Poly(N-vinylimidazole) Gels as Insoluble Buffers that Neutralize Acid

(2-5) Thus, when immersing gels of cross-linked poly(N-vinylimidazole) (PVI) into acidic aqueous solutions with pH in the range 4.5−6.5, the acid is...
0 downloads 0 Views 288KB Size
Download PDF
4226

J. Phys. Chem. B 2009, 113, 4226–4231

Poly(N-vinylimidazole) Gels as Insoluble Buffers that Neutralize Acid Solutions without Dissolving Arturo Horta* and Ine´s F. Pie´rola Departamento de Ciencias y Te´cnicas Fisicoquı´micas, Facultad de Ciencias, UniVersidad a Distancia (UNED), 28040 Madrid, Spain ReceiVed: NoVember 2, 2008; ReVised Manuscript ReceiVed: January 9, 2009

Typical buffers are solutions containing weak acids or bases. If these groups were anchored to insoluble gels, what would be their behavior? Simple thermodynamics is used to calculate the pH in two-phase systems that contain the weak acid or base fixed to only one of the phases and is absent in the other. The experimental reference of such systems are pH sensitive hydrogels and heterogeneous systems of biological interest. It is predicted that a basic hydrogel immersed in slightly acidic solutions should absorb the acid and leave the external solution exactly neutral (pH 7). This is in accordance with experimental results of cross-linked poly(Nvinylimidazole). The pH 7 cannot be obtained if the system were homogeneous; the confinement of the weak base inside the gel phase is a requisite for this neutral pH in the external solution. The solution inside the gel is regulated to a much higher pH, which has important implications in studies on chemical reactions and physical processes taking place inside a phase insoluble but in contact with a solution. Introduction Weak acids or bases act as buffers that stabilize (within a certain range) the pH of the solution where they are dissolved.1 In their usual form, these buffers are solutes mixed with the solvent and other components forming a homogeneous solution. The behavior of such solutions is well known but, what would be the behavior when the acid or base is anchored to a crosslinked polymer? The system is then not homogeneous because the cross-linked polymer is insoluble in the solution; it only swells with such solution. The swollen polymer network and the acid or base groups bonded to the chains constitute a gel phase that is separate from the homogeneous solution that surrounds the gel. In such two phase systems, what is the influence on the pH of the surrounding solution due to the presence of the weak acid or base in the insoluble swollen gel? This question we try to answer here, by using standard relationships of acid-base equilibria and simple thermodynamics of phase equilibria involving charged species. If the behavior in two-phase systems were the same as in homogeneous single-phase solutions, then adding an insoluble weak acid or base to water would give an acidic or basic pH, respectively. However, the behavior is not that simple; experimentally it has been found that two-phase buffers follow a different pattern. There are experimental results showing that the pH of solutions free of buffer have a puzzling pH behavior when in contact with a separate phase that contains a nondiffusible buffer.2-5 Thus, when immersing gels of cross-linked poly(Nvinylimidazole) (PVI) into acidic aqueous solutions with pH in the range 4.5-6.5, the acid is neutralized and the final pH of the solutions is always very close to 7, regardless of the initial pH.2 (Lower initial pHs lead to final pHs that depend on the mass of gel immersed,3 the ionic strength of the bath to be buffered,4 and the size of the polymer particle immersed).5 Similar constancy of the final pH is observed also in solutions * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 34-91-3987378. Fax: 34-91-3987390.

of linear polybases6,7 and even in suspensions of spores able to act as proton reservoirs.8 So, it appears to be a rather general phenomenon. This neutralization of one phase (the solution) by another phase (the hydrogel) is intriguing and has not been interpreted up to now (to our knowledge). It is explained here by calculating the pH of the two phases through a simple, though general, formalism. The formalism is valid for any equilibrium between a solution free of acid or base groups and another phase containing those groups, but the numerical calculations will be given only for the PVI case, as an example of its applications. Theory The clue to obtaining the pH in the two phases, hydrogel and external solution, is the acid dissociation constant of the weak acid or the weak base, Ka (the explanation is given here only for basic groups, but for acid groups it would go entirely in parallel). The equilibrium defining Ka for a weak base, P, is the dissociation of its conjugate acid, PH+

PH+ ) P + H+

(1)

Ka ) (P)(H+)/(PH+)

(2)

