Handbook of Electrochemical Impedance · PDF fileHandbook of Electrochemical Impedance Spectroscopy DISTRIBUTED and MIXED IMPEDANCES LEPMI J.-P. Diard, C. Montella Hosted by Bio-Logic

Handbookof

Electrochemical Impedance Spectroscopy

DISTRIBUTEDand

MIXED IMPEDANCES

LEPMIJ.-P. Diard, C. Montella

Hosted by Bio-Logic @ www.bio-logic.info

September 7, 2015

2

Contents

1 Introduction 51.1 Lumped vs. distributed systems . . . . . . . . . . . . . . . . . . . 5

1.1.1 Lumped systems . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Distributed systems . . . . . . . . . . . . . . . . . . . . . 51.1.3 Mixed lumped-distributed systems . . . . . . . . . . . . . 5

1.2 Examples in electrochemistry . . . . . . . . . . . . . . . . . . . . 51.2.1 Lumped systems . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Distributed systems . . . . . . . . . . . . . . . . . . . . . 61.2.3 Mixed lumped-distributed systems . . . . . . . . . . . . . 6

2 Impedance containing th√

S√S

9

2.1th

√S√

S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Electrochemical reaction . . . . . . . . . . . . . . . . . . . 92.1.2 Electrochemical impedance . . . . . . . . . . . . . . . . . 92.1.3 Reduced impedance . . . . . . . . . . . . . . . . . . . . . 92.1.4 Graphs of the reduced impedance . . . . . . . . . . . . . . 92.1.5 Pole-zero map . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2th

√S√

S

1 + α th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Electrochemical reaction . . . . . . . . . . . . . . . . . . . 112.2.2 Reduced Faradaic impedance . . . . . . . . . . . . . . . . 112.2.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

(1 + α th

√S√

S

)th

√S√

S

1 + β th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Electrochemical reaction . . . . . . . . . . . . . . . . . . . 122.3.2 Reduced concentration impedance . . . . . . . . . . . . . 122.3.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 12

3 Mixed impedance 15

3.1th

√S√

S

1 + α S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Pole-zero map . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 15

3

4 CONTENTS

3.21 + α th

√S√

S

1 + β S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Electrochemical reaction: Volmer-Heyrovsky (V-H) . . . . 193.2.2 Reduced concentration impedance of adsorbed

species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 19

3.31

1 + α S + β S th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Electrochemical reaction: catalytic copperdeposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Reduced concentration impedance of adsorbed species . . 213.3.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 21

3.4S th

√S√

S

1 + α S + β S th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Electrochemical reaction: catalytic copperdeposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 Reduced concentration impedance of solublespecies Cl− . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 24

3.51 + α th

√S√

S

1 + β S + γ S th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Electrochemical reaction: E-EAR reaction . . . . . . . . . 263.5.2 Reduced concentration impedance of adsorbed species . . 263.5.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 26

3.6(1 + α S) th

√S√

S

1 + β S + γ S th√

S√S

. . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6.1 Electrochemical reaction: E-EAR reaction . . . . . . . . . 293.6.2 Reduced concentration impedance of soluble

species R . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6.3 Nyquist diagrams . . . . . . . . . . . . . . . . . . . . . . . 29

A Some rational fractions in√

S 33A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2

11 +

√S

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.31

1 + (√

S)3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.41 + (

√S)2√

S (1 + α(√

S)2). . . . . . . . . . . . . . . . . . . . . . . . . . . 35

A.51 − (

√S)2√

S (1 − α(√

S)2). . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B Reactions involving adsorbed and soluble species 39

Bibliography 42

Chapter 1

Introduction

1.1 Lumped vs. distributed systems

1.1.1 Lumped systems

The transfer functions of systems modeled by ordinary differential equations,often called lumped-parameter systems, are rational functions (i.e. a ratio oftwo polynomials in s, the Laplace variable) [1, 2].

1.1.2 Distributed systems

The transfer functions of distributed parameter systems are irrational functions.The analysis of rational and irrational transfer functions differ in a number ofimportant aspects. The most obvious differences between rational and irrationaltransfer functions are the poles and zeros. Irrational transfer functions oftenhave infinitely many poles and zeros [2].

1.1.3 Mixed lumped-distributed systems

The transfer functions of mixed lumped-distributed systems contain rationaland irrational functions in s.

1.2 Examples in electrochemistry

1.2.1 Lumped systems

Faradaic impedance Zf and impedance Z of electrochemical adsorption reaction(EAR) are lumped systems [3,4]. Eq. (1.1) is a rational fraction in s (1).

