Poisson-Nernst-Planck equations

Other resolutions of the Poisson Nernst Planck equations (i.e. using various simplifying assumptions) have been proposed that couple the adsorption, desorption and permeation of ions through a membrane (e.g. [273,274]) as might be observed for a carrier-mediated transport. For example, for a symmetrical membrane (identical electrolyte on both sides of the membrane) and variation in the electrical potential profile given by i//m, /int can be estimated from ... 

OTHER NONLINEAR PROBLEMS THE POISSON— BOLTZMANN AND POISSON—NERNST—PLANCK EQUATIONS... 

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

The bulk transport of ions in electrochemical systems without the contribution of advection is described by Poisson-Nernst-Planck (PNP) equations (Rubinstein, 1990).The well-known Nernst-Planck equation describes the processes of the process that drives the ions from regions of higher concentration to regions of lower concentration, and electromigration (also referred to as migration), the process that launches the ions in the direction of the electric field (Bard and Faulkner, 1980). Since the ions themselves contribute to the local electric potential, Poisson s equation that relates the electrostatic potential to local ion concentrations is solved simultaneously to describe this effect. The electroneutrality assumption simplifies the mathematical treatise of bulk transport in most electrochemical systems. Nevertheless, this no charge density accumulation assumption does not hold true at the interphase regions of the electric double layer between the solid and the Uquid, hence the cause of most electrokinetic phenomena in clay-electrolyte systems. 

Equations 12.39, 12.35, and 12.40 form a coupled system of equations describing the surface function S, charge concentrations Pa, and electrostatic potential. This coupled system differs from the original PNP equations through the coupling of the surface definition are to charge concentrations and electrostatics. We call this DG-based system the "Laplace-Beltrami Poisson-Nernst-Planck" (LB-PNP) model. 

This section describes the numerical techniques used for solving the set of differential equations that model the electrodiffusion of ions in solution. The method has historically been called the Poisson-Nernst-Planck (PNP) method because it is based on the coupHng of the Poisson equation with the Nernst-Planck equation. The basic equations used in the PNP method include the Poisson equation (Eq. [18]), the charge continuity equation (Eq. [55]), and the current density of the Nemst-Planck equation (Eq. [54]). 

Equations 8.44 and 8.45 form the basis of the Poisson-Nernst-Planck (PNP) theory, which is widely used in the context of ion transport through biological membranes [145-147]. In dimensionless form the equations are... 

The distribution of proton concentration Ch+ and potential in solution is governed by the Poisson-Nernst-Planck (PNP) model, widely used in the theory of ion transport in biological membranes (Coalson and Kurnikova, 2007 Keener and Sneyd, 1998). Oxygen diffusion is determined by Pick s law. Inside the pore, the continuity and transport equations for protons and oxygen are... 

Under conditions relevant for fuel cell operation, the reaction current density of the ORR is small compared to separate flux contributions caused by proton diffusion and migration in Equation 3.62. Therefore, the electrochemical flux termNjj+ on the left-hand side of the Nernst-Planck equation in Equation 3.62 can be set to zero. In this limit, the PNP equations reduce to the Poisson-Boltzmann equation (PB equation). This approach allows solving for the potential distribution independently and isolating the electrostatic effects from the effects of oxygen transport. 

The model of water-filled nanopores, presented in the section ORR in Water-Filled Nanopores Electrostatic Effects in Chapter 3, was adopted to calculate the agglomerate effectiveness factor. As a reminder, this model establishes the relation between metal-phase potential and faradaic current density at pore walls using Poisson-Nernst-Planck theory. Pick s law of diffusion, and Butler-Volmer equation... 

We shall treat each of these four contributions separately in the following sections, along with discussions of their effects in the context of experimental results. The same mathematical procedures described in Chapters 6 and 8, namely, the Fokker-Planck formalism, first passage times, Poisson-Nernst-Planck formalism, and the Goldman-Hodgkin-Katz equations, are implemented (Muthukumar 2010) to obtain the steady-state flux of the polymer chains and the probability of successful barrier crossing. 

