Experimental Investigation and Modeling of the Performance of Pure and Mixed Surfactant Inhibitors: Aggregation, Adsorption, and Corrosion Inhibition on Steel Pipe in Aqueous Phase

Two models for the prediction of corrosion inhibition efﬁciency of various pure surfactant and mixed surfactants in salt-containing solution are developed, compared, and validated. The two prediction models are based on a modiﬁed Langmuir adsorption (MLA) sub-model and a modiﬁed Quantitative Structure Activity Relation (MQSAR) sub-model. Each of the sub-models is combined with the critical micelle concentration (cmc) prediction sub-model to create an overall corrosion inhibition prediction model separately. sphere-to-rod transition. Descrip- tions of model derivation, model parameters including effective area of headgroup or headgroup-ion pair, distance from the surface of the micelle to the center of charged headgroup, Stern layer thickness, as well as related calculations are found in the supplementary informa- tion and existing literature. 25,27 With the predicted cmc values of various pure surfactant and mixed surfactants, MLA and MQSAR-1 can be used for the prediction of corrosion inhibition efﬁciency.

Carbon steel is widely used for production and transportation pipelines in the oil and gas industries. [1][2][3][4] However, carbon steel is easily corroded in environments that contain water and carbon dioxide (CO 2 ). [3][4][5][6][7] As one of the main corrosion types in the oil and gas industry, CO 2 related corrosion can cause tremendous damage to pipelines and structural components in water and crude oil transportation and thus threaten production and safety. [3][4][5][7][8][9] The annual direct cost of corrosion in United States has been estimated to be around 3.1% of the gross domestic product (GDP). About 3.7% out of the total cost comes from the oil and gas industry, 7,10 which is mainly due to the corrosion of carbon steel. Therefore, the cost of corrosion and safety has led to great interest in controlling CO 2 -related corrosion in various oilfields around the world.
The most popular control method is to use organic inhibitors that contain heterocyclic molecules to reduce CO 2 based corrosion on carbon steel. 1,2,[11][12][13][14] Many of the organic inhibitors are surfactants with hydrophilic and hydrophobic molecular sections.
The hydrophilic group of surfactant strongly prefers interaction with polar entities such as water, metals, and ions. These organic surfactants adsorb on the metal surface, block the active surface sites, and thereby reduce corrosion attack. 11,12 The presence and structure of specific atoms, such as N and O, in surfactants determine the adsorption mechanism and corrosion inhibition efficiency. 13,14 Surfactant mixtures have received wide attention in practical applications because of their superior physicochemical properties and capabilities in efficient solubilization, dispersion, suspension, and transportation. 15,16 Solutions containing mixed surfactants can often be conveniently tuned to achieve desired properties by adjusting the composition of the mixture. More surface-active and expensive surfactants are usually mixed with less surface-active and less expensive surfactants to reduce cost. 17 However, the authors are not aware of a completely established theory or model to adequately predict corrosion inhibition using mixed surfactants despite extensive research work. 3,[18][19][20][21][22][23][24] Because of hydrophobicity, surfactant molecules tend to adsorb at the air-liquid interface, liquid-solid interface, or liquid-liquid interface to escape from the aqueous phase by associating and aggregating hydrocarbon chains together. [15][16][17] The concentration at which a monolayer covers the solid-liquid interface is considered as the surface aggregation concentration (sac). Above the sac, surfactants will form aggregate structures to orient their hydrophobic tails toward those of neighboring surfactant molecules and their hydrophilic head groups toward water. The concentration at which surfactants start to form aggregates in solution is termed the critical micelles concentration (cmc). [21][22][23][24][25] It is often assumed that the corrosion rate in the presence of low concentration of surfactants (usually lower than sac) can be represented by the number of available surface sites remaining after limited surfactant adsorption. [21][22][23][24][25] As the concentration of surfactants increases, more and more active surface sites are covered by surfactants. Near the sac or cmc, the metal surface is assumed to be nearly covered by one monolayer or multilayers of surfactants, respectively, and metal is well protected from corrosion attack. [21][22][23][24][25][26] Thus, sac and cmc are important factors in the evaluation of the effect of surfactant concentration on surfactant adsorption and corrosion inhibition of metal. However, there is a lack of investigation and associated modeling work that illustrate how sac and cmc affect corrosion inhibition efficiency of surfactants in solutions with various dissolved salt contents.
