6804
Langmuir 1998, 14, 6804-6810
Dielectric Relaxation of Surfactant Micellar Solutions Toshiyuki Shikata* and Shin-ichiro Imai Department of Macromolecular Science, Osaka University, Toyonaka, Osaka 560-0043, Japan Received April 14, 1998. In Final Form: August 3, 1998 Dielectric relaxation behavior of spherical micelles of cationic surfactants, dodecyltrimethylammonium bromide, tetradecyltrimethylammonium bromide, and cetyltrimethylammonium bromide, in aqueous solution was examined in the frequency range from 106 to 109 Hz. Each surfactant solution shows pronounced dielectric dispersions with two distinct relaxation times above the critical micelle concentrations. A slow relaxation mode with a relaxation time τs ∼ 5 ns reduced its magnitude with increasing the concentration of surfactants (CD). On the other hand, the magnitude of a fast relaxation mechanism with the relaxation time τf ∼ 0.5 ns was proportional to CD. The slow relaxation mode was attributed to the fluctuation of a counterion distribution around the spherical micelle or that of a distribution of dissociated cationic surfactant headgroups on the micellar surface. The fast relaxation mode was assigned to the rotational motion of ionic pairs consisting of surfactant cations and counteranions in the spherical micelle.
Introduction Dielectric relaxation analysis (DRA) is known as a classical but useful technique to investigate molecular motions of functional groups with dipole moments.1 In some cases, DRA is also a powerful method to evaluate macroscopic dielectric properties of colloidal systems such as (oil in water (O/W)2 and/or water in oil (W/O)3) emulsions, vesicles,4 microcapsules,5 and biological cell dispersions,6 which have more than two electrically different components separated with insulation shells. The dielectric properties of these colloidal dispersion have been discussed with a Maxwell-Wagner theory7,8 based on macroscopic electric polarization on the interface between two different components. However, a few numbers of studies have been reported on motions of molecules which construct dispersed phases such as micelles. Molecular motions of headgroups of surfactant molecules in lecithin inverse micellar systems,9 in W/O microemulsions10 and also in aqueous lamella liquid crystalline systems11 have been studied with DRA. Recently, Barchini and Pottel12 reported dielectric relaxation behavior of aqueous surfactant micellar solutions. And they developed a theoretical model to explain experimental results. When one simply considers dispersion systems of spherical micelles formed with ionic surfactant molecules in aqueous media, the system has at least two components with electrically different properties. The first component is an aqueous medium containing dissociated counterions from surfactants, and the second one is a hydrophobic (1) For example: Daniel, V. V. Dielectric Relxation; Academic Press: London, New York. 1967. (2) Hanai, T.; Imakita, T.; Gotoh, R. Kolloid-Z. 1962, 184, 143-148. (3) Hanai, T.; Imakita, T.; Koizumi, N. Colloid Polym. Sci. 1982, 260, 1029-1034. (4) Sekine, K.; Hanai, T.; Koizumi, N. Bull. Inst. Chem. Res., Kyoto Univ. 1983, 61, 299-313. (5) Sekine, K. Colloid Polym. Sci. 1987, 265, 1054-1060. (6) Asami, K.; Hanai, T.; Koizumi, N. Biophys. J. 1980, 31, 215-228. (7) Wagner, K. W. Arch. Elektrotech. 1914, 2, 371. (8) Hanai, T. Kolloid-Z. 1961, 177, 57-61. (9) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Prog. Colloid Polym. Sci. 1997, 105, 204-208. (10) Firetto, D.; Freda, M.; Onori, G.; Santucci, A. Prog. Colloid Polym. Sci. 1997, 105, 256-259. (11) Kaatze, U.; Henze, R.; Eible, H. Biophys. Chem. 1979, 10, 351362. (12) Barchini, R.; Pottel, R. J. Phys. Chem. 1994, 98, 7899-7905.
