Electroneutrality Condition Research Articles

Comments on the Problem of Metal Ions Crossing through the Cations Membrane Electrolyte in Electrolysers and Redox Flow Cells

Redox flow Cells and hydrogen produced electrolysers are important green source of energy of the future. These cells can be activated to store energy (redox flow cells) and produce CO free hydrogen (electrolysers) by solar, wind or sea wave’s energy. In this Redox flow batteries and electrolysers a cations exchange membrane (e.g. Nafion from Du Point a cations exchange membrane selective to cations only) is used to separate the anodic regions from the cathodic regions. It is observed experimentally that during operations of these cells metal ions crosses from one region of the cell to the other which greatly lowers the efficiency of the process (irreversible capacity loss). This problem was partially solved in the case of Fe (II)/Fe(III) and Cr(II)/Cr(III) redox flow battery by utilizing the condition of electro-neutrality[1,2].The idea of mixed electrolyte introduced by Hagedorn [1] from NASA is based is using mixed cations in both anodic and cathodic regions so that during operation of the Fe (II)/Fe(III) and Cr(II)/Cr(III) redox flow cells the cations get reduced or oxidized instead of crossing the membrane and the condition of electro-neutrality is satisfied. The purpose of this publication is to generalize the application of the condition of electro-neutrality to other redox flow batteries and electrolysers and to give simple mathematical modeling of this effect [3]. [1] Introduction of mixed reactants, elevated temperatures and additive electrocatalysts NASA TM-83401. [2] Gahn, R. ; Hagedorn, N.; and Ling, J. S.: Single Cell Performance Studies on the Fe/Cr Redox Energy Storage System Using Mixed Reactant Solutions at Elevated Temperature. Proceedings of the Eighteenth Intersociety Energy Conversion Engineering Conference, AIChE, 1983, pp. 1647-1652. [3] Rakib et al., Journal of Applied Electrochemistry 29: 1439, 1999.

On the role of saturation in modeling ionic transport in the electrolyte of (Lithium ion) batteries

Recent computational simulations of ionic conductivity across the electrolyte of commercial batteries by Salvadori et al. (2015) have shown that the concentration of ions exceeds half the saturation limit near the electrodes. This observation, which is in agreement with other approaches by Danilov and Notten (2008), implies that the widespread assumption of infinite dilution far from saturation is questionable. The present contribution is therefore devoted to investigate the role of saturation in modeling ionic transport in the electrolyte of Li-ion batteries. An important result is found, that saturation has no effect on the diffusivity, whereby the condition of electroneutrality is well approximated in the solution. However saturation affects the electric potential up to 40% near the electrodes for all charge rates.

The three-dimensional global numerical model CHARM-I: The incorporation of processes in the ionospheric D-region

We describe the three-dimensional numerical global photochemical model CHARM-I (CHemical Atmospheric Research Model with Ions) and the results of numerical calculations of global distributions of neutral and charged atmospheric trace gases (in the height range of up to 90 km), such as ozone, nitrogen oxides, electrons, and positive and negative ions. This model is an improved version of the CHARM three-dimensional photochemical model of neutral components with additional reactions with the involvement of ions (a total of 200 photochemical reactions). The model incorporates UV-radiation fluxes on the Lyman-α line and galactic cosmic rays as ionizing factors. The neutral components are calculated with the method of “chemical families” and the concentrations of charged components are calculated by the electroneutrality condition at each time step. The spatial transport of chemically active species is described in the model by the Prather scheme. The developed model makes it also possible to take into account solar flares and particle precipitations in the ionospheric D-region.

Ion-exchange equilibrium of N-acetyl-d-neuraminic acid on a strong anionic exchanger

N-acetyl-d-neuraminic acid (Neu5Ac) is a high value-added product widely applied in the food industry. A suitable equilibrium model is required for purification of Neu5Ac based on ion-exchange chromatography. Hence, the equilibrium uptake of Neu5Ac on a strong anion exchanger, AD-1 was investigated experimentally and theoretically. The uptake of Neu5Ac by the hydroxyl form of the resin occurred primarily by a stoichiometric exchange of Neu5Ac− and OH−. The experimental data showed that the selectivity coefficient for the exchange of Neu5Ac− with OH− was a non-constant quantity. Subsequently, the Saunders’ model, which took into account the dissociation reactions of Neu5Ac and the condition of electroneutrality, was used to correlate the Neu5Ac sorption isotherms at various solution pHs and Neu5Ac concentrations. The model provided an excellent fit to the binary exchange data for Cl−/OH− and Neu5Ac−/OH−, and an approximate prediction of equilibrium in the ternary system Cl−/Neu5Ac−/OH−. This basic information combined with the general mass transfer model could lay the foundation for the prediction of dynamic behavior of fixed bed separation process afterwards.