Here, (P), (PH+), and (H+) are the activities of P, PH+, and H , respectively. If the base is anchored to the cross-linked polymer chains, then P is present only in the hydrogel phase but not in the surrounding homogeneous solution. The equilibrium (eq 1) is established in the hydrogel phase, and the activities entering into Ka (eq 2) are those inside such gel phase. We now describe what would be the pH behavior of an aqueous solution which is put in contact with a cross-linked polymer whose monomer units carry a weak base. Initially, the polymer is dry and apart from the solution. If the polymer carries a weak base, its protonation is induced by contact with a solution having an acidic pH. This initial acidic pH is reached by adding +

10.1021/jp809682d CCC: $40.75  2009 American Chemical Society Published on Web 02/26/2009

Insoluble Buffers

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4227

a strong acid, H+A-, to the solution before contact with the dry gel. When the dry polymer is immersed into the solution, there is diffusion of the components of the solution (solvent plus strong acid) into the network, but the species bonded to the gel cannot migrate to the solution. The charged species PH+ are thus “fixed” to the gel phase, while all other ionic species, H+, OH-, A-, are mobile or “free” to cross the boundary between the two phases. The proton activity in the solution is called (H+)init in its initial state, before having contact with the dry gel, and it is called (H+)bath in its final state, after contact with the gel and equilibrium has been reached between the swollen gel and the solution surrounding it (the bath). The process of attaining equilibrium between the two phases can be described in the following way. The acid H+A-, initially contained in the solution, diffuses into the gel. Some H+ of this acid gets fixed in the form of PH+ by the protonation reaction with base P (eq 1). The corresponding amount of A-, that has entered into the gel, remains as mobile counterions. As a consequence, inside the swollen gel there is a deficit of free H+ versus A- ions. The diffusion of acid H+A- into the gel continues until its activity in the swollen gel equals its activity in the bath. The crucial point is to realize that the starting activity of acid for the protonation reaction in the gel is not equal to the initial acid activity in the solution, (H+)init, because protonation proceeds only with the acid that has diffused into the gel. Then, what is the starting activity of acid for the protonation reaction in the gel? We can answer this question by formally dividing the overall process into a diffusion of acid into the gel followed by the protonation reaction itself. The starting acid activity for the protonation is the activity corresponding to the concentration of H+A- that has diffused into the gel. This is equal to the concentration of A- that is present inside the swollen gel at equilibrium, [A-]gel, because the A- counterions are not involved in the protonation reaction and the initial gel (the dry polymer) contains no such ions. The electroneutrality of the swollen gel phase requires the following balance of molar concentrations:

[PH+]gel ) [A-]gel - [H+]gel + [OH-]gel

(3)

The task now is to relate this starting acid concentration inside the gel phase, [A-]gel, with the initial acid concentration in the solution, [H+]init, which is the proton concentration that is controlled experimentally. To this end, we resort to the mass balance of A-

[A-]gelVgel ) [A-]initVinit - [A-]bathVbath

(7)

where Vinit is the volume occupied by the solution before contact with the dry gel, and Vbath and Vgel are the volumes occupied by the surrounding solution (bath) and the swollen gel, respectively, after reaching equilibrium. The electroneutrality in the initial acid solution is expressed by

[A-]init)[H+]init - Kw /[H+]init

(8)

and the electroneutrality of the solution at equilibrium (bath) by

[A-]bath)[H+]bath - Kw/[H+]bath

(9)

Then, [A-]gel of eq 7 reads

[A-]gel ) ([H+]init - Kw /[H+]init)Vinit /Vgel - ([H+]bath Kw/[H+]bath)Vbath /Vgel (10) There is thermodynamic equilibrium between the two phases, so the activity of the common electrolyte, H+A-, has to be the same

[H+]gel[A-]gel ) [H+]bath[A-]bath

(11)

which, due to electroneutrality in the bath (eq 9), turns into

[H+]gel[A-]gel ) [H+]bath([H+]bath - Kw/[H+]bath) (12)

and, obviously

[P]gel ) Cgel - [PH+]gel

Solving for [H+]bath

(4) [H+]bath ) ([H+]gel[A-]gel + Kw)1/2

where Cgel is the molar concentration of total weak base in the swollen gel. We now substitute these [PH+]gel and [P]gel for the corresponding activities in Ka (eq 2) (committing thus the usual simplification of not distinguishing activities from molar concentrations), which yields

