J. Phys. Chem. B 1997, 101, 9833-9837
9833
Polyacids Self-Dissociation Model Augusto Agostinho Neto,* Elso Drigo Filho, Marcelo Andres Fossey, and Joa˜ o Ruggiero Neto Departamento de Fı´sica Instituto de Biocieˆ ncias, Letras e Cieˆ ncias Exatas UNESP, 15054-000-Rua CristoVa˜ o Colombo, 2265, Jd. Nazareth, Caixa Postal 136 Sa˜ o Jose´ do Rio Preto, SP, Brazil ReceiVed: June 2, 1997; In Final Form: August 28, 1997X
This work deals with a model to interpret pH measurements of solutions of weak rodlike polyacids, in the absence of added salts or titrating base. The polyacid is modeled as a series of point charges discretely distributed in a straight line with a distance of closest approach for the protons and an average distance between dissociable monomers, projected in the polymer chain axis. Aside from these two geometrical parameters, the dissociation constant for the isolated monomer that describes the proton dissociated monomer interaction forms the basis of the model. The assumption of cylindrical symmetry and the adoption of the cell model lead to a form written in terms of elementary functions for the mean electrostatic potential. Values of pH (related to the proton concentration in a region beyond the influence of the polyacid) as a function of polymer concentration are displayed graphically for some values of the geometrical parameters and of the dissociation constant. Theoretical predictions of pH values as a function of polymeric concentration are compared with measured values for poly-L-glutamic and polygalacturonic acids, and a good agreement is found. Theoretical values for the dissociation degree in terms of polymeric concentration are shown for the two experimentally investigated systems. These values are in a range appreciably smaller than what is usually studied as a result of titration.
Introduction The process of dissociation of ionizable groups in weak polyacids is in general followed by the onset of long-range interactions due to the electrostatic repulsion between charged groups in the polymeric chain. Proton relaxation time measurements carried out by Kotin and Nagasawa1,2 for weak polyacids showed that in the acid form the protons are tightly bound to the acid groups whose electric charges are neutralized in part by bound protons. However, despite the almost absent translational freedom of the protons bound to the weak acid groups, a relatively high dissociation of these groups is frequently observed for these polyacids in solution independently of any ion exchange process. The self-dissociation for dilute solution of these polyacids is on the order of 10% of the number of dissociating monomers in the polymer and also presents a strong dependence on the distance between dissociable monomers in the chain. The degree of dissociation in a self-dissociating process accounts also for the variation of the conductivity of a weak polyacid solution with the polymer concentration.3,4 The strong dependence of the conductivity values observed for the weak polyacids in aqueous solution with polymer concentration (σ ∝ Cp3/2)4,5 also suggests that the long-range interactions between counterions in solution and charged groups in the polymer chain play an important role in the self-dissociation process. The aim of the present work is to describe the self-dissociation process of weak rodlike polyacids taking into account the longrange interaction between the partially charged chain and protons in the solution. The long-range electrostatic interaction is described through a mean field approach, and the hypothesis of a local chemical equilibrium is used for a macroscopic description of the degree of ionization.
approach of the protons, measured from the charged line. A value b is used to measure the distance projected in the line between consecutive dissociating monomers. The single chain modeled in this way is similar to the model previously discussed.6-8 The monomers in the macromolecule are considered to be equivalent, distinguishable, and independent. The hypothesis of chemical equilibrium and the thermodynamic law of mass action lead to a relation for the degree of dissociation containing the equilibrium dissociation constant, K(T):
1 1+
Fp K(T)
where Fp is the proton concentration (number density) at a distance Rp from the chain axis. The effective electric charge on the line introduces inhomogeneities on the proton distribution around the polyacid chain. pH measurements give information about the protonic concentration in a region out of the influence of a particular polyacid chain. Let this protonic concentration be denoted by Fp0. The concentrations Fp and Fp0 are related by9
Fp ) Fp0 exp(-ω(r)/kT)
The polyacid is modeled as an extended line of point charges to which the protons are bound, Rp being the distance of closest * Corresponding author. Email:
[email protected] X Abstract published in AdVance ACS Abstracts, November 1, 1997.
