characterization and quantiﬁcation of electronic and ionic

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Kyle N. Grew

John R. Izzo, Jr.

Wilson K. S. Chiu2

e-mail: [email protected]

Department of Mechanical Engineering,University of Connecticut,

191 Auditorium Road,Storrs, CT 06269-3139

Characterization andQuantification of Electronic andIonic Ohmic Overpotential andHeat Generation in a Solid OxideFuel Cell Anode1

The development of a solid oxide fuel cell (SOFC) with a higher efficiency and powerdensity requires an improved understanding and treatment of the irreversibilities. Lossesdue to the electronic and ionic resistances, which are also known as ohmic losses in theform of Joule heating, can hinder the SOFC’s performance. Ohmic losses can result fromthe bulk material resistivities as well as the complexities introduced by the cell’s micro-structure. In this work, two-dimensional (2D), electronic and ionic transport models areused to develop a method of quantification of the ohmic losses within the SOFC anodemicrostructure. This quantification is completed as a function of properties determinedfrom a detailed microstructure characterization, namely, the tortuosity of the electronicand ionic phases, phase volume fraction, contiguity, and mean free path. A direct mod-eling approach at the level of the pore-scale microstructure is achieved through the use ofa representative volume element (RVE) method. The correlation of these ohmic losseswith the quantification of the SOFC anode microstructure are examined. It is found withthis analysis that the contributions of the SOFC anode microstructure on ohmic lossescan be correlated with the volume fraction, contiguity, and mean free path.�DOI: 10.1115/1.4002226�

Keywords: SOFC, ohmic losses, Joule, characterization, representative volume element,pore-scale

BackgroundSignificant losses can arise in the solid oxide fuel cell �SOFC�,

rising from the cell’s electronic and ionic resistances. Dependingn the cell design and operating conditions, resistive losses in theorm of Joule heating can be among the more substantial lossesithin the cell �1,2�. These losses need to be understood and mini-ized to develop a more efficient SOFC. In this work, a focus is

laced on the contributions of the SOFC anode microstructure tohese losses. An analysis of the impact of the pore-scale micro-tructure’s features on mass transfer process is being examined incomplementary effort �3–6�.The reduction of resistive losses in a state of the art SOFC

node, comprised of a porous nickel-yttria-stabilized zirconia �Ni-SZ� cermet is not just a mater of shortening the respective trans-ort path lengths, enhancing transport properties, or changing theolumetric composition to improve percolation. The optimizationf the SOFC anode requires the balance of the ohmic, activation,nd concentration losses with additional consideration of the me-hanical, thermal, and chemical stability requirements �7�. Thisalance necessitates sufficient ionic and electronic conductionathways, pore-space for the transport of fuel and product species,lectrocatalytically active three-phase boundary �TPB� regions,nd a robust and stable structure �7�. The complexity of the anodeicrostructure makes it difficult to optimize a particular aspect of

he SOFC anode design without impacting another. Numerous mi-

1Presented at the 5th ASME International Fuel Cell Science, Engineering andechnology Conference, Paper No. FuelCell2007-25162, Brooklyn, NY, June 18–20.

2Corresponding author.Manuscript received July 27, 2007; final manuscript received June 26, 2010;

ublished online February 15, 2011. Assoc. Editor: Abel Hernandez.

ournal of Fuel Cell Science and TechnologyCopyright © 20

aded 13 Sep 2011 to 129.49.56.80. Redistribution subject to ASME

crographs of the Ni-YSZ cermet anode microstructures show thiscomplexity �8,9�. These micrographs show a complex heteroge-neous microstructure to nanostructure, which has complex Ni andYSZ phase-networks, irregular pores, and unique features. To ex-amine the charge transfer processes, these complex features mustbe considered at the length scales of the features �i.e., pore-scalemicrostructure� to understand their effects. In a recent review,Reifsnider et al. suggest that the losses originate at nanometerlength scales in these complex structure �8�. These length scalesare characteristic of the grains structure and pore-scale features.The details of these effects are difficult to capture in aggregatedcell or electrode models, which use effective transport and aver-aged microstructural properties. Due to a lack of understandingregarding the impact that the features at these length scales haveon SOFC performance, we explore the effect that the anode mi-crostructure has on ohmic losses in this work.

