Low-Frequency Dielectric Relaxation of Polyelectrolyte Gels - The

Division of Biological Sciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan. J. Phys. Chem. B , 1998, 102 (27), pp 5246â...
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J. Phys. Chem. B 1998, 102, 5246-5251

Low-Frequency Dielectric Relaxation of Polyelectrolyte Gels Tetsu Mitsumata, Jian Ping Gong, Kazuo Ikeda, and Yoshihito Osada* DiVision of Biological Sciences, Graduate School of Science, Hokkaido UniVersity, Sapporo 060-0810, Japan ReceiVed: NoVember 5, 1997; In Final Form: March 18, 1998

The complex dielectric constant * of the poly(sodium 2-acrylamido-2-methylpropanesulfonate) (PNaAMPS) gels and their corresponding linear polymer solutions have been measured at 25 °C in a frequency range of 50-100 kHz. The polyelectrolyte gels showed a low-frequency dielectric relaxation in a frequency region lower than that of linear polymer solutions. Furthermore, the mean relaxation time τ0 of the gel decreases with increasing the cross-linking density or the concentration. This is quite different from the behavior of the linear polymer solution, which showed a constant value of relaxation time upon changing the polymer concentration. By comparing with the linear polymer solution, the effect of the cross-linkage on the dielectric relaxation of bound counterions has been discussed.

1. Introduction A polyelectrolyte gel consists of a charged network and a fluid filling the interstitial space of the network. Different from the polyelectrolyte solution, the charged polymer gel exhibits a variety of unique phenomena such as a repetitive current oscillation when a dc voltage is applied,1,2 piezoelectric property of some kinds of weak polyelectrolyte gels,3 chemomechanical behavior,4 and electrocontratile behavior.5-7 These intrinsic properties and behaviors observed in the polyelectrolyte gel should be characterized by the nature of the cross-linked charged network associated with their counterions and the solvents. We have revealed from the results of the calculation which is based on a two-dimensional stacking model that ionized polymer gels have large electrostatic potential energy wells at every cross-linking point.8 Contrary to this, linear polymer solutions have only potential energy valleys along the polymer chain and have no deep electrostatic potential wells. Therefore, we are able to easily predict that the nature of counterions, for example their fluctuation length and diffusion velocity, would be quite different in both systems. Furthermore, the difference in the electrostatic potential energy distribution between a gel and a linear polymer solution should affect their electrical properties. Nevertheless, we have recently found out no appreciable difference in the dc conductivity between the gel and the linear polymer solutions of poly(2-acrylamido-2-methylpropanesulfonic acid) (PAMPS) and its alkali-metal salts.9 According to the condensation theory,10-12 the electrical conduction of a polyelectrolyte solution is made by the electrical drift of a fraction (f) of counterions that are located in a Debye-Huckel atmosphere. The remaining (1 - f) fraction of counterions are considered to be bound to the macroions and do not contribute to the dc conductivity. In the latter case, the results of dc conductivity measurements are understandable if we take into account that the dc conductivity of gels and linear polymer solutions is largely dominated by the loosely bound counterions (free counterions). In other words, there should be no large difference in the state of free counterions between a gel and a linear polymer solution. Therefore, we have decided to focus our investigation on the strongly bound counterions (bound

