# Dynamics of the Galvanic Replacement Reaction of Silver by Gold: Phenomenological Models for Open Circuit Potential-Time Responsive Indicator

Bonito Aristide Karamoko, Yaovi Holade

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Bonito Aristide Karamoko, Yaovi Holade. Dynamics of the Galvanic Replacement Reaction of Silver by Gold: Phenomenological Models for Open Circuit Potential-Time Responsive Indicator. Chem-ElectroChem, 2025, 12 (22), pp.e202500338. 10.1002/celc.202500338. hal-05445835

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Submitted on 7 Jan 2026

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# Dynamics of the Galvanic Replacement Reaction of Silver by Gold: Phenomenological Models for Open Circuit Potential-Time Responsive Indicator

Bonito Aristide Karamoko and Yaovi Holade*

*In memory of Professor Jean-Michel LEGER*

Galvanic replacement reaction (GRR) is an oxidation-reduction process triggered by an electrochemical potential difference between two metal species, and involves the concerted motion of electrons, atoms, and ions at different times and spatial scales. Despite extensive research, a fundamental question remains unanswered: How can the driving force, that is, the electrochemical potential, be mapped in real time when existing microscopic, optical, and X-ray methods cannot capture it? In this article, the most widely used and fascinating system: silver-gold, in which three silver atoms are replaced by one gold atom, despite silver and gold having almost identical atomic radii, is interrogated. The experimental time-dependent open-circuit potential (OCP(t))

data, as well as phenomenological and mathematical models, are leveraged to describe the dynamics of the GRR. Specifically, modified sigmoidal kinetic functions are proposed based on autocatalytic networks and enzyme cascades performing logic gates, in order to account for the offset and sharpness of the OCP(t) responses at different input concentrations. This allows quantifying, for the first time, the two highly sought-after kinetic parameters of the apparent rate constant and the midpoint growth time. This knowledge can inspire new explorations in GRR-derived syntheses involving different galvanic exchange ratios for new functional nanostructured materials.

## 1. Introduction

Galvanic replacement reaction (GRR), also known as "galvanic displacement reaction" in materials chemistry, "immersion plating" in the plating industry, and "cementation" in hydrometallurgy, is a concerted chemical phenomenon comprising electrons, atoms, and ions motion triggered by the electrochemical potential difference between two metallic species.[1,2] For the two metals involved in the galvanic exchange process, the outgoing metal has the lower oxidation potential and the incoming metal has the higher oxidation potential. By manipulating experimentally, the galvanic exchange ratios between the involved redox pairs and the experimental conditions, different materials ranging from core-shell to alloys have been reported for various applications in

catalysis, sensing, electronics, and energy.[2-8] However, the lack of quantification of the driving force, that is, the time-dependent potential during GRR, has limited the possibility of extending the implementation the principle GRR to a mixture of multiple metals. Yet this was expected to enable the exploration of medium- to high-entropy materials,[9,10] where the effects of alloys, porosity, stain or contraction could coexist depending on the size of the metals entering or leaving the initial architecture. For example, Skrabalak's team recently showed that galvanic exchange between PdCu intermetallic nanoparticles and Au(III) or Pt(IV) ionic species produces exotic trimetallic particle architectures, ranging from core-shell particles to Janus particles, due to the competing effects of Ostwald ripening and Kirkendall effects.[5,11] Therefore, direct capturing and modeling of the dynamics of the GRR could pave the way for structural control exhibiting characteristics of synergy (multiple metals), long-range geometric deformation, and short-range electronic charge transfer effects. Here, we examined the case of the GRR of Ag by Au, which is the most widely used and fascinating system in which three silver atoms are replaced by one gold atom (similar atomic radii for Au and Ag), leaving behind a porous and alloyed structure for applications in catalysis, biology, and energy.[2,8,12-19]

The GRR of silver by gold can be described by the net reaction of Equation (1), which is a redox reaction (electron driven) characterized by the standard redox potential ($E°$) of 1.50 and 0.80 V versus standard hydrogen electrode (SHE) for $Au^{3+}$/Au and $Ag^{+}$/Ag redox couples at bulk to micros-scale. It should be noted that the complexation of the metal cations lowers the redox

B. A. Karamoko, Y. Holade
Institut Européen des Membranes IEM
UMR 5635
ENSCM
CNRS
Univ Montpellier
34090 Montpellier, France
E-mail: yaovi.holade@umontpellier.fr

Y. Holade
Institut Universitaire de France (IUF)
75005 Paris, France

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© 2025 The Author(s). ChemElectroChem published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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potentials and $E^0(AuCl_4^-)/Au) = 1.00$ V versus SHE while $E^0(AgCl/Ag) = 0.22$ V versus SHE. It is worth mentioning the report of the inversion of this bulk-like behavior at nanoscale when the replacement of gold by silver has been reported by Wu in 2012,[20] O'Mullane's group in 2013,[21] and Zamborini's group in 2020,[22] among others, wherein some results may arguably be explained by the seminal Plieth's model[23] that links the metal oxidation potential to the nanoparticle size and the surface stress.[24] This raises a fundamentally critical question: how does the structure of a nanoparticle of 2–5 nm survive when three silver atoms are replaced by one gold atom, and vice versa, given that silver and gold have almost identical atomic radii? Although directly beyond the scope of the present study, the likely answer to this question could lie in real-time knowledge of the temporal profile of the electrical potential of the GRR, hence the purpose of this study. One of the hypotheses is based on the determining the open circuit potential (OCP) of the Equation (1), which should allow capturing the collective and synchronous movements of participating electrons, ions, and atoms. Thus, monitoring OCP as a function of time and reagent concentration should provide a precise understanding of GRR events and fill gaps in real-time kinetic knowledge as we recently observed.[17] To date, Patsay et al.[25] proposed in 2022 to study the OCP(t) responses of the GRR of zinc by copper using a nonlinear potential scanning approach to reproduce open-circuit conditions. However, this methodology could not accurately mimic the real operation of the GRR, as it overlooks the possible atom rearrangement and the coupled kinetics of Zn(0)-to-Zn(II) oxidation and Cu(II)-to-Cu(0) reduction.

$$ \begin{gathered} \mathrm{Au}^{(\mathrm{III})}_{(\mathrm{aq})} + 3\mathrm{Ag}^{(0)}_{(\mathrm{s})} \rightleftharpoons \mathrm{Au}^{(0)}_{(\mathrm{s})} + 3\mathrm{Ag}^{(1)}_{(\mathrm{aq})/(\mathrm{s})}, \\ K_{\mathrm{eq},1} = \left[ \frac{(a_{\mathrm{Ag}^{(0)}})^3}{a_{\mathrm{Au}^{(0)}}} \right] \times \left[ \frac{a_{\mathrm{Au}^{(0)}}}{(a_{\mathrm{Ag}^{(0)}})^3} \right] \end{gathered} \quad (1) $$

where $K_{eq,1}$ is the equilibrium constant and a is the activity.

The GRR is in fact two concerted electrochemical reactions so that the main driving force, that is, the time-dependent redox potential cannot be captured by widely employed ex situ and/or in situ microscopic, optical, and X-ray techniques (scanning transmission electron microscopy, small angle X-ray scattering, X-ray absorption fine structure, and so on) attempting to interrogate the kinetics of the Equation (1).[1,13,18,22,26–30] Although these techniques can provide useful information by monitoring the structural evolution of the $Ag_{1-x}Au_x$ particles ($0 \le x < 1$, x = atomic fraction of gold in the newly formed structure), they cannot capture the dynamic electrical changes during the reaction. Another limitation is the inability to monitor the true OCP when studying GRR in liquid solution (i.e., using a colloidal suspension of Ag particles), which hinders the correct elucidation of GRR evolution in equilibrium and disequilibrium domains.[17,31,32] Knowledge of the OCP(t) can provide not only key experimental information on the time required to stop synthesis and target a specific structure-performance pair or prevent/limit serious material damage (corrosion, void formation, and so on), but also fundamental knowledge on the thermodynamic and kinetic data of the GRR. To date, a number of studies have reported the

monitoring of GRR by time-dependent OCP, for example, GRR of Fe by Cu,[33] GRR of Ni by Pd,[32] GRR of Ag by Pd,[31] GRR of Ag by Au,[17] and so on. Among these studies, a limited number of reports have witnessed sigmoid shape of OCP-time profiles[17,32] characteristic of autocatalytic/cascade reactions.[34–40] Furthermore, there is still limited knowledge about the numerical models that can account for the experimentally observed OCP(t) responses during GRR of silver by gold, which is widely used to design nanostructures for applications in catalysis, biology, energy, and environment.

