Langmuir 1996,11, 1559-1570
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Viscosity, Microstructure, and Interparticle Potential of AOTIHaOln-DecaneInverse Microemulsions Johan Bergenholtz, Aldo A. Romagnoli, and Norman J. Wagner" Colburn Laboratory, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received July 3, 1994. I n Final Form: February 21, 1995@ Single-phase AOT/HzO/n-decaneinversemicroemulsions have been studiedby using capillary viscometry and small-angle neutron scattering (SANS). This ternary mixture is treated as a polydisperse colloidal suspension,where the established overlap potential for the droplet interaction is approximated by a squarewell interaction. Viscosity measurements on both dilute and concentrated microemulsions show an anomalous maximum with increased swelling of the droplets. Predictionsfor the dilute viscosity from the effective interaction parameters extracted from the microemulsion structures measured by SANS also show a maximum with swelling, but of a smaller magnitude. The viscometry and the SANS measurements are combined to extract the interparticle potential. Comparison shows, however, that two different sets of interaction parameters are extracted, a consequence of the approximate nature of the square-well colloidal model. A molecular mechanism is proposed to explain the viscosity anomaly, whereby disorder at the surfactant/water interface leads to increased overlap and, hence, stronger effective interactions. The results are discussed within the context of earlier work on the percolation transition in these inverse microemulsion systems.
Introduction Under certain conditions mixtures of surfactant, water, electrolyte, oil, and cosurfactant form thermodynamically stable, isotropic, and optically transparent solutions called microemulsions. These show a wide variety of microstructures throughout the phase diagram. In particular, an inverse or water-in-oil microemulsion is formed, consisting of colloidal sized (-100 A) droplets distributed in the continuous oil phase, when the stabilizingsurfactant layer curves toward the water microphase. The doubletailed, anionic surfactant AOT (sodium bis(2-ethylhexy1)sulfosuccinate) together with water and an oil packs into ternary inverse microemulsions requiring neither electrolyte nor cosurfactant for thermodynamic stability. For this reason and the fact that the single inverse microemulsion phase persists over wide ranges of temperature and composition, this system is frequently quoted as a prototypical inverse microemulsion. This particular system has been the subject of extensive study. The equilibrium phase behavior as a function of composition and temperature is well-known (see, for instance, Kotlarchyk et a1.l and Toprakcioglu et a1.2).The equilibrium microstructure has also been established by a variety of techniques, revealing that the surfactant is present primarily in monolayers around spherical water cores with the hydrophobic tail-groups of the surfactant penetrating radially outward into the oil phase.lz3s4 In addition, numerous studies have demonstrated a linear relationship between the radius of the dispersed droplets and the water-to-surfactant molar ratio, defined as X = [H20]/[AOT].5-8 Dilution with oil leaves the radius unchanged, thus allowing for independent control of both
* Author to whom correspondence should be addressed. e-mail
[email protected]. @Abstractpublished in Advance A C S Abstracts, May 15, 1995. (1)Kotlarchyk,M.;Chen, S.-H.;Huang,J. S.;Kim,M.W. Phys. Rev. A . 1984.29.2054. (4)Kotlarchyk, M.; Huang R.9 --, 42x2
( 5 ) Day, R. A.; Robinson, B. H.; Clarke, J. H.; Doherty, J. V. J . Chem. Soc., Faraday Trans. 1 1979,75, 132.
(6) Zulauf, M.; Eicke, H.-F. J . Phys. Chem. 1979,83, 480.
droplet size and c o n ~ e n t r a t i o n This . ~ ~ ~colloidal ~~ droplet structure persists even when the system crosses the socalled cloud-point curve,l a lower critical solution temperature (LCST) where the solution segregates into two coexisting microemulsion phases. This phase transition is analogous to a gas-liquid phase transition in simple fluids and is thought to be driven by attractive interactions that are entropic in nature. Even far from the critical point and phase boundaries the attractive interactions result in deviations from hard-sphere behavior, as found in previous viscosity and small-angle scattering studies.10,11 The source of these attractive interactions is the desolvation and interpenetration of alkyl segments on the surfactant tails of neighboring aggregates.12J3 The resulting short-range attractive potential has been approximated by a square-well potential, where the attractive well depth is postulated to scale linearly with the radius of the aggregates.14 While deemed successful for the modeling of equilibrium properties, only a few studies have tested the applicability of the square-well potential for the modeling of transport properties of these inverse micro emulsion^.^^-^^ Therefore, one purpose of this study is to determine the applicability of the square-well potential for modeling of both the transport and equilibrium properties of the model system AOT/HzO/n-decane. (7) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S. J . Phys. Chem. 1982, 86. 3273.
(8) Robinson, B. H.; Toprakcioglu, C.; Dore, J. C. J . Chem. SOC., Faraday Trans. 2. 1984,80, 13. (9) Jahn, W.; Strey, R. J.Phys. Chem. 1988,92,2294. (10)Bedwell, B.; Gulari, E. J . Colloid Interface Sci. 1984,102, 88. (11)Huane. J.S.:Safran,S.A.:Kim.M. W.:Grest,G.S.:Kotlarchvk. . . _ . M.; Quirke, fi: Phys. Reu. Lett. 1984,'53,592. (12) Calje, A. A,; Agterof, W. G. M.; Vrij, A. In Micellization, Solubilization, and Microemulsions;Mittal, K., Ed.; Plenum Press: New York, 1977; Vol. 2, p 779. (13)Lemaire, B.; Bothorel, P.; Row, D. J . Phys. Chem. 1983,87, 1023. (14) Huang, J. S.J . Chem. Phys. 1985,82,480. (15)Finsy, R.; Devriese, A.;Lekkerkerker, H. J . Chem. Soc., Faraday Trans. 2 1980,76, 767. (16) Kim, M. W.; Dozier, W. D.; Klein, R. J . Chem. Phys. 1986,84, 591 R
(17) Genz, U.; Dhont, J. K. G.;Klein, R. Progr. Colloid Polym. Sci. 1987,73, 142.
0743-7463/95/2411-1559$09.00/0 0 1995 American Chemical Society
1560 Langmuir, Vol. 11, No. 5, 1995
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"i : 401
8
j
7lP 2o
1 I
OO
I
0
0
0
?
