The Stem Model

The interaction of an electrolyte with an adsorbent may take one of several forms. Several of these are discussed, albeit briefly, in what follows. The electrolyte may be adsorbed in toto, in which case the situation is similar to that for molecular adsorption. It is more often true, however, that ions of one sign are held more strongly, with those of the opposite sign forming a diffuse or secondary layer. The surface may be polar, with a potential l/, so that primary adsorption can be treated in terms of the Stem model (Section V-3), or the adsorption of interest may involve exchange of ions in the diffuse layer. 

In the Stem model, the surface charge is balanced by the charge in solution, which is distributed between the Stem layer at a distance d from the surface and a diffuse layer having an ionic Boltzman-type distribution. The total charge a is therefore due to the charge in the two layers ... 

Some Ions Stuck to the Electrode, Others Scattered in Thermal Disarray The Stem Model... 

When, however, charges are separated, potential drops result. The Stem model implies, therefore, two potential drops, i.e.,... 

Fig. 6.66. The Stem model, (a) A layer of ions stuck to the electrode and the remainder scattered in cloud fashion, (b) The potential variation according to this model, (c) The corresponding total differential capacity C is given by the Helmholtz and Gouy capacities in series.

After all this analysis, can we say that the Stem model is consistent with experimental results In other words, is the Stem model able to reproduce the differential capacity curves Under certain conditions, it is. So, to some extent, the Stem model was successful. However, what are the restrictions the model imposes Recall that in the Helmholtz-Perrin model the ions lay close to the electrode on the OHP. The condition for the Stem model to succeed is that ions not be in close proximity to the electrode they are not to be adsorbed. Thus the model proved to be valid only for electrolytes such as NaF (Graliame, 1947).45 Both of these ions, Na+ and F, are known to have a hydration layer strongly attached to them in such a way that even in the proximity of the electrode they are almost not interacting with the electrode surface. The Stem model works well representing noninteracting ions. 

Nevertheless, when other ions such as Cl", Br, F, or PO are in solution, they may come in close contact with the electrode and strongly interact chemically with it. They may specifically adsorb46 on the electrode. The Stem model is not applicable in... 

The contribution of the metal to the double layer was discussed in Sections 6.6.7 to 6.6.9. However, we have said little about the ions in solution adsorbed on the electrode and how they affect the properties of the double layer. For example, when presenting the Stem model of the double layer (Section 6.6.6), we talked about ions sticking to the electrode. How does an interface look with ions stuck on the metal What is the distance of closest approach Are hydrated ions held on a hydrated electrode i.e., is an electrode covered with a sheet of water molecules Or are ions stripped of their solvent sheaths and in intimate contact with a bare electrode What are the forces that influence the sticking of ions to electrodes ... 

Figure 3.31. Diagram of the Stem model (from Van Olphen, 1977, with permission).

The data in Figure 3.32 show the theoretical estimated concentration of Na+, Li+, and K+ or Rb+ between the Stem layer and the diffuse layer based on the Stem model. The distribution appears to be consistent, as expected, with the hydration energy of the cation. The greater the heat of hydration (see Chapter 4, Table 4.1) is, the greater the concentration of the cation in the diffuse layer in relationship to the Stern layer. The Stern model has been the basis for many variations of the model recently also known as the surface complexation model (Goldberg, 1992). 

On what basis does the Stem model distinguish the potential location of ions near a surface under the influence of an electrical potential ... 

The surface after adsorption will be chained with a potential, as in Figure 9.14, so that primary adsorption can be treated in terms of a capacitor model called the Stem model [43]. The other type of adsorption that can occur involves an exchange of ions in the diffuse layer with those of the surface. In the case of ion exchange, the primary ions are chemically bound to the structure of the solid and exchanged between ions in the diffuse double layer. 

Two best-fit Ns values in Table 5.29 are significantly higher than the highest Ns reported in Tables 5.1 and 5.3, and their physical sense is questionable. The log K (=SiOPb ) calculated for the best-fit Stern model (Table 5.29) are lower by about 0.2 than the corresponding log K for the diffuse layer model (Table 5.28), but higher by 2-3 than the log A)(=SiOPb ) calculated for the best-fit TLM (Table 5.27). The calculated uptake curves presented in Fig. 5.132 (diffuse layer model) are practically identical as corresponding curves calculated for the Stem model (Kosmulski. model... 

A. 19.9 The Stem model considers the finite size of ions and the interactions of the ions with the surface, which naturally leads to the Stern layer a layer of ions closely associated with the electrode surface similar to as found... 

The purely electrostatic diffuse layer model often underestimates the affinity of the counterions to the surface. In the Stem model, the surface charge is partially balanced by chemisorbed counterions (the Stem layer), and the rest of the surface charge is balanced by a diffuse layer. In the Stern model, the interface is modeled as two capacitors in series. One capacitor has a constant capacitance (independent of pH and ionic strength), which represents the affinity of the surface to chemisorbed counterions, and which is an adjustable parameter the relationship between a, and Vd in the other capacitor (the diffuse layer) is expressed by Equation 2.18. A version of the Stern model with two different values of C (below and above pHg) has also been used. The capacitance of the Stem layer reflects the size of the hydrated counterion and varies from one salt to another. The correlation between cation size and Stern layer thickness was studied for a silica-alkali chloride system in [733]. Ion specificity of adsorption on titania was discussed in terms of differential capacity as a function of pH in [545]. The Stern model with the shear plane set at the end of the diffuse layer overestimated the absolute values of the potential of titania [734]. A better fit was obtained with the location of the shear plane as an additional adjustable parameter (fitted separately for each ionic strength). Chemisorption of counterions can also be quantified within the chemical model in terms of expressions similar to the mass law (Section 2.9.3.3). 