In the present study, a new model for prediction of corrosion inhibition efficiency in salt solution using surfactants (both pure and mixture) is introduced based on previous work. [21][22][23][24][25]27 This model is based on utilization of either a Langmuir adsorption (LA) sub-model or a Quantitative Structure Activity Relation (QSAR) sub-model with a cmc prediction sub-model. 27 The developed model is referred to as a modified Langmuir adsorption (MLA) 25 or modified Quantitative Structure Activity Relation (MQSAR) model, respectively, and will be introduced in the following section. The predictive MLA and MQSAR models are validated using electrochemical data collected from X65 steel corrosion inhibition testing using mixed homologous benzalkonium chlorides (BAC) surfactants as well as the reported data on other testing systems (Table I). The chemical structure of various surfactant molecules discussed in the present work is given in Fig. 1. The predicted results from MLA and MQSAR agree well with experimental results. In addition, the effect of BAC concentration on steel corrosion inhibition as well as the associated adsorption mechanism on steel is discussed based on electrochemical measurements and density functional theory (DFT) calculations.

Model Derivation
MLA sub-model.-One of the widely accepted models which is used for the adsorption of surfactants at an electrode-solution interface is the Langmuir adsorption model, [21][22][23][24][25]30 in which the surface  20 22 coverage is represented by: where K ad is the equilibrium adsorption constant given by where C is the concentration of surfactant in the bulk solution, C wm is the molar concentration of water which is 55.5 M, G o ad is the standard free energy of adsorption, R is gas constant, and T is absolute temperature.
As mentioned in the former section, sac (represented using¯ ) and cmc (represented using ) are important parameters characterizing corrosion inhibition efficiency of surfactants. Therefore, MLA is introduced to evaluate corrosion inhibition efficiency of surfactants under various solution conditions by the incorporation of the cmc considering that the cmc is easier to measure and predict than sac. 25 The MLA is presented below or η (%) = 100θ = 1 − 1 1 + K C × 100 [4] C n TAB: where K is equal to the adsorption constant K ad multiplied by , and η is corrosion inhibition efficiency. Note homologous surfactants tend to achieve similar levels of surface coverage at similar ratios of surfactant concentration to surfactant cmc, so the value of K barely varies for homologous surfactants and can be used as a universal constant for such homologous surfactants. 25 Note that C can increase above the sac or the cmc, but the fitting will not be as good as the fitting for C below the sac as shown in the following sections, which show that the sac is a transition point in characterizing the effectiveness of surfactants as corrosion inhibitors. The essence of Eqs. 3 and 4 is that the incorporation of cmc can successfully adjust for the effect of solution conditions and surfactant properties, such as salt concentration, solution temperature, hydrocarbon chain length, lateral surfactant interactions, and counterion binding, on surfactant adsorption and thus on corrosion inhibition efficiency.
MQSAR sub-model.-It is reported that a nonlinear relationship, QSAR, exists between a series of quantum chemical descriptors, such HOMO and LUMO energies, and average corrosion inhibition efficiency for a specific surfactant: [31][32][33] where A is a vector of regression coefficients specific to surfactant and solution conditions (such as salt and temperature), Q is a vector of quantum chemical descriptors for a particular surfactant, andB is a regression constant. For mixture of homologous surfactants the quantum chemical descriptors are weight-based average values. Considering that QSAR in Eq. 5 was derived based on LA, it is reasonable that QSAR can also be modified to a general relation to predict η by the incorporation of cmc as presented below: where A is a modified vector of regression coefficients, andB is a modified regression constant. Eq. 6 is termed MQSAR-1, which is similar to MLA in essence and can be adjusted for the effect of solution conditions on corrosion inhibition. The correlation between salt concentration and cmc of surfactant is well described by the Corrin-Harkins relation as follows: 34 log 10 ( ) = a log 10 (C s ) + b [7] where a and b are regression constants, and C s is salt concentration. With Corrin-Harkins relation, MQSAR-1 can be further modified to a more general form by eliminated term: Eq. 8 is termed MQSAR-2 and is comparable to MLA and MQSAR-1 with respect to corrosion inhibition efficiency prediction. The advantage of MQSAR-2 is that it does not need the cmc as an input. cmc prediction sub-model.-To use MLA and MQSAR-1 for the prediction of corrosion inhibition efficiency as mentioned above, known value of the cmc for the associated surfactant or mixed surfactants is a prerequisite. A model for the cmc prediction is briefly introduced in this section. More details about model derivation and validation can be found in supplementary information and existing reference. 27 The cmc of surfactant is evaluated using the following equation: where k is Boltzmann constant, T is temperature, and μ o m is micellization free energy which is estimated from several contributing terms as described below.
