Fick-Nernst-Planck equation

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

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. 

However, in interdiffusion of ions of different mobilities, Fick s law fluxes would be unequal and disturb electroneutrality. Here, the first, minute deviation from local electroneutrality generates an electric potential gradient (diffusion potential) that produces electric transference of ions superimposed on diffusion. This is the mechanism by which the system manages to balance the fluxes so as to maintain electroneutrality (Schlogl and Helfferich, 1957 Helfferich, 1962a Helfferich and Hwang, 1988). The flux now obeys the Nernst-Planck equation (Nernst, 1888 1889 Planck, 1890)... 

Donnan dialysis The BAHLM systems with ion-exchange membranes, based on Donnan dialysis [18,19], will be considered below. Donnan dialysis is a continuously operating ion-exchange process. There are many theoretical models describing transport mechanisms and kinetics of DL) [18-26]. All transport kinetics models are based on Fick s or Nernst-Planck s equations for ion fluxes. In both cases, the authors introduce many assumptions and simplifications. 

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