{Experimental study and mathematical modeling of the corrosion inhibition of mild steel with an organic compound in 1 M HCl:} http://dx.doi.org/10.5599/jese.1050 227 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239; http://dx.doi.org/10.5599/jese.1050 Open Access : : ISSN 1847-9286 www.jESE-online.org Original scientific paper Experimental study and mathematical modeling of the corrosion inhibition of mild steel with an organic compound in 1 M HCl Wafia Boukhedena1,3,, and Samir Deghboudj2,3 1Department of Science Materials, Larbi Tebessi University, 12002 Tebessa, Algeria 2Department of Mechanics, Larbi Tebessi University, 12002 Tebessa, Algeria 3Mines Laboratory, Larbi Tebessi University, 12002 Tebessa, Algeria Corresponding author: wafia.boukhedena@univ-tebessa.dz; Tel.: +213 7 71 64 25 62 Received: July 10, 2021; Accepted: August 5, 2021; Published: August 20, 2021 Abstract In this paper, a synthesized organic compound from the family of ketene dithioacetal was studied as corrosion inhibitor for mild steel in 1.0 M hydrochloric acid by gravimetric measurements. The aim of this work is to study the effect of inhibitor concentration and temperature on the corrosion resistance, and to compare the experimental results with those obtained by mathematical models. The structural properties are characterized using the scanning electron microscopy technique. It has been found that the inhibition efficiency increases with increasing inhibitor concentration. The adsorption of studied compound on mild steel surface follows Langmuir’s isotherm. Taking into account the influence of inhibitor concentration and temperature on the corrosion inhibition efficiency, obtained data were analyzed by two mathematical models based on linear and quadratic regression. The obtained experimental results are in a good agreement with those predicted by the quadratic regression models. Keywords Ketene dithioacetal; hydrochloric acid; gravimetric measurements; linear regression; quadratic regression. Introduction The use of organic inhibitors for the control of corrosion of metals and alloys is of practical importance for many industrial applications, where acid solutions are commonly used in several processes. Among these, hydrochloric acid is one of the most widely used for pickling, and chemical and electrochemical etching of some metals and alloys [1]. Because of acid aggressiveness, the use of corrosion inhibitors is considered as the most effective method for the protection of many metals and alloys against acid attack [2-7]. Inhibitors are also employed to reduce the dissolution rate of metals. http://dx.doi.org/10.5599/jese.1050 http://dx.doi.org/10.5599/jese.1050 http://www.jese-online.org/ mailto:wafia.boukhedena@univ-tebessa.dz J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 228 Many authors have reported various types of organic inhibitors used as corrosion inhibitors for steel in hydrochloric acid solution [8-15]. These investigations revealed that organic compounds are acting as inhibitors due to heteroatoms such as nitrogen, sulfur, oxygen and phosphorus, which owing to their free electron pairs are capable to form coordinate covalent bonds with metals. In addition, π electrons in triple or conjugated double bond also exhibit good inhibitive properties [16-18]. The purpose of this work is to investigate the inhibitory action of the organic compound 1,3-dithianes, substituted with two electroactive groups A1 and A2: 3-(1,3-dithian-2-ylidene) pentane-2,4 dione (PDDY). The behavior of mild steel in acidic medium in the absence and presence of the inhibitor at different concentrations (5×10-6 - 10-3 M) and at various temperatures (20-60 oC) was explored. The chemical structure of PDDY is shown in Figure 1. The assessment of corrosion behavior is carried out using weight loss measurements and scanning electron micrograph (SEM) imaging at 20 °C. In the second part of this work, two mathematical models, based on linear and quadratic regression, are suggested to investigate the effect of concentration and temperature upon inhibition efficiency of PDDY. C S S C A1 A2 Figure 1. Chemical structure of 3-(1, 3-dithian-2-ylidene) pentane-2,4 dione compound (PDDY) Experimental Preparation of samples The composition of mild steel used in this study is: C - 0.09 wt.%; Si - 0.05 wt.%; Mn - 0.13 wt.%; S - 0.24 wt.%; P - 0.24 wt.% and Fe balance. The mild steel specimens were cut into 1 × 1 × 1 cm pieces for the mass measurements. The specimens were abraded with a series of SiC papers (grades 320, 400, 500, 800, 1000, 1200 and 2000), washed with distilled water, degreased with acetone, and dried with a cold air stream at room temperature before use in the experiments. Electrolytic solution The aqueous electrolyte solution (1 M HCl) was prepared by dilution of analytical reagent grade, 37 % w/w HCl (Merck) with bi-distilled water. The measurements were carried out in 1 M HCl in the absence and presence of the inhibitor within the concentration range 5×10-6 to 1×10-3 M. Preparation of inhibitor The inhibitor 3-(1,3-dithian-2-ylidene) pentane-2,4-dione was synthesized using K2CO3 (21 g, 0.15 mol) and active methylene compound, 5.2 ml (0.05 mol) in 25 ml of DMF. The mixture was stirred magnetically. 