In this article, we aim to disentangle the dynamics of the GRR of silver by gold events by using mathematical models to phenomenologically fit experimental data from the OCP-time sensitive indicator. Based on a model of bare, monodisperse silver nanoparticles (AgNPs) (average particle size of 20.5 ± 0.3 nm) that were deposited electrochemically on a glassy carbon electrode and various refutable mathematical models, we proposed a modified sigmoidal kinetic function inspired by autocatalytic reactions[34] and enzyme cascades performing an AND logic gate[36] phenomena. This phenomenological model accounts for the offset of the OCP(t)-responsive curves and the sharpness of the transition between the commencement and the steady state of the GRR of silver by gold at different concentrations of the input reagents.

## 2. Results and Discussion

### 2.1. Characterization of Experimental Model of Investigation

Before examining the various mathematical models that can explain the experimental observations of OCP as a function of time (OCP(t)) in the presence of a population of Au(III) species in an aqueous solution containing Ag(0) at the electrode, we first establish the framework of the study model in terms of the nature of the species and the properties of the silver particles. The OCP-time-responsive indicator of GRR of silver by gold was experimentally obtained by performing a two-electrode electrochemical measurement, wherein a GCE-Ag electrode was immersed in a saline solution containing $KAuCl_4$ and housing a reference electrode (Ag/AgCl) to record the potential through a potentiostat. Figure 1a depicts the true picture of GRR of silver by gold at the solid-liquid interface. The net reaction of Equation (1) results from two half-cell electrochemical reactions of Ag oxidation into $Ag^+$ according to Equation (2) and $Au^{3+}$ reduction into Au according to Equation (3) provided that the trend in the equilibrium Nernst potentials is not different from the standard conditions ($E^0(Au^{3+}/Au) > E^0(Ag^+/Ag)$). The reaction of Equation (4) is quantitative since the equilibrium constant is $K_{eq,4} = 3.2 \times 10^{-35}$ at 25 °C.

$$ \mathrm{Ag}_{(\mathrm{s})} \rightarrow \mathrm{Ag}^{+}_{(\mathrm{aq})} + \mathrm{e}^{-}, \quad E_{\mathrm{eq}}(\mathrm{Ag}^{+}/\mathrm{Ag}) = E^{0}(\mathrm{Ag}^{+}/\mathrm{Ag}) + \frac{RT}{F} \ln \left[ \frac{a_{\mathrm{Ag}^{+}}}{a_{\mathrm{Ag}}} \right] \quad (2) $$

$$ \begin{gathered} \mathrm{Au}^{3+}_{(\mathrm{aq})} + 3\mathrm{e}^{-} \rightarrow \mathrm{Au}_{(\mathrm{s})}, \\ E_{\mathrm{eq}}(\mathrm{Au}^{3+}/\mathrm{Au}) = E^{0}(\mathrm{Au}^{3+}/\mathrm{Au}) + \frac{RT}{3F} \ln \left[ \frac{a_{\mathrm{Au}^{3+}}}{a_{\mathrm{Au}}} \right] \end{gathered} \quad (3) $$
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<figure><img src="image_7.png" /><figcaption>Figure 1. a) Illustration of the different processes during the GRR of silver by gold in an aqueous solution near the solid–liquid interface. b) Effect of the initial concentration of AuCl₄⁻ on the proportion of aurous ions (Au³⁺) in water in the presence of excess chloride ions (right y-axis, 3.5 M NaCl, Equation (10)) and in the absence of chloride ions (left y-axis, 0 M NaCl, Equation (11) by the dichotomy method).</figcaption></figure>

$$ 
\begin{gathered}
\mathrm{Au}^{3+}_{(\mathrm{aq})} + 3\mathrm{Ag}_{(\mathrm{s})} \rightleftharpoons \mathrm{Au}_{(\mathrm{s})} + 3\mathrm{Ag}^{+}_{(\mathrm{aq})}, \\
K_{\mathrm{eq},4} = \frac{(\mathrm{a}_{\mathrm{Ag}^{+}})^{3} \times \mathrm{a}_{\mathrm{Au}}}{(\mathrm{a}_{\mathrm{Ag}})^{3} \times \mathrm{a}_{\mathrm{Au}^{3+}}} = \exp\left[\frac{3F}{RT}(E^{0}_{\mathrm{Au}^{3+}/\mathrm{Au}} - E^{0}_{\mathrm{Ag}^{+}/\mathrm{Ag}})\right] \quad (4)
\end{gathered}
$$

$$ \mathrm{AuCl}_{4}^{-}(\mathrm{aq}) + 3\mathrm{e}^{-} \rightarrow \mathrm{Au}_{(\mathrm{s})} + 4\mathrm{Cl}_{(\mathrm{aq})}^{-}, \qquad (5) $$

$$ \mathrm{AuCl}_{4}^{-}(\mathrm{aq}) + 3\mathrm{Ag}_{(\mathrm{s})} \rightleftharpoons \mathrm{Au}_{(\mathrm{s})} + 3\mathrm{Ag}^{+}_{(\mathrm{aq})} + 4\mathrm{Cl}_{(\mathrm{aq})}^{-}, \qquad (6) $$

$$ \begin{gathered} x\mathrm{AuCl}_{4}^{-}(\mathrm{aq}) + 3\mathrm{Ag}_{(\mathrm{s})} \rightleftharpoons \mathrm{Ag}_{(3-3x)}\mathrm{Au}_{x(\mathrm{aq})} + 3x\mathrm{Ag}^{+}_{(\mathrm{aq})} + 4x\mathrm{Cl}_{(\mathrm{aq})}^{-}, \qquad (7) \\ K_{\mathrm{eq},7} = \frac{(\mathrm{a}_{\mathrm{Ag}^{+}})^{3x} \times (\mathrm{a}_{\mathrm{Cl}^{-}})^{4x} \times \mathrm{a}_{\mathrm{Ag}_{(3-3x)}\mathrm{Au}_{x(\mathrm{aq})}}}{(\mathrm{a}_{\mathrm{AuCl}_{4}^{-}})^{x} \times (\mathrm{a}_{\mathrm{Ag}})^{3}} \end{gathered} $$

Since the commonly used gold (III) precursor in aqueous solutions is AuCl₄⁻ (hydrated KAuCl₄ or HAuCl₄), the reduction of gold (III) into gold (0) is supported by Equation (5). The overall reaction of GRR of silver by gold is thus modeled by Equation (6) ($K_{\mathrm{eq},6} = 1.4 \times 10^{-10}$ at 25 °C) or Equation (7), as it is impossible to replace all the silver atoms with gold atoms. From an experimental standpoint, the Ag⁺ precipitation side reaction by capturing Cl⁻ from AuCl₄⁻ is detrimental to the quality of silver-gold nanostructures. Either the insoluble AgCl ($K_{\mathrm{s}}(\mathrm{AgCl}) = 1.8 \times 10^{-10}$ at 20 °C[41,42]) contaminates the targeted materials, or the post-treatment used to dissolve AgCl does not accurately depict the GRR dynamics. To circumvent such an undesirable phenomenon and leveraging the speciation diagrams of silver-chloride,[41,42] the present study of the GRR of silver by gold was carried out in a brine solution that we have proven[17,18] to be effective in fostering the formation of soluble species, AgClₙ⁽ⁿ⁻¹⁾⁻ (n ≥ 3) according Equation (8) at [NaCl] ≥ 1 M.[41]

$$ \mathrm{Ag}^{+}_{(\mathrm{aq})} + n\mathrm{Cl}_{(\mathrm{aq})}^{-} \rightleftharpoons \mathrm{AgCl}_{n}^{(n-1)-}(\mathrm{aq}), \quad K_{\mathrm{e},8} = \frac{a_{\mathrm{AgCl}_{n}^{(n-1)-}}}{a_{\mathrm{Ag}^{+}} \times (a_{\mathrm{Cl}^{-}})^{n}}, \quad n \ge 3 \quad (8) $$

