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Article << Previous     |         Contents Vol 11(2)

Labilities of aqueous nanoparticulate metal complexes in environmental speciation analysis

Raewyn M. Town A C and Herman P. van Leeuwen B

A Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, Campusvej 55, DK-5230 Odense, Denmark.
B Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, NL-6703 HB Wageningen, the Netherlands.
C Corresponding author. Email: raewyn.town@sdu.dk

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Environmental Chemistry 11(2) 196-205 http://dx.doi.org/10.1071/EN13138
Submitted: 20 July 2013  Accepted: 10 October 2013   Published: 25 March 2014

Environmental context. Sorbing nanoparticles can have a significant effect on the speciation of small ions and molecules in the environment. The reactivity of nanoparticulate-bound species can differ significantly from that of their molecular or colloidal counterparts. We present a conceptual framework that describes the chemodynamics and lability of nanoparticulate metal complexes over a wide range of experimental timescales and environmental conditions.

Abstract. An inherent property of a dispersion of charged nanoparticles is that their charges and reactive sites are spatially confined to the particle body which is at a different potential from that in the bulk medium. This feature has important consequences for the reactivity of nanoparticulate complexants: the diffusive rate of reactant supply is lower as compared to molecular complexants, whereas the local concentration of reactant ions may be enhanced if the particle’s electric field has the opposite charge sign. These effects are most dramatic for soft nanoparticles for which the electrostatic accumulation mechanisms operate on a 3-D level. We show how the interplay of these effects governs the reactivity of charged nanoparticulate metal complexes (M-NPs) at the surface of an analytical speciation sensor. A theoretical framework is presented that describes the lability of M-NP species over a range of effective timescales for different electrochemical and other dynamic speciation analysis techniques. The concepts are illustrated by electrochemical stripping data on metal complexes with natural soft nanoparticles of humic acid.

Additional keywords: humic acid, kinetics, lability, stripping chronopotentiometry.


Nanoparticulate complexants are ubiquitous in aquatic systems. They form a diverse group including permeable nanoparticles (NPs) such as humic acids (HAs), impermeable NPs such as mineral oxides and clays, as well as an increasing number of engineered NPs released into the environment. The potential toxicological effect of the latter has been the subject of many recent studies, both directly[14] and by their effect on the speciation and ensuing bioavailability of small ions and molecules.[57] This latter aspect has long been recognised for natural NPs,[811] and generic approaches are being developed to understand and predict the chemodynamic properties of NPs based on their inherent features such as size, charge and permeability.[12] The measurement of fluxes and reactivity of compounds, i.e. dynamic speciation analysis, is increasingly recognised as fundamental to understanding the fate and behaviour of pollutants in environmental systems. The uptake and release kinetics of small ions and molecules by NPs can differ significantly from those with molecular ligands, or larger colloidal entities. Accordingly, appropriate theory is needed to describe and predict the lability of nanoparticulate complexes, i.e. the extent to which nanoparticulate compounds may dissociate to release free target compounds which then contribute to signals measured by dynamic speciation sensors, uptake by organisms or accumulation at reactive interphases. The notion of lability is inherently linked to the timescale of the process under consideration. For example, various analytical speciation techniques each have their own characteristic timescale; Fig. 1.[13]

Fig. 1.  Schematic representation of diffusional timescales for a range of dynamic electrochemical and non-electrochemical speciation sensors and environmental processes. PLM, permeation liquid membrane, DGT, diffusive gradients in thin film; DMT, Donnan membrane technique.

Here we present a theoretical framework for describing the lability of nanoparticulate complexes with small ions and molecules. It is applicable to NPs bearing an arbitrary charge, and an independent amount of complexing sites. The utility of the approach is illustrated for metal complexes with negatively charged NPs. We show that the conceptual framework, combined with knowledge of the timescales of interfacial processes, enables sound interpretation of speciation measurements in nanoparticulate dispersions with dynamic sensors. In doing so, we place the timescale of widely used electroanalytical techniques in context with other dynamic metal speciation methods (see Fig. 1).


General reaction scheme for nanoparticulate metal ion complexation

A differentiated form of the original Eigen mechanism, applicable to aqueous metal ion complexation by simple molecular ligands, has been developed for the case of soft nanoparticulate complexants.[14,15] It includes several distinguishable steps involved in the overall complexation process, as illustrated in Fig. 2.

