Handbook of electrochemical impedance spectroscopy.pdf

Handbookof

Electrochemical Impedance Spectroscopy

-0.2 Ecorr 0.2

-1

0

1

EV

i fÈi d

OÈ

ififMifO

0 Rct Rp

0

0.4

Re Z

-Im

Z

Ω = 1HRctCdlL

Ω = Α2Τ

CORROSIONREACTIONS LIBRARY

J.-P. Diard, C. Montella, N. MurerLEPMI, Bio-Logic

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

July 17, 2012

Contents

1 Reactions without adsorbed species 51.1 Reactions without mass transport limitation . . . . . . . . . . . . 5

1.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . . 51.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . . 6

1.2 Reactions with mass transport limitation . . . . . . . . . . . . . 81.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Example: Iron in aerobic solution [4] . . . . . . . . . . . . 81.2.3 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Steady-state conditions . . . . . . . . . . . . . . . . . . . 81.2.5 Faradaic impedance [3] . . . . . . . . . . . . . . . . . . . . 9

2 Reactions involving one adsorbate 112.1 Volmer-Heyrovsky (V-H) corrosion reaction . . . . . . . . . . . . 11

2.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . . 122.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . . 13

2.2 (V-H) corrosion reaction with mass transport limitation . . . . . 152.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . . 152.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . . 16

2.3 Volmer-Tafel (V-T) corrosion reaction . . . . . . . . . . . . . . . 192.3.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 Steady-state conditions . . . . . . . . . . . . . . . . . . . 192.3.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . . 20

3

4 CONTENTS

Chapter 1

Electrochemical reactionswithout adsorbed species

1.1 Electrochemical reactions without mass trans-port limitation

1.1.1 Mechanism

M Ko1−→ Mn1+ + n1 e−

O + n2 e Kr2−→ R

1.1.2 Kinetic equations

Butler-Volmer kinetic, kinetic irreversibility of electrochemical reactions (kr1 =ko2 = 0), no mass transport limitation (O(0, t) = O∗).

Ko1 = ko1 exp (αo1 n1 f E) , Kr2 = kr2 exp (−αr2 n2 f E) , f = F/(R T )

Current density vs. step rates

if(t) = F (n1 v1(t) − n2 v2(t))

Step rates

v1(t) = Ko1(t), v2(t) = Kr2(t) O∗

1.1.3 Steady-state conditions

Step rates

v1 = Ko1, v2 = Kr2 O∗

Current density

if = F (n1 Ko1 − n2Kr2 O∗) (1.1)

5

6 CHAPTER 1. REACTIONS WITHOUT ADSORBED SPECIES

1.1.4 Faradaic impedance

Faradaic impedance

Zf(s) = Rct (1.2)

Charge transfer resistance

Rct =1

f F (n22 O∗ Kr2 αr2 + n2

1 Ko1 αo1)(1.3)

Polarization resistance

Rp = Rct

Relation between Rct, Rp and icorr

Rct(Ecorr) = Rp(Ecorr) =1

f F (n22 O∗ Kr2(Ecorr)αr2 + n2

1 Ko1(Ecorr) αo1)

⇒ icorr =1

f (αo1 n1 + αr2 n2) Rct(Ecorr)

Stern-Geary relationship [5].

Electrode impedance

Z(s) =Zf(s)

1 + sCdl Zf(s)=

Rct

1 + sCdl Rct(1.4)

Ecorr

icorr

- icorr

E

i f

ififMifO

0 Rp = Rct

0

Re Z

-Im

Z

Ωc=1HRctCdlL

Figure 1.1: Typical if vs. E curve calculated with Eq. (1.1) and Nyquist impedancediagram calculated at the corrosion potential with Eqs. (1.3)-(1.4).

1.1. REACTIONS WITHOUT MASS TRANSPORT LIMITATION 7

Equivalent circuit

Rct

Cdl

Figure 1.2: Equivalent circuit of the electrode impedance (Eq. (1.4)).


