Percolation in Nonionic Water-in-Oil−Microemulsion Systems: A Small

Langmuir 1998, 14, 1041-1049

1041

Percolation in Nonionic Water-in-Oil-Microemulsion Systems: A Small Angle Neutron Scattering Study S. Lipgens, D. Schu¨bel, L. Schlicht, J.-H. Spilgies, and G. Ilgenfritz*,1 Institut fu¨ r Physikalische Chemie der Universita¨ t zu Ko¨ ln, Luxemburger Strasse 116, 50939 Ko¨ ln, Germany

J. Eastoe School of Chemistry, University of Bristol, Bristol, U.K. BS8 1TS

R. K. Heenan ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxon, U.K. OX110QX Received February 20, 1997. In Final Form: December 19, 1997 Nonionic water-in-oil (w/o)-microemulsions of the type water/c-hexane, n-hexane/Igepal were examined by SANS in both shell and core contrast. Variation of the water to surfactant ratio W0 in the droplet regime yields a linear relation between W0 and the droplet water core radius and R/nm ) 0.19 W0 + 0.70. Small angle neutron scattering (SANS) data for W0 ) 10 samples show that droplet aggregation occurs with a fractal dimension of D e 1.3 as a percolation-like transition is crossed by decreasing temperature. Data analysis and model calculations, done in both q and r space, are consistent with the formation of linear aggregates of discrete spherical droplets.

Introduction Mixtures of water, oil, and a suitable surfactant exhibit interesting experimental and theoretical aspects of selforganization. It is characteristic of the subtle balance of forces in these complex fluids that small changes in external parameters like temperature can result in drastic structural changes. The phase behavior of oil-water mixtures stabilized by ionic surfactants like AOT (Aerosol OT) and nonionic amphiphiles CiEj (alkyl glycol ethers) has been extensively studied.2-10 In particular, for oil-rich systems with a nonionic surfactant,6-10 at higher temperatures a water-in-oil microemulsion coexists with excess water (Winsor II system) and the surfactant is predominantly present in the oil phase. On lowering the temperature below the “solubilization boundary” a homogeneous single phase forms (L2). At lower temperatures a “haze boundary” is reached where a surfactant-rich phase separates from an oil-rich portion (Winsor I system). These changes reflect an increase in water solubility of the surfactant at lower temperatures. It is well established that at the solubilization boundary droplets are present that assume a natural radius of curvature determined by the molecular structure of the (1) To whom correspondence should be addressed. (2) Eicke, H. F. Chimia 1982, 36, 241. (3) Chen, S. H.; Chang, S. L.; Strey, R. J. Chem. Phys. 1990, 93, 1907. (4) Chen, S. H.; Chang, S. L.; Strey, R.; Samseth, J.; Mortensen, K. J. Phys. Chem. 1991, 95, 7427. (5) Koper, G. J. M.; Sager, W. F. C.; Smeets, J.; Bedeaux, D. J. Phys. Chem. 1995, 99, 13291. (6) Shinoda, K.; Kunieda, H.; Arai, T.; Saijo, H. J. Phys. Chem. 1984, 88, 5126. (7) Kahlweit, M.; Strey, R. Angew. Chem. 1985, 97, 655. (8) Aveyard, R.; Binks, P.; Fletcher, P. Langmuir 1989, 5, 1210. (9) Kahlweit, M.; Strey, R.; Busse, G. J. Phys. Chem. 1990, 94, 3881. (10) Lindman, B.; Olsson, U. Ber. Bunsenges. Phys. Chem. 1996, 100, 344.

surfactant and the composition of the mixture.11-13 Collision between the droplets leads to material exchange between the compartments.14-17 Systems stabilized by the ionic AOT exhibit the same sequence of phases but with an opposite temperature dependence. The transformation from a Winsor II to a Winsor I system with temperature in water-in-oil (w/o)-microemulsions is accompanied by a steep increase in conductivity near the haze boundary. This highly cooperative process, which has been studied in detail for the ionic AOT,18-23 is generally called a percolation transition since it has been shown that the conductivity curves can quite well be described by the scaling laws of the percolation theory.24 Irrespective of the detailed mechanism involved, the term percolation is often used in the literal sense to (11) Chen, S. J.; Evans, D.; Ninham, B.; Mitchell, D; Blum, F.; Pickup, S. J. Phys. Chem. 1986, 90, 842. (12) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (13) Kellay, H.; Binks, B. P.; Hendrikx, Y.; Lee, L. T.; Meunier, J. Adv. Colloid Interface Sci. 1994, 49, 85. (14) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1987, 83, 985. (15) Fletcher, P.; Horsup, D. J. Chem. Soc., Faraday Trans. 1992, 88, 855. (16) Almgren, M.; Johannsson, R. J. Phys. Chem. 1992, 96, 9512. (17) Mays, H.; Ilgenfritz, G. J. Chem. Soc., Faraday Trans. 1996, 92, 3145. (18) Eicke, H. F.; Kubik, R.; Hasse, R.; Zschokke, I. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; p 1533. (19) Cazabat, A. M.; Chatenay, D.; Guering, P.; Langevin, D.; Meunier, J.; Sorba, O. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; p 1737. (20) Bhattacharya, S.; Stokes, J. P.; Kim, M. W.; Huang, J. S. Phys. Rev. Lett. 1985, 55, 1884. (21) Van Dijk, M. A.; Castleijn, G.; Joosten, J. G.; Levine, Y. K. J. Chem. Phys. 1986, 85, 626. (22) Borkovec, M.; Eicke, H. F.; Hammerich, H.; Das-Gupta, B. J. Phys. Chem. 1988, 92, 206. (23) Cametti, C.; Codastefano, P.; Tartaglia, P.; Rouch, J.; Chen, S. H. Phys. Rev. Lett. 1990, 64, 1461.

