# Lability of Nanoparticulate Metal Complexes at a Macroscopic Metal Responsive (Bio)interface: Expression and Asymptotic Scaling Laws

Jérôme Duval, Raewyn Town, Herman van Leeuwen

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## To cite this version:

Jérôme Duval, Raewyn Town, Herman van Leeuwen. Lability of Nanoparticulate Metal Complexes at a Macroscopic Metal Responsive (Bio)interface: Expression and Asymptotic Scaling Laws. Journal of Physical Chemistry C, 2018, 122 (11), pp.6052-6065. 10.1021/acs.jpcc.7b11982. hal-02059661

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This item is the archived peer-reviewed author-version of:

Lability of nanoparticulate metal complexes at a macroscopic metal responsive (bio)interface : expression and asymptotic scaling laws

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Duval Jérôme F.L., Town Raewyn M., van Leeuwen Herman P.- Lability of nanoparticulate metal complexes at a macroscopic metal responsive (bio)interface : expression and asymptotic scaling laws

The journal of physical chemistry : C : nanomaterials and interfaces - ISSN 1932-7447 - 122:11(2018), p. 6052-6065

Full text (Publisher's DOI): https://doi.org/doi:10.1021/ACS.JPCC.7B11982

Full text (Publisher's DOI): https://doi.org/10.1021/ACS.JPCC.7B11982

To cite this reference: https://hdl.handle.net/10067/1500800151162165141
Article

# Lability of Nanoparticulate Metal Complexes at a Macroscopic Metal
Responsive (Bio)Interface: Expression and Asymptotic Scaling Laws

Jerome F.L. Duval, Raewyn M. Town, and Herman P. Van Leeuwen

J. Phys. Chem. C, **Just Accepted Manuscript** • DOI: 10.1021/acs.jpcc.7b11982 • Publication Date (Web): 19 Feb 2018

Downloaded from http://pubs.acs.org on February 25, 2018

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# Lability of Nanoparticulate Metal Complexes at a Macroscopic Metal Responsive (Bio)Interface: Expression and Asymptotic Scaling Laws

Jérôme F.L. Duval*,<fnref n="1" />,<fnref n="2" /> Raewyn M. Town,<fnref n="3" />,<fnref n="4" /> Herman P. van Leeuwen<fnref n="4" />

<footnote n="1">¹ CNRS, Laboratoire Interdisciplinaire des Environnements Continentaux (LIEC), UMR 7360, Vandoeuvre-lès-Nancy, F-54501 France.</footnote>

<footnote n="2">² Université de Lorraine, LIEC, UMR 7360, Vandoeuvre-lès-Nancy, F-54501 France.</footnote>

<footnote n="3">³ Systemic Physiological and Ecotoxicological Research (SPHERE), Department of Biology, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.</footnote>

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

<footnote>* Corresponding author. Email: jerome.duval@univ-lorraine.fr</footnote>

<footnote>Laboratoire Interdisciplinaire des Environnements Continentaux- UMR 7360 Université de Lorraine, 15 avenue du Charmois, 54501 Vandoeuvre-les-Nancy, France.</footnote>
**Abstract.**

The lability of metal complexes expresses the extent of the dissociative contribution of the complex species to the flux of metal ions toward a macroscopic metal-responsive (bio)interface, *e.g.* an electrodic sensor or an organism. While the case of molecular ligands is well established, it is only recently that a definition was elaborated for the lability of metal complexes with nanoparticles (NPs) in aqueous dispersions. The definition includes the thickness of the non-equilibrium reaction layer operational at the (bio)interface and the extent of geometrical exclusion of NPs therefrom. In this work, an explicit expression is derived for the lability of nanoparticulate metal complexes (M-NP) towards a macroscopic reactive (bio)interface. Interpretation accounts for the M-NP chemodynamic properties that depend on NP size, electrostatics, metal diffusion and dehydration rates, and density of metal binding sites for various NPs types, *i.e.* soft/core-shell and hard NPs having volume and surface sites distribution, respectively. Computational examples under practical conditions illustrate how these factors jointly determine the remarkable non-monotonous dependence of M-NP lability parameter on NP size. The analysis is supported by the formulation of asymptotic scaling laws clarifying how local M-NP dissociation dynamics affect the lability parameter for M-NP complexes at the scale of the macroscopic (bio)interface.
# 1. Introduction.

The nanoparticle (NP) size domain has been shown to exhibit unique reactivity features towards target ions and small molecules.<sup>1-3</sup> The modified reactivity of various target species upon their association with NPs has broad significance for e.g. robust risk assessments of NPs,<sup>4,5</sup> and design of NPs with tailored uptake/release kinetics for application in e.g. drug delivery.<sup>6-10</sup> The interpretation calls for coupling the physicochemical features of NPs, e.g. size, charge density, chemical functionality, with their chemodynamic behaviour under the conditions encountered in environmental and biological media. The conceptual framework we develop herein quantitatively describes the reactivity of NP-target associates at a fundamental mechanistic level. The concepts are applicable to any target species: without loss of generality, we take the example of metal ion (M) complexes with negatively charged NPs.

Recent work has elaborated a conceptual framework for evaluation of the chemodynamics of (nano)particulate metal complexes, i.e. the dynamics of interconversion between free and complexed metal forms.<sup>1,2,11-17</sup> The work extends the original Eigen mechanism<sup>18</sup> for formation of complexes between M and molecular ligands to the more intricate situation of charged complexing agents with nanoparticulate and colloidal dimensions. The confinement of metal binding sites within the volume of the permeable particle body or shell (3D soft/core-shell NPs) or at the particle surface (2D hard NPs) leads to metal complex chemodynamics that generally involve a balance between chemical reaction kinetic and diffusive metal transport contributions.<sup>1,2,11-17</sup> Several reports demonstrated that these contributions are strongly affected by the electrostatic conditions prevailing in the vicinity, at the surface and/or in the volume of the NP body.<sup>2</sup> Depending on medium salinity, NP charge and size, the particulate electric field may lead to a significant increase of the concentration of metal ions at the surface or in the volume of the NPs, and subsequently to a kinetics of M-NP complex formation that may be several orders of magnitude faster than that for complexes with molecular ligands.<sup>1</sup>

Given this knowledge on the chemodynamic properties of M-NP complexes, it was recently possible to elaborate a definition of their lability at a metal-responsive macroscopic interface such as a sensing electrode or the surface of an organism.<sup>13</sup> The lability parameter is a measure of the extent to which the dissociation of the complexes contributes to the flux of metal ions toward the reactive sensing interface.<sup>13</sup> Such formulation of M-NP lability is mandatory for e.g. a proper evaluation of the bioavailability of metal ions to organisms in the presence of complexants,<sup>19</sup> for interpreting measurements obtained by dynamic metal speciation techniques such as various voltammetries<sup>20-22</sup> and Diffusive Gradients in Thin Films (DGT),<sup>23-25</sup> as well as in predicting release kinetics in the context of nanodrug formulations.<sup>7</sup> At the labile limit, complexation of metal ions by nanoparticulate ligands is at equilibrium for any position from the sensing surface so that the overall flux of metal to the sensor results from the coupled diffusion-controlled transport of M-NP and free M.<sup>13</sup> In the other limit of non-labile complexes, the complexation equilibrium
is not maintained and the relevant metal flux to the sensor is governed by the rate at which metal ions are released from the particle body.¹³ For intermediate situations, the lability parameter is determined by the ratio between the kinetic dissociation flux of complexes and the coupled diffusion flux of M-NP and M (Figure 1). A convenient way to formulate the dissociative kinetic flux of M-NP complexes at the sensor surface is to invoke the concept of a reaction layer, originally developed by Heyrovský and Brdička²⁶⁻²⁸ for molecular complexing ligands and recently detailed for nanoparticulate ligands.²⁹ The thickness of this reaction layer, $\bar{\lambda}$, basically derives from the relative mobilities of free and complexed metal species in the medium and their respective lifetimes as determined by the rate of re-association of M with NPs and the rate of M-NP dissociation.²,²⁹ In turn, the spatial zone lying within a distance $\bar{\lambda}$ from the sensor defines a non-labile region where metal release from the complexes to the macroscopic interface is operational, i.e. $\bar{\lambda}$ denotes an effective reaction layer thickness at the macroscopic reactive interface.²,²⁹ By its very nature, a reaction layer can only exist in the presence of complexing sites.¹³,³⁰ Accordingly, for the case of NP complexants, the reaction layer at the sensing surface is an operational one that is related to the time-averaged presence of particle body volume/surface area and the corresponding time-averaged complexing site concentration. Unlike molecular ligands,²⁵⁻²⁸ the dissociation kinetic flux of M-NP complexes, or for that matter, the reaction layer thickness $\bar{\lambda}$, generally depends on the degree of geometrical exclusion of the complexing NP’s body from the reaction layer, i.e. on the fraction of metal binding sites effectively involved in the dissociation of the complexes at the metal-sensing interface.²,¹³,²⁹ Recent successful confrontation between the above lability framework and electrochemical data collected for Cd(II) in a dispersion of polymer nanoparticles evidenced the paramount importance of NP exclusion effects for a proper assessment of the lability of the nanoparticulate metal complexes considered.²,²⁹ In contrast, the application of the conventional reaction layer approach - which neglects such effects²⁵⁻²⁸ - was shown to overestimate the lability parameter of the complexes by more than 3 orders of magnitude.²⁹

In the present work, an expression is developed for the lability parameter of M-NP complexes at a macroscopic metal sensor with explicit integration of the local M-NP chemodynamic features in conjunction with the macroscopic scale kinetic and transport-mediated exchange of metal species at the sensor interface. Such elaboration generalizes previous work¹³ and is further complemented by the derivation of useful tractable asymptotic scaling laws for analysis of the cumbersome dependence of lability of nanoparticulate complexes on their size. In turn, the results allow for a clear identification of the multi-factorial processes that control the magnitude of the M-NP lability parameter over the practical range of particle size as a function of e.g. the particle electrostatic features, the density of metal binding sites located within the volume or at the surface of the particle, the dehydration kinetics of the complexing metal ion (Eigen type association) and the intrinsic stability of the nanoparticulate metal complex.
## 2. Theory: setting the stage.

In the following, we consider charged spherical nanoparticles of radius $r_p$ dispersed in a medium containing aqueous metal ions M (valence $z_M$) with bulk concentration $c_M^*$ and ions from a monovalent background electrolyte in large excess over M. In line with usual practice, the electric potential profile at the NP/medium interface is assumed to be pre-established by the indifferent electrolyte ions.² The binding of metal ions to NPs occurs *via* reactive sites S, that are present in excess over the metal ions in solution, and distributed either at the NP surface or within the particle body. The former case refers to 2D hard complexing NPs with surface local site density denoted as $\Gamma_S$ and the latter to soft/permeable and core-shell particulate systems with intraparticulate volume site density $c_s$.¹⁷,²⁹ We introduce the ratio $z = a/r_p$ with $a$ being the NP core radius so that soft, core-shell and hard complexing NPs refer to $z=0$, $0<z<1$ and $z=1$, respectively. The volume fraction of particles in dispersion is denoted as $\varphi$. For many practical cases with $\varphi \ll 1$, there is no overlap between diffusion layers and electric double layers of neighbouring particles.²⁹ For the sake of readability, a list of the symbols (with corresponding units) used in this work is provided at the end of the manuscript.

