# The Intrinsic Stability of Metal Ion Complexes with Nanoparticulate Fulvic Acids

Raewyn Town, Jérôme Duval, Herman van Leeuwen

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Raewyn Town, Jérôme Duval, Herman van Leeuwen. The Intrinsic Stability of Metal Ion Complexes with Nanoparticulate Fulvic Acids. Environmental Science and Technology, 2018, 52 (20), pp.11682-11690. 10.1021/acs.est.8b02896 . hal-02059655

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The intrinsic stability of metal ion complexes with nanoparticulate fulvic acids

Reference:

Town Raewyn M., Duval Jérôme F.L., van Leeuwen Herman P..- The intrinsic stability of metal ion complexes with nanoparticulate fulvic acids
Environmental science and technology / American Chemical Society - ISSN 0013-936X - 52:20(2018), p. 11682-11690
Full text (Publisher's DOI): https://doi.org/10.1021/ACS.EST.8B02896
To cite this reference: https://hdl.handle.net/10067/1539910151162165141
1 The Intrinsic Stability of Metal Ion Complexes with Nanoparticulate Fulvic Acids

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3 Raewyn M. Town<sup>1,4*</sup>, Jérôme F. L. Duval<sup>2,3</sup>, and Herman P. van Leeuwen<sup>4</sup>

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5 ¹ Systemic Physiological and Ecotoxicological Research (SPHERE), Department of Biology, University of
6 Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium. Email: raewyn.town@uantwerpen.be

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8 ² CNRS, Laboratoire Interdisciplinaire des Environnements Continenteaux (LIEC), UMR 7360, Vandoeuvre-lès-Nancy, F-54501 Nancy, France

9 ³ Université de Lorraine, LIEC, UMR 7360, Vandoeuvre-lès-Nancy, F-54501 Nancy, France

10 ⁴ Physical Chemistry and Soft Matter, Wageningen University & Research, Stippeneng 4, 6708 WE Wageningen, The Netherlands

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14 Abstract

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The electrostatic contributions to metal ion binding by fulvic acids (FAs) are characterised in light of recent
theoretical developments on description of the net charge density of soft nanoparticles. Under practical
electrolyte concentrations, the radius of the small, highly charged soft nanoparticulate FAs is comparable to
the electrostatic screening length and their electric potential profile has a bell shape that extends into the
surrounding aqueous medium. Consequently, accumulation of counterions in the extraparticulate zone can be
significant. By comparison of experimentally-derived Boltzmann partitioning coefficients with those
computed on the basis of (i) the structural FA particle charge and (ii) the potential profile for a nanoparticulate
FA entity equilibrated with indifferent electrolyte, we identify the thickness of the *extraparticulate* counter
charge accumulation shell in 1-1 and 2-1 electrolytes. The results point to the involvement of counterion
condensation phenomena and call into question the approaches for modelling electrostatic contributions to ion
binding that are invoked by popular equilibrium speciation codes. Overall, the electrostatic contributions to
Cd<sup>2+</sup><sub>aq</sub> and Cu<sup>2+</sup><sub>aq</sub> association with FA are weaker than those previously found for much larger humic acids
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(HA). The intrinsic chemical binding strength of CdFA is comparable to that of CdHA, whilst CuFA complexes are weaker than CuHA ones.

## INTRODUCTION

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Humic substances (HS) are heterogeneous complexants that are ubiquitous in environmental systems and play an important role in buffering concentrations of cations and small molecules.¹ On the basis of their solubility in water, HS are separated into a larger humic acid (HA) fraction and a smaller, more soluble fulvic acid (FA) fraction. A typical FA entity has a radius of the order of 1 nm, and contains *ca.* 5 to 10 ionisable carboxyl groups and up to one N- or S- containing functionality.¹⁻³ In contrast, the radii of HA particles ranges from a few nm for aquatic HA up to the order of 100 nm for soil HA aggregates, with each particle containing hundreds to thousands of chemically diverse groups.¹ The smaller FA are more conformationally rigid than the larger HA.⁴⁻⁶ Most literature on binding of aqueous transition metal ions $M^{2+}$ (M in short) by HS describes the strength of M-HS association in terms of an *apparent* stability constant, $\overline{K}_{app}$.⁷,⁸ That is, all M associated with the HS is considered to be chemically bound and concentrations of the complex species MHS are expressed as smeared-out over the entire volume of the solution/dispersion. The overbar notation signifies that $\overline{K}_{app}$ represents a weighted average of the stabilities over all associated forms of M with HS. $\overline{K}_{app}$ values for MHA are generally significantly greater than those for MFA.⁹,¹⁰

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Recent work has shown that a more differentiated approach is needed to describe ion association with HA: the extent to which *e.g.* $M^{2+}$ associates with negative HA entities is determined by a combination of electrostatic association and inner-sphere complexation.¹¹,¹² The electric field contribution to ion association by HA can be significant, leading some authors to propose that $Ca^{2+}$ association has an intrinsic chemical nature.¹³⁻¹⁵ Rather, the extent to which the predominantly electrostatically binding $Ca^{2+}$ ion associates with large HA particles is well described by a two-state Counterion Condensation-Donnan (CCD) model that combines (i) electric condensation of $Ca^{2+}$ in the strongly negatively charged intraparticulate double layer shell at the particle/medium interface, with (ii) Donnan partitioning in the net uncharged core of the particle body.¹¹ Even for metal ions such as $Cd^{2+}$ and $Cu^{2+}$, whose association with HA includes substantial inner-sphere complex
formation, a significant amount of the total M associated with the HA particle can be in the form of electrostatically condensed ions in the intraparticulate part of the double-layer shell.¹² Such detail in the intraparticulate speciation is fundamental for proper interpretation of $\bar{K}_{app}$ values, metal speciation dynamics and bioavailability.

