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 po int charges, with a charged plane is investigated on the basis of the Poiss on-Boltzmann equation under the Debye-Huckel approximation. During interact ions, the plate can be linearly regulated by the charged particle. The inte raction energy and force are derived in an integral expression for a string -like particle and an ion-penetrable object. Analytical results can be obta ined for a point charge and a fiber perpendicular to the plane. The interac tion depends on three characteristic lengths, including the Debye length ka ppa(-1), the separation, and the regulation length lambda. Though kappa lam bda much greater than 1 corresponds to a plane of constant surface charge d ensity, kappa lambda much less than 1 does not necessarily represent that o f constant surface potential. Two interesting results are observed. The int eraction for kappa lambda much less than 1 may change from repulsion to att raction, and eventually return to repulsion when the particle moves toward the like-charged plate. When the particle-plate distance is fixed, the inte raction can change from repulsion to attraction upon varying the concentrat ion of indifferent ions, such as adding salts or dilution. The effect of ch arge regulation on the change of both electric potential and charge density on the surface is also discussed.