Environ. Sci. Technol. 2007, 41, 7621-7631
Metal Flux and Dynamic Speciation at (Bio)interfaces. Part II: Evaluation and Compilation of Physicochemical Parameters for Complexes with Particles and Aggregates ZESHI ZHANG, JACQUES BUFFLE,* AND DAVIDE ALEMANI Analytical and Biophysical Environmental Chemistry (CABE), University of Geneva, Sciences II, 30 quai E. Ansermet, CH-1211 Geneva 4
The computation of metal flux in aquatic systems at consuming surfaces like organism membranes must consider the diffusion processes of metal ions, ligands, and complex species, as well as the kinetic and thermodynamic aspects of their chemical interactions. Many natural ligands, however, have complicated properties (formation of successive complexes for simple ligands, polyelectrolytic properties and chemical heterogeneity for macromolecular ligands, large size distribution and fractal structure for suspended aggregates). These properties should be properly modeled to get the correct values of the chemical rate constants and diffusion coefficients required for flux computations. The selection of the most appropriate models and parameter values is far from straightforward. In this series of papers, models and compilations of parameters for application to the three most important types of complexants found in aquatic systems, the small, simple ligands, the fulvic and humic compounds, and the colloidal “particles” or aggregates, are discussed. In particular, new approaches are presented to compute the rate constants of metal complex formation for both fulvics/humics and particles/aggregates. A method to include the site distribution of fulvics/ humics and the size distribution of particles/aggregates in metal flux computation at consuming interfaces is also discussed in detail. These models and parameters are discussed critically and presented in a single consistent framework, applicable to the computation of metal flux in presence of any of the above complexants or their mixtures. Part I of the series focuses on simple ligands and fulvic/ humic compounds. Part II deals with particulate and aggregate complexants.
1. Introduction Computations of metal flux and dynamic speciation are important in various fields, particularly, for understanding metal bioavailability and ecotoxicology. However, in environmental systems, the vital or toxic role of a metal depends not only on the equilibrium distribution between numerous metal complexes but also on the dynamic properties (chemical rate constants and diffusion coefficients) of the metals, ligands, and complexes (1-4). In addition, the * Corresponding author phone: ++4122/379.6053; fax: ++4122/ 379.60.69; e-mail:
[email protected]. 10.1021/es071117r CCC: $37.00 Published on Web 11/13/2007
2007 American Chemical Society
number of environmental complexants is vast with highly variable chemical natures. Such computations are made difficult, as discussed previously (5), mainly, for two reasons: (i) no user-friendly computer code has been available until recently (see 6 and 7 for such codes), and (ii) the required parameter values for making such computations in a single framework, applicable to all natural ligands, are difficult to obtain. It is the purpose of Parts I and II of this series to provide the values of these parameters or to explain how they can be obtained for the three most important types of environmental complexants, namely (5), (i) the so-called “simple” ligands, (e.g., OH-, CO3-2, NH3, salicylate) having low molar mass, (ii) the organic macromolecules, usually chemically heterogeneous and bearing negative electric charge, in particular the fulvic/humic substances, and (iii) the natural “particles”, with diverse structure and fairly broad size distribution. In part I of this series, the simple ligands and fulvic/humic substances were discussed. The present paper will focus on the natural particles and aggregates. The parameters required to compute the flux of a trace metal at a consuming interface such as an organism membrane (4) or the surface of a bioanalogical dynamic sensor (1, 8; Figure 1; see also discussion of Figure 1 in ref 5), are discussed here. As explained previously (5), the transport of trace metal ions, whose concentrations (typically e10-6M) are much lower than that of the major electrolyte, is not influenced by electrostatc migration (9). Hence the most important parameters required for flux computations are the thermodynamic equilibrium constants for complex formation, K, the rate constants for complex formation, ka, and dissociation, kd, and the diffusion coefficients of the hydrated ion M, DM, and of the complexes and complexants. A few aspects, relevant to this paper, are mentioned below. (a) The boundary conditions (9-12), in particular, the exact geometry of the consuming interface or the thickness of the diffusion layer, δ, significantly affect the metal flux. These conditions however are included in the code used for flux computation (e.g., 6, 7) and are independent of the above key parameters. Hence they will not be discussed here. It must be stressed that the diffusion layer (thickness ) δ, Figure 1) is a conceptual layer of solution, where the transport of compounds