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Study of Tetrabutylammonium Perfluorooctanoate Aqueous Solutions with Two Cloud Points by Dielectric Relaxation Spectroscopy Li-Kun Yang,† Kong-Shuang Zhao,*,† and Jin-Xin Xiao*,‡ Department of Chemistry, Beijing Normal UniVersity, Beijing 100875, China, and Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, College of Chemistry and Molecular Engineering, Peking UniVersity, Beijing 100871, China ReceiVed April 5, 2006. In Final Form: July 20, 2006 Dielectric relaxation spectra of tetrabutylammonium perfluorooctanoate (TBPFO), an anionic fluorocarbon surfactant with two cloud points in aqueous solution, were investigated in the frequency range from 40 Hz to 110 MHz. Striking dielectric relaxations were observed when both the temperature-dependent and concentration-dependent phase transitions in TBPFO aqueous solution occurred. The changes in dielectric relaxation and the distribution of dielectric parameters were consistent with the phase boundaries of the phase diagram. In the first homogeneous phase region, two relaxations of rodlike micelles appeared at about 100 kHz and 5 MHz, which originated from the diffusion of the free counterions in the directions of the long axis and the short axis of rodlike micelles, respectively. With increasing temperature, two relaxations gradually turned to one as a result of the formation of connected or entanglement points between the wormlike micelles. The lengths of the long half-axis and the short half-axis of the rodlike micelles, as well as the average distance of the connected or entanglement points of the wormlike micelles, were evaluated by the obtained relaxation times.
Introduction
* Corresponding authors. (K.-S.Z.) E-mail:
[email protected]. Tel: +8601058808283. (J.-X.X.) E-mail:
[email protected]. † Beijing Normal University. ‡ Peking University.
noninvasive way, can monitor the structural changes in many systems in situ and provide insights into structures and electrical properties on the molecular and macroscopic levels.9 The dipolar reorientational motion of molecules in a polymer has been successfully monitored by frequency-dependent dielectric sensing (FDEMS).10,11 The dielectric relaxation parameters that reflect the characteristics of DRS obtained by fitting the experimental data with proper equations can represent the interior properties of the system to some extent. Therefore, it is very important to determine whether DRS can be used to monitor the morphological changes in a micelle system. There are few reports in the literature on this topic at present. Moreover, DRS is very sensitive to all kinds of intermolecular interactions and dipole moment fluctuations.12 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) and provide important and even unique information on the dynamic and structural properties of materials. Useful information provided by DRS includes the movement state of the whole molecule in solution, the diffusion of counterions in the compact and the diffuse layers, the size and the distribution of particles estimated by the relaxation time, and the property 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,13,14 micelles,15-18 microemulsions,19 vesicles,20 and biological cell dispersions.21
(1) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980. (2) Barzykin, A. V.; Tachiya, M. Heterog. Chem. ReV. 1996, 3, 105. (3) Gehlan, M.; DeSchryver, F. C. Chem. ReV. 1993, 93, 199. (4) Fisicaro, E.; Compari, C.; Duce, E.; Contestabili, C.; Viscardi, G.; Quagliotto, P. J. Phys. Chem. B 2005, 109, 1744. (5) Luca, V.; Hook, J. M. Chem. Mater. 1997, 9, 2731. (6) Berlepsch, H. V.; Dautzenberg, H.; Rother, G.; Ja¨ger, J. Langmuir 1996, 12, 3613. (7) Jeong, B.; Windisch, C. F., Jr.; Park, M. J.; Sohn, Y. S.; Gutowska, A.; Char, K. J. Phys. Chem. B 2003, 107, 10032. (8) Holmqvist, P.; Alexandridis, P.; Lindman, B. Langmuir 1997, 13, 2471.
