Screening of a macroion by multivalent ions: correlation-induced inversion of charge

Screening of a strongly charged macroion by multivalent counterions is considered. It is shown that counterions form a strongly correlated liquid at the surface of the macroion. Cohesive energy of this liquid leads to additional attraction of counterions to the surface, which is absent in conventional solutions of the Poisson-Boltzmann equation. Away from the surface this attraction can be taken into account by a new boundary condition for the concentration of counterions near the surface. The Poisson-Boltzmann equation is solved with this boundary condition for a charged flat surface, a cylinder, and a sphere. In all three cases, screening is much stronger than in the conventional approach. At some critical exponentially small concentration of multivalent counterions in the solution, they totally neutralize the surface charge at small distances from the surface. At larger concentrations they invert the sign of the net macroion charge. The absolute value of the inverted charge density can be as large as 20% of the bare one. In particular, for a cylindrical macroion it is shown that for screening by multivalent counterions, predictions of the Onsager-Manning theory are quantitatively incorrect. The net charge density of the cylinder is smaller than their theory predicts and inverts the sign with a growing concentration of counterions. Moreover, the condensation loses its universality and the net charge linear density depends on the bare one.