1.2 Electrochemical reactions with mass trans-port limitation

1.2.1 Mechanism

M Ko1−→ Mn1+ + n1 e−

O + n2 e Kr2−→ R

1.2.2 Example: Iron in aerobic solution [4]

Fe → Fe2+ + 2e−

O2 +2H2O + 4e− → 4OH−


Butler-Volmer kinetic, kinetic irreversibility of electrochemical reactions (kr1 =ko2 = 0), mass transport limitation.

Ko1 = ko1 exp (αo1 n1 f E) , Kr2 = kr2 exp (−αr2 n2 f E) , f = F/(R T )

Mass balance equations

Flux of soluble speciesJO(0, t) = −v2(t)


if(t) = F (n1 v1(t) − n2 v2(t))

Step rates

v1(t) = Ko1(t), v2(t) = Kr2(t)O(0, t)


Steady-state equations

Soluble speciesJO(0) = −mO (O∗ − O(0))

mO = DO/δO, δO = 1, 611 D1/3O ν1/6 Ω−1/2, mO = 0, 620D

2/3O ν−1/6 Ω1/2

Steady-state solutions

Soluble species

O(0) =O∗mO

Kr2 + mO

Current density

if = F

(n1 Ko1 −

n2 O∗ Kr2 mO

Kr2 + mO

)(1.5)

1.2. REACTIONS WITH MASS TRANSPORT LIMITATION 9

if = ifM + ifO, ifM = n1 F Ko1, ifO = −n2 F O∗ Kr2 mO

Kr2 + mO

idO = limE→−∞

if = −n2 F O∗ mO

1.2.5 Faradaic impedance [3]

Faradaic impedance

Zf(s) = Rct + ZO(s) (1.6)


Rct =1

f F (n22 O(0)Kr2 αr2 + n2

1 Ko1 αo1)(1.7)

Concentration impedance

Soluble species

ZO(s) =Rct n2

2O(0)MO(s)K2r2αr2

n22O(0)Kr2αr2 + n2

1Ko1αo1 (MO(s)Kr2 + 1)(1.8)

MO(s) =1

mO

th√

τdOs√

τdOs, τdO =

δO

DO

ZO(s) = kO

th√

τdOs√

τdOs

1 + αth

√τdOs

√τdOs

,

α =αo1 n2

1 Ko1 Kr2

mO (αo1 n21 Ko1 + O(0)αr2 n2

2 Kr2), kO =

Rct n22O(0)K2

r2αr2

mO (αo1n21Ko1 + αr2n2

2Kr2O(0))

RO(s) = lims→0

ZO(s) =kO

1 + α


Rp = Rct

(1 +

n22O(0)K2

r2αr2

n22O(0)Kr2mOαr2 + n2

1Ko1αo1 (Kr2 + mO)

)Relation between Rct, Rp and icorr

Rct(Ecorr) =1

f F (n22 O(0, Ecorr) Kr2(Ecorr)αr2 + n2

1 Ko1(Ecorr)αo1)

⇒ icorr =1

f (αo1 n1 + αr2 n2) Rct(Ecorr)

icorr =1

f

(αo1 n1 + αr2 n2

(mO

mO + Kr2(Ecorr)

))Rp(Ecorr)

Limiting case

ko1 → ∞ ⇒ Ecorr → −∞ ⇒ icorr ≈1

f αo1 n1 Rp(Ecorr)= −idO


Electrode impedance

Z(s) =Zf(s)

1 + sCdl Zf(s)(1.9)

Equivalent circuit

Rct ZO

Cdl

∆O

Rct

R'O

Cdl

Figure 1.3: Equivalent circuits of the electrode impedance (Eq. (1.9)). ZWδO=

R′dO

th√

τdOs√

τdOs, R′

dO = kO, R′O = kO/α.

-0.2 Ecorr 0.2

-1

0

1

EV

i fÈi d

OÈ

ififMifO

0 Rct Rp

0

0.5

Re Z

-Im

Z

Ω = 1HRctCdlL

Ω = 2.54Τ

-0.2 Ecorr 0.2

-1

0

1

EV

i fÈi d

OÈ

ififMifO

0 Rct Rp

0

0.4

Re Z

-Im

Z

Ω = 1HRctCdlL

Ω = Α2Τ

Figure 1.4: Typical if vs. E curve calculated with Eq. (1.5) and Nyquist impedancediagrams calculated at the corrosion potential with Eqs. (1.6)-(1.8) [2, 1].

