Conceptualization, Methodology, Validation, Investigation, Data Curation,
Visualization, Writing - Original Draft

1]organization=Institute for Building Materials, ETH Zurich, city=Zurich,
postcode=8093, country=Switzerland

Conceptualization, Writing - Review & Editing

2]organization=Institute of Geophysics, ETH Zurich, city=Zurich,
postcode=8092, country=Switzerland

[1]

Conceptualization, Writing - Review & Editing, Supervision

[1]Corresponding author

# Nondestructive detection and quantification of localized corrosion rates by
electrochemical tomography

M.C. van Ede  [  A. Fichtner  [  U. Angst<EMAIL_ADDRESS>
###### Abstract

Localized corrosion is one of the most common causes of early degradation of
engineering structures. To non-destructively determine the location, size and
rate of localized corrosion in porous media, a new technique, electrochemical
tomography (ECT), has been theoretically and numerically formulated. The
current work shows the application of ECT to measure corrosion rates in a
controlled laboratory setup, with a stable electrolyte and well-defined macro-
cell. The results show that ECT is able to replicate the corrosion size and
location and can give a good estimation of the corrosion rate. Moreover, the
validation of ECT on a well defined localized corrosion system, brings the
technique closer to future field applications.

###### keywords:

Non-destructive testing localized corrosion corrosion rate electrical
potential kinetic parameters

##  1 Introduction

Non-destructive testing (NDT) techniques are becoming increasingly important
in the condition assessment of civil infrastructure [1, 2]. The use of NDT
enables detection of deterioration at an early stage, where it is often not
yet visible during conventional visual inspection. Combined with advances made
in automated inspection, e.g. with the use of robots, NDT techniques could
significantly reduce the economic impact related to ageing infrastructure,
which is a major challenge in all industrialized countries [3, 4]. Corrosion
is one of the most common causes of early degradation of engineering
structures, causing safety hazards and bringing along huge maintenance costs
[5]. In the United States, corrosion of infrastructure was estimated to cost
more than 22 billion dollars annually [6]. In Switzerland, a recent study
showed that more than half of the maintenance costs of road bridges is related
to corrosion [7].

Localized corrosion, or macro-cell corrosion, is the most dangerous type of
corrosion, as large cathodic areas can result in high corrosion rates at small
local anodes. This type of corrosion is especially problematic in porous
media, such as reinforced concrete [8, 9] and underground steel structures
[10, 11, 12]. To non-destructively detect localized corrosion in porous media,
half-cell potential mapping is a well established and reliable technique [13,
14], widely applied on-site, sometimes in the combination with electrical
resistivity measurements [15]. After the detection of areas with a high-risk
of corrosion, additional destructive testing, such as drilling cores to
determine chloride profiles, are used to assess if repair is needed [16]. Not
only are these destructive investigations costly, the measured chloride
content cannot always reliably tell if corrosion has initiated, as the
critical chloride content needed for initiation is influenced by a multitude
of factors, such as the cement type [17]. For the efficient scheduling of
maintenance works, it would be more effective to directly non-destructively
locate and quantify the localized corrosion.

However, as of now, there are no commercially available techniques that can
reliably quantify the rate of localized corrosion non-destructively in the
field [18]. Existing electrochemical techniques, that aim to measure corrosion
rates, mostly rely on the measurement of the polarization resistance, e.g., by
the linear polarisation resistance (LPR) technique [18, 19, 20],
electrochemical impedance spectroscopy [21] and the galvanic pulse technique
[18, 22]. A large source of error arises from the assumption of uniform
corrosion and the need to estimate the polarized area of the steel [23].
Therefore, these techniques are often found to yield unreliable estimations
[24, 25] and their applicability for localized corrosion has been questioned
[23, 26, 27].

These limitations may be overcome by the combined use of a numerical model,
simulating the potential field in the porous medium around the steel, and an
inverse method. This idea has been formulated theoretically and numerically in
previous work, and has been named electrochemical tomography (ECT) [28]. Using
electrical potential values, measured at the surface of the porous medium for
small externally applied currents, ECT determines the location and size of the
corroding steel, as well as the corrosion rate. As ECT explicitly takes into
account the geometry, electrical resistivity and the polarization behaviour of
the steel, ECT can be applied to quantify localized corrosion and is not
limited by the need for assumptions related to the (unknown) polarized area.

In van Ede et al. (2021) [28], the authors showed that the technique is able
to accurately estimate the corrosion location and size, as well as the
corrosion rate, and give insight about the uncertainty associated with the
solution. Moreover, this work also showed that the technique is able to study
fundamental parameters that describe the corrosion kinetics of the steel in a
macro-cell corrosion system, such as the Tafel slopes and exchange current
densities.

Others have also shown that the use of inverse techniques is promising to
increase the information obtained by potential mapping [29, 30, 31, 32, 33].
Marinier & Isgor (2013) [32] were the first to implement the Butler-Volmer
kinetics, describing the polarization behaviour of the steel, into their
inverse technique. Recently, Adriman et al. (2021) [33] applied their
deterministic inverse technique to the case study of a concrete pillar. They
applied a more simplified approach than used in ECT, concerning the assumed
corrosion kinetics and the required input data. Although their approach was
not able to quantify the corrosion rate, they showed it was possible to
accurately estimate the location and size of the localized corrosion area.

The current work further explores the technique of ECT and shows its first
validation in a controlled laboratory setting. This validation is essential to
evaluate the ability of the numerical model to correctly describe macro-cell
corrosion, and to assess the accuracy of ECT in determining the corrosion
location, size and rate in a real macro-cell corrosion setup. This work also
further optimizes the technique to increase its accuracy and efficiency, in
terms of the applied external currents during the measurement of the input
potential data, and the corrosion kinetics fitted during the inversion
procedure.

