Nernst-Planck Equation Research Articles

Sinusoidal motion of small particles through a Darcy-Brinkman-Forchheimer microchannel filled with non-Newtonian fluid under electro-osmotic forces

This article deals with the electro-osmotic flow of non-Newtonian fluid in the presence of small particles moving in the sinusoidal form in a Darcy-Brinkman-Forchheimer medium. A transverse magnetic field is applied by ignoring the effects of an induced magnetic field with a small magnetic Reynolds number. To examine the non-Newtonian effects, Jeffrey fluid model is considered. The proposed mathematical modeling is based on the Poisson–Boltzmann equation, Debye length approximation, and ionic Nernst Planck equation. The modeled problem is formulated in a fixed frame, which is further transformed into a wave frame. Lubrication theory is also taken into account. Analytical solutions of nonlinear equations are obtained by using perturbation method. The impact of all the parameters is discussed for both fluid- and particulate-phases. The present effects are beneficial in designing and manufacturing microfluidics devices to determine the transport mechanism.

A simulation study of steric effects on the anodic dissolution at high current densities

AbstractElectrochemical machining (ECM) and electropolishing are examples of anodic dissolution used in a wide variety of applications. Proper simulations of ECM require the establishment of a so‐called gel layer of liquid metal salt, responsible for the depletion of water close to the surface such that the oxygen evolution is reduced. However, this effect is neglected in many simulation efforts, although it is significant to improve the accuracy of the machining process. Some authors consider this effect by an engineered “water‐repelling” function that depends on the local metal ion concentration. As an alternative, we discuss the theoretical description and the implementation of modified Poisson–Nernst–Planck equations taking into account expressions in which each ion has its own maximum concentration and corresponding steric limit. We investigate a basic system of water, sodium chloride, and iron ions. Simulations pending on the local metal ion concentration for current densities show that the modified model works; steric effects become important for large current densities and the formation of a resistive gel layer is responsible for the reduction of water at the electrode. We observe that the electrostatic potential resulting from space charge effects is not affected by implementing the different species sizes. Further research is needed to explore the applicability of industrial simulation tools.

Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification.

The article is devoted to the development and creation of a multiphysics simulator that can, on the one hand, simulate the most significant physical processes in the IPMC actuator, and on the other hand, unlike commercial products such as COMSOL, can use computing resources economically. The developed mathematical model is an adjoint differential equation describing the transport of charged particles and water molecules in the ion-exchange membrane, the electrostatic field inside, and the mechanical deformation of the actuator. The distribution of the electrostatic potential in the interelectrode space is located by means of the solution of the Poisson equation with the Dirichlet boundary conditions, where the charge density is a function of the concentration of cations inside the membrane. The cation distribution was obtained by means of the solution of the equation system, in which the fluxes of ions and water molecules are described by the modified Nernst-Planck equations with boundary conditions of the third kind (the Robin problem). The cantilever beam forced oscillation equation in the presence of resistance (allowing for dissipative processes) with assumptions of elasticity theory was used to describe the actuator motion. A combination of the following computational methods was used as a numerical algorithm for the solution: the Poisson equation was solved by a direct method, the modified Nernst-Planck equations were solved by the Newton-Raphson method, and the mechanical oscillation equation was solved using an explicit scheme. For this model, a difference scheme has been created and an algorithm has been described, which can be implemented in any programming language and allows for fast computational experiments. On the basis of the created algorithm and with the help of the obtained experimental data, a program has been created and the verification of the difference scheme and the algorithm has been performed. Model parameters have been determined, and recommendations on the ranges of applicability of the algorithm and the program have been given.

Multi-ionic effects on the equilibrium and dynamic properties of electric double layers based on the Bikerman correction

Abstract Multi-ion electrolytes are used in many electrochemical applications; therefore, systematic analyses of the effect of multi-ion electrolytes on the capacitance and charging rate are required. In this study, the electrochemical behaviors of the electrical double layer of multi-ion electrolytes confined between parallel electrodes are predicted using a continuum approach. To study the characteristics of multi-ion electrolytes, modified Poisson–Boltzmann equations and modified Poisson–Nernst–Planck equations with the Bikerman correction term were used. Multi-valence and multi-size ion electrolytes with various compositions were selected, and the ion distribution, capacitance, and charging rates were obtained. We divided fluxes into thermal, electrical, and Bikerman-type fluxes for further investigation. In multi-valence cases, a relatively small number of high-valence ions might significantly enhance the equilibrium capacitance. However, the small number of the ions restricts the system from reaching equilibrium. In multi-size cases, smaller ions enhance the equilibrium capacitance, and the curves for the capacitance and charging rates gradually change as the fraction of the smaller ions increases. The results of this study should provide strategies for modifying the capacitance and charging rates using multi-ion electrolytes.

