Molecular Electrocatalysis in Confined Spaces: Analysis of the Cyclic Staircase and Cyclic Voltammetry Responses

Abstract

Molecular electrocatalytic processes in confined environments are becoming relevant processes with many applications in electrosynthesis, electroanalysis, and electrical energy generation and conversion. Nevertheless, the analysis of catalytic responses is mostly carried out with theoretical frameworks developed for semi-infinite linear diffusion conditions, which are not applicable for the adequate understanding of electrochemical processes in confined spaces. To fill the existing gap in the comprehension of these complex reactions, the analysis of a molecular catalytic process under finite diffusive conditions for cyclic staircase voltammetry (CSCV) and cyclic voltammetry (CV) techniques is presented in this work. The proposed model considers a finite diffusive field of thickness L under two configurations: bounded diffusion, where no mass renovation is allowed, and unbounded diffusion, where there is effective mass replenishment at L. Expressions for the current–potential responses under different particular cases have been obtained, leading to a kinetic zone diagram for limiting cases in terms of two key variables related to the thickness of the solution region and the catalytic rate constant. From the general expression of the current, it is observed that the electrochemical response of molecular electrocatalytic processes taking place in confined spaces is strongly dependent on the mass transport conditions. Thus, under bounded diffusion, a decrease of the catalytic current with L is observed, which is more pronounced when the diffusive field is narrower. On the other hand, unbounded conditions give rise to an enhancement of the catalytic current and, eventually, to the loss of the kinetic sensitivity of the response for small enough values of L. An experimental application of the theoretical results is performed for the conversion of isopropyl alcohol (IPA) to acetone mediated by the oxidation of 4-methoxy-2,2,6,6-tetra­methyl-1-piperi­dinyl­oxy (4-methoxy-TEMPO radical) at a glassy carbon electrode for both bound and unbounded configurations. The catalytic rate constant for this process has been obtained from the equations for the current, indicating that the accuracy of the result is strongly dependent on the correct understanding of the mass transport influences at play.

Introduction

Relevant real life chemical reactions are, in many cases, heterogeneous multistep processes. Electrochemistry is probably central to comprehend reactivity in these complex environments because electrochemical measurements occur at phase boundaries (i.e., the metal–electrolyte interface), and few other methods are as easily miniaturized as electrochemical techniques. − Nonetheless, in electrochemistry, achieving a deep understanding of the complexity of interfaces is a challenging task.

A key aspect in these studies is the role of mass transport and its influence on the electrochemical responses (that can be a chief factor in the efficiency and selectivity of the process under study). ,− This is of special importance in the case of electrochemistry taking place at confined spaces, that is, processes at spatially restricted domains, ,− for which there has been a growing interest in the last years.

Examples of recent studies dealing with confined spaces are the use of mesoporous membranes, , micrometric droplets, , the analysis of electrolysis under finite diffusion, , the use of screen-printed electrodes under thin-layer conditions, , or the possibility of combining several measurements beyond purely electrochemical ones with a control of the finite diffusion field. There are also previous studies devoted to the electrochemical responses of electrocatalytic films − and the response of a catalytic mechanism with rotating disc electrode voltammetry, where the influence of confinement is addressed through the assumption of finite diffusion.

Moreover, materials with intrinsic porosity, where mass transport occurs in confined spaces, are increasingly prominent in multiple applications. These materials include different types of carbonaceous matrices (carbon composites, carbon felts, etc.), metal–organic frameworks (MOFs), covalent organic frameworks (COFs), and polymers of intrinsic porosity (PIMs). ,−

In a general way, electrochemical processes at confined spaces are analyzed in terms of so-called thin-layer electrochemistry (TLE) or, in a more general way, in terms of finite diffusion (FD) mass transport. TLE studies were initially developed in the early 1960s, especially as an alternative approach for achieving electrolysis conditions at small volume cells. − In this type of experiment, the electrolyte solution is confined within a thin solution layer comparable in its thickness to that of the diffusion layer. This provides substantial area/volume improvement and the opportunity to overcome the limitations of conventional electrolysis cells; for example, the complete faradaic conversion of the species under study could be ideally achieved for short durations. ,,

Electrochemical reactions complicated with chemical steps, and especially molecular electrocatalysis (or molecular catalysis of electrochemical reactions , ), are among the most interesting systems under study due to their ever-growing applications. − Cyclic voltammetry (CV) is the routine technique for carrying out the analysis of these processes in order to determine the rate constant (or maximum turnover frequency, TOF) , of a given catalyst/substrate couple. ,,,, However, it should be noted that, in practice, this technique is not the one actually applied in most experimental situations. Rather, its digital analogue, cyclic staircase voltammetry (CSCV), is used, for which the potential waveform is a potential staircase instead of a linear ramp. , Depending on the potential resolution of the potential perturbation, the current–potential responses obtained in both techniques under transient conditions can be very different.

Determining the impact of finite diffusion on the response of charge transfer processes complicated with chemical reactions was the purpose of some of the original efforts of electrochemists interested in the thin-layer configuration, ,, owing to the greater simplicity of the theoretical treatment as well as the specific features of the electrochemical responses. However, the methodology discussed in these references was not followed in the literature until very recent contributions that put the focus on the influence of spatial confinement on electrochemical processes with chemical steps. Indeed, the analysis of molecular electrocatalytic reactions under confinement has not been extensively explored, and the methodologies deduced for CV under semi-infinite diffusion (SID) are typically applied to determine the catalytic rate constants of these reactions without a thorough analysis of their limits of applicability.

To fill the existing gap in the comprehension of electrocatalytic mechanisms at confined environments, our research group recently carried out the analysis of a molecular catalytic process under finite diffusion conditions for the case of the application of a constant potential. Here, this solution is extended to any multipotential pulse technique for which an arbitrary sequence of constant potentials is applied without restoring the initial conditions. This general solution has been applied to cyclic staircase voltammetry (CSCV) and cyclic voltammetry (CV) techniques, ,, both when the chemical step is reversible (bidirectional catalysis) , or irreversible. The proposed model considered a finite diffusive field of thickness L under two configurations: bounded finite diffusion, where no mass renovation is allowed at L, and therefore, a complete electrochemical conversion of the redox species can take place in a small volume; and unbounded finite diffusion, where there is effective mass replenishment at L. Both mass transport configurations seek to achieve a reasonable description of real systems, providing insightful information about the key factors that govern the response. From the general expression of the current–potential response obtained, a kinetic zone diagram for limiting cases in terms of two key variables has been built, in an analogous way to that previously reported by Savéant et al. for the catalytic mechanism under SID conditions. , Of special interest are the stationary limits, which allow for the simple determination of the catalytic rate constant under SID and FD conditions. These expressions clearly point out the fact that at confined spaces, the stationary current depends on the mass transport mode and deviates from those used for SID conditions.

An experimental application of the theoretical results is performed for the conversion of isopropyl alcohol (IPA) to acetone mediated by the oxidation of 4-methoxy-2,2,6,6-tetra­methyl-1-piperi­dinyl­oxy (4-methoxy-TEMPO radical) at a glassy carbon electrode in a basic aqueous medium (carbonate buffer 0.1 M, pH = 10) − for both bounded and unbounded configurations. Although the overall catalytic conversion mechanism with TEMPO derivatives and analogue species could be complex, , in the operating pH conditions, the overall reaction can be treated as a catalytic (EC′) mechanism. The catalytic rate constant for this process has been obtained from the equations for the current, with the accuracy of the result being strongly dependent on the correct understanding of the mass transport influences at play.

