Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T19:55:33.114Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
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: [email protected]
M.-T. WOLFRAM
Affiliation:
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Strasse 69, 4040 Linz, Austria email: [email protected]
G. RICHARDSON
Affiliation:
School of Mathematics, University of Southampton, SO17 1BJ Southampton, UK email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

[1] Andelman, D. (1995) Electrostatic properties of membranes: The Poisson–Boltzmann theory. Handb. of Biol. Phys. 1, 603–642.Google Scholar
[2] Ball, F., Milne, R. K. & Yeo, G. F. (2002) Multivariate semi-Markov analysis of burst properties of multiconductance single ion channels. J. of Appl. Probab. 39 (1), 179–196.Google Scholar
[3] Burger, M., Schlake, B. A. & Wolfram, M.-T. (2012) Nonlinear Poisson–Nernst–Planck equations for ion flux through confined geometries. Nonlinearity 25 (4), 961.Google Scholar
[4] Cervera, J., Schiedt, B., Neumann, R., Mafe, S. & Ramirez, P. (2006) Ionic conduction, rectification, and selectivity in single conical nanopores. J. Chem. Phys. 124 (10), 104706.Google Scholar
[5] Cervera, J., Schiedt, B. & Ramírez, P. (2005) A Nernst–Planck model for ionic transport through synthetic conical nanopores. Europhys. Lett. 71 (1), 35–41.Google Scholar
[6] Chapman, S. J., Norbury, J., Please, C. & Richardson, G. (2005) Ions in solutions and protein channels. In: Proceedings of the 5th Mathematics in Medicine Study Group, University of Oxford.Google Scholar
[7] Chen, W., Erban, R. & Chapman, S. J. (2014) From Brownian dynamics to Markov chain: An ion channel example. SIAM J. Appl. Math. 74 (1), 208–235.Google Scholar
[8] Constantin, D. & Siwy, Z. S. (2007) Poisson–Nernst–Planck model of ion current rectification through a nanofluidic diode. Phys. Rev. E 76, 041202.Google Scholar
[9] Corry, B., Kuyucak, S. & Chung, S.-H. (2000) Tests of continuum theories as models of ion channels. II. Poisson–Nernst–Planck theory versus Brownian dynamics. Biophys. J. 78 (5), 2364–2381.Google Scholar
[10] Courtier, N., Foster, J., O'Kane, S., Walker, A., & Richardson, G. (2018). Systematic derivation of a surface polarisation model for planar perovskite solar cells. Eur. J. Appl. Math. 1–31. doi:10.1017/S0956792518000207Google Scholar
[11] Dreyer, W., Guhlke, C. & Muller, R. (2013) Overcoming the shortcomings of the Nernst–Planck model. Phys. Chem. Chem. Phys. 15, 7075–7086.Google Scholar
[12] Foster, J. M., Kirkpatrick, J. & Richardson, G. (2013) Asymptotic and numerical prediction of current–voltage curves for an organic bilayer solar cell under varying illumination and comparison to the Shockley equivalent circuit. J. Appl. Phys. 114 (10), 104501.Google Scholar
[13] Foster, J. M., Snaith, H. J., Leijtens, T. & Richardson, G. (2014) A model for the operation of perovskite based hybrid solar cells: Formulation, analysis, and comparison to experiment. SIAM J. Appl. Math. 74 (6), 1935–1966.Google Scholar
[14] George, S., Foster, J. M. & Richardson, G. (2015) Modelling in vivo action potential propagation along a giant axon. J. Math. Biol. 70 (1–2), 237–263.Google Scholar
[15] Gillespie, D., Nonner, W. & Eisenberg, R. S. (2002) Coupling Poisson–Nernst–Planck and density functional theory to calculate ion flux. J. Phys.: Condens. Matter 14 (46), 12129.Google Scholar
[16] Gummel, H. K. (1964) A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electron. Devices 11 (10), 455–465.Google Scholar
[17] Hollerbach, U., Chen, D.-P. & Eisenberg, R. S. (2001) Two- and three-dimensional Poisson–Nernst–Planck simulations of current flow through Gramicidin A. J. Sci. Comput. 16 (4), 373–409.Google Scholar
[18] Horng, T.-L., Lin, T.-C., Liu, C. & Eisenberg, R. S. (2012) PNP equations with steric effects: A model of ion flow through channels. J. Phys. Chem. B 116 (37), 11422–11441.Google Scholar
