The AC Corrosion Mechanisms and Models: A Review

The alternating current (AC)-induced corrosion problem of metallic materials has already been studied since the early stage of the application of AC power.¹  However, it did not receive much attention at first because the corrosion rate caused by AC was believed to be much lower compared to that by direct current (DC) interference.²  After several pipelines were severely corroded by AC,³  it was realized that even though AC had less impact on corrosion rate than DC, it was still a serious threat to the safety of underground structures.^4-6 

As a result, the AC corrosion phenomena attracted more and more interest. Intensive experimental studies have been performed, through which the effect of AC on the corrosion rate,^7-8  corrosion potential,^9-10  and corrosion morphology^11-13  have been investigated.

Meanwhile, in order to reveal the principle of AC corrosion, theoretical studies also were performed.^14-15  Several researchers have proposed various mechanisms or models to explain the AC-induced corrosion phenomena. However, it is still commonly reported that there is no technical agreement on the AC corrosion mechanism.^16-18  Some authors have attempted to review the mechanisms of AC corrosion, but either in a very simplified way or just as one part of their research background.^19-22  A comprehensive review concentrating on the AC corrosion mechanism and model is still missing. However, this kind of review is crucial because it could provide an effective way to clearly understand the similarities and differences among various theoretical studies, help the further development of AC corrosion theory. Therefore, in this paper, the AC corrosion mechanisms and models in the published literature are systematically reviewed.

When AC is imposed on the corrosion system, it could produce significant effects on the anodic and cathodic electrode reactions, resulting in an impact on the corrosion process. In fact, the effect of AC on the electrode process has been discussed for a long time. Depending on the purpose of the study, different aspects were investigated, such as faradaic impedance which is the behavior at the fundamental frequency, faradaic distortion which is the behavior at the first harmonic frequency, and faradaic rectification which is the behavior of the DC component.²³  Among these aspects, the faradaic rectification is more important than the other concepts because the induced DC component could enhance the corrosion rate. Moreover, based on the faradaic rectification, several classical AC corrosion models have been proposed.^24-27 

Faradaic rectification refers to the shift of electrode potential of an electrochemical process under the effect of AC. This phenomenon was first observed by Doss and Agarwal.^28-29  They presented a circuit diagram, as shown in Figure 1, in which a transformer (T) was used to supply AC, a rheostat (R₁), and a resistor (R₂) were, respectively, used to adjust and measure the AC, and a capacitor (C) was added to block the DC circuit. With the experiments, they found that as two platinum electrodes (E₁, E₂) were immersed in the aqueous solution with AC passing across, a DC potential was generated on both of the two electrodes. The developed DC potential could be detected by a reference platinum electrode (Ref) with a large area. They referred it as redoxokinetic effect because its magnitude was relevant to the kinetics of the redox reaction. Later, Doss and Agarwal made a series of tests and verified that the redoxokinetic effect was general for redox systems.³⁰  In 1957, the term “redoxokinetic effect” was renamed as “faradaic rectification” by Oldham.³¹ 

Faradaic rectification was believed to be the result of the asymmetry of the polarization curve,³²  which was caused by the asymmetry of charge transfer reaction or the inequalities in mass transfer rates of oxidant or reductant.²³  In the negative half cycle of AC, the potential of the electrode decreases, as a result, the electrode reaction is promoted in cathodic direction. On the contrary, in the positive half cycle of AC, the potential increases, leading to the enhancement of the electrode reaction in the anodic direction. But due to the asymmetric of the polarization curve, in one period of AC, the increment in the anodic direction is not equal to that in the cathodic direction, the net result is the produce of the DC component. The case of the slope of the cathodic polarization curve is larger than that of the anodic branch is shown in Figure 2. As demonstrated that the asymmetry of the polarization curve transfers the AC into a new potential fluctuation with the average value shifted negatively, resulting the DC component. The theoretical analysis of faradaic rectification for the single electrode reaction was first performed by Doss and Agarwal within a small AC voltage, but their treatment was under the assumption that the concentrations of oxidant and reductant equaled to each other.²⁹  Later, Oldham developed the theory into a general one by considering the oxidant and reductant in different concentrations.³¹  Barker also derived an expression for the rectification potential, with which he developed a new electroanalytical technique, named square wave polarography.³³  In 1959, Rangarajan presented a more general expression and proved that the results of Oldham and Barker were the particular cases of the general one. His work demonstrated that the faradaic rectification was powerful for the quantitative study of the kinetic parameters of the single electrode process.³⁴  Sathyanarayana extended the theory of faradaic rectification for the mixed reactions in corrosion electrode and developed a new electrochemical method for testing the corrosion rate.^35-36  In Sathyanarayana’s work, the concentration polarization was not considered, and the applied peak amplitude of the alternating voltage on the electrode should be small enough, less than 30 mV. However, his study was the first one to relate the faradaic rectification with metal corrosion phenomena in theory.