Theory of indirect potentiometric determination of the acid number with the use of single-component reagent without titration

Based on the condition of solution electroneutrality, the theory of potentiometric determination of the acid number (AN) with the use of a single-component ionic or uncharged reagent without titration is being substantially advanced. The generalized pH (potential of the indicator glass electrode) vs. AN dependences are obtained. They take into account the overall completeness of all analytical reactions and the effect of reagent hydrolysis and also allow the linear region in the pH vs. logAN dependence with the slope equal to -1 to be found. The use of the generalized theory of potentiometric determination of AN in the viticulture products is illustrated.

Several ways to simulate time dependent liquid junction potentials by finite differences

Several ways of simulating time-dependent migration effects at an electrolytic liquid junction are explored, for a simple uni-univalent electrolyte, both with the full Nernst-Planck-Poisson (NPP) equation set and the reduced set from the electroneutrality condition (ENC) assumption. Using the NPP approach, the system can be simulated using all three variables (method ABψ), the two concentrations and the potential, or the two concentrations and the potential field (method ABE), or in principle by substituting for the potential field, thereby reducing to the two concentration variables (method AB). The two first methods are about equally efficient, whereas the latter method is seen to be quite inaccurate. Results at long times compare very well with the Henderson equation. Using the full NPP set, junction potentials are time-dependent but not when applying the ENC, where the potential rises to the Henderson value immediately. Results for KCl and HCl are presented, with left/right concentration ratios equal to 0.1 in both cases.

제설제 사용에 따른 콘크리트 포장의 염화물 침투 예측기법

제설제는 겨울철 동안에 콘크리트 포장에 결빙을 방지하기 위해서 사용되고 있으며, 그 사용량은 매년 증가하고 있다. 이러한 제설제는 콘크리트내로 침투하여 동결융해 작용과 함께 열화를 증가시켜서 내구연한을 감소시킨다. 따라서, 본 연구에서는 Nernst-plank식을 바탕으로 이를 수치해석방법을 이용하여 염화물 침투해석을 수행하는데 그 목적이 있다. 본 연구에서는 다양한 제설제(Nacl, <TEX>$Cacl_2$</TEX>, <TEX>$Mgcl_2$</TEX>)들의 콘크리트 내 확산 현상을 이온들의 전기적 중성도에 의해 정전기 포텐셜로 나타내었다. 이를 fick의 제 2법칙으로 정식화하였으며, 양생, 물-시멘트비, 온도, 습도, 자유 염소이온, 그리고 골재와 시멘트 혼합물에 의해 영향을 고려한 재료모델을 적용하였다. 이러한 수학적 물리모델을 유한요소법을 적용한 수치해석기법을 적용하여 해석을 수행하였다. 그 결과, 본 연구에서 제시한 모델은 실험결과와 일치하였으며, 염소이온 및 다양한 이온들의 시간에 따른 침투확산정도를 예측할 수 있었다. Deicing chemicals have been used to prevent the freezing of concrete pavement during winter season and the usage amount of the chemicals is increasing every year. Deicing chemicals gradually penetrate into concrete and exacerbate the concrete damage originating from freezing and thawing. The purpose of this study is to carry out the chloride penetration analysis with Nernst-plank equation and numerical model. In this study, the transport phenomenon of various deicing chemicals such as Nacl, <TEX>$Cacl_2$</TEX>, and <TEX>$Mgcl_2$</TEX> into concrete was expressed with electro-neutrality condition and electric potential. Nernst-plank equation was substituted into fick's second law and material model was considered with curing, water-cement ratio, temperature, moisture, and free chloride and so on. This mathematical model can be solved with finite element method, one of the numerical methods. As the results, this model was well agreed with experimental results and the diffusion analysis of other chemical ions into concrete can be predicted as well.

