Role of Linear Charge Density and Counterion Quality in

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Langmuir 2006, 22, 10963-10971

10963

Role of Linear Charge Density and Counterion Quality in Thermodynamic Properties of Strong Acid Type Polyelectrolytes: Divalent Transition Metal Cations Judit Horva´th* and Miklo´s Nagy Institute of Chemistry, Eo¨tVo¨s Lora´ nd UniVersity, P.O. Box 32, H-1518 Budapest 112, Hungary ReceiVed June 7, 2006. In Final Form: September 27, 2006 Thermodynamic properties of aqueous solutions of poly[(vinyl alcohol)-co-(vinyl sulfate)] (PVAS) copolymer polyelectrolytes with divalent transition metal (Co(II), Ni(II), and Cu(II)) counterions have been determined by the gel deswelling method in the concentration range of 0.0005-0.12 mol of counterion/kg of water (0.09-9 w/w% of the polymer). The influence of the chemical nature of the counterion as well as the effect of the composition of the copolymer from small to medium linear charge density have been systematically studied. Solvent activity, reduced osmotic pressure, the Flory-Huggins pair interaction parameter, rational osmotic coefficients, and degrees of dissociation were calculated from the measured data. No difference could have been observed between the three counterions. Reduced osmotic pressure curves are found to be convex from above, as for Na+ counterions studied previously, which is contrary to the usual behavior of neutral polymers. Intercepts are increasing, and the calculated apparent molar masses and degrees of dissociation at infinite dilution are decreasing with increasing linear charge density of the polyelectrolytes. The pair interaction parameters show a considerable negative deviation from linearity, except for the high volume fraction region. From the differences, concentration dependence of degrees of dissociation could have been calculated. The values at infinite dilution are in good agreement with those obtained from the intercepts of the reduced osmotic pressure curves. Degrees of dissociation seem to decrease approximately linearly with increasing concentration and reach zero at finite concentrations. Rational osmotic coefficients have been calculated in three different ways, both regarding and neglecting the change in the degrees of dissociation.

Introduction Synthetic polyelectrolytes often serve as model substances of biopolyelectrolytes in both experimental and theoretical studies. Natural polyelectrolytes have a varying composition (and purity) strongly depending on their origin, so their experimental study can hardly lead to the establishment of rules having general validity. On the other hand, nearly all the thoroughly investigated synthetic polyelectrolytes are very uniform materials: they are all homopolymers of high charge density carrying 100% carboxylate or sulfonate functional groups. The former type is represented in most cases by poly(acrylic acid) (PAA) or poly(methacrylic acid) (PMA) and the latter by poly(vinyl sulfonic acid) (PVS), poly(styrene sulfonic acid) (PSS), or poly(methyl styrene sulfonic acid) (PMSS). However, the crucial role of charge density and degree of polymerization in biological activity were already emphasized in an early work dealing with the relationship between physical and biological properties of synthetic copolymers having similar anticoagulant effects to that of heparin.1 Charge density of homopolymers can exclusively be varied through changing the degree of dissociation. In the case of weak polyacids, this can be most easily achieved by changing the pH (i.e., the degree of titration). Generally, altering the ionic strength (the amount of added low molar mass salt) or varying the ratio of different types of counterions are possibilities to influence the charge density of a homopolymer. Copolymer polyelectrolytes can be rarely found in the literature, and these are mainly styrene copolymers: partially sulfonated polystyrene or, for example, the alternating copolymer poly[(maleic acid)-alt-styrene]. The problem that only polyelectrolytes of high charge densities can be studied still holds in their case: the amount of charge carrier monomer units cannot be lowered arbitrarily (approximately >50%) because the copolymer becomes insoluble in water as polystyrene itself is insoluble, too. * Corresponding author. E-mail: [email protected]. (1) Patat, F.; Vogler, K. HelV. Chim. Acta 1952, 35, 128-138.

To overcome this restriction, derivates of poly(vinyl alcohol) (PVA) could serve as a solution, while PVA is the simplest and most studied water soluble polymer, and it offers numerous possibilities to polymer analogous reactions such as esterification. In the present work, poly[(vinyl alcohol)-co-(vinyl sulfate ester)] (PVAS) copolymer polyelectrolytes were investigated: this is the same type of copolymer already mentioned previously that was studied because of its similar anticoagulant property to that of heparin. But no systematic further investigation of the physicochemical properties of these copolymers was undertaken, apart from a very few early studies with a medium linear charge density sample.2-3 In our laboratory, the systematic study of these polyelectrolytes has lasted for more then two decades.4-10 Despite of the fact that PVAS has lost its significance as a heparin analogue since the development of the synthetic pentasaccharide ARIXTRA (Fondaparinux Sodium),11 it could remain an excellent model substance with well-controlled properties. These polyelectrolytes contain both nondissociating hydroxyl and dissociating strong acidic sulfate half ester functional groups, so the charge number of the polymer chain can be varied by the composition (the degree of esterification, DS) of the copolymer, too, entirely independently of the degree of dissociation. According to our synthesis method,5 polyelectrolytes of very low to medium linear charge density (charge number) could be obtained. This enables the detailed investigation of the effects of the charge distribution transition from the entirely separated (2) Nagasawa, M.; Kagawa, I. J. Polym. Sci.1957, 25, 61-76. (3) Ise, N.; Okubo, T. J. Phys. Chem. 1967, 71, 1886-1890. (4) Machovich, R.; Nagy, M.; Gyo¨rgyi-Edele´nyi, J.; Csomor, K.; Horva´th, I. Thromb. Haemostasis 1986, 56, 397. (5) Nagy, M. Magy. Kem. Foly. 1992, 98, 18-24. (6) Baba, M.; De Clercq, E. D. A.; Go¨ro¨g, S.; Lo¨w, M.; Nagy, M.; Gyo¨rgyi, S. U.S. Patent 5,152,978, 1992. (7) Varga, I.; Nagy, M. Spectrochim. Acta, Part B 2001, 56, 2229-2234. (8) Nagy, M. Colloids Surf., A 2004, 250, 467-471. (9) Nagy, M. J. Phys. Chem. B 2004, 108, 8866-8875. (10) Nagy, M. Manuscript in preparation. (11) Walenga, J. M.; Fareed, J.; Jeske, W. P.; Frapaise, F. X.; Bick, R. L.; Samama, M. M. Turk. J. Haematol. 2002, 19, 137-150.

