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DIVISION S-2—SOIL CHEMISTRY

Formulating the Charge-distribution Multisite Surface Complexation Model Using FITEQL

^a Dep. of Geological Sciences, Virginia Polytechnic Institute and State Univ., Blacksburg, VA 24061
^b Dep. of Crop and Soil Environmental Sciences, Virginia Polytechnic Institute and State Univ., Blacksburg, VA 24061

* Corresponding author (ctadanie{at}vt.edu)

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Sorptive interactions at the solid–water interface strongly influence the bioavailability of many important nutrient oxyanions and trace contaminant metals in both natural and engineered settings. Recently, the charge-distribution multisite complexation (CD-MUSIC) model has been developed to model ion adsorption behavior on variable-charge minerals. Although this model shows great promise, its use has been limited by lack of incorporation into commonly used computer codes. In this study formulation of the CD-MUSIC model in the surface complexation modeling program FITEQL 4.0 is described, and demonstrated using Cu²⁺- and orthophosphate (P_i)-goethite adsorption data. Mass-action and mass-balance expressions for Cu²⁺ and P_i adsorption on goethite were developed using a combination of monodentate and bidentate surface species. Pauling's rules were used to determine the charge of surface adsorption sites and adsorption site density (nm^-2) was calculated from crystallographic considerations. Electrostatic component coefficients in the mass-balance expressions were adjusted to reflect the actual charge of goethite adsorption sites, thereby satisfying both the local charge balance for adsorbed species and the global charge balance of the system as a whole. FITEQL 4.0 was used to determine the best-fit equilibrium constants for the Cu²⁺ and P_i surface adsorption mass-action expressions, and the associated speciation of adsorbed ions. The speciation of adsorbed Cu²⁺ ions was dominated by a single monodentate surface species; whereas, two monodentate and one bidentate surface species were required to adequately describe P_i adsorption. Formulating the CD-MUSIC model as outlined here provides a thermodynamically, electrostatically, and crystallographically consistent approach for solving surface adsorption equilibrium problems with FITEQL.

Abbreviations: SAI, specifically absorbed ion • CD, charge distribution • CD-Music, CD multisite complexation • DDL, diffuse-double layer • DL, diffuse layer • EDL, electrical double layer • NAI, nonspecifically absorbed ion • P_i, orthophosphate • PZC, point-of-zero charge • TLM, triple layer model

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
SORPTIVE INTERACTIONS at the solid–water interface strongly influence the bioavailability of many important nutrient oxyanions and trace contaminant metals in both natural and engineered settings. Because of the ubiquitous distribution and high specific surface area of colloidal minerals in the biosphere, mineral surfaces, particularly metal (hydr)oxides, often act as reservoirs for metals and oxyanions. Metals and oxyanions are concentrated on mineral surfaces through precipitation and both specific and nonspecific adsorption processes. The formation of surface microprecipitates as opposed to adsorbed species is controlled by a complex array of geochemical factors including pH, ionic composition and strength of embathing solution, chemical functionality and crystallography of mineral surfaces involved, and the total concentration of the metal or oxyanion present. However, most nutrient oxyanions and trace metals occur at relatively low concentrations in natural environments (micromolar or less) precluding precipitation. Therefore, adsorption processes ultimately control the phase distribution and potential bioavailability of these ions (McBride, 1994).

Because trace metals including Fe, Cu, Zn, and oxyanions such as P_i have central functions in many biochemical pathways, and occur at relatively low concentrations in pristine environments, organisms generally take up inorganic nutrient ions using transport systems specific for a particular chemical form. For example, both microorganisms and plants generally take-up P and Cu as free ions from solution—P_i (Frossard et al., 1995) and Cu²⁺ (Deighton and Goodman, 1995). However, Cu, Zn, and other trace metals may become toxic to microorganisms, phytoplankton, and plants at concentrations as low as 10^-9 M, with toxicity and free metal ion activity well correlated (Deighton and Goodman, 1995). Therefore, determining the distribution and speciation of metals and oxyanions between adsorbed and solution phases is essential for evaluating their potential bioavailability in the environment.

Mechanistic surface complexation models have been widely used to estimate the distribution of oxyanions and to a lesser extent metals between sorbed and solution phases, as well as their ionic speciation within these phases. The underlying physics and chemistry of adsorption phenomena are described in these models using equilibrium adsorption reactions which place adsorbed ions in specific electrostatic planes within an electrical double layer (EDL) adjacent to the mineral surface. The constant capacitance, diffuse double-layer (DDL), basic Stern, and triple layer models have been extensively applied to describe surface adsorption on mineral surfaces (Westall and Hohl, 1980; Hayes et al., 1991; Goldberg, 1992). These models typically require several user specified adjustable fitting parameters including capacitance of the EDL and types of reactive surface sites and their densities, and rely on an arbitrarily defined amphoteric description of ionizable surface functional groups. These shortcomings have recently been addressed through development of the CD-MUSIC model, which uses crystallographic information to describe surface acidity and reactive site density of specific minerals (Hiemstra et al., 1989a, 1989b, 1996; Hiemstra and Van Riemsdijk, 1996).

