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Impedance II :
Characterization of a corroding metal
N. Murer
Objectives
2
1. Understand to which mechanisms correspond the impedance graphs that can be obtained on corroding systems.
2. Be able to determine which relevant information for corrosion can be obtained from impedance graphs
/41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
3 /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
4 /41
Considering that a metallic electrode undergoes a dissolution following the oxidation reaction : M Mn1+ + n1e- (1) The reduction reaction of species O also takes place at this electrode and is written : O + n2e- R (2) The steady-state polarization curve of the electrode/solution interface can be written around the corrosion potential Ecorr using the Wagner-Traud or Stern relationship (assuming no mass transport limitation):
(3)
Icorr : corrosion current in A E = electrode potential in V Ecorr : corrosion potential in V
b1,2 = α1,2n1,2F/(RT) α1,2 are the symmetry factors of the oxidation and reduction reactions, respectively.
5
Expression of the Faradaic impedance
Wagner – Traud /41
The Wagner-Traud relationship being valid in the steady-state regime, we can assume it is valid in the dynamic regime. It can then be written :
(4)
To calculate the Faradaic impedance, first this relationship needs to be linearized using a Taylor series development, limited to the first order, around the steady-state current Icorr :
With the polarization Π(t) = E(t) – Ecorr
The Laplace Transform of ΔI(t) is :
The Faradaic impedance Zf = Δ
(p)/ΔI(p)
Rct is the charge transfer resistance
(5)
(6)
(7)
6
Expression of the Faradaic impedance
Wagner – Traud /41
The Faradaic impedance Zf
The electrode impedance is obtained considering the Faradaic impedance in parallel with the impedance of the double layer capacitance. Adding in series the electrolyte ohmic drop gives the total measured impedance :
This expression can be illustrated by the following circuit.
RΩ
Rct Electrolyte resistance
Double layer capacitance
Cdl
(8)
(9)
7
Expression of the Faradaic impedance
Wagner – Traud /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
8 /41
The impedance is a complex number: Z = a + jb = Re(Z) + jIm(Z) (with j2= - 1) Z = ρ(cosφ + jsinφ) with ρ the modulus and φ the phase In the Bode Plot, the modulus and the phase of the impedance are plotted against the frequency of the modulation. In the Nyquist plot, the impedance for each frequency is plotted in the complex plane -Im(Z) vs Re(Z).
You just need a ruler to deduce Bode plot from Nyquist plot.
9
Equivalent circuit
Wagner – Traud /41
In this case, the polarization resistance Rp = Rct= (RΩ + Rct) - RΩ
The electrochemical double layer capacitance can be determined at the frequency for which -Im(Zf) is maximum. At this characteristic frequency fc:
2πfc = 1/(RctCdl)
The Nyquist plot of the total impedance is a semi-circle of diameter Rct and centered around the Re(Z) axis.
