Dielectric Relaxation Behavior of Cationic Micellar Solutions: 2

around 1 ns, had relaxation strength proportional to CD up to CD values of 100 m mol kg-1, and ... havior of aqueous cetyltrimethylammonium bromide so...
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J. Phys. Chem. B 2001, 105, 4495-4502

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Dielectric Relaxation Behavior of Cationic Micellar Solutions: 2 Shin-ichiro Imai, Mamoru Shiokawa, and Toshiyuki Shikata* Department of Macromolecular Science, Osaka UniVersity, Toyonaka, Osaka 560-0043, Japan ReceiVed: June 30, 2000; In Final Form: February 26, 2001

Dielectric relaxation behavior was examined, in the frequency range from 1 MHz to 20 GHz, for an aqueous micellar system of cetyltrimethylammonium bromide (CTAB) with the concentration (CD) from 2 to 1000 m mol kg-1. The system exhibited three dielectric relaxation modes. The fastest mode, with a relaxation time around 1 ns, had relaxation strength proportional to CD up to CD values of 100 m mol kg-1, and showed a constant relaxation strength independent of CD above that concentration. The fastest relaxation mode attributed to the rotational relaxation mode of ionic pairs formed between CTA+ and Br- in spherical micelles is considerably depressed in contact areas on the micellar surface of two micelles in contact with each other at CD values higher than 100 m mol kg-1. This is likely the reason for the leveling-off in the magnitude of the fastest relaxation at high CD and the appearance of the slowest relaxation mode, which has a relaxation time of about 20 ns and increases in strength with increasing CD. The intermediate relaxation mode (with a relaxation time around 8 ns, and strength which decreases with increasing with CD) is attributed to the migration of electrically bound Br- ions around spherical micelles. The average number of Br- ions bound into an ionic cloud covering a micelle was estimated to be less than unity (from the values of direct current electric conductance and Br- concentration determined by use of a Br- selective ion electrode), and a small number of micelles possessed ionic clouds which showed dielectric relaxation behavior.

Introduction The spherical micelle, which is the simplest supermolecular assembly, is formed in aqueous solutions of many kinds of detergents above the critical micelle concentration (cmc). It is likely that there are many types of fast molecular motions of detergent molecules and additives in micelles, and that these determine dynamic structure of micelles. Several techniques for detecting the rates of fast molecular motions in micelles have been proposed, including relaxation time measurement (T1 or T2) by nuclear magnetic resonance (NMR),1-3 a fluorescence anisotropy relaxation measurement,3,4 and dielectric relaxation measurement.5-8 Dielectric relaxation measurement is a very sensitive method for detecting the presence of electric dipole moments and measuring their rates of rotation. In a previous study,6 we reported dielectric relaxation behavior of aqueous cetyltrimethylammonium bromide solutions at concentrations (CD) from 10 to 100 m mol kg-1. That system showed two distinct relaxation modes: fast mode with a CDindependent relaxation time (τ1) around 1 ns and relaxation strength (∆1) proportional to CD, and another mode, with a relaxation time (τ2) around 10 ns, which disappeared at CD values above 80 m mol kg-1. The fast relaxation mode was attributed to the rotational relaxation mode of an ionic pair consisting of CTA+ and Br- in micelles, because the τ1 value estimated was close to the rotational relaxation time of the detergent cation CTA+, as determined by fluorescence anisotropy analysis3,4 with a probe molecule. The slow relaxation mode was attributed to fluctuation of the distribution of electrically bound Br- in ionic clouds formed around micelles. However, in the previous study,6 we did not discuss the * Author to whom correspondence should be addressed. Fax: +81-66850-5538. E-mail: [email protected].