10.1021/la061642o CCC: $33.50 © 2006 American Chemical Society Published on Web 11/15/2006

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(isolated) randomly distributed point-like charges through the accumulation of the charges along the chain to a more or less continuous linear charge distribution, as well.9-10 Beside charge density of the polyion, valence and chemical nature of the counterions play decisive roles in the properties of polyelectrolytes. Literature dealing with ion binding between divalent counterions and polyelectrolytes was studied thoroughly from the beginning of this kind of investigation over more than 50 years. In the case of the weak polyacids,12-28 PAA15,19,21-28 and PMA,12-20,22,25 Cu(II) was the absolutely most studied divalent counterion12-14,18,25-28 since peculiar spectral changes indicate its interaction with the carboxylate groups. Copper is followed by other transition metal ions such as Zn(II),13,15-17,20 Cd(II),13,17,21,23 Co(II),13 Ni(II),13,27 and Pb(II).19 The study of these counterions was initiated mainly by environmental considerations, while the binding of alkaline earth metal counterions, Mg(II),13,25 Ca(II),22,25 and Ba(II),24 was investigated because of their biological relevance. In the case of strong polyacids,26-43 PSS,26,30-40,42,43 PMSS,29 and PVS,27,28,41 Cd(II)29-30,32-34,42,43 and Cu(II)26-28,32,33,38 were preferred as divalent counterions, but Mg(II),31,35,36,40,41 Ca(II),3,30-34 Ba(II),3,39,41 and Zn(II)29-30,37 were also often investigated. Besides them, one or two examples for Sr(II),30 Ni(II),30 and Pb(II)29,38 also occur in the literature. To study counterion binding by weak polyacids, potentiometry seems to be still the absolutely favored method,12,21-25 although (12) Mandel, M.; Leyte, J. C. J. Polym. Sci., Part A: Polym. Chem. 1964, 1, 2883-2899. (13) Mandel, M.; Leyte, J. C. J. Polym. Sci., Part A: Polym. Chem. 1964, 1, 3771-3780. (14) Leyte, J. C.; Zuiderweg, L. H.; van Reisen, M. J. Phys. Chem. 1968, 72, 1127-1132. (15) Esteban, M.; Casassas, E.; de Jong, H. G.; van Leeuwen, H. P. Anal. Chim. Acta 1990, 229, 93-100. (16) Dı´az-Cruz, J. M.; Esteban, M.; van den Hoop, M. A. G. T.; van Leeuwen, H. P. Anal. Chim. Acta 1992, 264, 163-175. (17) van den Hoop, M. A. G. T.; van Leeuwen, H. P. Anal. Chim. Acta 1993, 273, 275-287. (18) Dı´az-Cruz, J. M.; Arino, C.; Esteban, M.; Casassas, E. Anal. Chim. Acta 1993, 273, 289-296. (19) Dı´az-Cruz, J. M.; Arino, C.; Esteban, M.; Casassas, E.; van Leeuwen, H. P. J. Electroanal. Chem. 1993, 344, 119-134. (20) van den Hoop, M. A. G. T.; van Leeuwen, H. P.; Benegas, J. C. Biophys. Chem. 1995, 54, 35-42. (21) Miyajima, T.; Mori, M.; Ishiguro, S.; Chung, K. H.; Moon, C. H. J. Colloid Interface Sci. 1996, 184, 279-288. (22) Iida, S. Biophys. Chem. 1996, 57, 133-142. (23) Benegas, J. C.; Cleven, R. F. M. J.; van den Hoop, M. A. G. T. Anal. Chim. Acta 1998, 369, 109-114. (24) Pochard, I.; Couchot, P.; Foissy, A. Colloid Polym. Sci. 1998, 276, 10881097. (25) Porasso, R. D.; Benegas, J. C.; van den Hoop, M. A. G. T. J. Phys. Chem. 1999, 103, 2361-2365. (26) Kruczala, K.; Schlick, S. J. Phys. Chem. B 1999, 103, 1934-1943. (27) Rivas, B. L.; Schippacasse, N.; Basaez, L. Polym. Bull. 2000, 45, 259265. (28) Rivas, B. L.; Schippacasse, N. L.; Pereira, U. E.; Moreno-Villoslada, I. Polymer 2004, 45, 1771-1775. (29) Oman, S.; Dolar, D. Z. Phys. Chem. Neue Folge 1967, 56, 13-19. (30) Reddy, M.; Marinsky, J. A.; Sarkar, A. J. Phys. Chem. 1970, 74, 38913895. (31) Kozak, D.; Kristan, J.; Dolar, D. Z. Physik. Chem. Neue Folge 1971, 76, 93-97. (32) Oman, S. Makromol. Chem. 1974, 175, 2133-2140. (33) Oman, S. Makromol. Chem. 1974, 175, 2141-2148. (34) Oman, S. Makromol. Chem. 1977, 178, 475-484. (35) Skerjanc, J.; Dolar, D.; Leskovsek, D. Z. Physik. Chem. Neue Folge 1967, 56, 218-222. (36) Dolar, D.; Skerjanc, J. J. Chem. Phys. 1974, 61, 4106-4109. (37) Skerjanc, J. J. Phys. Chem. 1975, 79, 2185-2157. (38) Peregudov, Y. S.; Amelin, A. N.; Perelygin, V. M. Vysokomol. Soedin. 1995, 37, 302-305. (39) Dolar, D.; Bester, M. J. Phys. Chem. 1995, 99, 4763-4767. (40) Kogej, K.; Skerjanc, J. J. Chem. Soc., Faraday Trans. 1996, 92, 31093115. (41) Hen, J.; Strauss, U. P. J. Phys. Chem. 1974, 78, 1013-1017. (42) Rivas, B. L.; Moreno-Villoslada, I. J. Phys. Chem. B 1998, 102, 1102411028. (43) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1999, 466, 129-143.