Ka ) {(Cgel - [A-]gel+[H+]gel - Kw /[H+]gel)/([A-]gel [H+]gel + Kw /[H+]gel)}[H+]gel (5) where Kw has its usual meaning of constant for water autodissociation. We can solve for [A-]gel, as a function of [H+]gel

[A-]gel ) {[H+]gel3 + (Cgel + Ka)[H+]gel2 - Kw[H+]gel KwKa}/([H+]gel2 + Ka[H+]gel) (6)

(13)

Additionally, solving for [H+]init in eq 10

[H+]init ) (1/2)[x + (x2 + 4Kw)1/2]

(14)

x)[A-]gelVgel /Vinit + ([H+]bath - Kw /[H+]bath)Vbath /Vinit (15) Then, we can obtain all the proton concentrations, “init”, “bath”, and “gel” from Ka, by calculating backward. We start with a hypothetical [H+]gel and obtain [A-]gel (eq 6), then from [A-]gel and [H+]gel we obtain [H+]bath (eq 13), and finally from [A-]gel and [H+]bath we obtain [H+]init (eqs 14 and 15). Thus, we can obtain the pH in the bath solution and inside the gel, both as functions of the initial pH.

4228 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Horta and Pie´rola

Cgel ) V2F2 /M0

(16)

where F2 is density (g/L) of the xerogel and M0 molecular weight of the N-vinylimidazole monomer unit. Cgel varies with pHinit because the swelling of the gel depends on pH. Experimentally, there is a broad range of pHinit, extending over two pHinit units (pHinit ) 4.3-6.4), where Cgel remains constant (see Figure 2).3,21,22 In most of this range (pHinit ) 4.9-6.4) is where the plateau of pHbath close to 7 occurs (Figure 1). At lower pHinit, from pHinit ) 4.2 to pHinit ) 2 there is a volume transition of the gel to a much lower Cgel (the gel dilates to four times its original volume, see Figure 2). The ratios of volumes Vgel/Vinit and Vbath/Vinit are also needed in the calculations. They are Figure 1. Dry samples of a cross-linked poly(N-vinylimidazole) gel are immersed in acidic water solutions having different pH values (pHinit), and the pH of the solution is measured after reaching equilibrium with the gel (pHbath). The strong acid added to fix each pHinit value is HCl. The amount of N-vinylimidazole moieties (moles) added per liter of the initial acid solution is C ) 0.01 M. Data from Horta et al.3 (Details on the synthesis and purification of the gels and on the protocol for measuring pH can be found in Supporting Information).

The Poly(N-vinylimidazole) Puzzle Imidazole (Im) is an example of weak base giving a buffer solution when dissolved in acidic water. Its basic behavior is due to the protonation, Im + H+ ) ImH+, with a pKa ) 7.9,10 When incorporated into a polymer chain11-18 or a polymer network,3-5,19,20 by polymerization of the monomer N-vinylimidazole, the resulting poly(N-vinylimidazole) is a weak polybase that can suffer the same protonation equilibrium. As dry sample, cross-linked PVI is neutral, but in contact with an acid solution it swells by absorbing water and ionic species present in the solution, giving a swollen hydrogel that contains ionic charges created on its chains by protons that get fixed to the imidazole moieties. When the dry PVI is immersed in aqueous acidic solutions that contain variable amounts of HCl, the resulting pH of the solutions, after the gel is fully swollen and equilibrium has been attained, is higher than the initial one. Let us call pHinit to the pH value of the initial acidic solution before PVI is immersed, and pHbath to the pH value of the solution that remains after PVI is added and equilibrium between the swollen gel and the solution surrounding the gel (the bath) has been attained. Experimental results for pHbath, as a function of pHinit, are shown in Figure 1.3 It sounds surprising the plateau of pHbath, that remains close to the value 7, irrespective of the acidity of the initial solution when PVI is added to acid solutions in the range 4 < pHinit < 7. Since the imidazole groups in PVI are bases, initial solutions with pHinit not far from 7 would give a pH clearly above 7 in the final solution if the system were homogeneous (see Appendix) instead of being two-phase. This puzzling constancy of pHbath at value 7 is what we try to understand here. To this end, we use the formalism of the Theory section. For the calculation of pHbath and pHgel, as functions of pHinit, according to the equations in the Theory section, we need also the concentration of base in the swollen gel, Cgel. It is obtained from experimental results of the degree of swelling, which allows the determination of the volume fraction of polyelectrolyte in the hydrogel, V2. From this V2, the molar concentration Cgel is obtained as