magnitude of relaxation strengths for these two modes in a quantitative way with any theoretical bases. In this study, we expanded the range of CD to 2 to 1000 m mol kg-1. According to experimental results obtained in previous studies, using a small angle neutron scattering technique, the shape of CTAB micelles begins to change from an ellipsoid resembling a sphere to an obvious ellipsoid with a major-tominor axis ratio higher than 2 when CD > 200 m mol kg-1.9,10 However, the value of the axis ratio is still 3 at CD ) 400 m mol kg-1.9 A distinct increase in viscosity was also observed at CD > 200 m mol kg-1.11 These imply that ellipsoidal micelles which are not so long contact and collide with one another very frequently at high CD isotropic micellar conditions. Because the aqueous CTAB system generates a hexagonal liquid crystalline (H1) phase above 700 m mol kg-1, the solution is completely in the liquid crystalline state at 1000 m mol kg-1. At CD > 100 m mol kg-1, deviation from proportionality was found in the relationship between ∆1 and CD - cmc. Moreover, a new relaxation mode, with a relaxation time longer than τ1 and τ2 and strength which steeply increased with increasing CD, was found at CD > 100 m mol kg-1. In the present paper, to explain the fast τ1 relaxation mode quantitatively, we propose a simple model. The τ2 relaxation mode is also discussed quantitatively with a simple model. Furthermore, we present an interpretation for the slowest relaxation mode which is based on contact between the surfaces of two individual spherical micelles. Experimental Section Materials. Supergrade cetyltrimethylammonium bromide (CTAB) was purchased from Wako Pure Chemical Ltd. (Osaka, Japan), and was purified twice by recrystallization from a mixture of methanol and acetone. Highly deionized water

10.1021/jp002348m CCC: $20.00 © 2001 American Chemical Society Published on Web 04/17/2001

4496 J. Phys. Chem. B, Vol. 105, No. 19, 2001 possessing specific resistance higher than 16 MΩ cm, obtained using a MilliQ system, was used as the solvent. The concentration (CD) of CTAB ranged from 2 to 1000 m mol kg-1. Supergrade sodium 3-hydroxy-2-naphthoate (NaHN) and sodium salicylate (NaSal) were purchased from Wako Pure Chemical Ltd, and were used as fluorescence molecular probe molecules, without any purification. Dielectric Relaxation Measurement. Dielectric measurements were performed with an RF LCR meter (Agilent Technologies, 4287A), equipped with a homemade electrode cell,6 with a frequency range of 1 MHz to 3 GHz (6.2 × 106 to 1.9 × 1010 rad s-1 at angular frequency ω). A dielectric materials probe system (Hewlett-Packard, 85070B) which contained a network analyzer (Hewlett-Packard, 8720ES) as a signal analyzer was employed to measure dielectric behavior of samples in the frequency range from 50 MHz to 20 GHz (3.1 × 108 to 1.3 × 1011 rad s-1, at ω). Temperature in the electrode cells was kept at 25 °C with circulating thermostated water in both systems. Open, short, and load correction at the standard resistance of 50 Ω were carried out prior to taking measurements with the LCR meter, to ensure that the system was at peak performance over the ω range examined. Dielectric spectra for pure water were obtained, and we confirmed that the LCR meter system was working correctly before each measurement of aqueous CTAB samples. Open, short, and load correction, using pure water as the standard material, were carried out (with computer software supplied by Hewlett-Packard) prior to taking measurements with the dielectric materials probe system, to ensure that the system was peak performance over the ω range examined. For samples in the system with the RF LCR meter, data were collected as an equivalent parallel electric circuit with capacitance C and conductance G. From these data, the dielectric constant (′) and dielectric loss (′′) were calculated as ′ ) C/C0 and ′′ ) G/(ωC0), using the vacant capacitance (C0 ) 0.17 pF) of the homemade electrode cell. In the dielectric materials probe system, the values of ′ and ′′ were estimated automatically. The reflection coefficient of the electric waves was determined at the terminal of the probe, and the electric impedance of matter contacting the surface of the terminal was estimated by the network analyzer. The measured electric impedance value was converted to values of ′ and ′′. Because estimated dielectric spectra contained significant contributions from pure water (to ′ and ′′) and free dissociated bromide anions (Br-) (to ′′), we must subtract those contribution from the ′ and ′′ values to obtain accurate values for the dielectric constant and loss caused by the presence of micelles (∆′ and ∆′′), as follows: ∆′ ) ′ - ′m and ∆′′ ) ′′ ′′m - G0/(ωC0), where ′m and ′′m stand for the contribution of medium (water) and G0 stands for the direct current conductivity due to free dissociated Br-. When CD was high, the contribution of the electrode polarization effect became noticeable. In such cases, a correcting procedure for the ∆′ value was necessary (especially at values of ω lower than 108 rad s-1), as was described in detail elsewhere.6,8 Br- Ion Concentration Measurement. The concentration of Br- ions was determined using a Br- ion selective ion electrode (8005, Horiba, Kyoto). A homemade Ag-AgCl electrode, inserted into a 3.3 M KCl aqueous solution with a salt bridge of 3% agar containing 1.0 M NH4Cl, was employed as a reference electrode. A potentiometer (pH meter F-22, Horiba, Kyoto) was used to monitor potential difference between

