Helmholtz free energy of a multilayer system of highly charged plates immersed in an electrolyte

The Helmholtz free energy of highly charged plates immersed in an electrolyte solution in a finite container is investigated using mean field theory. Exact solutions of the Poisson-Boltzmann equation under Dirichlet and Neumann boundary conditions lead to analytical expressions for the free energies. Adiabatic potentials of the charged plates, which are derived from the free energy, have long-range weak attractive parts and medium-range strong repulsive parts, irrespective of the type of boundary conditions. The repulsion results mainly from the osmotic pressure of the excess ions trapped between the plates by the large surface charges and the attraction arises essentially from an electric pull from the intermediate cloud of excess counterions shared by the plates.