S0743-7463(97)00179-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/05/1998

1042 Langmuir, Vol. 14, No. 5, 1998

describe the transformation from a state of isolated droplets to a system-spanning bicontinuous network. We use the term percolation in this sense, too. However, it has been pointed out that this use might be misleading since percolation implies a special mechanism of droplet aggregation and material exchange while bicontinuity is a structural concept.25 Temperature-induced percolation phenomena can also be observed in nonionic (w/o)microemulsions stabilized by CiEj26,27 or Igepal (alkylphenyl glycol ethers)28 when electrolyte is added to the aqueous phase. Due to the inverse temperature dependence of the phase behavior as compared with anionic AOT, in this case the conductivity rises with decreasing temperature. In the AOT system, percolation was originally observed when the droplet concentration of the aqueous (conducting) phase was increased at constant temperature.29-33 Here, comparison with the nonionic Igepal reveals a principle difference, because increasing the volume fraction of the aqueous phase can result in transferring the system from a highly conducting to a low-conducting state. The same behavior has been observed in systems stabilized by the cationic DDAB (didodecyldimethylammonium bromide) and has been termed a “percolation-like transition”34 or “antipercolation transition”.35 This observation clearly demonstrates that it is the spontaneous curvature of the surfactant film that is the primary factor determining the microstructure of the system. Comparison of ionic and nonionic amphiphiles shows that very similar temperature-induced conductivity enhancements, changes in viscosity, electric field induced shifts of the percolation curves,28 and droplet exchange kinetics in phosphorescence quenching experiments17 are observed in both systems except for an inverse temperature dependence. Therefore, there is strong evidence that the temperature-driven processes are of the same type for both systems. The existence of droplets at the solubilization boundary has been inferred from several types of experiments, like, e.g., light scattering,36 neutron scattering,37 and fluorescence and phosphorescence quenching.15-17 The driving force for the temperature-induced aggregation and phase inversion is the tendency of the surfactant film to achieve its natural curvature, which exhibits an inverse temperature dependence for ionic and nonionic amphiphiles. The development of the interactions between droplets and the structure of the resulting network itself are still the subject of discussion: While most authors interpret their data in terms of aggregating spheres and

Lipgens et al.

the formation of clusters,3,5,17,38-42 others discuss it in terms of dispersion inversion, i.e., the formation of a bicontinuous structure.10,26 Vollmer et al.43 have provided evidence from calorimetric measurements that in AOT systems a percolative aggregation process is followed by a transition to a bicontinuous structure. The cluster model has been reviewed by Safran42 and Koper et al.5 Since the sizes of microemulsion droplets are in the nanometer range, small-angle neutron scattering (SANS) has been widely applied, especially to study the AOT system.3,4,44-46 For the nonionic C12E5, Strey et al.37 have recently given a survey on the scattering profiles along the one-phase channel pointing out the analogy of the spectra, with inverse temperature dependence, on the oiland water-rich sides, respectively. In the present paper we investigate structural changes along the percolation curve for (w/o)-microemulsions with the nonionic surfactant Igepal-CO-520 (IG) by SANS measurements. In particular, we want to see whether the data are consistent with the picture of droplet aggregation. Evidence for the existence of droplet clusters along the temperature-induced transition derives from time-resolved phosphorescence quenching experiments17 and measurements of electric birefringence.47,48 Furthermore, freeze fracture micrographs49 of C12E5 systems reveal the formation of branched linear aggregates of individual droplets when the mixture is rapidly cooled. In the nonpercolated region, a relationship between the molar water-to-surfactant ratio W0 and the droplet radius is established. Three different approaches were used for the analysis of SANS data: model fitting, Fourier transformation, and generic model calculations. Experimental Section Materials. Deuterated solvents (c-hexane-d12 and n-hexaned14) were from Goss Scientific (both 99+% D-atom). D2O (99.9+% D-atom) was from Fluorochem. Protiated solvents were obtained from Merck in p.a. grade. The surfactant IG (Igepal-CO-520, n-nonylphenyl pentaethylene glycol ether) is a technical product obtained from Aldrich with an average molecular weight of Mw ) 440.6 corresponding to an average number of five ethoxy groups. The composition of a microemulsion may be characterized by two parameters, the weight fraction of hydrocarbon, R, and surfactant, γ, respectively