Imai et al. the electrodes. All the measurements were carried out at 25 °C. A calibration line between the potential difference and the Brconcentration ([Br-]) was obtained using aqueous standard NaBr solutions at 0.001, 0.01, and 0.1 M. Fluorescence Anisotropy Measurement. Rotational relaxation times (τφ) for fluorescence probe molecules (NaHN and NaSal) which completely intercalated into micelles of CTAB were determined using a fluorescence anisotropy relaxation technique. The excitation (λex) and emission (λem) wavelengths used for NaHN and NaSal were λex ) 355 and λem ) 512 nm and λex ) 298 and λem ) 404 nm, respectively. A fluorescence spectrometer (F4500, Hitachi, Tokyo) with polarizing plates placed in front of the excitation and emission windows was used to monitor the steady-state fluorescence intensities I0-0, I0-90, I90-0, and I90-90. The subscript numbers indicate the angles of the two polarizing plates. The value g ) I90-0/I90-90 is a constant dependent on emission wavelength, and was determined to be 1.32 for NaHN and 1.03 for NaSal. The fluorescence lifetimes (τlife) of NaHN and NaSal in micelles were estimated using a time correlation single photon counting fluorescence photometer (NAES-500, Horiba, Kyoto) equipped with a high-pressure hydrogen gas flash lamp which generated short excitation pulses with a half-width of ca. 2 ns. The values of τlife were calculated with a deconvolution program supplied by Horiba. A single τlife value (the average value) was determined for each condition with fairly high accuracy. Results Dielectric Relaxation Spectra. ′ spectra for aqueous CTAB solutions at various CD values are shown in Figure 1. From this figure, it is easy to visually determine relaxation mode around 1011 rad s-1. At this ω value, the relaxation strength decreases with increasing CD, and this is attributed to a mobile relaxation mode of water molecules in the bulk state. The second fastest relaxation mode is also easily visually determined around 109 rad s-1, where its strength increases with increasing CD. Steep increases in ∆′ spectra for high CD samples (especially at low values of ω) are caused by the electrode polarization effect, the contribution of which to the total dielectric behavior can be adequately corrected using a method reported elsewhere.6,8 Two more relaxation modes, which exist at ω values around 10-8 rad s-1, are discussed in detail below. Because the relaxation time for water molecules in the bulk state (τw ) 8.3 × 10-12 s) did not depend on the CD value in any of the solutions examined, the height of the plateau at 1010 rad s-1 corresponds to the contribution of water molecules in the bulk state. Assuming dielectric behavior for pure water as ′w ) 73.3/(1 + ω2τw2) + 5.1 and ′′w ) 73.3 (ωτw)/(1 + ω2τw2) at 25 °C,5 the dielectric contribution of the medium for each solution (given by the equations ′m ) f ′w and ′′m ) f ′′w) were determined by shifting both ′w and ′′w curves by a factor of f (to be superposed onto ′ and ′′ curves around 1011 rad s-1). Figure 2 shows the value of f for each solution as a function of CD. It is likely that the f value decreases in approximate proportion to the CD value. The solid line in Figure 2 represents a theoretical prediction,12 based on the molar volume of CTAB spherical micelles,13 which has been thought to be valid for aqueous solutions containing simple electrolytes with large ionic radius such as tetramethylammonium bromide (TMAB).14 The deviation of experimental data from the prediction suggests the existence of water molecules in spherical micelles possessing a rotational relaxation rate which is not very different from that of water molecules in the bulk state. This is discussed in detail below.

