Electrochemical Impedance Spectroscopy Library

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Electrochemical

Impedance Spectroscopy

Library

∆O ∆R

Rt

Cdc

ZO ZR

ER@SE

December 28, 2003



ER@SE/December 28, 2003 2



Contents

1 Reactions involving soluble species only 51.1 Redox reaction (E) . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 51.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 61.1.5 RDE (diffusion-convection) . . . . . . . . . . . . . . . . 6

1.1.6 Warburg conditions (semi-infinite linear diffusion) . . . 71.2 EE reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Kinetic equations, without

coupled homogeneous reactions . . . . . . . . . . . . . . 81.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 81.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 9

2 Reactions involving one adsorbate 112.1 Electroadsorption reaction (EAR) . . . . . . . . . . . . . . . . . . 11

2.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 112.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 122.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 12

2.2 Dissolution-passivation reaction . . . . . . . . . . . . . . . . . . . 122.2.1 Mechanism [7] . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 132.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 13

2.3 Volmer-Heyrovsky (V-H) reaction . . . . . . . . . . . . . . . . . . 142.3.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 152.3.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 15

2.4 Volmer-Tafel (V-T) reaction . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 162.4.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 17

2.5 Volmer-Heyrovsky-Tafel (V-H-T) reaction . . . . . . . . . . . . . 172.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 17

3



2.5.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 182.5.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 182.5.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 19

3 Reactions involving two adsorbates 213.1 Volmer-Heyrovsky with chemical desorption . . . . . . . . . . . . 21

3.1.1 Mechanism [6, 3, 4] . . . . . . . . . . . . . . . . . . . . . 213.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 213.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 223.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 22

3.2 Schuhmann dissolution-passivationreaction # 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Mechanism [7] . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 233.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 24

4 Reactions involving both adsorbed and soluble species 254.1 Electroadsorption reaction (EAR) with

limitation by mass transport . . . . . . . . . . . . . . . . . . . . 254.1.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 264.1.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 26

4.2 Electrosorption-desorption reaction . . . . . . . . . . . . . . . . . 264.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 274.2.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 274.2.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 28

4.3 (V-H) reaction with mass transport limitation . . . . . . . . . . . 284.3.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 28

4.3.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 294.3.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 30

4.4 Copper dissolution in HCl . . . . . . . . . . . . . . . . . . . . . . 314.4.1 Mechanism [5, 1, 2] . . . . . . . . . . . . . . . . . . . . . 314.4.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 314.4.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 324.4.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 32

4.5 (V-T) reaction with mass transfer limitation . . . . . . . . . . . . 334.5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . 334.5.3 Steady-state conditions . . . . . . . . . . . . . . . . . . 344.5.4 Faradaic impedance . . . . . . . . . . . . . . . . . . . . 35




Chapter 1

Reactions involving solublespecies only

1.1 Redox reaction (E)

1.1.1 Mechanism

O + eKr←→Ko

R

K r = kr exp(−αr f E ) = ko exp(−αr f (E −E ◦))

K o = ko exp(αo f E ) = ko exp(αo f (E −E ◦)) , f = F/(R T ), αo + αr = 1

1.1.2 Kinetic equations

Transformation rates

vO(t) =

−v(t), vR(t) = v(t)

Mass balance equations

Flux of soluble species : J O(0, t) = vO(t), J R(0, t) = vR(t)

Current density vs. reaction rate

if (t) = −F v(t)

Reaction rate

v(t) = −R(0, t) K o(t) + O(0, t) K r(t)

1.1.3 Steady-state conditionsSteady-state equations

Soluble species

J O(0) = − (O∗ −O(0)) mO, J R(0) = − (R∗ −R(0)) mR

5



Steady-state solutions

Soluble species

R(0) =R∗ + K r (R∗/mO + O∗/mR)

1 + K o/mR + K r/mO

, O(0) =O∗ + K o (R∗/mO + O∗/mR)

1 + K o/mR + K r/mO

Current density

if =F (K o R∗ −K r O∗)

1 + K o/mR + K r/mO

1.1.4 Faradaic impedance

1.1.5 RDE (diffusion-convection)

Faradaic impedance

Faradaic impedance

Z f (s) = Rct + Z O(s) + Z R(s)

Z f (s) = 1 + K r M O(s) + K o M R(s)f F (R(0) K o αo + O(0) K r αr)

Charge transfer resistance

Rct =1

f F (R(0) K o αo + O(0) K r αr)

Concentration impedances (with ∂ Xv =∂v

∂X )

Z O = −∂ Ov M O(s)

∂ Ev= K r Rct M O(s)

Z R = ∂ Rv M R(s)∂ Ev = K o Rct M R(s)

M O(s) =1

mO

th√

τ dO s√

τ dO s, M R(s) =

1

mR

th√

τ dR s√

τ dR s

Equivalent circuit (Fig. 1.1)

∆O ∆R

Rt

Cdc

ZO ZR

Figure 1.1: Equivalent circuit for the impedance of redox reactions (RDE).




1.1.6 Warburg conditions (semi-infinite linear diffusion)

Faradaic impedance

Z f (s) = Rct + Z O(s) + Z R(s)

only at the equilibrium potential:

E = E eq = E ◦ +1

f ln

O∗

R∗

Z f (s) =1 + K r M O(s) + K o M R(s)

f F (R∗ K o αo + O∗ K r αr)


Rct =1

f F (R∗ K o αo + O∗ K r αr)

Concentration impedances

Z O(s) = K r Rct M O(s), Z R(s) = K o Rct M R(s)

M O(s) =1√

s DO

, M R(s) =1√

s DR

Equivalent circuit (Fig. 1.2)

O RRt

Cdc

ZO ZR

Figure 1.2: Equivalent circuit for the impedance of redox reactions: Warburg condi-tions.