The goal of this microstructural or pore-scale approach is toprovide a tool for the quantitative characterization of the SOFCmicrostructure with respect to the ohmic losses. This is completedusing several characterization parameters that provide descriptionsof the cell’s microstructure �i.e., tortuosity of the conductingphase, volume fraction, contiguity, and mean free path�. A focus isplaced on the two-dimensional �2D� electronic and ionic chargetransfer processes within the anode. This effort includes the analy-sis of ohmic losses resulting from ionic conduction in the electro-chemically active regions of the anode. Phase-specific micro-graphs of the SOFC anode are collected using Auger electronspectroscopy �AES� on several samples. Conceptual microstruc-tures are used to supplement these AES samples. Both are used todevelop a better understanding of the microstructure’s role in
ohmic losses.
JUNE 2011, Vol. 8 / 031001-111 by ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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Method of ApproachThis paper begins by presenting a one-dimensional �1D� cellodel that is used to specify boundary conditions for a RVE. TheVE method is used to consider detailed electrode structure. Once

he RVE method is defined, AES is used to provide detailed RVEsf actual SOFC samples. A discussion on the structural character-zation and performance quantification method is then presented.

2.1 One-Dimensional Cell Level Model. A cell model iseeded to prescribe boundary conditions to a detailed RVE, whichermits the microstructural details contained in the RVE to besed for more detailed analysis. Experimental data published byiang and Virkar �10� is used to parameterize and validate a 1Dodel that is developed here. The work by Jiang and Virkar con-

iders a laboratory SOFC button cell operating on hydrogen �H2�uel diluted with water vapor �H2O� and oxygen �O2� taken fromtmospheric air as the oxidant. A schematic of the 1D model isrovided in Fig. 1. A Ni-YSZ cermet anode, bilayer YSZ-SDChin film electrolyte, and LSC-SDC cathode are used in this study.able 1 provides a summary of the conditions and cell properties

hat were used for the cell-level model that will be consideredithin this section �10�.The cell model is used to prescribe boundary conditions to a

ocalized and finite sized RVE. As a part of this effort, this modelust consider electrochemical reaction process within the elec-

rode structures so that the local electronic and ionic fluxes andotentials can be identified. Specifically, the Faradaic portion ofhe electrochemical oxidation processes must be considered,

ig. 1 Schematic of the 1D SOFC button cell model used toodel experimental work in literature †10‡. The electronic RVE

s at an arbitrary location within the anode support while theonic RVE is localized to the anode/electrolyte interface.

able 1 Parameters used in 1D cell-level model. Bulk conduc-ivities were taken from the provided references. Anode effec-ive conductivities are corrected based on percolation theory16‡.

arameter Value Reference

1073 K �10�an 1.1�10−3 m �10�lyte 2.0�10−5 m �10�cat 5.0�10−5 m �10�H2–H2O 7.5�10−4 m2 /sec �10�H2

k 11.2�10−4 m2 /sec �10�H2Ok 3.7�10−4 m2 /s �10�

pore 0.5�10−6 m �10�an=� /� 0.08 �10�eleff,an 3.33�105 S /m �11�ellyte 1.0�10−4 S /m �7�eleff,cat 25.5 S/m �12–14�ioneff,an 1.05 S/m �15�ionlyte 5.31 S/m �15�ioneff,cat 1.86 S/m �12–14�tref 11.36 mol /m3

TPBeff 5.0�105 m2 /m3 �16�H2O�1073 K� 454.9 kJ/mol �7�H2

�1073 K� 147.1 kJ/mol �7�O2

�1073 K� 228.0 kJ/mol �7�O2−�973 K� 236.4 kJ/mol �17�

31001-2 / Vol. 8, JUNE 2011


which represent the electronic neutralization of the ionic flux withthe consumption of hydrogen and generation of water. This allowsthe ohmic loss contributions to be examined in the electrochemi-cally active region of the anode. Similar processes may be con-sidered for the electrochemical oxygen reduction in the cathode.Experimental literature suggests this exchange region is of com-parable length to that of the electrolyte in the anode of an anode-supported cell �18�. The ionic current will be depleted in theseregions due to its electrical neutralization through the Faradaicportion of the electrochemical reactions; however, even with anionic current that is being neutralized as it extends into the anoderegion, the losses can be substantial and contribute to the cell’spolarization resistance. Structural complexities may magnify theselosses.

