Langmuir 1999, 15, 4069-4075
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Concentration Regimes in Polyelectrolyte Solutions† Christine Wandrey Laboratory of Polymers and Biomaterials, Department of Chemistry, Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerland Received July 15, 1998. In Final Form: November 5, 1998 A general description of the electrolytic conductivity behavior of highly charged strong polyelectrolytes in dilute and semidilute aqueous solutions is presented. For the first time this model considers the influences of the molar mass, charge density (for charge distances less than the Bjerrum length), polyelectrolyte concentration, and the ionic strength adjusted by both the polyelectrolyte concentration (cp in monomol L-1) and added simple salt concentration (cs in mol L-1). Above the overlap concentration (c*) the molar mass influence is weak. Below this overlap concentration the equivalent conductivity (Λ) increases strongly with decreasing cp, as long as cp > cs. Correlations could be established for the maximum of Λ, Λmax ∼ cs-1/5 and Λmax ∼ M-2/5. The conductivity behavior can be qualitatively explained in terms of Manning’s theory with an additional change of the interaction parameter fc. On the basis of this fact an empirical dependence of fc on the ratio of the Debye length to the contour length (lD/L) has been found. Three concentration regimes differing in the polyion-counterion interaction could be identified. These are characterized by lD/L located below (4π)-1/2, between (4π)-1/2 and unity, and above unity, respectively. The model description includes polyelectrolyte solutions without added salt as well as solutions with any ratio of polyelectrolyte to salt concentration. This is the first model which is based on sufficient experimental data in the highly diluted concentration regime. Therefore, it should stimulate further theoretical studies.
Introduction Polyelectrolyte effects include both deviations from the behavior of neutral polymers caused by the existence of charges along the polymer chain and deviations from the behavior of electrolytes caused by the fixation of one sort of the charges on the polymer chain. Therefore, in addition to the methods of macromolecular characterization, electrochemical techniques have also been applied to investigate polyelectrolytes. The specific conductance, κ, and the equivalent conductivity, Λ, are experimentally determinable parameters which are suitable to describe the electrolytic transport properties of both electrolyte and polyelectrolyte solutions. The conductivity behavior of low molar mass electrolytes can be explained theoretically very well. However, a completely satisfactory theory to describe the electrolytic conductivity of flexible polyelectrolytes in aqueous solution has not been developed. This situation has been summarized in recent publications.1-3 The difficulties of theoretical approaches result from the asymmetry of polyelectrolytes. The highly charged and often flexible polyion is surrounded by numerous small counterions and in some cases additionally by co-ions, which mostly carry only one charge or a few charges. For all theoretical approaches and calculations, therefore, simplifying assumptions have been introduced. In addition, polyelectrolyte solutions without added electrolyte, and with an excess of low molar mass electrolyte, have been treated separately. Generally, the equivalent conductivity of polyelectrolyte solutions is given by4-7 † Presented at Polyelectrolytes ‘98, Inuyama, Japan, May 31June 3, 1998.
(1) Mandel, M. In Encyclopedia of Polymer Science and Engineering, 2nd ed.; Mark, H. F., Bikales, N. M., Overberger, C. G., Menges, G., Eds.; New York, 1988; Vol. 11, p 739. (2) Leeuven, H. P. v.; Cleven, R. F. M.; Valenta, P. Pure Appl. Chem. 1991, 63, 1251. (3) Wandrey, C. Polyelektrolyte-Makromolekulare Parameter und Elektrolytverhalten; Cuvillier Verlag: Go¨ttingen, 1997. (4) Manning, G. S. Biopolymers 1970, 9, 1543.