1.2 EE reaction

1.2.1 Mechanism

RKo1←→Kr1

X + e−

XKo2←→

Kr2O + e−

K o1 = ko1 exp(αo1 f E ) = ko1 exp(αo1 f (E −E o1))

K r1 = kr1 exp(−αr1 f E ) = ko1 exp(−αr1 f (E −E o1)) , αo1 + αr1 = 1

K o2 = ko2 exp(αo2 f E ) = ko2 exp(αo2 f (E −E o2))

K r2 = kr2 exp(−αr2 f E ) = ko2 exp(−αr2 f (E −E o2)) , αo2 + αr2 = 1

1.2.2 Kinetic equations, withoutcoupled homogeneous reactions


vR(t) = −v1(t), vX(t) = v1(t)− v2(t), vO(t) = v2(t)


Flux of soluble species

J R(0, t) = vR(t), J X(0, t) = vX(t), J O(0, t) = vO(t)

Current density vs. step rates

if (t) = F (v1(t) + v2(t))

Step rates

v1(t) = R(0, t) K o1(t)−X (0, t) K r1(t), v2(t) = X (0, t) K o2(t)−O(0, t) K r2(t)

1.2.3 Steady-state conditions

Steady-state equations

J R(0) = − (R∗ −R(0)) mR, J X(0) = − (X ∗ −X (0)) mX, J O(0) = − (O∗ −O(0)) mO


Soluble species

R(0) = R∗ +R∗ K r2

mO

+X ∗ K r1

mR

+X ∗ K r1 K r2

mO mR

+

R∗ K o2mX

+R∗ K r1

mX

+R∗ K r1 K r2

mO mX

+O∗ K r1 K r2

mR mX

/

1 +K r2mO

+K o1mR

+K o1 K r2mO mR

+K o2mX

+K r1mX

+K r1 K r2mO mX

+K o1 K o2mR mX




X (0) =

X ∗ +

X ∗ K r2mO

+X ∗ K o1

mR

+X ∗ K o1 K r2

mO mR

+

R∗ K o1mX

+O∗ K r2

mX

+R∗ K o1 K r2

mO mX

+O∗ K o1 K r2

mR mX

/

1 +K r2

mO

+K o1

mR

+K o1 K r2

mO mR

+K o2

mX

+K r1

mX

+K r1 K r2

mO mX

+K o1 K o2

mR mX

O(0) =

O∗ +

X ∗ K o2mO

+O∗ K o1

mR

+X ∗ K o1 K o2

mO mR

+

O∗ K o2mX

+O∗ K r1

mX

+R∗ K o1 K o2

mO mX

+O∗ K o1 K o2

mR mX

/

1 +K r2mO

+K o1mR

+K o1 K r2mO mR

+K o2mX

+K r1mX

+K r1 K r2mO mX

+K o1 K o2mR mX

Current density

if =

K o1 R

∗

+ K o2 X

∗

+

K o1 K o2 X ∗

mR +

K o1 K r2 R∗

mO +

2 K o1 K o2 R∗

mX −

K r1 X ∗ −K r2 O∗ − K r1 K r2 X ∗

mO

− K o1 K r2 O∗

mR

− 2 K r1 K r2 O∗

mX

/

1 +K r2mO

+K o1mR

+K o1 K r2mO mR

+K o2mX

+K r1mX

+K r1 K r2mO mX

+K o1 K o2mR mX


Faradaic impedance

Z f (s) = Rct + Z O(s) + Z R(s) + Z X(s)

Z f (s) = (1 + K o1 M O(s)) (1 + K r2 M R(s)) + (K o2 (1 + K o1 M O(s)) + K r1 (1 + K r2 M R(s))) M X(s)/

(f F (X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))

+X (0) K o2 αo2 (1 + 2 K r1 M X(s)) + R(0) K r2 αr2 (1 + 2 K r1 M X(s))

+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s)))))


Rct =1

f F (O(0) K o1 αo1 + X (0) (K o2 αo2 + K r1 αr1) + R(0) K r2 αr2)


Z O(s) = Rct K o1 M O(s) (O(0) K o1 αo1 (1 + K r2 M R(s) + K o2 M X(s))

+K r1 (X (0) αr1 (1 + K r2 M R(s)) + X (0) K o2 (αo2 + αr1) M X(s) + R(0) K r2 αr2 M X(s))) /

(X (0) K r1 αr1 (1 + K r2 M R(s) + 2 K o2 M X(s))


+K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) + O(0) αo1 (1 + K r2 M R(s) + 2 K o2 M X(s))))




Z X(s) = Rct (K o2 −K r1) M X(s) (X (0) K o2 αo2 −X (0) K r1 αr1 (1 + K r2 M R(s))

+R(0) K r2 αr2 + K o1 ((X (0) K o2 αo2 + R(0) K r2 αr2) M O(s) − αo1 O(0) (1 + K r2 M R(s)))) /




Z R(s) = Rct K r2 M R(s) (R(0) K r2 αr2 (1 + K o1 M O(s) + K r1 M X(s))

+K o2 (X (0)αo2 + X (0)K r1 (αo2 + αr1) M X(s) + K o1 (X (0)αo2M O(s) + O(0)αo1M X(s)))) /







Chapter 2

Reactions involving oneadsorbate

2.1 Electroadsorption reaction (EAR)

2.1.1 Mechanism

A− + sKo←→Kr

A,s + e−


No mass transport limitations, Langmuir isotherm

A−(0, t) ≈ A−∗, K o = ko A−∗ exp(αo f E ) , K r = kr exp(−αr f E )


vA−(t) = −v1(t), vs(t) = −v1(t), vA(t) = v1(t)


Rate of production of adsorbed species

dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = F v(t)

Reaction rate

v(t) = θs(t) Γ K o(t)− θA(t) Γ K r(t)

11





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


Adsorbed species

θs =K r

K o + K r, θA =

K oK o + K r

Current densityif = 0


Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)

Z f (s) =s + K o + K r

f F Γ s (θs K o αo + θA K r αr)=

(K o + K r) (s + K o + K r)

f F s Γ K o K r


Rct =1

f F Γ (θs K o αo + θA K r αr)=

K o + K rf F Γ K o K r

Concentration impedancesAdsorbed species

Z A(s) =Γ K r Rct

s=

K o + K rf F s Γ K o

, Z s(s) =K o Rct

s=

K o + K rf F s Γ K r

2.2 Dissolution-passivation reaction

2.2.1 Mechanism [7]

M,sKo1←→Kr1

M2+ + s + 2 e−

A− + sKo2←→Kr2

A,s + e−



M 2+(0, t) ≈ M 2+∗, A−(0, t) ≈ A−∗

K o1 = ko1 exp(2 αo1 f E ) , K r1 = kr1 M 2+∗ exp(−2 αr1 f E )

