J. Phys. Chem. B 2008, 112, 10123–10129
10123
The pH Inside a Swollen Polyelectrolyte Gel: Poly(N-Vinylimidazole) Arturo Horta,† M. Jesu´s Molina,† M. Rosa Go´mez-Anto´n,‡ and Ine´s F. Pie´rola*,† Departamento de Ciencias y Te´cnicas Fisicoquı´micas, Facultad de Ciencias, and Departamento de Quı´mica Aplicada a la Ingenierı´a, ETSI Industriales, UniVersidad a Distancia (UNED), 28040 Madrid, Spain ReceiVed: February 7, 2008; ReVised Manuscript ReceiVed: May 13, 2008
The pH inside a dissolved polyelectrolyte coil or a swollen ionic polymer network is not accessible to direct measurement. It is here calculated through a simple model, based on Donnan equilibrium, counterion condensation (for charge density exceeding the critical value), and balance of mobile ions, without any assumption on the pKa of the ionizable groups. The data needed for the calculation with this model are polymer concentration, pH value in the initial solution, and pH value in the bath at equilibrium. All three can be determined experimentally by a batch method where the polymer is immersed in a different pot for each starting pH. The model is applied to a sample system, namely, chemically cross-linked poly(N-vinylimidazole) immersed in acidic baths of different pH values. The imidazole units are basic and become protonated by the acid, thus changing the pH of the initial bath. The model shows how the pH developed inside the swollen gel is several units higher than the pH of the bath at equilibrium, both with or without the correction for counterion condensation. Consequently, when the pKa of the polyelectrolyte is determined in the usual way (with the pH measured in the external bath), it gives an apparent value that is several units below the pKa determined from the actual pH inside the swollen gel at equilibrium. The inclusion of the counterion condensation decreases very slightly the polymer basicity. Surface effects and intramolecular association between protonated and unprotonated imidazole rings are discussed to explain the pKa behavior in the limit of low degree of ionization. Introduction Polyelectrolytes are polymers having charged groups in the macromolecular chain. This macromolecular charge is compensated by counterions, which are essentially mobile species that pervade the volume occupied both by solvent and polyelectrolytes. The number of charged groups on the macromolecule depends on the type of group that can be ionized and on the thermodynamic conditions. For example, if the groups are acidic (or basic), the number of charged groups depends on the proton activity (pH) of the medium. This is the case of chemically cross-linked (N-vinylimidazole), a pH-sensitive hydrogel.1 As such, it is neutral, but the imidazole rings are proton-accepting (basic) groups, which follow the protonation equilibrium:2–4
P + H+) PH+
(1)
This equilibrium is more or less displaced to the protonated species depending on pH. The imidazole molecule anchored to small size compounds has an equilibrium constant for the acid dissociation5 (reverse of eq 1), pKa, close to 7 (7.3 for N-ethylimidazole at 25 °C in salt free aqueous solution).5 This means that an acid medium is needed to substantially protonate the imidazole ring. When the ring is part of the monomer unit in a macromolecule, two differentiating features appear: its basicity may be modified with respect to that of the free molecule, and Ka is not a real thermodynamic constant, because its value depends on the polymer degree of ionization. There are many examples in the literature reporting that the acidity (or basicity) of a group * Corresponding author. Phone: 34-91-3987376. Fax: 34-91-3987390. E-mail:
[email protected]. † Departamento de Ciencias y Te ´ cnicas Fisicoquı´micas. ‡ Departamento de Quı´mica Aplicada a la Ingenierı´a.