and μ o elec are the free energy contributions from hydrocarbon transfer from water into micelle, formation of micellar core-water interface, hydrocarbon tail packing in the micelle, surfactant headgroup steric interaction, headgroupcounterion mixing, and electrostatic interaction, respectively. [35][36] μ o act comes from surfactant activity and counterion activity contribution.
Free energy micellization as a function of variables, including on micelle shape, micelle composition, micelle radius, and counterion binding coefficient, at given solution conditions is minimized using home-designed MATLAB code. The minimized micellization free energy is then used for the evaluation of cmc, aggregation number, counterion binding coefficient, and sphere-to-rod transition. Descriptions of model derivation, model parameters including effective area of headgroup or headgroup-ion pair, distance from the surface of the micelle to the center of charged headgroup, Stern layer thickness, as well as related calculations are found in the supplementary information and existing literature. 25,27 With the predicted cmc values of various pure surfactant and mixed surfactants, MLA and MQSAR-1 can be used for the prediction of corrosion inhibition efficiency.

Experimental
The homologous cationic benzalkonium chlorides (BAC) surfactants, including benzyl dimethyl dodecyl ammonium chloride (C 12 BzCl), benzyl dimethyl tetradecyl ammonium chloride (C 14 BzCl), and benzyl dimethyl hexadecyl ammonium chloride (C 16 BzCl), were supplied by Sigma-Aldrich Co. LLC with assay values higher than 99%. The molecular structure of the surfactants are optimized and quantum parameters calculated using Gasussian09 simulation package with the method of B3LYP and the basis set of 6-311G(d, p) based on DFT. The test samples for surface tension measurements were prepared by sequential dilution of concentrated aqueous solutions of surfactants using double deionized water, made through a water purification system (Simplicity UV made by EMD Millipore). The stock solution was prepared at a total surfactant concentration of 25 mM for electrochemical measurements using deionized water.
The surface of the X65 electrode was polished using SiC paper in the sequence of 400-600-800-1200 grit and followed by MicroCloth with grit size of ∼5 μm supplied by Buehler. A platinum ring electrode and a single junction saturated calomel electrode (SCE) were employed as counter electrode and reference electrode, respectively. Test solutions contained 0.171 or 0.599 M NaCl and were purged with Ar (>99.999%) for 2 hours (hrs) to remove oxygen followed by the purge of CO 2 (>99.999%) for 2 hrs to ensure CO 2 saturation prior to measurements. A flow of CO 2 was maintained during the experiments to keep a positive pressure inside the cell to avoid air ingress. The concentration of dissolved oxygen was monitored before electrochemical measurements using an Oxygen ULR CHEMets Kit and the concentration was measured to be below 20 ppb. The pH was adjusted to 4 -5 for different mixtures by the injection of 1.0 M NaHCO 3 or diluted HCl into the cell. The surfactants were added at the beginning of each measurement. The test solutions were then kept at open circuit potential (OCP), E corr , for 2 hours for equilibration. Experimental conditions for various testing systems are listed in Table I. For time and experimental resource conservation, only Testing System II is used as an example for the results discussion from electrochemical measurements and for the inhibition efficiency prediction model derivation.
