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Langmuir 2002, 18, 6404-6409
Dielectric Relaxations in Aqueous Polyelectrolyte Solutions: A Scaling Approach and the Role of the Solvent Quality Parameter F. Bordi Dipartimento di Medicina Interna, Universita` di Roma “Tor Vergata”, Rome, Italy, and Istituto Nazionale per la Fisica della Materia (INFM), Unita’ di Roma1
C. Cametti* and T. Gili Dipartimento di Fisica, Universita` di Roma “La Sapienza”, Piazzale A. Moro 5-I-00185 Rome, Italy, and Istituto Nazionale per la Fisica della Materia (INFM), Unita’ di Roma1
R. H. Colby Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received February 19, 2002. In Final Form: April 25, 2002 The radiowave dielectric behavior of sodium polyacrylate (NaPAA) aqueous solutions, in an extended concentration range and at various degrees of polymerization, was analyzed in the light of the scaling approach for polyelectrolyte solutions recently proposed by Dobrynin et al. [Macromolecules 1995, 28, 1859]. In the frequency range from 1 MHz to 1 GHz, the observed dielectric relaxation is attributed to the fluctuation of loosely bound counterions over a characteristic correlation length which depends on the polymer concentration. Assuming that free counterions fluctuate on a length scale that is proportional to the correlation length ξ0 of the semidilute solution, we derived power laws for the dielectric strength ∆ and the relaxation frequency ν0. In agreement with our recently published data on conductivity, the dielectric data indicate that the effective charge on the NaPAA chains increases as concentration is raised, until the correlation length reaches the size of the electrostatic blob. By comparing the dielectric behavior of a series of polymers exhibiting different affinities for water as a solvent, we showed that the scaling behavior furnishes a very good description of the observed concentration dependence of the dielectric parameters. These findings suggest a method for evaluating, from simple dielectric measurements, the solvent quality parameter τ for the uncharged polyelectrolyte chain, a parameter which up to now eluded a precise determination by means of other experimental techniques. Finally, a correlation between the fraction f of free counterions and the solvent quality parameter τ, in poor solvent conditions, was evidenced. This gives further support to the hypothesis that the effective polyelectrolyte-solvent entropic interaction is primarily dictated by the effective charge on the chain.
1. Introduction The dielectric behavior of aqueous polyelectrolyte solutions has been extensively investigated in the past few decades,1-7 and a common picture has been now established, consisting, at least, of the presence of three different, partially overlapping, relaxation regions, ranging from some hertz to microwave frequencies. This complex phenomenology is attributed to different molecular level mechanisms, originated or induced by the external electric field. Whereas in the low-frequency and in the high-frequency regions the observed dielectric dispersions are characterized by length scales related to the polymer chain size and to that of the solvent phase molecules, respectively, in the intermediate-frequency * To whom correspondence should be addressed. (1) Polyelectrolytes: Science and Technologies; Hara, M., Ed.; Marcel Dekker: New York, 1993. (2) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1970. (3) Schmitz, K. S. Macroions in solution and colloidal suspensions; VCH: New York, 1993. (4) Mandel, M. In Polyelectrolytes: Science and Technologies; Hara, M., Ed.; Marcel Dekker: 1993. (5) Bordi, F.; Cametti, C.; Paradossi, G. Biopolymers 1995, 36, 539. (6) Bordi, F.; Cametti, C.; Paradossi, G. Biopolymers 2000, 53, 129. (7) Bordi, F.; Cametti, C.; Motta, A. Macromolecules 2000, 33, 1910.
region, the mechanism causing the dispersion has been for a long time more controversial and only recently has been attributed to local fluctuation of counterions along some typical correlation length.8,9 Although various theoretical approaches have been developed, most of them neglect polyion-polyion interaction as well as the influence of the solvent on the polyion chain conformation. The scaling theory of polyelectrolyte solutions,10,11 based on the idea of de Gennes et al.12,13 of electrostatic blobs inside which Coulombic repulsion competes with polymersolvent interactions to determine the chain configuration, provides a description of the overall chain conformation in the different concentration regimes, which can be summarized as follows. When the electrostatic screening (8) Ito, K.; Yagi, A.; Ookubo, N.; Hayakawa, R. Macromolecules 1990, 23, 857. (9) Odijk, T. Macromolecules 1979, 12, 688. (10) Rubinstein, M.; Colby, R. H.; Dobrynin, A. V. Phys. Rev. Lett. 1994, 73, 2776. (11) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859. (12) de Gennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. France 1976, 37, 1461. (13) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1980.