##  2 Material and methods

The goal of the validation of ECT [28] is to test if the technique as a whole,
and the numerical model in particular, is accurate enough to estimate the
location, the size and the rate of localized corrosion in a well defined
geometry with a homogenous electrical resistivity. ECT was developed for
locating and quantifying localized corrosion in porous media. However, direct
validation in media such as soil or concrete is not ideal, as the steel
surface, and thus the location and size of the corroding steel, cannot be
observed during the experiments, and it is difficult to monitor and control
the conditions of the electrolyte.

Therefore we use a flow-cell with a pH neutral NaCl solution. The transparent
flow-cell allows us to observe the steel during the experimental runs.
Moreover, thanks to the continuous flow, the electrolyte around the steel
surface can be maintained constant, as the solution stays well aerated and
mixed during the experiment, and corrosion products are transported away from
the steel surface. The macro-cell corrosion is simulated by using a steel bar
consisting of two metals: carbon steel representing the corroding spot
(anode), and stainless steel representing the surrounding passive steel
(cathode). The galvanic interaction of these steels ensures that we have a
well defined macro-cell, where the anode size and location are known, and the
macro-cell current can be monitored by measuring the current flowing through
the steel between the anode and cathode.

In the following we describe the experimental setup, the numerical model and
settings of the ECT technique, and finally the experimental procedure.

Figure 1: The experimental setup for the validation of ECT. a) Schematic
overview showing the dimensions and the wire configurations to the working
electrode (WE), the counter electrodes (CE) and the reference electrodes (RE).
b) 3-dimensional overview of the flow-cell, ensuring quasi-laminar flow. c)
Photograph of the laboratory setup, showing the flow-cell and all electrodes.

###  2.1 Experimental setup

Figure 1a gives an schematic overview of the laboratory setup, indicating the
positions, dimensions and connections of the electrodes. A hollow cylindrical,
smooth steel bar with a diameter of 2 cm was placed in the middle of the
cylindrical flow-cell, consisting of an SAE J2340 carbon steel (anode) and an
X5CrNi18-10 stainless steel (cathode). The carbon steel was located in the
centre of the flow-cell and had a length of 5 cm. The carbon and stainless
steels were electrically connected with wire, that were guided through the
hollow bar, outside of the flow-cell. This allowed us to measure the macro-
cell current flowing between the anode and cathode with a zero resistance
Ammeter (ZRA).

Four high grade stainless steel rings, with a radius of 4.7 cm and width of 2
cm, were placed at the outer rim of the flow-cell and acted as counter
electrodes (CE). Two (CE1 & CE4) were placed in the region of the cathode, and
two (CE2 & CE3) in the proximity of the anode. The CE, as well as the steel
bar (working electrode WE), were connected to a Gamry
Potentiostat/Galvanostat, in a 2-electrode setup. This setup was used to apply
the external currents and polarize the system. Depending on the measurements,
only CE1 & CE4, only CE2 & CE3, or all CE were connected (see section 2.3).

At the surface of the flow-cell, 22 Ag/AgCl/Sat.KCl reference electrodes (RE)
could be positioned. The RE were connected to a Keithley multimeter with 22
channels, which was used to monitor the electrical potential between the steel
bar (WE) and each reference electrode, with 0.1 s intervals.

Figure 1b shows the 3-dimensional shape of the flow-cell and figure 1c the
flow-cell in the laboratory setup, highlighting the steel bar, the CE, the RE
and the flow direction. The flow-cell was connected to a 10 l reservoir, which
was stirred and bubbled with air to constantly homogenize the electrolyte.
From the reservoir, the solution was pumped to the flow-cell using a
peristaltic pump, with a flow speed of around 500 ml/min, resulting in the
solution in the flow-cell to be completely replaced around every 12 min. The
solution entered and exited the flow-cell through cones (figure 1b),
minimizing the turbulence of the flow and leading to a quasi-laminar flow
along the steel surface. From the other end of the flow-cell, the solution
flowed back into the reservoir. We used a $\sim$0.46 mM NaCl solution, pH 7,
with an electrical resistivity of 130-140 $\Omega$m.

The RE consisted of an AgCl wire in a glass capillary filled with saturated
KCl solution. At the junction, the capillary was placed in a second capillary
containing the flow-cell NaCl solution, to protect both the RE and the
solution in the flow-cell: The KCl solution was hindered from leaking directly
into the flow-cell, and the KCl in the RE had minimal influence on the flow-
cell solution, ensuring the stability of the RE. These reference electrodes
could be screwed on top of the flow-cell (see figure 1c), so that they could
be calibrated before each experiment against an external Ag/AgCl/Sat. KCL
Reference electrode. RE 5, 6, 10, 11, 13, 14, 18 and 19 were permanently fixed
on the flow-cell for a duration of 4 days maximum, due to space limitations.
These electrodes were calibrated directly in the flow-cell before the
measurements, placing the external reference electrode in the RE positions
closest to the fixed RE.

Figure 2: Overview of the geometry and mesh of the numerical model. The model
is 2D axi-symmetric around the z-axis. Surfaces on which specific boundary
conditions apply are labelled: the anodic and cathodic surface, and the
counter electrodes (CE). The model parameters $W_{an}$, the width of the
anode, and $z_{an}$, the location of the middle of the anode, are indicated.

###  2.2 ECT

ECT is described in detail in van Ede et al. (2021) [28]. It uses a stochastic
inverse technique to find probability distributions for a set of model
parameters that produce a good fit between the measured potentials at the
surface and a numerical model. This numerical model simulates the electrical
potential field, and thus the potential at the surface, for the investigated
geometry. Figure 2 shows a schematic overview of the geometry and the mesh of
the 2D axi-symmetric numerical model, used to represent the laboratory setup.
The mesh consists of tetrahedral elements, that have a maximum size of 5 mm at
the outer surface and get finer towards the steel surface, where they approach
a size of <1 mm.

