INTERNATIONAL JOURNAL OF ENERGY RESEARCH
Int. J. Energy Res. 2005; 29:1133–1151
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/er.1144
Analysis of electrochemical impedance spectroscopy
in proton exchange membrane fuel cells
Parthasarathy M. Gomadam and John W. Weidnern,y
Center for Electrochemical Engineering, Department of Chemical Engineering, University of South Carolina,
Columbia, SC 29208, U.S.A.
SUMMARY
A literature review of electrochemical impedance spectroscopy (EIS) analysis of proton exchange
membrane fuel cells (PEMFCs) is presented. Emphasis is placed on the papers that analyse the impedance
response of the cathode and anode half-cells of the PEMFCs based on a continuum-mechanics approach.
The other type of analysis, which is based on the equivalent-circuits approach, is addressed for comparison.
The relative advantages and disadvantages of the two approaches are discussed. Papers dealing with
continuum-mechanics-based EIS modelling of general electrochemical systems are briefly reviewed.
Copyright # 2005 John Wiley & Sons, Ltd.
KEY WORDS:
impedance; EIS; fuel cell; porous electrode; catalyst layer; Nyquist plot; linear kinetics; Tafel
kinetics; equivalent circuit
1. INTRODUCTION
A schematic of a proton exchange membrane fuel cell (PEMFC) stack is shown in Figure 1
along with blown-up views of the seven-layer membrane electrode assembly and the catalyst
region. The reactant gases, hydrogen and oxygen/air, are flown into the fuel cell through the
flow-fields. The flow-fields are normally made of a light, electronically conducting material
(e.g. graphite) with channels grooved in them for gas flow. From the flow-fields the gases diffuse
through the gas-diffusion layers (GDLs) into the catalyst layers, where electrochemical reactions
(hydrogen oxidation in the anode and oxygen reduction in the cathode) occur. The GDLs are
made of porous carbon paper or cloth allowing simultaneous gas and liquid flows. The catalyst
layers are made of a porous mixture of the ionomer and platinum-loaded carbon particles,
allowing contact between the solid, ionomer, and gas phases. The Nafion1 ionomer membrane,
made of immobile sulphonate anions and mobile protons, serves to conduct the protons
generated by hydrogen oxidation from the anode catalyst layer to the cathode catalyst layer. In
the cathode catalyst layer the protons combine with the oxygen entering through the GDL to
n
Correspondence to: John W. Weidner, Center for Electrochemical Engineering, Department of Chemical Engineering,
University of South Carolina, Columbia, SC 29208, U.S.A.
y
E-mail: weidner@engr.sc.edu
Copyright # 2005 John Wiley & Sons, Ltd.
Received 8 February 2005
Revised 14 March 2005
Accepted 17 March 2005
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P. M. GOMADAM AND J. W. WEIDNER
H2 flow
O2 flow
H2 flow
O2 flow
H2 flow
O2 flow
H2 flow
Catalyst
Catalyst
O2 flow
GDL
Membrane
Carbon
Platinum
Ionomer +
Gas Pores
Figure 1. A schematic of a PEMFC stack, showing the seven-layer membrane electrode assembly and the
catalyst pore structure in detail.
form water. The generated water is transported out of the fuel cell by diffusion/permeation
through the cathode GDL into the flow-field and by back-diffusion through the membrane
towards the anode side.
According to Gasteiger and Mathias (2003) PEMFCs show excellent performance given the
right composition and operating conditions. However, today’s PEMFCs do not operate well
under extreme conditions (e.g. low humidity, high temperature, excessive gas contaminants) nor
are they cost-effective. In order to compete with the currently used power sources (e.g. internal
combustion engines) PEMFCs must operate over a wide range of conditions and meet stringent
cost requirements. As laid out by the authors, some of the desirable properties of PEMFCs are
that the operating temperature should be made higher than 1208C for efficient heat rejection, the
ionic conductivity of the membrane should be increased by over ten times at low relative
humidities, the platinum loading in the catalyst layers should be steeply reduced, the GDL
should be better tailored and optimized to allow fast gas transport while facilitating fast water
removal, and the catalyst should be made more tolerant to poisons. These properties can be
achieved by altering the materials used to make the fuel cell components, their composition, cell
design, and processing and operating conditions. Therefore, numerous researchers are involved
in characterizing fuel cell materials and their components as functions of the above variables
Copyright # 2005 John Wiley & Sons, Ltd.
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ANALYSIS OF EIS IN PEMFCs
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Figure 2. Measured in-plane ionic and electronic resistivities for various treatment types used in catalyst
layer preparation: 1. Proton form, as prepared; 2. Proton form, extracted with water; 3. Proton form, acid
boiled; 4. Tetrabutylammonium form, unpressed, acid boiled; 5. Tetrabutylammonium form, pressed, acid
boiled. Reproduced by permission of The Electrochemical Society (Saab et al., 2003).
and have reported significant effects. For example, Saab et al. (2003) observed a strong
dependence of the electronic and ionic conductivities of composite fuel cell electrodes on their
composition and processing conditions (see Figure 2).
