The interaction between a charged wall and its counterions: a condensation theory

We use the theory of counterion condensation to calculate the potential of mean force between a charged planar wall and an oppositely charged ion of unsigned valence Z. We find two solutions to this problem, an outer potential when the Z-ion is relatively far from the wall and an inner one when it is relatively close. There is a discontinuous upward jump from one to the other branch, as the Z-ion approaches the wall. The jump must occur at the wall itself, and we attribute it to interaction with the condensed layer of counterions which resides on the wall. At an infinitesimal distance from the wall, the Z-ion encounters a repulsive force resisting penetration of the condensed layer. Then, when the Z-ion penetrates the condensed layer, Z univalent counterions are released from the condensed layer into the bulk solution, maintaining the effective charge on the wall at an invariant critical value. We discuss the possible connections of the theory with published data from molecular dynamics simulations on lipid bilayers.