[19] Lai, W. & Ciucci, F. (2011) Mathematical modeling of porous battery electrodes: Revisit of Newman's model. Electrochim. Acta 56 (11), 4369–4377.Google Scholar
[20] Markowich, P. A. (1985) The Stationary Semiconductor Device Equations, Springer Vienna, Vol. 1, Springer Science & Business Media.Google Scholar
[21] Markowich, P. A., Ringhofer, C. A. & Schmeiser, C. (1986) An asymptotic analysis of one-dimensional models of semiconductor devices. IMA J. Appl. Math. 37 (1), 1–24.Google Scholar
[22] Markowich, P. A., Ringhofer, C. A. & Schmeiser, C. (1990) Semiconductor Equations, Springer-Verlag, New York.Google Scholar
[23] Markowich, P. A. & Schmeiser, C. (1986) Uniform asymptotic representation of solutions of the basic semiconductor-device equations. IMA J. Appl. Math. 36 (1), 43–57.Google Scholar
[24] Matejczyk, B., Valiskó, M., Wolfram, M.-T., Pietschmann, J.-F. & Boda, D. (2017) Multiscale modeling of a rectifying bipolar nanopore: Comparing Poisson–Nernst–Planck to Monte Carlo. J. Chem Phys. 146 (12), 124125.Google Scholar
[25] Newman, J. & Thomas-Alyea, K. E. (2012) Electrochemical Systems, John Wiley & Sons.Google Scholar
[26] Pietschmann, J.-F., Wolfram, M.-T., Burger, M., Trautmann, C., Nguyen, G., Pevarnik, M., Bayer, V. & Siwy, Z. (2013) Rectification properties of conically shaped nanopores: Consequences of miniaturization. Phys. Chem. Chem. Phys. 15, 16917–16926.Google Scholar
[27] Richardson, G. (2009) A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials. Math. Med. Biol. 26 (3), 201–224.Google Scholar
[28] Richardson, G., Denuault, G. & Please, C. P. (2012) Multiscale modelling and analysis of lithium-ion battery charge and discharge. J. Eng. Math. 72 (1), 41–72.Google Scholar
[29] Richardson, G., O'Kane, S. E. J., Niemann, R. G., Peltola, T. A., Foster, J. M., Cameron, P. J. & Walker, A. B. (2016) Can slow-moving ions explain hysteresis in the current–voltage curves of perovskite solar cells? Energy & Environ. Sci. 9 (4), 1476–1485.Google Scholar
[30] Richardson, G., Please, C., Foster, J. & Kirkpatrick, J. (2012) Asymptotic solution of a model for bilayer organic diodes and solar cells. SIAM J. Appl. Math. 72 (6), 1792–1817.Google Scholar
[31] Richardson, G., Please, C. P. & Styles, V. (2017) Derivation and solution of effective medium equations for bulk heterojunction organic solar cells. Eur. J. Appl. Math. 28 (6), 973–1014.Google Scholar
[32] Schneider, G. F. & Dekker, C. (2013) DNA sequencing with nanopores. Nat. Biotech. 30, 326–328.Google Scholar
[33] Schöberl, J. (1997) NETGEN An advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1 (1), 41–52.Google Scholar
[34] Singer, A., Gillespie, D., Norbury, J. & Eisenberg, R. S. (2008) Singular perturbation analysis of the steady-state Poisson–Nernst–Planck system: Applications to ion channels. Eur. J. Appl. Math. 19 (5), 541–560.Google Scholar
[35] Singer, A. & Norbury, J. (2009) A Poisson–Nernst–Planck model for biological ion channels: An asymptotic analysis in a three-dimensional narrow funnel. SIAM J. Appl. Math. 70 (3), 949–968.Google Scholar
[36] Siwy, Z., Apel, P., Baur, D., Dobrev, D. D., Korchev, Y. E., Neumann, R., Spohr, R., Trautmann, C. & Voss, K. O. (2003) Preparation of synthetic nanopores with transport properties analogous to biological channels. Surf. Sci. 532, 1061–1066.Google Scholar
[37] Vlassiouk, I., Smirnov, S. & Siwy, Z. (2008) Nanofluidic ionic diodes. Comparison of analytical and numerical solutions. ACS Nano 2 (8), 1589–1602.Google Scholar
[38] Yang, Z., Van, T. A., Straaten, D., Ravaioli, U. & Liu, Y. (2005) A coupled 3-D PNP/ECP model for ion transport in biological ion channels. J. Comput. Electron. 4 (1), 167–170.Google Scholar
[39] Zheng, Q. & Wei, G.-W. (2011) Poisson–Boltzmann–Nernst–Planck model. J. Chem. Phys. 134 (19), 194101.Google Scholar