Poisson-Nernst-Planck Research Articles

Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the l∞(L2) and l∞(H1) norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.

Impact of surface charge density modulation on ion transport in heterogeneous nanochannels

The PNP nanotransistor, consisting of emitter, base, and collector regions, exhibits distinct behavior based on surface charge densities and various electrolyte concentrations. In this study, we investigated the impact of surface charge density on ion transport behavior within PNP nanotransistors at different electrolyte concentrations and applied voltages. We employed a finite-element method to obtain steady-state solutions for the Poisson–Nernst-Planck and Navier–Stokes equations. The ions form a depletion region, influencing the ionic current, and we analyze the influence of surface charge density on the depth of this depletion region. Our findings demonstrate that an increase in surface charge density results in a deeper depletion zone, leading to a reduction in ionic current. However, at very low electrolyte concentrations, an optimal surface charge density causes the ion current to reach its lowest value, subsequently increasing with further increments in surface charge density. As such, at Vapp=+1V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{app}=+1 \ ext{V}$$\\end{document} and C0=1mM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${C}_{0}=1 \ ext{mM}$$\\end{document}, the ionic current increases by 25% when the surface charge density rises from 5 to 20 mC.m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{mC}.{\ ext{m}}^{-2}$$\\end{document}, whereas at C0=10mM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${C}_{0}=10 \ ext{mM}$$\\end{document}, the ionic current decreases by 65% with the same increase in surface charge density. This study provides valuable insights into the behavior of PNP nanotransistors and their potential applications in nanoelectronic devices.

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.

Manipulating Ion Transport Regimes in Nanomembranes via a "Pore-in-Pore" Approach Enabled by the Synergy of Metal-Organic Frameworks and Solid-State Nanochannels.

Solid-state nanochannels (SSNs) have emerged as promising platforms for controlling ionic transport at the nanoscale. SSNs are highly versatile, and this feature can be enhanced through their combination with porous materials such as Metal-Organic Frameworks (MOF). By selection of specific building blocks and experimental conditions, different MOF architectures can be obtained, and this can influence the ionic transport properties through the nanochannel. Herein, we study the effects of confined synthesis of Zr-based UiO-66 MOF on the ion transport properties of single bullet-shaped poly(ethylene terephthalate) (PET) nanochannels. We have found that emerging textural properties from the MOF phase play a determinant role in controlling ionic transport through the nanochannel. We demonstrate that a transition from ion current saturation regimes to diode-like regimes can be obtained by employing different synthetic approaches, namely, counterdiffusion synthesis, where MOF precursors are kept separate and forced to diffuse through the nanochannel, and one-pot synthesis, where both precursors are placed at both ends of the channel. Also, by considering the dependence of the charge state of the UiO-66 MOF on the protonation degree, pH changes offered a mechanism to tune the iontronic output (and selectivity) among different regimes, including anion-driven rectification, cation-driven rectification, ion current saturation, and ohmic behavior. Furthermore, Poisson-Nernst-Planck (PNP) simulations were employed to rationalize the different iontronic outputs observed experimentally for membranes modified by different methods. Our results demonstrate a straightforward tool to synthesize MOF-based SSN membranes with tunable ion transport regimes.

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.

Suppression of current-induced membrane discharge of bipolar membranes by regulating ion crossover transport

Bipolar membranes (BPMs) exhibit the unique capability to regulate the operating environment of electrochemical system through the water dissociation-combination processes. However, the industrial utilization of BPMs is limited by instability and serious energy consumption. The current-induced membrane discharge (CIMD) at high-current conditions has a negative influence on the performance of anion-exchange membranes, but the underlying ion transport mechanisms in the BPMs remain unclear. Here, the CIMD-coupled Poisson-Nernst-Planck (PNP) equations are used to explore the ion transport mechanisms in the BPMs for both reverse bias and forward bias at neutral and acid-base conditions. It is demonstrated that the CIMD effect in the reverse-bias mode can be suppressed by enhancing the diffusive transport of salt counter-ions (Na+ and Cl−) into the BPMs, and that in the forward-bias mode with acid-base electrolytes can be suppressed by matching the transport rate of water counter-ions (H3O+ and OH−). Suppressing the CIMD can promote the water dissociation in the reverse-bias mode, as well as overcome the plateau of limiting current density and reduce the interfacial blockage of salt co-ions (Cl−) in the anion-exchange layer in the forward-bias mode with acid-base electrolytes. Our work highlights the importance of regulating ion crossover transport on improving the performance of BPMs.

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

An extended finite element method for the Nernst-Planck-Poisson equations

The Nernst-Planck-Poisson (NPP) system of equations is used to study the ion diffusion mechanism in materials, commonly used in storage batteries and solid oxide fuel cells. In particular, NPP equations are used to predict the concentration of charged defects and electric potential. Depending on the size of the computational domain, the distributions exhibit steep gradients near the boundaries. The traditional finite element method, when employed, requires extremely refined mesh to capture the steep gradient as they employ simple polynomials. To alleviate the mesh dependence, in this work, we propose to augment the traditional finite element approximation space with a suitable ansatz to capture the steep gradient within the framework of the extended finite element method. The robustness and the accuracy of the proposed framework is demonstrated by comparing it with an overkill finite element solution.

