AC Induced Corrosion of Underground Steel Pipelines . Faradaic Rectification under Cathodic Protection : II . Theoretical Approach with Electrolyte Resistance and Double Layer Capacitance for Bi-Tafelian Corrosion Mechanism

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Introduction
In Part I of this series of papers, 1 the mathematical development and digital simulations of AC voltage enhanced corrosion (or simply AC corrosion) on buried steel structures under the cathodic protection were reported for highly conducting electrolyte medium, that is, the contribution of the electrolyte resistance is negligible.In contrast, the cathodic process is more realistic than other publications, [2][3][4][5][6][7] since two reactions, the reduction of dissolved oxygen with partially diffusion limited process, and the water reduction according to a Tafel law were considered. 1,8In this part, we will examine the AC enhanced corrosion when the solution resistance is no longer negligible.Then, the sinusoidal perturbation is not applied directly at the interface, and the ohmic potential drop through the soil electrolyte resistance R E should be taken into account.Besides, the presence of the double layer capacitance C d plays a bypass circuit, diverting a substantial proportion of the high frequency AC current transferred to the interface.The equivalent electrical circuit will then be represented by R E -(C d //R p ) (cf. Figure 1b on Part I). 1 Therefore, there is a strong coupling between the actual total current flowing across the interface and the overall potential.Main entities that determine the electrical behavior of the interface are, in addition to the reaction rate equations, on one hand the electrolyte resistance, and on the other hand the double layer capacitance.
Montella 9 studied recently the validity of the polarization resistance (R p ) measurements in presence of AC signal by analyzing the nonlinear response to a low frequency sinusoidal voltage perturbation on the electrochemical reaction.He was interested particularly to the maximum voltage amplitude admissible for determining accurately the local polarization resistance.For this sake, he examined an electrochemical system controlled by the activation energy (Tafelian system) at the steady-state in presence of ohmic drop.Since he considered the steady-state conditions, he neglected the contribution of the charging current of the interface capacitance, and then the current-potential response can be modelled by using the Lambert W function.
Though highly interesting, his results cannot be applied to the AC enhanced corrosion of metallic equipment subjected to AC voltage perturbation because of the relatively high range of AC frequency operated on the industrial equipment, typically 16⅔, 50, or 60 Hz, thus the contribution of the charging current of the interface capacitance is no longer negligible.
A model describing the corrosion of metals subjected to its enhancement induced by alternating voltage, taking into account the electrolyte resistance R E (Ω cm 2 ) and the double layer capacitance C d (F cm -2 ), was developed by Lalvani and his group. 6,7In their approach, they considered an electrochemical interface controlled by activation processes for both anodic and cathodic reactions (bi-tafelian corrosion mechanism).These authors supplied an approximate solution valid in principle for relatively small amplitude of the AC perturbation.The validity of this approach will be discussed later in this paper.
As explained in Part I, 1 the enhancement of the corrosion current density is due to the faradaic rectification as consequence of nonlinear property of the currentpotential characteristics.From a qualitative point of view, the faradaic rectification is always present, and it is significantly influenced by both the electrolyte (solution) resistance R E and the double layer capacitance C d .
The complete model, without any approximation will be presented in this Part II for bi-tafelian corrosion mechanism.The anodic dissolution of steel is assimilated to that of iron.

Anodic reaction: Fe
The cathodic process is the reduction of water molecules.

Cathodic reaction: 2H
In Part III, we will deal with the corrosion mechanism where cathodic process is constituted of the reduction of dissolved oxygen partly controlled by diffusion and that of the water following a Tafel law; this mixed mechanism will be named, for simplicity, three reaction corrosion mechanism.
The corrosion current density may be markedly higher than that of the diffusion limiting current density of dissolved oxygen, because of soil aggressiveness, or due to a hindering of the diffusion process by the presence, for instance, of sand at the vicinity of corroding area.In a such case, the diffusion reaction is negligible with respect to the overall cathodic current density.The corrosion mechanism will then be assimilated as a bi-tafelian one.Nonetheless, the corrosion kinetic parameters corresponding to mildly corrosive soil will also be examined, to highlight the difference between bi-tafelian and three reaction corrosion mechanisms.This aspect will be shown in detail in the following paper, Part III of this series.

