Poisson-Boltzmann for oppositely charged bodies: an explicit derivation
Résumé
The interaction between two parallel charged plates in ionic solution is a general starting point for studying colloidal complexes. An intuitive expression of the pressure exerted on the plates is usually proposed, which includes an electrostatic plus an osmotic contribution. We present here an explicit and self-consistent derivation of this formula in the only framework of the Poisson-Boltzmann theory. We also show that, depending on external constraints, the correct thermodynamic potential can differ from the usual PB free energy. For asymmetric, oppositely charged plates, the resulting expression predicts a non trivial equilibrium position with the plates separated by a finite distance. The depth of this energy minimum is decisive for the stability of the complex. It is therefore crucial to obtain its explicit dependence on the plates charge densities and on the ion concentration. It happens that analytic expressions for the position and depth of the energy minimum have been derived in 1975 by Ohshima [Ohshima H., Colloid and Polymer Sci. 253, 150-157 (1975)] but, surprisingly, these important results seem to be overlooked today. We retrieve these expressions in a simpler formalism, more familiar to the physics community, and give a physical interpretation of the observed behavior.
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