Mathematical models predict corrosion inhibition

Mathematical models predict corrosion inhibition: History

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The use of corrosion inhibitors is an important method to retard the process of metallic attack by corrosion. The construction of mathematical models from theoretical-computational and experimental data obtained for different molecules is one of the most attractive alternatives in the analysis of corrosion prevention, whose objective is to define those molecular characteristics that are common in high-performance corrosion inhibitors.

corrosion inhibition
organic inhibitors
theoretical studies
molecular descriptors
mathematical models

1. Introduction

Corrosion is defined as the destructive attack of a material by reaction with its environment. This process is a worldwide, significant problem because of the economic damage and safety loss it may lead to. The losses originating from corrosion can be categorized as direct and indirect, and the latter includes economic losses caused by plant shutdowns, efficiency reduction, costly maintenance, and contamination of products whose overall impact will ultimately require over-design. The direct losses include the cost of replacing corroded structures, fixing damaged machinery, and substituting some of its components.

Corrosion control can be achieved by recognizing and understanding the mechanisms associated with this process. It is possible to prevent corrosion by using corrosion-resistant materials and protective systems.

The environments to which metals are exposed make them prone to corrosion. A less aggressive environment can be obtained by removing constituents that facilitate corrosion, modifying the temperature, dehumidifying the air, removing dissolved O₂ or solid particles, controlling the pH or adding corrosion inhibitors. The usage of inhibitors is one of the most practical methods for protection against corrosion in acidic, alkaline, saline, and other aggressive environments.

2. Corrosion Inhibitors

Corrosion inhibitors are chemical substances that are added to aggressive environments in small concentrations to decrease the corrosion rate. These substances can react with either a metal surface or its surroundings providing protection to the surface, although it has been observed that corrosion inhibitors generally work by forming an adsorbed film. Corrosion inhibitors are an appropriate option for metal protection when these are exposed to aggressive media, such as, acidic solutions (widely employed for industrial cleaning), oil well acidification, and petrochemical processes ^[1]. Corrosion inhibitors can be introduced into the aggressive environment in a single application or continuously by gradual and controlled additions. The single addition is possible in static systems at low temperature and where friction is negligible. Gradual addition is necessary for systems where variation in flow and temperature degrade the integrity of the protective film formed on the metal surface, either by physisorption or chemisorption. The stability of the corrosion inhibitor films can be compromised by the concentration levels of the inhibitor in the medium. Comparative studies performed to determine inhibition effects are generally performed without shaking. Therefore, the existence of a significant effect of immersion time on the quality of the formed film of an inhibitor is reasonable.

An important number of compounds have been used for the corrosion protection of metals exposed to aggressive media. Their effectiveness is associated to their chemical composition, molecular structure, alkyl chain length (or molecular volume), planarity, presence of lone pairs of electrons in heteroatoms (e.g., S, N, O, P), affinity for the metallic surface, dipole moment, presence of π-electrons (unsaturation or aromatic ring) and energy of frontier molecular orbitals. The efficiency (E%) of corrosion inhibitors is measured by applying the Equation (1) to experimental data obtained by techniques such as weight loss (WL), potentiodynamic polarization (PP), and electrochemical impedance spectroscopy (EIS):

\( E\% = \frac{{\left( {C{R_u} - C{R_i}} \right)}}{{C{R_u}}} \times 100 \). Equation (1)

where CR_u is the corrosion rate of the uninhibited system and CR_i is the rate of the inhibited system.

The mechanisms through which corrosion inhibitors can be bounded to the metal or the metal oxide surface may be physisorption, chemisorption, complexation or precipitation. The film formed by corrosion inhibitors prevents the access of oxygen to the cathode and the diffusion of hydrogen away from it, or simply inhibits metal dissolution (anodic inhibitors). Inhibition efficiency can be altered by modifying the system parameters such as pH, temperature, metal composition, type of inhibitor and molecular structure of the corrosion inhibitor ^[2]. An important factor associated with the corrosion process is surface roughness. The increase in the surface roughness of alloys (magnesium, titanium-based), stainless steels, copper, and aluminum increases the pitting susceptibility and the corrosion rate ^[3].

3. N-Containing Corrosion Inhibitors

An important number of molecules evaluated as corrosion inhibitors in acidic media contain N-atoms and the structural properties that are characteristic of high-performance corrosion inhibitors. Table 1 shows some examples of the N-containing corrosion inhibitors evaluated by different researchers. The Table indicates which corrosion inhibitors have been theoretically analyzed.