10.1021/la020175h CCC: $22.00 © 2002 American Chemical Society Published on Web 07/13/2002
Dielectric Relaxation in Polyelectrolyte Solutions
length rs is much larger than the electrostatic blob size D, Coulomb repulsion stretches the chain into a quite stiff cylinder. With the increase of the polymer concentration, when the screening length becomes smaller than the extended chain size, on length scales of the order of rs the chain is still a rodlike sequence of electrostatic blobs. On longer length scales, however, the chain is a random walk, where the screening length plays the role of a correlation length ξ0. In salt-free solutions, the concentration where rs becomes approximately equal to the length of a fully extended chain of electrostatic blobs, L, is in the same concentration range of the overlap concentration c*, when the distance between chains is equal to their extended size L. As a consequence, while for c < c* the polymer solution is in the dilute regime, for c* < c < ce the solution consists of space-filling correlation blobs of size ξ0 ≈ rs (semidilute-unentangled regime). As the concentration is further increased, a concentration ce is reached where a significant overlap of neighboring chains is attained and they begin to constrain each other (semidiluteentangled regime). Finally, above the concentration cD, electrostatic blobs overlap, and the chain configuration is expected to cross over to that of uncharged polymer. In the present work, we have analyzed the radiowave dielectric properties of sodium polyacrylate aqueous solutions in a wide concentration range, from the dilute to the concentrated regime, in particular from the crossover between dilute and semidilute-unentangled to semidilute-entangled and concentrated regimes. Within these concentration regimes, the polyion chain assumes different configurations, causing a different dielectric behavior observed in the intermediate relaxation process. From the dielectric relaxation point of view, the model of Ito et al.8 assumes that the observed intermediate relaxation is due to counterion fluctuations on a length scale that depends on polyelectrolyte concentration. In dilute solutions, the chains are very far apart and can be considered as pointlike charges in the center of a spherical cell. In this case, the “fluctuation length” is proportional to the cell diameter and scales as c-1/3. At higher concentrations, when the distance between the chains becomes comparable with their length, the cylindrical cell model applies and the fluctuation length scales as c-1/2. Along this scheme, we assume that in the semidilute regime, counterions might fluctuate on a length scale that is proportional to the correlation length ξ0 defined within the scaling theory of polyelectrolyte solutions,10,11 and we derive the power-law dependencies for the dielectric parameters in the semidilute regime, according to the different solvent quality. In this paper, we analyze the dielectric behavior of polyelectrolyte aqueous solutions in light of the scaling approach for polyelectrolyte solutions in good solvent conditions. Moreover, within the same scheme we have reanalyzed the dielectric behavior of a series of different polymers with different hydrophobicities, the water phase behaving, in this case, as a poor solvent. Also in this case, we will show that the scaling behavior furnishes a very good description of the observed concentration dependence of the dielectric parameters. These findings allow us to suggest a method for evaluating, from rather simple dielectric measurements, the solvent quality parameter τ for the chain, a parameter which is difficult to estimate experimentally by means of other techniques. Finally, the correlation between the free counterion fraction f and the solvent quality parameter τ is discussed. This gives further support to the hypothesis that the effective polymer-