g) (24) Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; Taylor and Francis: Bristol, PA, 1994. (25) Reference 10, p 353. (26) Kahlweit, M.; Busse, G.; Winkler, J. J. Chem. Phys. 1993, 99, 5605. (27) Sager, W.; Sun, W.; Eicke, H. F. Prog. Colloid Polym. Sci. 1992, 89, 284. (28) Schlicht, L.; Spilgies, J. H.; Runge, F.; Lipgens, S.; Boye, S.; Schu¨bel, S.; Ilgenfritz, G. Biophys. Chem. 1996, 58, 39. (29) Lague¨s, M. J.; Sauterey, C. J. Phys. Chem. 1980, 84, 3503. (30) Bug, A. L. R.; Gefen, Y. Phys. Rev. A 1987, 35, 1301. (31) Bisal, S. R.; Bhattacharya, P. K.; Moulik, S. P. J. Phys. Chem. 1990, 94, 350. (32) Ponton, A.; Bose, T. K.; Delbos, G. J. Chem. Phys. 1991, 94, 6879. (33) Boned, C.; Peyerelasse, J.; Saidi, Z. Phys. Rev. E 1993, 47, 468. (34) Evans, D. F.; Mitchell, D. J.; Ninham, B. J. Phys. Chem. 1986, 90, 2817. (35) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; VCH: Deerfield Beach, FL, 1994; p 475. (36) Nicholson, J. D.; Clarke, L. H. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; p 1663. (37) Strey, R.; Glatter, O.; Schubert, K. V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175.

m(oil) m(water) + m(oil)

(1.1)

m(surf) m(water) + m(oil) + m(surf)

(1.2)

R)

(38) Clark, S.; Fletcher, P. D. I.; Xilin, Y. Langmuir 1990, 6, 1301. (39) Johannsson, R.; Almgren, M.; Alsins, J. J. Phys. Chem. 1991, 95, 3819. (40) Ray, S.; Bisal, S.; Moulik, S. J. Chem. Soc., Faraday Trans. 1993, 89, 3277. (41) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. J. Phys. Chem. 1995, 99, 8222. (42) Safran, S. A. In Micellar Solutions and Microemulsions; Chen, S. H., Rajagopalan, R., Eds.; Springer: New York, 1990; p 161. (43) Vollmer, D.; Vollmer, J.; Eicke, H. F. Europhys. Lett. 1994, 26, 389. (44) Toprakcioglu, C.; Dore, J. C.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1 1984, 80, 413. (45) Radiman, S.; Fountain, L. E.; Toprakcioglu, C.; De Vallera, A.; Chieux, P. Prog. Colloid Polym. Sci. 1990, 81, 54. (46) Eastoe, J.; Hetherington, K. J.; Sharpe, D.; Dong, J.; Heenan, R. K.; Steytler, D. Langmuir 1996, 12, 3876. (47) Schlicht, L. Dissertation, Ko¨ln, 1996. (48) Schlicht, L.; Spilgies, J.H.; Ilgenfritz, G. J. Mol. Liq. 1997, 72, 295. (49) Jahn, W.; Strey, R. J. Phys. Chem. 1988, 92, 2294.

Percolation in Nonionic w/o-Microemulsions

Langmuir, Vol. 14, No. 5, 1998 1043

Table 1. Sample Compositionsa sample 1 2 3 4 5 6 7 8

contrast D/H/H D/H/H D/H/H D/H/H D/H/D D/H/D D/H/D D/H/D

W0

m(water)/g

m(oil)/g

m(surf)/g

R

γ

φ(water)

φ(disp)

TU/°C

TP/°C

TL/°C

10 24.5 40 55 10 24.5 40 55

0.1266 0.2809 0.4452 0.5773 0.0506 0.1163 0.1781 0.2309

1.5060 1.4124 1.3244 1.2490 0.6952 0.6520 0.6114 0.5766

0.2784 0.2611 0.2448 0.2309 0.1114 0.1044 0.0979 0.0924

0.923 0.829 0.748 0.684 0.932 0.849 0.774 0.714

0.146 0.133 0.122 0.112 0.130 0.120 0.110 0.103

0.046 0.102 0.162 0.210 0.046 0.106 0.162 0.210

0.159 0.211 0.260 0.303 0.158 0.211 0.260 0.303

47.2 27.5 20.6 14.1

19.0 18.5 13.8 10.8

8.8 16.7 12.8 9.4

a The contrast depicts the scattering length densities of the samples in the order aqueous phase/surfactant/oil phase. D stands for the fully deuterated and H for the protiated forms. The oil phase is a 1:1 mixture (w:w) of c- and n-hexane. The surfactant concentration is coil surf ) 0.3 M for all samples. W0 is the water-to-surfactant ratio to eq 2.2. The next three columns contain the masses. R and γ are the weight fractions of oil and surfactant, respectively, according to eqs 1.1 and 1.2. φ(water) and φ(disp) are the volume fractions of the aqueous and dispersed phases (water + surfactant), respectively. TU and TL are the upper and lower phase boundaries, respectively. TP is the percolation temperature.

