Dielectric Model and Theoretical Analysis of Cationic Reverse Micellar

8732

Langmuir 2007, 23, 8732-8739

Dielectric Model and Theoretical Analysis of Cationic Reverse Micellar Solutions in CTAB/Isooctane/n-Hexanol/Water Systems Likun Yang and Kongshuang Zhao* College of Chemistry, Beijing Normal UniVersity, Beijing 100875, China ReceiVed March 7, 2007. In Final Form: May 12, 2007 Dielectric relaxation spectra of CTAB reverse micellar solutions, CTAB/isooctane/n-hexanol/water systems with different concentrations of CTAB and different water contents, were investigated in the frequency range from 40 Hz to 110 MHz. Two striking dielectric relaxations were observed at about 104 Hz and 105 Hz, respectively. Dielectric parameters were obtained by fitting the data using the Cole-Cole equation with two Cole-Cole dispersion terms and the electrode polarization term. These parameters show different variation with the increase of the concentration of CTAB or the water content. In order to explain the two relaxations systematically and obtain detailed information on the systems and the inner surface of the reverse micelles, an electrical model has been constituted. On the basis of this model, the low-frequency dielectric relaxation was interpreted by the radial diffusion of free counterions in the diffuse layer with Grosse model. For the high-frequency dielectric relaxation, Hanai theory and the corresponding analysis method were used to calculate the phase parameters of the constituent phases in these systems. The reasonable analysis results suggest that the high-frequency relaxation probably originated from the interfacial polarization. The structural and electrical information of the present systems were obtained from the phase parameters simultaneously.

Introduction It is well-known that surfactant/cosurfactant/water ternary systems, depending on the amount of each component, form different structures such as micelles, reverse micelles, hexagonal, lamellar, and cubic liquid crystals, and other structures.1 Reverse micelles are nanometer-sized droplets of water or polar solvent, which are surrounded by a layer of surfactant molecules and dispersed in a nonpolar solvent or weakly polar solvent.2-4 The hydrophilic headgroups of the surfactant molecules are directed toward the core of the micelles, and the hydrophobic groups are directed toward the bulk organic solvent. The cosurfactant acts as a “spacer” that minimizes repulsions between the electriferous surfactant heads.5 The size of a reverse micelle in a suspension is characterized by W0, the molar ratio of water or polar solvent to surfactant S, W0 ) [H2O]/[S]; and W0 has been shown to be directly proportional to the micellar radius, RM ) 3Vs/∑s + 3VWW0/∑s,6 so the increase of water content in the system will cause the reverse micelles to enlarge. Confined environments in reverse micelles can be used to carry out a variety of reactions, either by modifying the properties of the encapsulated liquid or by bringing reactants into close contact.7,8 Therefore, the properties of water within the reverse micelles have been extensively studied by many techniques, such as infrared spectroscopy,9 NMR spectroscopy,10 Raman and inelastic light scattering,11 * Tel: +8601058808283. E-mail: [email protected]. (1) Laughlin, R. G. The Aqueous Phase BehaVior of Surfactants; Academic Press: London, 1994. (2) Luisi, P. L., Straube, B. E., Eds. ReVerse Micelles; Plenum Press: New York, 1984. (3) Luisi, P. L. Angew. Chem., Int. Ed. Engl. 1985, 24, 439. (4) Fendler, J. H. Annu. ReV. Phys. Chem. 1984, 35, 137. (5) Schulman, J. H.; Riley, D. P. J. Colloid Sci. 1948, 3, 383. (6) Cushing, B. L.; Kolesnichenko, V. L.; O’Connor, C. J. Chem. ReV. 2004, 104, 3893. (7) Menger, F. M.; Donohue, J. A.; Williams, R. F. J. Am. Chem. Soc. 1973, 95, 286. (8) Menger, F. M.; Yamada, K. J. Am. Chem. Soc. 1979, 101, 6731. (9) Temsamani, M. B.; Maeck, M.; El, Hassani, I.; Hurwitz, H. D. J. Phys. Chem. B 1998, 102, 3335. (10) Hauser, H.; Haering, G.; Pande, A.; Luisi, P. L. J. Phys. Chem. 1989, 93, 7869.

