Gouy-Chapman case

In the Gouy-Chapman case the differential capacitance (a area) follows as... 

Approximate solutions for the Gouy-Chapman case were presented by Maier [21], while more recently analytical solutions for the Gouy-Chapman as well as the Mott-Schottky cases were presented by litzehnan et al. [22, 23]. The relevant expressions for each partial conductivity are summarized for the Gouy-Chapman case ... 

The maximum of 2(x) is at A. At exactly this point C2 = 1 becomes imity, i.e. reaches the bulk value. Beyond A — as already in the neighbourhood of A because of the conditions imposed — the function is no longer permitted. In contrast to the Gouy Chapman case the extent of the space charge thickness (A ) is dependent on the interfacial parameters according to... 

R.h.s. Enrichment profiles in the Gouy-Chapman case as a function of the strength of the effect measured by t . 

Let us first consider a case where both defects are in spatial equilibrium, that is the Gouy-Chapman case (with zi = z2 = z). 

Let us consider the Gouy-Chapman case. If Of is the enriched majority charge carrier, then, according to Eq. (5.252), the surface charge density S is proportional to Cq (x=0). Hence, with Eq. (5.259) we obtain... 

The relationships are different in the Mott-Schottky case. There 91nco"(x=0)oc 5In E does not apply. Rather, because of E a [D ]A and Eqs. (5.215), (5.231), it follows that 51nco (x=0)oc 5 E, i.e. the absolute change is important (more than that The concentration effect is proportional to Ej9 El, in complete contrast to 5 S / E in the Gouy-Chapman case). A detailed example will be discussed in the next section. 

Fig. 6.45 Normalized flux emerging from a bi-crystal with perpendicular grain boundary calculated for a given chemical potential gradient (Gouy-Chapman case) (arbitrary units). Any core resistance was ignored. The decrease with increasing space charge potential (3kT, 6kT space charge effect in energy units) reveals the influence of the equilibrium space charge [448].

The fact that no potential dependence occurred in the first case ( Gouy-Chapman case ), is due to the assumption of small effects. The result for large effects is also easily obtained for the Gouy-Chapman case Because Se (5.251),... 

FIG. 7 Parsons-Zobel plot of 1/Q as a function of the inverse Gouy-Chapman capacitance 1 /Cqc- The plot is calculated analytically from Eqs. (54) and (85) at zero charge density. The straight line represents the case = a = For the upper... 

It is natural to consider the case when the surface affinity h to adsorb or desorb ions remains unchanged when charging the wall but other cases could be considered as well. In Fig. 13 the differential capacitance C is plotted as a function of a for several values of h. The curves display a maximum for non-positive values of h and a flat minimum for positive values of h. At the pzc the value of the Gouy-Chapman theory and that for h = 0 coincide and the same symmetry argument as in the previous section for the totally symmetric local interaction can be used to rationalize this result. 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

Similar considerations apply to situations in which substrate and micelle carry like charges. If the ionic substrate carries highly apolar groups, it should be bound at the micellar surface, but if it is hydrophilic so that it does not bind in the Stern layer, it may, nonetheless, be distributed in the diffuse Gouy-Chapman layer close to the micellar surface. In this case the distinction between sharply defined reaction regions would be lost, and there would be some probability of reactions across the micelle-water interface. 

The Gouy-Chapman theory treats the electrolyte as consisting of point ions in a dielectric continuum. This is reasonable when the concentration of the ions is low, and the space charge is so far from the metal surface that the discrete molecular nature of the solution is not important. This is not true at higher electrolyte concentrations, and better models must be used in this case. Improvements on the Gouy-Chapman theory should explain the origin of the Helmholtz capacity. In the last section we have seen that the metal makes a contribution to the Helmholtz capacity other contributions are expected to arise from the molecular structure of the solution. 

For simplicity, we will consider the case in which surface charge and potential are positive, and that only anions adsorb. Furthermore, the potential drop in the Gouy-Chapman layer will be assumed to be small enough that its charge/potential relation can be linearized. The V o/oo/pH relationship can then be derived parametrically, with the charge in the Gouy-Chapman layer cr4 as the parameter. The potential at the plane of anion adsorption can then be calculated and substituted in Equation 28 to give ... 

B , while for an n-type semiconductor the reverse is true. An analog to the SCR in the semiconductor is an extended layer of ions in the bulk of the electrolyte, which is present especially in the case of electrolytes of low concentration (typically below 0.1 rnolh1). This diffuse double layer is described by the Gouy-Chap-man model. The Stern model, a combination of the Helmholtz and the Gouy-Chapman models, was developed in order to find a realistic description of the electrolytic interface layer. 

The preceding discussion was limited to the artificial case of a single ion. When multiple ions are present, in addition to the issues discussed, there is the problem of ion-ion interactions and correlations. The main motivation for such studies is to come close to the realistic situation in which a finite concentration of ions exists near the metal surface that is in equilibrium with ions in the bulk. Another important specific goal is to investigate the applicability of continuum models, such as the Gouy-Chapman theory. " Although this has been the subject of several Monte Carlo... 

The electrical interactions can be easily calculated when the particle is close to the substrate and the ionic strength is low. In this case, we can simplify the problem to a conventional punctual particle with a charge q placed in the electrical field generated by the substrate in solution [9]. Using the Gouy-Chapman model for the calculation of the electrical field generated by the substrate leads to the following equation ... 

Ionic surfactants are electrolytes dissociated in water, forming an electrical double layer consisting of counterions and co-ions at the interface. The Gouy-Chapman theory is used to model the double layer. In conjunction with the Gibbs adsorption equation and the equations of state, the theory allows the surfactant adsorption and the related interfacial properties to be determined [9,10] (The Gibbs adsorption model is certainly simpler than the Butler-Lucassen-Reynders model for this case.). 

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