Chapter 2

Electrochemical reactionsinvolving one adsorbate

2.1 Volmer-Heyrovsky (V-H) corrosion reaction

2.1.1 Mechanism

H+ + s + e− Kr1−→ H,s

H+ + H,s + e− Kr2−→ H2 + s

M,s Ko3−→ M2+ + s + 2 e−


No mass transport limitations, Langmuir isotherm

H+(0, t) ≈ H+∗, M, s ≡ s

Kr1 = kr1 exp (−αr1 f E) , kr1 = k′r1 H+∗

Kr2 = kr2 exp (−αr2 f E) , kr2 = k′r2 H+∗, Ko3 = ko3 exp (2 αo3 f E)

Transformation rates

vs(t) = −v1(t) + v2(t), vH(t) = v1(t) − v2(t)


Rate of production of adsorbed species

dθs(t)dt

=vs(t)

Γ,dθH(t)

dt=

vH(t)Γ


if(t) = −F (v1(t) + v2(t) − 2 v3(t))

11

12 CHAPTER 2. REACTIONS INVOLVING ONE ADSORBATE

Step rates

v1(t) = Kr1(t) Γ θs(t), v2(t) = Kr2(t) Γ θH(t), v3(t) = Ko3(t) Γ θs(t)



Adsorbed speciesdθs/dt = 0, θH + θs = 1


Adsorbed species

θs =Kr2

Kr1 + Kr2, θH =

Kr1

Kr1 + Kr2

Current density

if =2F Γ (Ko3 Kr2 − Kr1 Kr2)

Kr1 + Kr2(2.1)

if = 0 ⇒ E = Ecorr =1

(αr1 + 2 αo3) fln

(kr1

ko3

)

icorr =2F Γ Ko3(Ecorr)Kr2(Ecorr)

Kr1(Ecorr) + Kr2(Ecorr)=

2 F ΓKr1(Ecorr) Kr2(Ecorr)Kr1(Ecorr) + Kr2(Ecorr)

=2F Γ kr1 kr2

kr1

(kr1

ko3

) αr2

2αo3 + αr1 + kr2

(kr1

ko3

) αr1

2αo3 + αr1

-0.4 -0.2 Ecorr

-0.02

0

0.02

EV

i fHA

cm-

2L

Figure 2.1: if vs. E curve calculated with Eq. (2.1) for kr1 = 1 s−1, kr2 = 1 s−1,ko3 = 101 s−1, αr1 = 0.8, αr2 = 0.3, αo3 = 0.4, Γ = 10−9 mol cm−2, f = 38.9 V−1,F = 96485 C mol−1.

2.1. VOLMER-HEYROVSKY (V-H) CORROSION REACTION 13


Faradaic impedance

Zf(s) = Rct + ZH(s) + Zs(s)

Zf(s) =(Kr2 + Kr1)(Kr2 + Kr1 + s)

fFΓKr2(αr1Kr1(2(Ko3 + Kr2) + s) + 4αo3Ko3(Kr2 + Kr1 + s) + αr2Kr1(−2Ko3 + 2Kr1 + s))


Rct =1

f F Γ (αr2 θH Kr2 + (4 αo3 Ko3 + αr1 Kr1) θs)=

Kr2 + Kr1

f F Γ Kr2 (4αo3 Ko3 + (αr1 + αr2)Kr1)


Rp =(Kr2 + Kr1)2

2 f F Γ Kr2 (2 αo3 Ko3(Kr2 + Kr1) + Kr1 (αr1 (Ko3 + Kr2) + αr2(Kr1 − Ko3)))

No simple relation between Rp and icorr.