(9) Asami, K. Prog. Polym. Sci. 2002, 27, 1617. (10) Bonnet, A.; Pascault, J. P.; Sautereau, H.; Rogozinski, J.; Kranbuehl, D. Macromolecules 2000, 33, 3833. (11) Se, K.; Takayanagi, O.; Adachi, K. Macromolecules 1997, 30, 4877. (12) Daniel, V. V. Dielectric Relaxation; Academic Press: London, 1967. (13) Chen, Z.; Zhao, K.-S. J. Colloid Interface Sci. 2004, 276, 85. (14) He, K.-J.; Zhao, K.-S. Langmuir 2005, 21, 11878. (15) Baar, C.; Buchner, R.; Kunz, W. J. Phys. Chem. B 2001, 105, 2906. (16) Baar, C.; Buchner, R.; Kunz, W. J. Phys. Chem. B 2001, 105, 2914. (17) Shikata, T.; Imai, S. Langmuir 1998, 14, 6804. (18) Imai, S.; Shikata, T. Langmuir 1999, 15, 8388.
Amphiphilic surfactant molecules can form micelles, a kind of self-organized molecular assembly in an aqueous environment above their critical micelle concentrations (cmc).1 The interest in micelle solutions stems from their potential as functional molecular assemblies for use in many fields in pure and applied science because they can be used as models for several biochemical and pharmacological systems and they can solubilize water-insoluble substances (including certain medicines) in their hydrophobic cores.2 When exterior conditions such as the concentration of surfactant, the temperature of the solution, or the presence of additives are changed, the micelle shape can change from spherical to rodlike to wormlike or more complex morphologies.3 These drastic structural changes in micelle solutions have attracted much attention to the study of molecular aggregates and selforganization. These micelle structures are characterized by various methods and techniques, such as the thermodynamic method,4 nuclear magnetic resonance,5 static and dynamic light scattering,6 small-angle neutron scattering and Raman spectroscopy,7 and small-angle X-ray measurement.8 Dielectric relaxation spectroscopy (DRS), which measures permittivity and conductivity as a function of frequency in a
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Figure 1. Partial phase diagram of the tetrabutylammonium perfluorooctanoate (TBPFO) aqueous solution.
It is well known that suspensions in conducting dispersion media can be characterized by two typical mechanisms of dielectric relaxationslow-frequency dielectric relaxation and Maxwell-Wagner dielectric relaxationsboth of which are sensitive to the polarization of the electric double layer of the disperse particles. Low-frequency relaxation was first given a theoretical explanation by Schwarz.22 Dukhin and Shilov23,24 developed a new approach to this problem based on the concept of a diffuse double layer. O’Konski25 found that the presence of a conducting shell (such as the double layer surrounding a nonconducting particle) can be taken into account by assigning a certain bulk conductivity to the particle material on the basis of the Maxwell-Wagner26,27 interfacial polarization theory of spherical particles. Grosse28 presented a simple model for the dielectric properties of suspensions of charged particles in an electrolyte solution. This has been successfully used to interpret the mechanism of dielectric relaxation of suspensions of colloidal particles29-31 (including the aqueous micelle solution16). Recently, an interesting aqueous solution composed of an anionic fluorocarbon surfactant, tetrabutylammonium perfluorooctanoate (TBPFO), has been investigated. The anomalous temperature-dependent phase behavior and structure transitions of this system have been studied by differential scanning calorimetry, electrical conductivity, static/dynamic light scattering (SLS/DLS), and rheology methods.32 The partial phase diagram of the TBPFO aqueous solution is shown in Figure 1. It was observed that the aqueous solution of TBPFO exhibited two cloud points. Upon increasing the temperature, the solution turned from a transparent homogeneous phase (phase I) to a two-phase (19) Bordi, F.; Cametti, C. J. Colloid Interface Sci. 2001, 237, 224. (20) Schrader, W.; Halstenberg, S.; Behrends, R.; Kaatze, U. J. Phys. Chem. B 2003, 107, 14457. (21) Bai, W.; Zhao, K.-S.; Mi, H.-L. Bioelectrochemistry 2006, 69, 49. (22) Schwarz, G. J. Phys. Chem. 1962, 66, 2636. (23) Dukhin, S. S.; Shilov, V. N. Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes; Halsted: Jerusalem, 1974. (24) Shilov, V. N.; Dukhin, S. S. Colloid J. 1970, 32, 293. (25) O’Konski, C. T. J. Phys. Chem. 1960, 64, 605. (26) Maxwell, J. C. A Treatise on Electricity and Magnetism, 3rd ed.; Clarendon Press: Oxford, England, 1891; Chapter 4. (27) Wagner, K. W. Arch. Elektrotech. 1914, 2, 371. (28) Grosse, C. J. Phys. Chem. 1988, 92, 3905. (29) Shilov, V. N.; Delgado, A. V.; Gonzalez-Caballero, F.; Grosse, C. Colloids Surf., A 2001, 192, 253. (30) Grosse, C.; Arroyo, F. J.; Shilov, V. N.; Delgado, A. V. J. Colloid Interface Sci. 2001, 242, 75. (31) Delgado, A. V.; Arroyo, F. J.; Gonza´lez-Caballero, F.; Shilov, V. N.; Borkovskaya, Y. B. Colloids Surf., A 1998, 140, 139. (32) Yan, P.; Huang, J.; Lu, R.-C.; Jin, C.; Xiao, J.-X.; Chen, Y.-M. J. Phys. Chem. B 2005, 109, 5237.