Although FAs carry a high negative charge density at near neutral pH, their small size implies that the CCD model is not an appropriate descriptor of their electrostatic features.⁹,¹⁶,¹⁷ Since $\kappa r_p$ (ratio between particle radius $r_p$, and electrostatic screening length, $1/\kappa$) is typically of order unity for FA under practical electrolyte concentrations, counterion accumulation in the *extraparticulate* double layer zone may constitute a significant fraction of the associated M.⁹,¹⁸,¹⁹ This feature of ion binding by FA differs significantly from that for the larger HA ($\kappa r_p \gg 1$) for which the accumulation of cations in the *extraparticulate* double layer zone is generally insignificant with respect to that inside the bulk particle body. The present work compares the *intrinsic* stability of inner-sphere metal complexes of FA with those of HA. In doing so, we elucidate the extent to which the differences in *apparent* stability constants for MFA and MHA can be attributed to inherent physicochemical differences in the electrostatic vs. the intrinsic covalent components of M binding.

## THEORY

**Electrostatic Characteristics of FA.** At near-neutral pH, an individual FA entity ($r_p$ ca. 1 nm) houses approximately 5 negatively charged sites.²⁰ Various types of humic substances are reported to have a density in the range *ca.* 1.4 – 2 kg m⁻³ and *ca.* 80% water content.²¹⁻²³ The FA used herein has a density of 1.5 kg m⁻³,²¹ which corresponds to a local, intraparticulate charge density of *ca.* -2,000 mol e m⁻³ (with *e* the elementary charge). For a homogeneous distribution of charged sites, the average site separation distance, $\ell_c$, is approximately 1 nm, and thus the possibility arises for overlap of the electric double layers around individual charged sites. The thickness of the electric double layer, *i.e.* the Debye screening length $\kappa^{-1}$, depends on the ionic environment. For charges within a soft nanoparticle body, the relevant screening length is the local intraparticulate one, $\kappa_p^{-1}$.²⁴,²⁵ Once $2\kappa_p^{-1}$ becomes comparable to or greater than $\ell_c$, the electrostatics can be
described by a mean-field approach, whereas for $2\kappa_p^{-1} \ll l_c$, the applicable potential is dominated by the local coulombic field around each site.¹⁶ In a 1-1 electrolyte, $\kappa_p$ can be evaluated from:²⁴⁻²⁶

$$\kappa_p = \kappa[\cosh(y_{\text{center}})]^{1/2} \qquad (1)$$

where $y_{center}$ is the dimensionless potential at the center of the soft charged particle computed via the Poisson-Boltzmann mean field approach,²⁶ and $\kappa$ is the conventional reciprocal Debye length of the extraparticulate medium. Thus, for FA in 1-1 electrolyte, e.g. KNO₃, at an ionic strength, $I$, of 10 mM, $\kappa_p^{-1} = 0.58$ nm, and at $I = 100$ mM, $\kappa_p^{-1} = 0.39$ nm. In a 2-1 electrolyte, $\kappa_p^{-1}$ is obtained via (see Supporting Information):

$$\kappa_p = \frac{\kappa}{\sqrt{3}} [2\exp(-2y_{\text{center}}) + \exp(y_{\text{center}})]^{1/2} \qquad (2)$$

Application of eq 2 for the nanoparticulate FA in 2-1 electrolyte, e.g. Ca(NO₃)₂, at $I = 10$ mM, yields $\kappa_p^{-1} = 0.26$ nm, and at $I = 100$ mM, $\kappa_p^{-1} = 0.24$ nm. Thus, the FA system is poised at the point where the double layers around individual charges begin to overlap and hence continuous electrostatic models become more appropriate. The use of discrete approaches further requires arbitrary choices about the relative positions of charged sites,²⁷,²⁸ which are expected to vary considerably between individual FA entities. Herein we analyse FA electrostatics with a continuous approach, and invoke discrete concepts merely for particle charge numbers and ratios. We note that the relative magnitudes of $\kappa_p$ and $r_p$ precludes the existence of a Donnan phase which requires $\kappa_p r_p \gg 1$.¹⁶,²⁴⁻²⁶

To date, the electrostatic features of nanoparticles have been described on the basis of structural charge characteristics.¹⁹,²⁹ Recent work has presented a framework for computing the potential profile of small charged nanoparticles equilibrated in aqueous electrolytes, accounting for charge compensation by the indifferent electrolyte ions.³⁰ The following outlines the fundamental aspects of the two approaches; the outcomes are then discussed in context of experimental results for ion association with FA.
**Structural Charge Features.** For a random distribution of charged sites in a soft, ion-permeable spherical nanoparticulate FA body of radius $r_p$, the radial distribution of the electrostatic potential $\psi(r)$ generally has a bell shape that increasingly extends into the medium as the ionic strength decreases.⁹,¹⁹,²⁶ [Hereafter, $r$ denotes the radial coordinate with $r = 0$ the particle center.] The equilibrium accumulation of M at a distance $r$ from the center of the FA can be described by a Boltzmann partitioning coefficient, $f_B$, computed via the mean field Poisson-Boltzmann approach:

$$f_B(r) = \exp[-z_M U(r)] \qquad (3)$$

where $U$ is the dimensionless electrostatic potential energy at radial distance $r$, i.e. $U = U(r)/kT = F\psi(r)/RT$, where $k$, $F$, $R$, and $T$ have their usual meanings. The extent of counterion accumulation is obtained by computing the integral Boltzmann partitioning coefficient, $\overline{f}_B^{(s)}$, over the spatial scale from the particle center to $r$, i.e.:

$$\overline{f}_B^{(s)}(r) = \left( \int_0^r 4\pi r^2 f_B(r') dr' \right) / V(r) \qquad (4)$$

where $r'$ is a dummy integration variable and $V(r) = 4\pi r^3 / 3$. Using eq 4, we distinguish the extent of counterion accumulation within the FA particle body, $\overline{f}_{B,intra}^{(s)} = \overline{f}_B^{(s)}(r=r_p)$, from that spanning over only the extraparticulate double layer ($r > r_p$), $\overline{f}_{B,extra}^{(s)}(r > r_p)$, i.e.:

$$\overline{f}_{B,extra}^{(s)}(r > r_p) = \left( \int_{r_p}^r 4\pi r^2 f_B(r') dr' \right) / \left( V(r) - V(r_p) \right) = \frac{V(r) \overline{f}_B^{(s)}(r) - V(r_p) \overline{f}_B^{(s)}(r_p)}{V(r) - V(r_p)} \qquad (5)$$

As a consequence of the particle potential profile, extraparticulate counterions are less strongly associated than those within the particle body. The $\overline{f}_B^{(s)}$ values for 2+ counterions computed via eq 4 for a particle with $r_p$ of 1 nm and charge density of -2,000 mol $e$ m⁻³ in 1-1 and 2-1 electrolytes at $I = 10$ and 100 mM are shown in Figure 1. It is evident that counterion accumulation in the extraparticulate zone is significant. To highlight this aspect, Figure S1 shows the ratio $[\overline{f}_{B,extra}^{(s)}(r) - 1] / \overline{f}_{B,intra}^{(s)}$ that compares counterion accumulation in the extraparticulate zone ($r > r_p$) to that within the particle body.
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**Figure 1.** Structural charge-derived average Boltzmann partitioning coefficient, $\bar{f}_{B}^{(s)}$ (eq 4) for 2+ counterions, as a function of the radial distance, $r$, from the center of a negatively charged soft FA particle with radius, $r_p$, of 1 nm and structural charge density of -2,000 mol e m⁻³ in various bulk electrolyte media.. All data are shown in (a); the inset (b) is an amplification at small $\bar{f}_{B}^{(s)}$. The curves correspond to 1-1 electrolyte at $I = 10$ mM (black solid curve) and 100 mM (black dashed curve), and 2-1 electrolyte at $I = 10$ mM (blue solid curve) and 100 mM (blue dashed curve). On the x-axis, $r = 0$ denotes the center of the particle body and $r_{acc}$ at 2 nm and 3 nm indicates the radii of the counterion accumulation volumes considered for interpreting data on electrostatic binding of metal ions by FA (see text for details).

The electrostatic contribution to association of M with a small charged nanoparticle generally involves both the intraparticulate and the extraparticulate electric field. For the extraparticulate component, it is necessary to determine a cut-off point, beyond which counterion accumulation is no longer significant. As arbitrary first estimates, we take cut-off points as the distance from the particle surface at which the derivative of $\bar{f}_{B,extra}^{(s)}(r > r_p)$ with respect to $r$ has decreased to 10% and 2% of its value at the particle surface, $r = r_p$. This criterion locates a distance of ca. 1 nm and 2 nm from the particle surface, respectively. That is, counterion
accumulation is practically significant over a volume of radius ca. 2 to 3 nm (denoted by $r_{acc}$ in Figures 1 and S1), comprising the FA particle body ($r_p = 1$ nm) plus an extraparticulate shell extending 1 or 2 nm from the particle/medium interface, practically independent of the ionic strength and electrolyte type (1-1 or 2-1) considered. This spatial scale is similar to the thickness of Manning's counterion condensation volume around linear polyionic charges (ca. 2 nm),³¹ and the intraparticulate counterion condensation layer thickness of 2 nm found for large soft HA nanoparticles.¹¹,¹²,³²

**Potential Profile at the Equilibrated FA/Medium Interface.** The potential profile at the equilibrated FA/medium interface accounts for the reduction in net charge density of the FA due to accumulation of counterions within the particle body *and* the relevant extraparticulate zone.³⁰ For the general nanoparticle case, starting from the structural charge density, an iterative process was recently elaborated to evaluate the potential profile such that the amount of accumulated counterion charges is in equilibrium with the net particle charge.³⁰ Evidently, the lower the ionic strength, the greater the extent to which counterions accumulate and thus the lower the net particle charge density.³⁰ The Boltzmann partitioning coefficient for the equilibrated situation, $\bar{f}_{B}^{(\text{eq})}(r)$, is obtained from the concentration ratio of counterions in the accumulation volume $V(r)$ and in the bulk medium.³⁰ Under the conditions considered herein, the concentration of the target M is orders of magnitude lower than that of the electrolyte cations, thus electrostatic partitioning of M is dictated by the electric field set by the electrolyte, i.e. by $\bar{f}_{B}^{(\text{eq})}$.