4.5 mL (0.075 mol) of carbon disulfide was then added in just one time at room temperature. The stirring was maintained for 10 min before dropwise addition of reagent dielectrophile, 7.2 ml (0.06 mol) 1,3-dibromopropane, during 20 min. After seven hours of stirring at room temperature, 250 mL of ice water was added to the reaction mixture. The formed precipitate was filtered, dried and then purified by recrystallization from ethanol and used directly in the experiments in the concentration range (5×10-6, 10-5, 5×10-5, 10-4, 5×10-4 and 10-3 ) mol L-1 [19]. The studied compound was recovered in the form of orange crystals, exhibiting the following W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 229 characteristics: molar mass, M = 216 g/mol, yield 83 %, M.P. =104 °C. FT-IR (FT-IR spectra of PDDY in its solid state, 1/ : 1630 cm-1 (C=O), 1725 cm-1 (H3C-C=O), 1173-1234 cm-1 (C-S-C), 1415 cm-1 (C=C). 1H NMR ((CDCl3), 250 MHz)):  = 2.32 ppm (s, 6H, 2CH3),  = 2.25 ppm (m, 2H, CH2),  = 2.95 ppm (t, 4H, 2CH2). The mass spectroscopy analysis revealed that the inhibitor produced fragment ions m/z 217. Scanning electron microscopy (SEM) The morphological structures of mild steel surfaces before and after total immersion for 3 h in a corrosive solution (1 M HCl) with and without addition of 10−3 M of PDDY at 20 °C, were determined by the scanning electron microscope, model JEOL JSM-6360 LV. Gravimetric measurements Gravimetric experiments were performed according to the standard methods [20-22]. Experiments were conducted under total immersion of mild steel specimens in the stagnant and aerated conditions, using 250 mL capacity beakers. After being weighed accurately with high sensitivity balance, the specimens were immersed in 100 mL of 1 M HCl with and without various concentrations (5×10-6 - 10-3 M) of PDDY at various temperatures (20, 30, 40, 50 and 60 °C) in aerated conditions. After 3 hours of immersion, the specimens were taken out, rinsed thoroughly with distilled water, dried and weighed accurately again. The average of three replicates was used to further process the data. The average weight loss ∆W was calculated using the equation (1): ∆W= W1 – W2 (1) where, W1 and W2 are respectively the average weight of specimens before and after immersion. The corrosion rate (CR) in mg cm-2 h-1, the surface coverage () and inhibition efficiency (IEW), obtained from gravimetric experiments were computed using the equations (2)-(4) [20,23,24]: CR W St  = (2) 0 i 0 CR -CR CR  = (3) 0 i 0 CR -CR 100IEw CR = (4) where ∆W is the average weight loss, S is the total surface area of the specimen and t is the immersion time. CR0 and CRi are corrosion rates in the absence and presence of various concentrations of PDDY, respectively. Corrosion rates and inhibitor efficiencies were evaluated and computed at different operating conditions. Results and discussion Mass loss measurements Corrosion of mild steel in 1 M HCl in the absence and presence of various concentrations (5×10-6, 10-5, 5×10-5, 10-4, 5×10-4 and 10-3 M) of PDDY was studied by weight loss experiments at various temperatures (20, 30, 40, 50 and 60 °C). The corrosion rate (CR) in mg cm-2 h-1 and values of inhibition efficiency, obtained by 30 test runs of weight loss measurement after 3 hours of immersion in 1 M HCl solution, are summarized in Table 1. It was found that addition of PDDY inhibits corrosion of mild steel at all concentrations used in this study. As presented in Table 1, it is clear http://dx.doi.org/10.5599/jese.1050 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 230 that corrosion rate increased with temperature and decreased with inhibitor concentration. Many similar results were obtained by other researchers, who already found that corrosion rate and inhibition efficiency depend on the concentration of the inhibitor and temperature of the medium [10,11]. Table 1. Corrosion parameters obtained from weight loss measurements of mild steel after 3 h of immersion in 1 M HCl solution in the absence and presence of different concentrations of PDDY at various temperatures C / M CR, mg cm-2 h-1 IEW, % t / °C 20 30 40 50 60 20 30 40 50 60 blank 0.897 1.983 4.751 9.774 13.65 -- -- -- -- -- 5×10-6 0.361 0.807 2.419 6.271 9.510 59.755 59.304 49.084 35.840 30.330 1×10-5 0.277 0.681 2.087 5.016 8.947 69.119 65.658 56.072 48.680 34.454 5×10-5 0.194 0.552 1.649 4.354 8.161 78.372 72.163 65.291 55.453 40.212 1×10-4 0.166 0.393 1.337 3.755 6.860 81.494 80.181 71.858 61.582 49.744 5×10-4 0.074 0.336 0.925 2.598 4.944 91.750 83.056 80.530 73.419 63.780 1×10-3 0.062 0.187 0.776 1.943 4.165 93.088 90.570 83.667 80.121 69.487 Inhibition efficiencies are for different concentrations of PPDY at different temperatures presented in Figure 2, where it is clearly seen that IEw reached the maximum value of 93.09 % for 10-3 M PPDY at 20 °C. It is well known that adsorption of inhibitor at steel surface is primarily responsible for the reduction of metal dissolution process in corrosive media [23]. The adsorption is enhanced by the presence of hetero atoms with lone pairs of electrons of the molecule inhibitors that facilitate the electrostatic adsorption on the steel surface by forming stable insoluble films. Data in Table 1 show that without inhibitor, the corrosion rate is as high as 0.897 (mg cm-2 h-1), while in the presence of 10-3 M of inhibitor, CR value is reduced to 0.062 (mg cm-2 h-1). From these measurements, we can conclude that PDDY is an effective inhibitor for mild steel in 1 M HCl solution. 