Chemodynamics of Soft Nanoparticulate Complexes: Cu(II) and Ni(II

Critical Review pubs.acs.org/est

Chemodynamics of Soft Nanoparticulate Complexes: Cu(II) and Ni(II) Complexes with Fulvic Acids and Aquatic Humic Acids Raewyn M. Town,*,† Herman P. van Leeuwen,‡ and Jacques Buffle§ †

Institute of Physics, Chemistry and Pharmacy, University of Southern Denmark, Campusvej 55, DK-5230 Odense, Denmark Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands § CABE, Section de Chimie, University of Geneva, Sciences II, Quai Ernest-Ansermet 30, CH-1211 Geneva 4, Switzerland ‡

S Supporting Information *

ABSTRACT: The dynamics of metal complexation by small humic substances (fulvic acid and aquatic humic acid, collectively denoted as “fulvic-like substance”, FS) are explored within the framework of concepts recently developed for soft nanoparticulate complexants. From a comprehensive collection of published equilibrium and dissociation rate constants for CuFS and NiFS complexes, the association rate constant, ka, is determined as a function of the degree of complexing site occupation, θ. From this large data set, it is shown for the first time that ka is independent of θ. This result has important consequences for finding the nature of the rate limiting step in the association process. The influence of electric effects on the rate of the association process is described, namely (i) the accelerating effect of the negatively charged electrostatic field of FS on the diffusion of metal ions toward it, and (ii) the extent to which metal ions electrostatically accumulate in the counterionic atmosphere of FS. These processes are discussed qualitatively in relation to the derived values of ka. For slowly dehydrating metal ions such as Ni(H2O)2+ 6 (dehydration rate constant, kw), ka is expected to derive straight from kw. In contrast, for rapidly dehydrating metal ions such as Cu(H2O)2+ 6 , transport limitations and electric effects involved in the formation of the precursor outer-sphere associate appear to be important overall rate-limiting factors. This is of great significance for understanding the chemodynamics of humic complexes in the sense that inner-sphere complex formation would not always be the (sole) ratelimiting step.

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• the degree of heterogeneity toward binding of the target M, due to the chemical diversity of the binding sites S both within an individual HS entity and among the entire aqueous dispersion. The typical size of HS ranges from dissolved macromolecular fulvic acid (FA) and small aquatic humic acid (HA) (molar mass 800−2000, rp ca. 1 nm),8−11 with properties close to macroions, to nanoparticulate peat HA (molar mass >5000, rp > 3 nm),8,10,12,13 with properties consistent with semipermeable spherical colloids.14,15 Here we focus on FA and smaller (aquatic) HA, which hereafter we jointly denote as a fulvic-like substance, FS. A typical FS molecule contains a number of charged, weakly binding sites (ca. 5−10 carboxyl groups, 2−4 phenolic groups) and 1 or no strong complexing site, denoted below by S.1,16 The S is not necessarily charged and is possibly based on amino or sulfhydryl groups.1,17−20 FS have been described as small branched nanoparticles (NPs).21 Structures obtained by energy

INTRODUCTION

Humic substances (HS) are ubiquitous heterogeneous complexants in the environment that play a key role in complexing and buffering vital and toxic metal species in aquatic systems.1−3 There is a large body of literature on the thermodynamic features of metal ion complexation by humic substances, including reviews of stability constant data, which are used as the basis for various speciation models.4−7 There is increasing awareness that the kinetic properties of metal complexes, i.e. rates of association and dissociation, are fundamental to determining their reactivity and bioavailability. However, to date the chemodynamic features of metal−humic complexes have generally not been explored in a rigorous quantitative manner. Several interrelated and interdependent factors impact on the dynamics of metal−humic complex systems, notably: • the size of the HS entities, which determines the relative importance of diffusion of the metal ion, M, to/from (and within) the particles, • the density and spatial distribution of charged sites within HS, • the density and spatial distribution within HS of reactive binding sites, S, which are not necessarily charged but which may form a chemical bond with the target M, and © 2012 American Chemical Society

