Thermodynamic Characterization of Polyanetholesulfonic Acid and Its

Aug 9, 2007 - Experimental and theoretical results for the thermodynamic properties of polyanetholesulfonic acid and its lithium, sodium, and cesium s...
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J. Phys. Chem. B 2007, 111, 10130-10136

Thermodynamic Characterization of Polyanetholesulfonic Acid and Its Alkaline Salts Irena Lipar, Petra Zalar, Ciril Pohar, and Vojko Vlachy* Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, Asˇkercˇ eVa 5, P.O.B. 537, 1001 Ljubljana, SloVenia ReceiVed: May 12, 2007; In Final Form: June 24, 2007

Experimental and theoretical results for the thermodynamic properties of polyanetholesulfonic acid and its lithium, sodium, and cesium salts in aqueous solution at 298 K are presented. The osmotic pressure was measured using membrane and vapor pressure apparatus in the concentration range cm ) 0.001-0.30 monomoles/dm3. The osmotic coefficients obtained from these measurements were low, from 0.2 to 0.45 in this concentration range, indicating a strong interaction between counterions and polyions. The osmotic coefficients of the polyacid and its lithium and sodium salts appeared to be equal within experimental error, but the results for the cesium salt were lower. This indicates a somewhat stronger binding of cesium ions to the polyanion. In addition, enthalpies of dilution, ∆HD, from a certain concentration, mm, to mm ) 0.0044 monomoles/kg were measured. The measured heats of dilution were exothermic, with the acid producing the strongest and the cesium salt the weakest effect. These results were compared with previously published data for polyelectrolytes of similar structure, namely, polystyrenesulfonic acid and its alkaline salts. The osmotic pressure results indicate that polystyrenesulfonates bind the counterions more strongly than polyanetholesulfonic acid and its salts. Consistent with this finding, the enthalpies of dilution reveal that more heat is released upon dilution of polyanetholesulfonates (stronger exothermic effect) in comparison with the corresponding solutions of polystyrenesulfonic acid in its alkaline salts. These findings can be explained in terms of the structural differences between the two polyions. The experimental results were analyzed in relation to popular electrostatic theories such as the Manning condensation theory and the Poisson-Boltzmann cell model approach, where the polyion is pictured as a uniformly charged line or cylinder. In addition, we performed Monte Carlo simulations for a model polyanetholesulfonic anion having discrete charges. In all of the calculations, the solvent was treated as a continuum with the dielectric constant of pure water under the conditions of measurement. The theoretical considerations mentioned above yield results in semiquantitative agreement with the measured quantities.

1. Introduction Polyelectrolytes are defined as macromolecules which possess charged groups that can ionize in water or another appropriate solvent. Their properties differ from those of neutral polymers, as also from the properties of low molecular weight ions in solution. Polyelectrolytes are characterized by very diverse physical behavior and are used in numerous technological applications.1 In spite of the great interest coming from industry and science over many years of investigations, the properties of polyelectrolyte solutions are still not understood sufficiently well.2-9 The polyelectrolyte examined in this study is polyanetholesulfonic acid (HPAS) and its alkaline salts. The sodium salt of this polyacid (NaPAS) is known to have anticoagulant properties and is used in medical applications,10,11 yet there has been very little work done so far to characterize its properties in solution. The only paper concerned with the physicochemical properties of polyanetholesulfonic acid and its salts of which we are aware is the contribution of Zhang and co-workers.12 These authors applied isopiestic measurements to determine the osmotic coefficient of the sodium salt of polyanetholesulfonic acid at concentrations above 0.27 monomoles/dm3. In the present work, we wished to contribute to filling this gap by providing a more complete thermodynamic characteriza* To whom correspondence should be addressed. Phone: +38612419406. E-mail: [email protected].

tion of polyanetholesulfonic acid and its alkaline salts in water. Solutions of lithium, sodium, and cesium salts of polyanetholesulfonic acid were studied (in addition to the polyacid) in the concentration range cm ) 0.001-0.30 monomoles/dm3. Only salt-free aqueous solutions with monovalent counterions were investigated in this first study, while we hope to extend the measurements to divalent counterions and to mixtures with a low molecular electrolyte in the near future. The focus of this study was on two thermodynamic properties: the first was the osmotic coefficient, which yields information about the activity of the solvent (water in this case) in solution. Alternatively, the osmotic coefficient can be interpreted as the fraction of “osmotically free” counterions in the system.2,3,5 The second important thermodynamic property investigated in the present paper was the enthalpy of dilution. This quantity (also called the heat of dilution) is known to be quite sensitive to the chemical nature of the counterionsseven for fully ionized polyelectrolytes, strong ion-specific effects have been documented in the literature. For this reason, the enthalpy of dilution is an important thermodynamic quantity, yielding information about solvent mediated interactions in polyelectrolyte solutions.13-21 Chainlike polyelectrolytes in solution are difficult to model in great detail (for a review of theoretical developments, see ref 9). Their behavior is determined by the strong interaction between the highly charged polyion and oppositely charged