HorVa´ th and Nagy

the operation principle of ion selective electrodes is rather complex. Application of other thermodynamic methods, such as isopiestic or direct osmotic pressure measurements, is rather scarce,44-46 and what is more, no one refers to divalent metal salts. Structural information has been obtained by various spectroscopic methods such as UV-vis,13,14 IR,14 NMR,21 and ESR26 spectroscopy, while formation constants could have been calculated from diffusion constants of the counterions obtained by electrochemical methods such as polarography15,19 or voltammetry.16-20,28 Strong acid type polyelectrolytes were also investigated by ESR spectroscopy26 and voltammetry;28,43 transference numbers40 and distribution ratios27,41,42 were measured, as well. Thermodynamic measurements were performed predominantly by the isopiestic method.3,30 In some works, colligative properties such as osmotic pressure32-34 or freezing point depression31 were determined directly. The isopiestic method could only be applied for solutions of very high (0.1-7 mol of counterion/kg of water) concentrations, while direct osmometry was applicable for the very dilute concentration region (0.0003-0.04 mol of counterion/kg of water). Unfortunately, there exists an intermediate concentration range unavailable by both methods, which makes checking of consistency of the results difficult. Direct potentiometry by ion selective electrodes and by electrodes of first order was also applied to determine activity coefficients of the counterions.29 Thermodynamic quantities were calculated from the heat of mixing37 or heat of dilution35,37-38 and from solubility data,39 as well. In the present work, thermodynamic measurements were carried out by a relative method, called gel deswelling,47-48 which enables determination of the thermodynamic activity of the solvent exclusively in polymer solutions. This method is based on the phenomenon that the degree of swelling (i.e., the solvent uptake) of a gel highly and explicitly depends on the activity of the solvent if it is placed in a mixture from which the solute cannot enter the network. The method has the advantage that measurements can be performed in a concentration region (0.001-0.1 mol of counterion/kg of water) that is too low for the isopiestic method but at the same time too high for the direct osmometry. In our previous work,9 a rather detailed thermodynamic characterization was given for PVAS polyelectrolytes with monovalent metal counterions with special regard to the chemical nature of the counterion and the linear charge density of the polyanion. PVAS polyelectrolytes of not less than six different linear charge densities and three different alkali metal counterions were included in that study. Although a few examples demonstrating the effect of the valence of the counterions can also be found in that work, a systematic investigation of divalent counterions has undoubtedly remained necessary. Therefore, in the present work, a detailed thermodynamic study of ten PVAS polyelectrolytes of four different linear charge densities and three different divalent transition metal counterions, Co(II), Ni(II), and Cu(II), was undertaken. Experimental Procedures Materials. Poly[(vinyl alcohol)-co-(vinyl sulfate)] (PVAS) copolymer polyelectrolytes were synthesized starting from the medium molar mass fraction of a hydrolyzed and partially fractionated poly(vinyl alcohol) (POVAL 420, Kuraray Co., Japan). Its number average degree of polymerization, DPn, was determined by an osmometer constructed in our laboratory10 and found to be DPn ) 1005 ( 20. (44) Ise, N.; Okubo, T. J. Phys. Chem. 1967, 71, 1287-1290. (45) Asai, K.; Takaya, K.; Ise, N. J. Phys. Chem. 1969, 73, 4071-4076. (46) Biswas, B.; Williams, P. A.; Phillips, G. O. Polymer 1992, 33, 12841288. (47) Nagy, M.; Horkay, F. Acta Chim. Acad. Sci. Hung. 1980, 104, 49-61. (48) Nagy, M. Phys. Chem. Chem. Phys. 2000, 2, 2613-2622.

Thermodynamic Properties of Polyelectrolytes

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Table 1. Composition of Prepared Poly[(vinyl alcohol)-co-(vinyl sulfate ester)] (PVAS) Polyelectrolytesa sample

xVAS ≡ DSb

nVAS/nVOHc

z-d

PVAS/0.93 PVAS/3.40 PVAS/6.43 PVAS/16.9

0.0093 ( 0.0004 0.0340 ( 0.0008 0.0643 ( 0.0016 0.169 ( 0.001

1:107 1:28.4 1:14.5 1:4.92

9.34 34.1 64.6 170

a DPn ) 1005 ( 20. b Vinyl sulfate content of the copolymer (degree of esterification) in mol fraction. DS ) nVAS/(nVAS + nVOH). c VAS: VOH monomer ratio. nVAS/nVOH ) DS-1 - 1. d Average charge number of a polymer chain. z- ) xDPn.

For the preparation of the sulfate half ester derivative of PVA, a method worked out earlier in our laboratory was used, which relies on equilibrium esterification by sulfuric acid at low temperature.5 Details of the synthesis are given in the Supporting Information. This way, copolymer polyacids of low and medium charge densities (vinyl sulfate/vinyl alcohol (VAS/VOH) ratio between approximately 1:110-1:5) could be obtained. The amount of acidic functional groups was determined by both conductometric and potentiometric titration, while the polymer content of the solutions was determined by gravimetry. Composition and other characteristics of the copolymers are summarized in Table 1. Stoichiometric salts of the PVAS polyacids with the appropriate divalent transition metal cations were prepared by ion exchange dialysis described previously in detail.9 Cobalt and copper salts of all four polyelectrolytes and the nickel salt of the two copolymers of higher DS (PVAS/6.43 and PVAS/16.9) were prepared. Details of the dialysis procedure are given in Supporting Information. The metal ion content of some polyelectrolyte salts was checked by complexometric titration.49 The deviation from the stoichiometric composition was (1.7% for the copper salts; a similar agreement can be estimated for the salts of the other two metal ions as well. Stoichiometry was proved earlier for such metal-polyelectrolyte systems by the ICP-AES method.7 Double distilled water was produced in a Muldestor SE quartz still (Wagner and Munz, Munich) and had a specific conductivity 0.8-1 µS cm-1. Methods. In all experiments that were conducted at strictly constant temperature, a highly reliable Haake F6 B12 circulating bath was used to keep the temperature at 298.00 ( 0.01 K even for several weeks. Densities of polyelectrolyte solutions were determined in 10 cm3 picnometers (BLAUBRAND) at 298.00 ( 0.01 K in the polymer concentration region 1-10 w/w%. Conductivity measurements were carried out using a Radelkis (Hungary) OK-102/1 conductivity meter connected with a Radelkis OK-0902P conductivity cell having a cell constant of 1 cm-1. Gravimetric measurements were performed by drying the polymer samples to constant weight in a Binder VD 23 vacuum-drying oven at 343.0 ( 0.5 K, which took about 4-7 days. Before drying the PVAS polyelectrolytes, approximate stoichiometric amounts of a dilute NaOH solution had to be added to the solution samples to avoid the acid-catalyzed degradation of the copolymers caused by heating. The amount of the samples was most suitable when the mass of the dried polymer was in the range of 0.040-0.100 g. Relative error of the polymer mass fractions of the solutions is estimated to be (0.5-2%. For solvent activity, a1, measurements in polymer solutions by the gel deswelling method, composition, degree of cross-linking, and size of the gel laminae had been optimized to achieve the highest sensitivity in a previous work.48 In the present work, measurements were carried out using chemically cross-linked poly[(vinyl acetate)co-(vinyl alcohol)] copolymer gels of the composition 8:10:400. The gel consisted of a copolymer carrying 8 mol % vinyl acetate units, the cross-linking reaction was performed by glutardialdehyde (GDA) in an aqueous solution containing 10 w/w% copolymer, and the ratio of the number of monomer units versus that of GDA molecules was 400. The actual procedure of the gel deswelling (49) Schwarzenbach, G. Die komplexometrische Titration, 3rd ed.; Ferdinand Enke Verlag: Stuttgart, 1957; pp 73-78.