in the monomer unit of a polymer chain is diminished with respect to that of the same group in a free molecule.4,6–15 This conclusion is reached because the pKa determined in the polyelectrolyte is different from the pKa determined in the free molecule. The pKa is ideally the pH at half-ionization, namely, when the ionization degree, R, is R ) 0.5. However, the interaction between charged groups in the chain makes an additional contribution to free energy which varies with ionization degree.11 Therefore, pKa changes with R, and an intrinsic value, pKo, is usually determined by extrapolating pKa to the limit of vanishing interaction between charges (R f 0).4 The comparison of pKa between polyelectrolyte and free molecule poses a difficulty not yet adequately solved. The ionization degree is varied by titration (with an acid or base). This titration is followed by determining the pH of the solution. In order to avoid complications from intermolecular interactions between charges of different molecules, the solution is kept dilute. But a dilute solution of polyelectrolyte is not a uniform system (in contrast to dilute solutions of simple electrolytes). In dilute polyelectrolytes, the macromolecular coils are nonoverlapping, such that the local concentration of ionizable groups is high inside the coil volume and nihil in-between coils. The activity of any electrolyte that is both inside and outside the coil has to be the same on both sides, but the activity of the individual ions is not necessarily so. The ionized groups of the monomer units are present only inside the volume of the coils, and hence, the mobile ions are distributed unevenly between the volume inside the coils and the volume in-between coils. The result is that the pH inside the coil may be different from the pH outside the coil. To determine the true pKa of the polyelectrolyte, it would be required to measure the pH inside the coils, something that, up to now, is not feasible by any experimental technique available. The difference in ion concentrations between the inside and the
10.1021/jp801145g CCC: $40.75 2008 American Chemical Society Published on Web 07/26/2008
10124 J. Phys. Chem. B, Vol. 112, No. 33, 2008
Horta et al.
outside of macromolecular coils has to be small, because the coil, in itself, is a dilute system. Thus, typical concentrations for coil overlap are 10-2 g/mL, when the chains are random coils, but much less when they are polyelectrolyte chains expanded by the osmotic action of the free ions.16 It is possible to raise the concentration of ionized groups (monomer units) above this overlap value, while still having a nonuniform system (an inside with polymer and an outside without polymer). This is achieved by cross-linking the polymer chains to form a network. In the network swollen with solvent, the ionizable groups can be at concentrations above that corresponding to overlap of the independent chains. In the resulting gel, which now is a macroscopic entity, we can distinguish between the inside, containing the ionizable groups of the polymer (the swollen gel), and the outside, containing only mobile ions (swelling solution). With these larger concentrations of fixed ionized groups, the differences between mobile ion concentrations inside and outside of the macromolecular domain can be greatly enhanced. The problem mentioned above, that of determining the true pKa of the polyelectrolyte through titration, but monitoring the pH of the external solution, could then be serious. That the true pH inside the gel should be used has been repeatedly recognized, but very few attempts to calculate such pH can be found, and all of them use some drastic assumptions.17–28 This pH inside the gel is again not amenable to experimental determination with current techniques, but it can be calculated by taking into account the distribution of individual ions that satisfy Donnan equilibrium, phase electro-neutrality, and balance of species across the boundary. This is the task we pursue here by using experimental pH data for poly(N-vinylimidazole) (PVI) hydrogels, under different conditions of a strong acid being added to the external solution. PVI is a very interesting case to study basicity and to attempt to determine its true pKa, because it is useful as a buffer that can neutralize large quantities of strong acids.29 Also, changing the pH has a large effect on the swelling capacity of this gel.2,3 Therefore, PVI is attractive as well, because it enters into the class of the much studied pHsensitive gels.