10.1021/jp073641q CCC: $37.00 © 2007 American Chemical Society Published on Web 08/09/2007

Thermodynamic Characterization of HPAS

Figure 1. Structural drawing of the monomer unit of the (a) polyanetholesulfonate and (b) polystyrenesulfonate anions. The dimensions are here given in angstroms (1 Å ) 0.1 nm).

small ions (counterions) in solution. This interaction, however, may be affected by the conformation of the polyion in solution and vice versa.9,22,23 The interaction between charges on the backbone and the ions in solution is long-range (Coulomb interaction), and is supposed to play a major role in these systems. It has long been known,24 however, that some of the properties are ion-specific, so not only the charge but also the chemical nature of the counterions and the charged groups on the polyion can be important. This is the reason that we decided to examine the behavior of polyions with different counterions, from the strongly hydrated lithium (structure maker) to the cesium (structure-breaker) ion. Polyanetholesulfonic acid and its salts have a very similar structure to that of polystyrenesulfonic acid (HPSS). The difference is in the -O-CH3 group (see Figure 1) and in the different position of the sulfonic group, as well as the two -CH3 groups on the main chain. This has two consequences: First, the average conformation of the HPAS can be considered more rigid than it is in the case of HPPS. Note that experimental data for HPSS and its salts indicate that it assumes extended conformations in dilute solutions. This suggests that both polyelectrolytes can be studied within the rodlike cell model approximation. Second, such a configuration obstructs the approach to sulfonic groups, which are consequently less accessible for counterions than in the case of polystyrenesulfonic acid. We will show that this results in an increase of the osmotic coefficient, and also in more exothermic heats of dilution for HPAS and its related salts. The experimental results will be interpreted in the light of simple models treating polyions as infinite rodlike bodies, counterions as pointlike charges, and water as a continuous dielectric. In addition to the Poisson-Boltmann approach, where the polyion charge is uniformly smeared over the surface, we also applied the Monte Carlo method to examine a model of polyanetholesulfonic anion having embedded discrete charges. In this case, the neutralizing counterions were treated as charged hard spheres. The layout of this paper is as follows. After this Introduction, we first describe the experimental methods used in the study. Next, we outline the theoretical approaches applied to analyze the experimental data. The results of our measurements are given in the Results and Discussion. In this section, the experimental data for polyanetholesulfonic acid and its salts are compared with the results of electrostatic theories, and also with some previous investigations of polystyrenesulfonic acid and its alkaline salts. We summarize the findings in the Conclusions. 2. Materials and Methods 2.1. Materials. Aqueous solutions of lithium (LiPAS) and cesium (CsPAS) polyanetholesulfonates were prepared by

J. Phys. Chem. B, Vol. 111, No. 34, 2007 10131

Figure 2. Experimental values of osmotic coefficients of solutions of NaPAS (4), LiPAS (O), and CsPAS (0) in water at 298 K as a function of the logarithm of polyelectrolyte concentration, cm (in monomoles/ dm3). The results taken from ref 12 (NaPAS) are denoted by plus signs (+).