To consider these processes, a volumetric source term is used totreat the electrochemical conversion of species that results fromthe Faradaic process and/or due to Joule heating �16,19�. Thesource terms in the respective transport equations effectivelycouples the physics within the ionic, electronic, heat, and masstransport equations. We begin to describe the approach that isused, we begin with the governing transport equations. Thesetransport equations take the form of Poisson’s equation.

� • �− �eleff � �el� = in · ATPB

eff �1�

� • �− �ioneff � �ion� = − in · ATPB

eff �2�

� • �− Dieff � Ci� =

ATPBeff

zF· in �3�

� • �− keff � T� = Qjoule + Qreaction + Qactivation �4�

where the active area, per unit volume, in the vicinity of the TPBATPB

eff is prescribed using percolation theory �16,19�. Similarly, thelocal Faradaic current density resulting from the electrochemicaloxidation in, electronic potential �el, ionic potential �ion, concen-tration Ci, temperature T, effective electronic or ionic conductivity�eff, effective molecular diffusivity Deff, and effective thermalconductivity keff are used. The signs on the source terms for theelectronic and ionic conduction in Eqs. �1� and �2� are writtenwith respect to the anodic electrochemical oxidation process; theyare flipped for reduction in the cathode. The effective diffusivitiesfor gas-phase species i is expressed as

Dieff = �� Dij · Di

k

Dij + Dik� �5�

to treat the parallel continuum diffusion and Knudsen diffusioneffects, where the superscript k indicates the Knudsen diffusioncoefficient. In Eq. �5�, the porous electrode structure is scaled byan empirical parameter � that represents the porosity-tortuosityfactor of the structure.

The local Faradaic current densities in due to the respectiveanodic and cathodic �i.e., oxidation and reduction� reactions in theelectrodes are treated using Butler Volmer equations

inan = io

an�PH2

PH2

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PH2Oref exp��1 − ��

zF

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incat = io

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PO2

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zF

RT�cat�� 7�

for the respective electrodes. The signs for the overpotentials � inthe respective electrodes have been modified for consistency. Theexchange current densities io

ioan = 5.5 � 108 · �PH2

PHref��PH2O

PH Oref � · exp�− Ea

an

RT� �8�

2 2

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iocat = 7.0 � 108 · �PO2

PO2

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re experimental fits �20�. Consistent with previously reportedethods �16,19�, the electrode and total overpotentials are re-

orted in terms of icell or the operational current density.

�2�an = −RT

2F�2 ln� PH2

PH2O� + � 1

�eleff,an +

1

�ioneff,an� · ATPB

eff · icell

�10�

�2�cat = −RT

4F�2 ln�PO2

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1

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�11�

�cell = �an + �cat + icell

�ionlyteLlyte �12�

he overpotentials shown in Eqs. �10�–�12� are used to calculatehe Faradaic current density via the Butler Volmer equations �Eqs.6� and �7��. The source term in the transport equations, Eqs.1�–�4�, are prescribed using the local Faradaic current densities.he boundary conditions are provided in Table 2. The top andottom domain boundaries are considered periodic.

In these transport equations, the heat equation is provided; how-ver, it is not explicitly considered in this study. Preliminarynalyses produced small temperature gradients across the detailedVE �i.e., on the order of 0.001–0.01 K /m� due to its finite

ength. Should it be considered, it is rather straight-forward tomplement due to the consistency in the form of the equation tohe charge and mass transfer processes considered.

Results and validation for the cell-level model are provided inig. 2. In Fig. 2�a�, comparisons between the experimental datand the model results are provided. There are some deviationsetween the model and experimental data near the open circuitoltage �OCV�. There are several possible explanations for thiseviation, which can include experimental fuel leakage and/orinholes in the thin film electrolyte. Jiang and Virkar �10� pro-ided a similar explanation and note that the discrepancy from theheoretical Nernst potential is on the order of 50 mV to 100 mV,hich we incorporate into our model. Figure 2�b� provides a re-

ult from the cell-level model that shows the local distribution ofell voltage and current density at itotal=2.5 A /cm2 and a0%:50%, H2:H2O fuel ratio. The cell voltage is set so that itrovides the Nernst open circuit voltage at the anode current col-ector. Therefore, its value at the cathode current collector is rep-esentative of the operational cell voltage. The total current den-ity is conservative. As expected, the Faradiac neutralization ofhe ionic current in the form of an electronic current occurs over