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solvent energetic interaction is primarily dictated by the effective charge on the polyelectrolyte chain. 2. Experimental Section The dielectric spectrum of sodium polyacrylate aqueous suspensions has been measured in the frequency range from 1 MHz to 1.8 GHz where a well-defined dielectric dispersion due to counterion fluctuation along the characteristic correlation length ξ0 occurs. Six samples of sodium polyacrylate with nominal molecular weights of 2.1, 5.1, 20, 60, 140, and 225 kD were purchased from Polysciences Inc., Warrington, PA, as 20 or 25 wt % solutions in water and were used without any further purification. The polymers have a moderate polydispersity with a weight-averaged to number-averaged molecular weight Mw/Mn < 1.2 for all the samples investigated. The pH values of the solutions, for all the molecular weights investigated, show a continuous moderate decrease as the polymer concentration is increased, varying from about pH ) 9.0 at C ) 0.01 g/mL to about pH ) 8.0 at C ) 0.2 g/mL. At these pH values, the polymer behaves as a polyelectrolyte. Polymer solutions at the desired concentrations in the range from 0.001 to about 0.5 wt/wt were prepared with Q-quality water (Millipore) with an electrical conductivity at room temperature of 1-2 µΩ-1 cm-1. The polymer concentration investigated covers the dilute to concentrated regime, crossing the different crossovers between semidilute-unentangled, semidilute-entangled, and concentrated regimes. The dielectric spectra have been measured by means of a radio frequency impedance analyzer, Hewlett-Pachard model 4291A, at the temperature of 20 °C. Details of the dielectric cell and the calibration procedure have been reported elsewhere.14,15
3. Scaling of the Dielectric Parameters The polyelectrolyte solution contains Np flexible charged polyions per unit volume, each of them built up of N monomers of size b carrying an ionizable group of charge zpe and a counterion of charge z1e, where zp and z1 are their valences, respectively, and e is the electronic charge. Owing to the counterion condensation, an effective charge per polyion will appear, Q ) fNe, where f is the fraction of ionized groups (or conversely, of free counterions, after condensation takes place). In the semidilute regime (c > c*), when electrostatic and excluded volume interactions are screened on a length scale larger than the correlation length, each chain is represented by a random walk of Nξ0 ) N/g correlation blobs of size ξ0, containing an electric charge qξ0 ) fge, where g is the number of monomers inside a correlation blob. The dielectric relaxation, intermediate between that where the polarization due to the whole polyion length dominates and that where the relaxation of the pure water occurs, results from the counterion polarization on a length scale of the order of the correlation length ξ0. According to Ito et al.,8 this process causes a dielectric strength ∆ and a relaxation frequency ν0 given by
∆ ≈ fclBwξ02 ν0 ≈
D ξ02
(1) (2)
respectively, where lB ) e2/wkBT is the Bjerrum length and w is the permittivity of the aqueous phase. Here, D is the diffusion coefficient of free counterions, which we take to be identical to that of the counterion in a dilute (14) Takashima, S.; Casaleggio, A.; Giuliano, F.; Morando, M.; Arrigo, P.; Ridella, S. Biophys. J. 1986, 49, 1003. (15) Bordi, F.; Cametti, C.; Paradossi, G. Biopolymers 1996, 40, 485.
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simple salt solution, and kBT is the thermal energy. Combining eqs 1 and 2 to eliminate ξ0 gives
f≈
ν0∆ DclBw
(3)
which allows the fraction of free counterions to be determined from the measurement of the intermediate frequency dielectric relaxation alone, independently of the solvent quality. Moreover, eq 2 links the relaxation frequency ν0 to the correlation length ξ0, which, in turn, depends on the polyion chain configuration and on its specific electrostatic interaction with the solvent. Following the scaling approach proposed by Dobrynin et al.,10,11 the correlation length ξ0 is given by
{
τ1/2 Tθ f 2/7lB1/7b5/14c1/2
(4)
where τ ) (θ - T)/θ is the solvent quality parameter and θ is the temperature at which the net excluded volume for the monomer is zero. Their dependence on the polymer concentration c provides a simple means to estimate the solvent quality parameter, its influence on the dielectric parameters, and, more generally, the interaction with the solvent. Combining eqs 1 and 4 to eliminate ξ0 and substituting the expression for f given by eq 3 results in the following power-law dependencies:
τ-1ν01/3∆4/3 ≈ c1/3
(5)
in the case of poor solvent, T < θ,
ν0-1∆2 ≈ c-1
(6)
in the case of a θ solvent, T ) θ, and finally
ν0-1∆4/3 ≈ c-1
(7)