Table 1. N-containing corrosion inhibitors

Example	Corrosion Inhibitor	Ef_WL/Ef_PP/Ef_EIS	Ref
	(4-Chloro-benzylidene-pyridine-2-yl-amine)	99.5/99.6/99.6	^[4]
Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 2×10⁻⁴–1×10⁻²M, t: 24 h (WL), T: 25 °C, TA: A
	2,5-bis(2-hydroxyphenyl)-1,3,4-oxadiazole	98.2/94.7/99.1	^[5]
Exp. C. N_CI: 2, Met: mild steel, Media: HCl 1.0 M, 0.5 M H₂SO₄, Co: 20–80 mg•L⁻¹, t: 24 h (WL), 20 h (EIS), T: 30 °C, TA: NA
	Tributylamine	97.8/-/-	^[6]
Exp. C. N_CI: 23, Met: 13% Cr steel, Media: HCl (15%w/v), Co: 2%w/v inhibitor and 0.6% w/v formaldehyde (used to minimize hydrogen penetration), t: 3 h, T.: 60 °C, TA: A
	1,2-bisbenzylbenzimidazole	96	^[7]
Exp. C. N_CI: 11, Met: iron, Media: HCl 1.0 M. The analysis technique and the rest of the experimental conditions are not specified. TA: A
	3,4-Dichloro-acetophenone-O-1′-(1′.3′.4′-triazolyl)-metheneoxime (DATM)	92.8/98.7/98.7	^[8]
Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co:1×10⁻⁵–1×10⁻³M, t: 3 h (WL), 30 min (PP), T: 25 °C, TA: A
	* 2,6-bis-(2-benzimidazolyl) pyridine	-/94–96/97.0	^[9]
Exp. C. N_CI: 4, Met: mild steel, Media: HCl 1.0 M, Co: 0.1–1.0 mM, t: 1–96 h (WL), 45 min (PP), T.: 30 °C, TA: NA
	1-(4-Methyloxyphenylimino)-1-(phenylhydrazono)-propan-2-one	-/95.9/96.2	^[10]
Exp. C. N_CI: 5, Met: mild steel, Media: HCl 1.0 M, Co: 5×10⁻⁶–7.5×10⁻⁵M, t: 30 min, T: 25 °C, TA: NA
	1-Butyl-2-propylene-2-imidazoline	95.1/-/-	^[11]
Exp. C. N_CI: 34, Met: stainless steel, Media: HCl 5%, Co: ≈0.1 M, t: 48 h, T: T_R(room temperature), TA: A
	Aniline, N-(p-methoxybenzylidene)	-/94.0/-	^[12]
Exp. C. N_CI: 4, Met: aluminium, Media: HCl 2.0 N, Co: 10⁻⁵–10⁻²M, t:-, T: 30 °C, TA: NA
	Cyclohexanone oxime	93.9/98.4/95.2	^[13]
Exp. C. N_CI: 3, Met: aluminium, Media: HCl 1.0 M, Co: 0.2–2.0 mM, t: 2 h, T: 20–50 °C, TA: NA
	6-Bromo-1H-benzimidazole	-/-/89.4	^[14]
Exp. C. N_CI: 15, Met: carbon steel, Media: HCl 1.0 M, Co:0.1–10 mM, t: 30 min, T: T_R, TA: A
	2-Mercaptobenzimidazole	-/88.7/90.4	^[1]
Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 50–250 ppm, t: 30 min, T: 25 °C, TA: NA
	2-Chloromethylbenzimidazole	-/-/69.7	^[15]
Exp. C. N_CI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 1×10⁻³ M, t: -, T: T_R, TA: A
	2,4-Dimercaptopyrimidine	43.0/-/-	^[2] ^[16]
Exp. C. N_CI: 14, Met: Aramco iron, Media: HCl 2.0 M, Co: 1×10⁻³ M, t: 1 h, T: 40 °C, TA: A
Simbology: TA: theoretical analysis, A: applied, NA: not applied, Exp. C.: experimental conditions; N_CI: number of corrosion inhibitors, Co: inhibitor concentration, t: exposure time, T: temperature, Ef_WL/Ef_PP/Ef_EIS: corrosion inhibition efficiencies obtained by weight loss (WL), potentiodynamic polarization (PP), and electrochemical impedance spectroscopy (EIS), respectively.

Table 2. S-containing corrosion inhibitors

Example	Corrosion Inhibitor	Ef_WL/Ef_PP/Ef_EIS	Ref
	1-Phenyl thiosemicarbazide	-/100.0/-	^[17]
Exp. C. N_CI: 10, Met: mild steel, Media: H₂SO₄, Co: 1 mMdm⁻³, t: -, T: 30 °C. TA: A
	Phenylthiourea (PTU)	-/-/97.2	^[18]
Exp. C. N_CI: 3, Met: mild steel, Media: H₂SO₄ 0.1 M, Co: 1–10 mM, t:1 h, T: T_R, TA: NA
	2-Mercaptobenzothiazole	-/97/-	^[19]
Exp. C. N_CI: 1, Met: Steel (API 5L X52), Media: H₂SO₄ 1.0 M, Co: 10^−4-–10⁻³ M, t: 30 min, T: 25 °C, TA: A
	4-Hydroxybenzaldehyde thiosemicarbazone	-/97.0/90.0	^[20]
Exp. C. N_CI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 10⁻⁴–10⁻²M, t: 30 min, T: 25 °C, TA: NA
	2-Mercapto-1-methylimidazole	90.4 /95.5/-	^[21]
Exp. C. N_CI: 1, Met: carbon steel, Media: HClO₄ 1.0 M, Co: 7.510⁻⁵–2.5 10⁻³ M, t:1 h, T: 30 °C, TA: NA
Simbology: TA: theoretical analysis, A: applied, NA: not applied, Exp. C.: experimental conditions; N_CI: number of corrosion inhibitors, Co: inhibitor concentration, t: exposure time, T: temperature, Ef_WL/Ef_PP/Ef_EIS: corrosion inhibition efficiencies obtained by weight loss (WL), potentiodynamic polarization (PP), and electrochemical impedance spectroscopy (EIS), respectively.