*Chemodynamics of nanoparticulate metal complexes.* In the high charge density regime where the separation distance $\ell_c$ between structural charges carried by NPs is smaller than the electric Debye layer thickness $1/\kappa$,¹⁴ the formation of M-NP complexes basically comprises the following successive steps:²,¹³⁻¹⁷,²⁹⁻³⁰ (a) the conductive diffusion of $M_{aq}^{z_M+}$ from the bulk solution to the charged particle body (3D NPs) or surface (2D NPs), and the subsequent incorporation of $M_{aq}^{z_M+}$ as a free hydrated ion within the particle shell volume (relevant for 3D NP type) (step (i) in **Figure 1** with rate constant $k_{a,p}$ expressed in mol⁻¹ m³ s⁻¹); (b) the fast electrostatic pairing of $M_{aq}^{z_M+}$ with S, which leads to an outer-sphere M·S associate; and finally (c) the formation of an inner-sphere MS complex following the loss of water of hydration by $M_{aq}^{z_M+}$ and the creation of a chemical bond with S (step (ii) in **Figure 1** with rate constant $k_a^{is}$ in m³ mol⁻¹ s⁻¹). The dissociative counterparts of steps (i) and (ii) are shown schematically in **Figure 1** with the corresponding rate constants $k_{d,p}$ and $k_d^{is}$ (both in s⁻¹), respectively. Depending on a number of physicochemical parameters pertaining to the properties of the NP (size, site density, electrostatics), to the free metal ion (dehydration kinetics, valency) and to the stability of the complex MS, the rates for overall
formation and dissociation of M-NP complexes may be kinetically determined by step (i), step (ii) or by both. The conceptual framework has been extensively detailed in preceding theoretical reports, together with supporting experimental data.<sup>1,2,12-17</sup> The composite rate constants $k_a$ (mol<sup>-1</sup> m<sup>3</sup> s<sup>-1</sup>) and $k_d$ (s<sup>-1</sup>) for the overall association (step (i) to (ii) in **Figure 1**) and dissociation (step (ii) to (i)) thus generally involve both chemical kinetic and diffusive transport contributions according to $k_a = (1/k_a^{\text{is}} + 1/k_{a,p})^{-1}$ and $k_d = (1/k_d^{\text{is}} + 1/k_{d,p})^{-1}$, respectively. It is stressed that the intraparticulate diffusion of metal ions M has been shown to be fast compared to extraparticulate diffusion<sup>31</sup> so that the former can often be neglected with respect to the latter. In turn, $k_{a,p}$ and $k_{d,p}$ effectively represent the transport of M to and from the sites S, respectively.

*Lability of nanoparticulate metal complexes at a macroscopic metal-responsive (bio)interface.* A macroscopic metal sensor (with curvature radius $r_o$ much larger than $r_p$) converts or accumulates the free metal ions M (**Figure 1**). Examples of such sensors include bacteria,<sup>32-34</sup> algae,<sup>35-36</sup> gill surfaces,<sup>37-38</sup> electrodes,<sup>20</sup> polymeric gel layers and other ion-exchange resins.<sup>39-46</sup> The conversion or accumulation of M at the sensor surface or biointerface creates a concentration gradient of M from the bulk solution towards the sensor surface. Depending on the rate of interconversion between free and complexed metal forms as compared to the conversion/accumulation timescale, complexes may dissociate within the diffusion layer at the reactive sensor/medium interface (**Figure 1**), thus releasing M ions that enhance the metal transport flux towards the sensor. The lability parameter $\mathcal{L}$ for the nanoparticulate metal complexes at the metal-responsive (bio)interface is defined as the ratio between the maximum kinetic flux, $J_{\text{kin}}$, arising from the rate of dissociation of the complexed species, and the maximum diffusive flux, $J_{\text{diff}}$, that represents the rate at which free and complexed metal species arrive at the macroscopic reactive (bio)interface (**Figure 1**).<sup>2</sup> The expression of the maximum kinetic flux $J_{\text{kin}}$ for chemical dissociation of M-NP complexes may be written in the form<sup>2,13,29</sup>

$$J_{\text{kin}} = k_d^{\text{is}} \bar{c}_{\text{MS}}^{\lambda} \bar{\lambda} \qquad (1)$$

, where $\bar{\lambda}$ is the thickness of the macroscopic reaction layer at the medium side of the sensor/medium interface operational for nanoparticulate metal complexes,<sup>29</sup> and $\bar{c}_{\text{MS}}^{\lambda}$ is the concentration of MS complexes *averaged* of the entire dispersion volume (indicated by the overbar notation in $\bar{c}_{\text{MS}}^{\lambda}$) with proper consideration of NP geometrical exclusion from the reaction layer (indicated by the superscript $\lambda$ in $\bar{c}_{\text{MS}}^{\lambda}$, see §3).<sup>13,29</sup> Alternatively, $\bar{c}_{\text{MS}}^{\lambda}$ represents the averaged (maximum) concentration of MS species
within the macroscopic reaction layer volume. The meanings of the ‘overbar’ and ‘superscript λ’ notations as adopted in $\bar{c}_{MS}^{\lambda}$ also apply for the concentration term $\bar{c}_{s}^{\lambda}$ introduced below. Following the original idea by Koutecký and Koryta for molecular complexing ligands,47-48 the length scale $\bar{\lambda}$ derives from the diffusion coefficient $D_M$ of free metal M in the electrolyte medium, its mean free lifetime $1/(k_{a,p}\bar{c}_s^{\lambda})$ with $\bar{c}_s^{\lambda}$ the concentration of metal binding sites S smeared-out over the entire dispersion volume and corrected for the extent to which NP reactive shell/surface is excluded from the reaction layer (Figure 1), and from the diffusion coefficient $D_p$ and mean lifetime $1/k_d$ of the M-NP complex, with $D_p$ the particle diffusion coefficient. Taking all this together, we can write2,29

$$ \bar{\lambda} = 1 / \sqrt{k_{a,p}\bar{c}_s^{\lambda} / D_M + k_d / D_p} \quad (2) $$

, where $k_{a,p}$, $\bar{c}_s^{\lambda}$, $k_d$ and $D_p$ depend on particle size as detailed in previous reports.17,29

The total steady state maximum diffusive flux $J_{diff}$ of all metal forms from bulk medium to the sensor is given by2,13,29

$$ J_{\text{diff}} = \frac{D_M c_M^* + D_p (\bar{c}_M + \bar{c}_{MS})}{\bar{\delta}_o} \quad (3) $$

, where $\bar{c}_M$ and $\bar{c}_{MS}$ denote the concentrations of free and complexed metal forms associated to the NP and averaged over the entire dispersion volume (indicated by the overbar notation in $\bar{c}_M$ and $\bar{c}_{MS}$), respectively and $\bar{\delta}_o$ refers to the thickness of the macroscopic diffusion layer at the sensor/medium interface (indicated by the overbar notation similarly to $\bar{\lambda}$).2 A system is denoted as labile when $J_{kin} \gg J_{diff}$ ($\mathcal{L} \gg 1$), i.e. the metal flux at the sensor surface is determined by coupled diffusion of M and M-NP, whereas the inequality $J_{kin} \ll J_{diff}$ ($\mathcal{L} \ll 1$) applies to situations where it is the rate of dissociation of M-NP complexes that kinetically controls the contribution of M-NP to the flux of M. Equations (1)-(3) are valid for perfect metal sink condition at the sensor surface. However, previous report49 evidenced that boundary conditions do not influence the lability and relative contributions of metal species. It could be anticipated that this feature, derived for the case of molecular ligands L, also holds for the situation of NP complexants. Firm conclusion on this issue will be provided in a forthcoming study where the lability parameter evaluated from eqs. (1)-(3) will be compared to that derived from the numerical solving of the differential equations detailing the reactive transfers of M and NP at the sensor/solution interface.
Analogy with the situation of molecular ligands homogeneously distributed over the entire solution volume. Prior to further developments, it is useful to recall the basis of eqs (1)-(3) by drawing an analogy with the situation of molecular ligands (denoted as L hereafter) homogeneously distributed over the entire solution volume (in contrast to the binding sites S carried by the NP complexants, **Figure 1**). For such a situation, van Leeuwen unequivocally showed⁵⁰ that in the limit of labile complexes the metal flux expression at the sensor surface - as derived from rigorous solution of the relevant diffusion equation corrected for chemical kinetic terms - can be expressed in the form $J_{\text{diff}} = \bar{D}^{1/2} (c_M^* + c_{\text{ML}}^*) / (\pi t)^{1/2}$ with $c_M^*$ and $c_{\text{ML}}^*$ the bulk concentration of free M and ML complexes and $\bar{D}$ an operational diffusion coefficient defined by the sum of the diffusion coefficient of M ($D_M$) and ML ($D_{\text{ML}}$) weighted by the M and ML concentrations across the diffusion layer. The concentrations $c_M^*$ and $c_{\text{ML}}^*$ are interrelated via the stability constant $Kc_L^* = c_{\text{ML}}^* / c_M^*$ with $c_L^*$ the bulk concentration of ligands L in excess over M. Under steady-state transport conditions, $J_{\text{diff}}$ can be rearranged in the form $J_{\text{diff}} = \bar{D} (c_M^* + c_{\text{ML}}^*) / \bar{\delta}_o$ where $\bar{\delta}_o = (\pi \bar{D} t)^{1/2}$ is the thickness of the steady-state diffusion layer at the macroscopic sensor surface. For NP complexants hosting metal binding sites S within their body (soft NP case) or at their surface (hard NP), the term $c_M^* + c_{\text{ML}}^*$ in the expression $J_{\text{diff}} = \bar{D} (c_M^* + c_{\text{ML}}^*) / \bar{\delta}_o$ above is replaced by $c_M^* + \bar{c}_M + \bar{c}_{\text{MS}}$ where $\bar{c}_M$ and $\bar{c}_{\text{MS}}$ are defined below eq. (3). As both M and MS are confined within/at the reactive NP volume/surface, they diffuse toward the macroscopic reactive surface with the diffusion coefficient $D_p$ of the NP metal carrier. Following the analogy with the molecular ligand situation, the effective diffusion coefficient $\bar{D}$ introduced above then becomes for the case of NP complexants $\bar{D} = (D_M c_M^* + D_p (\bar{c}_M + \bar{c}_{\text{MS}})) / (c_M^* + \bar{c}_M + \bar{c}_{\text{MS}})$. Altogether, these elements lead to eq. (3).