measurement was also described earlier in detail.9,48 The gel lamina was placed in a semipermeable tube (Visking dialysis tubing, L 14.3 mm, Medicell Int. Ltd., London) and immersed into the polymer solution to ensure that only the solvent could enter the network.47 Being a relative method, gel laminae of the given composition had to be calibrated with polymer solutions of known solvent activities. The most suitable material for this purpose had been found to be poly(vinyl pyrrolidone) (PVP) (K-60, Fluka), as very reliable osmotic pressure data are available for its solutions in the literature.50 Recently, following the exhaustive dialysis of the PVP product lasting for 3 months, these osmotic pressure data were checked thoroughly employing an osmometer constructed in our laboratory, and excellent agreement was found.10 The gels were calibrated in the mass percent region of the PVP solutions between 1 and 15 w/w % that made water activity measurements in the region of -7 × 10-4 e ln a1 e -4 × 10-6 possible. Masses of the gels swollen in pure double distilled water (mw), in equilibrium with the polymer solution (me), and after drying (m0) were measured. Calibration curves were established as a function of the mass fraction of the polymer in the gel in equilibrium with the polymer solution (ln a1 vs m0/me) (Figure S1) and also as a function of the ratio of the masses of the gel swollen in polymer solution and in pure water (ln a1 vs 1 - me/mw) (Figure S2). The latter was found to give more precise data in the low concentration range.51 In the present work, solvent activity data in the range of - 4 × 10-4 e ln a1 e -4×10-6 were calculated based on both types of calibration curves, while for ln a1 data under this value (concentrated region), only the mass fraction curves gave acceptable results. Preparation of the gels is so reproducible that calibration points of gels of the same composition, but of different charges, fall exactly onto the same calibration curve even after 1-2 years, stored in double distilled water at 278 K (see Figures S1 and S2). The concentrations of the polyelectrolyte solutions in equilibrium with the gels were also determined by gravimetry. This way, 18 data points could be obtained per ∼3 weeks, as reaching (de)swelling equilibrium took 10 days, and the capacity of the circulating bath and the vacuum-drying oven limited the number of samples to 18.

Results and Discussion General Thermodynamic Characterization of Aqueous Solutions of PVAS Polyelectrolyte Salts. Thermodynamic activity of the solvent (water) (a1) was determined by the gel deswelling technique in solutions of 10 poly[(vinyl alcohol)co-(vinyl sulfate)] (PVAS) polyelectrolytes of four different linear charge densities and three different divalent counterions, cobalt, nickel, and copper. Dependence of ln a1 on mass concentration of the polyelectrolytes, cm,2, is shown in Figure 1. Data points of polyelectrolytes of the same DS fall on the same curve. Similar findings were made previously by us where different alkali or alkaline earth metal salts of PVAS of a given linear charge density could hardly be distinguished, as well.9 The order of the curves is in accordance with expectations: the uncharged parent polymer, PVA, lowers the solvent activity the least, and increasing the amount of ionic groups on the polymer chain causes a steep decrease in solvent activity. However, ln a1 values in solutions of divalent salts of PVAS are only less than half of those obtained for monovalent counterions. This indicates that ∆G of mixing with the solvent is much less negative in the divalent case. To characterize thermodynamic properties of polymer solutions, one of the further possibilities is to plot the reduced osmotic pressure, π/cm,2, as a function of cm,2 as it can be seen in Figure 2. Generally, this representation is used for the determination of the number average molar mass of the polymer from the intercept of the curve. In the case of neutral polymers, reduced osmotic pressure curves are usually concave from above and have a decreasing curvature approaching zero concentration; therefore, (50) Vink, H. Eur. Polym. J. 1971, 7, 1411-1419. (51) Csa´ki, K. F. Thermodynamic properties and interfacial behavior of polymer mixtures. Ph.D. Thesis, Eo¨tvo¨s Lora´nd University, Budapest, 2004; pp 42-43.

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Figure 1. Dependence of thermodynamic activity of the solvent (water) on polymer mass concentration: (9) PVAS/0.93/Co, (0) PVAS/0.93/Cu, (1) PVAS/3.40/Co, (3) PVAS/3.40/Cu, (2) PVAS/ 6.43/Co, (×) PVAS/6.43/Ni, (4) PVAS/6.43/Cu, (b) PVAS/16.9/ Co, (/) PVAS/16.9/Ni, and (O) PVAS/16.9/Cu. Curve A denotes pure PVA (data taken from ref 9). Curves B-E are fitted third-order polynomials on the overall data of polyelectrolytes of the same DS.

their intercepts can be relatively easily determined. Nevertheless, for PVAS polyelectrolytes, just the opposite shape could have been observed: curves have steep downward curvatures at very low polyelectrolyte concentrations (see Figure 2). However, PVAS is not the only polyelectrolyte showing this rather strange behavior. Reduced osmotic pressure curves of NaPSS in saltfree solutions have been found to take the same shape, while added NaCl has transformed the curve back to the usual form.52 Just as in the case of NaPVAS,9 polyelectrolytes with higher DS have higher intercepts in the case of divalent counterions, too. Scattering is somewhat greater in the present work, but it should also be noticed that scaling of the ordinate ends lower than half of the same axis in the figure representing NaPVAS.9 Reduced osmotic pressure data points of Co-, Ni-, and CuPVAS of the same DS cannot be distinguished from each other, although curves of Mg- and BaPVAS were found to run completely separated.9 Up to now, it remained a question as to whether the chemical nature of the counterion appears in the reduced osmotic pressure curve. Present findings suggest that the distance between the curves of the two alkaline earth metal counterions is caused not by the difference in their chemical nature but rather by the difference between their molar masses. The molar masses of Mg and Ba are so different that they cause considerable deviation in the Mn of their PVAS salts, as well. Mn of Ni- and CuPVAS/ 16.9 differs only by 0.6% (and this value is even smaller for lower DS), whereas it reaches 8.5% in the case of Mg- and BaPVAS/8.1. The Flory-Huggins theory is a well-proven, classical description of the thermodynamic behavior of solutions of neutral polymers. However, testing of its applicability for polyelectrolytes has just started recently.9 The key figure of the theory, the pair interaction parameter, χ1,2, characterizes the strength of the (52) Takahashi, A.; Kato, N.; Nagasawa, M. J. Phys. Chem. 1970, 74, 944946.