The existing double electrical layer can be described by the Stem model... 

Stem improved the Gouy-Chapman theory of the DDL by assuming that some ions are tightly retained immediately next to colloid surfaces in a layer of specifically adsorbed or Stem- layer cations. The double layer is diffuse beyond this layer. A satisfactory approximation of the Stem model can be made by assuming that the specifically adsorbed ions quantitatively reduce the surface density of the colloid. The diffuse portion of the double layer then is assumed to develop on a colloid surface of correspondingly reduced charge density. Sample Stem-modification calculations for a series of monovalent cations are shown in Fig. 8.10, Relatively few of the... 

Figure 26. Schematics of the electrical double layer at a solid-liquid interface, (a) the Helmholtz model, (b) the Gouy-Chapman model, and (c) the Stem model.

The method developed here for the description of chemical equilibria including adsorption on charged surfaces was applied to interpret phosphate adsorption on iron oxide (9), and to study electrical double-layer properties in simple electrolytes (6), and adsorption of metal ions on iron oxide (10). The mathematical formulation was combined with a procedure for determining constants from experimental data in a comparison of four different models for the surface/solution interface a constant-capacitance double-layer model, a diffuse double-layer model, the triplelayer model described here, and the Stem model (11). The reader is referred to the Literature Cited for an elaboration on the applications. 

The first unequivocal demonstration of this important, well-known constraint appears in O. Stern, Zur Theorie der elektrolytischen Doppelschicht, Z. Elektrochem. 30 508 (1924). The Stem model of the interfadal region was the first chemical model in the spirit of the present chapter. 

The physics of ILs at surfaces are important for a deeper understanding of the resulting properties and enables the design of appHcations. Each combination of cation and anion can lead to a different behavior on surfaces of sohds, because the molecular structure of each IL has a strong influence of the formation of layers at the interfaces. In aqueous electrolytes the Hehnholtz-model and its further developments are describing the physics in a sufficient way The Gouy-Chapman-model takes the diffusion into account, and the Stem-model combines the formation of a double layer with diffusion. Compared to aqueous solutions of salts, the situation in ILs is different The ions have no solvent environment Their next neighbors are also ions. As a consequence the physics at the interfaces between sohds and ILs cannot be described by the common models. 

FIGURE 4.4 Schematic representation of the Stem model of the stmcture of the double layer at the metal-electrolyte interface showing the ions and water moleeules. The inner and outer Helmholtz planes are labeled, along with the dilluse double layer. In the figure, the metal has been positively polarized. 

In Table 16.2, values of ccc s are collected for different aqueous colloidal dispersions. Within a series of counterions having the same valency, a small but systematic influence of the type of ion is observed. This nonelectrostatic and, therefore, specific influence seems to be related to the size of the hydrated ion. The larger the ions, the less they are hydrated and the smaller their hydrated size. These ions can approach the particle surface more closely. In view of the Stem model, presented in Section 9.4.3, a larger value of /j and, hence, a higher ccc is expected. The observation of... 

Two topical issues may be mentioned. The first is the definition of the potentials that are measured by different techniques, say by AFM, electrokinetically and externally imposed, and their relationships [11], The second is of a more theoretical nature and concerns modeling of the nondiffuse part of the double layer. The classical approach is through Stem theory [2], which in most cases is adequate, although it requires two additional parameters. A more recent development is in terms of ion correlations, essentially an advanced statistical theory whereby all coulombic ion-ion and ion-surface interaction pairs are counted and statistically summed [2]. This is a step forward over the smeared-out models of Gouy and Stern. The issue here is that cases must be found where deviations from Gouy theory cannot be interpreted on the basis of the Stem model... 

Model c is the Stem model, which combines an inner layer with a diffuse layer. It can cover a range of ionic strength. It allows for electrolyte specificity by the ion-pair formation between the ions of the electrolyte and the oppositely charged surface functional groups. Thus, it is the simplest model that is able (1) to cover a broad range of ionic strength and (2) to describe electrolyte-specific behavior. 

Model d is the popular triple-layer model, which has an additional layer compared to the Stem model. As the Stem model, it is electrolyte ion-specific. The additional layer lowers the potential at the head end of the diffuse layer and is intended to allow direct comparison between the diffuse-layer potential and measured zeta potentials. 

Compared to the constant-capacitance model, the Stem model allows variation of ionic strength via (1) the consideration of surface ion pairs with the electrolyte ions and (2) the difluse layer. The ion pairs permit a variation of the electrolyte in terms of composition (i.e., the electrolyte binding constant can be adjusted for each ion). The difluse layer is not ion-specific and, in principle, restrictions apply which have been pointed out for model c. Compared to a purely diffuse-layer model, violation of these restrictions will have less severe consequences for the surface-charge density because the Stem layer usually decreases the difluse-layer potential compared to the purely diffuse-layer model. There is one adjustable parameter pertaining to the electrostatic model (the capacitance). For the 1-pK formalism, in addition to the site density parameter, two electrolyte-binding constants are required. 

Model e adds a supplementary interfacial layer compared to the Stem model (model d). This supplementary layer merely has the piupose to sufficiently decrease the diffuse-layer potential and to have closer agreement with measmed zeta potentials. One additional adjustable parameter is introduced (the capacitance C2). Based on Eq. (15), the typically used value of C2 = 0.2 F/m will control the overall capacitance of the compact part to the electric double layer. Contrary to model d, model e uses a Gouy-Chapman approach rather than the HNC approximation to account for the diffuse layer, but this can, of course, be varied. Otherwise, the discussion of model d also applies to model e. 

The triple layer model does not significantly improve the description of acid-base properties of minerals when compared to a Stem model. The Stem model has fewer adjustable parameters (acid-base properties). For electrokinetic data, a detailed analysis should be conducted. 

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