The techniques used by researchers for characterizing fuel cells and their components involve
applying a current or voltage input and observing the response. While fuel cells are ultimately
characterized by their steady-state current–voltage responses, the great deal of extra information
obtained from transient responses is valuable for property measurements. The use of a variety of
transient techniques such as potential stepping, potential sweeping, current interrupt, and
electrochemical impedance spectroscopy (EIS) are reported in the literature (e.g. Saab et al.,
2002, 2003; Li and Pickup, 2003; Jaouen et al., 2003; Jaouen and Lindbergh, 2003). Among
these, EIS is the technique most widely used for measuring transport properties (e.g. ionic
conductivity of the membrane) in fuel cells. Researchers in PEMFCs use EIS to measure the
overall impedances of the cathode side, the anode side, as well as the whole fuel cell, through
which the fuel cell ‘health’ is diagnosed along with measurements of transport properties. While
EIS is a standard technique to obtain such information in other fields of electrochemistry such
as batteries, electrodeposition, and corrosion, fuel cells lend themselves especially to probing
with this technique because of their nearly steady-state operation.
Numerous experimental papers exist in the literature that obtain and analyse impedance data
on PEMFCs. However, on modelling impedance behaviour and on mathematically analysing
impedance data, papers are fewer. Nevertheless, mathematical analysis has been shown to be of
paramount importance if the extracted information about the materials or the transport
properties of the cell were to be accurate (Gomadam et al., 2003). In this work, we review
these papers for the variety of mathematical modelling and analyses of impedance data
Copyright # 2005 John Wiley & Sons, Ltd.
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P. M. GOMADAM AND J. W. WEIDNER
presented on PEMFCs and discuss the information obtained therein. While the popular
equivalent-circuits-based approach is outlined and briefly discussed, the more fundamental
continuum-mechanics-based approach to impedance modelling is reviewed in detail.
2. AN OVERVIEW OF EIS
The EIS experiment involves applying a small sinusoidal voltage or current perturbation around
a steady-state value and measuring the resulting current or voltage along with the phase angle.
Using this data the real and imaginary impedances are calculated and plotted against each other
for different perturbation frequencies in what are called Nyquist impedance spectra. Although
other types of plots}such as, the Bode plots, where the magnitude of the impedance and the
phase angle are plotted against frequency, the Cole–Cole plots, where the squares of the real and
imaginary impedances are plotted against each other}are also used in the literature, the
Nyquist plots are the most common way of analysing impedance data. However, in Nyquist
plots the frequency dependence of the impedance remains hidden. For this reason, Bode
plots, which provide explicit information on the frequency dependence of the impedance, are
often used.
The EIS technique forms a good diagnostic tool for evaluating performance owing to its
ability to separate the impedance responses of the various transport processes occurring
simultaneously in PEMFCs (Li and Pickup, 2003). Generally, the high-frequency region
(>100 Hz, in general) of an impedance spectrum reflects the charge transport in the catalyst
layer, whereas the low-frequency region (50.01 Hz, in general) represent mass transport in the
GDL, the catalyst layer, and the membrane. The relative importance of the transport processes
depend on the steady-state value of the overpotential at which the EIS experiment is conducted.
At low overpotentials when the mass transport resistance is not significant, the main contributor
to the impedance is the charge transport in the catalyst layer. At moderate overpotentials
proton, gas, and water transport begin to contribute to the total impedance of the cell. At high
overpotentials gas diffusion in the GDL and the catalyst layer become dominant, especially
when air is used as the oxidant.
In the various layers of the fuel cell one or more of the transport processes described above
are important depending on cell design and operating conditions. For example, electron
conduction, water transport, and gas diffusion are important in the GDL, and proton
conduction and water transport are important in the membrane. However, all these processes
coupled with distributed electrochemical reaction are important in the catalyst layer. This
distinguishes the catalyst layer from the other components in terms of the mathematical
complexity of the analysis, arising primarily due to its porous nature. Mathematical models of
PEMFCs existing in the literature differ fundamentally in the level of complexity with which the
porous catalyst layer is treated. With respect to EIS there are broadly two ways: (a) equivalentcircuits-based and (b) continuum-mechanics-based.
3. EQUIVALENT-CIRCUITS-BASED ANALYSES
The equivalent-circuits-based analysis involves reducing the transport processes into electrical
analogues made of networks of resistors and capacitors, and sometimes, inductors. The values
Copyright # 2005 John Wiley & Sons, Ltd.
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C dl,1
C dl
Re
Ri
R ct
R e,1
R ct,1
(a)
Re
R ct
Ri
Re
R ct
R ct
C dl
(b)
Re
C dl
Ri
R ct
C dl
Ri
C dl
Ri
Figure 3. Equivalent circuits for: (a) a PEMFC with catalyst layers in planar form; and (b) a half-cell with
catalyst layer in porous form. Ri ¼ ionic resistance, Re ¼ electronic resistance, Rct ¼ charge-transfer
resistance, Cdl ¼ double-layer capacitance. The resistances and capacitance of the electrode on the right are
distinguished with the subscript ‘1’.
of the resistances and capacitances are obtained by fitting the effective impedance of the network
to experimental data. This method provides us with a quick visualization tool to understand
system behaviour and limitations.