It is also convenient to use the molar concentration of surfactant with respect to the hydrocarbon coil surf and the molar water-tosurfactant ratio W0 defined as

n(surf) V(oil) + V(surf)

(2.1)

c(water) m(water)M(surf) ) c(surf) m(surf)M(water)

(2.2)

oil csurf )

W0 )

With nonionic surfactants, W0 should be corrected for the cµc (critical microemulsion concentration) to take into account the concentration of surfactant that is not incorporated in the film but is present as monomer in the oil and water phase, respectively. This is done by introducing the value W0*:

W0* )

c(water) c(surf) - cµc

etc.) to the experimental data, taking into account polydispersity and a structure factor. This was done with the model-fitting software FISH50 using the calibrated absolute intensity scale.

I(q) ) N(∆F)2

∫

∞

0

f(r) P(q,r) dr S(q)

(3.1)

with N the number density of the aggregates and ∆F the scattering length density contrast.

f(r) )

[t +R 1]

t+1

{

t+1 rt r exp R Γ(t + 1)

}

(3.2.1)

is the Schulz distribution, where

〈r〉 ) R )

∫

∞

0

r f(r) dr

(3.2.2)

(2.3)

Microemulsions were prepared by weighing in calculated amounts of oil, surfactant, and D2O (cf. Table 1). Electrical Conductivity. Samples were thermostated in a water bath ((0.05 deg), and the conductivity was measured with a WTW-LTA electrode and a Tinsley Prism LCR-Databridge 6458 at 1 kHz. Phase boundaries were determined in the course of these measurements as well, both through the instability of conductivity with time in the two-phase regions and the development of opacity. Conductivity measurements were performed without the addition of electrolyte since IG is a technical product and already contains ionic impurities. In order to estimate the impurity level, we measured the maximum conductivity in the percolated regime as a function of added KCl concentration. From the resulting linear dependence, the effect of the impurities is approximated to be equivalent to a 0.28 mM KCl solution. Small-Angle Neutron Scattering. SANS measurements were carried out at the LOQ instrument at the Rutherford Appleton Laboratory (pulsed spallation source). The incident white neutron beam had a wavelength distribution of 2.2 Å < λ < 10 Å, and a 64 cm × 64 cm position-sensitive detector, at 4.1 m from the sample, was used to measure the scattering. Evaluation in a time-of-flight analysis after correction for wavelength dependence of the incident flux, transmission of the samples, and detector efficiencies gave a useful q range of 0.008-0.2 Å-1 [q ) (4π/λ) sin(θ/2), where λ is the wavelength of the incident neutrons and θ is the scattering angle]. The samples were contained in stoppered Hellma cells and thermostated to (0.1 deg with a Lauda RUK50 Kryomat. Data normalization was done with standard software (COLETTE) available at ISIS. The scattering from an empty cell was subtracted from all samples. In the data analysis the background was input as a constant fraction of the measured incoherent level for the appropriate solvent. Data analysis can principally be performed in two ways. The first approach is to fit a model form factor (for spheres, cylinders,

is the average radius of the distribution and

p ) (t + 1)-1/2 )

σ R

(3.2.3)

is the polydispersity index (with σ the standard deviation). The form factors P(q) used were those for solid and hollow spheres, respectively. To account for aggregation, we used the following structure factors: -the Teixeira formalism for fractal aggregation SFr(q)51

SFr(q) ) 1 + DΓ(D - 1) 1 sin[(D - 1) arctan(qξ)] (3.3) D (rq) (1 + (1/q2ξ2))(D-1)/2 where r is the radius of the spherical droplets, D is the fractal dimension of the aggregates, ξ is the correlation length, and Γ is the gamma function. -the Ornstein-Zernicke (OZ) structure factor SOZ(q)

SOZ(q) ) 1 +

S0 - 1 1 + ξ2q2

(3.4)

with S0 the isothermal compressibility and ξ the correlation length. A hard sphere structure factor was used for nonpercolated systems. An alternative way of data analysis is the construction of the pair distance distribution function52 p(r) (densitiy correlation function C(r)). This was done by the indirect Fourier transformation method (software ITP from Glatter53), which is designed (50) Heenan, R.K. FISH data analysis program; Rutherford Appleton Laboratory Report RAL-89-129, 1989. (51) J. Teixeira, J. J. Appl. Crystallogr. 1988, 21, 781. (52) Glatter, O.; Kratky, O. Small Angle X-Ray Scattering; Academic Press: London, 1982. (53) Glatter, O. Progr. Colloid Polym. Sci. 1991, 84, 46.