Theory

Let us consider the application of a time-dependent potential to the study of the catalytic process , in which a heterogeneous fast (Nernstian) charge transfer process taking place at the interface between an electrode and an electrolytic solution (step E) is followed by a chemical regeneration of species R/O through a chemical second-order process (step C′) with rate constants k ₁′ and k ₂′ for the forward and backward chemical reactions:

$$\begin{array}{ll}\mathrm{O}+{\mathrm{e}}^{-}\rightleftarrows \mathrm{R}& \mathrm{E}\\ \mathrm{R}(+\mathrm{P})\underset{{k_2}^{\prime }}{\overset{{k_1}^{\prime }}{\rightleftarrows }}\mathrm{O}(+\mathrm{S})& {\mathrm{C}}^{\prime }\end{array}$$ I

O and R are the oxidized and reduced species with initial concentrations c _O and c _R , and S and P are the substrate and product of the chemical step, being present in the solution at concentrations c _S and c _P .

In the following, it will be considered that there is a great excess of species S and P in solution compared to the electroactive species O and R, such that their concentrations will remain practically unchanged and equal to their bulk values throughout the course of the experiment. Under these conditions, two first-order pseudo-constants can be defined:

$$\begin{array}{l}{k}_1={k_1}^{\prime }{c}_{\mathrm{P}}^{*}\\ k_2={k_2}^{\prime }{c}_{\mathrm{S}}^{*}\end{array}\}$$ 1

Moreover, it will be assumed that the diffusion coefficients of species O and R are equal:

$$D_{\mathrm{O}}=D_{\mathrm{R}}=D$$ 2

The electrochemical response of process under SID conditions is well-known, and expressions for its current-time-potential dependence can be found in the literature for different multipotential or potential sweep electrochemical techniques such as CSCV or CV. ,,,,,

Due to the relevance of the study of electrochemical processes under confinement, in this work, it will be considered that the diffusion field is restricted to a given spatial region, limited by a characteristic length L. The geometry of the mass transport model can be seen in Scheme , in which the solution is confined within a region of thickness L between the electrode surface, which is located at x = 0, and a wall located at x = L. Two different assumptions were made for this wall. The bounded diffusion (Scheme A) considers null flux of any species at this point such that the electrochemical and chemical conversions take place only for the species in the inner space of the diffusive confined region, without any possibility of matter exchange (no mass renovation). This situation is usually called “thin-layer” behavior. The second possibility, called unbounded diffusion, is that at the “wall”, there exists some very effective regeneration of species O and R such that at the distance x = L, their concentrations coincide with their bulk values (Scheme B).

1. Catalytic Reaction Taking Place at Bounded (A) and Unbounded (B) Diffusive Fields in a Solution Region Located at 0 ≤ x ≤ L .

^a The electrode surface is located at x = 0. Vertical line located at x = L corresponds to an electroinactive wall for bounded diffusion (solid line in (A)) and to an electroactive wall or a distance with very effective mass transport for unbounded diffusion (dashed line in (B)).

To obtain an expression for the current response of process , a diffusive-kinetic differential equation system must be solved. Under purely linear diffusive mass transport (that is, no migration or convection is present), this system is given by

$$\begin{array}{l}\mathrm{\delta ̂}{c}_{\mathrm{R}}^{\mathrm{B}/\mathrm{U}}=k_2{c}_{\mathrm{O}}^{\mathrm{B/U}}-k_1{c}_{\mathrm{R}}^{\mathrm{B}/\mathrm{U}}\\ \mathrm{\delta ̂}{c}_{\mathrm{O}}^{\mathrm{B}/\mathrm{U}}=-k_2{c}_{\mathrm{O}}^{\mathrm{B/U}}+k_1{c}_{\mathrm{R}}^{\mathrm{B}/\mathrm{U}}\end{array}\}$$ 3

where δ̂ = ∂/∂t – D∂²/∂x ² is the so-called linear diffusion operator. , Superscripts B and U refer to the bounded and unbounded diffusive cases, respectively.

The boundary conditions for the equation system (eq ) are the following:

t = 0, 0 ≤ x ≤ L

$$c_i(x,t=0)=c_i^{*},i=\mathrm{O},\mathrm{R}$$ 4

t > 0, x = 0

$$\begin{array}{lr}{\left(\frac{\partial c_{\mathrm{O}}^{\mathrm{B}/\mathrm{U}}}{\partial x}\right)}_{x=0}=-{\left(\frac{\partial c_{\mathrm{R}}^{\mathrm{B}/\mathrm{U}}}{\partial x}\right)}_{x=0}& \mathrm{(5)}\\ c_{\mathrm{O}}^{\mathrm{s}}={\mathrm{e}}^{\eta }{c}_{\mathrm{R}}^{\mathrm{s}}& \mathrm{(6)}\end{array}$$

where c _O and c _R are the “surface” concentrations of species O and R, that is, the concentrations of these species at x = 0 (electrode surface). Moreover,

$$\eta =\frac{F}{RT}(E(t)-E^{0\prime })$$ 7

with E ⁰′ and E(t) being the formal potential of the O/R pair and the time-dependent potential being applied, respectively. Furthermore, for the unbounded model, the following condition is fulfilled:

t > 0, x = L

$$c_i^{\mathrm{U}}(x=L,t)=c_i^{*},i=\mathrm{O},\mathrm{R}$$ 8

while for the bounded model, the condition of zero flux for species O and R at x = L is imposed:

$${\left(\frac{\partial c_i^{\mathrm{B}}}{\partial x}\right)}_{x=L}=0,i=\mathrm{O},\mathrm{R}$$ 9

The current for process is obtained from the surface flux of species O or R:

$$\frac{I^{\mathrm{B}/\mathrm{U}}}{FA}=D{\left(\frac{\partial c_{\mathrm{R}}^{\mathrm{B}/\mathrm{U}}}{\partial x}\right)}_{x=0}=-D{\left(\frac{\partial c_{\mathrm{O}}^{\mathrm{B}/\mathrm{U}}}{\partial x}\right)}_{x=0}$$ 10

Cyclic Staircase Voltammetry (CSCV)

In cyclic staircase voltammetry, the potential waveform is a cyclic staircase, and all the potentials applied have the same duration, τ, with the current measured at the end of each applied potential. The discrete potential waveform for an initial anodic scan is given by the following expression:

$$\begin{array}{lll}{E}_j=E_{\mathrm{initial}}+(j-1)\mathrm{\Delta }E& \mathrm{for}& j=1,2,\mathrm{...},N/2\\ E_j=E_{\mathrm{final}}-(j-\frac{N}{2})\mathrm{\Delta }E& \mathrm{for}& j>N/2\end{array}\}$$ 11

where ΔE is the constant difference between two potential pulses denoted as the “pulse amplitude” and N is the total number of potential pulses applied in the cyclic scan.

The scan rate is defined as

$$v=\frac{|\mathrm{\Delta }E|}{\tau }$$ 12

As discussed below, the case of sweep techniques (like linear sweep voltammetry, LSV, and CV) is considered here as a limiting behavior of the potential staircase given in eq , obtained when |ΔE| → 0 for a given scan rate. Note that for most commercial potentiostats, the potential–time waveform applied for CV is indeed a staircase of low pulse amplitude (i.e., |ΔE| in the range 0.1–2.0 mV; see Schemes S1 and S2 in Section S1 of the Supporting Information, SI).

