Fitting Procedures Example

A fitting example is given below to illustrate the fitting procedures and in relation to develop an adequate equivalent circuit model for further data analysis. A rep-resentative experimental data set is shown in Figure 5.36. Impedance spectra was acquired in-situ from ZIRLO^TM specimen at 200 days of exposure.

1^x10^-1 1 1^x10¹ 1^x10² 1^x10³ 1^x10⁴ 1^x10⁵

Frequency [Hz]

1^x10³ 1^x10⁴ 1^x10⁵ 1^x10⁶

|Z| [Ω‧cm²]

-80

-60

-40

-20

phase [degree]

|Z|

phase

ZIRLO 200 days^TM

Figure 5.36: Bode plot of impedance spectra collected from the 200 days ZIRLO^TM sample during in-situ experiment at 360^◦C.

By a close examine of the impedance data shown in Figure5.36, a diffusional term at the frequency range of 10⁻¹ - 10 Hz can be identified. At higher frequency end, the impedance response can be identified as the oxide response which can be described by the circuit in Figure 5.37.

R_e

R_ox

C_ox

Figure 5.37: Electrical circuit that describes the impedance response of an oxide layer, the equivalent circuit consists of a solution resistance R_e in series with an oxide resistance Rox in parallel with a capacitor Cox representing the dielectric

response of the oxide at high frequencies.

A Randles circuit shown in Figure5.38was proposed for the initial fitting procedure, which describes the high frequency oxide response and low frequency mass transport behaviour (i.e., diffusional). In Figure 5.39, by comparing the fitting result with the experimental data, the basic equivalent circuit can be established as it shown in Figure 5.38. However, at intermediate frequency, the equivalent circuit does not describe the experimental impedance response.

R_e

Z_D R_ox

C_ox

Figure 5.38: The Randles circuit that describes the impedance response of an oxide layer with diffusional influence.

1x10^-1 1 1x10¹ 1x10² 1x10³ 1x10⁴ 1x10⁵

Figure 5.39: Fitting results and experimental data from the 200 days ZIRLO^TM sample during in-situ experiment at 360^◦C.

At this point, it is useful to review some of other corrosion systems that may have similar impedance response. For example, a coated metal which share the similarity as the Zr-ZrO₂ system, since ZrO₂ can be seen as a coating materials that has been applied to the zirconium substrate. The impedance response for an impermeable coating can be described by a pure capacitor in series with a solution resistance.

However, most paint coatings degrade with time, and water penetrates into the coating and forms a new liquid metal interface under the coating, and corrosion can occur at this freshly formed interface.

R_e R_p

R_ct

Z_D

C_dl C_i

Figure 5.40: Equivalent circuit model for the impedance response measured on a failed coating system.

The equivalent circuit in Figure5.40can be used to describe the impedance response of a failed coating. The capacitance of the intact coatings is represented by C_i, which has a much small capacitance value than double layer capacitance. Units for these coating usually in the order of nF, due to the thickness of the coating (usually in the order of µm). Rp is the resistance of ion conducting paths (i.e., resistance of the electrolyte inside the pore) that developed in the coating. The actual paths can be porosities, cracks or grain boundaries, it may not be physical pores filled with electrolyte. The failed coating system, where a pocket of electrolyte is formed once the coating is penetrated, the bare metal would be in contact with the electrolyte. In this case, the electrolyte inside this pocket can be very different from the bulk electrolyte (e.g., pH level, resistivity, ionic concentration, etc). In addition, a kinetically controlled charge transfer reaction can be present between the pocket of electrolyte and the bare metal. Rct and Cdl are the interface components that

1^x10^-1 1 1^x10¹ 1^x10² 1^x10³ 1^x10⁴ 1^x10⁵

Figure 5.41: Fitting results obtained using the equivalent circuit shown in Fig-ure5.40.

describes this interfacial reaction that take place at the metal/electrolyte interface.

