2008 Leonide Sonn Weber

Journal of The Electrochemical Society, 155 �1� B36-B41 �2008�B36

Evaluation and Modeling of the Cell Resistance inAnode-Supported Solid Oxide Fuel CellsA. Leonide,z V. Sonn, A. Weber, and E. Ivers-Tiffée*

Institut für Werkstoffe der Elektrotechnik, Universität Karlsruhe, Karlsruhe (TH), Germany

The impedance of anode-supported single cells �Ni/8 yttria-stabilized zirconia �YSZ� anode; La0.58Sr0.4Co0.2Fe0.8O3−� cathode;8YSZ electrolyte; area 1 cm2� was characterized in a broad measuring range of temperature and air/fuel gas composition. The datahas been analyzed by calculating the distribution function of relaxation times �DRTs�. DRT computations enabled us to separatefive different loss mechanisms occurring inside the cathode and anode without the need of an equivalent circuit. Two processesexhibit a systematic dependency on changes in the oxygen partial pressure of the cathode gas and thus can be attributed todiffusional and electrochemical losses on the cathode side. The remaining three processes are very sensitive to changes in the fuelgas but are not affected by variations of the cathode gas. These resistances are classified as a gas diffusion polarization within theanode–substrate and as an electro-oxidation reaction at the triple-phase boundary, respectively.© 2007 The Electrochemical Society. �DOI: 10.1149/1.2801372� All rights reserved.

Manuscript submitted June 6, 2007; revised manuscript received September 26, 2007. Available electronically November 7, 2007.

0013-4651/2007/155�1�/B36/6/$23.00 © The Electrochemical Society

In order to develop a physicochemical model of solid oxide fuelcell �SOFC� single cells and to refine their efficiency and long-termstability, the performance-related polarization processes have to beidentified and proven. Electrochemical impedance spectroscopy�EIS� is one of the most promising methods for unfolding complexelectrochemical systems such as SOFCs.

Commonly the obtained impedance spectra are analyzed by acomplex nonlinear least-squares �CNLS� approximation to a modelfunction represented by an equivalent circuit.1 In this case theequivalent circuit model needs to be defined a priori without anyknowledge about the real number of polarization processes contrib-uting to the overall polarization loss of the cell. This leads very oftento a severe ambiguity of the adopted model.2

To overcome that disadvantage an alternative approach for ana-lyzing impedance spectra is used in this work. The equivalent circuitmodel and the optimal starting parameters for the CNLS algorithmwere obtained by a preidentification of the impedance response bycalculating and analyzing the corresponding distribution function ofrelaxation times �DRTs�, as reported in detail in Ref. 3. Our DRTapproach is particularly advantageous for the analysis of �anode-supported� SOFC single cells coupled to thin electrolytes �thicknessless than 20 �m�, where reference electrodes are not applicable forthe separation of anode and cathode losses.4

Experimental

Cell geometry.— The SOFC single cells analyzed within thisstudy were based on 50 � 50 mm anode substrates �Ni/8 yttria-stabilized zirconia �YSZ�� with an average thickness of about1.5 mm. On these substrates, an anode functional layer �Ni/8YSZ,approx. 10 �m� and an electrolyte �8YSZ, approx. 10 �m� weredeposited and cofired at 1400°C. A Ce0.8Gd0.2O2−� �CGO� interlayerwas screen-printed and sintered on the electrolyte �approx. 7 �mthick�. This interlayer was used to prevent a chemical reaction be-tween LSCF �lanthanum strontium cobalt ferrite� and 8YSZ, whichotherwise forms an insulating layer of SrZrO3. On top of this inter-layer a La0.58Sr0.4Co0.2Fe0.8O3−� cathode was applied by screen-printing, resulting in a thickness of approx. 45 �m after sintering.Details regarding the manufacturing procedures can be foundelsewhere.5,6

The active area of the working cathode was 10 � 10 mm. Twoauxiliary electrodes in gas flow direction in front of and behind thecathode were applied for open-circuit voltage �OCV� control �OCVprobes in Fig. 1b�. The electrodes were separated from the electro-

* Electrochemical Society Active Member.z E-mail: [email protected]

Downloaded 23 Feb 2012 to 131.215.51.245. Redistribution subject to E

lyte by a continuous CGO interlayer with lateral dimensions of12 � 30 mm. The cell geometry and microstructure are shown inFig. 1a and b.