In order to obtain the expression for the current, it has been previously reported that, under Nernstian conditions, the superposition principle holds for a pseudo-first-order catalytic process, ,− such that the current corresponding to the pth potential applied of an arbitrary potential waveform, I _p , can be written as a sum of contributions due to the different potentials of the sequence, which are only effective from the moment when each particular potential is applied. Under SID conditions, the general expression for the current, which will be denoted as diffusive-kinetic (DK), takes the form ,

$$\frac{I_{\mathrm{DK,}p}^{\mathrm{SID}}}{FAD{\zeta }^{*}}=\frac{1}{(1+K)}\sum _{m=1}^p[Z_m^{\mathrm{cat}}{f}_{\mathrm{DK}}^{\mathrm{SID}}({\chi }_{m,p})]$$ 13

where

$$Z_m^{\mathrm{cat}}=\{\begin{array}{ll}\frac{K{\mathrm{e}}^{{\eta }_1}-1}{1+{\mathrm{e}}^{{\eta }_1}}& m=1\\ (1+K)(\frac{{\mathrm{e}}^{{\eta }_m}}{1+{\mathrm{e}}^{{\eta }_m}}-\frac{{\mathrm{e}}^{{\eta }_{m-1}}}{1+{\mathrm{e}}^{{\eta }_{m-1}}})& m>1\end{array}$$ 14

$${\eta }_m=\frac{F}{RT}(E_m-E^{0\prime }),m\ge 1$$ 15

$$K=\frac{k_2}{k_1}=\frac{c_{\mathrm{R}}^{*}}{c_{\mathrm{O}}^{*}}=\frac{1}{K_{\mathrm{eq}}}$$ 16

$${\zeta }^{*}=c_{\mathrm{O}}^{*}+c_{\mathrm{R}}^{*}$$ 17

$$f_{\mathrm{DK}}^{\mathrm{SID}}({\chi }_{m,p})=\sqrt{\frac{k}{D}}[\frac{{\mathrm{e}}^{-{\chi }_{m,p}}}{\sqrt{\pi {\chi }_{m,p}}}+\mathrm{erf}(\sqrt{{\chi }_{m,p}})]$$ 18

$${\chi }_{m,p}=k{t}_{m,p}$$ 19

$$k=k_1+k_2$$ 20

with the time elapsed between the measurement of mth and pth currents, t _m,p , being given by

$$t_{m,p}=(p-m+1)\tau$$ 21

From the combination of eqs and , it is possible to write the expression of χ _m,p as

$${\chi }_{m,p}={\kappa }_{\mathrm{SCV}}(p-m+1)$$ 22

with

$${\kappa }_{\mathrm{SCV}}=k\tau$$ 23

Function f _DK (χ _m,p ) in eqs and is the inverse of the so-called “reaction layer” (or reaction-diffusion layer), ,, defined as the zone adjacent to the electrode surface in which the chemical equilibrium is disturbed. The reaction layer, which depends on the nature of the diffusive field and on the rate of the catalytic process, is decisive in characterizing the observed behavior of the studied systems.

Under FD conditions, there exists a solution for the application of a constant potential. By applying the superposition principle, it is possible to extend the applicability of this solution to an arbitrary sequence of potential pulses. ,− Therefore, in the particular case of a cyclic staircase (eq ), the following expression for the diffusive-kinetic CSCV current is obtained:

$$\frac{I_{\mathrm{DK,}p}^{\mathrm{FD,B}/\mathrm{U}}}{FAD{\zeta }^{*}}=\frac{1}{(1+K)}\sum _{m=1}^p[Z_m^{\mathrm{cat}}{f}_{\mathrm{DK}}^{\mathrm{FD,B}/\mathrm{U}}({\chi }_{m,p},\lambda )]$$ 24

where

$$f_{\mathrm{DK}}^{\mathrm{FD,B}/\mathrm{U}}({\chi }_{m,p},\lambda )=f_{\mathrm{DK}}^{\mathrm{SID}}({\chi }_{m,p})+\left[\frac{2}{\sqrt{\pi }}\lambda \sum _{n=1}^{\infty }{\mathrm{\Xi }}_n^{\mathrm{B}/\mathrm{U}}n{\int }_0^{t_{m,p}}{f}_{\mathrm{DK}}^{\mathrm{SID}}({\chi }_u)\frac{{\mathrm{e}}^{-{\chi }_{t-u}}{\mathrm{e}}^{\frac{-n^2{\lambda }^2}{{\chi }_{t-u}}}}{(t_{m,p}-u)\sqrt{{\chi }_{t-u}}}\mathrm{d}u\right]$$ 25

$$\begin{array}{l}{\mathrm{\Xi }}_n^{\mathrm{B}}={(-1)}^n\\ {\mathrm{\Xi }}_n^{\mathrm{U}}=1\end{array}\}$$ 26

$$\lambda =\sqrt{\frac{k}{D}}L$$ 27

$$\begin{array}{l}{\chi }_u=ku\\ {\chi }_{t-u}=k(t_{m,p}-u)\end{array}\}$$ 28

with f _DK (χ _m,p ) and χ _m,p given by eqs and , respectively.

Note that the function f _DK (χ _m,p , λ) in eqs and contains a term given as an infinite power series. This series is strongly convergent and, in practice, a maximum number of 10–15 terms in the sum are necessary for achieving errors below 0.1%.

The catalytic CSCV response for FD conditions depends on two parameters, $\lambda =L\sqrt{k/D}$ (eq ) and κ _SCV = kτ (eq ), which modulate the purely kinetic influence through χ _m,p = κ _SCV (p – m + 1), eq ). In a general way, the expression of the current given in eq is a sum of terms that contains a potential dependent function (Z _m , eq ), and a function f _DK (χ _m,p , λ) (eq ) that accounts for the dependence of the response on diffusion (for the different mass transport fields) and catalysis.

Different simplified expressions of the function f _DK can be obtained in terms of log­(λ) and log­(κ _SCV), as shown in Table . From these expressions it is possible to build a kinetic “zone diagram” similar to those proposed by Savéant for a catalytic process under SID conditions, where the different regions correspond to particular expressions of function f _DK (χ _m,p , λ), valid for a given interval of values of log­(λ) and log­(κ _SCV). This diagram is given in Scheme (with the different notations explained in the footnote).

1. Expressions for Function f _DK (χ _m,p , λ) in Terms of log­(λ) and log­(κ _SCV) (See Eq and Refs , , and − )­ .

^aWe have assumed a reference time equal to the unity, such that χ = κ _SCV. KSS refers to a kinetic steady state current; DK to a diffusive-kinetic current; D to a purely diffusive current; and SS to a steady state current.

2. Kinetic Zone Diagram and Simulated CSCV Curves for a Catalytic Mechanism Taking Place in Confined Spaces, Obtained from Eq .

^a KSS refers to a kinetic steady state current; DK to a diffusive-kinetic current; D to a purely diffusive current; and SS to a steady state current. Solid and dashed lines in DK, KSS, and D zones correspond to bounded and unbounded responses, respectively. κ_SCV and λ are given by eqs and , respectively. The zone boundaries (solid red lines) have been calculated by considering maximum differences in the peak current/plateau current below 5%.