Experimental
It is worth to recall that, as mentioned in Part I, 1 buried steel pipes are covered by a thick organic coating, and the corrosion protection is completed by the cathodic protection.It also postulated, as in Part I, 1 that the effect of stray AC signal is limited to its kinetic effect, and no side effect such as the modification of soil chemistry close to coating defect, such as pH change or formation of calcareous or magnesium-calcareous deposits will take place.In the corroding system where the electrolyte resistance is no longer considered negligible, the electrode potential E (in V) at the steel/electrolyte interface and the applied potential U (in V) with a reference electrode located at certain distance from the exposed steel area at coating defect are different.
where I stands for the current density (A cm -2 ).In presence of an AC strain potential DU at the pulsation ω (radian s -1 ) as a function of time t (s) is expressed by: With several mathematical operations, similar to those presented in Part I, the final equation to be solved, for the corrosion mechanism involving anodic and cathodic reactions, both of them controlled by the activation energy (bi-tafelian corrosion mechanism), can be expressed as follows: (5)   In this equation, U 0 represents DC applied potential corresponding to the cathodic protection potential, thus it will be represented also by U CP .The term between braces Vol. 27, No. 3, 2016   is the current passing through the interface involving that of the charge of double layer capacitance, the first term inside the braces.
Before resolving this equation, we will see the validity limit of the so-called "linear approximation" reported by Xiao and Lalvani, 7 regarding specifically the effect of the peak-voltage of AC perturbation, DU on the corrosion kinetics.

Comparison of linear approximation and fully developed equation on I-U curves under AC perturbation
With linear approximation, the current and potential in presence of the AC perturbation can be expressed analytically, whereas without approximation, digital resolutions are necessary.As presented in Part I, 1 when the electrolyte resistance is neglected, the faradaic rectification can be calculated by using a series of the modified Bessel functions of the first kind of order k (k = 0 to ∞).This is no longer possible in the present situation, then we employed Mathematica ® software. 10This method is rather time consuming, for instance, with a desktop PC equipped with Intel i5-4440 running at 3.1 GHz, two to three hours are necessary to obtain one curve with 500 points.With the linear approximation, one curve is obtained almost instantaneously.
Using digital integration, the mean DC interfacial potential E AV and the capacitive current were evaluated, then the average value of the faradaic anodic and cathodic current (I Fa,AV and I Fc,AV , respectively) flowing through the interface were estimated.In the model adopted to simulate the interface, the double layer capacitance is assumed to be equivalent to a pure capacitance, the mean DC capacitive current is therefore nil.Consequently, the mean DC total current transferred to the interface is equal to the sum of the mean DC anodic and cathodic faradaic currents.Hence, the DC mean polarization curves, U AV -I AV , were yielded for different values of the parameters as it was carried out in Part I. 1 The averaged values, marked by subscript "AV" is obtained by integrating through an entire period of the AC signal, both the interfacial potential and the sum of the faradaic anodic and cathodic currents signals.
Table 1 displays some corrosion kinetic parameters used for the comparison of two methods, linear approximation and the complete resolution.These values are essentially the same to those employed in Part I with negligibly small electrolyte resistance. 1 Figure 1 shows the comparison of the mean DC polarization curves of bi-tafelian corrosion mechanism obtained by the "linear approximation" method 7 and those without approximation by digital resolution.Note that the curve for DE = 0 V is the polarization curve for control obtained analytically.These curves allows one to evaluate the faradaic rectification effect on the corrosion potential shift (DE corr,AV = E corr,AV -E corr,0 ) and on the enhancement of corrosion current density (I corr,AV ).
It can be seen that, with linear approximation (Figure 1, dashed line), the anodic and cathodic reaction rates keep the same Tafel constant (expressed as V -1 ) whatever the value of the peak voltage of the AC perturbation.In other terms, if the correction of the ohmic drop term (E = U − R E I F ) was applied, not illustrated here, the Tafel slope becomes steeper, when DU increases.Even, for high current density, the polarization curves exhibit an overhanging shape "⊂" for cathodic branch and that of "⊃" shape for anodic branch.In other terms, two different current densities can be observed for a given Comparison of polarization curves calculated with the linear approximation (dashed line) and that of full development of equation 5 (solid line).The AC peak voltage DU is indicated near each curve.The corrosion parameters used are displayed in Table 1.
-1.0 -0.9 -0.8 -0.7 -0.6 10 potential, i.e., multiplicity of the steady states leading to an instability of the system under potential regulation.The ohmic drop allowed the stabilization of the system. 11n the contrary, with the calculation without approximation (solid line), the Tafel slope becomes significantly gentler when DU becomes greater.If the ohmic drop correction was applied, these curves show the Tafel constants for anodic and cathodic reactions (b a and b c , respectively), almost equal to those used for simulation calculations.One may remark also that, in presence of AC signal, the linear approximation overestimates markedly the corrosion potential shift and the corrosion current enhancement for high DU values.These effects are illustrated in more detail in Figure 2.
The dependence of the corrosion potential shift DE corr,AV , with respect to the peak voltage DU, for the two calculation methods, are shown in Figure 2a.It may worth to recall that, since the overall DC current is naught at the corrosion potential even in presence of the AC perturbation, U corr,AV = E corr,AV .The corrosion potential shift DE corr,AV is defined as: It can be seen that with the linear approximation, the corrosion potential decreases linearly above DU > 0.2 V whereas the complete resolution of equation 3 without approximation showed an inflexion point at about DU = 1 V. Above this point, the DE corr,AV variation is significantly slower with respect to DU.
Figure 2b shows the enhancement of the corrosion current density in presence of AC signal.In this Figure, the corrosion current density is normalized according to the following equation: (7)   For DU > 0.2 V, the corrosion current density increases exponentially (linear in semi-logarithmic scale) when evaluated by the linear approximation.On the contrary, when it was determined by digital simulation without approximation, two curves overlap almost up to DU ca.0.7 V, then diverge each other.The slope of log (<I corr,AV >) changes at about DU = 1 V, approximately at the same potential for DU − log (<I corr,AV >) and DU − DE corr,AV curves.At DU = 6 V, I corr,AV is increased by 750 times with respect to I corr,0 , according to the fully developed equation.The overestimation of the corrosion current density under AC voltage enhancement between two calculation methods is illustrated in Figure 3.
In Figure 3, I c,Lin and I c,Full represent, respectively, the averaged corrosion current density evaluated by linear approximation according to Xiao and Lalvani 7 and that by numerically resolving equation 5.It can be remarked that the overestimation by linear approximation reaches 10% at DU = 0.67 V, twice as much at 1.05 V, and ten times as great at 1.52 V.If the maximum error induced by the linear model is fixed arbitrary at 10%, the use of this method will be limited to DU = 0.67 V. AC peak-voltage of several volts is sometimes encountered in the field measurements.In other terms, for a reasonable evaluation of the corrosion enhancement in presence of AC signal, the use of the complete mathematical resolution by digital simulation is necessary.