is assumed to be controlled only by molecular diffusion and chemical reactions; at distances >δ, the solution is assumed to be homogeneous. δ can be rigorously related (9, 13) to the hydrodynamic flow and Fick’s laws of diffusionreaction. In reality, hydrodynamics affects the whole solution, up to the interface, but the above concept has been well established and is very beneficial to simplify flux computation at interfaces (1, 3, 9-12). In fact it is used in many codes VOL. 41, NO. 22, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Schematic representation of the diffusion of M toward a consuming interface and its reaction with complexing porous aggregates inside the diffusion layer. δ ) thickness of the diffusion layer (also called diffusion-reaction layer), at the boundary between the consuming surface and the solution. MS, S ) complex and complexing site inside the aggregate. ka, kd ) formation and dissociation rate constants of MS (5). DM, DaM ) diffusion coefficient of M in solution and inside the aggregate. Note that the scheme is not at scale. In particular, r is always much smaller than δ. (e.g., 6, 7, 12). It must be noted that, assuming molecular diffusion to be the only mode of transport within the diffusion layer, δ, implies that hydrodynamics does not affect the transport of M either outside or inside the aggregates, within this layer. (b) When the complexant is a particule or an aggregate (Figure 1), a few more complications than those discussed for simple ligands or fulvics (5) arise. To be consistent with the notation of Part I, the symbol L will be used for a complexing particle or aggregate. Each one, however, may contain and transport many complexing sites, S, and complexes, MS. In addition, a complexant L is much larger in size than M. Thus, DM >> DMS ≈ DL. Furthermore, because equilibrium is not always reached inside the diffusionreaction layer, δ, a particle/aggregate present in this layer may act as a consuming/releasing body, somehow competing for M, with the microorganism or sensor surface. In other words, the effective rates of association/dissociation of MS complexes in the diffusion-reaction layer will depend not only on the molecular rate constants ka and kd for the chemical reaction of M with a complexing site S but also on the diffusion of M toward and possibly inside the particle/aggregate (Figure 1). Such a problem has been discussed in the literature (14) for compact spherical and monodispersed particles. In nature, however, particles are more often aggregates of smaller subparticles, and their size distribution is very broad (15). (c) The size distribution of particles/aggregates, in natural systems, results from the formation (e.g., weathering, precipitation) and elimination (e.g., aggregation, sedimentation) processes (16, 17), which are not at equilibrium. Their kinetics, however, will not be considered here. In natural systems, it is often possible to assume that the whole of the above processes reaches a pseudo-steady-state, corresponding to a constant size distribution. In addition, when it is not the case, the time scales of these processes is greater than or equal to hours to days, while that required to reach a stationary-state flux at consuming interfaces is usually much smaller (> [MS] and CS,t ≈ CHS >> CMS inside the aggregate. These conditions hold when the free metal ion concentration, [M], is such that K′[M] () [MS]/[HS]) r
∂Cout M ) DM∇2Cout M ∂t
The initial conditions are
(28)
t ) 0, 0 e x e r
0 0 Cin M ) CM, CMS ) CMS
x>r
Cout M ) [M]
(29) (29′)
and the boundary conditions are out a Cin M ) CM , DM
t > 0, x ) r t > 0, x >> r
∂Cin ∂Cout M M ) DM ∂x ∂x
Cout M ) [M]
(30) (31)
C0M and C0MS (eq 29) are the initial concentrations of M and MS (assumed to be uniform) in the aggregate; [M] (eqs 29′ and 31) is the concentration of M in the solution located at the same distance from the consuming interface (Figure 1) as the aggregate of interest but far enough from any aggregate to not be influenced by the concentration gradients existing around the aggregates. Note that eqs 24-31 are valid within the diffusion layer of the consuming surface (distance < δ, Figure 1), where a gradient of M is established. Thus [M] is usually less than the bulk concentration, [M]*. Since CHS ≈ CS,t ) constant, the formation of MS (eq 24) is a pseudofirst-order reaction. ka is the association rate constant for reaction 17. It can be computed from the rate of formation of the outer-sphere complex (M,HS) and that for the elimination of a water molecule from the inner hydration shell of M (5, 30, 49), that is, ka t kMS (eq 9 in 5). Using a Laplace transforms and relating the species concentrations inside the aggregates (CMS, CHS, and CS,t) to the corresponding coarse-grained concentrations ([MS], [HS], and [S]t), eq 24 can be written as (section D in Supporting Information)
d[MS] eff ) keff a [M][HS] - k′d [MS] dt
FIGURE 3. Log{B(t)} vs log(t), for various values of aggregate radii, r. DM ) DaM ) 7 10-10 m2 s-1, ka ) 3 × 106 m3 mol-1 s-1, K′ ) 106 m3 mol-1, CHS ) 3.73 × 10-4 M. r ) (9) 893, (b) 89.3, and (2) 8.93 nm. For the mathematical expression for B(t), see Supporting Information, section D.