Λ ) fc(λp + λc0)
(1)
where λc0 is the equivalent conductivity of the counterion in an infinitely diluted solution in the absence of polyions, λp is the equivalent conductivity of the polyion, and fc is the interaction parameter which includes the electrostatic interactions between the polyion and the counterions and the degree of ionization. The various definitions of fc result from the differences in the theoretical derivations. On the basis of his polyelectrolyte theory and on the derivation of Huizenga5 and Kurucsev,7 Manning4 has calculated fc in salt-free solutions for charge distances b less than the Bjerrum length lB, and consequently lB/b ) ξ > 1 as 8
fc ) 0.866ξ-1
(2)
In contrast to Huizenga,5 where the counterions are regarded either as “bound” or as “free” implying that they are not influenced by the polyion, Manning assumes that the nonbounded counterions are influenced by the DebyeHu¨ckel potential of the polyion. The fraction of the condensed counterions is 1 - ξ-1 * 1 - f c. Hence, fc cannot be understood as the fraction of free counterions. Considering electrophoretic and relaxation contributions to the equivalent conductivity, λp has been theoretically calculated for the cylinder model of a polyelectrolyte.2,8 It follows for monovalent counterions
0.886H|ln λp )
r | lD
1 + (1 - 0.866)(λc0)-1H|ln
r | lD
(3)
In this model λp is determined by a temperature-dependent (5) Huizenga, J. R.; Grieger, P. F.; Wall, F. T. J. Am. Chem. Soc. 1950, 72, 2636. (6) Eisenberg, H. J. Polym. Sci. 1958, 30, 47. (7) Kuruczev, T.; Steel, B. J. Pure Appl. Chem. 1967, 17, 149. (8) Manning, G. S. J. Phys. Chem. 1975, 79, 262.
10.1021/la980895h CCC: $18.00 © 1999 American Chemical Society Published on Web 03/10/1999
4070 Langmuir, Vol. 15, No. 12, 1999
electrophoretic mobility factor
H)
Wandrey
2,8
Table 1. Molecular Characteristics of the NaPSS and PDADMAC Samples
40 RT 3η
(4)
parameter
NaPSS
PDADMAC
chemical structure
which contains the viscosity of the solvent η as well as its relative permittivity 0, λc0, the radius of the polymer chain r, and the Debye screening length lD. Generally, lD may be calculated from9
lD ) [4πNAlB(ξ-1cp + 2cs)]-1/2
(5)
Recently, lD is discussed from different points of view.10 This discussion is based on the conclusion that the description of the counterion distribution by lD is probably too simplified. In this work eq 5 will be used to calculate l D. Equation 3 contains the radius of the polyion cylinder r. This parameter is inaccurate because the dimensions of the hydration shell are not exactly known. Therefore, r has been eliminated by an alternative relationship Λ h) f(Λ) which is applicable to salt-free polyelectrolyte solutions8
(
)
3.12 × 10-3 Λ - λc0 A f c Λ h ) -|zc|-1 log cp + const ) 0.155 Λ 1- λc0 λ 0 fc c
(
)
(6)
with A ) 7.131 × 10-6 (S m2 mol-1). In terms of the Manning model, the plot of Λ h vs (log cp) yields a linear relationship with the slope related to the counterion valency. Several other models for pure polyelectrolyte solutions without added salt based on thermodynamics11-13 or scaling descriptions14 and also for polyelectrolyte solutions with an excess of low molar mass electrolyte15,16 have been published. At present, models are under discussion which are based on ionic transport processes in terms of the dynamic frictional formalism of nonequilibrium thermodynamics.17 Likewise, computer simulation techniques are being applied using mean spherical approximations.18-20 Another important aspect in more recent models is that the condensed counterions have been assigned a high degree of mobility along the polymer chain.21 The comparison of experimental results with theoretical predictions reveals only a partial agreement. However, the deviations between experiment and theory often result from the fact that the experimental conditions are not (9) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, C.; Stscherbina, D. Polylectrolytes: Formation, Characterization, Application; Carl Hanser Verlag: Mu¨nchen, 1994. (10) Schmitz, K. S. In Macro-ion Characterization; Schmitz, K. S., Ed.; ACS Symposium Series 548; American Chemical Society: Washington, DC, 1994. (11) Vink, H. J. Chem. Soc., Faraday Trans. 1 1984, 80, 507. (12) Schmitt, A.; Meullenet, J. P.; Varoqui, R. Biopolymers 1978, 17, 413. (13) Schmitt, A.; Varoqui, R. J. Chem. Soc., Faraday Trans. 2 1973, 8, 1087. (14) Colby, R. H.; Boris, D. C.; Krause, W. E.; Tan, J. S. J. Polym. Sci., Part B 1997, 35, 2951. (15) Manning, G. S. J. Phys. Chem. 1981, 85, 1506. (16) Schmitt, A.; Varoqui, R.; Meullenet, J. P. J. Phys. Chem. 1977, 81, 1514. (17) Vink, H. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 1472. (18) Turq, P. NATO, ASI Ser. 1987, C205, 409. (19) Blum, L. J. Phys. Chem. 1988, 92, 2969. (20) Sheng, W.; Kalogerakis, N.; Bishnoi, P. R. J. Phys. Chem. 1993, 97, 5403. (21) Vink, H. J. Colloid Interface Sci. 1995, 173, 211.