A Gamry reference 600 potentiostat was used for electrochemical measurements. Polarization resistance R p was measured using the linear polarization resistance (LPR) method by polarizing the working electrode +/− 0.010 V (SCE) vs. E corr with a sweep rate of 0.1 mV/s. Potentiodynamic scans were performed with a sweep rate of 1 mV/s from −0.9 V (SCE) to −0.35 V (SCE). Electrochemical impedance spectroscopy (EIS) measurements were made with an applied alternating current (AC) potential of +/−0.010 V rms vs. E corr in the frequency range of 100,000 -0.010 Hz. The direct current (DC) potential was set as zero relative to E corr . Each test was repeated at least three times as independent measurements within +/−4% deviation. The collected electrochemical data was analyzed with the Gamry Echem Analyst software package.
The surface tension of test solutions was measured within a precision of 0.1 mN/m by the platinum ring method using a Krüss K10 ST digital tensiometer, equipped with an isothermal vessel holder. All the measurements were performed at a constant temperature of 40 • C ± 0.2 • C, which has been shown to be higher than the Krafft point of the surfactants and their mixtures in aqueous media containing various concentrations of NaCl. The constant temperature was maintained through a water circulation bath using Polystat temperature controller, purchased from Cole-Parmer. The platinum ring was rinsed with water and heated to an orange color using a Bunsen burner between tests to ensure the complete removal of contaminants. Triplicate measurements were used to confirm reproducibility within +/−2% deviation.
Nova Nano scanning electron microscope (SEM) equipped with energy dispersive X-ray spectroscopy (EDS) system was used to observe the surface morphology.

Results and Model Validation
cmc measurement.-Upon the adsorption of surfactants at the airwater interface, the surface tension is reduced due to the amphiphilic nature of surfactants. Examples of surface tension versus surfactant concentration curves are given in Fig. 2 and the cmc is determined from the interception of the two solid lines in each curve. As can be seen, the surface tension decreases with the increase in surfactant concentration until the surface tension reaches a plateau value, which is a result of surfactant assembled into aggregates, such as micelles, bilayers, or multilayers. Beyond the cmc, additional micelles form but the surface tension remains constant. Fig. 2 also indicates that the cmc of C 12 BzCl decreases as the salt concentration increases because more surfactant molecules adsorb on the surface at the higher concentration of salt. The comparison between curves b and c reveals that the ternary mixture of C 12 BzCl, C 14 BzCl, and C 16 BzCl has a lower cmc value because the average hydrocarbon chain length is longer than that of pure C 12 BzCl.
Electrochemical measurements.-Considering E corr stability is important to electrochemical measurements, the X65 steel electrode was immersed in solution and kept at OCP for equilibration before measurement. Examples of the dependence of E corr of X65 steel electrode on time are given in Fig. 3a. The E corr stabilized at around  37 In the present research E corr of Testing System II stabilized between −0.720 V (SCE) and −0.620 V (SCE) after the addition of surfactants at a wide concentration range, which includes sac and cmc. The cmc is around 16.5 μM by measurement. E corr did not increase much at surfactant concentrations above cmc levels. The difference in E corr in the absence and presence of surfactants indicates that the steel surface was covered and protected by the surfactant adsorption. According to Riggs Jr., 38 it is possible to classify one surfactant as anodic or cathodic if E corr in the presence of surfactant shifts at least +85 mV or −85 mV, respectively, relative to E corr in the absence of surfactant. However, the positive shift of E corr of the investigated Testing System II at the highest concentration of 36 μM is only around 85 mV indicating that both the dissolution of iron at the anode and the hydrogen evolution at the cathode were affected. The LPR measurements were performed on Testing System II with various concentrations of surfactants and the results were used to evaluate corrosion inhibition efficiency, η (%), using Eq. 12: 39 where R po and R p are polarization resistances in the absence and presence of surfactants respectively. Fig. 3b shows selected potentiodynamic scan curves of Testing System II. The shape of the anodic branch does not change a lot when the surfactant concentration is less than 9 μM. Above 9 μM the anodic branch experiences a significant change. This phenomenon may be explained by coverage by the first monolayer on the steel where the monolayer effectively protects steel from corrosion. The concentration of 9 μM is interpreted as the sac of mixed surfactants in Testing System II. As the concentration continues to increase to the cmc or even higher values, the shape of the anodic branch does not shift much due to the fact that steel surface is already covered by monolayer before multilayers of surfactants form. The overall protection is slightly affected by the multilayers which form after the first monolayer. 