Nernst-Planck relation

This equation follows from continuity (Eq. 3.3.2) and the Nernst—Planck relation (Eq. 3.4.1). The overbars indicate that the variable has been averaged over the tube cross-section (Eq. 4.6.19), and R and R" are the molar rates of production due to chemical reactions and sorption, respectively. 

It is appropriate to indicate now an approximate form of flux-force relation in a binary isothermal liquid mixture in the presence of an electrical potential gradient. It is called the Nernst-Planck relation and is used for systems containing ions ... 

Since the membrane essentially has no free species, the process is controlled by the potential difference, A, driving the negatively charged species like OH, Br , etc. The flux of the ionic species may be obtained from the Nernst-Planck relation (3.1.106) for a membrane of thickness Ay in an integrated form as... 

Similar statements can be made about holes. They, too, have to be transported to the interface to be available for the receipt of electrons there. These matters all come under the influence of the Nernst-Planck equation, which is dealt with in (Section 4.4.15). There it is shown that a charged particle can move under two influences. The one is the concentration gradient, so here one is back with Fick s law (Section 4.2.2). On the other hand, as the particles are changed, they will be influenced by the electric field, the gradient of the potential-distance relation inside the semiconductor. Electrons that feel a concentration gradient near the interface, encouraging them to move from the interior of the semiconductor to the surface, get seized by the electric field inside the semiconductor and accelerated further to the interface. 

This is a three-component (A, B, C) system, and the transport or diffusion of each component can be described using the Nernst-Planck equation. The relation of the diffusion flux of the i component (i, j, k = A, B, C) can be obtained by combining three Nernst-Planck equations, for the respective components, under the conditions of zero electric current inside the resin particle ... 

Recall that v, is the mobility, which is related to the diffusivity by the relation D, = RTu,. Equations (3.4.1) and (3.4.2) are called the Nernst-Planck equations. 

The general form of this relationship agrees with the well-known Nernst-Planck-Einstein formula, according to which the diffusion coefficient is given by the relation... 

The contribution of electric field to lithium transport has been considered by a few authors. Pyun et argued on the basis of the Armand s model for the intercalation electrode that lithium deintercalation from the LiCo02 composite electrode was retarded by the electric field due to the formation of an electron-depleted space charge layer beneath the electrode/electrolyte interface. Nichina et al. estimated the chemical diffusivity of lithium in the LiCo02 film electrode from the current-time relation derived from the Nernst-Planck equation for combined lithium migration and diffusion within the electrode. 

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. 

Recall that u is the mobility, D is the coefficient of binary diffusion, related to with mobility by the equation D = ATo . Equations (5.85) and (5.86) are known as the Nernst-Planck equations. 

In electrode reactions, the reactant has to find the electrode surface where electrons are taken or released, and therefore the mass transport of reactants and products becomes very important in the description of electrode reactions where we will be interested in current-time and current-potential relations. The master equation for mass transport to an electrode surface is the Nernst-Planck equation ... 

In the case of nonhomovalent multiionic systems, the solution procedure is necessarily more complicated because there is no simple relation such as S2 = z Sq, a problem that was first encountered and solved in different ways by Schlogl [59] and Brady and Turner [60]. If the system contains N different ionic species, the N Nernst-Planck equations for the N flux densities J,- are replaced by... 

The derivative of U with respect to position is the regular force acting on the particle. However, this so-called stochastic approach results in similar Nernst-Planck type equations albeit with microscopic diffusion coefficient differing from the traditional macroscopic diffusion coefficient. The microscopic diffusion coefficient, D, is related to f, through D = k T)/ fim).)... 

The probability distribution functions in Eq. [59] applied to the trajectories of particles flowing into and out of a system provides a justification for using the Nernst-Planck equation (Eq. [54]) The net ionic directional fluxes can be expressed in terms of differences between the probability fluxes, normalized to the concentration at the sides of the region of interest. That ionic fluxes and differences in probability fluxes are related thus supplies a connection between the solution of the Nernst-Planck equation (Eq. [54]) and the Smoluchowski equation (Eq. [59]), and it provides a direct justification for using Eq. [54] for the study of ions subjected to Brownian dynamics in solution. 

In these calculations, the authors do not assume that the electric field strength U is constant. Rather, they examine this hypothesis, using the Nernst-Planck equation, which relates the fluxes of charged species to the electric field gradient. Even assuming local electric neutrality, that is, [A " "]-I-[T" ] — [C ], they find nonnegligible spatial variations in the electric... 

Obviously, a full formulation of governing equations should also include the mass transfer equation for each type of ions in the system. The ionic concentration distribution, ni(r), is generally described by the Nernst-Planck equation. According to Dukhin and Deryaguin [2], the time scale related to eleclromigration within the EDL is characterized by Dj, which is of the order... 

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... 

Now, combining the mass flux caused by diffusion, migration, and convection, the total flux is given by the relation known as the Nernst-Planck equation. 

Integrate the extended Nernst-Planck equation for any counterion i in the membrane in the z-direction perpendicular to the membrane surface when the membrane excludes co-ions perfectly. Determine the constant of integration in terms of counterion concentration and resin-phase potentitd just inside the membrane, CiR (0-t-) and r (0-I-). Use now the condition of thermodynamic equilibrium for the counterion at the interface (between C,t)(0-I-) and C, , via relation (3.3.118b)) and the assumption that , = 0 to relate Caj(z) to Ciw Obtain r z] by using the electroneutrality condition (3.3.30b). 

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... 

The Nernst-Planck-Poisson model relates the electric potential cf) to the space-charge density p, which is defined by the local concentration C of all ions i at the distance x. For monovalent ions (z +l or -1),... 

These relations, sometimes called the Nernst-Planck equations (Bard and Faulkner, 2000), could be written down directly as a definition for Z),-. If this were done, then the restriction to dilute solutions in Eq. 6.1-3 and the implicit neglect of a reference velocity in the first line of Eq. 6.1-5 would be hidden in the final flux equation, lumped into the experimental coefficient D,. I find the derivation a sensible, reassuring rationalization, even though I know that it is arbitrary. 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

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