An integrated approach to electrochemical impedance spectroscopy

Available online at www.sciencedirect.com
Electrochimica Acta 53 (2008) 7360–7366
An integrated approach to electrochemical impedance spectroscopy
Mark E. Orazem a,∗ , Bernard Tribollet b
a Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA
b UPR15 du CNRS, Laboratoire Interfaces et Systèmes Electrochimiques, Université P. et M. Curie,
CP 133, 4 Place Jussieu, 75252 Paris, France
Received 20 August 2007; received in revised form 25 October 2007; accepted 31 October 2007
Available online 21 December 2007
Abstract
A philosophy for electrochemical impedance spectroscopy is presented which integrates experimental observation, model development, and error
analysis. This approach is differentiated from the usual sequential model development for given impedance spectra by its emphasis on obtaining
supporting observations to guide model selection, use of error analysis to guide regression strategies and experimental design, and use of models to
guide selection of new experiments. These concepts are illustrated with two examples taken from the literature. This work illustrates that selection
of models, even those based on physical principles, requires both error analysis and additional experimental verification.
© 2007 Elsevier Ltd. All rights reserved.
Keywords: Impedance spectroscopy; Modeling; Error analysis
1. Introduction
While impedance spectroscopy can be a sensitive tool
for analysis of electrochemical and electronic systems, an
unambiguous interpretation of spectra cannot be obtained by
examination of raw data. Instead, interpretation of spectra
requires development of a model which accounts for the
impedance response in terms of the desired physical properties.
Model development should take into account both the impedance
measurement and the physical and chemical characteristics of
the system under study.
It is useful to envision a flow diagram for the measurement
and interpretation of experimental measurements such as
impedance spectroscopy. Barsoukov and Macdonald proposed
such a flow diagram for a general characterization procedure
(Figure 1.2.1 in both references [1] and [2]) consisting of two
blocks comprising the impedance measurement, three blocks
comprising a physical (or process) model, one block for an
equivalent electrical circuit, and blocks labeled curve fitting
and system characterization. They suggested that impedance
data may be analyzed for a given system by using either an
exact mathematical model based on a plausible physical theory
∗ Corresponding author. Tel.: +1 352 392 6207; fax: +1 352 392 9513.
E-mail address: meo@che.ufl.edu (M.E. Orazem).
or a comparatively empirical equivalent circuit. The parameters
for either model can be estimated by complex nonlinear
least squares regression. The authors observed that ideal electrical
circuit elements represent ideal lumped constant properties;
whereas, the physical properties of electrolytic cells are often
distributed. The distribution of cell properties motivates use of
distributed impedance elements such as constant-phase elements
(CPE). An additional problem with equivalent circuit analysis,
which the authors suggest is not shared by the direct comparison
to the theoretical model, is that circuit models are ambiguous
and different models may provide equivalent fits to a given spectrum.
The authors suggest that identification of the appropriate
equivalent circuit can be achieved only by employing physical
intuition and by carrying out several sets of measurements with
different conditions.
A similar flow diagram was presented by Huang et al. [3]
for solid-oxide fuel cells (SOFCs). The diagram accounts for
the actions of measuring impedance data, modeling, fitting the
model, interpreting the results, and optimizing the fuel cell for
power generation. The authors emphasize that the interpreting
action depends more on the electrochemical expertise of the
researchers than on a direct mapping from model parameters to
SOFC properties.
While helpful, the flow diagrams proposed by Barsoukov and
Macdonald [2] and Huang et al. [3] are incomplete because they
do not account for the role of independent assessment of exper-
0013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2007.10.075

M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366 7361
imental error structure and because they do not emphasize the
critical role of supporting experimental measurements. Orazem
et al. proposed a flow diagram (Figure 6 in reference [4]) consisting
of three elements: experiment, measurement model, and process
model. The measurement model was intended to assess the
stochastic and bias error structure of the data; thus, their diagram
accounts for the independent assessment of experimental error
structure. Their diagram does not, however, account for supporting
non-impedance measurements, and the use of regression
analysis, while implied, is not shown explicitly. The object of
this work is to formulate a comprehensive approach which can be
applied to measurement and interpretation of impedance spectra.
2. Integration of measurements, error analysis, and
model
A refined philosophical approach toward the use of
impedance spectroscopy is outlined in Fig. 1 where the triangle
evokes the concept of an operational amplifier for which
the potential of input channels must be equal. Sequential steps
are taken until the model provides a statistically adequate representation
of the data to within the independently obtained
stochastic error structure. The different aspects which comprise
the philosophy are presented in this section.
