Environ. Sci. Technol. 2009, 43, 7175–7183
Chemodynamics of Aquatic Metal Complexes: From Small Ligands to Colloids H E R M A N P . V A N L E E U W E N * ,† A N D J A C Q U E S B U F F L E ‡ Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands, and CABE, University of Geneva, Sciences II, 30 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland
Received March 25, 2009. Revised manuscript received June 9, 2009. Accepted June 9, 2009.
Recent progress in understanding the formation/dissociation kinetics of aquatic metal complexes with complexants in different size ranges is evaluated and put in perspective, with suggestions for further studies. The elementary steps in the Eigen mechanism, i.e., diffusion and dehydration of the metal ion, are reviewed and further developed. The (de)protonation of both the ligand and the coordinating metal ion is reconsidered in terms of the consequences for dehydration rates and stabilities of the various outer-sphere complexes. In the nanoparticulate size range, special attention is given to the case of fulvic ligands, for which the impact of electrostatic interactions is especially large. In complexation with colloidal ligands (hard, soft, and combination thereof) the diffusive transport of metal ions is generally a slower step than in the caseofcomplexationwithsmallligandsinahomogeneoussolution. The ensuing consequences for the chemodynamics of colloidal complexes are discussed in detail and placed in a generic framework, encompassing the complete range of ligand sizes.
Introduction For more than half a century, the Eigen mechanism has been used to describe the kinetics of metal ion complexation in aqueous systems (1, 2). It is composed of two essential steps, i.e., the formation of a precursor outer-sphere complex, followed by the release of water from the inner sphere of the metal ion to form a coordination bond with the ligand. The major importance of the basic Eigen scheme is that it allows for estimation of the rate of complex formation from the nature of the metal and the charge/size of the ligand, irrespective of the chemical nature of the latter. This greatly facilitates theoretical prediction of rate constants of environmental processes involved with dissociation of metal complexes. Such predictions are of importance since, in spite of existing compilations (2, 3), many rate constants have not yet been determined experimentally. One of the underlying reasons is that natural complexants are chemically heterogeneous, thus incorporating several possible chemical reactions in the overall complexation/decomplexation process. Consequently, the experimentally determined dissociation and association rate constants are not easily linked to each of the singular chemical reactions. A useful criterion to ensure that the determined constants are related to the same chemical reaction is to check that K ) ka/kd. Another * Corresponding author. † Wageningen University. ‡ University of Geneva. 10.1021/es900894h CCC: $40.75
Published on Web 07/15/2009
2009 American Chemical Society
complication is that many natural complexes exist in various forms with different degrees of protonation and different kinetic characteristics, especially in their dissociation. It is also important to emphasize that, even though the dissociation time scales of complexes are usually considered to be the relevant ones for environmental processes, both ka and kd are key parameters. First, for a well-defined chemical reaction, ka and kd are related via K, as mentioned above, and since theoretical predictions of ka values are much easier than for kd, this latter is usually obtained from kd ) ka/K. In addition, for the case of a metal flux at a consuming interface, e.g., in biouptake, both kd and ka govern the chemodynamic behavior of metal complexes. Indeed, the net flux resulting from the dissociation of a metal complex is due to the balance of its dissociation rate, controlled by kd, and its reassociation rate, controlled by kacL (where cL is the ligand concentration). The application of Eigen mechanism principles to large composite natural complexants requires a lot more than the mere electrostatics between two point charges. Therefore, the limits of validity of the basic Eigen scheme and the additional physicochemical concepts necessary for complexes with multidentate and colloidal ligands need to be clearly defined and developed. For instance, one may wonder whether or not the Eigen mechanism principle continues to hold for complexation with sites at solid surfaces in contact with an aqueous solution; or whether the transport of metal ions to/from the binding sites of colloidal complexants is more demanding than for a simple ligand in homogeneous solution. The goal of this paper is to provide a review of the major factors which affect the Eigen mechanism, in particular for the following types of ligands: • Well-defined mono- and multidentate ligands, which may incorporate remote charged groups that play a role in the electrostatics of the outer-sphere complex. The presence of such remote charged groups also calls for consideration of their protonation/deprotonation. E.g., the protonation of uncoordinated negative groups slows down the complex formation rate via a reduction of the stability of the outer-sphere intermediate. • Fulvics and other small sized branched