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Analytical theory for the voltammetry of the non-Nernstian catalytic mechanism at macro and microelectrodes: Interplay between the rates of mass transport, electron transfer and catalysis

T

A. Molina , J.M. Gómez-Gil, J. Gonzalez, E. Laborda ⁎

Departamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia, Spain

ARTICLE INFO

ABSTRACT

Keywords: Catalytic mechanism Non-Nernstian electron transfer Analytical solution Spherical electrodes Disc electrodes

An analytical theoretical treatment of the first-order catalytic mechanism with non-Nernstian electron transfer is presented for electrodes of different geometry and size. As a result, a general and rigorous expression for the transient current-potential response is obtained, applicable to any electrode size and to any electrochemical and chemical kinetics. From the general theoretical solution, the conditions under which the non-Nernstian catalytic mechanism reaches the steady state response are discussed and the corresponding analytical expression is derived. This has a very simple mathematical form, which holds for any electrode geometry as evidenced here for macroelectrodes, microhemispheres and microdiscs. The new analytical solutions reveal relevant analogies between the responses of the catalytic mechanism and a simple charge transfer, which enables us to establish simple procedures for the elucidation of the reversibility of the electrode reaction. The influence of the latter on the current-potential response of direct-current (normal pulse and cyclic voltammetries) and of differential (differential double pulse and square wave voltammetries) techniques are studied. Also, the dependence of the response on the catalytic kinetics and thermodynamics and on the electrode size are analyzed.

1. Introduction As defined in references [1–3], the molecular catalysis of electrochemical reactions refers to the study of a charge transfer reaction of a redox couple O/R in presence of a redox system Y/Z that shows a very slow heterogeneous kinetics so that the electro-reduction of Y at the electrode is only observed at more negative potentials than that corresponding to O (see Scheme I). Thus, Scheme I can be considered a catalytic process where the system O/R acts as the catalyst [1,2,4,5]. In these processes, the current obtained is always larger than that for the redox couple O/R, which gives rise to better sensitivity in any electrochemical method, both when species O/R are in solution and surface confined, as in the case of enzymes, proteins or biomolecules [3,6,7]. In previous papers we have deduced analytical solutions for the catalytic mechanism under Nernstian conditions applicable to any electrochemical technique at electrodes of different geometry and size [8–10]. Finding analytical solutions for the catalytic scheme when the kinetics of the charge transfer reaction is slow and the species are

⁎

soluble is not straightforward [11,12]. Here, we present a quite simple analytical solution for the current-potential response for transient conditions when the charge transfer reaction takes place under the above conditions, considering any degree of reversibility of the chemical catalytic reaction and any electrode size, from macro- to microelectrodes. The expression greatly simplifies under steady state conditions to a general mathematical form applicable to electrodes of any shape. In particular, the cases of (hemi)spherical and disc electrodes will be discussed in this work. The analytical solution obtained here points out that the electrochemical response of a catalytic process behaves, independently of the electrode geometry and size, as a slow charge transfer weighted by the chemical catalysis, i.e., the current is the product of two factors: one that is solely a function of the electrochemical kinetics (through a dimensionless rate constant defined in terms of the reaction layer intrinsically associated to this kind of mechanisms [9,13,14]), and another factor that is mainly (for the non-Nernstian case) or fully (for the Nernstian case) dependent on the homogeneous chemical kinetics. Moreover, it is demonstrated that the steady state current-potential

Corresponding author. E-mail address: [email protected] (A. Molina).

https://doi.org/10.1016/j.jelechem.2019.04.057 Received 1 March 2019; Received in revised form 16 April 2019; Accepted 24 April 2019 Available online 03 May 2019 1572-6657/ © 2019 Elsevier B.V. All rights reserved.

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

(6)

(r , t ) = cO (r , t ) + cR (r , t ) (r , t ) = [cR (r , t )

response is reached more easily (i.e., at lower catalytic rate constants) under non-Nernstian conditions. It is also demonstrated that the effective heterogeneous and homogeneous kinetics can be modified experimentally by changing the electrode size or by varying the pseudo-first order catalytic kinetics through the bulk concentrations of species Y and Z [15] since both variables affect the thickness of the linear reaction layer. Finally, it is worth highlighting that, unlike for the CE, EC, CEC… mechanisms [5,16], the use of macroelectrodes is always advantageous in the study of a homogeneous catalytic reaction since the electrochemical response shows greater sensitivity, unless the inference of capacitive and/or Ohmic drop effects recommends the use of microelectrodes.

(r , t ) =

2c (r , t ) O r2

+

2 r

c O (r , t ) r

c R (r , t ) =D t

2c (r , t ) R r2

+

2 r

c R (r , t ) r

t = 0, r t > 0, r

cO (r , t ) = cO; cR (r , t ) = cR

D

c O (r , t ) r

= kred cO (rs , t ) r = rs

rs

(11)

(r , t ) = 0

(12)

s NN

s N)

(13)

with being the value of ϕ at the electrode surface and refers to the corresponding surface value for a catalytic mechanism with a Nernstian charge transfer (CTN) [8]: s N

1 Ke e 1+e

=

(14)

with (15)

= t =

F (E RT

E0 )

(16)

0'

+ k1 cR (r , t )

where E is the formal potential of the redox couple O/R, and F, R and T have their usual meaning. Then, by introducing the following new variable:

k2 cO (r , t )

k1 cR (r , t ) + k2 cO (r , t )

r (r , t ) rs

u (r , t ) =

(17)

the boundary value problem reduces to:

u (r , t )

(3)

t

(4)

kox cR (rs , t )

(r , t ) r

s N

=D

t = 0, r t > 0, r r = rs

(r , t ) 2 + r2 r

s NN

t > 0, r = rs: c R (r , t ) r

(10)

( r, t )

t > 0, r = rs: (r , t ) (k + kox ) = red ( r D r = rs

(2)

r = rs

2

(r , t ) =D t

(1)

=

= cO + cR

By inserting Eqs. (7) and (10) into Eqs. (2)–(5), the bvp can now be expressed in terms of only the variable ϕ as follows:

Under conditions where convective and migrational transports are negligible, when a single constant potential (E) is applied to a spherical (or hemispherical) electrode of radius rs, the electrochemical response of the catalytic mechanism is defined by the following boundary value problem (bvp):

cO (r , t ) r

(9)

It can be easily demonstrated that the total concentration of electroactive species remains constant at any point of the solution and time of the experiment, regardless of the reversibility degree of the electrode reaction and of the electrode geometry [8], so that the solution for the variable ζ(r, t) is given by,

