Applicability of AC impedance method for measuring time-variant corrosion rate to cracked and crack-repaired reinforced concrete

Several models of the time-variant corrosion rate (CR) have been proposed to predict the service life of reinforced concrete (RC) structures based on polarization resistance (PR) monitoring. However, few of them have verified the accuracy by comparing the estimated corrosion weight loss (CWL) obtained from the PR monitoring with the actual CWL. In addition, few studies have measured the time-variant CR on crack-repaired RC specimens, and the evaluation of repair efficacy is clearly an essential step in the maintenance of the structures. This study aims to clarify the applicability of the alternating current (AC) impedance method, one of the PR methods, to cracked and crack-repaired RC. The CRs in such RC in an accelerated corrosion environment were measured every one to three months over three years using the AC impedance method. To confirm accuracy, the corroded area and CWL of the steel bars were measured. The results clarify that the measurement area of the AC impedance method was almost equal to the corroded area, which indicates that the corrosion area should be calculated in addition to the CWL for accuracy verification. The results also show that the AC impedance method can be applied to cracked and crack-repaired RC for measuring time-variant CR. The CR increases rapidly after crack initiation but becomes constant with time, and early crack repair can delay corrosion progress in cracked RC to the same extent as in uncracked concrete.


Introduction
In recent years, the number of aging reinforced concrete (RC) structures has been increasing, and the need has arisen for efficient repair and reinforcement measures to extend the service life at a low cost. RC structures near the sea particularly tend to deteriorate rapidly owing to contact with chlorides via direct seawater or seawater splashing, while deicing agents made of calcium chloride can also be a potential source of chloride. Steel corrosion in concrete occurs when the chlorides penetrate the concrete and reach the steel bar. The bar diameter is decreased due to the corrosion progress, which in turn decreases the load resistance of the RC structure. Therefore, it is crucial to measure the time-variant corrosion rate (CR) of steel bars in concrete to predict the service life of the structure.
Some studies have proposed prediction models of CR based on various environmental conditions, material properties, and exposure time before crack initiation [1][2][3][4][5][6][7]. Environmental conditions include chloride ion content, humidity, and temperature, and material properties include the water-binder ratio (W/ B), binder type, and concrete cover thickness on the steel. The CR also decreases with time because corrosion products gradually form in an interfacial transition zone between steel bars and concrete [2, 4,5].
Evaluation of the time-variant CR after cracking is becoming increasingly important. Steel corrosion in concrete may accelerate owing to bending cracks induced by mechanical causes such as earthquakes. In such cases, a decision needs to be made whether to simply fill the cracks or to provide other corrosion preventions. To address this issue, a method for evaluating the CR of steel bars after crack initiation is required. Several studies have experimentally evaluated the chloride-induced CR of the steel bar in cracked concrete [8][9][10][11][12][13]. These prediction models or factor analyses have been developed using polarization resistance (PR) monitoring.
However, few of them have had the accuracy verified by comparing the estimated corrosion weight loss (CWL) with actual CWL. The estimated CWL is obtained from the time integration of the CR, whereas the actual CWL is obtained by measuring the corrosion weight of a steel bar removed from the RC specimen. Law et al. [14] verified the accuracy of a PR method by actual CWL, but the PR method overestimated the CWL, especially in chloride-induced corrosion. This was because macrocell corrosion had occurred and the predetermined evaluation area did not match the actual evaluation area. Liu and Weyers [2] compared CRs calculated using two types of linear PR method to the mean CR obtained from actual CWL. Both estimated CRs significantly differed from the actual mean CR. Jung et al. [3] verified the accuracy of a PR method using 11 steel bars in concrete corroded for 1 year, but the difference due to the corroded area was unclear. Especially for RC after bending cracking, corrosion tends to progress around the transverse crack [11]. Thus, if a corrosion prediction model is proposed based on PR monitoring without accuracy verification, it may differ greatly from the actual CWL. It is necessary to clarify the verification method by performing the PR method experimentally. In addition, few studies have evaluated the effect of crack repair on the CR, and applicability of PR methods to such specimen remains unclear. Cracks increase the CR because oxygen and moisture can easily penetrate through them. However, it is expected that the CR can be reduced, if the cracks are repaired. Thus, this study verified both the case of introducing bending cracks and crack repair. In addition, as crack shape may affect the CR, the case of corrosion cracks was also verified.