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 110 t / °C CPDDY/ 10 4 M 10 M 5 M 1 M 0.5 M 0.1 M 0.05 M IE w , % Figure 2. Variation of inhibition efficiency with temperature after 3 hours of mild steel exposure in 1M HCl and various concentrations of PDDY Adsorption isotherm Corrosion inhibition process is characterized by adsorption of the inhibitor on the metal surface. Adsorption of inhibitor is a displacement reaction in which the adsorbed water molecule is removed from the metal surface according to [13,14]: W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 231 Org(sol) + nH2O(ads) ↔ Org(ads) + nH2O(sol) where n is the number of water molecules replaced by one organic molecule and H2O(ads) is the water molecule on the metal surface. Org(sol) and Org(ads) are organic molecules in the aqueous solution and adsorbed on the metal surface, respectively. For the effective adsorption of an inhibitor on the metal surface, the interaction force between metal and inhibitor must be greater than the interaction force of metal and water molecule [15]. Therefore, it is of great importance to find the appropriate adsorption isotherm that fits the experimental results. The experimental data have been tested with several adsorption isotherms including Langmuir, Temkin, Frumkin and Freundlich [16,17]. In this work, Langmuir adsorption isotherm was found as the best description of adsorption behavior of the studied compound. An expression of the Langmuir isotherm is given by [24,25]: ads 1C C K = + (5) where C is inhibitor concentration, Kads is equilibrium adsorption constant and  is fractional surface coverage. According to eq. (5), the plot of (C / ) versus C should yield straight line with nearly unit slope. Best results for Langmuir adsorption isotherm for PDDY on mild steel surface are presented in Figure 3. It is clear from Figure 3, however, that slopes of the straight lines are slightly greater than unity (cf. slope data referred in Figure 3). Therefore, it could be concluded that each PDDY unit occupies more than one adsorption site on the mild steel surface. 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 C  -1 / m o l L -1 C / mol L-1 20 °C R²= 0.9998; Slope= 1.07 30 °C R²= 0.9983; Slope= 1.11 40 °C R²= 0.9996; Slope= 1.19 50 °C R²= 0.9981; Slope= 1.24 60 °C R²= 0.9977; Slope= 1.42 Figure 3. Langmuir adsorption plots for mild steel in 1 M HCl containing different concentrations of PDDY at various temperatures To account for this phenomenon, a modified Langmuir adsorption isotherm could be applied, which is given by [7,19,20]: ads C n nC K = + (6) In eq. (6), n represents the slope of the respective line. The values of Kads obtained from the values of slopes at different temperatures (Figure 3) were computed and gathered in the second column of Table 2. The increased values of Kads for PDDY reflect the increasing adsorption capability due to structural formation on the metal surface [26,27]. Kads is related to the standard Gibbs free energy of adsorption (∆G°ads) by the following equation ΔGads0 = - RT ln (55.5 Kads) (7) C  -1 / m o l L- 1 http://dx.doi.org/10.5599/jese.1050 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 232 where R is the universal gas constant (8.134 J K-1 mol-1), T is temperature and 55.5 is molar concentration of water in the solution. The thermodynamic parameters derived from Langmuir isotherms are listed in Table 2. The values of ∆G°ads for PDDY are all negative, what indicates the stability and spontaneity of adsorption of the inhibitor on the metal surface. The effectiveness of corrosion inhibition increases with increasing of negative ∆G°ads values. Adsorption enthalpy and entropy ∆H°ads and ∆S°ads are determined graphically from the following equation ΔG°ads = ΔH°ads - ΔS°ads (8) Table 2. Thermodynamic parameters of adsorption of PDDY on mild steel in 1 M HCl at various temperatures It is well known that values of ∆G°ads less negative than -20 kJ / mol are associated with the physical adsorption, characterized by an electrostatic interaction between the charged molecule and the charged metal. ∆G°ads values around -40 kJ / mol or higher, are associated with the chemical adsorption where the sharing or transfer of organic molecules charge with the metal surface occurs [20-24]. The decrease observed for Kads (Table 2), and also for IEw (Figure 3) with increasing temperature, suggests that the PDDY molecules are physically adsorbed on the metal surface which favors their desorption processes. The value of ∆H°ads provides further information about the mechanism of corrosion inhibition. The negative value of ∆H°ads indicates that adsorption process is exothermic [3,7,23]. An exothermic adsorption process signifies chemical, physical or a mixture of both [24-27], whereas the endothermic process is attributed to chemisorption [28,29]. Results obtained from thermodynamic calculations for adsorption are in good agreement with the values of inhibition efficiency obtained from the weight loss. Thermodynamic calculations for corrosion reaction The stability of a corrosion inhibitor in an aggressive medium at some required operating temperature is very important for its practical applications. In this paper, the corrosion of mild steel in 1 M HCl was studied in the temperature range of 20-60 °C, in the absence and presence of different concentrations of PDDY, after 3 h of immersion time. Temperature dependence of corrosion rate (CR) is described by Arrhenius equation: a CR E RTAe − = (9) where A is the pre-exponential factor, Ea is activation energy for metal dissolution (corrosion) reaction, T is absolute temperature, and R is universal gas constant. The dependence of logarithm of the corrosion rate (ln CR) on the reciprocal of the absolute temperature (1 / T) for mild steel in 1 M HCl is presented Figure 4. The corrosion rate of mild steel in acidic solution increases with rise of the temperature, regardless to the presence of inhibitor in the corrosive solution. In the case of uninhibited solution, the increase in