Received: Revised: Accepted: Published: 10487

May 6, August August August

2012 7, 2012 30, 2012 30, 2012

dx.doi.org/10.1021/es3018013 | Environ. Sci. Technol. 2012, 46, 10487−10498

Environmental Science & Technology

Critical Review

Figure 1. Stepwise complexation of a hydrated metal ion, Mz+ aq , with FS (rp ≈ 1−1.5 nm) containing one charged or uncharged binding site (S) and ca. 5−10 charged sites (-). The steps are (1) diffusion of Mz+ aq from the bulk solution to the surface of the FS, (2) its penetration into the diffuse double layer of the FS and possible partial incorporation within the FS as a free hydrated ion (the dashed circle around Mz+ aq denotes its location somewhere within the extraparticulate spherical surface), (3) outer-sphere association of Mz+ aq with S, and (4) inner-sphere complex formation, including the loss of water of hydration by Mz+ aq and formation of a chemical bond with site S, either or not involving some conformational change.

Recently, a more rigorous conceptual framework has been presented for describing the chemodynamics, particularly electrostatics, of small and large soft nanoparticulate metal complexes.35 The framework elaborates the fundamental roles of the charge density, binding site density, and size of the nanoparticle in the various complex association and dissociation steps. It has been used to show that for large soil and peat humic acids ka is nonvariant over a range of metal-to-ligand ratios.36 However, in contrast to the usual assumption, the rate of diffusive supply of Mz+ aq was found to be significant in determining the overall rate for the case of the rapidly dehydrating Cu2+ ion.36 This is in agreement with recent reports that for HS the kd values that are derived via conventional Eigen ka values for simple homogeneous systems, are significantly greater than measured ones.37 It can be envisaged that other processes might be rate-limiting, e.g. the rate of dehydration of the HS molecule, conformational changes in the ligand, perhaps involving breaking of intramolecular hydrogen bonds, etc., but presently there is no evidence to invoke such factors. For simple ligands with special structures, notably macrocyles and chelates, the ring-closure step may be overall rate-limiting.38−41 A characteristic feature in such cases is a very low rate of association which is practically independent of the nature of the metal ion, even when kw varies over 4 orders of magnitude.42 In the present paper we show, via a comprehensive collation of literature experimental data, that ka is independent of the nature of the binding site. This result is discussed in terms of the electrostatic and chemodynamic features of metal binding by FS. The framework for describing metal complexation dynamics in soft nanoparticle dispersions35 is taken as the starting point for developing concepts applicable to the case of oligoionic FS, which has properties intermediate between those of simple multicharged ligands and gel-like charged nanoparticles. As compared to soil and peat HA, FS are smaller and have a higher charge density. Consequently, it will be crucial to consider the magnitude and impact of electric effects on the rate of association of metal ions with FS. By scrutiny of complexation data for different metal ions having either fast or slow dehydration rates we explore the potential role of electric effects in determining the rate limiting step. The approach is a differentiated version of the Eigen mechanism34 with four steps in the overall complexation