In this study, we measured the time-variant CR of the steel bar in cracked and crack-repaired concrete every one to three months over three years in an accelerated corrosion environment. An alternating current (AC) impedance method, one of the PR methods, was used for the CR monitoring. To confirm accuracy, the corroded area and CWL of the steel bars were measured. Then, the trend of the CR with time of the steel in such concrete was proposed.
2 Experimental procedure

Specimens
A corrosion experiment was conducted to consider the effect of cracks and crack repair on the corrosion progress using RC specimens. Figure 1 shows the geometry of the RC specimens and the arrangement of the electrodes when we applied an AC impedance method.
Each specimen was a 100 mm 9 100 mm 9 376 mm RC prism containing one 400-mm long deformed (ribbed) steel bar (SD295A) with a diameter D = 10 mm. The steel bar was used as received with mill-scale. This steel bar was cast in the specimen such that the minimum depth to the concrete surface c = 30 mm. The test length of the bar was 300 mm (test surface area: 89.5 (cm 2 )), with the remainder at each end coated with epoxy resin. A coaxial cable was connected to the steel bar. Compared to simple insulated copper wires, coaxial cables are less susceptible to external electromagnetic radiation and have the advantage of lower impedance noise. Table 1 provides the concrete mix design and Table 2 provides the experimental parameters of the specimens.
The RC specimens were casted in the laboratory for 28 d in a room at temperature of 20°C and 70% relative humidity by applying external vibration. Slump and air content were within the specified ranges in Table 1. These were cured by seal in the laboratory for 28 d in a room at a temperature of 20°C and 70% relative humidity. After curing, compressive strength tests were conducted on 3 cylindrical specimens with a 100 mm diameter and 200 mm height in accordance with JIS A 1108 [15], and the average compressive strength was 36.7 N/mm 2 . According to the experimental parameters reported in Table 2, a transverse crack was introduced in the test surface of each concrete specimen by applying three-point bending. To check the crack width, a distortion (Pi) gauge was placed at the center of the test surface in line with the steel bar inside the concrete. After the specimen was unloaded, each crack width was confirmed to be within ± 10% of the desired value.
The reported crack widths are the maximum crack widths on the specimen surfaces. The crack widths were larger closer to the center of the specimen and smaller at the edges. Specifically, the value was approximately 0.3 mm at the edge of the specimen when the maximum crack width was 0.4 mm. The depth of the cracks was visually confirmed to be approximately 50 mm.
Then, cracks of some specimens were repaired using an epoxy resin as per the experimental parameters. Crack repair was performed after the cracks were introduced but before the start of corrosion acceleration. The epoxy resin was introduced through low-pressure injection and the resin was a hard, lowviscosity epoxy resin. To ensure that the penetration of salinity progressed from only the test surface of the concrete, coal tar epoxy resin was coated on the remaining 5 surfaces. Then, the specimens were exposed to an environmental load device to accelerate corrosion. This device sprayed the specimens with 3% NaCl water solution for 3 min once a day, while maintaining the surrounding air at 40°C and 80% relative humidity.

AC impedance measurement procedure
Linear PR method has been used because of its simplicity of measurement and ease of interpretation, but with the recent miniaturization and enhancement of equipment, the AC impedance method is also being applied in many cases [16,17]. The method measures the target sample by changing the current frequency using a potentiometer/galvanostat with a frequency response analyzer (FRA). Impedance measurement has been susceptible to noise due to scattering of concrete, but this method is becoming applicable in the field due to the higher accuracy of the equipment. The impedance (Z) plotted on the complex plane are called Nyquist plots. Thus, Z is calculated as where Z Re is the real value and Z Im is the imaginary value of Z. Various equivalent circuit models have been proposed to interpret Nyquist plots [18]. In this study, an equivalent circuit model of RC is assumed as shown in Fig. 2a and b shows examples of Nyquist plots in our experiment. This model is the simplest among the various proposed models for the RC specimen. The capacitance (C dl ) and apparent PR (R p,app ) are formed at the interface between the steel and concrete because the charges from the steel and charges from concrete pore solution are separated by a potential difference, known as an electric double layer [19]. This ion flow is similar to a parallel circuit comprising C dl and R p,app . Here, R p,app is the PR not considering the measurement area of the steel. The PR (R p ) can be calculated by multiplying the R p,app by the measurement area (A) of the steel through which the measurement current flows. Also, the solution resistance of concrete (R s ) is expressed as an additional resistance. The impedance property of the equivalent circuit model is expressed as.