Table 2 1D cell-level m

Interface Physics

Anode/channel DiffusionElectronic conduction

Ionic conductionOverpotential

Anode/electrolyte OverpotentialCathode/electrolyte OverpotentialCathode/channel Diffusion

Electronic conductionIonic conduction

Overpotential

mall but discrete distances into the electrodes.

ournal of Fuel Cell Science and Technology


2.2 Representative Volume Element Method. The problemcan now be defined at the microstructural level. Representativevolume elements or RVEs are used in this effort. This concept isdemonstrated in Fig. 1 with the labeled squares of the anode. Thefirst RVE is associated with the electronic conduction processes.Because the electronic charge transfer process occurs through thebulk of the electrode, this RVE can be considered at any positionin the anode. The second RVE corresponds to ionic charge trans-fer process. The ionic current exchange occurs over a finite re-gion. Therefore, the ionic RVE should be located near the anode/electrolyte interface. In this study, the ionic charge transfer RVEwill be fixed with one face on the theoretical anode/electrolyteinterface from the cell model.

The RVE boundary conditions are set using the results obtainedfrom the cell model. A local ionic potential, electronic potential,and current density value are extracted from the cell model. Toproperly constrain the system, a potential and current densityboundary condition must be applied to each RVE. Assuming unitdepth for each 2D RVE is considered, the current density bound-ary condition must also be modified to account for the area frac-tion of the phase/region considered relative to the total area of theinterface for which it is prescribed. This modification is necessarybecause the experimentally measured current density is typicallydefined per unit area irrespective of phase or void fraction. Therespective current densities �i.e., electronic in the Ni and ionic inthe YSZ� are therefore modified to maintain consistency with thevalues prescribed from the cell model results. A potential bound-ary condition is used to properly constrain the system; however,the absolute potential is of little consequence for the RVE system.Potential differences within a RVE are indicative of losses in thesystem. Boundaries to phases being examined are insulated due to

el boundary conditions

Boundary condition

H2

ref, CH2Oref

el= �GH2O−GH2−GO2− / 2F �− RT / 2F ln� PH2

ref/ PH2O

ref �ion=�el− ��el−�ion�eq+�an=�an

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•��an=−icell /�ionan,eff

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ref

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ion=�el− �� 1 / 2GO2−GO2− / 2F �+ RT / 4F ln�PO2

ref��+�cat

•��cat= icell /�elcat,eff

Fig. 2 Validation of 1D SOFC button cell model „lines… againstexperimental data „symbols… †10‡. „a… Voltage versus currentdensity curves for different fuel feed streams. „b… Local cellpotential, normalized at anode current collector to Nernst OCVand local electronic, and ionic current densities. It can be notedthat total current density is conservative and current exchange

od

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occurs a finite distance into the respective electrodes.

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he large discrepancies in the electronic conductivity for the Nind YSZ phases; the ionic charge transfer process is treated in aimilar manner �7�.

The details of the transport processes within the RVE requireurther consideration for examining the effects that the micro-tructure has on the ohmic losses. An assumption that the pro-esses fall within the continuum transport regime is made for thisnalysis. This assumption is merited by the length scales of therains and features that are observed within the anode microstruc-ure. Furthermore, this assumption provides the use of the generalharge transport equations that take on the form of a Poisson’s oraplace equation. Experimental work with ionic conduction inpitaxial YSZ films has been shown to exhibit continuum behav-or down to lengths of 60 nm �15�. This length is nearly an orderf magnitude smaller than the minimum YSZ grain size consid-red in this work, 0.5 m �21,22�. A continuum assumption alsoaintains merit for electron conduction in Ni. Nickel is a metallic

onductor with a large number of electrons in the conductionands. This permits its approximation as an electron gas. Further-ore, Drude’s free electron theory of metals would suggest aean free path on the order of several nanometers �23�. This is

everal orders of magnitude smaller than the minimum Ni grainize considered in this work, 1 m �21,22�. Although the lengthcales are appropriate for the conduction processes to be assumedontinuum, the present model also neglects grain boundary diffu-ion processes. This could also be stated as the neglecting of spaceharge effects �24�. These types of effects have been recognized aseing important for ionic conduction as length scales approachanometer length scales �9�. A mean field approximation is usedo address this issue. The mean field approximation considers thatrain boundary diffusion effects are contained in the bulk chargeransport coefficients and processes. The model additionally doesot consider long range effects from the secondary conducting oras-phase potentials or the nature of the capacitive double layer.