Fig. 2.  Stepwise complexation of a hydrated metal ion (Mz+(aq)) from the bulk medium with a nanoparticulate complexant containing charged or uncharged binding sites (S) and mere charges (–). The plus symbols denote the extraparticulate counterionic part of the atmosphere. The association steps are (1) diffusion of Mz+(aq) from the bulk solution to the surface of the complexant, (2) crossing the solution–particle interface and incorporation within the particle as a free hydrated ion, (3) outer-sphere association of Mz+(aq) with S, Mz+(aq)•S and (4) inner-sphere complex formation, MS, including the loss of water of hydration by Mz+(aq) and formation of a chemical bond with site S. The various rate constants corresponding to the above steps are indicated with the subscript ‘a’ (association process) and ‘d’ (dissociation process). See main text for definitions of symbols used. Adapted from Town et al.[24]
Click to zoom

Any one of the steps outlined in Fig. 2, or underlying processes thereof, may be rate-limiting for the overall association or dissociation reaction. The nature of both the NP complexant (size, water content, charge density, binding site density, backbone flexibility), and the type of metal ion (charge, size, rate of dehydration) may influence the rates of the component processes.

Rate constants for association and dissociation of nanoparticulate complexes

The magnitude of the particulate electrostatic field has an effect on the various types of rate constants in the above scheme in Fig. 2. For the case of a positively charged metal ion interacting with a negatively charged NP, the electrostatic potential gives rise to (i) enhancement of the rate of diffusion of metal ions towards the NP, as described by a coefficient for conductive diffusion (fel,a)[16] and (ii) accumulation of metal ions within the NP body and its extraparticulate double layer, as described by a distributed Boltzmann partitioning factor (fB).

Expressions for the rate constants for the various steps in Fig. 2 have been derived in previous publications,[12,17,18] and are summarised below for the high charge density case. The rate constants are formulated in the usual manner as chemical reaction rate constants, i.e. per mole of sites (S, reactive site that covalently binds a metal ion), and refer to the maximum magnitudes under quasi steady-state conditions for diffusion in the medium (step 1, Fig. 2). The association step may be limited by the rate of diffusive supply of Mz+(aq) to the particle or by inner-sphere complex formation. Similarly, the dissociation step may be limited by diffusion of Mz+(aq) out of the particle or by dissociation of the inner-sphere complex. In each case, the nature of the various rate-limiting steps is identified by comparing the corresponding maximum rates of the overall association–dissociation process.

Complex formation

The limiting rate of diffusive supply (Ra,p, mol m–3 s–1), for transport of Mz+(aq) to the spherical NP body is generally given by:

with corresponding rate constant (ka,p, m3 mol–1 s–1):

where NAv is Avogadro’s number, rp (m) is the particle radius, DM (m2 s–1) is the diffusion coefficient of the metal ion Mz+(aq) in aqueous solution, fel,a is the electrostatic coefficient for conductive diffusion towards the NP, c*M (mol m–3) is the bulk concentration of free metal ion Mz+(aq), cS (mol m–3) is the smeared-out concentration of reactive sites per unit volume of solution in the bulk dispersion and NS is the number of reactive sites per particle; cS/NS is the concentration of particles in the bulk dispersion (cp).

Provided that the Boltzmann distribution of charged species between the medium and particle is at equilibrium, the limiting rate of inner-sphere complex formation (Rais, mol m–3 s–1) is given by:

where the pertaining rate constant (kais, m3 mol–1 s–1) is given by:

where kw is the rate of dehydration of the hydrated metal ion, fB is the Boltzmann equilibrium partitioning factor and Vos is the outer-sphere volume for an ion pair between Mz+(aq) and an individual site S (m3).