corrosion rate was significant compared to inhibited cases involving a decrease in inhibition efficiency with increasing temperature up to 60 °C. The decrease in inhibition efficiency reveals that the film formed on the T / °C Kads / 10-4 L mol-1 ∆G°ads / kJ mol-1 ∆H°ads / kJ mol-1 ∆S°ads / kJ mol-1 K-1 20 0.1374 -38.604 -24.36 0.049 30 8.935 -38.837 40 9.580 -40.300 50 5.424 -40.060 60 3.968 -40.434 W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 233 metal surface is less protective at higher temperatures, since the desorption rate of the inhibitor is greater at higher temperatures [30]. Thermodynamic parameters of the corrosion reaction, namely activation energy Ea, entropy (∆Sa) and enthalpy (∆Ha) were calculated using Arrhenius equation (9) and transition state theory equation [19]: a a CR e S H R RT RT A e Nh  − = (10) In eq. (10), N is Avogadro's number (6.022×1023 mol-1), and h is Planck's constant (6.63×10-34 J s). Eq. (10) is plotted in Figure 5 as ln (CR / T) against (1 / T). 0.0030 0.0031 0.0032 0.0033 0.0034 -3 -2 -1 0 1 2 3 CPDDY / 10 4 M 0 M 10 M 5 M 1 M 0.5 M 0.1 M 0.05 M ln ( C R / m g c m -2 h -1 ) (1 / T) / K-1 Figure 4. Plots for mild steel corrosion rates in 1 M HCl in absence and presence of different concentrations of PDDY 0.0030 0.0031 0.0032 0.0033 0.0034 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 (1 / T) / K-1 ln ( (C R / T ) / m g c m -2 h -1 K -1 ) CPDDY / 10 4 M 0 M 10 M 5 M 1 M 0.5 M 0.1 M 0.05 M Figure 5. Transition state plots for mild steel corrosion rates in 1 M HCl in absence and presence of different concentrations of PDDY The values of Ea and A, obtained from the slopes (-Ea / R) and intercepts (ln A) of the lines in Figure 4, are displayed in Table 3, together with values of ΔHa and ΔSa deduced from the slopes (-ΔHa / R) and intercepts ln (R / Nh) +ΔSa / R of the lines in Figure 5. Table 3. Thermodynamic activation parameters of mild steel dissolution in 1 M HCl with and without various concentrations of PDDY C / 104 M Ea / kJ mol-1 ΔHa / kJ mol-1 ΔSa / J mol-1 K-1 ln A blank 57.334 54.739 -58.37 23.483 0.05 69.877 67.283 -23.7 27.655 0.10 72.741 70.147 -16.0 28.584 0.50 77.604 75.009 -2.11 30.248 1.00 78.797 76.202 -0.0025 30.504 5.00 85.139 82.545 16.93 32.541 10.00 87.480 84.885 22.09 33.161 Examination of Table 3 shows that the activation energy values in the presence of PDDY ranged from 69.877 to 87.480 kJ mol−1, revealing that the activation energy for metal dissolution increased in the presence of inhibitor. Also, Ea values in the presence of inhibitor are all higher than in absence of inhibitor, which indicates physical adsorption (electrostatic interaction). It has already been reported in the literature that inhibitors for which the activation energy in the inhibited solution is greater than that of the blank solution, Ea(inh) > Ea, are adsorbed on the substrate by electrostatic http://dx.doi.org/10.5599/jese.1050 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 234 bonds (physisorption). At the other side, inhibitors where Ea(inh) < Ea, adsorb on the metal surface through strong bonds (chemisorption) [31,32]. The activation energy rises with increasing inhibitor concentration, suggesting strong adsorption of inhibitor molecules at the metal surface [33-37]. The values of ΔHa and Ea are nearly the same and are higher in the presence of the inhibitor, indicating that the energy barrier of the corrosion reaction increased in the presence of inhibitor without changing the mechanism of dissolution [19]. The negative sign of the activation entropy values either in absence, or presence of the inhibitor may be explained by activated molecules in a higher order state than that at the initial stage [38,39]. At higher inhibitor concentrations (5×10-4 and 10−3 M), positive values of the entropy of activation ΔSa were observed in the media. This indicates that the system passes from a more ordered state to a more random arrangement [40]. Morphological characterization The scanning electron micrographs (SEM) of the mild steel surface in the absence and presence of 10-3 M of PDDY are presented in Figure 6. Comparison of Figures 6a and 6b shows that the surface of the sample is heavily damaged and severely corroded after 3 hours of immersion in 1 M HCl. Damages appear uniform with some lines resulting from polishing made before the testing. In the presence of PDDY, however, Figure 6c shows that the external morphology appears softer, indicating a protected surface. These images suggest that protection comes from the formation of PDDY layer on the mild steel surface that prevents the attack of acids. a b c Figure 6. SEM micrographs of mild steel surface: before corrosion (a); after immersion in 1 M HCl solution for 3 h at 20 °C (b); after immersion in 1 M HCl containing 10-3 M of PDDY for 3 h at 20 °C (c) Mathematical modeling Mathematical regression analysis can be employed as a powerful tool for data representation [19]. The least squares method is the standard statistical approach in the regression analysis, used to find W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 235 the optimum fit for a set of data points by minimizing the sum of the squares of the residuals of points from the plotted curve. In this study, two mathematical models were applied to illustrate the inhibition efficiency of PDDY for mild steel in 1.0 M HCl. The proposed mathematical models are classified into two groups, linear and quadratic models. Experimental data obtained in this work and presented above were used to construct the models that describe the relationship between the inhibition