Upscaling the Poisson–Nernst–Planck equations for ion transport in weakly heterogeneous charged porous media. (March 2023)

• Record Type: Journal Article
• Authors: Klika, Václav; Gaffney, Eamonn A.
• Is Part Of: Applied mathematics letters. Volume 137(2023)
• Publication Date: 2023-03
• Subjects: Poisson–Nernst–Planck equations -- Non-periodic homogenisation -- Swelling pressure; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2022.108482
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: The Poisson–Nernst–Planck (PNP) equations govern the continuum level description of ions in electrolytes and especially the impact of charged surfaces. In numerous applications such surfaces are complex, varying on a small lengthscale compared to the overall scale of the system, often prohibiting the direct prediction of the osmotic swelling pressures induced by ion behaviours in Debye layers near surfaces. With periodicity, upscaling techniques can be readily used to determine the behaviour of the swelling pressure on large lengthscales without solving the PNP equations on the complex domain, though generalising to cases where the periodicity is only approximate is more challenging. Here, we generalise a method by Bruna and Chapman (2015) for upscaling a non-periodic diffusion equation to the PNP equations. After upscaling, we find a rational derivation of the swelling pressure closely resembling the classical, though phenomenological, use of Donnan membrane theory predictions for the swelling pressure in cartilage, together with a novel contribution driven by heterogeneous fixed (surface) charges. The resulting macroscale model is also shown to be thermodynamically consistent, though its comparison with a recent upscaled models for swelling pressure in cartilage mechanics emphasises the need to understand how macroscale models depend on differing upscaling techniques, especially in the absence of perfect periodicity. Highlights: A homogenisation of the Poisson-Nernst-Planck equation is presented. Periodicity except for a long lengthscale variation is considered. The predicted swelling pressure possesses bulk and boundary layer contributions. The boundary layer contribution matches Donnan theory in cartilage mechanics. The bulk term is only nonzero for heterogeneous fixed charge.