minimization/molecular modeling calculations suggest that the geometry of an FS molecule is closer to a 2D folded plane than to a 3D structure,18 and high conformational flexibility is predicted.22 The electric charges within the FS molecule can be significantly delocalized due to the many electron-withdrawing groups and the small length of aliphatic chains. Thus, one can speak in terms of an electric field specific to the organic molecule, but since FS molecules are not much larger than a z+ hydrated metal ion Mz+ aq , the Maq does not really fully penetrate into the FS body. In this respect, the nature of the reaction of Mz+ aq with fulvics and small humics lies between that for the reactions with a simple multicharged ligand and with a gel-like nanoparticle. Indeed, the term “oligoion” is a convenient descriptor of their intermediary nature.23 FS are chemically heterogeneous, and comprise an ensemble of a large number of different molecules.24,25 Indeed, ca. 5000 individual types of molecules have been identified in Suwannee River fulvic acid by mass spectrometry.26 At very high concentrations, greater than ca. 1−10 g dm−3, formation of aggregates has been observed.27−30 Molecular dynamics simulations of a model FS molecule suggest that aggregation occurs above concentrations of 70 g dm−3.31 Such aggregates are likely held together by weak noncovalent interactions, such as hydrogen bridging, and are easily disrupted by small amounts of small organic acids.32 However, FS concentrations in natural waters are typically 5−10 mg dm−3, and aggregation has been shown to be negligible at such low concentrations.10 Thus aggregation of FS will not be considered in this paper. The chemical heterogeneity of FS implies that its thermodynamic and kinetic properties are distributed. Usually a nonvariant association rate constant, ka, is assumed for the binding of Mz+ aq by the site S. The distribution of the stability constant K is then inversely reflected in that of the dissociation rate constant, kd,16,33 whose values are computed from measured stability constants at the same metal ion loading via kd = ka/K. In this, ka can be estimated from the Eigen mechanism, using either the conventional34 or the generalized16 approach. In the conventional approach, the inner-sphere complexation process, assumed to be metal ion dehydration, is rate limiting. The generalized approach considers that outer-sphere and inner-sphere processes may have similar rates for rapidly dehydrating metal ions, and the outer-sphere processes may even be rate-limiting under certain conditions. 10488



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reaction: (i) diffusion of Mz+ aq from the bulk solution to the surface of the FS, (ii) penetration of Mz+ aq into the diffuse double layer of the FS and possible partial incorporation within the FS as a free hydrated ion, (iii) formation of an outer-sphere reactant pair between Mz+ aq with S, followed by (iv) release of water from the inner-sphere of the metal ion and formation of a coordination bond between Mz+ and the binding site S. The overall reaction scheme is shown in Figure 1, and any step may be rate-limiting.

Mz+ aq , weaker and weaker sites are occupied so that K* decreases as the overall site occupation θ increases. Analogous to the definition of K*, the weighted average value of the dissociation constant k*d is a function of individual values of this constant for each site type kdi, and the corresponding values of Δχi and θi:47 n

