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Langmuir 2005, 21, 9032-9037
Articles Dielectric Relaxation in a Water-Oil-Triton X-100 Microemulsion near Phase Inversion Koji Asami Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Received January 27, 2005. In Final Form: July 13, 2005 A mixture of water (10 mM KCl), toluene and Triton X-100 (40:40:20 wt %) shows temperature-dependent phase inversion. The phase inversion has been studied by dielectric spectroscopy over a frequency range of 10 Hz to 1 GHz. At temperatures above about 37 °C, dielectric relaxation appeared around 10 MHz, which was due to interfacial polarization in a water-in-oil type emulsion. The dielectric relaxation drastically changed between 30 and 25 °C. With decreasing temperature, the intensity of dielectric relaxation increased steeply below 30 °C to attain a peak at 27 °C, where that change was associated with an increase in low-frequency conductivity by about three orders between 30 and 26 °C. The dielectric behavior has been interpreted in terms of interfacial polarization with a percolation model in which spherical water droplets, arranged in array in a continuous oil phase, are randomly connected with their nearest neighbors using water bonds.
1. Introduction Some microemulsions containing at least oil, water, and surfactants show temperature-dependent phase inversion; the types of emulsions invert around the temperature at which the “surfactant phase” appears.1,2 The surfactant phase is supposed to be bicontinuous or a lamella structure.1,2 The phase inversion where drastic structural changes occur has received much attention in studying molecular aggregation and self-organization and thus characterized by various techniques,3-6 such as NMR, light and neutron scattering, and electron microscopy. It is also particularly interesting in studying dielectric behavior of heterogeneous systems. The dielectric properties of the two types of emulsions, water-in-oil (W/O) and oil-in-water (O/W), have been well described by interfacial polarization; dielectric relaxation does not clearly appear in O/W emulsions but in W/O emulsions as expected from the interfacial polarization theories.7,8 However, there is little information on dielectric properties of microemulsions near the temperature-dependent phase inversion except Aerosol OT microemulsions9,10 although many authors have reported temperature-dependent changes in lowfrequency electric conductivity.2,9-12 * Corresponding author. E-mail:
[email protected]. Tel: +81-774-38-3081. Fax: +81-774-38-3084. (1) Shinoda, K. Prog. Colloid Polym. Sci. 1983, 68, 1. (2) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (3) Kahlweit, M.; Strey, R.; Haase, D.; Kunieda, H.; Schmeling, T.; Faulhaber, B.; Borkovec, M.; Eicke, H. F.; Busse, G.; Eggers, F.; Funck, TH.; Richmann, H.; Magid, L.; Soderman, O.; Stilbs, P.; Winker, J.; Dittrich, A.; Jahn, W. J. Colloid Interface Sci. 1987, 118, 436. (4) Bodet, J. F.; Bellare, J. R.; Davis, H. T.; Scriven, L. E.; Miller, W. G. J. Phys. Chem. 1988, 92, 1898. (5) Jahn, W.; Strey, R. J. Phys. Chem. 1988, 92, 2294. (6) Strey, R.; Winkler, J.; Magid, L. J. Phys. Chem. 1991, 95, 7502. (7) Hanai, T.; Zhang, H. Z.; Sekine, K.; Asaka, K.; Asami, K. Ferroelectrics 1988, 86, 191. (8) Hanai, T. In Emulsion Science; Sherman, P., Ed.; Academic Press: New York, 1968; p 353. (9) van Dijk, M. A.; Casteleijn, G.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1986, 85, 626. (10) Feldman, Y.; Kozlovich, N.; Nir, I.; Garti, N. Phys. Rev. E 1995, 51, 478.