**Intrinsic Stability Constants.** Measurement of the extent to which M associates with HA in indifferent 1-1 electrolyte as compared to that in 2-1 electrolyte has been established as a useful means to evaluate the electrostatic contribution to M binding.¹¹,¹² The electrostatic contribution will be greater in an indifferent 1-1 electrolyte such as KNO₃ as compared to that in a 2-1 electrolyte such as Ca(NO₃)₂, with the concentration of Ca²⁺ in vast excess over that of the electrostatically similar M²⁺. For large soil-derived HA, this approach was shown to give results consistent with the CCD electrostatic model.³²
The intrinsic stability constant, $\overline{K}_{int}$, represents the inherent chemical affinity between M and a binding site S, i.e. not including electrostatic contributions beyond the atom/atom bond distance. That is:

$$ \overline{K}_{int} = c_{MS} / c_{M}c_{S} \qquad (6) $$

where $c_{MS}$, $c_{M}$ and $c_{S}$ are the local intraparticulate concentrations of inner-sphere complexes, free metal ion and free reactive sites, respectively. For the chemically heterogeneous FA, the magnitude of $\overline{K}_{int}$ depends on the degree of site occupation, $\theta_{M}$, of the binding sites by M.¹ We note that many equilibrium speciation codes define $\theta_{M}$ as the ratio of concentrations of ‘bound metal’ and available reactive sites, where ‘bound metal’ is tacitly taken as comprising all forms of M that are associated with the particle, i.e. including the intraparticulate free ions, the outer-sphere complexes, the extraparticulate excess in the double layer and the condensed fraction in the intraparticulate double layer zone. For characterising the heterogeneity of $\overline{K}_{int}$ we define a true $\theta_{M}$ which reflects the ratio between the concentrations of inner-sphere complexes and total reactive sites, i.e.:

$$ \theta_{M} = c_{MS} / c_{S,t} \qquad (7) $$

## MATERIALS AND METHODS

**Reagents and Solution Conditions.** All solutions were prepared in ultrapure deionized water (resistivity > 18 MΩ cm) from a Milli-Q Gradient system. Metal ion solutions were prepared by dilution of standards (Aldrich). Ionic strength was maintained at 10 or 100 mM with $KNO_3$ or $Ca(NO_3)_2$ (BDH, Analar). Solutions were buffered to pH 6 with 1 mM MES buffer ((2-(*N*-morpholino)ethanesulfonic acid) prepared from the solid (Fluka, MicroSelect, ≥ 99.5%). A pH of 6 was chosen to prevent the formation of metal hydroxy species and/or ternary M-FA-OH complexes.³³ The FA sample was Suwannee River fulvic acid from the International Humic Substances Society (IHSS): it has a density of 1500 kg m⁻³,²¹ a charge density of -10 meq gC⁻¹ at pH 6,²⁰ a carbon content of 52% (IHSS data sheet), a hydrated particle radius of ca. 1 nm,³ and molar mass in the range ca. 550 – 1000 g mol⁻¹.²¹,³⁴⁻³⁶ In combination, these parameters yield a water content in the range 74-85% (see Supporting Information for details).

**Electrochemical Measurements.** The electrochemical measurements were performed with an Ecochemie μAutolab potentiostat (input impedance > 100 GΩ) coupled with a Metrohm VA stand. The working electrode
196 was a Metrohm multimode mercury drop electrode, the auxiliary electrode was glassy carbon, and the reference 197 electrode was Ag|AgCl|KCl(sat) encased in a 100 mM KNO₃ jacket. Solutions were initially purged with 198 oxygen-free N₂, and a nitrogen blanket was maintained during measurements. Total metal concentrations in 199 the range 2x10⁻⁴ to 7x10⁻³ mol m⁻³, and total FA concentrations in the range 10 to 40 g m⁻³ were used to generate 200 the range of θₘ values studied. The concentration of free M in the bulk medium, c*ₘ, was obtained from 201 equilibrium measurements in the foot of the stripping chronopotentiometry at scanned deposition potential 202 (SSCP) wave, which equates to the method of Absence of Gradients and Nernstian Equilibrium Stripping 203 (AGNES).³⁷ The heterogeneity parameter Γ was obtained from the shape of the experimental SSCP wave, as 204 described in previous work.³⁸,³⁹ The oxidation step was performed under complete depletion conditions 205 (stripping current = 2 nA), which renders the signal practically free from adsorption effects.⁴⁰⁻⁴² Equilibrium 206 was rapidly attained upon mixing the metal and fulvic solutions: the SCP stripping time, τ, measured ca. 15 207 mins after mixing M and FA remained constant over several hours. This result is expected for the small, non- 208 aggregating Suwannee River FA,⁴³ at a pH below that at which metal ion hydrolysis occurs,³³ and is in line 209 with the fast ionic partition processes in charged nanoparticles (nano to microsecond scales).¹⁷,⁴⁴ 210

211 **Determination of the Electrostatic Contribution to M-FA Association.** The electrostatic contribution 212 to ion association by FA is much smaller than that for HA and thus it is difficult to measure FA association 213 parameters for predominantly electrostatically associating ions such as Ca²⁺.⁴⁵⁻⁴⁹ Accordingly, a strategic 214 experimental approach was adopted to gain insight into the electrostatic contribution to M association with FA 215 by conducting measurements in 1-1 and 2-1 electrolytes of different ionic strength.¹¹,¹² At a given metal-to- 216 binding site ratio,, the difference between K̄ₐpp values in the various electrolytes can be ascribed to differences 217 in the magnitude of the electrostatic contribution. Ca(NO₃)₂ at I = 100 mM was taken as the ‘reference’ medium 218 with insignificant electrostatic binding of target M, i.e. f̄ₐ = 1. Experimentally-derived f̄ₐ values are 219 obtained from the ratio of K̄ₐpp in a given electrolyte to that in the reference electrolyte. Agreement between 220 the experimental values and the computed f̄⁽s⁾ₐ and f̄⁽eq⁾ₐ values is used to identify the spatial zone over which 221 counterion accumulation is significant. The K̄ᵢnt values are obtained via eq 6 using cₘ = f̄ₐc*ₘ. Further details 222 are given in the Supporting Information.
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RESULTS AND DISCUSSION