Yang et al.
liquid-liquid (similar to phase III), then to another homogeneous phase (phase II), and finally to another two-phase liquid-liquid (phase III). In phase I, the aggregates of TBPFO were rodlike micelles, whereas in phase II the aggregates of TBPFO were large wormlike micelles. In both of the two-phase regions (phase III), the system separated into a surfactant-rich phase in equilibrium with a surfactant-depleted phase. The TBPFO aqueous solution with these special properties of the temperature-dependent and concentration-dependent phases and structural transition is a suitable system to be monitored by DRS theoretically. We expect to determine whether the new system is suitable for investigating with DRS. At the same time, most of the earlier work on micelles by DRS has focused on high frequency33,34 between about 10 MHz and 10 GHz, sometimes including the dielectric relaxation of solvent.15 These investigations involved only spherical16-18 or threadlike35 micelles. The temperature-dependent and concentration-dependent phase transitions have hardly been investigated. In this article, the phase transitions originating from the variety of temperatures and concentrations of the TBPFO-water system were monitored by DRS over a frequency range of 40 Hz to 110 MHz. The obvious change in the dielectric relaxation in different phase regions was ascribed to the phase transitions. The distribution of several dielectric parameters obtained by fitting the experimental data is consistent with the phase boundaries of the phase diagram. The mechanism of the dielectric relaxation in the two homogeneous phases can be interpreted by the diffusion of the counterions of Grosse’s model.28 The mechanism of the dielectric relaxation in the two-phase system was also proposed. Experimental Section Materials. Tetrabutylammonium perfluorooctanoate (TBPFO) was prepared by neutralizing perfluorooctanoic acid with tetrabutylammonium hydroxide, as previously described.32 Deionized water possessing a specific resistance higher than 16 MΩ cm-1 was used as the solvent, which was obtained from an Aquapro P Series water purification system (Taiwan). The concentration of TBPFO ranged from 10 to 87 mmol L-1, exceeding the critical micelle concentration (cmc) of 2.3 mmol L-1 at 25 °C. No additional electrolyte was added to the purified aqueous solution of TBPFO. The TBPFO aqueous solution was 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. A dielectric measurement cell with concentrically cylindrical platinum electrodes36 was employed. The volume of the solutions 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.37 The stray capacitance and cell constant were determined with pure water, ethanol, and air at different temperatures. The temperature of the cell was controlled from 10 to 50 °C by circulating thermostated water in all aqueous solution systems. The deviation of temperature was less than (0.1 °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 (33) Imai, S.; Shiokawa, M.; Shikata, T. J. Phys. Chem. B 2001, 105, 4495. (34) Fernandez, P.; Schro¨dle, S.; Buchner, R.; Kunz, W. ChemPhysChem 2003, 4, 1065. (35) Shikata, T.; Imai, S. J. Phys. Chem. B 1999, 103, 8694. (36) Hanai, T.; Zhang, H.-Z.; Sekine, K.; Asaka, K.; Asami, K. Ferroelectrics 1988, 86, 191. (37) 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.