For carbon steel, we assume that the main anodic reaction is the oxidation of
iron. The main cathodic reaction for the stainless steel is the oxygen
reduction reaction (ORR), as the solution is well aerated and around neutral
pH. The strong-form formulations for the numerical model are derived in van
Ede et al. (2021) [28] and are given by:

$\nabla^{2}\phi=0\ \ \text{in the electrolyte, }\Omega$ (1)

with boundary conditions:

$\displaystyle\frac{d\phi}{d\overrightarrow{n}}=-\rho*i_{Fe,0}*\exp{\frac{2.303(\phi-\phi_{Fe}^{rev})}{\beta_{Fe,an}}}\
\ \text{at the anode, }\partial\Omega_{anode}\text{,}$ (2)
$\displaystyle\frac{d\phi}{d\overrightarrow{n}}=\rho*\left[\frac{\left(i_{O_{2},0}\exp{\frac{2.303(\phi_{O_{2}}^{rev}-\phi)}{\beta_{O_{2},cath}}}\right)^{\gamma}}{1+\left(\frac{i_{O_{2},0}}{i_{L}}\exp{\frac{2.303(\phi_{O_{2}}^{rev}-\phi)}{\beta_{O_{2},cath}}}\right)^{\gamma}}\right]^{\frac{1}{\gamma}}\
\ \text{at the cathode, }\partial\Omega_{cathode}\text{,}$ (3)
$\displaystyle\frac{d\phi}{d\overrightarrow{n}}=-\rho*i_{CE}\ \ \text{at the
CE, }\partial\Omega_{CE}\text{,}$ (4)
$\displaystyle\frac{d\phi}{d\overrightarrow{n}}=0\ \ \text{at all other
boundaries}\text{.}$ (5)

Here, $\phi$ is the electrical potential, $\rho$ the electrical resistivity
and $\overrightarrow{n}$ the normal to the surface. The reversible potentials,
$\phi_{Fe}^{rev}$ and $\phi_{O_{2}}^{rev}$, can be computed using the Nernst
equation [34]. The Tafel slopes of iron oxidation and oxygen reduction are
given by $\beta_{Fe,an}$ and $\beta_{O_{2},cath}$ respectively, and $i_{Fe,0}$
and $i_{O_{2},0}$ are the exchange current densities. The oxygen concentration
controlled plateau of the oxygen reduction reaction is described by the
limiting current density, $i_{L}$, which is, due to the aeration of the used
solution, assumed to be high in the current setup (2 A/m2) and $\gamma$ is a
curvature defining constant, controlling the sharpness of the cathodic
polarization curve in the transition from the activation-controlled to the
concentration-controlled plateau [35]. Finally, $i_{CE}$ is the current
density at the counter electrodes, which is obtained by dividing the applied
current by the total surface of the counter electrodes.

The numerical model simulates the potential field as a function of 6 model
parameters: The width of the anode, $W_{an}$, and the location (z-coordinate)
of the centre of the anode, $z_{an}$ (figure 2) and four kinetic parameters,
the Tafel slopes, $\beta_{Fe,an}$ and $\beta_{O_{2},cath}$, and the exchange
current densities, $i_{Fe,0}$ and $i_{O_{2},0}$. All other parameters are
fixed and given in table 1.

ECT uses a Bayesian inverse approach to obtain the posterior probability
distribution of the model parameters, given the measured potentials at the
surface and a-priori information about the model parameters [28]. The noise
associated with the error of the potential measurements was assumed to be a
Gaussian, with a variance of 0.005 V2, which corresponds to a standard
deviation of around 7 mV. The considered prior probability distributions for
each model parameters are given in table 1. As described in van Ede et al.
(2021) [28], the prior probabilities for $W_{an}$ and $z_{an}$ are determined
from the shape of the measured potential distribution at the surface, when the
steel is not polarized (see figure 4a). The priors for the kinetic parameters
are based on experiments on stainless and carbon steel in an RDE setup at pH
7.5. In these experiments environmental factors, such as the convection of the
solution and the condition of the steel surface (clean, corroded and
containing a passive film), was varied, to obtain a realistic spread of the
kinetic parameters that could be observed in a macro-cell setting (as
presented for the ORR kinetics in [36]). The exchange current densities were
assigned a Gaussian prior probability, while the Tafel slopes had a flat prior
probability distribution, meaning that all values within the indicated range
in table 1 are equally likely.

The marginal posterior distributions were sampled using a Markov chain Monte
Carlo (MCMC) algorithm [28]. These chains contained 60,000 samples, obtained
after 5,000 burn-in samples. The samples approximate the probability
distribution for each model parameter separately. Finally the probability
distribution for the corrosion rate was sampled, by integrating the current
density at the anode surface, as described in van Ede et al. (2021) [28].

Table 1: The model constants fixed in the numerical model and the model
parameters, considered in the inversion, with their respective prior
distributions. ’Flat’ refers to a constant probability distribution within the
indicated range, ’Gaussian’ to a normal distribution with an indicated mean,
$\mu$, and standard deviation, $\sigma$ Model constants
---
Parameter | Description | Value [unit]
$\phi_{Fe}^{rev}$ | Reversible potential Fe oxidation (pH 7 [37] | -0.61 [V vs SHE]
$\phi_{O_{2}}^{rev}$ | Reversible potential O2 reduction (pH 7) [37] | 0.82 [V vs SHE]
$i_{L}$ | Limiting current density O2 (well aerated) | 2.0 [A/m2]
$\gamma$ | Curvature defining constant [35] | 3
Model parameters (solved by the inverse method)
Parameter | Description | Prior [unit]
$W_{an}$ | Width of the anode |  | ’Flat’ [m]
---
0.01-0.3
$z_{an}$ | Location of the middle of the anode |  | ’Gaussian’ [m]
---
$\mu$ = 0.40, $\sigma$ = 0.075
$i_{Fe,0}$ | Exchange current density Fe oxidation |  | ’Gaussian’ log10([A/m2])
---
$\mu$ = -3.25, $\sigma$ = 0.880
$i_{O_{2},0}$ | Exchange current density O2 reduction |  | ’Gaussian’ log10([A/m2])
---
$\mu$ = -6.98, $\sigma$ = 0.957
$\beta_{Fe,an}$ | Anodic Tafel slope Fe oxidation |  | ’Flat’ [V/dec]
---
0.01-0.2
$\beta_{O_{2},cath}$ | Cathodic Tafel slope O2 reduction |  | ’Flat’ [V/dec]
---
0.01-0.3

