# Rigorous Physicochemical Framework for Metal Ion Binding by Aqueous Nanoparticulate Humic Substances: Implications for Speciation Modelling by the NICA-Donnan and WHAM Codes

Raewyn Town, Herman van Leeuwen, Jérôme F.L. Duval

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Raewyn Town, Herman van Leeuwen, Jérôme F.L. Duval. Rigorous Physicochemical Framework for Metal Ion Binding by Aqueous Nanoparticulate Humic Substances: Implications for Speciation Modelling by the NICA-Donnan and WHAM Codes. Environmental Science and Technology, 2019, 53 (15), pp.8516-8532. 10.1021/acs.est.9b00624. hal-02185102

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Supporting Information

for the manuscript

Rigorous Physicochemical Framework for Metal Ion Binding by Aqueous
Nanoparticulate Humic Substances: Implications for Speciation Modeling by the
NICA-Donnan and WHAM Codes

Raewyn M. Town<fnref n="1" />,<fnref n="2" />*, Herman P. van Leeuwen<fnref n="2" />, Jérôme F. L. Duval<fnref n="3" />

The Supporting Information comprises 5 pages, and includes an outline of the Ohshima formalism for the electrophoretic mobility of soft charged particles, description of the procedures used to obtain the intrinsic stability constants for metal ion binding by nanoparticulate complexants, Table S1 with definitions of the parameters used in the CCD-based model, Figure S1 showing the intraparticulate speciation of Cd species associated with Aldrich HA, Figure S2 showing the double logarithmic plot of $\overline{K}_{\text{int}}$ vs. $\theta_M$ and SSCP waves for Cd(II) and Cu(II) complexes with FA, and a list of the references cited in the SI.

Ohshima Formalism for the Electrophoretic Mobility of Soft Charged Particles

The rigorous theory for electrophoresis of hard (ion-impermeable) particles, valid for any ratio between the particle radius, r_p, and the Debye layer thickness in the bulk medium, κ⁻<fnref n="1" />, was established in the late 70's by O'Brien and White.<fnref n="1" /> The key electrokinetic quantity obtained from experimental data by such a theory is the particle zeta-potential, i.e. the potential located at the slip plane separating the stagnant liquid layer surrounding the particle from the 'bulk' mobile liquid phase.<fnref n="2" /> The zeta-potential is converted to particle surface charge density by means of Gouy-Chapman theory or more sophisticated double layer representations.<fnref n="3" /> However, the basic concept of a zeta-potential is not applicable to soft NPs due to the absence of a discrete slip plane at the interphase they form with the electrolytic medium (see ref 4 and references cited therein). The formalisms for electrokinetics of soft particles differ according to their treatment of electrostatics, in particular their ability to apply to thin double layer cases (κr_p ≤ 1),⁵⁻⁸ or to the degree of sophistication in integrating particle backbone material distribution which impacts on both the electrostatic and hydrodynamic flow field distributions and, in turn, the particle electrophoretic mobility, μ.⁷⁻⁹ The model by Ohshima⁵ has been successfully employed to determine the electrostatic surface properties of various particles, including microgels, erythrocytes, and bacterial surfaces.⁴,⁶ Specifically, the electrophoretic mobility μ of soft particles that meet the conditions for establishment of a Donnan phase, i.e. $\kappa r_p \gg 1$, reads as:

$$ \mu = \frac{\rho_p}{\eta \lambda_o^2} + \frac{\varepsilon \psi_o \kappa_p^{-1} + \psi_D \lambda_o^{-1}}{\eta \kappa_p^{-1} + \lambda_o^{-1}} \quad (S1) $$

where ρ_p represents the net density of charges (in C m⁻<fnref n="3" />) carried by the soft NP (and assumed to be homogeneously distributed therein), 1/κ_p the intraparticulate Debye layer thickness, 1/λ_o the characteristic penetration length of the electroosmotic flow within the soft NP, ε and η are the permittivity and dynamic viscosity of the electrolyte medium, respectively, ψ_o corresponds to the NP surface potential, and ψ_D to the

<footnote n="1">¹ Systemic Physiological and Ecotoxicological Research (SPHERE), Department of Biology, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.</footnote>

<footnote n="2">² Physical Chemistry and Soft Matter, Wageningen University & Research, Stippeneng 4, 6708 WE Wageningen, The Netherlands.</footnote>

