) 2 Å and charge ZI ) 1. Because the van der Waals forces are negligible in these systems, they were not included.20 The primitive model was solved by Monte Carlo (MC) simulations in the canonical ensemble by employing 40 macroions and the appropriate amount of counterions to achieve electroneutrality.21 The result of the primitive model was used (i) to determine effective macroion charges in the one-component model, (ii) to predict osmotic (20) Bagger-Jo¨rgensen, H.; Olsson, U.; Jo¨nsson, B. J. Phys. Chem. B 1997, 101, 6504. (21) Lobaskin, V.; Linse, P. J. Chem. Phys. 1999, 111, 4300.
pressure, and (iii) to predict microemulsion structure (radial distribution functions). The Cell Model. This model is a simplification of the primitive model in which a single macroion and its counterions are enclosed in a sphere with a radius adjusted to provide the correct electrolyte number density. The description of the particles and their interactions are the same as in the primitive model. The cell model is not able to predict macroion structure but provides many other properties of the system with high accuracy. The model can be solved by MC simulations, but often approximate methods such as the Poisson-Boltzmann (PB) equation are employed. The cell model was used (i) to determine effective macroion charges in the onecomponent model and (ii) to provide the osmotic pressure and the osmotic compressibility. The One-Component Model. In the one-component model, only the charged macroions are considered explicitly, now interacting through a screened Coulomb potential according to
e-κ(r-2Rhc) 1 U(r) ) U0 + Uscreen (1 + κRM)2 r
(4.1)
where U0 is the hard-sphere repulsion potential, Uscreen ) 2 (Zeff M e) /(4π0rkT), and the inverse screening length κ ) eff 2 (FM|ZM |e /0rkT)1/2 (monovalent counterions) depending on the effective charge on the macroion Zeff M . In eq 4.1, FM ) Φ/(4πRhc3/3) is the hard-sphere number density, e is the electronic charge, r is the dielectric constant of the solvent, and 0 is the dielectric permittivity of vacuum. The use of the bare macroion charge in eq 4.1 leads to a too-strong macroion repulsion, because the equation is derived using the linear (Debye-Hu¨ckel) theory. The effective charges Zeff M are state dependent. For Φ < 0.1, they were determined by means of a charge renormalization procedure using the cell model solved by the PB equation.22,23 For Φ > 0.1, a force fit procedure24 using the force on the macroions obtained from MC simulations of the primitive model was used, because the charge renormalization procedure yielded a too-strong effective repulsion.25 The effective potentials at the lowest volume fraction Φ ) 0.0128 accounted also for the incomplete surfactant absorption as found by Bagger-Jo¨rgensen et al.20 The amount of free surfactant in the solution at this concentration was ≈10% of its total content, and the corresponding charges of the macroions were hence reduced by 10%. The one-component model was solved by mainly employing a Brownian dynamics (BD) simulation of 400 macroions interacting with the potential given by eq 4.1 and ignoring hydrodynamic effects. Quantities extracted were (i) radial distribution functions, (ii) structure factors, and (iii) long-time self-diffusion coefficients of the macroions. The MC and BD simulations of the primitive and onecomponent models as well as the MC simulation of the cell model were performed using the simulation package MOLSIM.26 (22) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (23) Bitzer, F.; Palberg, T.; Lo¨wen, H.; Simon, R.; Leiderer, P. Phys. Rev. E 1997, 50, 2821. (24) Walzlawek, M.; Na¨gele, G. Phys. Rev. E 1997, 56, 1258. (25) Lo¨wen, H.; D’Amico, I. J. Phys.: Condens. Matter 1997, 9, 8879. (26) Linse, P. MOLSIM, version 2.5 ed.; Lund University: Lund, Sweden.
