Mathematical Relationship Based on Experimental Data, for Corrosion Inhibition Mechanism of Phenolic Compounds Obtained from Echium italicum L.

We highlight in this chapter the corrosion protection using phenolic extract. The building of mathematical models using experimental results obtained from the investigation of phenolic molecules or fractions extracted from Echium italicum L., used as corrosion inhibitors is one of the new trends in the study of steel protection. The evaluation of the corrosion inhibition of carbon steel (API 5 L-X60) in a solution 1 M of hydrochloric acid was performed using gravimetric method, potentiodynamic polarization and electrochemical impedance spectroscopy (EIS). The predicted mathematical relationships between the corrosion rate and the inhibitory efficiency in the presence of the butanolic extract of Echium italicum L. (BEEI), when increasing temperature proved a good agreement between experimental and mathematical studies.


Introduction
Secondary metabolites, derived from natural plants, of which polyphenols constitute the grand part, have a wide range of activities [1]: biological (antibacterial, antioxidant [2,3], anti-inflammatory [4], antidiabetic [5], anti-carcinogenic [6], etc.) and chemical (in connection with their chelating power with metals and their reducing properties generated by the hydroxyl functions of their aromatic rings). These molecules are used in several fields: as preservative food additives (constituting an alternative to the use of synthetic ones, such as buthylhydroxyanisol (BHA) and buthylhydroxytoluene (BHT), which have carcinogenic effects [7]; as flavoring in cosmetology [8]; as additives in electrolytic baths during the metals electrodeposition [9][10][11] and as corrosion inhibitors [12][13][14].

Corrosion protection
Corrosion control is the set of measures that can protect metallic materials from the harmful effects of the aggressive environment. Many of these methods were reported in the literature [28,29]. The first protection is the choice of pure metal or alloys resistant to these attacks [30].
After designing the equipment using the appropriate material, it must be protected against corrosion should be considered to avoid many problems and ensure a certain service life. For this, the preferred outlet must comply with environmental protection requirements and allow the recycling or disposal of the various components at the end of their use by applying the following choices [31]: • Prevention by an adapted shape of the parts; • Protection by coatings; • Electrochemical protection; • Protection by corrosion inhibitors.
All these solutions have drawbacks of efficiency over time, cost and environmental pollution, which is why other alternatives are exploited in research for the benefit of sustainable development that respects the environment. To this end, there have recently been some new alternatives products both environmentally friendly and less expensive as the use of inhibitors from naturals origin [32].

Corrosion inhibitors
Inhibitors have been successfully applied to prevent corrosion and damage in many and varied technical fields for a long time. These products have been frequently studied because they provide simple solutions to protect metals from corrosion in the aquatic environment. With the originality of being the only means of interference of the corrosive environment with steel, these compounds reduce the rate of corrosion of metals when added in appropriate amounts, without apparent change [9,14].

Materials
The vegetable material was collected during May 2017 from the East-North of Tebessa (Algeria). The extraction and purification of the n-butanolic extract obtained from Echium italicum L. (BEEI) were performed as reported in litérature [10,33,34].
The tested aggressive medium was a chlorhydric acid solution (1 M), and the investigated inhibitors were freshly prepared solutions of BEEI with different concentrations (from 100 to 500 ppm).
Before each mesurements, the substrates were abraded using emery paper with different grades (from 200 to 2000), cleaned with distilled water and then acetone. The used substrates were carbon steels with the following composition: (by weight %): C (0.26%), Mn (1.35%), P (0.03%), S (0.03%) and Fe (98.33%). Weight loss measurements were performed with specimens with dimensions of 1 cm x 1 cm x 1 cm. For the electrochemical experiments, only an exposed surface of 1 cm 2 was used. All measurements were conducted in an aerated area.

Electrochemical measurements
All electrochemical measurements were accomplished through a Voltalab (PGZ 301) potentiostat and controlled by software model (Voltamaster 4) under given conditions. The electrochemical characteristics of CS sample in uninhibited and inhibited solutions were realized in conventional three-electrode cell: CS as working electrode, platinum electrode as counter electrode and saturated calomel electrode (SCE) as a reference.

Potentiodynamic polarization
Potentiodynamic polarization curves were recorded after total immersion of the working electrode (CS) in 1 M HCl solution containing different concentrations of the tested inhibitor by changing the electrode potential from +250 to À250 mV vs. open circuit potential (OCP) with a scan rate of 1 mV/s. The linear Tafel segments of anodic and cathodic curves were extrapolated to corrosion potential E corr ð Þto get corrosion current densities i corr ð Þ. The η w % ð Þat different concentrations of BEEI were calculated using the formula (1) [35]: where i 0 corr and i corr are the corrosion current density in the absence and presence of BEEI, respectively.

Electrochemical impedance spectroscopy
Electrochemical impedance spectroscopy (EIS) experiments were conducted in the frequency range of 100 kHz-10 mHz with a signal amplitude perturbation of 10 mV at open circuit potential (OCP) measured during 60 min of immersion in the tested solutions. The percentage of the inhibition efficiency η w % ð Þwas calculated using the polarization resistance by the following relationship [36]: where R 0 p and R p are polarization resistances without and with inhibitor addition, respectively.