kd* =

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∑i = 1 θi(1 − θi)Δχi n

1

∑i = 1 k θi(1 − θi)Δχi di

The k*d values are obtained from measurements of free Mz+ aq liberated from MS complexes as a function of time.54−57 Since usually weak complexes are more easily dissociated than strong ones, short times correspond to complexes with a small K* (large θ) and large k*d values. The value of k*d (and θ) decreases with time along the dissociation kinetic curve. A kinetic spectrum is constructed by plotting the proportion of bound metal as a function of dissociation rate constant. The kd abscissa can be divided into arbitrary segments with corresponding surface area to yield a series of k*d ,θ couples. Each point corresponds to the effective kd in an infinitesimally small perturbation of equilibrium; consequently division of the kinetic spectra into different segments will give different k*d ,θ couples, but they will all belong to one and the same k*d ,θ curve. The above considerations highlight the utility of the differential functions: ka obtained from K*k*d at a given θ is the best estimate of the association rate constant for the specific reaction of Mz+ aq with the site type that is the most reactive at that particular θ, and it does not matter whether or not that particular complex has been formed via successive formation of other complexes. That is, K* and k*d enable computation of ka for a given type of site, as if this site type were alone in the solution. Electrostatic Characteristics of Humic Substances in Aqueous Electrolyte. The electrostatics of a soft nanoparticle derive from its total charge content as well as the charge and concentration of electrolyte ions. As usually occurs in practice for metal ion binding by FS, we shall assume that the electrolyte concentration is in large excess over that of the complexing z+ metal ion Maq . Different types of potential profiles are distinguished based on the relative magnitude of average separation between charged sites, SC , as compared to the Debye length, κ−1. A typical FS molecule with radius rp ≈ 1−1.5 nm, contains ca. 5 to 10 carboxyl groups, respectively.1,58 Given the probable sterical constraints, it is likely that in the ionic strength range of 0.001−0.1 mol dm−3, SC is ≪ (or 100 GΩ. The working electrode was a Metrohm multimode mercury drop electrode, the auxiliary electrode was glassy carbon, and the reference electrode was Ag|AgCl|KCl(sat) encased in a 0.1 mol dm−3 KNO3 jacket. Measurements were performed at 20 °C. The following conditions were used: deposition potential, Ed = −1.35 V, oxidizing current, Is = 2 × 10−9 A applied in quiescent solution until the potential reached a value well passed the transition plateau. Reagents. All solutions were prepared with deionized water from a Milli-Q Gradient system (resistivity >18 MΩ cm). Ni(II) solutions were prepared by dilution of a commercial certified standard from Aldrich. The ionic strength was maintained with KNO3 (TraceSelect), and the pH was maintained at 5.0, 6.0, or 7.5 by use of acetate, MES, or MOPS buffers, respectively. Solutions were initially purged with oxygen-free nitrogen ( −1.48,52 The compiled K̅ data for Ni exhibit low or no dependence on θ (data not shown), even for log θ ≪ 1.68,78,79 This result may be a consequence of a significant amount of the Ni(II) being counted as inner-sphere complexes, whereas in reality it is retained electrostatically, i.e. nonspecifically. The extent of electrostatic accumulation would be independent of θ (for a sufficient excess of charged sites) thus yielding a constant K̅ . As detailed above, for our purposes the appropriate stability constant format is the differential equilibrium function K*. However, none of the literature data are useful for computation of K* due to either too few measurement points and/or * ). Thus, we measurements at too low α values (α = cM,t/cM measured FS complexation of Ni2+ by SCP (see Experimental section for details). The resulting K* values applicable to pH 8.0 are shown as a function of θ in Figure 2. The slope corresponds to Γ ≈ 0.7, in line with the moderate heterogeneity reported for Ni−humic complexes.4 Selected k*d Values. The k*d values were computed from kinetic spectra published by Cabaniss54 (pre-equilibration and measurement at pH 8.4) and Lavigne et al.55 (pre-equilibration at pH 5 or 6.4, measurement at pH 7.8). The k*d values are shown as a function of log θ in Figure 3.

Figure 2. Log K* values extrapolated to pH 8.0 for NiFS as a function of log θ. Results are determined from SCP measurements at I = 0.1 mol dm−3 (blue ⧫) and I = 0.01 mol dm−3 (○). Following usual practice, the degree of metal ion binding, θ, is expressed in terms of the concentration ratio of bound metal to carboxyl groups in the same weight of FS sample. The error bars correspond to the standard deviation in the log K* values (see Supporting Information (SI) for details).

Figure 3. Dissociation rate constants, k*d for NiFS. Data correspond to SRFA (I = 0.1 mol dm−3, blue ■; I = 0.03 mol dm−3, blue □; I = 0.01 mol dm−3, blue ▲; I = 0.003 mol dm−3, blue △; I = 0.002 mol dm−3, blue ●; I = 0.001 mol dm−3, blue ⧫)54 and soil FA (I = 0.1 mol dm−3, pre-equilibration at pH 5, red ⧫, or pH 6.4, ●).55 Dashed line is best fit through all the data points except those pre-equilibrated at pH 5. From the linear least-squares regression, the standard deviation in the predicted log k*d values is ±0.17 (see SI for details).