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Figure 2. Three-dimensional representations of the temperature dependence of (a) the relative permittivity spectrum and (b) the dielectric loss spectrum of the TBPFO-water system at C ) 70 mmol L-1. of frequency f. The response of the polarization of the sample is reflected in terms of the complex permittivity spectrum. To obtain the parameters of dielectric relaxation, such as the limiting values of low- and high-frequency permittivity and conductivity and the characteristic relaxation frequency, the Cole-Cole empirical equation38 can be used to fit the experimental data in the applied frequency range, which includes one (i ) 1 for a part of the data in phase II) or two (i ) 1, 2 for the rest of the data in phase II and for all of the data in phases I and III) Cole-Cole terms. * ) ′ - j′′ ) h +
∆i
∑1 + (jωτ )
βi
i
(1)
i
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 permittivity increment, l and h are the low- and high-frequency limits of relative 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 conductivity of the TBPFO-water system is similar to that of the salt solution, both exceeding 0.01 S m-1. 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. The effect cannot be neglected, especially in the case of high concentration. To acquire accurate values of the dielectric parameters, the electrode polarization term is added to the Cole-Cole equation (eq 1)39 * ) h +
∆i
∑1 + (jωτ ) i
βi
+ Aω-m
(2)
i
where A and m are adjustable parameters determined by fitting the experimental data simultaneously. All of 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 investigated samples is obtained.
Results and Discussion Dielectric Relaxation Near the Phase Transition. Temperature-Dependent Phase Transition. Figure 2 shows 3D representations of the temperature-dependent dielectric spectra when (38) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341. (39) Asami, K. Langmuir 2005, 21, 9032.
Figure 3. Relative permittivity spectra extracted from Figure 2 at several temperatures, (O) 15, (4) 25, (3) 30, and (]) 40 °C, when the concentration is 70 mmol L-1. The solid lines represent the best fitting curves evaluated from eq 2.
the concentration of TBPFO is 70 mmol L-1. The number of dielectric relaxation determined by the relative permittivity in Figure 2a corresponds to the number of peaks represented by the dielectric loss in Figure 2b. At this concentration, the system can transfer from phase I to phase II and then to phase III with increasing temperature. It can be clearly observed that the dielectric spectra have changed significantly with temperature, as shown in Figure 2 by the arrows. With increasing temperature, at first, two relaxations appear and then approach and turn into one relaxation with a larger permittivity increment step by step, eventually changing to two relaxations with a smaller permittivity increment. The changes in dielectric relaxation reflect the transformation of the micelle aggregates in the TBPFO-water system. For the purpose of investigating the dielectric relaxation change with increasing temperature in different phase regions, as shown in Figure 2a in detail, relative permittivity spectra of 70 mmol L-1 TBPFO aqueous solutions are displayed in Figure 3 at several temperatures. 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 to be the dielectric parameters representing the real properties of the whole system because the effect of the electrode polarization has been eliminated. It is very obvious that the temperature of the phase transition is dependent on this condition. The dielectric spectrum at 15 °C shows two relaxations around 188 kHz and 6 MHz. Though there still are two relaxations at 25 °C, they are close to each other at about
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Table 1. Dielectric Parameters of the TBPFO-Water System at 70 mmol L-1a temp (°C)
l
m
h
∆l
∆h
βl
βh
15 25 30 40
110.05 307.28 392.58 95.04
95.02 215.51
80.36 73.53 73.17 68.40
15.03 91.77 319.41 20.70
14.66 141.98
0.96 0.99 0.83 0.99
0.85 0.70
74.34
5.94
0.91
f0l (kHz)
f0h (MHz)
188 247 277 138
6.13 0.413 14.6
The permittivity increment of low-frequency dielectric relaxation is defined as ∆l ) l - m, and that of high-frequency dielectric relaxation is defined as ∆h ) m - h. a
Figure 4. Temperature dependence of the dielectric relaxation parameters evaluated by fitting eq 2 to the observed dielectric spectra. (a) Low-frequency limit of the relative permittivity l, (b) low-frequency limit of the conductivity κl, (c) characteristic relaxation frequency f0, and (d) Cole-Cole parameter β. There are three sets of data in all graphs, which are indicated by squares, circles, and triangles. Broken lines represent the phase transitions. In c and d, the low-frequency dielectric parameters are represented by closed symbols, and the highfrequency parameters are represented by open symbols.
