Introduction

A common natural occurrence, corrosion is the slow breakdown of materials, mostly metals, as a result of chemical or electrochemical reactions with their surroundings [1,2]. Accordingly, this degrading process causes large financial losses and presents serious safety issues in a variety of businesses worldwide [3]. Effective corrosion control techniques are crucial because corrosion-related repairs, replacements, and preventive measures cost billions of dollars each year [4]. Corrosion can cause catastrophic infrastructure failures, such as pipeline ruptures and bridge collapses, which have serious repercussions on environmental integrity and human safety in addition to the financial costs [5-7].

Corrosion is ubiquitous in a variety of contexts, including as underground, industrial, marine, and atmospheric situations, each of which has its own accelerating variables and obstacles [8]. To develop and use effective prevention and mitigation strategies, engineers, scientists, and industry stakeholders must have a thorough understanding of the complex mechanics of corrosion [9]. A multidisciplinary approach to study the development of this sector is required since the factors like temperature, humidity, salinity, and the presence of certain chemical agents significantly affect the pace and type of corrosion [10]. In the past, attempts to prevent corrosion have resulted in the creation of a number of protective techniques, such as the use of corrosion-resistant alloys, improved coatings, and electrochemical protection techniques [11].

Despite these advancements, corrosion remains a persistent challenge, driving continuous research into novel and more sustainable solutions [12]. Recent innovations have focused on environmentally friendly approaches, such as the utilization of plant extracts as green corrosion inhibitors and the development of graphene-based coatings, which offer promising avenues for extending the lifespan of metallic structures and reducing maintenance costs [13].

Despite these developments, corrosion is still a problem that necessitates ongoing research into new and more sustainable solutions [14]. One of the most important methods for reducing corrosion is the application of corrosion inhibitors, which create a protective layer on the metal surface [15]. In industrial settings, organic compounds can be employed as inhibitors for identifying chemicals or plant extracts to inhibit the corrosive process of steel against metal surfaces [16,17].  Heterocyclic compounds having lone pairs of electrons in their heteroatoms, such as sulfur (S), nitrogen (N), and oxygen (O) delocalized electrons in their molecules or pi- bond conjugation are utilized to prevent iron from corroding in acidic solutions [18,19].

The use of imidazole derivatives as mild steel corrosion inhibitors was look into because of its capacity to adsorb onto metal surfaces and provide a barrier against corrosive chemicals, imidazole derivatives which are distinguished by their heterocyclic structure containing nitrogen atoms have demonstrated considerable promising as potent corrosion inhibitors [20-22].

Using modern computational techniques including Density Functional Theory (DFT) and Monte Carlo (MC) simulations, this theoretical work intends to clarify the basic principles underlying the molecular interaction between imidazole derivatives and mild steel surfaces, also to investigate the anti-corrosive potentials of some imidazole derivatives using calculated frontier molecular orbitals (FMOs), reactivity descriptors, and electrostatic potential (MESP) maps, Fukui indices, and adsorption energies of the inhibitors compounds.

Materials and Methods

Computational Methods

The theoretical study of imidazole derivatives as corrosion inhibitors was carried out using Density Functional Theory (DFT) with B3LYP functional 6-31G(d) bases sets in Spartan 10 [23]. DFT is widely applied because it provides accurate electronic properties at relatively low computational cost compared to ab initio methods [24]. All calculations were performed using the Spartan 10 software package. The data obtained were used to estimate the electronic property of chemical quantum property. The following calculation procedure were performed: Geometry definition, Conformational search, Geometric optimization and Calculation of electronic properties [25].

Geometry Definitions

The 2D chemical structure was drawn using Altra Bio Chem Draw. The definition was done by 3D chemical numbering structure in Spartan 10. The basic geometric parameters include: Bond length (distance between two atoms), Bond angle (three atoms A-B-C), Dihedral angle (torsion; four atoms A-B-C-D) [26].

 Conformational Search

Initial 3D geometries built and subjected to a conformational search using molecular mechanics force fields (e.g., MMF or UFF) to identify the most stable conformer [27]. This ensure that the optimized geometry represents the global minimum energy structure, which is crucial for reliable electronic property calculations [11]. The conformational search is to find the low-energy conformers and choose the lowest (or a representative set) for DFT optimization.