K o2 = ko2 A−∗ exp(αo2 f E ) , K r2 = kr2 exp(−αr2 f E )




Transformation rates (vA stands for vA,s)

vs(t) = −v2(t), vA(t) = v2(t)



dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = F (2 v1(t) + v2(t))

Step rates

v1(t) = θs(t) Γ K o1(t)− θs(t) Γ K r1(t), v2(t) = θs(t) Γ K o2(t)− θA(t) Γ K r2(t)





Adsorbed species

θs =K o2

K o2 + K r2, θA =

K r2K o2 + K r2

Current density

if =2 F Γ (K o1 −K r1) K r2

K o2 + K r2


Faradaic impedance

Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)

Z f (s) = (s + K o2 + K r2) /

(f F Γ (θs (K o2 (s + 2 K r1) αo2 + 2 K o1 (2 (s + K o2 + K r2) αo1 −K o2 αo2) +

4 K r1 (s + K o2 + K r2) αr1) + θA (s−

2 K o1 + 2 K r1) K r2 αr2))

Z f (s) =

(K o2 + K r2) (s + K o2 + K r2)

f F Γ K r2 (4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1)))





Rct =1

f F Γ (4 θs K o1 αo1 + θs K o2 αo2 + 4 θs K r1 αr1 + θA K r2 αr2)

Rct =K o2 + K r2

f F Γ K r2 (K o2 + 4 K o1 αo1 + 4 K r1 αr1)Concentration impedances

Z A(s) = K r2 Rct (θs K o2 αo2 + θA K r2 αr2)/

(θs (K o2 (s + 2 K r1) αo2 + 2 K o1 (2 (s + K o2 + K r2) αo1 −K o2 αo2) +

4 K r1 (s + K o2 + K r2) αr1) + θA (s− 2 K o1 + 2 K r1) K r2 αr2)

Z A(s) =K o2 K r2 Rct

4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1))

Z s(s) = −(2 K o1 + K o2 − 2 K r1) Rct (θs K o2 αo2 + θA K r2 αr2)/

(θs (−4 K o1 (s + K o2 + K r2) αo1 −K o2 (s− 2 K o1 + 2 K r1) αo2−4 K r1 (s + K o2 + K r2) αr1)− θA (s− 2 K o1 + 2 K r1) K r2 αr2)

Z s(s) =K o2 (2 K o1 + K o2 − 2 K r1) Rct

4 (s + K r2) (K o1 αo1 + K r1 αr1) + K o2 (s + K o1 (−2 + 4 αo1) + K r1 (2 + 4 αr1))

2.3 Volmer-Heyrovsky (V-H) reaction

2.3.1 Mechanism

A+ + s + e−Kr1←→Ko1

A,s

A+ + A,s + e−Kr2←→Ko2

A2 + s



A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗

2

K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )

K r2 = kr2 A+∗ exp(−αr2 f E ) , K o2 = ko2 A∗

2 exp(αo2 f E )


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






dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = −F (v1(t) + v2(t))

Step rates

v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t), v2(t) = −θs(t) Γ K o2(t) + θA(t) Γ K r2(t)



Adsorbed species

dθs/dt = 0, θA + θs = 1


Adsorbed species

θs =K o1 + K r2

K o1 + K o2 + K r1 + K r2, θA =

K o2 + K r1K o1 + K o2 + K r1 + K r2

Current density

if =2 F Γ (K o1 K o2 −K r1 K r2)

K o1 + K o2 + K r1 + K r2


Faradaic impedance

Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)


Rct =1

f F Γ (θA K o1 αo1 + θs K o2 αo2 + θs K r1 αr1 + θA K r2 αr2)


Z A(s) = (K o1 −K r2) Rct (θA K o1 αo1 − θs K o2 αo2 + θs K r1 αr1 − θA K r2 αr2)/

(θs K o2 (s + 2 K r1) αo2 + θs K r1 (s + 2 K o2 + 2 K r2) αr1 + θA (s + 2 K r1) K r2 αr2+

K o1 (θA (s + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))




Z s(s) = (K o2 −K r1) Rct (− θA K o1 αo1 + θs K o2 αo2 − θs K r1 αr1 + θA K r2 αr2)/

(θs K o2 (s + 2 K r1) αo2 + θs K r1 (s + 2 K o2 + 2 K r2) αr1 + θA (s + 2 K r1) K r2 αr2+

K o1 (θA (s + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))

2.4 Volmer-Tafel (V-T) reaction

2.4.1 Mechanism

A+ + s + e−Kr1←→Ko1

A,s

2A,skd2←→kg2

A2 + 2 s



A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗

2

K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E ) , kg2 = kg2 A∗

2


vs(t) = −v1(t) + 2 v2(t), vA(t) = v1(t)− 2 v2(t)



dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = −F v1(t)

Step rates

v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t), v2(t) = θA(t)2

Γ2 kd2 − θs(t)2

Γ2 kg2



Adsorbed species

dθs/dt = 0, θA + θs = 1





Adsorbed species

θA =

−

4 Γ kg2 + K o1 + K r1 −

8 Γ kg2 K o1 + 8 Γ kd2 (2 Γ kg2 + K r1) + (K o1 + K r1)2

4 Γ (kd2 − kg2)

Current density

if =F

4 (kd2 − kg2)

−

4 Γ kg2 K o1 + 4 Γ kd2 K r1 + (K o1 + K r1)2

+

(K o1 + K r1)

8 Γ kg2 K o1 + 8 Γ kd2 (2 Γ kg2 + K r1) + (K o1 + K r1)

2


Faradaic impedance

Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)

