Poisson Nernst Planck Equations Research Articles

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.

Breather, lump, M-shape and other interaction for the Poisson–Nernst–Planck equation in biological membranes

This paper investigates a novel method for exploring soliton behavior in ion transport across biological membranes. This study uses the Hirota bilinear transformation technique together with the Poisson–Nernst–Planck equation. A thorough grasp of ion transport dynamics is crucial in many different scientific fields since biological membranes are important in controlling the movement of ions within cells. By extending the standard equation, the suggested methodology offers a more thorough framework for examining ion transport processes. We examine a variety of ion-acoustic wave structures using the Hirota bilinear transformation technique. The different forms of solitons are obtained including breather waves, lump waves, mixed-type waves, periodic cross-kink waves, M-shaped rational waves, M-shaped rational wave solutions with one kink, and M-shaped rational waves with two kinks. It is evident from these numerous wave shapes that ion transport inside biological membranes is highly relevant, and they provide important insights that may have an impact on various scientific disciplines, medication development, and other areas. This extensive approach helps scholars dig deeper into the complexity of ion transport, illuminating the complicated mechanisms driving this essential biological function. Additionally, to show the physical interpretations of these solutions we construct the 3D and their corresponding contour plots by choosing the different values of constants. So, these solutions give us the better physical behaviors.

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-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.

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.

Accelerating charging dynamics using self‐driven optimizing porous structures

AbstractRelating the complex structures of electrodes to their charging dynamics is crucial for optimizing supercapacitors, which remains an experimental and theoretical challenge. Here, we construct a pore network model (PNM) that can be downward‐transformed into a well‐known transmission‐line model and a stack‐electrode model to describe the disordered porous structure of carbon‐based electrodes. A mathematical expression is derived using an equivalent circuit model of the PNM to quantify the relaxation times of the potential and concentration. The expression is then verified using numerical solutions based on the simplified Poisson–Nernst–Planck equations and experimental data. The structure of the PNM for experimental verification is directly extracted from a porous electrode reconstructed using a scanning electron microscopic image. A self‐driven optimization framework is proposed by coupling the derived expression with a genetic algorithm to generate an optimal porous structure that can be used to investigate the changing dynamics of the electrode. Our framework provides a general image–structure–performance optimization platform for understanding and accelerating charging dynamics in porous electrodes.

Modeling the Transient Behavior of Ionic Diodes with the Nernst–Planck–Poisson Equations

AbstractIonic circuitry formed from soft charged polymers promises to enable a new kind of analog computing and usher in a new age of human–computer interaction, but devices remain in their infancy. Ionic diodes, created by the interface between two oppositely charged polymers, form one of the most fundamental elements of ionic circuitry. However, it is currently not well understood how the boundary conditions and geometry of a diode influence its transient response and performance. A consistent set of metrics with which to analyze these diodes is first developed, then a continuum model based on the Nernst–Planck–Poisson equations is used to study how device construction and three types of boundary conditions affect performance. This model is solved using the finite volume method and gives insight into transient diode behavior not described by previous analyses. It is demonstrated how different parts of a diode's construction, thickness, and operating voltage act as the limiting factors for different regimes. To conclude, it is demonstrated how blocking electrodes limit both diode lifetime and rectification behavior and how neutral bath boundary conditions play an important role in the performance of some diodes, with recommendations for building higher performance devices in the future.

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.

Global existence of large solutions for the three‐dimensional incompressible Navier–Stokes–Poisson–Nernst–Planck equations

This work is concerned with the global existence of large solutions to the three‐dimensional dissipative fluid‐dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants such that if the initial data satisfies then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution.

Induced-charge electroosmosis flow of viscoelastic fluids under different voltage arrangements

Efficient mixing of chemical analysis reagents with laboratory samples at a microscale is a key issue in numerous biomedical and chemical analyses but hardly to implement due to the limited of the low diffusivity in laminar flow. Induced-charge electroosmosis flow, as an innovative mixing method, has been proved to be effective and simple in rapid mixing attributes to its mechanism of vortex generation. This work aims to propose a new strategy for chaotic induced-charge electroosmosis flow based on different voltage arrangements to improve the mixing of viscoelastic fluids. The Phan–Thien–Tanner constitutive model is applied to characterize the flow behavior of viscoelastic fluid in a microfluidic preparation mixer. The direct numerical simulation method is used to solve the fully coupled Navier–Stokes and Poisson–Nernst–Planck equations for a polarizable cylinder in a two-dimensional cavity filled with electrolyte solution. The impact of Weissenberg number (Wi), Debye parameter, voltage strength on the velocity, net charge density, and potential profiles is investigated. The simulation results indicate that a greater Wi leads to the decrease in the maximum velocity, and a large voltage strength can heighten the net charge density and potential, thus improve the peak velocity. Moreover, the classical theoretical prediction that the maximum velocity is proportional to the square of the applied voltage has been authenticated.