The fitting result is shown in Figure5.41using the failed coating model (Figure5.40).

The results seems in good agreement with the experimental data, however, the circuit parameters values determined by the fitting cannot make sensible justification of the real system (a table of the fitting parameters and determined values is shown in Table5.1). The R_ct value is much smaller than expected, which means the corrosion rate is much higher than the observed value from weight gain experiment.

Table 5.1: Fitting results for the experimental data collected from the 200 days ZIRLO^TM sample during in-situ experiment at 360^◦C.

R_p W_R W_T W_P R_ct C_dl C_i

92 kΩ 1.58 MΩ 40 k 0.17 1.2 kΩ 1×10⁻⁸ F 1.1×10⁻⁹ F

The model to describe the zirconium oxide system would be similar to the failed coating model (shown in Figure5.40. However, instead of a kinetic controlled charge transfer process and a diffusion process, an oxide layer maybe always present at the metal surface (i.e., the barrier oxide layer) if it was exposed to electrolyte.

In addition to the failed coating model, there are certain modifications should be con-sidered according to an adequate physical model of the zirconium corrosion system under investigation. The direct access of the metal substrate is absent in the case of zirconium oxidation compared with a metal covered with a failed coating, due to zirconium’s high oxygen affinity in water, a thin layer of barrier oxide would always be present in between the electrolyte and bare metal. The barrier oxide layer that can be described with an oxide resistance Rb and capacitance Cb. The oxide/metal interface response may be neglected due to its very small contribution to the overall resistance and capacitance (i.e., oxide/metal interfacial transfer resistance is consid-ered very small compared with oxide electrical resistance, so it has very small effect on the total oxide electrical resistance; the interfacial capacitance is very large, how-ever, when it is in series with other capacitances, the largest capacitor has very small contribution towards the total capacitance according to Equation 3.37).

As a result, a modified equivalent circuit (Figure 5.35) based on the failed coating model was developed and illustrated in Section 5.6.4. An addition oxide layer was introduced to model the barrier oxide response of the system. The fitted impedance curve is shown in Figure5.42using the modified equivalent circuit (Figure5.35) and the fitting results are shown in Table 5.2.

Table 5.2: Fitting results using equivalent circuit shown in Figure 5.35 for the experimental data collected from the 200 days ZIRLO^TM sample during in-situ

experiment at 360^◦C.

R_p W_R W_T W_P R_ct C_dl C_p R_b C_b

62 kΩ 552 kΩ 50 0.29 53 kΩ 42 nF 1.3 nF 69 kΩ 9.5 nF

1^x10^-1 1 1^x10¹ 1^x10² 1^x10³ 1^x10⁴ 1^x10⁵

Figure 5.42: Fitting results obtained using the equivalent circuit shown in Fig-ure5.35.

By further examine the fitting results, the total oxide capacitance and oxide resis-tance would dominate the high frequency response in the frequency range between 1 k - 100 k Hz. From the fitting results shown in Table 5.2, the total oxide resis-tance can be expressed as (R_total = R_ct + R_b ≈ 120 kΩ) at high frequency range;

and the total oxide capacitance can be expressed as (C_total = [C_p⁻¹ + C_b⁻¹]⁻¹ ≈ 1 nF). The corresponding characteristic frequency for the total oxide response at high frequency can be calculated as f_RC = (2πR_totalC_total) ≈ 1.3 kHz. The characteristic frequency of barrier layer oxide response can be calculated as fRC = (2πRbCb) ≈ 240 Hz which is indicated in Figure 5.42. The characteristic frequency of the cathodic reaction which take place at the oxide/electrolyte interface can also be calculated as f_RC = (2πR_ctC_dl) ≈ 72 Hz. Although the characteristic frequencies for each time constant are very close to each other, the changing of gradient of the |Z| plot in Figure 5.42 can be interpreted as dispersion of the impedance response due to the

appearing of transmission line behaviour as the frequency decreases.