Electric double-layer diffuse model

Equation 14.39 is an implicit equation, because / is a function of o, which, in turn, is directly related to the degree of surface coverage, 0. Furthermore, y is influenced by the ionic strength of the medium. Evaluation of /(a) requires a model for the charge distribution in the electrical double layer. The models most currently applied are presented in Section 9.4. The diffuse double layer according to Gony and Chapman gives... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

The model more generally accepted for metal/electrolyte interfaces envisages the electrical double layer as split into two parts the inner layer and the diffuse layer, which can be represented by two capacitances in series.1,3-7,10,15,32 Thus, the total differential capacitance C is equal to... 

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 second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

Various pc electrode models have been tested.827 Using the independent diffuse layer electrode model74,262 the value of E n = -0.88 V (SCE) can be simulated for Cd + Pb alloys with 63% Pb if bulk and surface compositions coincide. However, large deviations of calculated and experimental C,E curves are observed at a 0. Better correspondence between experimental and calculated C,E curves was obtained with the common diffuse-layer electrode model,262 if the Pb percentage in the solid phase is taken as 20%. However, the calculated C, at a Ois noticeably lower than the experimental one. It has been concluded that Pb is the surface-active component in Cd + Pb alloys, but there are noticeable deviations from electrical double-layer models for composite electrodes.827... 

The description of the ion transfer process is closely related to the structure of the electrical double layer at the ITIES [50]. The most widely used approach is the combination of the BV equation and the modified Verwey-Niessen (MVN) model. In the MVN model, the electrical double layer at the ITIES is composed of two diffuse layers and one ion-free or inner layer (Fig. 8). The positions delimiting the inner layer are denoted by X2 and X2, and represent the positions of closest approach of the transferring ion to the ITIES from the organic and aqueous side, respectively. The total Galvani potential drop across the interfacial region, AgCp = cj) — [Pg.545]

At present it is impossible to formulate an exact theory of the structure of the electrical double layer, even in the simple case where no specific adsorption occurs. This is partly because of the lack of experimental data (e.g. on the permittivity in electric fields of up to 109 V m"1) and partly because even the largest computers are incapable of carrying out such a task. The analysis of a system where an electrically charged metal in which the positions of the ions in the lattice are known (the situation is more complicated with liquid metals) is in contact with an electrolyte solution should include the effect of the electrical field on the permittivity of the solvent, its structure and electrolyte ion concentrations in the vicinity of the interface, and, at the same time, the effect of varying ion concentrations on the structure and the permittivity of the solvent. Because of the unsolved difficulties in the solution of this problem, simplifying models must be employed the electrical double layer is divided into three regions that interact only electrostatically, i.e. the electrode itself, the compact layer and the diffuse layer. 

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

As we have seen, the electric state of a surface depends on the spatial distribution of free (electronic or ionic) charges in its neighborhood. The distribution is usually idealized as an electric double layer one layer is envisaged as a fixed charge or surface charge attached to the particle or solid surface while the other is distributed more or less diffusively in the liquid in contact (Gouy-Chapman diffuse model, Fig. 3.2). A balance between electrostatic and thermal forces is attained. 

For present purposes, the electrical double-layer is represented in terms of Stem s model (Figure 5.8) wherein the double-layer is divided into two parts separated by a plane (Stem plane) located at a distance of about one hydrated-ion radius from the surface. The potential changes from xj/o (surface) to x/s8 (Stem potential) in the Stem layer and decays to zero in the diffuse double-layer quantitative treatment of the diffuse double-layer follows the Gouy-Chapman theory(16,17 ... 

For a long time, the electric double layer was compared to a capacitor with two plates, one of which was the charged metal and the other, the ions in the solution. In the absence of specific adsorption, the two plates were viewed as separated only by a layer of solvent. This model was later modified by Stem, who took into account the existence of the diffuse layer. He combined both concepts, postulating that the double layer consists of a rigid part called the inner—or Helmholtz—layer, and a diffuse layer of ions extending from the outer Helmholtz plane into the bulk of the solution. Accordingly, the potential drop between the metal and the bulk consists of two parts ... 