2.1. Impedance measurement are integrated with
experimental error analysis
All impedance measurements should begin with measurement
of a steady-state polarization curve. The steady-state
polarization curve is used to guide selection of an appropriate
perturbation amplitude and can provide initial hypotheses
for model development. The impedance measurements can
then make be performed at selected points on the polarization
curve to explore the potential dependence of reaction rate constants.
Impedance measurements can be performed as well as
a function of other state variables such as temperature, rotation
speed, reactant concentration. Impedance scans made at different
points of time can be used to explore temporal changes in
system parameters. Some examples include growth of oxide or
corrosion-product.
Fig. 1. Schematic flowchart showing the relationship between impedance measurements,
error analysis, supporting observations, model development, and
weighted regression analysis.
The impedance measurements should also be conducted in
concert with error analysis with emphasis on both stochastic
and systematic bias errors. The bias errors can be defined
to be those that result in data that are inconsistent with the
Kramers–Kronig relations. An empirical error analysis can
be undertaken using the measurement model approach suggested
by Agarwal et al. [5–7] and Orazem [8]. It should be
noted, however, that this approach is not definitive because
Kramers–Kronig-consistent artifacts can be caused by electrical
leads and the electronics. Use of dummy cells can help identify
artifacts that are consistent with the Kramers–Kronig relations.
As an alternative approach for identification of stochastic
part of the error structure, Dygas and Breiter have shown
that impedance instrumentation could, in principle, provide
standard deviations for the impedance measurements at each
frequency [9].
The feedback loop shown in Fig. 1 between EIS Experiment
and Error Analysis indicates that the error structure is
obtained from the measured data and that knowledge of the
error structure can guide improvements to the experimental
design. The magnitude of perturbation, for example, should be
selected to minimize stochastic errors while avoiding inducing
a nonlinear response. The frequency range should be
selected to sample the system time constants, while avoiding
bias errors associated with non-stationary phenomena. In short,
the experimental parameters should be selected so as to minimize
the stochastic error structure while, at the same time,
allowing for the maximum frequency range that is free of bias
errors.
2.2. Process model is developed in concert with other
observations
The model identified in Fig. 1 represents a process model
intended to account for the hypothesized physical and chemical
character of the system under study. From the perspective
embodied in Fig. 1, the objective of the model is not to
provide a good fit with the smallest number of parameters.
The objective is rather to use the model to gain physical
understanding of the system. The model should be able to
account for, or at least be consistent with, all experimental
observations. The supporting measurements therefore provide
a means for model identification. The feedback loop shown in
Fig. 1 between Model and Other Observations is intended to
illustrate that the supporting measurements guide model development
and the proposed model can suggest experiments needed
to validate model hypotheses. The supporting experiments
can include both electrochemical and non-electrochemical
measurements.
Numerous scanning electrochemical methods such as scanning
reference electrodes, scanning tunneling microscopy, and
scanning electrochemical microscopy can be used to explore
surface heterogeneity. Scanning vibrating electrodes and probes
can be used to measure local current distributions. Local
electrochemical impedance spectroscopy provides a means of
exploring the distribution of surface reactivity. Measurements
can be performed across the electrode at a single frequency to

7362 M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366
create an image of the electrode or, alternatively, performed at
a given location to create a complete spectrum. Other experiments
may include in situ and ex situ surface analysis, chemical
analysis of electrolytes, and both in situ and ex situ visualization
and/or microscopy. Transfer-function methods such as electrohydrodynamic
(EHD) impedance spectroscopy allow isolation
of the phenomena that influence the electrochemical impedance
response [10].
2.3. Regression analysis accounts for error structure
The goal of the operation represented by the triangle in Fig. 1
is to develop a model that provides a good representation of the
impedance measurements to within the noise level of the measurement.
The error structure for the measurement clearly plays
a critical role in the regression analysis. The weighting strategy
for the complex nonlinear least squares regression should be
based on the variance of the stochastic errors in the data, and the
frequency range used for the regression should be that which has
been determined to be free of bias errors. In addition, knowledge
of the variance of the stochastic measurement errors is essential
to quantify the quality of the regression.
Sequential steps are taken until the model provides a statistically
adequate representation of the data to within the
independently obtained stochastic error structure. The comparison
between model and experiment can motivate modifications
to the model or to the experimental parameters.