10
20
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40
X=[D?O]/[AOT]
Figure 1. Relative viscosity of AOTIDzOln-decane inverse microemulsions as a function of molar ratio at temperatures of 25.00 ( 0 )and 20.00 & 0.02 "C(0) for a fured droplet volume fraction of 40%. The line representsan estimateof the expected qualitative behavior based on literature scalings for the potential parameters. A transport property of both theoretical and practical interest is the microemulsion viscosity and its variation with both composition and temperature. It has been observed that at low molar ratios there is an anomalous maximum in the viscosity with increasing molar ratio at fixed droplet volume fraction.18-20 This maximum has been observed in connection with percolation studies and has accordingly been ascribed to the relative proximity of the microemulsion composition to the percolation threshold in the ternary phase diagramaZ0This percolation threshold has been identified with dramatic increases in c o n d u c t i ~ i t y ~and ~-~ a~ maximum in d In vId4 with concentration at fixed molar ratio.20,24,25 For example, Figure 1shows that the microemulsion viscosity, normalized by the solvent viscosity, increases by a factor of 5 with increased swelling ofthe microemulsion droplets at a fxed droplet volume fraction (4 = 0.40). This behavior is anomalous in that a predictive theory based on the squarewell colloidal together with the proposed linear scaling of the well depth with molar ratio, predicts that the viscosity is a monotonically increasing function of composition at fixed volume fraction (solid line, Figure 1). In terms of molecular characterization, this particular regime of the phase diagram has been mostly overlooked as previous studies on the AOTIHzOln-decanesystem have primarily focused on micelles and swollen systems at higher molar ratios (XL 41). Note also, as shown in Figure 1, the observed maximum in relative viscosity grows with decreasing temperature, while at higher molar ratios the reverse is true. This latter behavior has been studied p r e v i o u ~ l y ,while ~ ~ ~ ~the ~ presence of the former has apparently escaped notice except by a few investigat o r ~ . ~ ~ , ~ ~ In addition to this viscosity maximum, several other properties are known to change irregularly at low molar ratios. The apparent molar volume ofthe emulsified water has been found to deviate strongly from the bulk (18)Rouviere, J.; Couret, J.-M.; Lindheimer, M.; Dejardin, J.-L.; Marrony, R. J. Chim. Phys. 1979,76,289. (19)Huang, J. S. Private communication. (20)Peyrelasse, J.; Moha-Ouchane, M.; Boned, C. Phys. Reu.A 1988, 38,4155. (21)Lagiies, M.; Ober, R.; Taupin, C. J . Phys. Lett. 1978,39,L487. (22)Bhattacharya, S.; Stokes, J. P.; Kim, M. W.; Huang, J . S. Phys. Reu. Lett. 1985,55, 1884. (23)Kim, M. W.; Huang, J. S. Phys. Reu. A 1986,34, 719. (24)Peyrelasse, J.; Boned, C.; Saidi, Z. Progr. Colloid Polym. Sei. 1989,79,263. (25)Mathew, C.; Saidi, 2.; Peyrelasse, J.; Boned, C. Phys. Reu. A 1991,43, 873. (26)Bergenholtz, J.; Wagner, N. J. Ind. Eng. Chem. Res. 1994,33, 2391. (27)Berg, R.F.; Moldover, M. R.; Huang, J. S. J . Chem. Phys. 1987, 87,3687. (28)Majolino, D.; Mallamace, F.; Micali, N. Solid State Commun. 1990,74, 465.
va1ue,3,5,29,30 Further, studies have revealed that the molar head-group area of the surfactant initially decreases and then increases to a plateau as the microemulsion is swollen.3~30-32 This quantity is of importance in characterizing surfactant systems and plays an integral part in equilibrium packing model^.^^^^^ Also, the droplet refractive index becomes equal to that of the solvent at X x 18 for AOT/HzOln-decane due t o the fact that the polarizability of AOT is intermediate between those of the water and oiL6 This has been exploitedwith laser light scattering to study purely self-diffusional modes35 and also to accurately determine the size polydispersity of the dropl e t ~A . result ~ ~ of this study of relevance here is that the size polydispersity of these inverse microemulsion systems has been found to be independent of composition and to lie in the range 0.10 < so < 0.15,36,37 which is substantially more monodisperse than previously thought. These fundamental studies form a wealth of knowledge serving as a foundation for designing technologically important formulations. Inverse microemulsions are, for instance, candidates for applications in enhanced oil recovery,38carrier-mediated transport,39protein separation,4O and b i o ~ a t a l y s i s . ~ Therefore, ~ a more detailed understanding of the behavior of this model inverse microemulsion at low molar ratios is warranted. The goal of this paper is to investigate the strong sensitivity of the microemulsion viscosity to composition and, in particular, to elucidate the origin of the viscosity anomaly. The approach will be to treat the AOTIHzOln-decane inverse microemulsion phase as a colloidal suspension of squarewell particles, whose size, polydispersity, and interaction potential are determined from the molecular properties governing self-assembly. By viewing the inverse microemulsion phase as a colloidal suspension we can employ the well-developed theoretical and experimental techniques of colloid science to link the anomalous viscosity behavior to changes in the interparticle potential. In the following we show that dilution viscometry relates the viscosity anomaly directly to increased interparticle attractions. Further, SANS measurements and statistical mechanical models for polydisperse square-well fluids relate the square-well depth and particle radius to composition. This combination of thermodynamic and dynamic probes of the square-well interaction parameters also rigorously tests the applicability of the square-well colloidal model. As will be shown, this colloidal level approach leads to a further understanding of the molecular structure of the inverse microemulsion droplet and, in addition, provides a molecular explanation for previous correlative attempts to explain the viscosity anomaly and shifts in the percolation threshold. (29)Mathews, M. B.; Hirschorn, E. J . Colloid Interface Sei. 1963,8, 86. Mandell, L.; Fontell, K. J . Colloid Interface Sci. 1970, (30)Ekwall, P.; 33, 215. (31)Eicke, H.-F.; Rehak, J . J.Helu. Chim. Acta 1976,59,2883. (32)Maitra, A.J . Phys. Chem. 1984,88,5122. (33)Mitchell, D.J.;Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981,77,601. (34)Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1985. (35)Ricka, J.; Borkovec, M.; Hofmeier, U.; Eicke, H.-F. Europhys. Lett. 1990,11, 379. (36)Ricka, J.; Borkovec, M.; Hofmeier, U. J . Chem. Phys. 1991,94, 8503. (37)Almgren, M.; Johannsson, R.; Eriksson, J. C. J. Phys. Chem. 1993,97,8590. (38)Shah, D. 0. Surface Phenomena in Enhanced Oil Recouery; Plenum Press: New York, 1981. (39)Hebrant, M.; Mettelin, P.; Tondre, C.; Joly, J.-P.; Larpent, C.; Chasseray, X. Colloid Surf.A 1993,75, 251. (40)Leser, M. E.;Mrkoci, K.; Luisi, P. L. Biotechnol. Bioeng. 1993, 41, 489. (41)Stamatis, H.; Xenakis, A,; Provelegiou, M.; Kolisis, F. N. Biotechnol. Bioeng. 1993,42, 103.
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To proceed, we first review the underlying colloidal forces governing the structure of inverse microemulsions and discuss the theoretical models necessary for the ensuing analysis and discussion of the viscometric and scattering results. After presentingresults and discussing their interpretation, we conclude with some remarks regarding suggestions for future studies.