25,40 The Tafel slopes were estimated from potentiodynamic scan curves. For those curves without anodic Tafel dependence above the sac, the anodic Tafel slopes were derived from the cathodic branches and cathodic Tafel slopes. 39 The corrosion inhibition efficiency was calculated using Eq. 13 based on the Tafel slope method. 39 [13] where i ocorr and i corr are the corrosion current density without and with surfactants in solution respectively. The calculated Tafel slopes, polarization resistance, corrosion rate, and inhibition efficiency are summarized in Table II for Testing System II. Each electrochemical measurement was repeated at least three times within +/−4 % deviation. The corrosion inhibition efficiency results from potentiodynamic scans and LPR match very well. The inhibition efficiency increases rapidly to around 90% with the increase in surfactant concentration from 0 up to 72 μM. Further increase in concentration does not effectively enhance inhibition efficiency even when the concentration is much higher than the cmc, . As mentioned previously the concentration of 72 μM is interpreted as the value of the sac, F, at which a complete monolayer usually forms at the electrodesolution interface, and above which, bilayers or multilayers usually form at the electrode-solution interface. [41][42] Corrosion inhibition is usually directly related to the electrode surface coverage. Therefore, the monolayer is effective with respect to corrosion protection and the formation of bilayers and multiplayers do not contribute much to additional corrosion inhibition beyond the protection provided by monolayer coverage.

Determination of parameter K in MLA sub-model.-
The corrosion current density as a function of (C/ ) for various testing systems is presented in Fig. 5a. As can be seen the current density decreses rapidly with the increase in surfactant concentration in the range between 0 and the sac for each of the testing systems. Above the sac, the current density reaches a plateau value. Fig. 5a also indicates that the surfactants in Testing Systems IV and V are not as effective in corrosion inhibition as those surfactants in Testing Systems II and III due to relatively higher plateau values of current density in the concentration range studied. Fig. 5b presents the plots of 1/θ versus 1/C based on the regular LA in the concentration range between 0 and the cmc for various testing systems. The calculated adsorption free energies based on Eqs. 1 and 2 are −45.6 kJ/mol for Testing System II, −43.4 kJ · mol −1 for Testing System III, −44.1 kJ · mol −1 for Testing System IV, and −37.9 kJ · mol −1 for Testing System V. If adsorption free energy is more positive than −20 kJ · mol −1 , the interaction between surfactant and metal is usually dominated by physisorption. If adsorption free energy is more negative than −40 kJ · mol −1 , the interaction is usually dominated by chemisorption in which the adsorption involves charge sharing or transfer between surfactant molecules and metal surface to form coordination bonds. 43  adsorption in Testing Systems V is dominated by both physicorption and chemisorption. However, physisorption can sometimes be energetically favorable and significant whereas chemisorption may sometimes have relatively weak binding energy due to various factors that influence adsorption. 45,46 Over the entire range of surfacant concentration, the linear fitting is excellent based on the regular LA model for each investigated system in Fig. 5b. In contrast, the MLA model features a sharp transition around the sac as shown in Fig. 5c. Below the sac, there is a good linear correlation between 1/(1−θ) and (C/ ); above the sac, a sharp transition appears, and after that the plot is curved and gradually reaches a plateau as the concentration increases. The transitions indicate that below the sac the inhibition efficiency increases rapidly with the increase in surfactant concentration and that further increase in concentration above the sac does not effectively enhance inhibition efficiency even when the concentration is much higher than the cmc. These results are in accordance with electrochemical measurements. For each testing system, the linear part (C < sac) of the plot in Fig. 5c is presented in Fig. 5d, in which the value of the modified adsorption constant K is given by the slope of the linear fitting equation. The K values are 13.97, 15.73, 0.96, and 4.84 for Testing System II, III, IV, and V, respectively. The fitted K value will be used for the corrosion inhibition prediction model discussed in this paper. Note that homologous surfactants tend to achieve similar levels of surface coverage at similar ratios of surfactant concentration to surfactant cmc, so the value of K does not vary a lot for homologous surfactants. 25 Considering the surfactant used in Testing Systems I and II are homologous, K = 13.97 can be directly used in Testing System I.