Gravimetric method
For these measurements, the prepared and pre-weighed CS substrates were totally immersed in beakers containing 1 M HCl without and with the addition of diverse concentrations of BEEI. The substrates were taken out after two hours, washed with 20% NaOH solution containing 200 g/l of zinc dust with a brush, rinsed severally with bidistilled water, dried with acetone, washed again with bidistilled water, dried and reweighted [34]. From the weight loss data, the corrosion rate CR ð Þ was calculated according to Eq. (3) [37]: where w is the average weight loss, A is the total area of one CS sample, and t is the immersion time. The inhibition efficiency η w can be calculated by Eq. (4) [36]: where CR 0 and CR are the values of corrosion rate in absence and presence of BEEI respectively.

Mathematical regression
A mathematical model is used to correlate variables by fitting an equation to experimental data. When using two variables, one of them is considered as explanatory, whereas the other is considered as a dependent. The linear regression of y is given as a function of x in the equation where a and b can be calculated as follow: 6. Results and discussion

Electrochemical impedance spectroscopy
EIS was performed to estimate CS corrosion behavior in the presence of 500 ppm of BEEI at 298 K.
The inspection of Nyquist plots presented in Figure 1 shows only one depressed capacitive imperfect semicircle at the higher frequency range, indicating that the corrosion reaction is controlled by the charges transfer process on a heterogeneous and irregular steel surface electrode [39][40][41].
The electrical equivalent circuit model (EEC) adjusted by fitting from the resulting impedance diagrams for Nyquist plots is reported in literature [12].
From Table 1, a noticeable increase in R ct values and a decrease in C dl at 500 ppm of BEEI due to the formation of protective diapers on the CS surface are observed. This phenomenon can be explained by the higher adsorption of phytochemical components above CS upon BEEI addition which ultimately reduces the charge transfer between the CS surface and the corrosion medium [42]. Consequently, we noticed that theη R reached 74% at 500 mg.L À1 .

Potentiodynamic polarization measurement
To further evaluate the efficiency of the expected green inhibitor, the polarization technique was studied due to its excellent reliability. Figure 2 shows the parameters given in Table 2 and extracted from the CS electrode polarization curves for 500 ppm BEEI in the presence of 1 M HCl solution, at 293 K.
As can be seen in this figure, at 500 ppm of BEEI, the cathodic and anodic corrosion current densities will decrease. This comportment can be attributed to the adsorption of the inhibitor at the carbon steel interface [43] by reducing the dissolution of steel and delaying the hydrogen evolution reaction [44].
In our study, the maximum displacement in E corr value for the optimum concentration of BEEI was À13.3 mV (< 85 mV), which exhibits that the inhibitor acts   as a mixed type [43,45]. The highest inhibition efficiency was 75% at a concentration of 500 ppm.

Weight loss measurements
To study the behavior of the steel protection in the presence of an aggressive solution, the weight loss method was used. Based on this approach, the CR, the η w and the degree of surface coverage θ were calculated and are very useful for discussing adsorption properties.
From Table 3, it can be observed that the CR values reduced progressively with increasing BEEI concentration, reaching 0.0334 g cm À2 h À1 for 500 ppm at 293 K. This result indicates that the addition of this inhibitor slows down the dissolution process of the CS. There was also a CR increase with increasing temperature.
This behavior can be attributed to the removal of some adsorbed molecules contained in the BEEI, through thermal energy-induced mechanical vibration [46]. As the inhibition efficiency is derived from the weight-loss method, the highest inhibition efficiency was 72.88% for 500 ppm of BEEI at 293 K.

Thermodynamic parameters
The good fitting for experimental data of gravimetric measurements at all measured temperatures supports the applicability of Freundlich adsorption model expressed by the subsequent Equation [47,48]: where, K ads is the adsorption constant which could be taken as a measure of the strength of the adsorption forces between the inhibitor molecules and the metal surface [49].  Table 3.

Corrosion parameters obtained from weight loss measurements of CS in 1 M HCl with different concentrations of BEEI at different temperatures.
α is the parameter taking into account the heterogeneity of the intermolecular interactions in the adsorbed layer and the steel surface.
K ads represent the values calculated from the intercept lines of logθ versus Log C. The standard adsorption free enthalpy ΔG°a ds can be calculated from the equation given below [50,51]: where R is the gas constant, T is the absolute temperature, C H 2 O is the water concentration expressed in mg L À1 with an approximate value of 10 6 . [52].
The standard adsorption enthalpy ΔH°a ds À Á can be calculated by the relationship (10) [53]: where I is the integration constant. The ΔH°a ds values were calculated from the intercept lines of ln K ads versus 1=T . The standard adsorption entropy ΔS°a ds À Á was calculated using the equation below: The obtained results for K ads , ΔG°a ds , ΔH°a ds and ΔS°a ds are presented in Table 4.
From Table 4, we noticed the following: 1. A decrease in K ads values with increasing temperatures, indicating that the adsorbent molecules contained in the BEEI adsorbed from the metal surface [54].