In the actual system, polydispersity in size dictates that the square-well interaction parameters will depend upon each individual pair of interacting particles. However, in the following we will simplify matters by only considering interactions in terms of an average well depth uo and a fixed range A, a common a p p r o x i m a t i ~ n . ~ ~ ~ ~ ~ Dilute-LimitingViscosity of the Square-WellColloidal Fluid. The dilute-limitingviscosity is an important physical characterization measurement in colloid science. The relative suspension viscosity (normalized by the solvent viscosity p ) can be expanded in powers of mass concentration c as
Theory Colloidal Interactions. It is well established by model calculations that the contribution to the interparticle potential from long-range van der Waals attractions is negligible in these inverse m i c r o e m ~ l s i o n s . ~As ~ Jthe ~~~~ qlp = 1 [qlc [ q l 2 k s 2 ... (3) charged surfactant head-groupsare confinedto the interior of the electrically neutral water cores,the relevant colloidal In this virial expansion [q]is a single particle property, forces are entropic or Brownian forces, hydrodynamic known as the intrinsic viscosity, which depends on the forces, and short-ranged attractive forces due to the molecular weight of the individual particle. The Huggins interpenetration or overlap of the surfactant tails on coefficient k~ depends on colloidal interparticle forces neighboring parti~1es.l~ The former two are well underanalogously to the second virial coefficient in the virial stood for colloidal spheres (see Russel et al.,43for example), expansion for the osmotic pressure. Hence, a theory while the latter has received considerable a t t e n t i ~ n . l l J ~ > ~relating ~ the Huggins coefficientto the interparticle forces The interparticle attraction, known as the “overlap enables, in principle, the extracting of the interparticle potential”, is physically attributed to the inherent alignpotential from viscosity measurements. We have recently ment of surfactant tails with respect to other tails on developed an exact theory for the dependence of the neighboring particles, competing with the more randomly Huggins coefficient on the potential parameters of the oriented solvent m01ecules.l~This allows for the expulsion square-well fluid for both monodisperse and polydisperse of entrained solvent molecules in favor of neighboringtails systems.26 As shown therein, attractive interactions with a relative gain in entropy and, hence, a net attractive greatly increase the Huggins coefficient compared to interaction. The increase in attractive interactions due repulsive interactions of similar magnitude. Also, size to both increased temperature, which eventually leads to polydispersity has only a small influence on the suspension a LCST, and increased oil chain-length, can be qualitaviscosity for the range of parameters determined in this tively explained by this m0de1.l~ work and will not be considered further. Smeets and coThe overlap potential is characterized by a strongly w o r k e r ~have ~ ~ established that, hydrodynamically, inrepulsive part due to the excluded volume of the particle verse microemulsions behave as colloidal suspensions of and a short attractive tail. Thus, the simple square-well rigid spheres with stick boundary conditions; hence, our potential, which captures these two essential features, is colloidal theory for the viscosity of the square-well fluid used in place of the more complicatedoverlap potential.12J4 is applicable to the inverse microemulsion systems of The square-well potential interest in this study. Huggins coefficient measurements can thus be used to determine the depth of the squarewell interaction provided the range of the interparticle potential is known. Small-AngleNeutron Scattering Model. The differential cross section for coherently scattered neutrons is described by three parameters: a well depth uij, a range by a polydisperse system of sphericallysymmetricparticles of interaction Aij, and a hard-sphere diameter oi. The can be written advantage of this approximation is that the potential parameters of the square-well can be related to the (4) molecular structure of the surfactant (and cosurfactant).13J4 The range of the potential is set to the distance where n is the total particle number density and (P(q)) of surfactant-tail interpenetration, which is the distance and (Se&)) are form and structure factors, respectively. from the end of the tail to the end of the ethyl branch of These are both functions of q , the magnitude of the the AOT molecule (as beyond this, the tails are more scattering vector, which is defined as densely packed). Thus, the hard-sphere diameter is set by the end of this branch as well. Further, the square4x q = - sin 0 / 2 (5) well depth increases linearly with the volume of interil penetration. For two equal spheres of radius a A where A is the incident neutron wavelength and 0 is the interpenetrating a distance A, the well depth scales as14 scattering angle. The form factor describes intraparticle scattering and is a function of the shape and size of the udkT individual particle. For N types of uniform spherical 2 particles the form factor is given by The proposed linear scaling of the well depth with radius is obtained when the radius is large in comparison with the overlap distance. The well depth is seen to be a strong function of the overlap distance.
+
+
+
+
-
(42)Brunetti, S.;Roux, D.; Bellocq, A. M.; Fourche, G.;Bothorel, P. J. Phys. Chem. 1983,87,1028. (43)Russel, W. B.; Saville, D. A,; Schowalter, W. R. Colloidal Dispersions; Cambridge, 1989. (44)Kaler, E.W.; Billman, J. F.; Fulton, J. L.; Smith, R. D. J.Phys. Chem. 1991,95,458. (45)Smeets, J.;Koper, G. J. M.; van der Ploeg, J. P. M.; Bedeaux, D. Langmuzr 1994,10, 1387.
where f l q ) is the single particle form factor and ni is the number density of particles of diameter ai and volume u,, (46) Chen, S-H.; Lin, T. L.; Kotlarchyk, M. In Surfactants in Solution; Mittal, K., Bothorel, P., Eds.; Plenum Press: New York, 1985;Vol. 6, p 1315.
1562 Langmuir, Vol. 11, No. 5, 1995
e is a mean particle scattering length density, and es is the solvent scattering length density. The structure factor, frequently referred to as a measured or effectivestructure factor when defined as below, describes the scattering due to interparticle correlation^^^
j l ( x ) is the first order spherical Bessel function,
i
N N
i
Bergenholtz et al. is again the mean particle diameter. Equating the moments of the histogram with those of the continuous distribution results in N
where xi is the number fraction of particles of type i and (Xm) is the mth normalized moment of the continuous distribution given by N N
772
where the definition of the partial structure factors Sij in terms of the total correlation functions h&)
S J q ) = 62,
+ q & q )= dij + n i 1 1 2 n j 1 1 2 hqr m 4 n ~ h i , ( dr r ) r(28 )
has been used with dij the Kronecker delta. It is evident from eq 7 that there are form effects present in this definition of the structure factor. Only in a truly monodisperse system can form and interparticle effects be cleanly separated. The total correlation functions remain unspecified in eq 7 and to evaluate these a statistical mechanical theory for interparticle correlations needs to be incorporated. As an alternative, past investigators have frequently used effective one-component approximation^.^^ These, however, do not accurately capture the subtle effects of even moderate polydispersity, especially when the particles exhibit strong attractive interactions. Therefore, in Appendix A we follow the procedure of Menon et al.48and derive the microstructure of an N-component system of spherical particles interacting via a short-ranged squarewell potential of uniform well width. Briefly, this procedure involves solvingthe Ornstein-Zernicke relation with the Percus-Yevick closure relation by using a perturbation in the relative well width. The results obtained are as the lowest order approximation in the perturbation parameter A/((a) A), where (a)is the mean particle diameter. Consequently, this model is only valid when the well width is small in comparison with the hardsphere diameter. The model gives an accurate representation of the (monodisperse) structure as compared to Monte Carlo simulations when Ma < 0.1 and the volume fraction is low (4 < 0.2).49 These requirements should be fulfilled for the systems considered in this study where [hl(o)l,,, x 0.05. The N-component model obtained is general in the sense that any discrete distribution of size and interaction strength can be incorporated. We employ a histogrammatic representation of a Schulz distribution to account for size polydispersity following the procedure developed by DAguanno and Kleinso and consider only systems interacting via an average well depth uo. The continuous Schulz distribution for a system with a size polydispersity equal to the standard deviation s, is given by
+
where z = (1 - sg2)/sU2,a is the particle diameter, and (a) (47)Kotlarchyk, M.;Chen, S-H. J . Chem. Phys. 1983,79, 2461. (48)Menon, S.V.G.; Manohar, C.; Srinivasa Rao, K. J . Chem. Phys. 1991,95, 9186. (49)Bergenholtz, J.;Wu, P.; Wagner, N. J.;D'Aguanno,B. Mol. Phys. Submitted for publication. (50)DAguanno, B.; Klein, R. Phys. Rev. A. 1992,46, 7652.
