Counterion distribution enclosed in a cylinder and a sphere

The electric potential and counterion distributions within a cylinder and a sphere have been obtained analytically by solving the Poisson - Boltzmann equation exactly. The existence of the upper limit of the counterion concentration at the center of the aqueous core has been observed regardless of the surface charge density. This result indicates that, at very high surface charge density, most of the counterions are strongly attracted to the surroundings of the surface and to balance the charges on the surface.