S1089-5647(97)01784-7 CCC: $14.00
(2)
where ω(r) is the mean force potential. Electrostatic interactions play the main role as far as proton concentration around the chain is concerned, and the mean electrostatic potential Ψ(r) is taken as the mean force potential. Poisson’s equation
∇2Ψ(r) ) -
Model
(1)
R)
F*(r)
(3)
relates the charge distribution (F*) and the electrostatic potential in a medium with dielectric permittivity ( is the product of the dielectric constant of the medium and the vacuum permittivity 0). An additional assumption of the cylindrical symmetry © 1997 American Chemical Society
9834 J. Phys. Chem. B, Vol. 101, No. 47, 1997
Agostinho Neto et al.
and the use of the eq 2 give the form below for the Poisson Boltzmann equation:
1 d dΦ χ ) -2 exp(-Φ) χ dχ dχ
(4)
written in terms of dimensionless parameters
Φ) κ2 )
eΨ kT
Fp0e2 2kT
and
χ ) κr with r representing the distance from the polyacid chain (line charge). The cell model establishes a distance Rc (Rc ) (πCpb)-1/2) as the boundary of the influence of a particular chain over the protons in the neighborhood of the polymer, being Cp the polymer concentration. The boundary conditions are
dΦ ) 0 at χ ) κRc dχ
(5)
implying overall electroneutrality inside the cylinder of radius Rc and
dΦ -e2 1 ) -δ ) dχ 2πbkT [1 + Fp0 exp(-Φ(χ))/K(T)] at χ ) κRp
{
}
χ2 sinh2[c ln(dχ)] 2 c
(7)
with c and d as integration constants evaluated with the boundary conditions (5) and (6). From eqs 5 and 6 result two equations relating the integration constants c and d:
1 - c2 1 + c coth[c ln(Rc/Rp)]
(8)
1 + c coth[c ln(dκRc)] ) 0
(9)
1 - c2 ) κ2Rc2
(10)
δ)
with that previously assumed; therefore the process is repeated until coincidence. Results
(6)
The solution of eq 4 was previously presented and described by Katchalsky,5,6 and it is
Φ ) ln
Figure 1. Theoretical plots of pH vs Cp for different values of the distance between acid groups (b). The curves presented are for b ) 1.0 Å (s), b ) 2.0 Å (- - -), b ) 3.0 Å (‚‚‚), b ) 5.0 Å (-‚-), b ) 7.0 Å (-‚‚-), and b ) 10.0 Å (---). In each case the pKm value is equal to 3.5 and the distance of closest approach (Rp) is fixed as 8 Å.
and
with
δ, as can be seen in eq 6, depends on the value of the potential at r ) Rp, and the potential depends on the value of δ. The set of coupled nonlinear equations must be solved iteratively. The solution is obtained according to the following steps: (1) The parameters Rp, Rc, b, and K(T) are given data, and an initial value for δ is assumed; (2) with the values of c and d calculated, Φ(χ) (and Φ(κRp)) can be calculated and the proton concentration at r ) Rp determined; (3) with this value and the data a value for δ can be obtained; (4) this value may not coincide
The proposed model requires three parameters as input data: the dissociation constant of the monomer weak acid group (pKm), the distance between acid groups in the polyacid, and the distance of closest approach of the proton to the chain axis. Of these parameters, the distance between dissociable monomers, and consequently the maximum value for the linear charge density, can be obtained from experimental data such as potentiometric and conductometric titrations10,11 or from crystallographic data.12 The monomer dissociation constant can also be obtained from the experimental titrations of the simple acid in solution. The distance of closest approach is the factor that is more difficult to obtain from simple experimental assays. However this parameter is more easily inferred from theoretical simulations of the polymer chain through commercially available software. In view of the dependence of the model on these parameters, it is interesting to explore how these parameters influence the theoretical values of pH as function of the polymer concentration C p. Plots of the theoretical values of the pH vs Cp obtained for different values of the distance between acid groups in the polymer are shown in Figure 1. These values are calculated using the same pKm value (pKm ) 3.5) and a fixed distance for the proton closest approach to the polymer axis (Rp ) 8 Å). The theoretical pH values are related to the proton concentration in regions where the electric mean field due to the charges in the chain and to the surrounding protonic distribution vanishes, that is, in the region corresponding to the border of the “cell”. The results are consistent for a fixed polymer concentration Cp; the increase in the distance between consecutive dissociable monomers results in a lower effective linear charge density and consequently leads to a greater concentration of protons in the bulk.
Polyacids Self-Dissociation Model
Figure 2. Theoretical plots of pH vs Cp for a fixed monomer dissociation constant pKm ) 3.5 and two different values for the distance of ionizable monomers (b). The curves are related to b ) 2.0 Å and Rp ) 5.0 Å (s); b ) 2.0 Å and Rp ) 10.0 Å (---); b ) 10.0 Å and Rp ) 5.0 Å (‚‚‚); and b ) 10.0 Å and Rp ) 10.0 Å (-‚-).