titrating polyanetholesulfonic acid (HPAS) with the corresponding metal hydroxide until the pH was about 5.5. As in previous studies, this pH value (and not 7.0) was chosen to avoid possible overneutralization of the acid. HPAS was prepared by the usual ion-exchange and dialysis techniques14 from sodium polyanetholesulfonate (NaPAS), with a weight average molecular weight of about 10 000 and a degree of sulfonation of 1.0, obtained from Aldrich, CAS 52993-95-0. The concentrations of the salts were determined from the optical density at 284.4 nm of solutions prepared by diluting the sample solutions with a known excess of a 5% KCl solution. The monomolar extinction coefficient in 5% KCl, obtained from calibration with an HPAS solution of known concentration, was found to be 2510 dm3 cm-1 mol-1. 2.2. Methods. The osmotic coefficient measurements were performed at 25 °C with a Knauer Vapor Pressure Osmometer (model K-7000) and a Knauer Membrane Osmometer (model 7310100000). A detailed description of the osmometers and the experimental procedures has been given elsewhere.25,26 The vapor pressure osmometer was calibrated with standard potassium chloride solutions. In the membrane osmometer measurements, we used cellulose triacetate membranes with MCO 10000 (Sartorius, AG, Goettingen, Germany). The results of the osmotic pressure measurements for various counterions (Li+, Na+, and Cs+) are shown in Figure 2. The data for concentrations lower than 0.02 monomoles/dm3 were obtained by the membrane osmometer, and the others, by the vapor pressure instrument. The reason for using the membrane osmometer is the low accuracy of the vapor pressure apparatus for dilute polyelectrolyte solutions. In the region where accurate results could be obtained by both methods, the measurements agreed within the experimental uncertainties, i.e., about (4%. The enthalpy measurements were performed using an LKB Batch and Flow Microcalorimetry System 10700 and a Thermo Metric 2277 Thermal Activity Monitor. 2.3. Theoretical Part. Theoretical analysis of experimental data on polyelectrolye solutions is most often based on the Manning condensation theory6,27 and/or the Poisson-Boltzmann cell model (see, for example, refs 2, 3, and 5). Condensation theory provides an elegant way of explaining the general features of highly diluted polyelectrolyte solutions. According to this approach, the osmotic coefficient (in the limit of infinite dilution) depends on the single parameter λ

10132 J. Phys. Chem. B, Vol. 111, No. 34, 2007

λ)

e02 4π0kBTb

Lipar et al.

(1)

where e0 is the elementary charge, kB is Boltzmann’s constant,  is the relative permittivity equal to that of pure water at T ) 298 K, and b is the length of the unit containing one elementary charge (the distance between two charges of the polyion projected to the z-axis). For polyanetholesulfonic acid, we estimated (cf. Figure 1) b ) 0.26 nm and λ ) 2.740. For these parameters (λ > 1), Manning theory predicts the osmotic coefficient to be Φ ) (2λ)-1 ≈ 0.183. In this paper, we will discriminate between the value of λ, which is calculated on the basis of the structure (cf. Figure 1) of a particular polyion, and the effective λeff, used in calculations to fit the experimental data. Another theoretical approach is based on solution of the Poisson-Boltzmann equation for cylindrical symmetry.2,3 According to this model, the polyion is represented as a uniformly charged and infinitely long cylinder placed in a cylindrical cell of radius Rc. All of the cells are assumed to be equivalent and independent of each other, and the polyelectrolyte concentration (cm in mol of monomer units/dm3) is defined as

cm )

1 πRc2bNA

(2)

where NA is Avogadro’s number. The counterions, neutralizing the charge on the polyions, are distributed within the cell according to Boltzmann statistics. In this model, the molecular nature of the solvent is ignored, as are all other than Coulomb interactions. The osmotic coefficient is calculated as the concentration of counterions at the cell boundary, c(Rc), divided by the average concentration of monovalent counterions, which in this case is just equal to cm:

Φ)

c(Rc) cm

(3)

Using the analytical solution of the Poisson-Boltzmann equation in cylindrical symmetry,2,3 the osmotic coefficient, Φ, can be written as

Φ)

1 - β2 (1 - e-2γ) 2λ

(4)

where γ ) ln(Rc/a) and β is an integration constant following from the solution of the Poisson-Boltzmann equation. In this approach, the counterions are treated as pointlike charges and their size is reflected merely in the polyion-counterion contact distance (in this case, the polyion radius), a. In addition, the counterion-counterion correlation in the electrical double layer around the polyion is neglected in the Poisson-Boltzmann approach, an approximation which is quite acceptable in the case of monovalent (but not divalent) counterions, as probably first pointed out by Fixman28 and most recently examined in refs 29 and 30. The enthalpy of dilution, ∆HD, in the Manning condensation theory is given by the simple equation

∆HD ) -

(

)( )

cm2 RT d ln  1+ ln 2λ d ln T cm1

(5)

where cm2 is the final and cm1 is the initial concentration in monomoles/dm3. The term [d ln /d ln T] is for water3,13-21 at

T ) 298 K equal to -1.37. This term is very important, since it determines the sign of the enthalpy of dilution in eq 5, as also in eq 6 in the next paragraph. For the models used in this work, where no other temperature-dependent interaction is included but the Coulomb interaction, the enthalpies of dilution can only be negative. From the solution of the nonlinear Poisson-Boltzmann equation in cylindrical symmetry, the excess enthalpy of a polyelectrolyte solution can be calculated as3,13-15