fluorescence upconversion,12 and molecular dynamics (MD) simulations.13 Dielectric relaxation spectroscopy (DRS), which measures permittivity and conductivity as a function of frequency in a noninvasive way, can detect the structural changes in many systems in situ and provide insights into structures and electrical properties on the molecular and macroscopic levels.14 Moreover, DRS is very sensitive to all kinds of intermolecular interactions and dipole moment fluctuations,15 so it can be used to investigate the relaxation processes that originate from different polarization mechanisms over an extremely wide range of characteristic frequency (10-6-1011Hz). The information provided by DRS includes the diffusion of counterions in the compact layer and the diffuse layer, the size and the distribution of particles estimated by the relaxation time, and the properties of each phase in the heterogeneous system obtained by dielectric analysis with the proper dielectric model. Therefore, DRS is very useful in analyzing the physical and chemical properties of heterogeneous systems such as colloidal particles,16,17 micelles,18,19 reverse micelles,20,21 microemulsions,22 vesicles,23 and biological cell dispersions.24 The four-component system, CTAB/isooctane/n-hexanol/ water, contains anisotropic structures with a high dielectric permittivity dispersed in a solvent of low permittivity. It can be expected that some properties of this system can be disclosed (11) D’Aprano, A.; Lizzio, A.; Liveri, V. T.; Aliotta, F.; Vasi, C.; Migliardo, P. J. Phys. Chem. 1988, 92, 4436. (12) Willard, D. M.; Levinger, N. E. J. Phys. Chem. B 2000, 104, 11075. (13) Faeder, J.; Ladanyi, B. M. J. Phys. Chem. B 2000, 104, 1033. (14) Asami, K. Prog. Polym. Sci. 2002, 27, 1617. (15) Daniel, V. V. Dielectric Relaxation; Academic Press: London, 1967. (16) Chen, Z.; Zhao, K. S. J. Colloid Interface Sci. 2004, 276, 85. (17) He, K. J.; Zhao, K. S. Langmuir 2005, 21, 11878. (18) Yang, L. K.; Zhao, K. S.; Xiao, J. X. Langmuir 2006, 22, 8655. (19) Shikata, T.; Imai, S. Langmuir 1998, 14, 6804. (20) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Phys. ReV. E 1998, 57, 6875. (21) Angelico, R.; Palazzo, G.; Colafemmina, G.; Cirkel, P. A.; Giustini, M.; Ceglie, A. J. Phys. Chem. B 1998, 102, 2883. (22) Bordi, F.; Cametti, C. J. Colloid Interface Sci. 2001, 237, 224. (23) Schrader, W.; Halstenberg, S.; Behrends, R.; Kaatze, U. J. Phys. Chem. B 2003, 107, 14457. (24) Bai, W.; Zhao, K. S.; Mi, H. L. Bioelectrochemistry 2006, 69, 49.

10.1021/la700665s CCC: $37.00 © 2007 American Chemical Society Published on Web 07/17/2007

Cationic ReVerse Micellar Solutions

by DRS. However, most of the earlier work on reverse micelles by DRS only involves the systems which consist of anionic surfactant or nonionic surfactant,25-27 and the researches have always focused on the condition of little water content.28,29 The information obtained is very limited, because most of the previous studies only rest on the description of the dielectric parameters. In fact, the most potential application of DRS consists of its explanation of the experimental data based on the proper model. Therefore, in this paper, the dielectric spectra of CTAB reverse micellar solutions, CTAB/isooctane/n-hexanol/water systems, have been investigated over a frequency range from 40 Hz to 110 MHz. The concentration of CTAB changed from 0.02 to 0.1 mol L-1, and the water content of CTAB reverse micelles changed from 5 to 40. It is surprising that two dielectric relaxations, low-frequency relaxation and high-frequency relaxation, have been observed under all of the conditions in this system. This is similar to those in the dispersing systems of particles.30,31 This phenomenon is seldom found in other studies of reverse micelles except for the soybean lecithin/water/perdeuterated cyclohexane system.20,21 In order to clarify the two relaxations systematically, a phenomenalistic electrical model has been considered. The two relaxations were reasonably explained by the radial diffusion of free counterions in the diffuse layer with Grosse model32 and the interfacial polarization with Hanai theory and the corresponding analysis method.33,34 On the basis of the dielectric parameters, the phase parameters of the high-frequency relaxation have been calculated, which was consistent with the real system. Experimental Section Materials. Cetyltrimethylammonium bromide (CTAB), analytical grade, purchased from Boao Biologic Science and Technology Ltd Co. (Shanghai, China), was recrystallized three times from methanol/ ether, and the recrystallized CTAB was without minima in its surface tension plot. Isooctane and n-hexanol used in the syntheses were obtained from Bodi Chemical Ltd Co. (Tianjin, China) without further purification. Highly deionized water possessing specific resistance higher than 16 MΩ cm-1 was used, which was obtained from an Aquapro P Series water purification system (Taiwan). Preparation of Reverse Micelles. Reverse micelles were prepared by mixing CTAB, isooctane, n-hexanol, and water. Certain quantities of CTAB were dissolved in a constant volume ratio (9:1) of isooctane and n-hexanol. Solutions of different concentrations were taken in different topped conical flasks. Certain volumes of water were added to the mixed solutions under stirring until they became clear. The concentration of CTAB ranged from 0.02 to 0.1 mol L-1, and W0 ranged from 5 to 40. These solutions were used for dielectric measurements. Dielectric Measurement. Dielectric measurements were carried out on an HP 4294A Precision Impedance Analyzer with a 16047E spring clip fixture (Agilent Technologies) over a continuous frequency range of 40 Hz to 110 MHz. The amplitude of the applied alternating field was 500 mV, and the measurement temperature was 20 ( 0.1 °C. A dielectric measurement cell with concentrically cylindrical platinum electrodes35 was employed. The volume of the solutions (25) Gestblom, B.; Sjoblom, J. Langmuir 1988, 4, 360. (26) Asami, K. Langmuir 2005, 21, 9032. (27) Middleton, M. A.; Schechter, R. S.; Johnston, K. P. Langmuir 1990, 6, 920. (28) Peyrelasse, J.; Boned, C. J. Phys. Chem. 1985, 89, 310. (29) Freda, M.; Onori, G.; Paciaroni, A.; Santucci, A. J. Non-Cryst. Solids 2002, 307-310, 874. (30) Blum, G.; Maier, H.; Sauer, F.; Schwan, H. P. J. Phys. Chem. 1995, 99, 780. (31) Rolda´n-Toro, R.; Solier, J. D. J. Colloid Interface Sci. 2004, 274, 76. (32) Grosse, C. J. Phys. Chem. 1988, 92, 3905. (33) Hanai, T.; Koizumi, N. Bull. Inst. Chem. Res. Kyoto UniV. 1975, 53, 153. (34) Hanai, T.; Koizumi, N.; Gotoh, R. Bull. Inst. Chem. Res. Kyoto UniV. 1962, 40, 240. (35) Hanai, T.; Zhang, H.-Z.; Sekine, K.; Asaka, K.; Asami, K. Ferroelectrics 1988, 86, 191.