in the case of good solvent, T > θ. Whereas eqs 6 and 7 clearly indicate a dependence on the polymer concentration as c-1, in the case of poor solvent, the dependence is less obvious owing to the presence of the further parameter τ which in turn varies with the concentration c. On the other hand, in the case of poor solvent, if expressions 1 and 2 for dielectric parameters ∆ and ν0 are conveniently rearranged and the appropriate dependence for ξ0 (T < θ) (eq 4) is taken into account, a relationship between τ and f can be easily obtained in the form
τ)
() fb lB
1/3∆
w
(8)
suggesting that there may be some correlation between the solvent quality factor and the fraction f of free counterions, for a polyelectrolyte in a poor solvent. A further comment is in order. Ito et al.8 have demonstrated that for a polyelectrolyte solution in a given solvent, in the semidilute regime, ∆ is proportional to the concentration c, to the fraction f of free counterions,
and to the square of the correlation length ξ0, while the relaxation frequency ν0 is proportional to the inverse of ξ02 (eqs 1 and 2). On the other hand, the dependence of ξ0 on the concentration c predicted by de Gennes12 (eq 4) has been confirmed for different polyelectrolytes by smallangle neutron scattering experiments.16,17 With this dependence, eq 2 predicts a relaxation frequency that increases with concentration as c and, if the fraction f of free counterions is independent of concentration, a constant value of ∆ should be obtained (eq 1). This expected behavior has been actually experimentally verified for NaPSS in the concentration range from 10-4 to 10-1 monomol/L.8 However, when the concentration is sufficiently high, these dependencies are no longer true, and while eq 2 still describes the concentration dependence of ν0, the dielectric increment shows a significant increase, reflecting a dependence of the fraction f of free counterions on the polymer concentration. The polymer investigated in this work clearly evidences this behavior in the semidilute regime (see data analysis, Figure 2). Dielectric parameters (∆ and ν0) provide a simple means to estimate f from exclusively experimental quantities. As a final remark, it must be noted that for a polymer in a poor solvent ξ0 depends on c both explicitly and through the fraction f (eq 4). Since only the expected dependence of ξ0 on c-1/2 is experimentally observed, this implies that the dependence on f has to be somehow compensated. Actually, as we have recently demonstrated, in the case of poor solvent conditions,18 there is a correlation between f and the solvent quality parameter τ, so that the expected power law ξ0 ∼ c-1/2 is satisfied. 4. Results and Discussion 4.1. Data Analysis. Representative dielectric spectra of sodium polyacrylate aqueous solutions of two different molecular weights (5.1 and 140 kD) at a selected polymer concentration (C ) 0.7 wt %) are shown in Figure 1. All the polymers investigated display a well-defined dielectric dispersion with relaxation times falling in the interval 10-8 to 10-10 s, intermediate between the low-frequency dispersion associated with motions involving the whole polymer chain and the high-frequency dispersion due to the orientational polarization of water. In the frequency range investigated, the complex dielectric constant *(ω) has been analyzed on the basis of the following expression:19
(
)
σ0 σ(ω) ) ) ′(ω) - i ′′diel + 0ω 0ω ∞ - ∞H2O σ0 - ∞ (9) + -i β 1 + iωτH2O 0ω 1 + (iωτ)
*(ω) ) ′(ω) - i
where ∆ ) - ∞, ν0 ) ω0/2π, and β (the dielectric strength, relaxation frequency, and relaxation time spread, respectively) are the dielectric parameters of a Cole-Cole relaxation function, ∆H2O ) ∞ - ∞H2O and ν0H2O ) ω0H2O/ 2π are the dielectric parameters (dielectric strength and relaxation frequency, respectively) of a Debye type relaxation function taking into account the contribution from polarization of water, σ0 is the dc electrical conductivity, (16) Nierlich, M.; et al. J. Phys. France 1979, 40, 701. (17) Essafi, W.; Lafuma, F.; Williams, C. E. Eur. Polym. J. B 1999, 9, 261. (18) Bordi, F.; Cametti, C.; Tan, J. S.; Boris, D. C.; Krause, W. E.; Plucktaveesak, N.; Colby, R. H. Macromolecules, in press. (19) Grant, E. H.; Sheppard, R. J.; South, G. P. Dielectric Behaviour of Biological Molecules in Solutions; Clarendon Press: Oxford, 1978.
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Figure 2. The dielectric increment ∆ of sodium polyacrylate aqueous solutions as a function of the polymer concentration, for different molecular weights: (O) 225 kD; (0) 140 kD; (]) 60 kD; (4) 20 kD; (3) 5.1 kD; (b) 2.1 kD. The temperature is T ) 20 ° C.
