Using Impedance Spectroscopy to Assess the Viability of the Rapid Chloride Test for Determining Concrete Conductivity

The suitability of using the initial current from the rapid chloride test (ASTM C 1202) to determine specimen conductivity is tested using impedance spectroscopy with a frequency spectrum of 10 Hz to 1 MHz. The specimen conductivity has an analytical relationship to specimen diffusivity and so is a useful quantity in service life prediction. Measurements made on specimens of different lengths indicate that the total charge passed during the six hour conduction test carried out according to ASTM C 1202 is not a direct measure of specimen conductivity. Further, ohmic heating during the 6 hour test makes it nearly impossible to directly measure any specimen transport property from the results. The total charge passed during the 6 hour conduction test is, therefore, not a reliable quantity for service life prediction. Results indicate that the direct current (dc) measurement of resistance using a voltage of 60 V is sufficient to overwhelm polarization effects, thereby yielding an accurate estimate of the true specimen conductivity. Impedance spectroscopy measurements also indicate that corrosion may form on the brass electrodes, adding bias to a conductivity estimate based upon a dc measurement.

The suitability of using the initial current from the rapid chloride test (ASTM C 1202) to determine specimen conductivity is tested using impedance spectroscopy with a frequency spectrum of 10 Hz to 1 MHz. The specimen conductivity has an analytical relationship to specimen diffusivity and so is a useful quantity in service life prediction. Measurements made on specimens of different lengths indicate that the total charge passed during the six hour conduction test carried out according to ASTM C 1202 is not a direct measure of specimen conductivity. Further, ohmic heating during the 6 hour test makes it nearly impossible to directly measure any specimen transport property from the results. The total charge passed during the 6 hour conduction test is, therefore, not a reliable quantity for service life prediction. Results indicate that the direct current (dc) measurement of resistance using a voltage of 60 V is sufficient to overwhelm polarization effects, thereby yielding an accurate estimate of the true specimen conductivity. Impedance spectroscopy measurements also indicate that corrosion may form on the brass electrodes, adding bias to a conductivity estimate based upon a dc measurement.

Introduction
There has been considerable research interest in studies of diffusive ion transport through the saturated pore space of portland cement paste. Measurements of conductivity have provided a useful insight into characterizing relevant transport properties [1][2][3][4][5][6][7][8]. This effort has been motivated by the direct relationship between the concrete conductivity and both the steel reinforcement corrosion rate and the ion diffusivity. The concrete conductivity is directly proportional to the reinforcement corrosion current, and it can also be related to the ion diffusivity through the Nernst-Einstein equation for electrical potential gradients or the Nernst-Plank equation for both electrical and chemical potential gradients (see Ref. [9], Chap. 4). Therefore, concrete conductivity can be an important transport property in determining concrete service life in corrosive environments.
Both the American Association of State Highway and Transportation Officials (AASHTO) and the American Society for Testing and Materials (ASTM) have a standardized test of electrical conduction through concrete, referred to here as the rapid chloride test (RCT). This test measures the cumulative electrical charge passing through a specimen subjected to a direct current (dc) voltage of 60 V over a period of 6 h. However, changes in the pore fluid conductivity due to ohmic heating [10,11], and changes in the microstructure due to elec-tromigration [12] prevent the standardized six hour test from yielding a direct measure of specimen dc conductivity and diffusivity.
The work described herein explores whether specimen diffusivity can be determined from some other aspect of the RCT. Experiments have been performed elsewhere that demonstrate the direct relationship between specimen conductivity and diffusivity [13][14][15]. Experiments have also been performed which purport to demonstrate a causal relationship between measurements of diffusivity and either the total charge passed or the initial current using the RCT cell [16][17][18][19][20][21][22]. However, no direct relationships between parameters of the RCT and diffusivity have been proven. Therefore, a means to accurately determine specimen conductivity from RCT data would be a crucial step towards establishing a relationship between RCT data and specimen diffusivity.
Accurate measurements of specimen conductivity can be made using the experimental techniques of impedance spectroscopy (IS). Using an alternating current (ac) that varies over a wide range of frequencies, effects such as electrode polarization and ion transfer can be eliminated, yielding the true specimen resistance. The specimen conductivity may then be calculated from the measured resistance and a knowledge of the specimen geometry. However, under certain conditions, these electrode effects may be negligible. There is a small range of frequencies over which the electrode effects have an insignificant contribution to the overall conductivity, leading a number of investigators to report specimen impedance at a fixed frequency. Further, it may be possible to estimate specimen conductivity in a situation where the electrode effects are relatively small compared to the applied potential. Experiments have shown that the combined electrode effects generate a voltage drop of less than 2 V [3,23]. Therefore, one may conjecture that if a specimen is subjected to a dc electrical potential of 60 V, and if the combined electrode effects generate a negligible voltage drop, the specimen conductivity calculated from the resultant dc current may be sufficiently accurate for most purposes.
Reported herein are the results from IS measurements performed on specimens just prior to the application of the 60 V specified by the RCT. A frequency spectrum of 10 Hz to 1 MHz was used to determine the true specimen resistance, from which the conductivity was calculated. The RCT setup used here recorded the current at 60 s intervals, including the instantaneous initial current, until the completed test at 6 h. Results show that the initial RCT current can be used to directly and accurately determine the specimen conductivity. Implications of this result for possible future "rapid" tests are discussed.