micellar interior filled up with surfactant alkyl tails. The surface of a micelle, which may be the third component, has electric charges arising from dissociated surfactant headgroups. Thus, an electrical condition of each component in a spherical micellar solution is not so different from that in the O/W emulsion, except for differences in sizes of droplets of the emulsion and the micelle. However, DRA has not been conducted successfully for physicochemical research on the aqueous spherical micellar system. The reason for difficulty in the dielectric research should be high electric (direct current) conductivity of the aqueous ionic micellar solution and electrode polarization. The high electric conductivity remarkably screens off small dielectric dispersions resulting from what one wants to detect. The electrode polarization sometimes strongly distorts frequency dependencies of the dielectric constant (′) and the dielectric loss (′′) especially in the lowfrequency side. In this study, dielectric relaxation behavior was examined for aqueous spherical micellar solutions of cationic surfactants such as dodecyltrimethylammonium bromide, tetradecyltrimethylammonium bromide, and cetyltrimethylammonium bromide. Because the frequency range covered in this study was rather high, 6.28 × 106 to 6.28 × 109 rad s-1 in angular frequency (ω), the contribution of the electrode polarization to the dependencies of ′ and ′′ on ω was not so serious. Therefore, DRA in the high ω range allows us precise discussion on molecular motions of surfactants in micelles. Obtained dielectric spectra were analyzed with two ideas. One is based on the fluctuation of a counterion distribution around charged spherical micelles. The other is that ionic pairs between surfactant cations and counteranions exist, and the ionic pairs possess fast molecular motions in the spherical micelle. Experimental Section Materials. Surfactants, dodecyltrimethylammonium bromide (DTAB), tetradecyltrimethylammonium bromide (TTAB), and cetyltrimethylammonium bromide (CTAB), were purchased from Wako Pure Chemicals, Ltd. (Osaka), and were purified twice by recrystalization with mixed solvents of methanol and acetone or methanol and tetrahydrofuran. Highly deionized water with the specific resistance more than 18 MΩ cm-1 was obtained with a Milli Q SP system (Millipore). The concentration of cationic surfactant (CD) for dielectric and electric conductivity measurements was up to 100 mM.
10.1021/la980421i CCC: $15.00 © 1998 American Chemical Society Published on Web 11/06/1998
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Figure 2. Dependencies of dielectric constant, pw′, and loss, pw′′, for pure water on frequency, ω, at room temperature estimated from highly deionized water with a measuring system used in this study: 9, ′′; 0, pw′′ ) ′′ - GdcCo-1ω-1. Solid lines are mean values of pw′ and pw′′ determined by Kaatze.13
Figure 1. Illustration for a handmade electrode cell used in this study. Method. An impedance analyzer (rf impedance analyzer 4191A, Hewlett-Packard) equipped with a handmade electrode cell was used to measure the dielectric spectra in the frequency (ω) range from 6.28 × 106 to 6.28 × 109 rad s-1 at room temperature, ca. 25 °C. A schematic illustration for the handmade electrode cell is depicted in Figure 1. Data were collected in the form of parallel connection of capacitance (C) and conductance (G) as functions of ω. The ω dependencies of the (relative) dielectric constant and loss, ′ and ′′, were evaluated as ′ ) CC0-1 and ′′ ) GC0-1ω-1, respectively: C0 meant the capacitance of the vacant electrode cell and was estimated to be 0.17 × 10-12 F. The increment in ′ from the value of water (w′), ∆′ ) ′ - w′, was estimated as the pure contribution to the dielectric constant of cationic surfactant micelles. We used ω independent w′ values from 78 to 74, because it is known that the dispersion due to the micelle was well separated from that due to free water molecules with a relaxation time of 8.3 ps.12 The w′ value used decreased slightly with CD as reported by Barchini and Pottel. They reported dielectric relaxation behavior for aqueous CTAB solutions and also for aqueous sodium dodecyl sulfate (SDS) solutions in the ω range from 6.28 × 106 to 1.88 × 1011 rad s-1.12 The trend of a change in the w′ value with CD for CTAB systems in this study reasonably agreed with that obtained by them. They expressed the w′ value in the form of w′ ) rw′ + ∞, where rw′ and ∞ (ca. 5.3) were the magnitude of a relaxing and a nonrelaxing parts of water molecules, respectively. On the other hand, pure contribution to the dielectric
loss was estimated in the form of ∆′′ ) ′′ - w′′ - GdcC0-1ω-1, where Gdc and w′′ were ω independent (and non-Faradeic) direct current conductivity of micellar solutions and the dielectric loss of water molecules, respectively. Changing the Gdc value carefully, we estimated reasonable ω dependence of ∆′′ curves which were consistent with those of ∆′ curves, because ∆′ and ∆′′ were not independent, but must satisfy the KramersKroning’s dispersion formula.1 The w′′ value used for estimation of ∆′′ was evaluated as fpw′′, where f and pw′′ were a factor of contribution of the pure water and the dielectric loss obtained from the pure water. The factor f was determined as to be rw′/ (78 - 5.3) and was not so different from unity. Most of ∆′ spectra obtained in this study were fairly reproduced with a model which is the connection of two Debey type elements1 possessing two relaxation times, τf and τs, and strengths, ∆f and ∆s, as given by eq 1. The Gdc value for each system was determined by fitting the spectrum of ∆′′ to calculated loss curves with eq 2.
∆′ )
∆′′ )
∆f 2
2
1 + ω τf ∆fωτf
1 + ω2τf2
+
∆s 1 + ω2τs2
+
∆sωτs 1 + ω2τs2
(1)
(2)
The electric conductivity (G0) of the micellar solution in the low ω side around ∼104 rad s-1 was measured by using a conventional electric conductivity meter (ES-14, Horiba, Kyoto) at room temperature. In principle, the physical meaning of G0 is the same as that for Gdc, if dielectric dispersions complete in the low ω side where G0 is measured.
Results Calibration of the Measuring System. First of all, the performance of the measuring system was tested by estimating dielectric constant, pw′, and loss, pw′′, of the pure water. Figure 2 shows ω dependencies of obtained ′ and loss ′′ for highly deionized water. Because the water used still has a small amount of ionic impurity, steep increasing in ′′ in the low ω side owing to the direct current, Gdc, is observed as shown in Figure 2 (closed symbols). The Gdc value can be estimated to be 26.5 ×
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Figure 3. Dependence of the dielectric increment, ∆′ ) ′ w′ and ∆′′ ) ′′ - w′′, on ω for CTAB aqueous solutions at several concentrations at room temperature. Thick and thin solid lines respectively represent the best fitting curves for ∆′ and ∆′′ evaluated with the two relaxation model given by eqs 1 and 2 (see text).