It is interesting to note that the achievement of the GRR of Ag by Au in such a chlorinated medium can, in theory, make the redox process more quantitative, since the Ag⁺/Ag redox couple ($E° = 0.8$ V vs SHE) is supplanted by a much weaker redox couple, AgClₙ⁽ⁿ⁻¹⁾⁻/Ag (n ≥ 3, $E° ≤ 0.2$ V vs SHE). In order to clearly identify the nature of the gold species (III) species (either AuCl₄⁻ or Au³⁺) in the GRR medium, we studied the effect of the initial concentration of AuCl₄⁻ on the proportion of aurous ions (Au³⁺) in water in the absence of chloride ions (0 M NaCl) and in the presence of an excess chloride ions (3.5 M NaCl). The percentage of free Au³⁺ can be determined from the dissociation reaction of Equation (9) by solving Equations (10) and (11). Figure 1b shows that for an initial AuCl₄⁻ concentration ranging from 1 nM to 1 M in the brine solution, [Au³⁺]eq is $2.65 \times 10^{-38}–2.65 \times 10^{-29}$ M ($C_{\mathrm{initial}}/C_{\mathrm{dissociated}} \approx 4 \times 10^{28}$), indicating that all gold(III) species are in the form of AuCl₄⁻. In the absence of free chloride ions in the starting solution, [Au³⁺]eq ranges from 1 nM to 1.73 µM. This indicates the presence of multiple populations of free Au³⁺ species such as 100% ([AuCl₄⁻]initial = 0.1–1 nM), 27% ([AuCl₄⁻]initial = 0.3 µM), 11% ([AuCl₄⁻]initial = 1 µM), 2% ([AuCl₄⁻]initial = 10 µM), and 0.04% ([AuCl₄⁻]initial = 1 mM). Therefore, if NaCl had not been introduced under our conditions of GRR of silver by gold ([KAuCl₄] = 0.25, 1.27, and 12.73 µM), the free Au³⁺ species would have represented 2%–30% of the initially introduced gold (III) complex. In theory, this would result in different galvanic replacement kinetics, given the significant difference in standard potentials of gold (III) species ($E°(\mathrm{Au}³⁺/\mathrm{Au}) = 1.50$ V vs SHE and $E°(\mathrm{AuCl₄}⁻/\mathrm{Au}) = 1.00$ V vs SHE) because the difference of 0.5 V makes it difficult to offset by considering the equilibrium Nernst potentials.

It is noteworthy that the investigative model underlying this study is contingent upon GRR's 3:1 stoichiometry, as outlined in Equation (1), given the exclusion of the hypothesis pertaining to the presence of Au(I). Indeed, the starting point is Au(III), and more specifically with KAuCl₄, which is supposed to be chemically
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To irrefutably rule out the hypothesis that the sigmoidal profile of the OCP(t) for GCE-Ag samples does not result from the contact between the carbon electrode and the $AuCl_4^-$ solution, an initial comparative control study was conducted using GCE and GCE-Ag materials for 0.25 µM $KAuCl_4$. **Figure 3a** shows the results in terms of OCP(t) profiles, and Figure 3b shows the results in terms of cyclic voltammograms (CV). These studies were supplemented by elemental analysis using inductively coupled plasma optical emission spectrometry (ICP-OES). For Figure 3a, region (1) corresponds to the state of the electrode-solution interface before the injection of gold (III) species, and region (2) corresponds to the stage after the injection of gold (III) species at a molar ratio $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.25$. For bare GCE without AgNPs, the spontaneous (presumably exponential) increase in OCP upon injecting gold (III) species represents the Nernst response according to Equation (5) to convey the potential where the electrochemical reduction reaction of Au(III) to Au(0) should experimentally begin. We recently observed that using a concentration of 142 µM of $KAuCl_4$ results in a rapid response, with stabilization occurring in less than a

minute.[17] Here, according to Equation (5) for 0.25 µM $KAuCl_4$, the thermodynamic potential at which the reduction of $Au^{(III)}Cl_4^-$ to $Au^{(0)}$ would begin is $E_{eq}(AuCl_4^-/Au) = 0.86$ V versus SHE = 0.66 V versus Ag/AgCl, which is consistent with the experimental value. After stabilization of the OCP, elemental analysis of the GCE revealed no traces of gold, which is corroborated by the CV in 0.1 M KOH electrolyte (Figure 3b). When Ag nanoparticles were deposited on the GCE (i.e., the GCE-Ag sample), a radical change in the OCP(t) and CV profiles can be observed. In Figure 3a, the OCP(t) profile for the GCE-Ag can be decomposed into three regions. The induction period or lag phase with an OCP plateau around 0 V versus Ag/AgCl and saturation at a potential above 0.6 V versus Ag/AgCl. These two states are bridged by an exponential phase. **Figure 4a** shows that each of these phases is influenced by the population of $AuCl_4^-$ species near the GCE-Ag surface and the amount of AgNPs on the GCE surface when the concentration of $KAuCl_4$ is poised.

For GCE-Ag, the increase in OCP toward 0–0.1 V versus Ag/AgCl upon introducing $AuCl_4^-$ is attributed to an increase in Ag(I) concentration due to the diffusion of $AuCl_4^-$ ions near

<figure><img src="image_9.png" /><figcaption>**Figure 3.** Electrochemical validation of GRR of Ag by Au. a) Time-dependent OCP, OCP(t), profiles for the bare glassy carbon electrode (GCE), and the AgNPs decorated glassy carbon electrode (GCE-Ag, 8 µg$_{Ag}$ cm⁻²) before the injection of Au(III) species [region (1): 3.5 M NaCl] and after the injection of gold (III) species corresponding to molar ratio $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.25$ [region (2): 3.5 M NaCl + 0.25 µM $KAuCl_4$]. b) Cyclic voltammogrammes (CV, 50 mV s⁻¹) of the GCE and GCE-Ag (referred to as 'GCE-Ag_GRR' after experiments of panel (a)). c–e) CVs of the GCE-Ag_GRR in the absence of glucose (0.1 M KOH) and in the presence of glucose (0.1 M KOH + 0.1 M D-glucose): c) Current density normalized by geometric surface area of the electrode, d) zoom in of the current density normalized by geometric surface area of the electrode, and e) current density normalized by the mass of gold (the active component. Note: all the solutions were outgassed with N₂ and all the experiments were recorded at 25 °C.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 4.** a) Effect of Ag and Au(III) amounts on the dynamics of GRR of Ag by Au quantified by the time-dependent OCP, OCP(t), profiles for GCE-Ag-3cycles\_KAuCl₄ [corresponding to molar ratio R<sub>GRR</sub> = *n*(KAuCl₄)/*n*(Ag) = 0.25–12.50] and for GCE-Ag-6cycles\_KAuCl₄ [corresponding to molar ratio R<sub>GRR</sub> = *n*(KAuCl₄)/*n*(Ag) = 0.62–6.25]: before the injection of Au(III) species [region (1): 3.5 M NaCl, before about 10 min] and after the injection of gold (III) species [region (2): 3.5 M NaCl + 0.25 µM KAuCl₄, after about 10 min]. b,c) Representative SEM images of silver-gold structures after GRR (3.5 M NaCl + KAuCl₄). (b) Au(III) species [R<sub>GRR</sub> = *n*(KAuCl₄)/*n*(Ag) = 0.25] reacted with Ag nanoparticles from GCE-Ag synthesized under the conditions of: GCE: 5 mm diameter GCE disk (0.196 cm² geometric surface area), Ar-saturated solution, 25 °C, 200 mM KNO₃, 60 µM AgNO₃, 8.2 mM NH₄OH, ON step: *j*<sub>ON</sub> = -0.5 mA cm⁻² for *t*<sub>ON</sub> = 5, OFF step: OCP for *t*<sub>OFF</sub> = 5 s; *N*<sub>cycles</sub> = 3 (8 µg<sub>Ag</sub> cm⁻²). (c) Au(III) species [R<sub>GRR</sub> = *n*(KAuCl₄)/*n*(Ag) = 1.27] reacted with Ag nanoparticles from GCE-Ag synthesized under the conditions of: GCE: 5 mm diameter GCE disk (0.196 cm² geometric surface area), Ar-saturated solution, 25 °C, 200 mM KNO₃, 60 µM AgNO₃, 8.2 mM NH₄OH, ON step: *j*<sub>ON</sub> = -0.5 mA cm⁻² for *t*<sub>ON</sub> = 5, OFF step: OCP for *t*<sub>OFF</sub> = 5 s; *N*<sub>cycles</sub> = 6 (17 µg<sub>Ag</sub> cm⁻²).</figcaption></figure>

