Journal of Molecular Liquids 251 (2018) 94–99
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Exploring the interactions in binary mixtures of polyelectrolytes: Influence of mixture composition, concentration, and temperature on counterion condensation Ranjit De a, Hohjai Lee a,⁎, Bijan Das b,⁎ a b
Department of Chemistry, Gwangju Institute of Science and Technology (GIST), Gwangju 61005, South Korea Department of Chemistry, Presidency University, Kolkata 700 073, India
a r t i c l e
i n f o
Article history: Received 30 August 2017 Received in revised form 8 December 2017 Accepted 16 December 2017 Available online 18 December 2017 Keywords: Electrical conductivity Polyelectrolyte Counterion condensation Fraction of free counterions Polyion transference number
a b s t r a c t The counterion condensation in the binary mixtures of like-charged polyelectrolytes with two different molecular weights, where both are negatively charged, has been studied by conductance measurement in aqueous medium. Sodium polystyrenesulfonate with two different chain lengths have been selected as model polyelectrolyte and the investigation has been carried out in the absence of any added salt. The specific conductancepolyelectrolyte concentration data were analyzed in the light of the scaling argument for the conformation of polyion chains in solutions to quantify the fractions of free counterions. Effects of polyelectrolyte concentration, molecular weights of the components of the mixture, and solution temperature on the condensation of counterions have been investigated to explore polyion–counterion interaction, and inter- and intra-chain interactions in these polyelectrolyte mixtures. This investigation revealed that the extent of counterion condensation is significantly influenced by the mixture composition, the polyelectrolyte concentration and the temperature. Evaluation of the polyion transference numbers provided important information as to the direction of the motion of the counterions under the action of the applied electric field. © 2017 Published by Elsevier B.V.
1. Introduction Polyelectrolytes are a special class of polymers with dissociable groups which ionize in polar media (e.g., water) to produce polyions and counterion bearing opposite charges. Owing to the strong electric field of the polyion, a certain proportion of the counterions becomes highly correlated with the polyion, i.e., a fraction of the counterions remains condensed onto the polyion chain. This phenomenon is known as the “counterion condensation”. Now, the extent of counterion condensation determines the charge on the polyion chains. It is obvious that the more the polyion charge, the stronger would be the intrachain electrostatic repulsion and hence there would be more stretching of the polyion chain. Thus both polyion charge and polyion size are governed by counterion condensation. Since polyelectrolytes present a ubiquitous class of materials and find widespread applications, a quantitative understanding of counterion condensation is of fundamental importance. While the behavior of counterion condensation in solutions containing single polyelectrolyte samples is now well-understood [1–6], the situation is far from satisfactory for multi-component polyelectrolyte solutions. Pairs of oppositely charged polyelectrolytes typically form complexes in aqueous solutions [2–6]. However, when both polyions have the ⁎ Corresponding authors. E-mail addresses:
[email protected] (H. Lee),
[email protected] (B. Das).
https://doi.org/10.1016/j.molliq.2017.12.088 0167-7322/© 2017 Published by Elsevier B.V.
same charge, they exhibit electrostatic repulsion which could lead to phase separation caused by incompatibility. Although the former case has been extensively investigated in the field of interpolymer complexes [3–5], the later has been paid relatively little attention both from experimental and theoretical points of view. Various applications exhibit the tendency towards phase separation in solutions of similarly-charged polyelectrolytes, such as aqueous two-phase partitioning of biological molecules [4,5]. A complete characterization of such polyelectrolyte mixtures is, therefore, of utmost importance to understand the underlying mechanisms and/or molecular interactions governing the phase behavior for practical applications. Studies exploring the interactions between the similarly-charged polyelectrolytes in a mixture are still in their infancy. Previous dynamic light scattering studies on aqueous binary mixtures of sodium polystyrenesulfonate with different molecular weights revealed coupled diffusion of polyions and counterions, which has been shown to be independent of the total polyelectrolyte concentration, mixture composition, and molecular weights of the two components [6]. Invariance of coupled diffusion of polyions and counterions implies an invariance of polyion-counterion interactions. Results from our previous conductometric studies, however, indicated a definite dependence of polyion-counterion interactions on both polyelectrolyte molecular weight and polyelectrolyte concentration in binary solutions [7–9]. These contradictory observations indicate that the nature of polyion-
