The influence of nearest- and next-nearest-neighbor interactions on

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J. Phys. Chem. 1993,97, 5745-5151

5745

The Influence of Nearest- and Next-Nearest-Neighbor Interactions on the Potentiometric Titration of Linear Poly(ethylenimine) R. G. Smits,' G. J. M. Koper, and M. Mandel Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratory, Leiden University,

P.O.Box 9502, 2300 RA Leiden, The Netherlands Received: October 8, 1992; In Final Form: February 26, 1993

Potentiometric titrations of linear poly(ethy1enimine) were performed in the presence of, as well as in the absence of, added NaC1. In the NaCl concentration range of 0.1-1 .O M, the titration curves, with an inflexion point at 9, = O S , have a total pH change of about 6 units and a shape which was independent of the NaCl concentration. The potentiometric behavior of LPEI in an excess of added salt is explained by taking into account the nearest- and next-nearest-neighbor interactions between charges on the LPEI chain. The experimental titration curves of LPEI solutions in 0.1 M NaC1,O.S M NaCl, and 1.0 M NaCl could well be described by the set of potentiometric equations derived from a statistical mechanical treatment of thedissociation equilibrium, based on the Ising model with three parameters: the intrinsic dissociation constant KOand the excess energies ed and Et determined by the nearest and next-nearest-neighbor interactions, respectively. The intrinsic viscosity of LPEI in 0.2 M NaCl, 0.5 M NaC1, and 1.0 M NaCl showed a strong charge density dependence, which is consistent with the assumption of nearest- and next-nearest-neighbor interactions between the charges on the chain.

Introduction

Numerous studies of the potentiometric behavior of weak polyelectrolyte solutions have shown that the pH depends considerably on the electrostatic potential arising from the large number of charges on a single polyelectrolyte chain interacting with the small ions in solution. As a consequence, the apparent pK of the polyion is a function of the fraction, j3, of monomeric units of the polyelectrolyte which are actually charged. Addition of low molecular weight salt to the solution screensthe electrostatic potential around the polyelectrolytes and reduces the pK dependence on the charge of a single chain. However, some weak polyelectrolytes in solutions with salt concentrations of 0.1 M and higher still manifest a considerable change of pK in their potentiometric titration curves, which have an inflection point around j3 = 0.5. Such a behavior is also observed in the potentiometric titration of linear poly(ethy1enimine) (LPEI), (-CHzCHzNH-) N. Linear poly(ethy1enimine) was first synthesized by Saegusa et al.1 in 1972, and in 1984, a method for the synthesis of higher molar mass LPEI was published by Tanaka et al.* Although the monomeric unit of LPEI is identical to that of the much more familiar branched poly(ethy1enimine) (BPEI), the physical properties of both polyelectrolytes differ considerably. In the linear form, branching of the chain is completely absent, with no substituents on the flexible backbone of the chain. In aqueous solution, LPEI can be charged (e.g., upon addition of HCl) by the protonation of the secondary amino groups, the only type of amino groups present in this electrolyte:

-NH-+ H+ e -NH;Thus, all the charges on the LPEI chain are positioned directly on the backbone of the chain, which is not very common for a linear flexible polyelectrolyte. As a consequence, the maximum distancebetween the neighbor chargeson the LPEI is considerably smaller than for other polyelectrolytes on which the charges are usually situated on bulky substituent groups. This small maximum distance between the nearest-neighborcharges on a LPEI chain appears to us to be the probable cause of the non-fully-screened interactions revealed by the titration curves of LPEI when dissolved in an excess concentration of low molar mass salt. Also,

the branched form of poly(ethy1enimine) dissolved in an excess of low molar mass salt shows a peculiar titration behavior, which may also be attributed to nonscreened local interactions as suggested by Shepherd and Kit~hener.~ The interpretation of the BPEI titration curves is much more complicated because about 50%of thechargeablegroupsareprimaryor tertiaryamines. Several theoretical approaches have been proposed" I to explain these special potentiometric titration curves for linear polyelectrolytes. Most of them assume short-range interactions between nearby charges on the polyelectrolyte chain to be responsible for this behavior. The repulsive interaction between charged nearestneighbor monomers ("1) on the polyelectrolyte chain has been taken into account by Katchalsky et ala4They derived a set of two nonlinear equations, describing the potentiometric titration of a polybase with a fully screened electrostatic potential and NNI only. This model predicts an S-shaped titrationcurve which is completely symmetrical around j3 = 0.5 for all positive values of the interaction energy between nearest-neighbor charges. Experimental titration curves of poly(viny1amine) (PVA) were, however,asymmetricalwith respect to themidpointof totalcharge, and only the potentiometric data for /3 < 0.5 could satisfactorily be explained by this NNI model. In more recent theoretical work, the influence of short-range interactions, including eventually higher than nearest neighbors, has been investigated particularly for polyacids. Some of them were based on an Ising-type model: others on a mean-field approach.6 Also, Monte Carlo studies have been used.' Some of them take hydrogen bond formation5 and also the influence of trans and gauche conformations of the charged groupsll into account. Generally, these theories do not satisfactorily explain the experimental asymmetric titration curves. In this paper, we present an extension of the Ising model used for linear polyelectrolytes, including a next-nearest-neighbor interaction (NNNI). This leads to an analytically derived set of three nonlinear equations. They satisfactorily describe the potentiometric titration curves for aqueous solutions of linear poly(ethylenimine), which we have measured over the complete charging range in the presence of 0.1,0.5, and 1 M NaCl. Only three parameters are needed to fit the experimentaldata: (1) the intrinsic dissociation constant, KO,of the protonated amino group, (2) the excess energy resulting from the interaction between the