The faradaic rectification can be observed not only on a single electrode but also between two different electrodes. In fact, the corrosion of underground metallic structures induced by AC due to the faradaic rectification had been reported as early as in 1945 by Sherer and Granbois.³⁷  They found that when AC passed through two electrodes which had different surface conditions, a direct current could be measured due to the process of rectification, which responded for the enhancement of corrosion. Although they did not give the theoretical explanation of this phenomena, it could infer from their experimental results that the asymmetric electrochemical characteristic of the connected electrodes may be responsible. This type of faradaic rectification was also reported by Bruckner³⁸  and Jiang.²⁰ 

The applications of faradaic rectification on the theoretical analysis of AC corrosion characteristic on single metal electrodes were performed by numerous researchers.⁸  Their conclusions about the effect of the ratio (r = β_a/β_c) of anodic (β_a) and cathodic (β_c) Tafel slopes on the shift of corrosion potential were that: (1) if r < 1, the potential shifts negatively; (2) if r > 1, the potential shifts positively; and (3) if r = 1, the potential remain the same, which were qualitatively verified by some experiments.^10,17,39  However, the faradaic rectification mechanism assumed that the kinetic parameters were independent from AC²⁷  and the corrosion interface was unchanged during the AC corrosion process.⁴⁰  Therefore, it cannot reveal the AC corrosion characteristics under more general environmental conditions.

The mechanism of AC on the corrosion process is complicated due to both the characteristics of AC and the electrochemical essence of the corrosion process. AC is quite different from DC because its flow direction constantly changes. Therefore, when it is imposed on the electrode, both the anodic and cathodic reactions at the corrosion surface suffer a periodic anodic and cathodic polarization, keeping the corrosion process in a transient state. Moreover, it has been reported that more than one electrode reaction would occur during both the anodic and cathodic polarization.⁴¹  Besides, the waveform and frequency of AC also could influence the corrosion process, bringing more difficulties to understand the corrosion mechanism. Several corrosion mechanisms were proposed to reveal the complex periodic electrochemical process.

It has been discovered early on that AC could cause the dissolution of metal electrodes in the measurement of electrolyte conductivity. In the review presented by Venkatesh and Chin, it was reported that the effect of AC on the electrochemical system was similar to that of a depolarizer, resulting in the reduction of the anodic and cathodic polarization.⁴²  Subsequently, they observed in the polarization experiments that AC could increase the dissolution current of mild steel in the active region in Na₂SO₄ solution.⁴³ 

In the study of the effect of AC on the corrosion behavior of low alloy and carbon steel in the aerated and deaerated 0.1 N NaCl solutions, Jones attributed the AC inducing potential shift to the different polarization rate between the anodic and cathodic process.⁴⁴  He found that when the electrode polarized more rapidly in the cathodic process than that in the anodic process, the positive half-cycle of AC would not have enough time to compensate the potential shift caused by the negative half cycle, resulting in the decrease of the anodic Tafel slope which he referred as the depolarization of anodic reaction, as shown in Figure 3. Consequently, a net active potential shift happened and the corrosion current density increased. He inferred that the depolarization of the anodic reaction by AC may be due to the anion desorption or surface film reduction in the negative half cycle of AC.

Lazzari, et al., reported that when AC was imposed on the cathodic and anodic polarization, it could cause the reduction of both cathodic and anodic overpotential, which are demonstrated in Figure 4.⁴⁵  Moreover, in the experiments performed by Goidanich, et al., they also found that the Tafel slopes change under the presence of AC; however, they did not present more detailed information on whether the Tafel slopes increase or decrease with AC.⁸  Wang, et al., investigated the effect of AC on the polarization properties of carbon steels with a specially designed AC voltammetry technique. In contrast to previous studies, they found that the anodic Tafel slope increase with the imposed AC.¹⁴  However, in the recent works of the authors, it was found that the anodic Tafel slopes of the pipeline steel in 0.5 M Na₂SO₄ remain constant under different AC current density, which is shown in Figure 5.

