The Triple Layer Model

The triple layer modeP offers a molecular description of surface complexation reactions that differs from the constant capacitance model in several fundamental respects. These differences can be brought into clear relief through a comparative listing of the principal chemical assumptions that underlie the triple layer model  

The proton and the hydroxide ion form inner-sphere surface complexes. AH other adsorbed metal cations and inorganic or organic anions form outer-sphere complexes. 

The relationship between surface charge and inner potential is specified through the following equations  

Moreover, the charge-balance condition, Eq. 5.59, is imposed explicitly and the full DDL theory expression for cr = aj) is used. Note that Eqs. 5.58 and.5.59 are not consistent for arbitrary values of the surface charge densities and inner potentials unless ions are present in the plane under all circumstances. 

THE NET PROTON CHARGE. In the triple layer model, the protonation and proton dissociation reactions in Eq. 5.41 are described by the conditional equilibrium constants 

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Wu CH, Shang LL, Cheng FL, Chao YK (2001) Modeling competitive adsorption of molybdate, sulfate and selenate on y-Al203 by the triple-layer model. J Colloid Interf Sci 233 259-264... 

To be useful in modeling electrolyte sorption, a theory needs to describe hydrolysis and the mineral surface, account for electrical charge there, and provide for mass balance on the sorbing sites. In addition, an internally consistent and sufficiently broad database of sorption reactions should accompany the theory. Of the approaches available, a class known as surface complexation models (e.g., Adamson, 1976 Stumm, 1992) reflect such an ideal most closely. This class includes the double layer model (also known as the diffuse layer model) and the triple layer model (e.g., Westall and Hohl, 1980 Sverjensky, 1993). 

The Triple Layer Model. This model developped by Yates et al. (1974) and Davis et al. (1978) uses a similar idea as the Stern model the specifically adsorbed ions are... 

The structure of the interface according to the Stern model and several limiting-case approximations is presented in Figure 4. The electrostatic models of the interface will be introduced in terms of the most complete one, the triple layer model (Figure 4a). Then the relationship of the triple layer model to the simplified models in Figures 4b-d will be discussed. 

Stern used this simplification in his calculations. The simplified model with only one Helmholtz capacitance is commonly referred to as the Stern model (Figure 4b), while the "extended" Stern model (Figure 4a) is designated the triple layer model. 

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. 

An extension of this method has been developed for the triple layer model which allows data obtained at several values of ionic strength to be considered simultaneously (7, 13.). However, this "double extrapolation technique" involves the same sort of approximation. 

The discussion above pertains to the diprotic acid chemical model and the constant capacitance electrostatic model. It is interesting to note that in some applications of the triple layer model with site binding of electrolyte ions at the IHP, the... 

The triple layer model has been described in detail elsewhere (11, 16, 17) however, the model as reported here has been slightly modified from the original versions (11, 15) in two ways (i) metal ions are allowed to form surface complexes at either the o- or 8-plane insted of at the 8-plane only, and (ii) the thermodynamic basis of the TLM has been modified leading to a different relationship between activity coefficients and interfacial potentials. The implementation and basis for these modifications are described below. 

Microscopic Subreactions and Macroscopic Proton Coefficients. The macroscopic proton coefficient may be used as a semi-empirical modeling variable when calibrated against major system parameters. However, x has also been used to evaluate the fundamental nature of metal/adsorbent interactions (e.g., 5). In this section, macroscopic proton coefficients (Xj and v) calculated from adsorption data are compared with the microscopic subreactions of the Triple-Layer Model ( 1 ) and their inter-relationships are discussed. 

A specific example of the relationship between the microscopic subreactions required to model experimental observations of metal removal and the macroscopic proton coefficient is shown for the case of Cd(II) adsorption onto a-A f (Figure 3). One variation of the surface coordination concept is used to describe the system subreactions the Triple Layer Model of Davis et al., (1,20). The specific subreactions which are considered, the formation constants and compact layer capacitances, are shown in Table IV. Protons are assigned to the o-plane (the oxide surface) and Cd(II) surface species and electrolyte ions to the 8-plane located a distance, 8, from the o-plane. 

In the triple layer model, the potential determining ions are located at the oxide surface with the specifically adsorbing ions and the ion pairs in the inner Helmholz... 

Sahai, N. Sverjensky D.A. (1997 a) Evaluation of externally consistent parameters for the triple layer model by the systematic analysis of oxide surface titration data. Geochim. Cosmochim. Acta 61 2801-2826 Sahai, N. Carroll, S.A. Roberts, S. O Day,... 