nanoparticles, which can be seen as intermediate between the welldefined multidentate ligands and the colloidal polyelectrolytes. • Colloidal complexants in the typical size range beyond 10 nm, which may bind metal ions at the solid surface or inside the soft colloidal body. Colloidal complexants typically include compact or aggregated inorganic solids (e.g., clays, silica, FeOOH) and soft macromolecular particles and films such as biological cell walls, biofilms, VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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aggregates of humics, etc., as well as combinations of III and IV (core-shell particles). The Basic Eigen Mechanism and Its Rate-Limiting Steps. The two basic steps in the Eigen scheme of complex formation between an aqueous metal ion, M(H2O)2+ 6 , and a small ligand L-, composed of one or a few atoms (e.g., Cl-, CN-, CO32-) are the following (1, 2): k aos
+ M(H2O)2+ 6 + L {\} M(H2O)6 • L
the total volume of the outer-sphere complex positions and the volume of the remaining solution. For sufficiently dilute systems, the latter can be taken as the total volume of solution. In the stationary state of the overall complex formation process, the outer-sphere complex concentration can be taken as essentially time-independent. Then the overall rate constant, ka, for inner-sphere complex formation from M(H2O)62+ and L- is given by (3)
(1)
k dos
ka )
k ais
M(H2O)6 • L+ {\} M(H2O)5L+ + H2O
(2)
k dis
where kaos and kdos are the formation and dissociation rate constants for the outer-sphere complex, M(H2O)6•L+, and kais and kdis are the rate constants for substitution of H2O for Lin the inner-sphere complex and vice versa, respectively. The rate-limiting step for inner-sphere complex formation from M(H2O)6•L+ is the elimination of a water molecule from the inner hydration shell of M(H2O)62+. Its rate constant kais is usually denoted as kw. In the formation of successive complexes M(H2O)4L2 etc., the value of kw depends on the nature of L and often increases with decreasing inner-sphere water (3, 4). The rate constants for the formation/dissociation of the outer-sphere complex (5, 6) are derived from the dimensionless electrostatic pair interaction energy, Uos (equal to the electrostatic energy over kT), and the mobilities of M(H2O)2+ 6 and L as represented by their respective diffusion coefficients DM and DL: k os a ) 4πNAva(DM + DL) k dos )
U os exp(U os) - 1
3(DM + DL) U osexp(U os) a2 exp(U os) - 1
(3a)
(3b)
where a is the charge center-to-center distance of closest approach between the intact M(H2O)62+ and L-. For a single ion pair M · L in an electrolyte solution, Uos is given by U os )
zMzLe 2 κa 14πεε0akT 1 + κa
(
)
(4)
where εε0 is the permittivity of the medium, and zM and zL are the algebraic numbers of charges of M and L, respectively, and κ is the reciprocal Debye length (see section on Electrostatics and Mass Transport in the Colloidal Size Domain for details). The first term between brackets in eq 4 represents the primary Coulombic energy between the metal and the ligand, and the second term corrects the energy for the electrostatic screening effect due to the presence of electrolyte ions (see, e.g., ref 7). The term Uos/(exp (Uos) - 1) in eq 3a and b approaches unity as Uos approaches zero. The limit of zero Uos corresponds with the purely diffusion-controlled encounter between the reactants M and L. The term is larger than 1 for Uos < 0 (the most common case of attraction), and less than 1 for Uos > 0 (repulsion). The ratio kaos/kdos is the outer-sphere stability constant Kos, which is related to Uos via Boltzmann statistics (8): K os )
[M(H2O)6 • L+] 4πNAva3 exp(-U os) ) 3 [M(H2O)2+ 6 ][L ]
(5)
For uncharged ligands Uos is 0, whereas Kos then corresponds to its statistically determined value based on the ratio between 7176
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k os a kw
(6)
k dos + k w
which implies that the Eigen mechanism has two limiting cases: (i) the first step (1) is fast compared to the second (2) and the outer-sphere complex is effectively at equilibrium with free M. Then ka tends to the well-known limit: kdos > > kw: ka ≈ K oskw
(7)
(ii) the formation of the outer-sphere complex is the ratelimiting step and we have k dos < < k w: k a ) k os a
(8)
In general, two types of conditions may lead to situations where kdos values are similar to (or even smaller than) kw. The first case is that of metal ions with very fast dehydration (kw values in the upper range, e.g., Pb2+, Hg2+, Cu2+). The second case is for ligands with a highly negative charge, leading to a large attractive Uos and a relatively low kdos. In general, it is preferable to use the more rigorous eq 6 rather than eqs 7 or 8. The explorative comparison between kdos and kais (kw) is straightforward. For complexes with simple, small ligands, DL and DM are O(10-9) m2 s-1 (O denotes order of magnitude) and the separation distance, a (i.e., rM + 2rH2O + rL), is typically 0.5-1 nm. Hence the basic diffusion-controlled rate constant for bringing the reactants M and L together, i.e., (DM + DL)/ a2 (see eq 3a and b), comes to O(1010) s-1. This value is comparable to that for the kw of very easily dehydrating metal ions such as Pb(II) and Hg(II). For such metal ions, especially in their complexes with relatively small ligands of high charge, ka should be computed from eq 6. A compilation of constants and a quantitative discussion for each metal is given in ref 3. Multidentate Ligands. Uos for Multicharged Ligands. For a multidentate ligand L, with several charged sites i, eq 4 for Uos has to be generalized to include all electrostatic terms involved in its interaction with the metal ion (9): U os )