HorVa´ th and Nagy

Figure 2. Dependence of the reduced osmotic pressure on polymer mass concentration: (+) PVAS/0.93/Co, (×) PVAS/0.93/Cu, (1) PVAS/3.40/Co, (2) PVAS/3.40/Cu, (3) PVAS/6.43/Co, (0) PVAS/ 6.43/Ni, (4) PVAS/6.43/Cu, (b) PVAS/16.9/Co, (9) PVAS/16.9/ Ni, and ([) PVAS/16.9/Cu. Curve A denotes pure PVA (data taken from ref 9). Curves B-E are third-order polynomials with their 95% confidence bands fitted on the overall data of polyelectrolytes of the same DS.

polymer-solvent interaction and can be calculated as follows:

χ1,2 )

ln a1 - ln(1 - φ2) 2

φ2

+

(

1 V1 -1 φ2 V2,R)0

)

(1)

where φ2 is the volume fraction of the polymer in the solution, V1 is the molar volume of the solvent, and V2,R)0 is that of the polymer. Molar volumes were calculated from solution density data. On the basis of elementary theoretical considerations, it is known that the following linear relationship holds between the F density of a solution and the w2 mass fraction of the solute:

(

)

V1 V2 1 V1 ) + w F M1 M2 M1 2

(2)

where M1 and M2 are the molar masses of the solvent and the solute, respectively. Strictly speaking, V1 and V2 denote partial molar volumes here, but in the case of polymer solutions, molar and partial molar volumes equal each other as the additivity of the volumes is assumed by the Flory-Huggins theory. Validity of the assumption is supported by the fact that 1/F versus w2 curves are strictly linear for the investigated polyelectrolytes in the mass fraction range w2 ) 0.01-0.11; therefore, partial molar volumes are independent of concentration in the concentration range studied. The ratio of the molar volumes can be easily obtained from the intercept (A) and the slope (B) of the straight lines 1/F versus w2

V1 A M1 ) V2 A + B M2

(3)

Thermodynamic Properties of Polyelectrolytes

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Table 2. Comparison of Data Characterizing Dissociation Behavior of PVAS Polyelectrolytes with Transition Metal Counterions Co(II), Ni(II), and Cu(II)a xVAS (%) 0.93 3.40 6.43 16.9

Mn (kg mol-1) ( 2%

(π/c)c)0 (105 Pa cm3 g-1)

45.3 48.0 51.3 62.8

2.1 ( 0.2 4.3 ( 0.1 6.6 ( 0.3 7.4 ( 0.3

Mn,app (kg mol-1)

Rc)0b

V2,R)0/V1 ( 2%

Rc)0c

[(Rc)0xDPn)/z] + 1

12 5.8 3.7 3.3

0.60 ( 0.07 0.43 ( 0.02 0.39 ( 0.02 0.21 ( 0.01

1.85 × 1.89 × 103 1.89 × 103 2.01 × 103

0.62 ( 0.02 0.44 ( 0.01 0.403 ( 0.008 0.220 ( 0.004

3.9 8.5 13.8 19.2

103

a Difference between data of the three counterions is less than 1%. Entire data set is available from Table S1. b By the π/c-method. c By the χ-method.

Figure 3. Dependence of the experimentally determined pair interaction parameter on the volume fraction of the polyelectrolytes. (×) PVAS/0.93/Co and Cu; ([) PVAS/3.40/Co and Cu; (O) PVAS/ 6.43/Co, Ni, and Cu; and (b) PVAS/16.9/Co, Ni, and Cu. Straight lines were fitted to the data points falling on the linear sections of the curves. Thick straight line: pure PVA (data from ref 48). Insert: same figure with larger scaling of the ordinate.

In our case, M2 equals the number average molar mass of the polymer, Mn. Mn and molar volume ratio data of the investigated divalent metal ion - PVAS salts can be found in Table 2 (and in a more detailed version in Table S1), while χ1,2 versus φ2 curves are represented in Figure 3. Like those of neutral polymers, χ1,2 curves are approximately linear in the φ2 ) 0.03-0.06 volume fraction range but show a steep downward curvature extending down to as low as about -20 approaching zero polymer concentration. Such low negative values would not be explicable for neutral polymers, but the classical Flory-Huggins theory could have been extended to polyelectrolytes in such a way that the degree of dissociation of the counterions was integrated into it.9 To demonstrate the reliability of these pair interaction parameters, χ1,2 values from the linear sections were plotted as a function of the composition (DS) of the copolymer at given volume fractions in Figure 4. Extrapolating these curves to zero VAS content, χ1,2 values of pure PVA could have been obtained at each concentration. The tendency of the curves agrees with expectations (i.e., with increasing polarity of the polymer chain, the interaction between solute and solvent becomes stronger). Degree of Dissociation. Previously, two methods were developed for determination of the effective number of osmotically active entities in polyelectrolyte solutions.9 On the basis of this quantity, a kind of degree of dissociation, R, could be defined assuming that the counterions can be either fixed (associated) to the polyion and become osmotically inactive or can freely mix with the solvent and can contribute additively to the total osmotic pressure of the polymer solution. Certainly, the real situation is not so polarized, and these R values cannot give information about the distribution of counterions around the

Figure 4. Dependence of the χlin pair interaction parameter on the vinyl sulfate (VAS) content (mol fraction) of the copolymers at various volume fractions. φ2 ) (9) 0.080, ([) 0.060, (1) 0.040, (2) 0.020, and (b) 0.000. Data at xVAS ) 0 (pure PVA) are taken from ref 48.