Impedance Spectroscopy
Specimen conductivity can be most accurately determined using the principles of IS [24]. In an IS measurement, the specimen is subjected to an ac voltage over a range of frequencies, and the phase (with respect to the applied potential) and amplitude of the current are measured at each frequency. In order to interpret the IS results, the impedance response of a specimen is approximated by an equivalent circuit. The components of the circuit have a physical correspondence to components of the specimen. For metal electrodes against a saturated porous material composed of an insulating solid framework and a pore space filled with electrolyte, the impedance response can be approximated by an equivalent circuit composed of resistances R and C [24,25]. A simple equivalent circuit for the RCT is shown in Fig. 1. The subscripts "E" and "B" represent the electrode and the bulk components, respectively. The term bulk represents the porous solid and electrolyte pore solution composite, and is interchangeable in meaning with specimen in this context. The resistance R B in Fig. 1 represents the dc resistance of the RCT specimen. The capacitance C B represents the specimen capacitance due to the electrolyte in the pore space. The electrode elements R E and C E represent the impedance response of the electrodes due to polarization and charging effects.
The equivalent circuit in Fig. 1 is composed of parallel resistors and capacitors connected in series. The impedance of a resistor Z R and a capacitor Z C are complex quantities (i = ͙Ϫ1) that are parametrized by the ac angular frequency [26]: The complex nature of the impedance corresponds to a phase difference between the current and the voltage through these devices; a resistor, having a pure real impedance, does not contribute to a change in the phase. The circuit in Fig. 1 can exhibit both a capacitive and resistive response. For some values of , the current and the voltage are nearly in phase. In this case, the impedance has no complex component, and the entire system behaves like a purely resistive element. These values of for which the system is purely resistive can be determined from the total impedance of the equivalent circuit in Fig. 1: The quantities Z' and Z" represent the real and imaginary components of Z , respectively. The constants E and B are equal to (R E C E ) Ϫ1 and (R B C B ) Ϫ1 , respectively, and are of the same dimension as (i.e., have the same unit). In a system like saturated concrete, the quantity E may be several orders of magnitude smaller than B ( E <<< B ). Given this information, there are three ranges of values for that are of interest: The third relationship expresses mathematically the fact that for intermediate values of , orders of magnitude from either E or B , the entire system becomes purely resistive; the phase difference between the applied voltage and the resultant current is negligible. Most importantly, this value of Z is equal to the bulk specimen resistance R B that is used to calculate the specimen conductivity. A schematic representation of Z ( ) is shown in Fig.  2 for E <<< B . The figure is an impedance plane plot, typically referred to as a Nyquist plot, and is parametrized by , where = 0 is at the right hand side of the curve, and = ϱ is at the left. The values of at the maximum values of ϪZ" are shown. Experimentally, impedance analyzers can only produce a finite range of frequencies, and typically only the portion of the curve near Z' = R B is measured. For an estimate of bulk conductivity, this is all that is required. However, only under ideal conditions does the imaginary component of Z go to zero at Z' = R B . In practice, the response of a specimen is more like that shown in Fig. 3, where the bulk resistance R B must be estimated from the value of Z' at the minimum of ϪZ" . The data collected for Fig. 3 consist of 10 data points per decade of frequency. The datum at each decade is shown as a filled circle along the curve. Therefore, the value of R B in Fig. 3 was  components of the total impedance for a typical specimen in the rapid chloride test (RCT) cell; the independent parameter is the angular frequence . The minimum of the curve is used to determine the sample bulk resistance RB. Solid circles represent data at decade frequencies; the sampled frequencies ranged from 10 1 to 10 6 Hz. determined at a frequency between 10 kHz and 100 kHz.
Although a more complete equivalent circuit for the bulk and electrode response of the RCT cell would be more complicated than that shown in Fig. 1 [27,28], this simple circuit captures the major behavior. However, there is an additional component of the bulk impedance that is not represented in Fig. 1. A schematic cross section of the RCT with sample and holders is shown in Fig. 4. Typically, between the specimen and each brass electrode, there is a 1 mm to 5 mm gap that is filled with aqueous electrolyte: either 3 % mass fraction of NaCl or 0.3 mol/L NaOH. The contribution of this resistive component to the total resistance can be calculated from the solution conductivities found in published tables [29]: NaCl = 4.4 mS/mm and NaOH = 5.7 mS/mm. For the geometry of the RCT cell, the resistances divided by length are 0.029 ⍀/mm and 0.022 ⍀/mm for the NaCl and the NaOH solutions, respectively. Therefore, a gap of 10 mm between each electrode and the specimen contributes less than 1 ⍀ to the bulk resistance. Since the bulk resistance for concrete is typically in the range of 100 ⍀ to 1000 ⍀, the contribution by the electrolyte between the electrodes and the sample can be neglected.

