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# Impedance of Corroding Metal – Tafelian System

Latest updated: August 21, 2024## Introduction – Corrosion series

*Corroding systems can show different impedance graphs depending on the nature of the corrosion mechanism. In this series of articles, we will consider first Tafelian system in this article and then a system following the Volmer Heyrovský mechanism (Part 2). You will learn how to express the faradaic impedance and what is the equivalent circuit. *

## Part I

Let’s take the case of a Tafelian system where the rate of corrosion is controlled by the rate of electron transfer and *I vs. E *follows the Wagner-Traud relationship.

### How to express the faradic impedance?

Considering that a metallic electrode undergoes a dissolution following the oxidation reaction:

$$M \rightarrow M^{n_1^{+}} + n_1e^-$$

The reduction reaction of species O also takes place at this electrode and is written:

$$\text{O} + n_2e^- \rightarrow \text{R}$$

The steady-state polarization curve of the electrode/solution interface can be written around the corrosion potential $E_\text{corr}$ using the Wagner-Traud or Stern relationship (assuming no mass transport limitation):

$$I=I_\text{corr} (\exp(b_1 (E-E_\text{corr}))-\exp(-b_2 (E-E_\text{corr})))$$

Where $I_\text{corr}$ is the corrosion current in ampere, $E$ is the electrode potential in volt, $E_\text{corr}$ is the corrosion potential in volt, $b_{1,2}=\alpha_{1,2}\nu_{1,2}F/(RF)$ and $\alpha_{1,2}$ are the symmetry factors of the oxidation and reduction reactions, respectively.

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:

$$I(t)=I_\text{corr} (\exp(b_1 (E(t)-E_\text{corr}))-\exp(-b_2 (E(t)-E_\text{corr})))$$

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 $I_\text{corr}$:

$$\Delta I(t) =I(t) -I_\text{corr} =\frac{\partial I}{\partial \Pi}\Delta \Pi (t)$$

With the polarization$\Pi(t)=E(t)-E_\text{corr}$

The Laplace Transform of is $\Delta I(t)=\frac{\partial I}{\partial \Pi}\Delta \Pi(p)$

The Faradaic impedance $Z_f=\Delta(p)/\Delta I(p)$

The charge transfer resistance is $R_\text{ct}=\frac{1}{\frac{\Delta I (p)}{\Delta \Pi(p)}}=\frac{1}{\frac{\partial I}{\partial \Pi}}$

The Faradaic impedance $Z_f$ is

$$Z_f=R_\text{ct}=\frac{1}{I_\text{corr} (b_1\exp (b_1 \Pi)-b_2 \exp(-b_2\Pi))}$$

### What is the equivalent circuit?

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:

$$Z_\text{tot}=R_\Omega+\frac{R_\text{ct}}{1+R_\text{ct}C_\text{dl}j2\pi f}$$

The ohmic drop is expressed by the resistance .

The electrochemical double layer capacitance can be determined at the frequency for which -Im(Zf) is maximum. At this characteristic frequency $f_\text{c}=2\pi f_\text{c} =\frac{1}{R_\text{ct}C_\text{dl}}$

This expression of electrode impedance can be illustrated by the following circuit. The Nyquist plot of the total impedance is a semi-circle of diameter $R_\text{ct}$ and centered around the Re(Z) axis.

In our case, the polarization resistance is $R_\text{p}=R_\text{ct}=(R_\Omega + R_\text{ct}) – R_\Omega$

You can use ZSim to understand how the Nyquist plot is modified when each value of equivalent circuit element varies. Here is an example below.

### How to calculate the corrosion current?

The Stern – Geary relationship is used to determine the corrosion current and is valid at the corrosion potential:

$$I_\text{corr} = \frac{B}{R_{\text{p},E_\text{corr}}}$$

Where $B=b_1b_2/(2.3(b_1+b_2))$ with $b_{1,2}$ the Tafel slopes,

And $R_{\text{p},E_\text{corr}}$ the polarization resistance at the corrosion potential $R_\text{p}=1/\left(\frac{dl}{dE}\right)$. The Stern – Geary relationship implies that the system follows the Wagner-Traud relationship for which $R_\text{p}$ (value of the impedance for f→ 0 or / ) is equal to $R_\text{ct}$.

$B$ is determined by using Tafel approximation or by harmonic analysis (CASP technique – AN #37) and $R_\text{p}$ can be determined either by impedance or micropolarization around $E_\text{corr}$ (Corrosion current measurement for an iron electrode in an acid solution (Tafel plot LPR) – AN #10).

Note: If the system has an inductive behaviour, it does not follow the Wagner-Traud relationship.

### Conclusion

For a Tafelian system, where the system follows the Wagner-Traud relationship, the Impedance equivalent circuit is RΩ + Cdl/Rct and the Stern-Geary relationship can be used to determine the corrosion current.

In the next article, we will take the case of a corroding systems following the Volmer Heyrovský corrosion mechanism.

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