<footnote n="3">³ CNRS – Université de Lorraine, Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), UMR 7360 CNRS, 15 avenue du Charmois, 54500 Vandoeuvre-les-Nancy, France.</footnote>
Donnan potential. The parameters $\psi_D$, $\psi_o$ and $1/\kappa_p$ all depend on the space charge density $\rho_p$ and on the electrolyte concentration according to:

$$ \psi_D = \frac{RT}{F} \operatorname{asinh} \left( \frac{\rho_p}{2Fc_1^*} \right) \qquad (S2) $$

$$ \psi_o = \psi_D - \frac{RT}{F} \tanh \left( \frac{F\psi_D}{2RT} \right) \qquad (S3) $$

$$ \kappa_p = \kappa \left\{ \cosh \left( \frac{F\psi_D}{RT} \right) \right\}^{1/2} \qquad (S4) $$

which are written here for a 1-1 electrolyte with concentration $c_1^*$.

In agreement with experimental data, Ohshima's model quantitatively describes how the particle mobility $\mu$ decreases in magnitude with increasing solution ionic strength due to particle charge screening. Most remarkably, it demonstrates that the mobility $\mu$ of soft Donnan particles (Figs. 1A-B in main text), unlike that of hard particulate systems (Fig. 1C in main text), asymptotically reaches a non-zero plateau value at sufficiently large electrolyte concentrations, which is a direct consequence of their defining ion permeability features.4,6,7

## Computation of $\overline{K}_{\text{int}}$ Values

The expression for the intrinsic stability constant $\overline{K}_{\text{int}}$ is given as eq 15 in the main text, and repeated here for convenience:

$$ \overline{K}_{\text{int}} = \frac{c_{\text{MS}}}{c_M c_S} \qquad (S5) $$

where $c_{\text{MS}}$, $c_M$ and $c_S$ are the local average concentrations of inner-sphere complexes, free metal ion and reactive sites in the particle body, respectively. In the present case, the total intraparticulate concentration of reactive sites, $c_{S,t}$, corresponds to the concentration of charged sites and $c_S = c_{S,t} - c_{\text{MS}}$.

For small NPs, with $\kappa_p r_p \approx 1$, $c_M$ is obtained from $\overline{f}_{B,M} c_M^*$, where $c_M^*$ is the concentration of the free metal ion in the bulk solution and $\overline{f}_{B,M}$ is the Boltzmann factor for $M^{2+}$ computed via the mean-field Poisson-Boltzmann approach based on the equilibrated potential profile over the intraparticulate and relevant extraparticulate zones. See ref 10 for details. The smeared-out concentration of metal ions associated with the NP, $\overline{c}_{M,b}$, simply corresponds to the difference between the total metal ion concentration in the dispersion and the free metal ion concentration in the bulk solution. The intraparticulate concentration of all forms of M associated with the particle (= $c_M + c_{\text{MS}}$) is given by $\overline{c}_{M,b}$ divided by the aqueous particle volume fraction in the dispersion; subsequent subtraction of $c_M$ yields $c_{\text{MS}}$.

For large NPs, with $\kappa_p r_p >> 1$, the CCD electrostatic model is used to compute the intraparticulate speciation in the intraparticulate double layer (DL) and Donnan volume.11 The defining parameters are given in Table S1. In practice, an iterative process is used to determine the intraparticulate speciation. In a first step, the particle-associated M is distributed to satisfy the electrostatic demands of counterion condensation and Donnan partitioning, and the remaining M is ascribed to inner-sphere complexes, MS. The presence of MS in the DL reduces the net charge therein, and thus the proportion of condensed ions is correspondingly decreased. Accordingly, the initial intraparticulate metal speciation is iterated with respect to the concentrations of inner-sphere complexes $c_{\text{MS}}$ versus condensed metal ions in the DL zone $c_{\text{M,cond}}^{\text{DL}}$ until a consistent intraparticulate distribution over the Donnan bulk and DL zone is attained. Typically 4 iterations are required to obtain consistent values for $c_{\text{MS}}$ and $c_{\text{M,cond}}^{\text{DL}}$.
<table><caption>Table S1. CCD-Based Model Parameters Defining the Intraparticulate Metal Speciation in Highly Charged, Soft Nanoparticulate Complexants(a)</caption><thead><tr><th>M(II) species</th><th>Intraparticulate double layer</th><th>Donnan volume</th><th>Governing physicochemical features</th></tr></thead><tbody><tr><td>Free M</td><td>c<sup>DL</sup><sub>M</sub> = f&#773;<sup>DL</sup><sub>B,M</sub>c<sup>*</sup><sub>M</sub></td><td>c<sup>D</sup><sub>M,f</sub> = f&#773;<sup>D</sup><sub>B,M</sub>c<sup>*</sup><sub>M</sub></td><td>Donnan</td></tr><tr><td>Condensed M</td><td>c<sup>DL</sup><sub>M,cond</sub> = f&#773;<sub>C</sub>c<sup>DL</sup><sub>S</sub></td><td>0<sup>(b)</sup></td><td>Counterion condensation</td></tr><tr><td>Inner-sphere MS<sup>(c)</sup></td><td>c<sub>MS</sub> = K&#773;<sub>int</sub>c<sub>S</sub>c<sub>M</sub></td><td></td><td>Covalent binding</td></tr></tbody></table>