Although the CD-MUSIC model may more explicitly depict the underlying physicochemical processes involved in ion adsorption on mineral surfaces, its use has been limited by its exclusion from commonly used surface complexation modeling codes. The objective of this study was to demonstrate formulation of the CD-MUSIC model for describing both metal ion and oxyanion adsorption on mineral surfaces using the readily available and widely used surface complexation modeling program FITEQL 4.0 (Herbelin and Westall, 1999). Here we describe how to specify the surface sorption equilibrium problem in terms of the CD-MUSIC model within the existing framework of the FITEQL 4.0 code, and illustrate this modeling approach using Cu²⁺ and P_i equilibrium sorption data for goethite.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Goethite Preparation and Characterization
Goethite was synthesized by oxidation of FeSO₄ in an aerated solution buffered to pH 7 with 0.1 M NaHCO₃ (Schwertmann and Cornell, 1991). Residual salts were removed by washing the goethite precipitates with ultrapure water (distilled-deionized, R > 18.2 M) until the rinse solution and clean water conductivities were equal. Amorphous Fe-(hydr) oxides were removed by soaking the goethite sample in 0.01 M HCl for 1 h, centrifuging the suspension to separate the colloidal goethite crystals, and discarding the supernatant. The goethite sample was resuspended in 0.01 M NaCl (pH 6) and then washed with ultra-pure water again as previously described. The cleaned goethite precipitates were freeze dried for storage prior to ion adsorption experiments.

The identity and purity of the goethite sample was verified by powder x-ray diffraction analysis. No other crystalline mineral phases were detected, and broad weak features indicative of amorphous material were also not present in the diffraction pattern. No surface contamination was detected on the cleaned goethite colloids by energy dispersive and x-ray photoelectron spectroscopic analyses. Scanning electron microscopy indicated that the size and shape of goethite crystals in our synthetic sample were very similar to those found in natural environments (Cornell and Schwertmann, 1996); colloids were relatively uniform euhedral acicular crystals approximately 400 to 600 nm in length. Specific surface area was 66 m² g^-1 as measured by a five-point N₂ Brunauer-Emmett-Teller (BET) analysis. The point-of-zero-charge (PZC) of the prepared goethite occurred at pH 9.2, as determined by the intersection of proton titration curves in 0.001, 0.01, and 0.1 M NaClO₄. The surface charge of the prepared goethite in 0.01 M NaClO₄ was measured as a function of pH using laser-doppler velocimetry photon-correlation spectroscopy, also resulting in a PZC at pH 9.2. Details of the preparation and characterization of goethite used for previous Cu²⁺ equilibrium adsorption studies modeled here is given in the literature (Grossl and Sparks, 1994).

Equilibrium Sorption Experiments
Equilibrium adsorption of P_i on goethite was studied by generating a series of adsorption edges over the pH range of 3 to 12, with 0.01 M NaClO₄ as the background electrolyte. Adsorption edges were collected over a wide range of P_i loadings at a goethite suspension concentration of 0.5 g L^-1. However, for the purpose of demonstrating the use of the CD-MUSIC model in FITEQL 4.0 only the 40 µM total P_i loading is reported and used here. Equilibrium adsorption experiments were conducted in a water-jacketed 500-mL Teflon reaction vessel, at a constant temperature of 25 ± 0.1°C. Orthophosphate (as Na₂HPO₄) was added to an initial sample volume of 400 mL at pH 12, and the P_i-goethite suspension was allowed to equilibrate for 24 h. The suspension was mixed at 300 rpm with a three-bladed impeller and sparged with N₂ gas continuously throughout the experiment.

Following the initial 24 h equilibration period, the pH of the suspension was lowered in 0.5 pH unit increments by adding 0.1 M HCl, with an 8-h equilibration time between successive adjustments. The suspension was maintained at the desired pH setpoint during each 8-h equilibration period by a Brinkmann 716 Stat-Titrino pH-stat (Brinkman Instruments, Westbury, NY). A 20-mL aliquot was removed prior to each pH adjustment, filtered through a 0.1-µm Whatman cellulose-ester filter (Whatman Inc., Clifton, NJ), and residual P_i in the filtrate was measured by the molybdate-blue method (APHA et al., 1995). Sorbed P_i was determined by difference between total initial and residual P_i concentrations. Although the HCl volume added for pH adjustment was small (<1% of suspension volume), the effect of dilution was included in all calculations. The equilibrium adsorption behavior of Cu²⁺ on goethite was studied in a similar manner, as described elsewhere (Grossl and Sparks, 1994).