R. De et al. / Journal of Molecular Liquids 251 (2018) 94–99
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counterion interactions in these systems in particular, and in polyelectrolyte mixtures, in general, is thus still unclear. In order to unravel the nature of these interactions in solutions, we have undertaken a study to probe various interactions occurring in these solutions using electrical conductivity measurement. 2. Theory Colby et al. [8] put forwarded a theory for the electrical conductivity of semidilute solutions of polyelectrolytes without added salt using the scaling description for the configuration of a polyelectrolyte chain proposed by Dobrynin et al. [9]. In semidilute solutions, the polyion chain is modeled as a random walk of Nξ correlation blobs of size ξ0, each of them containing g monomers. Each blob bears an electric charge qξ = zcefg (zc being the counterion valence, e is the electronic charge and f represents fraction of uncondensed counterions) and the complete chain, of contour length L = Nξ ξ0, bears a charge Qp = Nξ qξ = zcefg Nξ. Due to the strong electrostatic interactions within each correlation blob, the chain is a fully extended conformation of ge electrostatic blobs of size ξe. This means that for length scales less than ξ0, the electrostatic interactions dominate (and the chain is a fully extended conformation of electrostatic blobs of size ξe), and for length scales greater than ξ0, these interactions are screened and the chain is a random walk of correlation blobs of size ξ0. Following this model, in absence of an added salt, the equivalent conductivity of a polyion in a semidilute solution is given by λp ¼
2 Fzc efcξ0 ξ ln 0 3πη0 ξe
ð1Þ
where F is the Faraday constant, η0 is the absolute viscosity of the solvent and c is the concentration of the polyelectrolyte solution expressed in the number of monovalently-charged groups per cm3. In accordance with this model, Colby et al. [8] arrived at the following expression of the equivalent conductivity of a polyelectrolyte in a solution as a function of its concentration: " Λ ¼ f λ0c þ
# 2 Fzc efcξ0 ξ ln 0 3πη0 ξe
ð2Þ
where λ0c is the limiting equivalent conductivity of the counterion. This model treats the parameter f as an adjustable quantity which could be estimated from experiments in conjunction with this new approach. 3. Experiments 3.1. Materials Two sodium polystyrenesulfonate (NaPSS) samples with molecular weights of 70,000 g mol−1 (designated as P1) and 1,000,000 g mol−1 (designated as P2) (each with a degree of substitution of 1) were procured from Aldrich Chemical Company, Inc. and were used without further purification. The chemical structure of NaPSS is shown in Fig. 1. The polymer samples were dried well for a prolonged period of time immediately prior to use for solution preparations. The puriss grade NaCl was purchased from Fluka and was well dried prior to use. To weigh the materials, a digital balance (Shimadzu) with weighing sensitivity of ±0.2 mg was used. Ultrapure water with specific conductance below 1 μS cm−1 at 298.15 K, used for the preparation of all solutions, was obtained from a Milli-Q (resistivity 18.2 MΩ cm at 298.15 K) water purifying system. The uncertainty for the preparation of the sample solutions by dilution for investigations was assessed to be ±0.2%. NaPSS mixtures with different values of x(P2) = cp(P2) / (cp(P1) + cp(P2)), where cp(P1) and cp(P2) are the concentrations of component 1 (M = 70,000 g mol−1) and component 2 (M = 1,000,000 g mol− 1),
Fig. 1. Chemical structure of sodium polystyrenesulfonate. Molecular weight of each monomer unit is 206 g mol−1.
respectively with cp being the number of monovalently charged groups per liter i.e. equivalents per liter, were prepared. Total polymer concentration (cp) was varied from 0.005 through 0.121 equiv L−1, and mixtures with x(P2) values of 0 (pure P1), 0.17, 0.42, 0.58, 0.83, and 1 (pure P2) were used for the present study.
3.2. Viscosity measurements Viscosity measurements were carried out using a suspended level Ubbelohde viscometer kept vertically in a water thermostat which was placed on a vibration-free bench. The absolute viscosity (η) was obtained using the relation η ¼ ðCτ− Kτ Þρ where C and K are viscometer constants. The values of these constants were estimated to be 1.646 × 10−5 cm2 s− 2 and − 0.02331647 cm2, respectively, using the known density (ρ) and viscosity data of methanol and acetonitrile from the literatures [10,11]. The flow time (τ) was determined by using a digital stopwatch with capability of measuring time within ± 0.01 s. The combined expanded uncertainty in viscosity measurements was estimated to be ±0.003 mPa s. The experimental temperature was maintained within ±0.02 K.
