一个排除体积效应的Poisson-Nernst-Planck模型的理论与数值实现
A Modified Poisson--Nernst--Planck Model with Excluded Volume Effect: Theory and Numerical Implementation
摘要：
Poisson-Nernst-Planck（PNP）方程已被广泛应用于描述离子通道、纳米流体装置和许多电化学系统中的离子输运。尽管PNP方程有着广泛的应用，但由于忽略了排除体积效应，PNP方程在预测受限环境中离子浓度的动态和平衡态时是失败的。在这项工作中，一个简单而有效的修正PNP（MPNP）模型与排除体积效应的基础上，离子的扩散系数的修改。在稳态下，借助Lambert-W特殊函数，得到了修正的Poisson-Boltzmann（MPB）方程。建立了MPB方程弱解的存在唯一性。通过对弱静电势和强静电势极限的进一步分析，分别得到了两种修正的德拜屏蔽长度。对MPNP模型发展了一种既能保持总离子浓度又能满足能量耗散的数值格式。数值分析证明了该方案遵守离子质量守恒并满足相应的离散自由能量耗散定律。数值解的正性也进行了讨论和数值研究。数值试验表明，该格式在空间离散上具有二阶精度，并具有预期的性能。大量的数值模拟表明，排斥体积效应对离子浓度和通量的动力学有显着的影响。此外，通过研究自由能和扩散电荷的演化，系统地研究了体积排阻对电荷扩散时间尺度的影响。
英文摘要：
The Poisson--Nernst--Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson--Boltzmann (MPB) equation is obtained with the help of the Lambert-W special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.
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