Nernst Planck Poisson System Research Articles

Error analysis of mixed finite element method for Poisson‐Nernst‐Planck system

To improve the convergence rate in L2 norm from suboptimal to optimal for both electrostatic potential and ionic concentrations in Poisson‐Nernst‐Planck (PNP) system, we propose the mixed finite element method in this article to discretize the electrostatic potential equation, and still use the standard finite element method to discretize the time‐dependent ionic concentrations equations. Optimal error estimates in norm for the electrostatic potential, and in and norms for the ionic concentrations are attained. As a by‐product, the electric field can also achieve a higher approximation order in contrast with the standard finite element method for PNP system. Numerical experiments are performed to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1924–1948, 2017

An energy-preserving discretization for the Poisson–Nernst–Planck equations

The Poisson---Nernst---Planck (PNP) equations have recently been used to describe the dynamics of ion transport through biological ion channels besides being widely employed in semiconductor industry. This paper is about the design of a numerical scheme to solve the PNP equations that preserves exactly (up to roundoff error) a discretized form of the energy dynamics of the system. The proposed finite difference scheme is of second-order accurate in both space and time. Comparisons are made between this energy dynamics-preserving scheme and a standard finite difference scheme, showing a difference in satisfying the energy law. Numerical results are presented for validating the orders of convergence in both time and space of the new scheme for the PNP system.

Modelling compression sensing in ionic polymer metal composites

Ionic polymer metal composites (IPMCs) consist of an ionomeric membrane, including mobile counterions, sandwiched between two thin noble metal electrodes. IPMCs find application as sensors and actuators, where an imposed mechanical loading generates a voltage across the electrodes, and, vice versa, an imposed electric field causes deformation. Here, we present a predictive modelling approach to elucidate the dynamic sensing response of IPMCs subject to a time-varying through-the-thickness compression (‘compression sensing’). The model relies on the continuum theory recently developed by Porfiri and co-workers, which couples finite deformations to the modified Poisson–Nernst–Planck (PNP) system governing the IPMC electrochemistry. For the ‘compression sensing’ problem we establish a perturbative closed-form solution along with a finite element (FE) solution. The systematic comparison between these two solutions is a central contribution of this study, offering insight on accuracy and mathematical complexity. The method of matched asymptotic expansions is employed to find the analytical solution. To this end, we uncouple the force balance from the modified PNP system and separately linearise the PNP equations in the ionomer bulk and in the boundary layers at the ionomer–electrode interfaces. Comparison with FE results for the fully coupled nonlinear system demonstrates the accuracy of the analytical solution to describe IPMC sensing for moderate deformation levels. We finally demonstrate the potential of the modelling scheme to accurately reproduce experimental results from the literature. The proposed model is expected to aid in the design of IPMC sensors, contribute to an improved understanding of IPMC electrochemomechanical response, and offer insight into the role of nonlinear phenomena across mechanics and electrochemistry.

Non-localness of Excess Potentials and Boundary Value Problems of Poisson–Nernst–Planck Systems for Ionic Flow: A Case Study

Poisson–Nernst–Planck (PNP) type systems are basic primitive models for ionic flow through ion channels. Important properties of ion channels, such as current-voltage relations, permeation and selectivity, can be extracted from solutions of boundary value problems (BVP) of PNP type models. Many issues of BVP of PNP type systems with local excess potentials (including particularly classical PNP systems that treat ions as point-charges) are extensively examined analytically and numerically. On the other hand, for PNP type systems with nonlocal excess potentials, even the issue of well-posedness of BVP is poorly understood. In fact, the formulation of correct boundary conditions seems to be overlooked, even though complications of ionic behavior near the boundaries (locations of applied electrodes) have been long experienced in experiments and simulations. PNP type systems with nonlocal excess potentials can be viewed as functional differential systems and, for many approximation models of nonlocal excess potentials, as differential equations with both delays and advances. Thus PNP type systems with nonlocal excess potentials have infinite degree of freedoms and BVP with the traditional “two-point-boundary-conditions” would be severely under determined. The mathematical theory for PNP with nonlocal excess potential would be significantly different from that for PNP with local excess potentials. Taking into considerations of experimental designs of ionic flow through ion channels and in a relatively simple setting, we present a form of natural “boundary conditions” so that the corresponding BVP of PNP type systems with nonlocal excess potentials are generally well-posed. This work, at an early stage toward a better understanding of related issues, provides some insights on interpretations of experimental designs of imposing boundary conditions and for correct formulations of numerical simulations, and hopefully, will stimulate further mathematical analysis on this important issue.

A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson–Nernst–Planck systems

We design an arbitrary-order free energy satisfying discontinuous Galerkin (DG) method for solving time-dependent Poisson–Nernst–Planck systems. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivity of numerical solutions is enforced by an accuracy-preserving limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the high resolution of the numerical algorithm and to illustrate the proven properties of mass conservation, free energy dissipation, as well as the preservation of steady states.

