impedance ii : characterization of a corroding metal echem/lierature...in the bode plot, the modulus...

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




Outline


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



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






Outline


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


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


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


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


/Ohm

1000 0.1 100

1000 0.1 2000

Rct 100 2000

From 7 MHz to 1 mHz

12

Equivalent circuit

HF BF



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


/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



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


/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






Outline


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


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



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



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 : 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


/41




Outline


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


Volmer-Heyrovský /41


23






Outline


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


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





Outline


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


/41

The Faradaic impedance has the same structure as in the VH reaction but with different expressions for R et C .

29


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


Additional components





Outline


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


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


-Im(Z) vs. Re(Z)


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





Outline


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




Outline


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)


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

http://www.bio-logic.info/



http://www.bio-logic.info/potentiostat/notesifil.html





http://www.bio-logic.info/potentiostat/notes/20081210 - VH-ZmmaP.nbp









http://www.bio-logic.info/potentiostat/notes/20080328 - VHCor-ZmmaP.nbp








impedance ii : characterization of a corroding metal echem/lierature...in the bode plot, the modulus...

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