Table 3. O-containing corrosion inhibitors

Example	Corrosion Inhibitor	Ef_WL/Ef_PP/Ef_EIS	Ref
	2-(4-Methylbenzylidene)-3-oxo-2,3-dihydro-1H-indene-1-carboxylic acid	92.0/93.9/92.6	^[22]
Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 10⁻⁶–10⁻³ M, t: 6 h, T: 30 °C, TA: NA
	1-((4-Chlorophenyl)(2-hydroxynaphtalen-1-yl)(phenyl) methyl)urea (CPHU)	90/92.0/-	^[23]
Exp. C. NCI: 3, Met: mild steel, Media: H₂SO₄ 0.5 M, Co: 2–10 ppm, t: 24 h (WL) and 30 min (PP), T: 20 °C, TA:A
	1,3-Dibromo-5,5-dimethylhydantoin	-/89.8/91.4	^[24]
Exp. C. NCI: 6, Met: mild steel, Media: HCl 0.5 M, Co: 10–50 ppm, t: 30 min, T: 30 °C, TA:A
	L-ascorbic acid	69.0/-/-	^[25]
Exp. C. NCI: 1, Met: mild steel, Media: H₂SO₄ 0.01 M, Co: 10⁻⁷–10⁻³ M, t: -, T: T_R, TA: NA
Simbology: TA: theoretical analysis, A: applied, NA: not applied, Exp. C.: experimental conditions; N_CI: number of corrosion inhibitors, Co: inhibitor concentration, t: exposure time, T: temperature, Ef_WL/Ef_PP/Ef_EIS: corrosion inhibition efficiencies obtained by weight loss (WL), potentiodynamic polarization (PP), and electrochemical impedance spectroscopy (EIS), respectively.

Table 4. Multi-heteroatom containing corrosion inhibitors

Example	Corrosion Inhibitor	Ef_WL/Ef_PP/Ef_EIS	Ref
	(2,3,8,9-Dibenzo-4,7-dioxa-13-thia-11,12-diazabicyclo[8.2.1]trideca-10,12-diene (1-MCTH)	97.7/-/-	^[26]
Exp. C. NCI: 5, Met: mild steel, Media: 1.0 M HCl, Co: 110⁻⁶–110⁻⁴ M, t: 24 h (WL), T: 30 °C, TA: A
	Sulphadimethoxine	93.8/84.0/92.1	^[27]
Exp. C. NCI: 5, Met: mild steel, Media:1.0 M HCl, Co: 1–510⁻⁵ M, t: 30 min, T: 30–50 °C, TA: A
	5-Benzoyl-4-tolyl-6-phenyl-1,2,3,4-tetrahydro-2-thioxopyrimidine	-/93.0/93.0	^[28] ^[29]
Exp. C. NCI: 2, Met: stainless steel, Media:1.0 M HCl, Co:110⁻⁴–5 10⁻³ moldm⁻³, t: 30 min, T: 25 °C. The experimental data were obtained by Caliskan, et al. ^[28]. TA: A
	L-Cysteine	82.2/-/-
Exp. C. NCI: 4, Met: mild steel, Media:0.1 M HCl, Co: 0.1–0.5 gL⁻¹, t: 24, 168 h, T: 30, 60 °C, TA: A
	(E)-N-methyl-2-(1-phenylethylidene)hydrazinecarbothioamide PHCARB	80.7/-/-	^[30]
Exp. C. NCI: 4, Met: mild steel, Media:0.1 M HCl, Co: 1–510⁻⁴ M, t: 168 h, T: 30 °C, TA: A
	Methionine	72.7/84.6/85.8
Exp. C. NCI: 16, Met: Armco iron, Media: HCl 1.0 M, Co: 10⁻³ M, et: 6 h (WL), 10 y 30 min (PP), T: 35 °C, TA: A
Simbology: TA: theoretical analysis, A: applied, NA: not applied, Exp. C.: experimental conditions; NCI: number of corrosion inhibitors. Co: inhibitor concentration, t: exposure time, T: temperature, Ef_WL/Ef_PP/Ef_EIS: corrosion inhibition efficiencies obtained by weight loss (WL), potentiodynamic polarization (PP), and electrochemical impedance spectroscopy (EIS), respectively.

Table 5. Examples of descriptors used in the theoretical analysis of corrosion inhibitors.

