Charge inversion on membranes induced by multivalent-counterion fluctuations

Based on the two state model, we study the condensation of counterions on oppositely charged membranes in the presence of monovalent salts. Using the Gaussian approximation, we evaluate the contribution of two-dimensional charge fluctuation to the free energy, from which the number of condensed counterions is determined self-consistently. It is shown that charge inversion can occur upon the addition of divalent ions of millimolar range, or of trivalent ions of micromolar range. The driving force for the overcharging mechanism is the charge correlation induced by thermal fluctuation, in which the multivalency of the counterions plays a crucial role. (Some figures in this article are in colour only in the electronic version) Counterions in watery solution dominantly distribute near an oppositely charged surface of macroions, forming a diffuse electric double layer. The structure of a double layer of counterions, which has been usually described by the mean-field Poisson–Boltzmann theory, is of great importance in interaction and phase behaviours of macroions [1, 2]. When th