Electroneutrality and ionic interactions in the modeling of mass transport in dilute electrochemical systems (original) (raw)
Related papers
Russian Journal of Electrochemistry, 2001
Numerical methods that are used for modeling non-steady-state ion transfer in electrochemical systems and account for the diffusion, migration, convection, and homogeneous chemical reactions are analyzed. It is shown that the violation of the electroneutrality condition (ENC) in the process of numerical solution is due to the difference equations being inconsistent with the initial differential equations. Difference schemes for numerical calculation of transfer processes, which make it possible to split a set of coupled equations, are designed and conditions for their stability are determined. The explicit difference scheme is self-consistent, i.e. it ensures that ENC is rigorously met. In the implicit difference scheme, ENC is probably violated when splitting the set of equations. To restore electroneutrality of the medium, it is proposed to use a physically substantiated analytical relation for the space charge relaxation under the action of a strong electric field.
A nonlinear equation for ionic diffusion in a strong binary electrolyte
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
The problem of the one dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson-Nerst-Planck (PNP) system, consists of a diffusion equation for each species augmented by transport due to a self consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics such as electron and hole diffusion across semi-conductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here we derive a more general theory by exploiting the ratio of Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation which provides a better approximation than the classical linear equation for ambipolar diffusion but reduces to it in the appropriate limit.
Conductivity and electrophoretic mobility of dilute ionic solutions
Journal of Colloid and Interface Science, 2010
a b s t r a c t Two complementary continuum theories of electrokinetic transport are examined with particular emphasis on the equivalent conductance of binary electrolytes. The ''small ion" model [R.M. Fuoss, L. Onsager, J. Phys. Chem. 61 (1957) 668] and ''large ion" model [R.W. O'Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 ] are both discussed and the ''large ion" model is generalized to include an ion exclusion distance and to account in a simple but approximate way for the Brownian motion of all ions present. In addition, the ''large ion" model is modified to treat ''slip" hydrodynamic boundary conditions in addition to the standard ''stick" boundary condition. Both models are applied to the equivalent conductance of dilute KCl, MgCl 2 , and LaCl 3 solutions and both are able to reproduce experimental conductances to within an accuracy of several tenths of a percent. Despite fundamental differences in the ''small ion" and ''large ion" theories, they both work equally well in this application. In addition, both ''stick-large ion" and ''slip-large ion" models are equally capable of accounting for the equivalent conductances of the three electrolyte solutions.
Geochimica et Cosmochimica Acta, 2014
This study investigates the effects of Coulombic interactions during transport of electrolytes in heterogeneous porous media under steady-state flow and transport conditions. We performed flow-through experiments in a quasi two-dimensional setup using dilute solutions of strong 1:1 and 1:2 electrolytes to study the influence of electrochemical cross-coupling on mass transfer of charged species in saturated porous media. The experiments were carried out under advection-dominated conditions (seepage velocity: 1 and 1.5 m/day) in two well-defined heterogeneous domains where flow diverging around a low-permeability inclusion and flow focusing in high-permeability zones occurred. To quantitatively interpret the outcomes of our laboratory experiments in the spatially variable flow fields we developed a two-dimensional numerical model based on a multicomponent formulation and on charge conservation. The results of the multicomponent transport simulations were compared with the high-resolution concentration measurements of the ionic species at the outlet of the flow-through domain. The excellent agreement between the measured concentrations and the results of purely forward numerical simulations demonstrates the capability of the proposed two-dimensional multicomponent approach to describe transport of charged species and to accurately capture the Coulombic interactions between the ions, which are clearly observed in the flow-through experiments. Furthermore, the model allowed us to directly quantify and visualize the ionic interactions by mapping the Coulombic cross-coupling between the dispersive fluxes of the charged species in the heterogeneous domains.
Journal of Power Sources
A novel approach in modeling the ionic transport in the electrolyte of Li-ion batteries is here presented. Diffusion and migration processes govern the transport of ions in solution in the absence of convection. In the porous electrode theory [1] it is common to model these processes via mass balance equations and electroneutrality. A parabolic set of equations arises, in terms of a non constant electric field which is afflicted by the paradox of being generated without electrical charges. To remedy this contradiction, Maxwell's equations have been used here, coupled to Faraday's law of electrochemical charge transfer. The set of continuity equations for mass and Maxwell's equations lead to a consistent model, with distinctive energy characteristics. Numerical examples show the robustness of the approach, which is well suited for multi-scale analyses [2,3].