(10)
10
Equivalent circuit
Wagner – Traud /41
Influence of the parameters on the impedance : introduction of Z Sim (cf Part I)
From 7 MHz to 1 mHz
RΩ /Ohm Cdl /µF Rct
/Ohm
1000 0.1 100
11
Equivalent circuit
Re(Z)/Ohm
3 0002 5002 0001 5001 0005000
-Im
(Z)/
Oh
m
1 400
1 200
1 000
800
600
400
200
0
-200
-400
HF BF
Wagner – Traud /41
Re(Z)/Ohm
3 0002 5002 0001 5001 0005000
-Im
(Z)/
Oh
m
1 400
1 200
1 000
800
600
400
200
0
-200
-400
RΩ /Ohm Cdl /µF Rct
/Ohm
1000 0.1 100
1000 0.1 2000
Rct 100 2000
From 7 MHz to 1 mHz
12
Equivalent circuit
HF BF
Influence of the parameters on the impedance : introduction of Z Sim (cf Part I)
Wagner – Traud /41
Re(Z)/Ohm
3 0002 5002 0001 5001 0005000
-Im
(Z)/
Oh
m
1 400
1 200
1 000
800
600
400
200
0
-200
-400
RΩ /Ohm Cdl /µF Rct
/Ohm
1000 0.1 100
1000 0.1 2000
100 0.1 2000
RΩ 1000 100
From 7 MHz to 1 mHz
13
Equivalent circuit
Rct 100 2000
HF BF
Influence of the parameters on the impedance : introduction of Z Sim (cf Part I)
Wagner – Traud /41
Re(Z)/Ohm
3 0002 5002 0001 5001 0005000
-Im
(Z)/
Oh
m
1 400
1 200
1 000
800
600
400
200
0
-200
-400
RΩ /Ohm Cdl /µF Rct
/Ohm
1000 0.1 100
1000 0.1 2000
100 0.1 2000
100 0.0001 2000
RΩ 1000 100
Cdl 0.1 0.0001
From 7 MHz to 1 mHz
14
Equivalent circuit
Rct 100 2000
HF BF
Influence of the parameters on the impedance : introduction of Z Sim (cf Part I)
Wagner – Traud /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
15 /41
From S. E. Hernandez, A. J. Griffin, Jr., F. R. Brotzen, J. Electrochem. Soc., 142, (1995), 1225
Pure Al (99,9%) in 0.04 M NH4Cl Pure Ni (99,9%) in 1 M H2SO4
NA_CASP_PEIS.mpr
-Im(Z) vs. Re(Z)
Re(Z)/Ohm
5 0000
-Im
(Z)/
Oh
m2 000
1 000
0
From Application Note #37 : www.bio-logic.info
Re(Z)/Ohm cm2
-Im
(Z)/
Oh
m c
m2
16
Experimental results
Wagner – Traud /41
ZFit using R1 + R2/C2 equivalent circuit :
The deviation gives an estimation of the confidence interval of the value. If it is large, it means the parameter is not critical in the fitting process. The quality of the fit can be estimated using the
2 parameter. It is related to the distance between the measured points and the calculated points.
Using R1 + C2/R2, 2 is equal to ≈ 1x106. The semi-circle is not centered around the Re(Z) axis (« depressed » semi-circle). A component called a Constant Phase Element (CPE) can be used to replace C.
17
Experimental results
Wagner – Traud /41
Zfit using R1 + R2/Q2 equivalent circuit :
Using R1 + Q2/R2, 2 is equal to ≈ 3x105, which is better. Using the CPE adds another parameter α that is used in the minimization. It is possible to calculate the equivalent capacitance (pseudo-capacitance).
18
Experimental results
Wagner – Traud /41
Z Fit using R1 + R2/Q2 equivalent circuit
The total impedance now writes :
RΩ
Rct Electrolyte resistance
Constant Phase Element Qdl
The frequency is used to calculate the pseudo capacitance.
(11)
19
Experimental results
Wagner – Traud /41
Experimental results : Parameters determination
NA_CASP_PEIS.mpr
-Im(Z) vs. Re(Z)
Re(Z)/Ohm
5 0000
-Im
(Z)/
Oh
m2 000
1 000
0
It seems that the semi-circle is not only depressed but shows the beginning of a loop at lower frequencies. This is due to the nature of the corrosion reaction. In anaerobic acidic media, corrosion only involves the proton reduction 2H+ + 2e- H2
This reaction according to the Volmer-Heyrovský mechanism, takes place in two steps.
20 Wagner – Traud
Experimental results
/41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
21 /41
H+ + s + e- H,s H+ adsorption H+ + H,s + e- H2 + s H2 release
s is an adsorption site H,s is a proton adsorbed on the electrode surface.