Samples
The mixture proportions of the samples prepared for this experiment were designed to yield a moderate range of transport properties and are based upon mixture proportions from previous experiments [30]. The wide range of transport properties were achieved through variations in water-cement mass ratio m w /m c (commonly denoted by w/c), cement replacement by pozzolanic mineral admixtures, aggregate gradation and volume fraction, and dosage of high range water reducing admixture (HRWRA). The cement was ASTM Type I; the chemical composition is given in Table 1. The pozzolanic mineral admixture was silica fume in slurry  form. The aggregate type was natural silica and two gradations were used: silica sand conforming to the ASTM C 778 Graded Sand designation; and a "micro" concrete aggregate composed of natural silica sand. The gradation for the "micro" concrete aggregate, based upon a report by Fuller [31], is shown in Table 2. The HRWRA was an aqueous solution with a naphtalenesulfonate mass fraction of 40 %. The mixture proportions of the samples used in this experiment are shown in Table 3. The mixtures were prepared according to the procedures in ASTM C 109. The samples were cast in 100 mm diameter and 200 mm long cylindrical molds, covered, and stored in a 100 % relative humidity chamber. At 24 h of age, the specimens were removed from the molds and stored (50 d to 80 d) in a saturated calcium hydroxide bath until they were tested. Although no temperature controls were used, the laboratory temperature could be characterized by the interval (20 Ϯ 2) ЊC. Table 3. Sample mixture proportions. Relevant quantities are the sand gradation, the sand to cement mass ratio ms/mc, the water to cementitious materials mass ratio mw/(ms + mc), the fraction of mc replaced by silicon, high range water reducing admixture (HRWRA) dosage with respect to cement mass, and the air volume percentage as measured using a procedure based upon the specification ASTM C 185.