Langmuir, Vol. 23, No. 17, 2007 8733 used in the experiment was 1 mL in order to submerge the electrodes. The experimental data were corrected for the residual inductance arising from the terminal leads and measurement cell by using Schwan’s method.36 The stray capacitance and cell constant were determined with pure water, ethanol, and air at 20 °C. The permittivity and conductivity were calculated from the corrected capacitance and conductance. Determination of the Relaxation Parameters. DRS determines the polarization of the sample resulting from an applied electric field of frequency f. The response of the polarization of the sample is reflected in terms of the complex permittivity spectrum. In order to obtain the parameters of dielectric relaxation, such as the limiting values of low and high frequency of permittivity and conductivity and the characteristic relaxation frequency, the Cole-Cole empirical equation37 (eq 1) with two Cole-Cole dispersion terms can be used to fit the experimental data in the applied frequency range * ) ′ - j′′ ) h +

l - m 1 + (jωτl)

βl

+

 m - h 1 + (jωτh)βh

(1)

where * is the complex permittivity, ′′ ()(κ - κl)/ω0) is the dielectric loss, ω ()2πf) is the angular frequency, κl is the lowfrequency limit of conductivity, ∆ ()l - h) is the relative permittivity increment, l and h are the low- and high-frequency limits of permittivity, respectively, 0 is the permittivity of vacuum equal to 8.854 × 10-12 F m-1, f0 is the characteristic relaxation frequency, τ ()(2πf0)-1) is the relaxation time, β (0 < β e 1) is the Cole-Cole parameter indicating the dispersion of the relaxation time τ, and j ) x-1. However, the effect of electrode polarization due to the accumulation of spatial charges on the electrode surface often obscures the dielectric dispersion when it is measured in the low-frequency range. To acquire accurate values of the dielectric parameters, the electrode polarization term is added to the Cole-Cole equation (eq 1)26 * ) h +

l - m 1 + (jωτl)

βl

+

m - h 1 + (jωτh)βh

+ Aω-m

(2)

where A and m are adjustable parameters, respectively, determined by fitting the experimental data simultaneously. All the data were eventually fitted with eq 2. By using this method, the influence of electrode polarization can be subtracted from the experimental data, and the real dielectric response of the samples is obtained.