Z f (s) =s + 4 θA Γ kd2 + 4 θs Γ kg2 + K o1 + K r1

F f Γ (s + 4 θA Γ kd2 + 4 θs Γ kg2) (θA K o1 αo1 + θs K r1 αr1)


Rct =1

f F Γ (θA K o1 αo1 + θs K r1 αr1)


Z A(s) =K o1 Rct

s + 4 θA Γ kd2 + 4 θs Γ kg2, Z s(s) =

K r1 Rct

s + 4 θA Γ kd2 + 4 θs Γ kg2

2.5 Volmer-Heyrovsky-Tafel (V-H-T) reaction

2.5.1 Mechanism

A+ + s + e−Kr1←→Ko1

A,s

A+ + A,s + e−Kr2←→Ko2

A2 + s

2A,skd3←→kg3

A2 + 2 s






A+(0, t) ≈ A+∗, A2(0, t) ≈ A∗

2

K r1 = kr1 A+∗ exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )

K r2 = kr2 A+∗

exp(−αr2 f E ) , K o2 = ko2 A∗

2 exp(αo2 f E ) , kg3 = k

g3 A∗

2


vs(t) = −v1(t) + v2(t) + 2 v3(t), vA(t) = v1(t)− v2(t)− 2 v3(t)



dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = −F (v1(t) + v2(t))

Step rates

v1(t) = −θA(t) Γ K o1(t) + θs(t) Γ K r1(t)

v2(t) = −θs(t) Γ K o2(t) + θA(t) Γ K r2(t)

v3(t) = θA(t)2

Γ2 kd3 − θs(t)2

Γ2 kg3





Adsorbed species

θA =1

4 Γ (kg3 − kd3)(4 Γ kg3 + K o1 + K o2 + K r1 + K r2−

8 Γ (kd3 − kg3) (2 Γ kg3 + K o2 + K r1) + (4 Γ kg3 + K o1 + K o2 + K r1 + K r2)2

Current density

if = F 4 (kd3 − kg3)

4 Γ kd3 (K o2 −K r1)− (K o1 + K r1)2 + 4 Γ kg3 (−K o1 + K r2) +

(K o2 + K r2)2

+ (K o1 −K o2 + K r1 −K r2)

×

8 Γ (kd3 − kg3) (2 Γ kg3 + K o2 + K r1) + (4 Γ kg3 + K o1 + K o2 + K r1 + K r2)2





Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)

Z f (s) = (s + 4 θA Γ kd3 + 4 θs Γ kg3 + K o1 + K o2 + K r1 + K r2) /

(f F Γ (θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2

+θs K r1 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αr1

+θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) K r2 αr2

+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2))))


Rct =1

f F Γ (θA K o1 αo1 + θs K o2 αo2 + θs K r1 αr1 + θA K r2 αr2)


Z A(s) = (K o1 −K r2) Rct (θA K o1 αo1 − θs K o2 αo2 + θs K r1 αr1 − θA K r2 αr2)/

(θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2



+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))

Z s(s) = (K o2 −K r1) Rct (− θA K o1 αo1 + θs K o2 αo2 − θs K r1 αr1 + θA K r2 αr2)/

(θs K o2 (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K r1) αo2



+K o1 (θA (s + 4 θA Γ kd3 + 4 θs Γ kg3 + 2 K o2 + 2 K r2) αo1 + 2 (θs K o2 αo2 + θA K r2 αr2)))







Chapter 3

Reactions involvingtwo adsorbates

3.1 Volmer-Heyrovsky with chemical desorption

3.1.1 Mechanism [6, 3, 4]

A+ + s + e−Kr1−→ A,s

A+ + A,s + e−Kr2−→ A2,s

A2,skd3−→ A2 + s


No mass transfer limitations, Langmuir isotherm

A+(0, t) ≈ A+∗

K r1 = kr1 A+∗ exp(−αr1 f E ) , K r2 = kr2 A+∗ exp(−αr2 f E )


vs(t) = −v1(t) + v3(t), vA(t) = v1(t)− v2(t), vA2(t) = v2(t)− v3(t)



dθs(t)dt = vs(t)Γ , dθA(t)dt = vA(t)Γ , dθA2(t)dt = vA2(t)Γ


if (t) = −F (v1(t) + v2(t))

21



Step rates

v1(t) = θs(t) Γ K r1(t), v2(t) = θA(t) Γ K r2(t), v3(t) = θA2(t) Γ kd3



Adsorbed species

dθs/dt = 0, dθA/dt = 0, θA + θA2+ θs = 1


Adsorbed species

θA =kd3 K r1

K r1 K r2 + kd3 (K r1 + K r2), θA2

=K r1 K r2

K r1 K r2 + kd3 (K r1 + K r2)

Current density

if = −2 F Γ kd3 K r1 K r2K r1 K r2 + kd3 (K r1 + K r2)


Faradaic impedance

Z f (s) = Rct + Z A(s) + Z s(s)


Rct =1

f F Γ (θs K r1 αr1 + θA K r2 αr2)


Z A(s) =K r2 Rct (− (θs (s + kd3) K r1 αr1) + θA (s + kd3 + K r1) K r2 αr2)

θs (s + kd3) K r1 (s + 2 K r2) αr1 + θA (s (s + kd3) + (s + 2 kd3) K r1) K r2 αr2

Z s(s) =K r1 Rct (θs K r1 (s + K r2) αr1 − kd3 (−θs K r1 αr1 + θA K r2 αr2))

θs (s + kd3) K r1 (s + 2 K r2) αr1 + θA (s (s + kd3) + (s + 2 kd3) K r1) K r2 αr2

3.2 Schuhmann dissolution-passivationreaction # 1

3.2.1 Mechanism [7]

M,sKo1←→

Kr1X,s + 2 e

X,sKo2←→Kr2

Q,s + 2 e

X,s + AKo3−→ X,s + B + 2 e





No mass transfer limitations, Langmuir isotherm

A(0, t) ≈ A∗

K o1 = ko1 exp(2 αo1 f E ) , K r1 = kr1 exp(−2 αr1 f E )K o2 = ko2 exp(2 αo2 f E ) , K r2 = kr2 exp(−2 αr2 f E ) , K o3 = ko3 exp(2 αo3 f E )


vs(t) = −v1(t), vX(t) = v1(t)− v2(t), vQ(t) = v2(t)



dθs(t)

dt

=vs(t)