	SURFACE COMPLEXATION MODELING

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
General Approach for Mechanistic Models
Mechanistic surface complexation models describe surface adsorption phenomena using a thermodynamic approach in which surface adsorption reactions are described by mass-action and mass-balance expressions. Although the mathematical details of various mechanistic surface complexion model formulations vary somewhat, they all share two common attributes: (i) the Gibbs free energy of surface adsorption reactions is partitioned into chemical and electrostatic components, and (ii) an EDL adjacent to the surface with adsorbed ions placed in one or more discrete electrostatic planes, in which the electrical potential of each plane is related to its charge by an electrostatic model. For the purpose of formulating the CD-MUSIC model in chemical equilibrium computer codes, a brief discussion of the general approach for solving multicomponent chemical equilibrium problems including surface adsorption follows. Several thorough reviews of the computational formulation of surface adsorption equilibrium problems are present in the literature (Westall and Hohl, 1980; Westall, 1980, 1986).

Computational Scheme for Multicomponent Equilibrium Problems
Solution of multicomponent surface adsorption equilibrium problems has been facilitated by describing the chemical system in terms of mass-action and mass-balance expressions formulated in a manner consistent with available computational schemes (Gibbs, 1957; Westall and Hohl, 1980). In this formalism, a set of species is defined to include every chemical entity in the equilibrium problem, and a set of components is defined such that every species can be expressed as the product of a stoichiometric reaction involving only the system components. In addition, no component can be expressed as the product of a reaction involving the other components. A more thorough discussion of the background underlying this formalism and guidelines for defining system species and components is given elsewhere (Morel and Hering, 1993).

Mass-action expressions for each species in the system can be written from the stoichiometric reactions describing each species as

[1]

where K_i is the equilibrium constant, {S_i} is the activity of species i, X_j are the activities of the components, and a_ij are the stoichiometric coefficients of component j in species i. If the activity of species i in Eq. [1] is expressed as the product of the concentration and activity coefficient of species i, rearranging gives

[2]

where [S_i] is the concentration of species i. Defining the activity corrected equilibrium constant K^'_i for the reaction that describes species i in terms of all components as

[3]

and substituting Eq. [3] into Eq. [2] gives

[4]

Finally, the nonlinear mass-action expressions given by Eq. [4] can be written in linear form by taking the logarithm of both sides

[5]

where for notational convenience the concentration S_i of species i is replaced by C_i. The set of mass-action expressions defined by Eq. [5] is compactly and conveniently represented using matrix notation

[6]

where the chemical system has m species described by n components and the mass-action stoichiometric coefficients matrix is termed the A matrix in keeping with the notation used in FITEQL (Herbelin and Westall, 1999).

Mass-balance expressions are written for each component in the system to constrain the mass-action expressions to a unique equilibrium solution. These mass-balance expressions take the form

[7]

where b_ij are the stoichiometric coefficients for species i in component j, C_i is the concentration of species i, T_j is the total (analytical) known concentration of component j, and Y_j is the difference between the concentrations of component j calculated from species stoichiometry and the known total. The mass-balance expressions for a chemical system with m species and n components are expressed in matrix notation as

[8]

where the stoichiometric coefficient matrix is termed the B matrix (Herbelin and Westall, 1999), and the rows and columns of B are transposed prior to matrix multiplication. The stoichiometric coefficients in the A and B matrices are identical in the formulation of many surface complexation models; however as described in Results and Discussion below, this is often not the case when using the CD-MUSIC model.

The solution to the chemical equilibrium problem described by Eq. [6] and [8] is that set of component activities X which corresponds to the set of species concentrations C such that the differences between calculated and total component concentrations Y are less than a user defined tolerance value. An iterative strategy is used to solve the equilibrium problem: species concentrations C are calculated from initial guesses for component activities X using Eq. [6], mass-balance differences Y for all components are then calculated from Eq. [8], and if all mass-balance differences Y are not less than their preset tolerances new values for component activities X are calculated using the Newton-Raphson approximation method. This process is repeated until mass balances for all components are satisfied in Eq. [8] or convergence to a unique equilibrium solution is not obtained in a prescribed number of iterations. Convergence of the Newton-Raphson approximation method may be improved in some circumstances as outlined in Appendix 1 below.

Energetics of Ion Adsorption on Charged Surfaces
Mechanistic models consider the free energy change accompanying the adsorption of an ion on a charged mineral surface to be divided into intrinsic and electrostatic components

[9]

The intrinsic component G_intrinsic reflects the chemical energy change due to reaction of the adsorbing ion with a surface functional group, whereas the electrostatic component G_{electrostatic} represents the energy required to move the ion from bulk solution to the surface which has an electrostatic potential ₀ relative to the bulk solution. The third term on the right-hand side of Eq. [9], in which Q is the reaction quotient, represents the dependence of adsorption free energy on system composition. The intrinsic contribution of the free energy of adsorption is given as

[10]

where K^S_int is the intrinsic surface adsorption equilibrium constant and the electrostatic contribution is given by

[11]

where z is the net change in charge on the surface due to exchange of species that define the adsorption reaction, F is Faraday's constant, and ₀ is the surface potential (Stumm et al., 1970). Substituting Eq. [10] and [11] into Eq. [9], invoking the equilibrium condition that G_adsorption = 0, and simplifying and rearranging the resulting expression gives

[12]

where K^S_app is the apparent surface adsorption equilibrium constant. The exponential term in Eq. [12] is the Boltzmann factor, which mathematically describes the influence of surface potential on ion adsorption by charged (hydr)oxide minerals.