Vgel /Vinit ) C/Cgel

(17)

Vbath /Vinit ) 1 - (1 - V2)C/Cgel

(18)

where C is the amount (moles) of N-vinylimidazole units contained in the dry PVI gel added to the initial acidic solution per liter of such solution. The results calculated for pHbath, as a function of pHinit, using the value pKa ) 7, are shown in Figure 3. This value of pKa is very close to the one for the model molecule N-ethylimidazole (pKa ) 7.3 for N-ethylimidazole at 25 °C in salt-free aqueous solution)9 and also to some recent estimations of the intrinsic pKa corresponding to cross-linked PVI.4 For the concentration of N-vinylimidazole monomer units in the swollen gel, we use Cgel ) 1.9 M, which is valid in the range 4.3 e pHinit e 6.4, where the volume of the hydrogel remains practically constant. Therefore, the curves shown in Figure 3 have strict validity only in this same range, but this poses no limitation on our results because we are looking for the values of pHbath where the experimental results remain practically constant in a plateau close to the value pHbath = 7, and this happens precisely within such a range of constant degree of swelling (Figures 1 and 2). As we can see, the calculated values of pHbath reach a plateau pHbath ) 7, which is in excellent agreement with the experimental results. The calculated plateau is not so extensive as the experimental one however, but the value of pHbath is matched perfectly. If the system were treated as being homogeneous, it would be impossible to obtain the value 7 because the groups carried by the PVI chains are basic. The pH of a solution containing PVI would then be much higher than 7 if there were no boundary separating those basic groups from the solution. Thus, calculating pH as in a homogeneous solution (see Appendix) the plateau would be at pH ) 9.5 with pKa ) 7 instead of being at pH 7, which is the experimental value. So, we can say that the simulated description of pH regulation of an external solution by the action of groups in a different gel phase is satisfactorily accomplished and gives the main features of the behavior observed. Another interesting result from this calculation is that the pH inside the gel phase is very much higher than in the external solution. In the gel phase, there occurs also buffering and a plateau of constant pHgel, but this plateau occurs at pHgel ) 10.6, three and one-half-units above the pH of the external solution. Although the chemical potential of the electrolyte H+ A- is equal in the hydrogel phase and in the external phase, the activity of the individual ion H+ is more than 3 orders of magnitude lower inside the hydrogel than it is in the external bath. This result is

Insoluble Buffers

Figure 2. Reciprocal polymer volume fraction, V2, of cross-linked PVI hydrogel swollen at equilibrium after immersion of the dry gel in acidic solutions of different initial pH, pHinit. The amount of N-vinylimidazole moieties per liter of initial solution is C ) 0.01 M. Experimental data from Horta el al.3 (Details on the experimental determination of the swelling degree and the calculation of the polymer volume fraction can be found in Supporting Information).

Figure 3. Calculated pH for the external bath (pHbath, solid line) in equilibrium with a cross-linked gel of PVI when the gel is immersed in acidic aqueous solutions of different initial pH (pHinit) and calculated pH of the solution inside the swollen hydrogel (pHgel, dashed line). Results obtained with pKa ) 7 and C ) 0.01 M.

of great relevance for the chemical reactions and physical processes taking place inside swollen gels. The external bath is neutral, but the solution inside the hydrogel is clearly basic. Obviously, this basicity comes because PVI is a base and this base is entirely contained within the hydrogel phase. But it seems rather surprising that this basicity is not transmitted at all to the external bath in equilibrium with the hydrogel (which is predicted to be “exactly” neutral: pHbath ) 7). An understanding of this point comes when analyzing the influence of the degree of basicity of the polybase, namely by comparing the results for different values of the pKa of the polybase. In Figure 4 are shown the results for pKa ) 7 and pKa ) 10. Our interpretation of these results is as follows. The external solution becomes always neutral, regardless of the stronger or weaker basic character of the polybase, because the hydrogel absorbs any minor solute present in the initial solution. Although the initial solutions are acid, the component H+A- is always in minute amount compared to PVI, namely [H+]init , C. The hydrogel absorbs the ions of the acid and “purifies” the water, thus leaving pH ) 7. Each one of the two ions composing the acid absorbed by the hydrogel follows a different path. The protons are fixed by the N-vinylimidazole groups, thus the