Thus, according to these theories, all univalent (1 1) electrolytes should behave the same way. However, this is not what was observed experimentally. Solutions of different 1 1 electrolytes (e.g., NaCl, NaBr, Nal, KI) show species-specific behavior. In order to interpret this specific behavior, Grahame (5) proposed a new model of the interphase the triple-layer model. The basic idea in the interpretation of the ion-specific behavior is that anions, when attracted into the interphase, may become dehydrated and thus get closer to the electrode. Each anion undergoes this to a different extent. This difference in the degree of dehydration and the difference in the size of ions results in the specific behavior of the anions. Ions that are partially or fully dehydrated are in contact with the electrode. This contact adsorption of ions allows short-range forces (e.g., electric image forces) to act between the metal elec-... 

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). 

S. Goldberg, Inconsistency in the triple layer model description of ionic strength dependent boron adsorption, J. Colloid Interface Sci. 285, 509-517 (2005). 

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. 

A more sophisticated model is the triple-layer model, allowing the surface reaction of the background electrolyte (Hayes et al. 1991). The potential-determining ions (hydrogen and hydroxide) are directly on the surface (inner Helmholtz layer), the other ions are at a certain distance from the surface (outer Helmholtz layer), and there is a diffuse layer, also. 

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... 

In fig. 3.20b specific adsorption Is also accounted for. The notion of specific adsorption has been defined In sec. 3.3. In disperse systems, its occurrence is de facto Inferred from the dependence of certain double layer properties on the natures of counter- and co-lons Generally, ions interacting specifically (non-electrostatlcally) with the surface approach it to shorter distance p < d). The plane where these specifically adsorbed ions reside is called the inner Helmholtz plane (iHp) In colloid science, the model of fig. 3.20b Is also known as the triple layer model. In this model three charges and three capacitances can be distinguished. For the two inner layer differential capacitances... 

In the triple-layer model, at high c . C vanishes but in (3.6.31) the two capacitances and da /6a° remain. In addition, it has to be verified to what extent these capacitances are Independent of a°. At least three adjustable parameters remain, which seem more than adequate to explain the most recalcitrant surface charge curve. For the mercury system, where very refined capacitance measurements can be made, this is appropriate, but for unmly systems like oxides we easily mn the risk of overinterpretation. 

The Triple-Layer Model (TLM) This model, first developed by Yates et al. (1974) and Davis (1978), is essentially an expanded Stem-Grahame model the specifically adsorbed ions are placed as partially solvated ions at a plane of closest approach. Additional capacitances are introduced. Subsequently, Hayes (1987) had to modify the earlier TLM by moving specifically adsorbed ions to the mean plane of the surface. 

Figure 3. Schematic representation of coordinative surface complexes and ion pairs formed between inorganic ions and hydroxyl groups of an oxide surface in the triple layer model. Reprinted from Hayes (1987) with permission.

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. 

Because the various SCM s have different formulations for treating adsorption reactions and the electrostatic terms, parameters fit to one model may not he applicable to other models (Morel et al, 1981). For example, Gao and Mucci (2001) determined different Log K s for As(V) adsorption by goethite when the data were fit to the Constant Capacitance Model, the Basic Stem Model, and the Triple Layer Model. 

Current surface complexation models were developed with a focus on minor and trace ions and hence do not consider sorption in the diffuse layer. Even the triple-layer model (34), which can include electrolyte sorption as outer-sphere complexes, does not consider sorption in the diffuse layer. To... 

Figure 10.19 Schematic plot of surface species, charge (cr), and potential tfr) relationships versus distance from the surface, used in the triple-layer model. Integral capacitances C and C2 are assigned to regions between the 0 and /3 and the fi and d planes, respectively. Electrolyte ions are adsorbed at the plane. Reprinted from Adv. Colloid Interface ScL 12,...

The triple-layer model calls for capacitances C and C2, corresponding to zones between the zero and beta planes and beta and d planes, respectively. The capacitances are related to the net charge and potentials of those planes through the expressions... 

The following adsorption reactions and intrinsic constants are available for the triple-layer model, where FeO", FeOH, and FeOHJ are surface sites. 

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. 

The schematic picture of the triple layer model is shown in Fig. 13. According to the... 

Figure 14. (A) Diagram of the charge distribution in the triple layer model. (B) Flat capacitors connected in series as equivalent of a triple layer model at the aqueous solution/metal oxide interface. Charge distribution on capacitor plates is obtained from the electroneutrality condition written in the form 6g = (— o) + ( d).

The triple layer model can also be presented by using the capacitors scheme shown in Fig. 14. As within the compact layer, there must be fulfilled the electroneutrality condition ... 

The triple layer model assumes two values of the parameter C] to exist, depending on the sign of the charge of the surface [111] ... 

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 ... 

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