In the (pseudo-)first order catalytic mechanism with a nonNernstian (NN) charge transfer (CT) and a reversible first-order chemical reaction shown in Scheme (I), kred and kox are the potential-dependent rate constants of electro-reduction and electro-oxidation, respectively, and k1′ and k2′ are the first-order rate constants of the direct and reverse homogeneous chemical reactions, respectively (see also Appendix A). Under excess concentration of species Y and Z, it is possible to define the following pseudofirst-order rate constants:

rs

(8)

= k1 + k2

2.1. Rigurous solution

t = 0, r t > 0, r

c k2 = R k1 cO

K=

2. Theory

c O (r , t ) =D t

(7)

where ζ corresponds to the total concentration of electroactive species and ϕ is related to the magnitude of the perturbation of the chemical equilibrium by the non-Nernstian electrode reaction; K and κ parametrize the chemical thermodynamics and kinetics, respectively, and they are given by:

Scheme I. Non-Nernstian (pseudo)first order catalytic mechanism.

k1 = k1 cZ k2 = k 2 cY

KcO (r , t )] e

t

2u

(r , t ) r2

rs

(18)

u (r , t ) = 0

(19)

t > 0, r = rs: u (r , t )

(5)

r

where cO(r, t) and cR(r, t) are the concentration profiles of species O and R, respectively, cO∗ and cR∗ the bulk concentration of species O and R, respectively, D the diffusion coefficient of the redox species and the other parameters have their usual meaning. The rigorous resolution of the problem (Eqs.(2)–(5)) greatly simplifies with the introduction of the following variable changes [8]:

= r = rs

uϕs

us rs

+

kT s (u D

u sN )

(20)

s

where and uϕN are the values of uϕ at the electrode surface for nonNernstian and Nernstian CT, respectively, and kT corresponds to the sum of the electrochemical kinetic constants:

kT = kred + kox The current is defined as 2

(21)

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

I NN = FAD

cO (r , t ) r

= r = rs

e 1+K

us

u (r , t )

rs

r

r = rs

t = 0, r t > 0, r

(22)

t > 0, r = rs: d ss (r )

and by following the treatment detailed in Appendix B, an analytical expression for the current is obtained as the product of a potential dependent function identical to that of a simple non-Nernstian CT (ENN mechanism) under steady-state conditions “CTNNd” and another factor “Cat-CTNN” that relates to the “catalytic contribution” (see Section 3.1 for further discussion):

I NN FAD

=

kTd 1 Ke (1 + K )(1 + e ) 1 + kTd

1 + rs

e D 2

d

dr

kTd f j ( , wsp ) 1 + kTd

s ss, N

j

Fj (wsp )

j!

i+1 ( 1)iwsp i p m = 0 2j + m

i=0

Fj (wsp) =

( 1)i

1+

i=1

if wsp

i m = 1 p2j m

s ss, NN

t D

kTd + 1 kTd

kTr =

(25)

(26)

where the dimensionless parameter kT corresponds to the total effective electrochemical rate constant referred to the linear diffusion layer at micro(hemi)spheres (rs)

=

Il, a FAD

=

1 + rs

K 1+K

e

+ erf ( )

D

1 + rs

e D

+ erf ( )

sp r

(29)

ss (r ) dr 2

2 d ss (r ) + = r dr

ss (r )

kTr 1 1 Ke 1 1 + K 1 + e kTr + 1 r

=

(39)

Cat

1 1 rs

+

(40)

D

1 + 1.3650 rs 4

1 + 2.0016 rs

( ) + 0.96367 (r )

D

+ 0.8826 rs2 D + 0.32853 rs2 D

D

+ 1.8235 rs2 D

( )

2 s D

3/2 3/2

( ) + 0.307949 (r )

+ 0.063566 rs2 D

2 s D

2 2

5/2

(41) which simplify for very small electrodes to

(31)

D

(38)

+ 0.049925 rs2 D

sp r,micro (rs

Under these conditions, the bvp is defined by (see Eqs. (2)–(5) and (31)):

d2

=

= rs

KcO (r )

kT r D

(30)

For sufficiently fast chemical kinetics (see below), it can be assumed that the perturbation of the chemical equilibrium (i.e., the ϕ-value) is independent of time [8,9]. Thus, the equilibrium perturbation function can be written as

= cR (r )

(37)

disc r

2.2. Steady-state conditions

ss (r )

kTr

which, in a similar way to the general expression (Eq. (23)), is the product of two functions: “CTNNr” that is formally identical to that of a simple non-Nernstian CT (ENN mechanism) under steady-state conditions but now referred to the reaction layer and “Cat” that, under these stationary conditions, is only dependent on the reaction layer, δr, and thus, independent of the CT conditions (see Section 3.1). In Eqs. (38) and (39), the expression of the thickness of the linear reaction layer, r , is given for (hemi)spherical electrodes, δrsp, and for disc electrodes, rdisc , by [17,18]:

(28)

1 1+K

(36)

kTr +1

1 Ke 1+e

=

r

From Eq. (23), the expressions for the cathodic and anodic limiting currents (Il, c and Il, a) can be obtained considering the limits η → − ∞ and η → ∞, respectively:

Il, c FAD

/ D (r rs )

CTNN

(27)

t D

(35)

rs e r

s ss, NN

=

IssNN FAD

Note that when a macroelectrode is employed (rs → ∞), Eq. (26) simplifies to

wpl = 2kT

1 Ke 1+e

where δr is the thickness of the linear reaction layer that depends on the catalytic kinetics and on the electrode size and shape (see below). From Eqs. (17), (22) and (34)–(37), it can be easily demonstrated that the expression for the current of the catalytic mechanism under steady-state conditions for any electrochemical reversibility and electrode geometry is given by:

d

kT rs D

(34)

where the dimensionless parameter kT corresponds to the total effective electrochemical rate constant referred to the linear reaction layer:

χ (see Eq. (15)) and wsp are the dimensionless variables that account for the chemical and electrochemical kinetics, respectively:

wsp = 2kT

s ss, N )

r

1 if wsp > 1

i wsp

s ss, NN

with

(24)

with pi being given in Appendix B and:

kTd =

=

ss (r )

j=0

r = rs

kT ( D

(33)

Thus, the solution of the bvp given by Eqs. (32)–(34) is:

where

f j ( , wsp) =

=

)=0

with being the value of ϕss at the electrode surface for nonNernstian CTs and sss , N the corresponding surface value for a Nernstian charge transfer [8].