Cell testing.— The single cells were mounted into ceramic hous-ings. Cathode and anode were contacted by gold and nickel meshes,respectively. Gold rings were used for sealing. The cells were oper-ated under ambient pressure with different N2/O2 mixtures at the

Figure 1. �a� A scanning electron micrograph of the polished cross section ofa fractured cell showing part of the porous anode, the dense electrolyte, theCGO buffer layer, and the porous cathode. �b� Design of the working andauxiliary �OCV probe� electrodes at the cathode side of the anode-supportedSOFC single characterized cell.

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B37Journal of The Electrochemical Society, 155 �1� B36-B41 �2008� B37

cathode side and varying H2O/H2 mixtures at the anode side. Highcontents of water vapor could be realized by feeding oxygen into anupstream combustion chamber.

The total anodic and cathodic gas flow rates were maintained ata constant value of 250 mL/min during all experiments. The cellswere tested over a range of temperatures between 650 and 850°C.

Impedance measurements were carried out with a Solartron 1260frequency response analyzer in a frequency range from 100 mHz to1 MHz. The amplitude of the current stimulus was chosen in orderto achieve a voltage response not higher than 12 mV. All experi-ments within this study were conducted under open-circuit condi-tions �OCC�.

To investigate the parameter dependence of each single polariza-tion process, a series of impedance measurements was carried out inwhich only one cell parameter at a time was varied �oxygen partialpressure, water partial pressure, temperature�.

Theoretical Background

Distribution of relaxation times.— The distribution function ofrelaxation times �� is associated with the complex impedanceZ�� by the following expression3

Z�� = R0 + Zpol�� = R0 + Rpol�0

��

1 + j��d�

with �0

�

��d� = 1 �1�

Here R0 represents the ohmic and Zpol�� the polarization part ofZ��. Rpol is the total dc polarization resistance and j the imaginaryunit. The term ��/�1 + j��d� specifies the fraction of the over-all polarization with relaxation times between � and � + d�.

The real and imaginary parts of the impedance data of a linear,time-invariant system are connected by Kramers–Kronig �KK�transformations.7 Therefore, it is sufficient to consider the imaginarypart of the impedance only. This gives rise to the following equation

Im�Z�� = Z�� = − Rpol�0

��

1 + ��2��d� �2�

A logarithmic variable substitution transforms expression 2 into asimple convolution equation which can be solved by Fourier trans-formation. From this the distribution function g� f� �with f = 2�/�= 2�� of the measured impedance data can be computed.3

The necessary compliance of the measured data with theKramers–Kronig transformation rules is verified by using the “KKtest for Windows” software.8,9 Figure 2 shows a Kramers–Kronig

Figure 2. Kramers–Kronig test residuals of a typical impedance spectrum.


validation for a typical impedance spectrum. For the most part of thespectrum the relative errors of both real and imaginary data lie be-low an acceptable value of 0.4%.

As can be envisaged from the infinite integration range in Eq. 2,the calculation of the distribution function of relaxation times re-quires that the analyzed impedance data are measured over an infi-nite range of frequencies �from f = 0 to ��. However, in reality thefrequency range of measurement data is fixed by the experimentalconditions �noise, limited frequency range of the frequency responseanalyzer �FRA�, etc.�.

Therefore, in order to reduce numerical errors due to this finitefrequency range, an extrapolation of the data �in this case the imagi-nary part Z�� of the impedance� at each end of the frequencyspectrum is necessary. In our case Z�� was extrapolated by astraight line in the plot of log�Z�� vs log� f�.3 This can be done on theassumption that on the edges of the spectrum only one polarizationprocess is still active.10

The theoretical distribution function g� f� of an ideal resistance–capacitance �RC� element is represented by a Dirac peak �� f-fmax�at the respective summit frequency �max = 2�fmax. For a subcircuitresistor R connected in parallel with a constant phase element Q�RQ� the peak broadens out with decreasing exponent nRQ �Fig. 3b�.

Another impedance element often encountered in impedancestudies, describing diffusion-related processes, is the generalized fi-nite length Warburg element �G-FWS�.11 The impedance expressionfor the G-FWS element is given by

ZG−FWS�� = RWtanh��j�TW��

�j�TW� �3�

For a perfect one-dimensional diffusion limitation is equal to 0.5and TW corresponds to ld

2/Di with ld the effective diffusion thicknessand Di the effective diffusion coefficient of the diffusing species i.RW denotes the dc diffusion resistance. In this study, values of ca.0.46 were obtained. In Fig. 3c and d the Nyquist plot and the cor-responding theoretical DRT of a simulated G-FWS element withRW = 21.8 m and TW = 0.0783 s and = 0.465 are shown. In thiscase the theoretical distribution function is characterized by a largepeak located at the characteristic frequency followed by smallerpeaks at higher frequencies.