10-6 S; therefore, the pure contribution, pw′′, of water molecules is evaluated by subtracting GdcC0-1ω-1 values from the ′′ spectrum as shown with open symbols in the same figure. The dependencies of pw′ and pw′′ on ω perfectly coincide with those found in the literature (solid lines in Figure 2).13 Thus, we conclude the measuring system used here performs correctly. Micellar Systems. The dependencies of ∆′ and ∆′′ spectra on the surfactant concentration (CD) for CTAB and TTAB are shown in Figures 3 and 4. The critical micelle concentrations (cmc’s) for CTAB and TTAB are 0.8 and 3.8 mM at 25 °C,14 respectively. Because the CD values for both systems are higher than cmc’s, all ∆′ spectra shown in Figures 3 and 4 mean the contribution of spherical micelles. It is quite clear that both spherical micellar systems have remarkable dielectric relaxation behavior because of steep decreases in ∆′ spectra in the ω range higher than 109 rad s-1. Except for systems at 100 mM, ∆′ spectra are rather broader and can be described fairly well with the two-relaxation model cast into eqs 1 and 2. Solid lines drawn for systems at CD ) 10, 30, and 70 mM in Figures 3 and 4 for ∆′ spectra are the best fitting curves with adequate τs, τf, ∆s, and ∆f. The Gdc values were determined by very careful value setting to provide the best agreement between obtained ∆′′ and the dielectric loss increment curves calculated with eq 2 which are shown with thin solid lines. On the other hand, spectra for solutions at CD ) 100 mM in both CTAB and TTAB systems also show pronounced dielectric relaxation behavior in the high ω side, which possess the clear maximum in the ∆′′ curves. These are well described by a relaxation model with a single relaxation time. The (fast) relaxation times (τf) are easily estimated to be τf ∼ 0.5 ns from the reciprocal of the maximum frequencies of the ∆′′ spectra. Solid lines which (13) Kaatze, U. J. Chem. Eng. Data 1989, 34, 371-374. (14) Zielinski, R.; Ikeda, S.; Nomuran H.; Kato, S. J. Colloid Interface Sci. 1989, 129, 175.
Shikata and Imai
Figure 4. Dependencies of ∆′ and ∆′′ on ω for TTAB aqueous solutions at several concentrations obtained at the room temperature. Thick and thin solid lines respectively represent the best fitting curves for ∆′ and ∆′′ evaluated with the tworelaxation model.
fit well to ∆′ and ∆′′ for 100 mM systems in these figures are the best fitting curves based on the single relaxation model. Increase in ∆′ curves in the low ω side is observed more or less in all the solutions examined in this study. Because the handmade electrode used in this study is made with brass, the electrode polarization effect to dielectric spectra is not removed completely. Especially, in the case of a solution with small dielectric dispersions in magnitude, the effect of the electrode polarization looks pronounced in ∆′ spectra. Increases in ∆′ curves in the low ω side were also observed commonly in the aqueous solution of NaBr with the same measuring system, in which ∆′ was proportional to the square of ω and increased with the concentration of NaBr. Thus, the increase in ∆′ in the low ω side is not the contribution of the dielectric relaxation behavior owing to the presence of micelles in the solution but to the electrode polarization.15 Therefore, the Gdc values necessary to evaluate ∆′′ for all the solutions examined were estimated by neglecting the increase in ∆′ in the low ω side. Because cmc of DTAB is 14.6 mM at 25 °C,14 the ∆′ spectrum for a solution at 10 mM (Figure 5) means those of a micelle free solution, which has weak ω dependence. Since reliable resolution for ∆′ and ∆′′ estimated by the measuring system is ca. 0.2, the ∆′′ values less than 0.2 are not plotted in Figure 5. In the solution at 10 mM, every DTAB molecule dissociates into a free cation (DTA+) and an anion (Br-). They are hydrated as in an aqueous solution of a simple salt such as LiBr. It has been reported that in the aqueous solution of simple salts significant decreasing in ′ with the salt concentration, which means negative ∆′, is observed at high salt concetration.16,17 Although electrical conditions that occur in the DTAB solution below cmc are very close to those in the simple (15) Cirkel, P. A.; van der Ploeg, J. P. M.; Koper, G. J. M. Physica A 1997, 235, 269-278. (16) Hasted, J. B.; Riston, D. M.; Colli, C. H. J. Chem. Phys. 1948, 16, 1. (17) Pottel, B. R.; Lossen, O. Ber. Bunsen-Ges. Phys. Chem. 1967, 71, 135-146.
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Figure 5. Dependencies of ∆′ and ∆′′ on ω for DTAB aqueous solutions at several concentrations obtained at the room temperature. Thick and thin solid lines respectively represent the best fitting curves for ∆′ and ∆′′ evaluated with the tworelaxation model.