the surface of silver particles,[18,31,32,48] where the electron transfer kinetics of the cathodic Au(III)-to-Au(0) and anodic Ag(0)-to-Ag(I) reactions compete. The height of the first plateau as well as the OCP increase after this first plateau can then be explained by a progressive change in the active sites of the anodic reaction from silver atoms to Ag<sub>1-x</sub>Au<sub>x</sub> wherein silver and gold atom diffusion and/or rearrangement occur such as the hidden Kirkendall effect.[27,49] The presence of gold after contact between GCE-Ag and KAuCl₄ solution is confirmed by electrochemistry, as shown in Figure 3b–e. The extended CV characterization of a GCE-Ag in an alkaline electrolyte in the presence and absence of glucose is reported in Ref. [17]. Indeed, the catalytic waves in the presence of glucose (Figures 3c–e) confirm the presence of gold within the electrode materials because silver is inactive for glucose oxidation at potentials below 1 V versus reversible hydrogen electrode (RHE). We note that the oxidation peak observed during the backward scan is due to the oxidation of glucose molecules on the

surface of metallic gold, which is released by the AuOₓ-to-Au reduction process centered at ca. 1 V versus RHE.[50] Elemental analysis by ICP-OES of the materials of Figure 3 yields a value of 8 µgAg cm⁻² before the GRR and 6 µgAg+Au cm⁻² (2.3 µgAg cm⁻² and 3.7 µgAg cm⁻², i.e., about 53 at.% Ag and 47 at.% Au) after the GRR gives, thus confirming the gold loading in the particles of Figure 4b,c. Furthermore, these SEM images highlight the porosity structure after the GRR. This indicates a greater number of active sites, considering the high peak current density during glucose electrooxidation by considering either the geometric surface area of the electrode or the mass of gold. However, the latter could lead to an overestimation of the mass activity given the relatively low metal loading. Furthermore, Figure 4b,c illustrates heterogeneity in particle size, which would influence GRR dynamics, given the disparity in reactivity between small and large nanoparticles. This underscores the necessity for ongoing efforts to refine the control of nanoparticle size, particularly in scenarios where elevated metal
<figure><img src="image_12.png" /></figure>

loading levels are prevalent, as these conditions tend to favor the
formation of larger particles.

## 2.3. OCP-Responsive Indicator of GRR from Phenomenological Analytical Analysis of OCP(t) Data

The above studies enabled us to obtain clear physicochemical and electrochemical pictures of the GCE-Ag material before and after the GRR between silver (0) and gold (III). These studies also

established unambiguously that the time- and concentration-
dependent OCP profiles of a GCE-Ag electrode in the presence
of an Au(III) population in an aqueous solution result from the gal-
vanic replacement of silver by gold, as described by Equation (1).
Consequently, we next embarked on an exploratory endeavor, the
first of its kind, to delve into a series of mathematical models that
could elucidate such experimental observations. The objective of
our methodology is to ascertain a trend in GRR rate constant data
such as the rate constante.

<figure><img src="image_13.png" /><figcaption>**Figure 5.** Refutable screening models for fitting experimental data, the OCP(*t*<sub>cor</sub>) recorded on the course of GRR of silver by gold [GCE-Ag (8 µgAg cm<sup>-2</sup>), *R*<sub>GRR</sub> = *n*(KAuCl<sub>4</sub>)/n(Ag) = 1.25, 25 °C]. a) Linear, b) sigmoidal kinetics from autocatalytic reactions,<sup>[34]</sup> c) logistic function, d) Sigmoid function for RDE mass transport limited current,<sup>[58]</sup> e) Privman and Katz's model for enzymatic cascade realizing an AND logic gate,<sup>[36]</sup> f) Langevin function, g) DoseResp function, h) Boltzmann function, and i) modified Sigmoidal kinetics from autocatalytic reactions. Note: Extended models are reported in Figure 6 and the extracted fitting values are reported in Table 1. *t*<sub>cor</sub> = corrected time, starting from the beginning of the GRR (herein, introduction of Au(III) species in the reactor).</figcaption></figure>
<figure><img src="image_14.png" /></figure>

report[56] points out that the simultaneous transfer of more than one electron is highly unlikely. This implies that the GRR between Ag(0) and Au(III) is actually a cascade of multielectron steps (similar to proton-coupled electron transfer reactions[57]) that produce several reaction intermediates and energy barriers. Thus, the establishment of an analytical solution for the OCP(t)-sensitive curves is mathematically and physically challenging. One possible solution is to use a phenomenological method based on model functions describing key GRR steps, such as self-growth, autocatalysis, enzyme cascades performing logic gates, and current limited by mass transport.[34–40,58]

We primarily selected the case $R_{GRR} = 1.25$ based on the OCP(t) response profiles in Figure 4a for the galvanic replacement between Ag(0) and Au(III) to test different mathematical models. The results are presented in **Figure 5a–i**, and the adjustment parameters are summarized in **Table 1**. The selected models for fitting the experimental OCP(t) data during the GRR of silver (particles size between 10 and 30 nm) by gold (KAuCl₄ in saline solution) show that the linear function (Model-1, Figure 5a) and the exponential function (Model-2) are unsuitable. This is despite the fact that such behavior was considered when modeling the growth and aggregation of metal particles when the reducing and oxidizing agents are in the same physical state, the liquid phase.[59] Furthermore, the logistic function (Model-3, Figure 5b) and the sigmoidal kinetics of autocatalytic reactions (Model-4, Figure 5c)[34] provide an acceptable phenomenological description of the dependance of the OCP on time during the exponential and saturation phases. These models give a threshold potential limit of 0.65 V versus Ag/AgCl and an apparent rate constant of $k_{app} = 0.64 \text{ min}^{-1}$ (Table 1). However, these models poorly describe the lag phase (i.e., the induction period, which

is characterized by the first plateau at ≈0.05 V vs Ag/AgCl) and cannot explain the shift in OCP(t) responses by varying the input reactant concentrations, as Figure 4a clearly shows. Indeed, there is a significant discrepancy between the model and the experimental data for $t_{cor} \le 10$ min in Figure 5b,c. We note that, the Model-4, allows us to determine an important kinetic parameter: the midpoint of growth time, or the time when half of the saturation phase potential has been reached. Here, $b = 13.4$ min (Table 1).