Dielectric Relaxation of Cationic Micellar Solutions: 2

J. Phys. Chem. B, Vol. 105, No. 19, 2001 4497

Figure 1. Frequency (ω) dependence of dielectric constant (′) for aqueous CTAB micellar solutions, at several concentrations.

Figure 3. Typical examples of dielectric spectra (∆′ and ∆′′, vs ω) of aqueous CTAB solutions at various values of CD.

Figure 2. Relationship between the factor (f) of the contribution of pure water to dielectric spectra of aqueous CTAB solutions and the concentration of CTAB (CD).

∆′ and ∆′′ spectra evaluated as functions of ω were analyzed by fitting to Debye type model relaxation functions with three relaxation times (τi, i ) 1 to 3) and strength (∆i), as in our previous studies.6 In Figure 3, some typical dielectric spectra and their best fitted curves are shown as functions of ω. Figure 4 shows the dependence of τi on the value of CD. The concentration normalized relaxation strength for each mode, ∆i(CD - cmc)-1, is plotted in Figure 5 as a function of CD. Modes 1 and 2 are observed when CD < 100 m mol kg-1, but at CD > 200 m mol kg-1, mode 2 disappears and mode 3 suddenly appears. In a previous study,6 we found that dielectric spectra for an aqueous CTAB solution at CD ) 100 m mol kg-1 had a single relaxation time of τ1. However, in the present study, we found that the solution spectra include a small contribution from the τ2 mode, as seen in Figure 5. Since the ratio ∆2/∆1 for the solution at CD ) 100 m mol kg-1 is only 0.2, the shapes

of dielectric spectra for the solution are not very different from those of the single relaxation spectra, as observed in Figure 3. Because the G0 values are not very high in solutions at CD < 200 m mol kg-1, the effect of the electrode polarization on ∆′ was not very noticeable. Thus, the standard error included in calculated τ2 and ∆2 values was not very large. However, in solutions at CD g 200 m mol kg-1, the effect of the electrode polarization due to high G0 values was large. The data for τ3 and ∆3 in Figures 4 and 5 have error bars. The values included in the error bars can yield curves which fit the dielectric spectra reasonably well, such as those in Figure 3. The magnitudes of relaxation strengths and times for each mode are discussed in detail below as functions of CD. In the procedure to obtain the best fitting curves for ∆′ and ∆′′ spectra, the value of direct current conductance (G0) was automatically determined for each sample. The values of G0 for aqueous CTAB solutions estimated using a network analyzer and those estimated using an LCR meter are shown together in Figure 6 as a function of CD. The G0 values appear roughly proportional to CD, except at lower values of CD.

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Figure 4. Dependence of dielectric relaxation times (τi, i ) 1, 2, 3) on CD, for aqueous CTAB solutions.

Imai et al.

Figure 6. Relationship between direct current conductance (G0) and CD, for aqueous CTAB solutions.

Figure 7. CD dependence of the degree of dissociation (R) of CTAB estimated with a Br- ion selective electrode, degree of electrical dissociation (Re) estimated from the direct current conductance (G0), and the degree of dissociation in micelles (Rm) for aqueous CTAB solutions. Figure 5. Dependence of reduction of dielectric relaxation strength [∆i(CD - cmc)-1, i ) 1, 2, 3] on CD, for aqueous CTAB solutions.