Figure 6. Relationship between relaxation times, τs and τf, estimated with a curve-fitting technique and CD - cmc for each aqueous surfactant solution at room temperature.
salt aqueous solution, the decreasing in ∆′ is not observed due to very low CD as 10 mM. The ∆′ and ∆′′ spectra for a solution of DTAB at CD ) 100 mM in Figure 5 can be expressed reasonably well with the single relaxation model with τf ∼ 0.5 ns and those at 70-30 mM with the two-relaxation model with τf ∼ 0.5 ns and τs ∼ 5 ns as well as CATB and TTAB systems shown in Figures 3 and 4, respectively. Barchini and Pottel12 first reported dielectric dispersion in aqueous CTAB and SDS systems, and they used a socalled Cole-Cole arc type analysis18 with modified Debey formulas, eqs 3 and 4, to understand the broad ω dependence of ′ and ′′ due to the micelle with an exponential parameter of 1 - h. They reported decrease in the h value from 0.3 to 0.1 with varying CD from 10 to 120 mM for the CTAB system.12 This well corresponds to decreasing in the magnitude of the slow relaxation mode with increasing CD.
∆′ ) ∆′′ )
∆ 1 + (ω2τ2)1-h
(3)
∆ ωτ 1 + (ω2τ2)1-h
(4)
The parameter h obtained from the Cole-Cole arc type analysis can express the broadness of the dielectric relaxation spectrum in view of the average. However, it does not indicate anything on changes in the magnitude of two relaxation modes, when a system has two distinct relaxation modes at fixed relaxation times as observed in this study. Relaxation Times and Strength. Since free surfactants and couterions (such as DAT+ and Br-) essentially do not contribute the dielectric relaxation behavior, the magnitude of dielectric relaxation strength for two relaxation modes (slow, ∆s; fast, ∆f), should be discussed as functions of CD - cmc. Two relaxation times, τs and (18) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341.
Figure 7. Relationship between relaxation strengths, ∆s and ∆f, estimated with the curve fitting technique and CD - cmc for each aqueous surfactant solution at room temperature.
τf , estimated in all the systems in this study are essentially independent of CD - cmc as shown in Figure 6. The dependencies of ∆s and ∆f on CD - cmc for each system are plotted in Figure 7. In every micellar system, proportionality between ∆f and CD - cmc is clearly recognized; however, ∆s does not increase with CD - cmc. Electric Conductivity of Micellar Systems. The specific electric conductivity (κ0 ) G00C0-1) obtained at the low ω (∼104 rad s-1) for each micellar system are plotted in Figure 8 as functions of CD - cmc: 0 is the electric permittivity (dielectric constant) of the vacuum. The κdc ()Gdc0C0-1) values are also plotted in the same figure. Because agreement between κ0 and κdc is perfect, relaxation modes are not there any more in the ω range lower than 104 rad s-1. As is well-known, break points of the plots in Figure 8 correspond to cmc’s for each surfactant.
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Figure 8. Dependencies of the specific electric conductivity, κ0, and the specific direct current conductivity, κdc, on the surfactant concentrations, CD, for each aqueous surfactant solution at room temperature.
