Mean field description for Helmholtz free energy of highly charged plates in an electrolyte

The Helmholtz free energy of highly charged plates immersed in an electrolyte solution in 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. The adiabatic potential of the charged plates, which is derived from the free energy, is ascertained to have long-range weak attractive parts as well as medium-range strong repulsive parts irrespective of the type of boundary conditions. While the repulsion originates mainly in the osmotic pressure of the excess ions trapped between the plates by the large surface charges, the attraction is interpreted to arise from essentially an electric pull from the intermediate cloud of excess counterions shared by the plates.