(32)
with
ka
keff a ) 1+
ka[S]t
(33)
4 3 πr cPB(t) 3
Since [H] ) CH, k′d ) kd[H] and k′deff ) keff d [H] is the pseudo-
k′d
k′deff ) 1+
ka[S]t
(34)
4 3 πr cPB(t) 3
first-order effective dissociation rate constant at a given pH for reaction 17. B(t) varies with time and also depends on the parameters kaCHS, K′ ) ka/k′d , r, DM, and DaM. It includes two complicated Laplace transform functions, and analytical solution is not possible. As explained below, this however can be circumvented by using the results of numerical inversion. A systematic study of the effect of the above parameters on B(t), in the ranges of values encountered in environmental conditions, has provided curves with shapes always similar to those of Figure 3 (Supporting Information, section D). They level off to a constant value, Bss, at large times, corresponding to steady-state conditions at the aggregate/solution interface, that is, when the flux of M, inside or outside the aggregate, is independent of time. eff In theory, the time-dependent parameters, keff a and k′d (eqs 33 and 34), can be incorporated within codes allowing the computation of metal flux at the consuming surface (Figure 1) by means of a numerical methods, provided that eff the values of keff can be changed at each time step a and k′d
FIGURE 4. Plot of log(Bss) vs log{3DM/r2} at t ) 0.1 (0, y ) 0.989x + 0.00984), 1 (O, y ) 0.992x - 0.0333), 10 (4, y ) 0.984x - 0.0325), and 100 s (3, y ) 0.979x - 0.00307), for values of DaM and r ranging from 7 × 10-11 to 7 × 10-10 m2 s-1 and 8.93-893 nm, respectively. Note that many points are superimposed. to follow the B(t) function. In the existing codes, however, eff (6, 7, 50), keff a and k′d should be kept constant, independant of time. In this case, eqs 33 and 34 should be used only in a time domain where B(t) ) Bss ) constant. The useful time domain for flux computations can be estimated as follows: Most often steady-state metal fluxes at the consuming interface are of interest (Figure 1), and they are reached at t > δ2/πDM, that is, typically t g 0.1 s for δ ) 10 µm. Thus, in experiments lasting not more than an hour, the relevant time domain for flux computation is 10-1 to 103 s. In this time scale, Bss is found to be independent of ka, K′ (eq 19), and DaM (Supporting Information, section D), at least in the following ranges: 100 e ka e 1012 M-1 s-1, 107 e K′ e 1012 M-1, and 0.1 e DaM/DM e 1. These limits arise partly from the mathematical difficulties in the invertion of Laplace transforms at extreme values of the parameters. Thus, the real domains of application might be larger. Nevertheless these ranges cover much of the useful environmental conditions. Then, Bss only depends on DM and r, as follows (Figure 4)
Bss ) 3DM/r2
(35)
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diffusion coefficient, DaM, of a free cation in a porous medium, may be lower than that in the solution, DM, in particular because of steric hindrance (51, 52). The corresponding value of DaM/DM can be estimated, on the basis of the ratio, σ, of the metal ion and pore radii (53). For example, the values of DaM/DM are 0.79 and 0.98 for σ ) 0.1 and 0.01, respectively. In a fractal aggregate, the “pore size” decreases from the outside to the center of the aggregate. But in most of the aggregate, σ < 0.2, and eq 35 usually holds. 5.2. Effective Rate Constant at Steady-State and the Impact of Particle/Aggregate Size Distribution. Under conditions where B(t) ) Bss, the effective formation rate constant of MS in the size class j of polydispersed aggregates is obtained by combining eqs 21, 33, and 35 j eff ka
ka
) 1+
ka[S]t j∆Ab/Ab,t
)
ka 1 + jΘ
(36)
4π jrDMjcp
The corresponding pseudo-first-order dissociation rate constant, at constant pH, is obtained via jk′deff ) jkeff a /K′. Note that eq 36 is analogous to that obtained for monodispersed compact particles (14; the term [S]tj∆Ab/Ab,t in eq 36, is called CL in ref 14). Equation 36 is valid when the particle/aggregate contains only one type of reactive site, with a total coarsegrained concentration [S]t (eqs 15 or 16). Multiple site type reactivity is discussed briefly in section 6. The ka value can be obtained as discussed earlier (section 2.3, eqs 10,14, and 15 in 5), where HS is a complexing site bearing no charge. The parameter DL in eqs 14 and 