Van Leeuwen further showed⁵⁰ that for molecular ligands L taken in the extreme $Kc_L^* \gg D_M / D_{\text{ML}}$, the metal flux at the macroscopic surface in the non-labile limit is given by $J_{\text{kin}} = k_d c_{\text{ML}}^* (D_M / k'_a)^{1/2}$ with $k'_a = k_a c_L^*$. The latter expression is generally written $J_{\text{kin}} = k_d c_{\text{ML}}^* \mu$ where $\mu = (D_M / k'_a)^{1/2}$ is the thickness of the conventional reaction layer involving the kinetic constant $k'_a$ for M association with L, and $k_d$ identifies with $k_d^{\text{is}}$ according to Eigen association scheme between M and L.¹⁸ In the case where the transport of the released M from inside to outside of the NP is rate-limiting the overall M-NP
dissociation process, the system is labile, as argued in previous work.³⁰ This expression of $J_{\text{kin}}$ can be used for the situation of NP complexants *provided that* $c_{\text{ML}}^*$ is replaced by $\bar{c}_{\text{MS}}^{\lambda}$, the concentration of inner-sphere complexes MS averaged over the entire dispersion volume with account of the only fraction of the reactive shell volume located within the reaction layer of thickness $\bar{\lambda}$ (Figure 1, see §3). This thickness is the pendant of $\mu$ defined above for molecular L (see §3).²,¹³,²⁹ It is important to realize that the condition $Kc_L^* \gg D_M / D_{\text{ML}}$ argued to derive $J_{\text{kin}}$ (and subsequent $\mu$) is generally not satisfied for NP complexants because NP diffusion coefficient $D_p$ (the analogue of $D_{\text{ML}}$) is lower than $D_M$ to an extent that depends on the NP radius $r_p$. This implies that the *dissociative* term $D_M / k_a'$ alone is not sufficient for defining the reaction layer thickness $\bar{\lambda}$ relevant for NP complexants at a reactive sensor/solution interface. In turn, an MS formation term should be included in the definition of $\bar{\lambda}$ to account for the association between free M and S. This additional (*associative*) term corresponds to $k_d / D_p$ in eq. (2). In that equation, $k_d$ (the reciprocal life time of MS) is the composite kinetic constant $k_d = (1/k_d^{\text{is}} + 1/k_{d,p})^{-1}$ that should be considered in order to cover the entire range of M-NP dissociation dynamics, i.e. from slowly (e.g. Ni) to rapidly (e.g. Pb) dehydrating metal ions.¹²,¹⁷ Finally, the *dissociative* term $D_M / k_a'$ involved in the defining expression of $\mu$ should be replaced - for the NP case - by $D_M / (k_{a,p}\bar{c}_s^\lambda)$. This term involves the concentration $\bar{c}_s^\lambda$ of binding sites effectively present within the reaction layer (see §3) and smeared out over the entire dispersion volume, and the diffusion-component $k_{a,p}$ for M-NP association. It is stressed here that it is $k_{a,p}$ and not the composite constant $k_a$ that must be considered because the relevant timescale is the one required to get M inside the volume of the NP complexant (Figure 1). All these elements motivate the form of eq. (2) that defines $\bar{\lambda}$.

It is further emphasized that the original paper⁵¹ that elaborated an extended definition of the reaction layer thickness used a notation based on the involved rate constants, which has carried into subsequent work.²,¹⁷,²⁹ Herein we adopt a process-based nomenclature and define the association reaction layer thickness as $(k_d / D_p)^{-1/2}$ and dissociation layer thickness as $(D_M / (k_{a,p}\bar{c}_s^\lambda))^{1/2}$. The latter enters the original Heyrovsky formulation²⁶ of the kinetic flux due to dissociation of MS. The former involves $k_d$ that defines the lifetime of the complex MS. The corresponding flux is an *associative* one because it counts the total production of MS in the layer of thickness $(k_d / D_p)^{-1/2}$.
Finally, the basis of eqs. (1)-(3) is supported by the excellent agreement between the lability parameter measured by electrochemistry for Cd(II)-latex particles (Table 3 in Ref. [29], and Figure 10 in Ref. [2]) and the treatment of the lability parameter outlined above for NP complexants at a macroscopic metal sensor surface. It is stressed that this treatment did not require any adjustable parameters given the known physicochemical properties of the latex NPs considered, both in terms of size and density of sites S derived from independent electrokinetic analysis.²,²⁹ In particular, the comparison between experiments and theory highlighted the dramatic impact of the geometric exclusion of NP from the reaction layer (thickness $\bar{\lambda}$) on the lability parameter $\mathcal{L}$. Adopting the standard reaction layer formulation $\mu = (D_M / k'_a)^{1/2}$ with ignoring NP exclusion process, $\mathcal{L}$ was found to exceed by more than 3 orders of magnitude the experimental value. In addition, evaluation of $\mathcal{L}$ for Cd-latex NP from eqs. (1)-(3) with arbitrarily ignoring the $D_M c_M^*$ term in eq. (3) results in a lability parameter that is ca. 4 times larger than the experimental value. These elements further comfort our defining expressions here for $J_{\text{kin}}$, $J_{\text{diff}}$ and $\bar{\lambda}$. Within the context of lability evaluation for molecular complexes ML, some authors⁵² define $J_{\text{diff}}$ as the maximum diffusive supply of the complexes without any contribution from free M (i.e. $D_M c_M^* = 0$ in eq. (3)). As previously argued, such definition would lead to a poorer comparison to experimental data on Cd-latex NP system, and to ignoring the necessarily coupled diffusions of free M and complexed metal species to the sensor surface as evidenced by van Leeuwen.⁵⁰ Last, other operational definitions of the reaction layer thickness are proposed in the literature for the case of molecular ligands.²⁵ At this stage, it is difficult to appreciate whether or not the proposed definitions can be easily transposed to the intricate situation of NP complexants with proper integration of their defining physicochemical features and of e.g. the paramount contribution of NP exclusion from the reaction layer.²,²⁹ The solution proposed in this report and elsewhere²,¹³,²⁹ achieves such a transition between molecular to NP complexant situations.

In the sections below (§3-4), starting from the eqs. (1)-(3) an expression for the lability parameter $\mathcal{L} = J_{\text{kin}} / J_{\text{diff}}$ of M-NP complexes at a reactive (bio)interface is explicitly derived as a function of (i) the relevant physicochemical features of the nanoparticulate ligands, which includes their size, their electrostatic characteristics and the density of metal binding sites S they carry, (ii) the metal diffusion and dehydration kinetic properties, and (iii) the stability of the intraparticulate or surface metal complexes MS. For that purpose, differentiation is made between the case of 3D soft/core-shell complexing nanoparticles (§3) and that of 2D hard nanoparticulate ligands (§4). It is emphasized that the developments below hold for cases in line with the applicability of the generalized Eigen
scheme<sup>1,2,12,14,16,17</sup> (Figure 1) in the high NP charge density regime ($\kappa l_c \ll 1$) and for metal binding sites S homogeneously distributed within the reactive NP shell (3D case) or at the NP surface (2D). In addition, dilute NP suspensions are exclusively considered ($\varphi \ll 1$).

<figure><img src="image_3.png" /><figcaption>Figure 1. Schematic representation of a soft nanoparticulate metal complex M-NP (radius $r_p$) in the vicinity of a metal (bio)interface that is reactive toward aqueous metal ions $M_{aq}^{z_M^+}$ (M for short). The chemodynamic processes governing the formation/dissociation of the M-NP complexes are schematically depicted with the corresponding rate constants (see text for details). The volume component of the nanoparticle body lying within the operational reaction layer of thickness $\bar{\lambda}$ is indicated in grey shading. The figure clarifies the typical concentration profiles for free metal ions M and for NP-supported MS complexes from bulk solution to the responsive (bio)interface. Meaning of symbols and nomenclature are detailed in the text.</figcaption></figure>

### 3. Theory: Lability of nanoparticulate metal complexes M-NP at a (bio)interface which is reactive towards the free metal ion M.

#### 3.1. The case of 3D (soft/core-shell) nanoparticulate ligands.

The averaged concentrations $\bar{c}_{MS}^\lambda$ and $\bar{c}_S^\lambda$ involved in the definition of the kinetic flux $J_{kin}$ (eqs. (1)-(2)) depend on the NP volume fraction $\varphi$ in the entire dispersion according to

$$ \bar{c}_i^\lambda = c_i \varphi (1-\gamma) (1-z^3) \quad (4) $$

, where the index i refers to MS or S. $c_{MS}$ and $c_S$ represent the concentrations of MS and S in the NP shell volume and the dimensionless term $1-\gamma$ specifies the averaged fraction of nanoparticle shell
volume within the reaction layer zone (Figure 1).²⁹ It is noted that the concentrations $\bar{c}_S^\lambda$ is related to the concentration of S averaged over the entire dispersion, denoted as $\bar{c}_S$, via $\bar{c}_S^\lambda = \bar{c}_S(1-\gamma)$ so that eq. (2) can equally be written in the form $\bar{\lambda} = 1/\sqrt{k_{a,p}\bar{c}_S(1-\gamma)/D_M + k_d/D_p}$ given in eq. (2) of Ref. [29].

As demonstrated in Ref. [29], in the absence of interactions between the NP and the sensor surface, $\gamma$ solely depends on the ratios $\bar{\lambda}/r_p \equiv \Lambda$ and $z=a/r_p$. Qualitatively, $\gamma$ increases from 0 to 1 with increasing $\Lambda^{-1}$ with the limits $\gamma=0$ and $\gamma=1$ corresponding to situations where the geometrical exclusion of the reactive volume of the NPs from the reaction layer is insignificant ($\Lambda^{-1} \ll 1$) and complete ($\Lambda^{-1} \gg 1$), respectively. The magnitude of $1-\gamma$ is obtained from numerical evaluation of the NP shell volume lying in the reaction layer and averaged over all positions of the particle centre from the sensor, i.e.²⁹

$$ \gamma(\Lambda, z) = 1 - \frac{\Lambda^3}{4(1-z^3)} \int_0^1 F(\Lambda, z, x) dx \quad (5) $$

, where the cumbersome function $F$ was derived in Ref. [29] and is recalled in Supporting Information (eq. (S1) therein, §I), $x = H/\bar{\lambda}$ and $H$ is the separation distance between the NP and the (bio)interface as specified in Figure 1. In turn we obtain $J_{\text{kin}} = k_d^{\text{is}} c_{\text{MS}} \varphi(1-z^3)[1-\gamma(\Lambda,z)]\bar{\lambda}$ with $\bar{\lambda} = \Lambda r_p$. Further realizing that the concentrations $\bar{c}_M$ and $\bar{c}_{\text{MS}}$ involved in the expression of the diffusion flux $J_{\text{diff}}$ (eq. (3)) are defined by $\bar{c}_M = c_M \varphi(1-z^3)$ and $\bar{c}_{\text{MS}} = c_{\text{MS}} \varphi(1-z^3)$, respectively, the lability parameter $\mathcal{L} = J_{\text{kin}}/J_{\text{diff}}$ may be rearranged according to

$$ \mathcal{L} = \frac{k_d^{\text{is}} c_{\text{MS}} \varphi(1-z^3)[1-\gamma(\Lambda,z)]}{D_M c_M^* \left( 1 + \frac{\ell}{r_p} \varphi(1-z^3) \sum_{j=\text{MS,M}} \frac{c_j}{c_M^*} \right) r_p \bar{\delta}_o \Lambda} \quad (6) $$