Complex dissociation

For the case of κrp > 1 (κ–1 is the Debye length in the bulk solution (m)) at a high charge density (fB >> 1), the diffusion-limited rate of release of free Mz+(aq) from the particle body is given by the conventional rate for diffusion from a sphere, modified for the electric field (Rd,p, mol m–3 s–1):

where fel,d is the coefficient for conductive diffusion away from the NP (in the present case of a negatively charged NP and a positively charged metal ion, fel,d is less than unity) and cM,p (mol m–3) is the smeared-out concentration of the intraparticulate free metal ion Mz+(aq). The expression for the corresponding diffusion-controlled rate constant (kd,p, s–1) is obtained by accounting for the buffering of the free M within the NP by Mz+(aq)S, the reactive site that covalently binds a metal ion, and MS, the inner-sphere complex of M with a nanoparticulate reactive site, to give:

where Kint (m3 mol–1) is the intrinsic stability constant of the inner-sphere complex applicable in the absence of an electric field, and the term KintcS reflects the contribution of MS to the rate of release of Mz+(aq) from the particles.

In the other extreme of rate control by inner-sphere dissociation (Rdis, mol mol–3 s–1) the rate derives straightforwardly from kais and Kint:

where kais is given by Eqn 4, and the rate constant for inner-sphere dissociation (kdis, s–1) follows as

The above expressions are formulated in terms of smeared-out concentrations and constants that reflect parameters applicable at the level of the bulk solution. In this way, the computed parameters can be directly compared with experimental data. Still, it is essential to realise the link with the local conditions within the NP body. The intraparticulate free metal ion concentration is fB times greater than that in the bulk solution, which gives rise to an apparently stronger complex and a correspondingly greater cMS. The enhanced free metal ion concentration inside the NP will be counted as ‘bound metal’ by techniques that measure the bulk concentration of free M, e.g. ion selective electrode potentiometry. For computation of the stability constant, the intraparticulate free and bound M is set against the non-local bulk free metal ion concentration. As a consequence the measured stability of the nanoparticulate MS complex seems much stronger than its intrinsic value. Recognition of these parameters is important with respect to kinetic issues: Kint and fBcM are determinants of the nature of the dissociation rate-limiting step (Eqns 57).

For a given system, the rate limiting step can be identified by comparing the experimental value of the overall rate constant for association–complex formation (ka, m3 mol–1 s–1) or dissociation (kd) with the computed ka,p (rate constant for diffusive supply of Mz+(aq) to a NP (m3 mol–1 s–1)) and kais or kd,p and kdis values. Furthermore, for a given NP, the nature of the rate limiting step may change from inner-sphere association or dissociation to diffusive supply or efflux as the magnitude of the electrostatic field increases, because of the relative changes in magnitude of fel and fB.

Lability of nanoparticulate complexes

At the level of the bulk solution, we can distinguish two limiting situations for the equilibria between a metal ion and a nanoparticulate complexant. If the timescale is much longer than the characteristic lifetimes of free M and the nanoparticulate–metal complex (M-NP), there is frequent interchange between these two species and then the system is denoted as dynamic. At the other limit, if any change in species concentrations is not followed by significant re-equilibration then the system is inert. The terminology of dynamic v. inert refers to the rate characteristics of a volume reaction, i.e. whether or not the system can attain bulk equilibrium within a certain time. Even at this level, nanoparticulate metal complexes may exhibit reactivity that is significantly different from their molecular or colloidal counterparts. The consideration is relevant for verifying whether bulk equilibrium is maintained in speciation techniques such as competing ligand exchange – adsorptive stripping voltammetry.[19]

We are concerned with the dynamic nature of the interfacial reactions for a system comprising a dispersion of M-NP, in contact with the surface of a body (e.g. a sensor, or a biological organism) that converts or accumulates the metal ion after its release from the NP. Lability quantifies the extent to which the M-NP dissociates to release the free metal ion on the timescale of the diffusion of the M-NP entity towards the consuming interface. Thus, the notion of lability is inherently always coupled to the characteristic timescale of that diffusional process. The various processes involved during conversion of free metal ions at a macrointerface from a dispersion containing nanoparticulate complexants are shown schematically in Fig. 3.

Fig. 3.  Schematic view of the concentration profiles of free metal ions (Mz+(aq)) and complexes with nanoparticulate binding sites (MS) at a more macroscopic reactive interface. The concentration gradients in the diffusion layer (δ) and the reaction layer (λ) are shown for the case of labile complexes. For clarity, the size of the nanoparticle complex is exaggerated; λ and δ are just arbitrary. See main text for definitions of symbols used.