(a) The concentrations are denoted by superscript D or DL for local intraparticulate concentrations in the pertaining part of the particle volume; the superscript * denotes the free metal ion concentration in the bulk solution. The condensation limit for 2+ counterions in the DL with $\rho = \bar{\rho}_{DL}$ is denoted by $\bar{f}_C$.

(b) In the high charge density regime, with $\kappa_p r_p \gg 1$, counterion condensation is confined to the intraparticulate double layer shell. See main text.

(c) See eq S5.

<figure><img src="image_2.png" /></figure>

**Figure S1.** Intraparticulate speciation of Cd species associated with Aldrich HA, for a total Cd concentration of 4.6×10⁻³ mM and an HA concentration of 50.5 g m⁻³, in 10 mM KNO₃ at pH 6. Data correspond to the local concentrations determined via the (A) CCD model, intraparticulate double layer: condensed M (pale blue) and inner-sphere complexes (solid black); (B) CCD model, Donnan volume: free M (dark blue) and inner-sphere complexes (solid black); (C) and (D) NICA-Donnan model using (c) generic and (d) Aldrich optimised parameters: free M (dark blue) and inner-sphere complexes with nominal carboxylic groups (diagonal stripes) and phenolic groups (horizontal stripes); and (E) WHAM: free M (dark blue) and inner-sphere complexes that are monodentate (solid dark grey), bidentate (black dots), and tridentate (vertical black stripes).
<figure><img src="image_3.png" /></figure>

**Figure S2.** (A) Intrinsic stability constant, log K̄<sub>int</sub>, as a function of the degree of inner-sphere complex formation, log θ<sub>M</sub>, for Cd(II) and Cu(II) complexes with FA. The mean-field Poisson-Boltzmann (PB)-based results were obtained from experimental data for SRFA,<sup>10</sup> obtained in 10 mM KNO<sub>3</sub> (solid black squares), 100 mM KNO<sub>3</sub> (solid black diamonds), 3.33 mM Ca(NO<sub>3</sub>)<sub>2</sub> (solid blue squares), and 33.33 mM Ca(NO<sub>3</sub>)<sub>2</sub> (solid blue diamonds). The NICAD-Donnan computations were done using the generic parameters for FA,<sup>12,13</sup> and c<sub>MS</sub> was taken as the sum of the complexes with the nominal carboxylic and phenolic sites. For WHAM, c<sub>MS</sub> was taken as the sum of the mono-, bi-, and tri-dentate complexes. The computations with both speciation codes were performed for the background electrolytes 10 mM KNO<sub>3</sub> (NICA-Donnan: open grey squares; WHAM: open red squares), 100 mM KNO<sub>3</sub> (NICA-Donnan: open grey diamonds; WHAM: open red diamonds), 3.33 mM Ca(NO<sub>3</sub>)<sub>2</sub> (NICA-Donnan: open grey circles; WHAM: open red circles), and 33.33 mM Ca(NO<sub>3</sub>)<sub>2</sub> (NICA-Donnan: open grey triangles; WHAM: open red triangles). The SSCP waves<sup>10</sup> for (B) CuFA and (C) CdFA correspond to the experimental (solid diamonds) and computed curves for the indicated Γ values (curves) of the normalised reoxidation time, τ, as a function of deposition potential, E<sub>d</sub>. The experimental data correspond to c̄<sub>M,b</sub> / c̄<sub>S,t</sub> ≈ 0.03 for Cu and 0.01 for Cd, and were measured in Ca(NO<sub>3</sub>)<sub>2</sub> electrolyte at pH 6 and an ionic strength of 100 mol m<sup>-3</sup> for CuHA and 10 mol m<sup>-3</sup> for CdHA.
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