zMe 2 4πεε0kT
∑ a [1 - 1 + κa ] n
zi
i
i
κai
(9)
i
where n is the total number of charged groups in L, and ai is the charge center-to-center distance between the metal ion and site i. Computation of Uos now requires knowledge of the detailed molecular structure and orientation of the ligand in the outer-sphere complex. The charged sites generally are at various positions and not necessarily all will be involved in eventual inner-sphere interaction with the coordinating metal ion. In such cases it may be useful to express the summation over all charged sites i in eq 9 by an effective outer-sphere ligand charge parameter zˆL (9), defined such that it represents the electrostatically equivalent situation of one ligand with charge zˆL in the outer-sphere. If L is not rigid and its conformation in the outer-sphere complex differs from that of free L in solution, it may be necessary to include in eq 9 a term that accounts for the pertaining change
in intramolecular electrostatic energy. In detailing outersphere intermediates and their stabilities, it should be borne in mind that even for 1:1 complexes with monatomic ligands the uncertainty is appreciable. It has been estimated to be at least (0.5 in log Kos (9). Outer-Sphere Ligand Protonation. In complex systems where the ligand occurs in both protonated and unprotonated forms, the scheme of precursor outer-sphere complex formation will be inherently more differentiated. Let us consider the case of a ligand L that is present in various protonated forms HiL. For the formation of 1:1 inner-sphere complexes, which may be monodentate or multidentate (chelate), the Eigen scheme then becomes
where we have assumed that (i) the dehydration step is the rate-limiting one, i.e., the system is in the limiting case of eq 7, (ii) all protonation/deprotonation reactions are at equilibrium, and (iii) the rate of the dehydration step is essentially unaffected by the presence or absence of protons in the ligand, i.e., kw is a constant for the different species M(H2O)6 · HiL(2+m-i). The overall rate of inner-sphere complex formation, Ra, equals the sum of the contributions from the unprotonated and the protonated outer-sphere complexes. For example, for the simplest case of n ) 1, i.e., L- and HL only: Ra ) kw{[M(H2O)6 · L+] + [M(H2O)6 · HL2+]} ) os os cMcL + kwKM•HL cMcHL (11) kwKM•L
where cM, cL, and cHL are the concentrations of M(H2O)62+, L, and HL, respectively. Kos is the crucial parameter for the actual activity of the reactive species that produce the eventual inner-sphere complex. One may easily encounter cases where the protonated form dominates the ligand speciation (cHL > cL), but where the complex formation rate is governed by the unprotonated form with its higher charge, i.e., larger Kos. Scheme 10 shows that protonation of the ligand may have two effects on the kinetics of complex formation. First the electrostatic energy, and thus the Kos value, of an outer-sphere complex will decrease when the protonation of L increases, since its net negative charge decreases. Second, the proton may compete with the metal ion for the inner-sphere binding sites of the ligand. Confirmation for Cu(II) and Ni(II) binding by partially protonated macrocyclic ligands has been reported (10). Inner-Sphere Ligand Protonation. Ligand protonation of outer-sphere complexes impacts on the complex formation rate constant, ka, whereas ligand protonation/deprotonation at the level of inner-sphere complexes affects the dissociation rate constant, kd. The latter occurs with complexes of the type MHiL in which the proton(s) are in competition with the metal ion for inner-sphere binding sites. Very common complexes such as those with EDTA at modest pH fall in this category (11). The overall rate of inner-sphere formation of these complexes sums the contributions from all possible outer-sphere complexes in accordance with eq 11. For EDTA, the total number of protolytic groups, n, is 6, and the charge in fully protonated form, m, equals +2. The overall dissociation rate comprises the individual dissociation rates of the various forms of inner-sphere complexes:
n
Rd )
∑k
d,icMHiL
(12)
i)0
Generally, the overall dissociation rate of the complex system increases with decreasing pH. This is due to protonation of inner-sphere binding sites and consequent decrease in the number of bonds in the chelate, i.e., decrease in strength of the complex. For instance, in the case of Cd(II)/Y (Y ) EDTA), the thermodynamic stability of CdHY- is some 7 orders of magnitude lower than that of the unprotonated form, CdY2-. The correspondingly higher dissociation rate constant of the protonated form governs dissociation of Cd(II)/Y up to pH ca. 9 (where the equilibrium concentration of CdHY- is insignificant compared to that of CdY2-). These kinetic features are consistent with measured labilities of the Cd(II)/Y system (11). For many types of complexes an analysis such as that for Cd(II)/Y is desirable. This will