The electrostatic interaction of an assemblage of charges with a charged surface: The charge-regulation effect

The electrostatic interaction of a particle, modeled as an assemblage of point charges, with a charged plane is investigated on the basis of the Poisson-Boltzmann equation under the Debye-Huckel approximation. During interactions, the plate can be linearly regulated by the charged particle. The interaction energy and force are derived in an integral expression for a string-like particle and an ion-penetrable object. Analytical results can be obtained for a point charge and a fiber perpendicular to the plane. The interaction depends on three characteristic lengths, including the Debye length kappa(-1), the separation, and the regulation length lambda. Though kappa lambda much greater than 1 corresponds to a plane of constant surface charge density, kappa lambda much less than 1 does not necessarily represent that of constant surface potential. Two interesting results are observed. The interaction for kappa lambda much less than 1 may change from repulsion to attraction, and eventually return to repulsion when the particle moves toward the like-charged plate. When the particle-plate distance is fixed, the interaction can change from repulsion to attraction upon varying the concentration of indifferent ions, such as adding salts or dilution. The effect of charge regulation on the change of both electric potential and charge density on the surface is also discussed.