Scaling regimes for unsteady diffusion across particle-stabilized fluid interfaces

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.

Figures (4)

• Figure 1: Diffusion across particle-laden interfaces in multiphase systems. Diffusion between the inner and outer phases occurs through the interstitial pores between adsorbed particles, leading to processes such as (a) solute exchange in colloidosomes (reprinted with permission from dinsmore2002colloidosomes, copyright 2002 AAAS), (b) evaporation in liquid marbles (reprinted with permission from bormashenko2011liquid, copyright 2011 Elsevier), and (c) gas diffusion in armored bubbles (reprinted with permission from subramaniam2006mechanics, copyright 2006 ACS). The effect of a complex interface on diffusion can be modeled as (d) a layer of particles reducing the available cross-sectional area, depending on the surface coverage $\phi_0$, or (e) a homogeneous layer characterized by a locally reduced effective diffusivity.
• Figure 2: Numerical results for unsteady diffusion across a particle-laden interface. (a) Concentration profiles evolve identically at short times ($\tau\ll 1$, dotted curves), but steeper gradients develop within the layer for higher surface coverage $\phi_0$at later times (dashed and solid curves). (b) The diffused solute mass normalized by the layer volume, $\nu$, is initially reduced by a factor of $1-\phi_0$ compared to the bare interface case. At long times, all curves converge to the bare interface solution $\sqrt{\tau/\pi}$. (c) The regime transitions are not well captured by simply rescaling the diffused mass $\nu$ by the short-time asymptotes.
• Figure 3: Unsteady diffusion across a homogeneous layer with reduced diffusivity as an effective model for a particle-laden interface. (a) Analytical solutions for a layer with reduced diffusivity [Eq. \ref{['eq:NaAC']}]. (b) Numerical solutions of the Fick-Jacobs equation [Eq. \ref{['eq:FickJacobs_dimless']}] for a particle-laden interface. In both plots, time is normalized by the characteristic timescale that separates the intermediate and long-time regimes. The vertical axis in (b) is scaled by $(1-\phi_0)^{3/2}$ to account for the reduced fluid volume within the layer, aligning the curves with the analytical results. Black dashed lines represent first-order approximations for the transport regimes based on Eqs. \ref{['eq:NaAC_approx']} and \ref{['eq:nu_phi0_approx']}.
• Figure 4: Data from the literature compared against the criterion for long-term hindrance to diffusion. Experimental data includes evaporating droplets (triangles) laborie2013coatingsasaumi2020effectsaczek2024impactprakash2025evaporation and solute diffusion in emulsions (other symbols) kim2007colloidalyow2009releasethompson2010covalentlysjoo2015barrierschroder2019canjia2023efficientliu2024diffusion, colored by the observed effect of the particle-laden interface: significant hindrance (black), minor effect (gray), or no effect (white). Dashed lines represent the criterion from Eq. \ref{['eq:criterion']} and offsets by factors of ten, reflecting the typical sensitivity of our analysis. Together, these lines mark the transition from unaffected to strongly hindered diffusion.