3.3. Fourier transform-infrared (FT-IR) spectral measurements The purity and complete sulfonation of these polyelectrolytes were confirmed via Fourier Transform Infrared (FT-IR) spectroscopy. FT-IR spectra of both the NaPSS samples were recorded using an attenuated total reflectance (ATR) technique using a Jasco FTIR-4200 spectrometer with frequency resolution of 2 cm−1; the details of measurements are reported elsewhere [12]. Briefly, five characteristics peaks indicated by asterisk (*) symbol in Fig. 2, are observed at about 669, 1010, 1040, 1128 and 1182 cm− 1; these agreed well with the earlier literature values [13,14]. The peaks at 1010 and 1128 are assigned respectively to in-plane bending vibration of benzene ring and the in-plane skeleton vibration of benzene ring, whereas the peaks at 1040 and 1182 cm−1 can be assigned to the symmetric and antisymmetric vibration absorption peaks of SO− 3 group, respectively. The peak at 669 represents aromatic C\\H out of plane bending (deformation) vibration [14]. Absence of peaks at around 699 and 759 cm−1 confirms complete sulfonation (degree of substitution of 1) of NaPSS as reported by the manufacturer. These two bands are to represent out-of-plane skeleton bending vibrations of benzene ring and out-of-plane bending vibration of the five \\CH\\ groups in the benzene ring characteristic of monosubstituted benzene ring, respectively which are the characteristic bands of polystyrene samples [13].
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coefficients (η) of water measured in this study. These values were utilized during the conductance data analysis. 4.2. Experimental specific conductances The representative figure (Fig. 3) shows the specific conductivity vs. concentration profiles of aqueous mixtures of NaPSS with molecular weights of 70,000 g mol−1 and 1,000,000 g mol−1 at 298.15 K. This figure indicates clearly the effect of polyelectrolyte molecular weight on the specific conductance vs. concentration profiles. As the amount of the polyelectrolyte sample with higher molecular weight increased, the specific conductance of the solution was found to decrease. 4.3. Fraction of free counterions
Fig. 2. FT-IR spectra of sodium polystyrenesulfonate samples with average molecular weights (a) 70,000 g mol−1 and (b) 1,000,000 g mol−1. Absence of spurious peaks confirms the complete sulfonation and molecular grade purity.
Eq. (2) has been employed for the analysis of the present conductivity data. The electrostatic blob size (ξe) and the correlation blob size (ξ0) appearing in Eq. (2) are known to depend upon the quality of the solvent, and for poor solvents, these are given by [1,8,9,17] −1=3 2 ξe ¼ b ξf
3.4. Conductance measurements The specific conductivity of NaPSS solutions were measured using an Orion Star A212 benchtop four-electrode AC conductivity meter (Orion Research Inc.) which has the conductivity range from 0.00 to 3000 mS cm−1 with up to four significant figure resolution. The solutions were taken in a wide mouth hard glass test tube fitted with a dip-type cell (cell constant = ±1 cm−1) having a relative accuracy of ± 0.1. The cell was calibrated with aqueous KCl solutions using the method as described by Lind et al. [15] prior to the initiation of conductance measurements. Once the polyelectrolyte solutions were prepared, they were left for about 30 min to attain homogeneity at the experimental temperature prior to conductance measurements. The conductance values were found to be time independent during the period of measurements. Each measurement was replicated for 5–6 times and the average values were considered for analysis. The reported specific conductivities of solutions were always solvent corrected. All the experiments were carried out in a dehumidified room to avoid moisture pickup. Once the conductance measurement started, care was taken not to take the cell out of solution to avoid formation of air bubbles. The experimental relative uncertainties were always found to be within ±0.5%.