Models of the EDL may contain more than one electrostatic plane associated with ion adsorption (e.g., Stern, triple layer, and CD-MUSIC models). In this case, a Boltzmann factor is defined for each electrostatic plane using the ion charge in that plane and the potential of the plane relative to that of bulk solution. Inclusion of Boltzmann factors in the mass-action expressions for each species in a chemical system (Eq. [6]) requires that each factor be designated as a system component. Thus, the A matrix in Eq. [6] has one column of stoichiometric coefficients for each Boltzmann factor, which are also termed electrostatic components. The mass-balance B matrix in Eq. [8] also has one column of stoichiometric coefficients for each electrostatic component; however, there is no actual chemical entity in the system to ascribe a total concentration T_j. Rather, if T_j is taken as the total charge on a given electrostatic plane, then the mass-balance expression for electrostatic component j actually represents a charge balance for that electrostatic plane. The T_j value on each electrostatic plane is calculated using a charge-potential relationship appropriate for the given plane. Electrostatic plane potentials are adjusted simultaneously with chemical component activities at each iteration of the solution sequence.

Electrostatics of 3-Plane Models
The spatial separation of charge into discrete electrostatic planes in the EDL produces a potential difference between the mineral surface and bulk solution. In 3-plane models, the electrostatic planes define two interfacial regions which are treated as molecular capacitors arranged in series with unique capacitances C₁ and C₂ (Levine, 1971), as illustrated in Fig. 1 . The overall capacitance of the EDL C is calculated by series addition of the molecular capacitances

[13]

View larger version (45K): [in this window] [in a new window]   Fig. 1. Electrical double layer defined by electrostatic planes in 3-plane surface complexation models. Charge separation due to adsorbed and electrolyte ions located in the electrostatic planes defines two molecular capacitors which are termed the inner- and outer-Helmholz layers. These layers are collectively known as the Stern Layer. A layer of diffuse electrolyte ions separates the Stern layer and bulk solution. Electrical potential decreases from some positive value at the mineral surface to zero in the bulk solution.

[14]

where ₀ is the permittivity of free space, is the solution dielectric constant, and d is the thickness of the molecular capacitor. Solution composition influences the EDL capacitance through the affect of ionic strength on the relative dielectric constant. Because they cannot be directly measured or calculated, EDL molecular capacitances are typically treated as adjustable fitting parameters, within the bounds of Eq. [14].

Charge on electrostatic planes is calculated from charge-potential relationships for interfacial regions in the EDL and the global charge balance (Davis and Leckie, 1978). Charge-potential relationships for the electrostatic planes that define the EDL in 3-plane models are

[15]

[16]

and

[17]

where C₁ and C₂ are the molecular capacitances of the inner- and outer-Helmholz layers and ₀, ₁, and ₂ are the electrical potentials of the electrostatic planes that define these near-surface regions. The net charge of electrolyte ions in the diffuse layer (DL) is calculated from the Guoy-Chapman relationship

[18]

where R is the universal gas constant, T is absolute temperature, ₀ is the permittivity of free space, is the solution dielectric constant, c is molar electrolyte concentration, z is the charge of a symmetrical electrolyte, and F is Faraday's constant.

The CD-MUSIC Model
The CD-MUSIC model differs from other widely used surface complexation models (e.g., constant capacitance, DDL, Stern, and triple-layer models) in three principle ways; representation of surface acidity, placement of ions and charge in the EDL, and representation of reactive surface adsorption sites. The background and development of this model have been extensively described (Hiemstra et al., 1989a, 1989b, 1996; Hiemstra and Van Riemsdijk, 1996; Rietra et al., 1999a). The sections below outline the differences between the CD-MUSIC and other surface complexation models and describe how formulation of the equilibrium problem must be altered to use the CD-MUSIC model. The adsorption behavior of Cu²⁺ and P_i on goethite are used as examples to illustrate formulation of the CD-MUSIC model in the equilibrium adsorption computer code FITEQL.

Mineral Surface Acidity and Reactivity
Spectroscopic studies of (hydr)oxide and clay minerals have shown that the reactivity of surface hydroxyls varies with bulk crystal cation identity and local crystal environment. The goethite crystal structure consists of double chains of FeO₃(OH)₃ octahedra that run parallel to the c axis, as shown in Fig. 2 . Thus, goethite contains surface hydroxyls with the oxygen atom coordinated to either one, two, or three Fe³⁺ atoms in the bulk crystal, which are termed singly, doubly, and triply coordinated groups, respectively. The number of available coordination sites and relative proton affinity of these surface oxygens controls the extent of surface protonation, and hence surface acidity.