Numerical analysis of two-phase electrohydrodynamic flows in the presence of surface charge convection

The mechanism of surface charge convection (SCC) reveals inherently nonlinear coupling between the electrostatic problem and the hydrodynamic flow in the Melcher–Taylor electrohydrodynamic (EHD) model. Considering that most previous numerical models are based on decoupled leaky dielectric equations, the quantitative effect of SCC on two-phase EHD under different parameters remains unclear. In the present study, we propose a new numerical scheme to solve the two-phase EHD problems in the framework of the lattice Boltzmann method. The fully coupled equations including the Navier–Stokes equations, the Nernst–Planck equations, and the Poisson equation are solved using three well-designed lattice Boltzmann equations. The problem of droplet deformation under a uniform electric field is studied. By neglecting SCC at a small electric Reynolds number ReE ≪ 1, our model successfully reproduces previous theoretical and numerical results. When considering the SCC mechanism at finite values of ReE, the intensity of the EHD flow is reduced. Consequently, oblate droplets are predicted to be less deformed, while prolate droplets are enhanced. In addition, the SCC effect increases as the values of both ReE and the electric capillary number, Ca, increase. In addition, a sharp variation in surface charge density is observed near the equator of the droplet due to SCC.

A general theory for coupled chemo-electro-thermo-mechanical heterogeneous system

Many transport and rate processes in chemical, physical, mineral, material and biological fields are controlled by the coupled chemo-electro-thermo-mechanical (CETM) process. Though many literatures discussed these coupled problems, but a unified rigorous theory and a unified method based on the chemical thermodynamics are lacked. In this paper on the basis of electrochemistry, the non-equilibrium thermodynamics and modern continuum mechanics we modify some previous theories and give a general theory including mass conservation equation, the electric charge conservation equation, complete energy conservation equation, entropy equation, evolution equations and the complete governing equations of these couple CETM systems. An extension of Nernst–Planck equation is derived for the CETM system. This theory gives a theoretical foundation and a universal method to improve and develop engineering theories, especially for the gradual failure components and cells. In appendix we also discuss the interdiffusion problems in solids with vacancies shortly as a complement of the continuum diffusion.

Charge Properties and Electric Field Energy Density of Functional Group-Modified Nanoparticle Interacting with a Flat Substrate.

Functionalized nanofluidics devices have recently emerged as a powerful platform for applications of energy conversion. Inspired by biological cells, we theoretically studied the effect of the interaction between the nanoparticle and the plate which formed the brush layer modified by functional zwitterionic polyelectrolyte (PE) on the bulk charge density of the nanoparticle brush layer, and the charge/discharge effect when the distance between the particle and the plate was changed. In this paper, The Poisson–Nernst–Planck equation system is used to build the theoretical model to study the interaction between the nanoparticle and the plate modified by the PE brush layer, considering brush layer charge regulation in the presence of multiple ionic species. The results show that the bulk charge density of the brush layer decreases with the decrease of the distance between the nanoparticle and the flat substrate when the interaction occurs between the nanoparticle and the plate. When the distance between the particle and the plate is about 2 nm, the charge density of the brush layer at the bottom of the particle is about 69% of that at the top, and the electric field energy density reaches the maximum value when the concentration of the background salt solution is 10 mm.

Impact of finite ion size, Born energy difference and dielectric decrement on the electroosmosis of multivalent ionic mixtures in a nanotube

The electroosmotic flow (EOF) in a nanotube is analyzed by considering the finite ion size and dielectric decrement effects. The ion transport is governed by the modified Nernst-Planck equation, which takes into account the ion steric repulsion, Born energy difference, and dielectrophoretic force. The dielectric permittivity is considered to depend on the ionic concentration and ion size. The governing equations for fluid flow, ion transport, and electric field are solved numerically in their full form. The steric interaction is accounted, based on the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model, which considers the nonuniform ion size for monovalent as well as multivalent salts. We made a quantitative comparison with the standard PNP model for the lower range of charge density and highlighted the impact of ion size and dielectric decrement effects. For a highly charged surface, the impact due to finite ion size is non-negligible when the Debye length is in the order of the nanotube radius. However, the dielectric decrement is found to have a strong impact on a higher range of surface charge density or bulk ionic concentration. The EOF and the ion selectivity of the mixed electrolyte solution of different counterion size are considered in the present study. Present study provides an in-depth analysis of EOF through a cylindrical nanotube beyond the dilute solution and low charge density limits, which has not been addressed before.