Results and Discussion

Equivalent circuit analysis.— Figure 4a shows a classic examplefor the impedance spectrum of an anode-supported single cell andFig. 4b the corresponding DRT computation result. Whereas thepolarization processes overlap in the impedance curve, processescan be clearly distinguished in the distribution function of relaxationtimes.

From an accurate analysis of all recorded impedance data �andtheir corresponding DRTs� an equivalent circuit model composed offive impedance elements connected in series was developed �Fig.5a�. The processes P1C, P2A, and P3A are modeled by a RQ element,whereas P1A and P2C are modeled by a G-FWS element and aGerischer element, respectively. Special attention needs to be givento the two peaks at around 10 and 100 Hz �in Fig. 4b denoted as�P1A + P2C��. We believe that these two peaks are both ascribable tothe Warburg process P1A �compare with Fig. 3d�. However, as wesee later, depending on the operating conditions a further process�P2C� can overlap with the two peaks caused by P1A.

Figure 5b shows the CNLS fit applied to the imaginary part ofthe impedance curve depicted in Fig. 4b. The residuals �relativeerrors� are distributed uniformly around the frequency axis, showingno systematic deviation �Fig. 5c�. For most of the spectrum therelative errors lie below an absolute value of 0.25%, thus demon-


B38 Journal of The Electrochemical Society, 155 �1� B36-B41 �2008�B38

strating the validity of the model. Only from about 300 kHz upwarddo inductive artifacts caused by the electrical wiring become notice-able.

The CNLS fit of the impedance data was carried out with thecommercial program ZView.12

Cathodic oxygen partial pressure.— The oxygen content of thegas mixture supplied to the cathode was varied between 21 �air� and1% �balance nitrogen�, whereas the composition of the fuel gas waskept constant at a ratio of 62.5% H2O to 37.5% H2. The high watercontent was used to reduce the anodic polarization losses to a mini-mum, thereby making the deconvolution of the cathodic contribu-tions simpler.

Figure 6a shows the influence of the oxygen partial pressure�pO2�cathode�

� on the distribution of relaxation times. It is clearly vis-

ible how a process evolves in the frequency range below 10 Hz�denoted as P1C� at oxygen contents �0.05 atm. At the same time asecond process �here denoted as P2C� becomes visible between 100and 10 Hz, partly overlapping the low-frequency peak related toprocess P1A �see arrows in Fig. 6a�. This observation confirms thatthe overall polarization loss caused by the cathode is strongly relatedto the two processes P1C and P2C. However, P1A,P2A, and P3A showno dependency on the pO2�cathode�

, demonstrating that these three

losses are ascribable to the anode.The polarization resistances obtained by the CNLS fit �using the

equivalent circuit depicted in Fig. 5a� are plotted over the partialpressure of oxygen in Fig. 6b. In this case the resistance R1A waskept fixed during the entire fit procedures. This approach was essen-tial because otherwise the similar summit frequencies of P1A andP2C �at lower pO2�cathode�

� would have destabilized the fit algorithm.

As expected, the losses R2A and R3A are independent of thechange in O2, whereas the two resistances R1C and R2C both showan almost linear trend in the double-logarithmic plane with slopes of


−1.08 and −0.26, respectively. The high-frequency process of thecathode �P2C� is probably associated with the oxygen surface ex-change kinetics of LSCF as well as the diffusivity of oxygen ionsthrough the LSCF bulk. The low-frequency process P1C probablyreflects the mass-transfer resistance caused by the gas-phase diffu-sion in the pores of the LSCF electrode.14

In the following discussions we focus our attention on processP1C. It can easily be shown that the cathodic gas diffusion resistancecan be described by the following equation15

RD�cathode� = RT

4F2

lc1

DO2,N2

�c

VV,c 1

pO2�cathode�

− 1�1.0133 � 105 Pa

atm−1

�4�

where lc is the cathode thickness, DO2,N2is the binary diffusion

coefficient for a mixture of oxygen and nitrogen, �c is the tortuosityfactor, and VV,c is the volume fraction porosity related to the porouscathode structure.