A semicircle with a radius of R p /2 and center coordinates of (R s ? R p,app /2, 0) is shown in the complex plane, as shown in Fig. 2b. Using this procedure, the AC impedance method is able to calculate the R p,app and R s separately. In addition, because the measurement current is expected to flow into the corroded area in the case of localized corrosion, there is no need to consider a complicated equivalent circuit, and the equivalent circuit in Fig. 2a can be used. However, although the Randles circuit was shown to be sufficient in this experiment, complexity may require setting up a more complex circuit for corrosion rate measurements due to non- ideal capacitance behavior and current dispersion effects. For our experiment, the steel bar in the specimen was used as the working electrode (WE), a 50 mm 9 100 mm thin stainless-steel board (SUS304) was used as the counter electrode (CE), the Pb reference electrode (PRE) was the reference electrode (RE), and a high-water-absorption sponge made of polyvinyl alcohol was placed between the CE and the concrete surface, and between the RE and the concrete surface. Note there are two types of CE: one with and one without guard rings. Guard rings were developed to solve the complexity of current dispersion; Sehgal et al. [20] used the method of placement from experimental and analytical work. Kranc et al. [21] used numerical analysis to evaluate the accuracy of the PR method when guard rings are used. However, Hu et al. [22] stated that in localized corrosion, the current is concentrated at the corroded area even with guard rings.
In this study, the CE without guard rings was applied by adopting the experiment setup employed by Liu and Weyers [2], who stated that the PR method without guard rings was able to estimate the CR more accurately. A PRE is an electrode consisting of a lead rod surrounded by a solid electrolyte. It is advantageous as the obtained potential is stable over a long period of time. In addition, since the electrode does not use an aqueous solution, there is no chance of contamination of the concrete due to leakage. The CE was located at the center of the concrete surface and the RE was located 10 mm from the edge of the CE. The concrete was wet (6-8% water content) in every measurement.
In addition, to consider the current dispersion within the specimen, measurements were made using the CEs of 50 mm 9 100 mm and 300 mm 9 100 mm, and the measured values of R p,app were compared. If the values of R p,app are different, the area where the current flows into the rebar is different. On the other hand, when they are equal, it means that the areas are equal.
The measurement conditions were as follows. The potential difference was DV = ± 30 mV. The frequency range of the current was 10 to 10 kHz. The R p,app and R s , were calculated by performing curve fitting using analytical software for electrical measurements. The actual measurement is not an ideal semicircle on the Nyquist plot; it has a smaller shape in the Imag(Z) axis. In this case, curve fitting was performed by shifting the center position of the semicircle in the Imag(Z) axis direction, and R p,app was calculated from the two intersections with the Real(Z) axis. The R p,app was measured every one to three months until the specimens were split for verification. This type of approach is often referred to as monitoring.

Calculation of CR and estimated CWL
From the apparent PR (R p,app ) obtained from the procedure described in the previous section, the corrosion current density i corr (lA/cm 2 ) can be calculated as follows: where B is the constant value (mV) proposed by Stern and Geary [23], R p is the PR (kXÁcm 2 (= R p,app 9 A)), R p,app is the apparent PR (kX), and A is the measurement area of the steel (cm 2 ). The value of B has been calculated experimentally, and some studies have determined it to be 26 mV [24][25][26], so we used this value in this study. Subsequently, the corrosion current density i corr can be converted to the CR (mg/cm 2 /d) by.
where M is the atomic weight of iron (= 55.85 g/mol), F is the Faraday constant (= 96,485 C/mol), and a is a constant that converts from days to seconds (= 8.64 9 10 4 s/d). This equation is based on a halfreaction equation in which iron releases electrons to become divalent iron ions, and calculates the dissolution rate of iron from the electrons generated. Based on this procedure, the PR method can be used to quantitatively evaluate the CR. One of the problems of the PR method is the change in the measurement area owing to the current dispersion in the concrete.
