Membranes Nernst-Planck model

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... 

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. 

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... 

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. 

Simulation Results. A onc-dimensional simulation model based on the Nernst-Planck and Poisson equations [14, in which all the acid-base reactions occurring in the membrane are taken into account, has been used to give a qualitative description of the pH step titration process. In these simulations, a pH step is applied outside a 2 mm thick stagnant layer, which is assumed to be present in front of an 8 mm thick membrane. Diffusion coefficients in the membrane are assumed to be 4/10 of those in water (this value is based on experience with ion step experiments). Lysozyme, used as a model protein, is assumed to contain 11 carboxylic groups (pKa = 4.4), 2 imidazole groups (pKa = 6.0), and 9 amino groups (pKa = 10.4) per molecule. Concern... 

Donnan dialysis The BAHLM systems with ion-exchange membranes, based on Donnan dialysis [18,19], will be considered below. Donnan dialysis is a continuously operating ion-exchange process. There are many theoretical models describing transport mechanisms and kinetics of DL) [18-26]. All transport kinetics models are based on Fick s or Nernst-Planck s equations for ion fluxes. In both cases, the authors introduce many assumptions and simplifications. 

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. 

The Nernst-Planck approach to ion conduction, which is the model most frequently used, treats the membrane as a homogeneous, macroscopic dielectric of specified thickness. The ions have infinite mobility in the aqueous phase, activity coefficients are unities, there are no bound charges and the ion concentrations at large distances from the membrane are constant. For... 

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 ... 

Last but not least, the fundamental understanding of the permeation mechanisms within perovsldte and more extensively MIEC membranes is still in its infancy. The most extended models are based on the Nernst-Planck equations (e.g., the Wagner equation) providing a macroscopic view of the permeation process itself. These models usually cannot afford the description of heterogeneous materials including impurities and occluded bubbles, as is the case for most real perovskite layers. To this aim, the development of meso- or microscale models with a proper description of diffusion effects and vacancy generation would be desirable. 

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... 

Figure 1 depicts the electric displacement eE [where E is the field and e the dielectric constant] across the vesicle membrane of thickness 6 on the assumption that the media of both sides have equal ionic strengths. The resting potential difference across the membrane, is of the liquid-junction type but is usually close to a Nernst potential. The Debye lengths in both the ionic media are X. Inside the membrane E = / the constant Planck field. Outside the membrane, the field is assumed to decline in accord with a linear Goiiy-Chapman model, E = Epe, so that Fy = + 2A). If we define... 

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