Γ

,dθX(t)

dt

=vX(t)

Γ

,dθQ(t)

dt

=vQ(t)

Γ


if (t) = 2 F (v1(t) + v2(t) + v3(t))

Step rates

v1(t) = θs(t) Γ K o1(t)− θX(t) Γ K r1(t)

v2(t) = θX(t) Γ K o2(t)− θQ(t) Γ K r2(t)

v3(t) = θX(t) Γ K o3(t)



Adsorbed species

dθs/dt = 0, dθX/dt = 0, θQ + θs + θX = 1


Adsorbed species

θQ =K o1 K o2

K r1 K r2 + K o1 (K o2 + K r2)

, θX =K o1 K r2

K r1 K r2 + K o1 (K o2 + K r2)

Current density

if =2 F Γ K o1 K o3 K r2

K r1 K r2 + K o1 (K o2 + K r2)





Faradaic impedance

Z f (s) = Rct + Z Q(s) + Z s(s) + Z X(s)


Rct =1

4 f F Γ (θs K o1 αo1 + θX K o2 αo2 + θX K o3 αo3 + θX K r1 αr1 + θQ K r2 αr2)


Z Q(s) = K r2 Rct (K o2 (θX (s + K r1) αo2 + K o1 (θs αo1 + θX αo2) +

θX K r1 αr1) + θQ (s + K o1 + K r1) K r2 αr2) /θX s2 K o3 αo3 + θX s K o3 K r1 αo3 + θX s K o3 K r2 αo3 + θX K o3 K r1 K r2αo3+

θX s2 K r1 αr1 + θX s K o3 K r1 αr1 + θX s K r1 K r2 αr1 + θX K o3 K r1 K r2 αr1+

θX s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +

θQ s2 K r2 αr2 − θQ s K o3 K r2 αr2 + 2 θQ s K r1 K r2 αr2+K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX s K o3 αo3+

θX K o3 K r2 αo3 + θX K o2 ((s−K o3) αo2 + K o3 αo3) + θQ s K r2 αr2 − θQ K o3 K r2 αr2))

Z s(s) = K o1 Rct (θs K o1 (s + K o2 + K r2) αo1+

K r1 (θX K o2 αo2 + θX (s + K o2 + K r2) αr1 + θQ K r2 αr2)) /θX s2 K o3 αo3 + θX s K o3 K r1 αo3 + θX s K o3 K r2 αo3 + θX K o3 K r1 K r2αo3+

θX s2 K r1 αr1 + θX s K o3 K r1 αr1 + θX s K r1 K r2 αr1 + θX K o3 K r1 K r2 αr1+

θX s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +

θQ s2 K r2 αr2

−θQ s K o3 K r2 αr2 + 2 θQ s K r1 K r2 αr2+

K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX s K o3 αo3+

θX K o3 K r2 αo3 + θX K o2 ((s−K o3) αo2 + K o3 αo3) + θQ s K r2 αr2 − θQ K o3 K r2 αr2))

Z X(s) = (K o2 + K o3 −K r1) Rct (θX s K o2 αo2 − θX K r1 (s + K r2) αr1 + θQ s K r2 αr2+

K o1 (− (θs (s + K r2) αo1) + θX K o2 αo2 + θQ K r2 αr2)) /

(θX (s K o2 ((s−K o3 + 2 K r1) αo2 + K o3 αo3 + 2 K r1 αr1) +

(s + K r2) (K o3 (s + K r1) αo3 + (s + K o3) K r1 αr1)) + θQ s (s−K o3 + 2 K r1) K r2 αr2+

K o1 (θs (2 s K o2 + (s + K o3) (s + K r2)) αo1 + θX K o2 (s−K o3) αo2+

θX K o3 (s + K o2 + K r2) αo3 + θQ (s−K o3) K r2 αr2))




Chapter 4

Reactions involvingboth adsorbedand soluble species

4.1 Electroadsorption reaction (EAR) withlimitation by mass transport

4.1.1 Mechanism

A− + sKo←→Kr

A,s + e−


Langmuir isotherm : K o = ko exp(αo f E ) , K r = kr exp(−αr f E )


vA−(t) = −v1(t), vs(t) = −v1(t), vA(t) = v1(t)



J A−(0, t) = vA−(t)


dθs(t)dt = vs(t)Γ, dθA(t)dt = vA(t)Γ


if (t) = F v(t)

25



Reaction rate

v(t) = A−(0, t) θs(t) Γ K o(t)− θA(t) Γ K r(t)



Soluble species

J A−(0) = −

A−∗ −A−(0)

mA−



Soluble speciesA−(0) = A−∗

Adsorbed species

θs =K r

A−∗ K o + K r, θA =

A−∗ K oA−∗ K o + K r

Current densityif = 0


Faradaic impedance

Z f (s) = Rct + Z A−(s) + Z A(s) + Z s(s)


Rct =

1

f F Γ (A−∗ θs K o αo + θA K r αr)

Concentration impedancesSoluble species

Z A−(s) = θs Γ K o Rct M A−(s), M A−(s) =1

mA−

th√

τ A−s√

τ A−s

Adsorbed species

Z s(s) =A−∗ K o Rct

s, Z A(s) =

Γ K r Rct

s

4.2 Electrosorption-desorption reaction

4.2.1 Mechanism

A− + sKo1←→Kr1

A,s + e−

A,skd2−→ A + s





Langmuir isotherm: K o1 = ko1 exp(αo1 f E ) , K r1 = kr1 exp(−αr1 f E )


vA−(t) =

−v1

(t), vs(t) =

−v1

(t) + v2

(t), vA

(t) = v1

(t)−

v2

(t)



J A−(0, t) = vA−(t)


dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ


if (t) = F v1(t)