Phase-field study of dendritic morphology in lithium metal batteries

Lithium metal is a promising anode candidate for high-energy-density secondary batteries due to its high theoretical capacity and low electrochemical potential, but the uncontrolled lithium dendrite growth causing the poor cycling performance and safety concerns becomes the main drawback in the lithium metal batteries (LMBs). This study presents a phase-field method for modeling the lithium dendrite formation in LMBs. In this model, the electrochemical kinetics is described by a modified Butler–Volmer equation for deposition non-uniformity, and the Nernst–Planck equations and charge neutrality condition are used to solve the electric potential field and the ionic concentration fields. Our simulation results show that this model is able to accurately describe the growth of mossy dendrites, which is missing in previous phase-field works. We found that the lithium dendrite growth is strongly affected by the deposition non-uniformity originated from the competition of SEI growth and lithium growth and the charge conditions. Under high deposition rates, the depletion of the lithium ionic concentration takes place near the anode, and the dendrite growth undergoes a transition from the reaction-limited mossy growth to the diffusion-limited fractal growth.

Quantifying the Influence of the Electrode Porous Structure on Lithium-Ion Battery Performance

The growing demand of lithium-ion batteries (LIBs) on portable electronics, electric vehicles urgently requires higher energy density, longer cycling life, and better safety to satisfy the needs of environmental factors and economics. In order to have higher energy density, simply increasing the volume of the single electrodes of LIB results in poor capacity due to inefficient mass transport of Li ions through the irregular microstructure of the electrode. Since understanding the influence of geometrical properties electrodes is of considerable significance to battery performance, we investigate two basic cells: (i) an irregular microstructure intercalation host and (ii) a periodically aligned microchannel intercalation hosts. Our in-house developed simulation tool solves Nernst-Planck and electron conduction equations with proper interfacial reactions by considering experimentally verified ageing mechanism (2-layered SEI and lithium-plating) into account and compute the associated characteristic voltage vs. state of charge curves under various geometric and design parameters. The proposed model is parameterized and optimised for mass transport based on fresh cell and validated against experimental results available on the literature. We show that the nanostructured electrode design yields better battery performance because the uniformity of electrodes yields elimating the nonuniform intercalation and side reactions on active material electrode surface. This work presents a design concept for higher energy and power densities that are not limited to LIBs.*This work has been supported by The Warwick Research Development Fund, UK, through the Grant No. RDF19035. Figure 1

A Novel Approach for Understanding Heterogeneous Lithium-Ion Battery Aging Using Phase-Field Model

Aging due to capacity fade is a major cause of concern in the fast growing lithium-ion battery (LIB) industry. In particular, the evolution of Solid Electrolyte Interface (SEI) have been on the focus of battery scientist as one of the primary aging mechanism affecting the cycle life of the battery. According to the recent experimental studies, the formation of SEI possesses a heterogeneous morphology and it is attributed to the (i) anisotropic ionic mobility in the crystal and (ii) surface reactions governing the flux of ions across the electrode/electrolyte interface, depending on the local free energy difference (1, 2). In contrast to most previous study which utilise rough simplification on modelling of SEI growth, we employed a novel aging formulation that captures the heterogeneous multi-layered SEI morphology by including Cahn-Hilliard interstitial diffusion modelling approach coupled with Nernst-Planck and Butler-Volmer equations for mass transport and surface chemistry, respectively. Since the Cahn-Hilliard is able to captures the anisotropic surface intercalation reactions with depending the local free energy difference of active material, our numerical theory shows promising results for better explaining the cause of the heterogeneous SEI growths on active material surfaces than the classical diffusion-limited model. W. Huang, P. M. Attia, H. Wang, S. E. Renfrew, N. Jin, S. Das, Z. Zhang, D. T. Boyle, Y. Li and M. Z. Bazant, Nano letters, 19, 5140 (2019). P. Guan, L. Liu and X. Lin, Journal of The Electrochemical Society, 162, A1798 (2015). Figure 1

Phase-Field Simulation of Stress Corrosion Cracking in Stainless Steel

Stress corrosion cracking (SCC) sometimes occurs in stainless steels, which are used for steam turbines of geothermal power plants. In order to prevent SCC that causes serious accidents in the plants, understanding the SCC behavior through experiments and numerical simulations is essential. Phase-field (PF) method is one of the widely used numerical simulation methods to analyze various microscale-material phenomena including electrochemical reactions such as corrosion. In this study, we have developed a new PF model for simulating a SCC behavior in a stainless steel, which includes the stress concentration at the propagating crack tip, diffusion and electrophoresis of ions in the liquid solution. The PF model is based on the PF model for general and pitting corrosion in a pure iron [C. Tsuyuki et al., Scientific Reports, Vol. 8 (2018), 12777.], which incorporates the Butler-Volmer equation. The stress evolution during the SCC is calculated by solving the mechanical equilibrium equation by means of fast Fourier transform. The diffusion and electrophoresis of ions are analyzed by solving the Nernst-Planck equation. Through the numerical simulations of SCC in a polycrystalline stainless steel which have a notch, we revealed that the crack growth rate increase in accordance with the stress intensity factor. The results have proved the crack proceeds through the grain when the elastic strain energy at the crack tip is high, and it propagate through the grain boundary when the low energy.

