Nernst-Planck equation, membrane potential

Ion transport across membranes can be evaluated by using mucosal and serosal electrodes to read transepithelial current (I) and potential difference OP). With these parameters, equivalent circuit analysis can be utilized to account for the relative contributions of transcellular and paracellular pathways. Ionic flux (J) is defined by the 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... 

If there is a net transport of charge across the membrane, the membrane potential will influence the solute transfer and also be affected by it, complicating the data treatment. The starting point for most descriptions of the internalisation flux of permeant ions, i, is the one-dimensional Nernst-Planck equation (cf. equation (10)) that combines a concentration gradient with the corresponding electric potential gradient [270] ... 

Other resolutions of the Poisson Nernst Planck equations (i.e. using various simplifying assumptions) have been proposed that couple the adsorption, desorption and permeation of ions through a membrane (e.g. [273,274]) as might be observed for a carrier-mediated transport. For example, for a symmetrical membrane (identical electrolyte on both sides of the membrane) and variation in the electrical potential profile given by i//m, /int can be estimated from ... 

Equation 3.8, which is often called the Nernst-Planck equation, is a general expression for the one-dimensional flux density of species j either across a membrane or in a solution in terms of two components of the driving force — the gradients in activity and in electrical potential. 

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

One of the most general description of ion flow across a membrane is given by the Nernst-Planck equations. This is a partial differential equation where the independent variables represent space (x) and time (t). The main dependent variable is the concentration of the ion (c(x, t)). The potential (u(x, t)) is usually a fixed function but can be made a dependent variable in which case an additional equation is required. The Nernst-Planck equations can be written as ... 

In the next section we use the Nernst-Planck equation to show that the electrostatic potentials across membranes depend not only on the difference in ion concentrations, but also on the ion mobilities. 

The Nernst-Planck equations can be used for modeling mass transfer within a single-phase dense ceramic membrane with neither external diffusion and surface exchange effects nor occluded porosity in the dense layers (Figure 14.1c) [25]. The flux of each charged species i (i.e., vacancies or other charged species), y,-, can be modeled as a function of the electrochemical potential gradient, V/i ... 

Assuming that the active layer of the nanofiltration membrane is porous in nature, the extended Nernst-Planck equation is apphcable to describe the transport of multieomponent systems in nanofiltration membranes. It represents transport due to diffusion, eleetrieal potential gradient and convection. The equations can be written as ... 

Equation (3) suggests that the membrane potential in the presence of sufficient electrolytes in Wl, W2, and LM is primarily determined by the potential differences at two interfaces which depend on charge transfer reactions at the interfaces, though the potential differences at interfaces are not apparently taken into account in theoretical equations such as Nernst-Planck, Henderson, and Goldman-Hodgkin-Katz equations which have often been adopted in the discussion of the membrane potential. 

In general it can be said that the experimental material is not extensive. The experimental material concerns the application of irreversible thermodynamics, the application of refined Nernst-Planck flux equations and the application of quasi-thermodynamics. The latter is used to derive equations for membrane potentials. 

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. 

The distribution of proton concentration Ch+ and potential in solution is governed by the Poisson-Nernst-Planck (PNP) model, widely used in the theory of ion transport in biological membranes (Coalson and Kurnikova, 2007 Keener and Sneyd, 1998). Oxygen diffusion is determined by Pick s law. Inside the pore, the continuity and transport equations for protons and oxygen are... 

Concerns with liquid junctions—that is, electrolytes with different ionic concentrations or different ionic species meeting at a junction, such as a membrane or simply a small hole in a Luggin capillary, go back at least to the works of Nernst [4, 5], Planck [6] in the 1880s and 1890, and that of Henderson [7] in 1907. It is Henderson who is credited with the derivation of the equation named after him, for the potential difference across such a junction, see below, although we find essentially the same equation in the 1890 work of Planck [6]. These works were concerned with steady state solutions. Helfferich (in 1958) [8] and Cohen and Cooley [9] computed, by finite differences, time-dependent behaviour at liquid junctions. Many subsequent works were of course published since then, including the recent work of Strutwolf et al. [10, 11], Dickinson et al. [12] and Britz and Strutwolf [13],... 

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