Gouy-Chapman-Stem model

The charging curves of goethite can for instance be described very well with this approach, in combination with a Stem-Gouy-Chapman model as has been shown by Venema et al.(35) and the only fitting parameter in this case is the Stem layer... 

What are the implications of Eq. (6.132) The Stem synthesis of the two models implies a synthesis of the potential-distance relations characteristic of these two models [Fig. 6.66(b)] a Z/ncar variation in the region from. v = 0 to the position of the OHP according to the Helmholtz-Perrin model (see Section 6.6.2), and an exponential potential drop in the region from OHP to the bulk of solution according to the Gouy-Chapman model (see Section 6.6.4), as shown in Fig. 6.67. 

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],... 

It is evident now why the Helmholtz and Gouy-Chapman models were retained. While each alone fails completely when compared with experiment, a simple combination of the two yields good agreement. There is room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stem brings theory and experiment close enough for us to believe that it does describe the real situation at the interface. Moreover, the work of Grahame shows that the diffuse-double-layer theory, used in the proper context (i.e., assuming that the two capacitors are effectively connected in series), yields consistent results and can be considered to be correct, within the limits of the approximations used to derive it. 

The jigsaw puzzle was put together by Stem in 1926. Agreement between theory and experiment can be achieved once it is realized that both the Helmholtz and the Gouy-Chapman models are valid and exist simultaneously. Thus, there is a layer of ions on the surface that... 

The StShlberg model is based on the Langmuir isotherm model, which describes the competition of the ions for available surface area but it assumes that ions are adsorbed as separate, individual ions that xmdergo electrostatic interactions. The intensity of these interactions is calculated using the Stem-Gouy-Chapman theory. The isotherm equation obtained [92] is... 

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.

FIGURE 1.4 Double-layer models (a) Helmholtz model, (b) Gouy-Chapman model, (c) Stem model, and (d) Grahame model. (With kind permission from Springer Science+Business Media Electrochemical Supercapacitors Scientific Fundamentals and Technological Applications, 1999, Conway, B.E. Originally published by Kluwer Academic/ Plenum Pubhshers, New York in 1999.)... 

This model, shown in Figure 10.7, divides the doublelayer into two parts, i.e. (i) a fixed layer of strongly adsorbed counterions, adsorbed at specific sites on the surface, and (ii) a diffuse layer of ions similar to that of the Gouy-Chapman model. The fixed layer of ions is known as the Stem layer, and the potential decays rapidly and linearly in this layer. The potential decay is much more gradual in the diffuse layer. In the case of specifically adsorbing ions (multivalent ions, surfactants, etc.) the sign of the Stem potential may be reversed. 

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 1. Gouy-Chapman model of the electrical double layer and the potential distribution where S is the Stem plane within which counterions are adsorbed close to the suface and d is the diffuse layer of counterions.

The Guy-Chapman model can be applied to EDLs with Stem layers when the surface potential ij/o is replaced by the diffuse layer potential and when the thickness of the Stern layer is properly accounted for. Additionally, Eq. (3.19) would refer to the total net charge of surface and Stem layer. If, however, a quantitative description of the complete EDL is required, the Gouy-Chapman model of the diffuse layer has to be supplemented with according models for the ion adsorption in the Stem layer and the charging of the particle surface. Note that such models inherently depend on material properties which considerably increases the effort for model parametrisation (e.g. Sonnefeld et al. 2001). 

The Eq. (3.33) is based on the Gouy-Chapman model (i.e. the Stem layer is ignored) and on charge regulation with constant surface potential (CP). It applies to small surface potentials ( i/ o, l < 25 mV, to thin double layers (xa 1) and to small surface distances (h x,). Equation (3.33) provides only rough estimates for moderate and thick double layers (xa < 5). Yet, Sader et al. (1995) showed that by replacing the termxi +X2 withxi +x2+2 /i(i.e. twice the particle centre distance ri2), the HHF-equation is applicable to arbitrary values of h. 

Clay and grain surface phenomena create a double layer or interface conductivity . The electrical double layer can be described with the Gouy-Chapman model or the Stem model. A detailed description of the physical properties and processes in the interface region is given by Revil et al. (1997) and Revil and Glover (1998). 

Gouy-Chapman and Stem Models of the Double Layer... 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

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 ... 

The interfacial capacitance increases with the DDTC concentration added. The relationship among potential difference t/ of diffusion layer, the electric charge density q on the surface of an electrode and the concentration c of a solution according to Gouy, Chapman and Stem model theory is as follows. 

The earliest models used to describe the distribution of charges in the edl are, besides the Helmholz model, the Gouy-Chapman diffuse layer model and the Stem-Graham model. Details of these models are given in Westall and Hohl (1980), Schindler (1981, 1984) and Schindler and Stumm (1987). 

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