=
1,2, ... (11)
with ( X O ) = 11. Equations 10 and 11 represent a set of 2Ncoupled equations to be solved for the number fractions and the diameters of the histogram. This is done by requiring that moments 0 through 2N - 1 are matched between the histogram and the continuous distribution. This set of equations was solved by using a multidimensional secant method.51 To compare these predictions for the scattering intensity with the SANS measurements, instrument resolution effects must be accounted for. These appear primarily as a smearing of the features in the intensity spectra, which, if not properly accounted for, may be interpreted as due to a higher degree of polydispersity. Since the desmearing of the experimental data is significantly more difficult and an inherently ill-posed problem, we instead smear the model intensity spectra according to an algorithm developed by Barker and P e d e r ~ e n .This ~ ~ algorithm accounts for three sources of instrument smearing: the divergence of the incident beam, the finite resolution of the detector, and the polychromaticity of the incident beam. Smearing due to multiple scattering was not accounted for. The inputs to the routine are the dimensions of the experimental setup, the detector pixel width, and the polychromaticity of the beam (AYA), all of which have been determined. In summary, the inputs to the SANS model are the mean diameter (a),the polydispersity s,, the potential parameters of the square-well udkT and A/(o), the scattering length densities e and es,and the total volume fraction of particles, computed assuming additive volumes 0 . many of the dispersed components: 4 = ~ A O T 4 ~ ~ As of these quantities are known or can be measured independently, the only true fitting parameters are the potential parameters and the mean particle diameter. In particular, we set s, = 0.15, as determined by Ricka et al.36 using light scattering and Almgrsn et al.37 using fluorescence quenching, and A = 2.4 A, as determined from SANS.4J1Thus, the fits of the SANS measurements determine udkT and (a).
+
Experimental Section The surfactant AOT (Fluka '98% purity) was dried in uucuo. The water used for t h e viscosity measurements was deionized and filtered. The DzO (Cambridge Isotope Laboratories, 99.9% D),used in t h e preparation of both viscosity and SANS samples, was used as received. The solvents n-decane (Aldrich '99% purity) and CC4 (Aldrich '99% purity) were also used as received. Stock solutions were prepared by weighing appropriate amounts ofAOT and dissolving it in weighed amounts ofn-decane. Water or DzO was t h e n weighed into the solution to give the desired molar ratio. Actual samples were then obtained by dilution with n-decane. Finally, all samples were filtered by using 0.2-pm Teflon filters (Gelman). ~
(51) Press, W. H.; Flannery, B. P.; Teukolsly, S. A,; Vetterling, W.
T. Numerical Recipes; Cambridge, 1986. (52) Barker, J.; Pedersen, J. J . Appl. Crystallogr., in press.
AOTIH20 In-Decane Inverse Microemulsions
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11
1
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1 OO
28.6
10
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0
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Figure 2. Einstein coefficient as a function of molar ratio at a temperature of 25.00 f 0.02 "C: (0) AOT/HzO/decane; ( 0 )
AOT/DzO/decane;( 0 )AOT/HZO/CC14. Density measurements (Paar Model DMA 48) were made on dilution series of both waterless micellar solutions and microemulsion samples. The component densities necessary for the SANS analysis were computed from the linear slopes of density versus weight fraction of dispersed material assuming ideal mixing. Dilution viscosity measurements were carried out by using an Ubbelohde viscometer (Cannon)in a constant-temperaturebath (Cannon Model CT1000) controlled to within f0.02 "C. The relative viscosities as functions of dispersed material mass concentrationwere fitted to the expansion in mass concentration (eq 3) truncated after the &term by using a weighted X-square minimization routine,51from which the intrinsic viscosity [ r ] ] and the Huggins coeficient k~ were extracted. The fit was optimized by successively omitting data points at the highest concentration, thus minimizing the O(c3)effects on the extracted parameters. All samples showed Newtonian behavior in the shear rate range investigated (1 s-l < p < 100 s-l) as found by using a constant stress rheometer (Bohlin CS) with cone-andplate geometry. The small-angleneutron scattering experiments were carried out on the NG3 SANS instrument at the National Institute of Standards and Technology (NIST) in Gaithersburg, MD. All experiments were performed by using an incident neutron wavelength of 6 A and a wavelength spread (MlA)of 0.147 at full width half-maximum (FWHM). Samples were sealed in 1-mm quartz cells. Th? sample-to-detectordistances gave a composite q-range of 0.01A-l q < 0.24 The scattered intensities in arbitraryunits were converted to absolute cross sectionsby using the standard NIST pr0cedure.~3
Results As shown in Figure 1, the trend predicted from interpolating physical parameters based on property measurements of micellar and swollen micellar (X = 41) systems fails to qualitatively capture the observed 5-fold increase and then decrease in viscosity as the micelles are swollen at fixed overall volume fraction of surfactant and water. Dilution capillary viscometry was performed to rule out influences of many-body interactions and to connect this viscosity anomaly with the interparticle potential. Figure 2 shows the measured intrinsic viscosity (made dimensionless with the particle density) as a function of molar ratio at a temperature of 25.00 f 0.02 "C. For spheres, this quantity, known as the Einstein Coefficient, is 2.5. As a consequence of solvation of the surfactant tail layer, entrainment of solvent molecules effectively increases the molecular weight of the particles and the observed Einstein coefficients, as previously observed.54 This effect has also been attributed to a nonspherical particle shape;ls however, this has been ruled out by earlier SANS analyses' and depolarized light scattering and flow birefringence measurement^.^^ Extracting the Einstein coefficient from the intrinsic viscosityrequires knowledge of the variation ofthe particle (53) Barker, J.; Krueger, S.; Hammouda, B. SANS Data Reduction andlmaging Software;National Institute of Standardsand Technology, 1993. (54) Pen, J. B. J. Colloid Interface Sci. 1969,29,6.
I
20 30 X=[H2O]/[AOT]
40
Figure 3. Apparent molar volume of core-HzO (0) and DzO ( 0 ) as a function of molar ratio at a temperature of 25.00 f 0.02
"C as extracted from density measurements.
kn
ob
I 10
20
30
40
X=[H*O]/[AOT]
Figure 4. Huggins coefficient as a function of molar ratio at a temperature of 25.00 f 0.02 "C: (0) AOTMzOldecane; ( 0 ) AOT/DzO/decane;(0) AOT/HzO/CC4. The dotted line is a model
prediction based on literature scalings for the potential parameters.
density with molar ratio. As has been shown in previous it is not correct to simply assume bulk densities for the dispersed components. Density measurements, as shown in Figure 3, reveal that the apparent molar volume of the emulsified water increases with increasing molar ratio, eventually approaching the bulk value. However, even at a molar ratio of 41 the molar volume of the water is lower than that of bulk water. This trend is supported by the fact that there is a volume contraction associated with ion-water interaction^.^^ Upon substitution of DzO for H20, we observe an isotope effect resulting in a similar but less pronounced trend for DzO-based microemulsions. Figure 4 shows the Huggins coefficient as a function of molar ratio a t a temperature of 25.00 f 0.02 "C. It increases by an order of magnitude with increased swelling ofthe microemulsion droplets followed by an almost equal decrease with further swelling. This effect is not simply a consequence of the slight minimum in the intrinsic viscosity, even though this minimum is coincident with the maximum in the Huggins coefficient. The variation of the Huggins coefficient with the overall variation in intrinsic viscosity is generally within the error bars of the individual Huggins coefficient measurements. A comparison with Figure 1 shows that this behavior is consistent with the viscometry data on the concentrated microemulsions. Again, interpolations based on a linear scaling of the well depth with radius lead to the wrong qualitative description of the observed behavior (Figure 4,dashed line). We find a moderate isotope effect upon substitution of DzO for H20, resulting in a slightly lower maximum in the Huggins coefficient. We also observe that substitution of CCl4 for decane removes the maximum in the Huggins coefficient, in agreement with previous work.56 For reference, note that hard-sphere suspensions have a Huggins coefficient close to 1. Figure 1 shows that the microemulsion viscosity is a complicated function of temperature. In an attempt to (55)Onori, G. J . Chem. Phys. 1988,89,510. (56) Onon, G.;Santucci, A. J. Colloid Interface Sci. 1992,150,195.