Figure 1. The permittivity ′(ω) (A) and the dielectric loss ′′(ω) (B) of sodium polyacrylate aqueous solutions for two molecular weights (top curve, MW ) 140 kD; bottom curve, MW ) 5.1 kD) as a function of frequency. The polymer concentration is C ) 0.7 wt %. The temperature is T ) 20 °C. The full lines are the calculated values on the basis of a simultaneous fitting (the real and imaginary part) of the relaxation function (eq 9) to the experimental values.
and, finally, 0 is the dielectric constant of free space and ω is the angular frequency of the applied electric field. We have assumed that the low-frequency process will finish at frequencies lower than 1 MHz, so that we can exclude, in the low-frequency window investigated, any small contribution from this process. The deconvolution of a dielectric spectrum covering an extended frequency range, where different relaxation mechanisms overlap, is a very difficult task. This difficulty is particularly severe when the high electrical conductivity of the samples to be studied gives rise to a frequency-dependent loss process whose tail falls within the frequency window investigated. In the present case, with dc electrical conductivities of the order of 1 Ω-1 m-1, the contribution to the total (dielectric) loss deriving from the dc conductivity dominates that from the dielectric relaxation process. The term σ0/0ω is about 2 × 104, whereas we expect a dielectric loss of the order of ′′ ≈ 10. Moreover, the intrinsic phase angle φ of the conducting solution defined according to the following expression (assuming the dielectric loss contribution ′′(ω) to be negligible)
′ν tan φ ) 2π0 σ0
(10)
at a frequency as high as ν ) 1 MHz is of the order of 0.005 rad, that is, less than 0.3°. These difficulties of a purely experimental nature make accurate evaluation of the dielectric parameters quite difficult and make a rather elaborate analysis necessary.
As we have previously stated, the deconvolution of the whole dielectric spectrum into its components in the case of high-conductivity aqueous solutions is very difficult, in this case, being further complicated by the fact that due to the conductivity the experimental resolution is low and that in our frequency window we can observe only the high-frequency tail of the dc conductivity losses and only the low-frequency tail of the polarization of water. In particular, the localization of the main frequency relaxation by observing separately the permittivity and the dielectric loss spectra may be doubtful, thus influencing the evaluation of the whole set of the dielectric parameters. To overcome this difficulty, we have performed a multiplestep fitting procedure based on the Marquardt algorithm for complex functions.19 Since the shape of the dielectric loss spectrum ′′ strongly depends on the value of the dc electrical conductivity σ0 to be subtracted from the total loss σ(ω)/0ω, we have made a preliminary simultaneous fit of ′(ω) and the total loss σ(ω)/0ω with five free parameters ∆, τ, β, ∞, and σ0 with the only constraint that all parameters should be positive. The value of σ0 thus obtained is then subtracted from the measured conductivity σ(ω), the dielectric loss ′′(ω) is evaluated, and a new set of parameters from the simultaneous fit of both the permittivity ′(ω) and ′(ω) are now obtained. This procedure is iterated until a reasonable minimization is reached. We have successfully applied this procedure to the analysis of the dielectric spectra of poly-L-lysine aqueous solutions at different polymer concentrations.6,7,20 In the present case, the dielectric parameters characterizing the intermediate dielectric dispersion determined by the nonlinear least-squares minimization on the basis of the modified Cole-Cole relaxation function (eq 9) are shown in Figures 2-4. 4.2. Scaling Analysis. Here, we discuss the dependence of the dielectric parameters in light of the scaling approach. To this end, the polymer concentration regimes we have investigated must be assessed. In a previous work,21 we have estimated from viscosity measurements the different characteristic concentrations that define the limits of the different concentration regimes for the sodium polyacrylate aqueous solutions investigated. These values are approximately c* ) 0.0005 monomol/L, ce ) 0.2-0.3 monomol/L, and cD ) 1-2 (20) Bordi, F.; Cametti, C.; Paradossi, G. Phys. Chem. Chem. Phys. 1999, 1, 1555. (21) Bordi, F.; Colby, R. H.; Cametti, C.; De Lorenzo, L.; Gili, T. J. Phys. Chem., in press.
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Figure 3. The relaxation frequency ν0 of sodium polyacrylate aqueous solutions as a function of the polymer concentration, for different molecular weights: (O) 225 kD; (0) 140 kD; (]) 60 kD; (4) 20 kD; (3) 5.1 kD; (b) 2.1 kD. The temperature is T ) 20 °C.
