The contact theorem for charged fluids: from planar to curved geometries

When a Coulombic fluid is confined between two parallel charged plates, an exact relation links the difference of ionic densities at contact with the plates to the surface charges of these boundaries. It no longer applies when the boundaries are curved, and we work out how it generalises when the fluid is confined between two concentric spheres (or cylinders), in two and in three space dimensions. The analysis is thus performed within the cell model picture. The generalised contact relation opens the possibility to derive new exact expressions, of particular interest in the regime of strong Coulombic couplings. Some emphasis is put on cylindrical geometry, for which we discuss in depth the phenomenon of counterion evaporation/condensation, and obtain novel results. Good agreement is found with Monte Carlo simulation data.