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Lipgens et al.

Figure 1. Phase boundaries and transition temperatures of the D/H/H systems (cf. Table 1) as a function of water-tosurfactant ratio W0. The gray region represents the percolated part of the one phase regime. to minimize truncation effects, and by the model calculations described below. The scattering intensity I(q) and the density correlation function C(r) or p(r) are related by the Fourier transform

p(r) ) 4πr2C(r) )

2r π

∫

∞

0

q I(q) sin(qr) dq

(4)

Generic Model Calculations. We performed model calculations of form factors and correlation functions for linearly aggregated spheres. This was done by discretizing the objects into pixels of a cubic grid. Spheres with an average radius of 13 or 18 pixels were used. The correlation function p(r) with integer values of r ) 1, 2, 3, ... was calculated by counting the number of pair distances between r - 0.5 < rij < r + 0.5. The scattering intensity was obtained by summing up the sin(qr)/qr terms. Averages were calculated by introducing a discretized Schulz distribution. In the following discussion we will refer to this approach as the “pixel summation method”.

Results Phase Behavior and Percolation. The transition temperatures for the samples used in the SANS measurements are shown in Figure 1. Region 1 denotes the one-phase microemulsion range (L2 phase). At higher temperatures, as the surfactant becomes less soluble in water, phase separation occurs at the “emulsification-failure” or “solubilization” boundary into a water-rich and a surfactant/oil-rich phase (2h ) (Winsor II system). At lower temperatures, the surfactant becomes more and more soluble in water and the microemulsion separates at the “haze boundary” into a surfactant/water-rich and an oil-rich phase (2) (Winsor I system). We have no evidence for liquid crystal phases (LC) under these conditions, as judged by viewing the samples between crossed polarizers. The corresponding systems with H2O as aqueous phase show the same general behavior. The replacement by D2O merely shifts all transition curves approximately 2 °C to lower temperatures. Replacing protiated oils by their deuterated equivalents has no significant effect on the phase behavior. The transition temperatures are determined from the inflection points of log κ vs T plots, as shown in Figure 2 for D/H/H systems. The data show that with decreasing temperature there is a transition to a “percolated” state, where also the aqueous phase is continuous, even though it is present at a volume fraction as low as φaq ) 5% (cf. Table 1). It is also seen that increasing the water content (W0 value) shifts the transition curves to lower temperatures. As a

Figure 2. Electrical conductivity of the D/H/H systems (cf. Table 1) as a function of temperature for various W0 values. The arrows indicate the upper phase transition temperatures, while the point at the lowest temperature for each curve represents the lower phase boundary for the respective system.

consequence, addition of aqueous phase at constant temperature transfers the system from a highly conducting to a low conducting state. For the ionic AOT system the scaling laws of the percolation theory (eqs 5.1 and 5.2) have been applied,

T < Tp:

κ ∝ |Tp - T|-s

(5.1)

T > Tp:

κ ∝ |Tp - T|µ

(5.2)

and the exponents found were close to the universal values of the percolation theory (µstat ) 2.0 for static and µdyn ) 1.88 for dynamic percolation24). However, the percolation model for the temperature-induced structural changes might be inappropriate for systems where the conducting structure breaks down on isothermal addition of conducting material. Nevertheless, the above power laws give quite good fits to transition curves in nonionic systems, too. For the W0 ) 40 system made from water/IG/decalin with 10 mM KCl, we have found that eqs 5.1 and 5.2 account well for both temperature- and compositioninduced transition47 with the exponents being µ ) 1.71.8 in the highly conducting region. This also holds for the W0 ) 10 system in Figure 2 when only points in the vicinity of the transition temperature (inflection point) are taken into account. However, since these equations only apply over a narrow range, it is difficult to make any firm conclusion about their validity. The curves in Figure 2 were obtained without the addition of electrolyte for the reason given in the Experimental Section. The low-conducting branch is insensitive to the addition of salt, whereas the maximum conductivity depends linearly on salt concentration and scales with ∼φaq2 in all IG systems we have examined. The increase of conductivity in the nonpercolated region with temperature in Figure 2 reflects the properties of the droplet conduction mechanism as discussed, e.g., in the charge fluctuation model.54 Small-Angle Neutron Scattering. The first set of SANS experiments represent a W0 variation near the upper transition temperature where a spherical microstructure is known to exist.12 In the second set of experiments, W0 is kept constant at 10 and the effect of temperature is examined. (54) Eicke, H. F.; Borkovec, M.; Dasgupta, B. J. Phys. Chem. 1989, 93, 314.

