Finite-ion-size model

Debye-Hiickel Theory Finite-Ion-Size Model. If the approximation of the point... 

The Debye-Huckel Theory The Finite-Ion-Size Model. If the approximation of the point charge is removed, the extended form of the Debye-Huckel law is obtained ... 

This individual ionic-activity coefficient can be transformed into a mean ionie-activity coefficient by the same procedure as for the Debye-Hiickel limiting law (see Section 3.4.4). On going through the algebra, one finds that the expression for log/ in the finite-ion-size model is... 

If one compares Eq. (3.II9) of the fmite-ion-size model with Eq. (3.90) of the point-charge approximation, it is clear that the only difference between the two expressions is that the former contains a term 1/(1 + ku) in the denominator. Now, one of the tests of a more general version of a theory is the correspondence principle, i.e., the general version of a theory must reduce to the approximate version under the conditions of applicability of the latter. Does Eq. (3.119) from the finite-ion-size model reduce to Eq. (3.90) from the point-charge model ... 

Thus, when the solution is sufficiently dilute to make a k, i.e., to make the ion size insignificant in comparison with the radius of the ion atmosphere,the finite-ion-size model Eq (3.119) r uces to the corresponding Eq. (3.90) of the point-charge... 

Comparison of the Finite-Ion-Size Model with Experiment... 

The finite-ion-size model yielded agreement with experiment at concentrations up to 0.1 N. It also introduced through the value of a, the ion size parameter, a specificity to the eiectrolyte (making NaCi different from KCl), whereas the point-charge model yielded activity coefficients that depended only upon the valence type of electrolyte. Thus, while the limiting law sees only the charges on the ions, it is biind... 

Finally, the third aspect is the diffuse layer. This has traditionally been treated by the Gouy-Chapman theory. Recently, attempts have been made to develop an improved model of the diffuse layer which is analytical and considers the effects of finite ion size and change in dielectric properties. These aspects of the double layer are now considered in more detail. 

The GC results are compared in fig. 10.18 with Monte Carlo calculations of Boda et al. [32]. These were carried out assuming that the electrolyte ions are hard spheres with a diameter of 300 pm in a dielectric continuum. The estimates of < ) using the Monte Carlo technique fall below the GC estimates. They demonstrate the importance of including finite ion size in a model of the diffuse layer. 

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

Many attempts have been made to improve the classical Poisson-Boltzmann equation to include discrete charge effects, finite ion size, and so on (see, e.g., Refs. 35-37). At present some fundamental progress is being made on the basis of certain models in modem fluid-state theory, in which the hard-core repulsions of the ions are incorporated in a consistent way. " The Poisson-Boltzmann equation was found to be a limiting case of the hypernetted chain approximation at low densities. Also a computer simulation was reported." ... 

In the future, DESMO should be tested with finite ion size and compared to numerical solution of the LPBE using a cavity surface (defined by the van der Waals radii Ri) that does not coincide with the ion exclusion surface (defined by Rj + Rion)- Finite ion size has incorporated into Generalized Born models, however, via the ion exclusion factors in Eq. (11.31) [46], These models are discussed in the next section. 

Figure 12. Salt effect on the relative R 0H/R 0 quantum yield in two different solvents Water (left panel) and a 50/50% (by volume) mixture of met Hanoi/water [11c]. Circles are experimental data obtained from the relative height ratio of the two peaks in the steady-state fluorescence spectrum e.g., Figure 10. Dashed and full curves correspond to the Debye-Hiickel expression (with finite ion-size correction) [21] and the Naive Approximation [17, 11c], respectively. Both models employ the zero-salt kinetic parameters.

Franceschetti model involves relatively general boundary conditions at the electrodes and so includes the possibility of charge transfer reactions and specific adsorption. Because of its generality, however, the model prediction for Z,((o) is very complicated and, in general, cannot be well represented by even a complicated equivalent circuit. The Z,(m) expression, may, however, be used directly in CNLS fitting. Here, for simplicity, we shall consider only those specific situations where an approximate equivalent circuit is applicable. Idealizations involved in the model include the usual assumption of diffusion coefficients independent of field and position, the use of the simplified Chang-Jaff6 [1952] boundary conditions, and the omission of all inner layer and finite-ion-size effects. Some rectification of the latter two idealizations will be discussed later. 

Motivated by the need to consider finite size of ions in the PB model, Chu et al. employed a lattice gas model for the ionic system, where ions of finite sizes are placed on the grid cells. In this way, the ion size can be conveniently represented by the cell size and the system can be treated with the lattice gas approximation (Chu et al., 2007). The modified PB theory... 

However, the theory neglects the finite size of the ions, and it was Stern who postulated that ions could not approach the electrode beyond a plane of closest approach, thereby introducing in a crude way the ion size (Fig. 5.1c). Although formulated in a complex manner [1], the basis of Stern s model is a combination of the Helmholtz and Gouy-Chapman approaches. It may be noted that Fig. 5.1 also shows the potential and charge distributions resulting from the models. These will be discussed later. 

Since the discovery of the parton substructure of nucleons and its interpretation within the constituent quark model, much effort has been spent to explain the properties of these particles and the structure of high density phases of matter is under current experimental investigation in heavy-ion collisions [17]. While the diagnostics of a phase transition in experiments with heavy-ion beams faces the problems of strong non-equilibrium and finite size, the dense matter in a compact star forms a macroscopic system in thermal and chemical equilibrium for which effects signalling a phase transition shall be most pronounced [8],... 

It is noted that the finite size of ions may be taken into account using a modified Gouy-Chapman model of the diffuse double-layer. The finite size of ions limits the maximum concentrations of ions close to the particle surface. 

A lattice model for an electrolyte solution is proposed, which assumes that the hydrated ion occupies ti (i = 1, 2) sites on a water lattice. A lattice site is available to an ion i only if it is free (it is occupied by a water molecule, which does not hydrate an ion) and has also at least (i, - 1) first-neighbors free. The model accounts for the correlations between the probabilities of occupancy of adjacent sites and is used to calculate the excluded volume (lattice site exclusion) effect on the double layer interactions. It is shown that at high surface potentials the thickness of the double layer generated near a charged surface is increased, when compared to that predicted by the Poisson-Boltzmann treatment. However, at low surface potentials, the diffuse double layer can be slightly compressed, if the hydrated co-ions are larger than the hydrated counterions. The finite sizes of the ions can lead to either an increase or even a small decrease of the double layer repulsion. The effect can be strongly dependent on the hydration numbers of the two species of ions. 

The limitations of this simplified model have been immediately recognized, and the first criticism [3] even preceded the full development of the DLVO theory. Since then many improvements of the theory have been proposed, to account for the finite size of the ions [4], image forces [5], dielectric corrections [6], ion correlations [7], ion-dispersion [8] and ion-hydration forces [9], to name only a few. Despite the many corrections brought to the traditional DLVO theory, there are some experiments, such as those regarding the stability of neutral lipid multilayers, which could still not be explained within this framework. It is therefore commonly accepted that an additional repulsion occurs when two surfaces approach each other at a distance shorter than a few nanometers. Because this force was initially related to the structuring of water near surfaces, it is commonly named hydration force [10]. 

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