The coupled diffusion of M and MS and the kinetics of their interconversion determine the overall flux of free M towards the reactive interface. Thus the lability of a complex species is defined by the relative magnitudes of the dissociative kinetically controlled flux (Jkin, mol m–2 s–1) and its purely diffusive flux towards the reactive interface of interest (J*dif, mol m–2 s–1). The ratio Jkin/J*dif is often denoted as the so-called lability index (ℒ). For the case of nanoparticulate complexants, the concepts are analogous to those developed for molecular ligands,[13,20] albeit with a more differentiated meaning of the pertaining kinetic parameters, as elaborated below. Two important limiting cases are identified, corresponding to extreme values of ℒ. For ℒ >> 1, the system is labile (and thus diffusion-controlled), and the steady-state flux reduces to:

where δ (m) is the mean diffusion layer thickness for Mz+(aq) and M-NP as determined by the mean (weighted average) diffusion coefficient for the complex system (D, m2 s–1):

where ε = DMS/DM and cM,t is the total metal ion concentration (mol m–3). That is, for a labile complex the flux corresponds to the purely diffusion-controlled coupled transport of M and MS and the equilibrium is maintained on all relevant spatial scales.

At the other limit, ℒ << 1 and the rate of dissociation determines the extent to which MS complexes can dissociate and contribute to the metal ion accumulation flux. The flux now corresponds to its kinetically controlled value (Jkin), which for sufficiently large K′ (εK′ >> 1) is given by:

where kd is the rate constant for dissociation and λ (m) represents the thickness of the layer of solution adjacent to the consuming interface, where the equilibrium between free and bound M is distorted (often denoted as the ‘reaction layer’).[21] The magnitude of λ depends on the mobility of the free M in the medium (represented by its diffusion coefficient, DM) and its mean free lifetime (1/ka; where ka is the product of the rate constant for association and the concentration of reactive sites (kacS), together with the mobility of the complex M-NP (≈Dp) and its mean free lifetime (1/kd)[13,22]:

Combining Eqns 9 and 11 yields the lability index (ℒ):

In Fig. 3, the radius of the NP is arbitrarily shown as being smaller than the thickness of the reaction layer. Irrespective of the relative rp and λ values, at a given time a NP may find itself located partly within the reaction layer and partly within the diffusion layer (as shown in Fig. 3). Under conditions of excess binding sites, there is a statistical contribution of complex species to the overall flux, and thus the above expressions remain valid.

The characteristic timescale of a reactive interface is determined by its diffusion layer thickness (Eqn 13). That is, a given sensor, organism or any species-converting interface will detect a certain proportion of the complex species present, as determined by their lability on the pertaining timescale. The kinetic features and lability criteria for several dynamic speciation sensors have been previously described and quantitatively compared.[13] For example, the timescale of voltammetric techniques at macroelectrodes is of the order of 10–102 s, whereas the timescale at a microelectrode is in the millisecond range, as determined by the electrode radius (see also Fig. 1).



The HA sample was the International Humic Substances Society reference Summit Hill HA (1R105H), which has a carboxy group content of 7.14 mol kg–1 C.[23] Applicable electrostatic parameters were taken from the literature,[24] and the particle radius (rp) was taken as 2.3 nm. No significant change in particle size is expected for the ionic strength range considered herein.[25] Solutions were buffered to pH 6.4 with MES buffer ((2-(N-morpholino)-ethanesulfonic acid) prepared from the solid (Fluka, MicroSelect, ≥99.5 %), and the ionic strength was maintained with KNO3 (BDH, AnalaR). PbII solutions were prepared by dilution of a standard (Metrohm). All solutions were prepared in ultrapure deionised water (resistivity >18 MΩ cm) from a Milli-Q gradient system.