0022-3654/93/2097-5745~04.00~0 0 1993 American Chemical Society

Smits et al.

5146 The Journal of Physical Chemistry, Vol. 97, No. 21, 1993

nearest-neighbor charges, and (3) the excess energy resulting from the interaction between the next-nearest-neighborcharges on the polyelectrolyte. The same set of nonlinear equations also satisfactorily describes the potentiometricdata over the complete charging range of PVA solutions in the presence of excess salt, as measured by Katchalsky et aL4 The importance of NNI and NNNI, as evidenced by the potentiometric behavior of LPEI, is also manifested in the viscosimetric titrations of LPEI, for various concentrations of added NaCl. The intrinsic viscosity of LPEI in 0.2,0.5, and 1 M NaCl progressively increases above j3 = 0.5, in spite of the high salt concentrations. Some polyacids in aqueous solutions with an excess low molecular salt also have potentiometric titration curves with a considerable asymmetric pK change and an inflection point at half-charging. This is, e.g., the case for poly(ma1eic acid) and poly(fumaric acid).*,lz However, as will be discussed later, the experimental data cannot be explained by taking into account NNI and NNNI only. Apparently for the description of the potentiometriccurves of poly(fumaric acid) and poly(ma1eicacid), a more detailed model is needed.

Er for the triplet formed by three consecutive charges, and consequently, the fourth term of the free energy is proportional to the number of triplets t on the chain. It is in principle possible to extend the analysis to higher order contributions; the experimental data, however, do not seem to require this. The thus simplified expression for HNisthe starting point for the evaluation of the (semi)grand partition function in the Appendix. Accordingly, for the potentiometric titration of a polybase under conditions where the titration is dominated by short-range repulsions between nearest and next-nearest charged neighbors and assuming the following equilibrium for the amino groups -B- on the chain:

-BH+-e -B- + H+ the following set of nonlinear equations has been derived:

We consider the free energy of a single polyelectrolyte chain ...,S N ) , where the variable si = 1 of N chargeable sites, HN(s~,sz, if site i is charged and si = 0 if site i is not charged. This free energy can be expanded exactly as N

Here KO is the dissociation constant of the charged basic group and pK0 the negative decimal logarithm of KO,B = Z / N is the fraction of charged monomers, xd = d / N , the number of doublets per monomer (doublet fraction), andxt = t / N ,the triplet fraction. The three adjustable parameters in the model are pKo and both of the reduced excess energies, Q = 0.434EdlkTand et = 0.434Et/

N-l

N-2

being excess quantities with respect to I$@), #AI), ..., q5(n-l). Here and in the following, we shall neglect end effects, and in this approximation,the chain is translation invariant. This reflects itself in the energy with

@)

N- I

N

N-7

The first term in the free-energy expansion is the part that is independent of the charges on the chain. At the singlet level, the second term in the free-energy expansion, we only have contributions from the charged sites. The constant #O) is considered to be determined for a fully uncharged chain. Hence, the second term is equal to the number Z of charged sites times the singlet energy. This contribution is absorbed in the chemical potential. In the third term of the free-energy expansion, the interactions between two sites are taken into account. Considering only electrostatic interactions, the excess energy for two consecutive charges will be dominant. This is because the electrostatic interaction between consecutive sites of LPEI is mostly along the chain and hence not fully screened due to the high ionic strength solvent. The pair of two charged nearest-neighbor monomeric groups formed on the polyelectrolyte chain is called a doublet and is attributed a doublet energy Ed. Consequently, the third term is proportional to the number d of doublets on the chain. The last contribution that we consider, the fourth term in the free-energy expansion, takes the interactions between three sites into account. The configuration that is expected to dominate is where three consecutive monomer sites (i, j = i I, k = i 2 ) arecharged, because such a configurationmakes the chain locally stiffer. The energy involved is partly of configurational rather than of purely electrostatic origin. Hence, we attribute an energy