Ion Transport in Multi-Nanochannels Regulated by pH and Ion Concentration.

Nanochannels are a powerful technique for detecting a wide range of biomolecules without labeling. The ion transport phenomena in nanochannel arrays differ from those in single nanochannels and are caused by interchannel communication. This study uses a fully coupled Poisson-Nernst-Planck (PNP) and Navier-Stokes model to investigate ion transport in nanochannel arrays. Instead of being set at a constant value, the surface charge density used in this study is established by the protonation and deprotonation of the silanol groups that are present on the walls of the silicon-based nanochannels. The surface charge density of the nanochannel walls varies with the number of nanochannels, the channel lateral distance, and the background solution properties, which consequently influence the ionic concentration distribution, flow velocity, and electric field strength. For example, in different numbers of nanochannel systems, the ion concentration in nanochannels is not much different, but it is different in reservoirs, especially near the openings of nanochannels. The number of nanochannels and the distance between nanochannels can also affect the formation of electro-convective vortex zones under certain conditions. These findings can aid in optimizing the nanochannel array design by regulating the number and distance of nanochannels and facilitating the construction of solid-state nanochannel arrays with any desired nanochannel dimensions.

Ion-concentration gradients induced by synaptic input increase the voltage depolarization in dendritic spines

The vast majority of excitatory synaptic connections occur on dendritic spines. Due to their extremely small volume and spatial segregation from the dendrite, even moderate synaptic currents can significantly alter ionic concentrations. This results in chemical potential gradients between the dendrite and the spine head, leading to measurable electrical currents. In modeling electric signals in spines, different formalisms were previously used. While the cable equation is fundamental for understanding the electrical potential along dendrites, it only considers electrical currents as a result of gradients in electrical potential. The Poisson-Nernst-Planck (PNP) equations offer a more accurate description for spines by incorporating both electrical and chemical potential. However, solving PNP equations is computationally complex. In this work, diffusion currents are incorporated into the cable equation, leveraging an analogy between chemical and electrical potential. For simulating electric signals based on this extension of the cable equation, a straightforward numerical solver is introduced. The study demonstrates that this set of equations can be accurately solved using an explicit finite difference scheme. Through numerical simulations, this study unveils a previously unrecognized mechanism involving diffusion currents that amplify electric signals in spines. This discovery holds crucial implications for both numerical simulations and experimental studies focused on spine neck resistance and calcium signaling in dendritic spines.

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.

Direct numerical simulation of electrokinetic transport phenomena in fluids: Variational multi-scale stabilization and octree-based mesh refinement

Computational modeling of charged species transport has enabled the analysis, design, and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Planck (PNP) equations coupled with the Navier-Stokes (NS) equation. Direct numerical simulation (DNS) to accurately capture the spatio-temporal variation of ion concentration and current flux remains challenging due to the (a) small critical dimension of the diffuse charge layer (DCL), (b) stiff coupling due to fast charge relaxation times, large advective effects, and steep gradients close to boundaries, and (c) complex geometries exhibited by electrochemical devices.In the current study, we address these challenges by presenting a direct numerical simulation framework that incorporates (a) a variational multiscale (VMS) treatment, (b) a block-iterative strategy in conjunction with semi-implicit (for NS) and implicit (for PNP) time integrators, and (c) octree based adaptive mesh refinement. The VMS formulation provides numerical stabilization critical for capturing the electro-convective flows often observed in engineered devices. The block-iterative strategy decouples the difficulty of non-linear coupling between the NS and PNP equations and allows the use of tailored numerical schemes separately for NS and PNP equations. The carefully designed second-order, hybrid implicit methods circumvent the harsh timestep requirements of explicit time steppers, thus enabling simulations over longer time horizons. Finally, the octree-based meshing allows efficient and targeted spatial resolution of the DCL. These features are incorporated into a massively parallel computational framework, enabling the simulation of realistic engineering electrochemical devices. The numerical framework is illustrated using several challenging canonical examples.

An unconditionally energy stable linear scheme for Poisson–Nernst–Planck equations

This paper proposes a linear, unconditionally energy-stable scheme for the Poisson–Nernst–Planck (PNP) equations. Based on a gradient-flow formulation of the PNP equations, the energy factorization approach is applied to linearize the logarithm function at the previous time step, resulting in a linear semi-implicit scheme. Numerical analysis is conducted to illustrate that the proposed fully discrete scheme has desired properties at a discrete level, such as unconditional unique solvability, mass conservation, and energy dissipation. Numerical simulations verify that the proposed scheme, as expected, is first-order accurate in time and second-order accurate in space. Further numerical tests confirm that the proposed scheme can indeed preserve the desired properties. Applications of our numerical scheme to the simulations of electrolyte solutions demonstrate that, as a linear energy stable scheme of efficiency, it will be promising in simulating complicated transport phenomena of charged systems.