Stripping chronopotentiometry (SCP)

The electrochemical lability of PbII–HA complexes was measured by stripping chronopotentiometry (SCP) with an Ecochemie μAutolab potentiostat in conjunction with a Metrohm 663 VA stand. The electrometer input impedance of these instruments is >100 GΩ.The working electrode was a Metrohm multimode mercury drop electrode (surface area, A = 5.2 × 10–7 m2), the auxiliary electrode was glassy carbon and the reference electrode was Ag|AgCl|KCl(sat) encased in a 100 mol m–3 KNO3 jacket. Measurements were performed at 20 °C. SCP, in the complete depletion regime, is the chosen electrochemical technique because the signal is not affected by adsorption of humic substances on the electrode surface.[26] Measurements were made under limiting deposition current conditions (deposition potential, Ed = –1.00 V) with an accumulation time of 600 s and a stripping current of 2 × 10–9 A, corresponding to complete depletion.[27] Measurements were performed at an HA concentration of ~4 g m–3 and PbII concentrations in the range 10–5 to 10–4 mol m–3. The experimental value of the lability index corresponds to the measured limiting stripping time in the presence of HA, as compared to the signal that would be obtained for the labile, diffusion-controlled case.[28] The latter is derived from the signal for the equivalent total Pb-only concentration, corrected for the difference between the diffusion coefficients of the free Pb2+ and the Pb–HA complex, with DPb–HA ≈ DHA = 5 × 10–11 m2 s–1[29] and DPb = 8.3 × 10–10 m2 s–1.[30]

Results and discussion

Interpretation of data for environmentally relevant systems

HA is chemically heterogeneous, and its effective thermodynamic and kinetic parameters depend on the degree of occupation of the reactive sites (θ). Here we use the differential equilibrium function (K*, m3 mol–1)[31] and the differential kinetic function (k*d, differential rate constant for complex dissociation (s–1)),[32] which are linked to a specific type of reaction and are determined without any assumption on the binding mode. The approach is detailed in our previous work.[33] It has been established that the overall rate constant for association of metal ions with humic substances is independent of θ.[24,33] Thus, the distribution in K* is reflected in that of k*d.[34] The directly measured k*d values are compared with those computed from ka/K* for CuII–HA and NiII–HA in Figs 4 and 5 respectively. The slopes of the lines reflect the degree of heterogeneity of the complexation (Γ) (0 < Γ ≤ 1; Γ = 1 corresponds to the homogeneous case). For both CuII and NiII, the experimentally measured k*d values are apparently rather independent of ionic strength, and this is consistent with the computed values, i.e. the electrostatic contributions in K* and ka compensate in the derived k*d. For CuII, ka,p and kais are of comparable magnitude, and thus so are kd,p and kdis, i.e. both processes are significant for the overall rate. For NiII, kdis is the rate-limiting step for dissociation.

Fig. 4.  Experimental (points) and computed (lines) values of the dissociation rate constant (kd) for CuII–humic complexes. Computed values correspond to kdis (solid lines) and kd,p (dashed lines) at I = 100 (black lines) and 10 mol m–3 (blue lines). Experimental k*d values, measured by ligand exchange, correspond to: Latacho soil humic acid, I = 1 (open squares); 20 (solid diamonds); and 200 mol m–3 (solid circles) and Elliot soil humic acid, I = 20 mol m–3 (open diamonds).[60] Following the usual practice, the degree of metal ion loading (θ) is expressed in terms of the concentration ratio of bound metal to carboxy groups in the same weight of humic sample. See main text for definitions of symbols used. The experimental k*d values were obtained from published kinetic spectra by the approach detailed in Town et al.[33]

Fig. 5.  Experimental (points) and computed (lines) values of the dissociation rate constant (kd) for NiII–humic complexes. Computed values correspond to kdis (solid lines) and kd,p (dashed lines) at I = 100 (black lines) and 10 mol m–3 (blue lines). Experimental k*d values, measured by ligand exchange, correspond to: soil fulvic acid, I = 100 mol m–3 (solid circles)[61]; Suwannee River fulvic acid I = 2 (solid diamonds) 100 (open circles); 30 (solid squares); 10 (solid trianges); 3 (open squares); and 1 mol m–3 (open diamonds).[62] See main text for definitions of symbols used. The experimental k*d values were obtained from published kinetic spectra by the approach detailed in Town et al.[33]

The results presented in Figs 4 and 5 validate our approach and establish its robustness over a wide dynamic range. Thus, if directly measured k*d values are not available for a given metal ion, they can be derived for the given conditions from the applicable K* and the computed overall ka values. The ensuing lability of the metal complexes can then be computed for a given timescale.

