Nernst Planck Poisson System Research Articles

Cubic-like Features of I–V Relations via Classical Poisson–Nernst–Planck Systems Under Relaxed Electroneutrality Boundary Conditions

We focus on higher-order matched asymptotic expansions of a one-dimensional classical Poisson–Nernst–Planck system for ionic flow through membrane channels with two oppositely charged ion species under relaxed electroneutrality boundary conditions. Of particular interest are the current–voltage (I–V) relations, which are used to characterize the two most relevant biological properties of ion channels—permeation and selectivity—experimentally. Our result shows that, up to the second order in ε=λ/r, where λ is the Debye length and r is the characteristic radius of the channel, the cubic I–V relation has either three distinct real roots or a unique real root with a multiplicity of three, which sensitively depends on the boundary layers because of the relaxation of the electroneutrality boundary conditions. This indicates more rich dynamics of ionic flows under our more realistic setups and provides a better understanding of the mechanism of ionic flows through membrane channels.

A new semi-analytic model for Stern-layer polarization in pore throats

SUMMARY To explain induced polarization, membrane polarization is often referred to as a relevant process taking place in granular media – particularly, when narrow pore throats are present. This polarization effect is based on the membrane-like behaviour of pore throats caused by the presence of an usually negative charge on the pore surface, that influences charge transport in the pore fluid. Existing analytical, 1D models describe the pore system as a series of cylindrical pores with different radii and lengths. The polarization response is calculated by solving the Poisson–Nernst–Planck system for the current densities of one single anion and one single cation species representing the charge transport in the electrolyte and the diffuse layer at the pore surface. To include charge transport in the Stern layer, cations in the Stern layer have so far simply been considered by increasing the concentration of the diffuse layer cations. As we know from numerical modelling, this approach fails to predict the polarization response when the Stern layer is significantly charged. Here, we present a new semi-analytical model that treats the Stern-layer cations as a separate ion species and allows the Stern layer to polarize individually. To validate our new model, we compare it to the previously used analytical model and numerical simulations for different relative charges in Stern- and diffuse layer. We also use electrostatic surface-complexation models for two mineral surfaces (quartz and montmorillonite) to simulate the response of real geologic material under varying chemical conditions. This work is a step forward for considering realistic pore properties in induced-polarization modelling.

A meshless stochastic method for Poisson-Nernst-Planck equations.

A plethora of biological, physical, and chemical phenomena involve transport of charged particles (ions). Its continuum-scale description relies on the Poisson-Nernst-Planck (PNP) system, which encapsulates the conservation of mass and charge. The numerical solution of these coupled partial differential equations is challenging and suffers from both the curse of dimensionality and difficulty in efficiently parallelizing. We present a novel particle-based framework to solve the full PNP system by simulating a drift-diffusion process with time- and space-varying drift. We leverage Green's functions, kernel-independent fast multipole methods, and kernel density estimation to solve the PNP system in a meshless manner, capable of handling discontinuous initial states. The method is embarrassingly parallel, and the computational cost scales linearly with the number of particles and dimension. We use a series of numerical experiments to demonstrate both the method's convergence with respect to the number of particles and computational cost vis-à-vis a traditional partial differential equation solver.

Effects of Brownian motion on solitary wave structures for 1D stochastic Poisson–Nernst–Planck system in electrobiochemical

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry.

A flux-based moving mesh method applied to solving the Poisson–Nernst–Planck equations

The moving mesh method is one of the important adaptive mesh methods which is practically useful when the mesh size and overall resolution are provided and fixed as seen in many engineering computations. We employ a moving mesh technique combined with a finite element method (FEM) to solve a widely-used charged carrier transport model, the Poisson–Nernst–Planck (PNP) equations, the solutions of which often have boundary or internal layers and sharp interfaces when the convection is dominated. Considering that flux is a significant physical quantity in designing stable and accurate numerical algorithm for transport problems such as the PNP system, we start from a relatively general functional and propose a flux-based monitor function and an adaptive step size-controlling algorithm for guiding the mesh movement in the FEM solution of the PNP equations. The numerical results illustrate that the moving mesh method is effectively addressing challenges arising from solution singularities and the convection-dominated effects. It also shows that the moving mesh finite element method we propose exhibits superior performance compared to both the traditional moving mesh finite element method and fixed mesh finite element method in some scenarios. Furthermore, it demonstrates better adherence to the physical properties of energy dissipation inherent in the PNP equation.