Results and Discussion Dielectric Relaxation Behavior of CTAB/Isooctane/nHexanol/Water Reverse Micelles. In order to detect the dielectric relaxation behavior under different conditions, the DRS of CTAB reverse micelles with different concentrations of CTAB and different water contents were measured. Figure 1 shows 3D representations of the dielectric spectra as the water content W0 of CTAB reverse micelles is 15. It can be seen that there are two dielectric relaxations in Figure 1a as shown by the arrows, and two peaks appear in Figure 1b indicating two relaxations too. The number of relaxations determined by the relative permittivity in Figure 1a corresponds to the number of peaks represented by the dielectric loss in Figure 1b. Similarly, Figure 2 shows 3D representations of the dielectric spectra as the concentration of CTAB reverse micelles is 0.04 mol L-1. There are also two relaxations in CTAB reverse micelles as shown by the arrows in Figure 2a and the peaks in Figure 2b. Therefore, it can be concluded that there are two relaxations in this system. (36) Schwan, H. P. In Determination of Biological Impedance; Physical Techniques in Biological Research; Nastuk, W. L., Ed.; Academic Press: New York, 1963; Vol. VI, Part B, p 434. (37) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341.

8734 Langmuir, Vol. 23, No. 17, 2007

Figure 1. Three-dimensional representations of the concentration dependence of (a) the relative permittivity spectrum and (b) the dielectric loss spectrum of CTAB/isooctane/n-hexanol/water reverse micelles at W0 ) 15.

In order to investigate the dielectric relaxation change with increasing the concentration or the water content as shown in Figures 1a and 2a in detail, the relative permittivity spectra of four conditions are displayed in Figure 3. The figure shows the best-fit results simultaneously, which are in good agreement with the experimental data. Some of the dielectric parameters obtained by fitting the experimental data with the Cole-Cole equation (eq 2) are listed in Table 1. The results should be considered as the dielectric parameters representing the real properties of the whole system, because the effect of the electrode polarization has been eliminated. From this table, it can be found that the two relaxations in CTAB reverse micelles are around 104 Hz and 105 Hz, respectively. Cole-Cole parameters βl and βh are relatively large, indicating that the dispersion of the relaxations is single, which is close to Debye-type relaxation. These results also indicate that it is reasonable to fit the experimental data with eq 2.

Yang and Zhao

Figure 2. Three-dimensional representations of (a) the relative permittivity spectrum and (b) the dielectric loss spectrum of CTAB/ isooctane/n-hexanol/water reverse micelles at C ) 0.04 mol L-1 with different water contents.

The dependence of dielectric relaxation parameters, lowfrequency permittivity l, and conductivity κl was shown in Figure 4a,b at a fixed water content. In Figure 4a, the value of permittivity increases with the concentration of CTAB. Counterions can dissociate in the water pool in ionic reverse micelles. Some of them form counterion clouds covering the surface of the water pool. If the shapes of the counterion clouds deviate from this symmetry, a dipole moment u0 will be induced. Therefore, the water pool with counterion clouds can be regarded as a dipole to some extent. The permittivity which characterizes the polarization capability of materials is related to the dipole moment u0 and the molecular number n in unit volume.38 At the beginning of the relaxation, l is close to the whole permittivity  of CTAB reverse micelles. When the water content is fixed, the number of the water pool in reverse micelles approximately increases (38) Skanavi, G. I. Dielectric Physics; National Publishing Institute of Techno-theoretical Literature, 1949.

Cationic ReVerse Micellar Solutions

Figure 3. Relative permittivity spectra extracted from Figures 1a and 2a on the condition of two concentrations of CTAB and two water contents, respectively. The symbols and the solid lines represent the experimental data and the best fitting curves evaluated from eq 2, respectively.

with the concentration of CTAB, while the size of the water pool remains constant. If the dipole moment u0 remains almost invariable as a result of the constant size of the water pool, l will increase with n. In Figure 4b, the value of κl increases linearly with the concentration of CTAB. This is with respect to the concentration of counterions, Br- anions, which increases with the concentration of CTAB in the water pool. The dependence of dielectric relaxation parameters, lowfrequency permittivity l and conductivity κl, was shown in Figure 4c,d at fixed concentration of CTAB. In Figure 4c,d, both of l and κl increase with the water content to reach a peak and then decrease. When the concentration of CTAB is fixed, the size of the water pool in the system approximately increases with the increase in the water content, while the number of the water pool basically remains constant. CTAB molecules may not dissociate thoroughly at a relatively small water content. At the primal stage of the increase of the water content, some undissociated CTAB molecules dissociated once again, so that l and κl will increase with the number of Br-. As all of the CTAB molecules dissociate with the further increase of the water content, the concentration of Br- will decrease a little as a result of the increase of the water content. Therefore, l and κl decrease eventually. Dielectric Relaxation Mechanisms in CTAB Reverse Micelles. In order to expatiate on the two relaxation processes shown in Figures 1 and 2 and analyze the inner properties of CTAB reverse micelles, the structural characterization and the distribution of counterions are presented in a structural sketch in Figure 5a. The outermost layer is composed of the bulk isooctane molecules. The surfactant molecules of CTAB and the cosurfactant n-hexanol are located together in the surfactant molecular layer. The dissociated counterions from the surfactants, Br- anions, and all of the water molecules exist in the water pool. It can be noticed that there are three components with different electrical properties in CTAB reverse micelles, which are the oil phase, the surfactant molecular layer, and the water phase with counterions, respectively. Generally speaking, there are two characteristic dielectric relaxations in a heterogeneous structure of dispersing systems measured under the usual frequency range (from 1 Hz to 1 × 108 Hz). They appear at the low-frequency range and highfrequency range, respectively. Both are sensitive to the polarization of the electric double layer of the dispersing particles as well as the electric properties of the constituent phase. The lowfrequency relaxation occurs below a few kilohertz, caused by the tangential current of bound counterions in the electric double layer or the diffusion of counterions in the bulk solution.32 The high-frequency relaxation appears at several megahertz, originated