Geometry Optimization

Geometry optimization is the process of adjusting the atomic positions in a molecule to find the most stable (lowest energy) structure. It ensures that the molecule’s geometry (bond lengths, bond angle, and dihedral angles) corresponding to a local minimum on the potential energy surface. The purpose of Geometry Optimization is to provide accurate input geometries for further property calculations (e.g., HOMO- LUMO, dipole moment, adsorption studies).

 Calculation of Electronic Properties

The Spartan ‘10, program was utilized to thoroughly optimize and calculate the molecular properties of the investigated inhibitor molecules. Based on density functional theory (DFT), quantum chemical parameters were conducted utilizing the B3LYP/6-31G(d) basic set in the aqueous phase (DFT) [28]. Following optimization, the quantum chemical descriptors of each compound were evaluated to provide insights into their electronic characteristics and inhibition efficiency. The following parameters were calculated: Band Gap ΔΕ (ev-1), Dipole moment (debye), Molecular weight (amu), Ionization potential (I) (ev-1), Electronic affinity (A) (ev), Global hardness  Global softness, Absolute electronegativity Global electrophilicity index (ω), Nucleophilicit
, Energy of back donation electron (ΔΕb-d), Fraction of election transfer (ΔΝ). Band Gap is between HOMO and LUMO

qk (N + 1) is defined as the charges on the atoms on the molecules in its anionic (N + 1) state

qk (N) is the charges on the atoms on the molecules in neutral (N) state

qk (N-1), the charges on the atoms on the molecules in its cationic (N-1) state

Fk + and fk − are the nucleophilic and electrophilic funki function respectively.

Monte Carlo Simulations

Monte Carlo (MC) 2017 simulation, Using the COMPASS force field, NVT Ensemble at 298K, 200,000 Cycle, 100,000 equilibration and 100,000 production step and Fe (110) surface were employed to investigate the adsorption behavior of some imidazole derivatives on the Fe (110) surface. This method provides insights into the most stable adsorption configurations and the interaction energies between the metal surface and inhibitor molecules under simulated environmental conditions.

The simulations were performed using Material studio 8.0 Package. The Fe (110) surface was cleaved from a body-centered cubic (BCC) iron crystal and optimized prior to use. Each inhibitor molecule was placed near the metal surface, and the adsorption process was simulated by random molecular movements, including translation, rotation, and torsional changes.

The adsorption energy (Eads) was calculated according to the following expression: 

E_ads=E_total-(E_Surface+E_inhibitor)                                    (13)

where:

• E_total is the total energy of the metal inhibitor system,

• E_surface is the energy of the clean metal surface

• E_inhibitor is the energy of the isolated inhibitor molecule.

Result and Discussion

Geometry Definition

The 2D chemical structure was drawn using Altra Bio Chem Draw [7]. The definition was done by 3D chemical numbering structure in Spartan 10, as shown in figure 1 and the names of the compounds were given in Table 1.

Figure 1: Structures of Some Derivatives of Some Imidazole, (A) (B) and (C) 

Table 1: Names of the Molecules Where A, B and C Represent the Following Compounds Respectively

 Conformational Search

Initial 3D geometries built and subjected to a conformational search using molecular mechanics force fields (e.g., MMF or UFF) to identify the most stable conformer [28]. This ensure that the optimized geometry represents the global minimum energy structure, which is crucial for reliable electronic property calculations [11]. The conformational search was set to find the lowest energy conformers and choose the lowest (or a representative set) for DFT optimization as shown in Table 2.

Table 2: Numbers of Conformer and the Most Stable Conformer Where A, B and C Represent the Foll

Figure 2: Conformers with the Lowest Energy

Molecular Geometry

The bond length and bond angle parameters provide important insights into the molecular geometry and electronic characteristics of the studied compounds. The computational method used for the molecular geometry was the DFT (Density functional theory) as shown in figure 2. The tables (3-5) below represent selected bond lengths and bond angles for four closely related compounds or molecular configurations (possibly structural isomers or substituted derivatives). Examining these values provides insights into how substitution, resonance, and hybridization influence the molecular geometry and electronic distribution of each system. The electro potential maps, HOMO and LUMO map in the geometry help to understand the electron-rich and electron-deficient parts of molecules [29]. From the legend in figure 3, the blue and green regions (positive parts) of the map indicate the sites that favor nucleophilic attack, while the red and orange regions (negative parts) are sites that favor an electrophilic attack,