+

+

k T. Equations 4 and 5 can be solved for given values of Q and et to yield the doublet and triplet fractions as functionsof the degree of charge 8. For each couple of values €d and et, there are two solutions of which only one is physically significant. The other solution 8 = xd = xt is meaningful only for B = 0 and 8 = 1 and may be discarded. Thus, depending on the values of Q and tt chosen, for each 8, the corresponding values for xd and xt can be derived and the titration curve as represented by eq 3 evaluated. Instead of plotting pH us 8, we shall use the negative logarithm of the apparent dissociation constant, defined by

(6) to eliminatethe “trivial”8 dependenceresulting from the entropy of distribution of charged and uncharged groups along the chain in the absence of interactions. By setting tt = 0, which eliminates the next-nearest-neighbor interaction, eq 5 reduces to xt 3: xd2/8, which is the fraction of triplets for a random doublet distribution. With this value of Xd, the equation derived by Katchalsky et aL4 is recovered. The potentiometric equations can be further simplified by taking tt = t d = 0, which eliminates all short-range interactions between charges on the chain. This reduces eq 4 to xd = B2, which is the fraction of doublets for random charge distributions. Equation 3 then becomes equal to the potentiometric equation for monomeric bases, because the long-range electrostatic effects arising from the basic groups being fixed on a (charged) chain have been neglected. In Figure 1, the variations of xd and xt as a function of /3 are shown for various values of t d and et. For low charge densities (8 < 0.5), xd decreases with but remains relatively small. The doublet fraction sharply increases for B values larger than 0.5,

Linear Poly(ethy1enimine)

The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5141

0

G P

0.5

2

0.0 0.0

0.5

1.0

0.0

P

0.5

1.0

P Figure 2. Theoretical change of the pK with the degree of charge j3, according to eq 4. The parameter values used are ed = 1, for the highest four curves, €d = 3, for the lowest four curves, and c, of 0,0.25,0.50, and 1.0, as indicated in the figure.

0.5

0.0 0.0

0.5

1.0

P Figure 1. Dashed and dotted lines represent the doublet fraction x d and the triplet fractions xI, respectively, as a function the degree of charge j3, calculated from eqs 2 and 3. (a, top) The parameter values used for €d are 0, 1, 2, and 3, from the highest to the lowest x d , and x, curves, respectively, with c, = 0.25. The continuous straight lines are the asymptotes of xd and x,, respectively, corresponding to infinite td and e, values. (b, bottom) The parameter values used are Cd = 1 and the e, values 0, 0.25, 0.50, and 1.0 from the highest to the lowest x d and xI curves, respectively. The continuous straight lines are the asymptotes of xd and x,, respectively, corresponding to infinite Cd and e, values.

because from thereon almost every increase of the polyelectrolyte chain charge leads to two additional doublets. It asymptotically approaches the line 28 - 1 at high charge densities and crosses it at the value fl = 1. The triplet fraction is always smaller than the doublet fraction, because at least two doublets are needed to form one triplet. The triplet fraction is small for small values of fl, as is the case for the doublet fraction, and then strongly increases but only for values of fl > 2/s. From that charging value, almost every additional charge on the chain results in three more triplets, and xt then asymptoticallyapproaches the line 38 - 2. The approach to the asymptotic value and the sudden increase at fl = 2/3 are the more pronounced the larger the triplet excess energy. In Figure 2, the variation of pK - pK0 as a function of the degree of charge according to eq 6 is presented for ed equal to 1 and 3, respectively, and various values of et. For e, = 0, the potentiometric curve is symmetrical around the half-charging

point fl = 0.5. This is true for every positive Ed value chosen, as already observed by Katchalsky et aL4 The increase of ed values changesthe pKinterva1 over the total titration range and sharpens the pK drop around the point fl = I/2. For positive values of the triplet excess energy, the total pK interval of the potentiometric curve then further increases, but this additional increase is limited only to the side of high degrees of charge (6 > 0.5). As a result, the curves become asymmetrical with respect to the point of half total charge. The increase of this additional drop of the pH caused by the triplet energy is the largest around fl = 2/3. For sufficiently large triplet excess energies, the drop in pK at the half?chargingpoint, caused mainly by the doublet excess energy, is followed by a second one around fl = 2/3. The total change in pK over the complete charging range depends on the sum 2ed 39, because for a fully charged chain every monomer takes part in two doublets and three triplets. It is important to note that the triplet excess energy causes the potentiometric curves no longer to be symmetrical around the half-charging point. The influence of this positive triplet excess energy starts to become effective at about fl = 0.5 and increases upon further charging of the polyelectrolyte chain. As a consequence, the pK values for j3 C ' 1 2 are not influenced by the triplet excess energy and, hence, are completely determined by the parameters pK0 and Ed alone.