AC corrosion enhancement with complete mathematical resolution
Using the numerical resolution of the differential equation related to the instantaneous interfacial potential, we will illustrate, now, the faradaic rectification induced on the corrosion of steel pipeline buried in a mildly or a highly aggressive soil.The difference between them is characterized mainly by the corrosion current density in absence of AC signal I corr,0 , and by the electrolyte resistance R E of the soil.

Case 1: mildly aggressive soil
The corrosion kinetic parameters used for the digital simulation for a mildly aggressive soil are grouped in Table 2.These values were mainly determined from the field experiences collected by the French gas and electricity utility company (GDFSUEZ).The results of simulation calculation will be shown first for the corrosion kinetic parameters presented in Table 2, then in reduced variables to be applied for more general cases.

Presentation with particular value of parameters
The corrosion current density of 4 µA cm -2 corresponds to the thickness-loss of steel, by uniform corrosion, of 46 µm y -1 .The electrode impedance according to the corrosion kinetic parameters displayed in Table 2 is presented in Figure 4 in the Nyquist plot.On this Figure, the frequency of AC signal f adopted (50 Hz) leading to the corrosion enhancement is indicated with an arrow in the insert corresponding to the high frequency part of the impedance in enlarged scale.It can be noticed that at this frequency, the impedance is essentially determined by the electrolyte resistance R E and double layer capacitance C d .The real part due to the contribution of the faradaic impedance represented by the diameter of the Nyquist plot is negligibly small (0.27 over 5927 Ω cm 2 ).
Figure 5 shows the polarization curves obtained for various DU values with solid lines.It can be remarked that, under the influence of the AC voltage perturbation, the Tafel constant (b in V -1 ), in absolute value, decreases when DU becomes greater (i.e., gentler Tafel slope), for both anodic and cathodic processes, as indicated above for Figure 1.As mentioned above, once the ohmic potential drop was corrected, the Tafel constants of anodic and cathodic   2, and dashed lines for highly aggressive soil with the corrosion kinetic parameters in Table 3.
Highly aggressive soil branches are almost equal to those given in Table 2. On the other hand, it can be seen, as expected, that E corr,AV shifts towards more negative values since b a > −b c , and I corr,AV increases with DU as predicted in Part I, where the contribution of the electrolyte resistance was neglected. 1ow the faradaic rectification will be related to the double layer capacitance C d , and how it will be simultaneously influenced by the corrosion current density I corr,0 determined without AC signal?These effects are illustrated in Figure 6.
Figure 6A depicts the variation of the corrosion potential DE corr,AV = E corr,AV − E corr,0 with respect to the AC amplitude DU.The corrosion potential shift DE corr,AV decreases steeply at low DU, then markedly slowly (Figure 6A, curve a) as it was observed in Figure 2a.The double layer capacitance C d is linked in parallel, and then this element intervenes in the faradaic rectification.When C d value is increased, from 100 to 250 µF cm 2 , the corrosion potential shift is presented by curve b in Figure 6A.Actually, one observes that the same DE corr,AV is obtained for higher DU as indicated by an arrow.The double layer capacitance shunts the AC perturbation, and then the actual perturbation signal at the interface becomes smaller.