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Osmotic Pressure and Osmotic Compressibility. The osmotic pressure was determined (i) from the MC simulations of the primitive model employing the virial route and (ii) from the PB equation applied to the cell model. The osmotic compressibilities were calculated using a differentiation of the osmotic pressure data from the numerical solution of the PB equation.27,28 Collective Diffusion. The collective diffusion coefficient was evaluated according to
DC(0) H(0) ) D0 S(0)
(4.2)
where H(q) and S(q) are the hydrodynamic function and the structure factor as obtained from the BD simulations, respectively. The value of H(q) in the long-wave limit was evaluated using its low-density expansion derived recently by Baur et al.29
j1(y) ∞ + 18Φ 1 x(g(x) - 1)(j0(xy) y j1(xy)/xy + j2(xy)/6x2) dx (4.3)
∫
H(y) ) 1 - 15Φ
where g(x) is the radial distribution function of the colloidal particles, x ) r/(2Rhc), y ) 2Rhcq, and ji are spherical Bessel functions of order i. This approach was only applicable for cases with a macroion density that was not too high. Long-Time Self-Diffusion. The long-time self-diffusion coefficients of the macroions were calculated using two different routes. In the first one, they were directly extracted from the BD simulations. However, because the hydrodynamic effects were ignored in the simulations but they are expected to be quite significant for these systems, different additional approximations were used.30 First, instead of using the (normal) free-diffusion coefficient D0 in the BD simulations, the short-time self-diffusion coefficient Dss according to24
Dss ) D0(1 - 2.59Φhc1.3)
(4.4)
was employed. This formula accounts for electrolyte friction effects and provides a good approximation for the diffusion coefficient at low macroion volume fractions and high surface charge densities.24 Hence, long-time selfdiffusion coefficients are determined as Ds ) DssD/, where D/ denotes the reduction of the self-diffusion owing to the electrostatic interactions as obtained from the BD simulations. The long-time self-diffusion coefficients were calculated using the mode coupling approximation (MCA)29 and the Medina-Noyola (MN) theory30 together with structural data from the BD simulations (or from MC simulations for the lowest macroion charge where the radial distribution function was taking nonzero contact values). The MCA approach gives the following self-consistent relation for the reduced self-diffusion coefficient D/29
[
D/ ) 1 +
(S(q) - 1)2
]
∫0∞ dqq21 + D/S(q)
1 6π2FM
-1
(4.5)
Within the MN theory, D/ is related to the radial (27) Kirchhoff, T.; Lo¨wen, H.; Klein, R. Phys. Rev. E 1996, 53, 5011. (28) Jo¨nsson, B. PBcell Computer Program; Lund University: Lund, Sweden. (29) Baur, P.; Na¨gele, G.; Klein, R. Phys. Rev. E 1996, 53, 6224. (30) Medina-Noyola, M. Phys. Rev. Lett. 1988, 60, 2705.
Figure 1. Excess Rayleigh ratio extrapolated to Θ ) 0 as a function of the total volume fraction Φ of surfactant and oil for (4) β ) 0 (data taken from ref 14), (O) β ) 0.01, (0) β ) 0.04, and (]) β ) 0.06 and calculated using the PB cell model (dashed curves, see text). The solid curve is the result of a two-parameter nonlinear least-squares fit of ∆R(0) for β ) 0 using the structure factor S(0) given by the Carnahan-Starling equation for the hard-sphere system.14
distribution function g(r) according to31
[
D/ ) 1 +
]
4πFM 6
∫0∞ drr2(g(r) - 1)2
-1
(4.6)
5. Results Osmotic Pressure and Compressibility. In Figure 1, we present the excess Rayleigh ratio extrapolated to q ) 0, ∆R(0), as a function of the total volume fraction Φ along the lower phase boundary L1/(L1 + O) for β ) 0.01, 0.04, and 0.06. Here, previously published data for β ) 0 are also included for comparison.14 ∆R(0) is related to the osmotic compressibility (∂Π/∂Φ)-1 according to
∆R(0) )
4π2nw2 ∂n 2 ∂Π Φ ∂Φ λ 4 ∂Φ 0
( ) ( )
-1 T
kBT
(5.1)
where nw and n are the refractive indices of the solvent (D2O) and the solution, respectively, and kBT is the thermal energy. ∆R(0) was also calculated theoretically using eq 5.1 with the refractive index of the solvent (D2O) nw ≈ 1.33, the refractive index increment14 ∂n/∂Φ ) 0.11, λ0 ) 488 nm, and (∂Π/∂Φ)-1 calculated by solving the Poisson-Boltzmann equation within the cell model for different volume fractions. The results from the calculations are shown as dashed curves in Figure 1. The solid curve in the figure is the result of a two-parameter nonlinear least-squares fit of ∆R(0) for β ) 0 using the structure factor S(0) given by the Carnahan-Starling equation for the hard-sphere system.14 In Figure 2, we show that the osmotic compressibility (∂Π/∂Φ)-1 calculated from the SLS data using eq 5.1 displays a monotonic decay with micelle volume fraction, which is fairly linear in the semilogarithmic scale. The (31) Schomaker, R.; Strey, R. J. Phys. Chem. 1994, 98, 1994.