Degree of Br- Ion Dissociation. The degree of dissociation (R) of Br- from CTAB in aqueous solution, as a function of CD, was determined from the Br- concentration ([Br-]) data obtained by a Br- ion selective electrode. The relationship between CD and R values determined from [Br-] data is plotted in Figure 7. Assuming all dissociated Br- anions behave as electric carriers which determine conductance G0, the degree of electrical dissociation (Re) of Br- can be estimated with the limiting molar conductance of Br- (λBr- ) 78.1 S cm2 mol-1) reported in the literature,15 along with the equivalent value for CTAB (λCTAB ) 83.1 S cm2 mol-1)6 estimated below the cmc, because the limiting molar conductance of spherical micelles should be negligibly lower than λBr-. In general, the molar conductance of ions is concentration dependent, and decreases gradually with increasing concentration. Therefore, the values of Re shown in Figure 7 may be the result of underestimation. However, agreement between Re and R looks reasonably good, especially at high values of CD. Logically, though R must be greater than Re, because Re only includes the contribution of direct current conductance. These fact strongly suggest that most of the dissociated Br- is not bound into ionic clouds around spherical micelles, but is free and functions as electric carriers which determine G0.

Rotational Relaxation Motion in Spherical Micelles. The rotational relaxation time (τφ) of the fluorescence probe molecules NaHN and NaSal in spherical micelles were calculated with eqs 1 and 2:

r ) (I0-0 - I0-90)/(I0-0 + 2gI0-90)

(1)

1/r ) (1/r0)(1 + 3τlife/τφ)

(2)

where r0 and τlife represent the instantaneous fluorescence anisotropy (0.34 for NaHN4 and 0.32 for NaSal3), and fluorescence lifetimes of the probes, respectively. At low CD, the τφ values are independent of CD, because dynamics of micelle forming CTA+ cations are not altered by changes in CD. The dependence of τφ on CD is shown in Figure 8. The τφ values clearly increase with increasing CD at CD values higher than 100 m mol kg-1, and then reach certain constant values which are independent of CD. These facts suggest that molecular motion of CTA+ in spherical micelles decreases with increasing CD. Discussion Dielectric Relaxation Behavior of Polar Solution. The general expression for (relative) static dielectric constant () is given by eq 3′

Dielectric Relaxation of Cationic Micellar Solutions: 2

J. Phys. Chem. B, Vol. 105, No. 19, 2001 4499 particle, i, dissolved in the solution, if τi is longer than the relaxation time of water molecules; τw ) 8.3 × 10-12 s, at 25 °C.

Ni µi2/(1 + ω2τi2) + f ′w ∑ i)0

(8a)

Ni µi2ωτi /(1 + ω2τi2) + f ′′w ∑ i)0

(8b)

′ ) 1/(2vkBT) ′′ ) 1/(2vkBT)

Figure 8. Dependencies of rotational relaxation times (τφ) on CD, for sodium salicylate (NaSal) and sodium 3-hydroxy-2-naphthoate (NaHN). The τ1 and τ3 values are also plotted in this figure.

D ) vE + P

(3)

vE ) vE + P

(3′)

where E, D, P, and v represent the electric field, the electric displacement, the electric polarization and the electric permittivity of a vacuum, respectively.16 It is difficult to calculate  for polar liquids using standard methods. However, if there is not strong interaction between dipole moments of different molecular species, P can be calculated from the total contribution of electric dipole moment (µi) and polarizability (βi) for each particle (molecule), i, in the system (eq 4):

P)

Ni [µi〈µi〉/(3kBT) + βi]F ∑ i)0

(4)

where Ni, kBT, 〈µi〉, and F represent the number density of a molecule, i, in the system, the product of Boltzmann’s constant and the absolute temperature, the average value of µi, and the local electric field, respectively. From this, one can estimate  as

-1)

Ni[µi〈µi〉 /(3kBT) + βi]F/(Ev) ∑ i)0

(5)

According to Fro¨hlich,17 F should be replaced by Onsager’s cavity electric field:18 F ) 3E/(2 + 1). A simple expression, eq 6, is obtained for  in the case of polar solutions for which  ∼ w ) 78.