Diffusion of Charged Species in Liquids
In this study the laws of mechanics for multi-component systems are used to develop a theory for the diffusion of ions in the presence of an electrostatic field. The analysis begins with the governing equation for the species velocity and it leads to the governing equation for the species diffusion velocity. Simplification of this latter result provides a momentum equation containing three dominant forces: (a) the gradient of the partial pressure, (b) the electrostatic force, and (c) the diffusive drag force that is a central feature of the Maxwell-Stefan equations. For ideal gas mixtures we derive the classic Nernst-Planck equation. For liquid-phase diffusion we encounter a situation in which the Nernst-Planck contribution to diffusion differs by several orders of magnitude from that obtained for ideal gases. The study of ion transport in fluids is an important topic with a wide range of applications. Some classic examples are batteries, fuel cells, electroplating, and protection of metal structures against corrosion 1. In addition to the traditional battery, the flow battery or rechargeable fuel cell 2,3 represents an important new technology involving the transport of ions. Ion exchange membranes have a wide range of applications 4 , and the underlying theory has been a matter of concern for several decades 5. Other examples of complex electro-chemical systems are the transport of charged particles in ion channels 6–8 , in protein channels 9 , and during the primordial conversion of light to metabolic energy 10. Often upscaling is necessary for a complete analysis of the transport of electrolytes in charged pores 11,12. Much of ion transport occurs at the nano-scale 13 , and most of the studies use the Nernst-Planck equation 14–16 to describe this type of phenomena. However, there are molecular dynamic simulations indicating that the Nernst-Planck equation does not always provide a complete description 17 , and there are experimental studies that lead to the same conclusion 18. The need to analyze the limits of the Nernst-Planck contribution to diffusion has been emphasized 19 in an exploration of the transport of divalent ions in ionic channels. The authors of this paper have not found a derivation of the Nernst-Planck equation that does not make use of the ideal gas assumption. To be precise we note that ideal gas behavior for mixtures is based on Dalton's laws (see page 114 in ref. 20) that we list as Some care must be taken with the interpretation of Eq. 2 since it applies to all Stokesian fluids and is therefore not limited to ideal gas mixtures (see Appendix B in ref. 21). Caution must also be used with the interpretation of Eq. 3 which is applicable to all ideal fluid mixtures. For example, Eq. 3 should provide reliable results for a liquid mixture of hexane and heptane, but it should not for a mixture of ethanol and water. Here it is important to emphasize that Eq. 1 can only describe a fluid composed of non-interacting particles. To provide a reliable framework in which a fluid is considered to be composed of interacting particles ̶ as in the case of a liquid ̶ it is necessary to understand the process of Nernst-Planck diffusion in a fluid different from an ideal gas. This is the problem under consideration in this paper. In this work we analyze the ion transport process from a fundamental point of view using an axiomatic mechanical perspective. Our analysis is for ideal fluid mixtures, and from the analysis we find that the classic
Ion size and valence effects on ionic flows via Poisson–Nernst–Planck models
Communications in Mathematical Sciences, 2017
We study boundary value problems of a quasi-one-dimensional steady-state Poisson-Nernst-Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through an ion channel, focusing on effects of ion sizes and ion valences. The flow properties of interest, individual fluxes and total flow rates of the mixture, depend on multiple physical parameters such as boundary conditions (boundary concentrations and boundary potentials) and diffusion coefficients, in addition to ion sizes and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical values that are explicitly expressed in terms of the physical parameters. We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.
Computational Modeling of Ionic Transport in Continuous and Batch Electrodialysis
Separation Science and Technology, 2004
We describe the transport of ions and dissociation of a single salt and a solvent solution in an electrodialysis (ED) stack. An ED stack basic unit is made of a two-compartment cell: dilute and concentrate. We use the fundamental principles of electrochemistry, transport phenomena, and thermodynamics to describe mechanisms and to predict the performance of the ED process. We propose and analyze three model formulations for a single salt (KCl). The first and second models are for a one-and two-dimensional continuous electrodialysis, and the third examines batch electrodialysis. The models include the effect of the superficial velocity in the boundary layer near the ion-exchange membranes. We examine the diffusion and electromigration of ions in the polarization region and consider electromigration and convection in the bulk region.
Journal of Membrane Science, 1997
Model calculations of the steady-state ion transport against its external concentration gradient when the driving force of this transport is a pH difference across a charged membrane are presented. We have solved numerically the exact Nernst-Planck equations without any additional simplifying approximation, such as the Goldman constant field assumption within the membrane. The validity of this assumption for a broad range of pH values, and salt and membrane fixed charge concentrations was analyzed critically. The membrane characteristics studied are the ionic fluxes and the membrane potential. Special attention is paid to the physical mechanism which leads to the ion transport against the concentration gradient, and the experimental conditions for which this transport can occur. The case of a system with ions of different charge numbers is also considered.