polyion. Despite this, R can characterize the overall extent of attachment of counterions to the polymer chain. The first calculation method makes use of the intercept of the reduced osmotic pressure curve that should serve for the determination of the number average molar mass, Mn, of the polymer chains. On the other hand, Mn values can also be calculated from the known degree of polymerization and composition of the copolymer polyelectrolyte. However, intercepts for PVAS solutions have given only apparent molar masses, Mn,app, that were in all cases much lower than real Mn values. Real Mn and Mn,app values for PVAS copolymer polyelectrolytes with Co, Ni, and Cu counterions are summarized in Table 2. The large difference between the two values was explained by dissociation of the counterions. Osmotic pressure, as a colligative property, detects dissociated counterions as individual particles; therefore, they are also included in the averaging. Degrees of dissociation at zero concentration can be calculated directly from the intercepts of the osmotic pressure curves

(

RT RT Rc)0xDPn π (cm,2 ) 0) ) ) +1 cm,2 M h n,app M hn z

)

(4)

where π/cm,2 is the reduced osmotic pressure, Rc)0 is the degree of dissociation at zero concentration, x is the mol fraction of the ionic groups in the copolymer (degree of esterification, DS), DPn is the number average degree of polymerization, and z is the charge number of the counterion. The product xDPn/z gives the number of stoichiometric counterions per polymer chain, and the constant 1 represents the polymer chain itself. As it can also be seen in Figure 5, eq 4 gives a linear relationship between reduced osmotic pressure and R at zero concentration. From the graphical representation, it is revealed that Rc)0 can be most precisely determined in the case of high DS because for high charge densities, Rc)0 is quite insensitive with respect to relative great scattering in (π/c)c)0. As R can take values only between 0 and 1, a minimum and maximum value of the intercept is

10968 Langmuir, Vol. 22, No. 26, 2006

Figure 5. Dependence of the reduced osmotic pressure on the degree of dissociation at zero polymer concentration calculated from eq 4. Lines: (s) PVAS/16.9/Cu, (- - -) PVAS/6.43/Cu, (- ‚ -) PVAS/ 3.40/Cu, and (‚‚‚) PVAS/0.93/Cu.

determined for each polyelectrolyte of a given composition and counterion. Rc)0 values of Co-, Ni-, and CuPVAS salts calculated by this method (further on π/c-method) are summarized in Table 2. The three different counterions have the same Rc)0 value in polyelectrolytes of the same DS since their reduced osmotic pressure curves run together, as well. In our previous work,9 MgPVAS and BaPVAS also have been found to have nearly the same degree of dissociation at zero concentration, although their reduced osmotic pressure curves run completely separated from each other. This agreement of the Rc)0 values supports further that the difference between the π/cm,2 curves can be explained by the difference between the real molar masses of the polyelectrolyte salts. The second method (χ-method) uses the linear and nonlinear sections of the χ1,2 versus φ2 curves and enables the calculation of degrees of dissociation not only at zero concentration but also in the whole concentration range. In this case, it is assumed that in the linear range of the χ1,2 versus φ2 curve, polyelectrolyte

HorVa´ th and Nagy

solutions are so concentrated that their degree of dissociation equals zero. Extrapolating the linear section to lower concentrations, where the steep downward curvature appears, and substituting these χlin values into the Flory-Huggins equation (eq 1), apparent molar volumes, V2,app, can be obtained. Molar volumes obtained from solution density data belong to the nondissociated state because density measurements were performed in the concentration range corresponding to the linear part of the χ1,2 curve. Degree of dissociation can be calculated not only from the ratio of the real and apparent number average molar mass but also from the ratio of the real and apparent (partial) molar volume of the polyelectrolyte by a relationship of the same form

R)

(

)

V2,c)0 z -1 V2,app xDPn

(5)

Expressing V2,app by the Flory-Huggins equation (eq 1) leads to eq 6.

V2,R)0 z V1 xDPn

R ) φ2(χlin - χ1,2)

(6)

It can be seen that degree of dissociation is directly proportional to the product of the difference between χ1,2 and χlin and the volume fraction of the polyelectrolyte in the solution. Other terms in eq 6 are independent of concentration for a polyelectrolyte of a given composition and counterion. Dependence of the degrees of dissociation on counterion concentration is depicted in Figure 6 for polyelectrolytes of different linear charge densities. Each curve consists of two distinct sections: above a given concentration (linear section of the χ curves), R equals 0 as it was fixed by definition, but at lower concentrations, a linear increase of

Figure 6. Dependence of the degree of dissociation calculated by the χ-method on counterion concentration (molarity). (b) Co, (×) Ni, and (O) Cu stoichiometric salts of PVAS.

Thermodynamic Properties of Polyelectrolytes

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Figure 7. Dependence of degree of dissociation at zero concentration on the vinyl sulfate (VAS) content (mol fraction) of the copolymers. Methods of calculation: (0) π/c-method and (9) χ-method.

dissociate to a greater extent. R values refer to the overall amount of functional groups (or counterions) present in the system, while a single polymer chain carries RDS number of charges. If R is not low enough, it cannot compensate a high DS value; therefore, the product RDS that determines the order of the curves in Figure 8 still can remain high. Finally, it should be noted that the effect observed is not an ionic condensation, simply because the average charge distance, b, according to counterion condensation theory53 is not less than the Bjerrum length for any of the polyelectrolytes investigated in the present work. b values between 1.5 and 27 nm result for the DS range 16.9-0.93%, respectively, as b ) 0.25 nm for fully charged vinylic polyelectrolytes.53 Condensation of divalent counterions, in the classical sense, would just begin at DS higher than 17.5%. Osmotic Coefficients. In works dealing with colligative properties of polyelectrolyte solutions, the nonideal behavior is usually characterized by the osmotic coefficient. Two types of osmotic coefficients are defined by IUPAC: the rational and the practical osmotic coefficient. In the present work, calculations and discussions are restricted to the rational osmotic coefficient since it does not contain any approximation. Rational osmotic coefficient Φ is defined as follows:

Φ)

ln a1 ln x1

(7)

where a1 is the activity and x1 is the mol fraction of the solvent in the mixture. The value ln a1 is directly obtained from the gel deswelling measurement, while x1 can be calculated from the composition of the solution: Figure 8. Dependence of the molality of free counterions ) Co2+, Ni2+, and Cu2+) on the molality of the polymer chains in PVAS solutions without added salt. DS ) (/) 16.9%, (O) 6.43%, (b) 3.40%, and (+) 0.93%. (M2+

the R values can be observed by decreasing concentration. The same linear relationship could be observed recently for PVAS polyelectrolytes with alkali and alkaline earth metal counterions, as well.9 The slight back-folding found on those curves at very low concentrations cannot be definitely seen on the present ones; no doubt, this region lies at the performance limit of the gel deswelling technique. Intercepts of the linear sections can be seen in Table 2. Rc)0 values obtained by the π/c- and the χ-method, respectively, were compared with each other in Figure 7, and good agreement was found between them. The order of the Rc)0 values corresponds to the expectations concerning linear charge densities, although just the opposite dependence resulted for NaPVAS.9 In Figure 8, the molality of free (dissociated) counterions was plotted as a function of the molality of the polymer chains, and maximum curves were obtained referring to the four different DS values. Although there is a 2-20 times difference in DS between the polyelectrolytes, interestingly enough, all the curves have maximum values at about the same polymer chain molality (∼0.5 mmol/kg), and they all reach zero value at ∼1 mmol/kg. The great scattering around zero at higher concentrations results from the initial scattering of the experimental R values in the R ≡ 0 region multiplied by the high concentration values. Polyelectrolytes of higher DS release more counterions at the same polymer concentration, which means that average charges of the polymer chains are not the same. At first glance, it seems to be a paradox that lower R values were obtained for higher DS, and at the same time, one can say that polymer chains having more dissociable functional groups

x1 )

n1 n1 + n2(1 + R(xDP/z))

(8)

where n1 and n2 are the amount of solvent (water) and solute (polymer chains), respectively, in the mixture; other symbols have the same meaning as in eq 4. As the mol fraction is based on the number of entities present in the solution, dissociated counterions also have to be taken into account in the calculation. The expression in parentheses in the denominator gives the number of entities that a single polymer chain dissociates to at a given R. The constant 1 in the sum represents the polymer chain itself, which is always present independently from the degree of dissociation. The second term of the sum stands for the dissociated counterions. In the case of homopolymer polyelectrolytes, for which most of the osmotic coefficient data can be found, x equals 1, and the constant 1 in the sum is usually neglected as the number of polymer chains is negligible to the number of counterions if DPn and R are great enough. If Φ gives the extent of deviation from ideality, then first the ideal mixture has to be defined (i.e., which degree of dissociation is considered as ideal). Three cases are possible: R can equal 1 or 0 or can be a function of the concentration R(c), as will be seen later. In the first two cases, deviation from ideality is caused both by polymer-solvent interaction and by association or dissociation, respectively. In the third case, changes in R are already taken into account in the calculation, and deviation from ideality due to dissociation is eliminated from Φ. For homopolymer polyelectrolytes, the first type of calculation is applied in the literature. In the ideal case, R is always taken to be 1, which means that the dissociation is complete and that deviation from ideality is caused mainly by counterion condensation onto the polymer chain. Osmotic coefficients for the 10 PVAS copolymer (53) Manning, G. S. J. Chem. Phys. 1969, 51, 924-933.

10970 Langmuir, Vol. 22, No. 26, 2006

Figure 9. Dependence of the rational osmotic coefficient on the molality of the counterions if the degree of dissociation is considered to be 1 in the ideal case. (×) PVAS/0.93/Co and Cu; ([) PVAS/ 3.40/Co and Cu; (O) PVAS/6.43/Co, Ni, and Cu; and (b) PVAS/ 16.9/Co, Ni, and Cu.

polyelectrolytes obtained by this type of calculation are shown in Figure 9. Data of PVAS salts of the same DS fall on the same curve in this representation, as well. Intercepts give back the Rc)0 values obtained by the two methods discussed previously with good agreement. Namely, nonideality due to the polymeric character disappears at infinite dilution, and deviation from the value of 1 is caused merely by association of counterions to the polymer chain. The authors are aware that Manning’s rod-like model53 cannot describe satisfactorily polyelectrolytes of a low charge density parameter, ξ; nevertheless, an attempt was made to calculate osmotic coefficients. The reason was that the sample of the highest DS (16.9 ( 0.1%) has a charge density very close to the critical value (17.5%) where condensation of divalent counterions may begin. According to Manning’s theory,53 osmotic coefficients of the polyelectrolytes in the present study should take much higher values at zero concentration than the experimental ones (see intercepts in Figure 9). Taking the case ξ < 1/z and z ) 2, and also ξ values between 0.0265-0.485 determined by the DS range covered in the present work, Φ should be between 0.97 and 0.52 at zero concentration, which is obviously not realized. Osmotic coefficients increase with increasing concentration, and curves have similar shapes to the ones obtained previously for sodium counterions.9 It should be noted that osmotic coefficients in that former work were calculated based on the practical osmotic coefficient and that the stoichiometric factor ν was arbitrarily taken to be 2. This may explain why those curves run at lower values than the curves shown in the present work, although osmotic coefficients of polyelectrolytes (e.g., PSS) with monovalent counterions are at least 2 times greater than those with divalent counterions.3,29-34 However, the rather steep increase of the NaPVAS curves at very low concentrations appears in the case of transition metal counterions, as well, except for the polyelectrolyte of the highest linear charge density (PVAS/16.9). This further affirms that osmotic coefficient curves may have an inflection in the very low concentration range and may have the usual concave form only at higher concentrations. Namely, all the osmotic coefficient curves of PSS (and also of PVAS) salts determined in the concentration range higher than 0.01 mol/kg by the isopiestic method3,30 or by freezing point depression31 look concave, but direct osmotic pressure measurements demonstrate that NaPSS and HPSS curves are definitely convex under that concentration.32,34 The inflection can be clearly seen even as high as ∼0.5 mol/kg in the case of

HorVa´ th and Nagy

Figure 10. Dependence of the rational osmotic coefficient on the molality of the polymer chains if the degree of dissociation is considered to be 0 in the ideal case. (×) PVAS/0.93/Co and Cu; ([) PVAS/3.40/Co and Cu; (O) PVAS/6.43/Co, Ni, and Cu; and (b) PVAS/16.9/Co, Ni, and Cu.