Quantum Descriptor	Symbol	Description	Reference
Energy of the highest occupied molecular orbital	E_HOMO	Associated with the electron-donating ability of a molecule.	^[4]
Energy of the lowest unoccupied molecular orbital	E_LUMO	Indicates the ability of the molecule to accept electrons.	^[21]
Ionization potential	IP	It is a descriptor of the chemical reactivity of atoms and molecules. IP is the minimum energy required to remove an electron from an atom.	^[29]
Electron affinity	A	It is a property that determines how susceptible a molecule is towards the attack of a nucleophile.	^[31]
Dipole moment	_μD	The dipole moment is considered a measure of the stability of the formed complex on a metal surface. It is an indicator of the asymmetry in the molecular charge distribution. It is related to the hydrophobic character of the molecules.	^[23] ^[32] ^[33]
Energy gap	ΔE	ΔE= E_{HOMO −}E_LUMO It has been mentioned that the successful adsorption and proper efficiency of an inhibitor have been characterized by a high E_HOMO, a low E_LUMO_,and a small energy gap. The lower the ΔE is, the higher the stability of the metal-inhibitor interaction.	^[9] ^[32] ^[34]
Fraction of electron transferred	ΔN	If ΔN > 3.6, the inhibition efficiency increases	^[9]
Global hardness	η	It is a parameter associated with the resistance of an atom to transfer its charge.	^[9]
Softness	σ	The σ shows the reactivity of the inhibitor molecules in terms of charge transfer.	^[9]
Electronegativity	χ	The electronegativity measures the power of a group of atoms to attract electrons towards itself.	^[9]
Electrophilicity index	ω	It is a reactivity descriptor that allows the quantitative classification of the global electrophilic nature of a molecule. This descriptor has been proposed as a measure of energy lowering due to maximal electron flow between donor and acceptor.	^[8]
Electrodonating power	ω^-	A descriptor associated with the ability of a species to donate electrons.	^[35]
Electroaccepting power	ω⁺	A descriptor associated with the ability of a species to accept electrons.	^[35]
Dipole polarizability	α	α is a measure of the mean polarizability. Higher values of α enable a strong adsorption process.	^[8]
Fukui functions	f_k⁺, f_k ^-	These functions indicate the part of the molecule where nucleophilic, electrophilic, and radical attack is most likely to occur.	^[31]
Partition coefficient	Log P	Log P is a hydrophobic parameter of a molecule.	^[36]

Table 6. Examples of mathematical models generated for corrosion analysis.