Charged Spherical Microemulsion System
Figure 2. Osmotic compressibility as a function of the total volume fraction Φ measured from SLS experiments (open symbols) and calculated using the experimental osmotic pressure data (filled symbols) or PB cell model (dashed lines). The solid line describes the osmotic compressibility calculated from the osmotic pressure fit for β ) 0 using the Carnahan-Starling equation of state for a hard-sphere dispersion together with a van der Waals contribution to the osmotic pressure.20
compressibility for β ) 0.01 is roughly 3 times higher than that for β ) 0.04 and 4 times higher than that for β ) 0.06. The compressibility obtained from the PB cell calculations displays also a rather linear decrease with the volume fraction in the semilogarithmic scale. The intercept and the slope are however less than those in corresponding experimental systems. The solid line describes the osmotic compressibility calculated from the osmotic pressure fit for β ) 0 using the Carnahan-Starling equation of state for a hard-sphere dispersion together with a van der Waals contribution to the osmotic pressure.20 Figure 3 shows the measured osmotic pressure20 as well as the corresponding results from the MC simulations of the primitive model and the PB cell model. All the curves display similar trends, and the experimental pressure is well approximated by a parabola. All the theoretical results are higher than the experimental pressures at low volume fractions Φ < 0.1. At higher volume fractions, the MC results agree well with the experiment but the PB cell model underestimates the pressure. The solid curve represents a fit of the total pressure for β ) 0, ΠHS + ΠvdW, calculated using the Carnahan-Starling equation of state for a hard-sphere dispersion together with a van der Waals contribution to the osmotic pressure.20 In Figure 2, we also plotted the inverse derivative of the osmotic pressure (open symbols), which was calculated from the parabolic interpolation of the experimental data displayed in Figure 3. It is seen that these data are closer to the PB cell results than to the compressibilities measured by SLS. The simulated radial distribution functions for the charged hard-sphere systems are presented in Figure 4. The curves show that the particles are well separated by the Coulombic repulsion. The initial rises of the distribution appear at distances larger than the hard-sphere diameter 150 Å, and the first maximums appear at the mean interparticle distance except for the lowest charge data at Φ > 0.1 in Figure 4a where we find a nonzero count of collisions. The structure is increasing with increasing micelle charge. Moreover, it is seen that the
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Figure 3. Osmotic pressure relative to pure water as a function of the total volume fraction Φ measured from osmometric experiments in ref 20 (open symbols), MC simulation of the two-component charged hard-sphere model (filled symbols), and the PB cell model (dashed curves). The solid curve represents a fit of the total pressure for β ) 0, ΠHS + ΠvdW, calculated using the Carnahan-Starling equation of state for a hard-sphere dispersion together with a van der Waals contribution to the osmotic pressure.20
systems corresponding to β ) 0.04 and β ) 0.06 have a pronounced structure already at Φ ) 0.01, indicating the presence of repulsive long-range interactions between the micelles. Collective Diffusion. The normalized collective diffusion coefficients Dc/D0 (where D0 is the diffusion coefficient at infinite dilution) were obtained from the DLS experiments describing the dynamics of concentration fluctuations. Figure 5 shows the collective diffusion coefficients together with previous data for β ) 0.14 In the uncharged system, the collective diffusion coefficient shows a monotonic but weak increase with increasing concentration. However, the behavior is different for the charged systems. Here, the diffusion coefficient shows a maximum at lowest volume fraction Φ ≈ 0.01 and then decreases with increasing concentration. The explanation for the observed maximum is that at very low concentrations the hydrodynamic effect is negligible and the collective diffusion coefficient can then be written as Dc/D0 ) 1/S(0). The structure factor, S(0), is in turn proportional to (∂Π/∂Φ)-1, causing an initial increase in Dc/D0. At a higher concentration, the hydrodynamic interactions begin to affect the system strongly, reducing the collective diffusion coefficient. At a high volume fraction, we observe a kink and a slight increase in the collective diffusion coefficient. Theoretically, we evaluated the collective diffusion coefficients only for the lowest concentrations Φ e 0.05 using eqs 4.2 and 4.3 (see Figure 5). At higher volume fractions, eq 4.3, which is the first-order expansion to the hydrodynamic function, is too rough as it gives unphysical negative values in low q limit. Calculated hydrodynamic functions are presented in Figure 6. Equation 4.2 predicts a maximum on the concentration dependence of Dc/D0 at Φ ≈ 0.01 for all three charged systems, and its height increases with the charge. The height of the maximum for β ) 0.04 and 0.06 amounts to 25-30 (not shown), which is higher than we can expect from extrapolation of the experimental data (≈10).
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Figure 5. Normalized collective self-diffusion coefficients Dc/ D0 as a function of the total volume fraction Φ for (4) β ) 0 (data taken from ref 14), (O) β ) 0.01, (0) β ) 0.04, and (]) β ) 0.06. Results from the charged hard-sphere model for (b) β ) 0.01, (9) β ) 0.04, and ([) β ) 0.06 are also given.