[ ) [

](

z1RT (1 - λ)2 - β2 (1 + β2)γ + ln +λ 1+ z2λ 1 - β2 z1RT d ln  2λe2γ d ln V 2d ln a + 1 - β2 - 2γ (6) d ln T 2z2λ d ln T e - 1 d ln T

He )

](

)

The enthalpy of dilution is defined as the enthalpy change which accompanies dilution from concentration cm1 to cm2, calculated per monomole of solute. It may formally be divided into nonelectrostatic and electrostatic contributions; here, only the electrostatic part is calculated and the non-electrostatic contribution is assumed to be negligible. In eq 6, z1e0 is the charge carried by the ionic group on the polyions and z2e0 is the charge of the counterions. Further, R is the gas constant, γ is a concentration parameter proportional to -ln cm, and β is here the constant related to γ and λ via the next equation:

λ)

1 - β2 1 + β coth βγ

(7)

Equation 7 has to be solved iteratively to obtain β. The term in eq 6 containing derivatives of the volume is usually assumed to be small (d ln V/d ln T ) -8.7 × 10-4), as is the contribution given by the derivative [d ln a/d ln T]. We will discuss other possible contributions to the heat of dilution in relation to eq 9 below. Complementary to the rodlike cell model calculations, where polyions were treated as uniformly charged cylinders, we also studied model polyions with discrete charges. In this model, the charges on the polyion were assumed to be positioned helically and embedded within the polyion of radius a. Their positions (and other parameters) were chosen on the basis of a quantum-chemical calculation (AM1 method; MOPAC 2002): the radius of the helix was taken to be 0.375 nm, step in the z-direction b ) 0.27 nm, and the radius of the polyion cylinder 1.1 nm. The radii of monovalent counterions, aion, were taken to be 0.1 nm, and we applied the computer simulation method to obtain the results for measurable quantities. In this approach, the solvent averaged pair potential, uij(r), between two charges reads

uij(r) ) uij*(r) +

zizje02 4π0rij

(8)

where rij is the distance between the pair of charges i and j and zi (zj) are the signed electrovalences of these charges. The positive charge is located on the monomer unit of a polyion (zp ) +1), and the negative charges in the centers of the hard spheres describing the counterions, zc ) -1. The hard-core short-range potential, uij*(r), prevents the counterions from approaching each other closer than permitted by their diameter, 2aion. It also ensures that the counterions are excluded from the domain of the polyion. The distance of closest approach of a counterion to the charge on the polyion is for this model of PAS- equal to 0.825 nm. The short-range part, uij*(r), modeled

Thermodynamic Characterization of HPAS

J. Phys. Chem. B, Vol. 111, No. 34, 2007 10133

here as an infinitely repulsive potential, does not carry any temperature dependence. The Metropolis Monte Carlo code, used to evaluate the osmotic coefficients and enthalpies of dilution, was the same as that applied before,25 most recently in the extensive calculations presented in ref 29. The portion of the polyion contained in the basic Monte Carlo cell of cylindrical symmetry was 2000 monomer units long. The monovalent counterions were assumed to be distributed uniformly in the cell, and as shown by eq 8, the system was treated as a continuous dielectric. The osmotic coefficient was calculated from eq 3 using Widom’s test particle method (see, for example, ref 31 and the references therein) to obtain the canonical average of the counterion concentration at the cell boundary. The enthalpy of dilution can be approximated by the difference between the excess internal energies of the final and initial polyelectrolyte concentration. In this respect, it is important to stress that the solvent averaged pair potential, uij(r), has the properties of a state function; uij(r) ≡ uij(r;P,T). It is equal to the free energy required to bring two charges (infinite dilution limit) from infinity to the distance r, allowing the solvent molecules to assume all possible configurations consistent with P and T. In this, the so-called McMillan Mayer level of description,32 the pair potential depends on the experimental conditions of the solvent dictated by the external pressure and temperature. The excess internal energy, Ee, therefore needs to be calculated via the Gibbs-Helmholtz expression: Ee ) [∂(Ae/T)/∂(1/T)]N,V, where Ae is the excess free energy.33 In Monte Carlo simulations, we therefore need to evaluate the canonical average (denoted by 〈...〉) in the form

〈 ∑ ∑ 〉 〈 ∑∑ [ ∂(uij/T)

Ee ) 1/2

i

j

∂(1/T)

) 1/2

N,V

uij 1 -

i

j

]〉

∂ ln uij ∂ ln T

N,V

(9)

