Charge inversion of colloids in an exactly solvable model

We study a two-dimensional model for a long cylindrical stiff charged macroion immersed in a charge-asymmetric electrolyte with charge ratio +2/-1. The model is integrable and it allows an exact analytical determination of the effective charge of the macroion, which characterizes the electrostatic potential at large distances (compared to the screening length) from the macroion. At at high Coulombic coupling, this model predicts charge inversion: for a highly negatively charged macroion, the effective charge could become positive, indicating an overscreening of the macroion by the divalent counterions. By studying the behavior of the coions and counterions density profiles close to the macroion, we show that the counterion condensation threshold is shifted to a lower value in absolute value. This plays an important role in the charge inversion phenomenon.