Figure 4. Simulated micelle radial distribution functions for the hard-sphere systems with charges (a) ZM ) -14.5 (corresponding to β ) 0.01), (b) ZM ) - 58 (β ) 0.04), and (c) ZM ) - 87 (β ) 0.06) at the indicated total volume fractions.
Self-Diffusion. The normalized long-time self-diffusion coefficients, Ds/D0, of surfactant and oil obtained along the emulsification failure boundary are presented in Figure 7 as a function of the total volume fraction Φ. The self-diffusion coefficients are normalized with respect to the same D0(T) as the collective diffusion coefficients. These D0(T) values were calculated using eq 3.2 and D0(T ) 23.5 °C) ) 2.1 × 10-11 m2 s-1. The time scale of the self-diffusion experiments is of order of 0.1 s, which is much longer than the structural relaxation time in the solutions. Hence, the self-diffusion coefficients measured here correspond to long-time selfdiffusion coefficients. As expected, Ds/D0 decreases with increasing concentration and charge on the micelles. The system with the lowest charge (β ) 0.01) exhibits quantitatively the same diffusive properties as the uncharged system except at the highest concentrations. The results of the BD simulations, the MCA theory, and the MN theory for the charged hard-sphere system
Figure 6. Hydrodynamic functions from the model systems with different micellar charges: ZM ) -14.5 (solid curve), -58 (dotted curve), and -87 (dashed curve) at the indicated hardsphere volume fractions as calculated using eq 4.3.
are compared with the experimental data in Figure 8. In contrast with the data in Figure 7, the theoretical models predict that Ds/D0 is considerably lower than 1 already at Φ ) 0.01 and gradually decreasing on increasing Φ. All the theoretical curves have similar slopes but the magnitudes are different. The BD simulations predict the lowest Ds/D0 values, the MCA results are slightly higher, and the MN theory predicts the largest ones. Despite significant approximations made, the predicted selfdiffusion coefficients Ds/D0 at moderate volume fractions 0.1 < Φ < 0.2 are in reasonable agreement with the experimental data.
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Figure 7. Normalized self-diffusion coefficients Ds/D0 as a function of the total volume fraction Φ of (2) surfactant and (4) oil for β ) 0, (b) surfactant and (O) oil for β ) 0.01, (9) surfactant and (0) oil self-diffusion for β ) 0.04, and ([) surfactant and (]) oil for β ) 0.06. The dashed curves are obtained using eq 6.6 with the [η] values obtained by the previous fit to the viscosity coefficients (at Φm ) 0.63).
Low-Shear Viscosity. The relative low-shear viscosity, η/η0, where η is the viscosity of the solution and η0 is the solvent (D2O) viscosity, was measured as a function of the total volume fraction, Φ, and the results are presented in Figure 9. The viscosity increases with increasing volume fraction of micelles and with increasing micellar charge. The system with the lowest charge β ) 0.01 has a similar behavior to the uncharged system, as in the case with the self-diffusion. 6. Discussion Static Properties. First, one can see that the osmotic pressure results shown in Figure 3 indicate the presence of long-range interactions between the microemulsion micelles. Because (i) the pressure in the charged systems is significantly higher than in the uncharged one and (ii) the two-component model with only Coulombic interactions predicts correct Π values at high volume fractions, we conclude that the increase of the pressure is due to screened Coulombic repulsion between the micelles. We also note a peculiarity which is not seen in the scale of Figure 3. At low volume fractions, the theories predict a higher pressure than what is observed experimentally. This observation was made by Bagger-Jo¨rgensen et al.20 and was interpreted in terms of incomplete surfactant adsorption. The fraction of adsorbed surfactant was estimated from the PB cell model as pmic ≈ 0.9-0.95 at volume fractions of Φ ≈ 0.01. At the same time, it is known that the osmotic pressure is determined mainly by the micelle-counterion interactions and local counterion distribution, which are accurately accounted for in the cell model.21 Hence, the agreement in the pressure does not guarantee the accuracy of the predicted micellemicelle correlations and related properties. Indeed, the data shown in Figure 2 indicate that the osmotic compressibility, which is determined by the micelle-micelle correlations, is underestimated by the
Figure 8. Normalized self-diffusion coefficient Ds/D0 as a function of Φ at different surface charge densities for (a) β ) 0.01, (b) β ) 0.04, and (c) β ) 0.06 as obtained from NMR experiments (symbols), BD simulations (solid curves), the MCA theory (dotted curves), and the MN theory (dashed curves).