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4229

Figure 4. Same as in Figure 3 (pHbath, solid lines; pHgel, dashed lines). Comparison of results obtained with two pKa values, pKa ) 7 and pKa ) 10, for C ) 0.01M.

activity of free protons drops and results a basic pH in the swelling solution, while the counteranions stay unmodified. The strength of the polybase determines how much the activity of free protons drops inside the hydrogel. So, as the pHgel is higher, the stronger is the polybase (the higher its pKa). But, in the external solution the plateau occurs always at neutral pH, irrespective of the strength of the polybase (irrespective of pKa), because the neutralization of the external bath is due not merely to an acid-base equilibrium involving only the H+ ions but to the whole acid (H+ and A-) being absorbed by the gel. For this same reason, the neutral plateau is more extended the stronger is the polybase (higher pKa). This means that the drop in free protons that occurs inside the gel, determined by the protonation of the polybase, induces the diffusion into the gel of more protons from the external solution. But protons do not diffuse alone; it is the whole acid H+A- that diffuses. Hence, the plateau extends over a wider range of pHinit the stronger is the polybase (the higher its pKa). At the extreme of very acid solutions, when enough protons are available to protonate practically all the imidazolyl groups in the gel, the difference between internal and external pH vanishes and the curves for pHbath and pHgel converge to a common line. This convergence could be seen in Figures 3 and 4 if the scale of abscissas were extended down to pHinit ) -1. The simple description developed here is valid to justify the neutralization and constancy of pH in the plateau that is observed experimentally when the immersion solutions are not very acid. For more acidic solutions, the experimental pH in the external bath departs from neutral and becomes increasingly acidic.3,4 The comparison with the experimental results is shown in Figure 5. We can see that the plateau is better represented with pKa ) 10, while pKa ) 7 is more adequate for more acidic solutions. Our purpose here was not to obtain the best pKa value but to show the need of treating the system as two-phase in order to understand the neutral plateau, since the behavior predicted if the system were treated as homogeneous is way apart from experiment, as seen in Figure 5. When an acidic or basic group is in the monomer unit of a polymer chain, its Ka varies with the degree of ionization, so that Ka of a polymer is not a true thermodynamic constant, but only an apparent one. This variation of Ka with the degree of ionization means that it is not possible to describe the whole curve of titration with a single value of pKa. No wonder then that in Figure 5 the curves calculated with a constant pKa (7 or 10) agree with experiment only in a limited range of pH. The

4230 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Figure 5. Comparison between experimental and calculated pH at equilibrium when cross-linked PVI is immersed in acidic water solutions of varying pH (pHinit). 9: experimental data from Horta et al.3 pHbath: pH for the bath that surrounds the swollen gel (two-phase system), calculated with two different values of pKa (pKa ) 7, 10) and C ) 0.01 M. pHsoln: pH for the homogeneous solution that would be obtained if PVI were soluble, calculated with pKa ) 7 and C ) 0.01 M.

relevant finding is that the plateau can be reproduced by treating the equilibrium as heterogeneous, with the base in a gel phase separate from the acid solution. The simplicity of using a constant pKa does not allow for a better fit in the rest of the pH range, but the nonconstancy of Ka is a different problem that has been analyzed elsewhere.3 The mechanism of heterogeneous buffering described here is an idealized one, because the system is treated as if a sharply defined boundary existed between the two phases and all the ionizable groups were contained within that boundary. In this way, it is assumed that all the groups are equilibrated with the protons that are inside the gel phase. A more realistic description would take into account that some ionizable groups are on the surface of the gel, and that these groups can be equilibrated with the protons of the external solution. For the moment, we do not dare to analyze these surface effects (in which the fraction of groups equilibrated with the bath would enter as a new unknown), since the success in describing the neutral plateau with the idealized two sharp phases model is a satisfactory first advance. Future consideration of the fuzzy interphase can probably be needed to explain other experimental results in which the plateau occurs not at neutral pH but at basic pH. Thus, with the same cross-linked PVI gel described here, the experimental plateau rises to pHbath ) 7.5 when the amount of xerogel immersed in the acidic solutions is increased 10-fold, from C ) 0.01 to 0.1 M.3 Also, the cross-linked gels of poly(isopropylacrylamide-1-vinylimidazole), studied by Kazakov et al.,5 have a similar effect of concentration; for nanogels in dilute suspension the plateau is at pH ) 7, but for microgels at a concentration 8-fold higher the plateau reaches almost pH ) 8. These pH values of the plateau higher than 7 cannot be explained with a model of two sharp phases, because in that model the external bath contains only water and added acid, and the electroneutrality condition of eq 9 determines that the pH of the bath cannot be higher than 7. For a higher pH, some other cationic species in addition to H+ has to be taking part in the equilibrium of the bath phase. This additional species can be the PH+ of the gel surface. The equilibrium is then established between these surface groups and the free protons of the bath (like an adsorption equilibrium between a solid and a solution). The consumption of free protons in the bath by these surface groups then gives pH above 7.