###  2.3 Experimental procedure

To ensure the reliability of our validation data, we performed four
independent experimental runs, where potentials were monitored for different
externally applied currents at the CE. These 4 runs were performed in freshly
prepared electrolytes, that had similar electrical resistivity, around 130-140
$\Omega$m (table 2). Before each experiment, the steel bar was ground and
degreased by hand, using grinding paper and ethanol, to remove corrosion
products and ensure similar starting conditions for each experimental run. The
22 RE were calibrated against an external Ag/AgCl/Sat. KCl reference
electrode, before placing them on the flow-cell. The RE that were fixed on the
flow-cell (see section 2.1), were calibrated directly in the flow-cell for run
1, run 2 and run 3. The NaCl solution was stirred and bubbled with air, before
filling the flow-cell. As soon as the flow-cell was filled and the reference
electrodes were placed, the flow was started and the open circuit electrical
potentials (OCP), as well as the current between the anode and cathode were
monitored.

Table 2:  The performed repetitions of the experimental validation, with slightly varying resistivity of the electrolyte and submerge time of the steel bar in the flow-cell, before the start of the ECT measurement procedure. Exp. |  | Electrical resistivity
---
$\rho$ [$\Omega$m]
| submerge time
---
[hours]
| CE configurations
---
run 1 | 140 +/- 2 | 16 | ’All CE’ and ’CE2 & CE3’
run 2 | 138 +/- 2 | 17 | ’All CE’ and ’CE2 & CE3’
run 3 | 133 +/- 2 | 16 | ’All CE’, ’CE2 & CE3’ and ’CE1
run 4 | 128 +/- 2 | 20 | ’All CE’ and’CE2 & CE3’
Table 3:  The applied currents and the duration of the pulse (equal to the pause after the pulse) of the Galvanostatic pulse procedure (figure 3) | Applied currents
---
[mA]
| Galvanostatic pulse duration
---
[min]
\+ 0.10 | 5
\- 0.10 | 5
\+ 0.15 | 8
\- 0.15 | 8
\+ 0.20 | 10
\- 0.20 | 10
\+ 0.30 | 10
\+ 0.50 | 10
\+ 0.70 | 10

After a submerge time of around 16-20 hours (table 2), the monitored
electrical potentials and current reached stable values (see figure A1 in the
supplementary materials) and the ECT measurement procedure was started. The
complete procedure is summarized in the flowchart in figure 3. For different
CE configurations (’All CE’, ’CE2 & CE3’ and ’CE1 & CE4’), 9 increasing
cathodic and anodic currents were applied alternately for a certain
polarization duration (table 3 and visualized in figure 3). After each
individual applied current, the OCP was monitored for the same length of time
as the polarization duration, before applying the next current step. Only in
run 3 all three CE configurations were measured, due to time restrictions
(table 2). During the experiments the resistivity of the solution was measured
several times and was found to be stable within a +/- 2 $\Omega$m deviation.

Figure 3: Flowchart of the ECT measurement procedure.

##  3 Results and Discussion

###  3.1 ECT validation

Figure 4 gives an overview of and compares the potential and macro-cell
current measurements obtained during the 4 experimental runs. These results
show a good reproducibility of the obtained data in the experimental setup.
The OCP over distance (figure 4a) and the macro-cell current measured between
the anode and the cathode (figure 4b), show only a small variation between the
4 runs. This variation can arise due to factors such as small drift of the RE
or the variation in the condition of the steel surface. Part can be explained
by the small differences in the electrical resistivity of the solution, as a
slight trend of the macro-cell current is visible as a function of the
resistivity (figure 4b), which correlates with the differences in potential
directly above the anode at RE 12 (figure 4a). Figures 4c and d show the
measured potential at RE 2 and RE 12, respectively, during the galvanic pulse
procedures (see figure 3), using CE configuration ’all CE’. Again the
variation in potential for the different runs is visible, but these figures
also show that the response to the external polarization is very similar in
magnitude.

A further important observation is that the results are stable over time.
After applying each current step, the potential generally reached a stable OCP
within the assigned time (see table 3), before the next pulse is applied
(figures 4c and d). Additionally, figure 4a shows the mean and standard
deviation of the OCP, before each galvanostatic current step during the
measurement procedure. The small standard deviation indicates that this OCP is
very stable over time. This lets us conclude that the applied currents during
the galvanic pulses are small enough, and applied for a sufficiently short
duration, so that the corrosion system is not affected by the external
polarization in the setup.

Figure 4: The experimental results of all 4 runs (table 2). a) The mean of the
OCP, measured when no current is applied directly before each current step (19
measurements for runs 1,2 and 4, 28 measurements for run 3). The shaded
regions represent the standard deviation. b) The macro-cell current flowing
between the cathode and the anode before the start of the measurement
procedure. c) The applied current and measured potential over time, for
reference electrode 2, during the measurement procedure with all CE used. d)
The same results for reference electrode 12, located directly above the anode.