Langmuir, Vol. 23, No. 17, 2007 8735

from the Maxwell-Wagner interfacial polarization effect in the conventional meaning.39,40 The two relaxations in this investigated system are in accordance with those of the ordinary dispersing systems of particles. Therefore, it can be speculated that the diffusion of Br- ions may result in the dielectric relaxation of low frequency because of the existence of large numbers of Brions in the diffuse layer. Simultaneously, the presence of the large interface between the reverse micelles and the oil continuous phase probably causes the other relaxation. In order to analyze the high-frequency relaxation, a dielectric model is considered in Figure 5b based on Hanai theory. In this model, the surfactant molecular layer and the inner water pool are considered as a whole, which are dispersed in the continuous oil medium. The radius of the whole sphere is R, which includes the thickness of the surfactant molecular layer (R - Rw) and the radius of the water pool (Rw). The apparent complex permittivity of this sphere is i* and the apparent complex permittivity of the continuous oil phase is a*. The apparent complex permittivity * of the whole reverse micellar system is defined as * )  - jκ/ω0. Low-Frequency Dielectric Relaxation. Low-frequency relaxation was first given a theoretical explanation by Schwarz.41 Subsequently, Dukhin and Shilov42,43 developed a new approach to this problem based on the concept of a diffuse double layer. Grosse32 presented a simple model for the dielectric properties of suspensions of charged particles in an electrolyte solution, and the Grosse model provides the calculation formula for the two mean relaxation times as follows, by which we can estimate the relaxation time in the present system

τ1 ) r2/Dt

( ) ( )

p +2 m τ2 ) 2λs +2 κm rκm

(3)

0m

(4)

where Dt is the diffusion coefficient of the counterions; r is the radius of a spherical particle; m, κm, p, and κp are the permittivity and conductivity of a medium and a particle, respectively; and λs is the surface conductivity arising from the tangential motion of bound counterions. It is well-known that the value of λs is about 10-9 Ω with respect to the order of magnitude for many systems;44 therefore, λs in CTAB reverse micelles can be estimated. The values of τ2 calculated from eq 4 under all conditions are equal to 10-9 s, which is much less than the experimental values. That is, the rapid tangential motion of the counterions along the surface of CTAB reverse micelles is not the natural reason for the dielectric relaxation. The diffusion coefficient Dt of Br- anions in the aqueous solution is about 2.1 × 10-5 cm2 s-1.45 The average radius Rw of the water pool of CTAB reverse micelle shown in Figure 5a is about 3 nm.46 So, the relaxation time τ can be estimated with eq 3, equal to about 5 × 10-9 s. This relaxation time τ is also less than the low-frequency relaxation time obtained from the (39) Maxwell, J. C. A Treatise on Electricity and Magnetism, 3rd ed.; Clarendon Press: Oxford, England, 1891; Chapter 4. (40) Wagner, K. W. Arch. Elektrotech. (Berlin) 1914, 2, 371. (41) Schwarz, G. J. Phys. Chem. 1962, 66, 2636. (42) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes; Halsted: Jerusalem, 1974. (43) Shilov, V. N.; Dukhin, S. S. Colloid J. 1970, 32, 293. (44) Baar, C.; Buchner, R.; Kunz, W. J. Phys. Chem. B 2001, 105, 2914. (45) Imai, S.; Shikata, T. Langmuir 1999, 15, 8388. (46) Das, P. K.; Chaudhuri, A. Langmuir 2000, 16, 76.