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**Characterisation of the Electrostatic Contribution to M²⁺-FA Association.** The values of the apparent stability constant, $\bar{K}_{\text{app}}$, for CdFA and CuFA are shown in Figure 2. $\bar{K}_{\text{app}}$ is defined in terms of smeared-out concentrations, without discrimination between the intraparticulate species, i.e. $\bar{K}_{\text{app}} = \bar{c}_{\text{M,b}} / c_{\text{M}}^* (\bar{c}_{\text{S,t}} - \bar{c}_{\text{M,b}})$, where $\bar{c}_{\text{M,b}}$ is the smeared out concentration of all forms of M associated with the FA particle, and $\bar{c}_{\text{S,t}}$ is the smeared-out total concentration of binding sites. Binding of Cd at pH 6 and I = 100 mM in 1-1 and 2-1 electrolyte was too weak for reliable measurements to be obtained by AGNES in the $\bar{c}_{\text{M,b}} / \bar{c}_{\text{S,t}}$ range of interest. From Figure 2, the $\bar{f}_{\text{B}}$ value relative to that for Ca(NO₃)₂ at I = 100 mM, is found to be 26 for I = 10 mM KNO₃, and 5 for both KNO₃ at I = 100 mM and Ca(NO₃)₂ at I = 10 mM (Table 1). For both Cd(II) and Cu(II), similar electrostatic contributions are in effect: at I = 10 mM the same factor difference was observed between $\bar{K}_{\text{app}}$ measured in KNO₃ and Ca(NO₃)₂. This observation supports the theoretical approach, i.e. the electrostatic contribution to M²⁺-FA association is independent of the nature of the ion.

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**Figure 2.** Apparent stability constants, $\overline{K}_{\text{app}}$, obtained *via* smeared-out concentrations for Cd(II) (red symbols) and Cu(II) (blue symbols) complexes with FA as a function of the degree of association of M with FA, $\overline{c}_{M,b} / \overline{c}_{S,t}$. Data were obtained from AGNES measurements in $Ca(NO_3)_2$ at an ionic strength of 100 mM (open circles) and 10 mM (open squares), and in $KNO_3$ at an ionic strength of 100 mM (solid circles), and 10 mM (solid squares). The error bars correspond to the experimental uncertainty of ± 0.1 in $\log \overline{K}_{\text{app}}$. The blue dashed lines through the Cu(II) data have a slope of -1.65 and the red dashed lines through the Cd(II) data have a slope of -1. For Cu(II), the standard error on the experimental data relative to the line with slope of -1.65 is 0.22 for data measured in I = 10 mM $KNO_3$, 0.26 for I = 100 mM $Ca(NO_3)_2$, and 0.14 for the data measured in I = 100 mM $KNO_3$ and I = 10 mM $Ca(NO_3)_2$. For Cd(II), the standard error on the experimental data relative to the line with slope of -1 is 0.19 for I = 10 mM $KNO_3$ and 0.20 for I = 10 mM $Ca(NO_3)_2$.

**Comparison of Computed and Experimental Boltzmann Partitioning Coefficients, $\overline{f}_B$.** For the FA conditions, i.e. $r_p$ of 1 nm and charge density -2,000 mol e m⁻³, the $\overline{f}_B$ values computed on the basis of the density of structural charges, $\overline{f}_B^{(s)}$, and from the equilibrated net FA charge, $\overline{f}_B^{(\text{eq})}$, are given in Table 1. The values depend on the thickness of the extraparticulate layer over which counterion accumulation is considered to be significant. Following the structural charged-based approach (eq 4), the $\overline{f}_B^{(s)}$ values computed for the spatial zone r = 0 to r = $r_p$ + 2 nm are in good agreement with experimental values for all electrolyte compositions considered. Despite this good agreement, the $\overline{f}_B^{(s)}$ values do not account for charge compensation by the indifferent electrolyte ions,³⁰ and thus the interpretation is far less rigorous than that based on the $\overline{f}_B^{(\text{eq})}$ values. In this context it is relevant to highlight that the timescale for ionic partitioning in soft charged nanoparticles is rather fast (nano- to microsecond scale).¹⁷,⁴⁴ Thus, background electrolyte ions will rapidly attain their equilibrium distribution between the bulk medium and the particle phase, in accordance with their respective concentrations and $\overline{f}_B$ values. This feature is simply accounted for by consideration of *equilibrated* net FA charges, which shows up feasible differences in the spatial zone relevant for counterion accumulation. For 1-1 electrolyte (I = 10 and 100 mM) and 2-1 electrolyte at I = 100 mM, the $\overline{f}_B^{(\text{eq})}$ values computed for the
spatial zone $r = 0$ to $r = r_p + 1$ nm are in best agreement with experimental data, whilst for a 2-1 electrolyte $I = 10$ mM the $\overline{f}_B^{(\text{eq})}$ computed for the spatial zone $r = 0$ to $r = r_p + 2$ (or 2.5) nm yields the best agreement (Table 1). These observations can be explained by considering the number, hydration state, and charge density of the counterions. At $I = 100$ mM in Ca(NO$_3$)$_2$ the $\overline{f}_B^{(\text{eq})}$ value corresponds to ca. 1 Ca$^{2+}$ associated with each FA entity, and in 100 mM KNO$_3$ $\overline{f}_B^{(\text{eq})}$ corresponds to 1.6 K$^+$ per FA. In both cases, electrostatically associated cations occupy the spatial zone from $r = 0$ to $r = r_p + 1$ nm. On the other hand, at the lower ionic strength of 10 mM in Ca(NO$_3$)$_2$ there are ca. 2.3 Ca$^{2+}$ accumulated per FA, and in 10 mM KNO$_3$ there are 4.8 K$^+$ per FA, i.e. practically full compensation of the structural charge in both cases. The experimental results are consistent with the 4.8 K$^+$ occupying a spatial zone from $r = 0$ to $r = r_p + 1$ nm whilst the 2.3 Ca$^{2+}$ occupy space from $r = 0$ to $r = r_p + 2$ nm (Table 1). This observation points to (a greater extent of) partial dehydration of the larger K$^+$ ion in the outer-sphere configuration, and is suggestive of counterion condensation. In a soft particle body such phenomena generally occur at larger $\ell_c$ than that needed to invoke condensation around line charges.⁵⁰ In 2-1 electrolyte at $I = 10$ mM, the larger accumulation zone is consistent with the greater electrostatic attraction of the 2+ ion over larger distances and the greater affinity of the 2.3 Ca$^{2+}$ for their associated H$_2$O molecules: the charge density of the bare Ca$^{2+}$ and Gibbs energy of hydration of Ca$^{2+}$ is ca. 4 times greater than that of K$^+$.⁵¹

