Model electrodiffusion

Valent I, Neogrady P, Schreiber I, Marek M (2012) Numerical solutions of the full set of the time-dependent Nernst-Planck and Poisson equations modeling electrodiffusion in a simple ion channel. J Comput Interdiscip Sci 3 65-76... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

The same term is sometimes used to describe the potential-distance relations in semiconductors with a low concentration of surface states (hence the term Schottky barrier model ). However, as can be understood by a reconsideration of the mechanism there (see Figs. 10.6 and 10.7), the so-called barrier is either used for the acceleration of electrons in p-type cathodes or the electrodiffusion of holes to the surface in n-type anodes. Nevertheless, the term barrier is still applied. 

Kinetic Model for Chromatographic Dispersion and Electrodiffusion, J. C. Giddings, J. Chem. Phys., 26, 1755 (1957). 

Most information needed may be inferred from analyzing the following simple problem modeling the one-dimensional steady state electrodiffusion in a solution layer flanked by two perfectly cation-selective walls (all variables and parameters are dimensionless, listed at the end of this entry) ... 

One continuum model for electrodiffusion of ions between regions of different concentration is based on the combination of Pick s law that describes the diffusion of ions along a concentration gradient and Kohl-rausch s law that describes the drift of ions along a potential gradient. Nemst and Planck combined these two laws to obtain the electrodiffusive equation, now known as the Nernst-Planck equation, and which can be written in the Stratonovich form as... 

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]). 

Models for the transport of ions across BLMs are usually based upon the fundamental laws of electrostatics and electrodiffusion. Traditionally, these treatments consider the membrane to be a uniform macroscopic phase, devoid of chemistry (such as incorporation of cholesterol shown in Figure 7.3). This is a very serious fault, but at least the simple derived expressions provide a qualitative picture of the problem. In broad terms, it is possible to consider two separate factors which determine the mechanism of charge transport, electrostatic and steric (lipid fluidity), although there may well be a synergism between the two influences. The potential energy barriers to introduction of ions into the membrane can then be included into generalized expressions for electrodiffusion. 

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