Molecules Evaluated	Theoretical Calculation	Mathematical Model	R²	Ref.
Schiff base	AM1 semi-empirical method	\( {\small I{E_{exp}}\left( \% \right) = 2.084{E_{HOMO}} - 3.041{E_{LUMO}} + 115.772 } \)	1.00	^[4]
Schiff base	AM1 semi-empirical method	\( {\small IE_{Theor}=\frac{\left(-604.90E_{HOMO}+5864.86E_{LUMO}+1190.06\mu_D-642.67\right)C}{\left[1+\left(-604.90E_{HOMO}+5864.86E_{LUMO}+1190.06\mu_D-642.67\right)C\right]} } \)	0.98	^[4]
Amines, thioureas, acetylenic alcohols	AM1 methodology was used for most descriptors, PC model provided the volume calculations	\( {\small \begin{matrix}lnK_{ads}=-0.93N-7.64P+6.74C-2.27C_{12}+0.94C_{13}-1.06C_1\\-7.22C_2-2.17E_{HOMO}-1.17DP+7.45V-1.15A_1\\-1.81A_2+7.12NCS-1.869NOH-1.085NCR\\\end{matrix} } \)	0.98	^[6]
		\( {\small lnK_{ads}=1.52M-0.79P+0.53C_{12}+0.80NT-0.66NOH } \)	0.86
		\( {\small \begin{array}{*{20}{c}} {{\rm{ ln}}{{\rm{K}}_{{\rm{ads}}}}{\rm{ = }}}&{ - 2.688 \times {{10}^{ - 2}}{A_1} + 0.115{A_2} + 4.530 \times {{10}^{ - 2}}{A_3} + 8.762 \times {{10}^{ - 2}}NB - 1.305 \times {{10}^{ - 2}}NC}\\ {}&{ + 0.102NCS - 4.518 \times {{10}^{ - 2}}NT - 5.172 \times {{10}^{ - 2}}NOH - 1.099 \times {{10}^{ - 3}}NCR + 4.996 \times {{10}^{ - 2}}NR}\\ {}&{ - 6.338 \times {{10}^{ - 2}}N - 0.114ED + 0.184M + 0.180P - 6.838 \times {{10}^{ - 2}}C + 8.092 \times {{10}^{ - 2}}{C_{12}}}\\ {}&{ - 4.783 \times {{10}^{ - 2}}{C_{13}} - 1.130 \times {{10}^{ - 2}}{C_{14}} - 0.167{C_1} - 0.152{C_2} - 0.122{E_{HOMO}}}\\ {}&{{\rm{ - 7.838}} \times {\rm{1}}{{\rm{0}}^{ - 2}}{{\rm{E}}_{{\rm{LUMO}}}}{\rm{ - 7.004}} \times {\rm{1}}{{\rm{0}}^{ - 3}}\Delta {\rm{E + 0.109}}\mu {\rm{ + 0.141V}}} \end{array} } \)	0.85
Triazole derivatives	Density Functional Theory (DFT), B3LYP/6-31G	\( {\small IE_{Theor}=\frac{\left(18.38E_{HOMO}-7.28E_{LUMO}-0.012V+123.36\right)C}{\left[1+\left(18.38E_{HOMO}-7.28E_{LUMO}-0.012V+123.36\right)C\right]} } \) Derived QSAR equation in gas phase	0.94	^[8]
Triazole derivatives	Density Functional Theory (DFT), B3LYP/6-31G	\( {\small IE_{Theor}=\frac{\left(-2.19E_{HOMO}-1.24E_{LUMO}-0.014V-10.94\right)C}{\left[1+\left(-2.19E_{HOMO}-1.24E_{LUMO}-0.014V-10.94\right)C\right]} } \) Derived QSAR equation in aqueous phase	0.95	^[8]
Imidazole derivatives	Restricted Hartree–Fock level (RHF) using MINDO/3, MNDO, PM3 and AM1 semi-empirical SCF-MO methods.	\( {\small I{E_{exp}}\left( \% \right) = 1174.95 + 214.612{E_{HOMO}} - 16.793{E_{LUMO}} } \) Gas phase (series 1)	0.90	^[7]
		\( {\small I{E_{exp}}\left( \% \right) = 517.7 + 53.8{E_{HOMO}} - 1.97{E_{LUMO}} } \) Aqueous phase (series 1)	0.82
		\( {\small I{E_{exp}}\left( \% \right) = 2420.86 + 295.67{E_{HOMO}} - 30.08{E_{LUMO}} } \) Gas phase (series 2)	0.97
		\( {\small I{E_{exp}}\left( \% \right) = 601.53 + 63.07{E_{HOMO}} - 1.405{E_{LUMO}} } \) Aqueous phase (series 2)	0.99
Imidazole and benzimidazole derivatives	---	\( {\small I{E_{exp}}\left( \% \right) = 38.47 + 20.21n\left( N \right) - 7.98N\left( {O + N{H_2}} \right) + 14.94{\eta ^ + } - 17.93{\eta ^ - } } \)	0.97	^[11]
Indole derivatives	DFT, B3LYP functional and 6-31G(2d,2p) basis	\( {\small R_{ct}=150+\left(-359539E_{HOMO}+1585825E_{LUMO}\right)C } \)	0.97	^[26]
Indole derivatives	DFT, B3LYP functional and 6-31G(2d,2p) basis	\( {\small R_{ct}=150+\left(-402535E_{HOMO}+960146E_{LUMO}\right)C } \)	0.97	^[26]
Imidazole and benzimidazole derivatives	DFT: PBE/6-311++G **	\( {\small {E_{exp}}\left( \% \right) = 5130.95 - 32.03\chi + 533.4b{q^{ISO}} + 0.37V + 1433.78{q_{N1}} } \)	0.92	^[37]