Analysis of the Poisson–Nernst–Planck equation in a ball for modeling the Voltage–Current relation in neurobiological microdomains

The electro-diffusion of ions is often described by the Poisson–Nernst–Planck (PNP) equations, which couple nonlinearly the charge concentration and the electric potential. This model is used, among others, to describe the motion of ions in neuronal micro-compartments. It remains at this time an open question how to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a domain bounded by an impermeable dielectric membrane. The purpose of this paper is to construct an asymptotic approximation to the solution of the stationary PNP equations in a d-dimensional ball (d=1,2,3) in the limit of large total charge. In this geometry the PNP system reduces to the Liouville–Gelfand–Bratú (LGB) equation, with the difference that the boundary condition is Neumann, not Dirichlet, and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. These differences replace attraction by repulsion in the LGB equation, thus completely changing the solution. We find that the voltage is maximal in the center and decreases toward the boundary. We also find that the potential drop between the center and the surface increases logarithmically in the total number of charges and not linearly, as in classical capacitance theory. This logarithmic singularity is obtained for d=3 from an asymptotic argument and cannot be derived from the analysis of the phase portrait. These results are used to derive the relation between the outward current and the voltage in a dendritic spine, which is idealized as a dielectric sphere connected smoothly to the nerve axon by a narrow neck. This is a fundamental microdomain involved in neuronal communication. We compute the escape rate of an ion from the steady density in a ball, which models a neuronal spine head, to a small absorbing window in the sphere. We predict that the current is defined by the narrow neck that is connected to the sphere by a small absorbing window, as suggested by the narrow escape theory, while voltage is controlled by the PNP equations independently of the neck.

On local weak solutions to Nernst–Planck–Poisson system

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well as electrons and holes transport in semiconductors. The considered boundary conditions allow the physical system to be not only closed but also open. Theorems on existence, uniqueness, and nonnegativity of local weak solutions are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

Global weak solutions in three space dimensions for electrokinetic flow processes

For a Navier–Stokes–Nernst–Planck–Poisson system we construct global weak solutions in a three-dimensional bounded domain. A special feature of our approach is that we allow for nonconstant diffusion coefficients which may vary from species to species as well as for \({L^2}\)-initial data without any further constraints. Our approach is based on the intrinsic energy structure, Aubin–Simon compactness arguments, and maximal \({L^p}\)-regularity.

Gevrey class regularity of solutions to the Nernst–Planck–Poisson equations with generalized dissipation

In this article, we consider Nernst–Planck–Poisson system with generalized dissipation. First, we prove the Gevrey class regularity of local solutions to system with large rough initial data in modulation spaces . Secondly, applying so-called Gevrey estimates, which is motivated by the works of Foias and Temam, we establish Gevrey class regularity of solutions to the system with initial data in a certein critical Fourier–Besov spaces . The results of us particularly imply that the solution is analytic in the spatial variable and obtain temporal decay rates of higher Fourier–Besov norms of solutions.

An energy preserving finite difference scheme for the Poisson–Nernst–Planck system

In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson–Nernst–Planck system. Although the Poisson–Nernst–Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergent in space and first order convergent in time. Finally we point out that our method can be easily extended to the case of multi-ions.

Hierarchical electrochemical modeling and simulation of bio-hybrid interfaces

In this article we propose and investigate a hierarchy of mathematical models based on partial differential equations (PDE) and ordinary differential equations (ODE) for the simulation of the biophysical phenomena occurring in the electrolyte fluid that connects a biological component (a single cell or a system of cells) and a solid-state device (a single silicon transistor or an array of transistors). The three members of the hierarchy, ordered by decreasing complexity, are: (i) a 3D Poisson–Nernst–Planck (PNP) PDE system for ion concentrations and electric potential; (ii) a 2D reduced PNP system for the same dependent variables as in (i); (iii) a 2D area-contact PDE system for electric potential coupled with a system of ODEs for ion concentrations. The backward Euler method is adopted for temporal semi-discretization and a fixed-point iteration based on Gummel’s map is used to decouple system equations. Spatial discretization is performed using piecewise linear triangular finite elements stabilized via edge-based exponential fitting. Extensively conducted simulation results are in excellent agreement with existing analytical solutions of the PNP problem in radial coordinates and experimental and simulated data using simplified lumped parameter models.

Energetically stable discretizations for charge transport and electrokinetic models

A finite element discretization using a method of lines approached is proposed for approximately solving the Poisson–Nernst–Planck (PNP) equations. This discretization scheme enforces positivity of the computed solutions, corresponding to particle density functions, and a discrete energy estimate is established that takes the same form as the energy law for the continuous PNP system. This energy estimate is extended to finite element solutions to an electrokinetic model, which couples the PNP system with the incompressible Navier–Stokes equations. Numerical experiments are conducted to validate convergence of the computed solution and verify the discrete energy estimate.