Figure 3 shows typical equivalent circuits of (a) a fuel cell treating the catalyst layer in planar
form, and (b) the catalyst layer in porous form. In the planar form, the catalyst layers are
described by the parallel combination of the charge-transfer resistance, Rct and Rct,1, with the
double-layer capacitances, Cdl and Cdl,1, respectively. The resistances (Re and Re,1) at the ends
describe the electronic resistance due to the GDLs and the flow-fields, while the resistance (Ri) in
the middle describes the ionic resistance of the membrane. The boxed region shows the
equivalent circuit of a half-cell obtained by considering a reference electrode placed in the
membrane. In Figure 3(b) the equivalent circuit of this half-cell is shown with the catalyst layer
in porous form. The parallel combination of Rct and Cdl, used to represent the entire catalyst
layer in the planar form, now represents the local interface in every volume element of the
porous catalyst layer. Two adjacent volume elements are separated by electronic and ionic
resistances. This results in the ladder-like circuit called the transmission-line model, which is
widely used to capture the impedance response of porous electrodes. A more empirical way of
achieving this is to replace the ‘normal’ double-layer capacitance (Cdl) in the planar form with a
‘distributed’ capacitance. The distributed nature is introduced by expressing the impedance of
the double-layer capacitance as the impedance of the ‘normal’ capacitance raised to the power of
an empirical parameter called the constant phase element (Lasia, 1999).
The Nyquist impedance responses of the half-cell in its planar form, the half-cell in its porous
form, and the full cell are shown in Figures 4(a)–(c), respectively. Figure 4(a) shows a semi-circle
Copyright # 2005 John Wiley & Sons, Ltd.
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P. M. GOMADAM AND J. W. WEIDNER
Figure 4. Sample Nyquist plots presented for: (a) the half-cell in planar form shown boxed in Figure 3(a);
(b) the half-cell in porous form shown in Figure 3(b); and (c) the full cell shown in Figure 3(a).
with a high-frequency intercept equal to the ionic resistance of the membrane, and a lowfrequency intercept equal to the sum of the ionic resistance of the membrane and the chargetransfer resistance of the catalyst layer. The frequency dependence of the impedance depends on
the double-layer capacitance. In the case of the transmission-line model for the porous catalyst
layer (Figure 4(b)) the high-frequency intercept is the sum of the ionic resistance of the
membrane and a function of the ionic and electronic resistances of the catalyst layer. On the
other hand, the low-frequency intercept is the sum of the ionic resistance of the membrane and a
function of the ionic, electronic, and charge-transfer resistances of the catalyst layer. Further,
the frequency dependence of the impedance is qualitatively different from the planar case,
showing a high-frequency straight line of unit slope and an asymmetric semi-circle (Lasia, 1999).
These features of the impedance spectrum of a porous electrode will be discussed in greater
detail when reviewing the theoretical analyses published by Gomadam et al. (2003). Note that
while charge-transfer resistances are often described normally as pure resistances, complex
reaction mechanisms result in non-resistive behaviours. For example, Antoine et al. (2001) show
that the oxygen reduction reaction occurring in fuel cells involves an adsorption step giving rise
to an inductive behaviour at low frequencies (around 0.6 Hz).
Using the model based on equivalent circuits, one can extract the above resistances and
capacitances from an EIS experiment conducted on a half-cell. However, because of the lack of
a reliable reference electrode, measurements are often made only on full cells. The impedance
response of a full cell is a combination of the responses of the anode and cathode half-cells as
shown in Figure 4(c). The figure shows that if the time-constants of the two half-cells are very
different, two distinct semi-circles appear such that the high-frequency intercept gives the ionic
plus electronic resistances, the moderate-frequency intercept gives the charge-transfer resistance
with the lower time-constant, and the low-frequency intercept gives the charge-transfer
resistance with the higher time-constant. On the other hand, if the time-constants of the
Copyright # 2005 John Wiley & Sons, Ltd.
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Figure 5. Capacitance vs resistance plots of a porous electrode with: (a) uniformly distributed ionic
resistance; and (b) linearly increasing ionic resistivity. Reproduced by permission of The Electrochemical
Society (Lefebvre et al., 1999).
half-cells are comparable, the two semi-circles merge and, therefore, the two charge-transfer
resistances are not separable.
Numerous papers have been published using the equivalent-circuits analysis to measure
properties of PEMFCs from impedance data. For example, Saab et al. (2002, 2003) studied the
effect of ionic and electronic conductivities of catalyst layers as functions of processing and
operating conditions, Abe et al. (2004) studied the effect of humidity in oxygen gases on the
performance of a PEMFC, Ciureanu and Wang (1999) studied hydrogen oxidation at a PEMFC
anode with and without CO, Wang et al. (2001) studied CO poisoning in the anode, and Li and
Pickup (2003) studied the effect of Nafion1 loading in the cathode catalyst layer. All these
papers use the analysis explained above to extract physical properties from EIS data. Only very
few papers (Lefebvre et al., 1999; Waraksa et al., 2003) provide alternative methods of analyses
based on the equivalent-circuits approach.