Imidazole, benzimidazole and pyridine derivatives	DFT: PBE/B3LYP/M06, using the orbital basis 6-31G* and 6-311++G**	\( {\small I{E_{exp}}\left( \% \right) = 92.965 + 0.152V + 35.337{\omega ^ - } + 3.592b{q_{ANS}} } \)	0.75	^[14]
Pyrimidine derivatives	DFT BLYP/DNP	\( {\small I{E_{exp}}\left( \% \right) = - 4.324\mu - 46.527{\rm{\Delta }}E + 376.48{q_{N1}} } \)	0.98	^[16]
Thiosemicarbazides	PM3 and MNDO method.	\( {\small IE_{Theor}=\frac{\left(-6.7E_{HOMO}-5.9E_{LUMO}-3.5\mu-43.7\right)C}{\left[1+\left(-6.7E_{HOMO}-5.9E_{LUMO}-3.5\mu-43.7\right)C\right]}\times100 } \)	0.84
Thiosemicarbazides	PM3 and MNDO method.	\( {\small IE_{Theor}=\frac{\left(227.2+23.5E_{HOMO}-3.8\mu\right)C}{\left[1+\left(227.2+23.5E_{HOMO}-3.8\mu\right)C\right]}\times100 } \)	0.92
Thiosemicarbazides/ Thiosemicarbazones	Data obtained from literature	\( {\small IE_{exp}=\frac{e^{-2.5219E_{HOMO}-0.5119\mu_D-18.1761}C}{\left(1+e^{-2.5219E_{HOMO}-0.5119\mu_D-18.1761}C\right)} } \)	0.89	^[17]
Thiosemicarbazides/ Thiosemicarbazones	Data obtained from literature	\( {\small {\rm{I}}{{\rm{E}}_{{\rm{exp}}}}{\rm{ = }}\frac{{{{\rm{e}}^{{\rm{263.12}}\Delta {\rm{E - 17.26}}\Delta {\rm{E2 - 999.22}}}}{\rm{C}}}}{{{\rm{1 + }}{{\rm{e}}^{{\rm{263.12}}\Delta {\rm{E - 17.26}}\Delta {\rm{E2 - 999.22}}}}}} } \)	0.88	^[17]
2-Mercaptobenzothiazole	DFT: B3LYP/6-31+G*	\( {\small I{E_{Theor}} = \frac{{\left( {2.23{E_{HOMO}} - 8.37{E_{LUMO}} + 5.47\Delta E + 1.76\Delta N + 6.47\mu + 119.8V + 2.07} \right)C}}{{\left[ {1 + \left( {2.23{E_{HOMO}} - 8.37{E_{LUMO}} + 5.47\Delta E + 1.76\Delta N + 6.47\mu + 119.8V + 2.07} \right)C} \right]}} \times 100 } \)	0.96	^[19]
Urea derivatives	DFT: B3LYP/6-31G	\( {\small ∆E=-0.0024IEexp-4.3145} \)	0.95	^[23]
Urea derivatives	DFT: B3LYP/6-31G	\( {\small {E_{HOMO}} = - 0.0107I{E_{\exp }} - 4.88596 } \)	0.93	^[23]
Hydantoin derivatives	DFT: B3LYP/6-31G+(d,p)	\( {\small IE_{Theor}=129.054-0.181Mwt-6.550∆E-174.884∆N+0.484BE } \)	1.00	^[24]
Hydantoin derivatives	DFT: B3LYP/6-31G+(d,p)	\( {\small I{E_{Theor}} = \frac{{\left( {9.1 \times {{10}^{13}}Mwt + 4.9 \times {{10}^{14}}\Delta N + 9.7 \times {{10}^{13}}\Delta E + 2.4 \times {{10}^{13}}BE - 5.9 \times {{10}^{13}}} \right)}}{{\left( {1 + 1.8 \times {{10}^{13}}Mwt + 1.8 \times {{10}^{16}}\Delta N + 8.1 \times {{10}^{14}}\Delta E - 1.2 \times {{10}^{13}}BE - 7.2 \times {{10}^{15}}} \right)}} } \)	0.99	^[24]
Polyether compounds	DFT B3LYP/6-31G(d,p)	\( {\small R_t=5+\left(2.710^7E_{HOMO}-7.610^7E_{LUMO}+3.910^6\mu\right)C } \)	0.81	^[26]
Sulphonamide derivatives	DFT: B3LYP/6-311+G(d,p)	\( {\small I{E_{\exp }}\left( \% \right) = \frac{{\left( {2.47\omega + 8.56 \times {{10}^{ - 2}}{E_{LUMO}} - 5.27 \times {{10}^{ - 2}}\mu + 6.20\eta - 4.11 \times {{10}^{ - 2}}LogP - 21.81} \right) * 5000}}{{\left( {1 + \left( {2.47\omega + 8.56 \times {{10}^{ - 2}}{E_{LUMO}} - 5.27 \times {{10}^{ - 2}}\mu + 6.20\eta - 4.11 \times {{10}^{ - 2}}LogP - 21.81} \right) * 50} \right)}} } \)	1.00	^[27]
Sulphonamide derivatives	DFT: B3LYP/6-311+G(d,p)	\( {\small I{E_{\exp }}\left( \% \right) = \frac{{\left( { - 1.44\Delta E + 3.48{E_{LUMO}} - 3.50{E_{HOMO}} + 1.07\eta + 1.75{\omega ^ + } - 17.85} \right)5000}}{{\left( {1 + \left( { - 1.44\Delta E + 3.48{E_{LUMO}} - 3.50{E_{HOMO}} + 1.07\eta + 1.75{\omega ^ + } - 17.85} \right)50} \right)}} } \)	0.99	^[27]
Pyrimidine compounds	B3LYP/6-311++G(d,p)	\( {\small ∆E=-0.0024IEexp-4.3145} \)