One may remark also that these two curves approaches when DU increases.At high DU, the current density becomes greater, that is, the mean polarization resistance R p,AV (= DU/DI AV ) becomes smaller.The actual AC amplitude at the interface will then diminishes due to the "bypass" effect brought about by the circuit constituted of R E −C d //R p,AV .Now, let us see the effect of the corrosion current density.When I corr,0 is increased from 4 to 100 µA cm -2 , the results of calculation are presented by curves a and c in Figure 6A.The difference of DE corr,AV between these two curves is small at low DU.However, the difference appears clearly when DU becomes greater than 1 V.At low DU, C d plays predominant role on the faradaic rectification, whereas at high DU, R p,AV becomes smaller, then it comes to impact on E corr,AV , leading to a decrease of the corrosion potential shift.If C d and I corr,0 both increase (from 100 to 250 µF cm -2 and 4 to 100 µA cm -2 , respectively), the comparison of curves a and d in Figure 6A shows the combination of both effects, as observed on individual effect of these two parameters.Now, let us see the influence of the peak AC amplitude, DU, on the averaged corrosion current density I corr,AV due to the faradaic rectification.As the increase of this current density is rather huge, several orders of magnitude, we will present it in logarithmic scale.Curve a (Figure 6B) shows S-shaped feature.log (I corr,AV ) increases slowly for small DU values, up to DU ca.0.2 V, then linearly to DU ca.2.5 V in semi-logarithmic scale.Beyond this AC potential, the increase of I corr,AV becomes slower.When C d is increased from 100 to 250 µF cm 2 (curve b, Figure 6B), the slope of log (I corr,Av ) − DU is much gentler indicating a smaller corrosion enhancement effect of the faradaic rectification.Now, if I corr,0 is increased, from 4 to 100 µA cm -2 , curve c in Figure 6B exhibits almost parallel shift with respect to curve a in Figure 6B (in semi-logarithmic scale) up to DU ca. 1 V, and beyond this value the two curves approach.The same remark can be formulated when both I corr,0 and C d are augmented.The contribution of C d to the faradaic rectification is therefore strongly dependent on the averaged current density flowing through the interface, since this capacitance is linked in parallel to R p,AV , which becomes smaller with the increase of I corr,AV .Note that, when the corrosion process progresses, an accumulation of corrosion products, namely, carbon, iron oxides, and manganese ones, leads to an increase of the true surface area of electrolyte/ electronic conducting species, sometimes called as roughness factor, consequently the C d value becomes greater.With time, therefore, DI corr,AV becomes smaller for a given DU as can be noticed by the comparison of curves a and b in Figure 6B.
In what extent the cathodic protection will be effective in presence of AC signal? Figure 7 presents the partial  2).As stated above, it was postulated in this calculation no change of the electrochemical parameters characterizing the steel/soil interface induced by the corrosion process itself.If, arbitrarily, the maximal admissible DC anodic faradaic current density I Fa,AV , i.e., the corrosion current density under cathodic protection, is assumed to be 10 µA cm -2 , which corresponds to a uniform thickness-loss of 116 µm y -1 , cathodic protection potential (U CP ) of ca.-1 V is necessary for DU = 1.5 V and more than −1.2 V for DU = 2 V.For higher amplitude of AC signal (for instance DU = 2.5 V), an efficient cathode protection (according to the protection criteria arbitrarily considered here) is no longer possible, because in the cathodic range beyond U CP = −1.2V the hydrogen evolution reaction becomes the major cathodic process, i.e., the cathodic current density becomes higher than 1.5 mA cm -2 when U CP is in the range of U CP = −1.2V.This high rate of hydrogen evolution reaction in presence of the AC perturbation may make the soil pH at the vicinity of corroding area highly alkaline, and may prevent any stabilization of an iron oxide protecting layer at steel surface.This aspect of the AC corrosion mitigation challenge on coated and cathodically protected underground pipelines will be specifically addressed in Part IV of this series of papers.