−1=2
ξ0 ¼ ðcbÞ
−1=3 2 ξf
ð6Þ
ð7Þ
with the charge density parameter ξ defined as ξ¼
e2 εkTb
ð8Þ
where b is the spacing between the charged groups taken along the axis of the polyion chain, c is the number density of monomers as mentioned earlier, k the Boltzmann constant, and T the temperature in absolute scale. For good solvent cases, on the other hand, these are given by [1,8,9, 17] −3=7 2 ξe ¼ b ξf
ð9Þ
4. Results and discussion 4.1. Physical properties of water and limiting equivalent conductance of Na+ ion at different temperatures The physical properties of water at different experimental temperatures are presented in Table 1. Relative permittivity (ε) values of water and the equivalent conductances of the counterion (Na+) at infinite dilution in water (λ0c ) are obtained from literature at the experimental temperatures [16]. Also included in this table are the viscosity Table 1 Temperature dependent physical properties of water: absolute viscosity (η), relative permittivity of media (ε) and the limiting ionic equivalent conductance values of Na+ ion at infinite dilution (λ0c ) used for data analysis. Temperature (K)
η (mPa s)
ε
104 λ0c (S m2 equiv−1)
298.15 303.15 308.15 313.15 318.15
0.8903 0.7975 0.7194 0.6531 0.5963
78.30 76.55 74.82 73.15 71.51
50.10 55.71 61.54 67.55 73.73
Fig. 3. Specific conductances (σ) of aqueous solutions of mixtures of NaPSS with molecular weights of 70,000 g mol−1 and 1,000,000 g mol−1 as a function of the polyelectrolyte concentration at 298.15 K for different mass fractions of the higher molecular weight species, x(P2). Dashed lines connecting the data points are used to guide the eyes.
R. De et al. / Journal of Molecular Liquids 251 (2018) 94–99
−1=2
ξ0 ¼ ðcbÞ
−1=7 2 ξf
ð10Þ
For the present system, poor solvent correlations were used as pointed out by Colby et al. [8]. In the individual NaPSS solutions as well as in NaPSS mixtures, the fractions of uncondensed counterions (f) were found to increase with polyelectrolyte concentration (Table 2 and Fig. 4). This indicates that the effective charge on the polyion chains is NaPSS concentration dependent. As polyelectrolyte concentration increases, the relative permittivity of the medium is known to increase, owing to the polarizability of polyelectrolytes [18,19]. While the increase in the effective relative permittivity of the medium is not fully understood yet [20, 21], the experimental result is well established [18,22]. Specifically, for NaPSS, the frequency-dependent relative permittivity is known to increase with NaPSS concentration [23]. The increase in relative permittivity will cause less electrostatic interaction between the polyion and the counterions thus resulting in more counterion dissociation. So there will be more uncondensed counterions as the relative permittivity of the medium increases. The fraction of free counterions (f) will not approach zero as the polyelectrolyte concentration tends to zero since at that limit the relative permittivity of the medium approaches that of pure water still high enough to cause dissociation of counterions. The non-zero value of the fraction of free counterions at zero polyelectrolyte
Table 2 The fractions of free counterions (f) in different aqueous binary mixtures of two sodium polystyrenesulfonate samples with molecular weights of 70,000 and 1,000,000 having different total polyelectrolyte concentrations (cp) at various temperatures (T). x(P2)a
cp (equiv L−1) 0.097
0.073
0.048
0.036
0.024
0.012
0.005
T = 298.15 K 0 0.53 0.17 0.51 0.42 0.50 0.58 0.47 0.83 0.44 1 0.41
0.121
0.51 0.50 0.48 0.45 0.42 0.39
0.49 0.48 0.46 0.43 0.40 0.37
0.47 0.46 0.44 0.42 0.39 0.36
0.46 0.44 0.43 0.41 0.38 0.35
0.45 0.43 0.41 0.39 0.37 0.34
0.44 0.42 0.40 0.37 0.35 0.32
0.43 0.41 0.39 0.36 0.33 0.30
T = 303.15 K 0 0.47 0.17 0.46 0.42 0.44 0.58 0.41 0.83 0.39 1 0.37
0.46 0.44 0.42 0.40 0.37 0.35
0.45 0.42 0.41 0.38 0.36 0.33
0.42 0.40 0.38 0.37 0.34 0.31
0.41 0.39 0.37 0.36 0.33 0.31
0.41 0.38 0.36 0.35 0.32 0.30
0.39 0.37 0.35 0.33 0.30 0.28
0.38 0.36 0.34 0.31 0.28 0.26
T = 308.15 K 0 0.43 0.17 0.41 0.42 0.39 0.58 0.37 0.83 0.35 1 0.33
0.41 0.39 0.37 0.35 0.33 0.31
0.40 0.38 0.36 0.34 0.32 0.30
0.38 0.36 0.34 0.33 0.30 0.28
0.38 0.35 0.33 0.32 0.29 0.27
0.37 0.34 0.32 0.31 0.29 0.27
0.36 0.33 0.31 0.30 0.27 0.25
0.35 0.32 0.30 0.28 0.26 0.23
T = 313.15 K 0 0.39 0.17 0.37 0.42 0.35 0.58 0.33 0.83 0.32 1 0.30
0.37 0.35 0.33 0.32 0.30 0.28
0.36 0.34 0.32 0.31 0.29 0.27
0.35 0.33 0.31 0.29 0.27 0.25
0.34 0.32 0.30 0.28 0.26 0.24
0.34 0.31 0.29 0.27 0.25 0.24
0.32 0.30 0.28 0.27 0.24 0.22
0.32 0.29 0.27 0.27 0.23 0.21
T = 318.15 K 0 0.36 0.17 0.33 0.42 0.32 0.58 0.30 0.83 0.29 1 0.27
0.34 0.32 0.30 0.29 0.27 0.25
0.33 0.30 0.29 0.28 0.26 0.24
0.32 0.29 0.28 0.26 0.25 0.23
0.31 0.29 0.27 0.25 0.24 0.22
0.31 0.28 0.26 0.25 0.23 0.21
0.30 0.27 0.25 0.23 0.22 0.20
0.29 0.27 0.24 0.23 0.21 0.19
a Mass fraction of the sodium polystyrenesulfonate sample with a molecular weight of 1,000,000.