Z(s) =1 + RctCadss

s ((Cdl + Cads) + sRctCdlCads)(1.1)

Z(s) = K1 + α s

s (1 + β s),K =

1Cdl + Cads

, α = RctCads, β =RctCdlCads

Cdl + Cads(1.2)

1Replacing a capacitor, for example Cdl, by a CPE [5] transforms a lumped impedance ina distributed impedance. This case is not subsequently envisaged.

5

6 CHAPTER 1. INTRODUCTION

Rct

Cdl

Cads

Figure 1.1: Equivalent circuit for electrochemical adsorption reaction (EAR).

1.2.2 Distributed systems

The Faradaic impedance of a corroding electrode with mass transfer limitation(Fig 1.2)) is a rational fraction in

√s, i.e. an irrational fraction in s.

Zf(s) =R σ

σ + R√

s(1.3)

Zf(s) =K

1 + α√

s, K = R, α =

R

σ(1.4)

Rct

Figure 1.2: Equivalent circuit for a corroding electrode with mass transfer limitation.W: Warburg element for semi-innite linear diffusion [6].

1.2.3 Mixed lumped-distributed systems

The impedance of the Randles equivalent circuit [4, 6, 7] (Fig. 1.3) is a mixedlumped and distributed system:

Z(s) =Rct + Rd

th√

τds√

τds

1 + RctCdl s + Cdl s Rdth

√τds

√τds

(1.5)

Z(s) = K

1 + αth

√τds

√τds

1 + β s + γ sth

√τds

√τds

, K = Rct, α =Rd

Rct, β = RctCdl γ = CdlRd

(1.6)

1.2. EXAMPLES IN ELECTROCHEMISTRY 7

∆

Rct

Cdl

Figure 1.3: Randles equivalent circuit for a redox reactions studied on a rotating diskelectrode. Wδ: bounded diffusion impedance [6].

8 CHAPTER 1. INTRODUCTION

Chapter 2

Impedance containing th√

S√S

2.1th

√S√

S

2.1.1 Electrochemical reaction

Redox reaction [4,8, 9]:

O + e ↔ R

studied on a rotating disk electrode with mass transfer limitation.

2.1.2 Electrochemical impedance

ZWδ(s) = Rd

th√

τ s√τ s

(1) (2.1)

2.1.3 Reduced impedance

ZWδ(s) = Rd

th√

τ s√τ s

⇒ Z(S) =ZWδ

(s)Rd

=th

√S√

S, S = τ s = Σ + iu (2.2)

2.1.4 Graphs of the reduced impedance

2.1.5 Pole-zero map

Infinite product expansion [12–16]:

th√

S√S

=1

1 +4S

π2

∞∏k=1

1 +S

(k π)2

1 +4S

((2 k + 1) π)2

= P∞ (2.3)

Thanks to Eq. (2.3)1This expression could be replaced by a more accurate one [10, 11]. This case is not

subsequently envisaged.

9

10 CHAPTER 2. IMPEDANCE CONTAINING TH√

S√S

• infinity of interlaced real poles and zeros (Fig. 2.1).

sp = −14((2k + 1) π)2, k = 1 · · ·∞ (2.4)

sZ = −(k π)2, k = 1 · · ·∞ (2.5)

Figure 2.1: Pole-zero map ofth

√S√

s.

Figure 2.2: 3D plot of the modulus ofth

√S√

S.

2.2.

TH√

S√S

1 + α TH√

S√S

11

2.2

th√

S√S

1 + α th√

S√S


Corroding electrode with mass transfert limitation:

M → Mn+ + n e−

O2 + 4 e− + 4 H+ → 2 H2O

2.2.2 Reduced Faradaic impedance

Z(S) =

th√

S√S

1 + αth

√S√

S

, uc1 = 2.54, uc2 = α2 (2.6)

2.2.3 Nyquist diagrams

Figure 2.3: Nyquist diagrams calculated from Eq. (2.6). Red dots : uc1 = 2.54, blackdots : uc2 = α2.

• uc1 ≫ uc2, 2.54 ≫ α2 ⇒ quarter of a lemniscate,

• uc1 ≪ uc2, 2.54 ≪ α2 ⇒ quarter of a circle (see Annex A.2).


S√S

2.3

(1 + α th

√S√

S

)th

√S√

S

1 + β th√

S√S


EE reaction [17]:

R ↔ X + eX ↔ O + e

studied on a rotating disk electrode with DR = DX = DO.