Therefore, it could be inferred that even though the depolarization caused by AC could explain the AC enhanced corrosion on the metal electrodes, due to the complication of corrosion process, it may not be enough to represent the whole AC corrosion mechanism and further research is needed.

The AC-induced corrosion on pipeline is a very complicated process which is influenced not only by the AC interference properties, but also by the conditions of the surroundings, such as the implementation of the cathodic protection, the species of the cations as well as the pH in the soil nearby the coating defects, which could result in a specific modification of the double layer interface and the final corrosion products.²  In order to understand the effect of AC in the interface modification, the role of the irreversibility of anodic dissolution in the AC-induced corrosion process due to the change in the corrosion products that formed in the positive and negative cycles of AC should be noted first.

As early as in 1916, McCollum and Ahlborn¹  discovered that the corrosion rate enhanced by AC was small, but it was larger in the stirred electrolyte than in the undisturbed one. They explained that most of the metal ions corroded in the positive cycle of AC could be electroplated back on the surface in the negative cycle, and the redeposited metal on the surface acted as new anode in the next positive cycle protecting the uncorroded metal beneath.¹  Therefore, when the dissolved metal ions were taken away from the corroded surface by the movement of the electrolyte or reacted with other chemicals to form insoluble compounds making the reverse process of the dissolution difficult to occur, the corrosion rate was enhanced. Their view was supported by Goidanich, et al.,³  who also reported the AC enhanced corrosion was due to that the metal dissolution process occurring in the positive-half cycle was irreversible in the negative half-cycle. However, they attributed the irreversible process to the kinetic characterization of the corrosion process that the metal dissolution was most prone to occur in the positive half-cycle, while oxygen reduction or hydrogen evolution was most prone to occur in the negative one.

In fact, before the study of Goidanich, Yunovich, and Thompson illustrated the oxidation current waveform in one period of AC when the dissolution of iron was supposed completely irreversible.⁴⁶  They showed that the magnitude of the current which increased in the positive half-cycle of AC was much greater than that decreased in the negative half-cycle, resulting in a net increase of the corrosion current. Based on this method, they made a successful prediction of AC-induced corrosion rate on the experimental results.

Eventually, by considering the influence of AC on the corrosion interface, alkalization theory, potential vibration and passivity destruction, and pitting mechanisms have been developed.

When the underground pipeline is under cathodic protection, the applied cathodic current causes production of OH⁻ near the coat defects where the potential is the most cathodic. Consequently, the increased amount of OH⁻ was believed to have different impacts on the spread resistance, which refers to the ohmic resistance caused by the soil in the vicinity of the surface based on the ratio between alkaline earth (i.e., Mg²⁺, Ca²⁺) and alkali metal (i.e., K⁺, Na⁺) cations. When alkaline earth metal cations are abundant, insoluble hydroxides (Ca(OH)₂, Mg(OH)₂) would form protective layers which may be converted into carbonates (CaCO₃, MgCO₃) with the presence of CO₂, increasing the spread resistance. In addition, the insoluble products may deposit inside the coating defect, weakening the effect of AC on corrosion.⁴⁷  On the other hand, the absence of the earth alkaline cations makes it impossible to form the protective produce layer, which means that the resistance for AC to reach the coating defects is not limited; therefore, the AC corrosion process continues.⁴⁸  Based on the experimental and field studies, Nielsen, et al., supported the alkalization theory which indicated that the combined action of the AC resulting potential cycling and the accumulation of hydroxyl ions that produced by CP (Figure 6) was the cause of the AC-induced corrosion.^49-50  They suggested that under the effect of AC and the condition of elevated pH, the corrosion potential cycled between the immunity and passivity region illustrated in the Pourbaix diagram, as presented in Figure 7. Due to the metal dissolution being more rapid than the formation of the product layer, in a cycle of AC, a net dissolution occurred, increasing the corrosion rate.

