A Faradaic impedance study of E-EAR reaction

Accepted Manuscript A Faradaic impedance study of E-EAR reaction M.B. Molina Concha, M. Chatenet, C. Montella, J.-P. Diard PII: DOI: Reference:

S1572-6657(13)00094-5 http://dx.doi.org/10.1016/j.jelechem.2013.02.013 JEAC 1197

To appear in:

Journal of Electroanalytical Chemistry

Received Date: Revised Date: Accepted Date:

10 September 2012 11 February 2013 15 February 2013

Please cite this article as: M.B. Molina Concha, M. Chatenet, C. Montella, J.-P. Diard, A Faradaic impedance study of E-EAR reaction, Journal of Electroanalytical Chemistry (2013), doi: http://dx.doi.org/10.1016/j.jelechem. 2013.02.013

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A Faradaic impedance study of E-EAR reaction

M. B. Molina Concha, M. Chatenet, C. Montella, J.-P. Diard ∗, a LEPMI,

UMR 5279 CNRS - Grenoble INP - Universit´e de Savoie - Universit´e Joseph Fourier, BP 75, 38402 Saint Martin d’H`eres.

Abstract Recently we have proposed a new mechanism describing the borohydride oxidation reaction (BOR) on gold electrode. Despite its apparent simplicity, this mechanism, which consists of one E step and one Electrochemical Adsorption Reaction step (E-EAR) deserves to be studied for its own. The Faradaic impedance of one-step reactions involving adsorbed and soluble species has been studied in the past. Multistep mechanisms have also been under focus, but theoretical studies remain scarce, and they often rely on oversimplistic assumptions (e.g. fast mass transport). Once recalled the basics of Faradaic impedance calculation for an electrochemical reaction, the paper will describe how difficult the interpretation of Faradaic impedance diagrams can be when the reaction involves both adsorbed species and mass transport limitation for soluble species. The paper introduces a method to interpret the theoretical graphs of Faradaic impedance. Key words: Redox Electroadsorption Reaction, EIS, Faradaic impedance, Concentration impedance, Borohydride Oxydation Reaction (BOR)

Preprint submitted to J. Electroanal. Chem.

11 February 2013

1

Introduction

In these times of uncertain future climate, the hydrogen vector is often presented as a relevant alternative to oil for the energy supply chain for automotive and portable applications [1]. Nevertheless, the use of H2 as a fuel in energy-efficient fuel cells conveys intrinsic problematic of storage and distribution, which can be overcome by directly feeding the fuel cells with liquid or solid fuels [2, 3]. In that latter case, the direct oxidation of sodium borohydride was recently shed to light [4], thanks to the unambiguous theoretical advantages of this strategy [5]. However, although in principle very interesting, the direct conversion of NaBH4 into electricity at the anode of a so-called direct borohydride fuel cell (DBFC) via the borohydride oxidation reaction (Eq. (1)), is often hardened because the practical reaction at stake on most electrode materials is not trivial. − − − BH− 4 + 8OH → BO2 + 6H2 O + 8e

(1)

In a recent example, Chatenet et al. [6] proposed that an E-EAR reaction scheme could be proposed to account for the complete BOR on smooth gold electrodes. Although the authors concluded that this simplistic scheme could not exactly represent the physical pathway, notably because at least two electron-transfer steps are necessary to account for the stationary jf vs. E plots of the BOR [7], the E-EAR is a reaction scheme that deserves being studied for its own. The E-EAR reaction consists of a redox reaction (E reaction) occuring at ∗ Corresponding author. Email address: [email protected] (J.-P. Diard).

2

the free sites of the electrode surface, and competing with an electrosorption reaction (EAR reaction) K

o1 R + s −→ O + s + n1 e

K

o2 A,s + e− A− + s ←→

Kr2

(2)

This reaction scheme is similar to that of metal oxidation previously studied in the absence of limitation by mass transport [8–12] : K

o1 Mn1 + + s + n1 e− M,s −→

K

o2 A,s + e− A− + s ←→

Kr2

(3)

The Faradaic impedance for one-step electrochemical reactions involving both adsorbed species and soluble species has been studied for a long time [13–15]. Several multistep reactions have been also envisaged to fit experimental EIS data, see [16–27] as non-exhaustive examples, but theoretical investigation of the structure of Faradaic impedance in relation to the structure of reaction mechanism was only dealt with in a few works, mainly in the works of Harrington [28–31], Harrington and van den Driessche [32, 33] and Naito et al. [34, 35]. If considered separately, both E and EAR reactions are simple (one-step) electrochemical reactions with well-known equivalent circuits for the electrode impedance. The complexity of the E-EAR reaction, compared to the one-step E and EAR reactions, firstly arises because of the presence of the free sites of adsorption (s) in the two reactions. Therefore the Faradaic impedance of E-EAR reaction is not the parallel association of the two Faradaic impedances pertaining to E and EAR reactions. The second difficulty comes from the reaction stoichiometry for the diffusing species involved. Looking at the E-EAR reaction mechanism in Eq. (2), it is 3

evident that the diffusion fluxes of soluble species at the electrode | electrolyte interface are not linearly proportional to the Faradaic current. As a consequence, the influence of mass transport processes does not manifest simply as additional terms in the Faradaic impedance Zf , as previously noted by Armstrong for other reaction schemes [36]. As discussed later in this article, diffusional terms will be present both in the numerator and the denominator of the rational expression for Zf , which weights on the number of possible shapes for EIS diagrams and renders more difficult their interpretation. Owing to these difficulties, simplifications are made in this article. Firstly we assumes that the first step of reaction has irreversible kinetics in the direction of oxidation, and secondly the interfacial concentration of species A− is nearly constant with respect to space and time, as in the case of the BOR, where OH− is the supporting electrolyte [6, 7]. Despite the above simplifications, the Faradaic impedance of E-EAR reaction has a complicated mathematical structure due to the coupling effects of adsorption and mass transport processes. Hence, the structure and the main features of Zf for the E-EAR reaction deserve to be investigated and analyzed for their own, which is the purpose of this work. This article is organized as follows. First, the kinetic equations modelling EEAR reaction dynamics are derived in Section 2. Section 3 deals with steadystate conditions and derivation of the steady-state current-potential characteristic of the electrode. Different limiting behaviours can be predicted for the E-EAR reaction at steady-state. The approach based on the use of kinetic zone diagrams is employed in Section 4 to analyze of such behaviours. The calculation of the Faradaic impedance of E-EAR reaction is attempted in Sec4

tion 5, while the mathematical structure of Zf is discussed in Section 6. The main features of the Nyquist diagrams for Zf are analyzed in Section 7. Some interpretation of Zf diagram shapes in terms of concentration impedances of electroactive species is also presented in this Section. The Nyquist diagrams for the concentration impedances of adsorbed species and soluble species involved in the E-EAR reaction are plotted and discussed in Section 8. Finally, some concluding remarks are reported in Section 9.

2

Kinetic equations

The diffusion flux of species R at the electrode | electrolyte interface and the production rate of the adsorbate A are related to the reaction step rates, v1 and v2 , through JR (0, t) = −v1 (t)

(4)

dθA (t) = v2 (t) dt

(5)

Γ

where Γ is the total number of electrosorption sites per unit of electrode surface area, and θA is the fractional surface coverage of the electrode by adsorbed species. Moreover, the Faradaic current density is given by jf (t) = F (n1 v1 (t) + v2 (t))

(6)

It is assumed that oxidation of R occurs only at the free site s, i.e. at the active sites non blocked by the adsorbate A. The same assumption was made for the Volmer-Heyrovsk´y reaction competing with ferricyanide reduction [19,23]. On this basis, the step rates of E-EAR reaction can be formulated as follows, assuming kinetic irreversibility for the first step of reaction in the direction of 5

oxidation v1 (t) = Ko1 (t) R(0, t) Γ θs (t)

(7)

v2 (t) = Ko2 (t) Γ θs (t) − Kr2 (t) Γ θA (t)

(8)

with θs being the fractional coverage of the electrode surface by free sites, which satisfies θs (t) + θA (t) = 1

(9)

while R(0, t) denotes the interfacial concentration of species R. The interfacial concentration of A− is assumed to be constant, for the sake of simplicity A− (0, t) ≈ A−∗