2.3.2 Reduced concentration impedance

Concentration impedances of soluble species:

ZXi(S) =

(1 + α

th√

S√S

)th

√S√

S

1 + βth

√S√

S

, α, β ≷ 0 (2.7)


Figs. 2.4, 2.5 and 2.6.

Figure 2.4: Some amazing Nyquist diagrams calculated from Eq. (2.7), α = −1,β = −10 (left), β = −104 (right). Red dots : uc1 = 2.54, black dots : uc2 = β2.a: α = −1, β = −10, b: α = −1, β = −10−4, c: α = −2.5, β = −10−5, d: α =−2.5, β = −10 (Hokusai’s great wave).

2.3.

(1 + α TH

√S√

S

)TH

√S√

S

1 + β TH√

S√S

13

Figure 2.5: Array of impedance diagrams calculated from Eq. (2.7). Red dots :uc1 = 2.54, black dots : uc2 = β2.


S√S

Figure 2.6: Array of impedance diagrams calculated from Eq. (2.7). Red dots :uc1 = 2.54.

Chapter 3

Impedance containinglumped and distributedelements

3.1

th√

S√S

1 + α S

Z(S) =th

√S√

S

1 + α S(3.1)

Characteristic frequencies:

• uc1 = 2.54

• uc2 = 1/α

3.1.1 Pole-zero map

• Same zeros asth

√S√

S

• Same poles asth

√S√

Splus one real pole (− 1

α) (Figs. 3.4-3.1).


Figs. 3.4-3.1.

• uc1 ≪ uc2 ⇒ Z(S) ≈ th√

S√S

• uc1 ≫ uc2 ⇒ Z(S) ≈ 11 + α S

15

16 CHAPTER 3. MIXED IMPEDANCE

-

Π2

4-

H3 ΠL2

4-

H5 ΠL2

4-

H7 ΠL2

4

0

-Π2

-H2ΠL2-H3ΠL2-H4ΠL2

Re sIm

s

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re H

-Im

H

Figure 3.1: Pole-zero map and Nyquist diagram of1

1 + α S

th√

S√S

. α = 10−4.

Red dot: uc1 = 2.54, black dot: uc2 = 1/α.

-

Π2

4-

H3 ΠL2

4-

H5 ΠL2

4-

H7 ΠL2

4

0

-Π2


Re s

Ims

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re H

-Im

H


1 + α S

th√

S√S

. α = 10−2.

Red dot: uc1 = 2.54, black dot: uc2 = 1/α.

3.1.

TH√

S√S

1 + α S17

-

Π2

4-

H3 ΠL2

4-

H5 ΠL2

4-

H7 ΠL2

4

0

-Π2


Re s

Ims

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re H

-Im

H


1 + α S

th√

S√S

. α = 1

(− 1α

> −π2

4). Red dot: uc1 = 2.54, black dot: uc2 = 1/α.

-

Π2

4-

H3 ΠL2

4-

H5 ΠL2

4-

H7 ΠL2

4

0

-Π2


Re s

Ims

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Re H

-Im

H


1 + α s

th√

S√S

. α = 103

(− 1α

> −π2

4). Red dot: uc1 = 2.54, black dot: uc2 = 1/α.


Figure 3.5: Change of Nyquist diagram of1

1 + α S

th√

S√S

with increasing value

of α from 10−3 to 100.9. Decimal logarithm of α reported on the Nyquistdiagrams. Red dots: uc1 = 2.54, black dots: uc2 = 1/α.

3.2.1 + α TH

√S√

S

1 + β S19

3.21 + α th

√S√

S

1 + β S

3.2.1 Electrochemical reaction: Volmer-Heyrovsky (V-H)

Electrochemical reaction: Volmer-Heyrovsky (V-H) [4,8]

A+ + s + e− → A,sA+ + A,s + e− → A2 + s

3.2.2 Reduced concentration impedance of adsorbedspecies

Z∗θ (S) =

1 + αth

√S√

S1 + β S

=1

1 + β S+

αth

√S√

S1 + β S

(3.2)

α → ∞ ⇒ Z∗θ (S) ≈ α

th√

S√S

⇒ uc1 = 2.54 (3.3)

α → 0 ⇒ Z∗θ (S) ≈ 1

1 + β iu⇒ uc2 =

1β

(3.4)


Figs. 3.6 and 3.7.