counterions) of gels and linear polymer solutions by studying their dielectric properties, and this forms the case of the present work. It is well-known that the dielectric relaxation spectrum gives us information about ions that move under an alternative electric field. The dielectric relaxation spectroscopy of the polyelectrolyte solutions shows two kinds of relaxation processes: the high-frequency relaxation in the MHz frequency range and the low-frequency one in the radio-wave frequency range. The high-frequency relaxation is due to the fluctuation of free counterions which exist far from macroions.13-16 The lowfrequency relaxation arises from the fluctuation of bound counterions which exist in the vicinity of macroions.13,14,17 The low-frequency relaxation that is caused by the bound counterion fluctuation along the polymer chain is expected to be observed in the polyelectrolyte gels as well as in the polyelectrolyte solutions. Moreover, we are able to speculate that the polyelectrolyte gel shows a slower relaxation process than the linear polymer solution, because of the strong networkcounterion interaction. To the authors’ knowledge, there are no reports of the lowfrequency relaxation of the polyelectrolyte gel, partly because of the experimental difficulty that originates from their high conductivity and the large electrode polarization effect.18 In this paper, we have attempted to investigate the low-frequency dielectric relaxation of the fully ionized sulfonate gels (sodium salts of poly(2-acrylamido-2-methylpropanesulfonic acid), PNaAMPS) of various cross-linking density and compare with their linear polymer solutions. 2. Experimental Section (i) Sample Preparation. The NaAMPS monomer was obtained by neutralization of 2-acrylamido-2-methylpropanesulfonic acid (AMPS) solution by adding an appropriate amount of hydroxide solution in methanol. The solution was precipitated in acetone and dried under a reduced pressure of 10-2 Torr. The PNaAMPS gels with various cross-linking density Fm (2-6 mol %) were prepared by radical polymerization using potassium persulfate as a radical initiator in the presence of a calculated amount of N,N′-methylenebisacrylamide (MBAA)

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Dielectric Relaxation of Polyelectrolyte Gels

J. Phys. Chem. B, Vol. 102, No. 27, 1998 5247

TABLE 1: Samples of the PNaAMPS Gels Used in the Present Measurement gelsa

Fm (mol %)b

qc

c (mol/L)d

polymer solutionse

S1G S2G S3G S4G S5G

2 3 4 5 6

479 180 85 49 48

9.1 × 10-3 2.4 × 10-2 5.1 × 10-2 8.1 × 10-2 9.1 × 10-2

S1P S2P S3P S4P S5P

a Abbreviations of gel samples. b Fm is the cross-linking density. c q is the degree of swelling. d c is the monomeric concentration. e Abbreviations of linear polymer solution samples.

used as a cross-linking agent. Polymerization was carried out at 50 °C under a N2 atmosphere for more than 12 h in a test tube. To remove any unreacted monomer and initiator, the gels were immersed in a large amount of purified water at least for 1 week until the gels had reached an equilibrium state. The linear polymer solutions of PNaAMPS were prepared by a similar procedure but in the absence of MBAA. Purification was performed by dialysing for a week and then freeze-drying. The molecular mass Mw of the linear PNaAMPS was determined from the intrinsic viscosity η in 0.5 N NaCl at 25 °C using the relation η ) K Mνw. Here, the parameters K and ν are 1.95 × 105 and 0.83, respectively for PNaAMPS.19 Mw was estimated to be 7.2 × 105 g/mol, which corresponds to a degree of polymerization N ) 3100. All of the measurements of the dielectric properties of gels were made at the equilibrium swelling state. The degree of swelling q was calculated using q ) (swollen sample weight)/ (dry sample weight). The monomeric molar concentration c of a gel in the equilibrium state was obtained using the relation c ) 1000/qFw, where Fw is the formula weight of the sodium salt of AMPS monomer. c of a gel is inversely proportional to the degree of swelling and is varied by changing the crosslinking density of the gel. Linear polymer solutions with the same monomeric concentrations of gels were prepared as reference samples. The degree of swelling of gels with various cross-linking density, the monomeric concentration, and the sample name of their corresponding linear polymer solutions are listed in Table 1. (ii) Measurements. The complex dielectric constant * measurements were carried out by an ac two-terminals method using an LCZ meter. The frequency range was from 50 Hz to 100 kHz, and the applied voltage was 1 V. The PNaAMPS gel was cut into a 2 × 2 × 1 cm3 rectangular solid shape and was sandwiched between a pair of parallel-plate electrodes with an electrode distance of about 1 cm. The accurate distance between two electrodes was determined using a microscope. The measurement of the linear polymer solutions was carried out using the conventional method by using the parallel-plate condenser type cell. The electrodes were black-platinized platinum with an apparent dimension of 1 × 1 cm2. The effective dimension of the electrodes and the cell constant were determined by using a KCl aqueous solution. The sample assembly was placed in a water bath which was composed of two cylindrical cylinders. All of the dielectric measurement system, i.e., sample space and electrical leads, was electrically shielded. We could not obtain an accurate dielectric constant below 300 Hz without this shield. In the measurement of the cross-linking density (monomeric concentration) dependence, the temperature in the sample space was maintained at 25.0 ( 0.1 °C. The * measurements were started when the sample reached the thermal equilibrium state. We considered the thermal equilibrium state as an electric conductivity change per second ∆σ/σ less than