(23)

p2j

ss (r

s ss, NN

Cat CTNN

CTNN

rs

<

disc r,micro (rs

0) 0) >

sp d,micro

<

(42)

= rs

disc d,micro>

= rs

4

(43)

When macroelectrodes are employed, the expression for δr for any electrode shape is:

(32) 3

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

r,macro

=

D

for an ENN mechanism under stationary conditions, regardless of the electrode geometry:

(44)

The expressions for the cathodic (Ilss, c ) and anodic (Ilss, a ) limiting current under steady-state conditions are given by:

Ilss, c

1 1 = 1+K r

Ilss, a =

IssNN ,E

(46)

IssNN ,K =0 Ilss, c

RT F

=

kTr 1 Ke 1 1 e 1+K r 1 + e kT + 1 1 + K r 1 + e 1 Ke

E e (1 + e ) (kTr + 1)(1 + e ) (47)

, k ox IssNNBV Ilss, c

Note that this equation also describes the responses in differential double pulse and multipulse techniques (such as Differential Double Pulse Voltammetry, DDPV, and Square Wave Voltammetry, SWV) provided that small amplitudes are considered, in particular: |ΔEDDPV| ≤ 10mV for DDPV and ESW≤ 5 mV for SWV. From the rigorous general solution (23), it is found that the steady state current is attained when (k1 + k2)t ≥ 1.5 for reversible CTs at macroelectrodes [8] and at lower values of the catalytic rate constants as the CT is more sluggish (e.g., (k1 + k2)t ≥ 1for k0 = 10−5cms−1 at macroelectrodes). Thus, it can be concluded that the current-potential response of the catalytic mechanism is typically stationary. It is worth noting that under steady-state conditions the currentpotential response in normal pulse voltammetry (NPV) and cyclic voltammetry (CV) are totally equivalent, and so are the stationary signals in differential techniques, whenever equivalent pulse amplitudes are considered (as indicated above). Thus, they will be hereafter referred to as steady-state voltammetry (SSV) or steady-state differential voltammetry (SSDV). All the analytical expressions that can be derived from the general solution (23) are compiled in Table I.

(50)

0

e 1+e

=

1/2

(51)

1/2

with 1/2

=

F (E RT

E1/2)

(52)

where E1/2 is the half-wave potential for a non-Nernstian catalytic mechanism with an irreversible CT and an irreversible chemical catalysis (K = 0):

E1/2 = E 0 +

RT ln(kr0 ) F

(53)

kr0

with being the effective heterogeneous rate constant referred to the thickness of the linear reaction layer:

kr0 =

k0 D

(54)

r 0

where k is the standard heterogeneous rate constant (k = kT(E = E0')/ 2) and δr is given by Eq. (40) for (hemi)spherical electrodes and by Eq. (41) for disc electrodes. For an irreversible CT (kr0 < 10−1 [5,20,21]), it is easily deduced from Eq. (51) that a linear relationship between the applied potential E and ln[(Il,c − INN)/INN] is expected for any electrode geometry in a similar way to an Eirrev mechanism (see Fig. 1c):

3. Results and discussion 3.1. Current-potential relationship of the catalytic response. Interplay between the electrode and catalytic kinetics for any electrode geometry

0

E = E1/2 +

As indicated previously, the general Eq. (23) points out that, in a similar way to the Nernstian case (see below), the current under trand sient conditions is the product of the potential-dependent factor (CTNN ) formally-equivalent to that of a non-reversible CT under steady-state conditions, and another factor (Cat-CTNN) that reflects the “catalytic contribution” and, unlike in the Nernstian case, it is influenced by the electrode kinetics in a complex form. For χ ≥ 1.5, a stationary response is obtained (Eq. (39)). Under these conditions, the influence of the electrode kinetics (and therefore, of the applied potential) on the second factor disappears and it becomes only dependent on the catalytic kinetics and the electrode geometry (Cat, see Eqs. (23) and (39)). From Eqs. (39) and (45), it is easily deduced that for any degree of reversibility of the CT, the ratio IssNN /Ilss, c for any electrode size and geometry fulfils

IssNN kTr 1 Ke = Ilss, c 1 + e 1 + kTr

kTr 1 1 + e 1 + kTr

=

which is equivalent to Eq. (33) of Ref. [19] when only species O is initially present. All the above is valid for any electrochemical kinetic formalism. Hereafter, the Butler-Volmer model will be considered and thus, Eq. (50) can be rewritten for a totally irreversible CT (i.e.: kox → 0)1 as

(IssNN / FADcT )

=

(49)

where is now the total effective electrochemical rate constant referred to the linear diffusion layer (Eq. (27)) and K is replaced by the ratio of the bulk concentrations of the electroactive species: µ = CR /CO . In the case of an irreversible chemical catalysis (K = 0), Eq. (48) becomes

From Eq. (39), the expression of the derivative current-potential response under steady-state conditions can be obtained in a compact form for any electrode geometry and size: NN ss

kTd 1 µe 1 + e 1 + kTd

kTd

(45)

K 1 1+K r

=

IlE, c

Il, c I NN RT ln F I NN

(55) 0

Regarding the Nernstian case (k → ∞), one of the most important characteristics of the catalytic mechanism is that the current under transient and steady-state conditions is the product of a factor that corresponds to the potential-dependent function of a reversible charge transfer reaction (EN mechanism) independently of the electrode geometry “CTN”, and another function that is independent of the applied potential and it is associated with the catalytic contribution “Cat”,

IN FAD

=

1 Ke 1 + (1 + e )(1 + K ) rs CTN

e D

+ erf ( )

Cat

Eq. (56) becomes independent of time for (k1 + k2)t ≥ 1.5: 1

(48)

In the Butler-Volmer formalism:

kred = k 0e kox = k 0e (1

where kTr is given by Eqs. (38) (see also Eqs. (40)–(41)). It is important to highlight that Eq. (48), independently of the catalytic equilibrium and rate constants, is formally identical to Eq. (32) deduced in Ref. [19] 4

)

(56)

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

Table I Expressions for the different particular cases of a first-order catalytic mechanism with non-Nernstian (black) or Nernstian (red) CT. rG and dG, micro are given by Eqs. (40) and (42), respectively, for (hemi)spherical electrodes and by Eqs. (41) and (43) for disc electrodes, and by Eq. (44) for macroelectrodes (δr, macro). kTd and kTr are defined by Eqs. (27) and (38), respectively. All the parameters are given in Section 2. *Kinetic steady-state responses in this mechanism are truly steady-state responses unlike in the CE or EC mechanisms. **The chemical kinetic-insensitive limit is easily attained at ultramicroelectrodes (UMEs). ***This limit can be attained more easily as the chemical kinetics or the electrode radius increase.