, where the index j refers here to MS and M species, $\ell = k_B T / (6\pi\eta D_M)$ is the length scale defined by $\ell/r_p = D_p/D_M$ with $D_p$ provided by the Stokes-Einstein equation, and $\eta$ is the dynamic viscosity of the medium. We define the intrinsic stability constant of the inner-sphere complexes MS in terms of the intraparticulate conditions, i.e. $K_{\text{int}} = k_a^{\text{is}}/k_d^{\text{is}} = c_{\text{MS}}/(c_M c_s)$ (in mol$^{-1}$ m$^3$). Recalling that in the high charge density regime $\kappa\ell_c \ll 1$ and for a relaxed electric field (large excess of background electrolyte),
$c_M$ is related to the bulk concentration $c_M^*$ via $c_M = c_M^* \bar{f}_B$ with $\bar{f}_B$ the Boltzmann equilibrium partitioning factor averaged over the particle shell volume,¹²,⁵³ we obtain after some rearrangements

$$ \mathcal{L} = \frac{k_a^{\text{is}} c_s \varphi \bar{f}_B (1-z^3) [1-\gamma(\Lambda, z)]}{D_M \left( 1 + \frac{\ell}{r_p} \varphi \bar{f}_B (1-z^3) (1+K_{\text{int}} c_s) \right)} r_p \bar{\delta}_o \Lambda. \quad (7) $$

In details, the quantity $\bar{f}_B$ corresponds to the integral average of the position-dependent Boltzmann factor termed $f_B$ according to $\bar{f}_B = \int_a^{r_p} 4\pi r^2 f_B(r) dr / V_{\text{shell}}$ with $V_{\text{shell}}$ the shell volume. Under Donnan condition, this expression reduces to $\bar{f}_B = f_B = \exp(-z_M F \psi^D / RT)$ with $\psi^D$ the applicable Donnan potential.

Equation 7 involves the rate constant for formation of inner-sphere complex MS (Figure 1), $k_a^{\text{is}}$, expressed in terms of the intraparticulate conditions by¹⁷,²⁹

$$ k_a^{\text{is}} = \frac{3D_M}{\ell_a^2 c_s} \quad (8) $$

, where we define $\ell_a = \sqrt{3D_M / (k_w V^{\text{os}} N_{\text{Av}} c_s)}$ (m) as the thickness of an intraparticulate dissociation reaction layer ('dissociation' qualification given here on a process-based nomenclature and not on the kinetic constant-based nomenclature adopted in earlier work¹⁷,²⁹) defined as the average distance travelled by a free M within the NP shell at zero intraparticulate potential (i.e. in the absence of electrostatic conduction contribution) before it (re)associates with a site S to form MS, $k_w (s^{-1})$ is the metal dehydration rate constant (tabulated values are available for a number of metal ions⁵⁴), $V^{\text{os}}$ is the outer sphere volume and $N_{\text{Av}}$ is the Avogadro number.

Combining eqs. (7)-(8), the final result for $\mathcal{L}$ reads as

$$ \mathcal{L} = 3\varphi \bar{f}_B (1-z^3) [1-\gamma(\Lambda, z)] \frac{r_p \bar{\delta}_o}{\ell_a^2} \Lambda \left( 1 + \frac{r_p^*}{r_p} \right)^{-1} \quad (9) $$

, where $\gamma(\Lambda, z)$ is defined by eq. (5) and we introduced the length scale $r_p^* = \varphi \bar{f}_B \ell (1-z^3) (1+K_{\text{int}} c_s)$ for dimensioning the diffusive flux according to $J_{\text{diff}} = (D_M c_M^* / \bar{\delta}_o) (1+r_p^* / r_p)$.
The remaining task is the formulation of the dimensionless macroscopic reaction layer thickness $\Lambda = \bar{\lambda} / r_p$ operative at the sensor/medium interface. Starting from eq. (2) and substituting therein the expressions of $k_{a,p}$ and $k_d$ derived elsewhere$^{2,17}$ for core-shell particles with a given $c_s$, it is shown that $\Lambda = \bar{\lambda} / r_p$ identifies with the physical root of the transcendental equation$^{29}$

$$ \Lambda^2 \Phi_{\text{el,a}} \left\{ 1 - \gamma(\Lambda, z) + \left\{ \ell r_p^{-1} \Phi_B \left[ 1 + \left( \frac{\mu_d}{r_p} \right)^2 \right] \right\}^{-1} \right\} - 1 = 0 \quad (10) $$

, with $\Phi_{\text{el,a}} = 3\varphi \bar{f}_{\text{el,a}}$, $\Phi_B = \varphi \bar{f}_B / (1 + K_{\text{int}}c_s)$ (both dimensionless). The length scale $\mu_d = \sqrt{\frac{3D_M(1+K_{\text{int}}c_s)K_{\text{int}}}{k_w V^{\text{os}} N_{\text{Av}}}} \bar{f}_{\text{el,d}}$ in eq. (10) corresponds to the thickness of the *intraparticulate* reaction layer (corrected for particle electrostatics on transport rates) with regard to the association process (process-based nomenclature). It is straightforward to realise that for zero electrostatic potential within the NP shell, we have $\mu_d / \ell_a = \sqrt{(1+K_{\text{int}}c_s)K_{\text{int}}c_s}$ or $\mu_d / \ell_a = K_{\text{int}}c_s$ for $K_{\text{int}}c_s \gg 1$. Previous work$^{17,29}$ elaborated diagnosis for chemodynamic behaviour for M-NP association and dissociation from criteria involving the ratios $\ell_a / r_p$ and $\mu_d / r_p$, respectively. Unlike the choice originally made in Ref. [17], the term $\sqrt{(1+K_{\text{int}}c_s)K_{\text{int}}c_s}$ could have been left outside the defining expression of $\mu_d$, then resulting in $\mu_d / \ell_a = 1$. The dimensionless scalars $\bar{f}_{\text{el,a}}$ involved in $\Phi_{\text{el,a}}$ and $\bar{f}_{\text{el,d}} = \bar{f}_{\text{el,a}} / \bar{f}_B$ appearing in $\mu_d$ correct the convergent (subscript 'a') or divergent (subscript 'd') diffusion of M to or from the NPs for particle electrostatic effects, respectively. Quantitatively, these scalars are obtained from solution of the non-linear Poisson-Boltzmann equation as detailed elsewhere,$^{17,53}$ and their deviation from unity indicates the extent to which diffusion of M is accelerated or impeded by the particle electric double layer field. For practical situations where NPs are negatively charged, in contrast to the behaviour of $\bar{f}_B$, the factor $\bar{f}_{\text{el,a}}$ does not exceed $c_a$. one order of magnitude,$^{12,17}$ so that the inequality $\Phi_{\text{el,a}} \ll 1$ is always satisfied in the limit of dilute NP dispersions ($\varphi \ll 1$).$^{29}$

The set of eqs. (5), (9) and (10) fully defines the lability parameter of a given nanoparticulate metal complex and elegantly pinpoints the nature of the various physicochemical terms that come into play both at the local nanoparticulate (NP) and the macroscopic (sensor) levels. The nonlinear coupling between the exclusion factor $\gamma$ (eq. (5)) and the dimensionless reaction layer thickness $\Lambda = \bar{\lambda} / r_p$ (eq. (10)) makes it impossible to formulate general expressions for $\gamma$, $\Lambda$ and $\mathcal{L}$ as a function of particle size. Such a
formulation is however possible for the specific case of soft nanoparticulate ligands, as shown in the next section. In the following, the symbol ‘ ∩ ’ means ‘scales with’.

3.2. Asymptotic scaling laws for the lability parameter of soft nanoparticulate metal complexes versus particle size.

For soft NPs devoid of impermeable core component ( z = 0 ), eq. (5) exactly reduces to 29

$$
\gamma(\Lambda, z=0) = \begin{cases} 1 + \frac{\Lambda^2}{4} \left( \frac{\Lambda}{4} - 1 \right), & \text{for } \Lambda < 2. \\ \Lambda^{-1}, & \text{for } \Lambda \ge 2 \end{cases}. \qquad (11)
$$

Substituting eq. (11) into eq. (10) and subsequently solving of the so-obtained polynomial equations in
Λ = λ̄ / rₚ in the practical limit of dilute NP suspensions ( i.e. φ << 1 or Φel,a << 1 ), 29 we obtain the
expressions for the reaction layer thickness λ̄ elaborated elsewhere 29 (Table S1 therein) and rearranged
here in the concise form

$$
\bar{\lambda} = \begin{cases} r_p \left(1 + \sqrt{1 + 4 / \Phi_{el,a}}\right) / 2, & \text{for } r_p \ll r_{p,\text{max}} \\ r_{p,\text{int}}^{3/2} / \sqrt{r_p}, & \text{for } r_{p,\text{max}} \ll r_p \ll \mu_d \\ \sqrt{\tilde{r}_p r_p}, & \text{for } r_p \gg \mu_d \end{cases} \tag{12}
$$

, where the length scales $r_{p,max} = (2\ell\Phi_B\mu_d^2)^{1/3}$, $r_{p,int} = (\ell\Phi_B\mu_d^2 / \Phi_{el,a})^{1/3}$ ($> r_{p,max}$) and $\tilde{r}_p = \ell\Phi_B / \Phi_{el,a}$ are introduced for proper dimensioning and reduction of eq. (12). As shown below, these length scales offer a pragmatic way to rationalise and parameterize the multifactorial lability parameter of nanoparticulate metal complexes at macroscopic reactive (bio)interfaces and, in particular, explain the non-monotonous dependence of $\mathcal{L}$ on NP size. It is stressed that no tractable analytical solution exists for $\Lambda = \bar{\lambda} / r_p$ (and thus for $\mathcal{L}$) over the whole range of particle size $r_p$.