efficiency IEW as a response, and temperature t and inhibitor concentration C as factors. Both models take into account the individual effect of each variable and the interaction between them. Linear model A linear mathematical model was used as a statistical tool aiming to predict the effect of the variables xi and yi which are respectively the temperature t and the inhibitor concentration to the function zi which is the inhibitor efficiency IEW. The proposed model is given by the equation (11): z = a1x + a2y + a3xy + a4 +  (11) where  is the error term. The optimal model parameters ai within the meaning of the least-squares method are these that minimize the quantity: ( ) N 2 1 2 3 4 1 2 3 4 i i=1 , , , ( )S a a a a a x a y a xy a z= + + + + − (12) In this expression, N is the number of experiments, ai are constants representing the model para- meters. The minimum of this expression is found when the partial derivatives (∂S/∂ai) are equal to zero: i 0 S a  =  i= 1, 2, 3, 4 (13) This leads to a system of equations. Parameters of the model were estimated based on the least square method. The data were analyzed using computer program MATLAB that performs these calculations. Since x = t, y = C and z= IEW, the final model equation is: IEw = -0.865 t + 17324.609 C + 312.357 TC + 92.125 (14) Based on the proposed linear model, the predicted values of inhibition efficiency as a function of temperature and inhibitor concentration are computed and listed in the first part of Table 4. Quadratic model The second proposed quadratic model equation was obtained by representing the inhibition efficiency IEW by the response function z which can be expressed by the following equation: z = a1x² + a2y² + a3x + a4y + a5xy + a6+  (15) where,  is the error term, x the temperature, °C and y the inhibitor concentration (mol L-1). Similarly, the optimal model parameters ai within the meaning of the least-squares method are these which minimize the quantity: ( ) 52 25 6 6 N 2 1 2 3 4 1 2 3 4 i i=1 , ,, , , ( )a a yS a a a a a x a y a x a a xy a z+= + + + + + − (16) The minimum of this expression is found using the eq. (13). Using MATLAB software, parameters a1-a6 of the model were estimated and the final model equation is defined as: http://dx.doi.org/10.5599/jese.1050 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 236 IEw = -0.01143 t² - 5.19673×107 C² + 0.13615 t + 80173.4333 C + 0.01667 TC + 69.6171 (17) As for the linear model, the predicted values of inhibition efficiency were computed using the quadratic regression model and gathered in the second part of Table 4. For analysis of the accuracy of the predicted results obtained with linear and quadratic regression models, we estimated the coefficient of determination known as R-squared (or R2), using the equation below ( ) ( ) N 2 i mi i=1 N 2 i i=1 ² 1 z z R z z − = − −   (18) In this relationship zi is experimental inhibition efficiency, zmi predicted inhibition efficiency and z̅ the average. The determination coefficient was found equal to 0.802 for the linear model and 0.902 for quadratic regression model, respectively. In comparison with experimentally determined IEW given in Table 1, data in Table 4 show similar IEW values at all temperatures and concentrations of PDDY. Figures 7 and 8 show the relationship between the values of the inhibitor efficiency obtained from the experimental work and those predicted by linear and quadratic regression models. It is obvious that almost 80 and 90 % of the data obtained from the linear and quadratic regression models are located on the line of equality, which means that the experimental and predicted values of inhibitory efficacy are close. Table 4. Inhibition efficiencies values computed by linear and quadratic regression models for different inhibitor concentrations and temperatures C / 104 M IEW, % Linear regression Quadratic regression t / °C 20 30 40 50 60 20 30 40 50 60 0.05 74.944 66.310 57.676 49.042 40.408 68.168 63.814 57.185 48.249 37.038 0.01 75.062 66.443 57.825 49.207 40.588 68.565 64.211 57.572 48.646 37.435 0.50 76.005 67.511 59.018 50.524 42.031 71.647 67.293 60.654 51.728 40.517 1.00 77.183 68.846 60.509 52.171 43.834 75.266 70.912 64.273 55.347 44.136 5.00 86.612 79.524 72.436 65.348 58.261 94.863 90.510 83.870 74.945 63.734 10.00 98.398 92.872 87.346 81.820 76.294 95.975 91.621 84.982 76.057 64.845 30 40 50 60 70 80 90 100 40 50 60 70 80 90 100 R²= 0.8294 P re d ic te d E I w , % Experimental EIw, % Figure 7. Fitting curves of predicted against experimental inhibition efficiency obtained by the linear regression model 30 40 50 60 70 80 90 100 30 40 50 60 70 80 90 100 R²= 0.90259 P re d ic te d IE w , % Experimental IE w , % Figure 8. Fitting curves of predicted against experimental inhibition efficiency obtained by the quadratic regression model W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 237 This is quite true because the determination coefficients were R²= 0.829 for linear model and R² = 0.920 for quadratic regression model. According to [19], when R2 is < 0.30 the relationship is weak, for R2 = 0.50 and 0.70, the relationship is important, while for R2 > 0.90, the relationship is powerful. Based on correlation coefficients, the present results indicate a strong relationship between experimental and mathematical data, indicating the concordance between the experimental and the predicted results. As shown in Figure 9, examination of data provided by linear and quadratic regression models and presented as surface plots, reveals that the addition of PDDY at different concentrations decreases the corrosion rate