Panossian, et al., also hold the view that the AC corrosion mechanism was related to the pH of the environment, in which the oscillation of the potential could make the thermodynamic state of the corroded surface vibrated between the active and immune domains or between passive and immune domains.⁵¹  However, they argued that the reasons for the AC-induced corrosion in different alkaline electrolytes were different. In acid, neutral or slight alkaline electrolytes, it was due to the irreversibility of the metal dissolution process. In more alkaline electrolytes, it was due to the reduced kinetics of the formation of a passive layer. Moreover, in extra alkaline electrolytes, it was due to the continuous formation and dissolution of the oxide film. Tang, et al., investigated the effect of pH on the corrosion behaviors of carbon steel under various AC and proposed a mechanism for AC corrosion in both the near-neutral and alkaline environment.⁵²  They argued that AC corrosion was a result of the irreversible oxidation of iron occurring at the period when the corrosion potential was nobler than the equilibrium potential of anodic reaction. In highly alkaline solutions where the passive film could form on the carbon steel surface, the step before AC corrosion was the mechanical breakdown of the passive film caused by the alternating electric field. The roles of OH⁻ played in the AC corrosion process were different in different alkaline electrolytes, demonstrating that the AC-induced corrosion process is related to the conditions of the surroundings.

Olesen, et al., presented a map of corrosion rate under different AC/DC current densities and discovered that the AC corrosion in the presence of cathodic protection could be categorized into high and low CP levels, in which the effect of AC on the corrosion are different. At high CP level, the AC-enhanced corrosion process is consistent with alkalization theory. At low CP level, AC could produce a dealkalization effect on the steel. However, they believed that both the AC corrosion mechanisms of the high and low CP level can be explained by Pourbaix diagram and attributed the effectiveness of CP on mitigating the AC corrosion to its ability to maintain the pH at the passivity region.⁵³  They found that the stability region of dihypoferrite ion (HFeO₂⁻) in the Pourbaix diagram was enhanced by AC, producing a corrosive condition.⁵⁴  Particularly, they presented three AC corrosion models based on spans of the AC-induced potential fluctuation, which are shown in Figure 8.

Büchler, et al., did not support the alkalization theory.^55-56  They emphasized the roles of the potential cycling induced by AC and presented a mechanism to describe its effect on the corrosion process of steels, which is illustrated in Figure 9. In the positive half-cycle, the dissolution of the steel surface caused the formation of a passive film, which was converted into a porous rust layer in the negative half-cycle. In the following positive half-cycle, a new passive film was formed underneath the rust layer. Meanwhile, a part of the Fe(II) in the rust layer could also be oxidized. Moreover, the oxidizing process of Fe(II) would be reversed in the reduction half-cycle, which combined with the dissolution of the passive film, causing the increase of the thickness of porous rust. The process repeated, resulting in the enhancement of the corrosion rate.

The potential cycling mechanism is supported by many researchers.^5,17,22  However, the theory relied on the experimental results of cyclic voltammetry test, which was used to simulate the effect of AC-induced potential vibration on the electrochemical process.^51-55  But due to the limitation of the test instrument, cyclic voltammetry test could only simulate low-frequency potential vibration, such as 0.31 Hz (equals to a high scanning rate of 1,000 mV/s), which was much lower than 50 Hz or 60 Hz, the frequency of AC power commonly used in the society. Therefore, the AC corrosion mechanism inferred from cyclic voltammetry test might not be suitable for the high-frequency one.

It has been reported that AC could induce pitting, indicating that the AC mechanism may be associated with the localized corrosion process. French discussed the mechanism of the AC corrosion of aluminum, a metal which could be oxidized in the air to form a protective film.⁵⁷  He illustrated a process in which the concentration of OH⁻ in the solution near the aluminum surface was increased by both half cycles of AC. As a result, the increased OH⁻ concentration dissolved the oxide film causing caustic corrosion. Moreover, at the location where a flaws were present in the oxide, the positive half-cycle of AC resulted in pitting. Tan and Chin investigated the anodic and cathodic polarization of aluminum under the effect of AC in sulfate solutions.⁵⁸  Their experimental results confirmed that AC exerted influence on the aluminum corrosion process through the flaws of the oxide film.