Faradaic rectification effects in generalized presentation
As in Part I, we will calculate the faradaic rectification effect with dimensionless variables. 1For practical use of these Figures (8, 9, 11, and 12) for a given system, therefore, it is sufficient to determine, in absence of AC signal, the corrosion parameters (I corr,0 , E corr,0 , b a , and b c ) which allow converting Figures below (8, 9, 11, and 12) to dimensioned curves, in V and A cm -2 .
The dimensionless peak voltage of AC perturbing signal <DU> was represented by dividing DU by so called Stern-Geary coefficient B SG . 12) In the example given here, if the corrosion current density without AC signal is fixed to be equal to the case illustrated above, B SG remains constant and equal to 23.6 mV.Besides, instead of individual Tafel constants, their ratio r was used.(9)   Note that the value of r is positive.The reduced corrosion potential shift <DE corr,AV > in presence of AC signal was defined as follows: (10)   Figure 8 illustrates the effect of r ratio on the corrosion potential shift.In this calculation, the kinetic constants used correspond to those presented in Table 2 except for the two Tafel constants b a and b c .
On Figure 8, it can be noticed firstly that <DE corr,AV > are symmetrical for r and 1/r with respect to the <DU>-axis as it was also reported in Part I for bi-tafelian corrosion mechanism. 1Secondly, these curves exhibit a peak which E corr,0 , I corr,0    was not observed in Part I where the electrolyte resistance and consequently the double layer capacitance were neglected. 1No such behavior was observed neither in Figure 2a with fully resolved equation 5 expressed with the dimensioned variables.The peak observed is, indeed, only due to the fact that these curves are presented with respect to dimensionless variables, according to equation 8 and equation 10.If <DE corr,AV > was multiplied by DU, i.e., dimensioned magnitude, monotonically decreasing (when r < 1) or increasing (when r > 1) curves, were observed.As this was the case also in Part I, when r = 1, no corrosion potential shift was observed. 1It can also be noted that more asymmetrical the polarization curve is, the greater the corrosion potential shift is. Figure 9 shows the relative variation of the reduced corrosion current density, <I corr,AV >, for various r values with respect to the normalized peak amplitude of AC signal <DU>.
As it was observed in Part I for the system for bi-tafelian mechanism with a negligibly small electrolyte resistance, the most marked corrosion enhancement was observed for r = 1. 1 Also, as observed in Part I, when considering a given r ratio value and its reciprocal value (1/r), the two curves are completely superimposed.On this Figure, one can remark also an inflexion point depending on the r ratio.These curves are different to those observed in Part I where the electrolyte resistance R E is neglected.Indeed <I corr,AV > increased linearly with <DU>. 1 The ohmic drop term and the attenuation of AC signal make therefore the corrosion enhancement smaller.For high <DU> values, the corrosion current enhancement with increasing values of <DU> attenuate markedly.The electrolyte resistance shows therefore a beneficial effect on the AC induced corrosion.Now, let us see the faradaic rectification effects on steel pipe buried in a highly aggressive soil.The kinetic constants used for this system, based upon the field experiments, are given in Table 3.In this Table , it can be noticed that the "free corrosion" current density is considered as 37.5 times as great as in a mildly corrosive medium presented above.In other terms, the corrosion rate corresponds to 1.74 mm y -1 for the uniform thickness loss.The open circuit corrosion potential is 0.1 V more positive, and b a is ca.1.66 fold greater than that used in a mildly aggressive soil.The solution resistance is slightly smaller whereas the double layer capacitance is 1.5 times as great.