Entropy generation in electromagnetohydrodynamic water based three Nano fluids via porous asymmetric microchannel

Nanofluids are of immense importance to the researchers as they have significant uses industrially due to their high heat transfer rates. This new model is employed to examine the electroosmotic flow of magnetohydrodynamic nanofluids through an asymmetric microfluidic channel. Microchannel flow is driven by the electroosmosis and peristalsis mechanisms. Nanoparticles (copper, alumina and titania) with water as a base fluid have been considered. The effects of Joule heating, mixed convection, permeability of the porous medium and dissipation of energy are considered. Besides, velocity slip conditions are also encountered at the walls. Electroosmotic phenomenon is modeled through Poisson equation and Nernst–Planck equation. Debye–Hückel approximation is considered to obtain Boltzmann distribution of electric potential across electric double layer. The emerging non-linear mathematical model is solved numerically by the built-in scheme of working software. The table is inserted for numerical values of heat transfer rate at the upper wall for three different type of water-based metallic nanofluid. To verify the accuracy of results, the comparison of solutions is also shown. Entropy generation in terms of Bejan number under different parameters is also investigated. The influence of physical factors on the characteristics of heat transfer has been pointed out. Water-based copper and titania nanofluids have relatively higher temperature and heat transfer rate when compared to the alumina-water nanofluid. Porosity decays the temperature throughout the microchannel. Moreover, temperature is strongly increased for higher values of the Joule heating parameter γ, in presence of Cu-water nanofluid. The current analysis is related to bio-inspired electrokinetic nanofluidic micropump models and incipient nanomedicine technologies.

A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

A model based on the code CrunchClay is presented for a fracture-clay matrix system that takes electrostatic effects on transport into account. The electrostatic effects on transport include those associated with the development of a diffusion potential as captured by the Nernst-Planck equation, and the formation of a diffuse layer bordering negatively charged clay particles within which partial anion exclusion occurs. The model is based on a dual continuum formulation that accounts for diffuse layer and bulk water pore space, providing a more flexible framework than is found in the classical mean electrostatic potential models. The diffuse layer model is obtained by volume averaging ion concentrations in the Poisson-Boltzmann equation, but also includes the treatment of longitudinal transport within this continuum. The calculation of transport within the bulk and diffuse layer porosity is based on a new formulation for the Nernst-Planck equation that considers averaging of diffusion coefficients and accumulation factors at grid cell interfaces. Equations for function residuals and the associated Jacobian matrix are presented such that the system of nonlinear differential-algebraic equations can be solved with Newton’s method. As an example, we consider a 2D system with a single discrete fracture within which flow and advective transport occurs that is coupled to diffusion in the clay-rich matrix. The simulation results demonstrate the lack of retardation for anions (e.g., 36Cl−) of the contaminant plume within the fracture flow system because they are largely excluded from the charged clay rock, while the migration of cations (e.g., 90Sr++) is more strongly attenuated. The diffusive loss of divalent cations in particular from the fracture is accentuated by their accumulation in the diffuse layer within the clay-rich matrix.

A Supersmall Single-Cell Nanosensor for Intracellular K + Detection

Intracellular potassium ions (K+) play pivotal roles in many physiological processes. Several K+ sensors have been developed for probing cellular K+ fluctuations. Nevertheless, the existing solutio...

Electro-osmotic flow and the limiting current in alkaline water electrolysis

Under alkaline conditions, hydroxide ions can deplete at the anode of a water electrolyser for hydrogen production, resulting in a limiting current density. We found experimentally that in a micro-porous separator, an electro-osmotic flow from anode to cathode lowers this limiting current density. Using the Nernst-Planck equation, a useful expression for the potential drop in the presence of diffusion, migration, and advection is derived. A quasi-stationary, one-dimensional model is used to successfully describe the transient dynamics. Electro-osmotic flow-driven cross-over of dissolved oxygen is argued to impact the hydrogen purity.

