---
abstract: 'Effective Poisson-Nernst-Planck (PNP) equations are derived for macroscopic ion transport in charged porous media. Homogenization analysis is performed for a two-component periodic composite consisting of a dilute electrolyte continuum (described by standard PNP equations) and a continuous dielectric matrix, which is impermeable to the ions and carries a given surface charge. Three new features arise in the upscaled equations: (i) the effective ionic diffusivities and mobilities become tensors, related to the microstructure; (ii) the effective permittivity is also a tensor, depending on the electrolyte/matrix permittivity ratio and the ratio of the Debye screening length to mean pore size; and (iii) the surface charge per volume appears as a continuous “background charge density". The coefficient tensors in the macroscopic PNP equations can be calculated from periodic reference cell problem, and several examples are considered. For an insulating solid matrix, all gradients are corrected by a single tortuosity tensor, and the Einstein relation holds at the macroscopic scale, which is not generally the case for a polarizable matrix. In the limit of thin double layers, Poisson’s equation is replaced by macroscopic electroneutrality (balancing ionic and surface charges). The general form of the macroscopic PNP equations may also hold for concentrated solution theories, based on the local-density and mean-field approximations. These results have broad applicability to ion transport in porous electrodes, separators, membranes, ion-exchange resins, soils, porous rocks, and biological tissues.'
author:
- 'Markus Schmuck[^1]'
- 'Martin Z. Bazant [^2]'
bibliography:
- 'porous-extra.bib'
- 'pmSPNP\_Martin2.bib'
title: 'Homogenization of the Poisson-Nernst-Planck equations for ion transport in charged porous media [^3]'
---

diffusion, electromigration, porous media, membranes, Poisson-Nernst-Planck equations, homogenization

Introduction {#sec:Intro}
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Poisson-Nernst-Planck equations for homogeneous media  {#sec:BaGr}
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Homogenized Poisson-Nernst-Planck equations for porous media {#sec:pmPNP}
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Optimizing the conductivity for parallel straight channels {#sec:optCond}
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Conclusion  {#sec:Disc}
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[^1]: Departments of Chemical Engineering and Mathematics, Imperial College, London, UK

[^2]: Departments of Chemical Engineering and Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

[^3]: This work was supported by the Swiss National Science Foundation (SNSF) under the grant PBSKP2-12459/1 (MS) and in part by the National Science Foundation under contract DMS-0948071 (MZB).