3. Examples
The systems described below illustrate the approach outlined
in Fig. 1.
3.1. Deep-level states in GaAs diodes
Orazem et al. [11] and Jansen et al. [12] described the
impedance response for an n-type GaAs Schottky diode with
temperature as a parameter. The system consisted of an n-GaAs
single crystal with a Ti Schottky contact at one end and a Au,
Ge, Ni Schottky contact at the eutectic composition at the other
end. This material has been well characterized in the literature
and, in particular, has a well known EL2 deep-level state
that lies 0.83–0.85 eV below the conduction band edge [13].
Experimental details are provided elsewhere [11,12].
The experimental data are presented in Fig. 2 (a) and (b)
for the real and imaginary parts of the impedance, respectively,
for temperatures ranging from 320 to 400 K. The logarithmic
scale used in Fig. 2 emphasizes the scatter seen in the imaginary
impedance at low frequencies. The impedance response
is seen to be a strong function of temperature. The impedanceplane
plots shown in Fig. 3 for data collected at 320 and 340 K
approach the classic semicircle associated with a single relaxation
process.
Jansen and Orazem showed that the impedance data could be
superposed as given in Fig. 4[12]. The impedance data collected
at different temperatures were normalized by the maximum
mean value of the real part of the impedance and plotted against
Fig. 2. Impedance data as a function of frequency for an n-GaAs/Ti Schottky
diode with temperature as a parameter: (a) the real part of the impedance; and
(b) the imaginary part of the impedance. Data taken from Orazem et al. [11] and
Jansen et al. [12].
a normalized frequency defined by
f ∗ = f ( ) E
f ◦ exp (1)
kT
where E = 0.827 eV, and the characteristic frequency f ◦ was
assigned a value of 2.964 × 10 14 Hz such that the imaginary part
of the normalized impedance values reached a peak value near
f ∗ = 1. The data collected at different temperatures are reduced
Fig. 3. Impedance data in impedance-plane format for an n-GaAs/Ti Schottky
diode with temperature as a parameter. Data taken from Orazem et al. [11] and
Jansen et al. [12]

M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366 7363
Fig. 4. Impedance data from Fig. 2 collected for an n-GaAs/Ti Schottky diode
as a function of frequency f ∗ = (f/f ◦ ) exp(E/kT ): (a) the real part of the
impedance; and (b) the imaginary part of the impedance. Data taken from Jansen
et al. [12]
to a single line. The extent to which the data are superposed is
seen more clearly on the logarithmic scale shown in Fig. 5. The
superposition shown in Figs. 4 and 5 suggest that the system is
controlled by a single-activation-energy-controlled process.
A closer examination of Fig. 4(b) reveals that the maximum
magnitude of the scaled imaginary impedance is slightly
less than 0.5; whereas the corresponding value for a singleactivation-energy-controlled
process should be identically 0.5.
Regression analysis using the traditional weighting strategies
under which the standard deviation of the experimental values
was assumed to be proportional to the modulus of the impedance,
σ r = σ j = α|Z|, to the magnitude of the respective components
of the impedance, σ r = α r |Z r | and σ j = α j |Z j |, or independent
of frequency, σ r = σ j = α, yielded one dominant RC time constant
with only a hint that other parameters could be extracted.
Orazem et al. [11] and Jansen et al. [12] used the measurement
model approach described by Agarwal et al. [5–7] and Orazem
[8] to identify the stochastic error structure for the impedance
data. When the data were regressed using this error structure for
a weighting strategy, additional parameters could be resolved
revealing additional activation energies. Thus, while the data
do superpose nicely in Figs. 4 and 5, the impedance data do in
fact contain information on minor activation-energy-controlled
Fig. 5. Impedance data from Fig. 2 collected for an n-GaAs/Ti Schottky diode
as a function of frequency f ∗ = (f/f ◦ ) exp(E/kT ): (a) the real part of the
impedance; and (b) the imaginary part of the impedance.
electronic transitions [11,12]. The information concerning these
transitions could be extracted by regression of an appropriate
process model using a weighting strategy based on the error
structure of the measurement.
Two models have been proposed for the data presented above.
Macdonald proposed a distributed-time-constant model which
accounts for distributed relaxation processes [14]. This model
fits the data very well and has the advantage that it requires
a minimal number of parameters. A second model, presented
in Fig. 6, accounts for discrete energy levels and provides a
fit of equivalent regression quality at each temperature [11,12].