A comparison of FEM results from the use of different governing equations in a galvanic cell part II: Impact of low supporting electrolyte concentration

Finite Element Method modeling has been widely used to simulate the distributions of potential, current density, and dissolved species for a wide range of electrochemical systems. The use of the Nernst–Planck–Poisson equations as governing equations is the most rigorous approach, but it brings with it a substantial computational time cost. Reduced-order models offer options in which different assumptions allow more rapid computational solutions. In Part I of this work, it was shown that the loss in accuracy of the reduced-order models is small in high supporting electrolyte concentrations, and there were substantial savings in computational time. The analysis is extended to lower supporting electrolyte concentrations. The most reduced-order modeling approach performs poorly in low supporting electrolyte to electrochemically reactive species (SER) ratios. A modified Laplace approach, in which the conductivity is recalculated at each position at each time step, improves the solution by updating the electrolyte conductivity at each time step, but the errors can still be significant in low SER ratios due to the absence of the diffusion potential term in the calculations.

Numerical study of rectified electroosmotic flow in nanofluidics: Influence of surface charge and geometrical asymmetry

The transport of ions in nanofluidic systems, specifically the rectified ion transport or the ionic diode phenomenon occurring in the presence of asymmetrical geometry and/or charge distribution, has drawn considerable attention due to its relevance in energy conversion and biosensing applications. However, previous numerical research has frequently overlooked the concurrent liquid flow within these systems, even though multiple experimental studies have highlighted intriguing flow patterns in ionic diode configurations. In the present study, we employ comprehensive numerical simulations to probe the influence of geometrical or charge asymmetry in a nanofluidic system on electroosmotic flow and ion transport. These simulations employ the Poisson–Nernst–Planck equation in conjunction with the Navier–Stokes equation. Our findings reveal that even when the current rectification trend is consistent between conical and straight nanopores, charge asymmetry and geometric asymmetry can generate significant variations in the rectification effects of electroosmotic flow. Furthermore, our research indicates that the direction of ion rectification and flow rectification can be independently manipulated by utilizing charge asymmetry in conjunction with geometric asymmetry, thereby facilitating advanced control of ions and flows within nanofluidic systems. Collectively, our findings contribute to a more profound understanding of the mechanisms underlying osmotic flow rectification and propose a novel approach for developing efficient ion and flow rectification systems.

Hydration kinetics of Portland cement shifting from silicate to aluminate dominance based on multi-mineral reactions and interactions

A theoretical model for cement hydration was proposed to study multi-mineral reactive transport processes under various mineral compositions and determine the effect of multi-mineral reactions and interactions on cement hydration kinetics shifting from silicate to aluminate dominance. The reaction rates of each mineral dissolution, product precipitation, ionic diffusion, and adsorption were calculated individually through the degrees of undersaturation and supersaturation associated with the ionic concentration, all of which were coupled in the modified Poisson–Nernst–Planck equation. The hydration heat flow was then theoretically calculated by the superposition of the reaction rates of silicate and aluminate phases, which was derived from the calculated ionic concentration. The model was validated by comparison with experimental data obtained under various conditions, showing consistency. The combined effect of multi-mineral reactions and interactions on hydration kinetics was investigated using the model, and the results indicated that (1) faster dissolution of gypsum or a higher ratio of tricalcium aluminate to tricalcium silicate leads to a larger time interval between silicate and aluminate peaks; (2) faster precipitation of calcium silicate hydrate results in a more significant difference between silicate and aluminate peaks; and (3) the sulfate ion retards cement hydration kinetics shifting from silicate to aluminate dominance.

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

The diffusiophoresis of a soft particle governed by an imposed ionic concentration gradient is studied based on a modified ion transport equation. The surface of the rigid core of the soft particle is considered to be hydrophobic, which is coated with a charged polyelectrolyte layer (PEL) whose dielectric constant is different from the dispersion medium. This step-change in the dielectric constant of the PEL and the dispersion medium creates an ion partitioning effect. The governing equations based on the coupled Navier–Stokes–Nernst–Planck–Poisson equations are modified to incorporate the ion partitioning effect. A simplified model based on the linear perturbation technique under a small applied concentration gradient in the scale of the particle radius is developed. The simplified model agrees well with the exact numerical solutions for a Debye length smaller than the particle size. We have analyzed the impact of the core slip length and PEL permittivity on the mobility of the soft particle. The interfacial slip enhances the impact of the core charge density on the diffusiophoresis. In this case, a reversal of mobility occurs at a thinner Debye length compared to a hydrophilic core. Ion partitioning attenuates the concentration of mobile counterions in PEL, leading to a reduction in screening of PEL fixed charge. We find that the diffusiophoresis of the soft particle in a non-z:z asymmetric electrolyte strongly depends on the valence of the counterions. Diffusiophoresis of the soft particle in a KCl-NaCl mixture is considered to achieve positive mobility.