The predicted trends in lability have been verified by measurement of the lability of PbII–HA complexes at a macroelectrode for ionic strengths of 100 and 10 mol m–3. The K* values computed from the literature at 10 mol m–3[35,36] were approximately a factor of 10 greater than those for 100 mol m–3, similar to the change in magnitude observed for CuII.[24] The experimental data are well described by the theoretical values (Eqn 13); Fig. 6. At the lower ionic strength, both K* and the rate-limiting ka,p are enhanced. The K* is enhanced because of the greater Boltzmann partitioning factor (fB), whereas ka,p increases because of an increase in the coefficient for conductive diffusion (fel,a). The increase in fB is much greater than that in fel,a,[24] with the net result that kd (= ka/K*), and consequently the lability is reduced as the ionic strength decreases. As expected from Eqn 13, the lability of M–HA complexes depends on the degree of metal ion loading, to an extent determined by the degree of heterogeneity (Γ), i.e. it parallels the trend in kd.

Fig. 6.  Comparison of measured (points) and computed (striped bands) lability of PbII–HA complexes, as determined at a macroelectrode, pH 6.4. The data are shown for an ionic strength of 100 mol m–3 (solid black diamonds, open black diamonds; black band) and 10 mol m–3 (solid blue circles, open blue circles; blue band). Solid and open symbols represent replicate measurements. The band of computed values encompasses the range of reported DHA values, 2 × 10–11 to 1 × 10–10 m2 s–1.[29,6365] See main text for definitions of symbols used.

In general, the lability of metal complexes with HA will depend on the operational timescale of the measurement technique, as well as on several experimental conditions such as the nature of the metal ion, the degree of occupation of binding sites, the pH and ionic strength. To illustrate this feature, we have calculated K* values for M–HA complexes from published titration curves for CuII,[3742] PbII[35,38,43,44] and CdII,[4446] and computed the lability index as a function of metal ion loading and pH. For the rapidly dehydrating CuII (kw = 4 × 109 s–1)[4750] and PbII (kw = 7 × 109 s–1),[51] under the conditions shown in Fig. 7, the association rate constants ka,p and kais are of comparable magnitude, whereas for CdII (kw = 4 × 108 s–1)[51] kais is overall rate-limiting. In each case the lowest rate constants are used for computation of the lability index. Over the pH and θ range considered, the electrostatic parameters are assumed to be approximately constant: the degree of protonation is strongly buffered in this region,[52] and for log θ values below –1, metal ion binding will have a negligible effect on the overall particulate charge. The results are presented as 3-D plots in Fig. 7 for the timescale of voltammetry at a macroelectrode (δ = 10–4 m). At a microelectrode (with radius r0 = ~5 × 10–6 m), the ℒ scale is shifted by 2 log units to lower values, and for diffusive gradients in thin film (DGT) (δ = ~10–3 m), the ℒ scale is shifted by 1 log unit to higher values. Fig. 7 shows that for a given pH and metal ion loading, the order of lability of the M–HA complexes is CdII > PbII > CuII.

Fig. 7.  Lability index (Eqn 13), plotted as log ℒ, for humic acid complexes with (a) CuII, (b) PbII and (c) CdII. The ℒ values are shown as a function of degree of metal ion loading (θ) and pH for an ionic strength of 100 mol m–3 and δ = 10–4 m (typical of a macroelectrode).

There is a paucity of systematic data for DGT measurements on well defined systems. One study found that PbII complexes with an aquatic HA were labile at pH 7, I = 10 mol m–3, log θ = –1.41 for δ = 8 × 10–4 m.[53] This is in agreement with a computed ℒ of ~1 for these conditions. Similarly, CuII complexes with a peat HA were labile by DGT (δ = 8 × 10–4 m) at pH 7.8, I = 100 mol m–3, log θ = –1.0,[54] which is consistent with the computed ℒ of ~4. Quantitative interpretation of DGT data is confounded by complications associated with the extension of the reaction layer into the accumulating resin phase, leading to higher than predicted lability,[55] and the tendency for HA to accumulate within the DGT gel.[56]

In more complex media such as natural waters[57] and soil solutions,[58] speciation measurements indicate that a greater concentration of metal species are measured as labile by DGT as compared to electrochemical stripping methods, consistent with the overall theoretical expectations presented herein. A quantitative interpretation of such samples is rather involved, encompassing detailed characterisation of the nanoparticulate complexants, as well as consideration of mixture effects.[59]

General trends in predicted labilities of M-NP complexes

The results presented above for HA can be generalised to metal complexes with any type of NP. The lability of M-NP complexes depends on the size of the NP, its charge density, the intrinsic stability of the complexes, the ionic strength of the medium, the rate of dehydration of the metal ion, as well as the effective timescale at the reactive interface. Some examples of the computed lability index are shown in Figs 8 and 9 for the case of the rapidly dehydrating Cu(H2O)62+ for which the association with and release from the particle body are rate-limiting under all conditions considered.

