Constant-capacitance double-layer

The method is based on the chemical equilibrium program MINEQL (2), modified to include the coulombic energy of adsorption caused by the charged surface. First the principles of MINEQL are presented through a short example from solution chemistry and then the extension of the method to the constant-capacitance double-layer model of the surface/solution interface used by Stumm, Schindler, and co-workers (3,4) is demonstrated. Finally, the use of the method in the triple-layer site-binding model introduced by Yates, Levine, and Healy (5), and used by Davis, James, and Leckie (6), is shown. In each case the mathematics are described in suflBcient detail to be reproducible. 

The method developed here for the description of chemical equilibria including adsorption on charged surfaces was applied to interpret phosphate adsorption on iron oxide (9), and to study electrical double-layer properties in simple electrolytes (6), and adsorption of metal ions on iron oxide (10). The mathematical formulation was combined with a procedure for determining constants from experimental data in a comparison of four different models for the surface/solution interface a constant-capacitance double-layer model, a diffuse double-layer model, the triplelayer model described here, and the Stem model (11). The reader is referred to the Literature Cited for an elaboration on the applications. 

Surface complexation models, on the other hand, account explicitly for the electrical state of the sorbing surface (e.g., Adamson, 1976 Stumm, 1992). This class of models includes the constant capacitance, double layer, and triple layer theories (e.g., Westall and Hohl, 1980 Sverjensky, 1993). Of these, double layer theory (also known as diffuse layer theory) is most fully developed in the literature and probably the most useful in geochemical modeling (e.g., Dzombak and Morel, 1987). 

It is possible on the basis of this model (arrangement O) to explain the constant capacitance region on the negative side of the C vs. E curve (Fig. 20.7), and why the capacitance in this region is independent of the nature of the cations in the solution. The model of the double layer is shown in Fig. 20.12 in which it can be seen that the surface of the electrode and the... 

The origin of the observed correlation was not established, and the relation was not interpreted as causal. It could be argued that a sustained elevated potential due to as-yet unknown microbial processes altered the passive film characteristics, as is known to occur for metals polarized at anodic potentials. If these conditions thickened the oxide film or decreased the dielectric constant to the point where passive film capacitance was on the order of double-layer capacitance (Cji), the series equivalent oxide would have begun to reflect the contribution from the oxide. In this scenario, decreased C would have appeared as a consequence of sustained elevated potential. 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

A Simplified Double Layer Model (Constant Capacitance)... 

The diffuse double layer model is used to correct for Coulombic effects. The constant capacitance model depends on the input of a capacitance but the result obtained is not very different. 

In addition to the diffuse double layer and the constant capacitance model dis-... 

Two models of surface hydrolysis reactions and four models of the electrical double layer have been discussed. In this section two examples will be discussed the diprotic surface group model with constant capacitance electric double layer model and the monoprotic surface group model with a Stern double layer model. More details on the derivation of equations used in this section are found elsewhere (3JL). ... 

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 equilibrium constants KNa+ and Kci- introduced here characterize the extent of counterion complexation that occurs. Two other constants characterize the potential generation that results from this complexation, namely the capacitances CNa+ and ( Cl-- These are the capacitances between the planes of counterion complexation and the surface plane where ao is located. The potentials rpNa+ and rpcl are the electrostatic potential at the location in the double layer where the ions adsorb and form a surface complex. 

The area of an electrode is finite and essentially constant. Similarly, the thickness of the electric double-layer does not vary by a large amount. As an empirical rule, we find that the double-layer capacitance has a value in the range 10-40 pF cm, where F is the SI unit of capacitance, the farad. Note that a capacitance without an area is not particularly useful - we need to know the complete capacitance. 

---