Sample
Sand Each specimen was prepared for testing according to the specifications of ASTM C 1202. A single cylinder mold was cast from each mixture and two specimens, one specimen 50 mm long (ASTM C 1202) and the other 100 mm long, taken from the middle 150 mm of the cylinder, were tested. The purpose of using two specimen lengths was to verify the validity of the conductivity measurement techniques. Conductivity, an intrinsic property, is independent of specimen geometry and size. A reliable technique for determining specimen conductivity should obtain equivalent results from replicate specimens with different lengths.

IS and Initial Current Measurements
The specimens were mounted into the RCT cells and the NaOH (0.3 mol/L) and NaCl (3 %) chambers were filled. A commercial impedance analyzer was then connected to the RCT cell banana jacks. Using a voltage of approximately 1 V (peak-to-peak), with a zero volt dc offset bias, across the RCT cell, the analyzer scanned frequencies from 10 Hz to 1 MHz, completing the test in approximately 1 min; results using ac potentials ranging from 0.1 V to 1.0 V yielded similar results. The data were stored in the computer for analysis to determine the value of R B .
At the completion of the IS measurement, the analyzer leads were removed from the cell and the cell was connected to a commercial dc power supply capable of two-way communication to a computer via an IEEE-488 interface. The computer program would initiate the test, query the power supply for the initial current, and subsequently query the power supply for the current every 60 s until the completion of the test. The power supply was equipped with remote sensing to compensate for the voltage drop along the power cable. This ensured that the voltage delivered to the RCT cell was always within 0.1 V of 60 V.

Calculated Quantities
The reported values are calculated quantities based upon the physical measurements. For clarity, these calculated quantities shall be defined explicitly. The total charge passed Q T is calculated from the measurements of the currents I i that were carried out at times t i . Each t i occurred at the i th minute of the test: This method conforms to the procedure specified in ASTM C 1202. The initial RCT resistance R 0 was calculated from the initial current I 0 : As indicated in Fig. 3, the bulk resistance R B measured using IS was determined from the value of Z' at the minimum value of ϪZ" . Specimen conductivities IS and RCT were calculated using the measured resistance and the specimen length L and area A : Since a number of studies reported in the literature have used a fixed frequency, it will be instructive to determine the validity of this approach. The quantity R 20 is the real component of Z ( ) at a frequency of 20 kHz, a frequency that has been used in similar experiments with a fixed frequency [16,18,19].

Results
The results of the experiment are summarized in Table 4 and are in qualitative agreement with values cited elsewhere [3,17,22,23]. The specimens are grouped by sample for direct comparison between specimens with different lengths from the same mixture. Although most of the specimens were more than two months of age at the time of testing, results of the total charge passed Q T indicate that specimens from a range of qualities were tested. In fact, some of the tests had to be terminated due to excessive heating. Unfortunately, there were also some specimens that have no IS data due to computer software problems. However, the comparison among R B , R 20 , and R 0 can be carried out for all but five specimens.
The issue of measurement uncertainty is addressed in Appendix A. Expanded uncertainties U reported in Table 4 represent an estimate of two standard deviations, or an estimated coverage of approximately 95 %. The uncertainties in the total charge passed Q T were all less than 2 C at this level, with most of the uncertainties less than 1 C. Appendix A addresses the calculations of the reported uncertainties, and concentrates on the complicated issue of the uncertainty in Q T .