redistribution of ions to recuperate the protonation and Donnan equilibria. If the ionic strength is larger than the concentration of fixed charges in the gel phase, the Donnan equilibrium does not modify significantly the distribution of ions, which have about the same concentration in the two phases. Definitions of Concentrations. A xerogel sample PVI (of mass m) is immersed in a much larger volume (VT) of an aqueous solution of a protonating acid HA. Each monomer in the polymer gel carries a ring that can be protonated. The number of proton-accepting units (or monomers) in the gel is m/M0 (M0 being the molecular weight of a monomer unit). We define an effective molar concentration of proton accepting units in the system as
Theoretical Basis
and the volume ratio of the surrounding bath to the swollen gel is
The immersion of a PVI sample in an aqueous solution of a strong acid (HA) produces, simultaneously to swelling, a rise of the solution pH from its initial value. The process can be visualized in the following sequence. Water and HA diffuse from the swelling bath into the gel. The imidazole groups become protonated by the protons of HA (eq 1). That creates (temporarily) a difference of free proton concentration between the swelling solution and the gel inside, without creating any difference in the concentration of counteranions. The deficit of free proton concentration inside the gel forces additional diffusion of HA into it. That creates an excess concentration of mobile counteranions in the gel, with respect to the external bath. The development of this excess counteranion concentration opposes the diffusion of additional HA acid into the gel and hence opposes further protonation of the gel. Excess HA diffusion and proton uptake by the gel continue until Donnan equilibrium is reached between the inside and the outside of the gel phase (the product of free protons and counteranions activities in the two phases are equal), and the protonation equilibrium inside the gel is also achieved. As a consequence of Donnan equilibrium in salt free baths, the pH inside the gel is higher than that outside. Above a certain degree of protonation, corresponding to the critical charge density, counteranions condensate on protonated rings,2,3 and that induces a new
C)
m M0VT
(2)
This C is a formal concentration, not a real one. (It is the concentration that the polyelectrolyte would have in a solution if it were soluble instead of being a cross-linked network). A given xerogel can have different values of the effective concentration, C, simply by putting different masses (m) in the same volume VT. Upon swelling of the xerogel, the initial aqueous solution is divided into the swollen gel phase and the remaining bath solution surrounding the gel. We call VG the volume of the swollen hydrogel and VS the volume of the surrounding bath. Inside the hydrogel phase, the volume of liquid is (1 - V2)VG, and the volume of polymer network is V2VG, where V2 represents the polymer volume fraction in the swollen gel. At a given degree of swelling, the molar concentration of proton-accepting units in the gel phase is
CG )
V2F2 m ) M0VG M0
(3)
where F2 (in g/L) is the xerogel density. (CG in a coil can be calculated with the intrinsic viscosity, [η] in dL/g, as 10/M0[η]). The volume ratio of the initial solution to the swollen gel is
VT CG ) VG C
(
(1 - V2)C VS CG ) 1VG C CG
(4)
)
(5)
Protonation Equilibrium. The protonation of the imidazole rings in the monomer units of the gel follows eq 1. The equilibrium constant for this protonation is the reciprocal of Ka, the equilibrium constant for the dissociation of the acid PH+:
Ka )
[P][H+]G [PH+]
(6)
All three concentrations, [P], [PH+], and [H+], refer to the gel phase, but only the protons need a subindex to specify in which phase they are. For simplicity, concentrations and activities are indistinctly employed in eq 6 and in the following. This is a standard use. The degree of protonation, R, is
R)
[PH+] [PH+] ) CG [P] + [PH+]
(7)
Therefore, the concentration of fixed protons, or charges on the polymer network, referred to the volume of the gel phase, is [PH+] ) RCG, and the concentration of free protons inside the gel is:
pH Inside a Gel
J. Phys. Chem. B, Vol. 112, No. 33, 2008 10125
[H+]G ) Ka
R 1-R
(8)
In order to calculate the pH inside the gel, the protonation degree, and the ionization constant, we have five variables that correspond to the concentrations of ionic species inside the hydrogel and in the external bath, namely, [PH+], [H+]G, [A-]G (inside the gel), and [H+]S and [A-]S (in the external bath). Several conditions must be met by these variables: Donnan equilibrium, mass balance, and electro-neutrality. Mass Balance. The balance of counteranions that enter into the gel is