Lefebvre et al. (1999) developed a new method of analysing impedance data so as to obtain
information about the ionic conductivity profiles in catalyst layers. Assuming a constant doublelayer capacitance, they showed that analysing impedance data in the form of effective
capacitance vs resistance plots, as shown in Figure 5, reveals semi-quantitatively how the ionic
conductivity varies across the catalyst layer. They demonstrated this approach by extracting a
non-uniform ionic conductivity profile of the catalyst layer from its measured impedance
response. The approach involved fitting measured impedance data with simulations using
various assumed ionic conductivity profiles. Waraksa et al. (2003) developed a transmission-line
model for the EIS response of porous oxygen electrodes with discrete particles. Their model was
based on an array of parallel, non-uniform, transmission lines, incorporating physically realistic
elements, such as discrete particles of variable sizes and adjustable multilayer stacking
geometries. Using this analysis Chen et al. (2003) studied the EIS response of porous oxygen
electrodes with discrete particles and compared two titanium-oxide-based catalyst supports.
Copyright # 2005 John Wiley & Sons, Ltd.
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The popularity of the equivalent-circuit approach stems from the fact that it is
mathematically simple and easy to use. However, this method normally involves oversimplified
representations of the components of electrochemical systems. Therefore, this method cannot
always be used to accurately describe the system and estimate parameters, especially in porous
electrodes. Further, because the various transport processes occur simultaneously in porous
electrodes, converting the estimated resistances and capacitances into physically meaningful
parameters is difficult. For these reasons, a more fundamental, continuum-mechanics-based
approach is preferred (Gomadam et al., 2003) while the equivalent-circuit approach is good for
a quick visualization tool and, perhaps, as a first approximation. However, when certain
simplifying restrictions (e.g. invariant physical properties, linear kinetics, no mass-transfer
limitations) are enforced, the continuum-mechanics approach reduces to the equivalent-circuit
approach (Springer and Raistrick, 1989; Eikerling and Kornyshev, 1999).
4. CONTINUUM-MECHANICS-BASED ANALYSES
The continuum-mechanics approach involves mathematical descriptions of phenomena derived
from conservation equations using first principles. A number of continuum-mechanics-based
impedance models have been published for electrochemical systems. However, a vast majority of
them are for planar electrodes, which are often not applicable to the porous catalyst layers of
PEMFCs since they do not predict the qualitative features of impedance spectra measured on
porous electrodes. Owing to the mathematical complexity associated, only a few papers have
been published that analyse the impedance response of porous electrodes based on continuummechanics. De Levie (1967) developed impedance models for porous electrodes by treating them
either as made of uniform cylindrical pores flooded with electrolyte or as a macrohomogeneous
superposition of the solid and solution phases. Following them, Darby (1966) developed an
impedance model for porous gas-diffusion electrodes. An error in Darby’s model predicted that
the occurrence of reactions inside the pores will result in inductance loops in the Nyquist
spectra, which was later corrected by Keddam et al. (1984). Rangarajan (1969) developed an
impedance model for porous electrodes with potential and concentration gradients and derived
analytical solutions for linear kinetics. Cachet and Wiart (1985) developed an approximate
solution for the impedance of a porous electrode with potential and concentration gradients
under conditions of Tafel kinetics. Paasch et al. (1993) extended the above models by including
time delay. Lasia (1995) developed an impedance model for porous electrodes with potential
gradients only in the solution phase and with Butler–Volmer kinetics for finite and semi-infinite
pores. Lasia (1997) later extended the work by including concentration gradients in the pore. In
a more recent paper, Lasia (2001) discussed the nature of the two semi-circles observed on the
Nyquist spectra in the presence of concentration gradients. Lasia (1999) summarizes impedance
modelling for planar and porous electrodes.
Continuum-mechanics-based EIS models using the macrohomogeneous representation
of porous electrodes have been published in many fields of electrochemistry, especially, in Li
and Li-ion batteries. In this representation the porous electrode is treated to be a continuous
superposition of electronic and ionic phases, with average properties (Newman and
Thomas-Alyea, 2004). Doyle et al. (2002) developed an impedance model for a Li-polymer
cell sandwich incorporating charge and mass transport in the solid and solution phases. Meyers
et al. (2002) developed an impedance model for a porous intercalation electrode by combining a
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single-particle impedance model to the impedance response of a porous electrode. Guo et al.
(2002) applied the model of Doyle et al. (2002) to estimate the solid-phase diffusion coefficient of
Li and analysed the effect of design variables on the accuracy of the estimated parameter. Devan
et al. (2004) developed an analytical solution to the impedance response of a porous Li-ion
battery electrode under the presence of concentration gradients in the solution phase.