Figure 1. (a) Typical example of the as-measured data of the real part (′) and imaginary part (′′) of the complex relative dielectric constant ′r of PNaAMPS gel (S3G) at 25 °C: (O) ′; (b) ′′. (b) Lowfrequency dielectric relaxation spectra of ′ (O) after deleting the electrode polarization component from the as-measured data and the calculated ′′ (b) of PNaAMPS gel (S3G) at 25 °C.

10-4%. The measurements were carried out at various crosslinking densities of the gels and their corresponding concentration of linear polymer solutions. The temperature dependence of dielectric loss was measured in the temperature range from 25 to 65 °C. Samples were a 4 mol % gel (S3G) and its corresponding linear polymer solution (S3P). The peak of the dielectric loss curve of the gel appeared around 200 Hz at 25 °C. We could not obtain reliable measurements either below 25 °C or above 65 °C in this system. (iii) Data Decomposition. As a typical example of results, Figure 1a shows as-measured data of the relative complex dielectric constant * ) ′ - i′′ at 25 °C for the sample of S3G. Filled circles in Figure 1a are the real part ′ of the complex dielectric constant. ′ is extremely large in the lowfrequency range and decreases with increasing frequency. As described in the Introduction, this large ′ is apparently caused by the large polarization between electrodes, which is called the “electrode polarization effect”. The ion-blocking electrodes give rise to a frequency-dependent polarization. Such a large electrode polarization was the most serious obstacle in the lowfrequency dielectric measurement and should be deleted. According to the literature, the electrode polarization term of a linear polyelectrolyte solution ′el can be expressed by the following empirical equation:18,20,21

′el ) Af-R

(1)

where A is a constant, f is the frequency of the electric field, and R is the fractional power. Mandel and Odijk have reported that the polarization impedance generally depends on the

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Mitsumata et al.

TABLE 2: Best-Fitting Parameters of the Low-Frequency Relaxation and the dc Conductivity of Gels and their Linear Polymer Solutions sample

τ0 (ms)a

∆b

Rc

βd

Fe

σ (mS/cm) f

S1G S2G S3G S4G S5G S1P S2P S3P S4P S5P

2.13 1.16 1.00 0.77 0.69 0.21 0.32 0.28 0.16 0.17

5000 3000 5790 19640 21100 1080 2240 4400 11200 13600

1.34 1.46 1.46 1.44 1.44 1.60 1.50 1.61 1.50 1.53

1.00 0.96 0.95 0.96 0.98 0.80 0.71 0.73 0.82 0.83

0.99632 0.99826 0.99978 0.99998 0.99998 0.99940 0.99987 0.99997 0.99999 1.00000

0.20 0.51 1.15 1.96 2.48 0.26 0.70 1.59 2.88 2.89

a τ is the mean relaxation time. b ∆ is the dielectric increment. c R 0 is the fractional power in eq 1. d β is the broadness of distribution of relaxation time. e F is the correlation coefficient. f σ is the dc conductivity.

electrode material, the adsorption on the electrode surface, the conductivity of the solution and some other factors which are not so well understood.13 R takes a variety of numbers even though in the same system.20,21 As shown by the solid line in Figure 1a, which represents the electrode polarization component, the frequency dependence of ′ in the lower frequency region is well explained by eq 1. Therefore, we use this empirical relation to delete the electrode polarization component from the as-measured data of gel by subtracting this term from the as-measured data. In this subtraction process, we have determined the R by letting the correlation coefficient F take the maximum value. Here, F indicates the correlation between the residual value of the observed data after the subtraction and the theoretical curve given by the following equations assuming that the relaxation obeys Cole-Cole’s circular arc law:22