Previous researchers conducted a current-dispersion analysis of concrete before crack initiation using the finite-element method [27], but only a few experimental studies have considered the current dispersion of RC specimens. The CWL was then calculated from the monitoring results by.
where G is the CWL per unit surface of the steel (mg/ cm 2 ), t is the number of measurements, t f is that of the last measurement, R p,app(t) is the apparent PR of the tth measurement (kX), and T (t) is the time from the start of exposure to the t-th measurement (d). Here, we determined CR (0) = 0 (mg/cm 2 /d) and T (0) = 0 (d).
Equation (5) first calculates the area of the trapezoid enclosed by the interval between two consecutive points in time and their corresponding PR values. Thereafter, the CWL is calculated by integrating the area of this trapezoid over the entire corrosion time; this is called the estimated CWL.

Measurement of actual CWL
The procedure for measuring the actual CWL was applied as follows. First, following the exposure period, each specimen was divided into two parts by split loading at the steel bar location. The steel bar was then removed from inside the specimen, and the corroded area and the CWL were measured. Figure 3 shows the calculation procedure of the corrosion area ratio. Six photographs of the removed steel bar, rotated every 60°around its longitudinal axis, were obtained to determine the corroded area. Then, the corroded and non-corroded areas were binarized using the Image J area calculation software [28]. The corrosion area ratio r ca (%) was calculated by.
where S is the total test area of the steel, calculated by multiplying half of the diameter of the bar, D/2, by the test length (30 cm) for all six surfaces, and S corr is the total corroded area, calculated by binarizing the noncorroded and corroded areas in the six images based on the color of the bar, given that the color of a rusted area is changed from silver to red or black. The weight loss due to corrosion was measured by weighing the steel after the corrosion products were removed by immersion in an aqueous solution of 10% diammonium hydrogen citrate at 60°C for approximately 24 h, as per JCI-SC1 [29]. Thereafter, the CWL per unit area at the corroded area DW (mg/cm 2 ) was calculated by.
where W 0 is the weight of the steel before placement (mg), W is the weight of the corroded steel after removing the rust (mg), and W p is the weight of the mill-scale on the steel (mg). Mill-scale is an oxidized film that forms on the surface during the manufacture of a steel bar. The same procedure for measuring CWL was performed on 10 non-corroded rebars with mill scale, and the average weight loss per unit surface area was used as the weight loss per unit area of the mill scale.

Comparison between estimated and actual CWL
The actual CWL is compared to that estimated by the PR monitoring in Fig. 4. As a consideration of the measurement area (A) of the PR method, Fig. 4a shows the comparison when A is set as the total test area (S), whereas Fig. 4b shows the comparison when A is set as the total corroded area (S corr ) of each specimen. The figure shows each specimen type using different plot points, and straight lines passing through 0 and 400 mg/cm 2 representing DW = G, DW = 0.5 9 G, and DW = 2.0 9 G. The C-N and C-R specimen types each had three different crack widths (0.2, 0.4, and 0.6 mm), but are not distinguished here. Thus, the G underestimated the DW by a factor of two or more when S was used, whereas the G estimated DW with relatively good accuracy regardless of the specimen parameters when S corr was used. The results clarify that the measurement area A was almost equal to the corroded area S corr . Figure 5 shows the results of comparative measurements of R p,app with changes in the counter electrode area. R p,app did not change with counter electrode area regardless of the factor (N-N, C-N, and C-R) and regardless of the corrosion state. This means that the measurement currents flowed in the same steel surface area even if the counter electrode area changed.