(10)

Assuming Langmuir isotherm conditions, in the absence of lateral interactions between adsorbed species onto the electrode surface, and introducing the constant concentration of A− in the oxidation rate constant, the rate constants for the two reaction steps can be explicited in terms of the electrode potential E(t) and the symmetry factors for electron transfer reactions (αo1 , αo2 and αr2 = 1 − αo2 ) as: Ko1 (t) = ko1 exp(αo1 n1 f E(t)), ′ ′ Ko2 (t) = ko2 A−∗ exp(αo2 f E(t)) = ko2 exp(αo2 f E(t)), ko2 = ko2 A−∗ ,

Kr2 (t) = kr2 exp(−αr2 f E(t))

3

(11)

Steady-state conditions

Under steady-state conditions, we have dθA = v2 = 0 dt 6

(12)

and the diffusion flux of R at the interface is given, using a rotating disk electrode (RDE), by [11] JR (0) = −mR (R∗ − R(0)) = −v1

(13)

where R∗ denotes both the initial and bulk concentration of species R, and mR = DR /δR

(14)

with DR and δR being respectively the diffusion coefficient and the characteristic length of diffusion-convection derived from Levich theory [43]. The interfacial concentration of R, and the surface coverages of s and A are obtained from Eqs. (7)-(9), (12) and (13) respectively as R(0) =

(Ko2 + Kr2 ) mR R∗ Ko2 Kr2 , θA = , θs = (Ko2 + Kr2 ) mR + Γ Ko1 Kr2 Ko2 + Kr2 Ko2 + Kr2 (15)

The steady-state current density is then given by jf =

n1 F mR R∗ ΓKo1 Kr2 = ΓKo1 Kr2 + mR (Ko2 + Kr2 )

jdR , j = n1 F mR R∗ mR (Ko2 + Kr2 ) dR 1+ Γ Ko1 Kr2 (16)

Different features of the steady-state current-potential curve jf vs. E can be observed depending on the values of the parameters αo1 and n1 . • αo1 n1 < 1 E → ∞ ⇒ jf ≈

n1 F Γ ko1 kr2 R∗ exp(−(1 − αo1 n1 ) f E) ko2

(17)

so limE→∞ jf = 0. The curve is peak shaped, and the electrode potential corresponding to the maximum current is given by (Figs. 1 and 2) : Ko2 (n1 αo1 − 1) + n1 Kr2 αo1

1 ko2 = 0 ⇒ Ep = − ln f kr2 7



1 αo1 n1

−1

!

(18)

This case corresponds to that experimentally observed for the BOR on Au [6]. • αo1 n1 > 1 lim jf = jdR

E→∞

(19)

and the curve jf vs. E shows a plateau corresponding to reaction rate limitation by mass transport of R, and varying linearly with the square root of the rotation rate of RDE, so-called Levich’s criterion (Fig. 3). • αo1 n1 = 1

jdR (20) mR ko2 E→∞ 1+ ko1 Γ kr2 and the curve jf vs. E shows a plateau corresponding to θA ≈ 1. Depending lim jf =

on the value K/Λ = (ko2 /kr2 )/(ko1 Γ/mR ), the limiting current satisfies Levich’s criterion (K ≪ Λ), or it does not depend on the rotation rate Ω of RDE (K ≫ Λ), or finally it depends more or less of Ω (K ≈ Λ).

4

Kinetic zones diagram

The kinetic study of E-EAR reaction at steady state is greatly simplified by introducing dimensionless variables (Tab. I). Here ψ is the dimensionless current density, Λ compares the oxidation rate constant Ko1 of the first step of reaction and the mass-transport rate constant mR , while K compares the oxidation and reduction rate constants of the EAR step. A kinetic zones diagram (KZD) summarizing the different steady-state behaviours of E-EAR reaction can be plotted using the representation log Λ vs. log K in Figs. 1-3. Seven domains of parameters values (kinetic zones) can be defined at the maximum from the theoretical formulation of the dimensionless current 8

density (ψ vs. K, Λ) in Table II. The most important zones are reported in the KZD of Figs. 1-3(a). First ψO/R pertains to the single E reaction occuring on a metal surface free of adsorbate, which corresponds to jf ≈ jf,O/R = n1 F Γ Ko1 R∗ /(1 + Ko1 Γ mR ). The first step of E-EAR reaction is assumed to be irreversible in the oxidation direction so two limiting expressions for ψO/R are the Tafel region ψct1 (due to charge transfer control on the free surface with jf ≈ jct1 = n1 F Γ ko1 R∗ exp(αo1 n1 f E)), and the Levich region ψdR (due to mass transport control on the RDE with jf ≈ jdR = n1 F mR R∗ ). In the kinetic zone labelled as ψct there is no depletion of species R at the electrode surface, i.e. R(0) ≈ R∗ , and therefore jf ≈ jct = n1 F Γ Ko1 R∗ /(1 + Ko2 /Kr2 ). No influence of the electrode rotation rate is then observed. Two limiting subcases can be readily predicted from ψct . The first one is ψct1 already discussed. The second behaviour ψct2 is less common. It corresponds to charge transfer control on the electrode surface covered by the adsorbate (θA ≈ 1) with jf ≈ jct2 = n1 F Γ ko1 (kr2 /ko2 ) R∗ exp((αo1 n1 − 1) f E). Depending on the product αo1 n1 , three limiting formulations are possible for the Faradaic current density, i.e. the decreasing exponential function in Eq. (17) when αo1 n1 < 1, the constant value n1 F Γ ko1 (kr2 /ko2 ) R∗ when αo1 n1 = 1, and finally the increasing exponential function n1 F Γ ko1 (kr2 /ko2 ) R∗ exp((αo1 n1 − 1) f E) corresponding to a second Tafel region when αo1 n1 > 1. Limiting conditions where the overall reaction rate is controlled by the kinetics of a reaction step or by mass transport of species R are achieved when the relative deviation of the Faradaic current density jf from the limiting current density jlim is less than or equal to a small value ǫ, which is fixed here at 5%, 9

i.e. when jlim − jf jlim ψlim ≤ǫ⇔ ≤1+ǫ⇔ ≤1+ǫ jlim jf ψ

(21)

Given a set of parameters and electrode potential values, the steady-state behaviour of electrochemical reaction is represented by a characteristic point in the KZD. At fixed values of parameters and increasing value of electrode potential, the characteristic point moves from left to right on a straight line, so-called electrode trajectory in KZD. The sequence of steady-state behaviours for the E-EAR reaction can be predicted by looking at the intersections of each electrode trajectory with the different zones of KZD.

The equation of the electrode trajectory is obtained by eliminating the electrode potential from the variables Λ et K, which gives

ko1 Γ log Λ = log λ + αo1 n1 log K, λ = m

kr2 ko2

!αo1 n1

(22)

Note that the straight-line slope is equal to αo1 n1 , while the slope of the zone boundary condition in rhs of Figs. 1-3(a) is unity, which denotes the influence of the sign of αo1 n1 −1, as discussed in Eqs. (17)-(20) above. The KZD and the thin solid lines plotted in Figs. 1-3(a) show four different possible sequences of steady-state behaviours for the E-EAR reaction depending on the steady-state potential.