IssN FAD

=

1 Ke 1 + (1 + e )(1 + K ) rs CTN

Cat

D

mechanism with a totally irreversible CT (see Fig. 1c). Unlike for reversible CTs, these curves are influenced by the catalytic rate constants so that they shift towards more negative potentials as the catalysis is faster, as predicted by Eq. (55). The intercept value coincides with the half-wave potential, determined by the effective heterogeneous rate constant (kr0) under steady state conditions (see Eqs. (53) and (54)). Also, it is important to highlight that Eq. (53) is formally equivalent to that of an ENN mechanism, replacing kr0 by 2.309k 0 t /D [5,19]. Regarding quasi-reversible CTs (Fig. 1b), nonlinear E vs ln [(Il,c – I)/ I] plots are observed, which shift continuously towards more negative potentials as κ increases. Hence, the linear or nonlinear behavior of the E vs ln [(Il,c – I)/I] plot and the value of the intercept with respect to E0' enable us to identify readily the degree of reversibility of the CT of a catalytic mechanism, in a similar way to a simple E mechanism. Eq. (39) shows that the current-potential response of the catalytic mechanism is influenced by k0 and δr (and thus, by κ and rs) through the dimensionless parameter kr0 (=k0δr/D with δr being given by Eq. (40) for (hemi)spherical electrodes and by (41) for disc electrodes). This points out the interplay between the rates of the electron transfer, the catalysis and the mass transport as well as their joint influence on the potential dependence of the current response shown in Fig. 2, where the effect of kr0 on the SSV and SSDV responses normalized with respect to the cathodic limiting current of the catalytic mechanism (see Eqs. (39), (45) and (47)) are studied. As can be seen, the decrease of k0 and/or δr (defined by the catalytic kinetics and by the electrode shape and size) have equivalent effects on the apparent reversibility of the CT, as will be discussed in the following section. According to Eqs. (48) and (49), the normalized steady-state (SSV) and steady-state differential (SSDV) voltammetric responses of the catalytic mechanism (which is attainable at any electrode size) are equivalent to those of a simple CT under stationary conditions (attainable at microelectrodes), that is, the SSV current and position and the SSDV peak height, peak potential and half-peak width are equals for both mechanism under stationary conditions.

(57)

N

Thus, the ratio I /Il, c only depends on the applied potential regardless of that the steady-state has been reached or not:

IN 1 Ke = Il, c 1+e

(58)

This implies that, in a similar way to a simple reversible CT (EN mechanism) under transient and stationary conditions, the plot E vs ln [(Il,c − IN)/(IN − Il,a)] of the Nernstian catalytic mechanism is linear, independently of the electrode geometry and size [5]:

E = E0 +

Il, c RT ln N F I

IN Il, a

(59)

Moreover, under these conditions, the current expression corresponding to multipulse techniques at any electrode geometry can be deduced since the superposition principle is applicable [8,22]. From the above results, it can be concluded that the transient and stationary current-potential responses of a catalytic mechanism can be considered as that of a simple CT under stationary conditions, modulated by a factor (more or less complex) related to the homogeneous catalytic reaction, whatever the kinetics of the chemical and electrochemical reactions. The above-mentioned behaviors of the E vs ln ((Il,c – I)/I) curves are shown in Fig. 1, considering a catalytic mechanism with an irreversible chemical reaction (K = 0) and different catalytic rate constants for three different degrees of the CT reversibility: Nernstian (Fig. 1a), quasi-reversible (Fig. 1b) and totally irreversible (see Fig. 1c). Regarding the Nernstian case (Fig. 1a), the E vs ln [(Il,c – I)/I] responses are linear, independent of the catalytic kinetics and the intercept value corresponds to E0', as expected from Eq. (59). These facts have been employed as simple criteria to identify a catalytic mechanism with a reversible CT [5,8]. Linear E vs ln [(Il,c – I)/I] plots are also obtained for the catalytic 5

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

potentials when k0 decreases or κ increases. This clearly indicates that the effective reversibility of the CT decreases as the electrochemical kinetics becomes slower and as the catalysis is faster. This suggests that (deliberately) coupling a catalytic reaction to the electrode reaction can be useful for the study of (very) fast electrode kinetics, as an alternative to more sophisticated approaches such as the use of nanoelectrodes or ultrafast voltammetry [20,23]. Regarding the limiting current in NPV, this increases with the catalytic kinetics, whereas it is independent of the electrode kinetics as expected. The peak height in differential techniques increases with both the chemical and electrochemical kinetics, and the peak becomes broader as k0 decreases and κ increases in the quasi-reversible regime up to reaching the irreversible regime where the half-peak width is independent of k0 and κ. The effect of the enhancement of mass transport as the electrode shrinks on the electrochemical and chemical effective kinetics is studied under stationary conditions in Fig. 4, where the current-potential responses in SSV and SSDV have been normalized with respect to the cathodic limiting current of a simple CT. The SSV (Fig. 4a) and SSDV (Fig. 4b) responses shift towards more negative potentials as rs decreases, and the SSDV peak becomes smaller and broader, which reflects the decrease of the apparent electrochemical reversibility as diffusion is enhanced. An analogous effect is observed for the effective catalytic kinetics, which decreases with rs (see inset of Fig. 4a where the catalytic contribution is plotted as a function of the electrode size). Note that this means that the sensitivity of the electrochemical response to the catalytic kinetics diminishes with the enhancement of the diffusive mass transport so that the use of macroelectrodes is more convenient for the study of the catalytic mechanism, even for fast chemical (catalytic) reactions. This behavior is contrary to that concluded for other reaction mechanisms such as EC, CE, CEC, ECE… where (ultra)microelectrodes are required for the determination of (very) fast homogeneous kinetics [16]. 3.3. Influence of the chemical equilibrium constant, K The influence of the chemical equilibrium constant, K, on the SSV and SSDV responses is considered in Fig. 5, where the SSV responses have been normalized with respect to the cathodic limiting current of a simple CT. Regarding the SSV response, a symmetry centre in the E0'value is always observed, independently of the reversibility of the CT. As expected, the anodic limiting current (Il,a) increases with K while the cathodic limiting current (Il,c) decreases. The ratio |Il,a/Il,c| yields the Kvalue for any electrode size and shape and for any degree of reversibility of the CT [5,9]:

Il, a =K Il, c

(60)

