The counterion-release interaction

The case of two oppositely charged planar surfaces interacting across a salt solution is reconsidered to explain why the release of counterions from the slit between the plates can lead to an attractive contribution to the effective interaction between the plates. We estimate the order of magnitude of the effective inter-plate potential by first considering a Gouy-Chapman layer in front of a single charged plate. The Gouy-Chapman problem is fully characterized by just one parameter, the dimensionless surface charge parameter s. Considering the number of ions in the double layer as a function of s, we can identify two regimes: one in which the electro-neutrality of the double layer is ensured through the exchange of co- and counterions, and another, at large s, where counterions are to be recruited from the reservoir for the system to remain electro-neutral. The counterion-release force plays a major role only for plates with surface parameters that belong to the second regime. Having at hand the results from the single-plate case we are then able to give a simple interpretation of the counterion-release interaction. The counterion-release force is dominant for systems with large s and at large distances. At small distances, when all extra ions have left the slit between the plates, the release mechanism ceases to work and the two plates attract each other due to an trivial electrostatic attraction.