Diffusion and Migration of Redox Active Molecules in Nano-Electrochemical Systems

The absence of supporting electrolyte leads to great variety in electrode responses when the charge numbers and diffusion coefficients of electroactive species vary. The variability in the voltammetric behavior of the redox species may be further increased after diminishing the size of- or the distance between the working electrodes to the nanometer range.In a system containing no added inert electrolyte, both diffusion and migration contribute to the total flux of the substance transported to the electrode. The flux for each of the three solutes (i.e. S, P, and Y- the substrate, the product, and the counterion of the substrate (if it is charged), respectively) can be represented by the Nernst-Planck equation.If the solution containing a redox species is placed between two closely-spaced plane electrodes (separated by a distance l) oriented parallel to each other, and both electrodes are independently polarized, then the mass transport will take place in the limited linear space and the steady-state conditions will be easily attainable. In such the electrochemical cell the product of the electrooxidation at one electrode is transported to- and re-electroreduced at the other electrode (communicating electrodes).The obtained analytical solutions of the transport equations [1] allowed derivation of the analytical expressions for the limiting electrostatic potential and the limiting faradaic current recorded in a thin layer dual electrode system [2]. Interestingly, the expression for the limiting electrostatic potential at the interface is identical to that derived for the semi-infinite diffusional-migrational transport to a hemispherical microelectrode in the absence of supporting electrolyte. The corresponding expression for the limiting current, iL , normalized with respect to the purely diffusional response, id L , is identical to its equivalent derived for the semi-infinite diffusional-migrational transport to a hemispherical microelectrode when either θ (DP / DS ) is equal to 1 (equal diffusion coefficients of electroactive species, any zS , zP , zS ≠ zP ) or zS is zero (uncharged substrate, any θ and zP , and zS ≠ zP ) or zP is zero (uncharged product, any θ and zS , and zP ≠ zS ). If both electroactive species bear charge a specific relation between iL / id L and θ is established for various zS , zP , zY values.The theoretical modeling revealed high sensitivity of the thin-layer electrode system response to the diffusion coefficients diversity. It is particularly pronounced for the charge reversal processes: the θ - dependence of the limiting current “blows up” at θ = 1 where the limiting conditions can not be reached and the transport is purely migrational.The theory allowed one to examine quantitatively the limitation of the derived expressions due to the application of the electroneutrality condition at any point in the system. The electroneutrality approximation in the mass transport modeling associated with the electron exchange implies that the electric field resulting from the flow of current through a solution is not big enough to induce the charge accumulation at any point in the solution. This assumption is generally valid if the electrical double layer occupies an insignificant fraction of the transport depletion layer. This may become questionable when the distance between the electrodes tends to nanometer range. The electrode system modeled in this work gave one the opportunity to test the validity of the electroneutrality condition. It was found that the departure from the electroneutrality diminishes with the increase in the distance between the electrodes. For a 100-nm separation of electrodes and comparable diffusion coefficients of the redox species, the error (expressed by the local net charge density in the solution, ρ exc ) does not exceed 1% of the initial substrate concentration (CS 0). For the electrode processes with charge sign retention (for which the limiting conditions can be reached) the error ρ exc does not exceed 10% of CS 0 down to 60-nm gap between the electrodes for the entire θ range examined (0.2 – 5.0). The calculations indicated also a strong dependence of ρ exc with respect to θ. However, the direction of this dependence is a function of the type of the electrode process.The theoretical predictions derived for ultra-thin layer dual electrode system provide quantitative characteristics of the redox cycling process affected by the migrational contribution. The results may be of great significance for the extension of applicability of many devices utilizing redox cycling, including the electrochemical scanning microscopy in low-supported media. Additionally, by decreasing the distance between the electrodes to the nanometer range it may be possible to perform electrochemical detection of individual redox-active molecules affected by migration [3].[1] W. Hyk, Z. Stojek Anal. Chem.74 (2002) 4805.[2] W. Hyk, Z. Stojek Electrochemistry Communications34 (2013) 192.[3] M.A.G. Zevenbergen, P.S. Singh, E.D. Goluch, B.L. Wolfrum, S.G. Lemay, Nano Lett. 11 (2011) 2881.

