Dispersion in two-dimensional periodic channels with discontinuous profiles

TL;DR

This paper develops asymptotically exact formulas for effective diffusivity in two-dimensional periodic channels with discontinuous profiles, accounting for entropic trapping effects and validating results with numerical analysis.

Contribution

It extends the Fick-Jacobs' approximation to discontinuous channels, providing explicit formulas for effective diffusivity at next-to-leading order, capturing entropic trapping effects.

Findings

Discontinuities reduce effective diffusivity.

Formulas are asymptotically exact at next-to-leading order.

Theory aligns with trapping rate concepts.

Abstract

The effective diffusivity of Brownian tracer particles confined in periodic micro-channels is smaller than the microscopic diffusivity due to entropic trapping. Here, we study diffusion in two-dimensional periodic channels whose cross-section presents singular points, such as abrupt changes of radius or the presence of thin walls, with openings, delimiting periodic compartments composing the channel. Dispersion in such systems is analyzed using the Fick-Jacobs' approximation. This approximation assumes a much faster equilibration in the lateral than in the axial direction, along which the dispersion is measured. If the characteristic width $a$ of the channel is much smaller than the period $L$ of the channel, i.e. $\varepsilon = a/L$ is small, this assumption is clearly valid for Brownian particles. For discontinuous channels, the Fick-Jacobs' approximation is only valid at the lowest order in $\varepsilon$ and provides a rough, though on occasions rather accurate, estimate of the effective diffusivity. Here we provide formulas for the effective diffusivity in discontinuous channels that are asymptotically exact at the next-to-leading order in $\varepsilon$. Each discontinuity leads to a reduction of the effective diffusivity. We show that our theory is consistent with the picture of effective {\em trapping rates} associated with each discontinuity, for which our theory provides explicit and asymptotically exact formulas. Our analytical predictions are confirmed by numerical analysis. Our results provide a precise quantification of the kinetic entropic barriers associated with profile singularities.