To estimate the influence of uncertainties in the values, we performed a sensitivity analysis to encompass the range of parameters applicable to Suwannee River FA, i.e. $r_p = 0.8$ to 1.5 nm (corresponding to extremes of the Gaussian distribution of particle sizes),³ and charge density = -312 to -4122 mol e m$^{-3}$ (arising from the range of reported molar masses and consequent water content; see Supporting Information). The outcomes are presented in Tables S1-S6, and the ensuing consequences for the $\overline{K}_{\text{int}}$ values computed from implementation of $\overline{f}_B^{(\text{eq})}$ are shown in Figures S2-S5 in the Supporting Information. An analysis of the root mean squared error (RMSE) between the modelled and experimental $\overline{K}_{\text{int}}$ values confirms that the data presented in Table 1 provide the best description of the experimental data, i.e. the RMSE for these conditions is lower than that for any of the other conditions considered, where $\text{RMSE} = \sqrt{\frac{1}{n}\sum_n[\log(\overline{K}_{\text{int}})_{\text{theory}} - \log(\overline{K}_{\text{int}})_{\text{experiment}}]^2}$, for the
summation over the number, *n*, of $\theta_M$ values at which $\overline{K}_{int}$ was determined for the combined Cd and Cu datasets. These RMSE values are given in the caption of Figure 3 and Figures S2-S5.

<table><caption>**Table 1.** Comparison of Computed and Experimentally-Derived Magnitudes of the Electrostatic Boltzmann Partitioning Coefficient, $\overline{f}_B$, for Divalent Cations. Values in parentheses correspond to different spatial zones of counterion accumulation. Experimental data correspond to Cd(II) and Cu(II) binding by FA.</caption><thead><tr><th>Electrolyte and ionic strength</th><th>Theoretical</th><th>Experimental</th></tr></thead><tbody><tr><td></td><td>equilibrated charge, $\overline{f}_B^{(eq)}$</td><td>Figure 3<sup>d</sup></td></tr><tr><td>KNO<sub>3</sub>, I = 10 mM</td><td>25.79<sup>b</sup> (7.42<sup>a</sup>)</td><td>27.21<sup>a</sup> (84.37<sup>b</sup>)</td></tr><tr><td>KNO<sub>3</sub>, I = 100 mM</td><td>1.69<sup>b</sup> (NA<sup>a</sup>)</td><td>3.06<sup>a</sup> (7.38<sup>b</sup>)</td></tr><tr><td>Ca(NO<sub>3</sub>)<sub>2</sub>, I = 10 mM</td><td>9.89 [$n_{Ca}$=2.25]<sup>a</sup><br>(36.45 [$n_{Ca}$=2.46]<sup>b</sup>)<br>(5.67 [$n_{Ca}$=2.05]<sup>c</sup>)</td><td>6.62<sup>a</sup> (17.19<sup>b</sup>)</td></tr><tr><td>Ca(NO<sub>3</sub>)<sub>2</sub>, I = 100 mM</td><td>1.40 [$n_{Ca}$=0.94]<sup>b</sup> (NA<sup>a</sup>)</td><td>1.84<sup>a</sup> (3.46<sup>b</sup>)</td></tr></tbody></table>

<footnote><sup>a</sup> integral from $r = 0$ to $r = r_p + 2$ nm, i.e. $\overline{f}_B^{(eq)}$ ($r = r_{acc} = 3$ nm) or $\overline{f}_B^{(s)}$ ($r = r_{acc} = 3$ nm)</footnote>

<footnote><sup>b</sup> integral from $r = 0$ to $r = r_p + 1$ nm, i.e. $\overline{f}_B^{(eq)}$ ($r = r_{acc} = 2$ nm) or $\overline{f}_B^{(s)}$ ($r = r_{acc} = 2$ nm)</footnote>

<footnote><sup>c</sup> integral from $r = 0$ to $r = r_p + 2.5$ nm, i.e. $\overline{f}_B^{(eq)}$ ($r = r_{acc} = 3.5$ nm)</footnote>

<footnote><sup>d</sup> experimental $\overline{f}_B$ values are obtained from the difference between the dashed lines shown in Figure 3. The range of values in parentheses is obtained from the standard errors on the regression lines (see Figure 2 caption).</footnote>

<footnote><sup>e</sup> reference electrolyte, for which the *a priori* $\overline{f}_B$ value of unity is supported by the corresponding theoretical</footnote>

<footnote>$\overline{f}_B^{(eq)}$ value</footnote>

<footnote>$n_{Ca}$ is the average number of Ca<sup>2+</sup> per FA</footnote>
NA: not applicable, because a 2 nm extension of the condensation volume from the FA surface significantly exceeds the 0.96 nm extraparticulate Debye length operational in 100 mM solution ionic strength (see ref. 30 for details)

