Electrostatic image force energy for charges in three-layer structures: exact formulas and their approximations

The problem of image force energyW(Z) in three-layer plane structures, whereZis the coordinate perpendicular to the layers, has been reconsidered. In the classical electrostatic limit, where the dielectric permittivitiesɛof all structure components (i= 1, 2, 3) are constants, the exact general dependencesW(Z) were obtained for each layer and anyɛ-combination in terms of the Lerch transcendent function. For certain combinations ofɛ, an ion adsorption minimum was found to arise in one of the covers far from the interlayer. Some other combinations ofɛcan lead to the appearance of a potential barrier, which does not permit a free charge existing in the cover to approach the interlayer, although it will be attracted to the interlayer in the close vicinity of the latter. For symmetric structures (ɛ₁=ɛ₃), the asymptotic behavior ofW(Z→∞)was shown to beZ-2rather thanZ-1, as it takes place in the two-layer case. Simple approximate analytical formulas that describeW(Z) and possess high accuracy for arbitrary relationships among theɛ-constants were proposed.