The structures of nonionic microemulsions were visualized by freeze-fracture electron microscopy, indicating that tubular structures with circular cross-sections or spongelike structures appear in bicontinuous microemusions.4,5 The phase inversion through the bicontinuous phase, therefore, would cause marked changes in dielectric behavior in view of interfacial polarization as follows. Water droplets in a continuous oil phase form clusters, in which droplets dynamically interact with each other to exchange charges between them or are statically connected with water paths. Since the shape and structure of the clusters affect the charge accumulation at the interfaces between water and oil phases, it seems likely that the interfacial polarization effect increases with the cluster growth. Once water channels are formed across the sample between the electrodes, the accumulated charges are partially discharged through the channels, and the conductivity of the sample increases. The evaluation of the interfacial polarization effect near the phase inversion is difficult to be performed by conventional analytical approaches with dielectric mixture equations. Recently, a number of numerical methods have been developed to calculate the complex permittivity of complicated composite systems using the finite difference method (FDM),13-15 the finite element method (FEM),16-21 (11) Ito, M.; Takisawa, N.; Shirahama, K. J. Phys. Chem. 1990, 94, 3726. (12) Cametti, C.; Codastefano, P.; Tartaglia, P.; Chen, S. H.; Rouch, J. Phys. Rev. A 1992, 45. R5358. (13) Calame, J. P.; Birman, A.; Camel, Y.; Gershon, D.; Levush, B.; Sorokin, A. A.; Semenov, V. E.; Dadon, D.; Martin, L. P.; Rosen, M. J. Appl. Phys. 1996, 80, 3992. (14) Calame, J. P. J. Appl. Phys. 2003, 94, 5945. (15) Asami, K. J. Colloid Interface Sci. 2005 (in press) (16) Sekine, K. Colloid Polym. Sci. 1999, 277, 388. (17) Brosseau, C.; Beroual, A. Eur. Phys. J. Appl. Phys. 1999, 6, 23. (18) Tuncer, E.; Gubanski, S. M.; Nettelblad, B. J. Appl. Phys. 2001, 89, 8092. (19) Tuncer, E.; Nettelblad, B.; Gubanski, S. M. J. Appl. Phys. 2002, 92, 4612. (20) Krakovsky, I.; Myroshnychenko, C. J. Appl. Phys. 2002, 92, 6743. (21) Zhao, X.; Wu Y.; Fan, Z.; Li, F. J. Appl. Phys. 2004, 95, 8110.
10.1021/la050235u CCC: $30.25 © 2005 American Chemical Society Published on Web 08/26/2005
Dielectric Relaxation of a Microemulsion
Figure 1. Coaxial cell with an APC-7 connector for dielectric measurement. The cell except the APC-7 connector is illustrated as a cross section. (a) Sample cavity, (b) Teflon plug, (c) inner conductor (stainless steel) of 3 mm in diameter, (d) outer conductor (stainless steel) of 7 mm in diameter, (e) spacer, (f) Pt-100 temperature probe, (g) water jacket (brass), and (h) APC-7 connector.
the boundary-integral equation method (BIEM),17 and the boundary element method (BEM).22 These methods solve the Laplace equation in arbitrary composite systems to provide their effective complex permittivity. For percolation phenomena that would be related to the phase inversion of microemulsions, dielectric relaxation has been simulated with two-dimensional binary mixture structures generated randomly in square networks19 and with threedimensional binary mixtures in which cubic conductive inclusions are randomly distributed in a cubic lattice.14 The simulations, however, are not directly applicable to account for the dielectric behavior of microemulsions near the temperature-dependent phase inversion, although some important properties were pointed out. In this paper, the dielectric spectra of a mixture containing 10 mM KCl, toluene, and Triton X-100 (a nonionic detergent with a polyoxyethylene chain) (40:40:20 wt %) have been measured by varying temperature. It was found that dielectric relaxation markedly depended on temperatures between 20 and 40 °C. The contribution of interfacial polarization effects to the dielectric relaxation has been evaluated by the threedimensional FDM with a percolation model in which a regular cubic array of spherical water droplets are randomly connected with their nearest neighbors with water bonds. 2. Materials and Methods 2.1. Materials. Triton X-100, p-(1,1,3,3-tetramethylbutyl)phenoxypolyoxyethylene glycole, containing an average of 9.5 oxyethylene units per molecule, was purchased from Sigma (St. Louis). Toluene and potassium chloride were from Wako Chemical Industry (Tokyo, Japan). All of the commercially obtained chemicals were used without further purification. A 10 mM KCl solution, toluene, and Triton X-100 were blended in a weight ratio of 2:2:1, respectively. The mixture was subjected to dielectric measurements. 2.2. Dielectric Measurement. Dielectric measurements were carried out over a frequency range of 10 Hz to 1 GHz using 4192A and 4191A Impedance Analyzers (Hewlett-Packard, Palo Alto, CA). Figure 1 shows a coaxial cell with an APC-7 connector. Its sample cavity has dimensions of 3 mm inner diameter, 7 mm outer diameter, and 2 mm height. The electric length of the cell was determined to be 3.33 cm as follows. The cell was shorted at the bottom of the sample cavity with a metal plug, and the optimum electric length was searched until the absolute value and the phase angle of reflection coefficient are respectively unity and 180° over the frequency range of 1 MHz to 1 GHz. Correction for the electrical length was made using a function equipped for electrical length compensation in the 4191A Impedance Analyzer. The dielectric cell constant of the cell was 0.1 pF, which was determined with pure water and air. The cell was filled with a liquid sample, and then a Teflon plug was inserted into the cell to contact with the end of the inner conductor. The (22) Sekine, K. Bioelectrochemistry 2000, 52, 1.
