Nemst-Planck

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

The flux ( J ) is a common measure of the rate of mass transport at a fixed point. It is defined as the number of molecules penetrating a unit area of an imaginary plane in a unit of time, and has the units of mol cm 2 s-1. The flux to the electrode is described mathematically by a differential equation, known as the Nemst-Planck equation, given here for one dimension ... 

The EMD studies are performed without any external electric field. The applicability of the EMD results to useful situations is based on the validity of the Nemst-Planck equation, Eq. (10). From Eq. (10), the current can be computed from the diffusion coefficient obtained from EMD simulations. It is well known that Eq. (10) is valid only for a dilute concentration of ions, in the absence of significant ion-ion interactions, and a macroscopic theory can apply. Intuitively, the Nemst-Planck theory can be expected to fail when there is a significant confinement effect or ion-wall interaction and at high electric... 

Exchange between counter-ions A in beds of resin and counter-ions B in a well-stirred solution may be represented by the Nemst-Planck equation as ... 

The situation is more complicated when charged ions rather than uncharged molecules are transferring. In this case, a Nemst-Planck equation which includes terms for both counter-ions and mobile co-ions must be applied. The problem may be simplified by assuming that the counter-ions have equal mobility, when the relationship is ... 

However, in bulk diffusion, ions cannot move independently of each other because electrical neutrality must be maintained. Consequently there is an electric potential between diffusing ions such that the faster ions tend to be slowed down by the slower ones and vice versa. The flux of a particular ion is therefore the sum of the diffusion due to its own concentration gradient and that due to the gradient of the diffusion potential arising from differences in the mobilities of the ions present. This is expressed by the Nemst-Planck equation along the x-axis ... 

In a dilute solution the activities can be replaced by concentrations and (2.6.1) yields the Nemst-Planck equation... 

For inter diffusion between same-valence ions (ionic exchange) in an aqueous solution, or a melt, or a solid solution such as olivine (Fe +, Mg +)2Si04, an equation similar to Equation 3-135c has been derived from the Nemst-Planck equations first by Helfferich and Plesset (1958) and then with refinement by Barter et al. (1963) with the assumption that (i) the matrix (or solvent) concentration does not vary and (ii) cross-coefficient Lab (phenomenological coefficient in Equation 3-96a) is negligible, which is similar to the activity-based effective binary diffusion treatment. The equation takes the following form ... 

This leads to the refined Nemst Planck equations (cf. 2). 

The equations (19) may be considered as refined Nemst-Planck equations. These equations combined with the M.S.T. model are treated by F. Helfferich (55) in a very exhaustive manner in the chapter on ion-exchange resin membranes of his book on ion-exchange resins. Extensive literature references are also given here. 

Previous to giving a quantitative elaboration of the Nemst-Planck equations for the different membrane processes, at first a qualitative treatment of membrane phenomena will be given here on the basis of M.S.T. theory and Donnan equilibrium. 

An expression for the diffusion potential can also be obtained by integration of the Nemst-Planck flux equations. This integration is generally very complicated, so that further approximations must be introduced. 

Neglecting osmotic flow, it is possible to integrate the Nemst-Planck equations including activity coefficients (55, 142) (ref. 55, p. 338), an expression being obtained for the diffusion potential. Adding the Donnan potentials, the result is ... 

In order to be able to distinguish between active and passive transport through biological membranes, P. Meares and H. H. Ussing (95) likewise made a study of the fluxes through a membrane under the influence of diffusion together with an electric current. They studied the influxes and the outfluxes of sodium- and chloride ions at a cation exchange resin membrane. They started from the Nemst-Planck flux equations of the type ... 

Starting from the Nemst-Planck flux equations Helfferich (ref. 55, page 359) obtained for the concentration ratio of the original solution and the solution pressed through, in the case of a cation-exchange membrane ... 

As long as this has not been done, the author (96) prefers, as regards the co-ions, the theoretical flux ratios based on the Nemst-Planck equations. 

Application of the Nemst-Planck Flux Equations Combined with the... 

I.2. Interdiffusion. F. Helfferich and H. D. Ocher (54) studied the interdiffusion of counterions through an ion-exchange membrane. Bases for their calculations were the Nemst-Planck flux equations combined with the M.S.T. model. Ion-fluxes and concentration profiles in the membrane were calculated. 

R. ScHLOGL and U. Schodel 146) have supplied another proof for the usability of the Nemst-Planck flux equations combined with the M.S.T. model, also for the case that an electric current flows through an ion-selective membrane. They determined the concentration profiles of the mobile ions for the case of a cation selective membrane on the basis of phenol sulfonic acid and NaCl solutions, under application of an electric current. 

In the Nemst-Planck equations used the activity coefficients were neglected a term accounting for the electro-osmosis, however, is present. Calculated and measured concentration profiles could be made to inter-correspond by adapting the term for water transport. The values indirectly determined by electro-osmotic flow were now found to agree with those measured directly. 

Equations (1.134) or (1.136) are called the Nemst-Planck equation [4, 13, 50, 52]. In the deduction of this equation it has been assumed that the flux of a species i... 

Buck, R.P. 1984. Kinetics of bulk and interfacial ionic motion Microscopic bases and limits of the Nemst-Planck equation applied to membrane systems. J. Membr. Sci. 17, 1-62. 

In dilute electrolyte systems, the diffusional interactions can usually be neglected, and the generalized Maxwell-Stefan equations are reduced to the Nemst-Planck equations (B3) ... 

An outline of Butler s theory for the terms of the low surface state (transport-controlled) case is given in Section 10.3.5. Uosaki s 1977 theory of kinetics in the high surface state case was developed in greater detail by Khan (1984). Here, the beginning equation for the steady state (dnx/dt = 0) in the space charge region has a flux independent of distance, so that from the Nemst-Planck equation (4.226) and with dJIdx = 0, one obtains ... 

Here, n x) is the carrier concentration at a distance x from the surface, R is the coefficient of reflection, lv is the intensity of illumination at a frequency v and a, is the absorption coefficient for photons, while De is the diffusion coefficient of electrons. This equation can be understood if one recalls Fick s second law, which gives the rate of change in concentration during diffusion in the absence of an electric field (Vol. 1, Section 4.2). The other terms on the left represent the application of the Nemst-Planck equation (Section 4.4). The term on the right represents the rate of absorption of light, taking into account a reflection coefficient, Rv at a frequency V. 

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