Figure 3.6: Impedance diagrams calculated from Eq. (3.2)). a : α = 1, β = 10−5,b : α = 103, β = 10−3, c : α = 10−3, β = 10−3, d : α = 103, β = 1. Characteristicdimensionless frequencies: red dots : uc1 = 2.54, black dots : uc2 = 1/β.


Figure 3.7: Array of impedance diagrams calculated from Eq. (3.2). Impedancediagrams are made of one or two arcs. Characteristic dimensionless frequencies: reddots : uc1 = 2.54, black dots : uc2 = 1/β.

3.3.1

1 + α S + β S TH√

S√S

21

3.31

1 + α S + β S th√

S√S

3.3.1 Electrochemical reaction: catalytic copperdeposition

Cu2++Cl− + s + e−→ CuCl,sCuCl,s + e− → Cu + Cl− + s

Hypotheses: no mass transfer limitation for Cu2+, (Cu2+(0, t) ≈ Cu2+), kineticirreversibility of the two steps [18,19].

3.3.2 Reduced concentration impedance of adsorbed species

Z(S) =1

1 + α S + β Sth

√S√

S

(3.5)

• Poles and zeros of Z∗(S) are real and interlaced.

• Zeros of Z(S) are the poles ofth

√S√

S: − 1

4 ((2k + 1) π)2, k = 1 · · ·∞.

• Characteristic dimensionless frequencies: uc1 = 2.54, uc2 = 1/α, uc3 =1/β


Figs. 3.8 and 3.9.


0

-Π

2

4-I3 ΠM2

4-I5 ΠM2

4-I7 ΠM2

4

Re s

Ims

0 0.5 10

0.5

Re Z

-Im

Z

ZΘ

Figure 3.8: Pole-zero map and impedance diagrams calculated from Eq. (3.5). α =10−2, β = 1, red dot : uc1 = 2.54, black dot: uc2 = 1/α, blue dot: uc3 = 1/β.

3.3.1

1 + α S + β S TH√

S√S

23

Figure 3.9: Graphics array representation of the impedance diagram, calculatedform Eq. (3.5) and plotted using the Nyquist representation (orthonormal scales)for catalytic copper deposition. Characteristic dimensionless frequencies: red dots :uc1 = 2.54, black dots: uc2 = 1/α, blue dots: uc3 = 1/β.


3.4S th

√S√

S

1 + α S + β S th√

S√S

3.4.1 Electrochemical reaction: catalytic copperdeposition

Cu2++Cl− + s + e−→ CuCl,sCuCl,s + e− → Cu + Cl− + s

Hypotheses: no mass transfer limitation for Cu2+, (Cu2+(0, t) ≈ Cu2+), kineticirreversibility of the two steps [19].

3.4.2 Reduced concentration impedance of solublespecies Cl−

Z(S) =S

th√

S√S

1 + α S + β Sth

√S√

S

(3.6)


• Poles and zeros of Z(S) are real.

• Zeros of Z(S) are the zeros ofth

√S√

S(SZ = −(k π)2, k = 1 · · ·∞) plus

one zero at the origine of the complex plane (derivator).

• Characteristic dimensionless frequencies: uc1 = 2.54, uc2 = 1/α, uc3 =1/β, uc4 = (β/α)2.

Fig. 3.10.

3.4.S TH

√S√

S

1 + α S + β S TH√

S√S

25

Figure 3.10: Graphics array representation of the reduced impedance diagram cal-culated from Eq. (3.6) and plotted using the Nyquist complex plane representation(orthonormal scales) for catalytic copper deposition. Characteristic dimensionless fre-quencies: red dots : uc1 = 2.54, black dots: uc2 = 1/α, blue dots: uc3 = 1/β, greendots: uc4 = (β/α)2.


3.51 + α th

√S√

S

1 + β S + γ S th√

S√S

3.5.1 Electrochemical reaction: E-EAR reaction [20]

R + s → O + s + n1 eA− + s ↔ A,s + n2 e−

3.5.2 Reduced concentration impedance of adsorbed species

Z(S) =1 + α

th√

S√S

1 + β S + γ Sth

√S√

S

, α, β, γ ≷ 0 (3.7)


Figure 3.11: Graphics array representation of the Nyquist diagram for the impedancecalculated from Eq. (3.7) and plotted using the Nyquist representation (orthonormalscales). Characteristic dimensionless angular frequencies: red dots: uc1 = 2.54, blackdots: uc2 = 1/β1. γ = 10−2.

