790
J. Phys. Chem. 1996, 100, 790-796
Evaluation of Distribution of Protons and Electrostatic Potential in Poly(allylamine hydrochloride) Solution by the pH Indicator Method Kimio Sumaru, Hideki Matsuoka, and Hitoshi Yamaoka* Department of Polymer Chemistry, Faculty of Engineering, Kyoto UniVersity, Kyoto 606-01, Japan ReceiVed: July 25, 1995; In Final Form: October 4, 1995X
The distribution of protons and the electrostatic potential in poly(allylamine hydrochloride) (PAAmHCl) aqueous solutions containing excess HCl were evaluated by measuring the visible light absorbance with a small amount of the pH indicator bromophenol blue (BPB), which coexisted in the system under various conditions. The experimental data, obtained as an apparent proton concentration ([H+]app), were reproduced quantitatively by a cylindrical Poisson-Boltzmann cell model with only one fitting parameter. It was proved that [H+]app could be expressed as a product of the mean proton concentration and a dimensionless factor, which sensitively reflects the uniformity in the electrostatic potential not only for polyanionic systems such as poly(vinylsulfonic acid) solutions but also for the polycationic systems studied here. The surface electric potential of the polyion was determined from the experimental data and found to be constant with increasing [PAAmHCl] although it was expected to decrease from the cell model. This result suggested that the local conformation of the polyion chain is influenced much more by the ionic atmosphere than by the concentration of the polyion itself.
1. Introduction The simple ion distribution in linear macroionic solutions is one of the most important factors in determining various characteristics of the systems,1-4 and attempts have been made to evaluate it by activity measurements such as potentiometry5-9 and osmotic measurements.1,8,10 The counterion activities obtained from these measurements have been reported to be well rationalized by simple approximate theories of counterion condensation proposed by Oosawa,11 Nagasawa,5 and Manning.12 Later, the counterion activity calculated from these theories was proved to agree well with that calculated from a rigorous solution of the nonlinear Poisson-Boltzmann equation (P.B. eq.).13,14 Recently, Outhwaite et al. proposed a modified P.B. eq., taking local correlations among the simple ions into account, and compared the results to those of Monte Carlo simulations and the classical nonlinear P.B. eq.14 They concluded that all these theories give a very similar simple ion distribution when all the simple ions are monovalent. Whereas theoretical treatments of the problem have become well established in this way, experimental methods for evaluating the whole profile of the simple ion distribution in macroionic systems are currently very few and limited; the measured activity does not reflect the ion distribution near the macroion directly. As an attempt to evaluate the counterion distribution around a linear macroion, Maarel et al. carried out small-angle neutron scattering (SANS) measurements on tetramethylammonium poly(styrenesulfonate) (TMAPSS) and extracted the scattering from the counterions only.15 By fitting the theoretical scattering intensities calculated from an isotropically oriented cylindrical cell model of finite length to the experimental data, they evaluated the counterion distribution around the PSS main chain. However, some ambiguity remained in the analysis since in the calculation they had to introduce many fitting parameters such as the polyion radius, the charge spacing along the backbone, a cell length, and a correlation factor and scattering base line in order to describe the SANS intensity from the system. Furthermore, the application of this method is very limited X
Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-0790$12.00/0