Studies on Ionic Flows via Poisson–Nernst–Planck Systems with Bikerman’s Local Hard-Sphere Potentials under Relaxed Neutral Boundary Conditions

We examine the qualitative properties of ionic flows through ion channels via a quasi-one-dimensional Poisson–Nernst–Planck model under relaxed neutral boundary conditions. Bikerman’s local hard-sphere potential is included in the model to account for finite ion size effects. Our main interest is to examine the boundary layer effects (due to the relaxation of electroneutrality boundary conditions) on both individual fluxes and current–voltage relations systematically. Critical values of potentials are identified that play significant roles in studying internal dynamics of ionic flows. It turns out that the finite ion size can either enhance or reduce the ionic flow under different nonlinear interplays between the physical parameters in the system, particularly, boundary concentrations, boundary potentials, boundary layers, and finite ion sizes. Much more rich dynamics of ionic flows through membrane channels is observed.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

A bubble model for the gating of Kv channels

Abstract Voltage-gated K$_{\mathrm{v}}$ channels play fundamental roles in many biological processes, such as the generation of the action potential. The gating mechanism of K$_{\mathrm{v}}$ channels is characterized experimentally by single-channel recordings and ensemble properties of the channel currents. In this work, we propose a bubble model coupled with a Poisson–Nernst–Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. We hypothesize that the stochastic behaviour of the gating phenomenon is due to randomness of the bubble and channel sizes. The predicted ensemble average of the currents under various applied voltage across the channels is consistent with experimental observations, and the Cole–Moore delay is captured by varying the holding potential.

The linearized Poisson–Nernst–Planck system as heat flow on the interval under non‐local boundary conditions

The linearized Poisson–Nernst–Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non‐local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take advantage of the representation of the resolvent operator and recover the heat kernel by applying the inverse Laplace transform.

A Second Order Accurate, Positivity Preserving Numerical Method for the Poisson–Nernst–Planck System and Its Convergence Analysis

A Second Order Accurate, Positivity Preserving Numerical Method for the Poisson–Nernst–Planck System and Its Convergence Analysis

Justification of a New Original Homogenized Model for Ionic Diffusion in Porous Media

Abstract In this work, a new original justification of an homogenized model for ionic diffusion in porous media is proposed. The approach used enables to specify clearly the domain of validity of this homogenized model, involving a source term characterizing the electrical double layer effect at the macroscale. This homogenized model is obtained from the formal periodic homogenization of a Nernst–Planck–Poisson system at the pore scale accounting for conductivity of the solid phase which is generally neglected. The Poisson equation is defined in both fluid and solid phases and the discontinuity of fluxes at the solid–fluid interface is modeled by a jump of the electrical field, linked to the surface electrical charge of the solid interface. Numerical simulations are carried out at the scale of the unit cell to underscore the influence of the contrast on the electrical permittivity between fluid and solid phases. The comparison of the concentrations and the electrical potential given at the macroscale by the homogenized model and by a direct pore scale model reveals the accuracy of the homogenized model which is very simple to use.

Existence of [formula omitted] singular solutions to Euler–Nernst–Planck–Poisson system on [formula omitted] with free-moving charges

We construct a special C1,α blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst–Planck–Poisson equation. Our construction lies on the framework established in Elgindi et al. (2021) and relies on a special solution to variant spherical Laplacian.

Mathematical Analysis on Current–Voltage Relations via Classical Poisson–Nernst–Planck Systems with Nonzero Permanent Charges under Relaxed Electroneutrality Boundary Conditions

We focus on a quasi-one-dimensional Poisson-Nernst-Planck model with small permanent charges for ionic flows of two oppositely charged ion species through an ion channel. Of particular interest is to examine the dynamics of ionic flows in terms of I-V (current-voltage) relations with boundary layers due to the relaxation of neutral conditions on boundary concentrations. This is achieved by employing the regular perturbation analysis on the solutions established through geometric singular perturbation analysis. Rich dynamics are observed, particularly, the nonlinear interplays among different physical parameters are characterized. Critical potentials are identified, which play critical roles in the study of ionic flows and can be estimated experimentally. Numerical simulations are performed to further illustrate and provide more intuitive understandings of our analytical results.

Effects of ion sizes on Poisson–Nernst–Planck systems with multiple ions

In this work, we analyze a one‐dimensional version of steady‐state Poisson–Nernst–Planck models for ionic flow through a membrane channel including ionic interactions modeled by the density functional theory. The model includes an arbitrary number of positively changed ions with the same valences and one negatively charged ion and ignores the permanent charge. The model can be viewed as a singularly perturbed differential system; therefore, our analysis is mainly based on the geometric singular perturbation theory. The existence and the uniqueness result for small ion sizes are established, and treating the sizes as small parameters, we also derive an approximation of the individual flux, the – (current–voltage) relation and the individual flux difference. Critical potentials are identified, and their roles in characterizing ionic flow properties are studied.

