Gouy-Chapman-Stem theory

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

Measurements of the surface tension and surface stress of solids are not easy. Some attempts have been made to measure the surface energy, or at least to determine the PZC, of solid electrodes attached to piezoelectric materials (36, 37). More often there is a reliance on studies of differential capacitance (Section 13.4.3) (35, 38). In principle, these measurements could provide all of the information needed to describe the surface charges and relative excesses however, one must first know the PZC. Evaluating it for a solid electrode/electrolyte system is not straightforward, and indeed, as discussed below, the PZC is not uniquely defined for a polycrystalline electrode. The most widely used approach is to evaluate the potential of minimum differential capacitance in a system involving dilute electrolyte. The identification of this potential as the PZC rests on the Gouy-Chapman-Stem theory discussed in Section 13.3,... 

Gouy-Chapman-Stem Theory for Nonoverlapping EDLs.432... 

In the Stem-Gouy-Chapman (SGC) theory the double layer is divided into a Stem layer, adjacent to the surface with a thickness d, and a diffuse layer of point charges. The diffuse layer begins at the Stem plane in a distance d, from the surface. In the simplest case the Stem layer is free of charges. In real cases the Stem layer is formed by specifically adsorbed ions. The condition of electroneutrality was given by Eq. (2.59) In addition to and Oj, the surface charge can be represented by the Stem potential. It transforms the conditions of electroneutrality into the equation for the determination of the Stem potential. 

Figure 13.3.6 (a) A view of the differential capacitance in the Gouy-Chapman-Stem (GCS) model as a series network of Helmholtz-layer and diffuse-layer capacitances. (b) Potential profile through the solution side of the double layer, according to GCS theory. Calculated from (13.3.23) for 10 M 1 1 electrolyte in water at 25°C. 

R. O. James and G. A. Parks, Characterization of aqueous colloids by their electrical double-layer and intrinsic surface chemical properties. Surface and Colloid Science 12 119 (1982). Perhaps the most complete review of the triple layer model from the perspective of Gouy-Chapman-Stem-Graham e double layer theory. 

FIGURE 3.11 Schematic representation of the Gouy-Chapman-Stern theory. Compare with Figure 3.3 an additional plane, the Stem plane, is defined as the distance of closest ionic approach to the surface. The region between the Stern plane and the surface behaves as a dielectric. 

Figure 1-6). Maxwell-Wagner, Helmholtz, Gouy-Chapman, Stem, and Gra-hame theories have been used to describe the interfacial and double layer dynamics [9]. 

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 second modification of the Stem theory consists in dividing the solution charge into two contributions. Thus, according to the Stem picture, part of the charge qs on the solution is immobilized close to the electrode in the OHP (the Helmholtz-Perrin charge or q ), and the remainder is diffusely spread out in the solution (the Gouy-Chapman charge or qc), i.e.,... 

In the Gouy-Chapman and Stem theories—equations showing the variation of potential with the distance and the dependence of capacitance on the potential—ions in the interphase are characterized by one parameter only, the valence z. 

Earlier theories by Gouy, Chapman, and Hcrzfeld discussed the double layer as wholly of this diffuse type but Stem points out that these give far too high values for the capacity of the double layer, partly because in them the ions are supposed mathematically to be able to approach indefinitely close to the solid surface, which is impossible physically owing to the size of the ions. Stern s theory gives a complicated expression for the capacity of the double layer, but accounts reasonably well for the experimental values. Though the layer is largely diffuse in many cases, the capacity is usually of the same order as if the layer were of the plane parallel type, because most of the ions are fairly close to the fixed part of the layer. 

Theories of colloid stability based on electrostatics go way back beyond the DLVO theory, to the Gouy-Chapman theory of the electrical double layer proposed in the early 1910s and the Stem theory of counterion condensation proposed in 1924. There was much weighty speculation about the counterion distribution around colloidal particles throughout the 20th century, but nobody succeeded in measuring it until our work in 1997. This work is described in detail in Chapter 8. 

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... 

O. Stem. Z. Elektrochem. 30 (1924) 508. Note that this improvement of the Gouy-Chapman theory is almost as old as the Debye-Huckel theory (1923). In 1943 Stem received the Nobel Prize in physics for work on the quantization of magnetic moments. 

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