Negative
ΔG°a ds values point out the spontaneity of the adsorption process. Moreover, the results obtained for ΔG°a ds adsorption values below À40 kJmol À1 confirm the physisorption mechanism [55].
3. The negative sign of ΔH°a ds indicates that the BEEI adsorption process on the steel surface is exothermic [56,57].
4. The positive sign of ΔS°a ds , indicates that the adsorption process is accompanied by increased disorder due to the substitution of water molecules during BEEI adsorption [58][59][60].

Activation parameters of the corrosion process
The Arrhenius-type process was used to calculate the activation energies E a ð Þ between the corrosion rate of CS in acidic solution, from the equation given below [61]: Where E a is the apparent activation energy of the CS dissolution. D is the Arrhenius pre-exponential factor. The logarithm of the CR versus 1=T can be branded by straight lines. The activation energy values were calculated from Arrhenius plots lnCR versus 1/T and registered in Table 5.
For the evaluation of the enthalpy ΔH a as well as the standard adsorption entropy ΔS a of the corrosion activation process, the alternative formulation of Arrhenius equation was employed [62,63]: where h is the Planck's constant, N a is the Avogadro's number. The plot of lnCR=T vs. 1=T gave a straight line with a slope of ÀΔH a =R ð Þand an intercept of ln R=hN a þ ΔS a ð /R), from which the values of ΔS a and ΔH a were calculated and registered in Table 5.
The activation energy values were higher in the presence of 500 ppm of BEEI than in its absence, which proved the adsorption of BEEI molecules on the substrate by electrostatic bonds (physisorption) [62,64]. The positive signs of ΔH a indicate the endothermic nature of the dissolution process [65].
The BEEI adsorption process is accompanied by a decrease in its entropy. This can be explained that before the adsorption of the extract on the steel, the disorder degree of the inhibitor molecules is high, but when the molecules are adsorbed on the surface of the substrate, there is a decrease the in the disorder (i.e. a decrease in the entropy) [66].

Mathematical regression
This investigation focused on how could the used inhibitor (BEEI) decrease the corrosion rate CR ð Þ and offer a good inhibition efficiency η w ð Þ. For that, a mathematical method was employed to predict the influence of a variable x ð Þ which is the BEEI concentration as a function of different variables y i À Á which are CR and η w for 293 K, 303 K, and 313 K [51,67]. The building of a mathematical model that can correlate between three variables (CR, η w , and C) was also highlighted. Based on experimental data of the CS behavior in the presence of BEEI at various temperatures, a linear regression relation between CR and η w , was performed.
As reported by Khadom et al. [67], when the correlation coefficient of the correlation is ˂ 0.30, the correlation is weak and when this coefficient is between 0.50 and 0.70, the correlation is important, while if it is >0.90, the correlation will be strong.
According to the plots shown in Figure 3, a decrease in the CR with a raise in the η w , when adding BEEI with various concentrations at different temperatures was distinguished, and based on correlation coefficients r 2 CR and r 2 η w given in Table 6, the concordance between the experimental and the simulated results is very strong.
As we can see from the obtained results, the correlation coefficients were almost >0.90 for considered temperatures. However, as the temperature rises the correlation coefficients go up. The Eqs. (12)- (14) construe the mathematical expression obtained for the relationship between the CR and the η w at different temperatures.
This mathematical model exhibits a decrease in the corrosion rate when inhibition efficiency rises, with reducing in temperature. These results indicate the powerful concordance between experimental and predicted results.

Conclusion
In order to estimate the multiple benefits of phenolic compounds, a phenolic Echium italicum L. extract was evaluated as an efficient corrosion inhibitor. The subsequent conclusions can be pointed out: • A concordance between the employed evaluating methods, suggesting that the phenolic extract could serve as an effective corrosion inhibitor for CS against corrosion.
• Depending on the gravimetric measurements, the investigated phenolic extract was spontaneously adsorbed on the CS surface following a physical model according to Freundlich isotherm.
• The prediction of the phenolic extract effect of as a corrosion inhibitor by the mathematical study was in good agreement with the experimental results, which confirms that the corrosion rate is affected by the temperature and the inhibitor concentration.
• The generated mathematical model shows a decrease in the corrosion rate with the increase of inhibition efficiency. These results indicate a strong agreement between experimental and mathematical results.
• The Echium italicum L. extract has prooved to be an effective corrosion inhibitor, and hence the phenolics can be useful on various existing industries to prevent losses caused by corrosion of steel without harming the environment.
For a more extensive valuation of phenolic products, many perspectives are envisaged, namely: • Evaluate the synergy between phenolic extracts extracted from different plants looking for a better efficiency.
• The study can be extended to analyze the effect of these phenolic compounds in other corrosive media and on other types of steel.
• Perform a theoretical simulation to highlight the active compound (s) responsible for this inhibition.