A dynamic mass transport method for Poisson-Nernst-Planck equations

A dynamic mass-transport method is proposed for approximately solving the Poisson–Nernst–Planck (PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally enforces solution positivity and the energy law as for the continuous PNP system. The fully discrete scheme is further formulated as a constrained minimization problem, shown to be solvable, and satisfy all three solution properties (mass conservation, positivity and energy dissipation) independent of time step size or the spatial mesh size. Numerical experiments are conducted to validate convergence of the computed solutions and verify the structure preserving property of the proposed scheme.

Finite Ion Size Effects on Ionic Flows via Poisson–Nernst–Planck Systems: Higher Order Contributions

Finite Ion Size Effects on Ionic Flows via Poisson–Nernst–Planck Systems: Higher Order Contributions

Anomalous Sorption Kinetics of Self-Interacting Particles by a Spherical Trap

In this paper we propose a computational framework for the investigation of the correlated motion between positive and negative ions exposed to the attraction of a bubble surface that mimics the (oscillating) cell membrane. The correlated diffusion of surfactants is described by a Poisson-Nernst-Planck (PNP) system, in which the drift term is given by the gradient of a potential which includes both the effect of the bubble and the Coulomb interaction between the carriers. The latter term is obtained from the solution of a self-consistent Poisson equation. For very short Debye lengths one can adopt the so called Quasi-Neutral limit which drastically simplifies the system, thus allowing for much faster numerical simulations. The paper has four main objectives. The first one is to present a PNP model that describes ion charges in presence of a trap. The second one is to provide benchmark tests for the validation of simplified multiscale models under current development [1]. The third one is to explore the relevance of the term describing the interaction among the apolar tails of the anions. The last one is to quantitatively explore the validity of the Quasi-Neutral limit by comparison with detailed numerical simulation for smaller and smaller Debye lengths. In order to reach these goals, we propose a simple and efficient Alternate Direction Implicit method for the numerical solution of the non-linear PNP system, which guarantees second order accuracy both in space and time, without requiring solution of nonlinear equation at each time step. New semi-implicit scheme for a simplified PNP system near quasi neutrality is also proposed.

An iteration solver for the Poisson–Nernst–Planck system and its convergence analysis

In this paper, we provide a theoretical analysis for an iteration solver to implement a finite difference numerical scheme for the Poisson-Nernst–Planck (PNP) system, based on the Energetic Variational Approach (EnVarA), in which a non-constant mobility H−1 gradient flow is formulated. In particular, the nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty. In the numerical design, the mobility function is explicitly updated, for the sake of unique solvability analysis. The logarithmic and the electric potential diffusion terms, which come from the gradient of convex energy functional parts, are implicitly computed. The positivity-preserving property for all the concentrations, an unconditional energy stability, and the optimal rate error estimate have been established in a recent work. A modified Newton iteration for the nonlinear and logarithmic part, combined with a linear iteration for the electric potential part, is proposed to implement the given numerical scheme, in which a non-constant linear elliptic equation needs to be solved at each iteration stage. A theoretical analysis is presented in this article, and a linear convergence is proved for such an iteration, with an asymptotic error constant in the same order of the time step size. A numerical test is also presented in this article, which demonstrates the linear convergence rate of the proposed iteration solver.

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert’s time. In this paper, we consider a diagram of the diffusion limit from the Vlasov–Poisson–Fokker–Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson–Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

Effects on I–V relations from small permanent charge and channel geometry via classical Poisson–Nernst–Planck equations with multiple cations

We analyse a quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel with nonzero but small permanent charge. The system includes three ion species, two cations with the same valences and one anion. The cross-sectional area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is critical for our analysis. We treat the model as a boundary value problem of a singularly perturbed system of differential equations. Under the framework of geometric singular perturbation theory, together with specific structures of the model, the existence and local uniqueness of solutions to the boundary value problem for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, via regular perturbation analysis, we are able to derive an approximation of the I–V (current–voltage) relations explicitly, from which the permanent charge and channel geometry effects on ionic flows are analysed in detail. Critical potentials are identified and their roles in characterising the ionic flow properties of interest are studied. Some critical potentials can be experimentally estimated, and this provides an efficient way to adjust the boundary pot to observe distinct dynamics of ionic flows. Mathematical analysis indicates that a short and narrow filter within which the permanent charge is confined is expected in order to optimise the effect of permanent charges. This is consistent with the typical structure of an ion channel.