The distance of closest approach of the proton to the polyacid axis is another parameter on which the theoretical pH values are dependent. The study of this dependence for polyacids bearing two different linear charge densities is shown in Figure 2. The theoretical values are calculated using a fixed monomer dissociation constant (pKm ) 3.5) and considering different values for the distance between dissociable monomers (b ) 2 Å and 10 Å) and the values for the proton closest approach (Rp ) 5 Å and 10 Å). From these plots it can be seen that there tends to be a slight increase in the difference between pH values with the polymer concentration for a given Rp value. This difference ranges up to one-tenth of a pH unit as the polymer concentration rises to 40mM, which is a concentration very difficult to use in an experimental assay. The differences in the pH values decrease with a decreasing linear charge density. It is noteworthy that the differences observed in the pH values are less than the experimental error in the pH measurement. The behavior displayed in Figure 2 is consistent with the fact that for a charged line in vacuum the electric potential is weakly dependent on the distance to the charged line (a logarithmic function) and directly proportional to the linear charge density. For a charged line in solution in the presence of surrounding protons the equilibrium is modulated by two opposing factors. One of the factors is the fact that the linear charge density is dependent on the value of the electric potential at the distance representing the distance of closest approach for the protons. Mentally representing the equilibrium approach, one has that a high charge density and a small distance (Rp) result in high electric potential, which determines a high value for the local protonic concentration. At equilibrium, this high protonic concentration results in a lower effective linear charge density and a lower electric potential. On the other hand, a low linear charge density and a greater distance (Rp) both determine a low potential, a low local concentration for the surrounding protons, and consequently more dissociated monomers in equilibrium. These effects will appear more intensively in the limit of high
J. Phys. Chem. B, Vol. 101, No. 47, 1997 9835
Figure 3. Theoretical plots of pH vs Cp for fixed values of the distance of ionizable monomers (b ) 5.0 Å) and the distance of closest approach (Rp ) 10.0 Å). Values for pKm are 3.0 (s), 3.5 (---), and 4.0 (‚‚‚).
polymer concentration, since there will be a lower volume to confine the protons, and this fact corresponds to a small decrease in pH values, as the value of Rp increases. The plots displayed in the Figure 3 show that the theoretical pH values have a strong dependence on the monomer dissociation constant. For instance a variation of one unit in the pKm value results in a variation of 0.25 in the theoretical pH values. However the pKm value can be accessed with high accuracy (around 3%), which reinforces the applicability of the model in the description of rodlike polyacid self-dissociation. A comparison of theoretical dissociation curve predictions with experimental values was made for two different rodlike weak polyacids bearing different values for the parameter b: polygalacturonic acid (PGL) and poly-L-glutamic acid (PGLU). These polymers were transformed in the acid form through dialysis against acetic acid and exhaustively against water to remove the excess acid. These experiments were carried out by adding polymer to water and measuring the pH of the solution; in these experiments the polymer concentration varied over a wide range. The experimental and theoretical curves for these polyacids are shown in Figure 4. The monomer dissociation constants were obtained titrating the monomer in the acid form. The values obtained from these titrations are in very good agreement with those reported in the literature: pKm ) 3.32 for galacturonic13 and pKm ) 4.07 for glutamic acid.14 The distances of the proton closest approach to the line charge were calculated from theoretical chains generated through molecular mechanics software using crystallographic parameters in these simulations. The values used are 5.87 Å for PGL and 9.7 Å for PGLU. The experimental and theoretical curves of pH vs Cp for these polyacids are shown in Figure 4. The parameter b ) 4.453 Å used for the polygalacturonic acid was obtained from crystallographic data.12 For poly-L-glutamic b ) 1.5 Å, the value used by Murai and Sugai15 in the simulations of the pK vs R curves. These values were also obtained from molecular mechanics simulations. One can see in Figure 4 a good agreement between values from experiment and theoretical prediction. At this point
9836 J. Phys. Chem. B, Vol. 101, No. 47, 1997
Figure 4. Values of pH as a function of Cp for poly-L-glutamic acid (PGLU) and polygalacturonic acid (PGL). The lines show the theoretical values where pKm ) 4.07, b ) 1.5 Å, and Rp ) 9.7 Å for PGL and pKm ) 3.32, b)4.453 Å, and Rp ) 5.87 Å for PGL are used. The discontinuous curves represent experimental data for PGLU (O) and PGL (dilution data ] and concentration data 9).
