Screening of a macroion by multivalent ions: a new boundary condition for the Poisson–Boltzmann equation and charge inversion

Screening of a macroion by multivalent counterions is considered. It is shown that ions form strongly correlated liquid at the macroion surface. Cohesive energy of this liquid leads to strong additional attraction of counterions to the surface. Away from the surface this attraction is taken into account by a new boundary condition for the Poisson–Boltzmann equation. This equation is solved with the new boundary condition for a charged flat surface and a long cylinder. For a cylinder Onsager–Manning theory looses its universality so that apparent charge of the cylinder is smaller than their theory predicts and depends on its bare charge. It can also vanish or change sign.