Transformed Poisson−Boltzmann Relations and Ionic Distributions

• Published in: The journal of physical chemistry. B Vol. 104; no. 48; pp. 11528 - 11540
• Main Authors: Qian, Hong, Schellman, John A
• Format: Journal Article
• Language: English
• Published: American Chemical Society 07.12.2000
• ISSN: 1520-6106; 1520-5207
• DOI: 10.1021/jp994168m

For many applications, charge distributions around macromolecules in aqueous solution are of greater interest than the electrical potential. We show that it is possible to use the Poisson−Boltzmann (PB) relation to develop differential equations for the ionic distributions. The solutions to these equations are the integral distribution functions whose derivatives give the charge density functions for counterions and coions. In this formalism the salt-free atmosphere of a cylindrical polyelectrolyte is very easily solvable for the counterion. Quantities such as the “condensation radius” (Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 287−312) and the “Bjerrum association radius” (Bjerrum, N. Investigations on Association of Ions, I. In Niels Bjerrum Selected Papers; Munksgaard:  Copenhagen, 1926; pp 108−19) appear naturally as inflection points in curves of the counterion distribution functions. Moreover, a number of the properties of condensation theory arise as scaling limits of the transformed PB equation. In the presence of added salt separate equations can be derived for the excess charge distributions of counterions and coions. In this case the total excesses of counterion φct and of coion φco are simply related to experiment. Various combinations of these two quantities lead to formulas for (1) the total charge, (2) Donnan exclusion, (3) counterion release (Record, M. T.; Lohman, T. M.; de Haseth, P. J. Mol. Biol. 1976, 107, 145−158), and (4) fraction of “condensed” ions. Bjerrum's theory of ion assocition and Manning's theory of counterion condensation are discussed in the context of the transformed Poisson−Boltzmann theory.