一类稳态变介电 Poisson-Nernst-Planck 方程的有限元计算

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 318-326.

一类稳态变介电 Poisson-Nernst-Planck 方程的有限元计算

张冰洁, 沈瑞刚^*()

桂林电子科技大学数学与计算科学学院, 广西应用数学中心 (GUET), 广西高校数据分析与计算重点实验室广西桂林 541004

收稿日期:2024-12-26 修回日期:2025-06-27 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 沈瑞刚 E-mail:rgshen@guet.edu.cn
基金资助:
广西科技基地和人才专项(Guike AD23026048);国家自然科学基金(12101595);国家自然科学基金(12161026)

Finite Element Calculation of a Steady-State Variable Dielectric Poisson-Nernst-Planck Equations

Bingjie Zhang, Ruigang Shen^*()

Guilin University of Electronic Technology, School of Mathematics and Computating Science, Guangxi Applied Mathematics Center (GUET), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guangxi Guilin 541004

Received:2024-12-26 Revised:2025-06-27 Online:2026-02-26 Published:2026-01-19
Contact: Ruigang Shen E-mail:rgshen@guet.edu.cn
Supported by:
Special Fund for Scientific and Technological Bases and Talents of Guangxi(Guike AD23026048);NSFC(12101595);NSFC(12161026)

摘要/Abstract

摘要：

变介电 Poisson-Nernst-Planck (VDPNP) 方程是一种描述电解质溶液中离子传输行为的模型, 该方程在经典 Poisson-Nernst-Planck (PNP) 方程的基础上引入了介电系数与离子浓度的依赖关系, 能更好地描述电解质溶液中复杂的动力学过程. 为了研究 VDPNP 方程对体系相互作用的影响, 该文中首先基于 Gummel 有限元算法, 对 VDPNP 模型进行了求解. 随后进行了纳米孔体系的数值模拟, 模拟结果表明, 在混合溶液中, 传统 PNP 方程在浓度上不能区分出钠、钾离子, 而 VDPNP 模型可以明显区分钠、钾离子, 且随着表面电荷越大时, 其区分效果越加明显.

关键词: 变介电 Poisson-Nernst-Planck 方程, 有限元方法, 数值模拟

Abstract:

The Variable Dielectric Poisson-Nernst-Planck (VDPNP) equations are models that describe ion transport behavior in electrolyte solutions. These equations extend the classical Poisson-Nernst-Planck (PNP) equations by introducing a dependence of the dielectric constant on ion concentration, which allows for a better description of the complex dynamic processes in electrolyte solutions. To investigate the impact of the VDPNP equations on system interactions, this paper first solves the VDPNP model using the Gummel finite element method. Subsequently, numerical simulations of nanopore systems are performed. The simulation results show that, in mixed solutions, the traditional PNP equations cannot distinguish between sodium and potassium ions based on concentration, whereas the VDPNP models can clearly differentiate between sodium and potassium ions. Furthermore, the distinguishing effect becomes more pronounced as the surface charge increases.

Key words: variable Dielectric Poisson-Nernst-Planck equations, finite element method, numerical simulation

中图分类号:

O241.82

张冰洁, 沈瑞刚. 一类稳态变介电 Poisson-Nernst-Planck 方程的有限元计算[J]. 数学物理学报, 2026, 46(1): 318-326.

Bingjie Zhang, Ruigang Shen. Finite Element Calculation of a Steady-State Variable Dielectric Poisson-Nernst-Planck Equations[J]. Acta mathematica scientia,Series A, 2026, 46(1): 318-326.

图/表 6

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参考文献 13

[1]	Lu B Z, Holst M J, McCammon J A, et al. Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions. J Comput Phys, 2010, 229(19): 6979-6994 pmid: 21709855
[2]	Eisenberg R S, Chen D P. Poisson-Nernst-Planck (PNP) theory of an open ionic channel. Biophys J, 1993, 64(2): Art 22
[3]	Constantin D, Siwy Z S. Poisson-Nernst-Planck model of ion current rectification through a nanofluidic diode. Phys Rev E, 2007, 76: Art 041202
[4]	Wei G W, Zheng Q, Chen Z, et al. Variational multiscale models for charge transport. SIAM Rev, 2012, 54: 699-754 doi: 10.1137/110845690
[5]	Eisenberg B. Ionic channels in biological membranes: Natural nanotubes. Acc Chem Res, 1998, 31: 117-123 doi: 10.1021/ar950051e
[6]	Booth F. Dielectric constant of polar liquids at high field strengths. J Chem Phys, 1955, 23: 453-457 doi: 10.1063/1.1742009
[7]	Wei Y Z, Chiang P, Sridhar S. Ion size effects on the dynamic and static dielectric properties of aqueous alkali solutions. J Chem Phys, 1992, 96: 4569-4573 doi: 10.1063/1.462792
[8]	Hess B, Holm C, van der Vegt N. Modeling multibody effects in ionic solutions with a concentration dependent dielectric permittivity. Phys Rev Lett, 2006, 96: Art 147801
[9]	Li B, Wen J Y, Zhou S G. Mean-field theory and computation of electrostatics with ionic concentration depenent dielectrics. Commun Math Sci, 2016, 14: 249-271 pmid: 26877718
[10]	Liu X J, Qiao Y, Lu B Z. Analysis of the mean field free energy functional of electrolyte solution with nonhomogenous boundary conditions and the generalized PB/PNP equations with inhomogeneous dielectric permittivity. SIAM J Appl Math, 2018, 78(2): 1131-1154 doi: 10.1137/16M1108583
[11]	Li H L, Lu B Z. An ionic concentration and size dependent dielectric permittivity Poisson-Boltzmann model for biomolecular solvation studies. J Chem Phys, 2014, 141: Art 024115
[12]	Liu B, Liu P, Xu Z L, et al. Ionic size effects: Generalized Boltzmann distributions, counterion stratification, and modified Debye length. Nonlinearity, 2013, 26: 2899-2922 pmid: 24465094
[13]	Qiao Y, Liu X J, Chen M X, et al. A local approximation of fundamental measure theory incorporated into three dimensional Poisson-Nernst-Planck equations to account for hard sphere repulsion among ions. J Stat Phys, 2016, 163: 156-174 doi: 10.1007/s10955-016-1470-7

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