According to the Chapman–Enskog theory, the binary diffusioncoefficient can be estimated by16

Djn = 1.858 � 10−3T1.5��Mj + Mn�/MjMn�1/2

P � � jn2 � D

�5�

where D is the collision integral �dimensionless�, � jn is the averagecollision diameter �in Angstroms�, Mj and Mn are the molecularweights of component j and n, respectively, and P is the total pres-sure �in atm�. Using D and � jn data from Ref. 16, DO2,N2

�T= 800°C� = 1.7855 cm2/s is estimated.

The model Eq. 4 was fitted to the experimental data with theporosity-to-tortuosity factor c = VV,c/�c representing the unknownvariable �l was set to 45 �m�. The fit result is shown in Fig. 6b

Figure 3. �a� Nyquist plot of an RQ ele-ment for five different nRQ values and �b�corresponding theoretical distributions ofrelaxation times ��max = 2�fmax�. �c� Ny-quist plot of a G-FWS element and �d�corresponding theoretical distribution ofrelaxation times. �The distributions werecalculated from the imaginary part of theimpedance using the formula derived inRef. 13.�

c



�dashed line�. As can be seen, the agreement between the model andthe experimentally obtained resistance R1C is quite good. From thefit a porosity-to-tortuosity factor c = 0.014 is estimated. This valueis in accordance with values reported in the literature for this type ofcathode structure.17

Anodic partial pressure of water vapor.— In order to analyzethe dependency of each anodic process on the partial pressure ofwater �pH2O�anode�

� in the fuel gas, the H2O content was varied step-

wise between 4.88 and 62.5%. Air was used as the cathode gas.Figure 7a shows the DRTs computed from the impedance spectra

recorded at four different pH2O�anode�values. The processes P1A and

P2A both show a significant dependency on the water content in thefuel gas, whereas process P3A is characterized by a minor depen-dency.

In Fig. 7b the polarization resistances obtained from the CNLSfit are plotted over the water partial pressure. During this fit proce-dure the resistance R2C was kept fixed in order to ensure a stable fit.The polarization contribution caused by the gas-phase diffusion inthe pores of the LSCF electrode can be neglected when air is used ascathode gas; thus, the resistance R1C was set fixed to zero. The tworesistances R2A and R3A both show an almost linear trend in thedouble-logarithmic plane with slopes of −0.44 and −0.20, respec-tively. R1A shows the highest dependence on changes in the watercontent. This resistance is characterized by a negligible thermal ac-tivation �compare next subchapter� and shows attributes that areexpected to be seen by gas diffusion processes.18

Figure 4. �a� Impedance spectra of a SOFC single cell recorded at T= 800°C, pO2�cathode�

= 0.01 atm, pH2O�anode�= 0.625 atm, and �b� correspond-

ing DRTs. Unlike the Nyquist plot, at least five processes are visible in thedistribution curve.


The two high-frequency processes P2A �1000 Hz� and P3A�10 kHz� are associated with the reaction at the triple-phase bound-ary involving the charge-transfer reaction.19

In the following discussions we focus our attention on processP1A. We assume that R1A describes the mass-transfer resistancecaused by the gas-phase diffusion in the pores of the Ni/YSZ-anodesubstrate. It can easily be shown that the resistance caused by dif-fusion limitations in a porous anode structure can be described bythe following equation15

RD�anode� = RT

2F2

la1

DH2O,H2

�a

VV,a 1

pH2�anode�

+1

pH2O�anode�

�1.0133 � 105 Pa

atm−1

�6�

where la is the anode thickness, DH2O,H2is the binary diffusion co-

efficient for a mixture of water and hydrogen, �a is the tortuosityfactor, and V is the volume fraction porosity related to the porous

Figure 5. �a� Equivalent circuit model used for the CNLS fit of impedancedata. �b� CNLS fit of the imaginary part of the impedance spectra shown inFig. 3. �c� Residual pattern of the fit.

V,a


B40 Journal of The Electrochemical Society, 155 �1� B36-B41 �2008�B40

anode substrate. Using Eq. 5 with D and � jn data from Ref. 16,DH2O,H2

�T = 757°C� = 7.1659 cm2/s is estimated.The model Eq. 6 was fitted to the experimental data with the

porosity-to-tortuosity factor a = VV,a/�a representing the unknownvariable �la was set to 1.5 mm�. The fit result is shown in Fig. 7b�dashed line�. As can be seen, the model approximates very well thetrend of the experimentally obtained resistance R1A. From the fit aporosity-to-tortuosity factor a = 0.0575 is estimated.