This entry is adapted from the peer-reviewed paper 10.3390/ma13245656

References

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1. Introduction

2. Corrosion Inhibitors

3. N-Containing Corrosion Inhibitors

Table 1. N-containing corrosion inhibitors

Example

Corrosion Inhibitor

EfWL/EfPP/EfEIS

Ref

(4-Chloro-benzylidene-pyridine-2-yl-amine)

99.5/99.6/99.6

Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 2×10−4–1×10−2 M, t: 24 h (WL), T: 25 °C, TA: A

2,5-bis(2-hydroxyphenyl)-1,3,4-oxadiazole

98.2/94.7/99.1

Exp. C. NCI: 2, Met: mild steel, Media: HCl 1.0 M, 0.5 M H2SO4, Co: 20–80 mg•L−1, t: 24 h (WL), 20 h (EIS), T: 30 °C, TA: NA

Tributylamine

97.8/-/-

Exp. C. NCI: 23, Met: 13% Cr steel, Media: HCl (15%w/v), Co: 2%w/v inhibitor and 0.6% w/v formaldehyde (used to minimize hydrogen penetration), t: 3 h, T.: 60 °C, TA: A

1,2-bisbenzylbenzimidazole

96

Exp. C. NCI: 11, Met: iron, Media: HCl 1.0 M. The analysis technique and the rest of the experimental conditions are not specified. TA: A

3,4-Dichloro-acetophenone-O-1′-(1′.3′.4′-triazolyl)-metheneoxime (DATM)

92.8/98.7/98.7

Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co:1×10−5–1×10−3 M, t: 3 h (WL), 30 min (PP), T: 25 °C, TA: A

* 2,6-bis-(2-benzimidazolyl) pyridine

-/94–96/97.0

Exp. C. NCI: 4, Met: mild steel, Media: HCl 1.0 M, Co: 0.1–1.0 mM, t: 1–96 h (WL), 45 min (PP), T.: 30 °C, TA: NA

1-(4-Methyloxyphenylimino)-1-(phenylhydrazono)-propan-2-one

-/95.9/96.2

Exp. C. NCI: 5, Met: mild steel, Media: HCl 1.0 M, Co: 5×10−6–7.5×10−5 M, t: 30 min, T: 25 °C, TA: NA

1-Butyl-2-propylene-2-imidazoline

95.1/-/-

Exp. C. NCI: 34, Met: stainless steel, Media: HCl 5%, Co: ≈0.1 M, t: 48 h, T: TR (room temperature), TA: A

Aniline, N-(p-methoxybenzylidene)

-/94.0/-

Exp. C. NCI: 4, Met: aluminium, Media: HCl 2.0 N, Co: 10−5–10−2 M, t:-, T: 30 °C, TA: NA

Cyclohexanone oxime

93.9/98.4/95.2

Exp. C. NCI: 3, Met: aluminium, Media: HCl 1.0 M, Co: 0.2–2.0 mM, t: 2 h, T: 20–50 °C, TA: NA

6-Bromo-1H-benzimidazole

-/-/89.4

Exp. C. NCI: 15, Met: carbon steel, Media: HCl 1.0 M, Co:0.1–10 mM, t: 30 min, T: TR, TA: A

2-Mercaptobenzimidazole

-/88.7/90.4

Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 50–250 ppm, t: 30 min, T: 25 °C, TA: NA

2-Chloromethylbenzimidazole

-/-/69.7

Exp. C. NCI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 1×10−3 M, t: -, T: TR, TA: A

2,4-Dimercaptopyrimidine

43.0/-/-

Exp. C. NCI: 14, Met: Aramco iron, Media: HCl 2.0 M, Co: 1×10−3 M, t: 1 h, T: 40 °C, TA: A

Table 2. S-containing corrosion inhibitors

Example

Corrosion Inhibitor

EfWL/EfPP/EfEIS

Ref

1-Phenyl thiosemicarbazide

-/100.0/-

Exp. C. NCI: 10, Met: mild steel, Media: H2SO4, Co: 1 mMdm−3, t: -, T: 30 °C. TA: A

Phenylthiourea (PTU)

-/-/97.2

Exp. C. NCI: 3, Met: mild steel, Media: H2SO4 0.1 M, Co: 1–10 mM, t:1 h, T: TR, TA: NA

2-Mercaptobenzothiazole

-/97/-

Exp. C. NCI: 1, Met: Steel (API 5L X52), Media: H2SO4 1.0 M, Co: 10−4-–10−3 M, t: 30 min, T: 25 °C, TA: A

4-Hydroxybenzaldehyde thiosemicarbazone

-/97.0/90.0

Exp. C. NCI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 10−4–10−2 M, t: 30 min, T: 25 °C, TA: NA

2-Mercapto-1-methylimidazole

90.4 /95.5/-

Exp. C. NCI: 1, Met: carbon steel, Media: HClO4 1.0 M, Co: 7.510−5–2.5 10−3 M, t:1 h, T: 30 °C, TA: NA

Table 3. O-containing corrosion inhibitors

Example

Corrosion Inhibitor

EfWL/EfPP/EfEIS

Ref

2-(4-Methylbenzylidene)-3-oxo-2,3-dihydro-1H-indene-1-carboxylic acid

92.0/93.9/92.6

Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 10−6–10−3 M, t: 6 h, T: 30 °C, TA: NA

1-((4-Chlorophenyl)(2-hydroxynaphtalen-1-yl)(phenyl) methyl)urea (CPHU)

90/92.0/-

Ef_WL/Ef_PP/Ef_EIS

Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 2×10⁻⁴–1×10⁻²M, t: 24 h (WL), T: 25 °C, TA: A

Exp. C. N_CI: 2, Met: mild steel, Media: HCl 1.0 M, 0.5 M H₂SO₄, Co: 20–80 mg•L⁻¹, t: 24 h (WL), 20 h (EIS), T: 30 °C, TA: NA

Exp. C. N_CI: 23, Met: 13% Cr steel, Media: HCl (15%w/v), Co: 2%w/v inhibitor and 0.6% w/v formaldehyde (used to minimize hydrogen penetration), t: 3 h, T.: 60 °C, TA: A