In figure 5, the estimated probability density functions (pdf) are shown for
all six model parameters, as well as the corrosion rate, obtained by using ECT
for all experimental data of run 3. For this run, the most information was
obtained, as it includes the electrical potentials at the surface of the flow-
cell, measured for 9 different applied external currents and for three
different CE configurations (table 2, figure 3). The results show that ECT is
well able to estimate the anode location and size for this experimental setup.
In figure 5a, we can see a large gain of information for the width of the
anode, $W_{an}$, and the location of the anode, $z_{an}$, when compared to the
prior information (indicated in red). The probability distributions are
unimodal and symmetric. The distributions can be approximated by a Gaussian,
with a standard deviation of 0.01 m for $W_{an}$, and 0.002 m for $z_{an}$.
Figure 5 also shows the maximum-likelihood model (dashed yellow lines), which
are the model parameters that produce the highest posterior probability, and
therefore are a function of the prior probability and the fit between the
observed data and the numerical model [28]. For $W_{an}$ and $z_{an}$, this
model is similar to the mean of the marginal distributions (dashed black
lines) and is a good estimate of the known, ground-truth values of $W_{an}$ =
0.05 m, and $z_{an}$ = 0.4 m (solid black lines).

Figure 5: ECT results of experimental run 3, obtained with all data from the
three measurements with 9 currents and different CE positions. a) The
probability density distributions (pdf, blue) of the 6 model parameters, with
the prior probability densities (red). b) The pdf of the corrosion rate. The
dashed black lines indicate the mean of the marginal distributions, the solid
black lines indicate the ground-truth values for $W_{an}$, $z_{an}$ and the
current density at the anode (obtained from the measured macro-cell current
between the anode and cathode). The maximum-likelihood models are indicated in
yellow.

Experimental runs 1, 2 and 4 only contained 2 CE configurations (table 2).
However using all data from these configurations, a similar quality of results
could be obtained for $W_{an}$ and $z_{an}$. Figure 6 compares the results for
$W_{an}$, $z_{an}$, as well as the corrosion rate, represented as box plots.
It also shows the results for run 3, if only the data obtained with 2 CE
configurations were used. The probability distributions can be found in
section B of the supplementary materials. The representation of a box plot
indicates the mean value in the centre of the box, the 25th and 75th
percentiles with the bottom and top edges of the box and the extreme values
with the whiskers. Figure 6 shows that all experimental runs are well able to
estimate the anode location, within a maximum offset of 0.006 m. Runs 1, 2 and
4, using only 2 CE configurations, have a tendency to estimate a lower
$W_{an}$ than run 3 (using all configurations), showing a slight
underestimation of 0.012 m maximum. These offsets are well within the accuracy
needed in engineering applications. The larger offset of the runs 1,2, and 4
is the result of a smaller amount of information due to missing CE
configuration ’CE1 & CE4’, as a similar underestimation is visible for run 3,
when only the data of 2 CE configurations are considered. Section 3.2.1 will
discuss this in more detail.

Figure 6: ECT results of all experimental runs obtained with the data of 2 CE
configurations (’CE2 & CE3’ and ’all CE’), compared to the results of
experimental run 3 obtained with all CE configurations (table 2), for a)
$W_{an}$, b) $z_{an}$ and c) the corrosion rate, represented as box plots, for
all experimental runs. In red the known values for $W_{an}$, $z_{an}$ and the
measured current density at the anode (obtained from the macro-cell current
between the anode and cathode). In yellow the maximum-likelihood model for the
corrosion rates. The outliers are omitted from the box plots. The probability
distributions can be found in figure 5 and section B of the supplementary
materials.

The ECT results for the corrosion rates are shown in figures 5b and 6c. It is
important to note that this corrosion rate is actually the macro-cell
corrosion rate, $i_{macro-cell}$. It will slightly underestimate the total
corrosion rate, as $i_{total}$ = $i_{self}$ \+ $i_{macro-cell}$. ECT neglects
$i_{self}$, which is the corrosion associated to local electrochemical cells
at the anode, i.e., when an additional cathodic reaction occurs at the anode.
Similarly, the experimentally determined corrosion rate in these figures also
represents $i_{macro-cell}$, as it is obtained by dividing the measured macro-
cell current (measured between the anode and cathode), by the anode surface.
The corrosion rate can be well estimated by $i_{macro-cell}$, when
$i_{self}<<i_{macro-cell}$. The measured $i_{macro-cell}$ in the experimental
setup up is around 0.1 A/m2. This high corrosion rate will lead to the
depletion of oxygen at the anode, and therefore $i_{self}$ can be assumed to
be on the order of corrosion rates measured for uniform corrosion in a neutral
de-aerated electrolyte on carbon steel. These rates may be around 0.02-0.03
A/m2 in high conductive solution [36], but are found to be even smaller in
porous media, such as concrete [38]. Therefore $i_{self}$ can be expected to
be smaller than 10 % of $i_{macro-cell}$. In the current work, the macro-cell
corrosion rate is referred to as the corrosion rate, assuming $i_{self}$ is
negligible small in relation to the error range found of ECT (figure 5b).

Figure 5b shows that if all CE configurations are considered, the
experimentally measured corrosion rate can be well estimated by the maximum
likelihood of the probability distribution obtained by ECT for run 3. It is
visible that the distribution of the corrosion rate cannot be accurately
represented by a Gaussian. Therefore, as opposed to $W_{an}$ and $z_{an}$, the
mean values of the marginal distributions and thus the boxplots in figure 6c
might give a false sense of the overestimation of the corrosion rate by ECT.
Instead of comparing the mean to the ground-truth, the maximum likelihood
should be used. In figure 6c it is visible that if only 2 CE configurations
are used, the maximum-likelihood model overestimates the corrosion rate up to
a factor of 2. More accurate estimations would be obtained by considering the
peak of the marginal distribution of the corrosion rate, which would reduce
the overestimation to a maximum of 25 % (Supplementary materials B). However,
if data of CE configuration ’CE1 & CE4’ is included, as is the case for run 3,
the offset becomes smaller than 0.004 A/m2, corresponding to an error smaller
than 5 %. This is substantially better than the error of a factor 2 of the
corrosion rate, generally accepted for existing techniques relying on the
polarization resistance [39], and much smaller than the error range obtained
with these techniques for localized corrosion [40]. Moreover, the probability
distribution for the corrosion rate also gives a good estimation of the error
associated with the solution.