247 kHz and 0.4 MHz, and the permittivity increment has increased. When the temperature is increased to 30 °C, only one relaxation appears around 277 kHz. When the temperature is increased further to 40 °C, two relaxations occur again with a lower permittivity increment. The inset of Figure 3 magnifies the part of the high-frequency dielectric relaxation at 15 and 40 °C in order to give a more explicit image. Figure 4 shows the temperature dependence of the dielectric relaxation parameters at concentrations of 28, 40, and 70 mmol L-1. In Figure 4a, with increasing temperature, the low-frequency limit of l, the relative permittivity, begins to increase slowly in phase I and then sharply to reach a peak in phase II, and then it subsequently decreases slowly in phase III. It is known that counterions dissociate in ionic micelle solutions. Some of them form counterion clouds that cover the surface of the micelles. If the shapes of the counterion clouds deviate from this symmetry, then a dipole moment u0 will be induced.17 Therefore, micelles with counterion clouds can be regarded as a dipole subject to some extent. The permittivity characterizing the polarization capability of materials is related to the dipole moment u0 and the molecular number n in unit volume.40 At the beginning of the (40) Ceˆa`´ıa`aˆÅ, A ˜ . EÅ . Dielectric Physics 1949, 106-248.
relaxation, l is close to the whole permittivity of the solution. The dipole moment u0 of the wormlike micelles is much larger than the rodlike micelles in the TBPFO-water system as a result of the large size of the wormlike micelles. When the concentration is fixed, n remains invariable; therefore, l shows the maximum value in phase II. In Figure 4b, the low-frequency limit of conductivity κl increases linearly in phases I and III. However, it decreases anomalously in phase II. Walden’s40 law describes the relationship between the conductivity and the viscosity of the dielectric liquid, which indicates that the product of the conductivity and the viscosity is a fixed value for a confined dielectric liquid. Increasing the temperature favors an increase in conductivity. However, the higher viscosity of the wormlike micelles leads to a decrease in κl in phase II. In Figure 4c, the relaxation frequency f0 decreases with temperature in phase I whereas it increases with the temperature in phase III. In phase II, two dielectric relaxations gradually turn into one. In Figure 4d, Cole-Cole parameters βl and βh are relatively large, which indicates that the dispersion of the relaxations is single close to Debye-type relaxation, except for the one for the phase transition. The distribution of the relaxation frequency f0 and the values of βl and βh suggest that dielectric relaxation strongly depends on
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Figure 5. Concentration dependence of the dielectric relaxation parameters evaluated by fitting eq 2 to the observed dielectric spectra. (a) Low-frequency limit of the relative permittivity l, (b) low-frequency limit of the conductivity κl, (c) characteristic relaxation frequency f0, and (d) Cole-Cole parameter β. There is only one set of data (T ) 20 °C) in all graphs, indicated by squares. Broken lines represent the phase transitions. In c and d, the low-frequency dielectric parameters are represented by closed symbols, and the high-frequency parameters are represented by open symbols.
the shape or morphology of the micelle aggregates. The temperature range of the first two-phase system is so small that the phase transition in this region is not very distinct. Concentration-Dependent Phase Transition. From the phase diagram Figure 1, it can be found that the concentration-dependent phase transition through the three phase regions appears merely in a small range of temperature. Therefore, only the concentrationdependent phase transition at 20 °C has been adopted in this article. At this temperature, the system goes through transformations from phase I to phase III and then to phase II. The concentration dependence of the dielectric relaxation parameters, obtained by fitting the experimental data with the Cole-Cole equation (eq 2) at 20 °C in the TBPFO-water system, is shown in Figure 5. The concentration-dependent phase transition is inconspicuous probably because the range of the first two-phase system is too limited. However, it still can be observed that the dielectric parameters undergo a delicate change with concentration. In Figure 5a, with the increase in concentration, l increases linearly in phases I and II with slopes of k1 and k2, respectively, but it is irregular in phase III. In phases I and II, the dipole moment u0 remains almost invariable; therefore, l is in proportion to n, which increases with concentration. However, the simultaneous changes in u0 and n lead to the anomalous variety of l in phase III. In Figure 5b, the transformation of κl is similar to that of l, which is consistent with the literature.32 Increasing the concentration results in an increase in conductivity. The different slopes of κl in phases I and II indicate that the aggregate structures of TBPFO are distinct in the two regions. In Figure 5c, f0 changes slightly with the increasing concentration. In Figure 5d, the value of βl for the low-frequency dielectric relaxation is comparatively high, which indicates a single-relaxation dispersion. These results strongly depend on the aggregate morphology of TBPFO.