Apart from aluminum, Chin and coworkers also performed intensive studies to understand the effect of AC on the corrosion of mild steel, stainless steel, and nickel.^9,43,59-60  They concluded that AC could destroy the passivity of these metals. Particularly, they pointed out that the effect of AC on the breakdown of the passivity of mild steel and nickel was similar to that of chloride ions; however, they did not propose a mechanism.

Zhu, et al., attributed the localized corrosion of pipeline steel in the solution to the alternating electric field force generated by the low frequency AC.⁶¹  They stated that when the electric field had sufficient time to accelerate the solvated ions in the solution such as Fe²⁺, OH⁻, and H⁺, it could reinforce collision of the ions to the corrosion surface, resulting in the pitting corrosion occurring on the steel. In the two-step AC corrosion mechanism of carbon steel in CP condition proposed by Brenna, et al., they also considered the effect of the alternating electric field, which was believed that could produce electrostriction stresses, resulting in the breakdown of the passive film that formed on carbon steel surface under CP conditions in the first step, then in the next step the corrosion continued under the effect of high-pH chemical.^62-64 

Kuang and Cheng discovered that AC could induce pitting on carbon steel in both high pH and neutral pH carbonate/bicarbonate solutions, and proposed two mechanistic models, respectively, which are illustrated in Figure 10.⁶⁵  In high pH solution, the passive film was able to form on the steel surface serves as a barrier to keep the iron ions dissolved in anodic polarization of AC when the current density was small. However, with the increase of AC, the passive film would be damaged during the cathodic polarization in the negative half-cycle of AC; therefore, the accumulated iron ions had the opportunity to escaped from the weak point of the film, causing the formation of pits. In the neutral pH solution, pitting was due to the porous structure of the corrosion product layer formed on the steel surface under the effect of AC.

Some researchers also used an equivalent electric circuit model to demonstrate the effect of AC on the corrosion process.^20,46,48,66 

Nielsen and Cohn proposed a comprehensive electrical equivalent circuit, which is shown in Figure 11, to discuss the physical and chemical aspects involved in the AC corrosion process.⁴⁸  In the circuit, the anodic dissolution of metal and its redeposition, as well as the cathodic process such as the reduction of oxygen or hydrogen ions were represented individually. The DC sources symbolled by E₀1 and E₀2, respectively. denoted the equilibrium potentials of the anodic and cathodic process, in addition, the activation kinetics of the two processes were represented by the diode components (VB1 and VB2). The Warburg impedance component (W) was used to include the influence of diffusion limitation. R_S was the spread resistance representing the ohmic resistance between pipe and remote earth, and C indicated the interfacial capacitance. There was also an AC source to stand for the induced interfere and a DC source to represent the cathodic protection. Additionally, Nielsen and Cohn divided the elements in the electrical equivalent circuit into two types, static and dynamic, based on whether the element was related to the frequency, and then presented a detailed analysis on each of the element on the AC corrosion process.

Another typical equivalent circuit is presented in Figure 12. As shown, the corrosion process was simplified by using a typical RC circuit, with the double layer capacitor (C_dl) in parallel with a charge transfer resistor (R_t), and a series resistor (R_s) to represent the solution resistance. This circuit was simpler than that of Nielsen and Cohn, which brought a convenience in the mathematic process, making it widely adopted in the build of AC model. Some researchers also tried to modify it by separating the anodic (R_a) and cathodic (R_c) parts from the charge transfer resistance,^20,67  which is shown in Figure 13. In the previous works of the authors’ research group, with intensive electrochemical impedance spectroscopy tests on X70 under the effect of AC, a more practical equivalent circuit (Figure 14) was proposed to represent the corrosion process.⁶⁸  In the circuit, R_l was added to stand for the resistance between corrosion products and solution. Moreover, constant phase element (CPE) was used to replace pure capacitance. Hence, CPE1 was the combined capacitance of product and solution and CPE2 was the double layer capacitance.

From the equivalent circuit, it can be concluded that most of the imposed AC pass through the double layer capacitor, only a small fraction is involved in the corrosion process.⁶⁹  This, it was believed, could well explain why the AC corrosion rate is much less than that of DC at the same magnitude. The impedance of the capacitor decreases with an increase of frequency; hence, the proportion of AC flowing across the double layer enlarges with the increase of frequency, which could cause even less AC participates in the corrosion process, causing the decrease of corrosion rate. The other elements, such as R_t and R_s, are used to demonstrate the resistance in the AC corrosion process; therefore, the effect of AC on the corrosion rate reduces with the increase of these two elements.