Regarding differential techniques, the SSD-voltammograms are independent of the thermodynamics of the chemical reaction for a reversible CT (Fig. 5b, [24]), being centered at the E0'-value. Note that, related to the symmetry centre in the SSV response, a symmetry plane is observed at the E0'-value, independently of the CT reversibility (see Fig. 5b, d and f). In the case of quasi-reversible CTs (Fig. 5d), the peak height increases and the peak width decreases as the K-value differs from K = 1, the peak potential shifting to more positive potentials as K increases. Also note that the peaks corresponding to different K-values (i.e., different cZ∗, cY∗ values) have a crossing point at the formal potential, which also corresponds to the position of the symmetry axis, provided that α = 0.5. From the expression for the derivative current under steady-state conditions, the current value of the crossing point can be obtained for any degree of reversibility and α = 0.5 by making E = E0' in Eq. (47):

Fig. 1. Effect of the chemical kinetics on the E vs ln [(Il,c – I)/I] curves of a chemically irreversible pseudofirst order catalytic mechanism (K = 0) for three different degrees of CT-reversibility (k0 and κ-values indicated on the graphs, α = 0.5), considering a macroelectrode (rs = 1 mm). Other conditions: t = 1 s, D = 10−5 cm2s‐1 and T = 298 K.

3.2. Influence of the experimental variables on the catalytic currentpotential response In Fig. 3, the effect of the electrochemical (k0) and chemical (κ) rate constants on the NPV and on the derivative voltammetry (DV) responses are shown, where the NPV is normalized with respect to the cathodic limiting current of a simple CT. As can be seen, the half-wave potential in NPV and the peak potential in DV of the catalytic mechanism with a non-Nernstian CT shift towards more negative 6

Journal of Electroanalytical Chemistry 847 (2019) 113097

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Fig. 2. Influence of the effective heterogeneous kinetic rate constant (kr0, values indicated on the graphs) on the normalized a) SSV (Eqs. (39)) and b) SSDV (Eqs. (47), −1 ss ss NN NN ). ΔEDDPV < 10 mV, r ≡ (∂(Iss /Il, c )/∂η)) responses of a chemically irreversible (pseudo)first order catalytic mechanism (K = 0 and κ = 10 s ad = ss ESW < 5 mV. Other conditions as in Fig. 1. NN ss, n (E

= E0 ) =

kr0 1 2 r 1 + 2kr0

3.4. Quantitative analysis of experimental data

(61)

Once the K-value is known from Eq. (60), the chemical rate constants (κ is included in δr) can be estimated from Ilss, c or Ilss, a . Note that this procedure is applicable whatever the electrochemical reversibility. Next, the determination of the electrode kinetics can be addressed. For this, differential techniques are more appropriate given that the peak-shaped signal allows for more accurate quantitative analyses than the sigmoid response in direct-current techniques. Thus, depending on the reversibility regime:

In the case of irreversible CTs (Figs. 5f), two peaks are always observed when K ≠ 0. As can be seen, neither the peak position nor the peak shape depend on K, while the ratio between the peak heights fulfils that: ss, irrev p, a ss, irrev p, c

=K

(62)

̵ Reversible electron transfer: this case can be confirmed by the lin-

Also note that the average of the peak potentials corresponds to E0' since both peaks are equally separated from the symmetry axis: RT ± F ln(kr0 ) . As indicated in previous sections, given that the voltammetric response of a simple CT under stationary conditions is formally equivalent to that obtained for the catalytic mechanism, a symmetry axis at the E0'value is also observed in the E mechanism when the ratio of the electroactive species at the bulk solution (i.e., the μ-value) is varied, independently of the electrode kinetics. Thus, the SSV current and position and the SSDV peak height, peak potential and half-peak width are equal for both mechanisms under stationary conditions.

(

I

I

)

earity of the E vs ln l, cI plots with the E0'-value as the intercept (see Section 3.1) or by comparing the experimental value of the halfpeak width (W1/2,N) with the theoretical predictions for reversible electron transfers for differential techniques (both criteria can be considered under transient and steady-state conditions):

j W1/2,N =

1 + e2 j + 4e j + RT ln F 1 + e2 j + 4e j

(1 + e2 j + 4e j )2 (1 +

e2 j

+ 4e

j )2

4e2

j

4e2

j

;

j

SWV, DDPV

(63) with

SW

=

FESW RT

and

DDPV

=

F | E| . 2RT

Note that the value in DV,

DV W1/2,N ,

Fig. 3. Influence of the electrochemical (a and b, κt = 5) and chemical (c and d, k0 = 10−3 cm/s) kinetics on the NPV (a and c, Eq. (39)) and the DV responses (b and d, from Eq. (47) for κt > 1.5, otherwise by numerical derivation of NPV) of a chemically irreversible pseudofirst order catalytic mechanism (K = 0) at a macroelectrode (rs = 1 mm). ΔEDDPV < 10 mV, ESW < 5 mV. IlE, c = FAD / Dt . Other conditions as in Fig. 1.

7

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Fig. 4. Influence of the electrode size (rs-values indicated on the graphs) on the normalized SSV (a, Eq. (23)) and in SSDV (b, from Eq. (47): (∂(IssNN /IlE, c )/∂η) ≡

NN d ss

with

d

=

(

1 rs

+

1 Dt

)

1

ss ad, E

=

for a spherical microelectrode) responses of a chemically irreversible (K = 0) non-Nernstian (k0 = 10−3 cm/s)

pseudofirst order catalytic mechanism for κ = 5 s−1. Ratio Ilss, c /IlE, c in NPV plotted in inset of Figure a) at different κt values (indicated on graph). Ilss, c plotted with Eq. (45) and IlE, c = FADζ∗/δd. Other conditions as in Fig. 1.

Fig. 5. Influence of the chemical thermodynamics (K-value) on the current-potential responses obtained in SSV (Figures a, c and e; Eq. (23)) and in SSDV (Figures b, d and f; Eq. (47)) for a chemically reversible pseudofirst order catalytic mechanism (κt = 2), considering three different k0-values: 0.1 (a, b), 10−3 (c, d) and 10−5 (e, f) cm/s. SSV responses normalized with respect to cathodic limiting current of a simple CT: IlE, c = FAD / Dt . Other conditions as in Fig. 1.