require determination of both the stability constants (K) of the pertinent protonated inner-sphere complexes, and the overall dissociation rate constants of the complex system in the relevant pH regime. The result can then be compared to the value predicted on the basis of the comprehensive Eigen scheme (eq 11). It is not only the ligand but also the coordinating metal ion that may be involved in protolytic effects, e.g., in case of highly charged ions such as Fe3+. At circumneutral pH, relevant Fe(III) species are Fe(H2O)5OH2+, Fe(H2O)4(OH)2+, Fe(H2O)3(OH)30 and Fe(H2O)2(OH)4-, simply formed by fast deprotonations of Fe(H2O)3+ 6 . All these species may react with a ligand L, according to the Eigen mechanism (eqs 1 and 2), with or without elimination of OH- as the second step. However, their associative kinetic properties vary immensely, primarily due to the large differences in dehydration rate constants, kw (values for Fe(III) vary from O(102) s-1 for 9 -1 for Fe(H2O)2(OH)Fe(H2O)3+ 6 to O(10 ) s 4 (4)). These features have important consequences on the overall complexation/ dissociation kinetics of Fe(III). The various deprotonated forms of Fe(H2O)63+, with the much faster complex formation rate constants, will often govern the chemodynamics of Fe(III) in aqueous systems, even under conditions where they are minor equilibrium species. Interestingly, the (de)protonation of the coordinating metal ion is particularly significant for high valency ions like Fe3+, Al3+, or Cr3+. Thus the weak dissociative reactivity of their relatively strong complexes is to some extent compensated for by the easy proton release from their hydration shell (4). Electrostatics and Mass Transport in the Colloidal Size Domain. The transition from a multidentate ligand to a colloidal complexant with a certain metal binding site density is a gradual one. It involves abandoning the computation of key parameters such as Kos on the basis of well-defined locations of sites (with respect to each other and the coordinating metal). The fundamental factors to be considered are briefly depicted in general terms below, and will be applied in following sections. Electrostatics. Figure 1 illustrates the typical electrostatic potential profiles as encountered in the nanoparticulate and colloidal size regimes, for negatively charged sites, S-. The schematic picture holds for 1D, 2D, and 3D reactions. A 1D reaction would be on a linear polyelectrolyte, a 2D reaction on a charged surface, e.g., a single hard colloidal particle or the hard subunits inside a porous aggregate, and a 3D reaction, e.g., inside a charged volume of a permeable soft colloid. Site densities are related to the separation distance, l S, between adjacent sites. The characteristic electrostatic parameter of the surrounding electrolyte solution is the Debye length, usually denoted as κ-1. It is a measure of the radius of the charge-screening ionic atmosphere around the charged site S, defined by (12) VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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κ-1 )
(∑ ) εε0RT
F
2
z i2c *i
1/2
(13)
i
where the summation over z2i c*i counts all ions in the medium by their charge zi and bulk concentration c*. i By comparing l S and κ-1, we can distinguish between three different types of potential profile (Figure 1): • The low charge density regime, with l S . κ-1, where the potential, ψ, is dominated by the local Coulombic field around each individual site S-. This case may hold especially in marine environments, where κ-1 may be as small as a few tenths of a nm (κa typically larger than unity). The electrostatic energy Uos for an ion pair between a hydrated metal ion M and a site S- is then approximately given by eq 4, provided that the screening term κa/(1 + κa) is not significantly modified by the presence of the colloidal entity. This may occur, for example, with a hard colloid where the surface outer-sphere complex [S · M(H2O)6]+ is not completely surrounded by a homogeneous electrolyte solution. Rigorous computation of the electrostatic energy of the corresponding dipole, at the interface between different phases with different permittivities and conductivities, is quite involved (13, 14). Thus, even for the straightforward case of individual interaction between a metal ion and a single colloidal surface binding site, the impact of screening must be carefully considered. • The high charge density regime, with l S , κ-1, where the potential ψ predominantly results from the cooperative electrostatic forces of many adjacent sites S-. Now the primary Coulombic interaction at a given site S- is small compared to the overall smeared-out potential fixed by the entirety of charges. For a given type of geometrysline, surface, or volumesthe magnitude of the potential ψ can be computed from the charge density. Well-known examples of such potential/charge relationships are those for (i) polyelectrolytes based on line charge density, with the phenomenon of ‘counterion condensation’ occurring for l S below the Bjerrum length zMzLe2/4πεε0kT (15, 16), (ii) charged surfaces (GouyChapman-Stern equations (12, 17), and (iii) volume charges (Donnan equations (18)) in, e.g., soft colloids and films. Uos, in 3a, is then computed by Uos ) zMeψ/kT. The major impact on Kos, kaos or kdos can be seen as a drastic change