Triple-layer model capacitance values

In the triple layer model, values of the capacitance, Ci, can be obtained experimentally from the slopes of plots used in linear (Davis et al., 1978), double (James et al., 1978), or electrokinetic (Sprycha, 1984) extrapolations as described... 

Sverjensky (2001) showed that the values for capacitance Ci obtained for a wide variety of oxides and electrolyte types within the framework of the triple layer model (Sahai and Sverjensky, 1997a) fell into two groups. For rutile, anatase, and magnetite, values of Ci increased with decreasing crystallographic radius of the electrolyte cation from Cs+ to Li+. For quartz, amorphous silica, goethite, hematite, and alumina, values of Ci increased with decreasing hydrated electrolyte cation radius from Li+ to Cs+. 

When represents a metal cation not in the background electrolyte, the intrinsic constants are determined by fitting the triple layer model to adsorption edge data. This fitting entails a surface speciation calculation with previously measured values of the intrinsic constants in Eq. 5.61, the capacitance parameters Ci and C2, and the parameter Af. The computation includes Eqs. 5.58, 5.59, and 5.69, as well as surface charge and mole balance equations imposed as constraints. " ... 

Equations (l)-(4) are the foundations of electrical double layer theory and are often used in modeling the adsorption of metal ions at interfaces of charged solid and electrolyte solutions. In a typieal TLM, the outer layer capacitance is often fixed at a lower value (i.e., C2 = 0.2 F/m ), whereas iimer layer capacitance (Ci) can be adjusted to between 1.0 and 1.4 F/m [25]. It should be noted that the three-plane model (TPM) is a variation of the classical triple-layer model, in which the outer layer eapaeitanee is not fixed. Although the physical presentations of the TLM and TPM are identical as shown in Fig. 2, i.e., both involve a surface layer (0), an inner Helmholtz plane (p), and an outer Helmholtz plane d) where the diffuse double layer starts, a one-step protonation process (i.e., 1 piC approach) is, in general, assumed in the TPM, in eontrast to a two-step protonation process (i.e., 2 p/C approach) in the TLM. Another distinct difference is that pair-forming ions are assumed to be on the outer Helmholtz plane in the TPM but on the inner Helmholtz plane in the TLM. In our study, the outer layer capacitance is allowed to vary while the pair-forming ions are placed on the iimer Helmholtz plane with a complete set of surface eomplexation reactions being considered. Therefore, our approach represents a hybrid of the TPM and TLM. 

Model h is the so-called four layer model, which, compared to the triple-layer model, introduces one supplementary layer. The major reason for this is that the electrolyte ions typically differ in size. It may be argued that the usually larger anion has to be placed further away from the surface than the more compact cation. The consequence of the additional layer is an additional capacitance value. Furthermore, constraints on electrolyte binding from experimental data, which indicate symmetrical electrolyte binding, become more complicated. This is because in addition to the stability constant for the formation of the outer-sphere complexes, the location of charge also affects the interaction of the ion with the surface. 

The electrostatic part of model i has originally been proposed by Smit [6] because of the need to invoke very low values for the outer-layer capacitance. The values of about 0.2 F/m typically used in the triple-layer model can thus be decreased, which according to work quoted by Smit, would correspond to experimental observations. 

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