because of the complexity of the analysis and the problem of low scattering intensity. Under these circumstances, we proposed a pH indicator method as a novel tool for evaluating the simple ion (proton) distribution in macroionic systems and applied the method to poly(vinylsulfonic acid) (PVS) solutions under various conditions to evaluate the distribution of protons as counterions.16 With this method, the distribution of protons and the electrostatic potential are estimated by measuring the visible light (vis) absorbance spectra of macroionic solutions containing a very small fraction of pH indicator (bromophenol blue, BPB). The BPB molecules, which are themselves thermodynamically distributed as mono- and divalent anions, attain dissociation equilibrium with the protons distributed in various local concentrations that are different from place to place. In other words, BPB plays the role of a micro pH meter that can be distributed throughout the system, even near a polyion chain. The results of our measurements on PVS solutions not only gave us considerable information about the physicochemical properties of this system but also suggested that the method could be applied to various other macroionic systems. In this study, we applied the pH indicator method to poly(allylamine hydrochloride) (PAAmHCl) solutions under various conditions in order to evaluate the distribution of protons as coions and the electrostatic potential in a linear polycationic system. As reported in several papers, poly(allylamine) is completely protonated even at neutral pH; positive charges are fully provided to the amino groups on the polymer chain by association with the protons.17 This behavior contrasts with that of poly(ethyleneimine)18,19 and poly(vinylamine),19-21 which are difficult to completely protonate. Poly(allylamine hydrochloride) is therefore expected to behave as a polycation in the presence of excess HCl, with protons as coions and chloride ions as counterions. By use of an anionic pH indicator such as BPB, it becomes possible to detect sensitively the electrostatic situation and local proton concentration near the polycation chain. Experimental data were collected as functions of added [NaCl], [PAAmHCl], degree of neutralization, and added [HCl] © 1996 American Chemical Society
Protons and Potentials in Poly(allylamine hydrochloride)
J. Phys. Chem., Vol. 100, No. 2, 1996 791
for PAAmHCl samples of two different molecular weights and analyzed with the cylindrical P.B. cell model. 2. Experimental Section 2.1. Preparation of Standard Solutions of PAAmHCl and BPB. The PAAmHCl samples of two different molecular weights (Nittobo, Tokyo, Japan, MW ) 60 000 and 10 000) were dissolved in Milli-Q (Millipore, Bedford, Warrington, PA) grade water and purified by dialysis. After freeze-drying, the PAAmHCl samples were again dissolved in Milli-Q water to give 25 mM standard solutions. An approximately 0.1 g/L of BPB standard solution was prepared by dissolving BPB (Wako Pure Chemical Industries, Ltd., Osaka, Japan) powder in Milli-Q water. Because BPB is not very soluble in water, the molecules may be associated with each other rather than individually dispersed, even if the solution appears to be homogeneous. In general, the sensitivity of BPB therefore depends on the method used in the preparation of the standard solution. It was accordingly prepared to have stable sensitivity by using a definite procedure that included ultrasonification for 3 h before each set of measurements. 2.2. Characterization of BPB. In order to check the sensitivity of the BPB, its vis absorbance was measured for HCl solutions of various concentrations containing a small amount of BPB using a UV-vis spectrometer (U-3400, Hitachi Ltd., Tokyo, Japan). BPB (H2BPB) molecules in aqueous solution exist in either acidic (HBPB-) or basic (BPB2-) forms that have λmax at 436 and 591 nm, respectively. Although these maxima are well separated, the absorbance peaks overlap each other slightly. To estimate the extent of this overlapping, the small absorbance of each particular state of the molecules at the λmax of the other state was measured for strongly basic and acidic BPB solutions. Taking this result into account, the concentration of BPB in its acidic and basic forms was related to the absorbance values A436 and A591 as follows. -
[HBPB ] ∝ A436 - 0.040A591
(1)
[BPB ] ∝ A591 - 0.015A436
(2)
2-
Hence, in uniform systems (HCl solutions), [H+] can be expressed in terms of the apparent dissociation constant Kobs and the absorbance values as16
[H+] ) Kobs(A436 - 0.040A591)/(A591 - 0.015A436) (3) Calibration of Kobs with normal HCl solutions was carried out before each set of measurements. To further check the influence of BPB molecules as a perturbation on the simple ion distribution, vis measurements were also carried out for samples containing PAAmHCl-HCl at various BPB concentrations. As a result, it was confirmed that in the range [PAAmHCl] > 3 mM, the BPB of the concentration required for the absorbance measurements does not purturb the proton distribution in the system. 2.3. Determination of [H+]app under Various Conditions. Visible spectra of BPB in the PAAmHCl-HCl systems were measured at various [NaCl], [PAAmHCl], [NaOH], and [HCl] for PAAmHCl’s of two different molecular weights (MW). In the measurements, the net vis absorbance of BPB was obtained by subtracting that of PAAmHCl, which is negligible anyway, from the original spectral profiles. From the absorbance data, the apparent proton concentration [H+]app, which is defined by