in the activity of the free metal ion, M(H2O)62+, from the bulk medium to its position in the outer-sphere complex. Estimation of the potential ψ, at that position, is the crucial step in the analysis. For highly charged oxidic surfaces, or polyelectrolytes such as fulvics, ψ can easily be -100 mV (3, 12, 18, 19), which gives rise to activity enhancement factors () e- zMFψ/kT) over 1000 for divalent metal ions. • The intermediate charge density regime, with l S ≈ κ-1, where both the primary local Coulombic term and the smeared-out potential due to the presence of adjacent sites count in the overall potential ψ. In that case, rigorous computation of the value of ψ at the individual site location is already complicated for well-defined systems, let alone for environmental colloids that are physically and chemically heterogeneous. As illustrated in the following sections, analysis of the chemodynamics of colloidal complexes in
natural systems is so far largely based on interpretation following the above limiting cases for low and high charge density. Distinction should be made between charged sites and metal binding sites: colloids can be highly charged, with only few of the charges being involved in the actual inner-sphere complexation. This is particularly true with natural chemically heterogeneous complexants. For instance, at the usually low trace metal/fulvic concentration ratio in natural waters, metals are bound to strong sites (minor in proportion) and not to the weaker phenolic and carboxylic groups (major in proportion) which fix the electric charge of the fulvics (3, 22). The same is expected with natural heterogeneous colloidal aggregates, even though, due to experimental difficulties, binding properties are often interpreted on the basis of the existence of a single site type. It has been shown that even pure, amorphous iron oxyhydroxide includes a range of different site types (20). Thus further experimental and theoretical developments of the field of colloidal metal complexes is highly dependent on separate manipulations of outer-sphere potential (ψos) and outer-sphere volume (Vos) by independently changing the density of the binding sites S- and that of the other charged sites (also see the section on Colloidal Complexants below). Mass Transport. For small ligands L, the mass transport involved in the complexation reaction with M(H2O)62+ is expressed in the rate constants for formation/dissociation of their outer-sphere complex, eq 3a and b, based on the steady-state diffusion of the two species. For larger ligands, DL becomes smaller than DM and for a colloidal complexant DL can be neglected with respect to DM. Then, the purely diffusive flux, JM, of M to/from a single colloidal complexant equals JM )
DMc *M [mol m-2 s-1] rp
(14)
where JM ) dnM/Adt (where nM ) number of moles of M arriving at the surface with area, A, of the colloid per unit of time, t), and rp is the radius of the supposedly spherical colloid. Since A ) 4πrp2, the total rate of supply of metal ions, RM, to all particles per unit dispersion volume is R M ) cp
dnM ) 4πrpDMc *Mcp [mol s-1] dt
(15)
where cp is the number concentration of colloids and V is the dispersion volume. This should be compared to the chemical association rate of M with sites S of the colloids: Ra ) -
dc *M ) kac *McS [mol m-3 s-1] dt
(16)
where cS is the equivalent smeared-out site concentration over the entire dispersion volume. If we consider the case of a hard colloid with binding sites at the surface only, the site number surface density nS (m-2) is related to cS and cp according to
FIGURE 1. Profile of electrostatic potential ψ, at the distance of closest approach of the hydrated metal ion, along a spatial arrangement of metal binding sites S- for low (a), intermediate (b), and high (c) site densities. 7178
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nS ) NAvcS /Acp
(17)
For a given type of colloid with a constant site density nS, eqs 15 -17 can be combined to yield the ensuing diffusioncontrolled complex formation rate constant as ka )
NAvDM rpnS
(18)
It will be seen below that the effective rate constant with colloidal complexants is a combination of the chemical rate constant given by eq 6 and its diffusion limit given by eq 18. In eq 17 the surface site density is defined for a reaction occurring at the surface of impenetrable colloids, which results in a rp-1 dependence of ka (eq 18). However, other situations may arise, e.g., for a soft colloid with a certain volume site density, where the dependence of ka on particle radius will follow a rp-2 behavior. For colloidal aggregates with fractal dimensions, and possibly adsorbed complexing layers, different dependencies may arise (see below), but the exponent of rp is always in the range 0 to -2 (21). Nanoparticulate Ligands: The Case of Fulvics. Fulvic substances can be described (3, 22, 23) as rather rigid, branched oligoelectrolytes bearing a large negative charge density at pH 6-9. Their molar mass (∼1000 Da) and radius (∼1 nm) is not much larger than that of hydrated metal ions. On average a single fulvic molecule includes not more than one strong complexing site and ca. five negatively charged groups (corresponding to an equivalent concentration of 2 mol dm-3 inside the fulvic molecule). A solution of fulvics includes a large number of similar