Temperature effect on multi-ionic species diffusion in saturated concrete

This study presents the mathematical model for predicting chloride penetration into saturated concrete under non-isothermal condition. The model considers not only diffusion mechanism but also migration process of chloride ions and other chemical species in concrete pore solution such as sodium, potassium, and hydroxyl ions. The coupled multi-ionic transport in concrete is described by the Nernst-Planck equation associated with electro-neutrality condition. The coupling parameter taken into account the effect of temperature on ion diffusion obtained from available test data is proposed and explicitly incorporated in the governing equations. The coupled transport equations are solved using the finite element method. The numerical results are validated with available experimental data and the comparison shows a good agreement.

Electrochemical modelling using electroneutrality equation as a constraint

This paper deals with ionic species migration based on classical mass balance equations in conjunction with a charge balance equation and the electroneutrality condition. Two new methods are proposed to apply the electroneutrality condition, avoiding the elimination of one of the conservation equations of chemical species and permitting to explicitly apply boundary conditions on all species. The first method is based on variational principles and the second on a Lagrange multiplier. It is shown that the new methods are analytically equivalent to the more standard method in which one conservation equation is eliminated without any other consideration. The variational method is more complex to implement than the Lagrange multiplier method. The new methods were applied to a multi-ionic problem together with the charge conservation equation without capacitive effects. Different kinds of boundary conditions were applied: Neumann, Dirichlet and a nonlinear case based on Butler–Volmer kinetics. All methods gave the same results for non-complex problems. In the case of complex problems including chemical equilibrium between each ionic species, it was found that more investigations are necessary even for more conventional methods.

Multiphasic Finite Element Framework for Modeling Hydrated Mixtures With Multiple Neutral and Charged Solutes

Computational tools are often needed to model the complex behavior of biological tissues and cells when they are represented as mixtures of multiple neutral or charged constituents. This study presents the formulation of a finite element modeling framework for describing multiphasic materials in the open-source finite element software febio.1 Multiphasic materials may consist of a charged porous solid matrix, a solvent, and any number of neutral or charged solutes. This formulation proposes novel approaches for addressing several challenges posed by the finite element analysis of such complex materials: The exclusion of solutes from a fraction of the pore space due to steric volume and short-range electrostatic effects is modeled by a solubility factor, whose dependence on solid matrix deformation and solute concentrations may be described by user-defined constitutive relations. These solute exclusion mechanisms combine with long-range electrostatic interactions into a partition coefficient for each solute whose value is dependent upon the evaluation of the electric potential from the electroneutrality condition. It is shown that this electroneutrality condition reduces to a polynomial equation with only one valid root for the electric potential, regardless of the number and valence of charged solutes in the mixture. The equation of charge conservation is enforced as a constraint within the equation of mass balance for each solute, producing a natural boundary condition for solute fluxes that facilitates the prescription of electric current density on a boundary. It is also shown that electrical grounding is necessary to produce numerical stability in analyses where all the boundaries of a multiphasic material are impermeable to ions. Several verification problems are presented that demonstrate the ability of the code to reproduce known or newly derived solutions: (1) the Kedem-Katchalsky model for osmotic loading of a cell; (2) Donnan osmotic swelling of a charged hydrated tissue; and (3) current flow in an electrolyte. Furthermore, the code is used to generate novel theoretical predictions of known experimental findings in biological tissues: (1) current-generated stress in articular cartilage and (2) the influence of salt cation charge number on the cartilage creep response. This generalized finite element framework for multiphasic materials makes it possible to model the mechanoelectrochemical behavior of biological tissues and cells and sets the stage for the future analysis of reactive mixtures to account for growth and remodeling.

Transport properties for an electroneutral Yukawa-type fluid in the MSA

In the framework of a linear response theory, in which Onsager’s continuity equations are combined with the Mean Spherical Approximation (MSA) from the theory of correlation functions in equilibrium, and using a Green’s function formalism, we consider the transport properties of electrolytes. The interaction between the ions in the electrolyte is represented by an intermolecular Yukawa potential which satisfies the electroneutrality condition. The model contains an adjustable parameter z which takes into account the effects of the solvent. Transport processes in an ionic solution are determined by two dominant forces: the relaxation and the electrophoretic forces; their contributions to the transport properties are calculated using the Fuoss–Onsager theory. We find the conductivity and the self-diffusion coefficient for a family of electrolytes using the linear response theory. The predictions of our model can be adjusted by means of the parameter z. The electrophoretic effect, due to the hydrodynamic interaction between the ions, is calculated using the Rotne–Prager tensor. Our theoretical results are in good agreement with experimental data for electrolytes 1–1, even for high concentrations. We applied this theory also to two unsymmetrical electrolytes, namely the aqueous solutions of MgCl2 and CaCl2, with results in good accord with experimental data.