charge distance b (nm) degree of polymerization n molar mass (g mol-1) molar mass distribution Mw/Mn contour length L (nm)
0.25 40-1540
0.5 74-2000
Mw: 8000-356000 ≈1.1
Mn: 12000-325000 cs. If the two concentrations become of similar order, the slope levels off. Finally, Λ decreases if cs > cp. Even in “salt-free” solutions at very low polyelectrolyte concentrations Λ decreases caused by the remaining impurities of water and its starting selfdissociation. On the maximum equivalent conductivity a similar influence of the ionic strength has been obtained for both polyelectrolytes, independent of the charge density and
Λmax ∼ M-0.17(0.01
(12)
To answer the question if the charge density or the broader molar mass distribution of the PDADMAC are responsible for the different exponents in eqs 11 and 12, experiments with narrow distributed samples having various charge distances are recommended. The regression results of the linear regressions log Λmax ) f(log cs) and log Λmax ) f(log M) are summarized in Table 2. The measurements without addition of NaCl have not been considered for the determination of the cs dependence since the ionic strength was not exactly known. Further, the contribution of the polyelectrolyte was taken into account to calculate the ionic strength if its contribution was expected to be more than 10%.
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Table 2. Linear Regressions (a) log Λmax ) a + R log cs for NaPSS and PDADMAC of Different Molar Masses in the Range 3 × 10-6 < cs < 10-4 mol L-1 polyelectrolyte NaPSS
PDADMAC
M 103 (g mol-1)
a
R
regr coeff
8 35 46 183 12 22 72 170 325
1.131 0.799 0.735 0.525 1.264 1.099 0.973 1.005 1.022
-0.174 -0.204 -0.196 -0.184 -0.166 -0.184 -0.192 -0.175 -0.166
0.977 0.992 0.972 0.979 0.965 0.979 0.993 0.949 0.965
(b) Λmax ) a + β log M for Different Ionic Strengths polyelectrolyte NaPSS
PDADMAC
ionic strength (mol L-1) 10-6
1× 3 × 10-6 1 × 10-5 5 × 10-5 3 × 10-6 5 × 10-6 1 × 10-5 2 × 10-5 5 × 10-5 1 × 10-5
a
β
regr coeff
3.77 3.66 3.63 3.26 2.75 2.86 2.80 2.79 2.65 2.56
-0.405 -0.392 -0.407 -0.353 -0.146 -0.176 -0.175 -0.185 -0.169 -0.162
0.971 0.921 0.971 0.973 0.975 0.972 0.963 0.972 0.932 0.941
Figure 4. Scaling of the interaction parameter fc,fit for NaPSS with different molar masses (“salt-free” conditions). M ) (b) 8000, (O) 35 000, (9) 46 400, (0) 183 000, (2) 356 000 g mol-1; (‚‚‚) fc ) 0.304, (-‚-) lD/L ) (4π)-1/2, (- - -) lD/L ) 1, linear regression for lD/L > 10. Table 3. Ratio lD/L at cs ) 10-6 mol L-1 for the Investigated Polyelectrolytes polymer NaPSS
Accordingly, the molar mass and the total ionic strength determine the value of Λmax. Its position, however, depends on the ratio cp/cs. Discussion The experimental findings cannot be explained by an existing theory. Their discussion in terms of Manning’s theory resulted in the conclusion that the electrolytic conductivity behavior can be understood permitting a change of fc, the fraction of counterions which take part in the charge transport process.36 Applying eq 2, a theoretical value of fc ) 0.304 results for a charge distance b ) 0.25 nm. Only the experimental equivalent conductivities of the high molar mass NaPSS sample (upper curve in Figure 2d) are very close to the theoretical curve.36 On the basis of the assumption that the change of Λ below c* results primarily from an increase of fc, the reason of the concentration dependence has to be explained. This shall be discussed in regards to the concentration regimes in salt-free polyelectrolyte solutions38,39 in the following paragraphs. At cp ) c*, the polyions should have reached their maximum expansion and not longer overlap. The persistence length Lp becomes the order of the contour length, Lp ≈ L. Substituting cp in eq 5 by c* (eq 8) for polyelectrolytes with a ) b and cs f 0 the ratio lD/L ) (4π)-1/2 ) 0.28 is obtained. Accordingly, at c* the Debye length is still less than the contour length. With further dilution, however, lD increases and has under the applied experimental conditions a limit of approximately 300 nm. Table 3 contains the maximum values of lD/L for the investigated NaPSS and PDADMAC samples. The Manning theory predicts a monotonical increase of λp with dilution resulting from the change of lD (eq 3), but fc is constant in terms of this theory and not influenced by lD. The assumption of a change of fc is in agreement with theoretical derivations from Ramanathan and Woodbury40 that the predictions of the counterion condensation (38) De Gennes, P. G.; Pincus, P.; Velasco, R. M. J. Phys. (Paris) 1976, 37, 1461. (39) Odijk, T. Macromolecules 1979, 12, 688.