Equation 12 highlights the existence of three distinct asymptotic regimes for the dependence of $\bar{\lambda}$ on particle size.²⁹ Briefly, the linear increase of $\bar{\lambda}$ with $r_p$ for $r_p \ll r_{p,\text{max}}$ corresponds to a reaction layer that is primarily governed by the term $k_{a,p}c_s^\lambda / D_M$ in eq. (2). Here, the exclusion process impacts on the extension of the reaction layer at the sensor/medium interface to an extent that depends on the deviation of $2^{-1}\sqrt{\Phi_{el,a}}(-1+\sqrt{1+4\Phi_{el,a}^{-1}})$ from unity.²⁹ In the regime $r_p \gg \mu_d$, $\bar{\lambda}$ increases with particle size according to the power law $r_p^{1/2}$ and the associative term $k_d / D_p$ in eq. (2) now significantly exceeds
$k_{a,p}c_s^{\bar{\lambda}}/D_M$, so that the exclusion process ($\gamma$-dependent term in eq. (2)) is immaterial for the magnitude of $\bar{\lambda}$ in this range of $r_p$. In the intermediate regime, i.e. $r_{p,max} \ll r_p \ll \mu_d$, $\bar{\lambda}$ decreases with $r_p$ according to $r_p^{-1/2}$ and both association and dissociation of M-NP complexes then contribute to $\bar{\lambda}$. The dependence of $\bar{\lambda}$ on particle size is therefore generally marked by the presence of a maximum at $r_p = r_{p,max}$ and a minimum at $r_p = \mu_d$.²⁹ Depending on e.g. the stability of MS complexes and the magnitude of the intraparticulate potential (as subsumed in the term $\Phi_B$), these extrema are more or less pronounced or may even disappear.²⁹ In this latter situation, $\bar{\lambda}$ monotonously increases with $r_p$ from $\bar{\lambda} \ll r_p$ to $\bar{\lambda} \gg r_p^{1/2}$ at sufficiently small and large NPs, respectively. The existence of extrema in $\bar{\lambda}$ versus $r_p$ is tied to the generic condition $\mu_d/(\Phi_B^\ell) > 54$ applicable for $r_{p,max}/\mu_d \ll 1$ (see Table S3 in Ref. [29]).

With the dependence of $\bar{\lambda}$ (or $\Lambda$) on $r_p$ clarified, the way in which the exclusion factor $\gamma$ varies with $r_p$ can be explicitly formulated after substitution of eq. (12) into eq. (11). These expressions may then be exploited in combination with eq. (9) to define the sought asymptotic scaling laws for the lability parameter $\mathcal{L}$ versus particle size. The results collected in **Table 1** systematically demonstrate the power-law dependence of $\mathcal{L}$ on $r_p$ depending on the magnitude of $r_p$ compared to the characteristic length scales $r_{p,max}$, $\tilde{r}_p$, $r_{p,int}$, $r_p^*$ and $\mu_d$. For the sake of completeness, the corresponding expressions of $\Lambda$ and $\gamma$ are further detailed in **Table 1** for the various asymptotic limits identified therein. It is emphasized that the electrostatic features of the NPs, the particle volume fraction, the stability of the MS complexes and the M-NP chemodynamic properties are all subsumed in the aforementioned length scales.

**Figure 2A** displays the typical dependence of the lability parameter $\mathcal{L}$ for M-NP complexes towards a macroscopic sensor on particle size. The practical conditions $\{K_{int}, c_s, D_M, \bar{f}_B, \bar{f}_{el,a}, k_w, \varphi, \bar{\delta}_o\}$ adopted are specified in the caption and the corresponding $r_{p,max}$, $\tilde{r}_p$, $r_{p,int}$, $r_p^*$ and $\mu_d$ are indicated in **Figure 2A** by vertical lines. The results are derived from rigorous solution of eqs. (5), (9) and (10) and further compared to predictions (i) from the applicable asymptotic power-law expressions reported in **Table 1**, and (ii) with ignoring the exclusion of NP body from the reaction layer, i.e. by arbitrarily setting $\gamma=0$ in eqs. (9) and (10). For the sake of formal comparison between exact and approximate asymptotic results over the whole range of chemodynamic M-NP dissociation features, $\mathcal{L}$ is given as a function of $r_p$.
normalized by the intraparticulate reaction layer $\mu_d$. The regimes $r_p / \mu_d \ll 1$, $r_p / \mu_d \gg 1$ and $r_p / \mu_d \square 1$ correspond indeed to an overall M-NP complex dissociation rate constant $k_d$ determined by inner-sphere complex MS dissociation ($k_d \rightarrow k_d^{is}$), by the conductive diffusion of M into the medium ($k_d \rightarrow k_{d,p}$) and by both ($k_d = (1/k_d^{is} + 1/k_{d,p})^{-1}$), respectively.¹⁷ This typology of M-NP dissociation chemodynamics stems from the expression $k_d^{is} / k_{d,p} = (r_p / \mu_d)^2$ that applies for 3D soft/core-shell NPs of given $c_s$:¹⁷ it compares the characteristic timescale for M diffusion from the intraparticulate NP reactive volume $(1/k_{d,p})$ to that for the chemical dissociation of MS complexes $(1/k_d^{is})$. It is stressed that the extreme $r_p / \mu_d \ll 1$ does not represent a physically realistic reaction shell layer (with thickness $\mu_d$) at the NP/medium interface as it would then extend well beyond the physical shell with thickness $d = r_p - a$. Under the condition $r_p / \mu_d \ll 1$, the whole particle shell is then contributing evenly to the dissociation of MS.

<table><thead><tr><th>Particle size range</th><th>Lability <i>L</i></th><th>&Lambda; = &lambda;&#773;/r<sub>p</sub> and &gamma;</th></tr></thead><tbody><tr><td>r<sub>p</sub> &lt;&lt; r<sub>p,max</sub><br/>r<sub>p</sub> &gt;&gt; r<sub>p</sub><sup>*</sup></td><td>3(&Phi;<sub>el,a</sub><sup>-1/2</sup> - 1/2)&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p</sub><sup>2</sup>} / (l<sub>a</sub><sup>2</sup>r<sub>p</sub><sup>*</sup>)<br/>3(&Phi;<sub>el,a</sub><sup>-1/2</sup> - 1/2)&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p</sub><sup>2</sup>} / (l<sub>a</sub><sup>2</sup>)</td><td>&Lambda; = 1/2(1 + &radic;(1 + 4&Phi;<sub>el,a</sub><sup>-1/2</sup>))<br/>&gamma; = &Lambda;<sup>-1</sup></td></tr><tr><td>r<sub>p,max</sub> &lt;&lt; r<sub>p</sub> &lt;&lt; &mu;<sub>d</sub><br/>r<sub>p</sub> &gt;&gt; r<sub>p,max</sub><br/>r<sub>p</sub> &lt;&lt; r<sub>p</sub><sup>*</sup><br/>r<sub>p</sub> &gt;&gt; r<sub>p</sub><sup>*</sup></td><td>3&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p,int</sub><sup>3/2</sup>r<sub>p</sub><sup>1/2</sup>} / (l<sub>a</sub><sup>2</sup>r<sub>p</sub><sup>*</sup>)<br/>3&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p,int</sub><sup>3/2</sup>r<sub>p</sub><sup>-1/2</sup>} / (l<sub>a</sub><sup>2</sup>)<br/>3&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p,int</sub><sup>9/2</sup>r<sub>p</sub><sup>-5/2</sup>} / (4l<sub>a</sub><sup>2</sup>r<sub>p</sub><sup>*</sup>)<br/>3&phi;&overline{f}<sub>B</sub> &overline{\delta<sub>o</sub>r<sub>p,int</sub><sup>9/2</sup>r<sub>p</sub><sup>-7/2</sup>} / (4l<sub>a</sub><sup>2</sup>)</td><td>&Lambda; ~ (r<sub>p,int</sub>/r<sub>p</sub>)<sup>3/2</sup><br/>&Lambda; &gt;&gt; 2<br/>&gamma; = &Lambda;<sup>-1</sup><br/>&gamma; = 1 + 1/4&Lambda;<sup>2</sup>(&Lambda;/4 - 1)</td></tr></tbody></table>
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<figure><img src="image_4.png" /><figcaption>**Table 1.** Asymptotic expressions of the lability parameter $\mathcal{L}$ in various regimes of soft particle radius $r_p$ ($z=0$) and corresponding expressions of the (dimensionless) macroscopic reaction layer thickness $\Lambda = \bar{\lambda} / r_p$ and of the factor $\gamma$ for geometrical exclusion of NP volume from the reaction layer. The indications **1**, **2**, **3**, **4** and **5** correspond to the particle size domains identified in **Figure 2A**. The length scales $r_{p,\text{max}}$, $r_{p,\text{int}}$, $\tilde{r}_p$, $r_p^*$ and $\mu_d$ are defined by $r_{p,\text{max}} = (2\ell\Phi_B\mu_d^2)^{1/3}$, $r_{p,\text{int}} = (\ell\Phi_B\mu_d^2 / \Phi_{\text{el},a})^{1/3}$, $\tilde{r}_p = \ell\Phi_B / \Phi_{\text{el},a}$, $r_p^* = \varphi\bar{f}_B\ell(1+K_{\text{int}}c_s)$ and $\mu_d = \sqrt{\frac{3D_M(1+K_{\text{int}}c_s)K_{\text{int}}}{k_wV^{\text{os}}N_{\text{Av}}}}$ with $\bar{f}_{\text{el},d} = \bar{f}_{\text{el},a} / \bar{f}_B$, respectively. See text for details.</figcaption></figure>

<figure><img src="image_5.png" /></figure>
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<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