a remark is made. The success of the nonlinear PoissonBoltzmann equation in this application, in contrast to the problems it has encountered in other contexts, with the necessity of adjusting the polymer radius to unrealistically high values, is connected to the low values for the resulting effective linear charge densities in the chain. The relatively high values for the reduced potential at Rp, in the range -4 to -1 for PGL and from -5 to -1 for PGLU, make the linearization of the equation, leading to the Debye-Hu¨ckel linear regime, a poor approximation, mainly in the region near the chain. Figure 5 shows the theoretical degree of dissociation for the two systems experimentally investigated. It can be noted that the dissociation degree resulting from a self-dissociation process is in a range of dissociation degrees appreciably smaller than what is usually studied as a result of a titration.8 The curve in Figure 5 and all the curves in previous figures can be understood by taking into account the energy (electrostatic) and entropy balance. An amount of dissociated protons has a number of accessible states that depends on polymer concentration; the lower the concentration value, the higher the entropy gain since a greater volume is reserved for dissociated protons. The electrostatic energy value, considering a fraction of dissociated monomers, depends on the value of the parameter b. For a given polymer concentration lower values of b correspond to higher values of the electrostatic energy. The dissociation degree depends primarily on the values of the parameter b and K(T), and the dependence on Rp is weak. Direct measurements of the degree of dissociation will be a decisive test for the model. The theoretical values of the dissociation degree cannot be compared directly with the experimental values obtained from pH measurements using the Henderson-Hasselbach equation because these values are only apparent ones. The true dissociation degree should be obtained from NMR experiments, light scattering, or osmotic pressure determinations.6 As a comment about the matters discussed, one can see that the model can be used to indirectly evaluate pK values of acids
Agostinho Neto et al.
Figure 5. Plots of dissociation degree as a function of Cp for polyL-glutamic acid with pKm ) 4.07, b ) 1.5 Å, and Rp ) 9.7 Å (s) and for polygalacturonic acid with pKm ) 3.32, b ) 4.453 Å, and Rp ) 5.87 Å (---).
that polymerize, just measuring pH values for a polymer concentration range with the knowledge of the geometrical parameters b and Rp. Conclusion The model developed, dependent on the geometrical parameters b and Rp and on the monomeric equilibrium dissociation constant, allows the precise description of the behavior of pH values as a function of polymeric concentration for rodlike polymer solutions in the absence of added salts or titrating base. The curves pH versus Cp show a strong dependence on K(T) and b, parameters that can be easily accessed experimentally and determined with precision. The measured pH values as a function of the concentration can also be obtained with good precision, and such values compared with the theoretically predicted values are in excellent agreement with the model presented. Acknowledgment. The authors thank Dr. Pedro Geraldo Pascutti, who provided geometrical parameters of galacturonic acid and poly-L-glutamic acid. E.D.F. and J.R.N. are grateful to Fundac¸ a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo (FAPESP) and to Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) and PADCT (J.R.N.) for financial support. The recommendations of a referee, to improve the clearness of the paper, were followed in the revision. The authors thank the referee for his constructive observations. References and Notes (1) Nagasawa, M.; Kotin, L. J. Am. Chem. Soc. 1961, 83, 1026. (2) Rice, S. A.; Nagasawa, M. Polyelectrolyte Solutions; Academic Press: New York , 1961; Chapter 9. (3) Eisenberg, H. J. Polym. Sci. 1958, 30, 47. (4) Vink, H. J. Chem. Soc., Faraday Trans. 1981, 77, 2439. (5) Katchalsky, A. Pure Appl. Chem. 1971, 26, 327. (6) Katchalsky, A.; Alexandrowicz, Z.; Kedem, O. In Chemical Physics of Ionic Solutions; Conway, B. E., Barradas, R. G., Eds.; Wiley: New York, 1966.
Polyacids Self-Dissociation Model (7) Cesaro, A.; Benegas, J. C. Makromol. Chem. Rapid Commun. 1989, 10, 547. (8) Manning, G. S. J. Phys. Chem. 1981, 85, 870. (9) Hill, T. L. An Introduction to Statistical Thermodynamics; AddisonWesley Publishing Co.: Reading, MA, 1962. (10) Ruggiero, J.; Vieira, R. P.; Moura˜o, P. A. S. Carbohydr. Res. 1994, 256, 275.
J. Phys. Chem. B, Vol. 101, No. 47, 1997 9837 (11) Milas, M.; Shi, X.; Rinaudo, M. M. Biopolymers 1990, 30, 451. (12) Walkinshaw, M. D.; Arnott, S. J. Mol. Biol. 1981, 153, 1055. (13) Morris, E. R.; Rees, D. A; Sanderson, G. R.; Thom, D. J. Chem. Soc., Perkin Trans. 2 1975, 2, 1418. (14) Voet, D.; Voet, J. G. Biochemistry J. Wiley & Sons: New York, 1990. (15) Murai, N.; Sugai, S. Biopolymers 1974, 13, 857.