In Fig. 6, C n is the space-charge capacitance, R n is a resistance
that accounts for a small but finite leakage current, and
the parameters R 1 ...R k and C 1 ...C k are attributed to the
response of discrete deep-level energy states. Parameters corresponding
to deep-level states were added sequentially to the
model, subject to the constraint that the 2σ (95.4%) confidence
interval for each of the regressed parameters may not
include zero. Including the space charge capacitance and leakage
resistance, four resistor–capacitor pairs could be obtained
from the impedance data collected at 300, 320, and 340 K; three
resistor–11/23/2007capacitor pairs could be obtained from the
data collected at 360, 380, and 400 K; and two resistor-capacitor
pairs could be obtained from the data collected at 420 K [12].

7364 M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366
Fig. 7. Electrochemical impedance results obtained for a single-cell PEM fuel
cell with current density as a parameter. Results taken from Roy et al. [18].
Fig. 6. Electrical circuit corresponding to the model presented by Jansen et al.
[12] in which C n is the space-charge capacitance, R n is a resistance that accounts
for a small but finite leakage current, and the parameters R 1 ...R k and C 1 ...C k
are attributed to the response of discrete deep-level energy states.
This model has the disadvantage that up to eight parameters
are required, depending on the temperature, as compared to
the three parameters required for the distributed-time-constant
model. The question to be posed then, “which is the better model
for the measurements”
If the goal of the regression is to provide the most parsimonious
model for the data, the model with the smallest number of
parameters and a continuous distribution of activation energies
is the best model. If the goal of the regression is to provide a
quantitative physical description of the system for which the data
were obtained, additional measurements are needed to determine
whether the activation energies are discrete or continuously distributed.
In this case, deep-level transient spectroscopy (DLTS)
measurements indicated that the n-GaAs diode contained discrete
deep-level states. In addition, the energy levels and state
concentrations obtained by regression of the second model were
consistent with the results obtained by DLTS [12]. Thus, the second
model with a larger number of parameters provides the more
useful description of the GaAs diode. The choice between the
two models could not be made without the added experimental
evidence.
reactants [18]. The measurements were conducted in galavanostatic
mode for a frequency range of 1 kHz to 1 mHz with a 10 mA
peak-to-peak sinusoidal perturbation. Roy and Orazem [19] used
the measurement model approach developed by Orazem and
coworkers [5–8] to demonstrate that, for the fuel cell under
steady-state operation, the low-frequency inductive loops seen
in Fig. 7 were consistent with the Kramers–Kronig relations.
Therefore, the low-frequency inductive loops could be attributed
to process characteristics and not to non-stationary artifacts.
Roy et al. [18] proposed that that the low-frequency inductive
loops observed in PEM fuel cells could be caused by parasitic
reactions in which the Pt catalyst reacts to form PtO and subsequently
forms Pt + ions. They also showed that a reaction
involving formation of hydrogen peroxide could yield the same
inductive features. A comparison between the two models and
the experimental results is shown in Fig. 8. and the corresponding
values for the real and imaginary parts of the impedance are
presented as a function of frequency in Fig. 9(a) and (b), respectively.
The model calculations were not obtained by regression
but rather by simulation using approximate parameter values.
Regression was not used because the model was based on the
assumption of a uniform membrane-electrode assembly (MEA);
whereas, the use of a serpentine flow channel caused the reactivity
of the MEA to be very nonuniform. The parameters were first
selected to reproduce the current–potential curve and then the
same parameters were used to calculate the impedance response
3.2. Side reactions in PEM fuel cells
Low-frequency inductive features [15–17] are commonly
seen in impedance spectra for PEM fuel cells. Makharia et al.
[15] suggested that side reactions and intermediates involved
in the fuel cell operation can be possible causes of the inductive
loop seen at low frequency. However, such low-frequency
inductive loops could also be attributed to non-stationary behavior,
or, due to the time required to make measurements at low
frequencies, non-stationary behavior could influence the shapes
of the low-frequency features.