 ) 3/(2v)

Ni[µi〈µi〉/(3kBT) + βi] ∑ i)0

(6)

We assume that the molecule i ) 0 is a water molecule with w )78 in the pure state, and that other dielectric molecules or particles have only µi. Moreover, we assume 〈µi〉 can be replaced by µi for simplicity, as done in Onsager’s model.18 We then obtain the final expression of  for polar solutions including solutes (eq 7), with a factor, f, which represents the number (or volume) fraction of water molecules in a solution relative to pure water.

 ) 1/(2vkBT)

Niµi2 + f w ∑ i)0

(7)

Equation 7 is converted to dielectric constant (′) and loss (′′) (eqs 8a and 8b) by introducing a relaxation time (τi) for each

To test the validity of eqs 7 to 8b, the value of the dipole moment of a typical polar molecule, trimethylamineoxide (TMAO), was estimated from dielectric data for aqueous TMAO solutions using eqs 8a and 8b.19 The obtained value was 4.9 × 10-27 Ccm, which was 3 times the value determined from data for TMAO solutions in nonpolar solvents such as benzene.19 We feel that eqs 7 to 8b are basically correct. However, an additional numerical front factor, A (≈ 9), is necessary for each equation to produce valid values of dipole moments for solutes in aqueous solutions, because some highly approximate assumptions were used in the derivation of these equations. When a simple salt with large ionic radii and small hydration numbers, such as TMAB,14 is dissolved in water, the first terms in eqs 8a and 8b can be neglected. Only the second terms are important elements in the relationship between dielectric behavior and the concentration of TMAB. Pottel pointed out that f could be described well by eq 9, with volume fraction (V) values ranging up to V ≈ 0.28, in an aqueous TMAB system.12

f ) (1 - V)/[1 + (1/2)V]

(9)

Given these considerations, we conclude that eq 9 correctly expresses the volume effect of solute molecules on the f value. Contribution of Water Molecules. The solid line in Figure 2 represents values predicted by eq 9, using V calculated from reported specific volume data of CTAB13 in aqueous CTAB solution. The present experimental data deviate from the solid line predicted by eq 9 (exceeding predicted values), as seen in Figure 2. This suggests that some water molecules are kept in micelles, and that they have very fast rotational motion, as well as a mobile relaxation mode equal to that of water molecules in the bulk state.20 Because the number of water molecules kept in micelles is not very large, hydrogen bonding between the water molecules does not develop. Therefore, it is possible that the rate of rotation of the water molecules in micelles is faster than their mobile relaxation mode in the bulk sate. Since the deviation of f from the predicted values in a hexagonal liquid crystalline state at CD ) 1000 m mol kg-1 is not very large, the number of water molecules per CTAB in rodlike micellar type structures21 in the liquid crystalline state is the same as or slightly larger than that in spherical micelles in an isotropic state, which is estimated to be approximately 5 (from the difference between the data and the predicted values, as seen in Figure 2). Belmajdoub et al. investigated the location of water molecules in CTAB spherical micelles using a heteronuclear Overhouser effect (hNOE) measurement, and they observed no hNOE between protons belonging to water molecules and any carbons of CTAB, including headgroups.22 That suggests the possibility that water molecules in the micelle are not strongly solvated to CTAB, and thus show neither dielectric saturation nor hNOE. Rotational Relaxation Mode of the Ionic Pair CTA+-Br-. In our previous study,6 the dielectric relaxation mode of i ) 1 was attributed to the rotational relaxation mode of an ionic pair formed between CTA+ and Br- in spherical micelles. When interaction between ionic pairs is not very strong, the concentration normalized relaxation strength [∆1(CD - cmc)-1] resulting