Step rates

v1(t) = A−(0, t) θs(t) Γ K o1(t)− θA(t) Γ K r1(t), v2(t) = θA(t) Γ kd2



Soluble speciesJ A−(0) = −

A−∗ −A−(0)

mA−



Soluble species

A−(0) =1

2 K o1 mA−

A−∗

K o1 mA−−K r1 mA−− kd2 (Γ K o1 + mA−) + 4 Γ kd2 K o1 (kd2 + K r1) mA−+ ((A−∗ K o1 + K r1) mA−+ kd2 (−Γ K o1 + mA−))

2

Adsorbed species

θA =1

2 Γ kd2 K o1

A−∗

K o1 mA−+ K r1 mA−+ kd2 (Γ K o1 + mA−)− 4 Γ kd2 K o1 (kd2 + K r1) mA−+ ((sA−∗ K o1 + K r1) mA−+ kd2 (−Γ K o1 + mA−))

2




Current density

if =F

2 K o1

A−∗

K o1 + K r1

mA−+ kd2 (Γ K o1 + mA−)−

4 Γ kd2 K o1 (kd2 + K r1) mA−+ A−∗

K o1 + K r1 mA−+ kd2 (−Γ K o1 + mA−)2


Faradaic impedance

Z f (s) = Rct + Z A−(s) + Z A(s) + Z s(s)

Z f (s) =s + kd2 + A−(0) K o1 + K r1 + θs Γ (s + kd2) K o1 M A−(s)

f F Γ (s + kd2) (A−(0) θs K o1 αo1 + θA K r1 αr1)

M A−(s) =1

mA−

th√

τ A−s√

τ A−s


Rct = 1f F Γ (A−(0) θs K o1 αo1 + θA K r1 αr1)


Z A−(s) = θs Γ K o1 Rct M A−(s)

Adsorbed species

Z s(s) =A−(0) K o1 Rct

s + kd2, Z A(s) =

K r1 Rct

s + kd2

4.3 (V-H) reaction with mass transport limita-tion

4.3.1 Mechanism

A+ + s + e−Kr1←→Ko1

A,s

A+ + A,s + e−Kr2←→Ko2

A2 + s


Langmuir isotherm

K r1 = kr1 exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )

K r2 = kr2 exp(−αr2 f E ) , K o2 = ko2 exp(αo2 f E )





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



J A+(0, t) = vA+(t), J A2(0, t) = vA2

(t)

Rate of productions of adsorbed species

dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ

Current density vs. step rate

if (t) = −F (v1(t) + v2(t))

Step rates

v1(t) =

−θA(t) Γ K o1(t) + A+(0, t) θs(t) Γ K r1(t)

v2(t) = −A2(0, t) θs(t) Γ K o2(t) + A+(0, t) θA(t) Γ K r2(t)



Soluble species

J A+(0) = −

A+∗ −A+(0)

mA+, J A2(0) = − (A2

∗ −A2(0)) mA2


Steady-state solutionSoluble species (1)

A0 mA K r1 K r2 mA2A

K o1 K o2 mA2 K o2 mA A

A2 mA2

4 K o1 K r2 mA2 mA K r1 K r2 A

K o1 A2 K o2 mA2

K o1 K o22

4 K r2 mA A K o1 2 K r2 mA mA2

K o2 2 K r1 mA2A2

K r1 A

K o1 mA 2 K o1 K r2 mA2

K o2 mA2

2 K r1 K r2 mA2mA 4 K r1 K r2 mA2

A20 K r1 mA24 K r2 mA A

2 A2 mA2

mA mA A 4 A2

mA2

mA K o1 K o2 mA mA24 K r2 mA2

A2

K r2 A K o1 A2

K o2 mA


K o1 A2 K o2 mA2

K o1 K o22 4 K r2 mA A

K o12 K r2 mA

mA

2 K o2

2 K r1 mA

2

A2

K r1 A

K o1

mA

2 K o1 K r2 mA

2 2 mA2K o2 mA

2 2 K r1 K r2 mA2

mA 4 K r1 K r2 mA2

Adsorbed speciesCurrent density

1→ stands for =.




Θ A 4 K o2 K r1 mA2A2

mA K o2 2 K r1 A

K o1 K r1 K r2 A K o1 A2

K o2 mA2


K o1 A2 K o2 mA2

K o1 K o22

4 K r2 mA A K o1 2 K r2 mA mA2

K o2 2 K r1 mA2A2

K r1 A


4 K o1 K r2 mA2 2 K o2 2 K r1 mA2

A2

K r1 A K o1 mA

i f

F mA

4 K r1 K r2 mA2

A mA

K o1 K o2

K r1 K r2

A

K o1 A2 K o2

mA2


K o1 A2 K o2 mA2 K o1 K o22 4 K r2 mA A

K o1 2 K r2 mA mA2 K o2 2 K r1 mA2A2

K r1 A


K o2 mA2

2 K r1 K r2 mA2mA 4 K r1 K r2 mA2


Faradaic impedance

Z f (s) = Rct + Z A+(s) + Z A2(s) + Z A(s) + Z s(s)

Z f s s A0 K r1 K r2 K o1 2 Θ A K r2 M A s Θ s K o2 M A2s 1 K o2 A20 Θ s s A0 K r1 M A2

M A

s

Θ A

s K r2

K r1

Θ s

s 2 A

0

Θ

A Θ

s K r2

Θ s

K o2

2 A20

Θ

ss M

A2 s f F A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2

A0 Θ s K r1 Αr1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s

K o1 2 A20 Θ s K o2 Αo2 A0 Θ A K r2 Αr2 Θ A Αo1 s 2 A0 K r2 K o2 2 A20 Θ s s M A2s

M A+ =1

mA+

th√

τ A+ s√

τ A+ s, M A2

=1

mA2

th√

τ A2s

√τ A2

s

Transfert resistance

Rct 1

f F Θ A K o1 Αo1 A20 Θ s K o2 Αo2 A0 Θ s K r1 Αr1 A0 Θ A K r2 Αr2


Z A s Θ s K r1 Θ A K r2 Rct M A s

Z A2s

Θ s K o2 Rct A0 Θ s K r1 Αr1 A20 K o2 K r2 A0 Θ A s M A s Θ A K o1 Αo1 K r2 Θ A s M A s A0A20 K o2 A20 Θ s K o2 Αo2 s K o1 K r1 A0 Θ s s M A s

