Ion Flow through Narrow Membrane Channels: Part I

This paper is devoted to the study of the electrostatic potential in a composite dielectric medium. The geometry consists of a membrane-like layer with one dielectric constant, perforated by a hole filled with a material with a different dielectric constant. The field is due to a known, but otherwise arbitrary, distribution $f(z)$ of charges in the hole, where z is the coordinate along the axis of the hole. The induced surface charge at the interface between the two media is found as an asymptotic series in the small aspect ratio $\alpha $ of the hole, for the case in which $f(z)$ vanishes at the endpoints and is $C^1 $. This analysis reveals the existence of a distinguished limit in which the small ratio of the permittivities $\epsilon $ is proportional to $ - \alpha ^2 $ In $\alpha $.