4500 J. Phys. Chem. B, Vol. 105, No. 19, 2001 from this mode will likely remain constant. Up to CD ) 100 m mol kg-1, the value of ∆1(CD - cmc)-1 appears constant, independent of CD, as seen in Figure 5. The value of τ1 also remains constant, at 0.7-0.6 ns, until CD reaches 500 m mol kg-1, as seen Figure 4.6 These results suggest that molecular dynamics of the ionic pair in spherical micelles are not changed much by changes in CD until CD reaches 100 m mol kg-1. The total degree of dissociation, R, of Br- in an aqueous CTAB solution is very important in evaluating the degree of dissociation of Br- for CTAB in micelles (Rm). The Rm data (evaluated from R) plotted in Figure 7 were determined by assuming the cmc of CTAB to be 0.8 m mol kg-1.23 Because 1 - Rm, which remains constant at approximately 0.77 until CD ) 600 m mol kg-1, represents the fraction of CTAB in the form of the ionic pair CTA+-Br- in micelles, (1 - Rm)(CD cmc) represents the total concentration of the ionic pair. Given eqs 8a and 8b, the relationship ∆1 ) 9N1µ12/(2kBT) obviously requires the additional front factor A ) 9. Because the equation N1 ) (1 - Rm)(CD - cmc)NA contains Avogadoro’s number (NA), the value of µ1 can be evaluated as

µ1 ) {2 ∆1vkBT/[9(1 - Rm)(CD - cmc)NA]}1/2 (10) The value µ1 ) 5.92 × 10-27 Ccm is obtained for the ionic pair in the range of concentration 5 e CD e 500 m mol kg-1. The ionic pair has an elementary charge, e0, on both an ammonium group and Br-. Therefore, separation between the centers of the two ions is estimated to be 0.37 nm. Since the Br- anion has a bare ionic radius of 0.20 nm and has a small number of hydration water molecules (equal to or less than unity),14 the longest distance (rl) for the headgroup of the CTA+-Br- ionic pair (from the center of a trimethylammonium group to the outer boundary of Br-) should be approximately equal to 0.57 nm. The bare ionic radius of tetramethylammonium and its hydration number in aqueous solution have been reported to be 0.35 nm and zero, respectively.14 Therefore, the shortest distance (rs) from the center of a trimethylammonium ion to its outer boundary should be close to 0.35 nm. The average radius of the headgroup of CTAB exposed on the surface of spherical micelles has been evaluated to be 0.45 nm,24 and this value is intermediate between the values of rs and rl shown above. Thus, we conclude that the value of µ1 estimated using this model is adequate, and that the model is essentially correct for this aqueous CTAB system. As soon as CD exceeds 100 m mol kg-1, mode 3 (with τ3 around 25 ns) suddenly appears and increases in strength with increasing CD, while the normalized strength of mode 1, ∆1(CD - cmc)-1, decreases in the same CD region, as seen in Figure 5. This suggests the molecular motion of mode 1 is strictly restricted by important factors which become significant in the high CD region, and a part of the dielectric relaxation strength belonging to mode 1 moves to that of mode 3. It is well-known that the frequency of contacts or collisions between two spherical micelles is proportional to CD2 in the high CD region. It follows that the total area of micellar surface on which contacts between two micelles occur is also proportional to CD2. Consequently, the number of CTA+-Br- ionic pairs in the total area of contacts between two micelles is proportional to CD2. Because the dielectric relaxation strength of mode 1 is proportional to the number of ionic pairs on the surface of micelles without contacts, the decrease in ∆1(CD - cmc)-1 in the high CD region (slope of -1) seen in Figure 5 seems reasonable. This kind of decrease in the dielectric relaxation strength of the rotational mode of ionic pairs in micelles also has been reported in a