From Table it can be seen that when log­(κ _SCV) ≫ 1, function f _DK becomes only dependent on the variable λ (eq ), so its expression is independent of time (KSS and SS regions). Therefore, under stationary conditions, the general response given in eq leads to the following steady state current–potential responses for bounded and unbounded configurations (see sigmoidal responses in Scheme ):

$$\begin{array}{l}\frac{I_{\mathrm{KSS}}^{\mathrm{FD,B}}}{FAD{\zeta }^{*}}=\frac{K{\mathrm{e}}^{\eta }-1}{(1+{\mathrm{e}}^{\eta })(1+K)}\sqrt{\frac{k}{D}}\tanh (\lambda )\\ \frac{I_{\mathrm{KSS}}^{\mathrm{FD,U}}}{FAD{\zeta }^{*}}=\frac{K{\mathrm{e}}^{\eta }-1}{(1+{\mathrm{e}}^{\eta })(1+K)}\sqrt{\frac{k}{D}}\coth (\lambda )\end{array}\}$$ 29

For values of λ ≥ 2.7, functions tanh­(x) and coth­(x) become equal to the unity with an error below 1%, such that the expressions of the current given by eq become independent of λ and transform into that corresponding to the SID limit:

$$\frac{I_{\mathrm{KSS}}^{\mathrm{SID}}}{FAD{\zeta }^{*}}=\frac{I_{\mathrm{KSS}}^{\mathrm{FD,B/U}}}{FAD{\zeta }^{*}}(\lambda \ge 2.7)=\frac{K{\mathrm{e}}^{\eta }-1}{(1+{\mathrm{e}}^{\eta })(1+K)}\sqrt{\frac{k}{D}}$$ 30

On the other hand, for λ ≤ 0.174, the hyperbolic functions in eq can be approximated by the first term of a polynomic expansion (tanh­(x) ≅ x and coth­(x) ≅ 1/x, with errors below 1%). Under these conditions, the TL, B/U limits are obtained, and the expressions of the currents become

$$\begin{array}{l}{I}_{\mathrm{SS}}^{\mathrm{TL},\mathrm{B}}=\frac{K{\mathrm{e}}^{\eta }-1}{(1+{\mathrm{e}}^{\eta })(1+K)}FA{\zeta }^{*}kL\\ I_{\mathrm{SS}}^{\mathrm{TL},\mathrm{U}}=\frac{K{\mathrm{e}}^{\eta }-1}{(1+{\mathrm{e}}^{\eta })(1+K)}\frac{FAD{\zeta }^{*}}{L}\end{array}\}$$ 31

For log­(κ_SCV) ≪ 1, a non-catalytic (that its, purely diffusive) response is obtained. In agreement with Table , f _D , f _D , and f _D depend only on parameter $L/\sqrt{D{t}_{m,p}}$ = $\lambda /\sqrt{{\chi }_{m,p}}$ , with the expression of these functions being again dependent on the nature of the diffusive field. Under these conditions, the expression for the diffusive CSCV current is the following: ,,

$$\frac{I_{\mathrm{D,}p}^{\mathrm{FD,B}/\mathrm{U}}}{FAD{\zeta }^{*}}=\frac{1}{(1+K)}\sum _{m=1}^p[Z_m^{\mathrm{cat}}{f}_{\mathrm{D}}^{\mathrm{FD,B}/\mathrm{U}}(L/\sqrt{D{t}_{m,p}})]$$ 32

where t _m,p = (p – m + 1)τ (eq , see typical forms of the voltammograms in Scheme ).

For intermediate values of both log­(λ) and log­(κ _SCV), no simplified expressions can be used for function f _DK (χ _m,p , λ) (eq ). In this case, an interplay between the diffusive and catalytic components of the current is observed (DK, FD, and B/U). For values of log­(λ) ≥ 0.4, function f _DK (χ _m,p , λ) transforms into f _DK (χ _m,p ) given by eq (see Scheme ).

Cyclic Voltammetry (CV)

The expression of the current given by eq corresponds to a staircase potential of pulse amplitude |ΔE|, and it becomes identical to that obtained in CV when ΔE ≤ 0.01 mV. Nevertheless, it is possible to solve this problem directly for CV by assuming a linear cyclic potential ramp given by

$$\begin{array}{lll}E(t)=E_{\mathrm{initial}}+vt& \mathrm{for}& t\le t_{\mathrm{inv}}\\ E(t)=E_{\mathrm{final}}-vt& \mathrm{for}& t>t_{\mathrm{inv}}\end{array}\}$$ 33

where t _inv is the time at which the scan is inverted, and the scan rate is v = dE/dt.

By following the procedure detailed in Section S2, the following integral expression for the current corresponding to the application of a time-dependent potential when only species R is initially present is obtained and k ₁ = 0 (irreversible chemical step):

$$\frac{FA{c}_{\mathrm{R}}^{*}\sqrt{\pi D}}{1+{\mathrm{e}}^{-\eta }}={\int }_0^t{I}_{\mathrm{DK}}^{\mathrm{FD,B/U}}(u)[\frac{{\mathrm{e}}^{-k_2(t-u)}}{\sqrt{t-u}}+\frac{2L{\mathrm{e}}^{-k_2(t-u)}}{\sqrt{\pi D}}\sum _{n=1}^{\infty }{\mathrm{\Omega }}_{n}n{\int }_0^{t-u}\frac{{\mathrm{e}}^{-n^2{L}^2/D(t-u-w)}}{\sqrt{w{(t-u-w)}^3}}\mathrm{d}w]\mathrm{d}u$$ 34

where η is given in eq and

$$\begin{array}{l}{\mathrm{\Omega }}_n^{\mathrm{U}}={(-1)}^n\\ {\mathrm{\Omega }}_n^{\mathrm{B}}=1\end{array}\}$$ 35

The explicit expression of the CV current and the numerical procedure used to calculate it are discussed in Section S2 (see eqs S48 and S49).

As in the CSCV case previously discussed, different particular expressions of the response given in eq can be obtained in terms of λ and κ _CV. Note that for CV, the latter dimensionless parameter is defined as

$${\kappa }_{\mathrm{CV}}=\frac{k_2}{a}$$ 36

$$a=\frac{Fv}{RT}$$ 37

The stationary expressions of the current, corresponding to high values of log­(κ _CV), are given by eqs – since, due to its time-independent nature, they are the same for any potential (and current) controlled technique.

When λ is large enough (L → ∞), the SID limit is obtained, and eq takes the form ,

$$\frac{FA{c}_{\mathrm{R}}^{*}\sqrt{\pi D}}{1+{\mathrm{e}}^{-\eta }}={\int }_0^t{I}_{\mathrm{DK}}^{\mathrm{SID}}(u)\frac{{\mathrm{e}}^{-k(t-u)}}{\sqrt{t-u}}\mathrm{d}u$$ 38

which is the response equivalent to eq in the CV limit.

On the other hand, for very small values of the catalytic rate constants, an expression for the current in the absence of chemical reaction (that is, under finite diffusion conditions D, FD) can be given in the integral form ,

$$\frac{I_{\mathrm{D}}^{\mathrm{FD,B/U}}}{FAD}L=\frac{c_{\mathrm{R}}^{*}}{1+{\mathrm{e}}^{-\eta (t=0)}}{\theta }_{\alpha }(0|Dt/L^2)+{\int }_0^{Dt/L^2}{\theta }_{\alpha }(0|Dt/L^2-u)\frac{\mathrm{d}{c}_{\mathrm{O}}^{\mathrm{s}}(u)}{\mathrm{d}u}\mathrm{d}u$$ 39

where θ _α refers to the theta functions, with α = 2 for the bounded case and α = 3 for the unbounded one:

$${\theta }_2(0|x)=2\sum _{j=0}^{\infty }{\mathrm{e}}^{-{\left(\frac{j+1}{2}\right)}^2\pi x}$$ 40

$${\theta }_3(0|x)=\frac{1}{\sqrt{\pi x}}[1+2\sum _{j=1}^{\infty }{\mathrm{e}}^{-j^{2}x}]$$ 41

A zone diagram similar to that shown in Scheme can be found in ref , corresponding to the CV response of an electrocatalytic film, which can be considered as an analogous case to that of bounded diffusion for an irreversible chemical step.