From the analysis above, it can be concluded that the dielectric parameters obtained from fitting the experimental curves have shown a regular distribution. This is consistent with the phase diagram when the temperature or concentration is changed. These results indicate that frequency-dependent dielectric measurements are useful not only for monitoring the phase and structure transitions in aqueous micelle solutions of TBPFO in situ but also for testifying to the accuracy of other experimental methods. Dielectric Relaxations in Different Phase Regions. There are at least two components with different electrical properties in dispersing aqueous micelle solutions formed by ionic surfactant molecules. One is the aqueous medium containing dissociated counterions from the surfactants. Another is the hydrophobic micelle interior composed of surfactant alkyl tails. The surface of a micelle with electric charges originating from surfactant headgroups may be the third component. Therefore, the classical picture of the micelle can be regarded as an oil droplet with a charged surface surrounded by a highly conductive layer of condensed or associated counterions. If the shape of the counterion layer deviates from a symmetric structure, then a dipole moment will be induced. Several theories and models have been used to explain the relaxation mechanism of these systems, such as the Schwarz theory,22 O’Konski theory,25 Grosse model,28 and Pauly and Schwan model.41 Dielectric Relaxations in Phase I. Buchner et al.16 suggested that two dielectric relaxations should originate from the diffusional relaxation of free and bound counterions in the spherical micelles. Cetyltrimethylammonium bromide (CTAB) micelles were used to investigate an effect of the well-known sphere-to-rod transition in their research. This micelle transition does not affect the relaxation behavior of bound counterions in the higher-frequency (41) Pauly, H.; Schwan, H. P. Z. Naturforsch., B 1959, 14, 125.
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Figure 6. Schematic representation of the generative process of the induced dipole moment by the radial diffusion of the free counterions in rodlike micelles. Ra and Rb are the lengths of the long half-axis and the short half-axis of the rodlike micelles, respectively.
range of 109Hz, which is governed by the enthalpy of activation ∆H. This process is not dependent on the shape of the micelles but the number of available free sites and the interaction strength between X- and the polar headgroup of a surfactant ion. In contrast, the effective radius will change considerably at this transition; therefore, the low-frequency relaxation process of free counterions at about 108 Hz depends on the shape of the micelle. However, the micelles cannot be regarded as simple spheres in our investigated systems. The rodlike micelles in phase I in the TBPFO-water system show two dielectric relaxations when the temperature is 15 °C. Grosse’s model28 provides the calculation formula of 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 κm +2 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 (κp ) 0) are the dielectric permittivity and conductivity of a medium and a particle, respectively, and λs is the surface conductivity arising from the tangential motion of bound counterions. Because τ2 is independent of the shape of the micelles, the radius of the rodlike micelles can be replaced by the corresponding radius of the spherical micelles with the same volume, namely, about 10 nm.32 The value of λs corresponds to the value in the literature,16 about 10-9 Ω with respect to the order of magnitude. However, the value of τ2 calculated from eq 4 under all conditions is equal to 10-9 s, which is much less than the experimental values. Therefore, the rapid tangential motion of the counterions along the surface of the rodlike micelles is not the natural reason for the dielectric relaxation in phase I in the TBPFO-water system. The difference between this investigated system and an ordinary ionic surfactant system is that the counterion of the micelles in this system, tetrabutylammonium ion (TBA+), is modeled as a tetrahedron with the charged nitrogen atom at its center and the
