Distribution of counterions around a cylindrical polyelectrolyte and manning's condensation theory

The distribution of counterions around a charged polyion cylinder is calculated by several methods. First, the Debye‐Hückel approximation is used, and it is shown that Manning's condensation hypothesis is necessry to avoid overneutralization of the polyion charges by the counterions when the linear‐charge‐density parameter, ξ, of the polyion exceeds the critical value of unity. However, it appears that this method of getting this result involves inconsistent application of Debye‐Hückel theory. Therefore, we turn to the analytical solution of the Poisson‐Boltzmann equation that was obtained by Alfrey, Berg, and Morawetz for a polyion cylinder plus a neutralizing number of counterions but without added salt. One of the integration constants of this solution is a radius, which we call RM, within which lies precisely the fraction of counterions that Manning assumes to condense in his theory. This radius can be rather large, however, so that the “Manning fraction” of condensed ions actually forms a diffuse cloud whose size varies with the polyelectrolyte concentration; RM varies as κ−1/2, where κ is the Debye‐Hückel screening parameter. The Manning fraction, 1 – 1/ξ, and its associated radius are unique in their behavior with dilution; smaller fractions stay within finite radii, while with larger fractions the corresponding radii increase as κ−1. Thus, the condensation hypothesis does have a simple mathematical foundation in the Poisson‐Boltzmann equation. Finally, by comparison with numerical solutions, we find that these conclusions are not significantly changed even when salt is added to the polyelectrolyte. A short table of numerical solutions of the Poisson‐Boltzmann equation in cylindrical geometry is given, together with tables of coefficients tht enable one to discover the particular solution that applies for a given polyion radius and charge density.

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