PB cell model at low volume fractions. Thus, the chargedsphere model overestimates the correlations. At the same time, the compressibility extracted from the experimental structure factor S(0) is influenced by polydispersity that increases the compressibility.32 Dynamic Properties. Low-Shear Viscosity. It was previously shown that uncharged microemulsion systems are described very well by the hard-sphere model.14,33 In the hard-sphere model, the flow is affected only by hydrodynamic (viscous) interactions and Brownian motion. The viscosity of a hard-sphere system at high dilution is given by the Einstein relation34 (32) D’Aguanno, B.; Klein, R. J. Chem. Soc., Faraday Trans. 1991, 87, 379. (33) Leaver, M. S.; Olsson, U. Langmuir 1994, 10, 3449. (34) Einstein, A. Investigation on the Theory of Brownian Motion; Dover: New York, 1956.
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Figure 9. Relative viscosity η/η0 as a function of the total volume fraction Φ for (4) β ) 0, (O) β ) 0.01, (0) β ) 0.04, and (]) β ) 0.06. The solid curve is calculated according to eq 6.3, and the dashed curves are calculated from eq 6.2 with fitted [η] values at Φm ) 0.63.
η/η0 ) 1 + 2.5Φ
(6.1)
At higher concentrations, interactions affect the rheological behavior. There is yet no accurate theory describing the viscosity of concentrated dispersions, because it is a many-body problem. However, simple semiempirical relations such as the Krieger-Dougherty relation35
(
)
ΦHS η/η0 ) 1 Φm
h exp(E//kT) b3
)
ΦHS Φm
-2
(6.3)
have been shown to describe experimental data reasonably well. Here, Φm is an empirical constant and [η] is the intrinsic viscosity. Fitting experimental data with eq 6.2 (dashed curve), we have used the intrinsic viscosity value [η] as the fitting parameter, because it described the data better than the fit obtained with Φm as the fitting parameter. De Kruif and co-workers37 found that the rheological behavior of submicron-size silica spheres could be nicely described by eq 6.3 with Φm ) 0.63. This volume fraction is very close to Φm ) 0.64, which is the volume fraction for the random close packing of the spheres. Recent experiments on monodisperse hard-sphere colloids have indicated that Φm should perhaps be smaller. Hard spheres have a glass transition at Φ ) 0.58, and it is rather at this volume fraction that the viscosity should diverge.38,39 (35) Krieger, I. M.; Dougherty, T. J. Trans. Soc. Rheol. 1959, 3, 137. (36) Quemada, D. E. In Lecture Notes in Physics Stability of Thermodynamic Systems; Springer-Verlag: Berlin, 1982. (37) van der Werff, J. C.; de Kruif, C. G. J. Rheol. 1989, 33, 421. (38) Meeker, S. P.; Poon, W. C. K.; Pusey, P. N. Phys. Rev. E 1997, 55, 5718.
(6.4)
where h is Planck’s constant, b is the center-to-center particle separation, and E/ is an activation energy. Self-Diffusion. It has previously been shown that charged hard-sphere dispersions do not obey the generalized Stokes-Einstein relation (GSE)41,44 which holds for uncharged hard spheres:
(6.2)
or the Quemada relation
(
∆ξ )
Ds(Φ) )
-[η]Φm
36
η/η0 ) 1 -
Leaver and Olsson33 found that the uncharged micellar microemulsion however could be described accurately by eq 6.3 with Φm ) 0.63, just as for the data of de Kruif et al.37 That can be seen in Figure 9, where the solid curve has been calculated using eq 6.3. It is evident from Figure 9 that the Krieger-Dougherty model as well as the Quemada relation describes even the charged system quite reasonably. The fact that Φm ) 0.63 turned out to describe our data better than the more physical value Φm ) 0.58 can possibly be due to the polydispersity. Now, if we “insert” negatively charged DS- ions into the micelles, the long-range repulsion between the spheres leads to a stronger dependence of η/η0 on concentration with increasing SDS concentration. Similar behavior was observed for the charged silica particles by Krieger and Eguiluz40 and by Imhof et al.41 However, it remains to be further investigated whether the Krieger-Dougherty model alone can be applied to the charged spheres. Krieger suggested that two other fully independent variables (ZM for particle charge and c for added ionic concentration) should be present in the equation of state compared to the corresponding hardsphere system.42 Goodwin43 suggested a model where he adds the extra contribution (∆ξ) of repulsion to the zeroshear-rate viscosity over and above the usual KriegerDougherty contribution
kT 6πηγf0(Φ)a
(6.5)
where a is the sphere radius, implying Ds/D0 ) (η/η0)-1. For hard-sphere systems, (Ds/D0)×(η/η0) ≈ 1 over a large range of volume fractions.45,46 For charged colloids, however, (Ds/D0)×(η/η0) ≈ 1 holds only for low volume fractions, whereas a slightly modified relation introduced by Imhof et al.41 seems to provide a reasonable data description at higher concentrations. Therefore, we have plotted eq 6.6 in Figure 7 (dashed curves) using Φm ) 0.63 and intrinsic viscosity values [η], obtained in the previous fit.
