The Nernst-Planck Equation

Ignoring the effects of non-ideality, the electrochemical potential, of an ion in solution may be defined  

The flux / through a plane is the product of the velocity of the ions undergoing transport, and the concentration of ions  

Typically, z, is used such that absolute speed rather than velocity (implying a direction) is considered. Hence from the definition of Di we can confirm that  

Clearly this equation is cumbersome by comparison to Pick s second law, which is the special case where d tjdx = 0 everywhere. 

Most importantly, the Nernst-Planck equation is non-hnear, whereas Pick s second law is linear. Both analytical theory and simulation using the full Nernst-Planck equation are hence much more demanding than diffusion-only theory, thus indicating one good reason why excess supporting electrolyte is often used to remove the influence of electric fields in a system. 

In dilute electrolyte systems the driving force reduces to 

The superscript ° signifies the infinite dilution limit. Equations 2.4.13 for the diffusion 

In the electrochemical literature it is traditional to use molar concentration gradient 

We have shown that the Nernst-Planck equation is only a limiting case of the generalized Maxwell-Stefan equations. Nevertheless, many ionic systems of interest are dilute and the Nernst-Planck equation is widely used. 

Even when the system is dilute, the diffusing ionic species are coupled to one another in a very interesting manner. This coupling arises out of the constraint imposed by the electroneutrality condition. Equation 2.4.3 can be differentiated to give 

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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... 

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,... 

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 ... 

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 ... 

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... 

In the above it is already said that due to drastic-simplification of the flux equations, the Nernst-Planck equations arise. To show this it is necessary to first write equation (1) in a somewhat different form. For the purpose, the velocities of the several components are split up in a velocity of the particles relative to one another and a velocity of trans-... 

I.I. Concentration gradient Across a Membrane. In the instance that a membrane separates two solutions of the same electrolyte, but with different concentrations F. Helfferich (ref. 55, page 319) calculated the ion-fluxes and the profiles of the internal concentrations, starting from the Nernst-Planck equations. Gradients of activity coefficients could be involved. However, convection (osmosis) had to be neglected. 

R. Schlogl (144) obtained, through his general integration of the Nernst-Planck equations, also values for the diffusion potential. The approximations in the calculations are the same as those used for the fluxes (cf. 3.4). 

From (6.44), we see that in order to obtain a selective membrane, the value of the ion-exchange constant must be small and the sodium ion mobility in the hydrated layer relative to that of the hydrogen ion must also be small. The expansion of the selectivity coefficient to include selectivity to other ions involves inclusion of more complex ion-exchange equilibria, and the use of a more complex form of the Nernst-Planck equation. This rapidly leads to intractable algebra that requires numerical solution (Franceschetti et al 1991 Kucza et al 2006). Nevertheless, the concept of the physical origin of the selectivity coefficient remains the same. Electrochemical impedance spectroscopy has been successfully used in analysis of the ISE function (Gabrielli et al 2004). 

Complex membrane transport phenomena are relevant to several areas in medicine, such as ion transport through neurons, ion exchange, and electrodialysis [50,51]. The Nernst-Planck equation is often the starting point for the interpretation of transport through both artificial and biological membranes [50,51]. Strictly speaking, the Nernst-Planck equation requires the... 

The Nernst-Planck equation constitutes the starting point for the electrotransport models [55-57], The overall flux of the ionic species i (/,) comprises the diffusion term driven by the chemical potential gradient (dc,/dx) and the electric transference term due to the electrical potential gradient (d /dx) ... 

As the skin is relatively thick compared to the space-charge layers at its boundaries, the bulk of the membrane may be expected to be electroneutral [56,57], The Nernst-Planck equation can be solved, therefore, by imposing the electroneutrality condition C,/C= C. /C, where the subscripts j and k refer to positive and negative ions, respectively, and C is the average total ion concentration in the membrane. In the case of a homogenous and uncharged membrane bathed by a 1 1 electrolyte, the total ion concentration profile across the membrane is linear and the resulting steady-state flux is described by... 

The flux of the species O, /0(x, t), is described in terms of the three components that constitute the Nernst-Planck equation (Equation 6.14). The parameters are defined below. 

The Nernst-Planck equations are only limiting laws for ideal systems. If activity coefficient gradients are present, an additional term in Eq. (5.5) is created such that... 

Even though the Nernst-Planck equations work well in many instances and their theoretical basis is sound, Helfferich (19B3) mentions several cases where they are not statisfactory. For example, they may not work well in situations where other processes besides mass transfer occur. This could occur if ion mobilities increase or decrease such that diffusion coefficients in the Nernst-Planck equations are not constant and thus particle size of the ion exchanger is affected. In zeolites, for example, which are rigid, the exchange of one counterion for another of different size affects the ease of motion and creates diffusion coefficient differences. 

Even where ion exchange is not affected by the above factors, the Nernst-Planck equations are not very useful for diffusion phenomena in the film. After all, the Nernst film is somewhat enigmatic and there is a combination of diffusive and convective mass transfer that changes from the bulk solution to the particle surface. Nernst (1904) originally defined the outer limit of film only as the point where the concentration profile, if linearly extrapolated from the particle surface, reaches the concentration level of the bulk solution. 

Another problem with the Nernst-Planck equations for studying ion... 

This is a three-component (A, B, C) system, and the transport or diffusion of each component can be described using the Nernst-Planck equation. The relation of the diffusion flux of the i component (i, j, k = A, B, C) can be obtained by combining three Nernst-Planck equations, for the respective components, under the conditions of zero electric current inside the resin particle ... 

The first term D dQ/dz) represents the diffusion, the second term D ZjQF/flT) (dcp/dz) the migration of a component. Thus, the Nernst-Planck equation is an approximation of the more general phenomenological equation. 

Species flux can be described by the Nernst-Planck equation,... 

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