Horta and Pie´rola But why is this noticed only at the higher gel concentrations? It seems clear that higher gel/solution ratios mean more surface groups capable of depleting the free protons in the solution, and hence higher pH in the bath. With our PVI gel, the degree of swelling stays low (constant) over the range of pH of the plateau (Figure 2). Such low and constant degree of swelling suggests that the acid in the initial solutions is neutralized by the gel without absorbing solution into the bulk of the gel phase. Hence, the surface groups seem responsible for that neutralization. The range of initial pH over which this protonation without bulk absorption occurs is wider when the concentration is higher. Thus, the range of constant swelling degree extends until about pHinit ) 4.5 when C ) 0.01 M (Figure 2), but it extends further until about pHinit ) 3.5 when C ) 0.1 M.3 Ten times more groups are present when C equals 0.1 M, and hence the depletion of solution protons can be also ten times higher (reaching one pH unit lower) with this concentration. Finally, two-phase buffer systems consisting of a sparingly soluble diprotic acid and its saturated aqueous solution were investigated,23 but in such systems, the buffering agent is present in both phases, by difference with those here considered. Conclusions The formalism applied here, although general, is only a naive approach. First of all, it neglects deviations from ideality (activity coefficients). Second, it treats the two phases as bulk entities, not considering surface effects. Other simplifications are to treat the Ka of the polyelectrolyte as constant, and the concentration of polyelectrolyte in the hydrogel phase also as constant (which can be justified within the pH range of interest). However naive, the formalism is good to demonstrate that the existence of a neutral plateau in the bath can be explained only by considering the system as two-phase with the buffering groups separate from the bath. This shows that the mechanism for heterogeneous buffering of solutions, when the buffer resides in a separate phase, is entirely different from the classical mechanism of homogeneous systems, where the buffer is a component of the solution. It is the solution inside the gel the one that follows the classical homogeneous mechanism, not the bath solution outside the gel. The crucial point for interpreting the two-phase mechanism is to recognize that the acid that participates in the protonation equilibrium is not the acid in the initial solution, but the acid that diffuses into the gel phase where the proton reaction takes place; in other words, the acid that is absorbed by the gel. Acknowledgment. This work received financial support from Ministerio de Ciencia e Innovacio´n (Spain) under Grant CTQ2007-61007/BQU. Appendix Homogeneous Solution We now describe what would be the pH behavior if the added PVI and the initial acidic solution could form a homogeneous solution. We continue applying the label “init” to the quantities in the initial acidic solution before PVI is added, and we use now the label “soln” for the quantities in the final solution containing PVI as solute. Let us now correlate [H+]soln with [H+]init, knowing Ka and Csoln (molar concentration of total weak base in the solution). To this end, we use eq 2 and the condition of electroneutrality of the homogeneous solution, which is

Insoluble Buffers

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4231

[PH+]soln ) [A-]soln - [H+]soln + Kw/[H+]soln (A1)

References and Notes

It is also obvious that

[P]soln ) Csoln - [PH+]soln

(A2)

Csoln ) C/(1 + CM0 /F2)

(A3)

[A-]soln ) [A-]initCsoln /C

(A4)

and

(this expression corrects for the volume added by the dry gel to the initial solution). Because of electroneutrality of the initial acidic solution

[A-]init ) [H+]init - Kw/[H+]init

and calculation of the polymer volume fraction. This material is available free of charge via the Internet at http://pubs.acs.org.