A typical result is presented in Fig. 7 for the impedance
response of a single 5 cm 2 PEMFC with hydrogen and air as
Fig. 8. Comparison of the impedance response for a PEM fuel cell operated at
0.2 A/cm 2 to model predictions generated using a reaction sequence involving
formation of hydrogen peroxide and a reaction sequence involving formation of
PtO. Results taken from Roy et al. [18]

M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366 7365
with a decrease in the active catalytic surface area and a loss of
Pt ions in the effluent. Cyclic voltammetry after different periods
of fuel cell operation could be used to explore reduction
in electrochemically active area. Inductively coupled plasma
mass spectroscopy (ICPMS) could be used to detect residual
Pt ions in the fuel cell effluent, and ex situ techniques could
detect formation of PtO in the catalyst layer. A different set of
experiments could be performed to explore the hypothesis that
peroxide formation is responsible for the inductive loops. Platinum
dissolution has been observed in PEM fuel cells, [21] and
peroxide formation has been implicated in the degradation of
PEM membranes [22–24]. Thus, it is likely that both reactions
are taking place and contributing to the observed low-frequency
inductive loops.
4. Discussion
Fig. 9. Comparison of the impedance response for a PEM fuel cell operated at
0.2 A/cm 2 to model predictions generated using a reaction sequence involving
formation of hydrogen peroxide and a reaction sequence involving formation of
PtO: (a) real part of the impedance; and (b) imaginary part of the impedance.
Results taken from Roy et al. [18].
for each value of current density. The potential (or current)
dependence of model parameters was that associated with the
Tafel behavior assumed for the electrochemical reactions.
While reaction parameters were not identified by regression
to impedance data, the simulation presented by Roy et
al. [18] demonstrates that side reactions proposed in the literature
can account for low-frequency inductive loops. Indeed,
the results presented in Figs. 8 and 9 show that both models
can account for low-frequency inductive loops. Other models
can also account for low-frequency inductive loops so long as
they involve potential-dependent adsorbed intermediates [20].
It is generally understood that equivalent circuit models are not
unique and have therefore an ambiguous relationship to physical
properties of the electrochemical cell. As shown by Roy
et al. [18], even models based on physical and chemical processes
are ambiguous. In the present case, the ambiguity arises
from uncertainty as to which reactions are responsible for the
low-frequency inductive features.
Resolution of this ambiguity requires additional experiments.
The processes and reactions hypothesized for a given model can
suggest experiments to support or reject the underlying hypothesis.
For example, the proposed formation of PtO is consistent
The example presented in Section 3.1 demonstrates the
importance of coupling experimental observation, model development,
and error analysis. The measurements conducted
at different temperatures allowed identification of discrete
activation energies for electronic transitions. Use of a weighting
strategy based on the observed stochastic error structure
increased the number of parameters that could be obtained from
the regression analysis. Thus, four discrete activation-energycontrolled
processes could be identified, but at the expense of a
corresponding model that required eight parameters. A regression
of almost the same quality could be obtained under the
assumption of a continuous distribution of activation energies,
and this model required only three parameters. Discrimination
between the two models requires additional experimental observations,
such as the DLTS identification of electronic transitions
involving discrete deep-level states.
The example presented in Section 3.2 demonstrates the
utility of the error analysis for determining consistency
with the Kramers–Kronig relations. In this case, the lowfrequency
inductive loops were found to be consistent with the
Kramers–Kronig relations at frequencies as low as 0.001 Hz so
long as the system had reached a steady-state operation. The
error analysis employed regression of a measurement model.
The mathematical process models that were proposed to account
for the low-frequency features were based on plausible physical
and chemical hypotheses. Nevertheless, the models are ambiguous
and require additional measurements and observations to
identify the most appropriate for the system under study.
The philosophy described here cannot always be followed
to convergence. Often the hypothesized model is inadequate
and cannot reproduce the experimental results. Even if the
proposed reaction sequence is correct, surface heterogeneities
may introduce complications that are difficult to model. The
accessible frequency range may be limited at high frequency for
systems with a very small impedance. The accessible frequency
range may be limited at low frequency for systems subject
to significant non-stationary behavior. The experimentalist
may need to accept a large level of stochastic noise for a
system with a large impedance. In cases where the models are
unable to explain all features of the experiment, the graphical

7366 M.E. Orazem, B. Tribollet / Electrochimica Acta 53 (2008) 7360–7366
methods presented by Orazem et al. [25] can nevertheless yield
quantitative information.
5. Conclusions
The philosophy embodied in Fig. 1 integrates experimental
observation, model development, and error analysis. It takes into
account the observations that impedance spectroscopy is not a
stand-alone technique and that other observations are required to
validate a given interpretation of the impedance spectra. Within
the context of the work presented here, the objective of modeling
is not to provide a good fit with the smallest number of parameters,
but, rather, to use the model to learn about the physics and
chemistry of the system under study.
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