Figure 7: ECT results of all experimental runs obtained with the data of 2 CE
configurations (’CE2 & CE3’ and ’all CE’), compared to the results of
experimental run 3 obtained with all CE configurations (table 2), for a)
$i_{0,Fe}$, b) $i_{0,O_{2}}$, c) $\beta_{an,Fe}$ and d) $\beta_{cath,O_{2}}$,
represented as box plots, for all experimental runs. The outliers are omitted
from the box plots. The probability distributions can be found in figure 5 and
section B of the supplementary materials.

ECT is also able to gain information about the kinetic parameters, the Tafel
slopes and exchange current densities. For run 3, the probability
distributions obtained by ECT for these parameters show unimodal
distributions, with a single peak value close to the mean (figure 5a). Similar
results are found for the other three runs (figure 7, supplementary materials
B). For the ORR kinetics, there is a large information gain compared to the
prior. The distributions show a Tafel slope, $\beta_{cath,O_{2}}$, of around
0.2 V/dec, and an exchange current density, $i_{0,O_{2}}$, of around 10-6.5
A/m2. These values correspond well to earlier documented values for ORR on
stainless steel in a near neutral environment [36]. The information gain for
the anodic iron dissolution reaction kinetics is smaller. The distribution of
$i_{0,Fe}$, follows the prior distribution, showing a range around 10-3 to
10-4 A/m2. The distribution for the anodic Tafel slope, $\beta_{an,Fe}$, shows
a peak around 0.08 to 0.11 V/dec. These values are in the range of documented
values for iron oxidation in literature [41, 34].

In summary, the ECT results show that the method can provide an accurate
estimation of the anode size, location and the (macro-cell) corrosion rate.
For run 3 all data were included in the inversion that could be obtained in
the experimental setup, giving ECT the maximum amount of information as input.
It shows that the relatively simple numerical model, based on Butler-Volmer
kinetics, the diffusion limited current density, and homogenous conductivity,
is accurate enough within the current range of the applied galvanostatic
pulses. Furthermore, the amount of data is sufficient to obtain unimodal
probability distributions, thus showing a single peak value for all six model
parameters. The drawback of using this large amount of input information is
that the measurement time to obtain all data, using the current procedure, is
around 7.5 hours. Moreover, the computational time of ECT, using 4 cores on
the ETH Zurich Euler cluster for the computation of the forward model, takes
around three weeks for this amount of data. Decreasing the amount of input
data would decrease both the measurement and computation time.

In the following sections, we first elaborate on the adjustments made to ECT
compared to the technique as described in van Ede et al. (2021) [28], that
were crucial to obtain the presented validation results. Secondly, we discuss
two possibilities to decrease the amount of input data needed for ECT, thereby
decreasing the measurement and computation time, while still obtaining
reliable results. Here, the measurements obtained during run 3 are used, as it
contains all three CE configurations.

Figure 8: The maximum offset of the potential from OCP to the minimum and
maximum applied external current respectively, measured by the RE at the
surface of the flow-cell setup. In blue, currents in the range of -0.08 to 0.2
mA, applied through counter electrodes (CE) in the region of the cathodes of
the macro-cell (CE1 & CE4) (a), as used in van Ede et al (2021) [28]. In
yellow, currents applied in the range of -0.2 to 0.7 mA, applied through CE in
the region of the cathodes (CE1 & CE4) (b), through CE located in the region
of the anode (CE2 & CE3) (c) and using all 4 CE (d), for experimental run 3
(table 2).

###  3.2 Adaptions to the ECT technique

####  3.2.1 Adjusting the current range & the positions of the CE

ECT as described in van Ede et al. (2021), used a current range between -0.08
and 0.2 mA, applied at CE1 & CE4 (see figure 1), positioned in the region of
the cathodes. Figure 8a shows the maximum variation in potential measured at
the reference electrodes in the flow-cell setup, when applying this current
range with CE configuration ’CE1 & CE4’. It shows that the external
polarization predominantly changes the potential at the cathode. However, in
the centre at the location of the anode, the stimulation of the system is very
small. Thus, in this configuration, most information is obtained about the
cathode.

In the current work we increased the range to a minimum anodic current of -0.2
mA and a maximum cathodic current of +0.7 mA (table 3). The presented ECT
results (figure 5) showed that our numerical model is accurate within this
current range. However, this increase in currents, using only the CE in the
regions of the cathode, does not increase the potential variation at the anode
(figure 8b). To stimulate the anodic region, counter electrodes closer to the
anode are needed. This is apparent from the results displayed in figure 8.
When applying the galvanostatic polarization through CE2 and CE3, positioned
closer to the anode, we see a larger potential offset in the region of the
anode. Thus, using the additional CE configuration ’CE2 & CE3’, ECT can obtain
more information about the anode, compared to the situation when only CE in
the regions of the cathode of the macro-cell are used.

Figure 9: ECT results of experimental run 3, obtained with data of the three
different CE configurations: ’CE1 & CE4’ in the region of the cathode, ’CE2 &
CE3’ near the anode, and all CE. a) the probability distributions of the model
parameters, with the prior probabilities in red. b) the probability
distribution of the corrosion rate. The solid black lines, for $W_{an}$,
$z_{an}$ indicate the ground-truth values, and the current density at the
anode (obtained from the measured macro-cell current between the anode and
cathode).

