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Asymptotic models for transport in large aspect ratio nanopores

Published online by Cambridge University Press:  06 June 2018

B. MATEJCZYK ,
J.-F. PIETSCHMANN ,
M.-T. WOLFRAM  and
G. RICHARDSON
B. MATEJCZYK
Affiliation:
Department of Mathematics, University of Warwick, CV4 7AL Coventry, UK email: b.matejczyk@warwick.ac.uk
J.-F. PIETSCHMANN
Affiliation:
Institute for Computational and Applied Mathematics, WWU Münster, Münster 48149, Germany and Osnabrück University, Institute of Mathematics, 49069 Osnabrück, Germany email: jan.pietschmann@wwu.de
M.-T. WOLFRAM
Affiliation:
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Strasse 69, 4040 Linz, Austria email: M.Wolfram@warwick.ac.uk
G. RICHARDSON
Affiliation:
School of Mathematics, University of Southampton, SO17 1BJ Southampton, UK email: g.richardson@soton.ac.uk
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Abstract

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Ion flow in charged nanopores is strongly influenced by the ratio of the Debye length to the pore radius. We investigate the asymptotic behaviour of solutions to the Poisson–Nernst–Planck (PNP) system in narrow pore like geometries and study the influence of the pore geometry and surface charge on ion transport. The physical properties of real pores motivate the investigation of distinguished asymptotic limits, in which either the Debye length and pore radius are comparable or the pore length is very much greater than its radius This results in a quasi-one-dimensional (1D) PNP model, which can be further simplified, in the physically relevant limit of strong pore wall surface charge, to a fully 1D model. Favourable comparison is made to the two-dimensional (2D) PNP equations in typical pore geometries. It is also shown that, for physically realistic parameters, the standard 1D area averaged PNP model for ion flow through a pore is a very poor approximation to the (real) 2D solution to the PNP equations. This leads us to propose that the quasi-1D PNP model derived here, whose computational cost is significantly less than 2D solution of the PNP equations, should replace the use of the 1D area averaged PNP equations as a tool to investigate ion and current flows in ion pores.

Keywords

Mathematical Modellingnanoscale poresPoisson-Nernst-Planck equationsAymptotic methods
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 3 , June 2019 , pp. 557 - 584
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

The work of JFP was supported by DFG via Grant 1073/1-2. MTW and BM acknowledges financial support from the Austrian Academy of Sciences ÖAW via the New Frontiers Grant NST-001. BM acknowledges the support from the National Science Center from award No DEC-2013/09/D/ST1/03692.

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