All the three tables (3-5) show C1-N2 (1.305Å) (A), C1-N2 (1.302Å) (B) C1-N5 (1.305Å) (C) bond lengths of C3-Cl6(1.733Å) (A), C1-Cl8 (1.873Å) (B), C5-Cl13 (1.43Å) (B) are in this range. This indicates that partial double-bond character exists in all compounds, which gave a proof that there is a presence of conjugation or delocalized π-electrons including nitrogen atoms. The Carbon–Carbon double bond lengths through all the tables range approximately between 1.37–1.47 Å, which is an indication that confirmed conjugated carbon systems discovered in a structure of semi-aromatic or aromatics structures. The results from these tables of the bond length revealed that all the compounds considered possess sp² hybridized carbon atoms, leading to partial planarity. In each case, there is an alternation between shorter and longer C–C or C–N bonds, which confirms resonance and π-delocalization in the molecular framework. The Tables of the bond angles (3-5) shows that it fall between 120° and 132°, which also indicate that the compound has sp² hybridization with some angular distortions. Angles that are nearer to 120° leads in all compounds, reflecting a planar trigonal bipyramidal geometry a pointproperties of aromatic and conjugated molecules. There are some angle that deviates from the normal given above (angles below 110° or above 125°) appear in all three, showing local strain or substituent effects around nitrogen and carbon centers.

Figure 3: Structure of Imidazole and its Derivatives (a) Optimized Geometry (b) Highest Occupied Molecular Orbital (c) Lowest Unoccupied Molecular Orbital (d) Electro Potential MAP

                                                   Table 3: Selected Bond Length and Bond Angle for (A)

                                                     Table 4: Selected Bond Length and Bond Angle for b

                                                            Table 5: Selected Bond Length and Bond Angle for c                   

Chemical Quantum Parameter or Calculation

Band Gap or Energy Gap

The Eg or Band gap value are in Table 6, 4.50ev (A), 4.28ev (B), and 4.23ev (C) respectively, indicating that there is electron transport from the inhibitor’s molecule to the metal vacant d-orbitals [29]. This is indicated by the increase in value of EHOMO -6.22ev, -6.01ev, and -5.45ev indicating that the inhibitors molecules has perfect or excellent donating ability. The value of the LUMO orbital is low -1.72ev, -1.73ev, -1.22ev and -1.70ev, showing that the inhibitor molecules have the capacity that can readily receive or accept electron from metals orbitals.

Global Hardness and Global Softness

The results of the Band gap (Eg) that are directly related to the molecule are represented by the global hardness. To enhance desorption of the inhibitor molecule and the metals using Density Functional Theory (DFT), hardness must be sufficiently low. Table 6 shows that when the metal hardness tends toward zero (extremely soft), the global softness is 0.44ev, 0.46ev and 0.47ev while the global hardness is 2.25ev, 2.14ev and 2.11ev It implies that it will react with a soft basis more easily. As a result, the inhibitor molecules show that the inhibitor's effectiveness and adsorption have improved.

Electronegativity

A basic quantum chemical characteristic called electronegativity quantifies an atom or molecule's propensity to draw electrons to itself. The molecules' electronegativity values are as follows: A (-3.97), B (-3.87), and C (-3.33). The greatest negative electronegativity value is found in molecule A (-3.97). As a result, it has the strongest ability to attract electrons and has a propensity to take electrons from the iron surface, creating a stable coordination complex through electron back-donation that encourages the building of a protective film on the mild steel surface and increases the effectiveness of corrosion inhibition. The weakest molecules are molecules B (-3.87) and C (-3.33), which may function as a better electron giver but a poorer electron receiver.

Table 6: Chemical Quantum Calculation

Electronic Parameters of the Molecules

 Molecular Weight of the Molecule

All of the investigated molecules are comparatively large organic compounds appropriate for surface adsorption on mild steel, as indicated by their molecular weight values of 431.727 (amu), 397.282 (amu), and 411.522 (amu), respectively. According to the molecular weight, the initial molecules (431.727 amu) have the best chance of forming a protective and dense adsorption layer, which would increase the efficiency of corrosion inhibition. Less surface area is occupied by the molecule with the lowest molecular weight (397.282 amu). However, in addition to molecular size, effective corrosion resistance also depends on the electrical characteristics and adsorption energy of the molecule.