Figure 4. The parameter β of the spread of the relaxation frequency of sodium polyacrylate aqueous solutions as a function of the polymer concentration, for different molecular weights: (O) 225 kD; (0) 140 kD; (]) 60 kD; (4) 20 kD; (3) 5.1 kD; (b) 2.1 kD. The temperature is T ) 20 °C.
monomol/L, corresponding to c* ≈ 10-5 wt/wt, ce ≈ 2-3 × 10-2 wt/wt, and cD ≈ 0.1-0.2 wt/wt, respectively. On the basis of these values, the polyion solution is in the semidilute-unentangled regime for c* < c < ce, in the semidilute-entangled regime for ce < c < cD, and in the concentrated regime for c > cD, the dilute regime being well below the lowest concentration investigated. As can be seen (Figure 2), the dielectric increment ∆, for the higher molecular weights studied (20, 60, 140, and 225 kD), increases up to c ≈ ce; it remains approximately constant in the semidilute-entangled regime (ce < c < cD) and then markedly decreases for c > cD. This finding is qualitatively in agreement with the model of Ito et al.8 that predicts a dielectric increment independent of concentration in the semidilute regime and an increasing behavior at lower concentrations. The lowest molecular weights (2.1 and 5.1 kD) behave similarly, even if the semidilute-entangled regime extends over a narrow interval, shifted toward higher concentrations. As far as the relaxation frequency is concerned (Figure 3), in the whole semidilute regime (c* < c < cD), an approximately linear dependence on c is observed, marked deviations occurring only at concentrations larger than cD. In the present case, the fraction f of free counterions, calculated on the basis of the dielectric parameters ∆ and ν0 (eq 3), does not have a constant value in the whole concentration range where the system is in the semidilute regime and the independence of c for the quantity ν0∆/c is no longer verified. On the contrary, if eq 7 holds (good solvent conditions), a very good agreement with the experimental behavior is
Bordi et al.
Figure 5. The scaling behavior of the dielectric parameter ∆ and ν0 as a function of the concentration c for sodium polyacrylate aqueous solutions of different molecular weights at the temperature of 20 °C: (O) 225 kD; (0) 140 kD; (]) 60 kD; (4) 20 kD; (3) 5.1 kD. The dependence 1/c is that expected for good solvent conditions.
obtained. Figure 5 shows the dependence of ∆4/3/ν0 on the concentration c, which displays the slope of -1 as expected. These findings give strong support to a description of dielectric relaxation based on the Ito et al. model,8 with the characteristic distance on which the counterions fluctuate being the correlation length ξ0 (or proportional to this length) obtained from the scaling theory of polyelectrolyte solutions.10,11 On the basis of these findings, the intermediate dielectric dispersion in highly charged polyelectrolyte solutions and the consequent extraction of the dielectric strength and relaxation frequency provide a simple and unique means to evaluate the interaction of the polyelectrolyte with the solvent phase through the solvent quality parameter. Further support for this picture emerges when different polymers are considered, for which water represents a “poor” solvent. In our previous work,18 we have analyzed the intermediate dielectric dispersions of different polymers in aqueous solution in the case of poor solvent conditions. We investigated four different polymers with decreasing chain charge density, that is, the sodium salt of sulfonated polystyrene [NaPSS], the sodium salt of poly(2-acrylamido-2-methylpropane sulfonate) [NaPAMS], a random copolymer made from 80% 2-acrylamido-2methylpropane sulfonate and 20% acrylamide [NaPAMS/ PA20], and a random copolymer made from 45% 2-acrylamido-2-methylpropane sulfonate and 55% tert-butylacrylamide [NaPAMS345/PtBA55], and we have analyzed the scaling power laws given in eqs 5-7. A plot of ν0-1∆4/3 as a function of 1/c (eq 7) for these four polymers at the temperature of T ) 20 °C is shown in Figure 6. In this case, a good solvent condition is clearly inappropriate to describe the dependence on the concentration c. As can be seen, deviations from a straight line with a slope of -1 appear well below the concentration cD, suggesting that eq 7 fails and, owing to the hydrophobicity of the polymers, a poor solvent condition should be more appropriate. This trend is further confirmed by the data shown in Figure 7, where the quantity ∆4/3/ν0 for the same polymer solutions is shown at a higher temperature (T ) 40 °C). In this case, the polymer-solvent interactions are favored and the good solvent conditions are better approached. As can be seen, deviations from a slope of -1 are less evident for all the four polymers investigated and eq 7 works over the whole semidilute concentration regime. As we have stressed above, eq 5 in the case of a poor solvent is unable to furnish a well-defined concentration dependence owing to the dependence of the solvent quality
Dielectric Relaxation in Polyelectrolyte Solutions
Figure 6. The scaling behavior of the dielectric parameter ∆ and ν0 as a function of the concentration c for different polymers at the temperature of 20 °C: (O) NaPAMS45/PtBA55; (0) NaPSS; (]) NaPAMS; (4) NaPAMS80/PA20. The data are shifted along the y-axis to have them coincide with the calculated curve 1/c that is expected for good solvent conditions. Deviations in the high concentration range indicate that good solvent conditions are inappropriate.