Percolation in Nonionic w/o-Microemulsions

Langmuir, Vol. 14, No. 5, 1998 1045 Table 2. Fitting Results for the W0 Variation (Figure 4)a W0 10 24.5 40 55

T/°C 35.0 27.1 20.0 13.6

〈Ra〉/nm 4.20 7.47 10.75 12.13

〈Ri〉/nm 2.71b

(3.35) 5.35 (5.21) 8.54 (6.37) 9.76 (7.11)

σ(Ri)

dshell/nm

0.25 0.28 0.25 0.28

1.49 2.12 2.21 2.37

a 〈R 〉 and 〈R 〉 are the average outer and inner radii, respectively; a i dshell is the thickness of the shell obtained from the shell contrast data. The values in parentheses are the results from fits of the core contrast. b Fit without structure factor.

Figure 4. Droplet radii determined by fits of the data in Figure 3 (bottom) as a function of W0. The solid lines are linear fits not taking into account the W0 ) 55 values. The dotted lines refer to W0* (corrected for cµc, according to log(cµc) ) -1830‚1/T (K) + 4.48).15

Figure 3. (Top) SANS data for the core contrast (D/H/H) (cf. Table 1) for different W0 values near the upper phase transition boundary. (Bottom) shell contrast (D/H/D) under the same conditions. The lines are fits obtained from FISH (for W0 ) 10 without a structure factor).

W0 Variation. The results are depicted in Figure 3. The measurements were carried out 0.5 deg below the upper phase boundary except for the W0 ) 10 system, where the high value of Tu makes this point difficult to establish reliably. Here we measured at 35 °C, which is still reasonably far away from the percolation temperature (cf. Figures 2 and 5). The volume fraction of the dispersed phase increases with increasing W0. This leads to a more intense zeroangle scattering and also gives rise to the growing influence of interparticle interference manifesting in the decay of the scattering curves at low q values. Model fitting of the data was carried out using FISH according to eq 3 for discrete spheres taking into account polydispersity (Schulz distribution) and an effective hard sphere interparticle structure factor. The data and fits are presented in Figure 3; the fitting results are summarized in Table 2. The fit curves show deviations from the experimental data, especially in the low q range where the structure factor is high and the hard sphere model fails. Since these deviations are more pronounced in the core contrast fits, we put less weight on that contrast. The radii obtained from the core fits, put in parentheses in

Figure 5. Detail of Figure 2. Electrical conductivity of the W0 ) 10 system as a function of temperature. The arrows mark the temperatures that were examined in by SANS.

Table 2, nevertheless demonstrate the principal consistency of the two data sets. Figure 4 and eq 6 show the relation of droplet radius and W0 value derived from the shell contrast data.

Ri/nm ) 0.19W0 + 0.70

(6.1)

Ra/nm ) 0.22W0 + 2.05

(6.2)

Temperature Variation. To examine the temperature dependence of the phase transition in detail we have deliberately used the W0 ) 10 system, which has a broad “percolation” transition, as shown in Figure 5. The SANS data and fits as a function of temperature are presented in Figure 6. The figure clearly demonstrates an increase in scattering at low q with decreasing temperature, reflecting aggregation to larger structures. We have fitted the data using either the Teixeira fractal

1046 Langmuir, Vol. 14, No. 5, 1998

Lipgens et al.

Figure 6. (Top) SANS data for the core contrast of the W0 ) 10 system at different temperatures. (Bottom) shell contrast (D/H/D) of the same system under the same conditions. The solid lines are fits according to eq 3 with an OZ structure factor (cf. Table 3). The dotted line (top) is a fit with a fractal structure factor rescaled and compared to the OZ fit for the lowest temperatures (T ) 9.4 °C) in the low q range. Table 3. Fitting Results for the Temperature Variation of the W0 ) 10 System Obtained from FISH Using Eq 3 with the Teixeira Fractal Structure Factor [σ(R) Fixed at 0.25] (Core Contrast D/H/H) T/°C

〈Ri〉/nm

ξ/nm

D

9.4 12.8 16.1 18.2 25.0 35.0

3.03 3.10 3.13 3.07 3.17 3.00

14.9 13.6 10.8 11.9

1.3 1.2 1.1 1.1 1.0 1.0

or an Ornstein-Zernicke “critical” structure factor. The results of the fractal fit are presented in Table 3; those of the OZ fit, in Table 4. In Figure 6 (top) the two fits are compared for the lowest temperature (T ) 9.4 °C) in the low q range. Comparison of the two methods shows that the OZ fit describes the data better and gives lower residuals. The core radius 〈Ri〉 is only slightly affected by the choice of S(q). To assist in the discussion of microemulsion structure, Guinier and cross-section Guinier plots are also included (Figure 7). In these plots a linear dependence on q2 for x ) 0, 1, and 2 indicates the existence of spherical, cylindrical, and lamellar structures, respectively.52 However, from these plots alone it is hard to decide which case applies. In the percolated state (top), only lamellar structures can be ruled out. In the nonpercolated regime (bottom), the plot shows