The effect of the use of different CE configurations can also be observed in
figure 9. Here the ECT results are shown, obtained by using the data of the
three CE configurations: ’CE1 & CE4’ in the region of the cathode, ’CE2 & CE3’
near the anode, and ’all CE’. If using only the data obtained with the ’CE2 &
CE3’ configuration, we underestimate the width of the anode, and overestimate
the corrosion rate. However, we obtain more accurate results for the anodic
reaction kinetics (closer to the results presented in figure 5). If we only
use the ’CE1 & CE4’ configuration, we overestimate the anode size and
underestimate the corrosion rate. Using ’all CE’ will give us the better
estimations of the anode width and the corrosion rate. The most accurate ECT
results will be obtained by combining the data off all different CE
configurations, as was shown in figure 5. Therefore, leaving out the data for
the ’CE1 & CE4’ configuration, results in a slightly poorer estimation for the
corrosion location, size and rate as was shown in figure 6.

Figure 10: ECT results of experimental run 3, obtained with data of CE
configurations ’CE1 & CE4’, for four model parameters, using constant values
for the anodic and cathodic Tafel slopes, $\beta_{an,Fe}$ and
$\beta_{cath,O_{2}}$. The probability densities (pdf) of the priors are given
in red. The solid black lines for $W_{an}$, $z_{an}$ indicate the ground-truth
values.

####  3.2.2 Optimizing the number of model parameters

A possible goal of ECT when used as an NDT method, is to estimate the anode
size and location, and the corrosion rate of a localized corrosion system.
Theoretically, all other parameters in the numerical model may be considered
to be constant for a known stable corrosion environment. Here, we want to
assess if the number of model parameters (degrees of freedom) could be reduced
without compromising the performance of ECT as an NDT method. A first aspect
is that accurate data on the kinetic parameters, exchange current densities
and Tafel slopes, is scarce. A large spread can be observed for documented
values in literature, first because of the subjectivity in the estimation of
these parameters from measured potential-current curves [42], and second
because of the dependency of these parameters on the measurement methodology
[36]. For this reason, in van Ede et al. (2021) [28], the exchange current
densities were included as model parameters in the inversion, while the Tafel
slopes were considered constants in the numerical model.

Figure 11: 2D histograms of the sampled model parameters for the ECT results
of experimental run 3 (figure 5). a) The anodic Tafel slope, $\beta_{an,Fe}$,
versus the exchange current density, $i_{0,Fe}$. b) The cathodic Tafel slope,
$\beta_{cath,O_{2}}$, versus the exchange current density, $i_{0,O_{2}}$.

In the current study, we observed that fixing the Tafel slopes can lead to a
wrong indication of the exchange current densities and the anode size, as is
visible in figure 10. This figure shows the ECT results of experimental run 3,
using the data for CE configuration ’CE1 & CE4’ and fixing the anodic and
cathodic Tafel slopes, $\beta_{an,Fe}$ and $\beta_{cath,O_{2}}$, to the values
indicated in figure 10. The highest values for the Tafel slopes lead to a
clear overestimation of the width of the anode. The probability distributions
for the exchange current densities are even more dependent on the chosen
values for the Tafel slopes, especially the distribution of $i_{0,O_{2}}$.
These results are caused by the trade-off between the Tafel slopes and
exchange currents densities, which is visualized in figure 11. Here, the Tafel
slopes are plotted versus the exchange current densities of the anodic and
cathodic reaction kinetics in a 2D histogram, showing the number of samples
during the MCMC sampling of the probability distribution (section 2.2).
Especially for the cathodic reaction kinetics, it shows that the chosen value
for $\beta_{cath,O_{2}}$ directly influenced the probability distribution of
$i_{0,O_{2}}$. Therefore, we suggest to include Tafel slopes as model
parameters in the inversion of ECT. Figure 5 shows that including the
different CE configurations, we obtain sufficient information for ECT to be
able to solve for all six model parameters.

###  3.3 Optimizing ECT

At this stage, the largest limitation of ECT is the measurement and
computation time. This may be largely improved by decreasing the measured
input data needed. Figure 12 investigates the number of CE configurations
needed to obtain reliable results. The figure compares the results that were
presented in figure 5, for which all three CE configurations were used, to ECT
results using only configurations ’CE1 & CE4’ and ’CE2 & CE3’, and results
using the configuration ’all CE’. For the former, the configuration ’CE1 &
CE4’ should obtain most information for the cathode, while ’CE2 & CE3’
focusses more on the anode (see figure 8). ’All CE’ on itself should contain
information about both the anode and the cathode.

Figure 12: ECT results, comparing the amount of CE configurations used for
experimental run 3 (table 2). a) The probability density distributions (pdf)
of the model parameters, with the prior probabilities in red. b) The pdf of
the corrosion rate. The solid black lines, for $W_{an}$ and $z_{an}$ indicate
the ground-truth values, and the current density at the anode (obtained from
the measured macro-cell current between the anode and cathode).

As was shown in figure 6, leaving out configurations ’CE1 & CE4’, reduced the
accuracy of the estimations of ECT. However, figure 12 shows that using
configurations ’CE1 & CE4’ and ’CE2 & CE3’, would result in probability
distributions very similar to the use of all three CE configurations. The need
for only 2 configurations would reduce the measurement time by one third.
Figure 12 also shows that a single CE configuration (’All CE’) is not
sufficient to obtain accurate results.

A further means to decrease both the measurement and computational time, would
be to decrease the amount of externally applied currents. Instead of applying
9 different currents (table 3), the number of galvanostatic pulses may be
decreased to only the 5 highest pulses, or even to only include the maximum
anodic and cathodic pulses. This is investigated in figure 13. From this
figure, we can conclude that decreasing the number of galvanostatic pulses to
the 2 largest, does not significantly affect the accuracy of the estimation
for the anode width, location and corrosion rate. Only the cathodic kinetic
parameters are slightly affected. The obtained results show that most
information is carried by the highest applied pulses.

Figure 13: ECT results using data obtained with two CE configurations, ’CE1 &
CE4’ and ’CE2 & CE3’ for experimental run 3 (table 2), comparing the number of
galvanostatic pulses used. N = 9 refers to all pulses listed in table 3, N = 5
to [0.2;-0.2;0.3;0.5;0.7] mA, and N = 2 to [-0.2,0.7]. a) The probability
density distributions (pdf) of the model parameters, with the prior
probabilities in red. b) The pdf of the corrosion rate. The solid black lines,
for $W_{an}$ and $z_{an}$ indicate the ground-truth values, and the current
density at the anode (obtained from the measured macro-cell current between
the anode and cathode).