Results and Discussion

Theoretical Results

The CSCV and CV responses have been calculated by using eqs and , respectively, which assume that only linear diffusion takes place (eq ). To check the accuracy of the results and determine the extent of the impact of radial diffusion on the current obtained under these conditions, COMSOL Multiphysics v6.3 software has been used to perform different simulations of CV curves under transient and stationary conditions (see Figures S3–S5 in Section S3). In agreement with the obtained results, for the conditions under study here, radial diffusion is not relevant.

For the sake of simplicity, the current has been denoted with the technique used (CSCV or CV) and the diffusive mode considered (bounded or unbounded).

The evolution of the current–potential curves from a cyclic staircase to a cyclic linear ramp waveform depends on the pulse amplitude of the staircase, ΔE. As stated above, for amplitudes below 0.01 mV, the results obtained in both techniques can be considered as equivalent (see also Section S1). To verify this, in Figure the response corresponding to a catalytic process in CSCV has been plotted, calculated from eq under bounded (panel a) and unbounded (panel b) conditions for ΔE = 10, 1, 0.1, and 0.01 mV. The true CV response, calculated from eq , has also been included for comparison. These curves have been calculated for values of log­(λ) = 0 and log­(κ _SCV) = log­(κ _CV) = −0.3, for which the CSCV and CV curves show a DK, FD B/U behavior, in agreement with the kinetic zone diagram shown in Scheme .

1.

Theoretical CSCV curves calculated from eq for an anodic catalytic process with c _O = 0 and k ₁ = 0 (irreversible chemical step). (a) Bounded conditions and (b) unbounded conditions. Values of the pulse amplitude ΔE (in mV) are 10 (dashed blue line), 1.0 (dashed red line), 0.1 (dashed green line), and 0.01 (dashed pink line). c _R = 1.0 mM, D = 10^–5 cm² s^–1, A = 0.0707 cm². log­(λ) = 0, log­(κ _SCV) = log­(κ _CV) = −0.3, T = 298 K. The true CV response (black circles), calculated from eq , has also been included.

As can be seen, in the case of bounded conditions (Figure a), the maximum differences between the CV and CSCV responses correspond to the peak currents, being below 5% for ΔE ≤ 0.1 mV. For the unbounded case, the CSCV-CV coincidence is higher since the response is almost stationary. Thus, maximum differences below 5% are obtained for ΔE ≤ 1.0 mV under these conditions. In all of the cases shown in Figure , stationary current plateaus are observed for positive enough potentials. These plateaus are logically coincident for all the pulse amplitudes used since stationary currents are independent of the particular waveform employed, in line with eq , valid for KSS, FD B/U conditions. However, it is necessary to emphasize that one must be sure of the potential resolution of the potential waveform used since, under transient conditions, significant errors may be made in the estimation of kinetic parameters from the measurement of peak currents for pulse amplitudes above 1.0 mV (as an example, see Figure S6 in Section S4).

As stated above (see Table and kinetic zone diagram in Scheme ), the shape and main features of the corresponding catalytic CV and CSCV responses depend on parameters λ and κ _SCV/κ _CV. To clarify this, Figures and show the CV curves calculated from eq for both bounded (Figure ) and unbounded (Figure ) conditions for fixed values of log­(κ _CV) and different values of log­(λ). In practice, these CV curves point out the effect of a decrease of distance L for a fixed value of the dimensionless catalytic rate constant, κ _CV, when we move from SID to TL diffusive fields (from right to left in the kinetic zone diagram in Scheme ).

2.

Theoretical CV responses, calculated from eq under bounded conditions, for an anodic catalytic process with c _O = 0 and k ₁ = 0 (irreversible chemical step). The values of log­(λ) and log­(κ _CV) are indicated in (a–c). c _R = 1.0 mM, D = 10^–5 cm² s^–1, A = 0.0707 cm², T = 298 K.

3.

(a, b) Theoretical CV responses, calculated from eq under unbounded conditions, for an anodic catalytic process with c _O = 0 and k ₁ = 0 (irreversible chemical step). The values of log­(λ) and log­(κ _CV) are indicated in (a) and (b). c _R = 1.0 mM, D = 10^–5 cm² s^–1, A = 0.0707 cm², T = 298 K. (c) Evolution of the ratio I _KSS /I _SS with log­(λ), calculated from eq .

In the case of bounded diffusion (Figure ), the decrease in log­(λ) affects both the shape and magnitude of the CV response. For log­(κ _CV) = −1 (Figure a), the response is scarcely catalytic, and the voltammogram evolves from a distorted duck shape obtained under DK, SID conditions (see Scheme ) to a distorted quasi-symmetrical peak shape obtained under a TL field for log­(λ) < −1 (TL, B regions in Scheme ). When log­(κ _CV) increases, the catalysis effect on the current shape and magnitude is more evident, leading to quasi-sigmoidal (for log­(κ _CV) = −0.3, Figure b) or sigmoidal (for log­(κ _CV) = 0, Figure c) CV curves, with the current reaching a constant plateau at positive enough potentials, the value of which decreases with L. When log­(λ) ≥ 1, SID conditions are reached for any value of log­(κ _CV), and the current–potential response becomes independent of λ (see red lines in Figures a–c for curves with log­(λ) = 1 and 2).

For the unbounded CV curves shown in Figure a,b it can be seen that the influence of λ is the opposite to that observed in Figure , that is, the catalytic current increases as λ decreases since, for small values of the distance L (log­(λ) < 0.4), the current becomes dependent on coth­(λ) in the first place (eq ) and on the inverse of L for even smaller values of λ (eq ). Note also that the TL limit for small values of log­(λ) and log­(κ _CV) (that is, for scarcely catalytic responses, Figure a) is sigmoidal and therefore a symmetrical peak-shaped response, as that observed for the bounded case (D, TL, B region in Scheme ) is never obtained under these conditions.

The current amplification in the unbounded CV responses is always greater as the distance L is shorter (see curves with log­(λ) ≤ −0.3 in Figure a,b). Nevertheless, for small enough values of λ, the current becomes independent of the catalytic step (see eq for I _SS and Figure c) in a similar way to what happens when microelectrodes are used. , The reason for this fact is the same in both situations, that is, the diffusion layer becomes much smaller than the reaction layer such that the catalytic influence on the response vanishes.

This evolution can be seen in Figure c, which shows the ratio between the unbounded stationary currents I _KSS and I _SS in terms of log­(λ), calculated from eqs and and given by

$$\frac{I_{\mathrm{KSS}}^{\mathrm{FD},\mathrm{U}}}{I_{\mathrm{SS}}^{\mathrm{TL},\mathrm{U}}}=\lambda \coth (\lambda )$$ 42

In agreement with Figure c and eq , the TL, U limit for which the sensitivity of the current to the catalysis is lost is reached for values of log­(λ) < −1.0.