For the solvent averaged potential, given by eq 8, where the only temperature-dependent term is the dielectric constant, (P,T), the term in angular parentheses reads [1 + d ln /d ln T]. This explains the appearance of this term in eqs 5 and 6.3,13-17 The theoretical methods outlined above all treat the solvent as a continuum dielectric and are therefore unable to treat subtle ion-specific effects. The only way to discriminate one ion from another in these models is by its size (distance of closest approach to the polyion), and this can only explain part of the effect. Though some recent and very interesting computer simulations of short oligoelectrolytes treat water explicitly,34,35 thermodynamic properties such as the osmotic coefficient or heat of dilution seem to presently be out of reach for such detailed approaches. In the present state of the theory, more approximate explicit solvent models have to be used to interpret ion-specific effects reflected in thermodynamic quantities.22,36 Explicit water computer simulations34,35 should help us in constructing more realistic solvent averaged potentials. 3. Results and Discussion 3.1. Osmotic Coefficients. In this subsection, we present the results of the osmotic pressure measurements. As explained in the experimental part, we used both a membrane osmometer (for dilute solutions) and a vapor pressure apparatus to determine the osmotic coefficient defined as Φ ) Π/Πid, where Πid is the ideal contribution to the osmotic pressure. The measurements are presented in Figure 2 as a function of the negative logarithm of concentration given in monomoles/dm3; open triangles (4) denote the results for the sodium salt of polyanetholesulfonic

Figure 3. Comparison between the theory based on the PoissonBoltzmann equation (continuous lines) and experimental results for the osmotic coefficient for NaPAS (4) and NaPSS (2).37 The PoissonBoltzmann calculation for NaPAS applies to a ) 0.89 nm and λ ) 0.9λstr, while the Poisson-Boltzmann curve for NaPSS is obtained for a ) 0.80 nm and λ ) 1.3λstr. We present the Monte Carlo simulation results (a ) 1.2 nm and λ ) λstr) on the same figure by the dashed line. Note again that, at 298 K, λstr ) 2.638 for PAS- and 2.82 for PSS- solutions, respectively.

acid (NaPAS), circles (O) for the lithium salt (LiPAS), and open squares (0) the data for the corresponding cesium salt (CsPAS). We have also included in this figure some points taken from ref 12. They apply to concentrations slightly above the domain investigated in the present study. We found these results, which are shown in Figure 2 by plus signs (+), to be consistent with our measurements. The values of the osmotic coefficients shown in Figure 2 are low, indicating that only a fraction (from 20 to 45%) of the counterions are “osmotically active”. This finding is consistent with the osmotic pressure results for polystyrenesulfonic acid and its alkaline salts. The ion-specific effects observed are relatively small; the experimental results for HPAS and its sodium and lithium salts seem to be equal within the experimental error. The osmotic coefficient data for the polyacid (HPAS) are not shown in Figure 2 to keep the picture as transparent as possible. It is the cesium salt which gives a distinctly lower osmotic coefficient. This is not a surprise, since the hydrated cesium ions lose water easier than the other ionic species studied here, and may therefore approach closer to sulfonic groups on the polyions. This may account for the stronger binding of cesium ions. Altogether, the measurements seem to be consistent with the results for polystyrenesulfonic acid and its alkaline salts.3 There is, however, a quantitative difference; the osmotic coefficients of polyanetholesulfonic acid and the salts studied here are higher (smaller fraction of bound counterions) than those for the corresponding solutions of polystyrenesulfonates. We can attribute this result to the difference in structure between the monomer units of the two polyacids. In the case of the polystyrenesulfonic polyion, the sulfonic groups seem to be more easily accessible to counterions than they are in the case of the polyanetholesulfonic anion (cf. Figure 1). The experimental results presented above were analyzed in relation to electrostatic theories. As already mentioned, the Manning condensation approach gives a value of the osmotic coefficient of Φ ) 1/2λ ≈ 0.18, which is too low in comparison with the measurements. This is not surprising in view of the fact that the theory applies to very dilute solutions and besides in addition considers a relatively “thick” polyion as a line charge. In Figure 3, we present a comparison between the experimental results for sodium salts of polyanetholesulfonic acid (NaPAS;