Figure 7. Example Guinier and cross-section Guinier plots for the W0 ) 10 system: (top) T ) 9.4 °C in the percolated state (near the haze boundary); (bottom) T ) 35 °C in the nonpercolated state. Table 4. Fitting Results for the Temperature Variation of the W0 ) 10 System Obtained from FISH Using Eq 3 with an Ornstein-Zernicke Structure Factor Shell Contrast D/H/D T/°C

〈Ra〉/nm

〈Ri〉/nm

σ(Ri)

dshell/nm

ξ/nm

S0

9.4 12.8 16.1 18.2 25.0 35.0

4.28 4.44 4.43 4.31 4.35 4.20

1.71 1.73 1.96 2.24 2.37 2.71

0.29 0.31 0.29 0.26 0.26 0.25

2.57 2.71 2.47 2.07 1.98 1.49

19.0 20.0 20.0 16.0

8 6 4 2

Core Contrast D/H/H T/°C

〈Ri〉/nm

σ(R)

ξ/nm

S0

9.4 12.8 16.1 18.2 25.0 35.0

2.67 2.73 2.78 2.88 2.90 3.0

0.29 0.27 0.29 0.27 0.26 0.25

18.0 23.0 27.0 25.0

8 6 4 2

that the scattering is inconsistent with cylinders. Owing to interdroplet interactions, the expected linear dependence is not well obeyed either and the Guinier approach proves to be insufficient for the nondilute systems we examined. Additional information can be obtained from a consideration of the data in r space by determining the density correlation function. In Figures 8 and 9 the fitting results of the indirect Fourier transform method are shown. The curves clearly demonstrate the different structural patterns in r space occurring at different temperatures, and these are discussed in the next section.

Percolation in Nonionic w/o-Microemulsions

Figure 8. Results of the indirect Fourier transform method for the W0 ) 10 system at T ) 35 °C (nonpercolated): (top) experimental data (squares) and least squares fit (line) of the data in q space according to the ITP program; (bottom) corresponding pair distribution function in r space. The empty squares have not been used for fitting and transformation.

Discussion It is the aim of the present study to find out how far the SANS technique may add to the understanding of structural changes involved in the percolation transition for (w/o)-microemulsions with the nonionic surfactant IG. In particular, we concentrate on the question, to what extent does the droplet structure still remain in the percolated state? Our main reason to believe in the presence of discrete droplets comes from time-resolved Kerr effect measurements: Electric birefringence monitors both local asymmetries, like droplet deformation, and changes on the mesoscopic scale, like the formation of a percolated network. While the absolute values of the Kerr constants change strongly (being more than 2 orders of magnitude higher in the percolated than in the nonpercolated state), we found that the form contribution of the Kerr effect scales in exactly the same way (quadratic dependence) with the optical contrast (refractive index difference between water and oil phase), whether the system is percolated or not.47,48 The cluster model was also found to be appropriate to interpret the material exchange in phosphorescence quenching experiments.17 There is little doubt that near the solubilization boundary the structure of a microemulsion consists of discrete droplets. This is also true for the IG system, as we have demonstrated for the W0 ) 10 system at T ) 35 °C, since

Langmuir, Vol. 14, No. 5, 1998 1047

Figure 9. Results of the indirect Fourier transform method for the W0 ) 10 system at T ) 9.4 °C (percolated): (top) experimental data (squares) and least squares fit (line) according to the ITP program of the data in q space; (bottom) corresponding pair distribution function in r space. The empty squares have not been used for fitting and transformation.

the Guinier plots (Figure 7, bottom) rule out cylindrical and lamellar structures. Fits with the FISH software show that the assumption of spherical structures gives a good description of the data in terms of absolute intensities too. The ITP software gives an excellent fit of the data in q space for larger q values, and the corresponding correlation function clearly shows the structural pattern of spheres. The initial points at low q values, where interparticle structure factors become dominant, were not used for fitting and transformation. We also applied the pixel summation method for discrete spherical shells to approximate the scattering data. The points in Figure 10 show the correlation function derived from this method, which agrees best with that obtained from ITP. The comparison yields values of 〈Ra〉 ) 4.1 nm and 〈Ri〉 ) 2.7 nm, which are in excellent agreement with the results obtained from FISH (4.20 and 2.71 nm, respectively; cf. Table 2). Using the W0 series (cf. Figure 3) it was possible to derive a linear relationship between W0 and droplet radii (eqs 6.1 and 6.2). Assuming that the surfactant occupies a constant area per molecule at the interface, independent of droplet curvature, a linear dependence of droplet radius on W0* is to be expected. Respective relations have been established for microemulsions stabilized by several nonionic CiEj15 and the ionic AOT.44

1048 Langmuir, Vol. 14, No. 5, 1998

Lipgens et al.