Relation between Rct and icorr

Rct(Ecorr) =1

f F Γ (αr2 θH(Ecorr)Kr2(Ecorr) + (4 αo3 Ko3(Ecorr) + αr1 Kr1(Ecorr)) θs(Ecorr))

⇒ icorr =1

(2αo3 + (αr1 + αr2)/2) f Rct(Ecorr)


0 0.5 1 RctRp

0

0.5

Re Z Rp

-Im

ZR

p

E = Ecorr

Zf

Z

0 0.5 1RctRp

0

0.5

Re Z Rp

-Im

ZR

p

EV = Ecorr-0.04Zf

Z

-2 0 2 4

-1

0

logHΩHradsLL

logÈ

ZR

pÈ

E = Ecorr

Zf

Z

0 5

-0.2

0

logHΩHradsLL

logÈ

ZR

pÈ

EV = Ecorr-0.04Zf

Z

Figure 2.2: Two typical Nyquist impedance and Bode diagrams for the Volmer-Heyrovsky (V-H) corrosion reaction (Thick lines: Faradaic impedance, thin lines:electrode impedance). Parameters value as in Fig. 2.1 and Cdl = 10−5 F.

2.2. (V-H) CORROSION REACTION WITH MASS TRANSPORT LIMITATION15

2.2 Volmer-Heyrovsky (V-H) corrosion reactionwith mass transport limitation

2.2.1 Mechanism

H+ + s + e− Kr1−→ H,s

H+ + H,s + e− Kr2−→ H2 + s

M,s Ko3−→ M2+ + s + 2 e−


Butler-Volmer kinetic, Langmuir isotherm, with mass transfer limitation,M, s ≡ s.

Kr1 = kr1 exp (−αr1 f E) , Kr2 = kr2 exp (−αr2 f E) ,

Ko3 = ko3 exp(αo3 2 f E), f = F/(R T )


Flux of soluble species

JH+(0, t) = −v1(t) − v2(t)


dθs(t)dt

=−v1(t) + v2(t)

Γ,dθH(t)

dt=

v1(t) − v2(t)Γ


if(t) = −F (v1(t) + v2(t) − 2 v3(t))

Step rates

v1(t) = Kr1(t)H+(0, t) Γ θs(t), v2(t) = Kr2(t)H+(0, t) Γ θH(t), v3(t) = Ko3(t) Γ θs(t)



Soluble speciesJH+(0) = −mH+(H+∗ − H+(0))

mH+ = DH+/δH+ , δH+ = 1, 611D1/3

H+ ν1/6 Ω−1/2, mH+ = 0, 620 D2/3

H+ ν−1/6 Ω1/2




Soluble species

H+(0) =H+* (Kr1 + Kr2) mH+

2ΓKr1Kr2 + (Kr1 + Kr2)mH+,

Adsorbed species

θs =Kr2

Kr1 + Kr2, θH =

Kr1

Kr1 + Kr2

Current density

if =2FΓKr2

(mH+ (Kr1 + Kr2)

(Ko3 − H+*Kr1

)+ 2ΓKo3Kr1Kr2

)(Kr1 + Kr2) (mH+ (Kr1 + Kr2) + 2ΓKr1Kr2)

(2.2)

0 0.1 0.2 0.3-0.04

0

0.04

EV

i fHA

cm-

2L

Figure 2.3: if vs. E curve calculated with Eq. (2.2) for kr1 = 107 mol−1 cm3 s−1,kr2 = 107 mol−1 cm3 s−1, ko3 = 10−2 s−1, αr1 = 0.8, αr2 = 0.3, αo3 = 0.5, DH+ = 10−6

cm2 s−1, ν = 10−2 cm2 s−1, Ω = 100 rpm, Γ = 10−9 mol cm−2, f = 38.9 V−1,F = 96485 C mol−1.


Faradaic impedance

Zf(s) = Rct + ZH+(s) + Zθ(s), Zθ(s) = Zs(s) + ZH+(s)

Zf(s) =af1 + saf2 + af3MH+(s) + saf4MH+(s)bf1 + sbf2 + bf3MH+(s) + sbf4MH+(s)

af1 = (Kr1 + Kr2) H+(0), af2 = 1, af3 = 2ΓKr1Kr2H+(0), af4 = Γ (Kr2θH + Kr1θs )

bf1 = 2fFΓH+(0)(θH Kr2αr2

(Kr1H

+(0) − Ko3

)+ θs

(Kr1αr1

(Kr2H

+(0) + Ko3

)+ 2Ko3αo3 (Kr1 + Kr2)