Electrochemical advanced oxidation for treating ultrafiltration effluent of a landfill leachate system: Impacts of organics and inorganics and economic evaluation

We treated landfill leachate using electrochemical advanced oxidation processes (EAOPs) with a boron-doped diamond (BDD) anode. Before EAOPs, the leachate was first treated using biological treatment (anoxic–aerobic) and then ultrafiltration (hereafter referred as biopretreated UF landfill leachate). We explored impacts of organics and inorganics (nitrogen – and chloride – species) for various current densities (1–80 mA/cm2). We found that EAOPs enhanced biodegradability (BOD/COD ratio) significantly from < 0.03 to up to 0.78. COD destruction followed a pseudo-first order model, with the rate constant and electric energy per order ranging from 0.91 to 28.29 h−1 and 17.1 to 127.5 kWh m−3, respectively. The current efficiency (CE) was 100% for current density 1 mA/cm2 and CE decreased to 57.8% as the current density increased to 30 mA/cm2. The oxidation priority was in this order: COD > nitrogen – species > chloride – species. The evolution of nitrogen – and chloride – species and their oxidation mechanisms were also investigated. The impact of Fickian diffusion and electrical field driven charged ion transport (EFDCIT) on the total mass transport was examined using the Nernst-Planck equation. We found that the contribution of EFDCIT was similar to diffusion for current densities higher than 64 mA/cm2. Finally, we compared the costs for standalone EAOPs and EAOPs integrated with subsequent biological treatment for biopretreated UF landfill leachate and found that the lowest cost for the latter one that meets the effluent discharge criteria was only 0.76 USD/m3. Our study proved that EAOPs are technically and economically promising technology for leachate treatment.

Modeling and Experiments of Binary Electrolytes in the Presence of Diffusion, Migration, and Electro-Osmotic Flow

Combined diffusion, migration, and advection of ions in a binary electrolyte plays a role in various applications, including water electrolysis, electrodeposition, deionization, and electrophoresis. Here we analyze a dilute binary electrolyte with arbitrary ion valencies in a porous or nonporous medium using the one-dimensional Nernst-Planck equations. We examine how advection influences the limiting current, diffusion potential, and overall potential, deriving broadly useful analytical expressions. We provide experimental results for the electro-osmotic flow through a submicroporous separator in an alkaline water electrolysis setup. The time evolution of the potential is followed from the initial timescale of double-layer charging, followed by the diffusional timescale, to the time at which a limiting current is reached. For the longer timescales, a quasisteady model is shown to predict the time evolution of the advection-modified potential drop reasonably well. Additional interesting features arising due to electro-osmotic drag and unsteady diffusion are observed and explained.

A unified description of macroion diffusiophoresis, salt osmotic diffusion and collective diffusion coefficient

Diffusiophoresis describes the net migration of a macroion within an aqueous fluid due to a concentration gradient of salt, while salt osmotic diffusion describes the net motion of small ions toward Donnan equilibrium due to a macroion concentration gradient. A new theoretical description of macroion diffusiophoresis, salt osmotic diffusion and the light-scattering collective diffusion coefficient of macromolecules conforming to Onsager Reciprocal Relation (ORR) is developed. Specifically, symmetric transport equations for macroion-salt coupled transport in isothermal conditions are derived. These equations, which can be regarded as a generalization of Nernst–Planck equations, incorporate an electrokinetic factor for diffusiophoresis, salt thermodynamic non-ideality and concentration-dependent ionic transport coefficients. Due to the electrokinetic factor, thermodynamic driving forces must be modified to ensure that ORR is not violated. A mathematical expression for the collective diffusion coefficient compatible with ORR was then extracted. Diffusiophoresis and salt osmotic diffusion equations are discussed in relation to experimental transport data on lysozyme in aqueous NaCl. These transport equations represent a more appropriate theoretical reference model, compared to multicomponent-diffusion limiting laws for describing protein-salt coupled diffusion in aqueous mixtures.

Numerical modeling of the effect of concrete porosity evolution on electrochemical chloride removal from concrete structures

Porosity evolution caused by electrochemical chloride removal (ECR) process can affect the transport behaviour of concrete and vice versa. The main task in this investigation is to quantitatively understand the porosity evolution effect on ECR process, and find a more accurate model that gives good correlation with the experimental results concerning the ionic transport in concrete during the ECR. The porosity of the concrete near the cathode and anode were measured, before and after ECR treatment (mainly 7 days, 15 days and 30 days). A method to analyse and calculate the porosity profiles was proposed according to the ECR experimental data. The penetration and leaching of ionic species in concrete are studied by using a finite element procedure based on the Nernst–Planck equations. The proposed method is capable of reducing the numerical modelling error of the chloride profile in concrete during the ECR.