Effective interaction of highly charged plates in an electrolyte

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Abstract
Thermodynamic electrostatic interaction between highly charged plates immersed in an electrolyte solution is investigated. The Poisson-Boltzmann equation is solved under the assumption that counterions dominate coions between the plates, and an analytical formula for the total electrostatic energy of the solution is derived. The electric part of the Helmholtz free energy is obtained rigorously by Debye's charging-up procedure, and the non-electric part is calculated by describing the small ions as ideal gases. The resulting Helmholtz free energy leads to an adiabatic pair potential between the plates which has a long-range weak attraction as well as a medium-range strong repulsion. It is shown that, while the repulsion arises mainly from the osmotic pressure of excess ions trapped between the plates by the large surface charges, the attraction originates in an electric pull from the intermediate cloud of the excess ions.