Langmuir, Vol. 21, No. 20, 2005 9033 temperature of the cell was controlled by a thermostated water jacket and was measured by a Pt-100 probe fixed in the Teflon plug. The cell was connected to the 4191A Impedance Analyzer and to the 4192A Impedance Analyzer with 16085A Terminal Adapter, via a 20-cm airline, and admittance was measured by scanning the frequency of the applied ac field. Correction for the airline extension was performed by the auto-calibration function of the 4191A Impedance Analyzer with three kinds of terminations (0 ohm, 0 S, and 50 ohm) and was made with a distributed-parameter network model 23 for data obtained with the 4192A Impedance Analyzer. After the correction, the admittance of the sample was converted to the complex relative permittivity *, which is defined as * ) jκ/ω0, where is relative permittivity, κ conductivity, ω angular frequency (ω)2πf, f is frequency), 0 the permittivity of vacuum, and j2 ) -1. 2.3. Calculation of the Complex Permittivity by the Finite Difference Method. The effective complex permittivity of three-dimensional binary-composite structures was calculated by a finite-difference quasielectrostatic procedure described previously,15 which was similar to that adopted by Calame et al.12 We consider a cubic system that is a parallel capacitor filled with a binary mixture of oil and water. The system has n × n × n cubic element cells and (n + 1) × (n + 1) × (n + 1) lattice points. In this calculation, a cubic lattice with n ) 53 was used. Each cubic element cell has the complex relative permittivity of either oil or water. The potential at a lattice point is calculated from the potentials of the surrounding six lattice points and the complex relative permittivities of the surrounding eight cubic elements. The potentials on the top and bottom of the system, which correspond to electrodes or equipotential surfaces, are fixed at values of 1 and 0 Vrms, respectively. For the four sides, we assume symmetry boundary conditions for complex permittivity and potential. It means that electric flux through every side is zero. The potentials at all lattice points except those on the top and bottom were solved by the successive over-relaxation method with a relaxation factor of 1.8. The complex capacitance of the system can be calculated from the total charge accumulated on the electrode. The charge is determined from the total electric flux through a plane cut parallel to the electrode. The effective complex relative permittivity of the system is calculated from the complex capacitance. The applicability of this calculation method to binary composite systems was confirmed with a regular array of spherical particles in a continuous phase,15 for which analytical mixture equations were available. 2.4. Determination of Relaxation Parameters from Dielectric Spectra. The dielectric relaxation parameters were obtained by fitting the following equation including one (g ) 1) or two (g ) 1, 2) Cole-Cole’s terms24 to experimental and calculated dielectric spectra:
* ) ′ - j′′) h +
∆g
∑1 + (jωτ ) g
(1)
βg
g
where ′′ ) (κ - κl)/ω0, κl is the low-frequency limit of conductivity, h is the high-frequency limit of relative permittivity, ∆ is the relaxation intensity, τ is the relaxation time, and β is the ColeCole parameter (0 < β e 1). The curve-fitting was carried out by the Levenberg-Marquardt method to minimize the sum of the residuals for the real part ′ and the imaginary part ′′ of the complex permittivity
χ)
∑[′ (ω ) - ′(ω )] + ∑[′′(ω ) - ′′(ω )] 2
e
i
i
t
i
e
i
t
i
2
(2)
i
where the subscripts e and t respectively refer to experimental and theoretical values, and ωi is ith angular frequency. (23) Asami, K., Irimajiri, A., Hanai, T., Shiraishi, N., Utsumi, K. Biochim. Biophys. Acta 1984, 778, 559. (24) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341.