A0 Θ A K r2 Αr2 s K o1 K r1 A0 Θ s s M A s M A2s

A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2



Adsorbed species




Z As

K o1 A0 K r2 Rct A20 Θ s K o2 Αo2 2 Θ s K r1 M A s 1 A0 Θ A K r2 Αr2 2 Θ s K r1 M A s 1 Θ A K o1 Αo1

2 Θ A K r2 M A s Θ s K o2 M A2s 1 A0 Θ s K r1 Αr1 2 Θ A K r2 M A s Θ s K o2 M A2

s 1 A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2



Z s s

A20 K o2 A0 K r1 Rct A20 Θ s K o2 Αo2 2 Θ s K r1 M A s 1 A0 Θ A K r2 Αr2 2 Θ s K r1 M A s 1 Θ A K o1Αo1 2 Θ A K r2 M A s Θ s K o2 M A2

s 1 A0 Θ s K r1 Αr1 2 Θ A K r2 M A s Θ s K o2 M A2s 1

A20 Θ s K o2 s 2 A0 K r1 Αo2 A0 Θ A s 2 A0 K r1 K r2 Αr2



4.4 Copper dissolution in HCl

4.4.1 Mechanism [5, 1, 2]

A

−

+ M,s

Ko1

←→Kr1 MA,s + e

−

A− + MA,skd2←→kg2

MA−

2 + s


Langmuir isotherm: K o1 = ko1 exp(αo1 f E ) , K r1 = kr1 exp(−αr1 f E )


vA−(t) = −v1(t)−v2(t), vMA2−(t) = v2(t), vs(t) = −v1(t)+v2(t), vMA(t) = v1(t)−v2(t)



J A−(0, t) = vA−(t), J MA2−(0, t) = vMA2

−(t)


dθs(t)

dt=

vs(t)

Γ,

dθMA(t)

dt=

vM (t)

Γ


if (t) = F v1(t)

Step rates

v1(t) = A−(0, t) θs(t) Γ K o1(t) − θMA(t) Γ K r1(t)

v2(t) = A−(0, t) θMA(t) Γ kd2 −MA2−(0, t) θs(t) Γ kg2






Soluble species

J A−(0) =

−A−∗

−A−(0) m

A−, J

MA2−(0) =

−M A2−∗

−MA2

−(0) mMA2

−

Adsorbed species

dθs/dt = 0, θMA + θs = 1


Soluble species

A0 mA K r1 A k d2 K o1 mMA2 k g2 mA A

K r1 MA2 mMA2

k g2 K r1 mA 4 k d2 K r1 k d2 K o1 A MA2

k g2 K r1 mA mMA2 2

4 2 k d2 K r1 k d2 A K r1 mA mMA2

k g2 2 K o1 mMA2 MA2

K o1 A

K r1 mA

2 k d2 K r1 mMA2 k g2 mA

2 2 2 k d2 K o1 k d2 K o1 mA mMA2

MA20 k g2 K r1 MA2

mMA2

mA2


k g2 K r1 mA mMA2 2



K o1 A

K r1 mA 2 k d2 K r1 mMA2 mA

mMA2 4 2 k d2 K o1 k d2 K o1 mA mMA2

MA2

mA 4 k d2 K o1 A k d2 K o1 A

K r1 mA

4 2 k d2 K o1 k d2 K o1 mA mMA2

2 2 k g2 mA

2 mMA2

Adsorbed species

Θ MA k g2 2 K o1 A K r1 mA k g2 4 K o1 mA MA2

k d2 K o1 A

K r1 mA mMA2

k g2 K r1 mA

4 k d2 K r1

k d2 K o1

A

MA2

k g2 K r1

mA

mMA2

2

4

2 k d2 K r1

k d2 A K r1 mA mMA2


K o1 A

K r1 mA 2 k d2 K r1 mMA2

4 k d2 K r1 mMA2 2 k g2 2 K o1 mMA2

MA2

K o1 A K r1 mA

Current density

i f F mA k g2 K r1 mA 4 k d2 K o1 A k d2 K o1 A

MA2 k g2 K r1 mA mMA2


k g2 K r1 mA mMA2

2



K o1 A

K r1 mA 2 k d2 K r1 mMA2

2 k g2 mA2

4 2 k d2 K o1 k d2 K o1 mA mMA2


Faradaic impedance

Z f (s) = Rct + Z A−(s) + Z MA(s) + Z s(s)




Z f s s MA20 k 2 K 1 K 1 A0 Θ s s 2MA2

0 k 2 M A s

k 2 A0 2 A0 Θ MA Θ s K 1 Θ MA s 2 K 1 M A s

Θ s k 2 s K 1 K 1 A0 Θ s s M A s M MA2 s

f F Θ MA K 1 Α1 A0 Θ s K 1 Α1 s k 2 A0 Θ MA s M A s k 2 MA20 Θ s s M MA2

s

M A− =1

mA−

th√τ A−s√τ A−s

, M MA−2

=1

mMA−2

th

τ MA−2 s τ MA−2

s


Rct =1

f F Γ (A−(0) θs K o1 αo1 + θMA K r1 αr1)


Z A−(s) =θs Γ K o1 Rct M A−(s)

s + 2 A−(0) kd2 + kg2

2 M A2

−(0) + θs s Γ M MA2−(s)

s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2

M A2

−(0) + θs s Γ M MA2−(s)

Adsorbed species

Z MA(s) =K r1 Rct

1 + 2 θMA Γ kd2 M A−(s) + θs Γ kg2 M MA2−(s)

s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2

M A2

−(0) + θs s Γ M MA2−(s)

Z s(s) =

A−(0) K o1 Rct

1 + 2 θMA Γ kd2 M A−(s) + θs Γ kg2 M MA2

−(s)

s + kd2 (A−(0) + θMA s Γ M A−(s)) + kg2

MA2−(0) + θs s Γ M MA2−(s)

