Complete asymptotic solution of cylindrical and spherical Poisson-Boltzmann equations at experimental salt concentrations

We report an exact analytic representation of the nonlinear Poisson-Boltzma nn (PB) potential as a function of radial distance from a cylindrical or sp herical polyion in solutions containing a symmetrical electrolyte, in the f orm of an asymptotic series in elementary functions, generally valid at rad ial coordinates larger than Debye length. At sufficiently high salt concent rations, where the ratio of Debye length (kappa(-1)) to the polyion radius (a) is sufficiently small ((kappa a)(-1) less than or equal to 1), the asym ptotic series is valid at any distance from the polyion surface. This analy tic representation satisfies exactly the complete nonlinear Poisson-Boltzma nn equation, subject to the boundary condition on the derivative of potenti al at infinity, and therefore contains one integration constant, which in t his salt range we determine to an accuracy of order (kappa a)(-2). Because it explicitly introduces for the first time all the terms which arise due t o nonlinearity of the PB equation, this analytic representation clarifies t he connection between the exact solution of the PB equation and various app roximations including the Debye-Huckel approximation (the solution of the l inearized PB equation). From these considerations we obtain a new approxima te solution designated "quasi-planar" and expressed in elementary functions , which we show to be accurate at any distance from the polyion surface at typical experimental salt concentrations (e.g., 0.1 M 1:1 salt concentratio n for double-stranded DNA, where the PB equation retains its accuracy by co mparison to Monte Carlo simulations). We apply our analysis to the calculat ion of the electrostatic free energy and the salt-polyelectrolyte preferent ial interaction (Donnan) coefficient (Gamma).