View larger version (54K): [in this window] [in a new window]   Fig. 2. Goethite structure showing a double chain of FeO3(OH)3 octahedra that run parallel to the c axis. Singly, doubly, and triply coordinated surface oxygens are indicated.

[19]

where X is the predominant cation in the bulk mineral (Boehm, 1971; Schindler et al., 1976; Sposito, 1984; Schindler and Stumm, 1987). Surface complexation models that use this approach to describe (hydr)oxide mineral acidity are termed 2-pK models. However, in stable (hydr)oxide structures electroneutrality requires that the charge of a cation is balanced by oppositely charged surrounding oxygens. If the neutralization of cation charge is expressed on a per bond basis according to Pauling's rules (Pauling, 1929), then surface hydroxyls in (hydr)oxide minerals may have noninteger charge (Bolt and Van Riemsdijk, 1982; Westall, 1986, 1987; Goldberg, 1992; Hiemstra and Van Riemsdijk, 1996). For example, applying Pauling's rules to the goethite structure results in surface hydroxyls with charges in multiples of one-half, as demonstrated for singly and triply coordinated oxygens in Fig. 3 .

View larger version (53K): [in this window] [in a new window]   Fig. 3. Charge on singly, doubly, and triply coordinated surface oxygen functional groups in goethite using Pauling's rules. All Fe atoms are octahedrally coordinated and symmetrical distribution of charge is assumed.

View larger version (39K): [in this window] [in a new window]   Fig. 4. Placement of ions in the electrostatic double layer of the CD-MUSIC model. Protonated surface oxygens lie in the 0-plane, ligands of specifically adsorbed ions are distributed between the 0- and 1-planes, and ion pairs are placed in the 2-plane.

Apportionment of certain types of ionic charge to more than one electrostatic plane is a unique feature of the CD-MUSIC model, and in combination with the possibility of a nonzero reference state for ionizable mineral-surface functional groups (e.g., FeOH^1/2- and Fe₃O^1/2- for goethite) represents the fundamental challenge when formulating the CD-MUSIC model in chemical equilibrium computer codes such as FITEQL. By extension of the Pauling bond valence principle to charge neutralization in specifically adsorbed oxyanions and metal-ion hydroxide complexes, a certain fraction f of the central cation charge of the adsorbed species is attributed to common bridging ligands which lie in the 0-plane and the remaining portion (1 - f) is attributed to ligands located in the 1-plane (Hiemstra and Van Riemsdijk, 1996). If the central cation charge is assumed to be symmetrically neutralized by all surrounding ligands in accordance with Pauling's rules, the charge distribution (CD) factor f simply represents the fraction of charge neutralizing ligands located in the 0-plane. The CD factors for several monodentate and bidentate P_i and Cu²⁺ surface species are shown in Fig. 5 . It should be noted that nonsymmetrical neutralization of central cation charge is possible, for instance due to a shift in electron density in the adsorbed species caused by protonation of an oxygen ligand; however, is not considered in this study.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Charge distribution (CD) f values for Cu2+ and Pi surface complexes with goethite assuming symmetrical distribution of central cation charge among surrounding ligands according to Pauling's rules. The CD f values represent the portion of central cation charge which is neutralized by surface ligands, and for symmetrical neutralization of cation charge are calculated as the fraction of total ligands situated in the surface 0–plane.

[20]

where z_i is the net change in charge on electrostatic plane i and _i is the potential on plane i relative to bulk solution potential (Hiemstra and Van Riemsdijk, 1996). In the case of SAIs, apportionment of charge between the 0- and 1-planes is accomplished by defining the changes in charge z₀ and z₁ on these planes as

[21]

and

[22]

where n_H is the net change in protons on the 0-plane, z_H is proton charge, f is the CD factor, z_Me is the charge of the central atom in the adsorbed species, m_j is the number of adsorbed species ligands positioned in the 1-plane, and z_j is the charge on adsorbed species ligands m_j in the 1-plane. Table 3 and Table 4 list the Boltzmann factors for adsorption of Cu²⁺ and P_i on goethite.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Formulating the CD-MUSIC Model in FITEQL
The CD-MUSIC model may be formulated in the standard FITEQL code by modifying the 0-plane electrostatic component coefficients specified in the mass-balance B matrix. These coefficients must be modified whenever a nonzero charge is assigned to the reference state for surface adsorption sites (e.g., FeOH^1/2- and Fe₃O^1/2- for goethite). Because the coefficients for each electrostatic component in the B matrix (Boltzmann factors in Eq. [8]) represent ion charge rather than species stoichiometry, an accurate charge balance for the 0-plane requires that the electrostatic component coefficients in the B matrix reflect the actual charge of the reference surface adsorption site(s). For goethite, 0-plane electrostatic component coefficients of those species which have one or more surface adsorption sites as reactants must be adjusted by a multiple of the adsorption site reference charge of -1/2 to generate the correct ion charge coefficients in the B matrix. Comparison of Table 3 with Table 5 and Table 4 with Table 6 illustrates the differences between A and B matrix 0-plane electrostatic component coefficients for Cu²⁺ and P_i adsorption on goethite.