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2PA71
Bergenholtz et al.
I
A
4
0'
25
20
I
35
30
T ("C)
Figure 5. Einstein coefficient for AOT/HzO/decane as a function of temperature for various molar ratios: (O)X=0; (+)
0
0.0s
4
X = 6;(COX= 10;(x)X= 14;(a)X= 41. Note thatX= 41 phase separated inside the capillary at T = 20 and 30 "C as did
X = 14 at T = 20 and 34 "C.
Figure 7. Measured scattering cross sections as functions of = 41: (0)4 = 0.075;(0)4 = 0.14; (x)4 = 0.2;(A)4 = 0.4. Solid lines are model fits. Bold line is a Teubner-Strey
q for
x
fit to 4 = 0.4.
40
I
30
kn
(A-7
I
I
0.1
n. I rI
20 20
10
%S(d (cm-')
0
20
30
25
35
10
T ("C)
Figure 6. Huggins coefficient for AOT/HzO/decane as a functionof temperature for various molar ratios: (O)X=0; (+) X = 6;(0)X= 10;(x)X= 14;(A)X=41.
n 0
0.05
0.2
?!
isolate the effect of temperature on the interparticle potential, we also measured the dilute microemulsion viscosity as a function of temperature. Figures 5 and 6 show the variation in the Einstein coefficient and the Huggins coefficient with temperature, respectively. In these measurements we find a strong sensitivity of the dilute viscosity to the temperature as manifested in the increase in the Huggins coefficient with both increasing and decreasing temperature. Note that these increases at higher temperatures are correlated with decreases in the Einstein coefficient, suggesting some residual O(c3) contributions to the viscosity remain a t these temperatures. This complex temperature dependence of the Huggins coefficient precludes an accurate determination of any activation energy associated with the overlapforces. The temperature study does illustrate, however, that the data taken at 25 "C are "well-behaved" and accurately represents the bulk properties of the fluid. It was observed that microemulsions of higher molar ratios exhibited the strongest sensitivity to temperature, leading eventually to phase separations. TheX=4l phase separated inside the capillary a t temperatures of both 20 and 30 "C, as didX=14 a t temperatures of 20 and 34 "C. The phase transitions a t the higher temperatures were signaled by a clouding over of the microemulsion, indicating that these were of the same type as observed normally in bulk, i.e., leading to two coexisting microemulsion phases, but a t somewhat lower temperatures. No such clouding was observed for the lower temperature transition, perhaps indicatingthat these were of the type leading to a microemulsion coexistingwith an excess water phase, normally seen in bulk a t much lower temperatures.2These results suggest that the stability of these microemulsions is influenced by confinement in the capillary. The detailed structure of the inverse microemulsion phase was investigated a t 25 "C by using SANS. Noting that microemulsionscorrespondingto a composition range 5 < X < 15 exhibited an anomalous Huggins coefficient, scatteringspectra were collected on samples of molar ratios 6,10,14, and 41. These were modeled as suspensions of
Figure 8. Measured scattering cross sections as functions of
q for X = 14: (0) 4 = 0.075;(+) 4 = 0.1;(0)4 = 0.14;(x)?= 0.2;(A)4 = 0.4. Solid lines are model fits. Bold line is a Teubner-Strey fit to 4 = 0.4.
spheres of a uniform scattering length density e, which was calculated from the atomic scattering lengths and the apparent molar volume of the D20 in the droplet core. The contribution to the scattering from the surfactant was neglected as the scattering length density of the surfactant tail is close to that of the solvent and the scattering from the surfactant head group is expected to be weak3owing to the presence of some bound (protonated) water. A constant size polydispersity of 0.15 was assumed in accordance with recent finding^.^^,^^ Furthermore, as a fir$, approximation, a constant absolute well width of 2.4 A4J1was assumed. The procedure for modeling the SANS data employed scattering spectra from higher concentration samples (0.1 I4 I0.4) of molar ratios identical to dilute samples (4 = 0.075) as guides, since the interference peak present a t high concentrations is representative of the size of the individual droplets. Following the disappearance of the peak with decreasing concentrationenabled a reliable way of fitting the comparatively featureless scattering spectra corresponding to the lower concentration samples. Since the goal is to compare the SANS results to the viscometry, these dilute samples are the relevant ones as their compositions are close to those used in the viscometry. Figures 7- 10 show the measured absolute cross sections and the resulting model fits for four separate concentration series, with the fit parameters listed in Table 1. These are shown on linear scales to draw attention to the behavior a t low q where the effect of attractive interactions is dominant and the quality of the model fit is of primary concern. Experimentally,the measured interference peak present for the highest concentration samples can only be qualitatively represented by our model. Considerable difficultywith the modeling of concentratedmicroemulsion SANS data using dispersed droplet models has also been
AOTI H20 1n-Decane Inverse Microemulsions
Langmuir, Vol. 11, No. 5, 1995 1565
n I
40
I
e
Table 2. Correlation Length and Domain Periodicity d as Extracted from the Teubner-Strey Model Fits to the SANS Data on Concentrated Microemulsions (6 = 0.4)
4 = 0.4 6 10 14 41
15 21 30 59
51 66 82 181
encountered p r e v i ~ u s l y .This ~ ~ is not surprising as the structures present at high concentrations are characterized by a significant amount of clustering of the microemulsion droplet^.^ These structures are thus more reminiscent of bicontinuous structures with alternating water and oil domains rather than dispersed droplet structures. We therefore expect that the scattering no longer obeys eq 4 and we employ the model of Teubner and st re^^^ to fit the scattering from the concentrated microemulsions. The values for the correlation length E and the domain periodicity d as extracted from the Teubner-Strey fits are given in Table 2. Note that there appears to be no obvious correlationbetween the TeubnerStrey phenomenological fit coefficients and the viscosity (shown in Figure 1). As seen in Figure 10, the SANS model overpredicts the peak height in the scattering from microemulsions cor-
responding toX=6. To see whether this discrepancycould be due to deviations in particle shape, we attempted to model the data as a collection of prolate spheroids using the formalism of Kotlarchyk and Chen.47 The orientational averages of the form factor were found not to result in a diminishment of the interference peak of the magnitude seen in the data for physically reasonable axial ratios. The discrepancy is more likely due to multiple scattering, which was not accounted for in the analysis, as the contrast is high and there is significant attenuation of the incident beam due to incoherent scattering (transmissions were on the order of 50%). Improvements in the data quality for molar ratios this low can be achieved by employing a deuterated solvent. We note that one particular microemulsion, with a composition correspondingtoX= 10and @ = 0.2, exhibited an anomalousscattering (seeFigure 9, arrow) qualitatively similar to that observed in the critical scattering from these types of microemulsions (see, for instance, Kotlarchyk et a1.l). As this composition is far from the phase boundary at this temperature, it is inferred that the source of the strong interference a t low q observed in the scattering is the result of clustering of microemulsion droplets indicative of a microemulsion composition near the percolation threshold. As this scattering is due to a qualitatively different type of droplet correlation than those our scattering model is founded on, we formulate a new expression for the differential scattering cross section of a system composed of clusters of a uniform size with negligible cluster-cluster correlations based on the work of Chiew et al.59(see Appendix B). As the scattering from a sample nearing the percolation transition is expected to have contributions from clusters of a wide range of sizes, we allow for a multiplicative shift of the model to obtain agreement with the experimental data. The model correctly reproduces the qualitative features in the scattering: the strong interference a t low q, primarily due to the form effects of the clusters, and the slight peak coinciding with the position of the correlation peaks observed in the scattering from the other X = 10 microemulsions. The model fit to the SANS data yields an average cluster size of 6, which indicates that the microemulsion has not reached the percolation transition, yet the scattering is strongly influenced by even this moderate degree of clustering. We note that the parameters extracted from this model fit are consistent with the parameters extracted from the model fits to the scattering spectra from the remainder of the X = 10 concentration series (see Table 1). A mean radius of the D20 core and an average well depth were used as independent adjustable parameters in the modeling of the scattering data. We note that the same radius can be used to describe the entire data set of volume fractions for a fixed molar ratio. Figure 11shows that the mean radius of the polar cores indeed increases linearly with increasing molar ratio with good agreement compared to values reported p r e v i o ~ s l y . ~However, ~~*~~
(57) Kotlarchyk, M.; Huang, J. S.; Kim, M. W.; Chen, S.-H. In Surfactants in Solution. Mittal, K., Bothorel, P., Eds.; Plenum Press: New York, 1985; Vol. 6, p 1303. (58)Teubner, M.; Strey, R. J . Chem. Phys. 1987,87, 3195.
