Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study

Yiwei Wang; Mingji Zhang; Yiwei Wang; Mingji Zhang

doi:10.3934/mbe.2024084

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 1899-1916. doi: 10.3934/mbe.2024084

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Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study

Yiwei Wang ¹,
Mingji Zhang ^{1,2
,
,}

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
2.
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Academic Editor: Yang Kuang

Received: 22 November 2023 Revised: 28 December 2023 Accepted: 02 January 2024 Published: 05 January 2024

We consider a quasi-one-dimensional Poisson-Nernst-Planck model with two cations having the same valances and one anion. Bikerman's local hard-sphere potential is included to account for ion size effects. Under some further restrictions on the boundary conditions of the two cations, we obtain approximations of the I-V (current-voltage) relations by treating the ion sizes as small parameters. Critical potentials are identified, which play critical roles in characterizing finite ion size effects on ionic flows. Nonlinear interplays between system parameters, such as boundary concentrations and diffusion coefficients, are analyzed. To provide more intuitive illustrations of our analytical results and better understanding of the dynamics of ionic flows through membrane channels, numerical simulations are performed.
- PNP,
- finite ion sizes,
- I-V relations,
- critical potentials,
- Bikerman's LHS
Citation: Yiwei Wang, Mingji Zhang. Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 1899-1916. doi: 10.3934/mbe.2024084

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Abstract

We consider a quasi-one-dimensional Poisson-Nernst-Planck model with two cations having the same valances and one anion. Bikerman's local hard-sphere potential is included to account for ion size effects. Under some further restrictions on the boundary conditions of the two cations, we obtain approximations of the I-V (current-voltage) relations by treating the ion sizes as small parameters. Critical potentials are identified, which play critical roles in characterizing finite ion size effects on ionic flows. Nonlinear interplays between system parameters, such as boundary concentrations and diffusion coefficients, are analyzed. To provide more intuitive illustrations of our analytical results and better understanding of the dynamics of ionic flows through membrane channels, numerical simulations are performed.

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