Conductivity
For all but three specimens, the values RCT and IS for an individual specimen differed by less than 5 %. This fact is demonstrated graphically in Fig. 5. Immediately apparent from graph is range over which the agree-  ment is valid. The data shown on the graph represent samples that passed total charges ranging from 679 C to 10 865 C. In addition, there are data shown that represent specimens that did not finish the 6 h test due to excessive heating. Since IS gives the most accurate estimate of conductivity, and since the conductivities of replicate specimens of different lengths are not equal, the difference in IS between these specimens of different lengths must be due to specimen-specimen variability. Since the values of RCT and IS for each specimen are nearly equal, the initial current measurements yield a direct and accurate measure of specimen conductivity. Therefore, the ratio of I 0 values for the two specimens is directly proportional to the ratio of specimen conductances (inverse of resistance) However, the ratio of Q T values does not equal the ratio of I 0 values. Therefore, Q T is not a direct measure of specimen conductivity.
With few exceptions, the resistance measurements at 20 kHz R 20 are within a few percent of R B . This suggests that there may exist a constant intermediate frequency one could use with the RCT cell to determine sample conductivity to within an acceptable level of uncertainty. However, a suitable frequency should be chosen with care. Using a frequency of 100 Hz, as was used elsewhere [22], would not be advisable due to the substantial contribution by the imaginary portion of the impedance. A fixed frequency test would reduce the cost of an ac test that incorporated the RCT cell. However, with few exceptions, the initial dc resistances are a more accurate estimate than the 20 kHz resistances R 20 measured here.

Total Charge
Although the total charge passed Q T increased as the specimen conductance increased, the values are not linearly proportional to one another. Samples for which there are values of Q T for both specimen lengths, the values are not inversely proportional to the specimen length; the value of Q T for the 50 mm specimen was always significantly greater than twice the value for the 100 mm specimen. Figure 6 shows the measured current for the two specimens from Sample F. The current through the 100 mm specimen is nearly constant throughout the test. The current through the 50 mm specimen appears to increase linearly after approximately 100 min. Although the initial currents differed by a factor of two, the final currents did not. The difference is due to ohmic heating. Since fluid conductivity increases by approximately 2 % per degree Celsius [10,11], and assuming that specimen pore fluid conductivity behaves in a similar manner, one could estimate sample temperature from the relative change in current. Unfortunately, the specimen temperatures were not measured at the time of the tests and a duplicate experiment that included temperature measurements could not be conducted because the specimens had been discarded.
To demonstrate the effect of temperature, an additional test was performed on a completely different sample from the Federal Highway Administration. For this specimen, the temperature was monitored using an immersion temperature probe incorporating a precision thermistor. The probe was demonstrated to be accurate to within 1 ЊC. The temperature of the solutions at each end of the specimen was measured periodically throughout the RCT test. The data are shown as filled symbols in Fig. 7. Along with the symbols is a curve that shows the estimated specimen temperature based upon the increase in current passing through the specimen. Although the measured and estimated values are not equal, they agree to within 10 ЊC throughout the test. This is reasonable since the heat generated within the specimen is lost to the solutions and to the room. Also, the results are in general agreement with laboratory experience and with results reported elsewhere [20,21].

Current at Early Times
It is interesting to note the early conductivity response of the sample to the application of the 60 V required by the standardized RCT. Figure 8 is a plot of effective specimen conductivity calculated from the RCT current for the two specimens from Sample F during the first 4 min of the RCT. The solid lines shown are only meant to guide the reader's eyes. The dashed lines delineate the sample conductivities using IS. The figure shows the effects of two phenomena: a decrease in current due to the accumulation of polarization charge, and an increase in current due to ohmic heating.
Using these data from Specimen F, it is difficult to distinguish the proper time at which to measure the RCT current to ensure an accurate estimate of IS . One could argue that the proper value should be determined from an extrapolation to zero time using the values measured at 1 min and later. Although this method would be relatively accurate for the 50 mm specimen, it would not be justifiable for the 100 mm specimen. The estimate using the value at t = 0 seems to be a reasonable compromise. Since the values in the figure only vary by approximately 4 % during the first 4 min of the test, it would seem as though sufficiently accurate estimates of sample conductivity could be made using the dc current measurements at any time during the first few minutes.