10134 J. Phys. Chem. B, Vol. 111, No. 34, 2007 4) and the corresponding salt of polystyrenesulfonic acid37 (NaPSS; 2). As we can see, the osmotic coefficients of NaPAS over the whole range of concentrations are higher than those of NaPSS. The experimental results can be nicely fitted by the Poisson-Boltzmann cell model results if λeff * λ is used. Good agreement between the Poisson-Boltzmann calculations (these results are represented by continuous lines) in the case of NaPSS can be obtained for a ) 0.8 nm and λeff ) 3.68. Note that the “structural” λ value is 2.83 in this case. On the other hand, for NaPAS solution, equally good agreement between the theoretical and experimental values can be obtained using the values a ) 0.89 nm and λeff ) 2.47 (the structural value of λ is 2.74 for PAS- solutions). This result is not in line with previous findings for polyelectrolyte solutions! In our experience, good agreement between the measured results and the Poisson-Boltzmann results could always be obtained for some effective value, λeff, which was larger than the structural value of λ. Various explanations for this “rule” can be found in the literature: most often, departures from the fully extended configurations (causing beff < b) and/or dielectric saturation next to the polyion were blamed for the fact that λeff > λ. Here, we again demonstrated that polyelectrolyte solutions possess diverse physicochemical properties, which cannot be explained by a single parameter, λ. We have to emphasize that, even for stiff polyions, where modeling of the polyion as an infinite cylinder is much more justified than here, good agreement between the PoissonBoltzmann theory and experiment could only be obtained if the charge density parameter, λ, were treated as an adjustable variable.38 A comparison of the experimental data with PoissonBoltzmann and Monte Carlo calculations for the osmotic coefficient is shown in the same figure (Figure 3). The computer simulations apply to the model sodium salt of polyanetholesulfonic acid, and are shown by the dashed line. The following values of parameters were used to represent the model NaPAS: the radius of the helix on which the charges were embedded was 0.375, b ) 0.27 nm, the diameter of counterions, aion, was taken to be 0.2 nm, and the radius of the polyion cylinder was 1.1 nm. The Monte Carlo values of the osmotic coefficient in our experience were always lower than the Poisson-Boltzmann results.39,40 From Figure 3, we see that the Monte Carlo results are in no better agreement with the experimental data than the Poisson-Boltzmann results. This seems to be consistent with the fact that a value of λeff smaller than the structural value λ ) 2.74 must be used in the Poisson-Boltzmann approach to fit the measurements. In Figure 4, the osmotic coefficient measurements are shown for the cesium salts of both polyacids. The experimental data for CsPAS are shown by open squares (0), and those for CsPSS41 (these measurements are taken at 0 °C), by filled squares (9). Again, the corresponding Poisson-Boltzmann calculations are shown by continuous lines fitting the symbols. The parameters for this calculation are the same as above, though we could take the distance of closest approach of the Cs+ ion to the polyion to be slightly smaller than that for the (more solvated) Na+ ion. In this case, good agreement between the Poisson-Boltzmann calculations and the measurements (lines) can be obtained for λeff ) 4.41 in the case of CsPSS and for λeff ) 3.97 in the case of the cesium salt of polyanetholesulfonic acid. In both cases, the effective values, λeff, substantially exceed the structural values of λ, as has already been seen in many earlier studies. In conclusion to this subsection, we offer a possible explanation for the result leading to λeff < λ in fitting the experimental

Lipar et al.

Figure 4. Comparison between theory based on the PoissonBoltzmann equation (lines) and experimental results for the osmotic coefficient for CsPAS (0) and CsPSS (9; ref 39). The model parameters for the Poisson-Boltzmann calculation are the same as those for Figure 3. Measurements for CsPSS solutions are taken at zero degrees centigrade, while those for CsPAS are at 298 K.