When one very carefully looks at the shapes of curves above cmc’s in Figure 8, the slopes of κ0 and κdc for each micellar system decrease first and are followed by increases. These changes in the slopes of κ0 and κdc suggest the change in the fraction of electrically bound counterions to micelles in all the dissociated counterions and/or the change in the degree of dissociation of surfactants with increasing CD. Discussion Two-Phase Model. A two-phase model based on the interfacial polarization might be first applied to the ∆′ and ∆′′ spectra obtained in this study to understand mechanisms which govern the micellar system. According to the general theory of the two-phase model especially for an O/W type emulsion consisting of an oil particle phase with the complex dielectric constant of *p ) p - iκdpω-1 dispersed in an aqueous matrix phase with *m ) m iκdmω-1, the complex dielectric constant, *, for the system can be given by eq 5 with volume fraction, φ, of the oil phase: κdp and κdm mean direct current conductivity for the particle and the matrix, respectively.8
( )
* - *p *m *m - *p *
1/3
)1-φ
(5)
In this condition, one can assume κdp , κdm and gets approximation forms for the asymptotic dielectric constants, l and h, at low and high ω sides as below.8
l )
3 3 + m - p (1 - φ)3/2 2 p 2
(
)
()
h - p m m - p h
1/3
)1-φ
(6) (7)
The φ value for a solution at CD ) 100 mM is ca. 0.04, and m ()′w) ) 78. We do not know the precise value of p for this condition, however, to assume p to be 3 is plausible, which is a typical value for aliphatic oily phase. Then, one can roughly estimate the relaxation strength as l h ∼ 0.008 for the spherical micellar system at CD ) 100
Figure 9. Schematic representation of (a) the generative process of an induced dipole moment by the fluctuation of counterion distribution covering the spherical micelle, (a′) the generative process of an induced dipole moment by the fluctuation of dissociated ionic surfactant distribution on the spherical micelle, and (b) rotational and translational molecular motions of the ionic pair formed with CTA+ and Br- in the spherical micelle.
mM. Since the estimated relaxation strength is much lower than the experimental values (∆′ ∼ 20) and is also much lower than the resolution of the measuring system, we do not conclude that the contribution of interfacial polarization effect based on the two-phase model is essential in the spherical micellar system. Slow Relaxation Mode. In aqueous polyelectrolyte solutions, the fluctuation of a counterion distribution around a dissociated polyion is well-known as one of the most important contribution to dielectric relaxation behavior.19,20 In the case of ionic micellar solutions, counterions dissociate and some of them form counterion clouds covering spherical micelles. If the shape of the counterion cloud deviates from a symmetric spherical shape as schematically shown in Figure 9a, a dipole moment will be induced. In this mechanism, a certain induction time is necessary to generate the dipole moment, because the counterion (Br- in this study) has a finite translational diffusion constant (Dt) in the aqueous phase. Dt of Br- in aqueous solution at 25 °C can be found as 2.1 × 10-5 cm2 s-1 in the literature.21 The time (τ) necessary for a counterion to migrate the surface of a (19) Ito, K.; Yagi, A.; Ookubo, N.; Hayakawa, R. Macromolecules 1990, 23, 857. (20) Oosawa, F. Biopolymer 1970, 9, 689. (21) Robinson, R. A.; Stokes, R. H. Electrolyte Solution; Butterworths: London, 1959.
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spherical micelle with a radius (r) can be roughly estimated as follows.
τ ∼ r2/Dt
(8)
Since the r value of DTAB, TTAB, and CTAB micelles should be range from 3 to 5 nm,22 τ can be estimated to be 4-12 ns. This value is a measure for the time constant of the dielectric relaxation due to the fluctuation of the couterion distribution. Since the experimental τs values shown in Figure 6 reasonably agree with the τ value, the fluctuation of the counter ion distribution possibly is the essential reason for the slow dielectric relaxation mechanism in the spherical micellar system as well as in the aqueous polyelectrolyte system. According to the fluctuation of the counterion distribution, the magnitude of the dielectric relaxation strength must be proportional to the number density of spherical micelles and of (dissociated) counterions electrically bound into the ion cloud covering micelles. The fact that ∆s is not proportional to CD - cmc in every micellar system as seen in Figure 7 suggests decrease in the degree of dissociation of counterions in the micelle and/or decrease in the fraction of counterions electrically bound into the ionic cloud