A Mathematica demonstration program is available on the Mathematica website [44]. This file can be interactively used with free Mathematica cdf player [45]. An exhaustive investigation of the steady-state curve jf vs. E for E-EAR reaction is then possible by varying the different parameters involved. 10

5

Calculation of the Faradaic impedance

From Eqs. (6)–(8) and (11), jf (t) can be written formally in terms of the electrode potential E(t), the interfacial concentration R(0, t) of species R, and the surface coverage ratios of free sites θs (t) and adsorbed species θA (t), as jf (t) = jf [E(t), θs (t), θA (t), R(0, t)]

(23)

The Faradaic impedance, Zf (s) = ∆E(s)/∆jf (s), where s denotes the complex Laplace variable, is calculated according to the method introduced by Gerischer and Mehl [46] and developed by Schuhmann [8] and Epelboin and Keddam [47], using a first-order Taylor series expansion of the Faradaic current density with respect to the different variables in Eq. (23). It can be shown that the Faradaic impedance is the sum of the charge-transfer resistance, Rct , the concentration impedance of the dissolved species R, ZR , and the concentration impedance of the adsorbed species Zθ (s) (with Zθ (s) = Zs + ZA ), Zf (s) = Rct + ZR (s) + Zθ (s)

(24)

with Rct =

fFΓ

(αo1 Ko1 θs R(0)n21

1 + αr2 Kr2 θA + αo2 Ko2 θs )

(25)

ZR (s) = 2 θ M (s) (n α θ (K + K + s) − K α θ − θ K α ) Γn1 R(0)Rct Ko1 s R 1 o1 s o2 r2 o2 o2 s A r2 r2 s (θA Kr2 αr2 + Ko2 αo2 θs ) (ΓKo1 θs MR (s) + 1) − n1 R(0)Ko1 (θA Kr2 αr2 + Ko2 αo2 θs ) + n21 R(0)Ko1 αo1 θs (Ko2 + Kr2 + s)

(26) Zθ (s) = Rct (n1 R(0)Ko1 + Ko2 + Kr2 ) (θA Kr2 αr2 + Ko2 αo2 θs ) (ΓKo1 θs MR (s) + 1) s (θA Kr2 αr2 + Ko2 αo2 θs ) (ΓKo1 θs MR (s) + 1) − n1 R(0)Ko1 (θA Kr2 αr2 + Ko2 αo2 θs ) + n21 R(0)Ko1 αo1 θs (Ko2 + Kr2 + s) (27)

11

and √ 1 tanh( τ s) δ2 1/3 √ MR (s) = , τ = R , δR = 1.611 DR ν 1/6 Ω−1/2 mR τs DR

(28)

where ν is the kinematic viscosity of the electrolytic solution, and Ω is the angular rotation rate of RDE. The steady-state quantities R(0), θA and θs are given by Eq. (15). From Eqs. (24)-(27), the Faradaic impedance of E-EAR reaction can be derived as Zf (s) =

(Ko2 + Kr2 + s)(1 + Ko1 θs ΓMR (s))
f F Γ αo1 Ko1 (Ko2 + Kr2 + s)θs R(0)n21 + (αr2 Kr2 θA + αo2 Ko2 θs ) (−Ko1 R(0)n1 + s(1 + Ko1 θs ΓMR (s))) (29)

Some details of the calculation procedure of Faradaic impedance are given in the Appendix A. Finally, the electrode impedance is calculated assuming that (i) the double layer capacitance Cdl is associated in parallel with the Faradaic impedance, and (ii) Cdl is independent of the electrode potential and the coverage ratio of adsorbed species, so Z(s) =

Zf (s) 1 + s Cdl Zf (s)

(30)

A Mathematica demonstration program is available on the Mathematica website [48], which makes it possible to investigated all Nyquist diagrams for the E-EAR reaction by varying the parameters involved. The coupling effects of adsorption and mass transport processes are clearly put into evidence through the formulations of the impedances ZR and Zθ in Eqs. (26) and (27). Indeed, the concentration impedance ZR depends on the mass transport parameters of R, but also on the kinetics of the EAR reaction step. In the same way, the coverage impedance Zθ depends on the kinetics of the EAR reaction step, but also on the mass transport parameters of R. 12

6

Structure of the Faradaic impedance

Whatever the reaction mechanism considered, it is possible to predict the mathematical structure of Zf without detailed calculation. Two concepts are useful for this purpose. First, in a series of articles, Harrington [28–31] and Harrington and van den Driessche [32,33] pointed out that the number of reaction steps and the number of electroactive species involved in the electrochemical reaction are not the critical parameters to discuss the impedance structure. It is rather the number L of reaction steps with linearly independent stoichiometry that determines the complexity of impedance spectra. Looking at the E-EAR reaction scheme in Eq. (2), the two reaction steps are linearly independent, and therefore L = 2. Next, the concept of “concentration/rate” transfer function at the electrode surface, introduced in the pioneering work of Rangarajan [37–40] and developed by Montella [41, 42] is helpful. The transfer function Hi (s) = 1/s applies to adsorbed species (i is the running index), while Hi (s) = Mi (s) is valid for 1-D mass transport of soluble species, with the characteristic function Mi (s) depending on the electrode geometry, the mass-transport process and the boundary conditions away from the electrode surface. The Faradaic impedance takes the form of a rational function with respect to the different transfer function Hi (s). The numerator and the denominator of Zf should be limited at maximum to the products of L and L − 1 transfer functions Hi (s) respectively. Hence, in the case where L = 2, and N ≥ 2 electroactive species (both adsorbed species and soluble species) are involved in the reaction (e− and s species are not taken into consideration here), the 13

Faradaic impedance has the mathematical structure Zf (s) = Rct

1+

PN

i=1

N −1 N αi Hi (s) + i=1 k=i+1 αik Hi (s) Hk (s) PN 1 + i=1 βi Hi (s)

P

P

(31)

where the coefficients αi , αik and βi can be calculated from the partial derivatives of reaction steps rates with respect to the interfacial concentrations of electroactives species and the electrode potential. The E-EAR reaction involves four electroactive species {R, O, A− , A}. Due to the assumptions of (i) kinetic irreversibility of the first step of reaction, and (ii) constant concentration of A− , the list of active species can be reduced to N = 2, i.e. {R, A}. The adsorbate A and the soluble species R are numbered as species 1 and 2, respectively, so the Faradaic impedance of E-EAR reaction, as investigated in this work, is predicted to have the formal expression α1 MR (s) + α2 MR (s) + α12 s s Zf (s) = Rct (32) β1 + β2 MR (s) 1+ s √ √ with MR (s) = (1/mR ) tanh( τ s)/ τ s for 1-D diffusion-convection of R to 1+

the RDE surface. Now, let us consider the whole calculation procedure in the previous Section of this article. Looking at Eq. (29), the Faradaic impedance can be written formally as √ √ tanh( τ s) tanh( τ s) √ √ 1 + a1 s + a2 + a3 s τs τs √ Zf (s) = k tanh( τ s) √ 1 + b1 s + b2 s τs

(33)

with k=

(Ko2 + Kr2 ) (Kr2 (ΓKo1 + mR ) + Ko2 mR ) Γf F n1 R∗ Ko1 Kr2 mR (Ko2 (n1 αo1 − 1) + n1 Kr2 αo1 ) 14

(34)

a1 =

b1 =

ΓKo1 Kr2 1 , a2 = , a3 = a1 a2 Ko2 + Kr2 mR (Ko2 + Kr2 )

(35)

Ko1 (Ko2 (ΓKr2 + n21 R∗ mR αo1 ) + n21 R∗ Kr2 mR αo1 ) + Ko2 mR (Ko2 + Kr2 ) n1 R∗ Ko1 mR (Ko2 + Kr2 ) (Ko2 (n1 αo1 − 1) + n1 Kr2 αo1 ) (36)

b2 =

ΓKo2 Kr2 (Kr2 (ΓKo1 + mR ) + Ko2 mR ) (Ko2 + Kr2 ) 2 (Ko2 (n1 αo1 − 1) + n1 Kr2 αo1 )

n1 R∗ m2R

(37)

in total agreement with the prediction from Eq. (32) above, though the notation is different, which is a useful validation test for the calculation of Zf . It may be noticed here that no general equivalent circuit can be derived from the impedance formulation of Eqs. (32) and (33), which clearly shows the limits of the approach to EIS experimental data based on the use of equivalent circuits for the investigation of multi-step electrochemical reactions. The Faradaic impedance in Eq. (33) depends on the values of five parameters k, a1 , a2 , b1 and b2 (because a3 = a1 a2 ), as well as on the diffusion-convection time constant τ . The parameter a1 only depends on the charge transfer parameters of the second (EAR) reaction step, while the other parameters depend both on charge transfer and mass transport parameters. Due to the term Ko2 (n1 αo1 − 1) + n1 Kr2 αo1 , in the denominator of Eqs. (34), (36) and (37), the parameters k, b1 and b2 can take positive or negative values. The three parameters have the same sign. In contrast, the parameters a1 , a2 and a3 are strictly positive. √ √ The presence of the term tanh( τ s)/ τ s, corresponding to a bounded dif15

fusion impedance [15], renders possible the existence of a quarter-lemniscate impedance loop with the characteristic angular frequency ωc1 = 2.54/τ at the apex of the Nyquist diagram of Zf for certain values of parameters within some frequency range, i.e. when √ tanh( τ s) √ Zf (s) ≈ a + b τs

(38)

The presence of the term 1 + b1 s in the denominator of Zf renders possible the observation of a semi-circle arc with the characteristic angular frequency ωc2 = 1/|b1 | in the Nyquist diagram of Faradaic impedance, for certain values of parameters within some frequency range, i.e. when Zf (s) ≈ a′ +

b′ 1 + b1 s

(39)

Therefore the two characteristic angular frequencies, ωc1 and ωc2 , will be reported on each Nyquist diagram plotted in upcoming Sections.