We note that our findings are in strong contrast to the much larger electrostatic contribution to ion binding that is invoked by equilibrium speciation codes, such as WHAM⁵² and NICA-Donnan.⁵³ For example, at *I* = 10 mM in 1-1 electrolyte, the NICA-Donnan model predicts, on the basis of an inappropriate Donnan potential profile, a Boltzmann partitioning coefficient of *ca*. 10⁴ for FA. This picture is not physicochemically sound: the size of FA is far too small to generate a true Donnan phase within the particle body, let alone to accommodate the required massive amount of counterions. Indeed, indiscriminate implementation of Donnan volumes in equilibrium speciation codes as descriptors for electrostatic interactions between metal ions and humic substances is a generally erroneous strategy, as pointed out in our previous work.¹¹,¹²,⁵⁰

**The Intrinsic and Heterogeneous Stability of Cd(II) and Cu(II) Complexes with FA.** The $\overline{K}_{\text{int}}$ values for FA are shown as a function of $\theta_{\text{M}}$ in Figure 3. The magnitude of $\overline{f}_{\text{B}}$ corresponds to a modest contribution of electrostatics to M association with FA. For both metal ions, over the entire $\theta_{\text{M}}$ range studied, the ratio between the intraparticulate concentrations of free M and inner-sphere complexed M is well below unity. Details of the procedure used to obtain the intraparticulate concentrations and $\overline{K}_{\text{int}}$ are given in the Supporting Information. As an illustrative example, for CdFA in 10 mM KNO₃ at a total Cd concentration of 4.7x10⁻³ mol m⁻³ and an FA concentration of 30 g m⁻³, the measured free M in the bulk solution is 1.92x10⁻³ mol m⁻³, which translates to an intraparticulate free M concentration of 5x10⁻² mol m⁻³, a corresponding concentration of inner-sphere complexes of 48.9 mol m⁻³ (99.9% of the intraparticulate M) and $\theta_{\text{M}}$ = 0.024. This is in strong contrast to the intraparticulate speciation found for soil HA, e.g. for CdHA at $\theta_{\text{M}}$ = 0.02, where *ca*. 40% of the particle-associated Cd was found to be in the form of inner-sphere complexes.³² Evidently, electrostatic cooperativity is manifest to a much greater extent in the large soil HA particles with Donnan behavior and counterion condensation phenomena. Figure 3 also compares the $\overline{K}_{\text{int}}$ values for FA with those reported previously for HA.³² For the two types of HS, $\overline{K}_{\text{int}}$ for Cu(II) is greater than that for Cd(II) over
the considered θₘ range. K̄<sub>int</sub> for Cd(II) complexes is similar for FA and HA, and is comparable to that for
Cd(II) complexes with simple carboxylic acids such as malonic acid.⁵⁴ In contrast, significant differences in
binding behavior are observed for Cu(II). Notably, for θₘ values below ca. 0.03, K̄<sub>int</sub> for CuHA is greater
than that for CuFA, with the difference in magnitude increasing with decreasing θₘ, which may reflect
differences in the nature of heterogeneity for the two types of HS. As typical for heterogeneous complexants,
K̄<sub>int</sub> for both FA and HA is dependent on θₘ, i.e. the extent to which the reactive sites are involved in inner-
sphere complexation. For many types of HS the double logarithmic plot of K̄<sub>app</sub> vs. θₘ is a straight line with
slope equal to -1/Γ, where Γ is the heterogeneity parameter (0 < Γ ≤ 1; Γ = 1 for the homogeneous case).¹ The
heterogeneity of metal complexes with HS generally follows the order Cu > Pb > Cd,¹,³⁸,⁵⁵ which reflects the
different chemical binding features of the metal ions as elaborated in environmental context by Turner and
coworkers.⁵⁶ For CdFA, our data (Figure 3) yield a Γ value close to unity, i.e. CdFA complexation is rather
homogeneous. In contrast, Cu(II) complexation is more heterogeneous, and different behavior is found for FA
as compared to HA: a Γ value of 0.6 is obtained for CuFA, cf. 0.2 for CuHA. For the range of θₘ values
considered herein, the effective number of Cu²⁺ ions associated with an individual FA entity is less than one.
Accordingly, the heterogeneity of FA is largely inter-particulate, i.e. the Γ value of 0.6 represents the particle-
to-particle heterogeneity of individual FA entities.⁵⁷,⁵⁸ For HA, the much greater number of reactive sites per
individual entity means that θₘ corresponds to many metal ions associated with an individual HA, and thus Γ
reflects the intra-particulate heterogeneity, i.e. the range of chemical functionalities within each HA entity.⁵⁹
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Comparison of the intrinsic stability constants, $\bar{K}_{\text{int}}$, determined for Cd(II) and Cu(II) complexes with fulvic acid (FA) and humic acid (HA) as a function of the degree of occupation of reactive sites by inner-sphere complexes, i.e. the true $\theta_{\text{M}}$ (eq 7). The coloured points for CdFA and CuFA are experimental data obtained using the experimentally derived $\bar{f}_{\text{B}}$ values (Table 1), a charge density of -2000 mol e m⁻³ and water content of 74% (see Supporting Information for details). The dashed lines have slopes of -1/Γ for each of the indicated Γ values. The standard error of the CdFA data relative to the line for Γ = 1 is 0.12, and the standard error of the CuFA data relative to the line for Γ = 0.6 is 0.07. The black asterisks correspond to the modelled values obtained using the equilibrated charge, $\bar{f}_{\text{B}}^{(\text{eq})}$, values in Table 1 (the values not in parentheses in each case), and for visual convenience the shaded grey bar encompasses the range of modelled values. The RMSE between the modelled and experimental $\bar{K}_{\text{int}}$ values = 0.257 (see main text). The various symbols for the experimental data correspond to measurements at ionic strengths of 10 mM in KNO₃ (solid circles), 100 mM in KNO₃ (solid squares), 10 mM in Ca(NO₃)₂ (open circles) and 100 mM in Ca(NO₃)₂ (open squares). The different colors denote CuHA (dark blue), CuFA (light blue), CdHA (dark green), and CdFA (red). The HA</figcaption></figure>
363 data are obtained from a previous publication (standard error of the CdHA data relative to the line for $\Gamma = 0.6$)
364 is 0.11; standard error of the CuHA data relative to the line for $\Gamma = 0.2$ is 0.27).32