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Asami
Figure 2. Photographs of a mixture of 10 mM KCl, toluene and Triton X-100 (40, 40, 20 in wt %) at temperatures from 40 to 23 °C as indicated in the figure. When the sample become conductive and thus there is a considerable electrode polarization effect, the electrode polarization term is added to eq 1 as
* ) h +
∆g
∑1 + (jωτ )
βg
g
+ Aω-m
(3)
g
where A and m are adjustable parameters. In this case, only the real part of eq 3 was fitted to experimental data to minimize the following residual because of large errors in the imaginary part:
χ)
∑[log ′ (ω ) - log ′(ω )] e
i
t
2
i
(4)
i
3. Results and Discussion 3.1. Dielectric Relaxation near Phase Inversion. A mixture containing 10 mM KCl, toluene, and Triton X-100 (40, 40, 20 in wt %) was a bluish transparent liquid at temperatures between 30 and 28 °C. The sample turned to a milky turbidity above 31 °C, and it became slightly turbid below 27 °C (Figure 2). The sample was subjected to dielectric measurements over a frequency range of 10 Hz to 1 GHz. Figure 3 shows three-dimensional representations for the temperature dependence of the dielectric spectra in the temperature range of 40-20 °C. It is clearly seen that the dielectric spectra change markedly at temperatures between 30 and 25 °C. To examine the dielectric spectra in detail, typical ones cut at temperatures of 38, 30, 27, and 25 °C are shown in Figure 4. The dielectric spectrum at 38 °C showed definite dielectric relaxation around 10 MHz and an ambiguous increase in relative permittivity below 10 kHz. It is not clear whether the latter indicates another dielectric relaxation because of uncertainty of measurements at frequencies below 10 kHz. At 30 °C, there were two dielectric relaxation terms locating at frequencies below 10 kHz and around 1 MHz. The sample became conductive at 27 °C, so that the electrode polarization effect appeared below 1 kHz in addition to the two relaxation terms. At 25 °C, the electrode polarization effect became large compared with dielectric relaxation of the sample itself, thereby making it difficult to accurately characterize the dielectric relaxation. These dielectric spectra were all well represented by eq 1 or eq 3 with the best-fit dielectric relaxation parameters. Figure 5 shows the temperature dependence of the dielectric relaxation parameters. With decreasing temperature the low-frequency limit of relative permittivity l starts to increase gradually below about 36 °C and then
Figure 3. Three-dimensional representations of temperaturedependence of (a) the relative permittivity spectrum and (b) the conductivity spectrum of a mixture of 10 mM KCl, toluene, and Triton X-100 (40, 40, 20 in wt %).
Figure 4. Typical dielectric spectra extracted from Figure 3 at temperatures of (a) 38.4, (b) 30.3, (c) 27.0, and (d) 25.0 °C. The solid lines are the best-fit curves calculated from eq 1 for (a) and (b) and from eq 3 for (c) and (d).