{

}

1 sinh βx ′ ) ′∞ + (′0 - ′∞) 1 2 cosh βx + cos(βπ/2) sin(βπ/2) 1 ′′ ) (′0 - ′∞) 2 cosh βx + cos(βπ/2)

(2) (2′)

where x ) ln ωτ0. ′∞ and ′0 are the high-frequency value and the low-frequency value of the real part of the dielectric constant, respectively. ∆ ()′0 - ′∞) is the dielectric increment, ω is the angular frequency of applied electric field, τ0 is the mean relaxation time, and β is a parameter representing the broadness of distribution of relaxation times. The fitting using the nonlinear least-squares method was carried out in the frequency region that is not influenced by the high-frequency relaxation (for the sample S3G, for example, below 5 kHz). The obtained best-fitting parameters, τ0 and β and the correlation coefficient F are listed in Table 2. As listed in Table 2, the β of gels is independent of the crosslinking density, and it is higher than 0.95 and very close to 1, indicating that this relaxation is very close to Debye type. On the other hand, the β of linear polymer solutions is lower than that of gels. This suggests that the relaxation time of linear polymer solutions has a wider frequency distribution compared with gels. Open circles in Figure 1a show the as-measured data of the imaginary part ′′ of the complex dielectric constant. The broken line in Figure 1a has a slope of -1. ′′ monotonically decreases with increasing frequency, and it is almost inversely proportional to the frequency in the whole frequency region, which indicates that ′′ is dominated by the dc conductivity of

Figure 2. Low-frequency dielectric relaxation spectra of ′ (a) and ′′ (b) of PNaAMPS gels with various cross-liking density at 25 °C: (O) S1G; (0) S2G; (]) S3G; (4) S4G; (3) S5G.

free ions. The dc conductivity contribution can be written by the following equation:

′′dc )

σ 2πf

(3)

where σ is the dc electric conductivity. We have estimated the conductivity of samples from the slope of ′′ vs f plot, and the results are also listed in Table 2. Despite the high conductivity, we have also tried to evaluate the dielectric relaxation component of ′′ by eliminating the dc conductivity term from the as-measured data. However, the residual component after subtraction was less than 0.03% of the as-measured value and was concluded to be in the range of error. Hence, we could not rely on ′′ thus obtained and calculated the ′′ spectrum by using eq 2′. Figure 1b shows the corrected data of real and imaginary parts of the relative complex dielectric constant of S3G. The solid and broken lines in Figure 1b show the results of ColeCole fitting. All of the as-measured data of the gels and linear polymer solutions have been processed using the above method. To see the validity of the data decomposition, the error range of the residual component is discussed. The error mainly originates from the accuracy of the impedance meter and subtraction of the data. Below 100 Hz, the ratios of the residual values after subtraction to the as-measured values are 1.6% and 0.9% for gels and linear polymer solutions, respectively. The accuracy of the impedance meter at the low-frequency end is at least 0.15%. Accordingly, the residual values after subtraction are within the accuracy of the measurement and are worth discussing.

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J. Phys. Chem. B, Vol. 102, No. 27, 1998 5249

Figure 5. Concentration dependencies of the dielectric increments ∆ of PNaAMPS gels and their corresponding linear polymer solutions: (O) polyelectrolyte gels; (b) linear polymer solutions.

Figure 3. Low-frequency dielectric relaxation spectra of ′ (a) and ′′ (b) of PNaAMPS linear polymer solutions with various concentrations at 25 °C (O) S1P; (0) S2P; (]) S3P; (4) S4P; (3) S5P.

Figure 6. Schematic illustration of a possible explanation for the lowfrequency relaxation of the polyelectrolyte gel.

and the Cole-Cole fitting are considerably good. However, below the sample S3G or S3P, F decreases, and this means that the obtained parameters of the relaxation time and β contain large errors. The errors that are caused by the subtraction are shown in Figure 4, Figure 7, and Figure 8. 3. Results and Discussion Figure 4. Concentration dependencies of the mean relaxation time τ0 of PNaAMPS gels and their corresponding linear polymer solutions: (O) polyelectrolyte gels; (b) linear polymer solutions.