(59) Chiew, Y. C.; Stell, G.; Glandt, E. D. J . Chem. Phys. 1985,83, 761. (60) Kotlarchyk, M.; Stephens, R. B.; Huang, J. S. J . Phys. Chem. 1988,92, 1533.
0
0.05
0.1
0.1;
0.25
0.2
4
Figure 9. Measured scattering cross sections as functions of q for X = 10;(0) 4 = 0.075;(+) 4 = 0.1; ( 0 )4 = 0.14;( x ) 9 = 0.2; (A)4 = 0.4. Solid lines are model fits. Bold line is a
Teubner-Strey fit to 4 = 0.4. Note the strong scattering at low q from the sample correspondingto 4 = 0.2 (see arrow) due to the influence of the percolation threshold.
0
0.05
0.1
0.15
0.25
0.2
4
Figure 10. Measured scattering cross sections as functions of q for X = 6: (0)4 = 0.075; (+) 4 = 0.1;( 0 )4 = 0.14;( x ) 9 = 0.2; (A) 4 = 0.4. Solid lines are model fits. Bold line is a Teubner-Strey fit to 4 = 0.4. Table 1. Data for the Average Microemulsion Radius and the Square-Well Depth as Functions of Volume Fraction $ and Molar Ratio X as Extracted from the Model Fits to the SANS Data AsFuming a Fixed Well Width of 2.4 A 4 = 0.075 4 = 0.1 4 = 0.14 4 = 0.2 4 = 0.4 X 6 10 14 41 a
rlA udkT rlA udkT
r f A udkT rlA udkT
rlA udkT
29 33 38.5 77.5
28 35 38 76.5
27.5 33 39 77.5
-2.8 -2.9 -2.8 -3.1
28 -2.7 33 -2.75 38 -2.7
-2.5 -2.7 -2.6 -2.9
28 32" 38.5 77
-2.4 -2.4" -2.4 -2.7
-2.3 -2.4 -2.2 -2.4
Fit by using the percolation cluster scattering model in Appendix
B.
1566 Langmuir, Vol. 11, No. 5, 1995
Bergenholtz et al.
i A
351' 23
10
20 30 X=[D2O]/[AOT]
50
40
1 10
20 30 X=[D?O]/[AOT]
40
50
Figure 13. Surfactanthead-group area as a function of molar Figure 11. Radius of the polar core as a function of molar ratio: ( 0 )AOT/DzO/decane (present study); (A) AOT/HzO/ ratio: ( 0 )AOT/DzO/decane (present study); (0) AOT/D20/ ; ~ ~AOT/HzO/is~octane;~~ (0) AOT micelle^.^ d e ~ a n e(0) ; ~ AOT micelles in d e ~ a n e(A) ; ~ AOT/DzO/de~ane.~~ i s ~ o c t a n e(0) -4
0
J 0:
10
20 30 X=[DsO]/[AOT]
40
Figure 12. Square-welldepth as a function of molar ratio: ( 0 ) AOT/DzO/decane (present study); ( 0 )AOT/DzO/d-octane and decane.ll The line represents the postulated dependence of the well depth on radius. as shown in Figure 12,to within the accuracy ofthe results, the square-well depth is found to be independent of composition for the molar ratios investigated in this study. This contradicts the postulated linear dependence of the well depth on molar ratio.ll The following mass balance is performed to extract molecular parameters from the radii determined from the SANS measurements. Assuming that the surfactant molecules are confined to the interface yields the following relations:
where rDzo is the radius of the D2O core and 1 ~ is0the ~ length of a surfactant molecule. Since I$D~oand I$AOTare known from independent measurements, the surfactant length can be directly calculated once the radius is determined from the model fit. The surfactant length was found to be 10 f0.5 A for all compositions investigated. The surfactant molar head-group area is another calculable quantity once the radius is determined. It is given by
where n, is the surfactant number density and dH is the diameter of the surfactant head-group, determined assuming the head-group constitutes 14%of the surfactant volume.60 Figure 13 shows that the head-group area increases with increased swelling of the droplets toward a limiting value at high molar ratio, in good agreement with results obtained from other technique^.^^^^^ Discussion As shown by the Huggins coefficient measurements, the anomalous viscosity effect is present even at dilute conditions. Furthermore, the scattering analysis suggests that the droplets are spherical with radii independent of
composition. In addition, the droplet size polydispersity has been found to remain constant with c o n ~ e n t r a t i o n . ~ ~ It follows that we can safely disregard any anomalous changes in the droplet shape or polydispersity and instead conclude that the source of the viscosity anomaly is to be found in the interactions between pairs of particles. In this pair-limit the square-well theory, which provides an exact relationship between square-well parameters and the Huggins coefficient,clearly shows that only attractive interactions can produce this order of magnitude increase in the Huggins coefficient.26 The Huggins coefficient values obtained on the AOT/ H20/CC14 system (Figure 4) are, in contrast, an order of magnitude lower, in agreement with previous The considerable size of the CC4molecule prevents penetration of the surfactant tail-layer. The difference in microemulsion behavior between solvents is thus consistent with the theory that the interparticle attraction primarily results from the gain of configurational degrees of freedom associated with the desolvation of the surfactant tails in the overlap region. This qualitatively explains the observed difference in the magnitude of the measured Huggins coefficients of the AOTk20KC14 and AOTMaOI n-decane systems. To complete the characterization of the system with the colloidal square-wellformalism, SANS is used to relate the square-well parameters to microemulsion composition. The SANS measurements require the substitution of D2O for H2O for the purpose of improving the contrast between particle and solvent. Note that care was taken to actually evaluate the physical properties of DZO-based microemulsions (viscosity and density) instead of simply assuming the equivalence of H2O and DzO microemulsion systems. The extracted well depths demonstrate that, for these relatively low molar ratios, the interparticle potential is significantly more attractive than that expected from interpolating between micelle and microemulsion ( X = 41) measurements. Since the effect of well depth on the structure factor in our model is nearly indistinguishable from that of the well width, we fix, as a first approximation, the absolute well width to 2.4 A.4J1 This enables the determination of the dependence of the well depth and particle size on composition. Substitution of this relationship into the square-well theory yields a prediction for the Huggins coefficient of the microemulsion as a function of composition in terms of the molar ratio. As seen in Figure 14, there is a significant improvement in the agreement between prediction and the viscometry data over that obtained previously by using the linear scaling of the well depth with droplet size. A maximum in the Huggins coefficient is predicted, the location coinciding with the maximum in the viscometry data. There is, however, quantitative disagreement between the model and the viscometry data. A similar disagreement has also been observed in comparing viscometry with static light scattering results, where the second vinal coefficient was used
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Langmuir, Vol. 11, No, 5, 1995 1567
10,
7.5
kH
!
j.