steeply after 30 °C to attain a peak at about 27 °C. Whether the second increase exits below 25 °C is not clear because of considerable errors in correction for electrode polarization. The low-frequency limit of conductivity κl increases by about three orders between 32 and 26 °C. The dielectric spectra between 30 and 27 °C include two Cole-Cole terms. The β of the low-frequency relaxation is smaller
Dielectric Relaxation of a Microemulsion
Langmuir, Vol. 21, No. 20, 2005 9035 Table 1. Electrical Phase Parameters (P, Ka, Ep, and Kp) of W/O Emulsions Determined from Dielectric Relaxation Parameters (Kl, El, Eh, and Kh) Measured at 40 °Ca κl specimen µS m-1 1 2 a
Figure 5. Temperature dependence of the dielectric relaxation parameters estimated by fitting either eq 1 or eq 3 to the observed dielectric spectra. (a) The low-frequency limit of the relative permittivity l, (b) the low-frequency limit of the conductivity κl, (c) the characteristic frequency fc (fc ) (2πτ)-1), and (d) the Cole-Cole parameter β. In (c) and (d), note that there are points corresponding to two dielectric relaxation terms in the temperature interval between 32 and 26 °C (marked by broken lines). The solid curves are drawn for guiding eyes. Three sets of measurements are shown, which are indicated by open circles, closed circles, and open triangles.
than that of the high-frequency one, indicating a wider distribution of relaxation times. 3.2. Analysis for W/O Emulsion far above the Transition Temperature. The dielectric relaxation at temperatures above about 37 °C is characteristic of waterin-oil type emulsions. Hanai and his colleagues proved that the dielectric relaxation of emulsions was excellently simulated by Hanai’s equation25 that was an extension of Wagnar’s equation26 to high volume fractions along the Bruggeman’s effective medium approach27
( )
* - p* a* a* - p* *
1/3
)1-P
(5)
where p* (p* ) p - jκp/ω0) and a* (a* ) a - jκa/ω0) are respectively the complex relative permittivities of the particles and the medium and P is volume fraction. Since we can assume κp . κa for water-in-oil type emulsions, the phase parameters (P, p, κp, and κa) are approximately related to the dielectric relaxation parameters (l, h, κh, and κl) as7,8,25
P)1-
p ) a +
κ p ) κh
() a l
1/3
h - a 1 - (h/1)1/3
3 - (2 + a/h)(h/1)1/3 3[1 - (h/1)1/3]2 κa ) κ1(1 - P)3
(6)
(7)
(8) (9)
Hence, we can determine the phase parameters from the dielectric relaxation parameters using eqs 6-9, if the relative permittivity of the oil phase a is given. Since the determination of the phase parameters was not so sensitive (25) Hanai, T. Kolloid Z. 1960, 171, 23. (26) Wagner, K. W. Arch. Electrotech. (Berlin) 1914, 2, 371. (27) Bruggeman, D. A. G. Ann. Phys. (Leipzig) 1935, 24, 636.
7.06 8.11
l
h
37.3 12.8 33.3 12.3
κh mS m-1
P
κa µS m-1
p
κp mS m-1
14.9 14.7
0.53 0.51
0.72 0.93
33.8 33.9
76.6 82.3
The value of a was assumed to be 3.8.
Figure 6. Dielectric relaxation obtained at 40 °C. The data points of relative permittivity ′ (open circle) and loss factor ′′ (closed circle) were in good agreement with the theoretical curves (solid lines) calculated from eq 5 with the phase parameters of specimen 1 in Table 1.
to the value of a, we used a value of 3.8 for a that was obtained for a mixture of toluene and Triton X-100 in a weight ratio of 2:1. Table 1 shows the phase parameters determined in this analysis. The theoretical curves calculated from the estimated phase parameters are in good agreement with the measured data (Figure 6). The estimated volume fraction of about 0.5 is reasonable because the volume ratio of water/(toluene + water) is 0.46-0.47 in the system. The values of κp and p are smaller than those of 10 mM KCl, indicating that the droplet phase is regarded as a mixture of a 10 mM KCl solution and polyoxyethlene moieties of Triton X-100. 3.3. Evaluation of Interfacial Polarization Effects in Phase Inversion by FDM. In the preceding subsection, it was clear that the dielectric relaxation at temperatures above 37 °C was due to interfacial polarization in the W/O type emulsion. Next we will consider the drastic dielectric changes in a temperature interval below 36 °C, where the transition from the W/O type emulsion to the O/W type one through the bicontinuous phase may occur. For the sake of simplicity in modeling the bicontinuous phase, we suppose a bicontinuous cubic structure, but not lamellar structure. Since there was little difference between the dielectric spectra of the bicontinuous cubic structure and the O/W type emulsion,15 we confined our calculation to the transition from the W/O type emulsion to the bicontinuous cubic structure. For the transition, a percolation model was adopted, in which a regular array of water spheres (4 × 4 × 4) was placed in an oil phase and adjacent spheres were randomly connected by water bonds whose diameter was about one-third of the diameter of the spheres (Figure 7). To keep the volume fraction of the water phase constant, the excess water volume caused by introducing bonds was subtracted from the spheres. Here, the ratio of occupied bonds to the total available bonds (240 bonds) is referred to as the bond probability Pb. Thus, the states with Pb ) 0 and 1 correspond to the W/O emulsion and the bicontinuous phase, respectively. In this model, the surface conductance at the interface between the oil and water phases (10 mM KCl) was not taken into account because its contribution may be negligibly small in a 10 mM KCl solution.