+

Experimental Section Syntheses of the PolymerSamples. Two batches of LPEIDHC1 have been synthesized by different methods. The high molar mass batch with M, = (42.0 f 2.0) X lo3g/mol(520 monomeric units) was prepared by polymerization of 2-phenyl-2-oxazoline following the method of Tanaka et aL2 The low molar mass batch with M, = (5.85 i 0.4) X lo3g/mol(74 monomericunits) was polymerized using 2-ethyl-2-oxazoline according to the method of Saegusa et al.1 For both batches, methyl trifluoromethanesulfonate was used as the initiator. The IH NMR spectra of the LPEI batches in chloroform show two peaks due to the methylene proton and the nitrogen proton, respectively, as expected for a linear chain of this composition. In the high molecular weight LPEI spectrum, a small third peak corresponding to protons of benzoyl residues in the polymer is also observed but to an amount of less than 2.5%. This was also found by Tanaka et a1.2

Smits et al.

5748 The Journal of Physical Chemistry, Vol. 97, No. 21. 1993

2 1 0.0

I

I

I

I

,

0.5

I

I

1 .o

P Figure 3. Potentiometric titration of LPEI (c = 0.022 mol/L, M, = 42 X lo3 g/mol) at various concentrations of added salt. (dvo) 0 M NaCl, (A)0.1 M NaCI, (0)0.5 M NaC1, and in (0)1 M NaC1. The continuous lines are the calculated curves using e q s 1-3. The parameter values used aretd = 2.08, t, = 0.46, and thepKovalues 9.1,8.9, and 8.65 for the added NaCI concentrations of 1, 0.5, and 0.1 M, respectively.

Mass spectrometry for both LPEInHCl batches gave the following composition: 31.3% C, 7.5% H, 15.7% N, 38.5% C, 7.91% 0. After the sample was dried for 24 h at 100 OC, practically all the water was removed, resulting in a composition close to the theoretical one: 32.9% C, 7.4% H, 17.3% N, 39.3% C1, and 0% 0. Molecular weights were determined in aqueous 1.0 M NaCl solutions by static low-angle laser light scattering (LALLS) with a Chromatix KMX-6 and a 4-mW He-Ne laser. The small scattering angle, less than 6 O , allows us to use unity a$ the value of the molecular structure factors. Prior to the measurements, the solutions were first filtered through a 0.22 p M gv Millipore filter. The refractive index increment, determined with a Chromatix KMX-16differential refractometer operating at the same wavelength as the LALLS, was found to be (0.20 f 0.01) mL/g* Concentration Determination. The LPEI-HCl concentration was adjusted by weight, taking into account 10 mass %water in the samples. The concentrationof LPEIOHCL solutions was also checked by potentiometric titration with NaOH at a solution temperature of at least 55 OC, because this is the minimal temperature at which the LPEI is soluble in water over the complete charging range. For the potentiometric titrations at very low ionic strength (without added salt), the base form of LPEI was used, prepared by neutralizing the HC1 salt with an excess of NaOH and then washing the precipitated LPEI over a glass filter until the pH of the filtrate was neutral and did not contain C1- ions any more (detection with AgNO3 solutions). This LPEI is, however, very hygroscopic and may contain a variable amount of water between 20% and 40%, thereby seriously interfering with the concentration adjustment by weight. Because uncharged LPEI is insoluble in water at temperatures below 55 OC, it was dissolved in an aqueous HCl solution of a known concentration such that at least 20% of the monomeric units are charged. The concentrations of these LPEI solutions without added salt were determined by titration with HCl, followed by back titration with NaOH. The latter was compared with the direct titration of the LPEI-HCl solution without added salt of supposedly the same concentration. The results of both titrations were in excellent agreement. The

