Effects of salt addition on the ion distribution enclosed in a cylinder and a sphere

The electric potential and ion distributions within a cylinder and a sphere at various salt concentration have been obtained analytically by solving the Poisson-Boltzmann equation in terms of series solutions. At a fixed ratio of co-ion to counterion concentrations at the center of the aqueous core (α<1) or a given amount of added salt, the existence of the upper limit of the counterion concentration at the center has been observed regardless of the surface charge density. It is a consequence of counterion condensation on the charged surface. The upper limit depends on the value of α and increases rapidly when α→1. At high surface charge density, the counterions are strongly attracted to the surrounding of the surface. A boundary layer with the thickness inversely proportional to the surface charge density is formed. Both the effects of curvature and salt addition hence become unimportant, and the surface potential is primarily determined by the surface charge density.