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Figure 7. Percolation model used for FDM calculations. (a) Spherical water droplets are arranged in a continuous oil phase in array, (b) water bonds are formed between adjacent droplets at random, and (c) a bicontinuous structure is formed after all droplets are interconnected. The cross sections of the three states of the model are illustrated.
Asami
Figure 9. Typical dielectric spectra at bond probabilities Pb of 0, 0.3, and 1. Dielectric spectra are well represented by a Cole-Cole function for Pb ) 0 (open circles) and 1 (closed circles) and by a Davidson-Cole function for Pb ) 0.3 (closed triangles). The dielectric relaxation parameters used for the theoretical curves: for Pb ) 0, h ) 12.3, ∆ ) 11.1, fc ) 23.1 MHz, β ) 0.97; for Pb ) 1, h ) 13.8, ∆ ) 2.1, fc ) 32.9 MHz, β ) 0.99; for Pb ) 0.3, h ) 9.4, ∆ ) 129.7, fc ) 156 kHz, βD ) 0.48.
) 0 was almost the same as that expected from Wagner’s equation or Hanai’s one. The dielectric relaxation at Pb ) 1 had a very small intensity, being similar to that calculated for the bicontinuous cubic structure represented by Schwarz’s P surface.15 Both the dielectric spectra at Pb ) 0 and 1 were represented by a Cole-Cole function with β ≈ 1 or a Debye function. On the other hand, the dielectric spectra at Pb ) 0.3 spread over a wide frequency range, and some of them were better represented by a DavidsonCole function 28 than by a Cole-Cole function
* ) h +
Figure 8. Dielectric relaxation parameters in percolation are plotted against bond probability Pb. (a) The low-frequency limits of relative permittivity l, (b) the low-frequency limit of conductivity κl, (c) the characteristic frequency fc (fc ) (2πτ)-1) and (d) the Cole-Cole parameter β. The solid curves are drawn for guiding eyes.
The dielectric spectra for the model with varied values of Pb from 0 to 1 were calculated by FDM. The parameter values used in the calculation were taken from those estimated in the preceding subsection, i.e., ) 4 and κ ) 1 µS/m for the oil phase; ) 35 and κ ) 0.1 S/m for the aqueous phase; P ) 0.48. Dielectric relaxation parameters were obtained by fitting a Cole-Cole function to the calculated dielectric spectra. Figure 8 shows the dielectric relaxation parameters l, κl, fc, and β plotted against Pb. Around Pb ) 0.3, the low-frequency limit of relative permittivity l became largest, and both the characteristic frequency fc and the Cole-Cole parameter β had the lowest values. The low-frequency limit of conductivity κl steeply changed at Pb between 0.2 and 0.3. The data points of the dielectric relaxation parameters were widely distributed at Pb between 0.2 and 0.4, especially around 0.3, indicating that dielectric relaxation seriously depended on the shape or morphology of clusters created by connecting water spheres at random. This simulation qualitatively accounts for the temperature dependence of the dielectric relaxation parameters obtained experimentally, supposing that the bond probability increases with decreasing temperature. The peak height of l was about 10 times smaller than the experimental one, depending on the number of water droplets in the cubic lattice. Actually, preliminary calculations provided several times larger values for l at Pb ) 0.3, when the number of water droplet and the size of the lattice system were both increased eight times. Typical dielectric spectra calculated at Pb ) 0, 0.3, and 1 are shown in Figure 9. The dielectric relaxation at Pb
∆ (1 + jωτ)βD
(10)