accuracy of this method is about 5%, because at the equivalence point only a small pH jump is observed, even in a 2 M NaCl solution. Potentiometric Titration and Viscosity Measurements. Potentiometric titrations were performed with rather concentrated 2.0 M HCl or fresh 2.0 M NaOH solutions to minimize the dilution effect. A Radiometer Vit 90 video titrator, with a Radiometer GK240IC combined glass/reference electrode, was used. Every titration curve consists of at least 120 points up to 220 points, with a minimal delay between titrant additions of 1 min. The electrode was calibrated with buffer solutions of pH 4 and 7 (Titrisol, Merck). We used both molar masses of LPEI in solutions of various polyelectrolyte concentrationsand at several salt concentrations. The titration curves did not significantly depend on the molar mass nor on the polyelectrolyte concentration for concentrations higher than 0.01 mol/L LPEI. There were no noticeable effects of the temperature in the range between 25 and 60 OC, besides the increasing solubility with increasing temperatures at charge fractions below 20%. Variation of the monovalent counterions, changing to a 0.1 M NaI or a 0.1 M NaF solution instead of 0.1 M NaCl, had no influence on the shape of the titration curves, except for the buffering at the low pH values by NaF, which is the salt of a weak acid. The same conclusion for other counterions was already reached by Weyts and Goetha1s.l' 3! , is calculated using

8=

c~ - c f f +

+

CP

with c, being the concentration of added HCL in the solution in mol/L, cp being the concentration of LPEI in the solution in mol/L, and CH+ and COW being the concentration of free H+and free OH-ions in mol/L as determined by the p H of the solution. For the viscosity measurements,capillary viscosimetersof type KPG-Ubbelohde were used. No shear rate dependence of the viscosity of the LPEI solutions was observed in the shear rate rangebetween 1.2X 1Vand2.0 X 1 V s - * . Theintrinsicviscosity was obtained by extrapolating the reduced viscosity to zero LPEI-HCl concentration. The LPEbHCl concentrations were varied between 0.5 and 6 g/L, with a varying NaClconcentration from 0.2 to 1.0 M. About 10 different polymer concentrations were used for each extrapolation to determine the intrinsic viscosity.

Results and Discussion Thepotentiometric titration of LPEI in aqueous solutions with and without added sodium chloride was investigated. The monomeric concentration of the LPEI solution used was 0.022 M, and the final concentrations of added NaCl were 0,0.1,0.5, and 1 M. The attainable charge density of a solution of LPEI is limited to the range between 8 = 0.15 and 0.9 at the temperature of 25 OC. Thelowerlimitofthechargingrangeis theconsequence. of the minimum charge density which is needed to dissolve the LPEI in water of 25 OC, and the upper limit is determined by the large autodissociationof LPEI at high charge densities. Some experimental results of the potentiometrictitration of LPEI with various concentrationsof NaCl added are shown in Figures 3 and 4. As is usual for polyelectrolyte solutions, the pK of the LPEI solution is clearly influenced by the low molecular weight salt concentration added, due to the screening of the electrostatic potential around the LPEI chain by the salt. For a polybase such as LPEI, the screening of the electrostatic potential results in a lesser repulsion of the protons in the solution from the positively charged chains, which causes the pK of the solution to shift to higher values when increasing the NaCl concentration. For the LPEI solutions in the presence of excess salt concentrations of 0.1, 0.5, and 1 M NaCI, however, still a large pK dependence on

Linear Poly(ethy1enimine)

The Journal of Physical Chemistry, Vol. 97,No. 21, 1993 5749

9

$6

3

0.0

0.5

1.0

P Figure 4. pK dependence on the degree of charge of LPEI (c = 0.022 mol/L) at various concentrations of added salt. ( 0 ) 0 M NaCl, (A)0.1 M NaC1, (0)0.5 M NaCl, and in (0)1 M NaC1. The continuous lines

are the calculated curves using eqs 2 4 . The parameter values used are Cd = 2.08, c, = 0.46, and the pK0 values 9.1, 8.9, and 8.65 for the added NaCl concentrations of 1, 0.5, and 0.1 M, respectively.

3

0.0

0.5

1.0

B Figure 5. pKdependenceon the degree of charge of LPEI (0)(c = 0.022 mol/L) in 0.5 M NaCl. First the data are fitted for the charging range 0.15 C @ < with the parameter values pK0 = 8.9, Cd = 2.08, and cy = 0. The total @ range is fitted by taking cy = 0.46.

the degree of charge of about 6 units is observed. The shape of the potentiometric curves of LPEI is not changed significantly in the concentration range of added salt between 0.1 and 1 M NaCl, but it is only lifted some tenths of a pKunit when increasing the NaCl concentration from 0.1 to 1 M. The total pKvariation of about 6 units in the potentiometric titration of LPEI over the complete explored charging range is somewhat larger for systems with NaCl than for the casewithout added salt, but this difference is rather small taking into account the large variation in ionic strength. This is in contrast to what has been observed with many other linear flexible polyelectrolytes, such as, e.g., poly(acrylic acid).*J4 The shapes of the potentiometric curves of LPEI with and without added NaCl, however, clearly differ. All curves show an inflectionpoint aroundB = 0.5,but for the former, the slope of the pK curve around the half-charging point is much larger, corresponding to a significant drop in the pK values. The

