Double layer, capacitance diffuse

There is a range of equations used describing the experimental data for the interactions of a substance as liquid and solid phases. They extend from simple empirical equations (sorption isotherms) to complicated mechanistic models based on surface complexation for the determination of electric potentials, e.g. constant-capacitance, diffuse-double layer and triple-layer model. 

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

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-... 

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 addition of the inert electrolyte affords other advantages. The most important point is that the conductivity of the solution increases (and thus the ohmic drop decreases through a decrease of the resistance of the cell, Rccw see Sect. 1.9). Moreover, the diffuse double layer narrows, being formed mainly by the ions of the inert electrolyte (with a sharp potential drop over a very short distance from the electrode surface). This makes the capacitance more reproducible and the Frumkin effects less obtrusive. Activity coefficients of the electroactive species are also less variable (and, therefore, quantities like formal potentials and rate constants), since... 

The elegance of the surface complexation approch lies in the fact that it can be incorporated into the thermodynamic speciation models used for soluble complexes. Consequently many of the computer models, e.g. SOILCHEM, HYDRAQL, MINTEQA2 and ECOSAT, include several different SCMs. Some commonly used SCMs are the diffuse-double-layer model, DDLM (Huang and Stumm, 1973 Dzombak and Morel, 1990), the constant capacitance model, CCM (Stumm et al., 1970 1976 1980 Schindler et al., 1976), the triple-layer model, TLM (Davis etal., 1978 Davis and Leckie, 1978,1980 Hayes and Leckie, 1987 Hayes et al., 1988) and the 1 pK basic Stern model (Bolt and Van Riemsdijk, 1982 Van Riemsdijk et al., 1986 1987). 

In an attempt to rationalize the measured capacitance values, and especially the low value for the basal plane (ca. 3pF/cm2), these authors first concluded that space charge within the electrode is the dominant contribution (rather than the compact double layer with ca. 15-20 pF/cm2, or the diffuse double layer with >100 pF/cm2). They then applied the theory of semiconductor electrodes to confirm this and obtained a good agreement by assuming for SAPG a charge carrier density of 6 x 1018/cm3 and a dielectric constant of 3 for GC, they obtained 13 pF/cm2 with the same dielectric constant and 1019 carriers per cubic centimeter. 

On this basis, three models will be discussed, which enable a calculation of the electrical potential, namely the constant-capacitance, the diffuse-double-layer, and the triple-layer model. 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

A more mechanistic and robust depiction of reversible metal adsorption is provided by SCMs that account explicitly for competitive speciation reactions using an equilibrium thermodynamic framework. Examples of SCMs in current use include the constant capacitance model (CCM), the diffuse double-layer model (DDLM), and the triple-layer model (TLM) (Stumm Morgan, 1996 Koretsky, 2000). Each of these models envisages... 

The potential ([i is called the potential of the outer Helmholtz plane. The diffuse double layer starts at the outer Helmholtz plane, where the potential is ( ). It is this value of the potential, rather than (j), that must be used in Eqs. 14G and 15G, to relate the surface charge density and the diffuse-double-layer capacitance to potential. 

With both the measured capacitance and the diffuse-double-layer capacitance known as functions of E, the capacitance of the compact double layer can be calculated from Eq. 19G as a function of potential. 

How can we judge whether there is agreement between experiment and theory. For one thing, the capacitance of the compact double layer should be independent of concentration. Second, in concentrated solution, the diffuse double layer should have very little effect on the observed capacitance except at, or very close to, E. Thus, repeating these five steps in solutions of different concentrations should yield the same plot of C versus E, and this should coincide with the capacitance measured in concentrated solutions. 

Many methods have been used to determine the value of the PZC on solid electrodes. The one that seems to be most reliable, and relatively easy to perform, is based on diffuse-double-layer theory. Measurement of the capacitance in dilute solutions (C < 0.01 M) should show a minimum at , as seen in Eq. 15G and Fig. 4G. Lowering the concentration yields better defined minima. Modem instrumentation... 

It is impossible to write an advanced text in any area of physical chemistry without resort to some mathematical derivations, but these have been kept to a minimum consistent with clarity, and used mostly when several steps in the derivation involve approximations, or some other physical assumption, which may not be obvious to the reader. Thus, the theories of the diffuse-double-layer capacitance and of electrocapillary thermodynamics are derived in some detail, while the discussion of the diffusion equation is limited to the translation of the conditions of the experiment to the corresponding initial and boundary conditions and the presentation of the final results, while the sometimes tedious mathematical methods of solving the equations are left out. The mathematical skills needed to comprehend this book are minimal, and it should be easily followed by anybody with an undergraduate degree in science or engineering. An elementary knowledge of thermodynamics and of chemical kinetics is assumed, however. 

Tafel slopes for the anodic and the cathodic process double-layer capacitance ( lF/cm ) capacitance of the Helmholtz double layer capacitance of the diffuse double layer double-layer capacitance at 0 = 0 double-layer capacitance at 0 = 1 adsorption pseudocapacitance (llF/cm ) adsorption pseudocapacitance derived from the Langmuir isotherm... 

The differential capacitance of a diffuse double layer is found directly from differentiation of [3.5.13] with respect to As we are measuring the capacitance for positive charge In a negative potential field, or the other way around, we need a minus sign in... 

Figure 3.5. Difleren-tlal capacitance of a flat diffuse double layer (1-1) electrolyte at 25°C.

Figure 3.42. DifTerential capacitance of the double layer on silver iodide in KF. Temperature, 25°C. Solid curves, experiments dashed curves, diffuse double layer capacitance according to 13.5.17). (Redrawn from J. Lyklema and J.Th.G. Overbeek. J. Colloid ScL 16 (1961) 595.)...

O C m-2 V" ) (differential) electric capacitance of diffuse double layer... 

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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