For consideration of the relationship between the measurement area A and corroded area S corr , Fig. 6 shows a conceptual diagram of the current dispersion in the cases of almost non-corroded and locally corroded specimens. When the steel bar inside the concrete is almost non-corroded, the current disperses over its entire surface because the PR of the steelconcrete interface is uniform. However, in the case of localized corrosion, the measurement current is concentrated in the area with low PR, or corroded area. Hu et al. [18] stated that the CE with guard ring have been developed as a method of determining the measurement area, but that in localized corrosion, the current is concentrated at the corroded area even with guard rings. Elsener et al. [30] also showed that when the Fig. 3 Calculation procedure of the corrosion area ratio corroded area is smaller than the CE, the corrosion rate is underestimated due to current concentration. Based on these facts, the measurement area should be set considering the current concentration at the corroded area. The results and consideration indicate that the corroded area should be calculated in addition to the CWL for accuracy verification. In addition, the AC impedance method is highly accurate and feasible for the cracked and crack-repaired concrete specimens.
In this experiment, the concern is that many specimens underestimated the CWL. It was found that 80% of the specimens had DW [ G, and 40% of the specimens had DW [ 2.0 9 G. There are two possible reasons for this. Firstly, it is difficult to detect the minor corrosion occurrence such as point rust by the AC impedance method, while the actual CWL was calculated to be large due to the small corrosion area ratio. On this basis, there were a great number of specimens that were underestimated at 70 mg/cm 2 or less. Secondly, the measurement area was set as the corroded area of the steel bar at the time of splitting the specimens. This method can accurately determine the CR at the split time. However, the corroded area generally increases with the corrosion progress, and therefore, the corroded area during the exposure period is expected to be smaller than that at the split time. In other words, CR may be measured lower than the actual rate due to the determined measurement area being larger than the actual corroded area during the exposure period. However, when the actual amount of corrosion is large (DW [ 200 (mg/cm 2 )), the estimation accuracy was found to be high even with such a measurement area setting. This is presumably because the corroded area does not increase much when the CWL is relatively large. How to effectively set the actual corroded area during the exposure presents an important avenue for a future work.
As a consideration of current dispersion, if the cracks were filled with water, the concrete resistance would be expected to be extremely low. However, the fact that no such impedance was exhibited suggests Fig. 4 Comparison of actual and PR-estimated CWL: a Comparison when A is set as the total test area; b Comparison when A is set as the total corroded area Fig. 5 Comparison R p,app using 100 mm 9 50 mm and 100 mm 9 300 mm counter electrodes (CE) that the current flowed through concrete. In addition, since the crack repair point is insulated, it is considered that the current did not flow to the repair area and thus the current flowed to the concrete. Therefore, even if there are cracks or crack repair points, the impedance can be interpreted as if there were none. This study is characterized by the fact that even in the case of localized corrosion, a simple equivalent circuit can be quantitatively evaluated by identifying the fact that almost all the measured current was concentrated in the corroded area. Figure 7 shows the mean time-variant corrosion current density (I corr ) of specimens 0.2C-N, 0.4C-N, and 0.6C-N calculated by the AC impedance method. I corr can be converted to CR by simply multiplying by a constant as shown in Eq. (4). For each factor, the CR increased rapidly for approximately 50 weeks after exposure and thereafter decreased slightly before finally remaining at an almost constant value. To consider the difference of these CRs due to different factors, Fig. 8 shows the mean data of the current densities and their standard deviation after 50 weeks for 0.2C-N, 0.4C-N, and 0.6C-N. The large standard deviation compared to the difference in the mean values indicates that there is no significant difference for the evaluated crack range (0.2-0.6 mm). The reason for the slight CR decrease from 60-80 weeks to after 100 weeks was thought to be that oxygen and water from outside were prevented from entering due to the accumulation of corrosion product on the steel surface. Yuan et al. [5] also considered that the CR after the concrete cracking would become steady because corrosion products gradually filled up the cracks and oxygen access retarded. Figure 9 shows the mean time-variant corrosion current density for the C-R and N-N specimens. The corrosion can be observed to have occurred approximately 20-30 weeks after exposure as the chloride ions introduced by the salt spray gradually penetrated the concrete. Furthermore, presence or absence of cracks on the surfaces of the specimens were observed after each PR measurement, then the occurrence of corrosion cracks was confirmed after approximately 70-80 weeks. Because the N-N and C-R specimens showed similar increasing trends in CR, we assumed that corrosion of the C-R specimen occurred due to chloride ion penetration from concrete areas that were not crack repair areas, and that crack occurred due to corrosion expansion. Therefore, the chloride ion contents at the time of crack initiation are similar for the C-R and N-N specimens.