[A-]iVT - [A-]SVs ) [A-]GVG
(9)
or, by using molar concentrations instead of volumes (eqs 4 and 5),
(
[
[A-]G ) [A-]i - [A-]S 1 -
)]
(1 - V2)C CG CG C
(10)
Charge Balance. The condition of electro-neutrality holds in all phases. For the initial solution, for the hydrogel, and for the surrounding bath, it is
[A-]i ) [H+]i′
(11)
[A-]G ) [H+]G′ + [PH+]
(12)
[A-]S ) [H+]S′
(13)
where the primed quantities are defined as
[H+]′x ) [H+]x - Kw ⁄ [H+]x
(14)
Here, Kw/[H+]x is the concentration of hydroxyls in phase x, with x being either i, S, or G. These primed [H+]x′ are not actual proton concentrations but the difference between the concentration of protons and hydroxyls in each phase. When the solution is acid, with pH below 6, the concentration of hydroxyls is negligible compared to that of protons, and the primed [H+]x′ equals the unprimed [H+]x, so that eqs 11–13 can be written without primes. But when the solution approaches neutrality and the concentration of hydroxyls becomes comparable to that of protons, the primed [H+]x′ is noticeably smaller than the unprimed [H+]x, and eqs 11–13 should to be used as written above. Donnan Equilibrium. The electrolyte HA has the same chemical potential, or activity, in the gel phase and in the external solution; hence,
[H+]G[A-]G ) [H+]S[A-]S
[H+]G([H+]G′ +[PH+]) ) [H+]S[H+]S′ In eq 12, we substitute counteranions (eq 10)
[H+]G′ + [PH+] )
[
[A-]G
(16)
with the mass balance of
(
)]
(1 - V2)C CG [A ]i - [A ]S 1 CG C -
-
(17) and the conditions of electro-neutrality (eqs 11 and 13)
)]
(1 - V2)C CG CG C (18)
When substituted in eq 16, this yields for the free proton concentration inside the gel, [H+]G
[H+]G )
[H+]S[H+]S′ (1 - V2)C CG [H+]i′ - [H+]S′ 1 CG C
(
)
(19)
The experimental data needed to calculate the pH inside the gel are, therefore, the following: from the xerogel, its mass (m) and density (F2) (eqs 2 and 3); from the initial solution in which the xerogel is immersed, its pH (pHi) and volume (VT) (eq 2); from the swollen hydrogel, its degree of swelling (V2) (eq 3); from the external solution in contact with the gel, its pH (pHS). This calculation of proton concentration inside the gel is not feasible when the polyelectrolyte is studied in the usual way, by continuous titration, because then, the only proton concentration being recorded is that of the external bath ([H+]S), whereas the initial concentration ([H+]i) is not determined separately. The protocol that we follow here is a batch or discontinuous way (detailed in next section), where both concentrations, ([H+]S) and [H+]i), are determined separately for the same solution, and thus, all data needed in eq 19 are available. Degree of Protonation. The same experimental data serve to calculate also the fraction of total imidazole rings that are protonated in the polyelectrolyte gel. Thus, the concentration of fixed charges in the polyelectrolyte, [PH+], can be obtained from eq 17
[
(
[PH+] ) [H+]i′ - [H+]S′ 1 -
)]
(1 - V2)C CG - [H+[H+]G′ CG C (20)
with [H+]G′ given by eqs 14 and 19. The degree of protonation, R, is then obtained by using this [PH+] in eq 7:
R)
[
(
[H+]i′ - [H+]S′ 1 -
)]
(1 - V2)C CG - [H+]G′ CG C (21) CG
pKa of the Polyelectrolyte. The pKa is obtained from this R by rewriting eq 8 in the form of a Henderson-Hasselbalch expression:
(15)
(diprotic acids would have a quite similar treatment). pH Inside the Gel. By combining the above relationships, one can get the proton concentration inside the gel phase, [H+]G. In eq 15, we apply the conditions of electro-neutrality to the gel phase and to the external solution (eqs 12 and 13), getting
(
[
[H+]G′ +[PH+] ) [H+]i′ - [H+]S′ 1 -
pKa ) pHG - log
1-R R
(22)
This eq 22 yields an apparent value for the dissociation constant of the protonated imidazole groups in the polyelectrolyte network. The point of discussion is whether the pH to be used in conjunction with R, when applying eq 22, can be the pH measured in the surrounding bath, pHS, as is usually done, or it has to be the pH inside the gel, pHG, that can be calculated with eq 19 by using the available experimental data. Samples The specimens of the PVI hydrogel here employed were synthesized by radical cross-linking copolymerisation of Nvinylimidazole (VI) and N,N′-methylenebis(acrylamide) (BA) in aqueous solution, with 2,2′-azobis(isobutyronitrile) (AIBN) (6 × 10-3 M) as initiator, with total comonomer concentrations,
10126 J. Phys. Chem. B, Vol. 112, No. 33, 2008
Figure 1. Equilibrium pH outside the gel (pHS) and polymer volume fraction (V2), reached in HCl aqueous solutions with different initial pH (pHi) upon immersion of PVI40(2) with an effective concentration (C) indicated in the insert.
