Nernst-Planck approximation

Here C is the specific differential double layer capacitance. The two terms on the left side of Eq. (4) describe the capacitive and faradaic current densities at a position r at the electrode electrolyte interface. The sum of these two terms is equal to the current density due to all fluxes of charged species that flow into the double layer from the electrolyte side, z ei,z (r, z = WE), where z is the direction perpendicular to the electrode, and z = WE is at the working electrode, more precisely, at the transition from the charged double layer region to the electroneutral electrolyte. 4i,z is composed of diffusion and migration fluxes, which, in the Nernst-Planck approximation, are given by... 

R. Schlogl (144) obtained, through his general integration of the Nernst-Planck equations, also values for the diffusion potential. The approximations in the calculations are the same as those used for the fluxes (cf. 3.4). 

The first term D dQ/dz) represents the diffusion, the second term D ZjQF/flT) (dcp/dz) the migration of a component. Thus, the Nernst-Planck equation is an approximation of the more general phenomenological equation. 

The model upon which this equation is based neglects potential differences that may occur due to IR inside the membrane and assumes that the entire membrane potential consists of the difference of the two interfacial potential differences. In some cases, this may be a good approximation.4 In other cases, the potential difference through the membrane (determined by fluxes considered in the Nernst-Planck-type equations) may dominate. A comparison of Eq. (14.5) with (14.6) shows that both models lead to equations that have the same form. 

To what concentration level of the salt can one safely use the Nernst-Planck dilute solution approximation Eq. 2.4.18, instead of the generalized Maxwell-Stefan diffusion equations ... 

A brief comment should be made concerning the use of the Nernst-Planck equations for ion transport across the liquid film (e.g., Copeland and Marchello [1969], Kataoka et al. [1987]). This is a nonlinear, three-ion problem because of the presence of at least one coion at comparable concentration. The Nernst film model relies on the assumption of a linear concentration gradient in the liquid film. The film has no physical reality, and the calculation of nonlinear concentration profiles in it overburdens the model and offers little improvement over the much simpler linear driving force approximation. For higher accuracy, more refined and complex hydrodynamic models would have to be used (Van Brocklin and David, 1975). 

With respect to ion-exchange kinetics, the great complexity of soils has resulted in a penchant for the easy approximations of homogeneous models, even where the premise of a quasi-continuum of liquid and solid is hard to accept. The theory of heterogeneous systems offers the Nernst-Planck equations, but these also can provide no more than an approximation for migration of ions in soil constituents. 

Goldman [42], Kakiuchi derives the following approximate solution to the Nernst-Planck equation for the ion transport across the interface and the concomitant current density ... 

The Nernst-Planck equation [Eq. (88)] is suitable to describe the direct anion passage quantitatively in the constant field approximation. The mobile carrier mechanism is treated in the same way. The relevant diagram is shown... 

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. 

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. 

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 ... 

This approximation pertains to systems where the ohmic losses within the electrolyte and the kinetic limitations on the electrode are considered to be negligible as compared with mass transport limitations. Instead of solving the Laplace equation (25) for the potential, which is a common approximation to the more general Nernst-Planck equation (10), we need to solve the latter (10) for the case when the potential gradients are negligible as compared with concentration gradients, i.e.,... 

The modeling of the current distribution in a general-geometry cell nearly always requires a numerical solution. The following discussion focuses on the thin boundary layer approximation, with the overpotential components lumped within a thin boundary layer which may be of a varying thickness. The Laplace equation for the potential with nonlinear boundary conditions must be solved. Similar considerations typically apply to the more comprehensive solution of the Nernst-Planck equation (10) however, the need to account for the convective fluid flow in the latter case makes the application of the boundary methods more complex. We focus our brief discussion on the most common methods the finite-difference method, the finite-element method, and the boundary-element method, schematically depicted in Fig. 4. Since the finite-difference method is the simplest to implement and the best known technique, it is discussed in somewhat more detail. 

In the derivation of the formula for calculating the liquid junction potential, the electric work done in separating the charges is set equal to the work of diffusion that is, the change in chemical potential arising from the diffusion of the ions. Only after making certain approximations can one arrive at the so-called Henderson solution [56] of the Nernst-Planck equation [57] ... 

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