Kr1
Kr2
22
Expression of the Faradaic impedance
Volmer-Heyrovský /41
Expression of the Faradaic impedance
23
Expression of the Faradaic impedance
Volmer-Heyrovský /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
24 /41
The polarization resistance : Rp = Rct + R
Equivalent circuit for the Faradaic impedance : Rct + R /C
Rct
R Charge transfer resistance
C
Z : Impedance related to the adsorbed species
RΩ
Cdl
Electrolyte resistance
25
Equivalent circuit
Additional components compared to Wagner-Traud
Volmer-Heyrovský /41
Re(Z)/Ohm
2 5002 0001 5001 000
-Im
(Z)/
Oh
m
800
700
600
500
400
300
200
100
0
-100
-200
-300
Re(Z)/Ohm
2 0001 5001 000
-Im
(Z)/
Oh
m
500
400
300
200
100
0
-100
-200
Nyquist diagram evolution with the electrode potential E (αr1 ≠ αr2)
E 0
I
Eic (Kr1 = Kr2) R ,C > 0
R ,C < 0
Inductive behaviour
Capacitive behaviour 2H+ + 2e- H2
26
Equivalent circuit
Volmer-Heyrovský /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
27 /41
s is an adsorption site H,s is a proton adsorbed on the electrode surface. M,s is a surface metal atom.
H+ + s + e- H,s H+ adsorption H+ + H,s + e- H2 + s H2 release M,s Mn+ + ne- + s Metal corrosion
Kr1
Kr2
Ko3
28 Anaerobic corrosion
Expression of the Faradaic impedance
/41
The Faradaic impedance has the same structure as in the VH reaction but with different expressions for R et C .
29
Expression of the Faradaic impedance
Anaerobic corrosion /41
The polarization resistance : Rp = Rct + R
Equivalent circuit for the Faradaic impedance : Same as VH: Rct + R /C
RctR
R Charge transfer resistance
C
Z : Impedance related to the adsorbed species
RΩR
Cdl
Electrolyte resistance
30
Expression of the Faradaic impedance
Additional components
Anaerobic corrosion /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
31 /41
Re(Z)/Ohm
2 5002 0001 5001 000
-Im
(Z)/
Oh
m
800
700
600
500
400
300
200
100
0
-100
-200
-300 Re(Z)/Ohm
2 0001 5001 000
-Im
(Z)/
Oh
m
500
400
300
200
100
0
-100
-200
Nyquist diagram evolution with the electrode potential E
E
I
Eic (Kr1 -Kr2-Ko3 ,n = 0)
R , C > 0 R , C < 0
Inductive behaviour Capacitive behaviour
2H+ + 2e- H2
Net current
M M2+ + 2e- Ecorr
32
Equivalent circuit
Anaerobic corrosion /41
430 stainless steel in 1 M HCl
RΩ+Qdl/(Rct + C /R )) : equivalent circuit R and C < 0 Rp = Rct + R ≈ 667 – 154 ≈ 513 ohm
-Im(Z) vs. Re(Z)
PEIS stainless steel in 1M HCl .mpr PEIS stainless steel in 1M HCl _zfit.mpp #
Re(Z)/Ohm
6004002000
-Im
(Z)/
Oh
m
300
250
200
150
100
50
0
-50
x: 4,70322 Ohm
y: 29,254 Ohm
11,2572 Hz
Point 0
x: 547,897 Ohm
y: -48,3921 Ohm
4,73485e-3 Hz
Point 20
33
Equivalent circuit
Anaerobic corrosion /41
-Im(Z) vs. Re(Z)
PEIS stainless steel in 1M HCl .mpr PEIS stainless steel in 1M HCl _zfit.mpp #
Re(Z)/Ohm
6004002000
-Im
(Z)/
Oh
m
300
250
200
150
100
50
0
-50
x: 547,932 Ohm
y: -49,0661 Ohm
4,73485e-3 Hz
Point 48
x: 4,55796 Ohm
y: 1,09885e-3 Ohm
0,6e6 Hz
Point 0
Other equivalent circuit with components having positive values : RΩ + Qdl/(Rct/(L + R )). A capacitive behaviour could be fitted with negative values of L + R . (cf Mathematica demo :
http://www.bio-logic.info/potentiostat/notes/20080328%20-%20VHCor-ZmmaP.nbp)
430 stainless steel in 1 M HCl
34
Equivalent circuit
Anaerobic corrosion /41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
35 /41
The Stern – Geary relationship is used to determine the corrosion current and is valid at the corrosion potential Ecorr :
Icorr = B/Rp,Ecorr
B = b1b2/(2.3(b1 + b2)), with b1 and b2 the Tafel slopes. Rp,Ecorr is the polarization resistance at the corrosion potential Ecorr
Rp = 1/(dI/dE)
B is determined by using Tafel approximation or by harmonic analysis (CASP). Rp can be determined either by impedance or micropolarization around Ecorr
(cf Application Note #10 http://www.bio-logic.info/potentiostat/notes/20110224%20-%20Application%20note%2010.pdf).