8

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p,ss Fig. 6. Variation of the peak potential (a) and peak height (b, from Eq. (47) with ad = ssp, NN (δr)) in differential techniques with the effective heterogeneous rate 0 constant (kr , Eq. (54)). Solid line refers to the variation for a chemically irreversible catalytic mechanism (K = 0), dashed line to the variation for a chemically reversible catalytic mechanism with K = 1 where only the cathodic peak is considered in the irreversible regime. Other conditions as in Fig. 1.

DV corresponds to the limit ΔE→0, ESW → 0 in Eq. (63) (W1/2,N = 90.6 mV at 25 °C).

and steady state current-potential responses of the chemically reversible first-order catalytic mechanism when the electron transfer is not Nernstian and for electrodes of different size and shape. The analytical expressions obtained evidence that the response of the catalytic mechanism shows a dependence with the applied potential equivalent to that of an E mechanism, while the catalytic contribution (mainly) enhances/modulates the magnitude of the current. In other words, the shape of the voltammograms is determined by the effective electrochemical kinetics, while the effective chemical kinetics affects their magnitude. From this outcome, it is evident that the current-potential curves can be linearized as in the case of reversible and irreversible charge transfers, while not when the electron transfer is quasi-reversible. This offers simple criteria and procedure to categorize the electrode reversibility as well as for the determination of the formal potential or the effective electrode kinetics. The latter has been demonstrated to be a function, not only of the standard heterogeneous rate constant, but also of the catalytic kinetics and of the electrode size and shape. Thus, the faster the catalysis and/or the smaller the electrode are, the less reversible the electrochemical behavior will be. This has been illustrated in both direct-current and differential techniques making use of the theoretical solutions. In all cases, simple expressions and procedures for the complete characterization of the catalytic mechanism have been provided. First, the catalytic reaction can be readily studied from the limiting current, and then the electrode kinetics can be quantified from the position of the current-potential response.

̵ Fully irreversible electron transfer: In this case (i.e., kox → 0), the expression for the cathodic peak potential of the SSDV response obtained for differential techniques (e.g., DDPV, SWV) can be deduced from Eq. (51), being given by:

Ec,peak = E 0 +

RT k0 RT ln + ln( r ) F D F

(64)

which coincides with the half-wave potential and so it is equivalent to Eq. (53). Thus, for totally irreversible CT, this expression enables us to determine simultaneously the values of k0 and α from experimental measurements of the peak potential at different δr values, that is, different electrodes and/or cY∗ and cZ∗ values. ̵ Quasi-reversible electron transfer: the electrode kinetics (k0 and α in the widely-used Butler-Volmer formalism) can be studied in these situations from the analysis of the variation of the peak potential and the peak current with kr0 as illustrated in the working curves for DV shown in Fig. 6. It is worth noting that the value of kr0 can be modified experimentally by changing the electrode geometry and/ or by varying the pseudo-first order catalytic kinetics (κ) through the bulk concentration of species Y and Z [15]. As shown in Fig. 6, the conditions that fulfil that k 0r ≤ 1.0 are D adequate to carry out quantitative kinetic studies, that is, r k 0 . Thus, 0 −5 for example, the study of a CT with k ≈ 1 cm/s and D = 10 cm2/s can be attained with a catalytic regeneration process of κ ≈ 105 s−1 instead of with a hemispherical nanoelectrode of rs ≈ 100 nm. Note that the former approach would enable the use of conventional electrodes and instrumentation, and also that it could be of interest for the critical assessment of electrode kinetic formalisms [25,26].

Acknowledgements The authors greatly appreciate the financial support provided by the Fundación Séneca de la Región de Murcia (Project 19887/GERM/15) as well as by the Ministerio de Economía y Competitividad (Project CTQ2015-65243-P). JMGG thanks the Ministerio de Educación, Cultura y Deporte for the fellowship ‘Ayuda de Formación de Profesorado Universitario 2015’.

4. Conclusions Rigorous analytical expressions have been deduced for the transient Appendix A. Glossary A ci(r, t) ci∗ D δr

Electrode area Concentration profile of species i (i ≡ O, R). Bulk concentration of species i (i ≡ O, R). Diffusion coefficient of species O and R. Thickness of the linear reaction layer. 9

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A. Molina, et al.

E EO0/ R E1/2 Ec, peak |ΔEDDPV| ESW F IN INN Il, a Il, c Ilss, a Ilss, c IlE, c IssN IssNN NN ss ss ad

ss ad, E

K κ k1' k1 k2' k2 kox kred k0 kd0 kr0 kT kTd kTr ζ(r, t) ζ∗ ϕ(r, t) ϕss(r) s NN s ss, NN

s N s ss, N

R r rs t T

Applied potential. Formal potential of the redox coupled O/R. Half-wave potential. Cathodic peak potential. Pulse height in differential double pulse voltammetry. Pulse amplitude in square wave voltammetry. Faraday constant. Current response of a catalytic mechanism with a Nernstian charge transfer. Current response of a catalytic mechanism with a non-Nernstian charge transfer. Anodic limiting current response of a catalytic mechanism. Cathodic limiting current response of a catalytic mechanism. Anodic limiting current response of a catalytic mechanism under stationary conditions. Cathodic limiting current response of a catalytic mechanism under stationary conditions. Cathodic limiting current response of a simple charge transfer. Current response of a catalytic mechanism with a Nernstian charge transfer under steady-state conditions. Current response of a catalytic mechanism with a non-Nernstian charge transfer under steady-state conditions. Dimensionless differential current response of a catalytic mechanism with a non-Nernstian charge transfer under steady-state conditions. Dimensionless differential current response of a catalytic mechanism with a non-Nernstian charge transfer under steady-state conditions, normalized with respect to the cathodic limiting current of a catalytic mechanism. Dimensionless differential current response of a catalytic mechanism with a non-Nernstian charge transfer under steady-state conditions, normalized with respect to the cathodic limiting current of a simple CT. Conditional chemical equilibrium constant of the homogeneous reaction. Sum of the chemical rate constants of the homogeneous reaction. True chemical kinetic constant of the chemical conversion of species R to O. Pseudofirst-order kinetic constant of the chemical conversion of species R to O. True chemical kinetic constant of the chemical conversion of species O to R. Pseudofirst-order kinetic constant of the chemical conversion of species O to R. Kinetic constant of the oxidation of species R to O. Kinetic constant of the reduction of species O to R. Standard rate constant of the heterogeneous reaction. Dimensionless k0 rate constant referred to the thickness of the linear diffusion layer for (ultra)microelectrodes. Dimensionless k0 rate constant referred to the thickness of the linear reaction layer. Sum of the electrochemical rate constants. Dimensionless kT referred to the thickness of the linear diffusion layer for (ultra)microelectrodes. Dimensionless kT referred to the thickness of the linear reaction layer. Variable related to the total concentration of electroactive species at a certain point of the solution and time of the experiment. Sum of the bulk concentrations of the electroactive species (O and R). Variable related to the perturbation of the chemical equilibrium due to the electrode reaction at a certain point of the solution and time of the experiment. Variable related to the perturbation of the chemical equilibrium at a certain point of the solution under stationary conditions. Value of the ϕ-variable at the electrode surface when the chemical equilibrium is perturbed by a non-Nernstian charge transfer. Value of the ϕ-variable at the electrode surface when the chemical equilibrium is perturbed by a non-Nernstian charge transfer under stationary conditions. Value of the ϕ-variable at the electrode surface when the chemical equilibrium is perturbed by a Nernstian charge transfer. Value of the ϕ-variable at the electrode surface when the chemical equilibrium is perturbed by a Nernstian charge transfer under stationary conditions. Constant of ideal gases. Radial coordinate. Radius of the (hemi-)spherical electrode. Time of application of the potential pulse. Temperature of the electrolyte solution.