Given the aforementioned limitations, we tested the sigmoid function for the rotating disk electrode (RDE) mass transport-limited current[58] to account for both kinetic parameters and quantify the offset of the OCP(t) responses at different Au(III) concentrations. The corresponding Model-5 (Figure 5d) provides more accurate much between the model and the experimental data for reaction time $t_{cor} \le 10$ min. The determined kinetic parameters are similar in magnitude to those of Models 3 and 4. For example, the midpoint of the growth is $b = 13.7$ min and the apparent rate constant is $k_{app} = 0.62 \text{ min}^{-1}$ (Table 1). Since the three-electron exchange GRR described by Equation (1) can be associated with a cascade of multielectron steps, we next made an analogy with the seminal works of Privman and Katz regarding bimolecular systems with enzymatic cascades that realize logic gates.[36–38] Specifically, we modeled our experimental OCP-time response data using a simplified mathematical function (Model-6, Table 1, Figure 5e), which was derived from a phenomenological expression for fitting sigmoid signal shapes. These shapes were obtained through intensity filtering of an enzymatic AND logic gate.[36] The results in Figure 5e demonstrate a close correspondence between the model and the experimental data, where the newly introduced apparent rate constants are nearly identical:

<figure><img src="image_15.png" /><figcaption>**Figure 6.** Models of DoseResp versus BiDoseResp functions for fitting experimental data the OCP($t_{cor}$) recorded on the course of GRR of silver by gold. Other models are reported in Figure 5.</figcaption></figure>
<figure><img src="image_16.png" /></figure>

$k_1 = 0.82 \text{ min}^{-1}$ and $k_2 = 0.877 \text{ min}^{-1}$. In fact, compared to the original model, these two rate constants quantify both the offset of the response curve and the sharpness of the transition from a reduced response at low inputs to a convex response at larger inputs.[47]

We tested the usual functions (Figure 5f-h), and taking into account the specific features of Models 4-5-6, we proposed Model-10 (Figure 5i). Model-10 is a modification of sigmoidal kinetics based on autocatalytic network reactions and enzyme cascades

performing logic gates. This modification takes into account the offset of the OCP(t) response curves and the sharpness of the transition between the start and stable state of the GRR. Besides, the Langevin function (Figure 5f), which is derived from statistical thermodynamics, describes how magnetic dipoles align with an applied field. However, it is difficult presently to associate the Langevin function with the GRR phenomenon. Conversely, the DoseResp function (Model-8, Figure 5g) and the Boltzmann function (Model-9, Figure 5h) can be used to analyze the dynamic

<table><caption>**Table 2.** Detailed phenomenological fitting functions equations and parameter values for the experimental data of the OCP($t_{cor}$) recorded for GCE-Ag-3cycles_KAuCl₄ [corresponding to molar ratio $R_{GRR} = n(KAuCl₄)/n(Ag) = 0.25-12.50$, 25 °C] and for GCE-Ag-6cycles_KAuCl₄ [corresponding to molar ratio $R_{GRR} = n(KAuCl₄)/n(Ag) = 0.62-6.25$, 25 °C]. $E(t_{cor}) = E_{WE} - E_{REi}$ = OCP (V vs. Ag/AgCl $\equiv V_{Ag/AgCl}$). $t_{cor}$ = corrected time, starting from the beginning of the GRR (herein, introduction of Au(III) species in the reactor).</caption><thead><tr><th>Entry</th><th>GCE-Ag-3cycles<br>0.25µM-KAuCl₄<br>R<sub>GRR</sub> = 0.25</th><th>GCE-Ag-3cycles<br>1.27µM-KAuCl₄<br>R<sub>GRR</sub> = 1.25</th><th>GCE-Ag-3cycles<br>12.73µM-KAuCl₄<br>R<sub>GRR</sub> = 12.50</th><th>GCE-Ag-6cycles<br>1.27µM-KAuCl₄<br>R<sub>GRR</sub> = 0.62</th><th>GCE-Ag-6cycles<br>12.73µM-KAuCl₄<br>R<sub>GRR</sub> = 6.25</th></tr></thead><tbody><tr><td>Model-3<br>Sigmoid kinetics from autocatalytic reactions<sup>[34]</sup><br>E(t<sub>cor</sub>) = <sup>a</sup>&frasl;<sub>1 + be<sup>-k<sub>cor</sub>t</sup></sub></td><td>a = 0.55<br>b = 64.73<br>k = 0.11<br>R<sup>2</sup> = 0.992</td><td>a = 0.65<br>b = 5443.38<br>k = 0.64<br>R<sup>2</sup> = 0.991</td><td>a = 0.72<br>b = 460.83<br>k = 6.94<br>R<sup>2</sup> = 0.973</td><td>a = 0.67<br>b = 16,050.30<br>k = 0.51<br>R<sup>2</sup> = 0.993</td><td>a = 0.76<br>b = 372.54<br>k = 2.34<br>R<sup>2</sup> = 0.962</td></tr><tr><td>a: threshold limit [V<sub>Ag/AgCl</sub>]<br>b: dimensionless number<br>k: rate constant [min<sup>-1</sup>]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model-4<br>Logistic function<br>E(t<sub>cor</sub>) = <sup>a</sup>&frasl;<sub>1 + e<sup>-k(t<sub>cor}-b</sup></sub></td><td>a = 0.55<br>b = 37.65<br>k = 0.11<br>R<sup>2</sup> = 0.992</td><td>a = 0.65<br>b = 13.37<br>k = 0.64<br>R<sup>2</sup> = 0.991</td><td>a = 0.72<br>b = 0.88<br>k = 7.07<br>R<sup>2</sup> = 0.973</td><td>a = 0.68<br>b = 19.04<br>k = 0.51<br>R<sup>2</sup> = 0.993</td><td>a = 0.75<br>b = 2.53<br>k = 2.36<br>R<sup>2</sup> = 0.962</td></tr><tr><td>a: threshold limit [V<sub>Ag/AgCl</sub>]<br>b: midpoint of the growth [min]<br>k: rate constant [min<sup>-1</sup>]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model-5<br>Sigmoid function for RDE mass transport limited current<sup>[58]</sup><br>E(t<sub>cor</sub>) = <sup>a</sup>&frasl;<sub>1 + e<sup>-k(t<sub>cor}-b</sup></sub> + c</td><td>a = 0.61<br>b = 35.57<br>c = -0.05<br>k = 0.09<br>R<sup>2</sup> = 0.9985<br>a + c = 0.56</td><td>a = 0.60<br>b = 13.71<br>c = 0.04<br>k = 0.82<br>R<sup>2</sup> = 0.998<br>a + c = 0.64</td><td>a = 0.61<br>b = 0.93<br>c = 0.10<br>k = 13.08<br>R<sup>2</sup> = 0.997<br>a + c = 0.71</td><td>a = 0.63<br>b = 19.31<br>c = 0.04<br>k = 0.62<br>R<sup>2</sup> = 0.998<br>a + c = 0.67</td><td>a = 0.62<br>b = 2.66<br>c = 0.11<br>k = 4.81<br>R<sup>2</sup> = 0.998<br>a + c = 0.73</td></tr><tr><td>a: constant [V<sub>Ag/AgCl</sub>]<br>b: half-wave point [min]<br>c: constant [V<sub>Ag/AgCl</sub>]<br>k: rate constant [min<sup>-1</sup>]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model-6<br>Privman and Katz's model for enzymatic cascade<br>realizing an AND logic gate<sup>[36]</sup><br>E(t<sub>cor</sub>) = <sup>a + be<sup>-k<sub>1</sub>t<sub>cor</sub></sup></sup>&frasl;<sub>c + de<sup>-k<sub>2</sub>t<sub>cor</sub></sup></sub></td><td>a = 0.64<br>b = -0.89<br>c = 1.08<br>d = 108<br>k<sub>1</sub> = 0.03<br>k<sub>2</sub> = 0.17<br>R<sup>2</sup> = 0.999</td><td>a = 0.53<br>b = 4885<br>c = 0.82<br>d = 155,750<br>k<sub>1</sub> = 0.81<br>k<sub>2</sub> = 0.88<br>R<sup>2</sup> = 0.999</td><td>a = 0.71<br>b = 15,642<br>c = 0.99<br>d = 160,186<br>k<sub>1</sub> = 12.83<br>k<sub>2</sub> = 12.85<br>R<sup>2</sup> = 0.997</td><td>a = 0.47<br>b = 5262<br>c = 0.71<br>d = 155,720<br>k<sub>1</sub> = 0.62<br>k<sub>2</sub> = 0.64<br>R<sup>2</sup> = 0.998</td><td>a = 0.68<br>b = 28,375<br>c = 0.93<br>d = 285,283<br>k<sub>1</sub> = 4.64<br>k<sub>2</sub> = 4.72<br>R<sup>2</sup> = 0.999</td></tr><tr><td>a: constant<br>b: constant<br>c: constant<br>d: constant<br>k<sub>1</sub>: apparent rate constant [min<sup>-1</sup>]<br>k<sub>2</sub>: apparent rate constant [min<sup>-1</sup>]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model-8<br>DoseResp<br>E(t<sub>cor</sub>) = a + <sup>c - a</sup>&frasl;<sub>1 + 10<sup>k(b-t<sub>cor</sub>)</sub></sub></td><td>a = -0.05<br>b = 35.57<br>c = 0.56<br>k = 0.04<br>R<sup>2</sup> = 0.998</td><td>a = 0.04<br>b = 13.72<br>c = 0.64<br>k = 0.35<br>R<sup>2</sup> = 0.998</td><td>a = 0.099<br>b = 0.93<br>c = 0.71<br>k = 5.68<br>R<sup>2</sup> = 0.997</td><td>a = 0.04<br>b = 19.31<br>c = 0.67<br>k = 0.27<br>R<sup>2</sup> = 0.998</td><td>a = 0.11<br>b = 2.66<br>c = 0.73<br>k = 2.09<br>R<sup>2</sup> = 0.999</td></tr><tr><td>a: lower threshold limit [V<sub>Ag/AgCl</sub>]<br>b: midpoint [min]<br>c: upper threshold limit [V<sub>Ag/AgCl</sub>]<br>k: constant [min<sup>-1</sup>]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Model-10<br>Proposed GRR: leveraging models 4-5-6<br>E(t<sub>cor</sub>) = <sup>a</sup>&frasl;<sub>1 + de<sup>-k(t<sub>cor}-b</sup></sub> + c</td><td>a = 0.61<br>b = 33.91<br>c = -0.05<br>d = 1.17<br>k = 0.09<br>R<sup>2</sup> = 0.995<br>a + c = 0.56</td><td>a = 0.60<br>b = 13.59<br>c = 0.04<br>d = 1.11<br>k = 0.82<br>R<sup>2</sup> = 0.998<br>a + c = 0.64</td><td>a = 0.61<br>b = 0.92<br>c = 0.10<br>d = 1.12<br>k = 13.09<br>R<sup>2</sup> = 0.998<br>a + c = 0.71</td><td>a = 0.63<br>b = 18.90<br>c = 0.04<br>d = 1.29<br>k = 0.62<br>R<sup>2</sup> = 0.998<br>a + c = 0.67</td><td>a = 0.62<br>b = 2.60<br>c = 0.11<br>d = 1.31<br>k = 4.76<br>R<sup>2</sup> = 0.998<br>a + c = 0.73</td></tr></tbody></table>
<figure><img src="image_17.png" /></figure>