Finite-element simulations are useful to understand the mechanism of NDR and its dependence on the composition in the internal and external solutions, pore geometry, and nanopore surface charge density. Similar to modeling flow effects on nanopore ICR described earlier, the Nernst-Planck equation governing the diffusional, migrational, and convective fluxes of ions (Equation 2.18), the Navier-Stokes equation for low-Reynolds number flow engendered by the external pressure and electroosmosis (Equation 2.20), and Poisson s equation relating the ion distributions to the local electric field (Equation 2.19) were simultaneously solved to obtain local values of the fluid... 

Simulation Results. A onc-dimensional simulation model based on the Nernst-Planck and Poisson equations [14, in which all the acid-base reactions occurring in the membrane are taken into account, has been used to give a qualitative description of the pH step titration process. In these simulations, a pH step is applied outside a 2 mm thick stagnant layer, which is assumed to be present in front of an 8 mm thick membrane. Diffusion coefficients in the membrane are assumed to be 4/10 of those in water (this value is based on experience with ion step experiments). Lysozyme, used as a model protein, is assumed to contain 11 carboxylic groups (pKa = 4.4), 2 imidazole groups (pKa = 6.0), and 9 amino groups (pKa = 10.4) per molecule. Concern... 

Einstein-Schmolukowski, 378, 405 Gibbs-Duhem, 262 LaPlace, 392 Leonard-Jones, 45 Nernst-Einstein, 456 Nernst Planck, 476 Onsager, 494 Planck-Henderson, 500 Poisson, 235, 344 Poisson-Boltzmann, 239 Sackur-Tetrode equation, 128 Setchenow s, 172 Tafel, 2... 

The main difficulty when working with thin conducting polymer membranes is the lack of quantitative theory of ion diffusion within the membrane. Various theoretical schemes and approximations have been suggested, but the most difficult problem seems to be in the analytical solution or even approximation for the boundary problem of the combined Nernst-Planck and Poisson equations. The latter equation comes from the fact that electroneutrality cannot be assumed to prevail inside the thin membrane. Doblhofer et al." have made an attempt to solve the problem numerically, but even then certain initial approximations were made. Also the brute force method of finite differences does not allow to see clearly the influence of different parameters. 

As can be concluded from Eqs. (7.14) and (7.21), the diffusion-migration problem is non-linear. The Newton-Raphson method has been applied successfully to the resolution of the Nernst-Planck-Poisson equation system although the convergence is slower than for the kinetic-diffusion problems studied in Chapter 6. Thus, the unknown vector x corresponds to... 

The Nernst-Planck and Poisson equations were used to describe the underlying physical processes (see Section 2.5.2 Eq. (2.38) (2.41).). 

M. Kh. Urtenov, Methods of Solution of the Nernst-Planck-Poisson Equation System [in russian], Kuban State University, Krasnodar, Russia, 1998. 

Figure 2. 8. (a) Transmission line representation of Nernst-Planck Poisson equation system for a binary electrolyte. Rp and R are charge transfer resistances for positive and negative charge species at the electrode, respectively, (b) General approximate equivalent circuit (full-cell, unsupported) for the [0. Ai> cases applying to a homogeneous liquid or solid material. 

The evolution of the numerical approaches used for solving the PNP equations has paralleled the evolution of computing hardware. The numerical solution to the PNP equations evolved over the time period of a couple of decades beginning with the simulation of extremely simplified structures " ° to fully three-dimensional models, and with the implementation of sophisticated variants of the algorithmic schemes to increase robustness and performance. Even finite element tetrahedral discretization schemes have been employed successfully to selectively increase the resolution in regions inside the channels. An important aspect of the numerical procedures described is the need for full self-consistency between the force field and the charge distribution in space. It is obtained by coupling a Poisson solver to the Nernst-Planck solver within the iteration scheme described. 

Sokalski T, Lingenfelter P, Lewenstam A (2003) Numetictil solution of the coupled Nernst-Planck and Poisson equations for liquid junction tind ion selective membiane potentials. Phys Chem C 107 2443-2452... 

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