Difference equation solutions for hopping transport of ions through membranes

Abstract

Difference equations provide a more natural representation than do differential equations for the hopping conduction of charged particles in material media. The discreteness of a material medium on an atomistic scale can lead to large physical effects such as nonohmic currents in the presence of large electric fields. A steady-state expression for the current has been obtained by summing the coupled difference equations for transport of a single species in the limit of arbitrarily large values of the space charge density due to the mobile carriers, with a concentration gradient across the membrane included. The transient response has also been worked out, but only in the homogeneous field limit. It is concluded that the combination of high electric fields with the natural microscopic discreteness of the diffusion medium can result in readily observable nonlinear electric field effects which increase approximately exponentially with the separation distances of the discrete energy barriers