))bf2 = fFΓ

(θH Kr2αr2H

+(0) + Kr1αr1θs H+(0) + 4Ko3αo3θs

)bf3 = 4fFΓ2Ko3Kr1Kr2θs H+(0) (θH (αr1 − αr2) + 2αo3) , bf4 = 4fFΓ2Ko3αo3θs (θH Kr2 + Kr1θs )

2.2. (V-H) CORROSION REACTION WITH MASS TRANSPORT LIMITATION17


Rct =1

fFΓ (θH Kr2αr2H+(0) + Kr1αr1θs H+(0) + 4Ko3αo3θs )


Rp = (mH+ (Kr1 + Kr2) + 2ΓKr1Kr2) /(2fFΓ

(mH+

(θH Kr2αr2

(Kr1H

+(0) − Ko3

)+ θs

(Kr1αr1

(Kr2H

+(0) + Ko3

)+2Ko3αo3 (Kr1 + Kr2))) + 2ΓKo3Kr1Kr2θs (θH (αr1 − αr2) + 2αo3)))

No simple relation between Rp and icorr.

Relation between Rct and icorr

Rct(Ecorr) =1

fFΓ (θH (Ecorr)Kr2(Ecorr)αr2H+(0, Ecorr) + Kr1(Ecorr)αr1θs (Ecorr)H+(0, Ecorr) + 4Ko3(Ecorr)αo3θs (Ecorr))

⇒ icorr =1

(2αo3 + (αr1 + αr2)/2) f Rct(Ecorr)


0 0.5 1

0

0.5

Re Z Rp

-Im

ZR

p

E = Ecorr

Zf

Z

0 0.5 1

0

0.5

Re Z Rp

-Im

ZR

p

EV = Ecorr-0.06Zf

Z

0 5

-0.2

0

logHΩHradsLL

logÈ

ZR

pÈ

E = Ecorr

Zf

Z

0 5

-0.2

0

logHΩHradsLL

logÈ

ZR

pÈ

EV = Ecorr-0.06Zf

Z

Figure 2.4: Two typical Nyquist impedance and Bode diagrams for the Volmer-Heyrovsky (V-H) corrosion reaction with mass transport limitation (Thick lines:Faradaic impedance, thin lines: electrode impedance). Parameters value as in Fig. 2.3and Cdl = 5 × 10−5 F.

2.3. VOLMER-TAFEL (V-T) CORROSION REACTION 19

2.3 Volmer-Tafel (V-T) corrosion reaction

2.3.1 Mechanism

H+ + s + e− Kr1−→ H,s

2 H,s kd2−→ H2 + 2 s

M,s Ko3−→ M2+ + s + 2 e−


No mass transport limitations, Langmuir isotherm, H+(0, t) ≈ H+∗, M, s ≡ s,Kr1 = kr1 exp (−αr1 f E) , kr1 = k′

r1 H+∗ Ko3 = ko3 exp (2αo3 f E) , ko1 =0, kr3 = 0.



dθs(t)dt

=−v1(t) + 2 v2(t)

Γ,dθH(t)

dt=

v1(t) − 2 v2(t)Γ


if(t) = −F (v1(t) − 2 v3(t))

Step rates

v1(t) = Kr1(t) Γ θs(t), v2(t) = kd2 (Γ θH(t))2, v3(t) = Ko3(t) Γ θs(t)





Adsorbed species

θs =4Γkd2 + Kr1 −

√Kr1

√8Γkd2 + Kr1

4Γkd2, θs + θH = 1

Current density

if =4FΓ2kd2 (2Ko3 − Kr1)