pK drop around = ‘ 1 2 in solutions of high ionic strength can possibly be explained by nonshielded repulsive local interactions between charges on the LPEI chain. The potentiometric data measured for all added salt concentrations are definitely not symmetrical around /3 = 0.5. These curves show a much larger decrease in pKvalues at the side corresponding to higher charge fractions. The experimental titration results of LPEI in solutions with salt concentrations of 0.1,0.5, and 1 M NaCl may be compared to the theoretical predictions calculated with the use of eqs 3-6. To this end, the titration curve in 0.5M NaCl in a first stage is fitted to the theoretical expression assuming the excess triplet energy value C, = 0. (This as well as all the other fits has been performed by searching for a titration curve generated by eqs 3-6 which best corresponds to the experimental one.) Without a triplet energy and adjusting the values of pKo and the excess doublet energy Ed, the potentiometric titration of LPEI in 0.5 M NaCl can only be satisfactorily fitted in the range of low charge fractions /3 C 0.48,as is shown in Figure 5. The value of pKo is found to be 8.9,which is about 1 pK unit lower compared to the pK0 of the low molecular weight compound analogue of the monomeric unit, but this is not unreasonable. The value of t d is found to be 2.08. It is difficult to assess this value of td, because a prediction for the excess doublet and triplet energies is hardly possible. Using a small reduced triplet energy e, = 0.46 and the full set of equations (3)-(6), we find a satisfactory fit for the potentiometric titration of the LPEI solution in 0.5 M NaCl over the completecharging range measured (Figure 5 ) . It should be noted that the triplet energy is about 4 times smaller then the doublet energy. As a consequence, the quartet energy is expected to be at least another 4 times smaller than the triplet energy so that no further extension of the model, including higher multiplets, is needed for describing the potentiometric data of LPEI in the presence of excess NaCl. In Figures 3 and 4,the fits for the potentiometric titration of LPEI with salt concentrations of 0.1, 0.5,and 1 M NaCl have been represented as continuous lines, all obtained in an analogous way. The values of Cd and et are not significantly dependent on the salt concentration in the NaCl range explored and may all be taken to be identical to the parameter values found for the potentiometric curve in 0.5 M NaC1. The pK0 values, on the contrary, had to be varied for the various NaCl concentrations. The best fits have been obtained with values of pK0 of 8.65 and 9.1 for the 0.1 and 1 M NaCl solutions, respectively. To find further support for the theory presented here, the same model has been applied to the potentiometric titration curve of PVA in 2.0 M NaCl taken from the literature: which is also asymmetric and has an inflection point around g = 0.5. The data points from the literature for the potentiometric titration of PVA are shown in Figure 6together with the theoreticalcurve calculated with eqs 3-6. The low charged sides (0 C 0.45) of the potentiometric titrations have been fitted without taking into account a triplet excess energy, as done for the LPEI curves. Then a triplet energy has been introduced to fit the high charge side of the measured curve as well. The potentiometric titration of PVA in 2.0 M NaCl is reasonably well described by eqs 3-6using pKo = 9.4,ed = 1, and e, = 0.46. It is not surprising that the doublet excess energy value for PVA should be smaller than the value for LPEI, because the distance between neighboring charges on the PVA chains is increased by a factor of 1.5. Thevalueof the triplet excess energy for PVA does not appear to be smaller than for LPEI. But this could be caused by an uncertainty of about 15% in the bd and et values of PVA, which is that large because the total pK change is rather small and because of the small number of data points measured.

Smits et al.

5750 The Journal of Physical Chemistry, Vol. 97, No. 21, 1993

io 2.8 1

&

b P

2.2

I I /

~

-

-

2

c'

U

1.6 -

1.01

5

0.0

I

!

I

I

0.5

1,o

B Figure 6. pK dependence on the degree of charge of poly(viny1amine) (0)( c = 2.67e-3 M) in 2.0 M NaCl from ref 4 fitted with pK0 = 9.4, fd = 1, and t, = 0.46.