365
366 To further investigate the heterogeneity features of MHS binding, $\Gamma$ values were also obtained from the shape
367 of experimental SSCP waves (Figure 4).38,39 Previous work on CuHA showed excellent agreement between
368 the magnitude of $\Gamma$ obtained from the slope of the log $\bar{K}_{int}$ vs. log $\theta_M$ plot and that obtained from the shape
369 of an SSCP wave.32 The consistent shape of the SSCP waves for CuHA at different $\theta_M$ values is shown in
370 Figure S2. For CuFA at $\theta_M = 0.03$, the $\Gamma$ value obtained from the slope of the log $\bar{K}_{int}$ vs. log $\theta_M$ plot was the
371 same as that corresponding to the shape of the SSCP wave, but significant discrepancies were found at lower
372 $\theta_M$ values. For example, at $\theta_M = 0.003$, the SSCP wave shape corresponded to $\Gamma = 0.2$ (Figure 4), i.e. CuFA
373 complexation is much more heterogeneous at very low $\theta_M$, in apparent contrast with the data in Figure 3. To
374 understand this feature we need to consider the nature of the SSCP wave. Notably, on passing from the foot to
375 the plateau, there is a progressive decrease in the concentration of free M at the electrode surface, and thus also
376 in the *local* metal-to-binding site ratio.38,39 As a consequence, increasingly more stable complexes become
377 significant in determining the flux of M towards the electrode. Nevertheless, the magnitude of the SSCP
378 plateau remained consistent with that predicted for the fully labile case. Thus, for a unimodal distribution of
379 CuFA entities, we would expect the shape of the SSCP wave, i.e. $\Gamma$, to be independent of $\theta_M$. The observation
380 that the SSCP wave for CuFA is more elongated at a bulk $\theta_M$ of 0.003 cf. 0.03 (Figure 4), points to a
381 multimodal distribution: the stronger and more heterogeneous binding observed at low $\theta_M$ likely reflects the
382 diversity of the fewer but stronger binding sites. Potential candidates are N- and S- containing functional
383 groups. This line of reasoning is in accordance with the concept of ‘major’ versus ‘minor’ sites proposed by
384 Buffle to explain metal binding features of HS,60,61 and the reported very low complexing capacity of the
385 strongest sites in FA.8,62 Literature data on CuFA complexation correspond to $\theta_M$ values greater than ca. 0.003,
386 and collated $\bar{K}_{app}$ vs. $\theta_M$ plots for CuFA yield a $\Gamma$ value of ca. 0.5, in accordance with the present work.1 This
387 outcome highlights the utility of flux based electrochemical speciation techniques, such as SSCP, which allow
388 a broad range of effective $\theta_M$ values to be probed for a single bulk solution composition.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Experimental and computed SSCP waves (normalized reoxidation time, $\tau$, as a function of deposition potential, $E_d$) for Cu(II) complexes with fulvic acid at pH 6 in Ca(NO$_3$)$_2$ electrolyte at $I = 100$ mM. Points are experimental data; dashed curves are computed for the $\Gamma$ values indicated on the figure.</figcaption></figure>

All in all, our strategy provides a robust means to identify the electrostatic and chemical contributions to metal ion binding by nanoparticulate FA, and thereby characterise the intrinsic heterogeneity of MFA complexes. The most rigorous description of the electrostatic contribution to M binding by small nanoparticles ($\kappa r_p \approx 1$) is obtained via the *equilibrated* potential profile. The extent of counterion accumulation, and the available spatial zone for that, point to the involvement of simultaneous Boltzmann partitioning and counterion condensation. The extent to which the electrostatic field contributes to M association with FA is much less significant than for HA, underscoring the reduced cooperativity in effect for small nanoparticles with dimensions of the order of the Debye screening length. These findings underscore the need for equilibrium speciation codes to incorporate dedicated electrostatic descriptors that robustly account for the size and charge density of nanoparticulate complexants such as humic substances. A detailed exploration of the implications of the present work in this context is the subject of forthcoming work. Knowledge of the intraparticulate
speciation is also significant for prediction of the lability and bioavailability of nanoparticulate metal species:

obviously, the dynamic features of the intraparticulate free M differ from those of the inner-sphere complexes.

407

408 ACKNOWLEDGEMENTS

RMT conducted this work within the framework of the EnviroStress Centre of Excellence at the University of

Antwerp.

411

The authors declare no competing financial interest.

413

414 ASSOCIATED CONTENT

415 Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: xxx

The derivation of eq 2, Figure S1 showing the relative magnitude of counterion accumulation in the

extraparticulate zone as compared to that within the FA body, explanation of the procedures used to

obtain the water content and charge density of FA, and the $\overline{K}_{int}$ values, Tables S1-S6 showing the

sensitivity analysis for the computed $\overline{f}_{B}^{(eq)}$ and $\overline{f}_{B}^{(s)}$ values, Figures S2-S5 showing the sensitivity

analysis for the $\overline{K}_{int}$ values, and Figure S6 showing the SSCP waves for CuHA (PDF)

422

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254x190mm (96 x 96 DPI)