data for the osmotic coefficient of NaPAS via the PoissonBoltzmann theory. The osmotic pressure results for PAS- could indicate that additional -CH3 groups (not present in PSS-) influence the hydration of different counterions (and consequently the distance of closest approach of counterion to polyion charge) differently. For the Na+ ion, the hydration seems to be increased by the presence of -CH3 groups on the polyion, which yields a higher value of the osmotic coefficient. The extra groups have no or little effect on the more hydrophobic Cs+ ion. This can explain relatively large differences between the Cs+ ion and other counterions for osmotic coefficients, and is consistent with the heats of dilution for these salts presented in the next chapter. In other words, the difference in the osmotic coefficient results between NaPAS and CsPAS salt is bigger than that for equivalent salts of the HPSS. The “speculation” above may be confirmed (or rejected) by the all atom computer simulations, which include water as separate species, and/or by the additional (for example, dielectric relaxation) measurements for both types of polyacids. 3.2. Heats of Dilution. The enthalpy of dilution, ∆HD, is an important thermodynamic property, which often reveals details of the solvent averaged interaction not shown by the osmotic pressure measurements. As demonstrated by Sˇ kerjanc and coworkers,13-15 polystyrenesulfonic acid and its alkaline salts release heat upon dilution; i.e., ∆HD < 0 at room temperature. The same conclusion about the sign of the effect (∆HD < 0) can be arrived at for polyanetholesulfonic acid solutions (and those of its salts) on the basis of Figure 5. In this figure, we present enthalpy changes upon dilution from mm to mm ) 0.0044 monomoles/kg. In this figure, from top to bottom, results are shown for the lithium salt by circles (O), for HPAS by diamonds (]), for the sodium salt by open triangles (4), and for the cesium salt by open squares (0). Theoretical curves obtained by the Poisson-Boltzmann (continuous line) and Monte Carlo simulations (dashed line) are shown in the same figure. In both cases, the structural value of lambda, λ ) 2.74, was used in the calculations. Further, the numerical value of the term d ln / d ln T, which is used in eqs 5, 6, and 9, was -1.37 at 298 K and normal pressure.3 In contrast to the osmotic coefficient measurements shown in Figure 2, and in agreement with previous investigations of polystyrenesulfonic acid and its salts in water, the dependence of ∆HD on concentration reveals some ion-specific effects. The latter, however, seem to be less

Thermodynamic Characterization of HPAS

Figure 5. Enthalpies of dilution, ∆HD, from the concentration m f m ) 0.0044 mol/kg for the alkaline salts of polyanetholesulfonic acid. From top to bottom: results for the lithium salt (O), acid (]), sodium (4), and cesium salt (0). Theoretical curves obtained by the PoissonBoltzmann equation (full line) and the Monte Carlo simulation (dashed line) are shown on the same figure.

J. Phys. Chem. B, Vol. 111, No. 34, 2007 10135 other than the Coulomb interaction. These deviatons may be ascribed to solvent rearrangement in the process of dilution. As seen in Figure 6, the strongly hydrated ions such as H+ and Li+ show positive deviations from the “zero line”, while the loosely hydrated cesium ion shows a negative deviation. A recent computer simulation of short-chain polystyrenesulfonates in water indicates formation of the solvent shared ion-pair configuration.34 In other words, the strength of the sulfonatelithium ion interaction is not great enough for these ions to form a contact pair. This finding is consistent with the experimental results found in our present study. The experimental and theoretical results presented in this (see Figures 2-4) and many other studies suggest that counterions persist close to the polyion in very high concentration. It is plausible therefore to believe that they have to share the water molecule, in between, and also with the charged groups fixed on the polyions.34 There are at least two (opposing) processes taking place when a polyelectrolyte solution is diluted from one concentration to another. The first effect (i) is endothermic (heat is consumed), since the average distances between the ions and polyions are increased. The second effect (ii) is exothermic: since more water molecules are available to the counterions in the diluted state, they have a chance to become better solvated. The latter effect seems to take place for LiPAS and HPAS solutions, but not for the cesium salt (CsPAS), where the measured heats of dilution are considerably less exothermic than those for the other polyelectrolytes studied here. How can these effects be explained at the McMillan-Mayer level32 of theory? In order to calculate the excess internal energy (see eq 9), we need to evaluate the canonical average of the terms

∑i ∑j Figure 6. Deviations between experimental values and the Monte Carlo cell model calculations for enthalpies of dilution, ∆HD. The notation and parameters are the same as those for Figure 5.

expressed than in the case of polystyrenesulfonate salts in water. Note also that the ∆HD values of the alkaline salts of polyanetholesulfonic acid (and the acid itself) are more exothermic than the values of the corresponding polystyrenesulfonates. This is in qualitative agreement with the predictions of the electrostatic theories (see, for example, Manning’s limiting expression given by eq 5) and consistent with the osmotic pressure measurements. The initial slope of ∆HD for the cesium salt (denoted by open squares) of polyanetholesulfonic acid is smaller than that for the other salts or the acid. Similar behavior, moreover, has previously been observed for the cesium salt of polystyrenesulfonic acid, as demonstrated before;14,42 the enthalpies of dilution are strongly temperature-dependent for this salt and can even be positive at low enough temperature. We did not make any attempt to fit the experimental results empirically by taking different d ln a/d ln T values as a parameter in eq 6. The deviations between the experimental values of heats of dilution, ∆HD, and Monte Carlo cell model calculations, ∆HMC, are presented in Figure 6. The notation and parameters are the same as those for Figure 5. We can assume that the electrostatic effect on dilution is correctly represented by the ∆HMC Monte Carlo values. In other words, we supposed that we can ascribe all of the deviations of ∆HD - ∆HMC from zero to interactions