Dipole Moment of the Molecule

The dipole moment of the molecules under consideration was found to be 0.99 debye, 4.86 debye, and 6.02 debye, respectively. This indicates that there is a strong dipole-dipole interaction between the molecule and the surface of metal, indicating that adsorption is enhanced by electronic force. The electron-rich molecular orbital is primarily localized on the imidazole moiety, according to the HOMO plot, whereas the LUMO is nearly completely delocalized across the molecular system and primarily distributed over the phenol and azomethine moieties. This suggests that intra molecular charge transfer may occur in the system in the future, according to preliminary research.

Ionization Potential of the Molecule

The ionization potential (I) sometimes called ionization energy which represent the energy needed to remove an electron from the highest occupied molecular orbital (HOMO). Lower (I) values indicate higher HOMO energies and increased ability to donate electrons. The ionization potential of the molecules are 6.22ev, 6.01ev, and 5.45ev respectively. This shows that molecule C (5.45ev) is the softest and best electron donor, while molecule B (6.01ev) has an intermediate donor ability and molecule A (6.22ev) has the least donating ability, so it is the weakest donor among the three molecules.

Electronic Affinity Analysis

A molecule's capacity to take in electrons from its surroundings, such a metal surface, is known as its electrical affinity (A). A molecule with a higher electron affinity is more likely to take electrons, whereas a molecule with a lower electron affinity is less likely to gain electrons. The compounds' electron affinity values are as follows: Molecules A (1.72 eV), B (1.73 eV), and C (1.22 eV). It reveals that molecule 3 has the lowest electron affinity value (1.22ev), while molecule B has the greatest value (1.73ev), followed by molecule A (1.72). This demonstrates that, in comparison to molecule C, molecules A and B are better able to take electrons from the mild steel surface. Molecules A and B may be able to effectively interact with the d-orbitals of iron atoms due to their somewhat larger electron affinities, which could result in the creation of coordinate bonds and more robust adsorption on the metal surface. Molecule 3's comparatively lower electron affinity suggests that it has a lesser ability to take electrons, which leads to a less successful back-donation contact between the inhibitor and the metal surface. Because of their greater surface contact potential and stronger electron-accepting tendencies, Molecules A and B are therefore anticipated to perform greater in corrosion inhibition than Molecule C.

 Global Electrophilicity Index

A quantum chemical descriptor called the global electrophilicity index evaluates a molecule's stabilization energy when it absorbs more electron density from its environment. A molecule's strong propensity to attract electrons is indicated by a greater electrophilicity score, whereas a lower value indicates less electron-accepting activity. Each molecule's electrophilicity index values are as follows: 3.52, 3.61, and 3.15 eV. In table 7, Molecule B exhibits the greatest electrophilicity value (3.61ev) among the molecules, indicating that it is the most efficient electron acceptor, making it more reactive toward the mild steel surface during adsorption. Molecules A (3.52ev) and C (3.15ev), which have the lowest values, followed one another.

Nucleophilicity Index

The nucleophilicity index measures a molecule's capacity to donate electrons. A chemical that is having high nucleophilic value will have low electrophilicity, and vice versa. The compounds' corresponding nucleophilicity values are A (0.28), B (0.28), and C (0.32) in table 7. It reveals that molecule C has the highest nucleophilicity (0.32), indicating that it is the most potent electron donor among the molecules and has a higher capacity to transfer electrons to the mild steel surface, resulting in it good adsorption.

Fraction of Electron Transfer

The degree to which the molecule bonds to the metal surface is explained by ΔN, which indicates that the molecule stopped metal corrosion and displays the proportion of the electron transferred. For an inhibitor to be deemed sufficient, its fraction of electron transfer must be less than 3.6. For the molecules under consideration, Table 7, ΔN is 0.673 (6-31 G*), 0.732 (6-31 G*), and 0.887 (6-31 G*). All of these positive values show electron transport from inhibitor to metal, which is advantageous. Declare that they are all capable of contributing and that they work well as corrosion inhibitors.