As shown in Table 2, the correlation coefficient F of gels and linear polymer solutions is close to 1 at high cross-linking densities or high concentrations, suggesting both the subtraction

The low-frequency relaxation spectra of PNaAMPS gels at 25 °C with various cross-linking density are shown in Figure 2a,b. All samples have a flat part on the ′ curve in the frequency region around 103-104 Hz. We can see at the higher frequency side a slight decrease in ′, which is considered to be the high-frequency relaxation of loosely bound counterions. Figure 2b clearly indicates that the dielectric increment tends to increase and the peak frequency of ′′ shifts toward higher frequency regions with increasing cross-linking density.

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Mitsumata et al. TABLE 3: Diffusion Constant and the Ratio of “Bound” to the Total Counterions in Linear Polymer Solutions sample

D (×10-10 m2/s)a

nB (m-3)b

n (m-3)c

nB/n (%)d

S1P S2P S3P S4P S5P

2.48 1.62 1.86 3.25 3.06

2.9 × 1022 6.1 × 1022 1.2 × 1023 3.1 × 1023 3.7 × 1023

5.5 × 1024 1.4 × 1025 3.1 × 1025 4.9 × 1025 5.5 × 1025

0.5 0.4 0.4 0.6 0.7

a D is the diffusion constant of bound counterions. b nB is the number concentration of bound counterions. c n is the number concentration of total counterions. d nB/n represents the ratio of “bound” to the total counterions.

Assuming a square type potential well along the polymer chain, τ0 of the linear polymer solution is given by eq 4.24 Figure 7. Linear plot of the observed mean relaxation time τ0 against the calculated fluctuation time τ1 between two neighboring cross-linking points.

τ0 )

Lc2 Lf2 ) D 12D

(4)

where Lf and D are the fluctuation length and the diffusion constant of the bound counterion, respectively. Lc is the contour length of the macroion. The dielectric increment ∆ is expressed as

∆ ≈

Figure 8. Arrhenius plots of the peak frequency fm of loss curves of PNaAMPS gel (S3G) and its corresponding linear polymer solution (S3P): (O) polyelectrolyte gel; (b) linear polymer solution.

The low-frequency relaxation spectra of PNaAMPS linear polymer solutions at 25 °C are shown in Figure 3a,b. The solid lines in both figures represent the results of Cole-Cole fitting. It is evident that the low-frequency spectra of linear polymer solutions are quite different from gels. All linear polymer solutions have a plateau part of ′ below 300 Hz and show relaxation around 1 kHz and higher. The figure shows that although the dielectric increment increases with increasing concentration as the gels showed, the peak frequency of ′ is almost independent of the concentration. Figure 4 shows the concentration dependencies of τ0 of gels and their corresponding linear polymer solutions. The relaxation time of gels decreases with increasing the concentration, while that of linear polymer solutions is almost independent of the concentration. Figure 5 shows the concentration dependence of ∆ of gels and their linear polymer solutions. The solid lines in Figure 5 show the exponential fitting. Both gels and linear polymer solutions satisfactorily showed the exponential change of ∆ against the concentration, except for S1G. According to the literatures,13,14 the observed low-frequency relaxation of the linear polymer solution is associated with the tightly bound counterion fluctuation along the polymer chain. This low-frequency relaxation strongly depends on the molecular weight but is almost independent of the concentration.23

nBe2Lf2 0kBT

(5)