Most of the above impedance models were developed for flooded porous electrodes, where the
pores are considered to be filled with electrolyte. However, the electrodes used in fuel cells have
pores filled partly with the electrolyte and partly with the reactant gas. Relatively fewer models
exist in the literature that deal with continuum-mechanics-based analysis of such gas-diffusion
electrodes. Such models as applicable to PEMFCs are reviewed below in detail.
Springer and Raistrick (1989) calculated the impedance of a differential element of the pore
wall based on a flooded-agglomerate model for the catalyst layer of PEMFCs. Assuming
negligible ohmic drops in the electronic and ionic phases and a uniform reaction distribution,
they integrated the differential impedance over the pore length to obtain the total impedance of
the gas-diffusion electrode. However, the reaction rate distribution in a porous electrode
depends on the perturbation frequency; lower the frequency, more uniform the distribution.
Consequently, their model predictions, accurate only at relatively low frequencies, show
qualitative errors at high frequencies. For example, the high-frequency straight line in the
Nyquist plots, characteristic of porous electrodes, cannot be predicted by this model. At low
frequencies, the model predicts a Warburg straight line arising due to semi-infinite diffusion
of oxygen in the agglomerate. They combined the flooded-agglomerate impedance model in
series with a model for a thin electrolyte film surrounding the agglomerate. Using this ‘complete’
agglomerate model they calculated the pore wall impedance for different steady-state
currents. They also developed transfer functions that simplified the numerical fitting procedure
of model predictions to experimental data. An important conclusion made in their work
was that the agglomerate itself does not lead to a steady-state-limiting current behaviour,
but rather to a doubling of the effective Tafel slope when current penetration of the agglomerate
region is incomplete. This is because until current penetration of the agglomerate is
complete there will be a region of the agglomerate with a non-zero reactant concentration
and, therefore, a limiting current does not occur. They showed that it is the thin electrolyte
film, either by itself or in conjunction with the agglomerate, that ultimately leads to a diffusionlimited current.
In a later paper, Springer et al. (1996) developed an impedance model for the cathode side of a
PEMFC. In contrast to their previous work, they based this model on macrohomogeneous
porous electrode theory for charge transport in the catalyst layer and used Stefan–Maxwell
multicomponent diffusion equations for oxygen–nitrogen–water transport in the GDL. The
predicted Nyquist impedance spectra showed two loops in general}the higher frequency loop
determined by charge transport in the catalyst layer and the lower frequency loop determined by
mass transport in the GDL. They showed that the lower frequency loop does not appear if pure
oxygen is fed to the cathode instead of air. Comparing the model predictions to impedance data
measured under large steady-state currents of operation, they concluded that for PEMFCs in
general: (a) a high catalyst utilization is achieved with thin-film catalyst layers made of
ionomer//Pt/C composites; (b) the limited protonic conductivity in the composite cathode
catalyst, in conjunction with the finite oxygen permeability in this layer, contributes significantly
to the cathode impedance in the medium current density domain; and (c) cell humidification has
a profound effect on the physical properties several cell components. They also showed that the
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effects of insufficient humidification on electrocatalysis and on catalyst layer protonic
conductivity may supersede the most commonly considered effects on membrane conductivity.
Eikerling and Kornyshev (1999) analysed the impedance behaviour of a PEMFC cathode
catalyst layer by considering it to be governed by two transport processes}proton migration
and oxygen diffusion. For the transport processes they identified characteristic lengths, whose
magnitudes relative to the catalyst layer thickness determines the relative importance of the
transport processes. For situations when only one of the transport processes is limiting, they
derived analytical solutions to the impedance response of the catalyst layer at open-circuit and
under load considering both linear and Tafel kinetics. However, when both processes are
limiting simultaneously, analytical solutions do not exist and only numerical solutions were
presented. Based on this analysis they identified the following diagnostic criteria characterizing
catalyst layers: (a) When the characteristic lengths of both the transport processes are large the
catalyst layer operates under kinetic control. This results in a semi-circle in Nyquist plot, the
diameter of which gives the kinetic resistance. Further, under Tafel kinetics, the kinetic
resistance decreases with increase in load current; (b) When proton transport is limiting the
Nyquist plot shows an asymmetric semi-circle with a 458 straight line at high frequencies; and (c)
When oxygen diffusion or both the transport processes are limiting the impedance spectra shows
two semi-circles overlapping or separate. The semi-circle occurring at lower frequencies arises
due to oxygen diffusion.