Appendix B. Application of the Koutecký's dimensionless parameter method to the non-Nernstian (pseudo-)first order catalytic mechanism equation section In order to obtain a rigorous solution of the problem defined by Eqs. (18)–(20), Koutecký's dimensionless parameters method has been applied as follows. First, the following dimensionless parameters are introduced in Eqs. (18)–(20):

r rs 2 Dt = t

s=

wpl = 2kT

t D

(A1.1)

such that the problem is now given by:

10

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al. 2u

a)

b) u ( s

u

4

s2

2wpl

u wpl

=0

(A1.2) (A1.3)

)=0

u

c)

2u

+ 2s

s2

s

= wpl u sNN 1 +

s=0

1 kTd

1 Ke 1+e

e

(A1.4)

where

kT rs D

kTd =

(A1.5)

It is assumed that the solution is a functional series of the form:

u (s, , wpl ) =

i, j (s )

j (w )i pl

(A1.6)

j=0 i=0

by inserting (A1.6) into (A1.2), one obtains the following homogeneous equation system: i, j (s )

+ 2s

i, j (s )

2(2j + i )

i , j (s )

(A1.7)

= 0; i , j = 0, 1…

The corresponding solution of the above differential equation have the following form: i, j ( s )

= a i, j

2j + i (s )

i, j ( s

+

)

lim L2j + i

L2j + i

(A1.8)

s

where ai, j are constants to be determined by application of the boundary conditions (see below), L2j+i are numeric series of potencies of s with [52, 53]: s

L 0 (s > 0) =

e

t 2dt

=

0 s

lim L 0 = ±

1 erf(s ) p0

1 p0

(A1.9)

and ψ2j+i are Koutecký's functions that have the following properties: i (0) i( i

=1 )=0

(s > 0) =

pi

0 (s > 0) = 1

i 1 (s

> 0)

erf (s > 0)

(A1.10)

′

with ψ being the first derivative and pi are given by:

pi =

(1 + ) ( ) i 2

2

1+i 2

(A1.11)

where Γ(x) is the Euler gamma function and:

p0 =

2 (A1.12)

pi pi + 1 = 2(i + 1) The initial and bulk conditions (Eqs. (A1.3)) establish that:

s

:

0, j (s

) = 0(

(A1.13)

j)

and from the surface condition, taking into account that j

e = j=0

j!

(A1.14)

the following expressions for the coefficients are obtained

s

0: a0, j = 0

ai

1, j

=

1 Ke 1+e

kTd 1 ( 1)i + 1 [(kTd + 1)/ kTd ]i i + 1 j! p2j + i

kTd

(A1.15)

m=1

so that the following expression for the surface value of

uϕ (uϕs)

is obtained (see Eqs. (A1.6), (A1.8)–(A1.13) and (A1.15)):

11

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al.

= +

kTd

1 Ke 1+e

kTd

1 Ke 1+e

u sNN (0, , wpl ) =

p1 p2 (kTd )2

p1 kTd

(wpl )2 (kTd + 1)2

wpl (kTd + 1)

p3 p4 (kTd )2

p3 kTd

i m = 1 p2j + m

i=1

(wpl )2 (kTd + 1)2

wpl (kTd + 1)

kTd + 1

( 1)i + 1(wpl )i [(kTd + 1) / kTd ]i

j

j=0 j !

kTd + 1

=

+…

+… +…

(A1.16)

Thus, uϕNN(s, χ, wpl) can be also calculated (see Eqs. (A1.6), (A1.8)–(A1.13) and (A1.15)):

u

NN

kTd +1

1 Ke 1+e

(s, , wpl ) =

( 1)i + 1 (wpl )i [(kTd + 1)/ kTd ]i

j

kTd

j =0

j!

i

i=1

m=1

=

1 Ke 1+e

wpl (kTd + 1)

+

wpl (kTd

kTd d kT +

p3 kTd

+ 1)

1 (s )

p1 kTd

1

(wpl )2 (kTd + 1) 2

3 (s )

(wpl )2 (kTd

p3 p4 (kTd )2

+ 1)2

p1 p2 (kTd ) 2

4 (s )

2j + i (s )

=

p2j + m 2 (s )

+…

+… +…

(A1.17)

Finally, considering the definition for the current:

I e = FAD 1+K

u sNN

u (rt )

rs

r

= r = rs

e wpl u sNN 1+K

1 Ke 1+e

e

(A1.18)

Eq. (23) is deduced:

I NN FAD

=

kTd 1 Ke 1 + (1 + K )(1 + e ) 1 + kTd rs

kTd f j ( wsp) 1 + kTd

e D 2

(A1.19)

with

p2j

f j ( , wsp) = j=0

j!

j

Fj (wsp ) = p0 F0 + p2 F1 +

p4 2 F2 + … 2

(A1.20)

and: i+1 ( 1)iwsp

i=0

Fj (wsp) =

1+

i m = 0 p2j + m

( 1)i i=1

=

wsp

wsp2

p2j

p2j p2j + 1

i m = 1 p2j m i wsp

=1

+… p2j 1 wsp

+

if wsp p2j 1 p2j 2 wsp2

1 …

if wsp > 1

(A1.21)

References [11]