where nB is the number concentration of the bound counterions, e is the elementary charge, 0 is the vacuum permittivity, kB is Boltzmann constant, and T is the absolute temperature. Therefore, using eq 4 and eq 5, we can estimate the diffusion constant of bound counterions as well as the number concentration of bound counterions for linear polymer solutions. Using the degree of polymerization N as 3100 and the monomer unit length b as 2.55 Å, we get Lc and Lf as 791 nm and 228 nm, respectively. The estimated results of D and nB using eq 4 and eq 5 are listed in Table 3. The mean diffusion constant of five samples Dm is determined to be 2.5 × 10-10 m2/s, which is an order of magnitude smaller than the diffusion constant D0 ()1.22 × 10-9 m2/s) of Na+ ion in a free medium. This result is in good agreement with the previous data of sodium salts of polystyrenesulfonate (PNaPSS) solution reported by N. Ookubo et al.25 The amount of bound counterions for the linear polymer solutions estimated from eq 5 is 0.3-0.8% of total counterions. Dependence of τ0 on the cross-linking density of the polyelectrolyte gels seems to suggest that the relaxation time of the gels corresponds to the fluctuation of bound counterions between two neighboring cross-linking points along the polymer chain. If this was true, we would get a relaxation time on the order of 10-7 s, which is 4 orders of magnitude smaller than those we have observed. Therefore, the above assumption is apparently not appropriate. On the contrary, we can consider that the observed relaxation of gels might correspond to the counterion fluctuation along the polymer network by crossing through the cross-linking points, as illustrated in Figure 6. If we assume that the observed relaxation time consists of the two relaxation times, i.e., the time τ1 for fluctuating along the linear part of the polymer network and the time τ2 for exceeding the energy barrier that is based on the electrostatic potential well at the cross-linking points, we can write the observed τ0 as a linear combination of τ1 and τ2.

τ0 ) m2τ1 + (m - 1)2τ2

(6)

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Here, the coefficient m stands for the number of polymer segments that the bound counterion can fluctuate and (m - 1) corresponds to the cross-linking points that the bound counterions can cross through. Because the fluctuation length correlates with the relaxation time in squared form, we have the squared form of m or (m - 1) in eq 6. Since the average number of monomeric units of a polymer chain segment between two neighboring cross-linking points is 1/2Fm, the contour length of a polymer chain segment is Lc ) b/2Fm. Assuming that the contour length of the energy barrier at the cross-linking points is 2b, we have

(

)

2 1 - 2 b2 2Fm τ1 ) 12D1

(7)

and

τ2 )

(2b)2 12D2

(8)

where D1 and D2 are the diffusion constant of bound counterions along the polymer chain segment and that at the cross-linking points, respectively. D1 should be the same as that of a linear polymer solution Dm. Thus, we can estimate τ1 from Fm and Dm. The relationship between τ0 and τ1 is shown in Figure 7. As seen in the figure, the mean relaxation times τ0 at various crosslinking density are well explained by the eq 7. From the slope and the intersection at the τ0 axis of the straight line shown in Figure 7, we get m ) 97 and D2 ) 10-13 m2/s, which is 3 orders of magnitude smaller than that of the bound counterion in the linear part of the polymer chain. The linearity of eq 7 indicates that the number of cross-linking points that the bound counterions cross over is independent of the cross-linking density. Accordingly, the energy barriers of the cross-linking points are sensitive neither to the degree of cross-linking density nor to the polymer concentration. This is in accordance with the fact that the diffusion constant D of bound counterions along a linear polymer chain is not sensitive to the polymer concentration. These results suggest that the state of bound counterions in the vicinity of the polymer chain or in the cross-linking points is not affected by the presence of other polymers or their counterions. To investigate the temperature effect on the electrostatic interaction between macroions and bound counterions, we have measured the temperature dependence of the low-frequency dielectric relaxation. Figure 8 shows the Arrhenius plots of the peak frequency fm of the dielectric loss curves of samples S3P and S3G. The peak frequencies are well fitted by the Arrhenius formula

( )

E fm ) f0 exp kBT

(9)

where f0 is the frequency factor, kB and T represent the same notation as in eq 3, and E is the activation energy. The activation energy of the gel and its linear polymer solution were determined as 15.0 kJ/mol and 13.7 kJ/mol, respectively. These values are nearly equal to those of electric conductivity9 and might be interpreted by the viscous drag between the bound counterions and water described by the Stokes’ law. Due to the small difference of E between the gel and the linear polymer solution, we could not determine an appreciable temperature

