Nernst-Plancks Equation

The Nemst-Planck equation below relates the unidirectional (x) flux of a species j to diffusion, migration, and convection  

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 

Here D is the diffusion coefficient, c is the ionic concentration, and is the electrostatic potential caused by the charges within the system and the external boundary conditions. The absolute temperature of the solution is T, whereas kg is Boltzmann s constant and q is the ionic charge. Integrating Eq. [53] once gives 

The set of three coupled equations (Eq.[18], Eq. [54], and Eq.[55]) is then solved numerically with an iterative procedure that will be discussed in a subsequent section. 

The remainder of this section is devoted to the derivation of Eq.[54]. Besides the mathematics we also define the range of applicability of simulations based on the Nernst-Planck equation. The starting point for deriving the Nernst-Planck equation is Langevin s equation (Eq. [45]). A solution of this stochastic differential equation can be obtained by finding the probability that the solution in phase space is r, v at time t, starting from an initial condition ro, Vo at time = 0. This probability is described by the probability density function p r, v, t). The basic idea is to find the phase-space probability density function that is a solution to the appropriate partial differential equation, rather than to track the individual Brownian trajectories in phase space. This last point is important, because it defines the difference between particle-based and flux-based simulation strategies. 

The derivation of a differential equation for p(r, v, t) is performed by first defining the diffusion process as an independent Markov process to write a Chapman-Kolmogorov equation in phase space  

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In ion-exchange resins, diffusion is further complicated by electrical coupling effec ts. In a system with M counterions, diffusion rates are described by the Nernst-Planck equations (Helfferich, gen. refs.). Assuming complete Donnan exclusion, these equations canbe written... 

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As demonstrated in the preceding section, an electric potential gradient is formed in electrolyte solutions as a result of diffusion alone. Let us assume that no electric current passes through the solution and convection is absent. The Nernst-Planck equation (2.5.24) then has the form ... 

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,... 

Doi and his coworkers have proposed a semiquantitative theory for the swelling behavior of PAANa gels in electric fields [14]. They have considered the effect of the diffusion of mobile ions due to concentration gradients in the gel. First of all, the changes in ion concentration profiles under an electric field have been calculated using the partial differential Equation 16 (Nernst-Planck equation [21]). 

Application of elementary conservation laws leads to formulation of a general expression for which is often denoted as 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... 

Equation (10) is known as the Nernst-Planck equation. This equation can be given in all kinds of formulations. Another common one is ... 

We consider, then, two media (1 for the cell-wall layer and 2 for the solution medium) where the diffusion coefficients of species i are /),yi and 2 (see Figure 3). For the planar case, pure semi-infinite diffusion cannot sustain a steady-state, so we consider that the bulk conditions of species i are restored at a certain distance <5,- (diffusion layer thickness) from the surface where c, = 0 [28,45], so that a steady-state is possible. Using just the diffusive term in the Nernst-Planck equation (10), it can be seen that the flux at any surface is ... 

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 ... 

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Turning back to field effects, they derive from the second terms on the right-hand sides of equations (4.22), (4.23), (4.25), and (4.26). It should be noted that they are different from the corresponding term in the Nernst-Planck equation, which depicts migration effects for free-moving ions as recalled in... 

Dilute Solution Theory. Equation 28 is the result of using dilute solution theory.Such an analysis yields the Nernst—Planck equation... 

If water movement in the membrane is also to be considered, then one way to do this is to again use the Nernst—Planck equation. Because water has a zero valence, eq 29 reduces to Pick s law, eq 17. However, it is also well documented that, as the protons move across the membrane, they induce a flow of water in the same direction. Technically, this electroosmotic flow is a result of the proton—water interaction and is not a dilute solution effect, since the membrane is taken to be the solvent. As shown in the next section, the electroosmotic flux is proportional to the current density and can be added to the diffusive flux to get the overall flux of water... 

The first model to describe the membrane in the above fashion was that of Bernardi and Verbrugge, "° which was based on earlier work by Verbrugge and Hill. " 214 model utilized a dilute solution approach that used the Nernst— Planck equation (eq 29) to describe the movement of protons, except that now v is not equal to zero. The reason is that, because there are two phases, the protons are in the water and the velocity of the water is give by Schlogl s equation ... 

The flnely-porous membrane model (, ) assumes that a substantial amount of salt is transported by convective flow through the narrow pores of the membrane. Integrating the Nernst-Planck equation for salt transport O) and using the appropriate boundary conditions, the following relationship is obtained between the salt rejection and the volume flux ... 

However, a distinction should be made in that Eq. (12) is purely phenomenological and does not require any transport mechanism model while the Nermst-Planck equation used in the previous finely-porous membrane model requires a specific pore model. Another difference is that the salt concentration in Eq. (12) is that in the membrane while the quantity appearing in the Nernst-Planck equation refers to the salt concentration in the membrane pores. 

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. 

For a steady state the Nernst-Planck equations could be divided by Diy yielding (4.1.1), whose summation leads to (4.1.4a,b) for constant N. The factor r in (4.1.4a) may be viewed as a modified steady state conductivity.) For N nonvanishing the factor r in (4.1.4a) may be evaluated as follows ... 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

There are three kinds of mass transport process relevant to electrode reactions migration, convection and diffusion. The Nernst—Planck equation... 

Since the fluxes of electrolyte ions are zero at the electrode surface (rc), the Nernst-Planck equation for the anion becomes... 

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