The electrostatic interaction of a charged particle with a surface: The effect of surface charge rearrangement

In this paper we investigate the electric interaction between a charged particle and a surface in which the charged ions are capable of moving in response to the electric potential disturbance caused by the approach of the charged particle. Such surfaces include ionic surfactants distributed in air-water interface and charged lipids in bilayer membranes. On the basis of the mean field theory, the free energy of the system, which includes the electrostatic internal energy and the entropy of the mobile ions and surface ions, can be written down. The surface charge-potential relation is then derived by the calculus of variation. When the potential disturbance is small enough, a linear charge regulation model is obtained. The interaction energy associated with a long rod parallel to the interface is studied and an analytical expression is obtained. When a rod approaches an oppositely charged surface, the interaction can change from attraction to repulsion, depending on the ratio of the characteristic regulation length to the Debye length. At low surface charge density, the surface behaves as under the condition of constant charge density and acts as that of constant potential for high enough charge density.