Adiabatic pair potential of highly charged plates in an electrolyte

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Abstract
The effective interaction between highly charged plates immersed in an electrolyte solution is investigated using mean field theory. The Helmholtz free energy is formulated as a functional of the mean electric potential which satisfies the Poisson-Boltzmann equation. Exact solutions of the Poisson-Boltzmann equation under Dirichlet and Neumann boundary conditions enable us to express the free energy analytically in terms of Legendre's elliptic integrals of the first and second kinds. The adiabatic potentials of the charged plates, which are determined as the parts of the free energy that depend on the interplate distance, turn out 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 arises from an electric pull from the intermediate cloud of excess counterions between the plates. The interplate separation swells upon dilution of electrolyte and the swelling is enhanced for small values of surface potential or surface charge density. While the adiabatic potential of the plates becomes shallower for increasing concentration c in the case of the Neumann boundary condition, it becomes deeper for increasing c in the case of the Dirichlet boundary condition.