Energy of Back Donation

One crucial quantum chemical parameter used to characterize the durability of the connection between an inhibitor molecule and a metal surface is the energy of back donation. The corresponding numbers are (0.56ev), (0.53ev), and (0.52ev) in Table 7. In comparison to the other two molecules, molecule A exhibits the largest back-donation energy, indicating a greater back-donation interaction with the mild steel surface. This indicates that molecules A (0.56 eV), B (0.53 eV), and C (0.52 eV) have less extensive back donation but still enough to stabilize the adsorption layer because electrons from the filled d-orbitals of iron can readily flow into their empty molecular orbitals (LUMO).

                                                                          Table 7: Electronic Properties

Fukui Indices and Mulliken Charge Distribution

The partition of charges on the corrosion inhibitors gives a good idea of the possible reactive sites on the Imidazole. Mulliken charge analysis performed on the neutral, anionic, and cationic species of the imidazole are reported in Tables 8 – 10 (excluding the hydrogen atoms) from which the Fukui indices were determined. The heteroatoms, especially Chlorine, Sulphur, amine nitrogen and coumarin oxygen atoms, display negative Mulliken charges, suggesting that they are probable sites for nucleophilic attack and adsorption on a metal surface. These further support the good inhibitory property of the molecules under consideration [30,31]. The HOMO plot of inhibitors A, B, & C. (Fig. 2) shows that the orbitals with abundant electron spread across the molecule is the carbonyl oxygen (O24) from compound of A. From Table 7-9, atoms having fk > 0 are electrophilic while those with fk < 0 are the nucleophiles. The reactive site were found to be on the following atoms, N (2), -0.076, Cl (6), -0.218, Cl (22) -0.069, S (23), -0.069, O (24), 0.066 by using fukil function.

                                   Table 10: Selected Calculated Fukui Functions and Mulliken Atomic Charges of C
Monte Carlo Simulation

The Adsorption energy (EAds), total energy (ETot), adsorption rigid energy (ERigid), and deformation energy (EDef) of the compounds under consideration of Fe (110) surface are available in (Table 11). EAds is the sum of the ERigid and EDef. The ERigid is the energy released when the unrelaxed adsorbate components are adsorbed on Fe (110), while the EDef is the energy released when the adsorbate components have relaxed on Fe (110) [25]. The ETot of the compounds from Monte Carlo follows: C > B > A. The best adsorption configuration is the one with the lowest (most negative) adsorption energy. The more negative the value, the stronger interaction between the adsorbate and surface. The adsorption mechanism of the compound under consideration is Chemisorption because of the extremely high adsorption energy. The compound's adsorption on the metal surface is thermodynamically stabilized, as demonstrated by all values, with C being the best. Additionally, the compounds exhibit minimal or high extremely adsorption energy, indicating their ability to halt or progressively reduce the corrosion process [32]. Adsorption energy is the sum of the energy of stiffness and energy of deformation. The EAds are as follows: C > B > A.

Adsorbed compound A, B, C on the metal surface in acidic medium are depicted in Figures 4a and b, On Fe (110), every molecule showed good surface coverage. The parallel organization of the compounds on Fe (110) via covalent and non-bonded interactions which supports the negative ΔEback-donation values from DFT calculation, which suggest that the inhibitors accept electrons via back donation [33].

                          Table 11: Monte Carlo Simulation Parameters Computed for the Imidazole Derivatives

Figure 4: Simulation on Adsorption of Compound A, B & C Using Monte Carlo on Iron Fe (110) Surface (a) Side View (b) Top View

Conclusion

The corrosion inhibition potentials of three derivatives of imidazole were investigated using DFT and MC simulations. The compounds displayed a low energy gap, global hardness, high electron affinity, and global softness. Sites that (i) donate electrons to vacant d-orbitals of the metal and (ii) accept electrons from the metals via back donation. were revealed by the Fukui local reactivity indices and the molecular electrostatic potential (MESP) map. The HOMO and LUMO maps revealed asymmetric distribution of charge. The quantum chemical investigations of the compounds showed that Molecule (C) is more nucleophilic and better adsorbed than A and B. Also, Strong adsorption of the compounds and effective surface coverage of the molecules on Fe (110) surface were revealed by the Monte Carlo (MC) simulation in the range C > B > A. The negative ΔE back-donation values from DFT calculation imply that the inhibitors accept electrons via back donation, buttressed by the parallel and proximal arrangement of the compounds on Fe (110) via covalent interactions [34-36].

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