but not identical molecules. Hence to represent the property of the entirety of fulvics in solution, any measured physicochemical parameter must be interpreted on a statistical basis (24). Thus, for each pH j , can be and ionic strength, an average electric potential, ψ computed on the basis of experimental data on their metal j is averaged (i) over binding sites and charges (3, 25). This ψ the different types of fulvic molecules with different electric fields and (ii) over space within a single fulvic molecule. Averaging over space within the fulvic entity seems to be a reasonable approach for freshwaters in which the Debye length κ-1 is generally larger than the fulvic radius (i.e., case of high charge density discussed above for nanoparticulate and colloidal ligands). Based on detailed electrophoretic measurements as a function of pH and ionic strength (I), the j ) f(pH, I) relationships have been reported (3) and used ψ j /kT (26, 27), and ultimately kaos to estimate Uos via Uos ) zMeψ and kos d (eq 3a and b). The overall chemodynamics of a metalj . On the other fulvic complex will thus strongly depend on ψ hand, as opposed to larger colloidal complexants, the radius of a fulvic particle is not much larger than that of simple ligands and the mass transport of M and L is correctly taken into account by eq 3a and b with the appropriate DL. Note j for fulvics and humics may be rather negative (e.g., that ψ -40 to -100 mV in the pH range 6-9) so that kdos may be smaller than kw, even for metals with not too fast dehydration. Thus, with fulvics the nature of the rate-limiting step in eq 1 should be carefully checked. Figure 2 shows the change of the formation rate constants of Cu(II)- and Ni(II)- fulvic complexes with pH, as computed j /kT, and by eqs 3a, b and 6 with Uos estimated by Uos ) zMeψ j obtained from the pH, I relationship mentioned above (3). ψ For both Cu(II) and Ni(II), ka increases with pH, largely due j which becomes increasingly negative; this effect levels to ψ off in the pH range 8-10. Between pH 4 and 8, the increase in ka for Cu(II) is significantly less than that for Ni(II). This is due to the fact that (i) condition 8 applies to the former, i.e., ka ) kaos and (ii) kaos is proportional to -Uos (eq 3a) for very negative values of Uos (as is the case here for the whole pH range). In contrast, condition 7 applies to Ni(II), so that ka
FIGURE 2. Formation rate constant of Cu(II)- (9) and Ni(II)- (O) fulvic complexes as a function of pH. T ) 25oC, I ) 0.01 mol dm-3. The values of the input parameters (kw for metal ions, D j for fulvics), and the pH for metal ions and fulvics, ψ dependencies of the latter two, are taken from ref 3. is much more dependent on Uos through the term exp(-Uos) (eqs 5 and 7). Simultaneously, inspection of the expressions for the overall dissociation rate constants, kd, (3) and those of the outer-sphere complexes, kdos, shows that kd is independent of Uos for Ni(II) while kd is proportional to -Uosexp (Uos) for Cu(II). Because Uos is strongly negative, the electric field may drastically decrease the value of kd for Cu(II) complexes. Thus, interestingly, the overall electric field slightly enhances the association rate constants and strongly decreases the dissociation rate constants for complexes with rapidly dehydrating metals, e.g., Cu(II). In contrast, the same electric field tends to increase the overall lability of complexes with slowly dehydrating metals, e.g., Ni(II). In principle, this observation applies to any ligand, but it is particularly important with fulvics because of their high electric charge density and the ensuing dependence of Uos on pH and ionic strength. Another important characteristic of fulvics is the intrinsic chemical heterogeneity of their binding sites. Due to their numerous types of reactive sites, the metal complexation has to be represented by a distribution of complexation stability constants, jK, where j denotes the number of the site type. Since the local potential at a given site type changes from one fulvic molecule to another, only the average j is meaningful in evaluating an average ka. Thus, potential ψ j , the average ka provided by eq 3a represents on the basis of ψ all types of sites. Consequently, the distribution of jK is paralleled by a similar distribution of the dissociation rate constants, jkd, equal to jkd ) ka/jK (3, 26, 27). At each site type j, jkd is a composite of jkdis (eq 2) for the particular complexing site j, and kdos (eq 1) which only depends on the global electrostatic properties of the fulvic molecule (3b with Uos j /kT). For a given metal under given conditions, jK ≈ zMeψ values often cover more than 4-5 orders of magnitude, hence the same is true for jkd. Thus a given solution of fulvics may include metal complexes with very different kinetic properties, ranging from inert to fully labile behavior. Colloidal Complexants. Metal Complexation at Solid Surfaces. When an aqueous metal ion, M, reacts with a complexing site, >-S-, at the surface of a solid colloid, a 2D analogue of the Eigen mechanism applies (28). For the case where the rate of dissociation of the outer-sphere surface complex is fast compared to its dehydration step, so that condition eq 7 applies to the dissociation rate constant kdos of the surface complex, the scheme reads VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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K sos