Thermodynamic and Kinetic Data for Cu. Selected K Values. All k*d values for CuFS are measured at pH 7.5 (see below). At the pH values above ca. 7, it is possible that, at equilibrium, some mixed Cu−FS−OH complexes form80 and influence the value of the measured equilibrium constant. Since it is not straightforward to quantify and account for this effect,81 we have chosen not to use titration data at pH values above 7, but rather to base our interpretation on K* values obtained by 10492



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extrapolation up to pH 7.5 of the data published in the pH range 5.5−7 at a given θ. This procedure maximizes the probability that the obtained K* values correspond to CuFS complexes, without the influence of Cu−FS−OH species. At each ionic strength, plots of K* as a function of pH were constructed from published data.81−96 From this collated information, log K* values for CuFS effective at pH 7.5 were obtained as a function of log θ for different ionic strengths. Further details of this procedure are given in the SI. Results are shown in Figure 4. The slopes correspond to Γ in the range

Figure 5. Dissociation rate constants, k*d , obtained from kinetic spectra for CuFS and CuHA. Data correspond to (i) estuarine FS fractions,56 I = 0.1 mol dm−3, downstream, large size fraction (i.e., between 0.45 μm and 3.1 nm ultrafiltration membrane pore size), blue ⧫; upstream large size fraction, blue ■; downstream, small size fraction (i.e., < 1.0 nm ultrafiltration membrane pore size), blue ◊; upstream, small size fraction, blue ◻), (ii) soil HA57 (Latahco I = 0.2 mol dm−3, ○, I = 0.02 mol dm−3, ●; Elliot I = 0.02 mol dm−3, ⧫; Latahco I = 0.001 mol dm−3, ◊), (iii) soil FA57 (Elliot, I = 0.02 mol dm−3, red ■), and (iv) Latahco soil HA pre-equilibrated at pH 5.5,57 I = 0.02 mol dm−3 (▲). Unless indicated otherwise, samples were pre-equilibrated and measured at pH 7.5. Dashed line is best fit through all data points. From the linear leastsquares regression, the standard deviation in the predicted log k*d values is ±0.34 (see SI for details).

Figure 4. log K* values extrapolated to pH 7.5 for CuFS as a function of log θ at, I = 0.1 mol dm−3 (⧫), 0.01 mol dm−3 (red ■), and 0.001 mol dm−3 (blue ▲). The K* values are obtained by extrapolation of values computed from published titration curves in the pH range 5.5− 7.0.81−95 The error bars correspond to the standard deviation in the log K* values (see SI for details).

from K* and k*d values (Figures 2 and 3, respectively) applicable at pH 8.0. The results are shown in Figure 6. To avoid significant extrapolation of the K* and k*d values outside the θ range under which they were determined (Figures 2 and 3), ka values were derived in the log θ range −3.0 to −2.0. Copper(II). The ka was obtained from K* and k*d values (Figures 4 and 5, respectively) applicable at pH 7.5. The results are shown in Figure 7. To avoid significant extrapolation of the K* and k*d values outside the θ range under which they were determined (Figures 4 and 5), ka values were derived in the log θ range −3.5 to −1.5.

0.4−0.5, in agreement with the significant heterogeneity reported for CuFS complexes.97−99 Selected k*d Values. The k*d values that were computed from published spectra are shown as a function of log θ in Figure 5. Data for FS as well as larger HA are shown since the values lie in the same range. Olson and Shuman56 measured size fractions of organic matter isolated from an estuary with a 12-h preequilibration of Cu and FS. Even though the nature of the organic matter may be different in the various size fractions, and in the upstream vs downstream samples, the k*d vs θ dependence is comparable across all samples. (These data appear to be the most complete interpretation of a data set originally published by Shuman et al.100 and analyzed by Buffle1 and Langford et al.101). Stanley et al.57 measured soil FA and HA with 12-h pre-equilibration of Cu and HS. The amount of the total bound Cu liberated during the kinetic experiments was not stated. Therefore, since conditions were the same, the average of 50% observed by Olson and Shuman56 was assumed to be applicable to the Stanley et al. data.57 In both cases most of the unmeasured Cu is the free fraction. The convergence of the data for the various pre-equilibration conditions (given above) implies that assuming a preset electric field is realistic in the θ range of the measurements.