Additional information related to catalytic behavior can be obtained by fixing the value of log­(λ) and changing the value of log­(κ _CV). This influence can be seen in Figure , corresponding to bounded conditions for a typical confined space (log­(λ) = −1). The curves in Figure correspond to the analysis of the CV scan rate for fixed values of L and k. In practice, the time window of the experiment is changed from a pure D, TL diffusive behavior (for log­(κ _CV) = −2, region D, TL, B in Scheme ) to a pure stationary response for log­(κ _CV) ≥ 0 (region SS, TL, B in Scheme ). It can be seen in this figure that, under these conditions, there is a striking behavior since the current decreases as log­(κ _CV) increases (i.e., as the time window increases, see curves for log­(κ _CV) = −2, 1, and 0). This happens for FD or TL diffusion, where a complete faradaic conversion of the redox species takes place. Thus, when log­(κ _CV) goes from −2 to 0 (blue and green curves in Figure ), the time window is an order of magnitude higher, which implies a higher catalytic influence but also a stronger depletion of species R. For small enough values of log­(κ _CV), the amount of R that disappears in the solution is not completely recovered through the chemical step, thus leading to a smaller current. Only for values of log­(κ _CV) above zero, the chemical reaction not only is able to completely recover the amount of species R that has disappeared in faradaic terms but also results in an increase of the response above the current due to a purely diffusive contribution (see red and black lines in Figure ).

4.

Theoretical CV responses, calculated from eq under bounded conditions, for an anodic catalytic process with c _O = 0 and k ₁ = 0 (irreversible chemical step), for log­(λ) = −1. The values of log­(κ _CV) are indicated in the curves. c _R = 1.0 mM, D = 10^–5 cm² s^–1, A = 0.0707 cm², T = 298 K.

This is a clear example of how the interplay between the operating mass transport mode and the chemical kinetics defines the voltammetric response. An experimental verification of this behavior for catalytic responses in confined spaces, where the finite diffusion influence leads to peak-shaped responses, is that corresponding to catalytic processes taking place at conductive nanopipettes.

The effect of bidirectional catalysis under FD conditions (that is, the situation corresponding to a molecular catalyst able to perform the direct and reverse conversion of the substrates), , is analyzed in Figure . In practice, this situation is equivalent to the existence of a chemical equilibrium previous to the electrochemical step (i.e., K ≠ 0 in eq for CSCV, with ΔE = 0.01 mV and therefore equivalent to CV). In Figure , we have plotted the theoretical CSCV responses under bounded (a) and unbounded (b) conditions, calculated for log­(λ) = −0.5 and log­(κ _CV) = −1.0. Different values of the inverse of the equilibrium constant K = (k ₂/k ₁) = 1/K _eq (eq ) have been used by fixing the value of k ₂ and changing k ₁ for achieving a given K. The direct scan corresponds to the oxidation of species R.

5.

Theoretical CSCV responses, calculated from eq under bounded (a) and unbounded (b) conditions for an anodic catalytic process. log­(λ) = −0.5 and log­(κ _CV) = −1.0. The values of the inverse of the equilibrium constant K = k ₂/k ₁ are indicated in the curves. ΔE = 0.01 mV, v = 50 mV s^–1, c _R = 1.0 mM, D = 10^–5 cm² s^–1, A = 0.0707 cm², T = 298 K.

As can be seen in these curves, the existence of a previous equilibrium gives rise to the appearance of a cathodic “tail” in the CSCV curves at the initial negative potentials (see the current values below −0.25 V) and to a shift of the response through the current axis toward more cathodic values as K decreases (that is, as k ₁ and therefore the rate of chemical formation of species O increase). Two clear stationary current plateaus can be obtained under these conditions at very negative and positive potentials, the values of which are

$$\begin{array}{l}{I}_{\mathrm{DK},E_p\gg E^{0\prime }}^{\mathrm{FD},\mathrm{B}}=FAD{\zeta }^{*}\frac{K}{1+K}\sqrt{\frac{k}{D}}\tanh (\lambda )\\ I_{\mathrm{DK},E_p\ll E^{0\prime }}^{\mathrm{FD},\mathrm{B}}=-FAD{\zeta }^{*}\frac{1}{1+K}\sqrt{\frac{k}{D}}\tanh (\lambda )\end{array}\}$$ 43

for the bounded case (Figure a) and

$$\begin{array}{l}{I}_{\mathrm{DK},E_p\gg E^{0\prime }}^{\mathrm{FD},\mathrm{U}}=FAD{\zeta }^{*}\frac{K}{1+K}\sqrt{\frac{k}{D}}\coth (\lambda )\\ I_{\mathrm{DK},E_p\ll E^{0\prime }}^{\mathrm{FD},\mathrm{U}}=-FAD{\zeta }^{*}\frac{1}{1+K}\sqrt{\frac{k}{D}}\coth (\lambda )\end{array}\}$$ 44

for the unbounded one (Figure b).

The ratio between the anodic and cathodic current plateaus in absolute value is equal to the value of K for both mass transport modes. Indeed, this ratio is independent of the mass transport mode. ,

The effect of bidirectionality under steady state conditions can be seen in Figure S7 of Section S4 for bounded and unbounded configurations. In both cases, a sigmoidal response is obtained, whose location along the current axis depends on the value of K.

Experimental Results

There have been recent attempts of analyzing the thin-layer behavior of molecular electrocatalytic reaction schemes, see for example refs , , and . However, the focus of these works has been put on electrosynthetic considerations, such that the TL cell has been specifically designed to this end, and no influence of the size of the diffusive field on the response has been considered.

To show an example of the influence of confinement in the CSCV and CV catalytic responses of a redox probe, a “proof of concept” experiment was carried out. An FD three-electrode single compartment cell with adaptable length for the diffusive field has been used. The working electrode (a glassy carbon disc electrode of 1.5 mm radius) is fixed to a piezoelectric device that moves up and down with a micrometric resolution, whereas the cell is a square-based 20 × 20 × 8 mm piece made of PVC with a total volume of 2.0 mL (see Figures S8 and S9 in Section S5 for photographs of the experimental setup). As a catalyst, 4-methoxy-2,2,6,6-tetra­methyl-1-piperi­dinyl­oxy (4MT) has been chosen, being the substrate for the catalytic conversion of isopropyl alcohol (IPA). 4MT can be oxidized in an aqueous mild medium (0.1 M carbonate buffer, pH = 10), giving rise to a practically reversible (Nernstian) one-electron process. −

The general reaction mechanism is listed in Scheme . The active form of the catalyst is the oxidized R-TEMPO⁺, which in the presence of a secondary alcohol is able to oxidize to the corresponding ketone derivative, being regenerated to the R-TEMPO-OH form, which at the working potential and pH conditions evolves immediately to the R-TEMPO form. In practice, although the overall number of transferred electrons is two, the voltammetric wave is defined at the potential interval of electrochemical stability of R-TEMPO and R-TEMPO⁺, such that the process could be considered as an irreversible EC′ process (k ₁ = 0 in reaction ). ,, The catalytic efficiency of the process is strongly dependent on the pH of the electrolytic media. Typical pH values in the range of 9–11 are used for achieving adequate alcohol conversion rates. Different functionalized TEMPO moieties have been analyzed in the literature, and values for the pseudo-first-order rate constant of the catalytic conversion (or maximum TOF) of a great number of alcohols have been reported. −

3. Reaction Scheme for the Catalytic Conversion of IPA to Acetone Mediated by the Oxidation of 4MT − .

Here, the 4MT/IPA system will be studied for a pH 10 solution with an IPA concentration two orders of magnitude greater than that of 4MT to assure pseudo-first-order conditions. The current–potential responses have been obtained under both SID and bounded/unbounded FD conditions by using distances of the diffusion field in the range 10–100 μm, with the aim of analyzing the influence of spatial restrictions on the catalytic current in both configurations. The CSCV technique with a typical pulse amplitude of ΔE = 0.5 mV and a scan rate of 0.025 V s^–1 has been selected. Under these conditions, the current range of the CSCV responses is 10–12 μA, such that no ohmic effects have been observed.