four butyl chains at its four apexes.42 Such an organic ion with 16 hydrocarbons possesses significant hydrophobicity. Some of these short chains may penetrate into the micelle core because of the hydrophobic effect. However, the geometric restriction makes it unfeasible for all four chains to penetrate into the micelle core. At least two more chains should be located at the micelle surface layer in contact with water. The structural characteristic of these micelles has also proven that the rapid tangential motion of bound TBA+ is impossible under this condition. However, the order of magnitude of the diffusion coefficient Dt of TBA+ in the aqueous solution is about 10-6 cm2 s-1.43,44 The average apparent hydrodynamic radius (Rh) of the rodlike micelles in phase I is about 10 nm, which is estimated from the results of dynamic light scattering (DLS).32 The order of magnitude of the relaxation time τ estimated from eq 3 is 10-6 s. It lies between the slow mean relaxation time τs (4 × 10-6 s) and the fast mean relaxation time τf (4.5 × 10-8 s) obtained in the experiment. Compared with ordinary spherical micelles such as CTAB micelles, the difference in the lengths of the long halfaxis and the short half-axis increases in the rodlike micelles, and the diffusion coefficient decreases, originating from the change of the counterions. Therefore, the dielectric relaxation accompanying the radial diffusion of the free counterions in the spherical micelles has separated into two relaxations in rodlike micelles.35 This may be the real reason for the occurrence of the two relaxations in the rodlike micelles of the TBPFO-water system, which is proven by the experimental data. Thus, it can be speculated that the radial diffusion of the free counterions surrounding the micelles is probably the essential mechanism of the dielectric relaxation in the TBPFO-water system in the range of this measurement frequency, as shown in Figure 6. It has been proven recently that a largely incomplete second layer of TBA+ is loosely attached to the outside of the polar shell of the TBPFO micelle.45 If this second layer is actually available in the case of TBPFO micelles, then a few TBA+ ions may be associated with TBA+ bound to the micelle surface via hydrophobic interactions rather than being directly bound to the micelle surface. Therefore, it can be deduced that the diffusion of TBA+ takes place in this second layer. In fact, this incomplete second layer is just about the diffuse layer formed by the counterions. (42) Zana, R.; Benrraou, M.; Bales, B. L. J. Phys. Chem. B 2004, 108, 18195. (43) Kambe, S.; Nakade, S.; Kitamura, T.; Wada, Y.; Yanagida, S. J. Phys. Chem. B 2002, 106, 2967. (44) Kim, H.; Revzin, A.; Gosting, L. J. J. Phys. Chem. B 1973, 77, 2567. (45) Benrraou, M.; Bales, B. L.; Zana, R. J. Phys. Chem. B 2003, 107, 13432.
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Table 2. Size of the Rodlike Micelles at Different Concentrations of TBPFOa concn (mmol L-1) Ra (nm) Rb (nm) a
20
28
40
57
64
70
72
83
87
23.66 2.69
22.69 2.15
19.86 1.89
14.14 1.70
13.47 1.65
9.20 1.61
12.18 1.62
14.93 1.62
16.03 1.62
Ra and Rb are the lengths of the long half-axis and the short half-axis of the rodlike micelles, respectively. Table 3. Average Distance of the Connected or Entanglement Points in the Wormlike Micelles at Different Concentrations of TBPFOa 28 40 57 64 70 72 83 87 concn (mmol L-1) d (nm) 19.94 19.66 12.53 8.42 7.59 6.79 5.67 5.40 a d is the average distance of the connected or entanglement points in the wormlike micelles.
Figure 7. Relationship between permittivity increments and the concentration of TBPFO, estimated with the curve-fitting technique at T ) 15 °C in the TBPFO-water system. ∆l values are represented by closed symbols, and ∆h values are represented by open symbols.
The lengths of the long half-axis and the short half-axis of the rodlike micelles are about 20 and 2 nm, respectively, which are evaluated on the basis of the relaxation time obtained by the curve fitting. The detailed data are listed in Table 2. It can be found that these data are quite reasonable compared with the average apparent hydrodynamic radius (Rh) obtained from DLS.32 Figure 7 shows the relationship of the permittivity increment ∆ calculated by fitting the experimental data and the concentration of TBPFO. From Figure 7, it can be seen that the permittivity increment ∆ increases with concentration. The expression (eq 5) for the permittivity increment ∆ of the dielectric relaxation originating from the diffusion of the free counterions has also been given by the Grosse model.28 The volume fraction φ of the micelles increases with the concentration of surfactant; therefore, the permittivity increment ∆ also increases with the concentration of TBFPO. This further proves that the two relaxations in the rodlike micelles are certainly ascribed to the diffusion of the free counterions.