CT ) 4.2 M and BA concentration [BA] ) 0.054 M, following the protocol described in ref 30. This particular sample composition will be denoted, as in previous reports,2–4 by PVI40(2). Xerogels of well-known mass (m) were immersed in the volume (VT) of HCl aqueous solution of known pHi, required to have a fixed effective polymer concentration (C). It should be noted that measurements were not made as in a continuous titration; each pHi was handled in a different pot with a different PVI specimen. Xerogels were used as unshaped pellets about 1 mm thick, in order to minimize the time required to reach equilibrium, that is, constant pH.4 The equilibrium pH was measured in the external solution (pHS), at room temperature, two days after preparation, and the polymer volume fraction in the swollen gel (V2) was determined as previously described.30 Experimental Results The degree of swelling and the pH reached at equilibrium by immersion of PVI40(2) in HCl aqueous solution with different polymer effective concentrations are shown in Figure 1, plotted as a function of the initial pH of the acid solution (pHi). In any case, pHS > pHi, and the difference pHS - pHi is larger for the larger C. The trend shown in Figure 1 for pHS was found for other polybases, such as uncross-linked poly(2vinylpyridine)6 and poly(2-vinylquinoline),7 as well as for suspensions of microgels and nanogels containing imidazole.31 It is remarkable that for pHi above 3.5 (C ) 0.1 M) or 4.5 (C ) 0.01 M), the equilibrium pH of the external solution remains constant irrespectively of the initial pH. Thus, in this region, the gel acts as a buffer for the solution in which it is immersed.29 The buffering extends over a rather long interval of pH, about 2-3 units of pHi, the range extending down to lower pH for larger effective concentration of the gel (C). Because measurements were performed in air atmosphere, the results corresponding to pHS in the range 5.5-7 should be analyzed by taking into account the dissolution of CO2 and the equilibrium dissociation of carbonic acid. Only protonation of PVI by HCl was considered in the model here proposed, and therefore, the experimental results corresponding to pHS between 5.5 and 7 will not be taken into account in the analysis shown below, which was applied only to pHi in the range 1-4. At all degrees of swelling studied here, the polymer concentration inside the gel phase is larger than the polymer effective concentration (as shown in Figure 2) and much larger than
Horta et al.
Figure 2. Ratio of the molar concentration of VI units in the gel phase (CG) to the effective molar concentration in the system (C): PVI40(2) immersed in HCl aqueous solutions with different initial pH (pHi) and with the effective concentration (C) indicated in the insert.
Figure 3. Degree of protonation of the PVI40(2) gel, R, calculated through eq 21 with the results of Figures 1 and 2, as function of the initial pH (pHi) attained with HCl.