4.5 (V-T) reaction with mass transfer limitation

4.5.1 Mechanism

A+ + s + e−Kr1←→Ko1

A,s

2A,skd2←→kg2

A2 + 2 s


Langmuir isotherm: K r1 = kr1 exp(−αr1 f E ) , K o1 = ko1 exp(αo1 f E )


vA+(t) = −v1(t), vA2(t) = v2(t), vs(t) = −v1(t) + 2 v2(t), vA(t) = v1(t)− 2 v2(t)



J A+(0, t) = vA+(t), J A2(0, t) = vA2

(t)

Rate of production of adsorbed species,

dθs(t)

dt=

vs(t)

Γ,

dθA(t)

dt=

vA(t)

Γ





if (t) = −F v1(t)

Step rates

v1(t) =−

θA(t) Γ K o1(t)+A+(0, t) θs(t) Γ K r1(t), v2(t) = θA(t)2

Γ2 kd2−

A2(0, t) θs(t)2

Γ2 kg2



Soluble species

J A+(0) = −

A+∗ −A+(0)

mA+, J A2(0) = (A2

∗ −A2(0)) mA2

Adsorbed speciesdθA/dt = 0, θA + θs = 1


Soluble species: A+(0) and A2(0) are solutions of cubic equations.A+(0):

2 kd2

A+(0)Γ K r1 +

−A+∗

+ A+(0)

mA+

2mA2

=

A+∗ −A+(0)

K o1 + A+(0) K r12

mA

kg2

Γ K o1 +

A+∗ −A+(0)

mA+

2 A+∗ −A+(0)

mA+ + 2 A2

∗ mA2

A2(0):

4 (A2∗ −A2(0))

2K r1 mA2

2

K o1 + A+∗

K r1

mA+ + (A2

∗ −A2(0)) K r1 mA2

+

kd2

2 (A2

∗

−A2(0)) mA+

mA2 + Γ K r1

A

+∗

mA+

+ 2 (A2

∗

−A2(0)) mA22

=

mA+2

−

(A2∗ −A2(0))

K o1 + A+∗

K r1

2mA2

+ A2(0) kg2 (Γ K o1 + 2 (−A2

∗ + A2(0)) mA2)

Adsorbed species: θA is solution of a cubic equation

2 A2∗ Γ kg2 mA2

(−mA+ + Γ K r1 (−1 + θA)) (−1 + θA)2

+

mA+

mA2

+ Γ2 kg2 (−1 + θA)2

A+∗

K r1 (−1 + θA) + K o1 θA

+

2 Γ kd2 mA2θA

2 (mA+ + K r1 (Γ− Γ θA)) = 0

Current density: if is solution of a cubic equation

2 F if (2 Γ kd2 + K o1) K r1 mA+

if + A+∗ F mA+

mA2+

if + 2 F Γ2 kd2

K r1

2

if + A+∗

F mA+

2mA2

+

mA+2

F if

2 if kd2 + F K o12

mA2+ kg2 (if − F Γ K o1)

2(if − 2 A2

∗ F mA2)

= 0





Faradaic impedance

Z f (s) = Rct + Z A+(s) + Z A(s) + Z s(s)

Z f s s K o1 K r1 A0 Θ s s M A s 4 Θ A k d2 Θ s K r1 M A s 1

Θ s k g2 4 Θ s K r1 M A s A20 A20 Θ s s K o1 K r1 A0 Θ s s M A s M A2s

f F Θ A K o1 Αo1 A0 Θ s K r1 Αr1 s 4 Θ A k d2 Θ s k g2 4 A20 Θ s s M A2s

M A+ =1

mA+

th√

τ A+ s√

τ A+ s, M A2

=1

mA2

th√

τ A2s

√τ A2

s


Rct =1

f F Γ (θA K o1 αo1 + A+(0) θs K r1 αr1)

Concentration impedances Soluble species

Z A+(s) = θs Γ K r1 Rct M A+(s)

Adsorbed species

Z A(s) =K o1 Rct

1 + θs

2 Γ2 kg2 M A2(s)

s + 4 θA Γ kd2 + θs Γ kg2 (4 A2(0) + θs s Γ M A2

(s))

Z s(s) =A+(0) K r1 Rct

1 + θs

2 Γ2 kg2 M A2(s)

s + 4 θA Γ kd2 + θs Γ kg2 (4 A2(0) + θs s Γ M A2

(s))







Bibliography

[1] Diard, J.-P., LeCanut, J.-M., LeGorrec, B., and Montella, C.

Copper electrodissolution in 1 M HCl at low current densities. i. Generalsteady-state study. Electrochim. Acta 43 (1998), 2469–2483.

[2] Diard, J.-P., LeCanut, J.-M., LeGorrec, B., and Montella, C.

Copper electrodissolution in 1 M HCl at low current densities. II. elec-trochemical impedance spectroscopy study. Electrochim. Acta 43 (1998),2485–2501.

[3] Diard, J.-P., LeGorrec, B., Montella, C., and Montero-Ocampo,

C. Second order electrochemical impedances and electrical resonance phe-nomenon. Electrochim. Acta 37 (1992), 177–179.

[4] Diard, J.-P., LeGorrec, B., Montella, C., and Montero-Ocampo,

C. Calculation, simulation and interpretation of electrochemical impedancediagrams. part i. second-order electrochemical impedances. J. Electroanal.

Chem. 352 (1993), 1–15.

[5] LeCanut, J.-M. Impedance faradique en presence d’un couplage elec-

trosorption-desorption, transport de matiere. C as de l’electrodissolution du

cuivre. PhD thesis, Institut National Polytechnique de Grenoble, Grenoble,1995.

[6] Montero-Ocampo, C. Impedances electrochimiques du second ordre. Ex-

emple du mecanisme de Volmer-Heyrovsky avec desorption chimique. PhDthesis, Institut National Polytechnique de Grenoble, Grenoble, 1988.

[7] Schuhmann, D. etude phenomenologique a l’aide de sch emas reaction-nels des impedances faradiques contenant des resistances negatives et desinductances. J. Electroanal. Chem. 17 (1968), 45–59.

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