**Figure 2.** Typical dependence of the lability parameter $\mathcal{L}$ on particle radius $r_p$ (normalized in panel A by the intraparticulate reaction layer thickness $\mu_d$). (A) $K_{\text{int}} = 10 \text{ mol}^{-1} \text{ m}^3$ (typical for Cu(II)-humic acid NPs for a degree of reactive site coverage of 0.03),$^{55}$ $c_s = 7 \times 10^2 \text{ mol m}^{-3}$ (typical for humic acid NPs),$^{56}$ $D_M = 7 \times 10^{-10} \text{ m}^2 \text{s}^{-1}$,$^{57}$ $\bar{f}_B = 10^{4.2}$ (reached for humic acids under salinity conditions between 1 and 10 mM),$^{12}$ $\bar{f}_{\text{el,a}} = 1$,$^{12,17}$ $z = 0$ (soft NPs), $k_w = 10^9 \text{ s}^{-1}$ (relevant for Cu(II)),$^{54}$ $V^{\text{os}} = 0.38 \text{ nm}^3$,$^{14}$ $\varphi = 5 \times 10^{-4}$, $\bar{\delta}_o = 2$ $\mu$m. The positioning of the key length scales $\tilde{r}_p$, $r_p^*$, $r_{p,\text{max}}$, $r_{p,\text{int}}$ and $\mu_d$ is explicitly indicated together with the applicable asymptotic (power-law) regimes in $\mathcal{L}$ (Table 1) and with the associated domains 1 to 5 detailed in the text, §3. For the sake of illustration the dependence of $\mathcal{L}$ on $r_p / \mu_d$ is provided with ignoring NPs volume exclusion effects (i.e. $\gamma = 0$, black dashed line). Dependence of $\mathcal{L}$ on $r_p$ for various values of $\varphi$ (B), for values of $k_w (=3 \times 10^4 \text{ s}^{-1}$, $7 \times 10^7 \text{ s}^{-1}$, $3 \times 10^8 \text{ s}^{-1}$, $10^9 \text{ s}^{-1}$ and $7 \times 10^9 \text{ s}^{-1}$ for Ni(II), Zn(II), Cd(II), Cu(II) and Pb(II), respectively)$^{54}$ (C), for values of $z = a / r_p$ (D), $K_{\text{int}}$ (in mol$^{-1}$ m$^3$) (E) and $\bar{f}_B$ (F), other model parameters as in (A). In all cases, the relevant characteristic asymptotic regimes in $\mathcal{L}$ (see Table 1) are given. In order to simplify the analysis of the impact of $k_w$ on $\mathcal{L}$ (panel C), it is stressed that $K_{\text{int}}$ is kept constant for all metal ions tested, which refers to model rather than practical situations.
**Figure 2A** highlights that the non-monotonous variation of $\mathcal{L}$ with particle size closely follows that predicted on the basis of the relevant asymptotic power-law expressions listed in **Table 1** given the applicable sequence $\tilde{r}_p < r_{p,\text{max}} < r_{p,\text{int}} < \mu_d < r_p^*$. Namely, 5 domains of particle sizes may be formally distinguished (see **Figure 2A**): domain 1 with $\mathcal{L} \Box r_p^2$ for $r_p \ll r_{p,\text{max}}$, domain 2 with $\mathcal{L} \Box r_p^{1/2}$ over the restricted range $r_{p,\text{max}} \ll r_p \ll r_{p,\text{int}}$, domain 3 with $\mathcal{L} \Box r_p^{-5/2}$ for $r_{p,\text{int}} \ll r_p \ll \mu_d$, domain 4 with $\mathcal{L} \Box r_p^{1/2}$ for $\mu_d \ll r_p \ll r_p^*$, and finally domain 5 with $\mathcal{L} \Box r_p^{-1/2}$ for $r_p^* \ll r_p$. Qualitatively, the intricate evolution of $\mathcal{L}$ from domain 1 to 4 is reminiscent of that previously discussed for the macroscopic reaction layer thickness $\bar{\lambda}$ because here $\mathcal{L}$ exhibits a local maximum and a minimum in the vicinity of $r_p = r_{p,\text{max}}$ and $r_p = \mu_d$, respectively. The dependence of $\mathcal{L}$ on $r_p$ thus basically reflects that of $\bar{\lambda}$ detailed by eq. (10). The shift between the two in the $r_p$-domains 1 to 4 stems from the contribution of the term $(1-\gamma)r_p$, which is inferred from eq. (9) taken in the limit $r_p \ll r_p^*$. It is stressed that the lability parameter measured and successfully predicted from eqs (1)-(3) for the system Cd(II)-latex NP (71 nm in radius)$^{2,29}$ corresponds to domain 3 identified above. Recognizing that domain 1 corresponds to the expected increase of $\mathcal{L}$ upon moving from the molecular to nanoparticulate ligand situations (due to decreasing $J_{\text{diff}}$ that follows decreasing $D_p$), the maximum between domains 1-2 and 3 is supported by experimental finding.

In detail, comparison to predictions obtained with $\gamma=0$ reveals that domain 1 is not affected by the exclusion of the NP body volume from the reaction layer. This feature is in agreement with the magnitude of $1-\gamma \Box 1$ at such low $r_p$ and with the poor dependence of $\Lambda$ on $\gamma$ in this particle size regime given $\Phi_{\text{el,a}} \ll 1$.${}^{29}$ Exclusion becomes significant only for $r_p > r_{p,\text{max}}$ via the term $1-\gamma$ in the lability expression (eq. (9)). This arises because exclusion has become immaterial in defining the reaction layer thickness (or $\Lambda$), i.e. with increasing the particle size the term $k_d/D_p$ becomes the only significant one in eq. (2). Domain 5 ($r_p/r_p^* \gg 1$) originates from the fact that the diffusion flux $J_{\text{diff}}$ then becomes independent of $r_p$ and reduces to the maximum achievable limiting flux $D_M c_M^* / \bar{\delta}_o$, which is, in fact, the limiting non-labile flux. The contribution of the metal-NP carriers to $J_{\text{diff}}$ thus becomes immaterial due to their sluggish diffusion at such large $r_p$ within the time scale of M accumulation at the sensor (i.e. NP
diffusive transport is effectively ‘frozen’ in this time window). Accordingly, the dependence of the
lability parameter on r<sub>p</sub> in domain 5 becomes solely dictated by that of the kinetic flux, i.e.
J<sub>kin</sub> □ (1-γ)λ&#773; □ r<sub>p</sub><sup>-1/2</sup>. It is noted that in the extreme D<sub>p</sub> → 0 reached at sufficiently large (infinite) r<sub>p</sub>
(upper limit of domain 5), L would go to infinity if the free metal contribution to J<sub>diff</sub> in eq. (3) is
ignored (the intuitive labile limit). However, in such an extreme case, the NP exclusion from the reaction
layer of thickness λ&#773; is quasi-complete<sup>29</sup> and the kinetic flux J<sub>kin</sub> should tend to zero (the reasoning
discards the obvious impact of particle sedimentation at D<sub>p</sub> → 0 or equivalently r<sub>p</sub> → ∞). In turn, the
lability parameter L becomes mathematically ill-defined in the limit D<sub>p</sub> → 0 when ignoring the term
D<sub>M</sub>c<sub>M</sub><sup>*</sup> in eq. (3). The current formulation with integration of the term D<sub>M</sub>c<sub>M</sub><sup>*</sup> does not reproduce such
ill-defined lability parameter at D<sub>p</sub> → 0. This feature, together with the elements provided in §2, further
support the correctness of eqs. (1)-(3).

3.3. Dependence of the lability of soft/core-shell nanoparticulate metal complexes on the physicochemical and chemodynamic factors pertaining to M-NP complexes.

In this section, the particle size-dependent lability parameter of soft/core-shell M-NP complexes is
analysed over a range of practical values for relevant physicochemical NP characteristics involved in eq.
(9) (e.g. f̄<sub>B</sub>) and of the factors therein that determine M-NP chemodynamics (e.g. k<sub>w</sub>). For the sake of
comparison between these various conditions, the corresponding L is now represented as a function of r<sub>p</sub>
and not r<sub>p</sub> / μ<sub>d</sub> (as in **Figure 2A**) because the reaction layer thickness μ<sub>d</sub> itself depends on these
conditions (see §3.1). The analysis is therefore restricted to NP size from 1 nm to 1 μm, which does not
necessarily cover the whole spectrum of r<sub>p</sub> / μ<sub>d</sub> values with all lability domains identified in **Figure 2A**.

The extent to which the lability parameter of M-NP complexes at the macroscopic sensor/medium
interface is affected by the NP volume fraction φ depends on the particle size (Figure 2B). In domain 1
identified in Figure 2A (L □ r<sub>p</sub><sup>2</sup>), increasing φ leads to a loss of lability in line with L
□ 3(Φ<sub>el,a</sub><sup>-1/2</sup>)φ/r<sub>p</sub><sup>*</sup> □ φ<sup>-1/2</sup> (see Table 1 with r<sub>p</sub><sup>*</sup> □ φ and Φ<sub>el,a</sub> □ φ). Qualitatively, this feature applies to
the lability of molecular complexes ML upon increasing φ or, equivalently, upon increasing the bulk
concentration of ligands L, c<sub>L</sub><sup>*</sup>. The decreasing lability parameter with increasing φ basically originates
from the associated decrease in the dissociation reaction layer thickness ($D_M / (k_{a,p} \bar{c}_s^\lambda)^{1/2}$, recalling that $\bar{c}_s^\lambda \square \varphi$ in domain 1 where $\gamma \square 0$ (eq. (4) and **Figure 2A**). In domains 2 and 3, $\mathcal{L}$ is independent of $\varphi$ (**Table 1**) and the transition between domains 1 and 3 takes the form of a sharp maximum with increasing $\varphi$. The onset of domain 4 is reached for $r_p > \mu_d$ ($\approx 200$ nm under the conditions of **Figure 2B**) and for sufficiently large $\varphi$ ($\ge 10^{-4}$). In this domain, $\mathcal{L}$ is nearly independent of $\varphi$, which is in agreement with the asymptotic result in **Table 1**. For decreasing $\varphi$ below $10^{-4}$ and $r_p > 200$ nm, the onset of domain 5 appears as domain 4 becomes immaterial due to the decreasing applicability of the inequality $r_p \ll r_p^*$ (recalling that $r_p^* \square \varphi$) in the particle size range $r_p > \mu_d$ (see **Table 1**). Altogether, at sufficiently large particle size, a gain in lability is achieved with increasing $\varphi$. The opposite impacts of $\varphi$ on $\mathcal{L}$ at sufficiently low (domain 1) and large particle sizes (domain 5) are explained by the differentiated dependences of $J_{\text{diff}}$ and $J_{\text{kin}}$ on $\varphi$ in these respective size windows: $J_{\text{kin}} \square \varphi^{1/2} (\square (\bar{c}_s^\lambda)^{-1/2} \times \bar{c}_{\text{MS}}^\lambda)$ and $J_{\text{diff}} \square \varphi (\square \bar{c}_{\text{M,MS}})$ in domain 1, whereas $J_{\text{kin}} \square \varphi (\square \bar{c}_{\text{MS}}^\lambda)$ in domain 5 (where the associative reaction layer term $k_d / D_p$ dominates) with $J_{\text{diff}}$ independent of $\varphi$ in this latter domain (the limiting non-labile flux).