Presentation with particular value of parameters
The polarization curves obtained by digital summations of equation 5 are superimposed in Figure 5 above.It is important to emphasize that, in the conditions used here, DU has no significant effect on the faradic rectification for high overvoltage.All curves tend to overlap for U < −0.75 or U > −0.4 V (not illustrated here).The anodic or cathodic reaction rate becomes faster, then R p,AV becomes smaller.As a result, the amplitude DU of the AC perturbation is damped essentially through the electrolyte resistance.The effective AC amplitude DE at the steel/electrolyte interface becomes therefore small.In this steady-state plot, the double layer capacitance does no longer intervene markedly in the electrical characteristics, and then only the electrolyte resistance and the polarization resistance are linked in series.For instance, for the slope of polarization curve, I F,AV -U (Figure 5) at the current density equal to 1 mA cm -2 , for DU = 2 V, one yields the apparent polarization resistance at the interface R p,AV = 14.8 Ω cm 2 .The effective amplitude  of AC signal, DE is 12.8% of applied DU, i.e., DE = 0.26 V.When DU = 3 V at the same potential (−0.802 V), the I F,AV is −0.93 mA cm -2 and the slope of the polarization curve gives the polarization resistance, including the electrolyte resistance equal to 105.8 Ω cm 2 .Then the R p,AV is evaluated to 5.8 Ω cm 2 .Thus, 5.5% of applied AC signal is effectively modulating the interface potential, i.e., 0.165 V.Because of higher corrosion enhancement, the effective AC signal is surprisingly small in this case.This is why the averaged current density depends little beyond I F,AV equal to 1 mA cm -2 .
Figure 10 presents the anodic component of the averaged current density in presence of various AC amplitudes.The open circuit condition without AC signal in this highly aggressive soil is indicated by a dot.As above, if the admissible current density is arbitrarily assumed to be equal to 10 µA cm -2 , one can see that, upon this protection criteria, U CP of ca.−0.8 V will be necessary to protect the steel pipe from the corrosion in absence of AC signal.At U CP = −1.2V, the cathodic protection will be efficient up to DU ca.0.75 V.
We will see now AC corrosion enhancement with reduced scale to generalize the results of calculations.

Faradaic rectification effects in generalized presentation
Figure 11 shows the reduced corrosion potential shift <DE corr,AV > under AC perturbation for various r values.As this was the case for a mildly aggressive soil, the <E corr,AV > is symmetrical with respect to the <DU> axis for two reciprocal values of r.A peak of <E corr,AV > change was also observed, and this feature is actually due to the dimensionless presentation used here, that is the potential shift divided by the peak AC amplitude DU.If E corr,AV was presented under dimensioned form, a monotonous change was observed.
For small <DU> values, <DE corr,AV > is greater when compared with the results illustrated in Figure 8.As can be remarked on Figure 5, the corrosion potential shift for a given DU is more marked in highly aggressive soil.From curves a and c of Figure 6A, the increase of C d from 100 to 150 µF cm 2 , may decrease the corrosion potential shift.The increase of I corr,0 from 4 to 150 µA cm 2 may induce also the similar effect.Therefore, a greater <DE corr,AV > change observed in this medium is essentially due to a smaller R E value used.Indeed, the comparison for the solution with negligible small electrolyte resistance presented in Part I corroborates this remark. 1he AC corrosion enhancement induced by the faradaic rectification by AC signal is presented in Figure 12.
The averaged corrosion current density <I corr,AV > under AC signal is expressed as above in a dimensionless entity, as can be seen in the equation 7.In semi-logarithmic scale, there are two segments.At low <DU>, the reduced corrosion   The cathodic protection potential necessary to mitigate the corrosion of buried steel pipe in presence of AC voltage, U CP was tentatively evaluated by plotting the anodic contribution of the overall faradaic current I Fa,AV .In this calculation, some restricted hypotheses were used, namely the AC corrosion process itself does not alter the kinetics parameters describing the steel/soil interface behavior.With typical corrosion parameters used in this paper, for mildly aggressive soil, and AC peak amplitude less than 2 V, it appears that U CP values in the range of −1.2 V would be sufficient.In contrast in a highly aggressive soil, with the same U CP values, the cathodic protection would only be efficient for DU < 0.75 V.
When the diffusion of dissolved oxygen is negligible compared with the reduction of water molecules, which is the case, for instance, in contact with a highly corrosion aggressive soil or when this cathodic reaction is strongly hindered by the presence of sand close to the defect area, the cathodic reaction due to the reduction of dissolved oxygen is small.In this situation, the cathodic current is essentially determined by the reduction of water leading to the hydrogen evolution validating the corrosion mechanism adopted here, i.e., a bi-tafelian one.
In the next part of this series of papers, we will present the results of digital simulations for the corrosion mechanism involving anodic dissolution obeying a Tafel law, and mixed kinetics for cathodic reactions, one for the reduction of dissolved oxygen partly determined by the convective diffusion and another describing the mechanism of cathodic reduction of water, postulated to follow a Tafel law.
b a : Tafel constant for anodic reaction; b c : Tafel constant for cathodic reaction; R E : electrolyte resistance; C d : double layer capacitance; f: frequency of AC signal; R p : polarisation resistance ; r: −b c /b a .

