Models for the Electrical Double Layer

Figure 10. Theoretical model for the electrical double layer at an electrode with a polycrystalline surface, (a) Model of independent diffuse layers [Eq. (S3)], and (b) model of common diffuse layer [Eq. (54)).

In the electrochemical literature one finds the Gouy-Chapman (GC) and Gouy-Chapman-Stern (GCS) approaches as standard models for the electric double layer [9,10]. 

Surface complexation models for the oxide-electrolyte interface are reviewed two models for surface hydrolysis reactions are considered (diprotic surface groups and monoprotic surface groups) and four models for the electric double layer (Helmholtz,... 

To complement the models for the surface reactions, a model for the electric double layer is needed. Current models for the electric double layer are based on the work of Stern (21), who viewed the interface as a series of planes or layers, into which species were adsorbed by chemical and electrical forces. A detailed discussion of the application of these models to oxide surfaces is given by Westall and Hohl (2). 

Standing of surface charge and surface potential, and their relationship to pH, ionic strength, and medium composition. Westall and Hohl (1980) provide an excellent review of alternative models for the electrical double layer, and James and Parks (1982) provide a detailed description zind explanation of the surface chemistry and electrostatics of hydrous metal oxides. Hayes et al. (1988) present an excellent discussion of the effect of the electrical double layer on the adsorption of inorganic anions. A similar approach is used by Zachara et al. (1990) to model the adsorption of aminonaphthalene and quinoline onto amorphous silica. 

Varying Polarization in Water (A Model for the Electric Double Layer and the Hydration Force). 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

Figure 14. Models for the electrical double layer at a metal surface...

W. Schmickler and D. Henderson, /. Ghent. Phys., 80,3381 (1984). The Interphase Between Jellium and a Hard Sphere Electrolyte. A Model for the Electric Double Layer. 

To evaluate the integrals in Equations 9.17 and 9.21 the relation between rpo and Oo should be known. This relation may be obtained from Oo(Cedi) represented by the titration curves (see Figure 9.2), provided that Nemst s law, Equation 9.14, applies. If not, a model for the electrical double layer is required to derive rpo (oo ) ... 

MODELS FOR THE ELECTRICAL DOUBLE LAYER 9.4.1 The Molecular Condenser... 

FIGURE 9.10 Applicability of different models for the electrical double layer in various natural and biological environments. 

In order attain measurable SHG signals, pulsed femtosecond lasers with large intensities are usually employed (Yan et al. 1998 Schneider et al. 2007). It was possible to show that the SHG scales with surface potential and the independently measured zeta-potentials can be reproduced by adopting appropriate models for the electric double layer (Yan et al. 1998). More generally, SHG is directly related to the surface excess of adsorbate as shown for malachite green on polystyrene (Eckenrode et al. 2005). This technique offers the opportunily for online and in situ characterisation of colloidal suspensions with particle sizes considerably larger than 5 nm (Schneider and Peukert 2007 Schiirer and Peukeit 2010). 

Figure 4.5 The Stern model for the electric double layer showing compression from (a) low to (b) high ionic strength...

Figure 3.2 Models for the electric double layer around a charged colloid particle (a) diffuse double layer model, (b) Stern model...

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

Gouy-Chapman, Stern, and triple layer). Methods which have been used for determining thermodynamic constants from experimental data for surface hydrolysis reactions are examined critically. One method of linear extrapolation of the logarithm of the activity quotient to zero surface charge is shown to bias the values which are obtained for the intrinsic acidity constants of the diprotic surface groups. The advantages of a simple model based on monoprotic surface groups and a Stern model of the electric double layer are discussed. The model is physically plausible, and mathematically consistent with adsorption and surface potential data. 

Diprotic Surface Groups. Most of the recent research on surface hydrolysis reactions has been interpreted in terms of the diprotic surface hydrolysis model with either the triple layer model or the constant capacitance model of the electric double layer. The example presented here is cast in terms of the constant capacitance model, but the conclusions which are drawn apply for the triple layer model as well. 

The representation of the data for TiC in terms of the monoprotic surface group model of the oxide surface and the basic Stern model of the electric double layer is shown in Figure 5. It is seen that there is good agreement between the model and the adsorption data furthermore, the computed potential Vq (not shown in the figure) is almost Nernstian, as is observed experimentally. 