Thin and ultra-thin layer dual electrode electrochemistry: Theory of steady-state voltammetry without supporting electrolyte

The electrode system consisting of two closely-spaced plane electrodes oriented parallel to each other was examined. The expressions for the steady-state voltammetric response of the generator electrode in a solution of an electroactive species with no supporting electrolyte have been derived. The theoretical modeling of the mixed diffusional–migrational transport in the thin-layer cells reveals a strong effect of the diffusion coefficient on the magnitude of the steady-state limiting current at communicating electrodes. In general, the form of this dependence is different from that derived for the semi-infinite transport to microelectrodes in the supporting-electrolyte free systems. The limitation of the derived expressions due to the application of the electroneutrality condition at any point in the system has been also examined quantitatively.

Modeling for cardiac excitation propagation based on the Nernst-Planck equation and homogenization

The bidomain model is a commonly used mathematical model of the electrical properties of the cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces. However, the equations contain self-contradiction such that the update of ion concentrations does not consider intracellular or extracellular ion movements due to the gradient of electric potential and the membrane charge as capacitive currents in spite of the fact that those currents are taken into account in forming Kirchhoff's first law. To overcome this problem, we start with the Nernst-Planck equation, the ionic conservation law, and the electroneutrality condition at the cellular level, and by introducing a homogenization method and assuming uniformity of variables at the microscopic scale, we derive rational bidomain equations at the macroscopic level.

A temperature dependent multi-ion model for time accurate numerical simulation of the electrochemical machining process. Part III: Experimental validation

The temperature distribution and shape evolution during electrochemical machining (ECM) are the result of a large number of interacting physical processes. Electrolyte flow, electrical conduction, ion transport, electrochemical reactions, heat generation and heat transfer strongly influence one another, making modeling and numerical simulation of ECM a very challenging procedure. In part I [1], a temperature dependent multi-ion transport and reaction model (MITReM) is put forward which considers mass transfer as a consequence of diffusion, convection and migration, combined with the electroneutrality condition and linearized temperature dependent polarization relations at the electrode–electrolyte interface. The flow field is calculated using the incompressible laminar Navier–Stokes equations for viscous flow. The local temperature is obtained by solving internal energy balance, enabling the use of temperature dependent expressions for several physical properties such as the ion diffusion coefficients and electrolyte viscosity. In part II [2], the temperature dependent MITReM is used to simulate ECM of stainless steel in aqueous NaNO3 electrolyte solution. The effects of temperature, electrode thermal conduction, reaction heat generation, electrolyte flow and water depletion are investigated and a comparison is made between the temperature dependent potential model and MITReM. In this third part, the theoretical model is validated against ECM experiments in a flow-channel cell. The model is further optimized by including the effect of metal hydration and non-linear polarization relations. A close match is obtained between experiment and simulation.

Coupled diffusion of multi-component chemicals in non-saturated concrete

A comprehensive simulation model for the transport process of fully coupled moisture and multi-species in non-saturated concrete structures is proposed. The governing equations of moisture and ion diffusion are formulated based on Fick's law and the Nernst-Planck equation, respectively. The governing equations are modified by explicitly including the coupling terms corresponding to the coupled mechanisms. The ionic interaction-induced electrostatic potential is described by electroneutrality condition. The model takes into account the two-way coupled effect of moisture diffusion and ion transport in concrete. The coupling parameters are evaluated based on the available experimental data and incorporated in the governing equations. Differing from previous researches, the material parameters related to moisture diffusion and ion transport in concrete are considered not to be constant numbers and characterized by the material models that account for the concrete mix design parameters and age of concrete. Then, the material models are included in the numerical analysis and the governing equations are solved by using finite element method. The numerical results obtained from the present model agree very well with available test data. Thus, the model can predict satisfactorily the ingress of deicing salts into non-saturated concrete.