properties of the output signal of the current GRR of silver by gold, that is, the time-dependent open-circuit potential responses. These mathematical functions have previously been used to describe the sigmoid responses of chemical reactions.[40,60] Basically, the DoseResp function was developed to model the dose-response relationship, more specifically the magnitude of an organism's response to exposure to chemical interactors.[60] When two mechanisms are at work, the BiDoseResp function better describes the experimental results, as it produces a biphasic curve. This assumption could be valid in our case based on the comparative fit in **Figure 6**. The BiDoseResp function provides a better adjusted R-squared value and a better fit at the beginning of the reaction for $t_{cor} \le 10$ min. From a mechanistic point of view, we can assume the presence of material heterogeneity, surface orientation, or preferential deposition sites, which could trigger competition between the replacement of silver atoms at the monometallic Ag sites and the bimetallic Ag<sub>1-x</sub>Au<sub>x</sub> sites after the initial stages of the GRR. After the first replacements, it is likely that subsequent gold deposits occur on existing gold rather than replacing silver positions in the Ag or Ag<sub>1-x</sub>Au<sub>x</sub> substructure due to a more favorable energy barrier, surface atom mobility, and the Kirkendall effect[5,10] driven by the difference in the diffusion coefficients of silver and gold atoms.

Considering the physical significance of the models examined in Figure 5 and Table 1, a reduced number was used to study the

OCP-time responsive indicator of the GRR of silver by gold for low to high inputs of Au(III) relative to Ag(0), quantified by the molar ratio $R_{GRR} = n(KAuCl_4)/n(Ag)$. The detailed equations of the fitting functions, the experimental data, and the resulting parameter values are summarized in **Table 2**. Figure 7b-d presents the different kinetic parameters for various molar ratios corresponding to two initial amounts of silver on the GCE substrate: $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.25, 1.25$, and $12.50$ for GCE-Ag-3cycles ($8 \ \mu g_{Ag} \ cm^{-2}$), and $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.62$ and $6.25$ for GCE-Ag-6cycles ($17 \ \mu g_{Ag} \ cm^{-2}$). Figure 7a specifically demonstrates that the proposed Model-6, inspired by Privman and Katz's model for enzyme cascades performing an enzymatic AND logic gate,[36] provides satisfactory results at various input reagent concentrations, particularly with regard to the offset of the OCP-time responses and the sharpness of the transition between the lag phase and steady state. Figure 7b shows that the apparent rate constant of the GRR between Ag(0) at the glassy carbon electrode and Au(III) in solution depends strongly on the initial population of silver atoms.

To simplify the analysis of the fitting data, we introduced the average apparent rate constant ($k_{average}$), similar to the inspiration model[47] to quantify the shift in the response curve and the sharpness of the transition between responses to low and large inputs. Specifically, the rate constant increases by a factor of 15 from $R_{GRR} = 1.25$ to $R_{GRR} = 12.50$ for a GCE-Ag that initially

<figure><img src="image_18.png" /><figcaption>**Figure 7.** a) Fitting experimental data (Model-6) of the time dependent OCP (after the offset time correction of the experiment data of Figure 4a) recorded on the course of GRR of silver by gold: the used model is the privman and Katz's model for enzymatic cascade realizing an AND logic gate.[36] b-d) Fitted kinetic parameters of the apparent rate constant [$k, \min^{-1}$] and the midpoint of the growth [$b, \min$]: b) Effect of $R_{GRR}$ on the apparent rate constant determined from Model-6, c) Effect of $R_{GRR}$ on the apparent rate constant determined from the most reliable models from Figure 5, and d) Effect of $R_{GRR}$ on the apparent rate constant determined (left y-axis, bar charts) and the midpoint of the growth (right y-axis, line + symbols) from Model-10.
Note: $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.25, 1.25$ and $12.50$ for GCE-Ag-3cycles ($8 \ \mu g_{Ag} \ cm^{-2}$); $R_{GRR} = n(KAuCl_4)/n(Ag) = 0.62$ and $6.25$ for GCE-Ag-6cycles ($17 \ \mu g_{Ag} \ cm^{-2}$); the extracted fitting values are reported in Table 2.</figcaption></figure>
<figure><img src="image_19.png" /></figure>

contains ≈8 µg<sub>Ag</sub> cm<sup>-2</sup>. For GCE-Ag with 17 µg<sub>Ag</sub> cm<sup>-2</sup>, a tenfold increase in the amount of Au(III) produced a much smaller increase in the rate constant (a factor of 7.4), likely due to the saturation of the initial reaction sites on the silver surface and the change in particle size when the quantity of silver increases. Analysis of the comparative data in Figure 7c shows that Model-8 (DoseResp function, Table 2) yields the lowest apparent rate constant among the tested models. Interestingly, three of the proposed models (Models 5, 6, and 10) produce similar values for the GRR rate constant. Among these models, we note that Model 10—derived from modifying sigmoidal kinetics to account for experimental constraints that are better modeled by autocatalytic network reactions and enzyme cascades—allows us to determine trends in the two kinetic parameters: the rate constant and the midpoint growth time (i.e., the point at which half of the saturation phase potential has been reached). Specifically, Figure 7c shows that the overall GRR duration can be reduced to t<sub>threshold</sub> = 2 × b = 110 s for R<sub>GRR</sub> = 12.50 with a loading of 8 µg<sub>Ag</sub> cm<sup>-2</sup>, and t<sub>threshold</sub> = 2 × b = 312 s for R<sub>GRR</sub> = 6.25 at GCE-Ag with a loading of 17 µg<sub>Ag</sub> cm<sup>-2</sup>. Overall, we provided for the first time, highly sought-after experimental information on the time required to stop the GRR for a specific structure-performance pair.