8736 Langmuir, Vol. 23, No. 17, 2007

Yang and Zhao

Table 1. Partial Dielectric Parameters of CTAB Reverse Micellesa concn (mol L-1) 0.04 0.1

W0

l

m

h

∆l

∆h

βl

βh

f0l (kHz)

f0h (kHz)

15 25 15 25

1.390 1.414 1.559 1.676

0.882 0.950 0.996 1.148

0.692 0.746 0.928 1.031

0.508 0.464 0.563 0.528

0.190 0.204 0.068 0.117

0.859 0.922 0.863 0.879

0.898 0.913 0.904 0.78

11.1 14.4 31.0 25.6

237.8 372.0 508.8 225.0

a Permittivity increment of low-frequency dielectric relaxation, defined as ∆l ) l - m, and that of high-frequency relaxation, defined as ∆h ) m - h.

Figure 4. Dependence of two dielectric relaxation parameters evaluated by fitting eq 2 to the observed dielectric spectra. (a,c) Low-frequency limit of the relative permittivity l; (b,d) low-frequency limit of the conductivity κl. There are two sets of data in all graphs, which are indicated by squares and circles.

Figure 5. (a) Structural sketch of CTAB/isooctane/n-hexanol/water reverse micelle. (b) Dielectric model of high-frequency relaxation.

experiment (about 8 × 10-6 s). The difference between the calculated relaxation time and the experimental data is due to the variation of the microenvironment in the water pool of reverse micelles compared with the surroundings of the aqueous medium. The variation of the microenvironment will change the diffusion coefficient indeed.47-49 For example, the obtained lateral surface

diffusion coefficient of the Br- anion is about 10-9 m2/s, roughly half of the bulk diffusion coefficient of Br-.47 On the other hand, the diffusion of the counterions taking place in the inner water pool is different from the typical model. On this condition, the (47) Hedin, N.; Furo, I. J. Phys. Chem. B 1999, 103, 9640.

Cationic ReVerse Micellar Solutions

Langmuir, Vol. 23, No. 17, 2007 8737

Table 2. Phase Parameters of W0 ) 15 in CTAB Reverse Micelles

limits are given by

concn (mol L-1)

κa (S m-1)

φ

i

κi (S m-1)

V1

0.02 0.04 0.06 0.08 0.1

1.041 × 10-6 2.817 × 10-6 3.901 × 10-6 4.976 × 10-6 5.999 × 10-6

0.1770 0.2257 0.2651 0.2927 0.3165

48.21 48.64 48.87 49.25 49.57

3.803 × 10-5 1.084 × 10-4 1.259 × 10-4 1.610 × 10-4 2.076 × 10-4

0.5577 0.5637 0.5669 0.5721 0.5766

Table 3. Phase Parameters of C ) 0.04 mol Reverse Micellesa

L-1

κa (S m-1)

φ

i

κi (S m-1)

V1

5 10 15 20 25 30 35 40

4.338 × 10-7 1.280 × 10-6 2.817 × 10-6 2.941 × 10-6 2.806 × 10-6 2.837 × 10-6 1.676 × 10-6 1.585 × 10-6

0.1952 0.2185 0.2257 0.2395 0.2607 0.2618 0.2788 0.2994

42.02 44.80 48.64 49.58 50.77 52.35 54.62 55.94

3.279 × 10-5 9.384 × 10-5 1.084 × 10-5 9.057 × 10-5 9.442 × 10-5 1.051 × 10-4 1.371 × 10-4 6.888 × 10-5

0.4718 0.5104 0.5637 0.5767 0.5932 0.6151 0.6466 0.6649

a The two relaxations are explained by the diffusion of counterions in the diffuse layer with the Grosse model and the interfacial polarization with Hanai theory.

diffusion of the counterions is counteracted in the confined space. Therefore, a considerable amount of bound water existing in the water pool can lead to the decrease of Dt of Br- ions. As a result, the term of Dt in eq 3 must be modified by a corrected factor A, which means the degree of the decrease of Dt, as shown in eq 5. So, it can be speculated that the radial diffusion of the free counterions in the water pool is the essential mechanism of the low-frequency dielectric relaxation in this system. This model is shown in Figure 5a. From the analysis above, it also can be concluded that the effect of the variation of the microenvironment on Dt is very obvious.

τ1 ) r2/(Dt - A)

a

i

()

1/3

)1-φ

(6)

where the subscripts a and i represent the continuous phase and the dispersed phase respectively, and φ is the volume fraction of the dispersed phase. The low- and high-frequency (48) Okamoto, K.; Hirota, N.; Tominaga, T.; Terazima, M. J. Phys. Chem. A 2001, 105, 6586. (49) Shah, D. M.; Davies, K. M.; Hussam, A. Langmuir 1997, 13, 4729. (50) Bruggeman, D. A. G. Ann. Phys. (Leipzig) 1935, 24, 636.