7

Main features and interpretation of Faradaic impedance diagrams

7.1 An apparently trivial case

The Nyquist diagrams for both Faradaic and electrode impedances can be readily plotted from Eqs. (15), (29) and (30). Fig. 4 shows a steady-state curve, jf /jdR vs. E, calculated for the typical values of parameters given in the figure captions, as well as the related Nyquist diagrams of Faradaic impedance, electrode impedance and concentration impedances calculated at −0.12 V. A small value (10 µF cm−2 ) has been assumed for the double-layer capaci16

tance Cdl . Indeed, at higher values of Cdl , the characteristic shape of Faradaic impedance diagram is masked by the HF impedance arc of the electrode impedance. The Nyquist diagram for the electrode impedance 1 is composed of three impedance arcs, a capacitive semi-circle in the low frequency (LF) domain with the characteristic angular frequency ωc2 = 1/b1 at the apex, a capacitive quarter of lemniscate in the middle frequency (MF) domain with the characteristic angular frequency close to ωc1 = 2.54/τ at the apex, such that ωc1 > ωc2 , and a smaller impedance arc in the high frequency (HF) domain. This arc is due to the charge transfer resistance and the double layer capacitance (Eq. (30)). The Nyquist diagram for Zf is composed of two arcs with typical shapes. Such typical shapes, as well as the related characteristic angular frequencies allow us to propose, at least temporarily, the following interpretation. The LF semi-circle is due to the so-called “relaxation” of adsorbed species while the MF quarter of lemniscate is due to the “relaxation” of soluble species. This interpretation can be easily verified by plotting separately the Nyquist diagrams for the concentration impedances ZR and Zθ using Eqs. (26) and (27) respectively. The Nyquist diagram for Zθ seems to be a semi-circle with the characteristic angular frequency ωc2 = 1/b1 at the apex, so the LF semi-circle of the Faradaic impedance diagram is due to Zθ . In the same way, the Nyquist diagram for ZR is a quarter of lemniscate with ωc1 = 2.54/τ at the apex, so the MF impedance arc of the Faradaic impedance is due to the concentration

1

For convenience, all impedances are divided by |Rp |, with Rp = limω→0 Zf being

the polarization resistance of the electrode.

17

impedance ZR . The one-to-one assignment of the two impedance arcs of Zf , respectively to the adsorbed and soluble species, seems to be correct in this case. Nevertheless by plotting the modulus of the concentration impedance Zθ in the Bode plane representation (Fig. 5a) two bends can be observed, which correspond to the angular frequencies ωc2 and ωc1 in the increasing direction. In addition, the Nyquist diagram for Zθ is not exactly a semi-circle in the whole frequency range explored (Fig. 5c). Despite these peculiarities, it is possible to approximate ZR et Zθ (s), at least in a limited frequency range, by √ tanh( τ s) 1 √ ZR (s) ≈ k1 , Zθ (s) ≈ k2 τs 1 + b1 s

(40)

for the parameters values used in Fig. 4. In that case it is wise noting that the characteristic angular frequency ωc2 = 1/|b1 | depends on the kinetic parameters of the two reaction steps, as well as on the mass transport parameters of species R. On the contrary, the characteristic frequency ωc1 = 2.54/τ only depends on the time constant for mass transport of R. Hence the Faradaic impedance of E-EAR reaction in Fig. 4 can be written approximately as √ tanh( τ s) 1 √ Zf (s) ≈ Rct + k1 + k2 τs 1 + b1 s

(41)

and the equivalent circuit Rct +(Rθ /Cθ )+ZR can be proposed for Zf , where (+) and (/) symbols denote respectively series and parallel association of components. Unfortunately, as intuited from the peculiarities noted aboved, this simple interpretation is not correct for all impedance diagrams pertaining to the E18

EAR reaction, as it is shown in the following examples.

7.2 A slightly less trivial case

Fig. 6 shows a steady-state jf /jdR vs. E curve, calculated for the typical values of parameters given in the figure captions, as well as the related Nyquist graphs of Faradaic impedance, electrode impedance and concentration impedances calculated for a value of electrode potential just before the maximum of the steady-state curve (a). As in Fig. 4, the Nyquist diagram of Faradaic impedance is still composed of two impedance arcs, first a capacitive semi-circle in the LF domain with the angular frequency at the apex being near the characteristic value, ωc2 = 1/b1 , and next a capacitive arc in the MF/HF domain, with its apex angular frequency near ωc1 = 2.54/τ . A priori the explanation previously used for the interpretation of the Faradaic impedance diagram in Fig. 4 seems to be also valid for the Faradaic impedance diagram in Fig. 6: the LF semi-circle could be the concentration impedance Zθ due to the “relaxation” of adsorbed species, and the MF/HF arc the concentration impedance ZR due to the “relaxation” of R species. However, plotting separately the Nyquist diagrams for the concentration impedances ZR and Zθ (Fig. 6(c) and (d)) clearly shows that this interpretation is incorrect. The Nyquist diagram for Zθ is still composed of a semi-circle with an apex angular frequency close to ωc2 = 1/b1 , but the Nyquist diagram for ZR is now composed of two impedance arcs, i.e. a capacitive semi-circle in the LF domain with the angular frequency at the apex being close to ωc2 = 1/b1 19

and a capacitive arc in the MF/HF domain, with the angular frequency at the apex being close to ωc1 = 2.54/τ . The frequency range corresponding to the LF arc of ZR is the same as the frequency range for observation of Zθ . The concentration impedances Zθ and ZR overlap in the frequency domain and the LF semi-circle part of the Faradaic impedance is due to both Zθ and ZR , so that the LF semi-circle of Zf can be attributed to the “relaxation” of concentrations of adsorbed species and soluble species simultaneously. In contrast, the MF/HF impedance arc of Zf can be attributed to the “relaxation” of concentration of soluble species only. Here, two remarks are of interest. Firstly the term “concentration relaxation” can be misleading. Secondly, the one-to-one assigment of an impedance loop to the “relaxation” of one electroactive species, as often encountered in the interpretation of EIS experimental data, is not correct for most multistep electrochemical reactions involving both adsorbed species and soluble species, in the presence of coupling effects between electrosorption and mass transport processes. The E-EAR reaction belongs to this class of reactions, as exemplified by the Nyquist diagram plotted for ZR in Fig. 6(c). Let us first consider controlled-current conditions. Within the LF condition (ω → 0) the concentration profile R(x, t) vs. x (x is the distance to the electrode surface) can be derived from the steady-state profile R(x) vs. x by setting E → E(t) and jf → jf (t) at each time instant t. The variation amplitude for R(0, t) at the electrode surface is then at its maximum. In the MF region between the two impedance arcs (i.e. the frequency domain where ZR is real), the interfacial concentration R(0, t) and the Faradaic current are in phase, but the variation amplitude for R(0, t) is lower than that in the LF 20

domain. Finally, when ω → ∞ (HF limit), |ZR | tends towards zero and the interfacial concentration R(0, t) is fixed at its steady-state value R(0). In other words, the concentration profile is “frozen” in the high frequency domain. Hence, two “relaxations” are observed for the same species R within two different frequency ranges. The LF relaxation can be attributed to the influence of electrosorption of A on the interfacial concentration of R. The MF/HF relaxation is more usually related to mass transport of R. The one-to-one assignment of the two “relaxations” of species R to the two physical processes involved in the reaction (electrosorption and mass transport processes) seems to be correct, at least for the parameters values used in Fig. 6. The interpretation of Zf diagram in terms of “relaxation” of concentration of electroactive species is more complicated under controlled-potential conditions. Indeed, from Eq. (64), we have: ∆R(0, s) ∝ ZR (s) ∆jf (s) =