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DISCUSSION For both Ni(II) and Cu(II), the overall association rate constant ka derived from the experimental K* and k*d values is virtually independent of θ (Figures 6 and 7). Hence ka does not vary with the nature of the reactive site in the chemically heterogeneous FS complexants. The result establishes that the site distribution of K* is inversely reflected in that of k*d , for both moderately heterogeneous Ni(II) and significantly heterogeneous Cu(II) complexes with FS. This outcome also highlights the utility of the differential parameters, in which both the thermodynamic and kinetic functions are defined in terms of the same conceptual basis. The results are rather consistent and derive from a large number of different independent data sources, indicating the robustness of the approach. As well as being independent of θ, the ka values obtained for Cu(II) are much greater than those for Ni(II). This result suggests that if inner-sphere complexation is overall rate-limiting,

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RESULTS Rate of Association of MFS Complexes. Nickel(II). The overall rate constant for complex formation, ka, was obtained 10493



Critical Review

acceleration, fel) as well as to increase the rate of inner-sphere complex formation (via the increase in the local Mz+ aq as determined by the Boltzmann equilibrium factor, f B). It has been previously demonstrated that upon decreasing ionic strength, at a given charge density, the extent of enhancement of the limiting rate of inner-sphere association by the increase in f B is far greater than the extent to which the transport of the metal ion toward the particle is accelerated by f el.36 Furthermore, for very small nanoparticles, as the ionic strength decreases, the extraparticulate double layer volume increases, and consequently a smaller portion of the incoming flux of Mz+ aq contributes to MS formation. This feature offsets the increase in the magnitude of the supply flux due to the increase in the electric field. Accordingly, as the electric field increases and/or as the ionic strength decreases, Risa increases to a far greater extent than does Ra,p, and thus it becomes increasingly likely that the rate of diffusive supply of Mz+ aq will play a role in the rate-limiting step in MFS complex formation, particularly for fast inner-sphere complex formation, that is rapidly dehydrating metal ions if dehydration is the rate-limiting step. Generally, one may expect that if inner-sphere complex formation is overall rate limiting, then the rate will increase as the ionic strength decreases, while if formation of the outer-sphere precursor is overall rate limiting then the rate will be only weakly dependent on the ionic strength. For complexation of Ni(H2O)2+ 6 by FS, the experimentally derived ka (m3 mol−1 s−1) is found to increase as the ionic strength decreases (Figure 6). This is in agreement with innersphere complex formation being the overall rate-limiting step for this slowly dehydrating metal ion (kw = 104.5 s−1 102−112). That is, the settling of the electric field and the attainment of Boltzmann equilibrium between the free Ni(H2O)2+ 6 inside the counterionic atmosphere of the FS entity and that in the sample solution, as well as conductive diffusion of Ni(H2O)2+ 6 from the bulk solution toward FS, is likely achieved on a time scale shorter than that of dehydration of the metal ion. In contrast, for Cu(H2O)2+ 6 complexation by FS, the experimentally derived ka is approximately independent of ionic strength (Figure 7), suggesting that the rate of diffusive supply of Cu(H2O)2+ 6 plays a role in the rate-limiting step for this rapidly dehydrating metal ion (kw = 109.6 s−1 113−116). That is, electric equilibration processes contribute to limiting the outer-sphere association rate for the fast dehydrating Cu(H2O)2+ 6 ion and the inner-sphere complex formation already occurs in the initial time domain of the evolving Boltzmann transient. This result is of great significance for interpretation of the chemodynamics of FS complexes with rapidly dehydrating aqueous metal ions, such as Cu2+, Pb2+, and Cd2+. For such metal ions, the assumption that inner-sphere complexation is rate-limiting would yield predicted dissociation rate constants (and hence predicted labilities of the MFS complexes) that are too high. Thus the present result may resolve a long-standing mystery (or apparent “failure of Eigen”) for ions such as Cu and Pb where the measured ka is always found to be too low by some orders of magnitude.