Asymptotic expansions and numerical simulations of I-V relations via a steady state Poisson-Nernst-Planck system

A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

A biomimetic underwater vehicle actuated by waves with ionic polymer–metal composite soft sensors

The ionic polymer–metal composite (IPMC) is a soft material based actuator and sensor and has a promising potential in underwater application. This paper describes a hybrid biomimetic underwater vehicle that uses IPMCs as sensors. Propelled by the energy of waves, this underwater vehicle does not need an additional energy source. A physical model based on the hydrodynamics of the vehicle was developed, and simulations were conducted. Using the Poisson–Nernst–Planck system of equations, a physics model for the IPMC sensor was proposed. For this study, experimental apparatus was developed to conduct hydrodynamic experiments for both the underwater vehicle and the IPMC sensors. By comparing the experimental and theoretical results, the speed of the underwater vehicle and the output of the IPMC sensors were well predicted by the theoretical models. A maximum speed of 1.08 × 10−1 m s−1 was recorded experimentally at a wave frequency of 1.6 Hz. The peak output voltage of the IPMC sensor was 2.27 × 10−4 V, recorded at 0.8 Hz. It was found that the speed of the underwater vehicle increased as the wave frequency increased and the IPMC output decreased as the wave frequency increased. Further, the energy harvesting capabilities of the underwater vehicle hosting the IPMCs were tested. A maximum power of 9.50 × 10−10 W was recorded at 1.6 Hz.

A singularly perturbed nonlinear Poisson–Boltzmann equation: uniform and super‐asymptotic expansions

A steady‐state Poisson–Nernst–Planck system is investigated, which is conformed into a nonlinear Poisson equation by means of the Boltzmann statistics. It describes the electrostatic potential generated by multiple concentrations of ions in a heterogeneous (porous) medium with diluted (solid) particles. The nonlinear elliptic problem is singularly perturbed with the Debye length as a small parameter related to the electric double layer near the solid particle boundary. For star‐shaped solid particles, we prove rigorously that the solution of the problem in spatial dimensions 1d, 2d and 3d is uniformly and super‐asymptotically approximated by a constant reference state. Copyright © 2015 John Wiley & Sons, Ltd.

A survey on properties of Nernst–Planck–Poisson system. Application to ionic transport in porous media

The aim of this paper is to study the properties of Nernst–Planck–Poisson system describing the evolution of ionic concentrations in porous media. In particular, the well-posedness character of the system is proved and some qualitative properties of the global solution are established (energy and entropy laws, stationary states, Boltzmann distribution, influence of an external electrical field). We prove also that a superficial accumulation of ions occurs when an important external electrical field is applied.

Comparison and numerical treatment of generalised Nernst–Planck models

In its most widespread, classical formulation, the Nernst–Planck–Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an appropriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter–Gummel finite volume discretisation scheme for drift–diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretised generalised Poisson–Boltzmann system describing the thermodynamical equilibrium is a stationary state of the discretised time-dependent generalised Nernst–Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach.

Global well-posedness for the Euler–Nernst–Planck–Poisson system in dimension two

In this paper, we study the Cauchy problem of the Euler–Nernst–Planck–Poisson system. We obtain the global well-posedness for the system in dimension two for any initial data in Hs1(R2)×Hs2(R2)×Hs2(R2) under certain conditions of s1 and s2.

Analytical Solutions to the Steady State Poisson-Nernst-Planck Equations in Electrobiochemical Systems

The Poisson-Nernst-Planck equations are relevant in numerous electrobiochemical applications. In this paper, we provide analytical solutions to the steady state Poisson-Nernst-Planck (PNP) systems of equations for situations relevant to applications involving bioelectric dressings and bandages. The PNP system of equations is analyzed for two ionic species (one positively charged and the other negatively charged) both in the one-dimensional and two dimensional cases. The equations were formulated, non-dimensionalized, and an order of magnitude analysis was performed. Additionally, the method of singular perturbations was utilized in the two dimensional case. In the one-dimensional case, an exact solution is obtained while in the two-dimensional case an asymptotic solution is obtained. Both analytical solutions are compared with numerical solutions of the equations, and exhibit good agreement. The analytical solutions for the benchmark problems presented here are useful for verifying numerical solutions to more complex problems, and may also enable simple interpretation of experimental data for electrobiochemical systems.

Exponential Decay Estimates for the Stability of Boundary Layer Solutions to Poisson--Nernst--Planck Systems: One Spatial Dimension Case

With a small parameter $\varepsilon$ approaching zero, Poisson--Nernst--Planck (PNP) systems over a finite one dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, and these profiles form boundary layers near boundary points and become flat in the interior domain. For the stability of 1D boundary layer solutions to (time-dependent) PNP systems, we estimate the solution of the perturbed problem with global electroneutrality. We prove that the $H^{-1}_x$-norm of the solution of the perturbed problem decays exponentially (in time) with exponent independent of $\varepsilon$ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of 1D boundary layer solutions at boundary points may blow up as $\varepsilon$ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.