Stern - Geary relationship 36
Wagner-Traud behaviour
/41
The Stern – Geary relationship implies that the system follows the Wagner-Traud relationship for which Rp (value of the impedance for f 0 or dE/dI) is equal to Rct.
If the system has an inductive behaviour, it does not follow the Wagner-Traud relationship and Rp ≠ Rct, Rp = Rct + R
The value of the impedance for f 0 cannot give us the value of the resistance Rct to be used in the Stern – Geary relationship.
37 Stern - Geary relationship
Wagner-Traud behaviour
/41
1. System following the Wagner-Traud relationship 1. Expression of the Faradaic impedance 2. Equivalent circuit 3. Experimental results : C and CPE
2. The Volmer-Heyrovský reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
3. Anaerobic corrosion : The Volmer-Heyrovský corrosion reaction 1. Expression of the Faradaic impedance 2. Equivalent circuit : capacitive or inductive ?
Outline
4. The Stern-Geary Relationship 1. In the case of a Wagner-Traud behaviour 2. In the case of an inductive behaviour
38 /41
Example : the Nyquist diagram plotted before shows that in the case of an inductive behaviour, R, to be used in the Stern-Geary relationship must be determined at a frequency of around 33 mHz.
-Im(Z) vs. Re(Z)
PEIS stainless steel in 1M HCl .mpr PEIS stainless steel in 1M HCl _zfit.mpp #
Re(Z)/Ohm
700600500
-Im
(Z)/
Oh
m
100
50
0
-50
x: 547,932 Ohm
y: -49,0661 Ohm
4,73485e-3 Hz
Point 48
x: 658,647 Ohm
y: 18,6579 Ohm
0,0331038 Hz
Point 43
Rct
Rp
At 33 mHz, the value of |Z| is Rct not Rp. In the example above, if Rp instead of Rct is used for the calculation of Icorr then the error is (Rct - Rp)/Rct ≈ (660-550)/550 = 20%. Icorr will be overestimated.
39 Stern - Geary relationship
Inductive behaviour
/41
What have we learned ?
Corroding systems can show different impedance graphs depending on the nature of the corrosion mechanism: • Tafelian : the rate of corrosion is controlled by the rate of electron
transfer I vs. E follows the Wagner-Traud relationship
The Impedance equivalent circuit is RΩ + Cdl/Rct
The Stern-Geary relationship can be used to determine the corrosion current
• Following the Volmer Heyrovský corrosion mechanism : valid in deaerated acidic medium, the rate of corrosion depends on the rate of H+ adsorption and H2 release
The impedance equivalent circuit has an inductive loop at low frequencies
Rp ≠ Rct so the Stern-Geary relationship is not valid anymore. It can be used but an error will be induced if Rp is used instead of Rct
40 /41
References
41
Bio-Logic website : www.bio-logic.info Faradaic Impedances http://www.bio-logic.info/potentiostat/notesifil.html Mathematica Player files V-H reaction : http://www.bio-logic.info/potentiostat/notes/20081210%20-%20VH-ZmmaP.nbp Anaerobic corrosion (V-H reaction + dissolution) http://www.bio-logic.info/potentiostat/notes/20080328%20-%20VHCor-ZmmaP.nbp
/41
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