It could be concluded that none of the proposed mechanisms can explain all aspects of AC corrosion behaviors. The concept of faradaic rectification and AC depolarization are useful to describe the effect of AC on the corrosion potential shift or corrosion rate, but it could not explain the influence of the surrounding medium on the corrosion process. On the contrary, alkalization theory has the ability to illustrate the effect of the composition of electrolyte on the corrosion process, but they could not predict the influence of AC on the corrosion potential shift. The potential vibration theory focused on illustrating the electrochemical process during one period of AC on the metal surface. The mechanisms of passivity destruction and pitting were designed solely for explaining the formation of localized corrosion. Equivalent circuit model was built to explain the reason that AC corrosion rate is much less than that of DC at the same magnitude and decrease with the increase of frequency. Therefore, it is evident that no consensus has been reached in AC corrosion mechanism, and it has been argued that AC corrosion may be a result of the combination of different mechanisms.³  In order to furtherly understand the AC corrosion mechanism in practical situations, it may be necessary to develop a new rapid measurement technique which can investigate the electrochemical response of the corrosion system in the positive and negative period of a single cycle of AC on the whole specimen. Moreover, the morphology of the corroded surface under the effect of AC could be studied by the real-time imaging technique or combined with electrochemical measurements to provide extra visual information for the understanding of the AC corrosion evolution.

The AC corrosion model is essential to reveal the influence of AC on the corrosion potential and corrosion rate quantificationally. It has been studied by many researchers for a long time.

Gellings was the first one who worked out an expression for the AC-induced corrosion rate with the combination of electrochemical principles and mathematical methods.⁷⁰  Bertocci presented a derivation to obtain the value of the rectified faradaic current resulting from the alternating voltage (AV) on an electrode under charge-transfer control.²⁷  His work may be the very first one that bridged the theory of faradaic rectification with AC-induced corrosion model. Almost at the same time, Chin, et al., modeled the corrosion potential shift, the rectified DC density, and passivity destruction behavior under the effect of both AV and AC.^9,43  They introduced the time-average concept, which was generally used in the study of fluid mechanics, into the mathematical treatment of their derivation to avoid the linearization process which was impossible for large alternating voltage. Lalvani and Lin developed a mathematical model to predict the AC-induced corrosion under the activation control.²⁶  With their model, they were the first to work out a closed solution for the AC-induced corrosion current and corrosion potential. Later, they revised the model by considering the double layer capacitance.²⁵  Based on the work of Lalvani and Lin, Bosch and Bogaerts extended the model for the case that the anodic reaction was under activation control, and the cathodic reaction was under mixed control.²⁴  However, the double-layer capacitance was not involved in their model. In 2008, in order to reveal the effect of AC on the corrosion system under activation control, based on the equivalent circuit Xiao and Lalvani established a differential equation model, in which the double layer capacitance and the solution resistance were considered.⁶⁶  Ibrahim, et al., presented a series of study on the AC-induced model for the underground steel pipelines under cathodic protection based on the works of former researchers.^40,71-73  They finally proposed a comprehensive differential equation model in which the cathodic process consisting of the reductions both of the dissolved oxygen that was under partly diffusion control and of the water. Ghanbari, et al., also proposed a model which was similar to that of Ibrahim.⁶⁷ 

The most critical improvement of the AC corrosion model is that the assumptions behind the models become more and more realistic. The different models are summarized as follows.

Bertocci was the first one to perform the derivation of the model. When the anodic dissolution and cathodic reduction in the corrosion process both follow Tafel relation, the anodic current (i_a), and the cathodic current (i_c) are given by where i_corr,DC and E_corr,DC are the corrosion current and corrosion potential without the effect of AC, respectively, and β_a and β_c correspond to the anodic and the cathodic Tafel slope, respectively. The potential (E) at the working electrode under the effect of AC is the sum of a DC potential (E_DC) and an alternating signal [E_p sin(ωt)], where E_p is the peak potential and ω is the frequency. Thus,

Substitution of Equation (3) into Equations (1) and (2) gives:

Rearranging:

The treatments of Equations (6) and (7) by different researchers were different. By using the modified Bessel function, the exponential components in Equations (6) and (7) can be expanded according to Equations (8) and (9): where I_n is a modified Bessel function of the first kind, where Γ(n) is the gamma function,

Hence,

When the time average effect of AC is considered, only I₀(z) in Equations (8) and (9) is not zero, known as the DC component. Thus, the time average anodic and cathodic current is given by:

The time average faradaic current (i_f,DC) is the difference between i_a,DC and i_c,DC, or: i_a,DC and i_c,DC would be equal to each other at a certain value of E_DC, known as the time average corrosion potential, E_corr,AV. Thus, equal Equation (15) to zero and the AC resulting corrosion potential shift (ΔE_corr) is given by:

And the time average corrosion current (i_corr,AV) is given by:

In the derivation of Chin and Venkatesh,⁴³  as they introduced the time average scheme directly into Equations (6) and (7) and gives: where

Similarly, by equaling i_a,DC and i_c,DC, the AC resulting corrosion potential shift (ΔE_corr) is given by:

In the work of Lalvani and Lin,²⁶  the exponential components in the integral parts of Equations (6) and (7) were expanded based on the following equation:

Thus, they proved that the average anodic and cathodic current could be given by:

The corrosion current (i_corr,DC) is defined as:

Hence, with some simple derivations, Equations (24), (25) could be given as:

Even though i_a,DC, i_c,DC, and ΔE_corr derived by different researchers are different, they can be unified. Substitute Equation (23) into Equations (20) and (21), with the proof of Lalvani and Lin,²⁶  

According to Equation (12), it is obvious that:

Substitute Equations (32) and (33) into the formulas of i_a,DC, i_c,DC, and ΔE_corr derived by different researchers, it can be shown that their results are consistent. This is because all of the models adopted the time average scheme to find the steady-state. In order to take account of double layer capacitance, Lalvani and Lin²⁵  also used the root mean square (RMS) scheme and derived a result for the RMS current (i_RMS) which is given as: where

The RMS faradaic current (i_f,RMS) is given as:

When the cathodic reaction is under mixed control, the effect of AC on the corrosion process becomes more complex. By considering the concentration fluctuation resulted by AC, Bosch and Bogaerts²⁴  derived an implicit equation for ΔE_corr, where i_l refers to the diffusion limited current.

In Bosch and Bogaerts’ model, only one cathodic reaction was considered and it was not specified. Later, by taking into account the reduction of water and that of dissolved oxygen in the cathodic reaction, Ibrahim, et al.,⁷²  developed a comprehensive expression for the time average faradaic current, which they expressed as:where i_DC,O2 and i_DC,H2O are the time average faradaic current of the reduction of oxygen and water, respectively.

In their model, a lot of effort was made on the derivation of the equation of the time average cathodic faradaic current of the oxygen reduction reaction, which was believed controlled partly by the diffusion in most of the practical situation. After a series of deduction, they worked out the expression of i_DC,O2, shown as: where i_corr,O2 is the partial cathodic current density of the dissolved oxygen reduction (i_l,O2) is the diffusion limited current of the oxygen reduction reaction, and β_c,O2 correspond to the cathodic Tafel slopes of oxygen reduction reaction.

At last, they presented the total expression i_f,DC:where where β_c,H2O correspond to the cathodic Tafel slopes of water reduction reaction, respectively. i_corr,DC,H2O represents the part of the water reduction to the overall corrosion current density.

Another way to model the effect of AC on the corrosion process was based on the equivalent electric circuit which is shown in Figure 2. The advantage of this method is that both the double layer capacitance and the solution resistance could be considered, making the model a more realistic representation of the AC-induced corrosion phenomenon. With the assumption that both of the anodic and cathodic processes were under activation control, Xiao and Lalvani⁶⁶  proposed a differential equation according to the distribution of current and voltage in the equivalent electric circuit:

To solve this differential equation, they linearized its exponential components under the condition that the applied potential was relatively low. Later, Zhang, et al., solved the same equation with the perturbation method, but the equation was still linearized.⁷⁴  The validity of the linear approximation method was discussed by Ibrahim, et al.,⁷²  who compared its results to the solution of the numerical method and stated that the numerical solution was more reasonable than the linear one. They also developed Equation (45) into a more realistic one by considering the cathodic processes composed of the reduction of water and that of dissolved oxygen partly controlled by diffusion. The differential equation they proposed was given as:where

In the differential equation model, the role of the equivalent electric circuit was to demonstrate the distribution of current and voltage. The essential principle was still the response of the electrochemical process to the imposed AC signal. In fact, almost all of the proposed models were based on the faradaic rectification. Even though the models based on the equivalent electric circuit is powerful, the derived differential equations are difficult to solve analytically without any linear approximation; therefore, the numerical methods were often adopted, consequently, no expressions of the solutions were available.