Figure 2 .Figure 3 .
Figure 2. Comparison of the corrosion potential shift in (a) and the enhancement of the corrosion current density in (b) for simulated data obtained by the two approaches, linear approximation and full development of equation 5.

Figure 4 . 2 .Figure 5 .
Figure 4. Impedance spectrum of corroding steel electrode in a mildly aggressive corrosive soil.Some frequencies in Hz are indicated near the impedance diagram.The corrosion kinetic parameters are presented in Table2.
Vol. 27, No. 3, 2016   anodic contribution in the total faradaic current density I Fa,AV for various AC amplitudes.The open circuit corrosion potential and the corrosion current density without AC signal (E corr,0 , I corr,0 ) are indicated by a dot on this Figure.

Figure 7 .
Figure 7. Averaged faradaic anodic components I Fa,AV in presence of AC signal of various amplitudes to illustrate the theoretical DC cathodic protection potential that would be necessary to mitigate the AC corrosion of carbon steel in mildly aggressive soil for the bi-tafelian corrosion model.

Figure 8 .
Figure 8.The variation of dimensionless corrosion potential shift <DE corr,AV > in a mildly aggressive soil with respect to the reduced AC amplitude <DU> for various r values indicated as parameter.Corrosion kinetics parameters are those given in Table 1 except b a and b c , but (b a − b c ) was kept constant.B SG = 23.6 mV.

Case 2 :
highly aggressive soil

Figure 9 .
Figure 9.Effect of faradaic rectification on the averaged corrosion current density, <I corr,AV > = I corr,AV /I corr,0 with various -b c /b a ratios with respect to the dimensionless AC amplitude (corrosion parameters are expressed in Table 2, except for b a and b c but (b a -b c ) is kept constant).B SG = 23.6 mV.0 500 1000 1500 0

Figure 12 .
Figure 12.Effect of faradaic rectification with various −b c /b a ratios r on the enhancement of the corrosion current, as a function of the peak AC voltage in a highly aggressive soil, with regards to the corrosion process.B SG = 17.4 mV.

Figure 11 .Figure 10 .
Figure11.Effect of <DU> upon <DE corr,AV > in a highly corrosive soil for various r ratios.The corrosion kinetic parameters are given in Table3.B SG = 17.4 mV.
0 , I corr,0 Vol. 27, No. 3, 2016 monitoring software of almost all commercial devices by parameter regression procedure.R E and C d , also important parameters will be determined by electrochemical impedance spectroscopy technique collected at E corr .The Figures presented in this paper do not allow, however, determining directly their effect, but some interpolation with various Figures presented in this paper may indicate general tendency.

Table 1 .
Values of some corrosion kinetics parameter used for calculating I-U curves and several useful parameters 0 : corrosion potential without AC signal; I corr,0 : corrosion current density without AC signal;

Table 2 .
Values of some corrosion kinetic parameters used for calculating I-U curves for mildly aggressive corrosive medium

Table 3 .
Values of some corrosion kinetic parameters and pertinent ones used for calculating I-U curves for highly aggressive corrosive medium