25
0
0
I
t i o
;
10
:
,
,
20
30
0.15,
,!I 40
X=(HIO]/[AOT]
~ Q I ~ T
Figure 14. Huggins coefficient as a function of molar ratio obtained from viscometry ( 0 )compared t o model predictions based on SANS results (0).Note, the prediction for the micellar value ( X = 0) was obtained by using the micellar parameters found from SANS by Huang et aL1l Dotted line is the literature prediction.
Figure 16. Family of solutionsin well depth versus well width corresponding to the measured Huggins coefficient (unshaded region) and scattering intensity (shaded region) for X = 14. Note, the solutions appear as bands due to the uncertainty associated with the measurements.
in the determination of the attractive strength instead of the structure factor.61 The conclusion, however, is that the anomalous viscosity maximum is a consequence of increased interparticle attractions. The macroscopic behavior of microemulsions is governed largely by the organization of the interface between the polar and nonpolar components. Hence, even a small change at the molecular level in the character of this interface may have a profound effect on the macroscopic behavior of the microemulsion. From the SANS results we were able to extract an important measure of this interface, the molar surfactant head-group area, which, as shown in Figure 13, changes markedly in a composition regime corresponding to the location of the viscosity maximum. As discussed by Eicke and Rehak,31the values obtained are lower than what is expected for the minimum packing area of AOT molecules. These results are reconcilable if the surfactant molecules are disordered at the interface; that is, some surfactant molecules pack into interstices present within the surfactant layer. This diffuse interface is accompanied by a more diffuse density of surfactant tails near the droplet surface. Consequently, the distance available for interpenetration of surfactant tails increases, resulting in an increased range of attraction. This would imply that the well width also has a compositional dependence that produces an overall increase in attractive interaction strength at low molar ratios. We note that the micellar valug for the head-group area has been found to be 51-52 A,4,62and thus, the Huggins coefficient measurements mirror the variations in the head-group area with molar ratio (Figure 13).Note that only small changes in the well width are required to have a significant influence on the well depth within the overlap model (see eq 2). The micelle well depth has indeed been found to be only -0.92 kT,ll which is consistent with the trend in the head-group area and results in the lower viscosity observed for the micellar system. A decrease in the head-group area must result in an expansion of the surfactant/DzO interface as the headgroup molar volume is not expected to vary this dramatically. We are unfortunately unable to resolve any broadening of the surfactant/DzO interface in the SANS measurements. Model calculations indicate that it is necessary to obtain a deuterated version ofAOT in addition to deuterated decane to focus on the structure and arrangement of the head-groups and to remove the background, which masks the high-q region, precluding quantitative study. Conceivably, a small-angle X-ray scattering experiment might resolve this structure since the contrast there results from differences in electron density between the head-groups and their surroundings.
Concerning the quantitative discrepancy between the Huggins coefficient extracted from the viscosity measurements and that from the SANS data, the approximate nature of the square-well potential must be considered. The structure factor does not uniquely define a particular well depth and well width as these two parameters are absorbed into the definition of a “stickiness” parameter in the theory (see eq A.lO). Thus, there is a family of well depths and well widths that explains the observed scattering intensity. Similarly, the Huggins coefficient does not provide a unique determination of the squarewell parameters. Rather, there is a family of solutions of well depth and width that yields the measured Huggins coefficient. Ifthe square-well potential is in fact the exact potential for this system, then for a given microemulsion composition there is a unique solution given by the intersection of these two solution families. That is, the combination of viscometry and SANS should uniquely define the pair-potential parameters if this is the correct pair-potential. Figure 15 shows the two solution families corresponding to fixed Huggins coefficient and structure factor. The solutions do not intersect to give a unique determination of the square-well parameters, rather they are roughly parallel over the physically meaningful range ofparameters. This result is typical of all the compositions studied in this work, where the Huggins coefficient systematically yields a stronger attraction than that extracted from the SANS data. Note that we have accurately accounted for both size polydispersity and instrument effects in the modeling of the SANS data. Further, we have performed detailed calculations of the effects of size polydispersity on the Huggins coefficient.26 For the 15% size polydispersity seen in these samples, the calculated Huggins coefficientis 1.98forX= 14versus the moqodisperse value of 1.89 for an assumed well width of 2.4 A. This change is within the uncertainty of the experimental measurements. Therefore,we discount size polydispersity as the source of the discrepancy. The conclusion is that the square-well model for the microemulsion interparticle potential is adequate to describe one particular property, such as scattering intensity or viscosity, with a given set of parameters. However, different sets of parameters are required to fit different types of experiments. This is, indeed, not entirely unexpected because the square-well potential is a rather drastic simplificationof the complex interparticle potential between swollen microemulsion droplets.13 Similar observations hold for simple fluids, where viscometry measurements systematically yield a stronger Lennard-Jones attraction than do virial coefficient meas u r e m e n t ~ .Here ~ ~ again the discrepancy arises from approximating the true intermolecular potential by a
(61) Batra, U.; Russel, W. B.; Huang, J. S. Unpublished results. (62) Assih, T.; Larche, F. Delord, P. J. Colloid Interface Sci. 1982, 89, 35.
(63) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley and Sons: New York, 1954.
Bergenholtz et al.