Energy dissipative and positivity preserving schemes for large-convection ion transport with steric and solvation effects

Ion transport plays a crucial role in biophysical and electrochemical applications. While being modeled by the classical Poisson–Nernst–Planck (PNP) equations, many ionic features are ignored in the course-graining modeling of ionic interactions. In this work, we study ion transport by modified PNP equations including effects arising from steric interactions and the Born solvation, which both could cause strong convection. The modified NP equations are first rewritten in a gradient-flow reformulation that involves mobility and chemical potential terms. The mobility is then approximated by upwind fluxes with various flux limiters on the half-grid points, and the chemical potential terms are then implicitly approximated by central differencing. The resulting first-order in time scheme is proved to be energy dissipative unconditionally. Furthermore, the unconditional existence of positive numerical concentration is proved by making use of the singularity of the entropy term, which prevents the concentration from approaching zero. The flux with the Minmod or Osher limiter is reduced to a second-order upwind flux in the strong convective limit. Numerical tests are performed to confirm the expected order of accuracy and the properties of the developed scheme at discrete level. In addition, our numerical experiments demonstrate that the developed schemes with limiters can achieve higher accuracy than the Scharfetter-Gummel scheme and classical upwind scheme in tests with strong convection. Finally, the developed schemes are applied to numerically assess the effects of Coulombic interaction, steric interaction, and Born solvation on the rectifying transport of ions through nanopores.

New mixed finite element methods for the coupled Stokes and Poisson–Nernst–Planck equations in Banach spaces

In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwardsviaa postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuška–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Nečas–Babuška theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme.A priorierror estimates are derived, and examples of discrete spaces that fit the theory, include,e.g., Raviart–Thomas elements of orderkalong with piecewise polynomials of degree ≤k. In addition, the latter yield approximate local conservation of momentum for all three equations involved. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the aforementioned balance-preserving properties and the applicability of the proposed family of methods.

Long-Range ACEO Phenomena in Microfluidic Channel

Microfluidic devices are increasingly utilized in numerous industries, including that of medicine, for their abilities to pump and mix fluid at a microscale. Within these devices, microchannels paired with microelectrodes enable the mixing and transportation of ionized fluid. The ionization process charges the microchannel and manipulates the fluid with an electric field. Although complex in operation at the microscale, microchannels within microfluidic devices are easy to produce and economical. This paper uses simulations to convey helpful insights into the analysis of electrokinetic microfluidic device phenomena. The simulations in this paper use the Navier–Stokes and Poisson Nernst–Planck equations solved using COMSOL to determine the maximum attainable fluid velocity with an electric potential applied to the microchannel and the most suitable frequency or voltage to use for transporting the fluid. Alternating current electroosmosis (ACEO) directs and provides velocity to the ionized fluid. ACEO can also mix the fluid at low frequencies for the purpose of dispersing particles. DC electroosmosis (DCEO) applies voltage along the microchannel to create an electric field that ionizes fluid within the microchannel, making it a cost-effective method for transporting fluid. This paper explores a method for an alternate efficient utilization of microfluidic devices for efficient mixing and transportation of ionized fluid and analyzes the electrokinetic phenomena through simulations using the Navier–Stokes and Poisson Nernst–Planck equations. The results provide insights into the parameters at play for transporting the fluid using alternating current electroosmosis (ACEO) and DC electroosmosis (DCEO).

Nano-scale solution of the Poisson-Nernst-Planck (PNP) equations in a fraction of two neighboring cells reveals the magnitude of intercellular electrochemical waves.

The basic building blocks of the electrophysiology of cardiomyocytes are ion channels integrated in the cell membranes. Close to the ion channels there are very strong electrical and chemical gradients. However, these gradients extend for only a few nano-meters and are therefore commonly ignored in mathematical models. The full complexity of the dynamics is modelled by the Poisson-Nernst-Planck (PNP) equations but these equations must be solved using temporal and spatial scales of nano-seconds and nano-meters. Here we report solutions of the PNP equations in a fraction of two abuttal cells separated by a tiny extracellular space. We show that when only the potassium channels of the two cells are open, a stationary solution is reached with the well-known Debye layer close to the membranes. When the sodium channels of one of the cells are opened, a very strong and brief electrochemical wave emanates from the channels. If the extracellular space is sufficiently small and the number of sodium channels is sufficiently high, the wave extends all the way over to the neighboring cell and may therefore explain cardiac conduction even at very low levels of gap junctional coupling.