Triple layer model equation

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. 

The models describing hydrolysis and adsorption on oxide surfaces are called surface complexation models in literature. They differ in the assumptions concerning the structure of the double electrical layer, i.e. in the definition of planes situation, where adsorbed ions are located and equations asociating the surface potential with surface charge (t/> = f(5)). The most important models are presented in the papers by Westall and Hohl [102]. Tbe most commonly used is the triple layer model proposed by Davis et al. [103-105] from conceptualization of the electrical double layer discussed by Yates et al. [106] and by Chan et al. [107]. Reviews and representative applications of this model have been given by Davis and Leckie [108] and by Morel et al. [109]. We will base our consideration on this model. 

Except for a few rare cases, the experimental titration curves corresponding to different concentrations of basic electrolyte have a conunon intersection point (CIP) in PZC. That intriguing feature made us study consequences of treating it in a fully rigorous way. For the triple layer model, that way has been outlined in the works by Rudzihski et al. [21,91]. A point of zero charge is determined by the condition <5o(pH = PZC) = 0. Taking equations (52-54) into account, for pH=PZC, Eq. (52) can be transformed as follows ... 

The full set of equations can be solved with a computer program using the mathematical approach outlined by Westall (1980). Caution must be used to ensure that the computer code does not implement the standard and reference states proposed by Hayes and Leckie (1987). A fit of the triple layer model to silver adsorption on amorphous iron oxide is presented in Figure 6.7. 

For this purpose it is possible to extend to a multiple oxide the one-site model of Johnson (1984), which provides a thermodynamic description of the double layer surrounding simple hydrous oxides. Briefly, in this model the double layer charge is divided into the charge inside the slip plane, slip plane a[d]. While occupied sites, a[tl] is obtained from the Poisson-Boltzmann equation. Note that unlike the triple-layer model (Davis et al., 1978) which allows ions to form surface complexes at two different planes (0 or / ) instead of al the slip plane only, this model does not distinguish between inner- and outer-sphere complexes. Expression of the... 

Equations 5.62c and 5.62d provide the basis for two independent determinations of the capacitance parameter C in the triple layer model. However, the check on internal consistency that these two separate evaluations of C could furnish has never been performed in any published study.In fact, C l has been taken universally as an adjustable... 

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. " ... 

ANION ADSORPTION. The triple layer model postulates that anions react with surface hydroxyl groups according to the general equation ... 

Table 5.3. Constraint equations and molecular hypotheses in the constant capacitance and triple layer models for a hydrous oxide suspended in a 1 1 electrolyte solution. 

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. 

The original Smit model separates the surface plane into two sections one fiuction [i.e., 1 — /], of the overall surface (respectively the fraction of the surface sites) where only uncomplexed surface groups are present, and another fraction /, where only ion pairs formed with the electrolyte. For both sections, a different electrostatic model concept is introduced a Stem model (obviously without electrolyte binding) for the fraction 1 —/ and a triple-layer model for the fraction /. This separation is, of course, artificial. A mean value of the zeta potential is calculated from the equation given in Fig. 17i. Application of the model to experimental surface-charge data requires very low values for C2 One advantage of this model can be seen in the closer agreement of the model with the experimental observations quoted by Smit. 

The three planes of the triple layer model divide the interfaee into three regions each with their own electrostatic potential gradients [31] (1) between the IHP and the SP the electrostatic potential decreases linearly, (2) between the SP and the beginning of the solution the electrostatic potential decreases linearly, and (3) in the solution the electrostatic potential decays asymptotically as described by the Boltzmann equation. 

The pseudo-Stem model (and Stem model) differs from the triple layer model in that (1) the eleetrostatie potential remains constant between the OHP and the Stem plane in the former but decays linearly in this region in the latter [31], and (2) the pseudo-Stem model uses the BPD mass action equation (429). In the pseudo-Stem and triple layer models separate adsorption equations are used for adsorption of protons and hydroxide ions at the IHP and for adsorption of all other solute moleeules at the OHP. The amount of adsorption at both planes is then adjusted in order to be self-consistent with an electrostatic balance over Pq, P j and the diffuse layer. 

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 triple-layer site-binding model now fits within the scheme of the general equilibrium problem given in Equations 1-3. Other adsorbed cations and anions can be included in the equilibria simply by adding the appropriate components and species. 

Most of the research on metal sorption at the mineral/water interface has dealt with equilibrium aspects. Numerous studies have used macroscopic approaches such as adsorption isotherms, empirical and semi-empirical equations (e.g., Freundlich, Langmuir), and surface complexation models (e.g., constant capacitance, triple layer) to describe adsorption, usually based on a 24 hour reaction time. 

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