Non-equilibrium steady states of electrolyte interfaces

The non-equilibrium steady states of a semi-infinite quasi-one-dimensional univalent binary electrolyte solution, characterised by non-vanishing electric currents, are investigated by means of Poisson-Nernst-Planck (PNP) theory. Exact analytical expressions of the electric field, the charge density and the number density are derived, which depend on the electric current density as a parameter. From a non-equilibrium version of the Grahame equation, which relates the total space charge per cross-sectional area and the corresponding contribution of the electric potential drop, the current-dependent differential capacitance of the diffuse layer is derived. In the limit of vanishing electric current these results reduce to those within Gouy-Chapman theory. It is shown that improperly chosen boundary conditions lead to non-equilibrium steady state solutions of the PNP equations with negative ion number densities. A necessary and sufficient criterion on surface conductivity constitutive relations is formulated which allows one to detect such unphysical solutions.

New insights into the effects of small permanent charge on ionic flows: A higher order analysis.

This study investigated how permanent charges influence the dynamics of ionic channels. Using a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model, we investigated the behavior of two distinct ion species-one positively charged and the other negatively charged. The spatial distribution of permanent charges was characterized by zero values at the channel ends and a constant charge $ Q_0 $ within the central region. By treating the classical PNP model as a boundary value problem (BVP) for a singularly perturbed system, the singular orbit of the BVP depended on $ Q_0 $ in a regular way. We therefore explored the solution space in the presence of a small permanent charge, uncovering a systematic dependence on this parameter. Our analysis employed a rigorous perturbation approach to reveal higher-order effects originating from the permanent charges. Through this investigation, we shed light on the intricate interplay among boundary conditions and permanent charges, providing insights into their impact on the behavior of ionic current, fluxes, and flux ratios. We derived the quadratic solutions in terms of permanent charge, which were notably more intricate compared to the linear solutions. Through computational tools, we investigated the impact of these quadratic solutions on fluxes, current-voltage relations, and flux ratios, conducting a thorough analysis of the results. These novel findings contributed to a deeper comprehension of ionic flow dynamics and hold potential implications for enhancing the design and optimization of ion channel-based technologies.

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.

Continuum and Equivalent Circuit Modelling of Porous Electrode Charging

The flow of electrolytes through pores is important in various fields of science and technology. Examples range from axons in the brain and plasmodesmata in plant cells to energy storage in batteries and supercapacitors. In the porous carbon electrodes of supercapacitors, ions move over micrometers through macropores until they arrive at nanopores, forming so-called electrical double layers. In this presentation, I discuss several models for this multi-scale charging process. First, I present analytical [1] and finite-element solutions [2,3] to the Poisson-Nernst-Planck (PNP) equation for the charging of cylindrical electrolyte-filled pores [Figure (a)]. With these methods, I will delineate the validity of the famous “transmission line” circuit model developed by de Levie in the 1960s [4,5] [Figure (b)]. The TL model assumes equipotential lines in a pore to be straight, which is not the case at a pore’s entrance and end, see Figure (a). As a result, the TL model does not accurately describe short-pore charging. Related, the impedance of short pores deviates from the impedance of the TL circuit (the “Warburg open” impedance Wo), especially at high frequencies, see Figure (c).Figure Caption: (a) The electrostatic potential in a cylindrical pore next to an electrolyte reservoir of resistance Rb shortly after applying a potential step. The data represent numerical solutions to the Poisson-Nernst-Planck (PNP) equations, which we solved with finite elements using a Newton solver from the FEniCS library. (b) The transmission line circuit is an equivalent circuit that describes the charging of pores such as the one shown in panel (a). In the circuit, the resistance R and capacitance C of the pore are cut into small pieces r and c. (c) The impedance of the TL circuit is given by Rb +Wo (blue dotted line), where Wo is the Warburg open impedance. Also shown is the impedance of a pore whose length is five times its diameter, as obtained from numerical PNP simulations (black). T. Aslyamov and M. Janssen, Electrochim. Acta 424, (2022)J. Yang, M. Janssen, C. Lian, R. van Roij, J. Chem. Phys. 156, (2022)C. Pedersen, T. Aslyamov, M. Janssen, in preparation R. de Levie, Electrochim. Acta 8, (1963)M. Janssen, Phys. Rev. Lett. 126, (2021) Figure 1

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 simple quantitative model of neuromodulation, Part I: Ion flow through neural ion channels

We develop a simple model of ionic current through neuronal membranes as a function of membrane potential and extracellular ion concentration. The model combines a simplified Poisson–Nernst–Planck (PNP) model of ion transport through individual ion channels with channel activation functions calibrated from ad hoc in-house experimental data. The simplified PNP model is validated against bacterial gramicidin A ion channel data. The calibrated model accounts for the transport of calcium, sodium, potassium, and chloride and exhibits remarkable agreement with the experimentally measured current–voltage curves for the differentiated human neural cells. All relevant data and code related to the ion flow models are available at Werneck et al. (2023).