common polymer concentrations employed for uncross-linked polymer solutions. Thus, the concentration of ionizable units in the PVI polyelectrolyte reaches almost 2 M at the lowest degrees of swelling. Because the volume of the bath solution is much larger than the volume of the swollen gel, the term 1 (1-V2)C/CG, that appears in eqs 10 and 17–21, can be approximated by unity without noticeable error. (However, this approximation was avoided in the following calculations). Calculations and Discussion Protonated Groups. The degree of protonation, R, is calculated through eq 21 with the experimental data of Figures 1 and 2. The results are shown in Figure 3, as function of pHi. The same results (within 1% incertitude), for R in the range 0.002-1.0, are obtained when R is calculated, with the simplified expression:2–4
R≈
[H+]i - [Η+]S C
(23)
The typical sigmoidal trend of R is visible with the lower effective concentration (C ) 0.01 M). With this smaller amount of polymer, there are enough protons in the more acid solutions to completely protonate the polymer. However, when the amount of polymer being immersed is greater (C ) 0.1 M), the acid can protonate the polymer only partially, and the ionization reaches hardly half of all the groups available. The molar concentration of charged groups inside the gel, [PH+], can be obtained as RCG. This RCG is shown in Figure
pH Inside a Gel
Figure 4. Concentration of fixed charges, RCG, in the PVI40(2) gel, as function of the initial pH (pHi) attained with HCl.
4, where it is plotted versus the concentration of protons in the immersion solution in the form -log [PH+] () -log RCG) versus -log [H+]i () pHi). It is seen that (a) the concentration of charged groups inside the gel is higher than the concentration of protons in the initial solution and (b) the difference between these two concentrations gets smaller when the amount of gel being immersed is larger (C ) 0.1 M). That the concentration of charged groups inside the gel is much higher than the concentration of protons in the initial solution means that most protons migrate to the swollen gel and get fixed (in the sense that they are much less mobile than counterions). In parallel, most of the Cl- counterions are also inside the swollen gel, as mobile ions. The external solution is then depleted from HCl. This depletion can be seen from the experimental results of the proton concentration that is left in the external solution at equilibrium (pHS) versus the initial concentration (pHi), in Figure 1. We can see that only in the case of the smaller amount of polymer (C ) 0.01 M) in the most acidic solutions does pHS approach pHi. The concentration of HCl that leaves the external solution and enters into the polymer phase induces the swelling of the gel (Figure 1). This occurs because of the osmotic force generated by the larger concentration of mobile species Cl- in the gel.2,3 The swelling increases with the acidity of the immersion solution; namely, ν2-1 increases as pHi decreases. However, a plateau is reached when pHi gets below 2.5 for C ) 0.01 M or below 1.5 for C ) 0.1 M. If we look at Figure 3, we can see that these are the pHi values where the degree of ionization, R, reaches approximately the value 0.33 where the threshold for the Manning condensation of counterions is reached in this type of polyelectrolyte chains.2 Hence, at these pHi values, further increase of proton concentration in the immersion solution affects neither the concentration of free ions in the swollen gel nor the number of net or uncompensated charges in the polymer chains. Concentration of Protons in the Gel. It is clear, from the above discussion, that most protons in the immersion solution get into the gel protonating the chains. But some of the protons entering the gel can also stay free as mobile ions and be responsible for the pH inside the swollen gel. Although such concentration of free protons inside the gel, [H+]G, is not measured, it can be calculated through eq 19 with the experimental data of Figures 1 and 2. The results are given in Figure 5 by plotting pHG as function of the pHi. These pHG values are compared with the pH in the external bath at equilibrium, pHS, in Figure 5. Calculated pHG is always larger than pHS. Qualitatively, this is an expected result, because Donnan exclusion explains that
J. Phys. Chem. B, Vol. 112, No. 33, 2008 10127 positive fixed charges reject partially mobile charges of the same sign to the outside of the gel phase. However, the magnitude of the effect is very large. Thus, the proton concentration gets almost five orders of magnitude lower inside the gel than it is in the outside bath at equilibrium. This, obviously, should have important consequences for the determination of the ionization constant of the polyelectrolyte gel, if such constant is determined from proton concentrations measured in the external solution. The effect is so large that the solution inside the gel gets even basic (when the initial bath is not extremely acid). This happens when pHi > 2.5, for C ) 0.01 M, and pHi > 1.5, for C ) 0.1 M. These two pHi values correspond to the same degree of ionization of the polymer, namely, R ≈ 0.3, which, as explained above, is where counterion condensation starts. Therefore, the pH that is developed inside the gel is always basic for the whole range of initial acidic solutions where the protons that enter into the gel create noncompensated charges in the chains and the counterions that enter along with the protons stay mobile in the swollen gel. The solution inside the gel becomes acid only when the pH of the immersion solution is so acid that counterion condensation is already taking place and the uncompensated net charges and free counterions do not grow further. The inclusion of the counterion condensation in our model modifies eq 19. Let us consider that, in the gel, counterions can be free or condensed, and protonated units can have net charge or be compensated by counterion condensation. [A-]G is now the concentration of the noncondensed free counterions in the gel, and [PH+] is the concentration of units with net charge. The condensed counterions and the protonated units with compensated charge form the species (PH+A-). All protonated units in excess of the threshold for condensation, RC, are in this form (PH+A-), the concentration in the gel of which is, therefore, [(PH+A-)] ) (R - RC)CG. The Donnan equilibrium of eq 15 continues to apply, with [A-]G being the concentration of noncondensed free counterions. The balance of eq 9 has now to be modified in order to include all the counterions that enter into the gel: free, [A-]G, plus condensed, (R - RC)CG.
