Scaling regimes for unsteady diffusion across particle-stabilized fluid interfaces

TL;DR

This paper develops a theoretical framework to understand how particles at fluid interfaces affect unsteady diffusion, revealing two regimes and providing criteria to reconcile conflicting experimental observations.

Contribution

It introduces a unified analytical and numerical approach to characterize diffusion regimes across particle-stabilized interfaces, clarifying when hindrance occurs.

Findings

Hindrance occurs only at short times

Derived expressions for transition times between regimes

Provided a criterion to distinguish hindering effects

Abstract

Colloidal particles at fluid interfaces can enhance the stability of drops and bubbles. Yet, their effect on mass transfer in these multiphase systems remains ambiguous, with some experiments reporting strongly hindered diffusion, while others show nearly no effect, even at near-complete surface coverage. To resolve this ambiguity, we solve the Fick-Jacobs equation for unsteady diffusion, allowing us to treat the particle-laden interface as a locally reduced cross-sectional area for mass transfer. Our numerical solutions reveal two limiting regimes, with the particle layer hindering diffusion only at short times. Guided by analytical solutions for a homogeneous layer with reduced diffusivity, we derive quantitative expressions for the transport regimes and associated transition times for diffusion across the particle layer. This analysis yields a simple criterion for long-term hindrance that accurately distinguishes between conflicting experimental results, providing a unifying framework for mass transfer in particle-laden multiphase systems.