dependence of the electrostatic interaction between macroions and counterions. Generally, the relaxation time associated with the rotation of the main chain of a linear polymer solution is reported in the range of milliseconds25,26 and coincides with the data we have observed in the polyelectrolyte gels. Accordingly, it seems probable that the observed relaxation of gels should be ascribed to this rotation. However, it is usual that the activation energy of rotational relaxation of a polymer chain is much higher than the observed values.27-29 Furthermore, since the end of the chains are fixed to the cross-liking points, the rotational activation energy of the gel might be even higher than that of the linear polymer solution. Therefore, these low values of activation energies E indicate that the observed low-frequency relaxation is not associated with the main chain rotation. In summary, we have observed the low-frequency dielectric relaxation of polyelectrolyte gels. The relaxation time of gels is much longer than that of polyelectrolyte solutions and strongly depends on the cross-linking density and/or the concentration. This behavior of gels is different from that of the linear polyelectrolyte solution, which showed a low-frequency relaxation independent of the monomeric concentration. The lowfrequency relaxation observed on the gels has been explained in terms of the fluctuation of the bound counterions along the polymer network by crossing through the cross-linking points. According to this explanation, the diffusion constant at the crosslinking point is 3 orders lower in magnitude in comparison with that along the polymer chain. This might be a proof of the presence of deep potential wells at cross-linking points. Acknowledgment. The authors wish to thank Professor R. Nozaki of Hokkaido University and Professor K. Ito and Dr. H. Furusawa of The University of Tokyo for their helpful discussions. This research is supported by a Grant-in-Aid for the Special Promoted Research Project “Construction of Biomimetic Moving Systems Using Polymer Gels” from the Ministry of Education, Science and Culture, Japan. References and Notes (1) Umezawa, K.; Osada, Y. Chem. Lett. 1987, 1795. (2) Miyano, M.; Osada, Y. Macromolecules 1991, 24, 4755. (3) Sawahata, K.; Gong, J. P.; Osada, Y. J. Macromol. Sci. 1991, 1189. (4) Osada, Y.; Okuzaki, H.; Hori, H. Nature 1992, 355, 242. (5) Osada, Y.; Hasebe, M. Chem. Lett. 1985, 1285. (6) Osada, Y.; Kishi, R. J. Chem. Soc., Faraday Trans. 1989, 85, 655. (7) Gong, J. P.; Nitta, T.; Osada, Y. J. Phys. Chem. 1994, 98, 9583. (8) Gong, J. P.; Osada, Y. Chem. Lett. 1995, 449. (9) Gong, J. P.; Komatsu, N.; Nitta, T.; Osada, Y. J. Phys. Chem. 1997, 101, 740. (10) Oosawa, F. J. Polym. Sci. 1957, 23, 421. (11) Manning, G. S. J. Chem. Phys. 1969, 51, 934. (12) Manning, G. S. J. Phys. Chem. 1975, 79, 262. (13) Mandel, M.; Odijk, T. Annu. ReV. Phys. Chem. 1984, 35, 75. (14) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971. (15) Minakata, A. A. Ann. N.Y. Acad. Sci. 1977, 107, 303. (16) Ito, K.; Yagi, A.; Ookubo, N.; Hayakawa, R. Macromolecules 1990, 23, 857. (17) Minakata, A.; Imai, N. Biopolymers 1972, 11, 329. (18) Scheider, W. J. Phys. Chem. 1975, 79, 197. (19) Fisher, L. W.; Sochor, A. R.; Tan, J. S. Macromolecules 1977, 10, 949. (20) Shaw, M. T. J. Chem. Phys. 1942, 10, 609. (21) F. J. Robert, J. Chem. Phys. 1954, 22, 1329. (22) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341. (23) Minakata, A. Biopolymers 1972, 11, 1567. (24) Mandel, M. Mol. Phys. 1961, 4, 489. (25) Ookubo, N.; Hirai, Y.; Ito, K.; Hayakawa, R. Macromolecules 1989, 22, 1359. (26) Takashima, S. J. Phys. Chem. 1966, 70, 1372. (27) Kauzmann, K. ReV. Mod. Phys. 1942, 12, 34. (28) Eyring, H. J. Chem. Phys. 1936, 4, 283. (29) Ewell, R. H. J. Appl. Phys. 1938, 9, 252.