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concentration physically signifies predominance of the electrostatic attraction over the entropy of dilution and condensation of a given fraction of counterions onto the macroion even at the highest dilution. Similar behavior was also observed by Colby et al. [8]. The NaPSS sample with a molecular weight of 70,000 g mol−1 always exhibited higher fractions of free counterions in comparison with the higher molecular weight sample in salt-free solutions. Similar behavior was also observed in one of our earlier studies where each of these polyelectrolytes was separately investigated in presence of an added electrolyte [12]. Bordi et al. also reported variation of the fraction of free counterions in aqueous salt-free solutions of polyacrylic acid and its sodium salt [21]. For the present NaPSS mixtures, the polyelectrolyte molecular weight has been found to have profound influence on the fraction of free counterions (Fig. 4). For example, at 298.15 K, the fraction of free counterions decreases by about 30–43% from pure NaPSS sample with a molecular weight of 70,000 g mol− 1 to a sample with a molecular weight of 1,000,000 g mol−1. This indicates that polyion-counterion interaction changes appreciably with polyelectrolyte molecular weight. Increase in the proportion of the higher molecular weight NaPSS fraction in the mixtures induces more coiling of the polyion chains thus causing less counterion dissociation. The above observations are in sharp contrast with what concluded earlier from the dependence of the fast diffusion coefficient on total polyelectrolyte concentration or on the polyelectrolyte molecular weight on aqueous NaPSS mixtures using dynamic light scattering measurements [6], where an independence of the polyion-counterion interactions was inferred. Further studies on different polyelectrolyte mixtures are, therefore, required in order to shed more light on polyion-counterion interactions. An increase in temperature favors counterion condensation for all the NaPSS mixtures over the entire total polyelectrolyte concentration range (Fig. 4). Raising the temperature has the effect of gradual desolvation of sodium counterions and/or polyions and reduction of relative permittivity of the medium. That is at elevated temperatures, the counterions would have lower effective diameter and hence higher surface charge density compared to the low temperatures. This effect along with the reduced relative permittivity of the medium causes more counterion condensation at higher temperatures. 4.4. Polyion transference number The transference number of the polyion (Tp) has been obtained from the equivalent conductivity of the polyion obtained earlier (λp) and the experimentally determined equivalent conductivity of the polyelectrolyte as a whole (Λ) using the following relationship Tp ¼
λp Λ
ð17Þ
Polyion transference numbers as a function of total polyelectrolyte concentration are displayed in Fig. 5. Tp values, in general, decrease monotonically with increasing total polyelectrolyte concentration at any given temperature and at a selected composition. It is noteworthy that the polyion transference number values are always found to be greater than unity over the entire concentration range investigated in this study. Accordingly, the transference numbers of the counterions are negative. The negative sign of the counterion transference numbers, which is a characteristic feature of solutions of many polyelectrolytes, indicates that a substantial proportion of the counterions is associated with the polyions and moves with them towards the anode. Reports on the Tp values larger than unity for several aqueous polyions are also available in the literature [24–28]. It is observed that while an increase in temperature, or an increase in the amount of the higher molecular weight polyelectrolyte fraction in the NaPSS mixtures facilitates counterion binding, an increase in the total polyelectrolyte
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Fig. 4. Influence of temperature and polyelectrolyte concentration on the fraction of uncondensed counterions (f) in aqueous solutions of mixtures of NaPSS with molecular weights of 70,000 g mol−1 and 1,000,000 g mol−1 with (a) 0.00, (b) 0.17, (c) 0.42, (d) 0.58, (e) 0.83 and (f) 1.00 mass fraction of the higher molecular weight species, x(P2). Dashed lines connecting the data points are used to guide the eyes.