Jaouen et al. (2003) developed an impedance model for the gas-diffusion cathode of PEMFCs
by combining the spherical agglomerate model for oxygen diffusion with the macrohomogeneous porous electrode model for charge transport. In this model, only pure-oxygen feed were
considered and concentration gradients across the GDL and the catalyst layer we assumed
negligible. For these conditions, they calculated the impedance under Tafel kinetics for ORR,
for which they defined a Tafel impedance, Zt dV=dðlog IÞ; and used it in place of the
traditional impedance, Z V=I or dV=dI: A distinguishing property of the Tafel impedance is
that its value is independent of load current under kinetic control. Using this model they
discussed the relative importance of kinetics, mass transfer, and proton migration for different
load currents of fuel cell operation. They predicted Nyquist spectra for conditions of diffusion
limitation, migration limitation, and diffusion–migration limitation, as shown in Figures 6(a),
(b), and (c), respectively. It was observed with Figure 6(a) that the low-frequency intercept at
high load currents (diffusion control) was identically equal to twice the value for low load
currents (kinetic control). Note that this was also predicted by Springer and Raistrick (1989).
When proton migration is limiting (Figure 6(b)) a straight line was observed at high-frequencies
although the low-frequency intercepts showed the same behaviour as the diffusion control case.
When both diffusion and migration are limiting (Figure 6(c)) the Nyquist curves show the same
general shape as migration control. However, the low-frequency intercept at high load currents
reaches four times that at low load currents. The authors address the common practice of using
the high-frequency intercept of the Nyquist plot to obtain the IR drop of the electrolyte. They
show that unless the IR drop is a linear function of the current density of operation, the highfrequency intercept cannot be directly used to obtain the IR drop. Instead, one has to use the
integral of the IR drop over the current range considered.
In the second part of their paper (Jaouen and Lindbergh, 2003), they use the above model to
extract charge- and mass-transport properties from impedance data measured on a PEMFC
under load currents ranging from 1 to 400 mA cm2. The parameters obtained indicated that the
electrode is limited by both diffusion and migration. Further, they concluded that the spherical
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Figure 6. Predicted impedance spectra at various current densities for: (a) diffusion limitation;
(b) migration limitation; and (c) diffusion and migration limitation. Reproduced by permission of
The Electrochemical Society (Jaouen et al., 2003).
agglomerate model used to describe oxygen diffusion is valid, although the model could not fit
data obtained at load currents higher than 200 mA cm2. This was attributed non-uniform
electrode thickness and/or the influence of the anode, which was not included in the model. They
also observed that at low humidities, a second loop appeared on the low-frequency side, which
was attributed to the membrane not behaving as an ideal resistor under dynamic conditions.
Guo and White (2004) extended the pure-oxygen-feed impedance model of Jaouen et al. to
consider air feed and including concentration variations across the GDL and the catalyst layer.
The transport of three components, namely, nitrogen, oxygen, and water, in the gas phase was
treated using the Stefan–Maxwell equations for multicomponent diffusion. The model
predictions showed significant concentration variations in the GDL and the catalyst layer.
Further, the effect of pure-oxygen feed and air feed were compared using the predicted
dimensionless Nyquist impedance spectra as shown in Figure 7. These comparisons were made
between high-pressure air feed and low-pressure oxygen feed so that the partial pressures of
oxygen in the feed were the same. For small operating current densities both air-fed and oxygenfed cathodes exhibited the same impedance profile and a normal Tafel slope, indicating the
absence of gas-phase transport limitations. However, with increase in load current the oxygen
cathode began to display an increasing Tafel slope. For very high load currents an almost
quadric Tafel slope was predicted on the oxygen-fed cathode, caused by a combined limitation
from slow ionic conduction and slow dissolved oxygen diffusion in the agglomerate. For the airfed cathode, besides the increasing Tafel slope observed from the high-frequency impedance
loop, a second impedance loop began to grow adding an extra Tafel slope.
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P. M. GOMADAM AND J. W. WEIDNER
Figure 7. Comparison of the simulated normalized impedance responses of an air cathode with those of an
O2 cathode. Both the air flow and O2 flow have the same partial pressure of O2. Reproduced by permission
of The Electrochemical Society (Guo and White, 2004).
In an earlier paper, Guo et al. (2003) developed an impedance model for a PEMFC cathode
with a non-uniform ionic conductivity. Porous electrode theory was used to derive the governing
equation for the impedance response of the cathode at open-circuit conditions. The influence of
an assumed ionic conductivity profile on the error in the estimation of total double-layer
capacitance of the catalyst layer was also investigated. Further, a characterization of the ionic
conduction of the catalyst layer was conducted using impedance measurements. The increase of
ionic conductivity in the catalyst layer of an air-fed cathode with increase in ionomer loading
was revealed from both impedance data and surface area measurements. A nonlinear parameter
estimation method was used to extract the ionic resistance from the high-frequency region of the
impedance data at open-circuit potential conditions. The assumed ionic conductivity
distribution in the active layer was found to vary with ionomer loadings.