4Γkd2 + Kr1 +√

Kr1

√8Γkd2 + Kr1

(2.3)

if = 0 ⇒ E = Ecorr =1

f (2αo3 + αr1)log

(kr1

2ko3

)


icorr =23−exp2FΓ2kd2ko3

(ko3kr1

)− exp2

2exp1kr1

(ko3kr1

)exp1 + 4Γkd2 +

√2exp1

(ko3kr1

)exp1kr1

√2exp1kr1

(ko3kr1

)exp1 + 8Γkd2

withexp1 =

αr1

2αo3 + αr1, exp2 =

2αo3

2αo3 + αr1

-0.4 -0.2 Ecorr

0

0.05

EV

i fHm

Acm-

2L

Figure 2.5: if vs. E curve calculated with Eq. (2.3) for kr1H+∗ = 10−3 s−1, kd2 = 108

mol−1 cm2 s−1, ko3 = 10−1 s−1, αr1 = 0.7, αo3 = 0.4, n3 = 1, Γ = 10−9 mol cm−2,f = 38.9 V−1, F = 96485 C mol−1.


Transfer resistance

Rct =1

fFΓ (4Ko3αo3 + Kr1αr1) θs

Concentration impedance

Zs(s) =Kr1 (Kr1 − 2Ko3)Rctαr1

2Ko3Kr1αr1 + Kr1 (s + 4Γkd2θH) αr1 + 4Ko3αo3 (s + Kr1 + 4Γkd2θH)

Faradaic impedance

Zf(s) = Rct + Zs(s)

Zf(s) = Rct

(1 +

Kr1 (Kr1 − 2Ko3)αr1

2Ko3Kr1αr1 + Kr1 (s + 4Γkd2θH) αr1 + 4Ko3αo3 (s + Kr1 + 4Γkd2θH)

)Polarization resistance

Rp = Rct

(1 +

Kr1 (Kr1 − 2Ko3)Rctαr1

2Ko3Kr1 (2αo3 + αr1) + 4Γkd2 (4Ko3αo3 + Kr1αr1) θH

)No simple relation between Rp(Ecorr) and icorr.

Relation between Rct(Ecorr) and icorr

icorr =1

f (2αo3 + αr1) Rct(Ecorr)

2.3. VOLMER-TAFEL (V-T) CORROSION REACTION 21

Equivalent circuit

Zs =Rs

1 + Rs Cs s, Cs =

4Ko3αo3 + Kr1αr1

Kr1 (Kr1 − 2Ko3) Rctαr1

Rs =Kr1 (Kr1 − 2Ko3)Rctαr1

2Ko3Kr1 (2αo3 + αr1) + 4Γkd2 (4Ko3αo3 + Kr1αr1) θA

0 0.5 1

0

0.45

Re Z Rp

-Im

ZR

p

E = Ecorr

Zf

Z

0 0.5 1

0

0.45

Re Z Rp

-Im

ZR

p

EV = Ecorr+0.04

Zf

Z

-2 0 2

-1

0

logHΩHradsLL

logÈ

ZR

pÈ

E = Ecorr

Zf

Z

-2 0 2

-0.5

0

logHΩHradsLL

logÈ

ZR

pÈ

EV = Ecorr+0.04

Zf

Z

Figure 2.6: Two typical Nyquist impedance and Bode diagrams for the Volmer-Tafel (V-T) corrosion reaction (Thick lines: Faradaic impedance, thin lines: electrodeimpedance). Parameters value as in Fig. 2.5 and Cdl = 1 × 10−5 F.


Rct

Cdl

Cs

Rs

Figure 2.7: Equivalent circuit for the impedance of (V-T) corrosion reaction.

Bibliography

[1] Diard, J.-P., B. Le Gorrec, and Montella, C. Hand-book of electrochemical impedance spectroscopy. Diffusion Impedances.www.bio-logic.info/potentiostat/notes.html (2012).

[2] Macdonald, J. R. Impedance spectroscopy: old problems and new devel-opments. Electrochim. Acta 35 (1990), 1483–1492.

[3] Montella, C., Diard, J.-P., and B. Le Gorrec. Exercices de cinetiqueelectrochimique. II. Methode d’impedance. Hermann, Paris, 2005.

[4] Orazem, M. E., and Tribollet, B. Electrochemical impedance spec-troscopy. John Wiley & Sons, Hoboken, 2008.

[5] Stern, M., and Geary, A. J. Electrochemical polarization. I. A theo-retical analysis of the shape of polarization curves. J. Electrochem. Soc 104(1957), 56–63.

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