Interaction of Potential Sources in Infinite 2D Arrays: Diffusion through Composite Membranes, Micro‐Electrochemistry, Entrance Resistance, and Other Examples

Abstract

AbstractRegular 2D arrays of potential sources on impervious “screens” are a mathematical idealization for the description of a number of natural and/or technological processes. At steady state, they are described by Laplace equation with suitable boundary conditions. This study explains the evolution of boundary conditions from a given potential to a given potential gradient at infinity with increasing size of arrays and provides a criterion for micro‐ and macro‐arrays in terms of distance to potential‐defining surfaces. For regular macro‐arrays, the problem is formulated and solved numerically by using cell models. The use of rigorous cell models requires 3D numerical simulations, but a simplified (cylindrical, 2D) cell model is shown to have an excellent accuracy. Numerical as well as theoretical analyses reveal a simple far‐field behavior described by an asymptotic expression for an additional potential drop (applicable for sources whose size does not exceed about 40% of the intersource distance) dependent on just one numerical constant. This expression is used for the derivation of useful approximate formulae for several applications (diffusion resistance of composite membranes, limiting current in arrays of microelectrodes, entrance diffusion resistance in arrays of scarce and short nanopores) and compared with relevant interpolation formulae available in the literature.