The potentiometric titration curves of some bicarboxylic polyacids such as poly(fumaric acid) (PFA) in 0.1 M NaCl show a big increase in the pK values over the complete titration range with an asymmetry of the pK around j3 = 0.5, even larger than for LPEI under the same conditions.* This occurs in spite of the fact that the distance between neighboring charges is larger for PFA than for LPEI or PVA, because for PFA they are situated on bulky substituent groups. It is possible to fit the data points of the potentiometric titration of PFA to eqs 3-6, albeit with a relatively large reduced triplet excess energy value, approximately the same value as that of the reduced doublet excess energy. This would point to the necessity of taking into account higher multiplets as well, but we cannot think of any physical argument to justify this. Probably a more detailed analysis is needed to interpret the titration data of such polyacids. Some attempts have been made in this direction, e.g., by taking into account the entropic effects of the carboxylic substituent groups, which can take a trans or a gauche conformationll and will be a function of the charge density on the chain. Such trans and gauche conformations of bulky substituent groups can obviouslynot play a part in the case of LPEI. Hydrogen bond formation, also proposed for some polyacids, however, may have some influence on the potentiometric behavior of LPEI, but it is probably incorporated into the excess doublet and triplet energy values. The potentiometrictitration curve of LPEI without added NaCl indicates an increased electrostatic potential around the polyelectrolyte chain because of lesser screening due to the absence of added salt. This is also commonly observed in the titration of polyelectrolytes. The quantitative analysis of the titration curves of LPEI without salt, however, appears to be more complicated than, e.g., those of poly(acry1ic acids)l4 because the titration curve still exhibits an inflection point around j3 = 0.5. Moreover, the pK change over the complete charging range is almost equal to that of the curves in the presence of an excess of salt, another difficulty which complicates the interpretation of the salt-free potentiometric titration curve of LPEI. Although at low values of j3 the pK decreases faster with j3 in the case of LPEI solutions without added salt than for the solutions with excess salt concentrations, above j3 = 0.5 the pH difference between the potentiometrictitration curves of both types of solutionsdecreases, leaving only a very small pH difference at j3 = 0.9, independent of the NaCl concentration. It seems that at high charge fractions the short-range NNI and NNNI interactions in LPEI dominate the global electrostatic potential effect, even when no NaCl is added. A possible explanation could be the strong autodissociation

,

0.5

0.0

1

I

I

I

1.0

B Figure 7. Intrinsic viscosity of LPEI (Mw = 5.85 X lo3 g/mol) on the degree of charge in (A) 0.1 M NaCI, (0)0.5 M NaCI, and (0)1 M NaCI.

of LPEI at j3 = 0.9. This autodissociation, as evidenced by the low pH value, implies that the ionic strength of the solution is determined by the concentration of the dissociated free HCl as well. Thus, in the absence of added salt, the ionic strength of a LPEI solution at j3 = 0.9 would correspond to an effective salt concentration of 3 mM at least. This salt concentration is apparently sufficiently high for allowing interactions between neighborsand next-nearest neighborstodominate the effect arising from the overall electrostatic potential around the charged chain. Further evidence for the importance of nonscreened NNI and NNNI of LPEI at high ionic strength comes from viscosity measurements at various polymer concentrations, the charge fractions, and concentration of added salt. A definitedependence of the intrinsic viscosity [ q ] on the degree of charge j3 is observed in 0.2,0.5, and 1 M NaCl solutions, as shown in Figure 7. The intrinsic viscositydoubles its value going from j3 = 0.5 to 0.9. This behavior of the intrinsic viscosity on the degree of charge clearly deviates from the behavior of, e.g., poly(acry1ic acid) in NaCl solutions14or of the branched form of PEI in 1 M KCl,15 for which no j3 dependence of [ q ] at all has been found. As shown in Figure 7, at all salt concentrations, the intrinsic viscosity of LPEI increases particularly rapidly with the degree of the charge above approximately j3 = 2/3. This corresponds to the value at which the doublet and the triplet fraction start to increase and supports the assumption that NNI and NNNI are responsible for the viscosity effects observed by their attribution to the local stiffening of the chain. The fact that [ q ] at constant j3 decreases with increasing salt concentrations is not necessarily contradicting the suggestion that these weakly screened, shortrange interactions dominate the change in the intrinsic viscosity. The effect of increasing salt concentrations on [ q ] may probably be ascribed to the decreasing solvent quality. Acknowledgment. We are gratefully indebted to Prof. Goethals and co-workers at the University of Gent, Belgium for assistance with the syntheses of the LPEI. We also thank P. Brain for doing the viscosity measurements and M. Kuil for stimulating discussions. Appendix

In this section, the potentiometricequations (3)-(5) are derived from the (semi)grand partition function for a single linear polyelectrolyte consisting of N ionizable groups N 2-1 d-l

The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5751

Linear Poly(ethy1enimine)

where Z is the number of charged groups and d the number of doublets and t the number of triplets on the polyelectrolytechain. Ed and El are the excess energies for the formation of doublets and triplets, respectively, and E, is a conformation-dependent energy. p, is the chemical potential of the protons and is equal to the difference between the chemical potential of the charged groups and the uncharged groups on the polyelectrolyte chain. qo and q+ are the single particle partition functions for uncharged and charged groups, respectively. The factor gN(Z,d,t) is the number of configurations of a polymer with N ionizable groups of which Z are charged such that there are d doublets and t triplets. In order to calculate the combinatorial factor gdZ,d,t), one considers the polymer as consecutive sequences of charged and uncharged groups. First consider the sequences of the charged groups, and let n k be the number of sequences consisting of k charged groups (k = 1 , 2 , ... and n k = 0, 1 , 2, ...). The number of charges can then be expressed as m