of mild steel. The inhibition efficiency increases with increasing inhibitor concentration (red zone) and decreases with increasing temperature (blue zone). This can be explained by a regression of adsorption induced by temperature rise. At concentration 10-3 M and temperature 20 °C, PDDY exhibits maximum inhibition efficiency for both models: 98.398 % (linear model) and 95.975 % (quadratic regression). A comparison with the experimentally found value of inhibition efficiency of 93.088 % shows more reliability of the quadratic regression model than the linear model. a b Figure 9. Surface plot for inhibition efficiency, temperature and inhibitor concentration based on: a) linear regression model; b) quadratic regression model Conclusion In this paper, the inhibition effect and adsorption behavior of the organic compound 3-(1,3-di- thian-2-ylidene) pentane-2,4 dione (PDDY) on mild steel in 1 M HCl medium were examined, using the weight loss method, morphological characterization and mathematical modeling. The following conclusions are drawn: • The inhibition efficiency of PDDY increases with increasing inhibitor concentration in the range of 5×10-6 to 10-3 M, and reaches the maximum value of 93.088 % in the presence of 10-3 M of PDDY. The thermodynamic study showed that adsorption of this inhibitor on the mild steel surface is spontaneous and follows the Langmuir adsorption isotherm model. The negative value of the Gibbs free energy of adsorption (ΔGads) is indicative for a strong interaction between inhibitor molecules and the surface of mild steel. • The evolution of the corrosion rate of mild steel in the corrosive solution alone (1 M HCl) shows a regular and rapid growth, confirming an increasing metallic dissolution with increasing temperature. The inhibitory efficiency of PDDY decreases, while corrosion rate increases with temperature in the range 20 to 60 oC for all inhibitor concentrations used. This behavior illustrates physisorption of PDDY molecules on mild steel surface. http://dx.doi.org/10.5599/jese.1050 J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 CORROSION INHIBITION WITH AN ORGANIC COMPOUND 238 • Formation of a protective layer on mild steel surface by the inhibitor observed by SEM, confirmed high performance of the inhibitive effect of PDDY. • Linear and quadratic mathematical regression models are found suitable to represent the experimental data with high correlation coefficients. Experimentally determined inhibition efficiency of 93.088 % is found closer to the result predicted by the quadratic regression model. The predicted result also confirm that inhibition efficiency is influenced by temperature, inhibitor concentration and their combined effect. Acknowledgements: The authors like to thank the Algerian general direction of research (DGRSDT) for their support. References [1] L. Afia, R. Salghi, A. Zarrouk, H. Zarrok, E. Bazzi, B. Hammouti, M. Zougagh, Transactions of the Indian Institute of Metals 66(1) (2013) 43-49. http://doi.org/10.1007/s12666-012-0168-z [2] S. Issaadi, T. Douadi, A. Zouaoui, S. Chafaa, M. A. Khan, G. Bouet, Corrosion Science 53(4) (2011) 1484-1488. http://doi.org/10.1016/j.corsci.2011.01.022 [3] H. Jafari, I. Danaee, H. Eskandari, M. RashvandAvei, Industrial & Engineering Chemistry Research 52(20) (2013) 6617-6632. http://doi.org/10.1021/ie400066x [4] O. Ghasemi, I. Danaee, G. R. Rashed, M. RashvandAvei, M. H. Maddahy, Journal of Materials Engineering and Performance 22(4) (2013) 1054-1063. http://doi.org/10.1007/s11665-012- 0348-3 [5] Z. Moallem, I. Danaee, H. Eskandari, Transactions of The Indian Institute of Metals 67(6) (2014) 817-825. http://doi.org/10.1007/s12666-014-0403-x [6] A. Döner, R. Solmaz, M. Özcan, G. Kardaş, Corrosion Science 53(9) (2011) 2902-2913. http://doi.org/10.1016/j.corsci.2011.05.027 [7] A. Fiala, W. Boukhedena, S. Lemallem, H. B. Ladouani, H. Allal, Journal of Bio- and Tribo- Corrosion 5(2) (2019) 42. http://doi.org/10.1007/s40735-019-0237-5 [8] D. Daoud, T. Douadi, H. Hamani, S. Chafaa, M. Al-Noaimi, Corrosion Science 94 (2015) 21-37. http://doi.org/10.1016/j.corsci.2015.01.025 [9] E. E. Abd El Aal, S. Abd El Wanees, A. Farouk, S. M. Abd El Haleem, Corrosion Science 68 (2013) 14-24. http://doi.org/10.1016/j.corsci.2012.03.021 [10] H. Gerengi, I. Uygur, M. Solomon, M. Yildiz, H. Goksu, Sustainable Chemistry and Pharmacy 4 (2016) 57-66. http://doi.org/10.1016/j.sc2016.10.003 [11] M. Mobin, I. Ahmad, M. Basik, M. Murmu, P. Banerjee, Sustainable Chemistry and Pharmacy 18 (2020) 100337. https://doi.org/10.1016/j.scp.2016.10.003 [12] A. K. Singh, S. K. Shukla, M. Singh, M. A. Quraishi, Materials Chemistry and Physics 129(1-2) (2011) 68-76. http://doi.org/10.1016/j.matchemphys.2011.03.054 [13] S. Cheng, S. Chen, T. Liu, X. Chang, Y. Yin, Materials Letters 61(14-15) (2007) 3276-3280. http://doi.org/10.1016/j.matlet.2006.11.102 [14] M. Hazwan Hussin, M. Jain Kassim, Materials Chemistry and Physics 125(3) (2011) 461-468. http://doi.org/10.1016/j.matchemphys.2010.10.032 [15] E. Ghali, V. S. Sastri, M. Elboujdaini, Corrosion Prevention and Protection: Practical Solutions, John Wiley & Sons, Chichester, England, 2007. p. 579. http://doi.org/10.1002/9780470024546 [16] S. Manimegalai, P. Manjula, Journal of Materials and Environmental Science 6(6) (2015) 1629- 1637. [17] M. A. Petrunin, L. B. Maksaeva, T. A. Yurasova, E. V. Terekhova, M. A. Maleeva, V. A. Kotenev, E. N. Kablov, A. Yu. Tsivadze, Protection of Metals and Physical Chemistry of Surfaces 51(6) (2015) 1010-1017. http://doi.org/10.1134/S2070205115060179 http://doi.org/10.1007/s12666-012-0168-z