Solution of the non-linear Poisson—Boltzmann equation in the interior of charged, spherical and cylindrical vesicles. I. The high-charge limit

Abstract

Abstract Solutions are obtained for the truncated Poisson—Boltzmann (TPB) equation, valid for high surface charges, in the interior of charged cylinders and spheres. The solution for the cylinder is analytic, for the sphere numerical. For the sphere a simple polynomial algorithm is presented which can approximate the exact solution to any desired accuracy. A plot of various physical quantities is given for vesicles of inner radii 150 and 500 A, respectively. For each geometry a single universal function exists which can accommodate any value ψ0 for the potential ψ at zero radius. The potential ψ is singular at the radius r∞′ related to ψ0 through: r∞′ = r∞ exp(−ψ0/2), where ψ(r∞) = ∞ for ψ0 = 0.