(
Ds/D0 ) 1 -
)
ΦHS Φm
[η]Φm
(6.6)
Good agreement is seen between eq 6.6 and the experimental data at lower concentrations; however, at higher (39) Phan, S. E.; Russel, W. B.; Cheng, Z.; Zhu, J.; Chaikin, J.; Dunsmuir, H.; Ottewill; R. H. Phys. Rev. E 1996, 54, 6633. (40) Krieger, I. M.; Equiluz, M. Trans. Soc. Rheol. 1976, 20, 29. (41) Imhof, A.; van Blaaderen, A.; Maret, G.; Mellema, J.; Dhont, J. K. G. J. Chem. Phys. 1994, 100, 2171. (42) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111. (43) Goodwin, J. The rheology of colloidal dispersions. In Solid/Liquid Dispersions; Tadros, Th., Ed.; Academic Press: London, 1987. (44) Banchio, A. J.; Bergenholz, J.; Na¨gele, G. Phys. Rev. Lett. 1999, 82, 1792. (45) van Bladeren, A.; Peetermans, J.; Maret, G.; Dhont, J. K. G. J. Chem. Phys. 1992, 96. (46) Olsson, U.; Schurtenberger, P. Prog. Colloid Polym. Sci. 1997, 104, 157.
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Figure 10. (Ds/D0)×(η/η0) as a function of the total volume fraction Φ for (2) β ) 0, (b) β ) 0.01, (9) β ) 0.04, and ([) β ) 0.06.
volume fractions the relative diffusion coefficient has a slightly weaker dependence on volume fraction than the relative viscosity does. The same deviation was observed by Imhof et al. for dispersions of charge-stabilized silica particles at high salt concentrations.41 Banchio et al.44 found as well that GSE did not hold for the chargestabilized suspension with a surface charge density ≈ 1570 Å2/e. However, in our system surface charge densities are lower, except for the β ) 0.06 system (1620 Å2/e) for which GSE did not hold either, almost within the whole micellar concentration range. For better visualization of the above-mentioned relation, we have plotted the product (Ds/D0)×(η/η0) as a function of the total volume fraction Φ in Figure 10. From the figure, we see that the product (Ds/D0)×(η/η0) remains close to unity up to Φlim ≈ 0.35 for β ) 0 (as was found by Olsson and Schurtenberger17); Φlim ≈ 0.35 also for β ) 0.01, Φlim ≈ 0.30 for β ) 0.04, and Φlim ≈ 0.12 for β ) 0.06. Here, once again we can clearly see that in all the cases microemulsion systems at higher concentrations do not obey the GSE relation any longer. Furthermore, Φlim decreases with increasing sphere charge (hence, with increasing long-range interactions). The values of Ds/D0 at Φlim are approximately 0.2-0.3 independent of micelle charge. Imhof et al.41 explained the measured data for a charged silica dispersion at high concentrations by representing the friction experienced by a particle as a sum of (i) the normal Stokes friction due to the solvent molecules and (ii) a viscosity increase of η - η0 because of neighboring particles. They pointed out that for the latter part there is no reason to expect a no-slip boundary condition because the particle is not surrounded by a stationary shell of neighboring micelles. Thus, the friction can be described as
f ) 6πη0a + νπ(η - η0)a
(6.7)
Here, ν is expected to have a value between 4 and 6, because for a slip boundary condition f should contain a factor of 4π which for no slip would be 6π. Using the
Figure 11. (D0/Ds - 1)/(η/η0 - 1) as a function of the total volume fraction Φ for (2) β ) 0, (b) β ) 0.01, (9) β ) 0.04, and ([) β ) 0.06.