9φm(2χλs/κm)4 ∆ ) 2χλs 2λs 16 +1 +2 κm Rκm
[ (
) ]
2
(5)
where χ-1() x0mDt/κm) is the Debye length and φ is the volume fraction of the micelles. Dielectric Relaxations in Phase II. With increasing temperature, the system transfers from phase I to phase II, passing through the narrow phase III, as shown in Figure 1. The shape of the aggregates correspondingly changes from rodlike micelles to large wormlike micelles. In this transformation, the system is still homogeneous. However, in phase II the wormlike micelles entangle or cross link with each other and gradually form a connected network. This has been confirmed by the SLS/DLS and rheology experiments.32 The value of τ2 calculated from eq 4 in wormlike micelles is much less than the experimental value. Therefore, the rapid tangential motion of the bound counterions along the surface of the wormlike micelles is not the natural reason for the dielectric relaxation in the TBPFO-water system, whereas the radial diffusion of the free counterions surrounding the wormlike
micelles may be the essential mechanism of the dielectric relaxation in this measurement range and also in the rodlike micelles. However, the gradual formation of the connected or cross-linked points between the wormlike micelles results in the diffusion of the free counterions only between these entanglement points, which is very different from the diffusion of the counterions in the rodlike micelles. With the formation of the connected micelle network, the uniformity between the entanglement points may result in the gradual propinquity of the low- and highfrequency relaxations in phase II until only one relaxation remains. Therefore, it can be seen from the experimental data that two dielectric relaxations gradually close up to each other and turn into one with temperature. When the temperature is 30 °C, only one dielectric relaxation remains. The average distance d of the two connected or entanglement points can be roughly evaluated by the relaxation time obtained from curve fitting at about 30 °C, which is listed in Table 3. It can be seen from the data in Table 3 that the average distance of the connected or entanglement points decreases with the concentration of TBPFO because the viscosity of the TBPFOwater system increases. Dielectric Relaxations in Phase III. At higher temperature, the system separates into a surfactant-rich phase in equilibrium with a surfactant-depleted phase. They are both liquidlike phases that are similar to the two phases of nonionic surfactants formed above the cloud-point temperatures.46 Two dielectric relaxations still appear under this condition in DRS. One relaxation may originate from the diffusion of the free counterions, such as the one in rodlike or wormlike micelles. The formation of the large phase interface between the surfactantrich phase and the surfactant-depleted phase leads to the occurrence of the interfacial polarization processes. This is a result of the accumulation of virtual charge at the interface between the two media that have different permittivities and conductivities. Therefore, the interfacial polarization relaxation may be the essential mechanism at higher frequency in phase III.
Conclusions In this article, dielectric relaxation spectra of TBPFO aqueous solutions were investigated at different temperatures and concentrations. We showed that the dielectric relaxations occurred in different structural micelles. Relaxations of this system changed drastically near phase transitions upon increasing the temperature and concentration. The dielectric parameters characterizing dielectric spectroscopy were obtained by fitting the experimental data with the Cole-Cole equation. The distribution of several (46) Laughlin, R. G. Handbook of Detergents Part A: Properties; Broze, G., Ed.; Marcel Dekker: New York, 1999; Chapter 4.
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parameters consistent with the phase boundaries of the phase diagram indicated that this system was suitable monitoring with DRS. With increasing temperature, the low-frequency limit of relative permittivity l gained a maximum value in phase II. The low-frequency limit of conductivity κl increased linearly in phases I and III. With increasing concentration, both l and κl increased linearly in phases I and II. The changes in these parameters mainly depended on the morphologies of the micelles. The relaxation mechanisms in the two homogeneous phases were explained by the diffusion of the free counterions of Grosse’s model. In phase I, two relaxations of the rodlike micelles appeared around 100 kHz and 5 MHz and were attributed to the diffusion of the free counterions in the direction of the long micelle axis and the short micelle axis, respectively. With increasing temperature, only one relaxation remained and was a result of the formation of the connected or entanglement points between
Yang et al.
the wormlike micelles at about 30 °C. The lengths of the long half-axis and the short half-axis of the rodlike micelles as well as the average distance of the connected or entanglement points of the wormlike micelles were evaluated using the obtained relaxation times. In summary, the dielectric study of TBPFO micelle solution proved that DRS is an effective method of detecting the inner properties and the structural characteristics of this kind of system. The dynamic information and the structural properties obtained by DRS will provide the theoretical foundation for other related studies. Acknowledgment. This work is financially supported by the National Nature Science Foundation of China (no. 20673014). LA060907W