[H+]app,exp ) Kobs(A436 - 0.040A591)/(A591 - 0.015A436) (4)
was determined using the Kobs values obtained in the previous calibration. All the measurements were carried out at room temperature (23-25 °C), and the experimental conditions are tabulated in Table 1. 3. Theory 3.1. Distribution of Simple Ions in PAAmHCl Solutions. The simple ion distribution in the region where the influence of the polyion is not dominant has been proven to be well described by the two-state model proposed by Oosawa11 and by the linear approximation model of Manning.12 However, the state of simple ions near the polyion surface was discussed only in terms of the mathematical condition, which avoids the divergence in the solution of the P.B. eq for linear systems; for polyionic systems whose charge density ξ is larger than unity, counterions are condensed in the vicinity of the polyion to produce an effective ξ equal to 1. Thus, the atmospheric distribution of the simple ion in this region is not treated in these theories and it is inadequate to analyze the experimental data with these models in the case studied here, where we discussed the distributions of the electrostatic potential and protons near the polyion and extracted them by the pH indicator method. In this situation, we solved the nonlinear cylindrical P.B. eq numerically under the conditions corresponding to the experimental systems and used the results for the analysis. The P.B. eq is expressed as
1 d dφ r ) -4πλB(n+0e-φ - n-0eφ) r dr dr
[ ( )]
(5)
where r is the distance from the cell axis. This does not imply a rodlike conformation for the polyion chain but supposes that cylindrical symmetry is established locally around the main chain.13
λB ) e2/4πkT
(6)
is the Bjerrum length where e, , k, and T are the elementary charge, the dielectric constant of water, the Boltzmann constant, and the absolute temperature, respectively. φ is the dimensionless electrostatic potential normalized by kT. n(0 is the activity of cation/anion and is related to the mean concentration nj( by
nj( ) n(0〈e-φ〉
(7)
where 〈 〉 indicates a volume average. nj( is given by the concentration of excess monovalent electrolyte ns and the monomer unit concentration of PAAmHCl np () [Cl-]) by
nj+ ) ns, nj- ) ns + np
(8)
Equation 5 was solved with the following boundary conditions, in addition to eqs 7 and 8.
2λB dφ |r)a ) dr ab
(9)
φ(Rc) ) 0
(10)
a is the radius of the polyion and will be the only fitting parameter used in the data analysis. b is the charge spacing on the polyion chain, which was fixed to be 2.5 Å, corresponding to a fully stretched polymethylene chain. Rc is the cell radius (boundary), which was defined to reproduce the experimental polyion concentration by setting
Rc ) (πnpb)-1/2
(11)
792 J. Phys. Chem., Vol. 100, No. 2, 1996
Sumaru et al.
3.2. Apparent Proton Concentration. By use of n(0 and φ, which is a function of r, the theoretical value of the apparent proton concentration [H+]app,the was calculated from the following equations, as discussed in ref 16.
[H+]app,the ) βnjH β)
(
〈e2φ〉〈e-φ〉 〈eφ〉
+
njNa K′
)
(12) -1
(13)
where njH and njNa are the mean concentrations of protons and sodium ions. K′ is the dissociation constant of BPB2- and Na+, and its value had been determined to be 28.9 [mM].16 Because of the large value of K′, the influence of Na+ is small in all the systems studied here. If one neglects this term, the dimensionless factor β becomes a functional of φ only. Whether the polyion charge is positive or negative, β is smaller than unity when φ varies with the position r, and it approaches unity as φ becomes uniform (independent of r). Thus, β reflects the uniformity of φ sensitively and [H+]app,exp is expected to depend on both β and njH. All the numerical calculations were performed by using supercomputer M-1800 (Fujitsu, Japan) at the Kyoto University data processing center.
Figure 1. Visible spectra of BPB in PAAmHCl-HCl systems at various NaCl concentrations: [PAAmHCl] ) 3 mM, [HCl] ) 6 mN, and MW of PAAmHCl ) 60 000.