Determination of parameters A andB in MQSAR sub-model.-
For better illustration of adsorption of BAC surfactants on steel surface, regular quantum chemical descriptors, 19,20,[31][32][33]47 including the energies of molecular frontier orbitals (E HOMO and E LUMO ), energy difference between HOMO and LUMO ( E), Mulliken charge distribution on the backbone atoms, dipole moment of the surfactant molecule (μ), surface electrostatic potential based on electron density of the molecule, and the fraction of electrons transferred from the surfactant to the steel surface ( N) 31,48,49 were determined based on DFT using Gaussian09. HOMO tends to donate electrons to suitable acceptor substances on the steel surface while LUMO tends to accept electrons from the steel surface and lower LUMO energy usually indicates stronger electron accommodation. 19,20,47 The energy gap E usually characterizes the stability of the complex of surfactant and metal surface. 31,48,49 The value of N describes the inhibition achieved from electron donation. 31,49 The Milliken charges and surface electrostatic potential usually shed light on the electron distribution in surfactant molecules and electrostatic interaction between surfactant and iron or steel. [30][31][32][33] The molar volume of surfactant molecule, V sm , is also needed to be considered due to its potential effect on surfactant packing/aggregation efficiency and steric interactions. However, the relationship between dipole moment and corrosion inhibition is still a controversial issue. 31,45 Many existing literature articles illustrate in detail how to calculate the fraction of electrons transferred from inhibitor to the metal surface, N. 31,49 The calculation process is briefly described below. The ionization potential E ip and the electron affinity A e are approximated to -E HOMO and -E LUMO , respectively. The absolute electronegativity χ and the global hardness ϒ are defined as [15] N is given by [16] where χ mel and ϒ mel are electronegativity and global hardness of metals (electrodes), respectively. χ inh and ϒ inh are electronegativity and global hardness of surfactant molecule, respectively. The reported electronegativity and global hardness of iron (4.06 eV/mol and 3.81 eV/mol, respectively) 50,51 are used. For Testing System V, the electronegativity and global hardness of copper are 5.59 eV/mol and 0.15 eV/mol, which were calculated using Gaussian09 following the same procedures for surfactants. The values of quantum descriptors of homologous BAC surfactants as well as surfactants in other testing systems are summarized in Table III. For qualitative illustration of homologous BAC surfactants, the calculated results of C 14 BzCl are presented in Fig. 6 considering that the electronic properties of homologous series should be similar.
Application of Eqs. 5, 6, and 8 based on electrochemical data of Testing System II in Table II and calculated quantum descriptors in  Table III yields the following semi-empirical equations of QSAR, MQSAR-1, and MQSAR-2, respectively:  through cmc. The prediction using LA and QSAR, however, deviates significantly from experimental data. Note that MQSAR-2 is similar to MQSAR-1 in essence and thus is not shown in the comparison in Fig. 7.
As discussed above, both MLA and MQSAR need the cmc value of surfactant or surfactant mixture of discussed as an input. The validation of the well-developed cmc prediction model is exemplified in Fig. 8. The predicted cmc for pure, binary-and ternary-mixed BAC surfactants in the solution containing various NaCl concentrations agree well with experimental results as shown in Figs. 8a-8c. For all of the discussed testing systems listed in Table I, there is excellent agreement in the value of cmc between prediction and experiment (Fig. 8d). The application of the cmc prediction model to the binary mixture of anionic and nonionic surfactants (Fig. 8e) and to the ternary mixture of cationic, cationic, and nonionic surfactants (Fig. 8f) is also successful, even though the agreement between the prediction and the experiment is not as excellent as that shown in Figs. 8a-8d. Note that the experimental data of Testing Systems III, IV, and V is cited from references. 21,[52][53][54][55][56] It is expected that the integrated model based on the MLA submodel and the cmc sub-model or the integrated model based on the MQSAR sub-model and the cmc sub-model should be successful in the prediction of corrosion inhibition efficiency of the discussed surfactant systems. The comparison of the predicted results from the integrated model and from the experimental measurements is shown in Fig. 9, based on the data of corrosion inhibition from all the five testing systems as summarized in Table IV. The prediction agrees very well with experiment for Testing Systems I, II, III, and IV. The deviation for Testing System V is slightly higher but still falls within a reasonable range. This may be due to less effective monolayer coverage of C 16 TAB on copper surface in presence of slightly complicated salt Fe(NO 3 ) 3 in Testing System V. Note that the MQSAR predicted corrosion inhibition of all the five testing systems are based on Eqs. 18 and 19 with unified fitting parameters, which indicates the vastly improved applicability and robustness of developed MQSAR over regular QSAR.