The charge transfer processes are inherently linked to the Fara-aic electrochemical oxidation and activation processes in theOFC. Only the ohmic contributions are considered at this time. Aiscrete electrochemical reaction mechanism is not considered. Aolumetric source term is used in these regions, where the Fara-aic processes are occurring. The magnitude of this source term isetermined using the observed change in ionic current densityver the RVE region that was identified in the 1D cell model. Thishange in current represents the electrical neutralization of theonic current density due to the electrochemical oxidation kinetics.his source term is scaled by the area of the ionic phase beingonsidered �i.e., YSZ� of the RVE interface.

Within the RVE microstructure, electronic conduction is con-trained to the form of Laplace’s equation. The current density isrescribed by Ohm’s law.

� • �− �el � �el� = 0 �13�

iel = − �el � �el �14�

he ionic conduction processes are governed by Poisson’s equa-ion, with a source term representing the Faradaic electrochemicalxidation processes, as well as Ohm’s law

� • �−�ion

zF� ion� = −

�iion,RVE

LRVE·

ARVE

AYSZ�15�

iion = −�ion

zF� ion �16�

here the change in ionic current density within the RVEiion,RVE is that due to Faradaic processes in the cell model. Thelectronic conduction processes shown in Eqs. �13� and �14� canlso be considered in terms of the electrochemical potential ̄;owever, the chemical potential associated with the Fermi-levelnergy is assumed spatially invariant in the metallic Ni phase.
his means a standard electrical potential gradient can be used to
31001-4 / Vol. 8, JUNE 2011


represent the force driving transport. Equations �13�–�16� are dis-cretized using a finite element analysis �FEA� method and solvedusing COMSOL multiphysics modeling suite �25�. The detailedFEA solutions are checked to ensure charge conservation is satis-fied, the solutions are independent of the grid size, and individualsolutions are independently verified with the 1D cell model. Thesedetailed FEA solutions were verified by comparing the averagecurrent density on the second boundary to that predicted by the1D model to ensure consistency.

2.3 Auger Electron Spectroscopy. Using the RVE concept,2D phase-specific structures are collected from anode samples us-ing AES. The samples examined with AES are used for the de-tailed RVE studies. AES is an electron based method that collectsAuger electrons that are emitted from the sample when excitedusing a focused electron beam. A detector analyzes the energiesassociated with the emitted Auger electrons and provides an el-emental map of a prepared sample surface. Discrete phases areinferred from this elemental map.

The samples used in this study are taken from a SOFC anodesample. The region that is examined is taken from a plane that liesparallel to the anode/electrolyte interface and nearly midwaythrough the electrode. The electrode that the samples are preparedfrom is composed of comparable volume fractions of Ni, YSZ,and pores. Preparation of these samples for AES analysis is com-pleted by using consecutive surface polishing steps. This processbegins with 1200 grit silicon carbide sandpaper. Four intermediatepolishing steps are completed with diamond paste slurry down toa 0.5 m particle size. The sample surfaces are prepared in thismanner to remove hills and valleys that can obstruct the electronbeam. These types of surface topographic effects can conceal partsof the microstructure �26,27�. A PHI 595 Auger electron spectros-copy unit with a 50 nm probe diameter using a 10 kV Argon ionsputter gun in a 10−10 Torr ultrahigh vacuum chamber is used.The AES maps are completed using an 8 keV focused electronbeam. A spatial resolution of better than 0.26 m is obtained. TheAES map resolution is limited by the resolution of the raster usedduring the elemental mapping process. This resolution is set priorto the elemental mapping of the sample’s surface with the AES.Greater spatial resolutions can be obtained for future efforts; how-ever, the increase in resolution scales with the time that is requiredto acquire the map.