Previous I 0 vs Data
The direct relationship between the initial RCT current I 0 and sample conductivity established in this experiment is in contrast to results reported elsewhere. Hansen et al. [32] reported both the initial current and the ac impedance at 1 kHz. However, their data showed that, for a given specimen, the ac impedance was considerably greater than the resistance calculated from the initial current. This may be explained using the IS data for the RCT cell that is shown in Fig. 3. The real component of the total impedance at 1 kHz is greater than R B . Also, if the output of the apparatus used in the experiment by Hansen et al. is the magnitude of the impedance | Z | = (Z' 2 + Z" 2 ) 1/2 , the difference is accentuated by the greater contribution of Z" at 1 kHz than of Z' = R B . However, the general relationship between initial current and charge passed given in Table 7  The experiments reported by Feldman and coworkers [20,21] also included measurements of specimen impedance and RCT initial current. The impedance measurements were conducted at 10 kHz. However, cal-culations of specimen resistivity using their RCT initial current differ from the IS measurements by 20 % or more in most cases. The source of the discrepancy cannot be explained by our results since the real component of Z at 20 kHz reported in Table 4 is typically within a few percent of R B .

Significance for Diffusivity
The significance of a rapid test for determining sample conductivity is the relationship between bulk conductivity and bulk diffusivity. The Nernst-Einstein [9] equation can be used to relate the bulk diffusion coefficient D i for ion species i to the bulk conductivity B : The quantity D f i is the diffusivity of ion species i in bulk water, and the quantity P is the sample pore fluid conductivity. Since the values of D f i can be obtained from tables [10], D i could be calculated explicitly from bulk conductivity measurements if the value of P could be determined using a technique such as the expression of pore fluid [28,33].
For the existing RCT apparatus to be useful to researchers and practitioners, it must be able to measure scientifically useful quantities such as diffusivity or it must be able to report empirical measurements that are directly related to physical processes. It has been shown here that the total electrical charge passed during the RCT 6 h conduction test is not directly proportional to the true dc resistance, and so is not directly related to diffusivity, which is the process of interest. However, the initial current may be used to accurately estimate B .
A second calculation shows that the RCT test does not simulate chloride transport through the sample, and so does not simulate real-world conditions. The magnitude of the drift velocity v D of the chloride ions through the RCT cell is calculated from a modification of the Einstein equation [9]: The relevant quantities are the ion valence z , the elementary charge e , the magnitude of the applied electric field E , the bulk diffusivity D , the Boltzmann constant k , and the absolute temperature T . This equation for the drift velocity v D can be used to determine the time required for chloride ions to traverse a specimen, and is in agreement with the experimental results of both McGrath and Hooton [34] and Sugiyama et al. [35].
The drift velocity equation can be simplified using the geometry of the RCT cell (E = 1.2 V mm Ϫ1 ) and assuming a constant temperature of 300 K: This equation can be simplified further by expressing the diffusivity as a ratio of the chloride ion bulk diffusivity at 25 ЊC (2.0 ϫ 10 Ϫ3 mm 2 s Ϫ1 [10]) to the formation factor F [36,37]. The chloride ion penetration depth ␦ during the standard 6 h RCT, as a function of the formation factor F , is simply the drift velocity v D times 21 600 seconds: Since typical values of the formation factor F for 28 d specimens range from 100 to 1000 [38,39], the chloride ions do not traverse the specimen during the rapid chloride test. Therefore, the standard 6 h RCT does not simulate chloride transport through the specimen because the chloride ions typically penetrate only a fraction of the specimen thickness during the test.