We recently developed the theoretical analysis required to extract charge-transport properties
from impedance or time-dependent polarization data on composite (i.e. porous) electrodes
(Gomadam et al., 2003). The experiments considered in this theory involved perturbing the
composite under three different configurations around open-circuit. Although these configurations have been considered before by researchers for measuring some transport properties, their
analysis allows using combinations of these configurations to accurately obtain all the transport
properties of composite electrodes. The three configurations considered are shown schematically
in Figure 8. In Configuration I, the reference electrode is placed in the electrolyte at a point near
the face of the composite working electrode (i.e. at x ¼ 0) and the other side of the composite is
supported by a metallic current collector (i.e. at x ¼ L). In Configuration II, the composite is
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ANALYSIS OF EIS IN PEMFCs
Figure 8. Schematic of a composite electrode (shown shaded) under three different configurations labelled
I, II and III. The electrolyte fills the unshaded area between the current collectors as well as the pores
in the composite. In all the three configurations, current flows between points A and B, while the
resulting potential difference is measured between points C and D. Reproduced by permission
of The Electrochemical Society (Gomadam et al., 2003).
bounded by current collectors on either side, while in Configuration III, it is bounded by two
reference electrodes. In all the three configurations, a current, I, flows between points A and B,
while the resulting potential difference, V, is measured between points C and D.
Using porous electrode theory, we obtained analytical solutions for the frequency response of
the three configurations in terms of dimensionless impedances as
2 þ ððs=kÞ þ ðk=sÞÞ cosh nAC
I
Z ¼ RO 1 þ
ð1Þ
nAC sinh nAC
k1 cosh n AC
12
s nAC sinh nAC
ð2Þ
s1 cosh n AC
Z III ¼ RO 1 2
k nAC sinh nAC
ð3Þ
II
Z ¼ RO
where
nAC
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aðk þ sÞ
ði0 nf þ Cdl ojÞ
¼L
ks
Copyright # 2005 John Wiley & Sons, Ltd.
ð4Þ
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P. M. GOMADAM AND J. W. WEIDNER
and
RO ¼
1
1 þ k=s
ð5Þ
Figure 9(a) shows Nyquist plots of the composite for the three configurations simulated using
Equations (1)–(4), with k=s ¼ 1:25 and n ¼ 10: Here, n is a dimensionless parameter signifying
the ratio of ohmic resistance of the electrode relative to its charge-transfer resistance at steadystate. It is equal to the steady-state value of nAC, obtained by substituting o ¼ 0 in Equation (4).
In each of the three configurations, the Nyquist plots are characterized by a high-frequency real
intercept equal to RO, a high-frequency straight line of unit slope, and a low-frequency real
intercept equal to the steady-state polarization resistance of the composite. The values of the
III
low-frequency intercepts, RI1 ; RII
1 ; R1 ; for the three configurations are obtained by substituting
o ¼ 0 in Equations (1)–(4), respectively. These intercepts and the Nyquist spectra are functions
of the two parameters, k/s and n. Nevertheless, the qualitative shapes of the Nyquist plots of
Configurations II and III do not change with n. In contrast, Figure 9(b) shows that the shape of
the Nyquist plot of Configuration I varies significantly with n. The low-frequency real intercept,
approaches infinity as n ! 0. At n ¼ 0; the Nyquist plot does not come back towards the real
axis at all. Rather, it behaves as a purely capacitive system, whose impedance tends to infinity.
Except for some limiting conditions, the impedance of a composite electrode is a combination
of s, k, i0, and Cdl. Therefore, these parameters must be obtained from a set of data rather than
a single data point. For example, these parameters can be obtained by fitting the analytical
expressions derived here to the entire impedance spectrum, measured on any one of the
configurations. Which configuration is chosen depends on how sensitive its response is to the
parameter of interest. When the effect of one unknown parameter on the impedance of a
configuration is much greater than another parameter, then that configuration cannot be used to
determine the latter parameter with confidence. One has to use alternative configurations,
wherein the response is dominated by the parameter of interest. Since one often does not know
the relative importance of the parameters a priori, it is prudent to determine the parameters
from the responses of multiple configurations. Such use of multiple configurations to determine
different parameters has been considered by Saab et al. (2002, 2003) for fuel cells and by
Shibuya et al. (1996) for Li-ion batteries.
Impedance measurements on ‘normal’ PEMFCs, which have large cross-sectional areas and
small path lengths for current flow, often show low signal-to-noise ratios. Secondly, based on
the time-constant of the process of interest, the measured impedance data may not be well
resolved over the range of frequencies allowed by the instrument or by competing processes. For
these reasons, our analysis considered thin and long electrode samples, which dramatically
improve the signal-to-noise ratio, as shown by Saab et al. (2002). Moreover, by varying the
length of the sample the time-constant of the process can be varied so as to obtain reliable
impedance data in the frequency range allowed by the instrument.
In a recent paper (Gomadam and Weidner, submitted), we extended the open-circuit
analysis, mentioned above, to allow for in-situ property measurements under load. Impedance
under load with Tafel kinetics for Configuration I has been analysed by many researchers
(Springer and Raistrick, 1989; Springer et al., 1996; Jaouen et al., 2003; Guo and White,
2004). However, we considered linear and Tafel kinetics for all the three configurations and
their combinations under load. Analytical solutions were derived for most of these
combinations, while simple numerical solutions were obtained for more complex situations
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ANALYSIS OF EIS IN PEMFCs
1147
Figure 9. (a) Nyquist plots obtained for the three configurations simulated using Equations (1)–(3), with
k=s ¼ 1:25 and n ¼ 10; and (b) for the same value of k/s, as n decreases or as charge-transfer resistance
increases, the low-frequency intercept of Configuration I also increases, eventually tending to infinity as
n ! 0. Reproduced by permission of The Electrochemical Society (Gomadam et al., 2003).