Einstein relation Ds ) kT/f, Imhof obtains after rearrangement
D0/Ds - 1 ) ν/6 η/η0 - 1
(6.8)
Then, we set ν ) 4 and use eq 6.8 together with the previously measured η/η0 values to calculate Ds/D0. In this way, we obtained a better correspondence with the measured self-diffusion coefficients for a large range of volume fractions. Figure 11 shows the data points plotted as (D0/Ds - 1)/(η/η0 - 1) versus Φ. At this point, we should also notice that at the lowest diffusion coefficients (e10-13 m2 s-1) Dsurfactant > Doil (see Figure 7). This discrepancy arises from the fact that the observed diffusion coefficient Dobs consists of two main contributions:
Dobs ) pmicDmic + (1 - pmic)Dfree
(6.9)
where Dmic and Dfree represent the translational diffusion coefficients of micellized and free surfactant, respectively, and pmic is the fraction of micellized surfactant.47 Selfdiffusion rates for a monomeric surfactant are typically more than an order of magnitude higher than D0. However, because of the very poor solubility of the surfactant monomers in water, pmic is much higher than pfree. Therefore, the free surfactant contribution to the observed self-diffusion coefficient can be neglected. Yet, at very high micellar concentrations the micellar self-diffusion coefficient becomes so low (10-13-10-14 m2 s-1) that the pmicDmic term begins to be comparable with (1 - pmic)Dfree. Hence, at high concentrations we observe a higher diffusion coefficient due to the monomer contribution. The same is valid even for the oil. Anyway, monomeric contribution there is not so pronounced because the free monomer concentration of decane is extremely low because of its very poor solubility in water and very good solubility in (47) So¨derman, O.; Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1994, 26, 445.
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Figure 12. (a) The inverse of normalized self-diffusion coefficients (0) (Ds/D0)-1 and relative low-shear viscosity (O) (η/η0) and (b) the inverse of osmotic compressibility (O) (∂Π/∂Φ) as a function of the micellar charge ZM at Φ ) 0.189.
the micellar interior. For this reason, Ds/D0 for decane and not for C12E5 has been plotted in Figure 10. By now, we can make a generalization and conclude that concentration dependence of such transport properties as viscosity and self-diffusion increase with an increasing micellar charge, because of enhanced long-range electrostatic interaction between micelles. On the contrary, the concentration dependence of osmotic compressibility and collective diffusion decreases with increasing charge number, because of increasing charge density with increasing counterion condensation (Figure 12). The above enumerated similarities in rheological and diffusional behavior of charged silica particles and our charged microemulsion system indicate that an easily prepared charged microemulsion can probably serve as a model for a charged hard-sphere system.
Evilevitch et al.
Comparison with Charged-Sphere Model. The initial drop in the self-diffusion seen in the model data at very low volume fractions is caused by the screened Coulombic interactions, which promote pronounced long-range order in the system and effective friction hampering of the Brownian motion. A similar or even larger drop is predicted by the MCA calculations29,48 and supported by BD simulations and experiments for highly charged latex systems.23,27 The surface charge density in the studied microemulsion system (270-1620 Å2/e) is higher than that used in refs 23, 29, and 27 (5000-8000, 5600, 1800 Å2/e). At the same time, according to Imhof et al.41 this drop is not observed for silica suspensions with surface charge densities (2600-12000 Å2/e) in the presence of salt. We note that the concentration behavior of the self-diffusion in the present work and ref 41 resemble diffusion in hardsphere suspensions without charge rather than the results for charged colloids. The comparisons made between the microemulsion system and the charged-sphere model show that at high dilution the latter has (i) higher osmotic pressure, (ii) lower osmotic compressibility, (iii) a higher short-time collective diffusion coefficient, and (iv) lower long-time self-diffusion coefficients. All these differences can be clearly attributed to the fact that the monodisperse charged-sphere model overestimates the effective interactions in the diluted charged microemulsions. We can speculate about the possible reasons for this modeling error. First of all, the degree of size polydispersity of the analogous uncharged microemulsion is estimated to be ≈16%,49 that is, the system is not perfectly monodisperse. As we already noticed, assumption of a symmetric micelle size distribution (e.g., of a Schultz type) with fixed colloid volume fraction leads to decreases in spatial correlation between the micelles and in the apparent electrostatic interaction. Second, an incomplete surfactant adsorption can lead to a difference between the charged-sphere model and the microemulsion. This implies not only a lower micelle charge but also an enhanced screening of the intermicellar interaction due to the presence of additional charges in the solution. We should mention however that other data obtained for lamellar phases with a similar composition give evidence of the nearly complete surfactant adsorption.31 We performed an additional series of calculations to estimate the effect of polydispersity combined with the incomplete surfactant adsorption. We note that for the charged microemulsions mainly charge polydispersity (although related to the size polydispersity) influences the system properties. Electrostatic repulsion keeps particles apart from each other and conceals the effect of the size polydispersity itself. We accounted for charge polydispersity according to the method used in refs 32, 50, and 51. In general, we found that the introduction of polydispersity indeed decreases the intermicellar correlation and thus leads to increases in compressibility and self-diffusion coefficients and reduction of collective diffusion. Assuming a size polydispersity of 16% for all experimental conditions, we get significant improvement of the theoretical results relative to the experimental ones for low volume fractions Φ < 0.1, whereas for Φ > 0.1 the agreement worsens with introducing polydispersity. A straightforward explanation of this fact is based on (48) Na¨gele, G.; Baur, P. Europhys. Lett. 1997, 38, 557. (49) Bagger-Jo¨rgensen, H.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 1413. (50) Phalakornkul, J.; Gast, A.; Pecora, R.; Na¨gele, G. Phys. Rev. E 1996, 54, 661. (51) Na¨gele, G. Phys. Rep. 1996, 272, 215.