https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11000448#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2011.01.022 https://pubs.acs.org/action/doSearch?field1=Contrib&text1=Hojatollah++Jafari https://pubs.acs.org/action/doSearch?field1=Contrib&text1=Iman++Danaee https://pubs.acs.org/action/doSearch?field1=Contrib&text1=Hadi++Eskandari https://pubs.acs.org/action/doSearch?field1=Contrib&text1=Mehdi++RashvandAvei https://pubs.acs.org/iecr https://pubs.acs.org/iecr http://doi.org/10.1021/ie400066x https://link.springer.com/article/10.1007/s11665-012-0348-3#auth-O_-Ghasemi https://link.springer.com/article/10.1007/s11665-012-0348-3#auth-I_-Danaee https://link.springer.com/article/10.1007/s11665-012-0348-3#auth-G__R_-Rashed https://link.springer.com/article/10.1007/s11665-012-0348-3#auth-M_-RashvandAvei https://link.springer.com/article/10.1007/s11665-012-0348-3#auth-M__H_-Maddahy https://link.springer.com/journal/11665 https://link.springer.com/journal/11665 http://doi.org/10.1007/s11665-012-0348-3 http://doi.org/10.1007/s11665-012-0348-3 https://link.springer.com/article/10.1007/s12666-014-0403-x#auth-Z_-Moallem https://link.springer.com/article/10.1007/s12666-014-0403-x#auth-I_-Danaee https://link.springer.com/article/10.1007/s12666-014-0403-x#auth-H_-Eskandari http://doi.org/10.1007/s12666-014-0403-x https://www.sciencedirect.com/science/article/abs/pii/S0010938X11002496#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11002496#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11002496#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11002496#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2011.05.027 https://link.springer.com/article/10.1007/s40735-019-0237-5#auth-Hamza-Allal https://link.springer.com/journal/40735 https://link.springer.com/journal/40735 http://doi.org/10.1007/s40735-019-0237-5 https://www.sciencedirect.com/science/article/abs/pii/S0010938X15000360#! http://doi.org/10.1016/j.corsci.2015.01.025 http://doi.org/10.1016/j.corsci.2012.03.021 http://doi.org/10.1016/j.sc2016.10.003 https://www.sciencedirect.com/science/article/abs/pii/S2352554120305763#! https://www.sciencedirect.com/science/article/abs/pii/S2352554120305763#! https://www.sciencedirect.com/science/article/abs/pii/S2352554120305763#! https://www.sciencedirect.com/science/article/abs/pii/S2352554120305763#! https://www.sciencedirect.com/science/article/abs/pii/S2352554120305763#! https://www.sciencedirect.com/science/journal/23525541 https://doi.org/10.1016/j.scp.2016.10.003 https://www.sciencedirect.com/science/article/abs/pii/S0254058411002574#! https://www.sciencedirect.com/science/article/abs/pii/S0254058411002574#! https://www.sciencedirect.com/science/article/abs/pii/S0254058411002574#! https://www.sciencedirect.com/science/article/abs/pii/S0254058411002574#! https://www.sciencedirect.com/science/journal/02540584 http://doi.org/10.1016/j.matchemphys.2011.03.054 https://www.sciencedirect.com/science/article/abs/pii/S0167577X0601367X#! https://www.sciencedirect.com/science/article/abs/pii/S0167577X0601367X#! https://www.sciencedirect.com/science/article/abs/pii/S0167577X0601367X#! https://www.sciencedirect.com/science/article/abs/pii/S0167577X0601367X#! https://www.sciencedirect.com/science/article/abs/pii/S0167577X0601367X#! https://www.sciencedirect.com/science/journal/0167577X http://doi.org/10.1016/j.matlet.2006.11.102 https://www.sciencedirect.com/science/article/abs/pii/S0254058410008631#! https://www.sciencedirect.com/science/article/abs/pii/S0254058410008631#! https://www.sciencedirect.com/science/journal/02540584 http://doi.org/10.1016/j.matchemphys.2010.10.032 https://ur.booksc.eu/g/Petrunin,%20M.%20A. https://ur.booksc.eu/g/Maksaeva,%20L.%20B. https://ur.booksc.eu/g/Yurasova,%20T.%20A. https://ur.booksc.eu/g/Terekhova,%20E.%20V. https://ur.booksc.eu/g/Maleeva,%20M.%20A. https://ur.booksc.eu/g/Kotenev,%20V.%20A. https://ur.booksc.eu/g/Kotenev,%20V.%20A. https://ur.booksc.eu/g/Kablov,%20E.%20N. https://ur.booksc.eu/g/Tsivadze,%20A.%20Yu. https://ur.booksc.eu/journal/20303 http://doi.org/10.1134/S2070205115060179 W. Boukhedena and S. Deghboudj J. Electrochem. Sci. Eng. 11(4) (2021) 227-239 http://dx.doi.org/10.5599/jese.1050 239 [18] N. A. Negm, F. M. Ghuiba, S. M. Tawfik, Corrosion Science 53(11) (2011) 3566-3575. http://doi.org/10.1016/j.corsci.2011.06.029 [19] A. A. Khadom, A. N. Abd, N. Arif Ahmed, South African Journal of Chemical Engineering 25 (2018) 13-21. http://doi.org/10.1016/j.sajce.2017.11.002 [20] F. Bentiss, M. Lebrini, M. Lagrenée, Corrosion Science 47(12) (2005) 2915-2931. https://doi. org/10.1016/j.corsci.2005.05.034 [21] G. Avci, Materials Chemistry and Physics 112(1) (2008) 234-238. https://doi.org/10.1016/j. matchemphys.2008.05.036 [22] D. Özkır, K. Kayakırılmaz, E. Bayol, A. Ali Gürten, F. Kandemirli, Corrosion Science 56 (2012) 143- 152. http://doi.org/10.1016/j.corsci.2011.11.010 [23] M. A. Hegazy, M.F. Zaky, Corrosion Science 52(4) (2010) 1333-1341. https://doi.org/10.1016/ j.corsci.2009.11.043 [24] D.K. Yadav, D.S. Chauhan, I. Ahamad, M. A. Quraishi, RSC Advances 3(2) (2013) 632-646. http://doi.org/10.1039/C2RA21697C [25] S. K. Ahmed, W. B. Ali, A. A. Khadom, International Journal of Industrial Chemistry 10(2) (2019) 159-173. http://doi.org/10.1007/s40090-019-0181-8 [26] A. A. Al-Amiery, A. A. H. Kadhum, A. B. Mohamad, A. Y. Musa, C. J. Li, Materials 6(12) (2013) 5466-5477. http://doi.org/10.3390/ma6125466 [27] A. S. Fouda, M. A. Ismail, A. S. Abousalem, G. Y. Elewady, RSC Advances 7 (2017) 46414-46430. http://doi.org/10.1039/C7RA08092A [28] P. P. Kumari, P. Shetty, S. A. Rao, Arabian