## 3 Results and Discussion
### Electrochemical Studies
In view of the EIS Bode curves acquired after 96 h of exposure to 3.5 wt % NaCl (Figure 20), it may be remarked that the addition of Na\({}_{2}\)HPO\({}_{4}\) inhibitor alone or together with _MBT_ into the electrolytic solution protects the mild steel from corrosion. The EIS Bode plots for both the untreated mild steel and the systems including MBT (_Steel-WSS-MBT_, _Steel-NaCl-MBT_) after exposure to either WSS or 3.5 wt % NaCl for 24 h can be described with one time constant in the middle-low-frequency range that is attributed to the corrosion process (Figures 19 and 20). However, as corrosion progresses from 24 h to 96 h, the relative relaxation process moves to lower frequencies because of the increase of the corrosion active area and the corresponding double-layer capacitance enhancement. Figure 19: Electrochemical impedance spectroscopy (EIS) plots recorded for mild steel panels after submission in WSS for (**a**) 24 h and (**b**) 96 h: (**filled square**) _Steel–WSS–blank_, (**filled circle**) _Steel–WSS–Na\({}_{2}\)HPO\({}_{4}\)_, (**filled triangle**) _Steel–WSS–MBT_, (**filled triangle**) _Steel–WSS–Na\({}_{2}\)HPO\({}_{4}\)–MBT_, (—**) Fitted curve. On the other hand, estimating the EIS Bode plots for the systems including both Na\({}_{2}\)HPO\({}_{4}\)-MBT or only Na\({}_{2}\)HPO\({}_{4}\) in either WSS or 3.5 wt % NaCl for 24 h, it can be noted that two time constants are illustrated: one in the high-middle frequency range that is ascribed to a film formation between the mild steel surface and the phosphate compounds, and a second one in the middle-low-frequency range that is assigned to corrosion process (Figures 19 and 20). As the exposure time elapses from 24 h to 96 h, the time constant assigned to corrosion process is shifted to lower frequencies due to the enhancement of the corrosion area and the corresponding double-layer capacitance increment. The interpretation of the obtained EIS results for the mild steel panels after subjection in the electrolytic solutions in the absence or presence of corrosion inhibitors was conducted by numerical fitting using the equivalent circuits depicted in Figure 21. According to these equivalent circuits, the pure capacitors have been replaced by constant phase elements (CPE). This adaptation is mandatory when the phase shift of a capacitor is different from \(-90^{\circ}\)[84]. The impedance of an R-CPE parallel connection is given by:  
\[Z_{R-CPE}=\frac{R}{1+RY_{0}(j\omega)^{n}}, \tag{3}\]  
where \(Y_{0}\) is the admittance of the CPE and \(n\) is the CPE exponent. The \(n\) takes the value of 1 when it corresponds to a capacitor, it is between \(0.5<n<1\) when a non-ideal capacitor behavior occurs, \(n\) equals 0.5 when the CPE corresponds to a Warburg impedance describing a diffusion process, and finally, \(n\) equals 0 when it represents a resistor [85]. Using the CPE together with the Cole-Cole approach [83, 86], the capacitance can be calculated from the fittings by:  
\[C=\sqrt[n]{\frac{RY_{0}}{R^{n}}}. \tag{4}\]  
Figure 20: EIS plots recorded for mild steel panels after submission in 3.5 wt % NaCl for (**a**) 24 h and (**b**) 96 h: (**filled square**) _Steel–NaCl–blank_, (**filled circle**) _Steel–NaCl–Na\({}_{2}\)HPO\({}_{4}\), (**a filled triangleD**) _Steel–NaCl–MBT_, (**\(\blacktriangledown\) filled triangleU**) _Steel–NaCl–Na\({}_{2}\)HPO\({}_{4}\)–MBT_, (-) Fitted curve. The equivalent circuit for both the untreated mild steel and the systems including MBT (_Steel-WSS-MBT_, _Steel-NaCl-MBT_) after exposure to either WSS or 3.5 wt % NaCl includes three components and corresponds to an EIS spectrum comprising of one time constant. In particular, it consists of the solution resistance (\(R_{\text{sol}}\)), the charge transfer resistance together with a double-layer CPE (\(R_{\text{ct}}\)-\(\text{CPE}_{\text{dl}}\)) that are attributed to the presence of corrosion active pits either at the mild steel surface or at the metal-inhibitor interface, as shown in Figure 21a.