###  3.4 Outlook

As highlighted in the introduction, the non-destructive measurement of
corrosion rates can be highly beneficial to increase the efficiency of, and
thus decrease the costs related to, the corrosion condition assessment and
repair. The experimental validation of ECT has shown that it is a promising
NDT technique to determine the location, size and rate of localized corrosion.
Because it requires the external polarization of the steel, mapping of the
corrosion rate over large areas of engineering structures, as is done commonly
for the electrical potential, is not realistic. Instead the measurement of the
corrosion rate is most valuable in combination with existing commercial
techniques, such as half-cell potential mapping. Corrosion rate measurements
could be performed at high-risk sites, to gain confidence about the need for
repair.

To apply ECT in the field, further developments should first of all be
focussed on decreasing the measurement and computation time. By using only two
CE configurations and two externally applied galvanostatic pulses, the
measurement time is reduced to 40 minutes, which is still too long for future
field applications. Ideally measurements do not take more than a few minutes,
even if it would only be applied on locations with a high suspicion of
corrosion. To decrease the measurement time, ideally, the technique should
move from steady state measurements, to pulse measurements.

The reduction of the needed input data would reduce the computational time of
the described stochastic inversion approach only by half. This can be reduced
by further parallelization of the MCMC sampling, the use of GPUs and/or
decreasing the amount of samples to 30,000 - 40,000, as was shown in the
convergence assessment in van Ede et al. (2021) [28]. However, to be able to
perform, fast, real-time time corrosion rate measurements in the field, ECT
should transition from its described stochastic inverse approach, to a
gradient-based deterministic approach. The stochastic inversion allowed us to
investigate the inverse problem and optimize for the measured input data
needed. It found well-behaved, unimodal probability distributions for all
model parameters, of which the maximum-likelihood models were well able to
estimate the corrosion location and rate. Instead of the estimation of the
complete posterior, deterministic approaches seek to find the maximum-
likelihood, by minimizing a misfit function. This misfit function is mainly
described by the difference in observed data and the data produced by the
numerical model with a set of estimated model parameters. Gradient-based
deterministic approaches are computationally fast, but their main limitation
is the danger of finding a local minimum of the misfit function, instead of
the minimum that corresponds to the maximum-likelihood model [43, 44].
However, the obtained posterior distributions in the current work have shown
that if any initial value is chosen from the prior distributions, the chance
of converging towards a local minimum is small. Therefore, the application of
deterministic inverse methods in ECT is expected to reliably find the maximum-
likelihood model, and could substantially decrease the computation time.

Finally, the current work showed the validation of a controlled macro-cell
corrosion system in solution. Further steps should include the validation in
porous media such as soil and concrete. ECT will most likely need further
refinements. First of all, in porous media, there are several influences on
the electrical potential that are not incorporated in the numerical model,
such as diffusion potentials [45] and a heterogeneous resistivity [46]. It is
to be investigated how much these influences decrease the accuracy of ECT, and
if adjustments of the numerical model are necessary. Secondly, the accuracy of
ECT will be dependent on the size of the anode. The larger the anode, the
smaller the effect of the applied galvanic pulses will be (as shown by the
authors in [47]), and possibly the currents have to be adjusted accordingly to
get sufficient potential responses at the reference electrodes. Finally, the
detected anode might consist of multiple smaller anodes. It would be valuable
to evaluate the error of ECT in relation to assuming that these form one
single anode.

##  4 Conclusions

The validation presented in the current work has shown that ECT is a reliable
non-destructive technique to determine the location, size and rate of
localized corrosion. The following major conclusions are drawn:

  1. 1.

The validation of ECT was presented for the study of a localized corrosion
system in a controlled laboratory setup, which used a flow-cell to keep the
electrolyte constant in terms of aeration and electrical resistivity, and
simulated macro-cell corrosion using the galvanic interaction of carbon and
stainless steel. Within the current range of the galvanostatic pulses, the
numerical model was found to be well able to describe the experimental
behaviour. Additionally, the corrosion system was not permanently influenced
by the external polarization, which means that the application of ECT can be
considered "nondestructive".

  2. 2.

ECT is able to accurately estimate the anode size, location and corrosion rate
in the controlled laboratory setup, when the steel was polarized using counter
electrodes positioned in the region of the cathode and in the proximity of the
anode. The anode size and location were estimated with a standard deviation of
less than 1 cm, well within the margin of the accuracy needed in engineering
practice. Corrosion rates could be determined within an offset of 5 % compared
to the measured macro-cell currents between the carbon and stainless steel, an
accuracy that is much smaller than existing techniques relying on the
polarization resistance.

  3. 3.

ECT can give us insight into fundamental parameters of macro-cell corrosion.
It was able to solve for all kinetic parameters, the Tafel slopes and exchange
current densities of the oxygen reduction reaction and iron dissolution,
finding well defined and reproducible probability distributions and estimating
values well within the range of the literature.

  4. 4.

Further optimization of ECT, in terms of the measurement time, was possible by
decreasing the number of applied galvanostatic pulses, and the amount of used
counter electrode locations.

Due to our ageing infrastructure, the need for non-destructive testing for the
condition assessment of civil infrastructure will keep increasing in the
coming years. ECT can be a valuable asset in the early detection and
quantification of localized corrosion, allowing for more efficient
maintenance, minimizing safety risks and reducing the economic impact of
corrosion.

##  5 Acknowledgements

This work was supported by the Swiss National Science Foundation [projects no.
PP00P2_163675 and PP00P2_194812].

##  6 Data availability

Raw measurement data of the 4 experimental runs will be made available in an
online repository.

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