kw
+ >-S- + M(H2O)2+ 6 {\} >-S • M(H2O)6 98
>-SM(H2O)+ 5 + H2O
(19)
with the stability constant of the outer-sphere surface complex defined as K sos )
{>-S • M(H2O)+ 6} [M(H2O)2+ 6 ]{>-S }
(20)
The brackets, { }, denote surface concentrations (mol m-2). Since the dehydration rate constant, kw, is practically independent of the nature of the complexing site (see section The Basic Eigen Mechanism and Its Rate-Limiting Steps), the overall surface complexation rate constant kas (which is often presented as an “adsorption rate constant”) is kas ) kwK sos
(21)
Dynamics of Complexes with Hard Colloids/Aggregates. When a solution of M and a dispersion of impenetrable colloidal complexants are mixed, M reacts at the particle surface, and a concentration gradient is created with respect to the bulk solution. The overall reaction thus includes two successive steps: (i) the diffusion of M toward the colloid, and (ii) the complexation reaction of M with the colloidal surface site(s), according to the 2D Eigen mechanism explained above (31). The same approach can be used to interpret the reaction of M with a porous colloidal aggregate formed with impenetrable smaller units (21, 26, 27, 32). In all cases the complexing sites are confined to the colloid or subunit surface, and this defines the boundary conditions of the reactive diffusion step. The crucial result is an apparent association rate constant kaeff which generally differs from its congener for a homogeneous solution, ka. For a given impenetrable colloidal complexant dispersion, the ratio kaeff/ka in the steady-state diffusion regime is essentially constant (31). For a sufficiently dilute dispersion it can be expressed as
(
)
The stability of the outer-sphere surface complex, >-S · M(H2O)6+, derives from its electrostatic free energy Usos. For a low charge density surface with charge separation distances much larger than the Debye length (see the Electrostatics and Mass Transport in the Colloidal Size Domain section), the impact of neighboring surface sites is still weak and Usos can be computed on the basis of eq 4. Following the Boltzmann statistical approach by Fuoss, it can be shown that for the case of excess of surface binding sites over metal ions M
where rp, cS and cp are defined above. When diffusion is very fast with respect to the chemical reaction, kaeff reduces to ka:
K sos ≈ NAvV sosexp(-U sos)
In the opposite case, when diffusion is the slow step, the quotient is much larger than unity and kaeff is much smaller than ka (and kdeff, kd):
(22)
The volume available for hydrated metal ions to form pairs with sites on a solid surface, Vsos, will generally be smaller than in solution. Thus Ksos and kas will be correspondingly smaller than their 3D counterparts. For the limiting case of surfaces with very high charge densities (case c in Figure 1), the hydrated M is no longer strictly bound to a specific site, but fairly free to move on the whole charged surface, while staying at the distance of closest approach of the surface sites. Usos then approaches zMe ψ, where ψ is the approximately constant potential in the plane of closest approach, which derives from the charge density through an appropriate double layer model such as Gouy-Chapman-Stern (12, 17, 29). The outer-sphere volume has largely evolved from local hemispheres around individual sites into a smeared-out layer, comparable to the cylindrical one around the line charge in the polyelectrolytic counterion condensation model. The exp (-Usos) term in eq 22 may be conceived as the approximate Boltzmann term that corrects metal ion activities for the surface potential ψ. It should be borne in mind that the surface charge and screening terms in Usos represent different electrostatic features. For small surface charge densities, the double layer potential is modest, and so is Usos. The screening term, however, depends on the bulk electrolyte concentration (via the Debye length κ-1) and can be large for low surface charge density. Furthermore, if only a fraction of the charged surface sites are complexing, this does not affect ψ and Usos, but should be accounted for in the corresponding value of Vsos. Pertaining experimental data, for both low and high charge density surfaces, are needed. As mentioned before, for surfaces with intermediate charge densities a rigorous electrostatic theory is still needed. Metal ion adsorption rates at solid oxidic surfaces, e.g., Cu2+ at crystalline goethite (30), have recently been shown to obey the 2D version of the Eigen mechanism, via eq 22 (28). It will be interesting to extend such analysis to other areas of surface reactions of aqueous metal ions, e.g., bioreactor technology, corrosion, etc. 7180
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ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 43, NO. 19, 2009
k eff a /ka ) 1 +
kacS 4πrpDMcp
-1
(23)
eff kacS /4πrpDMcp < < 1 and k eff a ≈ ka (and k d ≈ kd)
(24a)
kacS /4πrpDMcp > > 1k eff a ≈
4πrpDMcp cS
(24b)