Journal of Chemistry 10(5) (2017) 653-663. http://doi.org/10.1016/j.arabjc.2014.09.005 [29] Y. Wang, J. Hu, L. Zhang, J. Cao, M. Lu, Royal Society Open Science 7(5) (2020) 191692. http://doi.org/10.1098/rsos.191692 [30] H. Z. Al-Sawaad, N. T. Faili, A. H. Essa, Portugaliae Electrochimica Acta 37(4) (2019) 205-216. http://doi.org/10.4152/pea.201904205 [31] M. N. El-Haddad,Carbohydrate Polymers 112 (2014) 595-602. https://doi.org/10.1016/j. carbpol.2014.06.032 [32] L. Li, Q. Qu, W. Bai, F. Yang, Y. Chen, S. Zhang, Z. Ding, Corrosion Science 59 (2012) 249-257. http://doi.org/10.1016/j.corsci.2012.03.008 [33] T. M. Lv, S. H. Zhu, L. Guo, S. T. Zhang, Research on Chemical Intermediates 41(10) (2015) 7073- 7093. http://doi.org/10.1007/s11164-014-1799-y [34] M. A. Hegazy, M. Abdallah, M. K. Awadd, M. Rezk, Corrosion Science 81 (2014) 54-64. http://doi.org/10.1016/j.corsci.2013.12.010 [35] R. Solmaz, Corrosion Science 81 (2014) 75-84. http://doi.org/10.1016/j.corsci.2013.12.006 [36] S. Umoren, I. Obot, Surface Review and Letters 15(03) (2008) 277-286. https://doi.org/10. 1142/S0218625X08011366 [37] E. Ebenso, Materials Chemistry and Physics 79(1) (2003) 58-70. http://doi.org/10.1016/S0254- 0584(02)00446-7 [38] C. Kumar, K. Mohana, Ionics 21(1) (2015) 263-281. http://doi.org/10.1007/s11581-014-1178-0 [39] A. Y. Musa, A. A. H. Kadhum, A. B. Mohamad, A. R. Daud, M. S. Takriff, S. K. Kamarudin, Corrosion Science 51(10) (2009) 2393-2399. http://doi.org/10.1016/j.corsci.2009.06.024 [40] A. Al-Fakih, M. Aziz, H. Sirat, Journal of Materials and Environmental Science 6(5) (2015) 1480- 1487. ©2021 by the authors; licensee IAPC, Zagreb, Croatia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (https://creativecommons.org/licenses/by/4.0/) http://dx.doi.org/10.5599/jese.1050 https://www.sciencedirect.com/science/article/abs/pii/S0010938X11003477#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11003477#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11003477#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2011.06.029 https://www.sciencedirect.com/science/article/pii/S1026918517300598#! https://www.sciencedirect.com/science/article/pii/S1026918517300598#! https://www.sciencedirect.com/science/journal/10269185 http://doi.org/10.1016/j.sajce.2017.11.002 https://www.sciencedirect.com/science/article/abs/pii/S0010938X05002106#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X05002106#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X05002106#! https://www.sciencedirect.com/science/journal/0010938X https://doi.org/10.1016/j.corsci.2005.05.034 https://doi.org/10.1016/j.corsci.2005.05.034 https://www.sciencedirect.com/science/journal/02540584 https://doi.org/10.1016/j.matchemphys.2008.05.036 https://doi.org/10.1016/j.matchemphys.2008.05.036 https://www.sciencedirect.com/science/article/abs/pii/S0010938X11006147#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11006147#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11006147#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11006147#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X11006147#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2011.11.010 https://www.sciencedirect.com/science/article/abs/pii/S0010938X0900599X#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X0900599X#! https://www.sciencedirect.com/science/journal/0010938X https://doi.org/10.1016/j.corsci.2009.11.043 https://doi.org/10.1016/j.corsci.2009.11.043 https://pubs.rsc.org/en/results?searchtext=Author%3ADileep%20Kumar%20Yadav https://pubs.rsc.org/en/results?searchtext=Author%3AD.S.%20Chauhan https://pubs.rsc.org/en/results?searchtext=Author%3AI.%20Ahamad https://pubs.rsc.org/en/results?searchtext=Author%3AM.A.%20Quraishi http://doi.org/10.1039/C2RA21697C https://link.springer.com/journal/40090 http://doi.org/10.1007/s40090-019-0181-8 http://doi.org/10.3390/ma6125466 http://doi.org/10.1039/C7RA08092A https://www.sciencedirect.com/science/article/pii/S1878535214002056#! http://doi.org/10.1016/j.arabjc.2014.09.005 https://royalsocietypublishing.org/doi/full/10.1098/rsos.191692 https://royalsocietypublishing.org/doi/full/10.1098/rsos.191692 https://royalsocietypublishing.org/doi/full/10.1098/rsos.191692 https://royalsocietypublishing.org/doi/full/10.1098/rsos.191692 https://royalsocietypublishing.org/doi/full/10.1098/rsos.191692 http://doi.org/10.1098/rsos.191692 http://doi.org/10.4152/pea.201904205 https://www.sciencedirect.com/science/journal/01448617 https://doi.org/10.1016/j.carbpol.2014.06.032 https://doi.org/10.1016/j.carbpol.2014.06.032 https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X12001138#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2012.03.008 http://doi.org/10.1007/s11164-014-1799-y https://www.sciencedirect.com/author/23970515900/m-a-hegazy https://www.sciencedirect.com/science/article/abs/pii/S0010938X13005738#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X13005738#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X13005738#! https://www.sciencedirect.com/science/journal/0010938X http://doi.org/10.1016/j.corsci.2013.12.010 http://doi.org/10.1016/j.corsci.2013.12.006 https://www.worldscientific.com/worldscinet/srl https://doi.org/10.1142/S0218625X08011366 https://doi.org/10.1142/S0218625X08011366 http://doi.org/10.1016/S0254-0584(02)00446-7 http://doi.org/10.1016/S0254-0584(02)00446-7 http://doi.org/10.1007/s11581-014-1178-0 https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! https://www.sciencedirect.com/science/article/abs/pii/S0010938X09002856#! http://doi.org/10.1016/j.corsci.2009.06.024 https://creativecommons.org/licenses/by/4.0/)