ZR (s) ∆E(s) Zf (s)

(42)

so the variation amplitude for the interfacial concentration R(0, t) depends on the moduli of both ZR and Zf . In particular, it can be shown that the amplitude in the MF region (where both ZR and Zf take on real values) is larger than that in the LF limit (ω → 0) under the conditions of Fig. 6c. Here the term “concentration relaxation” for the LF arc of ZR diagram should be used with caution. In contrast, this term is more appropriate for interpretation of the MF/HF arc of ZR diagram. Whatever the conditions considered, controlled-potential or controlled-current, the Nyquist diagram for Zθ in Fig. 6d is composed of a single capacitive arc close to a semi-circle. This arc can be attributed to the “relaxation” of surface 21

coverage of adsorbed species. However, the LF arc of Zf diagram is the sum of Zθ and the LF part of ZR diagram. Hence, the LF arc of Zf is related to the modulations of the surface coverage of adsorbed species and the interfacial concentration of soluble species R, simultaneously.

7.3 A more complex Nyquist diagram

Fig. 7 shows a steady-state curve jf /jdR vs. E, calculated for the typical values of parameters given in the captions, as well as the related Nyquist diagrams of Faradaic impedance, electrode impedance and concentration impedances calculated for a value of electrode potential just beyond the maximum of the steady-state curve (a), where the slope of the steady-state curve jf /jdR vs. E is negative in agreement with the impedance plot. As for Figs. 4 and 6, the Nyquist diagram of Faradaic impedance in Fig. 7b is still made of two impedance arcs, i.e. a capacitive semi-circle in the LF domain, with its apex angular frequency near ωc2 = 1/|b1 |, and a capacitive arc in the MH/HF domain, with its apex frequency near ωc1 = 2.54/τ . In contrast to Figs. 4(b) and 5(b), the polarization resistance is negative in Fig. 7(b), as it was observed in the literature for other reaction schemes [49, 50]: Rp = lim Zf (ω) = k (1 + a2 ) < 0 ω→0

(43)

The Nyquist diagram for the concentration impedance Zθ is still made of a semi-circle arc (Fig. 7d) with its apex frequency near ωc2 = 1/|b1 | and limω→0 Re Zθ < 0. The Nyquist diagram for the concentration impedance ZR in Fig. 7c is still made of two impedance arcs, as in Fig. 6, but the LF capacitive arc is now replaced by an inductive one, with its apex frequency 22

near ωc2 = 1/|b1 |. The capacitive arc (quarter of lemniscate) in the MF/HF domain has an apex frequency near ωc1 = 2.54/τ . Note that the inductive arc of ZR is neither visible in the Faradaic impedance nor in the electrode impedance diagram. As in Fig. 6, the LF semi-circle of Zf diagram is due to both Zθ and ZR . The LF semi-circle can be attributed to the “relaxation” of concentration of adsorbed and soluble species simultaneously. Only the MF impedance arc of Zf can be solely attributed to the “relaxation” of concentration of R only. It may be noticed that the MF/HF quarter of lemniscate is masked in Fig. 7(b) by the presence of the double-layer capacitance, which results in the HF semicircle of Z diagram. In addition, the electrode impedance diagram in Fig. 7(b) shows that the EEAR reaction can exhibit a saddle-node bifurcation taking account of additive ohmic drop effects [51,52] in the special case where RΩ + Rp = 0 and therefore limω→0 Z = 0.

7.4 A even more complex Nyquist diagram

Fig. 8 shows a steady-state curve jf /jdR vs. E, calculated for the typical values of parameters given in the captions, as well as the related Nyquist diagrams of Faradaic impedance, electrode impedance and concentration impedances calculated for an electrode potential in the decreasing region of the steadystate current-potential curve, so Rp < 0. The Nyquist diagram of Faradaic impedance is made of two impedance arcs, 23

i.e. an inductive arc in LF domain, with its apex angular frequency near ωc1 = 2.54/τ , and a distorted capacitive arc in MF/HF domain with its apex frequency near ωc2 = 1/|b1 |, with ωc1 < ωc2 in this case. Plotting the Nyquist diagrams for the impedances ZR and Zθ separately is still necessary to explain the shape of the Nyquist diagram for Zf . The Nyquist diagram for ZR is a classical capacitive quarter of lemniscate, with the characteristic angular frequency ωc1 = 2.54/τ . In contrast the Nyquist diagram for Zθ is made of two impedance arcs in this case, i.e. an inductive arc with its apex frequency near ωc1 = 2.54/τ and a capacitive arc with its apex frequency near ωc2 = 1/|b1 |. The LF inductive impedance arc of Zf diagram, whose frequency domain is characteristic of mass transport of R, is the sum of ZR and of the LF inductive arc of Zθ . The electrode impedance diagram in Fig. 8 shows that the E-EAR reaction can exhibit a Hopf bifurcation taking account of additive ohmic drop effects [51,52], due to the possible condition Z = 0 at ω 6= 0 in the presence of ohmic drop.

7.5 A last unusual diagram

Fig. 9 shows a Nyquist diagram of Faradaic impedance made of only one impedance arc for ωc1 = 2.54/τ ≈ ωc2 = 1/|b1 |. The Nyquist diagrams for the concentration impedances Zθ et ZR are each made of one impedance arc with unusual shapes. The Nyquist diagram ZR is cocked hat shaped. The shapes of the Nyquist diagrams for the concentration impedances ZR and Zθ shown in Fig. 9, as well as their sum, are not easily predictable. 24

8

Discussion of the concentration impedances of electroactive species

8.1 Mixed lumped-distributed impedance

Plotting the Nyquist diagrams for the concentration impedances is necessary to explain the shape of the Nyquist diagram for the Faradaic impedance, but is not sufficient to well understand their shapes. The concentration impedances ZR and Zθ in Eqs. (26) and (27) are mixed lumped-distributed impedances, i.e. they contain rational and irrational terms in the Laplace variable s. Rational impedances have a finite number of poles and zeros, while irrational impedances usually have an infinite number of poles and zeros [53]. The study of lumped impedances (rational impedances) is straightforward. The shape of the Nyquist diagram, the number of arcs, the capacitive or inductive character are deduced from the position of the poles and zeros of the impedance [54]. To the best of our knowledge, there is no general method for studying distributed impedances or mixed lumped-distributed impedances (irrational impedances). In particular no general method is available to predict the different possible shapes of Nyquist diagrams for such impedances.

8.2 Concentration impedance of R species

The concentration impedance ZR (s) derived from Eq. (26) can be written in standard form as √ tanh( S) √ (1 + αR S) S √ , S = i u, u = τ ω ZR (S) = kR tanh( S) √ 1 + β1 S + β2 S S 25

(44)

with αR =

n1 αo1 θs , β1 = b1 /τ, β2 = b2 /τ τ (Kr2 (n1 αo1 θs − θA αr2 ) + Ko2 θs (n1 αo1 − αo2 )) (45)