15 in ref 5 is now the diffusion coefficient of the particle/aggregate, and U(a) ) zMeψ is the electrostatic energy of the ion M at the potential ψ prevailing at the surface site. This term is analogous to that applicable to the ion M inside the charged fulvic/humic compounds (5). However, the distances between charged sites, inside an aggregate, are much larger than those in a fulvic molecule. For instance, for an aggregate with r ) 1 µm and Df ) 2, the average distance, between the surfaces of two subparticles with radii of 3 nm can be estimated using eq 3 and is found to vary from a few nanometers at the center to >50 nm at the edge of the aggregate. It is usually larger than the double-layer thicknesses (e.g., 0.4, 3.0, and 10 nm at ionic strengths of 0.5, 10-2, and 10-3 M, respectively). Hence, contrary to fulvic/humic molecules, an average ψ value for the aggregate as a whole is not physically relevant, and the local ψ value of the surface site must be used to compute U(a). This is however not known, in most cases. In absence of better value, and if the chemical nature of the complexing compound (X or P) is known, ψ can be estimated from the data reported for the corresponding pure compound (e.g., Figures S1A-C in Supporting Information). Note that, at pH close to 8, the ψ values of iron and aluminum (hydr)oxides are very small and have only slight or negligible effects on the ka values. In eq 36, jΘ can be considered as a “corrective” term which accounts for the diffusion of M around and inside the aggregate. Even though ka is independent of the size class j, j the effective rate constant jkeff a depends on this size via r (or jD), jc , and possibly j∆A /A p b b,t (eqs 11 and 11′). Two limiting cases of eq 36 must be emphasized: (1) When jΘ >1, even if a low value has been used for {X}t/{P}t (1%). Thus, as mentioned above, quite low values of ka (typically > 1, that is, keff a and k′d
are usually much lower (often by orders of magnitude) than ka and k′d . The little experimental data available in this field support the fact that the time scale of dissociation of metal complexes formed with natural particles is much larger (>hours, 55) than that expected for pure chemical reaction (seconds-minutes).
List of Symbols Ab
surface area of the solid phase of the aggregates, per liter of solution
Ab,t
total surface area of solid phase of the aggregates, per liter of solution
6. Brief Comments on the Application of the Concepts Discussed in Parts I and II to Real Systems
A X, A P
surface area of the component X or P, in the aggregate, per liter of solution
Parts I and II of this series provide concepts and parameter values to evaluate the diffusion coefficients and rate constants required for trace metal flux computations at consuming interfaces. A few additional effects might influence the metal flux. Some of them have been briefly discussed in the introduction, namely, electric migration, hydrodynamics, and aggregation/dissolution of colloids. In many cases they are expected to play a negligible role, but their precise influence should be studied in more detail. A few other processes may require further studies: (i) A generalized version of the Eigens-Wilkins mechanism for the computation of association rate constants is presented in Part I. Its application to complexes with fulvics/humics is based on linear complexation Freundlich isotherms. Improvement of this approach would require a better knowledge of the nature of complexing sites and of the mechanism of complexation reaction. Similarly, a more detailed approach for particle/aggregates (Part II) would require a better knowledge of their chemical composition and physical structure. (ii) The reaction of the test metal, M, with sites already occupied by another metal, M′, in particular Ca++ (competition reaction), is not discussed in detail, in particular, because the corresponding rate constants are not available for natural complexants. It has been observed however, for example, for the binding of Pb(II), Zn(II), or Cd(II) by EDTA in presence of Ca(II) as competing ion (54), that the dissociation of the complex Ca-EDTA is much slower than the dehydration rate of M. Thus it can be expected that sites already combined with a competing ion M′ will behave as “nonreactive” sites in the time scale (