(e.g. Butler–Volmer kinetics, two-dimensional distributions). The impedance-under-load theory,
developed for conditions of no concentration gradients, was modified for the case when
concentration gradients exist. Analytical solutions were obtained in spite of relaxing this
assumption since we recognized that under mass-transfer-limiting conditions the potential
drops in the catalyst layer are negligible compared to the large overpotential for reaction. The
various conditions treated (linear kinetics, Tafel kinetics, presence of concentration gradients)
allow the use of the model to measure properties at all load currents or voltages of operation of
PEMFCs.
Copyright # 2005 John Wiley & Sons, Ltd.
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P. M. GOMADAM AND J. W. WEIDNER
Figure 10. Simulated Nyquist plots of impedance under load for Configuration I under Tafel conditions.
The same governing equations as open-circuit are applicable under load, except: (i) the initial
conditions for the dependent variables are their pseudo-steady-state profiles under load; (ii) the
governing equations, if nonlinear, are linearized around their steady-state values to obtain
the solutions for impedance. When linear kinetics applies, the mathematical analysis and the
solutions for impedance under load are identical to that at open-circuit. However, for Tafel
kinetics, the mathematical analysis is more complex, although analytical solutions were still
obtained as plotted in Figure 10. The figure shows sample Nyquist impedance spectra simulated
for Configuration I as a function of load current applied also on Configuration I. The Nyquist
curves follow the same qualitative trends as in open-circuit, namely, an asymmetric semi-circle
with a straight line of unit slope at high frequencies, a high-frequency real intercept equal to the
effective ohmic resistance of the composite, and a low-frequency real intercept equal to the total
resistance (i.e. ohmic plus kinetic). The figure also shows that as the load current is decreased the
low-frequency intercept or the total resistance of the composite increases. Considering that the
ohmic resistance is a constant, this means that the kinetic resistance increases with decrease in
load current; a distinguishing feature of Tafel kinetics.
For impedance under load with linear kinetics, the same parameter estimation technique as
open-circuit is applicable. However, with Tafel kinetics, the kinetic parameters obtained from
the low-frequency intercept are different. Under Tafel conditions, the kinetic resistance of the
composite depends on the load current and the transfer-coefficient (a). Therefore, the lowfrequency intercept obtained from an impedance experiment under a given load gives the
transfer-coefficient. Using this transfer-coefficient in the steady-state current–voltage relationship for the porous composite gives the exchange-current density ai0.
The impedance-under-load experiment can also be performed with two configurations}one
operating under load and the other being perturbed. Thus, with the three configurations
considered here, we obtain nine different combinations. Further, it is often more convenient to
arrange the perturbed configuration perpendicular to the load configuration, as we proposed in
the paper. However, the analysis then becomes two-dimensional with only numerical solutions
possible unless simplifying assumptions are enforced.
5. CONCLUSIONS
A literature review of EIS analysis of PEMFCs is presented. Papers that analyse the
impedance response of the cathode and anode half-cells of the PEMFCs based on the
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ANALYSIS OF EIS IN PEMFCs
continuum-mechanics approach are reviewed in detail. Papers dealing with EIS analysis based
on the equivalent-circuits approach are briefly addressed along with the relative advantages and
disadvantages of the two approaches. Papers dealing with continuum-mechanics-based EIS
modelling of general electrochemical systems are also briefly reviewed.
NOMENCLATURE
a
Cdl
f
F
i0
I
j
L
n
R
RO
Rct
Re
Ri
III
RI1 ; RII
1 ; R1
T
V
x
Z
Zi, Zimag
Zr, Zreal
Zt
Z I ; Z II ; Z III
=specific area for electrochemical reaction (m2 m3)
=double-layer capacitance (F m2)
=F/RT (V1)
=Faraday’s constant (C mol1)
=exchange-current density (A m2)
=current
pffiffiffiffiffiffiffi (A)
= 1
=electrode length (m)
=number of electron transfers in electrochemical reaction
=universal gas constant (J mol1 K1)
=as defined in Equation (5)
=charge-transfer resistance (O)
=electronic resistance (O)
=ionic resistance (O)
=low-frequency intercepts of Configurations I, II, III, respectively
=temperature (K)
=voltage (V)
=position (m)
=impedance (O, O m or no units, as applicable)
=imaginary part of Z (O, O m or no units, as applicable)
=real part of Z (O, O m or no units, as applicable)
=Tafel impedance (V)
=dimensionless impedance of Configurations I, II, III, respectively
Greek letters
k
nAC
o
s
=ionic conductivity (S m1)
=as defined in Equation (4)
=frequency (Hz)
=electronic conductivity (S m1)
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