4. Results and Discussion 4.1. Characterization of BPB. 4.1.1. Determination of Kobs. In the calibration of BPB, the value of (A436 - 0.040A591)/ (A591 - 0.015A436) obtained at various HCl concentrations turned out to be accurately proportional to [HCl] and the result gave the Kobs value as its reciprocal slope. According to each calibration, the typical Kobs value was 0.21 [mM]. On the other hand, as a result of vis measurements for a PAAmHCl-HCl solution at various BPB concentrations, [H+]app observed for the PAAmHCl samples of both MWs was not changed by [BPB], where [BPB]/[PAAmHCl] is less than 10-3. This implies that BPB does not perturb the simple ion distribution in this range of concentrations. Therefore, the BPB concentration for the measurements was determined to be ∼2 µM. 4.1.2. BPB Adsorption onto PAAmHCl. The possibility that the negatively charged BPB molecules might be adsorbed onto the positively charged PAAm surface was checked by measuring λmax of BPB2- for all the vis data. This λmax value is strongly influenced by the dielectric environment around the BPB2molecules, as reported in our previous paper.16 We found that, λmax was always within the range 591.8-592.8. This suggests that the hydration state of the BPB molecules is scarcely affected by the coexistence of PAAm and that the interaction of BPB with PAAm remains an atmospheric one. 4.2. [H+]app of PAAmHCl-BPB Systems. 4.2.1. Procedure of Analysis. The experimental data for [H+]app ([H+]app,exp) obtained as functions of [NaCl], [PAAmHCl], degree of neutralization, and excess [HCl] were analyzed by fitting to the theoretical values. The polyion radius a was used as the only one fitting parameter because a is the characteristic that is expected to be the least influenced by added electrolyte concentration or [PAAmHCl]. Furthermore, varying a does not give rise to any contradiction to the physical situations expressed by the boundary conditions. In order to evaluate the state of the electrostatic potential in the system, each theoretical value of [H+]app was also fitted with the corresponding experimental one by varying a as well as by fitting the theoretical curves to sets of the experimental data obtained as functions of [NaCl] and [HCl], and the surface potential of the polyion φs was calculated under the conditions
Figure 2. [H+]app of PAAmHCl-HCl systems plotted against [NaCl] where [PAAmHCl] ) 3 mM. (b) and (9) indicate the experimental values at 3 and 6 mN excess HCl, respectively. The solid lines are the corresponding theoretical curves calculated from the PoissonBoltzmann cell model.
of the fitted model. In this operation, the radius a was regarded as a mere fitting parameter rather than the practical polyion size. Since φs is very sensitive to the value of the radius a, fitting a theoretical value of [H+]app to an experimental one by varying a is almost equivalent to fitting by varying φs. 4.2.2. [NaCl] Dependence. Visible spectra obtained for a PAAmHCl-HCl system at various NaCl concentrations are shown in Figure 1. As [NaCl] increases, A436 increases and A591 decreases, corresponding to larger [H+]app,exp. This tendency is analogous to that reported for PVS solutions in ref 16. Figure 2 shows the [H+]app measured for 3 mM PAAmHCl solutions containing 3 and 6 mN HCl as functions of added [NaCl]. In each case, [H+]app increases linearly with the [NaCl]. Since njH in eq 12 is equal to the excess [HCl] and independent of [NaCl], this increase of [H+]app is attributed solely to an increase in β. Figure 2 shows the results obtained for the PAAmHCl samples of MW ) 60 000 and 10 000, respectively,
Protons and Potentials in Poly(allylamine hydrochloride)
J. Phys. Chem., Vol. 100, No. 2, 1996 793
TABLE 1: Experimental Conditions and Fixed Polyion Radius a no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
[PAAmHCl] [HCl] [NaCl] [NaOH] [mM] [mN] [mM] [mN] 3 3 3 3 3 20 3-20 3-20 3-20 3 3 10 3 10
3 6 3 6 2 2 12 12 2 12 0-12 0-12 0-12 0-12
0-6 0-6 0-6 0-6 0-4 0-4
0-12
MW 60 000 60 000 10 000 10 000 60 000 60 000 60 000 10 000 60 000 60 000 60 000 60 000 10 000 10 000
a [Å]
σ [mN]a
6.3 6.6 6.5 6.3 7.0 4.7 6.1 6.2
0.0014 0.0035 0.0023 0.0034 0.0027 0.0015
6.3 6.5 5.9 6.5 5.9
0.0230 0.0250 0.0082 0.0049 0.0074
a
σ is the standard deviation between experimental data and theoretical values.