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the intermicellar interaction, so that the apparent micelle charge is much lower than the expected one. Figure 13 shows the apparent micelle charge values obtained from a fit of the self-diffusion coefficient data calculated for a monodisperse model with adjustable charge to the experimental Ds/D0. One can see, for example, that at the volume fraction Φ ≈ 0.05 the apparent charge amounts to half of the nominal one, which is unreasonably low for the microemulsion system. The exact reason for this deviation can be given after analysis of structure data for these microemulsions as obtained by neutron scattering, which can give more direct evidence of actual interactions in the system and is now on the way. 7. Concluding Remarks
Figure 13. Apparent micelle charge obtained from a fit of the BD self-diffusion experiment for the charged hard-sphere model to the experimental data for charged microemulsions at different surface charge densities (solid curves). Also shown are the values calculated using the cell model calculations and the charge renormalization procedure (dashed curves).
limitations of the scheme used to account for hydrodynamic interactions. First, eq 4.4 for the short-time dynamics was derived for dilute systems, whereas for concentrated ones it probably leads to a progressively underestimation of hydrodynamic interactions and hence an overestimated self-diffusion. Second, the MCA expressions were derived for pairwise-additive hydrodynamic interactions, which can be far from reality for concentrated microemulsions. On the other hand, one cannot rule out that actual charge polydispersity is indeed as low as a few percent at volume fractions higher than 0.1 and increases dramatically on dilution, which can be related to an increase in the size polydispersity (the latter can be due to the higher temperature at which the measurements were performed). The increase of polydispersity on dilution was predicted for microemulsions52,53 but comes in contrast with the previous data obtained on nonionic systems.49 We note that the concentration dependence of charge polydispersity in microemulsions as well as its relationship with size polydispersity are not known and have not even been addressed in the literature. Finally, the discrepancy between the experiment and the charged-sphere model could be also influenced by impurities. The presence of even a small amount of salt impurity enhances the screening considerably at low concentrations. Taken together, the polydispersity, incomplete adsorption, and salt impurity seriously affect (52) Sicoli, F.; Langevin, D.; Lee, L. T. J. Chem. Phys. 1993, 99, 4759.
We have studied water-rich weakly charged spherical microemulsion micelles in order to describe their thermodynamic and hydrodynamic properties and to see whether this system obeys the behavior of a charged stabilized spherical colloidal system. We found that the self-diffusion coefficients and the viscosity coefficients obey the generalized Stokes-Einstein relation at lower concentrations, where Φlim is charge dependent. However, at higher concentrations the relation can still be used upon the addition of an extra frictional factor due to stronger interactions, especially in the charged case. The same behavior has been previously found for interacting silica particles as well. The collective diffusion coefficient has been measured by dynamic light scattering. The results differ completely from the hard-sphere collective diffusion by showing a maximum, which was absent in the uncharged case. This result is in qualitative agreement with predictions based on a charged hard-sphere model with hydrodynamic interactions. We found that static and dynamic properties of the microemulsion display pronounced effects of the charge. Increase in the charge causes stronger ordering, lower osmotic compressibility, faster collective diffusion, and slower self-diffusion of microemulsion micelles which qualitatively resembles the behavior of the charged hardsphere model. At high volume fractions, Φ > 0.1, the agreement is quantitative, whereas at Φ < 0.1 the effect of the charge is smaller than expected from the charged hard-sphere model. The source of this discrepancy can be attributed to a combination of particle size and charge polydispersity, incomplete surfactant adsorption, and salt impurities which are all present in this system. Acknowledgment. This work was supported by the Swiss National Science Foundation. LA0011883 (53) Gradzienski, M.; Langevin, D.; Farago, B. Phys. Rev. E 1996, 53, 3900.