More generally, the 0-plane electrostatic component coefficients in the A and B matrices are not numerically equal for any surface adsorption problem involving a mineral with nonzero adsorption site reference charge: the A and B matrix coefficients for the 0-plane electrostatic component differ by the surface adsorption site reference charge for species involved in monodentate complexation reactions, and by twice the surface adsorption site reference charge for species involved in bidentate surface complexation reactions. Reformulating the mass-balance B matrix in this manner results in an electrostatically consistent description of the surface adsorption equilibrium problem, as the sum of coefficients for 0- and 1-plane electrostatic components equals the net charge on adsorbed species (compare adsorbed species coefficients in Table 5 and Table 6 with charge on adsorbed species in stoichiometric reactions of Table 1 and Table 2).

In the FITEQL implementation of 3-plane electrostatic models, all charge associated with protons, SAIs, NAIs, and DDL ions s_d must be accounted for in one of the three electrostatic planes which define the EDL (0-, 1-, and 2- planes). Adsorbed ion charge is distributed among the 0-, 1-, and 2- planes in the CD-MUSIC model as previously described. Although the DDL electrolyte ion charge _d is not physically located in the 2-plane (see Fig. 1 and Fig. 4), it is computationally incorporated in the electrostatic description of the mineral-water interface as part of the 2-plane charge balance (₂ = _NAI + _d). The global charge balance is therefore

[23]

where for computational purposes _NAI and _d are superimposed on the 2-plane. The 2-plane charge values reported in FITEQL output (TABLE 4.6) must therefore be interpreted based on the presence or absence of ions other than DDL electrolyte ions in the 2-plane. In the classical triple layer model (TLM), in which no electrolyte ion pairs or other NAIs are placed in the 2-plane, the charge values reported for the 2-plane indeed represent _d. However, if NAIs are located in the 2-plane—electrolyte-mineral surface ion-pairs in the CD-MUSIC model for example—the two 2-plane charge reported in FITEQL output actually represents the sum of _NAI and _d. This superposition of charge is demonstrated in the FITEQL output of CD-MUSIC model surface adsorption equilibrium problems, as the sum of _NAI calculated from the equilibrium ion speciation (FITEQL output TABLE 4.3) and _d calculated from Eq. [18], is numerically equal to the reported 2-plane charge.

CD-MUSIC Model Fit of Cu²⁺ and P_i Adsorption on Goethite
The CD-MUSIC model adequately described Cu²⁺ and P_i adsorption on goethite for the range of pH and adsorption densities examined in this study, as seen by comparing the CD-MUSIC model fit to adsorption edge data shown in Fig. 6 and Fig. 7 . FITEQL 4.0 was used to fit the CD-MUSIC model to adsorption edge data for Cu²⁺ and P_i adsorption reactions as specified in Table 1 and Table 2 with corresponding mass-action and mass-balance A and B matrices given in Tables 3 through 6. The 3-plane EDL description of the CD-MUSIC model was incorporated by selecting the TLM option in FITEQL input files. The types and site densities of reactive surface functional groups were as previously described. Inner- and outer-Helmholz layer capacitances for both Cu²⁺ and P_i adsorption were 1.0 and 4.1 F m^-2, respectively. Optimized values for surface adsorption reaction equilibrium constants computed by FITEQL are listed in Table 1 and Table 2.

View larger version (15K): [in this window] [in a new window]   Fig. 6. Copper2+ adsorption on goethite with 0.01 M NaNO3 background electrolyte. CuT = 0.785 mM. Goethite concentration 10 g L-1, specific surface area 50 m2 g-1. Open symbols represent experimentally determined Cu adsorption. Filled symbols with dotted lines represent adsorbed Cu species based on optimized CD-MUSIC fit to experimental data. Solid line represents cumulative total of adsorbed Cu species predicted by CD-MUSIC model.

View larger version (25K): [in this window] [in a new window]   Fig. 7. Pi adsorption on goethite with 0.01 M NaClO4 background electrolyte. Pi,T = 40 µM. Goethite concentration 0.5 g L-1, specific surface area 66.5 m2 g-1. Open symbols represent experimentally determined Pi adsorption. Filled symbols with dotted lines represent adsorbed Pi species based on optimized CD-MUSIC fit to experimental data. Solid line represents cumulative total of adsorbed Pi species predicted by CD-MUSIC model.

