Transport across thin membranes: Effective solute flux jump

Abstract

A model to describe the transport across membranes of chemical species dissolved in an incompressible flow is developed via homogenization. The asymptotic matching between the microscopic and macroscopic solute concentration fields leads to a solute flux jump across the membrane, quantified through the solution of diffusion problems at the microscale. The predictive model, written in a closed form, covers a wide range of membranes behaviors, in the limit of negligible Reynolds and Peclet numbers inside the membrane. The closure problem at the microscale, found via homogenization, allows one to link the membrane micro-structure to its effective macroscopic properties, such as solvent permeability and solute diffusivity.After a validation of the model through comparison with the corresponding full-scale solution, an immediate application is provided, where the membrane behavior is a priori predicted through an analysis of its microscopic properties. The introduced tools and considerations may find applications in the design of thin microstructured membranes.

Homogenization theory captures macroscopic flow discontinuities across Janus membranes

“…In the case of Re and Pe up to O( −1 ), i.e. when the pore Reynolds and Péclet numbers Re I = ρU I l/μ = 2 Re and Pe I = U I l/D = 2 Pe, based on the typical pore size and velocity, l and U I , are up to O( ), the solution of (2.1a-c) and (2.2a,b) in the whole full-scale domain converges, on average, to the solution of the problem where the membrane is replaced by a smooth macroscopic interface C between two fluid regions where (2.1a-c) still apply; see figure 1 (Zampogna & Gallaire 2020;Ledda et al 2021;Zampogna et al 2022). The macroscopic interface conditions imposed on C stem from the application of homogenization to (2.1a-c) and (2.2a,b).…”

“…Inversely, the no-slip boundary condition, namely a hydrophilic solvent-membrane interaction, is realized by choosing α i = α i = γ i = 0 and β i = 1. The several types of solute-membrane interactions that can be realized by varying the values of ζ i , η and λ have been analysed in Zampogna et al (2022); the membrane has a chemostat-like behaviour when ζ i = 0 and η = λ = 1, an insulating behaviour when ζ i = n i and η = λ = 0, and an absorbing/desorbing behaviour for ζ i = n i , η = ±1 and λ = 0.…”

“…To develop the macroscopic interface conditions valid on C, the procedure initially developed in Zampogna & Gallaire (2020) focuses on the solution of the problem within the microscopic domain sketched in figure 1(b). The full-scale spatial variable x i is decomposed as x i → x i + x i , with x i = x i / , so that x i → 0 ± when x i → ±∞, and vice versa.…”

“…Following Zampogna & Gallaire (2020) and Zampogna et al (2022), the microscopic solution of (2.3a-c) can be written as a linear combination of the outer fluid stresses and solute fluxes that enter the microscopic problems in the boundary conditions on the sides U and D of the microscopic elementary cell (cf. figure 1b), i.e.…”

“…To upscale (2.4)-(2.6), Zampogna & Gallaire (2020) and Zampogna et al (2022) introduced the central average within the microscopic elementary cell, defined as…”