Painlevé solution of the poisson-boltzmann equation for a cylindrical polyelectrolyte in excess salt solution

Abstract

An analytic solution for the reduced potential y connecting the radial boundary conditions at the cylinder surface (r = a) and infinity (r=∞) for the uni-univalent Poisson-Boltzmann equation d²y/dr²+(1/r) dy/dr = sinh y is made available through new developments in isomonodromy theory. Exact connection formulate for a= 0 lead to an excellent analytic approximation for the surface potential and the effective charge at a distance, providing the basis for an asymptotically exact matched solution. Manning's condensation theory and Katchalsky's matched solution are assessed in the light of these new results. Thermodynamic quantities such as activity coefficients may be evaluated explicitly. The extension of the theory to various multivalent salts is obtained via non-linear transformations.