It should be noticed that even though the equivalent electric circuit was used to model the AC-induced corrosion, the theory developed for electrochemical impedance spectroscopy (EIS) analysis, which is also based on the equivalent electric circuit, can not be used to describe the AC corrosion behavior. Because in the analysis of EIS, it is assumed that the electrode is at equilibrium and AC signal is in small amplitude, such as ±10 mV, under which condition the polarization curve of the electrode should be linear. However, during AC corrosion, the potential amplitude is much higher than that.³⁹  Therefore, the rationalization of adopting equivalent electric circuit to build AC corrosion model is suspicious and should be discussed. Even if it is proved that the equivalent electric circuit is suitable to represent the AC inducing corrosion process, another question still exists. The used equivalent electric circuit is the simplest one without considering the effect of corrosion products form on the surface, which could influence both of the distribution and magnitude of the resistance and capacitance components in the circuit,⁷⁵  making the model more complicated. Moreover, to make the equivalent electric circuit more realistic, capacitance is normally replaced by constant phase angle element, which could make the model even more difficult to analyze.^39,76 

Based on AC corrosion models, the effects of AC on the shift of corrosion potential and the corrosion rate were revealed. Moreover, the models could provide a critical way to quantificationally analyzed the effect of parameters, such as the double layer capacitance and the solution resistance, on the AC corrosion process. In general, the models could establish an electrochemical fundamental to explain the AC corrosion process. However, most of the models have not been validated through the experiment. Only Ghanbari, et al.,⁶⁷  performed a series of experiments to verify their model. But the experiments were performed only in one type of solution lacking generalized verification. Hence, more experiments with the solutions even the soils that possess different composition and pH should be performed to test the models. Moreover, the derivation of these models more relied on mathematical treatment instead of emphasizing the essential effect of AC on the corrosion process. In addition, even though the waveform of AC was reported having influence on the corrosion process, it was not considered in any of the proposed models. Furthermore, the kinetic parameters, such as the anodic and cathodic Tafel slopes, were independent of AC magnitude in models, which is not consistence with some of the experiment results, in which the Tafel slopes varied under the effect of AC. Besides, the effect of environmental condition, such as the roles of OH⁻, on the AC corrosion process is not considered in all of the proposed AC corrosion models, indicating a coupled reactive-transport model is still required.

The mechanisms and models of AC corrosion were critically reviewed, presenting an overview of the development history of the AC corrosion theory. The following was concluded.

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  Even though many mechanisms were proposed, none of them can explain all aspects of the AC corrosion behaviors. Therefore, no consensus has been reached in AC-induced corrosion mechanism. It tends that the AC corrosion should be studied by combining different mechanisms.

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  The AC corrosion process is not comprehensively understood and still need further study and improvement. More experiments should be conducted to reveal how AC influences the corrosion kinetic parameters, such as anodic and cathodic Tafel slopes. Moreover, it is recommended that some new rapid measurement techniques should be developed to study the electrochemical response of the corrosion system in the positive and negative period of a single cycle of AC on the whole specimen. In addition, the evolution of the morphology on the corroded surface could be furtherly studied by the real-time imaging technique to investigate the interface modification of AC.

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  The significant improvement of AC corrosion models was the assumption underpinning them becoming more and more realistic. However, no new concepts were introduced. And it appears that most of the proposed models were not verified by experiments quantitatively. In addition, the waveform of AC was not considered in any of the proposed models. Furthermore, the kinetic parameters remained constant in all of the models, which is not consistence with some of the experiment results that the Tafel slopes varied under the effect of AC. In order to consider the effect of environmental condition, a coupled reactive-transport model is still required.