Figure 7.4. Schematic model of the Electrical Double Layer (EDL) at the metal oxide-aqueous solution interface showing elements of the Gouy-Chapman-Stern-Grahame model, including specifically adsorbed cations and non-specifically adsorbed solvated anions. The zero-plane is defined by the location of surface sites, which may be protonated or deprotonated. The inner Helmholtz plane, or [i-planc, is defined by the centers of specifically adsorbed anions and cations. The outer Helmholtz plane, d-plane, or Stern plane corresponds to the beginning of the diffuse layer of counter-ions and co-ions. Cation size has been exaggerated. Estimates of the dielectric constant of water, e, are indicated for the first and second water layers nearest the interface and for bulk water (modified after [6]).

Fig. 1.11 Behavior of Cd as a function of the electrolyte concentration for the Stem s model of the electrical double layer, calculated from Eq. (1.81) for a 1 1 electrolyte.

From the discussion so far it can be appreciated that the Stern model of the electric double layer presents only a rough picture of what is undoubtedly a most complex situation. Nevertheless, it provides a good basis for interpretating, at least semiquantitatively, most experimental observations connected with electric double layer phenomena. In particular, it helps to account for the magnitude of... 

Much work on these composite systems has to be done, For example, we have not considered yet the study of phenomena occurring at the surface of an electrified metal the version of PCM for ionic solutions [10] has been available for a longtime, but the modeling of the electric double layer has not been done yet (every new modeling requires considerable intellectual and computational efforts). 

Therefore, even in the absence of surface dipoles, the polarization model for the hydration/double layer predicts qualitatively different results from those of the traditional theory at moderate and high ionic strengths. At low electrolyte concentrations, the quantitative differences between the two models, far away from the surface, can be accounted for by suitable modifications of the surface charge. The shape of the electric field and polarization within a few Angstroms from the surface, predicted by the two models, are however different at all electrolyte concentrations. 

The composition of this chapter is based on a well-known and well-understood model of the electrical double layer and therefore does not pretend to enhance overall understanding. It does, however, aim to answer the question of whether a useful mathematical technique exists that may allow for a numerical, if not analytical, description of the double layer for a surface of arbitrary shape and topography. It is fair to say that the colloid scientist ultimately seeks a quantitative description of the electrical double layer for whatever reason. The task then now faced is to uncover the most appropriate theoretical method of calculating the electrical double layer properties for a given nonideal situation. Here we suggest a few methods that may help in this respect. 

The substantial parameter at the modeling of the electric double layer at metal oxide-electrolyte solution interface is a number of the hydroxyl group per surface unit of the oxide. For the titanium dioxide, although different crystalline faces form the surface [rutile 60% of the surface is formed by the face (110) whereas for anatase by (001)] the same density 12.8 of —OH group/nm2 is assumed [28]. That results from the very similar intersection of the elementary cells of the mentioned face, which have the highest density of the atoms in both oxides. 

Finally we shall argue that present-day theories of the nonprimitive models of the electric double layer have considerable difficulty in treating properly ion adsorption in the Stern inner region at metal-aqueous electrolyte interfaces and we suggest that this region is a useful concept which should not be dismissed as unphysical. Indeed Stern-like inner region models continue to be used in colloid and electrochemical science, for example in theories of electrokinetics and aqueous-non-metallic (e.g., oxide) interfaces. 

Figure 10. Electrical double layer models. Top right (a) typical type of potential vs. composition plot for a charged surface compared to (b) constant capacitance model. Top left Two double-layer models, (a) diffuse double layer, (b) part parallel plate capacitor and part diffuse layer.. Bottom left Stem layer model. Incorporation of adsorbed ions to surface. From Hiemenz and Rajagopalan (1997) Bottom right Comparison of Gouy-Chapman and Stem-Grahame models of the electrical double layer. From Davis and Kent (1990).

We propose a model of the electric double layer for ionized mono-layers at A—W or O-W interfaces, which is a compromise between strong... 

Fig. 12. Cyclic voltammogram and model of the electrical double layer at a silver electrode surface. Arrows indicate the direct-ions of molecular dipoles in the water (smallest circles) and pyridine (largest circles, Py) molecules, the arrow head being the positive end. The cations (solvated) could he Na+ or K+, the anions (unsolvated) Cl or SOJ-. IHP and OHP designate the inner and outer Helmholtz planes, respectively, and PZC is the potential of zero charge (see text for further explanations). (Reproduced with permission from ref. 14.)...

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