Figure 3. Calculated electrostatic potential in PAAmHCl solutions at various 1:1 electrolyte concentrations x as a function of distance d from the PAAm surface where [PAAmHCl] ) 3 mM and a ) 6.5 Å.
and no significant deviation between them was observed. The solid lines in the figures are the fitted theoretical curves, which agree well with the experimental values. The fitted a values for each series of data are tabulated in Table 1. The agreement between theory and experiment indicates that the behavior of [H+]app,exp can be reproduced by a model containing only one fitting parameter, and the validity of the assumptions in the model is strongly supported under these experimental conditions. Figure 3 shows the φ calculated numerically as functions of the distance from the polyion surface d at various 1:1 electrolyte concentrations x. This indicates that φ becomes more uniform and leads to the increase of β as x increases. When x > 2 mM, the large φ near the polyion surface decreases rapidly with d and becomes less than unity for d > 50 Å. In other words, the volume containing a large φ, which makes the major contribution to the reduction in the β value, is confined to within 50 Å of the polyion. Considering the typical persistence length of vinylic polyelectrolytes in aqueous solutions, the system is expected to be cylindrically symmetric in this range. Figure 4 shows the φs values calculated by fitting [H+]app,the to each [H+]app,exp value in Figure 2. φs is so large ((6-7)kT) that the β value is very small (∼0.01). This large φs is a result of the high charge density ξ on the PAAm main chain and indicates that the counterion (Cl-) is in a condensed state around the polyion. In ref 22, the |φs| value was estimated to be (5.25.8)kT for vinylic polyions with bulky ionizable groups on every monomer unit, which are each expected to have a larger radius than PAAmHCl, by determining the pK shift of a pH sensitive
Figure 4. Surface electrostatic potential calculated from the experimental [H+]app values of PAAmHCl-HCl systems at various NaCl concentrations where [PAAmHCl] ) 3 mM and [HCl] ) 6 mM. The solid lines are the corresponding theoretical curves.
Figure 5. [H+]app/[HCl] of PAAmHCl-HCl systems plotted against [NaCl] ([HCl] ) 2 mN). (b) and (9) indicate the experimental values at 3 and 20 mM PAAmHCl, respectively. The solid lines are the corresponding theoretical curves.
dye residue covalently attached to the main chain. Taking the difference in the polyion radius into account, the φs value determined here is consistent with the result in ref 22. As shown in Figure 3, the added NaCl not only makes the φ distribution more uniform, but also diminishes the value of φs. The same tendency was also observed in the experimental φs shown in Figure 4. Figure 5 shows the result of analogous measurements for the systems in which [HCl] ) 2 mN. In this figure, [H+]app normalized by excess [HCl] ()njH), which corresponds to β, is plotted against [NaCl]. [H+]app/[HCl] for two different [PAAmHCl] (3 and 20 mM) takes nearly the same value at [NaCl] ) 0 mM, while the rate at which it increases with β depends greatly on [PAAmHCl]. The values of the parameter a, which gave the best-fitted theoretical curve (solid lines), are quite different (7 and 4.7 Å for [PAAmHCl] ) 3 and 20 mM, respectively). In order to investigate the [NaCl] dependence of [H+]app/[HCl] in more detail, β values were calculated as functions of the 1:1 electrolyte concentration x holding a constant at different [PAAmHCl] (Figure 6). The behavior of β can be described concisely by regarding β as a first-order function of x; the intercept at x ) 0 increases and the slope decreases with increasing [PAAmHCl]. This interesting tendency will be discussed more precisely in the following section. 4.2.3. [PAAmHCl] Dependence. Figure 7 shows the [PAAmHCl] dependence of [H+]app/[HCl] obtained for the PAAmHClHCl systems that contain a certain concentration of HCl (12
794 J. Phys. Chem., Vol. 100, No. 2, 1996
Figure 6. Theoretical β values for PAAmHCl systems at various PAAmHCl concentrations as functions of the 1:1 electrolyte concentration x. a is fixed to be 6.5 Å.