**Figure 2C** highlights the modification of the dependence of the lability parameter of metal complexes on $r_p$ for the metal sequence Ni(II), Zn(II), Cd(II), Cu(II), Pb(II) with increasing values of the dehydration rate constant $k_w$ from $3 \times 10^4$ to $7 \times 10^9$ s$^{-1}$.$^{54}$ Obviously domain 1, where we have $\mathcal{L} \square r_p^2$, is shifted to lower particle sizes as $k_w$ increases from the Ni(II) to the Pb(II) case. In line with **Table 1**, this shift is due to the decrease of $r_{p,max} = (2\Phi_B \mu_d^2 \ell)^{1/3}$ with $k_w$ having in mind that $\mu_d^2 \square 1/k_w$ (§C1). At sufficiently low $r_p$ where domain 1 applies over the complete metal sequence examined, the lability of M-NP complexes increases from Ni(II) to Pb(II), which is consistent with $\mathcal{L} \square 1/\ell_a^2$ (**Table 1**) and $1/\ell_a^2 \square k_w$ (§3.1), and in qualitative agreement with data from literature.$^{58}$ From a physicochemical point of view, the larger is $k_w$, the faster is the rate constant $k_d^{\text{is}} = (k_a^{\text{is}} / K_{\text{int}} \square k_w)$ for chemical dissociation of MS complexes, and thus the larger is the dissociation kinetic flux $J_{\text{kin}}$ (eq. (1)) compared to $J_{\text{diff}}$ and
hence the greater is the lability parameter of the complexes. Putting aside the case of Ni(II) ions, the
aforementioned increase in lability with increasing k<sub>w</sub> is qualitatively observed in domain 4 (L □ r<sub>p</sub><sup>1/2</sup>)
which is reached at sufficiently large r<sub>p</sub> values as compared to μ<sub>d</sub>. This point marks the position of the
local minimum in L. In domain 3 (L □ r<sub>p</sub><sup>-5/2</sup>) that makes the transition between domains 1-2 and 4, the
lability decreases with increasing k<sub>w</sub>. This finding may be counter-intuitive given that an increase in k<sub>w</sub>
results in increasing values of k<sub>d</sub><sup>is</sup>,<sup>13</sup> from which it could be anticipated that L increases due to increasing
J<sub>kin</sub>. Instead, the observed decrease of L with k<sub>w</sub> is in agreement with the scaling L □ k<sub>w</sub><sup>-1/2</sup> obtained
after substituting the k<sub>w</sub>-dependent expressions of r<sub>p,int</sub> and l<sub>a</sub> in the asymptotic result displayed in
**Table 1** for domain 3. The physicochemical basis for this 'reversed' dependence of L on k<sub>w</sub> in domains
1,4 and in domain 3 lies in the distinct variations of the exclusion factor γ with changing k<sub>w</sub> in these
respective NP size regimes. In domains 1 and 4, NP exclusion is independent of k<sub>w</sub> because the limits of
insignificant and complete NP exclusion from the macroscopic reaction layer are nearly reached,
respectively. In contrast, domain 3 is that where γ increases with k<sub>w</sub>. Indeed, the larger is k<sub>w</sub> (or the
smaller is μ<sub>d</sub>) for this range of r<sub>p</sub>, the thinner is the reaction layer thickness λ&#773;<sup>29</sup> (see Figure 6 in Ref.
[29]) and thus the more significant becomes the NP exclusion from that layer. In turn, this leads to a
noticeable decrease of the kinetic flux J<sub>kin</sub> and consequently of the lability of the metal complexes. This
decrease overwhelms the k<sub>w</sub>-mediated increase of J<sub>kin</sub> and L which governs the trends found in
domains 1 and 4. For very slowly dehydrating Ni(II) ions, the corresponding k<sub>w</sub> value leads to a domain 3
that does not overlap the ones pertaining to the other metal ions, and to a domain 4 that is unachievable
over the range of particle sizes analyzed.

The impact of the core to particle radius ratio z = a / r<sub>p</sub> on M-NP lability is illustrated in **Figure 2D**.
Qualitatively, the lability parameter L [1-γ (eq. (9)) increases with increasing z in the intermediate
domain 3 identified for soft NPs in **Figure 2A** mainly because the extent of NP exclusion from the
macroscopic reaction layer decreases from soft (z = 0) to hard NP (z = 1) types at fixed r<sub>p</sub>.<sup>29</sup> Since
domain 1 is independent of γ (**Figure 2A**), lability is not impacted by z provided that r<sub>p</sub> is sufficiently
small compared to r<sub>p,max</sub>. In domain 3, L further evolves from [r<sub>p</sub><sup>-5/2</sup> for soft NPs (**Table 1**) to [r<sub>p</sub><sup>-1</sup>
for z → 1. The latter scaling is in agreement with the results given in §4 for hard NPs in the relevant size
regime $r_{p,max} \ll r_p \ll \mu_d$, $r_p \gg r_{p,int}$ and $r_p \ll r_p^*$ (see **Table 2**, §4). At sufficiently large $r_p$ well beyond $1$ µm, $\mathcal{L}$ becomes independent of $z$ (not shown) essentially because the asymptotic limit $\gamma \to 1$ is reached regardless of the NP type considered. Under the conditions of **Figure 2C**, the above influence of $z$-dependent NP exclusion on lability dominates the impact of the term $1-z^3$ that appears in eq. (9) following the required averaging of intraparticulate M and MS concentrations over the whole dispersion volume.

The way in which the stability constant $K_{\text{int}}$ of MS affects $\mathcal{L}$ is displayed in **Figure 2E**. The reported trends are basically similar to those detailed in **Figure 2C** where the effects observed upon increasing $k_w$ from Zn(II) to Pb(II) are qualitatively recovered in **Figure 2E** for decreasing values of $K_{\text{int}}$. The reason for such an equivalence is explained by realizing that $k_d^{\text{is}} = k_a^{\text{is}} / K_{\text{int}}$ ($k_w$ in eq. (9)). The transitions between domains 1-2, 3 and 4 (if applicable) are determined by the shifts of $r_{p,max}$, $r_{p,int}$, $\mu_d$ and $r_p^*$ that all depend on $K_{\text{int}}$ (§3.1 and §3.2). **Figure 2F** shows how increasing values of the Boltzmann partitioning factor $\bar{f}_B$ (with $\bar{f}_B > 1$, i.e. negatively charged NPs are considered) significantly increase M-NP complex lability, especially at large $r_p$ values (i.e. in domains 3, and 4 or 5 depending on the shift of $r_{p,max}$, $r_{p,int}$, $\mu_d$ and $r_p^*$ with $\bar{f}_B$) where $J_{\text{diff}} = (D_M c_M^* / \bar{\sigma}_o)(1+r_p^*/r_p)$ poorly depends on $r_p^* \square \bar{f}_B$. After analysing the contributions of all $\bar{f}_B$-dependent terms in eq. (9), it is found that this lability feature at large $r_p$ originates from the increase of $J_{\text{kin}}$ with $\bar{f}_B$ due to the associated increase in (i) the intraparticulate MS concentration that follows a significant electrostatically-mediated accumulation of M in the NP body, and (ii) the mean path $(D_p / k_d)^{1/2} \approx \bar{\lambda}$ of M-NP complexes due to retardation of the divergent diffusion of M away from the NP body with increasing $\bar{f}_B$ (we recall here that $k_d \to k_{d,p} \square 1 / \bar{f}_B$ at sufficiently large $r_p$, see Ref. [17]). Finally, the change in $\mathcal{L}$ with increasing concentration of metal binding sites density $c_s$ at fixed $\bar{f}_B$ (not shown) is found to qualitatively follow the trend discussed in **Figure 2F** with decreasing $\bar{f}_B$ at fixed $c_s$. Indeed, both situations lead to decreasing reaction layer thickness $(D_p / k_d)^{1/2} \approx \bar{\lambda}$ at large $r_p$ where $k_d \to k_{d,p} \square 1 + K_{\text{int}}c_s$,¹⁷ i.e. a decrease in $J_{\text{kin}}$ and a loss of M-NP lability.

**4. Theory: The case of 2D (hard) nanoparticulate ligands.**
Following the methodology detailed in §3.1 for 3D soft/core-shell NPs, and considering the proper chemodynamic properties of metal complexes formed with 2D hard NPs type ($z=1$) as derived elsewhere,¹⁷ we obtain after some algebra (see Supporting Information, §II, for details) the reduced expression of the lability parameter $\mathcal{L}$ for M-(2D hard) NP complexes

$$ \mathcal{L} = 3\varphi f_B[1 - \gamma(\Lambda)] \frac{\bar{\delta}_o}{\ell_a} \Lambda \left[ 1 + \left( \frac{r_p^*}{r_p} \right)^2 \right]^{-1} \quad (13) $$

, where $r_p^*$ is now defined by $r_p^* = \left( 3\varphi f_B \ell K_H \frac{1+K_{\text{int}}^{(z=1)}\Gamma_s}{K_{\text{int}}^{(z=1)}\Gamma_s} \right)^{1/2}$ with $K_{\text{int}}^{(z=1)}$ (mol⁻¹ m²) the intrinsic stability constant of MS surface complexes, $K_H$ (m) the Henry adsorption coefficient for M at the NP surface, and $\ell_a = D_M / (k_w V^{\text{os}} N_{\text{Av}} \Gamma_s)$.¹⁷ The exclusion factor $\gamma$ remains defined by eq. (5) in the limit $z \to 1$²⁹ and the dimensionless reaction layer $\Lambda = \bar{\lambda} / r_p$ by eq. (10) after systematic replacement of the quantities $\bar{f}_{\text{el},a}$, $\bar{f}_B$, $K_{\text{int}}$ and $K_{\text{int}}c_s$ by their 2D equivalents $f_{\text{el},a}$, $f_B$, $K_H / \Gamma_s$ and $K_{\text{int}}^{(z=1)}\Gamma_s$, respectively.²⁹ We specify that the reaction layer thickness $\ell_a$ relevant at the 2D NP level should be here conceived as the thickness of the reaction layer inside the surface layer containing all binding sites and where M may move freely in between the sites S in the 2D lattice layer.¹⁷ In that respect, it is stressed that the ‘surface layer’ in case of hard NPs has generally a given thickness. The volume of that layer is defined by two concentric surfaces defining the mean position of the outer binding sites (facing the outer electrolyte solution) and the inner binding sites close to the hard NP surface. The existence of such a surface layer with finite thickness is tied to the non-ideality of hard NPs due to their roughness (that varies with e.g. NP aging). Best experimental evidence of this feature is the fact that the electrophoretic properties of supposedly hard NPs are actually in line with that expected for soft NPs with nanometric shell layer permeable to ions and water.⁵⁹,⁶⁰

It is further found that the general expression eq. (5) defining $\gamma$ simplifies for 2D hard NPs according to²⁹

$$ \gamma(\Lambda, z=1) = \begin{cases} 1 - \frac{\Lambda}{4}, & \text{for } \Lambda < 2 \\ \Lambda^{-1}, & \text{for } \Lambda \ge 2 \end{cases}. \quad (14) $$
For the practical limit Φ<sub>el,a</sub> << 1 (i.e. φ << 1) where the exclusion factor γ may impact on the reaction layer thickness only at sufficiently small particle size r<sub>p</sub><sup>29</sup> λ&#773; further remains approximated for 2D hard NP types by eq. (12) pending use of the equivalence invoked above between factors involved in the chemodynamics of metal complexes formed with 3D soft/core-shell NPs and 2D hard NPs. As in §3.2, these analytical expressions for γ (eq. (14)) and λ&#773; (or equivalently Λ = λ&#773; / r<sub>p</sub>) are then substituted into eq. (13) to derive the sought scaling laws for the lability parameter L for surface metal complexes at a macroscopic medium/sensor interface. The results are collected in **Table 2** and may be used in conjunction with the rigorous eq. (13) to decipher the contributions of all processes governing L along the lines detailed in **Figure 2** for 3D soft/core-shell NPs. Given the strong analogy mentioned earlier between 3D and 2D NP cases, the results obtained for 2D hard NPs are provided in Supporting Information (**Figure S1**, §III). We report here **Figure 3** that supports the derivation of the exact (eq. (13)) and approximate (**Table 2**) lability expressions evaluated under the realistic conditions detailed in the figure caption. Comparison with predictions where geometrical exclusion of NP from the macroscopic reaction layer is discarded is further given and conclusions given in **Figure 2A** concerning the paramount important of that exclusion process still hold for 2D hard NPs. The conditions adopted in **Figure 3** are in line with shallow extrema of λ&#773; at r<sub>p</sub> = r<sub>p,max</sub> and r<sub>p</sub> = μ<sub>d</sub>, which explains the absence of extrema in the dependence of L on r<sub>p</sub> in domains 1 to 3 identified in **Figure 3**. Again, the domain reached at large particle size (r<sub>p</sub> >> r<sub>p</sub><sup>*</sup>) corresponds to that where J<sub>diff</sub> → (D<sub>M</sub>c<sub>M</sub><sup>*</sup> / δ&#773;<sub>o</sub>).
<figure><img src="image_10.png" /><figcaption>**Figure 3.** Typical dependence of the lability parameter $\mathcal{L}$ on hard particle radius $r_p$ normalized by the intraparticulate reaction layer thickness $\mu_d$.</figcaption></figure>