Here, the concentration impedance ZR (s) is not characterized by a simple convective diffusion impedance. It has been shown that the necessary and sufficient condition for obtaining a concentration impedance of soluble species in the form of a mass transport impedance is that the Faradaic current density should be linearly proportional to the interfacial flux of this species [41,55,56], which is not satisfied for species R in the E-EAR reaction. The choice of an electrical circuit from experimental Nyquist diagrams includes a step of pattern recognition, therefore it may be helpful to know the different possible shapes of Nyquist diagram for a given impedance. Using the Manipulate function of Mathematica [57] makes easier the investigation of the different shapes of Nyquist diagrams. A computable document file (.cdf) for plotting Nyquist diagrams of the concentration impedances ZR (s) and Zθ (s) is available as a supplementary file. It is possible to plot exhaustive graphics arrays of Nyquist diagrams when the impedance depends only on two dimensionless parameters, as it has been done previously to investigate the mechanism of two-step insertion reaction [58] or the mechanism of copper electrodeposition reaction [59]. This is no longer possible when the impedance depends on three dimensionless parameters, as in the present case (αR , β1 and β2 ). It is however possible to plot graphics arrays of Nyquist diagrams at fixed values of one of the three parameters. As an example, the Nyquist diagram for the concentration impedance of R species is plotted in Fig. 10 at different values of the dimensionless pa26

rameters αR and β1 , for β2 = 10−3 . Several very different shapes are shown in Fig. 10. As an example the Nyquist diagram for the impedance ZR (s) is composed of three impedance arcs, two capacitive and one inductive arcs, for αR = 10−1 and β1 = 10−2 . For the sake of generality the parameters αR , β1 and β2 are assumed to be independent of each other in Figs. 10 and 11. In fact they depend on the electrode potential in Figs. 4 and 6-8 for the E-EAR reaction. Hence Fig. 10 shows a rich diversity of Nyquist diagrams , which are not necessarily all encountered for the E-EAR reaction. The parameters αR , β1 and β2 can also be negative (Eq. 45). Such negative values increase the number of possible shapes of Nyquist diagrams for the concentration impedance ZR (s). Some diagrams are shown in Fig. 11. Moreover, plotting Nyquist diagrams of concentration impedances is not sufficient to well explain the shape of these graphs. Each unusual shape of impedance diagram observed in Fig. 10 and 11 should be studied for a better understanding. As an example, Fig. 12 explains one of the diagrams plotted for negative values of αR , β1 and β2 (Fig. 11) and |β2 | ≪ |αR | and |β1 |. The expression for ZR (S) can be simplified in this case as: √ tanh( S) √ (1 + αR S) S |β2 | ≪ |αR |, |β1| ⇒ ZR (S) ≈ kR 1 + β1 S

(46)

The two impedance arcs have the same size as: 1 + αR S 1 αR = = u→∞ 1 + β1 S β1 2 lim

(47)

and |β1 |, αR | ≫ 2.54 for the example dealt with in Fig. 12 that compares ZR∗ (S) in Nyquist and Bode representation for positive and negative values of 27

αR and β1 . The impedance diagrams shapes for ZR∗ (S) = ZR /kR can be more easily understood using the Bode plane representation [54] since the decimal logarithm of modulus and the phase in Eq. (46) can be factorized as: √ tanh( S) log |ZR (S)| = log |kR | + log |1 + αR S| + log | √ | − log |1 + β1 S| (48) S

φZR (S)

√ ! tanh( S) √ = arg(1 + αR S) + arg − arg(1 + β1 S) S

(49)

The moduli of 1 + αR S and 1 + β1 S do not depend on the sign of αR and β1 , in contrast to the phases of 1 + αR S and 1 + β1 S as it is shown in Fig. 12. √ √ The asymptotic plots for the moduli of 1/(1+β1 i u), 1+αR i u and tanh( i u)/ i u in Bode representation are also of interest. The log-log slope for the first modulus changes from zero at u ≪ 1/|β1 | to minus one at u ≫ 1/|β1 |. The slope for the second modulus is equal to zero at u ≪ 1/|αR | and one at u ≫ 1/|αR |. Finally the third modulus, which comes from mass transport (RDE), shows zero slope at u ≪ 2.54 and slope −1/2 at u ≫ 2.54. The Bode representation for the modulus of ZR∗ is then easily predictable by addition of the asymptotic slopes. At increasing frequencies, the sum of slopes follows the sequence: 0, −1 < slope < 0, 0 and −1/2. The slope −1 is not observed because the dimensionless angular frequencies 1/|β1 | and 1/|αR | are close to each other. A similar approach is possible using the Bode representation for the phase vs. log u. The asymptotic values of the arguments 0 and −π/2, 0 and π/2, and √ √ 0 and −π/4 for the complex numbers 1/(1+β1 i u), 1+αR i u and tanh( i u)/ i u, respectively in the same frequency ranges as above. The sequence of phases for ZR∗ is then predicted to be 0, −π/2 < phase < 0, 0, −π/4, as it is observed in Fig. 12. 28

8.3 Concentration impedance of adsorbed species

The concentration impedance of adsorbed species, Zθ , derived from Eq. (27) can be written in standard form as: √ tanh( S) √ 1 + αθ S √ Zθ (S) = kθ tanh( S) √ 1 + β1 S + β2 S S

(50)

with: αθ =

ΓKo1 θs , β1 = b1 /τ, β2 = b2 /τ mR

(51)

As an illustration, two arrays of Nyquist diagrams for the concentration impedance Zθ are plotted in Figs. 13 and 14, for positive or negative values of the dimensionless parameters β1 and β2 , respectively. The asymptotic plots proposed in Fig. 12 for ZR could also be used to explain some impedance graphs in Figs. 13 and 14.

9

Conclusion

This paper focused on the study of the Faradaic impedance of E-EAR reaction, a classical pathway encountered in electrochemical kinetics, i.e used in the past [6] to describe the BOR. In spite of the simplicity of this mechanism, which involves only two reaction steps and a single adsorbed species, the Faradaic impedance graphs show a great diversity, depending on the values of the kinetic parameters, the mass transport parameters, and the electrode potential. Several examples corresponding to different EIS diagrams have been presented and discussed. Moreover, the approach based on the use of equivalent circuits fails because no circuit corresponds to the impedance of the 29

E-EAR reaction.

Throughout the various examples under focus, we have demonstrated that plotting Zf diagrams is not necessarily sufficient to explain the impedance diagram shapes. On the contrary, plotting the concentration impedances ZR and Zθ diagrams often proves mandatory for such explanation, although it is not always sufficient. In particular, the one-to-one assigment of an impedance arc of Zf to the “relaxation” of concentration of one adsorbed or soluble species is incorrect for multistep reaction mechanisms in the presence of coupling effects of electrosorption and mass-transport processes. For the E-EAR reaction, situations exist where an impedance loop corresponds to the “relaxation” of both adsorbed and soluble species. Two “relaxations” are also possible in different frequency ranges for the electroactive (adsorbed or soluble) species involved in the E-EAR reaction.

Hence, only the detailled study of concentration impedances enable interpreting without ambiguousness the calculated Faradaic impedance. This strategy was exemplified by presenting different possible shapes for the impedance diagrams of E-EAR reaction in Nyquist and Bode representation.

A

Calculation of the Faradaic impedance

Eqs. (4)-(6) and (9) are linear, so each time-dependent variable can be replaced by its deviation from steady-state conditions. For example, ∆R(0, t) = R(0, t) − R(0). Next, using the Laplace transform method, we obtain the set 30

of equations ∆jf (s) = F (n1 ∆v1 (s) + ∆v2 (s))

(52)

Γ s ∆θA (s) = ∆v2 (s)

(53)

∆θs (s) + ∆θA (s) = 0

(54)

1 ∆R(0, s) = ∆JR (s) = −∆v1 (s) MR (s)

(55)

with, using the diffusion layer approximation, ∆R(δR , t) = 0, under dynamic conditions [11]: √ 1 tanh( τ s) δR2 1/3 √ , δR = 1.611 DR ν 1/6 Ω−1/2 MR (s) = ,τ = mR τs DR

(56)

where ν is the kinematic viscosity of the electrolytic solution, and Ω is the angular rotation rate of RDE. Linearization of Eqs. (7) and (8), using a first-order Taylor series expansion about steady-state conditions, and then application of the Laplace transform method, yields: ∆v1 (s) = ∂E v1 ∆E(s) + ∂θs v1 ∆θs (s) + ∂R v1 ∆R(0, s)

(57)

∆v2 (s) = ∂E v2 ∆E(s) + ∂θs v2 ∆θs (s) + ∂θA v2 ∆θA (s)

(58)

where ∂x y denotes the partial derivative of y with respect to x, which is evaluated at steady-state. From Eqs. (52), (57) and (58), Zf (s) can be splited as

2

: Zf (s) =

∆E(s) = Rct + Zs (s) + ZA (s) + ZR (s) ∆jf (s)

(59)