[1] Z. Galus, Fundamentals of electrochemical analysis, 2nd ed., Polish Scientific Publishers PWN, Chichester, UK, 1994. [2] J.-M. Savéant, Molecular catalysis of electrochemical reactions. Mechanistic aspects, Chem. Rev. 108 (2008) 2348–2378, https://doi.org/10.1021/cr068079z. [3] K.J. Lee, N. Elgrishi, B. Kandemir, J.L. Dempsey, Electrochemical and spectroscopic methods for evaluating molecular electrocatalysts, Nat. Rev. Chem. 1 (2017) 0039, https://doi.org/10.1038/s41570-017-0039. [4] R.G. Compton, C.E. Banks, Understanding Voltammetry, 2nd ed., Imperial College Press, London, 2010, https://doi.org/10.1142/p726. [5] A. Molina, J. Gonzalez, Pulse Voltammetry in Physical Electrochemistry and Electroanalysis, Monographs, Springer International Publishing, Cham, 2016. doi:https://doi.org/10.1007/978-3-319-21251-7. [6] J.-M. Savéant, Elements of Molecular and Biomolecular Electrochemistry, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2006, https://doi.org/10.1002/ 0471758078. [7] J. Gonzalez, C.M. Soto, A. Molina, Analytical I–E response for several multistep potential techniques applied to an electrocatalytic process at mediator modified electrodes, Electrochim. Acta 54 (2009) 6154–6160, https://doi.org/10.1016/j. electacta.2009.05.068. [8] A. Molina, C. Serna, J. Gonzalez, General analytical solution for a catalytic mechanism in potential step techniques at hemispherical microelectrodes: applications to chronoamperometry, cyclic staircase voltammetry and cyclic linear sweep voltammetry, J. Electroanal. Chem. 454 (1998) 15–31, https://doi.org/10.1016/ S0022-0728(98)00251-4. [9] A. Molina, I. Morales, Singularities of the catalytic mechanism in its route to the steady state, J. Electroanal. Chem. 583 (2005) 193–202, https://doi.org/10.1016/j. jelechem.2005.06.003. [10] A. Molina, J. Gonzalez, E. Laborda, Y. Wang, R.G. Compton, Catalytic mechanism in

[12] [13]

[14]

[15] [16]

[17]

[18]

12

cyclic voltammetry at disc electrodes: an analytical solution, Phys. Chem. Chem. Phys. 13 (2011) 14694, , https://doi.org/10.1039/c1cp21181a. R.S. Nicholson, I. Shain, Theory of stationary electrode polarography. Single scan and cyclic methods applied to reversible, irreversible, and kinetic systems, Anal. Chem. 36 (1964) 706–723, https://doi.org/10.1021/ac60210a007. J. Galvez, C. Serna, R. Saura, J. Zapata, Current-potential curves for a catalytic mechanism with non-Nernstian behavior, J. Electroanal. Chem. Interfacial Electrochem. 199 (1986) 27–35, https://doi.org/10.1016/0022-0728(86)87039-5. A. Molina, J. Gonzalez, C.M. Soto, Reaction layer thickness of a catalytic mechanism under transient and stationary chronopotentiometric conditions, J. Electroanal. Chem. 655 (2011) 173–179, https://doi.org/10.1016/j.jelechem.2011. 01.020. E. Laborda, J.M. Gómez-Gil, A. Molina, R.G. Compton, Spectroscopy takes electrochemistry beyond the interface: a compact analytical solution for the reversible first-order catalytic mechanism, Electrochim. Acta 284 (2018) 721–732, https:// doi.org/10.1016/j.electacta.2018.07.070. R. Gulaboski, V. Mirceski, New aspects of the electrochemical-catalytic (EC') mechanism in square-wave voltammetry, Electrochim. Acta 167 (2015) 219–225, https://doi.org/10.1016/j.electacta.2015.03.175. A. Molina, I. Morales, M. López-Tenés, Chronoamperometric behaviour of a CE process with fast chemical reactions at spherical electrodes and microelectrodes. Comparison with a catalytic reaction, Electrochem. Commun. 8 (2006) 1062–1070, https://doi.org/10.1016/j.elecom.2006.04.011. L. Rajendran, M.V. Sangaranarayanan, Diffusion at ultramicro disk electrodes: chronoamperometric current for steady-state EC‘ reaction using scattering analogue techniques, J. Phys. Chem. B 103 (1999) 1518–1524, https://doi.org/10.1021/ jp983384c. A. Molina, E. Laborda, J. Gonzalez, The reaction layer at microdiscs: a cornerstone for the analytical theoretical treatment of homogeneous chemical kinetics at nonuniformly accessible microelectrodes, Electrochem. Commun. 71 (2016) 18–22, https://doi.org/10.1016/j.elecom.2016.07.006.

Journal of Electroanalytical Chemistry 847 (2019) 113097

A. Molina, et al. [19] A. Molina, J. Gonzalez, E.O. Barnes, R.G. Compton, Simple analytical equations for the current–potential curves at microelectrodes: a universal approach, J. Phys. Chem. C 118 (2014) 346–356, https://doi.org/10.1021/jp409167m. [20] C. Amatore, Electrochemistry at ultramicroelectrodes, Phys. Electrochem. Princ. Methods Appl. 1995, pp. 131–208. [21] A.J. Bard, L.R. Faulkner, Electrochemical Methods. Fundamentals and Applications, 2nd ed., Wiley, New York, 2001, https://doi.org/10.1016/B978-0-12-381373-2. 00056-9. [22] A. Molina, C. Serna, L. Camacho, Conditions of applicability of the superposition principle in potential multipulse techniques: implications in the study of microelectrodes, J. Electroanal. Chem. 394 (1995) 1–6, https://doi.org/10.1016/00220728(95)04005-9.

[23] C. Amatore, E. Maisonhaute, G. Simonneau, Ultrafast cyclic voltammetry: performing in the few megavolts per second range without ohmic drop, Electrochem. Commun. 2 (2000) 81–84, https://doi.org/10.1016/S1388-2481(99)00150-2. [24] A. Molina, I. Morales, Comparison between derivative and differential pulse voltammetric curves of EC, CE and catalytic processes at spherical electrodes and microelectrodes, Int. J. Electrochem. Sci. 2 (2007) 386–405. [25] S.W. Feldberg, Implications of Marcus−Hush theory for steady-state heterogeneous electron transfer at an inlaid disk electrode, Anal. Chem. 82 (2010) 5176–5183, https://doi.org/10.1021/ac1004162. [26] E. Laborda, M.C. Henstridge, C. Batchelor-McAuley, R.G. Compton, Asymmetric Marcus–Hush theory for voltammetry, Chem. Soc. Rev. 42 (2013) 4894, https:// doi.org/10.1039/c3cs35487c.

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