concentration facilitates counterion dissociation (cf. Fig. 5). The values of Tp confirm our earlier conclusions derived from the measured fraction of the uncondensed counterions. 5. Conclusions A comprehensive and systematic investigation on the counterion condensation in aqueous solutions of the binary mixtures of sodium polystyrenesulfonate with two different molecular weights has been presented using conductometry as the probe. The data have been
analyzed on the basis of an equation for polyelectrolyte conductivity developed by Colby et al. [8] using the scaling description [9] for the conformation of polyions in solutions to quantify the fractions free counterions. Influences of a number of parameters, such as, polyelectrolyte concentration, molecular weight, composition of the system and the temperature on the polymer charge have been investigated extensively in terms of (i) fraction of uncondensed counterions, and (ii) polyion transference number to explore polyion–counterion interactions, inter- and intra-chain interactions in these polyelectrolyte mixtures. The extent of counterion condensation was found to be
Fig. 5. Influence of temperature and polyelectrolyte concentration on the polyion transference number (Tp) in aqueous solutions of mixtures of NaPSS with molecular weights of 70,000 g mol−1 and 1,000,000 g mol−1 with (a) 0.00, (b) 0.17, (c) 0.42, (d) 0.58, (e) 0.83 and (f) 1.00 mass fraction of the higher molecular weight species, x(P2). Dashed lines connecting the data points are used to guide the eyes.
R. De et al. / Journal of Molecular Liquids 251 (2018) 94–99
significantly influenced by polymer chain length. Increase in the proportion of the higher molecular weight component polyelectrolyte favored the counterion condensation causing the effective charge on the polyion chains to decrease. This leads to a reduction in the effective size of polyion chains. Counterion-condensation was found to be favored as the NaPSS mixtures become richer in the higher molecular weight fraction, or as the temperature is elevated, or as the total polyelectrolyte concentration is reduced. The evaluated polyion transference numbers indicated that a substantial proportion of the counterions is associated with the polyions and moves with them towards the anode, and amply corroborated the inferences concerning the counterioncondensation behavior in these polyelectrolyte mixtures. This study establishes a definite dependence of the polyion-counterion interactions on total polyelectrolyte concentration as well as on the composition of the polyelectrolyte mixtures, which is, however, not in agreement with what concluded earlier from the dependence of the fast diffusion coefficient on total polyelectrolyte concentration or on the composition of the mixtures of polyelectrolyte with different molecular weights in aqueous NaPSS mixtures using dynamic light scattering measurements [6]. Further studies on polyelectrolyte mixtures are, therefore, required in order to shed more light on polyion-counterion interactions. Acknowledgements R. De and H. Lee gratefully acknowledge the financial grant from the Basic Science Research Program (Grant Nos. NRF2016R1D1A1A02937339 and 2017R1D1A1B03031947) of NRF and a grant from the GIST Research Institute (GRI, 2016). B. Das acknowledges the generous financial support by the Presidency University through the Faculty Research & Professional Development Fund (FRPDF, Grant No. BD/FRPDF/2017-18). Authors also acknowledge Hyeong Yeol Oh for his help in cross-checking the reproducibility of some data. References [1] F. Bordi, C. Cametti, R.H. Colby, Dielectric spectroscopy and conductivity of polyelectrolyte solutions, J. Phys. Condens. Matter 16 (2004) R1423–R1463. [2] A.S. Michaels, Polyelectrolyte complexes, Ind. Eng. Chem. Res. 57 (1965) 32–40. [3] K. Petrak, Polyelectrolyte Complexes in Polyelectrolytes, Marcel Dekker, M. Hara, 1993 265. [4] B.Y. Zaslavsky, Aqueous Two-phase Partitioning, Marcel Dekker, New York, 1995 98. [5] P.A. Albertsson, Partition of Cells, Particles and Macromolecules, 3rd ed. John Wiley, New York, 1986. [6] M. Sedlák, Dynamic light scattering from binary mixtures of polyelectrolytes. I. Influence of mixing on the fast and slow polyelectrolyte mode behavior, J. Chem. Phys. 107 (1997) 10799–10804.
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