Multi-Ion and Temperature Dependent Numerical Simulation of Electrochemical Machining

A temperature dependent multi-domain model is presented for numerical simulation of the electrochemical machining process with a moving cathode tool. The method includes mass transfer as a consequence of diffusion, convection and migration, combined with the electroneutrality condition and linearized temperature dependent polarization relations at the electrolyte-electrode interface. The electrolyte flow field is calculated using the laminar Navier-Stokes equations for viscous incompressible flow. Heat is generated in the bulk solution and in the electrical double layer. The electrodes are cooled by natural convection. The level set method is used for tracking the anode interface. The model is applied to the electrochemical machining of steel in a supporting electrolyte of NaNO3. Hydrogen is formed at the cathode, and metal dissolution and oxygen evolution is considered at the anode. The effect of water depletion at the electrodes is modelled by limiting the oxygen and hydrogen evolution reaction rates depending on the surface water concentration. The heat conduction through electrodes and the heat production by the electrode reactions are found to play an important role.

Joule Heating Effect in Constant Voltage Mode Isotachophoresis in a Microchannel

Abstract Microchip ITP (isotachophoresis) is getting popularity as a preparative technique for preconcentration and separation of chemical species and/or ions in liquid phase. In constant voltage mode ITP, generally a high electric potential difference is applied to a discontinuous buffer for faster and higher resolution separation. However, the higher current from the applied electric field induces Joule heating in the buffer which modifies the mobility and diffusion coefficient of analytes in the system. This change in mobility and diffusion coefficient strongly influences the transient separation process in ITP. In this study the effect of Joule heating on separation behavior of analyte compounds has been presented in a constant voltage mode ITP where two chemical species are separated from an initial mixture. The model is based on mass, energy, and charge conservation and electroneutrality condition in the system. A set of nonlinear governing equations are solved numerically for temperature dependent properties such as diffusion coefficient, effective electrophoretic mobility, and thermal conductivity using a finite volume based model. Numerical results suggest that for temperature dependent properties of control parameters, the separation speed of analytes is significantly different from that of constant temperature case. In constant voltage mode ITP, the temperature peak forms at the location of trailing electrolyte, and its influence propagates toward the direction of band movement as separation proceeds.

Inhibition of electrokinetic ion transport in porous materials due to potential drops induced by electrolysis

In this work we present non-destructive measurements of sodium ion concentration profiles during the electrokinetic removal of sodium chloride from porous materials using Nuclear Magnetic Resonance (NMR). The effect of both protons and hydroxyl ions, generated due to the electrolysis of water, on the transport of the salt ions is studied by tracking the acidic and alkaline fronts using pH-indicator paper. In addition, the electrical potential distribution within the specimen is monitored to assess its influence on the process. To support the observations we compare the experimental results with a theoretical model based on the Poisson–Nernst–Plank equations. In this model we use the chemical equilibrium condition for the self-electrolysis of water in the description of the transport of protons and hydroxyl ions. In addition we use the electro-neutrality condition to compute the transport of salt ions through the material. At the edges of the system the electrical current is distributed over the chemical active species, i.e. protons, hydroxyl and chloride ions, according to the Butler–Volmer description for charge transfer at electrodes. Both the experimental and model results show in the final stage of the electrokinetic remediation process a sharp transition from the acidic to alkaline region at one third of the length of the specimen away from the positively biased electrode, i.e. the anode. From the model results we found that the formation of chlorine gas at the anode does not influence the position of this transition area. In this transition region we also observe a large gradient in electrical potential and a corresponding local deficit of ions. As a result of the large potential gradient in this small transition zone the electrical field in the acidic and alkaline region diminishes. Consequently, electrokinetic ion transport though the material will stagnate.

Multi-phase modelling of ionic transport in concrete when subjected to an externally applied electric field

Chloride-induced reinforcing steel corrosion is a worldwide problem. In order to predict how chlorides penetrate in concrete and how other ionic species in concrete pore solution affect the penetration of chlorides, this paper presents a numerical study on the two-phase modelling of ionic transport in concrete when subjected to an externally applied electric field. In most existing work, researchers tend to use the assumption of electro-neutrality condition to determine the electrostatic potential within an electrolyte in one-dimensional problems and consider only one phase. In contrast, this study employed a two-phase numerical model of concrete and the Poisson’s equation for determining the electrostatic potential. By solving both mass conservation and Poisson’s equations, the distribution profiles of each ionic species and electrostatic potential at any required time were successfully obtained. The results obtained from the present two-phase modelling reveal some important features which cannot be found from existing one-dimensional and one-phase models.