[A-]iVT - [A-]SVS ) {[A-]G + (R - RC)CG}VG (24) By reworking the equations as before, the following expression is obtained for the corrected concentration of free protons inside the gel
[H+]G )
[
[H+]S[H+]S′ (1 - V2)C CG [H+]i′ - [H+]S′ 1 - CG(R - RC) CG C
(
)]
(25)
which, indeed, can be applied only above the threshold for counterion condensation, R > RC, whereas eq 19 is strictly valid only for R < RC. The proton concentration inside the gel calculated with eq 25 is higher than that obtained if we neglect counterion condensation by use of eq 19. In our case, RC ≈ 0.33, and eq 25 affords slightly larger [H+]G values than eq 19 when pHi < 2.5, for C ) 0.01 M, and pHi < 1.5, for C ) 0.1 M. Figure 5 incorporates these values. Acid Dissociation Constant pKa. The behavior explained above highlights that cross-linked PVI is a rather basic species that displaces the equilibrium of protonation (eq 1) well to the right when no other competing equilibria are present. The development of a basic pH inside the swollen gel when the xerogel is immersed in an acidic solution would be a puzzling
10128 J. Phys. Chem. B, Vol. 112, No. 33, 2008
Horta et al.
Figure 5. pH inside (pHG) and outside (pHS) the PVI40(2) gel, swollen in HCl aqueous solutions with different initial pH (pHi). pHG was calculated with eq 19 (squares) and with eq 25 (pHGM, triangles), which includes the correction for counterion condensation.
Figure 6. Basicity of the PVI40(2) gel (pKa for its acid dissociation) as function of the degree of protonation, R, attained with HCl.
result, unless we recognize PVI as a basic species. The literature results for linear PVI indicate that this polymer is basic but a weak base (pKa ) 4-7), however.13 Let us see what happens if we use the actual concentration of free protons inside the gel, [H+]G, now that we can calculate such concentration. As mentioned above, the equilibrium constant for the acid dissociation, Ka, is usually determined4 through the Henderson-Hasselbach equation (eq 22). We now use it with the pH inside the gel (pHG) both with (pKaM) and without (pKa) the correction for counterion condensation. The resulting pKa values are shown in Figure 6. They are compared with the pKa that would be usually obtained by using the pH of the external bath, namely, by using the equation4,6–15
pKa(S) ) pHS - log
1-R R
(26)
The difference between pKa values determined through eqs 19 and 25 or eq 26 is quite significant. The values obtained with the pH of the external bath are about 3-5 units lower and would correspond to a less basic character of the imidazole units. The inclusion of the counterion condensation (pKaM in Figure 6) decreases, although very slightly, the polymer basicity. The high pKa values mean that substantial protonation is attained already at elevated equilibrium pHG values (e. g., around 7), but it must be emphasized that, in order to get such high equilibrium pHG values, acidic conditions are required in the initial solution, the more acidic the larger C is. At midionization (R ) 0.5), pKa is about 5.5. This R ) 0.5 is the point where pKa should be determined if it were a true equilibrium constant. The decrease of pKa values as the degree of ionization decreases is a behavior commonly observed for polybases, whereas for polyacids pKa increases with increasing
degree of dissociation.4,13,25,32,33 It was interpreted as due to repulsive interactions of fixed charges and protons approaching the polymer, which makes more difficult the protonation as R increases. The same argument, when applied to the pKa shown in Figure 6, which reaches values around 9.5 with decreasing R, would mean that the imidazole units are more basic in the gel than in the model compound, for which pKa ) 7.3.5,15 When trying to understand this puzzling result, surface effects must be considered in two ways. First, in the low R limit, the surface may entrap most of the protonated imidazole units, and the pKa of those units on the surface may be larger