Exp. C. N_CI: 11, Met: iron, Media: HCl 1.0 M. The analysis technique and the rest of the experimental conditions are not specified. TA: A

Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co:1×10⁻⁵–1×10⁻³M, t: 3 h (WL), 30 min (PP), T: 25 °C, TA: A

Exp. C. N_CI: 4, Met: mild steel, Media: HCl 1.0 M, Co: 0.1–1.0 mM, t: 1–96 h (WL), 45 min (PP), T.: 30 °C, TA: NA

Exp. C. N_CI: 5, Met: mild steel, Media: HCl 1.0 M, Co: 5×10⁻⁶–7.5×10⁻⁵M, t: 30 min, T: 25 °C, TA: NA

Exp. C. N_CI: 34, Met: stainless steel, Media: HCl 5%, Co: ≈0.1 M, t: 48 h, T: T_R(room temperature), TA: A

Exp. C. N_CI: 4, Met: aluminium, Media: HCl 2.0 N, Co: 10⁻⁵–10⁻²M, t:-, T: 30 °C, TA: NA

Exp. C. N_CI: 3, Met: aluminium, Media: HCl 1.0 M, Co: 0.2–2.0 mM, t: 2 h, T: 20–50 °C, TA: NA

Exp. C. N_CI: 15, Met: carbon steel, Media: HCl 1.0 M, Co:0.1–10 mM, t: 30 min, T: T_R, TA: A

Exp. C. N_CI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 50–250 ppm, t: 30 min, T: 25 °C, TA: NA

Exp. C. N_CI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 1×10⁻³ M, t: -, T: T_R, TA: A

Exp. C. N_CI: 14, Met: Aramco iron, Media: HCl 2.0 M, Co: 1×10⁻³ M, t: 1 h, T: 40 °C, TA: A

Ef_WL/Ef_PP/Ef_EIS

Exp. C. N_CI: 10, Met: mild steel, Media: H₂SO₄, Co: 1 mMdm⁻³, t: -, T: 30 °C. TA: A

Exp. C. N_CI: 3, Met: mild steel, Media: H₂SO₄ 0.1 M, Co: 1–10 mM, t:1 h, T: T_R, TA: NA

Exp. C. N_CI: 1, Met: Steel (API 5L X52), Media: H₂SO₄ 1.0 M, Co: 10^−4-–10⁻³ M, t: 30 min, T: 25 °C, TA: A

Exp. C. N_CI: 6, Met: carbon steel, Media: HCl 1.0 M, Co: 10⁻⁴–10⁻²M, t: 30 min, T: 25 °C, TA: NA

Exp. C. N_CI: 1, Met: carbon steel, Media: HClO₄ 1.0 M, Co: 7.510⁻⁵–2.5 10⁻³ M, t:1 h, T: 30 °C, TA: NA

Ef_WL/Ef_PP/Ef_EIS

Exp. C. NCI: 3, Met: mild steel, Media: HCl 1.0 M, Co: 10⁻⁶–10⁻³ M, t: 6 h, T: 30 °C, TA: NA

Exp. C. NCI: 3, Met: mild steel, Media: H₂SO₄ 0.5 M, Co: 2–10 ppm, t: 24 h (WL) and 30 min (PP), T: 20 °C, TA:A

Exp. C. NCI: 1, Met: mild steel, Media: H₂SO₄ 0.01 M, Co: 10⁻⁷–10⁻³ M, t: -, T: T_R, TA: NA

Ef_WL/Ef_PP/Ef_EIS

Exp. C. NCI: 5, Met: mild steel, Media: 1.0 M HCl, Co: 110⁻⁶–110⁻⁴ M, t: 24 h (WL), T: 30 °C, TA: A

Exp. C. NCI: 5, Met: mild steel, Media:1.0 M HCl, Co: 1–510⁻⁵ M, t: 30 min, T: 30–50 °C, TA: A

Exp. C. NCI: 2, Met: stainless steel, Media:1.0 M HCl, Co:110⁻⁴–5 10⁻³ moldm⁻³, t: 30 min, T: 25 °C. The experimental data were obtained by Caliskan, et al. ^[28]. TA: A

Exp. C. NCI: 4, Met: mild steel, Media:0.1 M HCl, Co: 0.1–0.5 gL⁻¹, t: 24, 168 h, T: 30, 60 °C, TA: A

Exp. C. NCI: 4, Met: mild steel, Media:0.1 M HCl, Co: 1–510⁻⁴ M, t: 168 h, T: 30 °C, TA: A

Exp. C. NCI: 16, Met: Armco iron, Media: HCl 1.0 M, Co: 10⁻³ M, et: 6 h (WL), 10 y 30 min (PP), T: 35 °C, TA: A

E_HOMO

E_LUMO

ΔE= E_{HOMO −}E_LUMO

It has been mentioned that the successful adsorption and proper efficiency of an inhibitor have been characterized by a high E_HOMO, a low E_LUMO_,and a small energy gap. The lower the ΔE is, the higher the stability of the metal-inhibitor interaction.