1568 Langmuir, Vol. 11, No. 5, 1995 Lennard-Jones form. The rather large discrepancy for the microemulsion square-well model suggests, however, that the model is indeed too crude to predict anything but the qualitative behavior of properties not directly used in determining the square-well parameters. Another possible discrepancy can arise from neglecting the influence of the surfactant tail ends on the fluid mechanics, which is implicit in the calculation of the Huggins coefficient. Accounting for the inhibited motion transverse to the direction along particle centers due to the overlapping of tails in the manner of Genz et al.I7 (leaving the parallel motion unperturbed, however) results in only slight quantitative changes in the Huggins coefficient. The Huggins coefficient generally increases for weaker attractions relative to the case of hard-sphere hydrodynamics but decreases for stronger attractions. The magnitude of these changes is insufficient to alter the conclusions that viscometry measurements yield a stronger attraction than do SANS measurements, and the square-well approximation for the interparticle potential of these inverse microemulsions requires the use of different sets of parameters for the modeling of different types of experiments. Finally, we consider the results of this study within the context of previous work on similar AOT/HzO/oil inverse microemulsions, which identified a percolation transition by c o n d ~ c t i v i t y . ~Correlated ~ ~ ~ ~ - ~ with ~ this increase in conductivity is an increase in viscosity with concentration and a maximum in d In Vld@,which has been ascribed to the existence of a percolation cluster.20 The viscosity maximum with composition has, until now, simply been taken to be a consequence of the dependence of the percolation transition on molar ratio (at fixed temperature), which indeed mirrors the isothermal viscosity plots shown here and elsewhere.20 This explanation, although it connects these two macroscopic probes, does not give any mechanism for the onset of the percolation threshold, its dependence on molar ratio, nor any connection to measurable molecular properties of the system and in that sense remains a phenomenological correlation of observations. Our measurements suggest that there is a direct molecular connection between the observed trends in the percolation transition, fluid structure, and viscosity with swelling of the microemulsion droplets. That is, disorder at the surfactant/water interface results in increased attractive interactions that both lower the percolation threshold and increase the dilute solution viscosity. Indeed, the observed dependence of the percolation threshold on molar ratio20can bepredicted from our dilute SANS measurements. The square-well potential parameters determined from these measurements can be used to to predict the average cluster size S (see eq B.6) as a function of both molar ratio and concentration and thus the percolation threshold (whenS --). We find, however, that the predictions for the percolation threshold are extremely sensitive to variations in the potential parameters. Thus, the uncertainties associated with our determination of the potential parameters preclude an exact determination of the cluster sizes used in the Huggins coefficient measurements. It is possible, however, that dimers or even trimers exist under these dilute conditions. The presence of larger, polydisperse, fractal aggregates of droplets, characteristic of systems strongly influenced by the percolation t r a n ~ i t i o n can , ~ ~ be ruled out. Such aggregates would “trap” a significant amount of solvent, leading to anomalously large intrinsic viscosities. We (64) Rouch, J.; Tartaglia, P. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S.-H., Huang, J. S., Tartaglia, P., Eds.;Kluwer Academic: Dordrecht, 1992; p 391.
observe uniform intrinsic viscosities well within values expected for spheres with a minimal amount of solvent bound to the surfactant tails. In addition, the scattered neutron intensity remains unaffected by the percolation transition until a volume fraction of 0.2 for X = 10, for which eq B.6 predicts the system should be close to percolation (conductivity measurements yielded a percolation transition at @ = 0.3 for this composition). Thus, we can conclude that our Huggins coefficient data reflect intrinsic properties of the potential of mean force acting between two microemulsion droplets and are not significantly corrupted by many-body microemulsion droplet interactions.
Conclusions Adetailed study ofthe fundamental droplet interactions in single phase AOT/HzO/n-decane inverse microemulsions has been undertaken, whereby the droplet interaction has been studied within the square-well colloidal framework. Predictions for the viscosity based on a proposed linear scaling of the square-well depth with droplet radius are shown to fail to account for a viscosity maximum with swelling of the microemulsion observed in capillary viscometry and are inconsistent with our SANS measurements. On the basis of the experimental observations, a molecular rearrangement of the surfactant/ water interface is proposed to qualitatively explain the increased attractive interactions leading to the viscosity maximum. This change in the organization of the interface also provides a direct molecular basis for the observed variation of the percolation transition with molar ratio. The square-well colloidal model for the microemulsion droplet interactions is rigorously tested and shown to suffice for the description of a single physical property. It is, however, found to be inadequate for simultaneously describing two different physical properties (viscosityand microstructure) with the same set of parameters, as is often found for simple molecular systems.
Acknowledgment. We thank U. Batra, W. B. Russel, and J . S. Huang for making their preliminary results available to us and E. W. Kaler for useful discussions. We also thank J. Barker, C. Glinka, and B. Hammouda of NIST for assistance with the SANS measurements. This material is based upon activities supported by the National Science Foundation under Agreement DMR-9122444. We acknowledge the support of the National Institute of Standards and Technology, U S . Department of Commerce, in providing the facilities used in this experiment. Funding is gratefully acknowledged from the National Science Foundation (CTS-9158164)and Eastman-Kodak. Appendix A Calculation of the Multicomponent Sticky HardSphere Structure Factor. This treatment is a generalization of the work of Menon et For fluid mixtures the direct correlation functions cLJ(r)are defined by the multicomponent form of the Ornstein-Zernicke (OZ) relation which can be expressed as h1J . .(r)= g 1J . .(r)- 1 = cij(r)
+
N
z n k S d tc&) hkj(lr- tl) (-4.1) k=l
where gij are the radial distribution functions, N is the total number of particle classes, r = Irl, and t = Iti. The multicomponent form of the Percus-Yevick (PY) closure relation is These two equations along with the square-well potential
AOTI HzO 1n-Decane Inverse Microemulsions
Langmuir, Vol. 11, No. 5, 1995 1569
(eq 1) completely determine the structure of the equilibrium square-well fluid. Solution of the above set of equations proceeds by Fourier transforming eq A. 1 and writing it in the matrix form
[I - C(q)l[I + H(q)1 = I
(A.3)
(A.15) Equation A.8 is the multicomponent analogue of eq ?;3of Menon et al.48 Finally, we obtain an expression for Q(q), here cast in a form equivalent to that of Robertus et a1.@
where I is the unit matrix. The N x N elements of the symmetric matrices H(k) and e ( k ) are found from
Following Baxter,@a set of ranges al, ..., U N are chosen in such a way that
cij(3c)= 0 if
+
3c
= rla > R i j
+
where Rij = (Ri RjY2 = (ai aj)/2a and the associated quantities Tij = (ai - aj)/2a. The Fourier transformed direct correlation functions can then be found via the matrix f a c t ~ r i z a t i o n ~ ~ N
1 - C ( 4 )= &T(-q)&(4)= c $ ; , i ( q ) $ k j ( q ) (A.6) k=l
where the elements of Q ( q ) are
+
+
= 12JRiJ RJ-E 3cgij(3c)&
(A.9) (A.10)
with the hard-sphere contribution
a.
q F ( x ) = >(x2 - R:j)
2
where j o and j , are the zeroth- and first-order spherical Bessel functions. For a given size distribution and a distribution in interaction strength, the N(N 1)/2 coupled quadratic equations given by eq A.8 for i l i j are solved. These & j values determine QiJtq)througheqA.16,yieldingI - C(q) from eqA.6. Finally, the structure is given by the solution to the linear matrix inversion problem defined by eq A.3, where S&) = d i j +&&) are the partial structure factors desired. We note that in the limit N 1, the problem reduces exactly to the monodisperse solution of Menon et al.,48and in the limits
+
-
€ 4 0
where a is a characteristic length scale to be chosen. In accordance with Menon et we choose ai = ui A and a = (a) A; hence, the well width is taken to be uniform and independent of particle type. Defining the perturbation parameter E = Ala, we obtain to leading order
1J’
(A.16)
+ P ~ ( x- RiJ
(A.17) with t i j kept constant, the multicomponent adhesive hardsphere the01-y~~~~’ is obtained by using the following renormalizations z,ja
where
AipZ
- lij4j
(A.18)
The advantage of the above theory over the adhesive sphere theory, as noted by Menon et a1.,48is that we obtain a direct relationship between the stickiness parameters and the square-well potential parameters via eq A.lO, alleviating the need for any ad hoc mapping onto the stickiness parameters. Appendix B Percolation in the Multicomponent Sticky HardSphere System. This treatment relies on the formalism developed by Coniglio et a1.,68Chiew and Glandt,69and Chiew et al.59 We consider a particle pair to be bound if they are within a distance aij A of each other and factor the Boltzmann factor accordingly
+
exp(-d:j(r)/kr)
(A.11)
- tijlaij
uij--m
=
O < r < o1J, oij< r < oij A ( B . l ) oij+A