The Faradaic impedance is the sum of the charge-transfer resistance, Rct , the concentration (or coverage) impedance of the adsorbed species 2 3

3

Zθ (s) =

All impedances have Ω cm2 units in this text. In Sluyters’ work [15], the various concentration impedances are denoted with

respect to the physicochemical process at stake. The Zf expression of Eq. (59) shows that it is more relevant to denote them as concentration impedances of the species

31

Zs (s)+ZA (s), and the concentration impedance ZR (s) of the soluble species R: Zf (s) = Rct + ZR (s) + Zθ (s), Zθ (s) = Zs (s) + ZA (s)

(60)

After linearization of the Faradaic current density in Eq. (23) by a Taylor series expansion limited to its first-order terms, and application of the Laplace transform, the different components of Zf (s) are given by: Rct =

1 ∂E jf

∆θs (s) ∂θs jf ∆θs (s) = −Rct ∂θs jf ∂E jf ∆jf (s) ∆jf (s) ∂θA jf ∆θA (s) ∆θA (s) ZA (s) = − = −Rct ∂θA jf ∂E jf ∆jf (s) ∆jf (s) ∂R jf ∆R(0, s) ∆R(0, s) ZR (s) = − = −Rct ∂R jf ∂E jf ∆jf (s) ∆jf (s) Zs (s) = −

(61) (62) (63) (64)

Solving the linear system of Eqs. (52)-(55) and taking account of Eqs. (57) and (58), yields: ∆R(0, s) = ∆jf (s) −

(−∂E v2 ∂θs v1 + ∂E v1 (sΓ − ∂θA v2 + ∂θs v2 )) MR (s) F ((−∂E v2 ∂θs v1 + ∂E v1 (sΓ − ∂θA v2 + ∂θs v2 )) n1 + sΓ (∂E v2 + ∂R v1 ∂E v2 MR )) (65)

∆θA (s) = ∆jf ∂E v2 + ∂R v1 ∂E v2 MR (s) F ((−∂E v2 ∂θs v1 + ∂E v1 (sΓ − ∂θA v2 + ∂θs v2 )) n1 + sΓ (∂E v2 + ∂R v1 ∂E v2 MR (s))) (66) ∆θs (s) ∆θA (s) =− ∆jf ∆jf

(67)

involved in the expression of the Faradaic current density, so-called electroactive species.

32

together with the partial derivatives expressions derived from Eqs. (7) and (8) at steady-state: ∂E v1 = αo1 n1 f Ko1 R(0) Γ θs , ∂R v1 = Ko1 Γ θs , ∂θs v1 = Ko1 R(0) Γ

(68)

∂E v2 = f Γ (αo2 Ko2 θs + αr2 Kr2 θA ) , ∂θs v2 = Ko2 Γ, ∂θA v2 = −Kr2 Γ

(69)

Substituting Eqs. (68) and (69) into Eqs. (65) and (66) yields Eqs. (26) and (27). Acknowledgments: Thanks are due to the referees for their valuable comments and suggestions.

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[42] C. Montella, J. Electroanal. Chem. 614 (2008) 121. [43] V. G. Levich, Physicochemical hydrodynamics, Prentice Hall, 1962. [44] http://demonstrations.wolfram.com/ RedoxElectroadsorptionReactionEEARAtARotatingDiskElectrodeRD. [45] http://www.wolfram.com/cdf-player/. [46] H. Gerisher, W. Mehl, Z. Elektrochem. 59 (1955) 1049. [47] I. Epelboin, M. Keddam, J. Electrochem. Soc. 117 (1970) 1333. [48] http://demonstrations.wolfram.com/ ImpedanceForAElectrochemicalRedoxElectroadsorptionReactionEE/. [49] R. D. Armstrong, J. Electroanal. Chem. 34 (1972) 387. [50] J. Gregori, J. J. Garc´ıa-Jare˜ no, M. Keddam, F. Vicente, Electrochim. Acta 52 (2007) 7903. [51] M. T. M. Koper, J. H. Sluyters, J. Electroanal. Chem. 371 (1994) 149. [52] F. Berthier, J.-P. Diard, C. Montella, Electrochim. Acta 44 (1999) 2397. [53] G. C. Temes, J. W. LaPatra, Introduction to Circuits Synthesis and Design, McGraw-Hill, New-York, 1977. [54] J.-P. Diard, B. Le Gorrec, C. Montella, C. Montero-Ocampo, J. Electroanal. Chem. 352 (1993) 1. [55] J.-P. Diard, J.-M. Le Canut, B. Le Gorrec, C. Montella, Electrochim. Acta 43 (1998) 2485. [56] C. Montella, J.-P. Diard, B. Le Gorrec, Exercices de cin´etique ´electrochimique. II. M´ethode d’imp´edance, Hermann, Paris, 2005. [57] S. Wolfram,

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Wolfram Media &

Cambridge University Press, Champaign-USA, Cambridge-UK, 2007.

36

[58] C. Montella, J. Electroanal. Chem. 497 (2001) 3. [59] J.-P. Diard, C. Montella, J. Electroanal. Chem. 590 (2006) 126.

37

Table I Definition of the dimensionless variables

Variable

Dimensionless variables

Kinetic constants

Λ = Ko1 Γ/mR K = (ko2 /kr2 ) exp(f E)

Current density

ψ=

jf 1 = jdR 1 + (1 + K)/Λ

Table II Kinetic zone boundary equations

Limiting behaviour

Conditions

Limiting current

E reaction on free surface

θs ≈ 1

ψO/R =

charge transfer (Tafel region)

R(0) ≈ R∗

ψct1 = Λ

Λ = −K + ǫ

mass transport (Levich region)

R(0) ≈ 0

ψdR = 1

Λ = (1 + K)/ǫ

no depletion of R

R(0) ≈ R∗

ψct =

charge transfer on free surface

θs ≈ 1

ψct1 = Λ

Λ = −K + ǫ

charge transfer on covered surface

θA ≈ 1

ψct2 = Λ/K

Λ=Kǫ−1

38

1 1 + 1/Λ

Λ 1+K

Zone boundary equation Λ = −1 + K/ǫ

Λ = (1 + K) ǫ

4

ΨdR

log L

2 0

ΨOR

-2 Ψct1

-4

Ψct

Ψct2 a

-4

0

-2

2

4

log K

Ψ

1

0.5

b 0

-0.4

-0.2

0

0.2

0.4

EV

Fig. 1. Kinetic zone diagram (a) and steady-state current density vs. electrode potential curve (b) (large dot: E = −0.12 V) plotted for DR = 1.6 × 10−6 cm2 s−1 , ν = 10−2 cm2 s−1 , Ω = 3250 rpm, Γ = 10−9 mol cm−2 , f = 38.9 V−1 , ko1 = 108.5 mol−1 cm3 s−1 , n1 = 1, αo1 = 0.6, ko2 /kr2 = 101.6 , (αo1 n1 < 1), at 25◦ C. The potentials in (a) are plotted every 0.1 V along the electrode trajectory (small dots).

39

log L

5

ΨdR

0

ΨOR

-5

Ψct1

Ψct

Ψct2 a

0

-5

5

log K

Ψ

1

0.5

b 0

0

-0.5

0.5

EV

Fig. 2. Kinetic zone diagram (a) and steady-state current density vs. electrode potential curve (b) (large dot: E = 0 V) plotted for αo1 = 0.4, ko1 = 1010.1 , ko2 /kr2 = 10−0.9 , Ω = 900 rpm (αo1 n1 < 1). Other parameters as in Fig. 1.

40

4

ΨdR

log L

2 0

ΨOR

-2 Ψct

Ψct1

-4 -4

Ψct2 a

0

-2

2

4

log K

Ψ

1

0.5

b 0

-0.4

-0.2

0

0.2

0.4

EV

Fig. 3. Kinetic zone diagram (a) and steady-state current density vs. electrode potential curve (b) plotted for (upper trajectory and steady-state curve, thin curves) αo1 = 0.7, n1 = 2 (αo1 n1 > 1), ko1 = 108.7 mol−1 cm3 s−1 , ko2 /kr2 = 10−1.2 , and (lower trajectory and steady-state curve, thick curves) αo1 = 0.5, n1 = 2 (αo1 n1 = 1), ko1 = 108.7 mol−1 cm3 s−1 , ko2 /kr2 = 102 , Ω = 4500 rpm. Other parameters as in Fig. 1.