Conductivity vs Total Charge Correlations
Since there have been previous attempts to correlate specimen conductivity to the total charge passed, it will be useful to study this behavior using the data from this experiment. Figure 9 contains a plot of the measured conductivities IS as a function of the total charge Fig. 9. Sample conductivity IS as measured using impedance spectroscopy (IS) versus the total charge passed QT during the 6 h rapid chloride test (RCT). The filled symbols denote measured data for both 50 mm and 100 mm specimens. The dashed curve is the regression equation of Berke and Hicks [19] for estimating sample conductivity based upon total charge passed during the 6 h test. passed Q T for the data in this experiment. When considering all the data for a given specimen length, the relationship is not linear over the entire range of Q T values. The experiment of Zhao et al. [40] correlated total charge to specimen resistance, but only for specimens passing less than 4500 C. Based upon the data shown in Fig. 9, one would expect a reasonable correlation between total charge passed and specimen conductivity for specimens passing fewer than 4500 C. However, extrapolating a linear correlation for fewer than 4500 C to specimens passing as much as 10 000 C could prove to be erroneous.
The dashed curve in Fig. 9 represents the empirical prediction of Berke and Hicks (BH) [19] that was developed from correlations between measurements of total charge and conductivity measurements using a lollipop apparatus shown schematically in Fig. 10. The apparatus consists of a 9.5 mm diameter reinforcing bar embedded along the axis of a 76 mm diameter, 152 mm long concrete cylinder, with the reinforcing bar positioned 38 mm from the far end of the cylinder. The top 25 mm of the rod penetrating the cylinder is coated with epoxy. Since the BH equation was developed using 50 mm specimens, it should not be expected to predict the response of 100 mm specimens. Also, the equation was developed using data with few values of Q T greater than 4000 C. For our experiment, the BH equation is a reasonably good predictor of the 50 mm data for Q T less than 4000 C, but is a poor predictor for values of Q T greater than 4000 C. This is to be expected, given the parameter space over which the equation was developed. However, there are two features worthy of note. Use of the equation for samples passing greater than 4000 C would introduce large errors. Also, the estimate is consistently greater than the conductivity values measured here. This artifact may be due to the longitudinal component of the current originating from the end of the reinforcement bar used in the BH experiment. This additional current would cause an overestimate of the specimen conductivity, as is demonstrated in Fig. 9.

Corrosion Arc
An interesting component in the Nyquist plots appeared near the completion of this experiment. Samples J through M were cast and tested approximately three months after samples A through I. The IS results from samples A through I were fairly consistent. However, samples J through M exhibited an additional feature in the Nyquist plot such as that shown in Fig. 11 from specimen K-2. The data shown in the figure were measured at the beginning of the RCT, just prior to the Fig. 11. Impedance plane plot showing the real Z' and imaginary Z" components of the total impedance for Specimen K-2 in the rapid chloride test (RCT) cell. The independent parameter is the angular frequence ; solid circles represent data at decade frequencies. The specimen bulk resistance RB and the sample resistance R0, which based upon the initial direct current measurement, are shown for comparison purposes.  [16,18,19] to measure specimen conductivity. Dimensions shown are in millimeters. The sample is placed into a 3 % mass fraction solution of sodium chloride that contains an additional electrode. application of 60 V dc. The bulk arc can be seen at Z' < 700 ⍀, and the electrode arc can be seen at Z' > 750 ⍀. The interval 700 ⍀ < Z' < 750 ⍀ has an additional arc, possibly due to corrosion accumulating over the duration of the entire experiment. However, the cell was damaged during an attempt to clean one of the brass electrodes in order to confirm the corrosion theory.
The "corrosion" arc presents a possible difficulty in using the RCT cell for determining the sample conductivity. The Nyquist plot of Specimen K-2 in Fig. 11 has two arrows delineating the values R B and R 0 . The value of R B is an accurate estimate of the sample resistance since one would expect the actual value to be at the intercept of the bulk arc and the Z' axis; one can show this by adding another RC pair in the equivalent circuit shown in Fig. 1 to approximate corrosion and observing the result. The measured value of R 0 is affected by the presence of the intermediate arc. Despite this, the values for IS and RCT differ by less than 11 % for specimens J through M.

Summary
The results demonstrated that the total charge passed during the 6 h ASTM C 1202 rapid chloride test (RCT) was not an accurate indication of specimen conductivity. For every mixture proportion studied, the shorter specimen had a disproportionately greater total charge passed. This would be expected based upon the effects of ohmic heating.
A measurement of the initial current (t = 0) during the ASTM C 1202 rapid chloride test provides an estimate of specimen conductivity which is typically within 5 % of the value determined from impedance spectroscopy using a frequency spectrum of 10 Hz to 1 MHz. These results were confirmed using otherwise similar specimens of different lengths. Based upon the Nernst-Einstein relationship between specimen conductivity and specimen diffusivity, these results imply that an instantaneous measurement of current can yield quantitative information about the diffusivity of the specimen. Further, data obtained during the start of the RCT suggest that the dc current measured at any time during the first few minutes of the test would yield similar results.
Impedance spectroscopy impedance-plane plots also revealed an electrochemical feature in tests performed later in the experiment. This feature may be attributable to corrosion of the brass electrodes. The feature was responsible for a small bias (less than a 10 % difference) between bulk conductivities measured by impedance spectroscopy and the initial RCT current. Therefore, this suggests that implementing a rapid test based only on the initial current may require frequent monitoring of the electrode surface condition, or the use of electrodes made from a material that does not corrode in the testing solutions.