PDADMAC
molar mass 10-3 (g mol-1)
contour length (nm)
lD/L
8 35 46 183 356 12 22 72 170 325
8.8 39 51 202 390 37 68 222 525 1005
34 7.7 5.9 1.5 0.8 8.1 4.4 1.4 0.6 0.3
are only applicable if lD/L < 1. Therefore, it can be concluded that the counterion condensation decreases if the experimental conditions permit lD > L. This should result in an increase of fc in eq 1. To evaluate the importance of the ratio lD/L all data were plotted as log(fc,fit) vs log(lD/L). Here fc,fit was calculated by transforming eq 1 to fc,fit ) Λ/(λp0 + λc0) with λc0 ) 45.4 S cm2 mol-1 for sodium at 20 °C and λp0 ≈ 84 S cm2 mol-1. The value λp0, the equivalent conductivity of the polyion at maximum dilution, was determined by comparing the theoretical and experimental concentration dependence of Λ. The appropriate theoretical curves for 0.3 < fc < 1 were obtained by combination of eqs 1 and 3.3,36 Figure 4 shows the log(fc,fit) vs log(lD/L) plots for different molar masses of NaPSS. The same procedure has been applied to the PDADMAC data.3 In the range lD/L > 1 the logarithmic plots fit on lines for the three lowest molar masses of NaPSS and the two of PDADMAC. The data of the high molar mass samples are outside of this range. For (4π)-1/2 < (lD/L) < 1 there exists a nonlinear dependence which passes over to an independence of fc.fit on lD/L at lD/L < (4π)-1/2 crossing the overlap concentration. Above c*, the plot in Figure 4 shows a constancy for fc.fit but lower than the value calculated by the theory. This is qualitatively in agreement with results from counterion activity measurements.3 Though the theory relates to salt-free conditions the same plot was applied to all data based on the consideration that the counterion condensation is determined mainly by the ratio lD/L. Figure 5 shows the results therefrom. (40) Ramanathan, G. V.; Woodbury, C. P., Jr. J. Phys. Chem. 1982, 77, 4133.
4074 Langmuir, Vol. 15, No. 12, 1999
Wandrey
Figure 5. Scaling of the interaction parameter fc,fit for all data of Figure 2: (a) 8000, (b) 35 000, (c) 46 400, and (d) 183 000 g mol-1.
By linear regressions the following relationships were obtained for lD/L > 1:
NaPSS (8000):
fc,fit ) 0.32(lD/L)0.34
NaPSS (35000):
fc,fit ) 0.36(lD/L)0.36
NaPSS (46400):
fc,fit ) 0.35(lD/L)0.35
PDADMAC (12000):
fc,fit ) 0.61(lD/L)0.25
PDADMAC (22000):
fc,fit ) 0.60(lD/L)0.26
In Figure 6 a schematic master curve is presented which shows three different ranges of the conductivity behavior of polyelectrolytes with b < lB. These regimes are characterized by: 1. (lD/L) < (4π)-1/2 Manning regime with predicted counterion condensation cp > c* and lD < L. The polyelectrolyte chains overlap. cp < c* is experimentally impossible for infinite chain length. 2. (4π)-1/2 < (lD/L) < 1 Transition regime. cp < c*. Polyelectrolyte chains do not overlap, however, lD < L. Slow decrease in the interaction between the polyion and the counterions. 3. (lD/L) > 1 Highly diluted regime cp , c* and lD > L. Defined decrease in the interaction between the polyion and the counterions. The absolute values of fc,fit depend on the charge parameter ξ. With decreasing ξ, the slope in the highly
Figure 6. The interaction parameter fc as a function of the ratio of the Debye length to the contour length: (‚‚‚) fc ) 0.886ξ-1; (-‚-) lD/L ) (4π)-1/2; (- - -) lD/L ) 1 (fc and model curve based on NaPSS data).