1/3

)1-φ

(7)

) )

l

i a a - i 1 3 )3 + κ l - κi κ l κa - κi κl - κi κa

(8)

κh

κi κa κa - κi 3 1 )3 + h - i h a - i h - i a

(9)

κl - κi κ a κa - κi κl

1/3

)1-φ

(10)

Since the dielectric properties of reverse micelles which can be characterized by the phase parameters are useful for the application in synthesis, it is necessary to evaluate these parameters of the inner phases from the dielectric parameters. On the basis of eq 6, Hanai provides a systematic method to calculate the phase parameters,51 which has been successfully applied to many actual systems.52,53 The expressions of the phase parameters are obtained after the cumbersome mathematical treatments. For simplicity, a formula has been defined

C≡

() h a

1/3

‚(1 - φ)

(11)

From eqs 7, 8, and 10, it can be known that

C)

-Q - xQ2 - 4PR 2P

(12)

where

P)

(5)

High-Frequency Dielectric Relaxation. Different from the dielectric parameters, the phase parameters refer to a series of parameters presenting the inner properties of the continuous phase and dispersed phase, such as the permittivity and conductivity of the two phases, and the volume fraction of the dispersed phase. CTAB reverse micelles can be considered as spherical droplets dispersed in a continuous oil phase. Therefore, Hanai theory and the relevant analysis method can be applied to the present systems. It has been proven that Hanai theory is appropriate to the ordinary emulsions,33,34 which are similar to the reverse micelles, and this theory is an extension of Wagner’s equation40 to high volume fractions along the Bruggeman’s effective medium approach.5050 It can be expressed by the following equation in a complex relative permittivity

* - /i /a / - / *

( (

in CTAB

W0

() ) ( ) ( ()

 h - i  a a - i h

( )

( )

κa κl + 2 lD - 3[hD - a(D - 1)]D + - 1 aD (13) κl κa

Q ) 3[2hD - a(D - 1)] κa κl + 2 D + 3 l - - 1 aD (14) κl κa

[( ) ] ( )

R ) 3(l - h)

(15)

and

D)

( ) aκl hκa

1/3

(16)

Eventually, the function C given by eq 12 is a complicated function of κa. Next, eqs 7 and 10 are substituted for eq 9 to eliminate i and κi. Thus, the following equation is obtained:

[ ( )]

a C (1 - DC)κh - 3{κl - [κa(D - 1) + h h κl]C}(1 - C) + κa 1 - C(1 - DC) ) 0 (17) a

J(κa) ≡ 3 - 2 +

( )

(51) Hanai, T.; Ishikawa, A.; Koizumi, N. Bull. Inst. Chem. Res. Kyoto UniV. 1977, 55, 376. (52) Ishikawa, A.; Hanai, T.; Koizumi, N. Bull. Inst. Chem. Res. Kyoto UniV. 1984, 62, 251. (53) Zhao, K. S.; Asami, K.; Lei, J. P. Colloid Polym. Sci. 2002, 280, 1038.

8738 Langmuir, Vol. 23, No. 17, 2007

Yang and Zhao

Figure 6. Dependence of several phase parameters calculated by eq 6 on the basis of the dielectric parameters in CTAB reverse micelles. (a,b) Volume fraction φ of the dispersed phase; (c,d) permittivity i of the dispersed phase; (e,f) conductivity κi of the dispersed phase. There are two sets of data in all graphs, which are indicated by squares and circles.

If l, h, κl, κh, and a are given through eqs 12 and 16, the left-hand side of eq 17, which is a formula abbreviated as J(κa), is a function of κa. Although eq 17 cannot be solved for κa due to the complicated functional form, computers have made it possible to search out a root for J(κa) ) 0 numerically. Using the calculated value of κa in the above way, the values of φ, i, and κi then can be derived.

φ)1-

1/3

‚C

(18)

 h - aC 1-C

(19)

κl - κaDC 1 - DC

(20)

i ) κi )