AES is a suitable characterization technique because it onlyexcites Auger electrons from the first few monolayers of the sur-face sample �26–28�. Furthermore, it has a fine electron probediameter on the order of 50–100 nm for practical incident beamenergies �26–28� and can provide an elemental map. A secondaryelectron detector �SED� is simultaneously used to form an elec-tron micrograph from secondary electron sources during the Au-ger electron excitation process. This SED micrograph is used toverify pore locations. The Auger map is further verified by check-ing the consistency of the locations, where only trace amounts ofthe considered elements are found with the pores identified in theSED image. Additional Environmental Scanning Electron Micro-scope �ESEM� micrographs are used for verification. A sampleAES map from the SOFC anode samples is provided in Fig. 3.

A 2D section of the structure is obtained using Auger maps torepresent the SOFC anode microstructure. The AES method isused because the full 3D characterization of the microstructure atspatial resolutions suitable for capturing the pore-scale features isnot trivial �5,6�. By using 2D slices of the heterogeneous 3Dstructure, the full connectivity of the three-phase structure cannotbe captured. This is a limitation of the methods used and requiresadditional examination in future work. To combat the limitation ofusing 2D representations of the structure, RVEs of finite size areused. It is assumed that a single phase with a continuous pathacross the RVE approaches the connectivity of the 3D structure.This is accomplished by using finite sized RVEs and signifies alimiting assumption of this analysis. However, it may not be un-
reasonable to interpret this as an averaged description of the pore-


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cale structure. The 3D connectivity of the phase�s� examined inhe 2D plane can be both enhanced and/or exacerbated through theut-of-plane connectivity for the broader structure. Likewise,imilar features such as constricted regions of transport, blind oread end paths, and isolated regions may be observed. The tortu-sity of a given phase must be larger in the full 3D structure toupport the connectivity of all three phases; however, this can beppropriately handled for these systems. The use of 2D structureo represent the full structure can be thought of as a first approacho the problem. This approach finds validity in the self-onsistency with both the characterization and transport analysisn the proceeding sections. The extension to the full 3D structureas been considered in �5,6�, and the methods used can be ex-ended to a 3D network to support these efforts.

2.4 Microstructure Characterization Methods. Microstruc-ural characterization of the samples considered in this study iseeded to understand the details and effects of geometry of theore-scale anode microstructure. In this study, the microstructures quantitatively characterized using the tortuosity, volume frac-ion, contiguity, and mean free path of the phase being examined29–32�. These parameters provide a unique approach to charac-erize the microstructure compared with more traditional meansuch as percolation theory. Percolation theory treats the micro-tructure as an effective medium, based on factors such as meanrain size and phase volume fractions, with bulk contributions forhe structure’s impact on ohmic losses �11,16,19,20�. The perco-ation limit for a three-phase media is the vicinity of a 30% vol-me fraction for any of the incorporated phases, barring substan-ial difference in grain size. When a sample contains a phase withvolume that falls below the percolation limit, the effective con-

uctivity asymptotes toward zero and transport losses dominate.owever, percolation theory lumps the microstructural description

nd cannot necessarily account for the localized microstructuralffects.

Structures that fall below the traditional percolation limit maye observed in this study when examining the problem at theore-scale; however, these structures still maintain phase connec-ivity. Because there is connectivity within the structures, an ef-ective conductivity asymptote toward zero is not observed. Thiss possible because the structures that are considered are of a finiteize. However, the sample RVEs also may not be representative ofhe larger anode structure. Therefore, several samples are consid-red. This is done so that a statistical basis can be formed, whichrovides a range of results due to localized features that may bebserved within the boarder structure. A variety of features arexamined by taking a statistical approach, but this does not mean

ig. 3 A sample Auger electron spectroscopy map of polishedOFC anode sample. Note distinct phases: red is Ni, white isSZ, and black is pore.

hat the analysis is representative of the full anode properties.



Instead, these analyses provide snapshots of localized regions andtheir effects. The individual results are self-consistent and providea range of features within the structure. For this study, 10�10 m2 RVEs are used. This cross-section is considerablylarger than that, which has previously been used in 3D to showvolumetric independence for porosity and tortuosity of the poreregions �5�. Furthermore, the correlations shown in the results anddiscussion section of this effort demonstrate an underlying trendindicating that RVE sizing is acceptable.