41

a jf jdR

0.5

0

0

-0.4

0.4

- Im ZRp

EV 0.3

Ωc2

Ωc1

0

0

0.5

b

1

- Im ZR Rp

Re ZRp

0.2

0

Ωc1

0

c

0.2

0.4

Re ZR Rp

- Im ZΘ Rp

Ωc2

d

0.2

0

0

0.2

0.4

Re ZΘ Rp

Fig. 4. Steady-state jf /jdR vs. E curve (a), Nyquist diagram of Zf (thick blue curve) and Z (thin fuchsia curve) (b), and Nyquist diagrams for the concentration impedances ZR (c) and Zθ (d) calculated from Eqs. (15), (25)-(30) for ko1 = 108.25 mol−1 cm3 s−1 , αo1 = 0.6, n1 = 1, ko2 = 10−3 s−1 , kr2 = 10−4 s−1 , DR = 1.6 × 10−6 mol cm2 s−1 , Ω = 3250 rpm, Γ = 10−9 mol cm−2 , R∗ = 10−6 mol cm−3 , f = 38.9 V−1 , Cdl = 10−5 F cm−2 . Small red dot: ωc1 = 2.54/τ , large black dots: ωc2 = 1/b1 , and E = −0.12 V, which corresponds to the dot on the steady-state curve. Arrows always correspond to increasing frequencies on the Nyquist diagrams.

42

log ÈZΘ ÈRp

0

a

-5

log Ωc2

log Ωc1 log Ω

0

ΦZΘ °

b

-90

log Ωc2

log Ωc1 log Ω

- Im ZΘ Rp

c 0.0005

0 -5. ´ 10-7

0

Re ZΘ Rp

Fig. 5. Bode diagram (modulus (a) and phase (b)) for the concentration impedance Zθ , and zoom (c) of the HF part of Nyquist diagram plotted in a non-orthonormal scale. Same parameter value as in Fig. 4.

43

a

jf jdR

1

0

0

-0.5

0.5

EV 0.4 - Im ZRp

b

0

0

1

- Im ZR Rp

Re ZRp

c 0.2

0

0

0.5 Re ZR Rp

- Im ZΘ Rp

d 0.1

0

0

0.2 Re ZΘ Rp

Fig. 6. Steady-state jf /jdR vs. E curve (a), Nyquist diagram of Zf (thick blue curve) and Z (thin fuchsia curve) (b), and Nyquist diagrams of the concentration impedances ZR (c) and Zθ (d) calculated from Eqs. (15), (25)-(30) for ko1 = 1010.1 mol−1 cm3 s−1 , αo1 = 0.4, ko2 = 10−3.9 s−1 , kr2 = 10−3 s−1 , Ω = 900 rpm and E = 0 V, which corresponds to the dot on the steady-state curve. Other parameters as in Fig. 4.

44

a

jf jdR

1

0

0

-0.5

0.5

EV 1 - Im ZÈRp È

b

0

0

-1

Re ZÈRp È 0.4 - Im ZR ÈRp È

c

0

-0.4 0

1 Re ZR ÈRp È

- Im ZΘ ÈRp È

d 1

0

0

-2 Re ZΘ ÈRp È

Fig. 7. Steady-state jf /jdR vs. E curve (a), Nyquist diagram of Zf (thick blue curve) and Z (thin fuchsia curve) (b), and Nyquist diagrams of the concentration impedances ZR (c) and Zθ (d) calculated from Eqs. (15), (25)-(30). Same parameter values as in Fig. 6 and E = 0.15 V which corresponds to the dot on the steady-state curve.

45

a

jf jdR

1

0

0

-0.5

0.5

EV b

- Im ZÈRp È

0.5

0

0

-1

- Im ZR ÈRp È

Re ZÈRp È c 0.5

0

0

1 Re ZR ÈRp È

- Im ZΘ ÈRp È

d 0

-1 0

-2 Re ZΘ ÈRp È

Fig. 8. Steady-state jf /jdR vs. E curve (a), Nyquist diagram of Zf (thin (thick blue curve) and Z (thin fuchsia curve) (b), and Nyquist diagrams of the concentration impedances ZR (c) and Zθ (d) calculated from Eqs. (15), (25)-(30). Same parameter values as in Fig. 6 and E = 0.4 V which corresponds to the dot on the steady-state curve.

46

a

jf jdR

1

0

0

-0.5

0.5

EV 1 - Im ZÈRp È

b

0

0

-1

- Im ZR ÈRp È

Re ZÈRp È

c

0.5

0

0

1

- Im ZΘ ÈRp È

Re ZR ÈRp È

d

1

0

0

-2 Re ZΘ ÈRp È

Fig. 9. Steady-state jf /jdR vs. E curve (a), Nyquist diagram of Zf (thin (thick blue curve) and Z (thin fuchsia curve) (b), and Nyquist diagrams of the concentration impedances ZR (c) and Zθ (d) calculated from Eqs. (15), (25)-(30) for ko1 = 109.5 mol−1 cm3 s−1 , αo1 = 0.4, ko2 = 10−0.4 s−1 , kr2 = 10−1.7 s−1 , Ω = 900 rpm. Small red dot: ωc1 = 2.54/τ = 5.0 rd s−1 , large black dot: ωc2 = 1/|b1 | = 3.2 rd s−1 (ωc1 ≈ ωc2 ). Other parameters as in Fig. 4 and E = 0.05 V which corresponds to the dot on the steady-state curve.

47

log Β1

3

-4

-2

log ΑR

4

Fig. 10. Graphics array representation of the Nyquist diagram for the impedance ZR (u) calculated from Eq. (44) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: small red dots: uc1 = 2.54, large black dots: uc2 = 1/β1 . β2 = 10−3 .

48

log ÈΒ1 È

6

-3 3.25

log ÈΑR È

5.75

Fig. 11. Graphics array representation of the Nyquist diagram for the impedance ZR (u) calculated from Eq. (44) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: small red dots: uc1 = 2.54, large black dots: uc2 = 1/|β1 |. β2 = −10−1 .

49

uc1 = 2.54

uc1 = 2.54

1 Β1

- Im ZR*

- Im ZR*

uc2 =

0 uc2 =

0 0

0.5 Re

1

0

Re

0

- log Β1

1

ZR*

0

log 2.54

- log ÈΒ1 È

log u

log 2.54 log u

- log ΑR

- log ÈΑR È 90

ΦZR* °

90

ΦZR* °

Β1

- log ÈΑR È

log ÈZR* È

- log ΑR

log ÈZR* È

0.5

ZR*

1

0

-90

0

-90 - log Β1

log 2.54

- log ÈΒ1 È

log u

log 2.54 log u

Fig. 12. Nyquist and Bode diagrams (modulus and phase) for the impedance ZR (u) calculated from Eq. (46). |β2 | ≪ |αR |, |β1 | and left : αR , β1 > 0, right αR , β1 < 0 with β1 = 2 αR in both cases. Thick lines: ZR , thin lines: 1 + αR i u, tiny dashed √ √ lines: 1/(1 + β1 i u), large dashed lines: tanh( i u)/ i u. The vertical dashed lines correspond to u = 1/|β1 |, u = 1/|αR | and u = 2.54 respectively.

50

log Β1

1

-5 -2

log ΑΘ

3

Fig. 13. Graphics array representation of the Nyquist diagram for the impedance Zθ (u) calculated from Eq. (50) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: small red dots: uc1 = 2.54, large black dots: uc2 = 1/β1 . β2 = 10−2 .

51

log ÈΒ1 È

2

-4 -1

log ÈΑΘ È

5

Fig. 14. Graphics array representation of the Nyquist diagram for the impedance Zθ (u) calculated from Eq. (50) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: small red dots: uc1 = 2.54, large black dots: uc2 = 1/|β1 |. β2 = −10−1 .

52

We recall the basis of Faradaic impedance calculation. We describe how difficult the interpretation of Faradaic impedance diagrams can be. We introduce a method to interpret the theoretical graphs of Faradaic impedance.