with CD - cmc, because the number density of the micelle is proportional to CD - cmc. This discussion on the change in the degree of dissociation and in the fraction of electrically bound counterions into the ionic cloud is also related to the relationship between κ0 and CD in Figure 8. The above discussion is based on the idea that the rate of counterion fluctuation is faster than that of dissociated surfactant cationic headgroups in the spherical micelle owing to the translational motion along the curvilinear micellar surface as seen in Figure 9a schematically. However, it was reported that the translational motion of the CAT+ cation in the micelle was considerably fast by use of fluorescence anisotropy analysis with a substitute fluorescence probe, cetylacridiniumorange bromide (CAOB), for CTAB in spherical and threadlike micelles.20 Because the transition moment of the CAO+ cation is placed parallel to the direction connecting nitrogen and 9-carbon atoms in an acridinium group, the values of the fluorescence anisotropy relaxation time (τφ, ca. 2 ns)23 of the CAO+ cation in the spherical micelle essentially correspond to its translational molecular motion along the curvilinear micellar surface. Thus, the fluctuation of the distribution of dissociated cationic surfactant on the micellar surface due to the translational molecular motion as shown schematically in Figure 9a′ is able to be the other reason for the slow relaxation mode in this study, when τ for the counterion fluctuation is longer than the relaxation time for the fluctuation of the distribution of surfactant cations on the micellar surface. At present, we cannot conclude which mechanism is essential for the slow relaxation mode, because the estimated relaxation times for both mechanisms are not so different. It is possible that both mechanisms are essential. Fast Relaxation Mode. The degree of disassociation of Br- from surfactant molecules is not high in the micelle (22) Goyal et al. have reported that the actual shape of the micelle formed by CTAB is not shperical but ellipsoidal with major and minor radii of 5 and 2.3 nm, respectively. When one approximately regards the shape of micelles of CTAB, TTAB, and DTAB as a sphere, the radii of the micelles are roughly estimated to be 3-5 nm. Goyal, P. S.; Menon, S. V. G.; Dasannacharaya, B. A.; Rajagopalan, V. Chem. Phys. Lett. 1993, 211, 559-563. (23) Shikata, T.; Morishima, Y. Langmuir 1997, 13, 5229.
(ca. 20%); therefore, a large number of ionic pairs between with CTA+ and Br- must be formed in the micelle. Pronounced dielectric relaxation behavior observed in the solutions suggests the existence of a number of dipole moments. When one is concerned with molecular motions of the ionic pair in the micelle, rotational and translational motions should be dielectrically active. The ionic pair has a dipole moment of µ ) Ql as schematically shown in Figure 9b: Q is the electric charge of the ionic pair; l is the separation between CTA+ and Br-. The ionic pair in aqueous micellar systems is not bare, but its polar headgroup must be highly hydrated with water molecules; therefore, the contribution of the hydration to the dipole moment may be significant. The rotational motion of the ionic pair at a constant position on the micellar surface can be a quite effective molecular motion for dielectric relaxation, since the direction of the dipole moment alters due to the rotation of the ionic pair. However, the ionic pair is not able to randomize its direction of the dipole moment completely only by the rotational motion, but the overall rotation of the spherical micelle and/or the translational motion of the ionic pair along curvilinear surface of the spherical micelle become other effective mechanisms to complete randomization of the direction of the dipole moment. The overall rotational relaxation time of the spherical micelle with a radius of 3-5 nm22 can be estimated with the Stokes-Einstein relationship24 to be 80-400 ns. If the rotation of the spherical micelle is one of important mechanisms for dielectric relaxation behavior, a distinctive relaxation time should be found around these values. However, even the slow relaxation time, τs, estimated in this study is much shorter than that. Then, the overall rotation of the micelle is not essential for the dielectric relaxation behavior. Thus, the translational motion of the ionic pair along the curvilinear micellar surface is essential for the dielectric relaxation of the ionic pair. Because the rotational motion is independent of the translational motion for the ionic pair in the micelle, experimental τf can be expressed by the harmonic average of relaxation times, τr and τt, for both motions as below.
