Electroneutrality condition interface

The difference between Equations (55) and (60) may be qualitatively understood by comparing the results with the Donnan equilibrium discussed in Chapter 3. The amphipathic ions may be regarded as restrained at the interface by a hypothetical membrane, which is of course permeable to simple ions. Both the Donnan equilibrium (Equation (3.85)) and the electroneutrality condition (Equation (3.87)) may be combined to give the distribution of simple ions between the bulk and surface regions. As we saw in Chapter 3 (e.g., see Table 3.2), the restrained species behaves more and more as if it was uncharged as the concentration of the simple electrolyte is increased. In Chapter 11 we examine the distribution of ions near a charged surface from a statistical rather than a phenomenological point of view. 

For other models of flow of electrolytes through porous media the reader is referred to [2], [5], [6]. To take into account FCD (fixed charge density) one has to impose additional condition on the interface T (w) and the electroneutrality condition. A challenging problem is to use homogenisation methods for the case of finitely deformable skeleton, even hyperelastic. The permeability would then necessarily depend on strains. Such a dependence (nonlinear) is important even for small strain, cf. [7]. It is also important to include ion channels [8]. 

Distribution (Nernst) potential — Multi-ion partition equilibria at the -> interface between two immiscible electrolyte solutions give rise to a -> Galvanipotential difference, Af(j> = (j>w- 0°, where 0wand cj>°are the -> inner potentials of phases w and o. This potential difference is called the distribution potential [i]. The theory was developed for the system of N ionic species i (i = 1,2..N) in each phase on the basis of the -> Nernst equation, the -> electroneutrality condition, and the mass-conservation law [ii]. At equilibrium, the equality of the - electrochemical potentials of the ions in the adjacent phases yields the Nernst equation for the ion-transfer potential,... 

When ions specifically adsorb at the interface, the excess surface charge density is divided into three parts, the charges in the two diffuse parts of the double layer, q and q, and the charge due to the specifically adsorbed ions, q° [25]. The electroneutrality condition of the entire interfacial region is... 

The influence of ion size asymmetry on the properties of IL-vapor interfaces was investigated using soft primitive model MD simulations [106], The ion size asymmetry resulted in charge separation at the liquid-vapor interface and therefore in a local violation of the electroneutrality condition [106],... 

Figure 14. (A) Diagram of the charge distribution in the triple layer model. (B) Flat capacitors connected in series as equivalent of a triple layer model at the aqueous solution/metal oxide interface. Charge distribution on capacitor plates is obtained from the electroneutrality condition written in the form 6g = (— o) + ( d).

The TPM is generally designated for the chemical stimulation and can provide local chemical as well as mechanical unknowns. In most cases, the gel phase only is investigated by prescribing the concentrations at the boundary of the gel (at the gel-solution interface) by using the Donnan Equation (7) together with the electroneutrality condition of (6). 

The driving force in this case is the difference between the concentration of the carrier-extracted species complex at the membrane/feed solution interface and the practically zero concentration of the complex at the membrane/receiver solution interface. In the case of extracted ionic species, the driving force for the uphill transport can be the potential gradient generated by the coupled transport of another ionic species across the membrane. The extracted ionic species, in this case, is transported to satisfy the electroneutrality condition within the membrane system. This coupled transport process can be countertransported (Figure 27.5b,c) and cotransporled (Figure 27.5a,d) with respect to the extracted ionic species. 

This contact theorem, as well as other sum rules that are valid for the charged interface will be discussed in the next section. The density profiles obtained from the Gouy-Chapman theory are monotonous, that is they show no oscillations. Since in this theory the contact theorem and the electroneutrality condition are satisfied, then, p,-(2) is pinned at the origin, and has a fixed integral, so that the density profile cannot deviate too much from the correct result. When the contact theorem is not... 

Integrate the extended Nernst-Planck equation for any counterion i in the membrane in the z-direction perpendicular to the membrane surface when the membrane excludes co-ions perfectly. Determine the constant of integration in terms of counterion concentration and resin-phase potentitd just inside the membrane, CiR (0-t-) and r (0-I-). Use now the condition of thermodynamic equilibrium for the counterion at the interface (between C,t)(0-I-) and C, , via relation (3.3.118b)) and the assumption that , = 0 to relate Caj(z) to Ciw Obtain r z] by using the electroneutrality condition (3.3.30b). 

The overlap between CDL and EDL and its effects on the voltanunetric responses of nanoscale electrodes were first theoretically treated by White and coworkers in the early 1990s using the Poisson-Nemst-Planck equations. The possible failure of the electroneutrality condition at the nanoscale electrode interface has also been studied by Oldham and Bond, Dickinson and Compton, and other researchers. Most of these studies, however, may have overestimated the EDL effect due to the inappropriate treatment of the compact part of the EDL, especially the dielectric screening of solvent in EDL (will be discussed later on). To rationally treat the EDL effect at... 

Fig. 2. HNC ionic profiles at the air-water interface within the PM description. 1-M NaNOs salt of Fig. 1. Solid lines with air-ion dispersion potentials dotted lines without. Note that, according to the electroneutrality condition, the excess adsorbed quantity is identical for cation and anion. (From Ref 7.)...

Earlier, Gavach et al. studied the superselectivity of Nafion 125 sulfonate membranes in contact with aqueous NaCl solutions using the methods of zero-current membrane potential, electrolyte desorption kinetics into pure water, co-ion and counterion selfdiffusion fluxes, co-ion fluxes under a constant current, and membrane electrical conductance. Superselectivity refers to a condition where anion transport is very small relative to cation transport. The exclusion of the anions in these systems is much greater than that as predicted by simple Donnan equilibrium theory that involves the equality of chemical potentials of cations and anions across the membrane—electrolyte interface as well as the principle of electroneutrality. The results showed the importance of membrane swelling there is a loss of superselectivity, in that there is a decrease in the counterion/co-ion mobility, with greater swelling. 

The condition of electroneutrality of the interface as a whole says that... 

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

Since in the steady state, it is necessary to maintain a condition of electroneutrality in any macroscopic part of the system, the total charge flux through all cross-sections of the circuit must be the same. In particular, the rate of electron flow in the external circuit is equal to the rate of charge transfer at each electrode/electrolyte interface. 

Rule 2 The second rule is the condition of electroneutrality. It means that in an electrochemical cell, the sum of positive charges must equal the sum of negative charges. Thus, separation of positive and negative charges occurs at every interface, but their sum is always zero. 

NB to W in order to maintain the electroneutrality in both phases. In other words, the positive current due to the electron transfer should be equal to the negative current due to the cation transfer in their absolute values, since the currents correspond to the rates of electron and ion transfers. Applying this condition to the present case, the potential difference established by the coupling of transfers of Na+ and electron at the W/NB interface is expected to be a in Figure 6.4, where the magnitude of the current (indicated as A) in polarogram 2 for the electron transfer is equivalent to that in polarogram 3 for the transfer of Na+. 

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