which is the same as eq 18 since cS ) 4πrp2cp/NAv (eq 17). The stability constant K is not affected by the diffusion step, eff because the ratio keff d /kd is the same as ka /ka, even for surfaces with large charge density where the rate constants are affected by the surface potential (21, 31). As a consequence, the rates of formation and dissociation of the impenetrable colloidal complex decrease with increasing particle radius according to rp-1. Experimental data for Pb(II) and Cd(II) binding by core-shell nanoparticles confirm the theoretical predictions (33). The above concepts have been extended (21) to polydispersed porous fractal aggregates of impenetrable subunits, e.g., clays, SiO2, CaCO3. In addition, the subunits can be partially covered by thin patches of complexing components, e.g., organic matter, FeOOH. Thus, more than one type of complexing site can be considered. The diffusion of metal ion outside and inside the aggregate is taken into account and the size distribution of aggregates is assumed to follow the ubiquitous Pareto law (21, 26). For each size class j, expressions for jkaeff and jkdeff can be obtained for steady-state conditions, which are shown to be achieved for times larger than fractions of seconds (21). Under steady-state conditions, j eff ka is given by j eff k a /ka
(
) 1+
kac jS∆A/At j 4πjrpD M cp
)
(25)
where jrp and jcp now are the external radius and number concentration of aggregates, j∆A/At is the fraction of surface area (available for adsorption) of size class j aggregates, with respect to that of the entirety of aggregates. It is worth noting that both jcp and j∆A/At are functions of jrp, the size
FIGURE 3. The effective rate constant, keff a , as a function of aggregate radius, rp, as computed on the basis of eq 25. Data are shown for Pb(II) (dashed curve), Zn(II) (solid curve), and Ni(II) (dot-dashed curve), and various fractal dimensions of the aggregates, 1.7, 2.3, and 3.0, as indicated on the curves. The ka values for Pb(II), Zn(II), and Ni(II) on the left-hand axis are computed from eq 6 for a ligand with z ) -1 and kw ) 109.8, 107.5, and 104.5 s-1 for Pb(II), Zn(II) and Ni(II), respectively. For Ni(II) the horizontal dot-dashed line corresponds to coincident curves for fractal dimensions of 1.7 and 2.3, as well as ka. The aggregates are spherical porous assemblies of unreactive spherical subunits (radius ) 5 nm) with an adsorbed layer (thickness ) 1 nm) of a complexing component X. Other parameters: site density on X ) 10 sites per nm2; pH 8; surface potential ) 0; DM ) 9.45 × 10-10, 7.03 - 10-10, and 7.05 - 10-10 m2 s-1 for Pb(II), Zn(II) and Ni(II), respectively; a ) 0.5 nm. distribution parameter, and the fractal dimensions of aggregates. The aggregate suspension can thus be considered as a mixture of complexants of different types, j, reacting with M, each one with its own specific rate constant. The larger the aggregate size, the more jkaeff differs from ka. For very large aggregates, jkaeff may be smaller than ka by orders of magnitude (Figure 3). When the charge density of the primary colloids of the aggregates is large, both rate constants ka and kd are affected by the surface potential, ψ, in the same way as for single hard particles. Under diffusion-controlled conditions, however, jkaeff is not influenced by ψ, since ka cancels in eq 25. For the usual values of the fractal dimension (i.e., well below 3), the average distance between subunit particles inside an aggregate is much larger than the Debye length (see ref (21) for more detail). Then a uniform Donnan potential throughout the aggregate body (as in soft colloids) does not exist. Figure 3 shows the change in kaeff with aggregate radius rp for three metals (Pb(II), Zn(II), and Ni(II)), reacting with aggregates comprised of nonreactive spherical subunits that are partly covered with complexing FeOOH patches with a thickness of 1 nm. For sufficiently small values of rp, the chemical kinetics at the complexing site are rate-limiting, and kaeff ) ka (as discussed in The Basic Eigen Mechanism and Its Rate-Limiting Steps section) for all metals. With increasing values of rp, diffusion becomes more and more important as an additional rate-limiting factor for all metals. Since the metal ions have very similar diffusion coefficients, they all tend toward the same curve for a given value of the aggregate fractal dimension. However, it is clear that the importance of the chemical kinetics of the binding step varies with the dehydration rate of the metal ion: it is almost never limiting for Pb(II), while it is limiting over almost the whole colloidal range for Ni(II). Zn(II) is an intermediate case. It must be emphasized that even Pb(II), which usually forms very labile complexes, may behave very slowly, due to the diffusion process discussed here. Note that aggregates with a fractal dimension approaching 3.0, remain porous and complexation still occurs at the surface of the subunits inside the aggregates. Interestingly,
FIGURE 4. Sketch of a core-shell particle with impermeable core (radius rc) and permeable shell (thickness d). the kaeff value of such aggregates depends on rp-2, i.e., it decreases much more strongly with rp as compared to impenetrable 3D particles (for which kaeff is proportional to rp-1; see above). This reflects the fact that, even though a surface (2D) complexation reaction occurs on the subunits, the site concentration in the aggregate volume is also important (see Electrostatics and Mass Transport in the Colloidal Size Domain section for more detailed discussion). At the other extreme, for aggregates with fractal dimensions