Fig. 8.  Lability index (ℒ) as a function of the diffusion layer thickness at the reactive interface (δ) for the case of ka,p and kd,p rate limiting. Computations are performed for Kint = 1 m3 mol–1, cS = 0.01 mol m–3. Results are shown for rp values of 3 (solid lines), 10 (dashed lines), 30 (dotted lines) and 100 nm (dot-dashed line) at ionic strengths of 1 (black lines) and 100 mol m–3 (blue lines). The typical δ range for practical sensors are indicated with shaded bands: microelectrode (Micro), macroelectrodes (Macro) and diffusive gradients in thin film (DGT). See main text for definitions of symbols used.

Fig. 9.  Lability index (ℒ) as a function of the particle radius (rp) for various δ and ionic strengths. Computations are performed for Kint = 10 m3 mol–1, cS = 0.01 mol m–3, for the case of ka,p and kd,p rate limiting. Results are shown for δ = 10–6 m (solid curves) and 10–4 m (dashed curves) at ionic strengths of 1 (black curves), 10 (blue curves) and 100 mol m–3 (red curves). See main text for definitions of symbols used.

For the conditions used in Figs 8 and 9 (ka,p and kd,p rate limiting), the lability of the M-NP complexes is found to decrease as the ionic strength decreases and as rp increases. This result is a consequence of the intricate and interrelated changes in fel and fB. As the ionic strength decreases, both fel and fB increase[16,24]; the enhanced fel offsets to some extent the decrease in efficiency of diffusion to the NP as rp increases, whereas fB increases with particle size (up to the Donnan limit), for a given charge density. The change in fB is larger than that in fel,[24] with the overall consequence that changes in kd,p (Eqn 6) govern the ionic strength and rp dependence of ℒ (Eqn 13) for the conditions shown in Figs 8 and 9. However, the trends in lability for M-NP complexes depend on many factors, including the nature of the rate-limiting step and whether comparisons are made on the basis of constant bulk concentration of reactive sites (cS) (decrease in number of particles as rp increases) or constant concentration of particles (cp) (increase in number of reactive sites as rp increases).[12] Independent manipulation of the concentration of reactive sites (cS v. cp), and mere charges (cS v. charge density), in combination with variation of the operational timescale of the measurement technique (δ) is a powerful means to probe the kinetic features of nanoparticulate complexes.

Conclusion and outlook

A theoretical framework for the chemodynamic reactivity of metal complexes with soft NPs is shown to provide a good description of the dynamic behaviour of metal complexes with the natural nanoparticulate ligand, HA. The measured electrochemical lability of M-NP species is in good agreement with predicted values. The results enable the lability of nanoparticulate complexes to be predicted over a wide range of timescales, environmental conditions (degree of metal ion loading, pH, ionic strength) and NP properties (particle size, charge density, reactive site density, heterogeneity). Key features involved in the reactivity of molecules, NPs, and colloids are summarised in Table 1.

Table 1.  Key features involved in the reactivity of molecules, nanoparticles, and colloids
See main text for definitions of symbols used

Click to zoom

Our present analysis is for the relatively straightforward case of sufficiently large NPs with negligible accumulation of metal ions in the extraparticulate counter-ion atmosphere. In aquatic systems, a significant proportion of the natural organic matter is typically present as fulvic acids, with radii of the order of 1 nm. Upon going from HA to fulvic acid, the dynamic behaviour transitions from that for soft NPs to the discrete case of a small ion with few charges in given positions and most of the counter charge, and even the inner-sphere bound metal, lying predominantly in the extraparticulate zone. Accordingly, it becomes crucial to make the fundamental link between dynamic speciation analysis and the coupled size distribution and reactivity of the involved complexants.


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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