Electrodiffusion Equation

The problem has been treated by solving the electrodiffusion equation for the univalent ions subject to a perturbation caused by Ca + ions which can enter the membrane but which cannot pass through it. Thus in the system sketched in Fig. 3 the surface a = 0 is impermeable to Ca + ions. The essential evidence for impermeability to Ca + at a = 0 is that when EPP s are lost by Ca deficiency in the cleft, they are not restored by Ca injection into the presynaptic region. All the boundary values are taken just inside the membrane, hence include distribution coefficients. 

To study non-steady-state systems, we applied the SCM to the problem of excitation, where, based on the macroscopic concentrations of ions and electrical potentials, it is generally accepted that the ion fluxes do not follow the classic electrodiffusion equations. When the surface concentrations and surface potentials at the charged membrane surfaces are used, the ion fluxes are given by the same electrodiffusion equations that apply to ions in solution (14). 

Under steady-state conditions the partial currents persist, so that // = const., and Eq. (88) is then a first-order nonlinear differential equation containing unknown functions q and E and unknown constant U This fundamental electrodiffusion equation, conventionally referred to as the Nernst-Planck equation, may be obtained by direct differentiation of the expression for the electrochemical potential of ion species / present in the dilute solution... 

It is never possible to obtain an exact analytical solution of the electrodiffusion equation, and the standard practice is to seek approximate solutions under some sort of assumptions. In macroscopic membranes whose thickness is many times greater than the electric double layer thickness Planck s approximation, assuming that the system obeys the local electric neutrality... 

The Ussing equation may be derived from the general electrodiffusion equations without the assumption of a constant electric field. To demonstrate this it may be convenient to represent the partial ionic flux in the form ... 

There are electrochemical processes in which little or no supporting electrolyte is used. This is the case, for instance, of the study of overlimiting currents [34, 41,42], and of microelectrode [43-45], and different voltammetric techniques [17, 18, 46-48], in which the absence of supporting electrolyte increases the sensitivity of the detection of the redox species. In this section, a general procedure for solving the electrodiffusion equations is presented. The procedure is based on determining... 

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 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. 

Moreover the electrodiffusion potential gradient is likely to cause electroosmotic transfer of the solution, whose local content is not in equilibrium with that of the counterions [5]. In this case, as it is pointed out in Ref. 5, the ion mobility and concentration depend on the prior history of the process which can bring about non-Fickian diffusion. The application of Nemst-Planck equations to the real system may require inclusion of additional terms that account for the effect of activity coefficient gradients which may be important in IE with zeolites [4,5]. 

Equation 5.88 predicts that the perturbation of surface tension, Ao(f) = o(f) — o,- relaxes exponentially. This is an important difference with the cases of adsorption under diffusion and electrodiffusion control, for which Ao(f) 1/x/f see Equations 5.70, 5.76, and 5.78. Thus, a test whether or not the adsorption occurs under purely barrier control is to plot data for ln[Aa(f)] vs. t and to check if the plot complies with a straight line. 

At a set temperature and concentration, the coefficient of diffusion D is constant and its temperature dependence, calculated by the Arrhenius law [36], can be ignored. However, in the presence of external forces F and F stipulating the appearance of an additional flow of the gaseous components by electrodiffusion and by thermodiffusion (Soret effect [37]), respectively, the coefficient d in Equation (2.7) describes the force of the convective diffusion as follows ... 

This equation describes the distribution of concentration of admixture introduced into the glass, expressed by the parameters of the refractive index profile n(x) and of the equilibrium concentration Cq. Integrating the expression (6) over the entire volume of the glass V, the total number of admixture ions N, introduced into the glass during the electrodiffusion process in the time t is obtained as follows. 

Table 2 summarizes the results of calculation product Scq based on equation (9) for several electrodiffusion processes carried out in the substrate of soda-lime glass, using a pure silver nitrate AgNOs as the source of admixture of Ag ions. It also presents the maximum changes in refractive index profile Anj at the surface of the glass calculated on the basis of the determined refractive index profiles. 

Defining a total of 2 n -I- 2 boundary conditions, we will obtain a final formulation of the problem. Apart from this basic form of the electrodiffusion problem, it is sometimes convenient to use the integral form or an equation with excluded electric field. A comprehensive review of all questions pertinent to this problem can be found in Arndt and Roper s book the nonsteady-state case is also discussed in detail in this reference. 

This formula, called the Ussing equation, is, furthermore, extremely important for the assessment of the fundamental principles of the electrodiffusion theory, although those unidirectional fluxes which satisfy the Ussing equation may, at the same time, not obey the independent principle. 

In the case of ionic surfactants the existence of a diffuse EDL essentially influences the kinetics of adsorption. The process of adsorption is accompanied by a progressive increase in the surface-charge density and electric potential. The charged surface repels the incoming surfactant molecules, which results in a deceleration of the adsorption process (54). Theoretical studies on the dynamics of adsorption encounter difficulties with the nonlinear set of partial differential equations, whieh deseribes the electrodiffusion process (55). 

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]). 

Equation 4.288 provides a boundary condition for the normally resolved flux, From another viewpoint. Equation 4.288 represents a 2D analogue of Equation 4.287. The interfacial flux, ji, can also contain contributions from the interfacial molecular diffusion, electrodiffusion, and thermodiffusion. A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface is given by Brenner and Leal [734-737], Davis et al. [669], and Stone [738]. If the molecules are charged, the bulk and surfaces electrodiffusion fluxes can be expressed in the form [651,739,740] ... 

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... 

Jasielec J, Filipek R, Szyszkiewicz K, Fausek J, Danielewski M, et al. (2012) Computer simulations of electrodiffusion problems based on Nernst-Planck and Poisson equations. Comput Mater Sci 63 75-90... 

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