than that for the same units inside the gel because of the different environmental dielectric constant.34 Second, the border between swollen gel and bath solution should not be sharp but a soft interface with a gradient of proton concentration between the inside and the outside of the gel, and that would lead to lower pKa values. These two surface effects could, at least in part, compensate each other. Alternatively, the large pKa values in the low R limit might also be due to sharing of the proton by two next imidazole units, which would increase the basicity, that is, pKa, with regard to the single imidazole ring of the model compound.35,36 Conclusions The model here proposed allows us to calculate the pH inside (pHG) a dissolved polyelectrolyte coil or a swollen ionic polymer network on the basis of two kinds of experimental data, the polymer concentration in the coil or the swollen gel (V2) and the pH in the bath surrounding them, both initial value (pHi) and equilibrium value (pHS). The main advantage of our model, with regard to others based on the titration of polyelectrolytes, is that here, we measure both the initial pH and the final pH of
pH Inside a Gel each solution, thus being able to calculate the balance of mobile ions going from one phase to the other as equilibrium is being established. Another advantage over usual titrations is that our model is free from any assumption concerning the pKa of the ionizable groups in the polyelectrolyte. The only simplification employed in developing our equations is that activities and concentrations can be interchanged, but this is a standard practice. The model was applied to chemically cross-linked PVI immersed in acidic baths of deionized water, the initial pH’s (pHi) of which were lowered with different concentrations of HCl. In all cases, it was found that the pH inside the gel (pHG) was several units higher than the pH in the external bath at equilibrium (pHS). The inclusion of the counterion condensation in our model lowers slightly pHG, but still, it remains larger than pHS. If pKa of PVI is calculated from results of pH inside the gel (pHG), the polyelectrolyte gel, in the low R limit, appears as a base stronger than the linear polymer, and the imidazole groups in the gel as more basic than the monomeric model compound. Probably, the influence of the gel surface or the intramolecular association between protonated and unprotonated imidazole rings may help in explaining these results. The inclusion of the counterion condensation decreases very slightly the polymer basicity. By use of cross-linked networks, the range of polyelectrolyte concentrations covered inside the gel is much larger than the concentrations usually available with linear polyelectrolyte solutions. Thus, some polyelectrolyte effects may appear more evident in the case of gels. In the present case, the polyelectrolyte concentration in the gel phase is 1.8 M when swollen in the less acid solutions (pHi above 4) and 0.4 M in the most acid solutions (pHi about 2). Some effects have been set aside in this study. Thus, it was not considered that carbon dioxide dissolves in deionized water giving a slightly acid pH close to 6, the contribution of which was not taken into account in eqs 11–13. Also not considered are the possibility of one proton bonding to more than one vinylimidazole unit, thus establishing another equilibrium besides that expressed by eq 1, and the surface effects. Further extension of the model to media of variable ionic strength would also be required to better understand these points. Acknowledgment. This work received financial support from Ministerio de Educacio´n y Ciencia (Spain) under Grant CTQ200761007/BQU. References and Notes (1) Brøndsted, H.; Kopecˇek, J. Polyelectrolyte Gels. Properties, Preparation and Applications. In ACS Symposium Series ; Harland, R. S., Prud’homme, R. K. Eds.; ACS: Washington DC, 1992; Vol. 480.
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