1 1 1 ) + τ f τ r τt
(9)
Recently, the rotation relaxation time (τr) of salicylate (Sal-) anions in spherical and threadlike micelles formed with CTAB in aqueous solution was investigated by use of fluorescence anisotropy analysis: In the case of Sal-, the fluorescence anisotropy relaxation time (τφ) of Salhas the same physical meaning as τr.25 If one assumes that CTA+ and Sal- form an ionic pair in the spherical micelle as CTA+ and Br- do, the obtained τφ (ca. 0.5 ns)25 of the Sal- anion directly corresponds to the rotational relaxation time of the ionic pair between CTA+ and Sal-. This perfect agreement between the τf value and the τr value of the Sal- anion in the spherical micelle strongly suggests that τf essentially reflects the average relaxation time of the rotational motion of the ionic pair in the spherical micelle. Figure 10 shows the dependence of τf on the length of alkyl tails of surfactant molecules. Because the τf values increase with the length (the number of carbon atoms) of the surfactant, molecular motions of the ionic pair in the spherical micelle become slower with increasing in the length of micelle forming surfactants. The values of the (24) Perrin, F. J. Phys. Radium 1934, 5, 497. (25) Shikata, T.; Morishima, Y. Langmuir 1996, 12, 5307.
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As described above, the value of τφ (ca. 2 ns)23 of the CAO+ cation in the spherical micelle essentially corresponds to its translational molecular motion along a curvilinear micellar surface. If one assumes that the ionic pair formed with CTA+ and Br- in the spherical micelle has the τt ()τφ) ) 2 ns due to the translational motion along the curvilinear micellar surface, the relaxation time due to the rotational motion can be evaluated to be τr ∼ 1.1 ns by means of eq 9 for the spherical micelle of CTAB with τf ) 0.7 ns. Therefore, the experimental τf values obtained by DRA look closer to the τr values than the τt ones. Barchini and Pottel developed a theoretical model to explain the dielectric relaxation behavior for CTAB and SDS aqueous systems.12 Their model has a relatively sharp dielectric relaxation behavior with essentially a single relaxation time, which becomes shorter with CD. Therefore, they interpreted the change in the dependence of dielectric spectra on ω with CD in accordance with a change in the relaxation time, not in the distribution of relaxation modes. Figure 10. Dependencies of the fast relaxation time, τf, evaluated with the dielectric relaxation analysis on the number of carbon atoms (length) of surfactant molecules for each aqueous surfactant solution with CD ) 100 mM at room temperature. The fluorescence anisotropy relaxation time, τφ, of a probe molecule, sodium hydroxynaphthoate (NaHN) incorporated in the spherical micelles is also plotted. The concentration of the fluorescence probe molecule was ca. 10-6 M.
fluorescence anisotropy relaxation time τφ of a probe molecule, sodium hydroxynaphthoate (NaHN),23 which is completely incorporated in the micellar interior of every surfactant system examined, are also plotted in the same figure. The physical meaning of τφ for NaHN in the micelle is again the same as that for τr. The trend of the dependence of τφ on the length of the surfactant looks similar to that of τf. This strongly suggests that rates of molecular motions of the ionic pair in the spherical micelle are controlled by the length of the surfactant.
Conclusions Pronounced dielectric relaxation behavior was found in aqueous solutions of cationic surfactants, CTAB, TTAB, and DTAB, containing spherical micelles in the frequency range 106-109 Hz as observed by Barchini and Pottel. Distinct two relaxation modes with different relaxation times, 0.5 and 5 ns, were recognized in the dielectric relaxation behavior. A slow relaxation mode was assigned to the fluctuation of counterion distribution in the ion cloud covering the micelle and/or to that of the distribution of dissociated cationic surfactants on the micellar surface. On the other hand, a fast relaxation mode was attributed to the rotational molecular motion of ionic pairs consisting of surfactant cations and counteranions in the micelle. Acknowledgment. Support by the Kurata Foundation is gratefully acknowledged. LA980421I