## 2.4. OCP(t)-Responsive Indicator of GRR: Model Significance

GRR is a redox reaction characterized by collective and synchronized movements of participating electrons, ions, and atoms to convert a metal structure into a new structure. Thus, the kinetics of GRR can be obtained by looking closely at the OCP(t) profiles associated with the Nernst equation at different time scales due to the fundamental change in the current amplitude corresponding to each half-cell.[17,61] To the best of our knowledge, no study of silver GRR by gold (the most studied system in the literature due to the different applications of Au–Ag materials[2–8,62]) has provided quantitative data on kinetics, for example, the rate constant. The values of these rate constant have been obtained using a variety of models derived from different mechanistic considerations. Consequently, the findings of this study represent a substantial advancement beyond the current state of the art. Moreover, the establishment of a temporal model quantifying the dependence between the GRR driving force and time would facilitate the optimization of experimental conditions, thereby enhancing the predictive capacity of synthesis methods over conventional trial-and-error approaches. It is evident that achieving this objective would necessitate the generation of substantial data across a range of parameters, including but not limited to various sizes of particles, temperatures, and concentrations of Au(III) (or Au(I)). A comprehensive array of studies is thus necessitated.

## 3. Conclusion

In this contribution, we propose, for the first time to our knowledge, numerical models for the experimentally observed

sigmoidal behavior of the OCP-time-responsive indicator of the GRR of silver by gold. We leverage phenomenological functions that describe key GRR features, such as the autocatalytic network, enzymatic cascades, and mass transport limitation. We favored this approach over the theory due to the complexity of developing and solving theoretical equations that should take into account such as the multiple electron transfer kinetics, two simultaneous electrochemical reactions—silver (0) oxidation and gold (III) reduction reaction—as well as associated mass transport limitations and atom diffusion. To this end, bare (surfactant-free) monodisperse AgNPs with an average particle size of 20.5 ± 0.3 nm were synthesized electrochemically on a flat glassy carbon electrode to study the GRR in a saline solution containing KAuCl₄ at 25 °C. Under these GRR conditions, it was demonstrated that gold (III) species principally exist as undissociated AuCl₄⁻ (with a negligible amount of free Au³⁺), so the involved redox couple is AuCl₄⁻/Au, with a standard redox potential of 1.00 V versus SHE.

The dynamics of this GRR in both the nonequilibrium and equilibrium domains result in a sigmoidal dependence of the OCP on time. The length of the induction period, also known as the lag phase, and the exponential phase are strongly influenced by the population of Au(III) ionic species near the silver surface. Of the ten phenomenological fitting functions tested on the OCP-time experimental data, the linear and exponential functions were ruled out. The logistic function and the sigmoidal kinetics of autocatalytic reactions, however, provide an acceptable description of the OCP dependence on time during the exponential and saturation phases. These functions have a threshold potential limit of 0.6–0.7 V versus Ag/AgCl and an apparent rate constant of k<sub>app</sub> = 0.1–7.0 min<sup>-1</sup>. However, these models poorly match with the lag phase and cannot explain the offset of OCP(t) responses as a function of Au(III) concentrations. This leads to significant discrepancies between the models and the experimental data in the early stages of GRR. A model inspired by Privman and Katz’s system for enzyme cascades performing an enzymatic AND logic gate (*J. Phys. Chem. B*, 2013, 117, 7559) provides satisfactory results at different input Au(III) concentrations, particularly with regard to the offset of OCP-time responses and the sharpness of the transition between the lag phase and steady state at low and high Au(III) concentrations.

Finally, a modified sigmoidal kinetic model proved to be better at providing the two expected kinetic parameters. The model was derived from autocatalytic network reactions and enzyme cascades and allow quantifying, for the first time, the two highly sought-after kinetic parameters of the apparent rate constant and the midpoint growth time. The midpoint growth time is the point at which half of the saturation phase potential has been reached. Specifically, for the first time, we have demonstrated that the dynamics of the galvanic replacement of silver by gold can be described by an apparent rate constant of k<sub>app</sub> = 0.1–13.1 min<sup>-1</sup> and the midpoint growth time of b<sub>GRR</sub> = 55–160 s at 25 °C for an initial silver nanoparticles loading of 8–17 µg<sub>Ag</sub> cm<sup>-2</sup>. These results open up the possibility of optimizing the GRR-derived synthesis of nanostructures by exploring other systems with different galvanic exchange ratios. In conclusion, quantifying the driving force behind the synchronous movement of electrons, atoms,
<figure><img src="image_20.png" /></figure>

and ions to form stable, porous structures in the silver-gold GRR system could inspire new advances in exploring innovative catalytic architectures, such as medium- to high-entropy materials, in which the effects of alloying, porosity, and strain could coexist, depending on the size of the metals entering or leaving the initial architecture.

# 4. Experimental Section

## Chemicals and Materials

Silver nitrate (AgNO₃, Premium, 99.9995% (metals basis), Alfer aesar), potassium nitrate (KNO₃, Sigma-Aldrich, 99.0% min), ammonia solution (NH₄OH, 30%, for analysis-ACS, Carlo Erba Reagents), potassium tetrachloroauric (III) (KAuCl₄·xH₂O, 99.9%, Alfer aesar), sodium chloride (NaCl, 100%, VWR), potassium hydroxide (KOH, 99.98% (trace metal basis), Acros Organics), D-(+)-glucose (99.5%, Sigma-Aldrich), hydrochloric acid (HCl, 37%, VWR International), nitric acid (HNO₃, 68%, VWR International), potassium permanganate (KMnO₄, ≥99%, Sigma-Aldrich), sulfuric acid (H₂SO₄, 96%, Thermo Scientific), and hydrogen peroxide solution (H₂O₂, 30%, Sigma-Aldrich), were used as received. The non-rotating stainless-steel rod connected to a Change Disk RDE PTFE tip (Glassy Carbon Disk Insert, 5.0 mm OD × 4.0 mm thick) was purchased from Pine Research Instrumentation (USA). Nitrogen (N₂, 4.5) gas was ultrapure (Air Liquide). Ultrapure water was provided by Milli-Q Millipore source (18 MΩ cm at 20 °C). All glassware used was cleaned from organic residues and metallic species by the methods of the potassium permanganate (immersion for at least 12 h in KMnO₄ (3 g L⁻¹) + H₂SO₄ (2.25 mol L⁻¹) followed by rinsing with acidified hydrogen peroxide solution (water + H₂SO₄ + H₂O₂ (1 vol. each)) and aqua regia (2 vol. HCl + 1 vol. HNO₃) as explained in Refs. [63–65].