Figure 6. Overall association rate constant for NiFS, ka, derived from experimental K* and k*d values applicable at pH 8. The data correspond to K*k*d at I = 0.1 mol dm−3 (red points and red dotted shading) and I = 0.01 mol dm−3 (blue points and blue dotted shading). The points are example ka values to guide the eye. The error bars correspond to the standard deviation in the individual log ka values (see SI for details).

Figure 7. Overall association rate constant for CuFS, ka, computed from K* and k*d values derived from experimental data and applicable at pH 7.5. The ka values for all ionic strengths lie within the black dotted band. Points, and corresponding error bars, are examples to guide the eye and correspond to K*k*d at I = 0.1 mol dm−3 (⧫), I = 0.01 mol dm−3 (blue ●), and I = 0.001 mol dm−3 (red ▲). The error bars correspond to the standard deviation in the individual log ka values (see SI for details).

then metal dehydration is the most likely process underlying kisa (eq 11). For example, if some other process, e.g. chelate ring closure, would be rate-limiting then the ka should be expected to be practically independent of the nature of the metal ion.42 Consideration of the Rate-Limiting Step in Metal Association with FS. If no other processes than those described in Figure 1 influence the association rate, then the step which is rate-limiting for association of Mz+ aq with FS depends on the relative magnitude of the rate of diffusive supply of Mz+ aq , Ra,p (eq 8) as compared to the rate of innersphere complex formation, Risa (eq 10). For the negatively charged FS, the electric field serves to enhance the diffusive z+ supply flux of Maq (via the coefficient for conductive

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OUTLOOK A model is proposed as a basic conceptual framework for describing metal ion binding by small humic substances. The approach distinguishes between free metal ions that are accumulated in the extraparticulate diffuse double layer, and metal species within the reaction zone containing the reactive sites. The results highlight the significance of electric 10494



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Γ kos a

degree of heterogeneity rate constant for outer-sphere ion pair association (m3 mol−1 s−1) is ka rate constant for inner-sphere complex formation (m3 mol−1 s−1) ka overall rate constant for complex formation (m3 mol−1 s−1) ka,p rate constant for diffusive supply of Mz+ aq to a spherical particle (m3 mol−1 s−1) kos rate constant for outer-sphere ion pair dissociation (s−1) d is kd rate constant for inner-sphere complex dissociation (s−1) kd overall rate constant for complex dissociation (s−1) k*d differential rate constant for complex dissociation (s−1) kw rate constant for water substitution (s−1) Kos stability constant for the outer-sphere reactant pair (m3 mol−1) K* differential equilibrium function (m3 mol−1) κ−1 Debye length (m) SC separation distance between charged sites (m) Mz+ aq hydrated metal ion NP nanoparticle S site that covalently binds a metal ion θ degree of occupation of sites, S Uos dimensionless interaction energy between Mz+ aq and a simple ligand Vos outer-sphere volume for an ion pair between Mz+ aq and an individual site S (m3) Vp volume of the FS entity (m3) Ra,p rate of diffusive supply of Mz+ aq toward a spherical particle (mol m−3 s−1) Risa rate of inner-sphere complex formation (mol m−3 s−1) rp particle radius (m) τrel characteristic time constant for electrostatic relaxation (s) ψ̅ p average electrostatic potential difference between the solution and the particle body (V)