Bounded Configuration

To achieve this configuration, the working electrode was lowered to the electroinactive bottom of the electrolytic cell using a piezoelectric system, allowing for a change in the length of the diffusive domain (L) with no possibility of mass renovation in the time scale of the experiment. Under these conditions, the value of L is unknown and must be determined. While the piezoelectric device can establish a reference distance, there is inherent uncertainty in the actual value of this distance, as the plastic material of the cell is not perfectly rigid. It is important to note that mass transport is assumed to occur solely through linear diffusion within a finite volume (Section S3). Firstly, from CSCV curves of a 0.75 mM 4MT solution recorded at different scan rates in the range 25–100 mV s^–1, a formal potential E ⁰′ = 0.712 V vs Ag/AgCl, KCl sat. has been obtained from the average of the forward and reverse peak potentials, and a diffusion coefficient D _4MT = 5.4×10^–6 cm² s^–1 has been determined from a Randles–Ševčík plot (see refs and and Figure S10).

Figure shows the CSCV responses under SID and bounded conditions of a 0.75 mM 4MT solution with an IPA concentration of 0.107 M (pH = 10 and 0.1 M carbonate buffer). In this electrolytic medium, the formal potential of 4MT changes to a value of E ⁰′ = 0.663 V vs Ag/AgCl, KCl sat.

6.

(a) Experimental CSCV curves for 0.75 mM 4MT in an aqueous solution (carbonate buffer 0.1 M, pH = 10) at a GC electrode of radius 1.5 mm for a bounded configuration in the presence of c _IPA = 0.107 M, ΔE = 0.5 mV, scan rate v = 0.025 V s^–1, T = 298 K. Inset: I ^CSCV,B/I ^CSCV,SID–E curves obtained from (a) for the potential range 0.70–0.90 V. (b) Comparison between experimental (solid lines) and theoretical (dashed lines) CSCV curves calculated from eq for the values of L (in μm) indicated in the curves with D _4MT = 5.4×10^–6 cm² s^–1. Dotted vertical black line marks the position of the formal potential of 4MT/4MT⁺ couple in this medium (E ⁰′ = 0.663 V vs Ag/AgCl, KCl, sat.).

As can be seen in Figure a, the CSCV responses are sigmoidal, located around the formal potential of 4MT (dotted vertical line), with a clear current plateau above 0.75 V for all of the values of L. Nevertheless, direct and reverse scans are not superimposable, a fact that indicates that the current is not completely stationary and points to a DK, FD, B or DK, SID behavior in line with Scheme . Under these conditions, it is clear that, in order to obtain the value of the catalytic rate constant, simple expressions for the stationary CSCV current as those corresponding to KSS or SS behavior (eqs or ) cannot be applied to the complete response but only to the part of the CSCV curve that is truly stationary, that is, to the current plateau located at potentials above 0.75 V. Therefore, by applying the expression of the stationary SID current to the current plateau (eq with c _O = 0 and k ₁ = 0, irreversible chemical step, and potentials that fulfil e ^η → ∞), the following is obtained: ,,

$$I_{\mathrm{KSS}}^{\mathrm{SID},\mathrm{plateau}}=2FA{c}_{\mathrm{4MT}}^{*}\sqrt{k_2{D}_{4\mathrm{MT}}}$$ 45

where k ₂ = k ₂′c_IPA is the pseudo-first-order rate constant (or maximum TOF) and k ₂′ is the second-order rate constant. From the application of eq , the values k ₂ = 0.2 s^–1 and k ₂′ = 1.87 M^–1 s^–1 are immediately obtained.

The SID currents are logically independent of the location of the working electrode (that is, the value of L, see eq ) such that the two assays carried out under SID conditions are coincident (see black and red lines in Figure a). When the electrode approaches the bottom of the cell and the FD effects appear, the current decreases as L decreases (with the current plateau obtained for the smallest distance being 30% of the SID value), and the separation between the forward and reverse currents becomes higher. These results indicate that the catalytic 4MT + IPA response is indeed in the DK, FD, B region (see Scheme ). Moreover, as L (and therefore λ) decreases, the CSCV curves become less stationary since the influence of the diffusive mass transport moves the response away from a purely kinetic stationary behavior (compare these results with the theoretical curves in Figure b,c).

Since the value of the rate constant for all the CSCV curves in Figure a,b is the same, only the value of distance L is unknown for each of the different FD responses. To determine these values, it should be taken into account that, under kinetic steady state conditions (KSS, FD and SID regions in Scheme ), the ratio I _KSS /I _KSS is given by (see eqs and )­

$$\frac{I_{\mathrm{KSS}}^{\mathrm{FD},\mathrm{B}}}{I_{\mathrm{KSS}}^{\mathrm{SID}}}=\tanh (\lambda )$$ 46

Therefore, the value of λ = arctanh­(I _KSS /I _KSS ) can be obtained directly without requiring any additional data. As stated above, the experimental responses in Figure a,b are not completely stationary, and therefore, eq should be applied only for the stationary part of the response, that is, the plateau region. By restricting the ratio I _KSS /I _KSS to the potential region 0.70–0.90 V, where the current plateau is observed, the horizontal lines shown in the inset of Figure a are obtained. Since the value of the rate constant of the catalytic oxidation of IPA is directly obtained for the SID plateau by applying eq , it is possible to obtain the values of L once the diffusion coefficient is known. The corresponding L values are shown in Figure a. By using these values, theoretical CSCV curves have been calculated from eq for the bounded case and compared with the experimental responses. In agreement with Figure b, the degree of coincidence is very good for the different distances in the range of 8–50 μm.

Unbounded Configuration

This configuration implies that at a given finite distance, the concentration of electroactive species is equal to its bulk value such that a very effective mass renovation takes place. To achieve this condition, the strategy followed has been to corrode a metal substrate located at x = L with cationic 4MT⁺ generated at the working electrode acting as the redox mediator. This is a usual approach in corrosion studies with scanning electrochemical microscopy (SECM, see for example ref ), although here it will be used with a more noble substrate such as gold.

For obtaining the response under unbounded FD conditions, a rectangular 25 × 5 mm piece of glass coated with a few layers of chemically deposited gold (10 nm thickness) has been placed at the bottom of the cell, centered at the working electrode position. In addition to the presence of a carbonate buffer to fix the pH value, 0.1 M KCl was added to the solution. The chloride media and the low amount of gold also make its oxidation process easier, in the same way to that observed for the stripping of gold nanoparticles. Thus, the 4MT⁺ voltammetrically electrogenerated at the GC electrode is not only involved in the catalytic conversion of IPA to acetone; since the thickness of the reaction layer for the catalytic process given in Scheme is large, 4MT⁺ also diffuses away from the working electrode surface and reaches the gold surface, causing the oxidation of gold atoms at open circuit potential (OCP) and thus being reduced to its original 4MT state, which moves again toward the GC. The regeneration process is so fast that, in practice, all the 4MT⁺ that reaches the gold surface is immediately reduced to 4MT. The OCP of gold at this medium is 0.532 V, such that 4MT is able to oxidize the gold atoms (in this medium, the formal potential of the 4MT/4MT⁺ couple is E ⁰′ = 0.684 V vs Ag/AgCl, KCl sat., see below).