with the Davidson-Cole parameter βD (0 < βD e 1). The Davidson-Cole type relaxation was reported in numerical calculations with some percolation models. 14, 19 To understand the widespread distribution of dielectric relaxation parameters at Pb between 0.2 and 0.4, we consider two extreme cases with the same value of Pb: (1) All water droplets are interconnected horizontally, i.e., water channels are aligned perpendicular to the electric field, and (2) those are vertically interconnected. The dielectric spectrum for the former case was almost the same as that for water droplets without interconnection. The intensity of dielectric relaxation for the latter was about five times larger than that of the former, whereas the characteristic frequency was about one-fourth that of the former. This result indicates that the parallel alignment of water channels to the electric field is much more effective for polarization than the perpendicular alignment. Hence, the direction of water channels is an important factor to determine the dielectric relaxation parameters as well as the shape and size of the water clusters. The fluctuations of these factors may cause the widespread distribution of dielectric relaxation parameters. As seen from Figure 8, l and κl gradually increased with increasing Pb for Pb < 0.2. In this region, where any water channels are not yet formed between the electrodes, the changes might result from the deformation of water droplets, i.e., from spherical to rodlike shapes. The effect of droplet shape on dielectric relaxation can be examined with an ellipsoidal model. For concentrated suspensions of randomly oriented ellipsoids, the complex permittivity was calculated by a method along Bruggeman’s effective medium theory (see the Supporting Information).29,30 To (28) Davidson, D. W.; Cole, R. H. J. Chem. Phys. 1951, 19, 1484. (29) Asami, K. Prog. Polym. Sci. 2002, 27, 1617. (30) Boned, C.; Peyrelasse, J. Colloid. Polym. Sci. 1983, 261, 600.
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Langmuir, Vol. 21, No. 20, 2005 9037
which is accompanied by decreases in fc and β. This calculation indicates that the dielectric properties with Pb < 0.2 can be interpreted in terms of deformation of water droplets. 4. Conclusion
Figure 10. Effects of water droplet shape on dielectric relaxation parameters for the W/O type emulsion. In a prolate spheroid model, the axial ratio q is changed from 1 (for spherical shape) to 4. (a) The low-frequency limit of relative permittivity l, (b) the low-frequency limit of conductivity κl, (c) the characteristic frequency fc (fc ) (2πτ)-1) and (d) the Cole-Cole parameter β. The parameter values used in the calculation are a ) 4, κa ) 1 µS/m, p ) 35, κp ) 0.1 S/m, and P ) 0.5.
simplify the calculation, only prolate spheroid was considered as the shape of water droplets. The dielectric relaxation spectra were calculated by varying the axial ratio q of the rotational axis to the radial one, and their dielectric relaxation parameters were obtained by fitting a Cole-Cole function. Figure 10 shows the dielectric relaxation parameters plotted against q. With increasing q from q ) 1 (for spherical shape), both κl and l increase,
Dielectric relaxation of a mixture containing 10 mM KCl, toluene, and Triton X-100 drastically changed in a temperature interval between 30 and 25 °C. The relaxation intensity had a peak at 27 °C, and the low-frequency limit of conductivity markedly increased with decreasing temperature from 30 to 26 °C. The experimental results were well interpreted in terms of interfacial polarization with a percolation model: water droplets in a continuous oil phase are interconnected to change their shape from spherical to tubular and then to form water channels between electrodes. Further, the simulation with the percolation model suggested that the dielectric behavior found between 30 and 25 °C is characteristic of the transition from the W/O type emulsion to the bicontinuous phase. Acknowledgment. I thank Dr. K. Sekine and Dr. R. Tanaka for his critical reading of the manuscript and helpful discussion. Supporting Information Available: Information on the calculation of complex permittivity for concentrated suspensions of ellipsoidal particles. This material is available free of charge via the Internet at http://pubs.acs.org. LA050235U