Figure 7. [H+]app/[HCl] of PAAmHCl-HCl systems plotted against [PAAmHCl] ([HCl] ) 12 mN). (b) and (9) indicate the experimental values for PAAmHCl’s whose MW are 60 000 and 10 000, respectively. The solid lines are the corresponding theoretical curves.
mN). β decreases with [PAAmHCl] monotonically, and the influence of MW on PAAmHCl seems to be negligible. The point we should draw attention to is that PAAmHCl does not provide the system with any protons because protons are hardly dissociated from the amino groups on the polyion chain. Therefore, the [PAAmHCl] dependence of [H+]app can be attributed solely to the change of φ distribution in the system. The theoretical curve calculated by holding a constant (solid line) did not reproduce the large [PAAmHCl] dependence of β satisfactorily. In order to analyze this aspect in terms of the φ situation, the surface electrostatic potential φs was calculated as previously referred to in section 4.2.1. (Figure 8). In contrast to the φs variation calculated when a is held constant (solid lines), the experimental values tend to remain constant. As observed in this figure, the larger the [PAAmCl] is, the smaller a value reproduces [H+]app,exp. This tendency agrees with that observed in Figure 8 (4.2.2.) Compared to the tendency of φs to decrease with increasing [NaCl], this result indicates that φs depends on the ionic strength rather than on the concentration of the polyion itself. Since φs was expected to decrease with increasing [PAAmHCl] from the model based on a constant a, we suggest that there are some structural factors that contribute to keep φs constant, such as a change in the chain conformation, in practical systems. In order to investigate the [PAAmHCl] dependence of [H+]app at a low ionic strength, a series of analogous measurements were carried out at [HCl] ) 2 mM (Figure 9). The experimental [H+]app, which is normalized by excess [HCl] and plotted against
Sumaru et al.
Figure 8. Surface electrostatic potential calculated from the experimental [H+]app values of PAAmHCl-HCl systems at various PAAmHCl concentrations. The symbols are as in Figure 7. The solid lines are the theoretical curves based on a fixed a.
Figure 9. [H+]app/[HCl] of PAAmHCl-HCl systems plotted against [PAAmHCl] for the low [HCl] condition ()2 mN). (b) indicates the experimental values for PAAmHCl whose MW is 60 000. The solid and dashed lines indicate the theoretical [H+]app/[HCl] ()β) and surface electrostatic potential based on a fixed a ()6.0 Å), respectively.
[PAAmHCl], has a shallow and broad minimum in the range 6 mM < [PAAmHCl] < 10 mM. The theoretical curve for β (solid line) based on a constant a ()6.0 Å) also has a minimum, but its deviation from the experimental data is quite large. The behavior of the theoretical β is a result of a balance between two factors that compete with each other: the increase in [PAAmHCl] and the decrease in φs. Initially, the former factor exceeds the latter, causing β to decrease, and the subsequent increase is brought about by the opposite effect. However, the minimum point of the experimental [H+]app/[HCl] is at a higher [PAAmHCl] than the theoretical value, and the extent of the following increase is also much smaller than that predicted by the theoretical curve for β. These discrepancies can also be explained similarly for the case in Figure 7 by taking into account the fact that the real φs does not decrease with [PAAmHCl] as much as that estimated by the cell model. 4.2.4. R-Dependence. Figure 10 shows the variation of [H+]app measured for the system with 3 mM PAAmHCl and 12 mN HCl neutralized with NaOH. [H+]app is accurately proportional to 1 - R (R is the degree of neutralization), which in turn is proportional to njH. Since what occurs in the process of neutralization is that Na+ takes the place of H+, the φ distribution, and therefore β, is not expected to change in so far as H+ and Na+ behave in the same way as monovalent cations. Consequently, the change in [H+]app reflects only the change in njH and is proportional to 1 - R. However, the
Protons and Potentials in Poly(allylamine hydrochloride)
J. Phys. Chem., Vol. 100, No. 2, 1996 795 to be negligible. All the experimental data have an evident concavity with respect to [HCl] at lower [H+]app. This tendency can be considered to reflect two independent effects of the added HCl; as an added electrolyte relaxes the nonuniformity in φ, it contributes to an increase in β, while, on the other hand, it supplies the system with H+ and increases njH. The excellent agreement between the experimental data and the theoretical curves in Figure 11 indicates that the variation of [H+]app can be explained quantitatively by considering these two effects of adding HCl. 5. Conclusion
Figure 10. [H+]app of PAAmHCl-HCl system plotted against 1 - R where [PAAmHCl] ) 3 mM, [HCl] ) 12 mN, and MW of PAAmHCl ) 60 000. (b) indicates the experimental values, and the solid line is the corresponding theoretical curve.