() a h

The permittivity a (0.3) and the conductivity κa (3 × 10-6 S/m) of the continuous phase, which have been measured in an individual experiment, are used as a known quantity during all of the calculations. By using eqs 17-20, the phase parameters of CTAB reverse micelles were calculated, and the partial results

are listed in Tables 2 and 3, respectively. The information about microstructure and interface between the oil phase and the water phase is acquired simultaneously. From these tables, it can be noticed that, if the water content is fixed, φ and i increase with the concentration of CTAB; as the concentration of CTAB is fixed, φ and i also increase with the water content. These relationships are approximately linear as shown in Figure 6a-d. Because the water pool and the surfactant molecular layer of CTAB reverse micelles are regarded as a whole by Hanai theory, the volume fraction φ obtained from this theoretical calculation comprises both the dispersed phase (water) and the CTAB layer (n-hexanol and CTAB). This model is also shown in Figure 5b. It is rational that the values of φ in Tables 2 and 3 are much higher than those of the water content in the experiments. Following the increase of the water content or the concentration of CTAB, the value of φ will also increase, as shown in Figure 6a,b. Similarly, since the molecules of water, n-hexanol, and CTAB are considered simultaneously, the value of permittivity of the dispersed phase i is much lower than that of the pure water (80.1 at 20 °C). So, the values of i which are between 42 and 57 in our calculated results are reasonable. As the water

Cationic ReVerse Micellar Solutions

Langmuir, Vol. 23, No. 17, 2007 8739

the concentration of CTAB or the water content, whereas the extent of the increase is obviously different. When the water content is fixed, the size of the water pool remains constant with the concentration of CTAB. So, the increase of V1 is very slow in Figure 7a, or it can be considered constant compared with Figure 7b. The concentration of CTAB is fixed; the size of the water pool increases with the water content. Thus, it can be seen that the increase of V1 in Figure 7b is very apparent. From the analysis above, it can be concluded that the highfrequency relaxation is probably ascribed to the interfacial polarization, which is a result of the accumulation of virtual charge at the interface between the two phases with different permittivities and conductivities. The structural and electrical information about CTAB reverse micelles is obtained by analyzing dielectric spectra with Hanai theory and the corresponding method.

Conclusion

Figure 7. Relationship of V1 calculated by eq 16 on the basis of i in CTAB reverse micelles with the water content (a) or the concentration of CTAB (b). There are two sets of data in all graphs, which are indicated by squares and circles.

content or the concentration of CTAB increases, the value of i increases too, as shown in Figure 6c,d. The concentration of counterions increases with the concentration of CTAB; therefore, the value of κi in Figure 6e also increases. However, the value of κi shows different variational tendency in Figure 6f. Because the partial counterions continue to dissociate at the beginning of the increase of the water content, the value of κi also increases. When the water content increases further, the concentration of counterions will decrease, resulting in the value of κi decreasing. These varieties are consistent with the dielectric parameters given in Figure 4. According to the phase parameter i of the whole micellar sphere presented in Tables 2 and 3, the volume ratios of water distributed in the dispersed phase of CTAB reverse micelles were approximately evaluated by eq 21 where the subscripts 1

i ) V1i1 + (1 - V1)i2

(21)

and 2 represent the two components of water and the mixture of n-hexanol and CTAB, and V1 ()Rw3/R3) is the volume ratio of water in the dispersed phase. The results of V1 in Tables 2 and 3 are also plotted in Figure 7. The values of V1 are coincident with the literature,46 basically indicating that the phase parameters calculated on the basis of the electrical model proposed above are suitable. The relationship between V1 and the concentration of CTAB or the water content in CTAB reverse micelles is also presented in Figure 7. From the figure, it can be seen that V1 increases almost linearly with

In this article, dielectric relaxation spectra of CTAB reverse micelles were investigated under different concentrations of CTAB and water contents, and the dielectric spectra characterizing two relaxation processes were found. The dielectric parameters reflecting the dielectric relaxation spectroscopy were obtained by fitting the experimental data with the Cole-Cole equation. Both of the low-frequency limits of relative permittivity l and conductivity κl increased with the concentration of CTAB, while they increased with the water content at first and then gained a maximum. The properties of the dispersed phase changed differently with the increase of the concentration of CTAB or the water content, so the two parameters showed different variation. A modified electrical model for explaining systematically the two relaxations occurring in our investigating systems was proposed. In light of the model combining the Maxwell-Wagner interfacial polarization theory and the Grosse model, the relaxation mechanisms of these two relaxations were interpreted reasonably. In addition, the phase parameters of the high-frequency relaxation which is consistent with the real system composed of the micellar phase and the oil medium were calculated by Hanai theory and the relevant analysis method. On the basis of the phase parameters, the structural and electrical information of CTAB reverse micelles formed under the conditions of varying water content and concentration of CTAB was obtained. The present study shows that it is feasible to obtain many local properties of reverse micelles just by dielectric measurements, while the acquisition of these properties maybe require more instruments in general. Furthermore, as another successful example, it also demonstrates the advantage of DRS in obtaining inner information of intricate heterogeneous systems once again. Nevertheless, it should be pointed out that the analysis of dielectric spectra based on an appropriate model and calculation approach is essentially indispensable. Acknowledgment. This work is financially supported by the National Nature Science Foundation of China (no. 20673014). LA700665S