(A5)

Substituting eqs A1-A5 into Ka (eq 2) and solving for [H+]init as function of [H+]soln, we get eq 14 but with x now defined as

x){[H+]soln3 + (Csoln + Ka)[H+]soln2 - Kw[H+]soln 2 + Ka[H+]soln)(Csoln /C) (A6) KwKa}/([H+]soln

Then, we can obtain the correlation between [H+]soln and [H+]init by calculating backward, starting with a hypothetical [H+]soln, and through eqs 14 and A6, obtaining the [H+]init that corresponds to it. The results shown in Figure 5 have been calculated in this way. Supporting Information Available: Details on how the experimental data shown on Figures 1 and 2 were obtained, namely, the cross-linking polymerization of poly(N-vinylimidazole), pH determination, measurement of degree of swelling,

(1) Perrin, D. D.; Dempsey, B. Buffers for pH and Metal Ion Control; Chapman and Hall: London, 1974. (2) (a) Go´mez-Anto´n, M. R.; Molina, M. J.; Morales, E.; Pie´rola, I. F. U.S. Patent 5,393,853, 1995. (b) Go´mez-Anto´n, M. R.; Molina, M. J.; Morales, E.; Pie´rola, I. F. Chem. Abstr. 1999, 52, 386754 r. (3) (a) Horta, A.; Molina, M. J.; Go´mez-Anto´n, M. R; Pie´rola, I. F. J. Phys. Chem. B 2008, 112, 10123. (b) Horta, A.; Molina, M. J.; Go´mezAnto´n, M. R; Pie´rola, I. F. J. Phys. Chem. B 2008, 112, 13166. (4) Horta, A.; Molina, M. J; Go´mez-Anto´n, M. R; Pie´rola, I. F Macromolecules 2009, 42, 1285. (5) Kazakov, S.; Kaholek, M.; Gazaryan, I.; Krasnikov, B.; Miller, K.; Levon, K. J. Phys. Chem. B 2006, 110, 15107. (6) Pie´rola, I. F.; Turro, N. J.; Kuo, P.-L. Macromolecules 1985, 18, 508. (7) Go´mez-Anto´n, M. R.; Rodrı´guez, J. G.; Pie´rola, I. F. Macromolecules 1986, 19, 2932. (8) Kazakov, S.; Bonvouloir, E.; Gazaryan, I. J. Phys. Chem. B 2008, 112, 2233. (9) Perrin, D. D. Dissociation Constants of Organic Bases in Aqueous Solution; Butterworths: London, 1965; pp. 190-194. (10) Goldberg, R. N.; Kishore, N.; Lennen, R. M. J. Phys. Chem. Ref. Data 2002, 31, 231. (11) Sakurai, M.; Imai, T.; Yamashita, F.; Nakamura, K.; Komatsu, T. Polym. J. 1994, 26, 658. (12) Kodama, H.; Miyajima, T.; Mori, M.; Takahashi, M.; Nishimura, H.; Ishiguro, S. Colloid Polym. Sci. 1997, 275, 938. (13) Kodama, H.; Miyajima, T.; Tabuchi, H.; Ishiguro, S. Colloid Polym. Sci. 2000, 278, 1. (14) Maeda, Y.; Yamamoto, H.; Ikeda, I. Langmuir 2001, 17, 6855. (15) Annekov, V. V.; Mazyar, N. L.; Kruglova, V. A. Polym. Sci. Series 2001, A43, 807. (16) Cabot, B.; Deratani, A.; Foissy, A. Colloids Surf., A 1998, 139, 287. (17) Roques-Carmes, T.; Aouadj, S.; Filiatre, C.; Membrey, F.; Foissy, A. J. Colloid Interface Sci. 2004, 274, 421. (18) Bisht, H. S.; Wan, L.; Mao, G.; Oupicky, D. Polymer 2005, 46, 7945. (19) Pich, A.; Tessier, A.; Boyko, V.; Lu, Y.; Adler, H.-J. P. Macromolecules 2006, 39, 7701. (20) Molina, M. J.; Go´mez-Anto´n, M. R.; Pie´rola, I. F. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 2294. (21) Molina, M. J.; Go´mez-Anto´n, M. R.; Pie´rola, I. F. Macromol. Chem. Phys. 2002, 203, 2075. (22) Molina, M. J.; Go´mez-Anto´n, M. R.; Pie´rola, I. F. J. Phys. Chem. B 2007, 111, 12066. (23) Pfendt, L. B. Analyst 1995, 120, 2129.

JP809682D