Figure 10. (Solid line) pair distribution function from Figure 8 (bottom) rescaled to the dimensions of the pixel summation method. The squares are the result of the pixel summation method for polydisperse, hollow spheres (parameters: 48 spheres, Schulz-distributed radii with p ) 0.20, 〈Ra〉 ) 18 pixel, 〈Ri〉 ) 12 pixel).

Although we do not know the cµc for IG, it has essentially five ethoxy groups like C12E5. Fletcher et al. have studied the temperature dependence of cµc for the C12E5/hexane system,15 so we used their values to obtain the dotted lines in Figure 4. The corresponding area per molecule is 0.45 nm2, which is very close to the value for C12E5 for water-in-hexane systems (0.41 nm2).15 The results of the temperature study deserve extensive discussion. In our view the formation of a system-spanning network must involve aggregation of individual droplets as a first step. This is reflected by the increase in low-q-scattering with decreasing temperature. For the ionic AOT, two main approaches have been used in the literature to account for the increased scattering: (a) fractal aggregates according to Teixeira51 and (b) the OZ approach. If the laws of the percolation theory hold, as suggested by the scaling behavior of the conductivity data (µ ) 1.8), a fractal dimension of D ) 2.5 is to be expected.24 It has, however, been noticed45 that there are difficulties in describing the SANS data in this way. In this case, good fits could be obtained by using the OZ structure factor. This structure factor, introduced in the theory of critical fluctuations, arises from the Fourier transform of a radial exponentially decaying pair correlation function, thus implying spherical symmetry. We applied both approaches to fit our data and Figure 6 demonstrates that both core and shell cases can be quite well described with an OZ structure factor and discrete droplet form factors. The fits give an idea of the length scales involved in aggregation, namely several sphere diameters (cf. Table 4). Fits with a fractal structure factor have been carried out for the core data only (cf. Table 3). It can be seen that the core radii are somewhat larger, the correlation lengths on the other hand turn out smaller. The most important result is that in this q range the fractal dimension turns out to be close to unity, and it is inconsistent with the theoretical value of D ) 2.5; this value gives results too steep to account for the data. However, even for AOT systems,3 data analysis in terms of fractal aggregation yielded a value of D ) 1.7, which is as close to unity as to the predicted value of D. These low values for D suggest linear aggregates. In order to test this idea further, we applied the pixel summation method.

Figure 11. Results of the pixel summation method for polydisperse, hollow spheres. 〈Ra〉 ) 13 pixel, 〈Ri〉 ) 7 pixel, p ) 0.16: (top) calculated values of scattering intensity for aggregates of 8, 4, 2, and 1 polydisperse hollow spheres, respectively, using a total of 48 spheres for each calculation [the squares are the rescaled data of the W0 ) 10 system at 9.4 °C (percolated)]; (bottom) corresponding correlation functions.

In Figure 11 we present generated scattering curves for linearly aggregated spheres. In total, 48 spheres with Schulz-distributed radii were randomly grouped in aggregates of 8, 4, 2, and 1. The average scattering intensity and average density correlation function were determined by discrete summation, as described in the Experimental Section. The OZ structure factor at q ) 0 here corresponds to the number of spheres in the aggregate. The squares in Figure 11 represent the experimental data for the percolated system rescaled to the pixel dimensions of the calculations. Again, as in the case of model fitting, there is no perfect agreement between calculated and experimental data. Still, it is shown that the data are consistent with the assumptions of the model. Taking into account the data and the calculations, it cannot be said what the average number of droplets per cluster is, except that it must be 4 or greater. Glatter and Strey (personal communication) have pointed out that the periodically varying diameter in a linear droplet chain should also appear in the density correlation function. This is in fact true for rather monodisperse systems. Figure 11 shows, however, that even a moderate polydispersity index of p ) 0.16 smears out any oscillatory features. We also applied the ITP software to the SANS data (cf. Figures 8 and 9). The oscillations observed in Figure 9 (bottom) give support to the idea of aggregated droplets.

Percolation in Nonionic w/o-Microemulsions

However, this result may be biased by the truncation conditions. On balance, the model fitting, ITP, and pixel summation approach give very similar pictures for the changes in structure as the percolation boundary is crossed. All these methods are consistent with the progressive formation of near-linear aggregates of discrete core/shell particles with decreasing temperature in the one-phase region. It has been pointed out by F. Ehrburger-Dolle (personal communication) that percolation of clusters (as opposed to that of single droplets) may only become apparent at

Langmuir, Vol. 14, No. 5, 1998 1049

lower q values. Thus, further important information is expected from experiments in this extended range. Acknowledgment. We thank O. Glatter for letting us have his ITP software package. EPSRC are thanked for supporting this work in terms of neutron beam-time, as well as consumables and travel grants. S.L. and D.S. received EC support for travel and subsistence while at ISIS. J.E. acknowledges the CIBA GEIGY ACE foundation for meeting costs of travel to Ko¨ln. LA9701790