Figure 11. [H+]app of PAAmHCl-HCl system plotted against excess HCl concentration. (b) and (9) indicate the experimental values at 3 and 10 mM PAAmHCl, respectively.
experimental data exhibit a slight concavity at lower [H+]app, which disagrees with the behavior observed in our previous study on the PVS system.16 This suggests that the fraction of H+ interacting with BPB is in practice smaller than that estimated assuming an identical distribution for H+ and Na+. Although this tendency may reflect a difference in the excluded volume of the hydrated cation or other factors, the deviation between the experimental data and the theoretical curve is negligible and indicates that the φ distribution is scarcely influenced by the cation species. 4.2.5 [HCl] Dependence. Figure 11 shows the excess [HCl] dependence of [H+]app obtained for the 3 and 20 mM PAAmHCl solutions. The solid lines indicate the theoretical curves fitted by the polyion radius a. As tabulated in Table 1, the experimental data obtained at [PAAmHCl] ) 20 mM were reproduced by a smaller a than in the 3 mM system, and this tendency agrees qualitatively with the result of the analysis in section 4.2.3. For this case also, the influence of MW appeared
In this study, the pH indicator method was applied to the PAAmHCl solutions and the distribution of protons and the electrostatic potentials were evaluated quantitatively. Most of the experimental results were well reproduced by the cylindrical Poisson-Boltzmann cell model with only one fitting parameter, and it has been proven in a unified manner for both polyanionic and polycationic systems that the [H+]app obtained by this method can be interpreted as a product of β and njH as expressed by eq 12. The electrostatic potential at the polyion surface φs was determined from the experimental [H+]app and seemed to be independent of [PAAmHCl] at constant [HCl]. In this case, the theoretical model could not quantitatively reproduce the observed behavior with one fitting parameter. By use of more fitting parameters, it may be possible to determine some other size parameters or the conformational structure of the polyions. It is to be anticipated that this method would be applicable to polyionic systems when there are large conformational transitions accompanying a change in temperature or ionic strength.23 On the other hand, the MW of PAAmHCl in this range of the measurements turned out to have little influence on the observed characteristics, even on the [PAAmHCl] dependence, which seems the most sensitive to MW. This is in contrast to the significant MW dependence detected in our recent experiments on PVS systems with MW ) 1300 and 20 000, suggesting that the cylindrical symmetry is well established locally in PAAmHCl solutions when the MW is larger than 10 000. In this way, the pH indicator method has been shown to be a very powerful tool for investigating the simple ion distribution not only for polyanionic systems but also for polycationic ones. However, the experimental range was only in the region where β is much smaller than unity. Generally, β is reduced effectively by the existence of the polycation because of the positively large φ near the main chain of the polyion. Of course, β is expected to increase with increasing added [NaCl] and with the decreasing concentration of PAAmHCl itself. However, it was impossible to realize the experimental condition in which β would be large compared to unity because of some practical problems. First, the excess amount of added NaCl would reduce the β value although β increases in the initial stage of adding