## Galvanostatic Electrodeposition of AgNPs onto Glassy Carbon Electrode (GCE-Ag)

To synthesize monodisperse AgNPs on a glassy carbon electrode (GCE-Ag) for GRR applications, we adapted our previously developed pulsed electrodeposition methodology (either on a gas diffusion electrode (GDE) where the driving force was the applied current density,¹⁸,⁶⁶) or on a GCE where the driving force was the applied potential¹⁷). First, the GCE was mechanically polished (using 3, 1, and 0.05 µm alumina slurry) and the remaining traces of alumina were removed by sonication in a water bath for 10 min. Second, the electrodeposition was performed in single compartment glass cell at controlled temperature of (25 ± 2) °C in a conventional three-electrode setup. The working electrode was the polished GCE (geometric surface area of 0.196 cm²) mounted onto a non-rotating stainless-steel rod connected to a Change Disk RDE PTFE tip. The counter electrode was a glassy carbon plate (geometric surface area larger than 5 cm²). The reference electrode was a mercury-mercurous sulfate electrode (Hg|Hg₂SO₄|K₂SO₄ saturated, MSE, OrigamiLys ElectroChem SAS). To prevent competitive reduction of dissolved oxygen, which could become more significant in the presence of the first silver clusters, we have outgassed the electrodeposition medium consisting of 200 mM KNO₃ (for ionic conductivity), 60 µM AgNO₃ (as precursor of AgNPs), 8.2 mM NH₄OH (to regulate the kinetics of the Ag(+)-I) electrodeeducation into Ag(0) via Ag(NH₃)₂⁺ complex (detailed elsewhere¹⁷,¹⁸)). The optimized pulsed electrodeposition process, using a VIONIC potentiostat (Metrohm), consisted of an ON step at *j*<sub>ON</sub> = -0.5 mA cm<sup>-2</sup> for *t*<sub>ON</sub> = 5 s (nucleation-growth) followed by an OFF step at OCP for *t*<sub>OFF</sub> = 5 s (relaxation), the two sequences were repeated for

$N_{\text{cycles}} = 3$, which led to ca. 8 µgAg cm⁻² (ca. 80 nmolAg cm⁻²). A control synthesis was performed by doubling the number of cycles. Gentle agitation was maintained throughout the experiments.

## Real-Time Monitoring of GRR of Silver by Gold: OCP(t) Responsive Indicator of GRR

The real-time monitoring of GRR by OCP was performed in a temperature-controlled glass cell at (25 ± 2) °C. We used VIONIC potentiostat (Metrohm) to record OCP(t). An aqueous solution of 3.5 M NaCl (to avoid the formation of insoluble AgCl by favoring the formation of soluble species, AgCl<sub>n</sub>(n⁻¹)(n ≥ 3)¹⁷,¹⁸,⁴¹,⁴²) was used for GRR, outgassed (to prevent competitive reduction of dissolved oxygen) and maintained under gentle agitation throughout the experiment. Typically, the previously synthesized GCE-Ag electrode was immersed in 14.809 mL of this saline solution and recording of the OCP relative to a small reference electrode (Ag/AgCl) was started. After stabilization of OCP(t), 191 µL of an aqueous KAuCl₄ stock solution (0.02, 0.1, 1 mM, prepared in the above saline solution) was introduced at a flow rate of 750 µL min⁻¹ using a TYD01-01 Laboratory Syringe Pump (Lead Fluid (Baoding) Intelligent Equipment Manufacturing Co., Ltd.). Here, the total duration is 15 s, sufficiently short to have little impact on the GRR events profiles that we assumed to occur from minute-hours scales. Under these conditions and GCE-Ag($N_{\text{cycles}} = 3$), the molar ratio quantifying the population of silver (0) and gold (III) species at the early stage of the GRR is $R_{\text{GRR}} = n(\text{KAuCl}_4)/n(\text{Ag}) = 0.25, 1.25$ and 12.50 for initial concentrations of KAuCl₄ of 0.25, 1.27, and 12.73 µM, respectively (bearing in mind that one mole of Au(III) should react with three moles of Ag(0) in Ag-by-Au GRR events). It is worth noting that we did not study a much lower $R_{\text{GRR}}$ value, as our previous study¹⁷ has demonstrated that it would take tens of hours, or even days, to reach equilibrium. Therefore, the present values surrounding the theoretical one ($R_{\text{GRR}} = 0.33$) should allow examining the case of Au(III) deficiency and Au(III) excess, especially since the inclusion of GCE-Ag with twice the number of cycles introduces two new $R_{\text{GRR}}$ values, that is, 0.62 and 6.25. Once the equilibrium potential was reached, the reaction was quenched by quickly removing the electrode from the medium and rinsing it several times with MQ water before drying it at ambient temperature and pressure.

## Phenomenological Analytical Analysis of Time-Dependent OCP during GRR of Silver by Gold

With the aim of analyzing the dynamic response properties of the GRR of silver by gold, the experimental data on OCP(t) responses were fitted to mathematical models using Origin 2018 software. This approach was motivated by the complexity of developing and solving theoretical equations, which would require taking into account the kinetics of multiple electron transfer and mass transport phenomena (migration, diffusion, chemical potential gradient, convection, steric effect, Frumkin correction, and so on).⁵¹⁻⁵⁵ Even the recently developed Frumkin-Butler-Volmer size-modified Poisson-Nernst-Planck for CO₂ electrolysis⁵⁴ is not straightforward to implement for such a combined homogeneous and heterogeneous cascade system of GRR. So, we leveraged some classic and complex events, wherein the phenomenological fitting functions include, linear, autocatalytic network,³⁴ logistic, sigmoid function for RDE mass transport limited current,⁵⁸ Privman and Katz's model for enzymatic cascade realizing an AND logic gate,³⁶ Langevin, DoseResp, and Boltzmann. Through a comparative analysis of the fitting results, we finally proposed a modified sigmoidal kinetics from autocatalytic network reaction mechanism to take into account the offset of the OCP(t)-response
<figure><img src="image_21.png" /></figure>

curves and the sharpness of the transition from the start to the steady-state of GRR.

## Electrochemical and Physicochemical Characterization

The as-synthesized Ag nanoparticles were characterized by SEM (Hitachi S4800). After counting manually (ImageJ software), about 250 isolated nanoparticles, the particles size distribution was established by fitting the experimental data with LogNormal function.[46] After GRR, the electrodes were analyzed by SEM (such a system has been previously analyzed extensively[17,18]). An ICP-OES (Agilent 5110 VDV spectrometer) was employed for elemental analysis. Electrochemical and electrocatalytic studies before and after GRR were performed using the above potentiostat (VIONIC, Metrohm). The counter electrode was a glassy carbon plate. The reference electrode was Hg|HgO|KOH 0.1 M (MOE, RE-61AP Reference Electrode for alkaline solution, BAS Inc.) and the potentials were converted versus RHE according to the calibration relationship $E(V \text{ vs RHE}) = E(V \text{ vs MOE}) + \Delta E$, $\Delta E = 0.85 \text{ V at } 25 \text{ }^{\circ}\text{C in 0.1 M KOH, as reported in Ref. [17].}$ The model electrocatalytic reaction was the glucose oxidation as reported in Ref. [17].

## Acknowledgements

This work was funded by French National Research Agency (project MASTERS, grant no. ANR-22-CE43-0004), by I-SITE Excellence Program of the University of Montpellier under the Investissements France 2030 (project NANOGATE), and Institut Universitaire de France (IUF). Electron microscopy analyses were carried out at "PAC Chimie Balard", analytical facilities of Univ Montpellier, CNRS, ENSCM.

## Conflict of Interest

The authors declare no conflict of interest.

## Author Contributions

**Bonito Aristide Karamoko:** data curation (equal); formal analysis (equal); investigation (equal); methodology (equal); validation (equal); writing—review and editing (equal). **Yaovi Holade:** conceptualization (lead); data curation (lead); formal analysis (lead); funding acquisition (lead); investigation (lead); methodology (lead); project administration (lead); resources (lead); supervision (lead); validation (lead); writing original draft (lead); writing—review and editing (lead).

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Keywords:** autocatalytic reaction networks · electrochemical kinetics · electron transfer · galvanic replacement reaction · sigmoid function

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