$K_{\text{int}}^{(z=1)} = 10^5 \text{ mol}^{-1} \text{ m}^2, \quad K_H = 2 \times 10^{-4} \text{ m},$
$\Gamma_s = 1.7 \times 10^{-7} \text{ mol m}^{-2} \text{ (i.e. 1 site S per 10 nm}^2\text{)},$
$D_M = 7 \times 10^{-10} \text{ m}^2 \text{ s}^{-1},$
$\bar{f}_B = 10^{3.5},$
$\bar{f}_{\text{el,a}} = 1,^{12,17}$
$k_w = 10^9 \text{ s}^{-1}$ (relevant for Cu(II)),$^{54}$ $V^{\text{os}} = 0.38 \text{ nm}^3,$$^{14}$
$\varphi = 5 \times 10^{-4},$
$\bar{\delta}_o = 2 \text{ µm}$. The positioning of the key length scales $\tilde{r}_p, r_p^*, r_{p,max}, r_{p,int}$ and $\mu_d$ is explicitly indicated together with the applicable asymptotic regimes in $\mathcal{L}$ (Table 2) associated with the domains 1 to 4 identified in Table 2. For the sake of illustration the dependence of $\mathcal{L}$ on $r_p / \mu_d$ is provided with ignoring NPs volume exclusion effects (i.e. $\gamma = 0$, black dashed line).

## 5. Conclusion.

In this work, reduced expressions are provided to describe the lability parameter of nanoparticulate metal complexes at a macroscopic metal-sensing (bio)interface. The analysis is supported by tractable scaling laws derived for the lability parameter $\mathcal{L}$ as a function of particulate ligand size in the extremes of volume and surface binding site distributions applicable for 3D soft NPs and 2D hard NPs, respectively. This rationale allows the identification of the respective contributions of the various factors that control the lability of M-NP complexes, namely: the (intertwined) exclusion of the NP reactive body/surface from the reaction layer, the chemodynamics of M-NP dissociation, the extension of the macroscopic reaction layer from the sensor surface, the defining NP physicochemical properties and the NP volume fraction in the dispersion. The results evidence that the lability parameter of M-NP complexes is generally not an increasing function of particle size, a trend that could be anticipated on the sole basis of the associated decrease in the diffusion flux of the particulate metal-carriers. This major remarkable result is a fundamental consequence of the geometric exclusion of the NP from the reaction layer. This phenomenon dramatically affects the dissociative flux of M released by the NP complexant to the sensor/medium interface. The exclusion process is a characteristic feature of NP complexants for which the metal binding sites are inherently confined within/at particle body/surface. Such phenomena were obviously not included in the conventional Koutecký-Koryta reaction layer approach that was developed for molecular systems.
In a forthcoming study, the here-detailed concepts leading to the derivation of the lability parameter for a M-NP system at a metal-responsive sensor surface will be compared with rigorous solution of the coupling between Nernst-Poisson-Planck equations formulated at the NP scale and corrected for chemical kinetic terms over the relevant intraparticulate shell spatial region, *and* at the scale of the solution adjacent to the macroscopic sensor. Such comparison could lead to *e.g.* refinement of the expression for the reaction layer thickness relevant for the M-NP system at the sensor/medium interface. The formulation proposed here offers a way to implement this difficult scaling from the relevant nanometric to macroscopic scales.

To date, attempts to describe the impact of NPs on biota are generally restricted to elaborations of *ad hoc* empirical or statistical relationships between measured bio-indicators and bulk metal concentrations.⁶¹⁻⁶³ Such formulations ignore the fundamental interplay between nanometric and macroscopic kinetic- and transport-determining processes that are intimately connected to the physicochemical features of the complexing nanoparticles or colloids. The theoretical lability framework detailed herein opens perspectives for understanding - on a *mechanistic* level - the extent by which environmental or anthropogenic nanoparticles and colloids in natural waters control (i) the speciation and bioavailability of metal ions toward biological organisms or compartments (*e.g.* bacteria, algae, fish gills, etc) and (ii) therewith, the magnitude of metal biouptake and ensuing bioavailability of essential or therapeutic compounds, or deleterious effects in the case of toxic compounds. The results reported here have thus far-reaching implications for understanding the reactivity of NPs in environmental and biological media beyond the trivial perspective of their large specific surface area.

**Supporting Information**

I. Definition of the function *F* involved in eq. (5). II. Detailed derivation of eq. (13). III. Illustrative dependence of the lability parameter of hard nanoparticulate metal complexes on the physicochemical and chemodynamic factors pertaining to M-(2D)NP complexes (Figure S1). The Supporting Information is available free of charge on the ACS Publications website.

**Conflicts of interest.**

There are no conflicts of interest to declare.

**Acknowledgements.**

RMT conducted this work within the framework of the EnviroStress Center of Excellence at the University of Antwerp and with financial support from the BELSPO AquaStress and the UA IOF-SBO Waterside projects. JFLD acknowledges the financial support from the French national programme EC2CO-Ecodyn/MicrobiEen (TOXSCALE project).

**Glossary of main symbols**

**Latin**

*a* Particle core radius [m]
d Particle shell thickness [m]

c_s Density of binding sites S within the particle shell volume (applicable for 3D NP complexants) [mol m⁻³]

$\bar{c}_s$ Density of binding sites S smeared-out over the sample volume [mol m⁻³]

$c_M^*$ Bulk concentration of metal ions M [mol m⁻³]

c_M Concentration of intraparticulate free metal ions M [mol m⁻³]

$\bar{c}_M$ Concentration of intraparticulate free metal ions smeared-out over the sample volume [mol m⁻³]

$c_{MS}$ Concentration of intraparticulate complex metal species MS [mol m⁻³]

$\bar{c}_{MS}$ Concentration of complexes MS smeared-out over the sample volume [mol m⁻³]

$D_M$ Diffusion coefficient of M [m² s⁻¹]

$D_p$ Diffusion coefficient of the nanoparticle [m² s⁻¹]

$f_{el,a}$ Factor for conductive diffusion of M to the surface of 2D hard nanoparticulate complexant

$\bar{f}_{el,a}$ Factor for conductive diffusion of M to the reactive component of 3D nanoparticulate complexant

$f_{el,d}$ Factor for conductive diffusion of M from the surface of hard nanoparticulate complexant

$\bar{f}_{el,d}$ Factor for conductive diffusion of M from the reactive volume of 3D nanoparticulate complexant

$f_B$ Factor for Boltzmann accumulation of M at the surface of 2D nanoparticulate complexant

$\bar{f}_B$ Factor for volume-averaged Boltzmann accumulation of M in the reactive component of 3D nanoparticulate complexant

$J_{kin}$ Kinetic flux arising from the rate of dissociation of complexed species [mol m² s⁻¹]

$J_{diff}$ Diffusive flux arising from the diffusion of all forms of the metal towards a reactive macrointerface [mol m² s⁻¹]

$k_a$ Kinetic constant for M-NP formation [mol⁻¹ m³ s⁻¹]

$k_d$ Kinetic constant for M-NP dissociation [s⁻¹]

$k_a^{is}$ Kinetic constant for inner-sphere MS formation [mol⁻¹ m³ s⁻¹]

$k_d^{is}$ Kinetic constant for inner-sphere MS dissociation [s⁻¹]

$k_{a,p}$ Kinetic constant for M conductive diffusion to intraparticulate or surface binding sites S [mol⁻¹ m³ s⁻¹]

$k_{d,p}$ Kinetic constant for M conductive diffusion from intraparticulate or surface binding sites S [s⁻¹]

$k_w$ Kinetic constant for inner-sphere dehydration of $M^{z_M^+}$ [s⁻¹]

$K_{int}, K_{int}^{(z=1)}$ Intrinsic stability constant for inner-sphere metal complexes in 3D complexing nanoparticles [mol⁻¹ m³] or at the surface of 2D nanoparticles (superscript 'z = 1', [mol⁻¹ m²])

$K_H$ Henry coefficient for adsorption of M at the surface of 2D complexing nanoparticles [m]

$\mathcal{L}$ Lability parameter of metal complexes (defined as the ratio between $J_{kin}$ and $J_{diff}$)

ℓ Defined by $ℓ = k_B T / (6\pi\eta D_M)$ (derived from Stokes-Einstein equation)

ℓ_a Thickness of intraparticulate dissociation reaction layer devoid of electrostatic contribution [m]

ℓ_c Characteristic separation distance between structural charges carried by NPs [m]

M Metal ion

M-NP Metal-nanoparticle complex

MS Inner-sphere complex

NP Nanoparticulate complexant

$r_p$ Particle radius $a + d$ [m]
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$r_{p,max}$ Defined by $r_{p,max} = (2\ell\Phi_B\mu_d^2)^{1/3}$ [m]

$r_p^*$ Defined by $r_p^* = \varphi \bar{f}_B \ell (1-z^3)(1+K_{\text{int}}c_s)$ [m] for $0 \le z < 1$ and $r_p^* = \left( 3\varphi \bar{f}_B \ell K_H \frac{1+K_{\text{int}}^{(z=1)}\Gamma_s}{K_{\text{int}}^{(z=1)}\Gamma_s} \right)^{1/2}$ [m]

for $z=1$

$r_{p,\text{int}}$ Defined by $r_{p,\text{int}} = (\ell\Phi_B\mu_d^2 / \Phi_{\text{el,a}})^{1/3}$ [m]

$\tilde{r}_p$ Defined by $\tilde{r}_p = \ell\Phi_B / \Phi_{\text{el,a}}$ [m]

S Metal binding site

$V^{\text{os}}$ Outer-sphere volume per site S [m$^3$]

$z_M$ Valence of M

$z$ Defined by the ratio $a / r_p$

## Greek

$\gamma$ Factor for NP body exclusion from the macroscopic reaction layer ($0 \le \gamma \le 1$)

$\Gamma_s$ Surface density of metal binding sites S (applicable for 2D complexing nanoparticles)

$\bar{\delta}_o$ Thickness of the macroscopic diffusion layer at the sensor/medium interface

$1 / \kappa$ Thickness of extraparticulate electric double layer [m]

$\bar{\lambda}$ Macroscopic reaction layer thickness [m]

$\Lambda$ Dimensionless ratio defined by $\bar{\lambda} / r_p$

$\mu_d$ Intraparticulate reaction layer thickness with regard to association process (including NP electrostatic effects) [m]

$\varphi$ Particle volume fraction

$\Phi_{\text{el,a}}; \Phi_B$ Dimensionless numbers involved in the expressions of $\bar{\lambda}$ and $\Lambda$

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