Time-variant CR
The CR immediately after the appearance of cracks was approximately 2-3 times higher than the previous CR owing to the rapid increase in the chloride ion and oxygen supply through the cracks. Focusing on the plot in Fig. 9, the CR can be observed to increase almost linearly between corrosion initiation and crack initiation. Indeed, it was determined that the CR was low immediately after corrosion initiation during the progress period, but increased with time as the chloride ion content in the concrete gradually increased at the location of the steel bar.
Comparing the results in Figs. 7 and 9, the cracked RC specimens in a severe salt environment reached a high CR after 50 weeks of exposure, indicating that corrosion occurred and progressed early. This is attributed to the faster salt penetration and higher oxygen permeability facilitated by the presence of cracks. Furthermore, it took a long time for corrosion to occur in both the N-N and C-R specimens, and their subsequent increases in CR were both minimal. It was therefore shown that corrosion initiation and progression could be inhibited in specimens with bending cracks to the same extent as in specimens without cracks by repairing the cracks before corrosion initiation.

Discussion
The contributions of this work are detailed through a comparison of the results of this study with those of previous studies. It is well-known that the PR method can quantify the CR of RC specimens without concrete cracks [23]. There have also been several examples of the application of electrochemical methods to specimens after cracking [7,8]. However, few studies have confirmed the accuracy of their results by comparing the actual CWL with that obtained from the timeintegrated value of the CR after crack initiation or crack repair. Thus, an important contribution of this study is its empirical demonstration of the applicability of the AC impedance method to RC after being cracked and repaired.
The corrosion initiation time and time-variant CR of steel bars in concrete vary depending on the presence or absence of cracks. Figure 10 shows conceptual diagrams of the time-variant CR, as determined by monitoring. Figure 10a shows the results of a specimen with bending cracks, and Fig. 10b shows the results for a specimen without cracks or a specimen with the cracks repaired.
Some previous studies have shown that the CR is higher when the crack width is larger [31,32] whereas others have shown that the crack width is almost unrelated to the CR [11,12]. In this study, the crack width was found to be irrelevant in the range of 0.2-0.6 mm. The reason is that degradation factors such as chloride ions, water, and oxygen can easily penetrate the concrete through the crack, regardless of its width. Sangoju et al. [11] and Okada and Miyagawa [12] suggested that the type of concrete affects the CR more than the crack width. Therefore, it remains necessary to conduct a parametric study of the timevariant CR under various environmental and concrete conditions to develop a versatile CR prediction method.
Furthermore, our results demonstrated that the increase in CR after crack initiation does not continue with time. Japan Concrete Institute (JCI) [32] and our previous research [33] proposed the prediction model with continuously increasing CR. Rather, as shown by Otieno et al. [8], Chen and Mahadevan [34], and Cao et al. [35], the increase in CR gradually slows with time until eventually reaching a constant value. We verified such time-variant CR through accuracy verification of the CWL.
If the AC impedance method is conducted after identifying the areas where the CR should be obtained, the deterioration state of the structure can be efficiently quantified. In this study, the AC impedance method was applied to a series of specimens of a single geometry design. To apply this method to actual RC structures, it will be necessary to take into account various factors such as the steel bar length, concrete cover, electrical resistance of the concrete, and shape of the CE. In addition, it is considered difficult to apply the AC impedance method to structures that have deteriorated to the extent that the concrete cover has delaminated after cracking, so it remains necessary to clarify the types of deterioration to which it can be applied. Thus, the application of the AC impedance method to the maintenance of existing structures still has many issues that need to be addressed, and further studies are necessary.
In summary, this study provides valuable information describing methods for the measurement of corrosion deterioration and the prediction of corrosion progress.

Conclusions
In this study, AC impedance measurements for RC specimens with and without cracks and crack repairs