3.5.1 + α TH

√S√

S

1 + β S + γ S TH√

S√S

27

Figure 3.12: Graphics array representation of the Nyquist diagram for the impedancecalculated from Eq. (3.7) and plotted using the Nyquist representation (orthonormalscales). Characteristic dimensionless angular frequencies: red dots: uc1 = 2.54, blackdots: uc2 = 1/β1. γ = −10−1.


Figure 3.13: Graphics array representation of the Nyquist diagram for the impedancecalculated from Eq. 3.7 and plotted using the Nyquist representation (orthonormalscales). α, β, γ < 0. Characteristic dimensionless angular frequencies: red dots:uc1 = 2.54, black dots: uc2 = 1/|β1|. α, β, γ < 0, γ = −10−1.

3.6.(1 + α S) TH

√S√

S

1 + β S + γ S TH√

S√S

29

3.6(1 + α S) th

√S√

S

1 + β S + γ S th√

S√S

3.6.1 Electrochemical reaction: E-EAR reaction [20]

R + s → O + s + n1 eA− + s → A,s + n2 e−

3.6.2 Reduced concentration impedance of solublespecies R

Z(S) =(1 + α S)

th√

S√S

1 + β S + γ Sth

√S√

S

, α, β, γ ≷ 0 (3.8)


Figs. 3.14 and 3.15.


Figure 3.14: Graphics array representation of the Nyquist diagram for the impedancecalculated from Eq. (3.8) and plotted using the Nyquist representation (orthonormalscales). Characteristic dimensionless angular frequencies: red dots: uc1 = 2.54, blackdots: uc2 = 1/β. γ = 10−3.

3.6.(1 + α S) TH

√S√

S

1 + β S + γ S TH√

S√S

31

Figure 3.15: Graphics array representation of the Nyquist diagram for the impedancecalculated from Eq. (3.8) and plotted using the Nyquist representation (orthonormalscales). α, β, γ < 0. Characteristic dimensionless angular frequencies: red dots:uc1 = 2.54, black dots: uc2 = 1/|β|. γ = 10−3.


Appendix A

Some rational fractions in√S

A.1 Introduction

The use of a rational fraction in√

S

Z(√

S) =∑N

m=0 bm(√

S)m∑Pp=0 ap(

√S)p

(A.1)

has been proposed by Pintelon et al. [21, 22]. Some rational fraction in√

S arestudied below.

A.21

1 +√

S

H(u) =1

1 +√

iu(A.2)

Re H(u) =√

2√

u + 22(u +

√2√

u + 1) , Im H(u) = −

√u√

2(u +

√2√

u + 1) (A.3)

|H(u) − (1/2 + i/2)| =√

(Re H(u) − 1/2)2 + (Im H(u) − 1/2)2 =√

22

⇒ circle, radius =√

22

(A.4)

Nyquist diagram: Fig. A.1.

A.31

1 + (√

S)3

H(u) =1

1 + (iu)(3/2)(A.5)

33

34 APPENDIX A. SOME RATIONAL FRACTIONS IN√

S

Figure A.1: Nyquist diagram of1

1 +√

iu. One quarter circle. uc = 1,

Im H(uc) = − 1√2(2 +

√2) .

Re H(u) =√

2u3/2 − 22(√

2u3/2 − u3 − 1) , Im H(u) = − u3/2

−2u3/2 +√

2u3 +√

2(A.6)

|H(u) − (1/2 − i/2)| =√

(Re H(u) − 1/2)2 + (Im H(u) − 1/2)2 =√

22

⇒ circle, radius =√

22

(A.7)

Remarkable frequencies

u1 =3√

3 − 2√

2, u2 = 1, u3 = 3√

2, u4 =3√

3 + 2√

2 (A.8)

Nyquist diagram: Fig. A.2.

Figure A.2: Nyquist diagram of1

1 + (iu)3/2. Three quarter circle.