Additional Considerations for Using the FITEQL Approach
The equilibrium constants of adsorption reactions that specify bidentate complexes between sorbate ions and surface functional groups must be numerically adjusted to accurately reflect the intrinsic chemical energy changes associated with any surface adsorption equilibrium problem solved using FITEQL. The mass-action expressions that describe a surface adsorption problem in FITEQL (Eq. [6]) are based on molar activities of system components, which may be approximated by molar concentrations in low ionic strength solutions. However, because the entropic contribution to the G_adsorption resulting from ion exchange at the solution–mineral interface is derived from statistical thermodynamic considerations, the concentrations of surface species and components are more appropriately expressed on a mole fraction basis (Nordstrom and Munoz, 1994). With surface species and components expressed in terms of mole fractions, the mass-action equations for surface adsorption reactions can be written

[24]

where _ads is the mole fraction of an adsorbed species (Cu²⁺ or PO^3-₄ for example), _ref is the mole fraction of the reference surface component (FeOH^1/2- for goethite), m is the stoichiometric coefficient for the surface reference site component (1 for monodentate and 2 for bidentate complexes), and C_sol are the concentrations of solution components with stoichiometric coefficients n (Hiemstra and Van Riemsdijk, 1996). The numerical scaling factor that relates the concentration dependent K values used in FITEQL to the mole fraction based intrinsic K values for bidentate reactions is given by setting m equal to 2 in Eq. [24]

[25]

where S_ads is the adsorbed complex species concentration (mol L^-1), S_ref is the reference surface component concentration (mol L^-1), is the solid-solution ratio (kg L^-1), A is the specific surface area (m² kg^-1), and N_S,j is the site density (mol m^-2) of the reference surface component j. For surface adsorption reactions that involve monodentate complexes, the AN_S,j terms in the numerator and denominator of Eq. 24 cancel and the K values used in FITEQL are therefore numerically equal to the actual K_intrinsic values.

It is always desirable to minimize the number of adjustable fitting parameters when using mechanistic complexation models to describe ion adsorption on mineral surfaces. Ideally, all the physical and chemical properties of the mineral–water interface required in a mechanistic model could be experimentally measured directly, and no adjustable fitting parameters would be necessary. At present the electrical capacitances resulting from segregation of charge into discrete planes in the EDL cannot be directly measured and are therefore treated as fitting parameters in electrostatic surface complexation models. The CD factors f which describe the segregation of adsorbed-ion charge between the 0- and 1-planes in the CD-MUSIC model are also commonly treated as fitting parameters, numerically adjusted from the values determined using Pauling's rules to improve the model fit to adsorption data. However, FITEQL has no automated optimization capability for either the EDL C or CD f factors, requiring a laborious and often very time-consuming trial and error process for selecting optimal values.

Selection of appropriate values for inner- and outer-Helmholz layer capacitances of 3-plane electrostatic models (see Eq. [15] and [16]) can be constrained to a range of plausible values using the fundamental relationship for a molecular capacitor at the mineral–water interface (Eq. [14]). The capacitance values of 1.0 and 4.1 F m^-2 used for the inner- and outer-Helmholz layers in this work correspond to dielectric constants of approximately 30 and 80 and layer thicknesses of 2.8 and 1.7 Å for the two molecular capacitors, respectively. These values are consistent with dielectric constant variation in the EDL near a mineral surface (Bockris and Reddy, 1970; Hiemstra and Van Riemsdijk, 1991) and the effective ionic radii of phosphate molecules and background electrolyte ions (Ottonello, 1997). Further refinement of the spatial arrangement and dielectric properties of both adsorbed ions and water molecules in the EDL may be possible by interpreting synchrotron x-ray standing wave and high-resolution x-ray reflectivity measurements of electromagnetic radiation incident on the mineral–water interface at shallow angles (Sturchio et al., 1997; Farquhar et al., 1999; Fenter and Sturchio, 1999; Fenter et al., 2000). The impact of nonsymmetrical charge distribution among oxygen ligands surrounding metal atoms on the CD f values could also be evaluated by using ab initio or semi-empirical quantum mechanical calculations to determine the local electronic environment of oxyanions and metal-hydroxide complexes adsorbed on mineral surfaces.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The CD-MUSIC model provides a thermodynamically, electrostatically, and crystallographically consistent formulation for surface adsorption equilibrium problems. The stoichiometry of surface adsorption reactions is described by mass-action expressions, the surface charge of adsorbed species and adsorption sites is described by mass-balance equations, and the density and reactivity of surface adsorption sites is derived from the crystallographic considerations of the mineral surface. The formulation of the CD-MUSIC model outlined here has two distinct advantages compared with previous implementations: (i) local charge balance of species involved in surface adsorption reactions is satisfied, and (ii) no alteration of computer code is required to satisfy global charge balance of the system as a whole. Our results indicate that the CD-MUSIC model adequately describes the adsorption behavior of both metals and oxyanions on goethites with different physical properties synthesized by different methods and in varying background electrolytes. Furthermore, the CD factor f does not need to be treated as an adjustable fitting parameter but may be determined from Pauling's rules. Nonsymmetrical distribution of charge could be incorporated in the CD-MUSIC model by using ab initio or semi-empirical calculations to determine f values from the local electronic environment of oxyanions and metal-hydroxide complexes adsorbed on mineral surfaces.