Figure shows the CSCV curves obtained for the catalytic process given in Scheme for the unbounded configuration for the four values of L indicated on the curves. The case corresponding to SID conditions (black line) has been included for comparison. A smaller amount of 4MT (0.2325 mM) and a higher IPA concentration (c _IPA = 0.161M) were used in this case. As in the bounded case, the values of the diffusion coefficient and the formal potential have been determined. Small differences have been observed in this case, probably due to the presence of KCl in the electrolytic medium. Thus, the values E ⁰′ = 0.684 V vs Ag/AgCl, KCl sat. and D = 3.3×10^–6 cm² s^–1 have been obtained from the analysis of the CSCV curves under SID conditions. Moreover, by applying eq to the SID current plateau (black line in Figure ), a value of the pseudo-first-order rate constant k ₂ = k ₂′c _IPA = 0.3 s^–1 is directly obtained, leading to a second-order rate constant of k ₂′ = 1.86 M^–1 s^–1.

7.

Solid lines: experimental CSCV curves for 0.2325 mM 4MT in an aqueous solution (carbonate buffer 0.1 M, pH = 10, KCl 1.0 M) at a GC electrode of radius 1.5 mm for an unbounded configuration in the presence of c _IPA = 0.161 M, ΔE = 0.5 mV, scan rate v = 0.025 V s^–1, T = 298 K. Symbols: theoretical CSCV curves calculated from eq for the values of L (in μm) indicated in the curves with D _4MT = 3.3 × 10^–6 cm² s^–1. Inset: I ^CSCV,U/I ^CSCV,SID–E curves obtained from the experimental CSCV curves for the potential range 0.75–0.85 V. Dotted vertical line marks the position of the formal potential of 4MT/4MT⁺ couple in this medium (E ⁰′ = 0.684 V vs Ag/AgCl, KCl, sat.).

In the bounded case, the response is almost stationary, with a clear plateau for all the distances used but with no superimposable currents for the direct and reverse scans. Moreover, in an opposite trend to that observed for the bounded case, the catalytic current increases as the electrode approaches the gold surface (such that the current is registered at smaller values of L). The thinner diffusion field leads to a mass transport enhanced sensitivity of the catalytic response, especially for shorter distances (see the green and red curves in Figure ).

Under steady state conditions, the ratio I _KSS /I _KSS is given by (see eqs and )­

$$\frac{I_{\mathrm{KSS}}^{\mathrm{FD},\mathrm{U}}}{I_{\mathrm{KSS}}^{\mathrm{SID}}}=\coth (\lambda )$$ 47

Therefore, the value of λ = arccoth­(I _KSS /I _KSS ) can be obtained directly without requiring any additional data. Since the response is not completely stationary, the application of eq is restricted to the plateau zone as in the bounded case, that is, to the region 0.75–0.85 V, for which the lines shown in the inset of Figure have been obtained (which lead to the corresponding λ values from the application of eq ). By using the value of the rate constant of the catalytic oxidation of IPA directly obtained for the SID plateau, we deduce the values of L shown in Figure . It is important to highlight the great sensitivity of the CSCV responses obtained for this configuration since the different responses in the figure correspond to a solution region of 10 μm thickness. The degree of coincidence between the theoretical (symbols) and experimental (lines) CSCV curves in Figure is quite good, with the plateau current being perfectly reproduced. Differences between the direct and reverse currents could be due to background/capacitive contributions to the overall response due to the complexity of the electrolytic medium under these conditions, as has been pointed out previously in ref .

Conclusions

From a practical point of view, it is very important to be able to measure accurately the rate constant of a given molecular catalyst–substrate couple to adequately benchmark a given compound (or a family of them). The quantitative characterization of electrochemical catalytic responses is carried out in a routine way by using CV, an easy-to-implement and easy-to-use tool that delivers results very quickly. However, the classical diagnosis of CV catalytic currents is based on SID conditions and for the application of a true linear potential waveform. Both conditions cannot be accurately fulfilled when FD conditions are the real operating ones, and the CSCV technique is used instead. As discussed here, the electrochemical response of molecular electrocatalytic processes taking place in confined spaces is strongly dependent on the mass transport conditions. The case under study in this manuscript, the oxidation of IPA to acetone mediated by the oxidation of 4MT, is a clear example of how this influence can affect in a very relevant way the efficiency of the catalytic conversion (by decreasing or enhancing it, depending on the bound or unbound mass transport modality, respectively). The characterization of catalytic processes under these conditions needs to be done in a very careful way since, if the SID criteria are applied to the determination of the rate constant (or maximum turnover frequency, TOF) of the electrochemical conversion, errors of the order of functions tanh­(λ) or coth­(λ) can be obtained for bounded or unbounded conditions, respectively (for the data obtained here, bounded currents up to 85 times smaller and unbounded currents up to 35 times higher than those predicted under SID conditions have been observed). This fact is of special relevance since the experimental use of porous materials has become more prominent in recent years, and the electrochemical community is still using tools developed for CV under SID and not under FD conditions.

The expressions deduced here for the different situations depicted in the kinetic zone diagram (Scheme ) allow us to describe the CSCV and CV responses in an accurate way. Indeed, the experimental situation analyzed corresponds to the most complex case (DK, FD, B, and U), for which no simplified expressions for the current are obtained. Even so, it was possible to obtain the rate constants and reproduce the full current–potential response. This fact confirms the sound applicability of the treatment developed.

Of special interest is the unbound case, for which two current amplifications are observed when compared with the simple charge transfer case: one due to the catalysis and the other to the mass transport. For the rate constants obtained for the experimental system under study, no loss of the kinetic sensitivity of the current is observed, although, for small enough values of the distance L, this loss is expected since the diffusion layer would be much smaller than the reaction layer, leading to a pure diffusive response. This fact should be considered when this configuration is used for analytical or chemical-physics purposes.

Glossary

Symbols/Abbreviations

IPA

isopropyl alcohol

4MT

4-methoxy-TEMPO

SID

semi-infinite diffusion

FD

finite diffusion

TL

thin-layer

DK

diffusive-kinetic

KSS

kinetic steady state

SS

steady state

TLE

thin-layer electrochemistry

CV

cyclic voltammetry

CSCV

cyclic staircase voltammetry

SCV

staircase voltammetry

L

length of the diffusive field

λ

= $L\sqrt{k/D}$ ; ratio between the length of the diffusion field under finite diffusion conditions and the reaction layer.

κ _SCV

= kτ; dimensionless rate constant for (cyclic) staircase voltammetry.

κ _CV

= k/(Fv/(RT)); dimensionless rate constant for cyclic voltammetry.

k

= k ₁ + k ₂; pseudo-first-order catalytic rate constant for a bidirectional catalysis.

k _i ′, i = 1, 2

Second-order rate constant in M^–1 s^–1 (i = 1, conversion of O to R; i = 2, conversion of R to O)

k _i , i = 1, 2

Pseudo-first-order catalytic rate constant in s^–1 (i = 1, conversion of O to R; i = 2, conversion of R to O)

ζ*

= c _O + c _R ; total concentration of electroactive species

K

= k ₂/k ₁; inverse of the equilibrium constant

η

= F(E – E ⁰′)/(RT); dimensionless applied potential

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acselectrochem.4c00241.

• Differences between cyclic staircase voltammetry and cyclic voltammetry; the theoretical deduction of the current-potential-time response in cyclic voltammetry for bounded/unbounded finite diffusion; comparison between analytical and numerical results done with a finite elements simulation; additional theoretical results; experimental section; and additional experimental results (PDF)

• Python code for the simulation of CV curves under bounded conditions (TXT)

• COMSOL report for finite elements simulations (PDF)

The authors declare no competing financial interest.