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Within a mean field treatment of the interaction between two oppositelycharged plates in a salt free solution, the distance at which a transition froman attractive to a repulsive regime appears can be computed analytically. Themean field description however breaks down under strong coulombic couplings,that can be achieved at room temperature with multivalent counter-ions andhighly charged surfaces. Making use of the contact theorem and simple physicalarguments, we propose explicit expressions for the equation of state in severalsituations at short distances. The possibility of Bjerrum pair formation isaddressed and is shown to have profound consequences on the interactions. Tocomplete the picture, we finally consider the large distance limit, from whichschematic phase diagrams discriminating attractive from repulsive regions canbe proposed.

Interaction regimes for oppositely charged plates with multivalent counterions