An adsorbed layer of water molecules at the interface separates hydrated ions from the solid surface. The interfacial electric double layer can be represented by a condenser model comprising three distinct layers a diffuse charge layer in the ionic solution, a compact layer of adsorbed water molecules, and a diffuse charge layer in the solid as shown in Fig. 5-8. The interfacial excess charge on the... 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

The variation of the electric potential in the electric double layer with the distance from the charged surface is depicted in Figure 6.2. The potential at the surface ( /o) linearly decreases in the Stem layer to the value of the zeta potential (0- This is the electric potential at the plane of shear between the Stern layer (and that part of the double layer occupied by the molecules of solvent associated with the adsorbed ions) and the diffuse part of the double layer. The zeta potential decays exponentially from to zero with the distance from the plane of shear between the Stern layer and the diffuse part of the double layer. The location of the plane of shear a small distance further out from the surface than the Stem plane renders the zeta potential marginally smaller in magnitude than the potential at the Stem plane ( /5). However, in order to simplify the mathematical models describing the electric double layer, it is customary to assume the identity of (ti/j) and The bulk experimental evidence indicates that errors introduced through this approximation are usually small. 

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]).

In the absence of specific adsorption of anions, the GCSG model regards the electrical double layer as two plate capacitors in series that correspond respectively, to two regions of the electrolyte adjacent to the electrode, (a) An inner compact layer of solvent molecules (one or two layers) and immobile ions attracted by Coulombic forces (Helmholtz inner plane in Fig. 2). Specific adsorption of anions at the electrode surface may occur in this region by electronic orbital coupling with the metal, (b) An outer diffuse region of coulombically attracted ions in thermal motion that complete the countercharge of the electrode. 

Fig. 1.10 Schematic view of the electrical double layer in agreement with the Gouy-Chapman-Stem-Grahame models. The metallic electrode has a negative net charge and the solvated cations define the inner limit of the diffuse later at the Helmholtz outer plane (OHP). There are anions adsorbed at the electrode which are located at the inner Helmholtz plane (IHP). The presence of such anions is stabilized by the corresponding images at the electrode in such a way that each adsorbed ion establishes the presence of a surface dipole at the interface...

With FITEQL numeric procedure Hayes et al. fitted edl parameters to the three models of electric double layer DLM (diffuse layer model), CCM (constant capacity model) and TLM (three layer model) for the following oxides a-FeOOH, AI2O3 and TiC>2 in NaNC>3 solutions [51]. The fitting was performed for surface reaction constants, edl capacity and the densities of the hydroxyl groups on the surface of the oxides. The quality of the fitting was evaluated by the minimization of the function of the sum of the square deviations of the calculated value from the standard error of measured charge. The lower value of the function the better was the fit... 

The simplest, self-consistent model of the diffuse-ion swarm near a planar, charged surface like that of a smectite is modified Gouy-Chapman (MGQ theory [23,24]. The basic tenets of this and other electrical double layer models have been reviewed exhaustively by Carnie and Torrie [25] and Attard [26], who also have made detailed comparisons of model results with those of direct Monte Carlo simulations based in statistical mechanics. The postulates of MGC theory will only be summarized in the present chapter [23] ... 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

In addition, with the development of electrodes with dimensions approaching the nanometre regime, the sizes of the electrical double layer and diffusion layer will become comparable. Whereas the simulation techniques described in Section 7 have assumed no coupling of these layers, this will no longer be appropriate in the case of nanoelectrodes, and the resulting development of more sophisticated theories and resultant models will be required. 

Cantwell and co-workers submitted the second genuine electrostatic model the theory is reviewed in Reference 29 and described as a surface adsorption, diffuse layer ion exchange double layer model. The description of the electrical double layer adopted the Stem-Gouy-Chapman (SGC) version of the theory [30]. The role of the diffuse part of the double layer in enhancing retention was emphasized by assigning a stoichiometric constant for the exchange of the solute ion between the bulk of the mobile phase and the diffuse layer. However, the impact of the diffuse layer on organic ion retention was danonstrated to be residual [19],... 

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).

Several SCM s have been described in the literature. The more commonly used models include the Constant Capacitance Model (Schindler and Stumm, 1987), the Diffuse Double Layer Model (Stumm et al., 1970) and the Triple Layer Model (Davis et al., 1978 Yates et al, 1974). All are based on electric double layer theory but differ in their geometric description of the oxide-water interface and the treatment of the electrostatic interactions. 

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