diluted regime will decrease and is expected to become zero for ξ < 1. Experimental deviations from the master curve may be caused by a low polydispersity of the investigated “monodisperse” standards or inaccurate molar mass determination or an error in electrochemical experiments. For deviations at the lowest dilutions one needs to take into account that the inaccuracy in cs and the missing knowledge on the type of the impurity ions may influence the calculation of lD and fc,fit. A plot according to eq 6 shows additionally that the Manning model is valid only for high molar mass samples (Figure 7). This is qualitatively in agreement with conductivity results from Ander et al.26 Their Λ and Λ h values for PSS with different counterions having a molar mass 70 000 g mol-1 were also found above the theoretical curves in the concentration range 10-3 to 2 × 10-2.
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Langmuir, Vol. 15, No. 12, 1999 4075
definition43 with fc ) Dc/Dc0
(
0 RT Dc fcp ) 0 -1 Dc Dc
)
(13)
an increase of fc results from decreasing fcp. However, measurements of the self-diffusion at extremely low concentrations are not known. A further point under discussion is the value of the dielectric constant near the polyion.44 Its uncertainty is a weak point in many theoretical approaches. The recently derived theory of dynamics of dilute and semidilute polyelectrolyte solutions may also be helpful for the understanding of the electrolytic conductivity behavior.45
Figure 7. Comparison of the theoretical and experimental concentration dependence of the equivalent conductivity parameter Λ h ; molar mass of NaPSS: (O) 8000, (4) 35 000, (b) 46 400, (0) 183 000, and 356 000 g mol-1.
There are several details which are useful to support the understanding of the findings, particularly the change of fc. They will be discussed in the following paragraph. The driving force to overcome the counterion condensation is the diffusion potential of the “bound” counterions, which is determined by the differences between the local counterion concentration around the polyion and the counterion concentration in solution.41 This local counterion concentration exhibits for 0.55 monomol L-1 NaPSS and is much higher than the applied experimental concentrations. Differences in Λ above c* may result from end group effects. The influence of the lower charge density at the chain ends is expected to be more significant for shorter chains. Recently, this has been analyzed theoretically for ionic oligomers.42 Another discussion, based on nonequilibrium thermodynamics, presents a further point of departure to understand a change of fc. However, a complete explanation is still lacking. For cp < c* it is expected that the friction coefficient counterion/polyion (fcp) decreases resulting from the increasing distance between the polyion and the counterions. Using the (41) Manning, G. S. Q. Rev. Biophys. 1978, 11, 179. (42) Manning, G. S.; Mohanty, U. Physica A 1997, 247, 196.
Conclusions Electrolytic conductivity is a powerful tool to investigate the solution behavior in highly diluted solution. Herein a strong influence of concentration, molar mass, and ionic strength on the equivalent conductivity has been shown to exist for flexible polyelectrolytes having charge distances less than the Bjerrum length. Different concentration regimes could also be identified. On the basis of a variety of experimental data and generalizable corrrelations, a universal model could be proposed in order to describe the electrolytic conductivity behavior of polyelectrolyte solutions with polyelectrolytes having any chain length and with any ratio of polyelectrolyte to salt concentration. Since a theory has not been developed to describe these experimental findings completely, we believe these data stimulate theoretical development. Acknowledgment. I thank M. Antonietti and H. Dautzenberg, Max Planck Institute for Colloid and Interface Research, Teltow, and D. Hunkeler, Laboratory of Polymers and Biomaterials, Swiss Federal Institute of Technology, Lausanne, for critical and helpful discussions. S. Mackowiak and B. Zilske are acknowledged for accurate conductometric and potentiometric measurements. The work was supported by the “Fonds der Chemischen Industrie”. LA980895H (43) Varoqui, R.; Schmitt, A. Biopolymers 1972, 11, 1119. (44) Mandel, M.; Odijk, T. Am. Rev. Phys. Chem. 1984, 35, 75. (45) Muthukumar, M. J. Chem. Phys. 1997, 107, 2619.