equilibration processes as a rate-limiting step in the complex formation of fast dehydrating metal ions. The concepts provide a basis for more in-depth analysis of the impact of electric effects on the association and dissociation rates of such complexes under a range of environmental conditions. Furthermore, the theory elaborated herein facilitates extension of the initial framework outlined for a wide range of types of particles, e.g. FS sorbed on mineral particles.21 Important issues that require further detailed consideration, include the following: • Development of new experimental approaches to simultaneously measure K* and k*d values over a wide range of metal-to-ligand ratios. • The variation of the particle charge with the degree of metal ion complexation, θ. The present analysis deliberately considers low θ values at which the particle electric field is not significantly affected by charge compensation upon complexation of Mz+. However, for larger (changes in) θ, the effective f B and fel will decrease with increasing extent of charge compensation. • Increased knowledge of the flexibility of FS molecules and the role of conformational changes in the overall complexation kinetics. • Development of a quantitative electrodynamic transport model. This should include the Boltzmann partitioning coefficient, f B(r), as well as the coefficient for conductive acceleration, fel(r), and the spatial domains over which it is operative, as a function of ionic strength and pH. • Characterization of the physical heterogeneity of FS in terms of the spatial distribution of its electric charges, and their impact on the chemical heterogeneity of reactive sites.

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ASSOCIATED CONTENT

S Supporting Information *

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Procedure for obtaining the log K* values and estimation of errors in the various parameters. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

(1) Buffle, J. Complexation Reactions in Aquatic Systems: An Analytical Approach; Ellis Horwood: Chichester, 1988. (2) van Leeuwen, H. P.; Town, R. M.; Buffle, J.; Cleven, R. F. M. J.; Davison, W.; Puy, J.; van Riemsdijk, W. H.; Sigg, L. Dynamic metal speciation analysis and bioavailability of metals in aquatic systems. Environ. Sci. Technol. 2005, 39, 8545−8556. (3) Saar, R. A.; Weber, J. H. Fulvic acid: modifier of metal-ion chemistry. Environ. Sci. Technol. 1982, 16, 510A−517A. (4) Milne, C. J.; Kinniburgh, D. G.; van Riemsdijk, W. H.; Tipping, E. Generic NICA-Donnan model parameters for metal-ion binding by humic substances. Environ. Sci. Technol. 2003, 37, 958−971. (5) Tipping, E.; Lofts, S.; Sonke, J. E. Humic Ion-Binding Model VII: a revised parameterisation of cation-binding by humic substances. Environ. Chem. 2011, 8, 225−235. (6) Cabaniss, S. E. Forward modeling of metal complexation by NOM: I. A priori prediction of conditional constants and speciation. Environ. Sci. Technol. 2009, 43, 2838−2844. (7) Dudal, F.; Gérard, F. Accounting for natural organic matter in aqueous chemical equilibrium models: a review of the theories and applications. Earth Sci. Rev. 2004, 66, 199−216. (8) Aiken, G. R.; Malcolm, R. L. Molecular weight of aquatic fulvic acids by vapor pressure osmometry. Geochim. Cosmochim. Acta 1987, 51, 2177−2184. (9) Beckett, R.; Zhang, J.; Giddings, J. C. Determination of molecular weight distributions of fulvic and humic acids using flow field-flow fractionation. Environ. Sci. Technol. 1987, 21, 289−295. (10) Balnois, E.; Wilkinson, K. J.; Lead, J. R.; Buffle, J. Atomic force microscopy of humic substances: effects of pH and ionic strength. Environ. Sci. Technol. 1999, 33, 3911−3917.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was performed within the framework of the BIOMONAR project funded by the European Commission’s seventh framework program (Theme 2: Food, Agriculture and Biotechnology), under grant agreement 244405.

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SYMBOLS AND ABBREVIATIONS center-to-center distance of closest approach between a hydrated metal ion and a complexing site (m) −3 cM * bulk concentration of free metal ion Mz+ aq (mol m ) cM,p concentration of free metal ion Mz+ inside a particle (mol m−3) aq z+ DM diffusion coefficient of the metal ion Maq in aqueous solution (m2 s−1) DFS diffusion coefficient of FS in aqueous solution (m2 s−1) FS fulvic-like substance (collective term for fulvic acid and small humic acid entities) f B Boltzmann equilibrium partitioning factor fel coefficient for conductive diffusion a

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