Closed formula for the transport of micro-nano-particle across model porous media

TL;DR

This paper reviews the derivation of the Fick-Jacobs equation for particle transport in porous media, providing a simple formula for symmetric channels and discussing higher order effects in non-symmetric channels.

Contribution

It derives a straightforward formula for particle flux in symmetric channels within the Fick-Jacobs approximation, enhancing design and analysis of experiments and simulations.

Findings

Flux in symmetric channels is fully captured by linear response.

A simple formula accurately predicts trends in symmetric channels.

Higher order corrections are relevant for non-symmetric channels.

Abstract

In the last decade the Fick-Jacobs approximation has been exploited to capture the transport across constrictions. Here, we review the derivation of the Fick-Jacobs equation with particular emphasis on its linear response regime. We show that for fore-aft symmetric channels the flux of non-interacting systems is fully captured by its linear response regime. For this case we derive a very simple formula that captures the correct trends and that can be exploited as a simple tool to design experiments or simulations. Finally, we show that higher order corrections in the flux may appear for non-symmetric channels.