Discussion
It can be seen in Table III that all the three surfactants display slight differences in quantum descriptors. In the context of simple corrosion inhibition it is believed that alkyl chains are chemically unreactive  Table I; (e) cmc of binary mixed anionic and nonionic surfactants (SDS and OG) as a function of mixed bulk solution composition of OG with 20 mM NaCl at T = 25 • C; (f) Pre. cmc vs. Exp. cmc of ternary mixed cationic, cationic, and nonionic surfactants (C 16 BzCl, C 16 TAB, and C 16 E 20 ) at various mixed molar ratios with 30 mM NaCl in solution at T = 25 • C. For Figs. 8(a, b, c, and e) symbols represent experimental data; lines represent model predicted data. Experimental data of testing systems III, IV, and V is cited from references. 21,[52][53][54][55][56] substituents and that the homologous surfactants which vary only in alkyl chain length have very similar quantum descriptors. In other words, these quantum descriptors, calculated based on frontier orbital theory with a very strong fundamental basis, describe the characteristics of the head groups and reflect the associated adsorption properties. It is feasible to get quantum descriptors from only one type of surfactant and then apply these parameters to a series of homologous surfactants or surfactants with similar headgroups. For the surfactants of different classes with considerable differences in quantum descriptors, molar-based average quantum descriptors are recommended. In  Table I. The associated corrosion inhibition efficiency is summarized in Table IV. Experimental data of Testing systems III, IV, and V is cited from references. 21,22,28,29 this study, the molar-based average was used to calculate the quantum descriptors of mixtures regardless of surfactant classes. The cmc model takes into account the ion/salt effect on aggregation/adsorption, the effect of chain length of surfactant, van der Waals interactions between surfactant molecules, steric interactions between head groups, electrostatic interactions at interfacial region of micelles, and the interactions between solvent and surfactant. 21,22,25,27,35,57 Therefore, the insertion of cmc into QSAR or LA can accurately describe the adsorption phenomena of surfactants on substrate (metal electrode) and associated effects of physical and chemical properties of surfactants and solvent environment. Mixtures of surfactants, which involve surfactant of the same or different classes, have been widely used in practical applications because of their superior physicochemical properties, capabilities, and/or economic viability. 15,16 Beyond the applicability of our model for pure surfactant and mixed homologous surfactants, the more valuable part lies in its potential to evaluate the corrosion inhibition of various surfactant mixtures of different classes at various solution conditions using only one set of experimental data.
The fitting of MQSAR only requires one set of experimental data just as MLA does. The only difference is that the fitting of MQSAR yields a set of regression parameters simultaneously but MLA only has one parameter. In terms of the number of regression parameters, the use of MLA is simpler. In other words, the MQSAR is an alternative to the MLA for corrosion inhibition modeling.
It is interesting to note that the regression parameters in QSAR/MQSAR for one class of surfactants may be transferred to other surfactants with similar head groups or similar quantum descriptors. Similarly, the MLA parameter (K ) for one surfactant can be used for other surfactants with similar head groups. In addition, the parameters of one surfactant mixture can be used for the mixtures of similar surfactants. The initial evaluation of transferability of regression parameters was performed by applying Eqs. 18 and 19 to the five testing systems discussed and the results in Fig. 9 indicates