A.4.1 + (

√S)2√

S (1 + α(√

S)2)35

A.41 + (

√S)2

√S (1 + α(

√S)2)

H(u) =1 + (

√iu)2√

iu (1 + α (√

iu)2)(A.9)

Re H(u) =u(α(u − 1) + 1) + 1√

2√

u (α2u2 + 1), Im H(u) =

−u(αu + α − 1) − 1√2√

u (α2u2 + 1)(A.10)

Three different limiting cases

• α ≪ 1, Nyquist and Bode diagrams: Fig. A.3

u1 = uIm H=0 =−√

α2 − 6α + 1 − α + 12α

≈ 1 (A.11)

u2 = uIm H=0 =+√

α2 − 6α + 1 − α + 12α

≈ 1α

(A.12)

• α = 1, H(u) =1√iu

• α ≫ 1, Nyquist diagram: Fig. A.4

u1 = uReH=0 =−√

α2 − 6α + 1 + α − 12α

≈ 1α

(A.13)

u2 = uReH=0 =√

α2 − 6α + 1 + α − 12α

≈ 1 (A.14)

A.51 − (

√S)2

√S (1 − α(

√S)2)

H(u) =1 − (

√iu)2√

iu (1 − α (√

iu)2)(A.15)

Re H(u) =u(α + αu − 1) + 1√

2√

u (α2u2 + 1), Im H(u) =

u(α + α(−u) − 1) − 1√2√

u (α2u2 + 1)(A.16)

Three different limiting cases

• α ≪ 1, Nyquist diagram: Fig. A.5

u1 = uRe H=0 =−√

α2 − 6α + 1 − α + 12α

≈ 1 (A.17)

u2 = uRe H=0 =√

α2 − 6α + 1 − α + 12α

≈ 1α

(A.18)

• α = 1, H(u) =1√iu

• α ≫ 1.


S

Figure A.3: Nyquist and Bode (modulus) diagram of1 + (

√iu)2√

iu (1 + α (√

iu)2).

a ≪ 1. Red dot: u2 =≈ 1/α, black dot: u1 = 0 ≈ 1.

Figure A.4: Nyquist and Bode (modulus) diagram of1 + (

√iu)2√

iu (1 + α (√

iu)2).

a ≫ 1. Red dot: uIm H=0 ≈ 1/α, black dot: uIm H=0 ≈ 1.

A.5.1 − (

√S)2√

S (1 − α(√

S)2)37

Figure A.5: Nyquist and Bode (modulus) diagram of1 − (

√iu)2√

iu (1 − α (√

iu)2).

a ≪ 1. Red dot: u2 =≈ 1/α, black dot: u1 = 0 ≈ 1.


S

Appendix B

Impedance structure ofreactions involving bothadsorbed and solublespecies

Table B.1: Impedance structure of reactions involving both adsorbed and solublespecies. First order denominator.

Expression Reaction Impedance

1 + αth

√S√

S1 + β S

(V-H)A+ + s + e− → A,s

A+ + A,s + e− → A2 + s Zθ

Table B.2: Impedance structure of reactions involving both adsorbed and solublespecies. Second order denominator.


1 + α s + βth

√S√

S+ γ s

th√

S√S

1 + δ S + ϵ S2

(V-H) with desorptionA+ + s + e− → A,s

A+ + A,s + e− → A2,sA2,s → A2 + s

Zθ

39

40APPENDIX B. REACTIONS INVOLVING ADSORBED AND SOLUBLE SPECIES

Table B.3: Impedance structure of reactions involving both adsorbed and soluble

species. Denominateur : 1 + α S + βth

√S√

SExpressions Reactions Impedances

1

1 + α S + βth

√S√

S

(catalytic)A2+ + B− + s + e− → AB,sAB,s + e− → A + B− + s

Hyp. A2+(0, t) = cteZθ

th√

S√S

1 + β s + γ Sth

√S√

S

(catalytic)A2+ + B− + s + e− → AB,sAB,s + e− → A + B− + s

Hyp. A2+(0, t) = cteZB−


species. Denominateur : 1 + β S + γ Sth

√S√

S.

Expressions Reactions Impedances

1 + α S

1 + β S + γ Sth

√S√

S

(V-H)#2A+ + s + e− → A,sA,s + e− → A− + s Zθ

1 + αth

√S√

S

1 + β S + γ Sth

√S√

S

(E-EAR)R + s → O + s + n1 eA− + s ↔ A,s + n2 e− Zθ

(1 + α S)th

√S√

S

1 + β S + γ Sth

√S√

S

(E-EAR)R + s → O + s + n1 eA− + s ↔ A,s + n2 e− ZR

(V-H)#2A+ + s + e− → A,sA,s + e− → A− + s ZA+


species. Denominateur : 1 + β s + γth

√S√

S+ δ S

th√

S√S

.


1 + αth

√S√

s

1 + β S + γth

√S√

S+ δ s

th√

S√s

(DP3)M,s → M2+ + s + 2 e−

M,s + A2− → MA,s + 2 e−

MA,s + B → MAB + sHyp. A2−(0, t) = A2−∗

Zs

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