[

uij -

] ∑ ∑[

T ∂uij ∂T

)

i

j

uij -

T ∂uijcoul ∂T

-

]

T ∂uij* ∂T

(10)

where uij(r,P,T), given formally by eq 8, contains the Coulomb term (the second term in eq 6) and the short-range term uij*(r;P,T). This holds true also for the Poisson-Boltzmann result given by eq 6. For the temperature-independent shortrange potential, uij*(r) * f(T), the term in parentheses reduces to [1 + d ln /d ln T], as already explained in the theoretical section. This is the term associated with the rearrangement of water in the process of dilution, and it provides (d ln /d ln T ) -1.37) the correct sign of the enthalpy of dilution. For the temperature-dependent short-range potential, derivatives of the type ∂uij*/∂T have to be included in eq 10. Such terms can change the slope of the curves in Figure 5, and in some cases make the heats of dilution positive.14,20,21,42 The positive deviations shown in Figure 6 can be modeled to take into account stronger hydration simply by making the radius of the counterion larger.43 Negative deviations are more difficult to account for quantitatively,36 and the simplest way seems to be to invoke additional short-range attractions between the polyions and counterions, as already suggested.44 4. Conclusions The osmotic pressure and enthalpy of dilution of polyanetholesulfonic acid and its alkaline salts were measured in a broad concentration range. The results were compared with calculations based on the popular cylindrical cell model and Manning

10136 J. Phys. Chem. B, Vol. 111, No. 34, 2007 theory. The theoretical approaches used here differed, among other things, in different modeling of the polyion. In the Poisson-Boltzmann calculation, the polyions were modeled as uniformly charged cylinders. This theory provided a semiquantitatively correct concentration dependence of the osmotic coefficient. Similarly, good agreement between the PoissonBoltzmann cell model approach and experimental data has previously been obtained for polystyrenesulfonic acid and its alkaline salts. It has been shown3 that osmotic coefficients obtained by the Poisson-Boltzmann theory can be brought into close agreement with experiment by increasing the value of λ from its structural value to some effective value, λeff > λ. We found the same rule to be valid for the cesium salt, but not for the sodium salt of polyanetholesulfonic acid (see Figure 3). The results indicated that, while the linear charge density parameter, λ, is clearly an important quantity, the immediate environment of the charged group (in particular, its accessibility to counterions and the presence of hydrophobic groups) also plays a role. The influence of the polyion structure was also reflected in the measured enthalpies of dilution, which were more exothermic than those for the corresponding solutions of polystyrenesulfonic acid and its salts. The heats of dilution of polyanetholesulfonates followed semiquantitatively the concentration dependence predicted by the electrostatic theories applied. The deviations from continuum calculations were the strongest for Cs+ and can be attributed to the properties of this ion, known to be a structurebreaker. Possible corrections to the rodlike cell model were discussed in view of the short-range (non-Coulomb) interaction and their temperature dependence. The deviations from the cell model theory were not as strong as those for ionene (tetramethyl6,5-ionene) bromide and chloride solutions,18 where even the sign of the enthalpies of dilution was not predicted correctly by the electrostatic theories and continuum models. A study of the transport properties of polyanetholesulfonic acid and its alkaline salts will be presented in a future investigation to contribute toward better understanding of this system. Acknowledgment. This work was supported by the Slovenian Research Agency through the Research Program 01030201 and Project J1-6653. The authors wish to thank Dr. J. Dolenc for providing the data for Figure 1. Part of the paper was written while V.V. was a Visiting Professor at the Institute of Physical and Theoretical Chemistry at the University of Regensburg, Germany; the generous support of the German Research Foundation through the Mercator Programme is gratefully acknowledged. References and Notes (1) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Phillip, B.; Seidel, C.; Stscherbina, D. Polyelectrolytes, Formation, Characterization and Application; Hanser: Munich, Germany, 1994. (2) Katchalsky, A.; Alexandrowicz, Z.; Kedem, O. Polyelectrolyte solutions. In Chemical Physics of ionic solutions; Conway, B. E., Barrados, R. G., Eds.; John Wiley & Sons, Inc.: New York, 1966; pp 295-346. (3) Dolar, D. Thermodynamic properties of polyelectrolyte solutions. In Polyelectrolytes; Selegny, E., Mandel, M., Strauss U. P., Eds.; D. Reidel Publ. Co.: Dordrecht, Holland, 1974; pp 97-113.

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