Measured Quantities
The evaluation of uncertainty requires assumptions about the magnitudes of the uncertainties attributed to the individual measurements. Unfortunately, none of the individual measurement uncertainties are based upon a statistical analysis. Rather, they are based upon engineering judgment, classified as Type B [41,42] by the International Organization for Standardization (ISO). Table 5 summarizes the various measurement uncertainties for this experiment. Since the uncertainty in each measured quantity must be quantified, a probability distribution must be chosen for each. Here, a Gaussian distribution is used for convenience. The uncertainties are characterized by an expanded uncertainty U = ku c , where u c is the standard uncertainty, that is, estimated standard deviation, and k is the coverage factor. The expanded uncertainties U in Table 5 correspond to two standard deviation estimates (k = 2), implying a coverage of approximately 95 %. The two standard deviation estimates are based upon engineering judgment. If a reputable electronics manufacturer specifies that a voltmeter has an "accuracy" of 0.054 V, it is assumed here that U = 0.054 V. This is a conservative estimate because one would generally expect a better than 95 % confidence that the device is within U = 2u c . In Table 5, the values of U for the equipment specifications are the accuracies specified by the manufacturer.
Two of the remaining uncertainties in Table 5 are for dimensional measurements. Unfortunately, statistical measurements of the corresponding dimensions of each specimen were not performed. Therefore, the uncertainty in the length and diameter of each specimen is based upon the tolerances specified in the corresponding ASTM specification, with the assumption that the specified tolerance represents a two standard deviation estimate.
The final quantity in Table 5 is the bulk resistance measured by IS. Here, the value of U is a "best guess" based solely on general observations. The minimum in the value of ϪZ" is not an exact quantity. The curvature at the minimum dictates the uncertainty of the determined quantity. The interval defined by U represents an overall characterization of the interval between adjacent values of Z' at the minimum of ϪZ" . Many of the adjacent values were considerably less than this quantity, but none was greater.

Calculated Quantities
To obtain the uncertainties of the calculated quantities requires an analysis of both measurement uncertainty and method uncertainty. The uncertainty in quantities such as the specimen conductivity are based upon standard propagation of uncertainty techniques [42,43]. However, the uncertainty of the total charge passed Q T requires an additional analysis of the method uncertainty. Equation (3) is not only a means of calculating Q T , it is also a discrete approximation of the continuous function of current that varies with time. As a numerical method, trapezoidal integration has inherent uncertainty that is a function of both the time interval and the curvature of the function being integrated [44].
Since the curvature in the function of current versus time differs from specimen to specimen, a general approach was needed for the analysis of the method uncertainty. In this experiment, the current through the specimen was measured every minute. From these measurements one can perform a propagation of uncertainty calculation based upon Eq. (3) to yield a measurement uncertainty for a time interval of 1 min. Also, one could extract every other datum, as if the current was measured every 2 min, and perform the same uncertainty calculation. A comparison of these two results would indicate the effect of changing from a time interval of 1 min to a time interval of 2 min. Fortunately, the number 360 (the number of minutes in 6 h) has many possible multiplicative factors. For Sample C-2, the extraction of every n th datum was repeated for a number of n values, the measurement uncertainty calculated, and the results plotted (see Fig. 12) with the uncertainty bars representing the expanded uncertainty U . Based upon these results, for measurement intervals of less than 10 min, there appears to be no significant method uncertainty contribution to the overall uncertainty.