Charge Transport in one Dimension:Dissipative and Non-Dissipative Space-Charge Limited Currents

TL;DR

This paper models charge transport in nanopores using one-dimensional equations, deriving exact solutions and analyzing dissipative versus non-dissipative regimes, with implications for understanding electric field confinement and charge dynamics.

Contribution

It introduces a simplified one-dimensional model for charge transport in nanopores and derives exact solutions, including a novel application of the admissibility condition as Poynting's theorem.

Findings

Exact solutions for charge transport in nanopores

Comparison between dissipative and non-dissipative models

Admissibility condition as Poynting's theorem

Abstract

We consider charge transport in nanopores where the dielectric constant inside the nanopore is much greater than in the surrounding material, so that the flux of the electric fields due to the charges is almost entirely confined to the nanopore. That means that we may model the electric fields due to charge densities in the nanopore in terms of average properties across the nanopore as solutions of one dimensional Poisson equations. We develop basic equations for an M component system using equations of continuity to relate concentrations to currents, and flux equations relating currents to concentration gradients and conductivities. We then derive simplified scaled versions of the equations. We develop exact solutions for the one component case in a variety of boundary conditions using a Hopf-Cole transformation, Fourier series, and periodic solutions of the Burgers equation. These are compared with a simpler model in which the scaled diffusivity is zero so that all charge motion is driven by the electric field. In this non-dissipative case, recourse to an admissibility condition is utilised to obtain the physically relevant weak solution of a Riemann problem concerning the electric field. It is shown that the admissibility condition is Poynting's theorem.