Li and tetraalkylammonium salts of PVS.54 However, there is one remarkable difference between curves in the present work and curves obtained for NaPVAS previously.9 Surprisingly enough, in the case of sodium counterions, all the curves referring to linear charge densities lower than 2% coincided with each other, and all the others of higher charge densities fell onto another common curve. On the contrary, in the case of divalent counterions, osmotic coefficient curves are clearly separated by linear charge density regarding either the 10 curves in the present or the four ones in the previous work (see Figure 5S in ref 9). It seems that for some reason only monovalent counterions show this peculiar behavior. On the other hand, chemical properties of the divalent counterions do not seem to affect the osmotic coefficient as much as the linear charge density of the polyanion. Co, Ni, and Cu salts of PVAS have been found to be indistinguishable in the present work. In the previous work,9 data points of Mg- and BaPVAS also fell exactly on the same curve at a given linear charge density, but that single observation was not enough for the establishment of a general rule. The second type of calculation, when R is considered to be 0 in the ideal case, is used for small molar mass salts for which the extent of dissociation (the stoichiometric factor) is to be determined. Applying this method for PVAS salts, the resulting osmotic coefficients are shown in Figure 10. Now, Φ is plotted as a function of molality of the polymer chains, not of that of the counterions. Intercepts give the number of ions that a single polymer chain dissociates to at infinite dilution. These values are in very good agreement with the values that can be calculated using Rc)0 values obtained either by the π/c- or the χ-method (see last column in Table 2). However, Φ versus mc curves have just the opposite tendency to that of R versus c curves. It can be concluded that the activity increasing effect caused by nonideality can overcompensate the activity decreasing effect of counterion association. This means that the shape of the osmotic coefficient curve cannot give information about the extent of dissociation at finite concentrations. In the third type of calculation, the two effects that are responsible for nonideality, dissociation, and polymeric character are separated from each other. Degrees of dissociation obtained by the χ-method as a function of concentration are taken into account in the calculation. A similar type of calculation was first (54) Ise, N.; Asai, K. J. Phys. Chem. 1968, 72, 1366-1369.

Thermodynamic Properties of Polyelectrolytes

Figure 11. Dependence of the rational osmotic coefficient on the molality of the polymer chains when R(c) calculated by the χ-method is taken into account. (×) PVAS/0.93/Co and Cu; ([) PVAS/3.40/ Co and Cu; (O) PVAS/6.43/Co, Ni, and Cu; and (b) PVAS/16.9/Co, Ni, and Cu. Solid line: pure PVA (data from ref 9).

applied for NaPAA by Ise and Okubo,44 who defined Φ on the basis of the free counterions, as well. The resulting osmotic coefficients for PVAS divalent metal salts are shown in Figure 11, together with the curve of the uncharged parent polymer PVA. PVA cannot dissociate; therefore, its Φ curve must have the usual shape of reduced osmotic pressure curves of neutral polymers (i.e., concave, as the former is obtained from the latter by dividing only by a constant, the intercept). Therefore, the curve of PVA must start from the value of 1 by definition and can differ from this ideal value only due to polymer-solvent interaction. As it can be clearly seen in Figure 11, all five curves start from exactly 1, which proves that the effect of dissociation has been correctly eliminated from Φ. It affirms that the degrees of dissociation obtained by the χ-method are reliable and that the linear sections of the χ-curves were properly determined. Deviation from ideality increases steeply by increasing concentration or linear charge density. However, the osmotic coefficient curve of the polyelectrolyte of the lowest DS (PVAS/ 0.93) still runs together or even slightly under that of PVA. This slight negative difference may be explained by the difference in the degrees of polymerization: data of PVA refer to a previous sample of DPn ) 1600. In the case of the other three copolymers, Φ has high values even at concentrations where R undoubtedly equals 0, and all the counterions are associated to the polymer chain. It clearly demonstrates that the hydrophilicity of the polymer chain and solute-solvent interaction increase with increasing sulfate content of the copolymer even in the undissociated state.

Conclusion The gel deswelling method has proven to be effective and precise enough to clearly distinguish PVAS polyelectrolytes by their linear charge density in the case of divalent counterions, as well, when solvent activity values are less than the half of those measured in systems with monovalent counterions. Not

Langmuir, Vol. 22, No. 26, 2006 10971

even the slightest difference has been revealed, however, between the three counterions, Co(II), Ni(II), and Cu(II), in any representations. The previously seen deviation between the π/c curves of Mg(II) and Ba(II) is caused merely by the difference in molar masses despite the enormous difference in their chemical nature (i.e., affinity to a simple sulfate ion). This indicates to what extent reduced osmotic pressure curves are sensitive to molar mass, also in the case of completely identical osmotic coefficient curves. It was already established for NaPVAS that the π/c curves are convex at very low concentrations; however, the shapes of the curves with divalent counterions show a new feature: they seem to have an inflection, a plateau region at medium concentrations, and they are concave above that, just as neutral polymers. Simplification of former equations9 led to a new linear relationship between intercepts of reduced osmotic pressure curves and degrees of dissociation at zero concentration, which enables calculation of Rc)0 directly, without calculation of the apparent number average molar mass. Concentration dependence of degrees of dissociation was calculated from Flory-Huggins pair interaction parameters by a new expression, eq 6, containing the difference in χ1,2 and χlin, and which was derived, based on previous ideas,9 in an alternative way. Consistency of these types of calculations is proven by the good agreement between Rc)0 values calculated by the two methods. Contrary to sodium counterions, lower Rc)0 values belong to higher linear charge densities, which does correspond to expectations. Three different possible methods were shown for calculation of rational osmotic coefficients in the case of dissociable polymers, both neglecting and taking into account degrees of dissociation. The type of osmotic coefficient curve usually used in the literature has a similar intercept to the appropriate Rc)0 value, but it cannot be determined reliably if the measurement was made by the isopiestic method. The shape of the curve shows no similarity to the R(c) function, so it is unsuitable for conclusions with respect to degree of dissociation. The third type of calculation can successfully separate dissociation from the overall deviation from ideality, leading to osmotic coefficients that demonstrate the presence of solute-solvent interactions. Acknowledgment. The authors are greatly obliged to Mrs. Jaksity for technical assistance. The financial support of the Hungarian Scientific Research Fund, OTKA (T 35100), is gratefully acknowledged. Supporting Information Available: Details concerning the preparation of PVAS polyacids and their stoichiometric metal salts. Two types of calibration curves used for evaluation of gel deswelling measurements (Figures S1 and S2). Detailed molar mass and molar volume ratio data of the 10 investigated PVAS polyelectrolytes (Table S1). This material is available free of charge via the Internet at http://pubs.acs.org. LA061642O