NaCl. Initially, the first term in the parentheses of eq 13 decreases rapidly, contributing to the increase of β. Then the second term becomes dominant and causes the decrease of β as [Na+] goes far beyond K′. Secondly, the concentration of PAAmHCl has a lower limit because of the possibility that BPB will perturb the proton distribution in the system as discussed in the section 2.2. With all these limitations, however, this method still has much possibility for the application to various polyionic systems. Very recently, we have intensively studied the aqueous solutions of poly(vinylamine) (PVAm), which is characteristic for its property of protonation, at various conditions. In contrast to the PAAm, PVAm is difficult to protonate completely because of the strong interaction between ammonium groups on the main
796 J. Phys. Chem., Vol. 100, No. 2, 1996 chain. In order to investigate this unique property of PVAm in detail, it was necessary to determine the concentration and the distribution of the free protons in the system quantitatively. For this purpose, the pH indicator method was applied together with the other techniques such as potentiometry and conductometry, and some significant information about the protonation of PVAm was obtained. The details of the results of these analyses on PVAm solutions will be described in a forthcoming paper. Acknowledgment. The authors express sincere gratitude to Dr. Martin V. Smalley for his valuable comments and discussions. K.S. gratefully acknowledges the support of this work by JSPS Research Fellowships for Young Scientists. References and Notes (1) (a) Bloomfield, V. A.; Wang, L. Macromolecules 1990, 23, 194. (b) Bloomfield, V. A.; Wang, L. Macromolecules 1990, 23, 804. (2) Noda, I.; Tsuge, T.; Nagasawa, M. J. Phys. Chem. 1970, 74, 710. (3) Oostwal, M. G.; Blees, M. H.; de Bleijser, J.; Leite, J. C. Macromolecules 1993, 26, 7300. (4) Nordmeier, E. Polym. J. 1993, 25, 19. (5) Kotin, L.; Nagasawa, M. J. Chem. Phys. 1962, 36, 873. (6) Nagasawa, M.; Murase, T.; Kondo, K. J. Phys. Chem. 1965, 69, 4005. (7) Katchalsky, A.; Spitnik, P. J. Polym. Sci. 1947, 2, 432. (8) Biswas, B.; Williams, P. A.; Phillips, G. O. Polymer 1992, 33, 1284.
Sumaru et al. (9) Marinsky, J. A. J. Phys. Chem. 1992, 96, 6484. (10) Takahashi, A.; Kato, N.; Nagasawa, M. J. Phys. Chem. 1970, 74, 944. (11) Oosawa, F. J. Polym. Sci. 1957, 23, 421. (12) Manning, G. S. J. Chem. Phys. 1969, 51, 924. (13) (a) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (b) Gueron, M.; Weisbuch, G. J. Phys. Chem. 1979, 83, 1991. (14) Das, T.; Bratko, D.; Bhuiyan, L. B.; Outhwaite, C. W. J. Phys. Chem. 1995, 99, 410. (15) van der Marrel, J. R. C.; Groot, L. C. A.; Hollander, J. G.; Jesse, W.; Kuil, M. E.; Cotton, J. P.; Jannik, G.; Lapp, A.; Farago, B. Macromolecules 1993, 26, 7295. (16) Sumaru, K.; Matsuoka, H.; Yamaoka, H. J. Phys. Chem. 1994, 98, 6771. (17) Itaya, T.; Ochiai, H. J. Polym. Sci., Part B: Polym. Phys. 1992, 30, 587. (18) Thiele, H.; Gronau, K.-H. Macromol. Chem. 1963, 59, 207. (19) Bloys van Treslong, C. J.; Staverman, A. J. Recl. Chim. Pays-Bas 1974, 93, 171. (20) Kobayashi, S.; Suh, K.-D.; Shirokura, Y. Macromolecules 1989, 22, 2363. (21) Katchalsky, A.; Mazur, J.; Spitnik, P. J. Polym. Sci. 1957, 23, 513. (22) Morishima, Y.; Kobayashi, T.; Nozakura, S. Macromolecules 1988, 21, 101. (23) (a) Strauss, U. P.; Barbieri, B. W.; Wong, G. J. Phys. Chem. 1979, 83, 2840. (b) Nagasawa, M.; Holtzer, A. J. Am. Chem. Soc. 1964, 86, 538. (c) Hermans, J., Jr. J. Phys. Chem. 1966, 70, 510. (d) Bednar, B.; Morawetz, H.; Shafer, J. A. Macromolecules 1985, 18, 1940.
JP9520963