Z =pkn, =I

The number of sequences, to be called K , can conveniently be expressed in Z and the number of doublets d, m

Then consider the sequences of uncharged groups. The total number of uncharged groups is N - Z . There are three possibilities, depending on the end groups of the polymer. These ends can be both uncharged, one can be charged and the other not, and both ends can be charged. Since the number of configurations of m groups subdivided in n sequences is given by (-:); the total number of configurations, summed over the above-mentioned end conditions is

(N - ZK- 1 ) = (N - ZK + 1 ) ( A * 1 0 ) Hence, the total number of configurations is

Rather than evaluating the grand partition function, we use the maximum term method." The derivative of the combinatorial factor is calculated by using Stirling's approximation for the factorial

- -&(tr

(A.12) = N! In N The first equation, eq 3, originates from the derivative of In X to Z and leads to the expression

-&p

0

-In a, =

Po - p +

because each k sequence (k L 2) has k - 1 doublets. Likewise, the number of triplets is fixed by Z , K , and the number of nl of monosequences, m

t

(k- 2)nk = z - 2K + " 1

=

(A.5)

k=3

because each k sequence ( k 1 3 ) has k - 2 triplets. The number of configurations of K sequences of total length Z and with given number nl of monosequences is given by m

m

U

This expression can be further evaluated by introducing a contour integral representation for the Kronecker delta

which is a special case of Cauchy's theorem for contour integrals. At all times, the contour is assumed to encompass all poles of the integrand. After introduction of this expression, one can perform the summations, and one finds

(A.8) The integrals can be evaluated using the general form of the above-mentioned theorem, and one finds

It is understood that nonexistent binomial coefficients are to be replaced by 1.

0

0

+ PP

kT

+

(1 - 28 - x d ) 2 (1 - 8 ) ( 8 - 2 x d

+ x,) -

(y) d,f

(A.13)

where we have taken the thermodynamic limit of large N and where we have set /3 = Z / N , x d = d / N , and x , = t / N . By relating the proton activity a, to the pH and the term with the standard chemical potentials to the pK0, one obtains eq 3, provided that the charge dependence of the conformational energy E,, not included implicitly in the doublet and the triplet excess energies, is neglected as well as thedifference between the mean electrostatic potentialon thesurfaceof thechain and in the bulkof the solution. This seems to be reasonable in the presence of a large excess of low molecular weight salt. The other two equations are obtained likewise.

References and Notes (1) Seagusa, T.; Ikeda, H.; Fujii, H. Macromolecules 1972, 5, 108. (2) Tanaka, R.; Ueoka, I.; Takaki, Y.; Kataoka. K.; Saito, S.Macromolecules 1983, 16,849. (3) Shepherd, E. J.; Kitchener, J. A. J . Chem. SOC.1956, 2448. (4) Katchalsky, A.; Mazur, J.; Spitnik, P. J . Polym. Sci. 1957,23,513. ( 5 ) Nishio, T. Biophys. Chem. 1991, 40, 19. (6) Kawaguchi, S.;Kitano, T.; Ito, K. Macromolecules 1990, 23, 73 1 . (7) Reed, C. E.; Reed, W. F. J. Chem. Phys. 1992, 96, 1609. (8) Kitano, T.; Kawaguchi, S.;Ito, K. Macromolecules 1987,20, 1598. (9) Schultz, A. W.; Straws, U. P. J . Phys. Chem. 1972, 76,1767. (10) Kawabe, H.; Yanagita, M. Bull. Chem. SOC.Jpn. 1970,43, 2706. (1 1) Kawaguchi, S.;Nishikawa, Y .; Kitano, T.; Ito, K. Macromolecules 1990, 23, 2710.

(12) Barone, G.; Rizzo, E. Gaz. Chim. It. 1973, 103, 401. (13) Weyts, K. F.; Goethals, E. J. Macromol. Chem., Rapid Commun. 1989, 10,299. (14) Mandel, M. Encycl. Polym. Sci. Eng. 1988, 11, 739. (15) Kobayashi, S.;Hiroishi, K.; Tokuwoh, M.; Seagusa, T. Macromolecules 1987, 20, 1496. (16) Tandford, T. Physical Chemistry of Macromolecules; John Wiley & Sons: New York, 1961; p 391. (17) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986.