Thermal activation.— The cells were tested at temperatures be-tween 650 and 850°C in steps of 10 K with a fuel gas compositionof 62.5% H2O and 37.5% H2. Air was used as cathode gas. TheDRT of the cell impedance at four different temperatures is shown inFig. 8a. It is evident that the processes P2C, P2A, and P3A are allcharacterized by a pronounced thermal activation. Process P1Ashows, by contrast, a negligible dependency on the operating tem-perature.

The Arrhenius plots of each single polarization resistance, R1A,R2A, R3A, and R2C, obtained by the CNLS fit of the impedancecurves, are represented in Fig. 8b.

In order to facilitate the fit procedure the resistance R1A was keptfixed �constant�. The resistances R2C and R3A can be approximatedwell with a linear fit, demonstrating good Arrhenius behavior. Fromthe slope of the fitted lines the activation energies Ea,2C = 1.39 eVand E = 1.30 eV are obtained.

Figure 6. �Color online� �a� Series of distribution curves at four differentpO2�cathode�

values. �b� Characteristic dependence of the fitted equivalent cir-

cuit elements on the cathodic oxygen partial pressure. The dashed line indi-cates the model prediction according to Eq. 3 �pH2O�anode�

= 0.625 atm �bal-

ance H2�, T = 800°C�.

a,3A


The Arrhenius plot of process P2A is a more complicated one,showing two ranges of different slopes with a discontinuity between750 and 760°C. Thus, these two ranges were fitted separately. Forthe low-temperature range �730–750°C� an activation energy of2.05 eV was calculated, whereas the higher temperature range�760–850°C� is characterized by an activation energy of 0.35 eV.

Conclusion

A high-resolution impedance study of anode-supported singlecells with thin electrolytes ��20 �m� was presented. The cells werecharacterized at OCV over a broad range of operating conditions,including different temperatures and various cathode and anode gascompositions. Calculating the DRT allowed us to identify up to fivedifferent processes contributing to the total polarization loss of acomplete anode-supported cell. The high resolution of the DRTcombined with the numeric accuracy of the CNLS fit enabled us toidentify oxygen surface exchange kinetics, diffusivity of oxygenions in the bulk, and gas-phase diffusion in the mixed conductingLSCF cathode. At the anode side three limiting processes were iden-tified. One process is attributed to the gas diffusion within the anodesubstrate, whereas the other two are related to the electro-oxidation,including the charge-transfer reaction and the ionic transport in theNi/YSZ anode structure.

To the best of our knowledge, these are unique results reportedfor the first time in the literature, demonstrating that an unambigu-

Figure 7. �Color online� �a� Series of distribution curves at four differen.pH2O�anode�

values. �b� Characteristic dependence of the fitted equivalent cir-

cuit elements on the anodic partial pressure of water �pO2�cathode�

= 0.21 atm �air�, T = 757°C�.


pO2�cathode�= 0.21 atm�air��.



ous and detailed electrochemical characterization of each singleelectrode of an anode-supported cell is possible, even if no referenceelectrode can be used.

Acknowledgments

The authors thank Dr. N. H. Menzler of the ForschungszentrumJülich for the generous supply of cells. Funding by the Umweltmin-isterium Baden Württemberg �Z04B 26013� is gratefully acknowl-edged.

Universität Karlsruhe (TH) assisted in meeting the publication costs ofthis article.

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Scriber Associates, Inc., Southern Pines, NC, USA �2000�.13. R. M. Fuoss and J. G. Kirkwood, J. Am. Chem. Soc., 63, p. 385 �1941�.14. S. B. Adler, J. A. Lane, and B. C. H. Steele, J. Electrochem. Soc., 143, 3554

�1996�.15. J. Kim, A. V. Virkar, K. Fung, K. Mehta, and S. C. Singhal, J. Electrochem. Soc.,

146, 69 �1999�.16. R. Reid, J. Prausnitz, and T. Sherwood, The Properties of Gases and Liquids, 3rd

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150, 783 �2003�.18. S. Prindahl and M. Morgensen, J. Electrochem. Soc., 146, 2827 �1999�.19. V. Sonn, A. Leonide, and E. Ivers-Tiffée, J. Electrochem. Soc., Submitted.

Figure 8. �Color online� �a� Series of distribution curves at four tempera-tures. �b� Characteristic dependence of the fitted equivalent circuit elementson the operating temperature �pH2O�anode�

= 0.625 atm �balance H2�,


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2008 Leonide Sonn Weber