Biased diffusion in confined media: Test of the Fick-Jacobs approximation and validity criteria

TL;DR

This paper investigates the validity of the Fick-Jacobs approximation for biased diffusion in confined, periodic structures, establishing a criterion based on structure shape and force, supported by analytical and numerical analysis.

Contribution

The study provides a new validity criterion for the Fick-Jacobs approximation considering force and structure shape, validated through analytical and numerical methods.

Findings

Critical force scales with the square of the structure's periodicity.

Validity region visualized in a force versus periodicity diagram.

Approximation accuracy depends on the ratio of time scales.

Abstract

We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one dimensional diffusion. The validity of this approximation, being based on the assumption of an instantaneous equilibration of the particle distribution in the cross-section of the structure, is analyzed by comparing the different time scales that characterize the problem. A validity criterion is established in terms of the shape of the structure and of the applied force. It is analytically corroborated and verified by numerical simulations that the critical value of the force up to which this description holds true scales as the square of the periodicity of the structure. The criterion can be visualized by means of a diagram representing the regions where the Fick-Jacobs description becomes inaccurate in terms of the scaled force versus the periodicity of the structure.