Imai et al. threadlike micellar system.8 In an aqueous threadlike micellar system consisting of CTAB and sodium salicylate (NaSal), the strength of a rotational relaxation mode of ionic pairs consisting of CTA+ and Sal- in the threadlike micelles (approximately at 1 ns) almost moves to a relaxation mode observed in the lower frequency region at around 10 ns under high CD conditions.8 However, for the examined spherical micellar system, the value of ∆3(CD - cmc)-1 of mode 3 found in the high CD region is smaller than the difference between ∆1(CD - cmc)-1 and the value obtained at CD ) cmc, as seen in Figure 5. This means that most dipole moments due to CTA+-Br- ionic pairs in the area of contacts diminish without any dielectric relaxation processes, but a small amount of the dipole moments survives, resulting in mode 3 relaxation. The existence of a slow rotational mode of ionic pairs in the high CD region can be confirmed using a fluorescence anisotropy technique with rotational fluorescence probe molecules incorporated in micelles. The relationship between CD and rotational relaxation times (τφ) of NaHN and NaSal fluorescence probe molecules incorporated in spherical micelles of CTAB is shown in Figure 8. It is well-known that these probe molecules strongly interact with CTA+ to make stoichiometric complexes such as salts in micelles.25,26 Rotational motion of the complexes can be sensitively detected using the value of τφ , because these molecules possess well-characterized transition moments parallel to the direction of the chemical bond between two carbons connected to OH and COO- groups of both molecules. These complexes can be expected to work well as substitutes for the CTA+-Br- ionic pair. Although the values of τφ for these two molecules are not identical (due to different molecular sizes), they exhibit similar trends in the value of τφ with changes in CD, as seen in Figure 8. The values of τφ appear constant in the low CD region (reflecting the fast rotational mode independent of contacts between two micelles), increase with increasing CD. These features imply that there exist slow rotational modes in spherical micelles of CTAB, and that the difference between the intensities of the slow rotational mode and the fast one increases with increasing CD. It is possible to directly compare dielectric relaxation times (τ1 and τ3) with τφ. Figure 8 shows these values, with arrows for comparison. Because the τφ value represents the average of all rotational relaxation times of the probe molecules, the difference between τ3 and τφ in the high CD region suggests the fraction of the probes in slow rotational mode is not very high, even at CD ) 500 m mol kg-1. In a liquid crystalline (H1 state) sample at CD ) 1000 m mol kg-1, ∆1(CD - cmc)-1 is lower than it is in isotropic micellar solutions at CD < 1000 m mol kg-1, as seen in Figure 5. The τ1 value of the liquid crystalline sample is a little less than it is in isotropic micellar solutions. In the H1 state, rodlike micelles are formed and are aligned in hexagonal arrangement. Because these rodlike micelles are probably not stretched perfectly straight, the surfaces of the micelles might have many contact points in liquid crystalline domains. Therefore, the resulting ∆1(CD - cmc)-1 and τ1 values (which are less than those of isotropic micellar solutions) are obtained at all CD values. On the other hand, the slowest mode (mode 3) that appears in the liquid crystalline sample has ∆3(CD - cmc)-1 and τ3 values which are not very different from those observed in the isotropic micellar solution at CD ) 600 m mol kg-1. Given these results, the dynamic features of CTAB in a rodlike micelle in a liquid crystalline sample are not much different from those observed in spherical micelles in isotropic micellar solutions.

Dielectric Relaxation of Cationic Micellar Solutions: 2

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Figure 9. Schematic representation of a spherical micelle with an electrically bound Br- anion.

Dielectric Relaxation Due to Br- Bound to Ionic Clouds. Here, we discuss a dielectric relaxation process with a relaxation time, τ2, of approximately 5 ns with a simple model. As we pointed out above, few dissociated Br- ions are electrically bound into the ionic clouds around micelles of CTAB. We defined a parameter, δ (