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Figure 8. Correlation between the solvent quality parameter τ and the fraction of monomers bearing an effective charge (or, conversely, the fraction of free counterions) f for different polymers at the temperature of 20°: (O) NaPAMS45/PtBA55; (0) NaPSS; (]) NaPAMS; (4) NaPAMS80/PA20.
The correlation shown in Figure 8 gives support to the ansatz that the correct solvent quality parameter in the scaling theory of polyelectrolyte solutions is not the energetic interaction between uncharged polymer and the solvent, but it is completely dwarfed by the interaction between a bare charge and water. 5. Conclusions
Figure 7. The scaling behavior of the dielectric parameter ∆ and ν0 as a function of the concentration c for different polymers at the temperature of 40 °C: (O) NaPAMS45/PtBA55; (0) NaPSS; (]) NaPAMS; (4) NaPAMS80/PA20. The data are shifted along the y-axis to have them coincide with the calculated curve 1/c that is expected for good solvent conditions.
parameter τ on c. However, the influence of the solvent and then the electrostatic interaction with the aqueous phase can be implicitly derived from eq 8, giving the dependence of the quality solvent parameter τ on the fraction f of free counterions. We plot τ as a function of f in Figure 8 for the four polymers at the temperature of T ) 20 °C (poor solvent conditions). As can be seen, there exists a strong correlation and furthermore a logical progression in both the fraction of monomers bearing a charge f (and consequently the fraction of free counterions) and the solvent quality parameter τ. As the fraction f of monomers having salt groups increases from 0.45 (NaPAMS45/PtBA55) to 0.8 (NaPAMS80/PA20) to 1 (NaPAMS and NaPSS), both f and τ steadily increase. Finally, values of the solvent quality parameter in the range of 0.05-0.25 compare reasonably well with those derived assuming a θ-temperature falling in the interval 45-55 °C, according to the different hydrophobicity of the polymers investigated.
Although dielectric and electrical conductivity properties of polyelectrolyte solutions have been extensively studied in highly diluted systems, only a few studies cover a wide concentration range, from semidilute to concentrated regimes. In this paper, we have shown that the relaxation process occurring at intermediate frequencies (1 MHz to 1 GHz) due to fluctuation of free counterions, according to the Ito et al. model,8 can be described within the scaling theory of polyelectrolyte solutions10,11 that furnishes a simple and effective description of the observed dependence of the dielectric parameters on the polymer concentration in the whole concentration range we have investigated. Moreover, we have also shown that this simple scheme allows the quality solvent parameter τ to be evaluated experimentally in the poor solvent case. This parameter in the scaling theory of polyelectrolytes takes into account the interaction between the solvent and the uncharged chain, that is, the effective interaction with solvent that the chain would exhibit if it were uncharged. This particular aspect of τ makes it difficult to measure this parameter, because the chain is, in effect, charged. The fact that dielectric parameters, for polymers that have a different affinity for water, show a different scaling behavior, in agreement with the known hydrophobicity of the chain, gives strong experimental support to the scaling approach. The observed correlation between the fraction f of free counterions and the solvent quality parameter τ at increasing polymer concentration opens an interesting perspective for studying the role played by electrostatic interactions between solvent and polymer charged groups in determining the effective overall energetics of polyelectrolyte-solvent interactions. LA020175H