APPENDIX 1

The multidimensional Newton-Raphson method is used to calculate improved guesses for all component concentrations after each iteration in the solution procedure. Updated values for component activities are calculated

[A.1]

where Y_j are the mass-balance error terms calculated from Eq. [8], X_j is the difference between the original and improved component activities, and Z_j is the partial derivative of Y_j with respect to X_j (Jacobian matrix). The updated X_j values are then used in Eq. [6] to calculate improved species concentrations, which are in turn used to generate updated mass-balance error values. The elements of the Jacobian Z are calculated

[A.2]

with terms as previously defined. Only the electrostatic components have nonzero T/X terms according to Eq. [A.2], because these T_j values are calculated from charge-potential relationships for the electrostatic planes as given in Eq. [15] through [17]. Note that molar charge values T are related to specific charges (C m^-2) by

[A.3]

where is the solid/solution ratio (kg L^-1), A is the specific surface area (m² kg^-1), and F is Faraday's constant.

There are subtle omissions in the FITEQL subroutine SURFZ, which calculates the T/X terms for the Jacobian elements that correspond to the electrostatic components X. These omissions are also present in all previous versions of FITEQL. The correct code for calculating these Jacobian elements in SURFZ is

CASE 4

Z(KO%, KO%)

Z(KO%, KO%) + CAP1 * RTF/X(KO%) * SAF

Z(KO%, KB%)

Z(KO%, KB%) - CAP1 * RTF/X(KB%) * SAF

Z(KO%, KD%)

Z(KO%, KD%) + 0#

Z(KB%, KO%)

Z(KB%, KO%) - CAP1 * RTF/X(KO%) * SAF

Z(KB%, KB%)

Z(KB%, KB%) + (CAP1 + CAP2) * RTF/X(KB%) * SAF

Z(KB%, KD%)

Z(KB%, KD%) - CAP2 * RTF/X(KD%) * SAF

Z(KD%, KO%)

Z(KD%, KO%) + 0#

Z(KD%, KB%)

Z(KD%, KB%) - CAP2 * RTF/X(KB%) * SAF

Temp#

ZEL * PSID * F2RT

Temp#

(EXP(Temp#) + EXP(-Temp#))/2#

Z(KD%, KD%)

Z(KD%, KD%) + (CAP2 + EERT8 * SQRTC* ZEL * F2RT * Temp#) * RTF/X(KD%) * SAF

END SELECT

where the omissions are underlined. It should be noted that the omitted terms in the T/X calculations above are technically required for any 3-plane model including the classical TLM. However, because the TLM places no charge in the 2-plane (d-plane), the omitted terms are all numerically equal to zero for TLM calculations, as the B-matrix coefficients used in their calculation are all zero. Therefore, although the T/X terms for the 2-plane (d-plane) are coded incorrectly in the all versions of FITEQL through v. 4.0, the omissions noted above have no effect on classical TLM calculations.

The omitted terms in the T/X calculations noted above do not affect the equilibrium solution to surface adsorption problems when 3-plane models such as the CD-MUSIC model are formulated in FITEQL, but only the numerical stability of the Newton-Raphson optimization procedure used to solve the B-matrix equations. In other words, this issue may possibly effect FITEQL's convergence to the equilibrium solution, but not the solution itself. We have implemented the Jacobian element modifications to subroutine SURFZ, and conducted extensive side-by-side comparisons with the standard FITEQL 4.0 code (Herbelin and Westall, 1999). For a variety of adsorbate-mineral systems including P_i, As(III/V), chromate, silica, and Cu adsorbed on goethite, and Zn-HFO (FITEQL Example 10) comprising 21 total data sets, the optimized surface equilibrium constants and WSOS/DF values calculated by the two codes were identical to the fifth decimal place (the limit reported in FITEQL output).

The FITEQL 4.0 source code is divided between the files FITEQL4.BAS and FQ4MOD1.BAS on the distribution disks. Modification of the FITEQL source code can be made in any convenient editor, and then recompiled using Microsoft BASIC Professional Development System Version 7.0 (Microsoft, Remond, WA). A modified executable module is generated by linking the compiled object files FITEQL4.OBJ and FQ4MOD1.OBJ, with the library BRT70ENR provided in the BASIC software.

	ACKNOWLEDGMENTS

 
The authors gratefully acknowledge constructive discussions with Drs. Michael F. Hochella, Lucian W. Zelazny, and John C. Westall. Financial support was provided by the United States Department of Energy (Project DE-FG02-99ER15002) and the ASPIRES program at VPI&SU. We also thank Dr. Paul Grossl for generously providing Cu²⁺-goethite adsorption data. The constructive comments of three anonymous reviewers are also appreciated.

Received for publication May 14, 2001.

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TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS SURFACE COMPLEXATION MODELING RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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