Regardless, the deviations observed in these pore-scale struc-tures from that of the bulk structure are of interest because theycan provide substantial contributions to the ohmic losses. Litera-ture has suggested that these types of pore-scale effects, in par-ticular, those associated with the YSZ phase, may be a primarycontributor to the overall cell resistivity �33�. This also motivatesthe pore-scale approaches to the problem.

To characterize the pore-scale structure, we begin with the defi-nition of the tortuosity of a primary conducting phase, for theregions of the microstructure which continuously traverse the mi-crograph. We refer to these regions as type-A clusters. The tortu-osity is required for domains that exhibit irregular structures thatresult in a conduction path length greater than that of the nominaldomain length. The ratio of the two defines the tortuosity. Thetortuosity is considered because the line counting methods thatwill be considered cannot account for the additional path length.The tortuosity of this region traversing the sample can be definedby considering the two complementary Laplace solutions as illus-trated in Fig. 4. An area weighted integration of the streamlines isperformed. To complete this analysis, a primary conducting phase

Fig. 4 Definition of the phase tortuosity, defined as the pri-mary type-A conductor, which represents a continuous path ofthe conducting phase across the structure. „a… 1D domain con-sidering type-A cluster continuity with remainder of domain.The gray region is the remainder of the domain that is consid-ered continuous with the light blue type-A conductor in the firstLaplace solution, the dotted line is a continuity boundary, andthe solid black lines at top/bottom of domain are insulationboundaries. „b… Local Laplace solutions, where the cross-hatched region is not considered, and the solid black lines rep-resent insulation boundaries between bulk type-A cluster andthe rest of the RVE.

devoid of any inclusions of the secondary solid phase that are

JUNE 2011, Vol. 8 / 031001-5


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nclosed by the continuous conduction path is examined. In otherords, a secondary solid phase surrounded by the primary con-ucting phase has temporarily been redefined as the bulk conduct-ng material. It is important to recognize that this redefinition ofhe secondary phases is completed only during the analysis of theortuosity. This is done to simplify the analysis such that even aisual estimation could be used as a rough approximation byomeone attempting to repeat or use this work. Among the goalsf this work is to be able to correlate the microstructural effects onhe ohmic losses within the structure without needing to performetailed meshing and analysis of the micrographs obtained. Morepecifically, that line counting methods can be used to repeat thenalysis. By removing the inclusions of secondary phases fromhe continuous path�s� being studied, the tortuosity identified isot a true tortuosity. The removal of these secondary regions re-uces the tortuosity and must be considered as a pseudo- orimplified-tortuosity. The effects of the portions that are redefinedre captured and treated by the additional characterization param-ters. As described, the tortuosity can be defined as

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+ � �

�y�2�1/2

dAInsulation

�17�

here ��A is the tortuosity of the type-A conducting path �i.e.,

ontinuous, traversing the sample� of solid phase �, is an arbi-rary scalar field in Laplace’s equation �i.e., �2 =0�, continuityefers to a continuous solution through the RVE for the firstaplace solution �Fig. 4�a��, and insulation refers to the boundaryondition of the primary regions being examined with the rest ofhe domain for the second Laplace solution �Fig. 4�b��. By apply-ng scalar boundary conditions of =1 and =0 on the left andight boundaries, continuity between phases effectively leads to aD solution. Using the same scalar boundary conditions, the insu-ated boundaries of that particular phase lead to the developmentf discrete streamlines within the region studied in the form ofollowing lines with a constant gradient of the scalar field. Byntegrating the magnitude of the vector field defined by the gradi-nt of the scalar over the area of the region examined and thenaking the ratio of the two definitions, the area weighted stream-ine integration is arrived at as defined in Eq. �17�. This is per-ormed analytically for the 1D solution.

In examination of the tortuosity, even seemingly simple varia-ions within a RVE can lead to a tortuosity substantial enough toarrant consideration. For example, even a phase path moving at

n angle to the primary direction of transport can result in anncreased tortuosity. The actual samples, which support a three-hase structure and are typically made from sintered and reducedowders, are quite complex. The method is readily extendable toD structures.

The second quantification parameter that will be defined is thehase volume fraction. Assuming unit depth for the 2D geometry,he volume fraction of phase � is defined as

V� =A�

ARVE�18�

ithin a RVE, where the area of phase �, A� is divided by theotal area of the 2D RVE ARVE. The ohmic losses associated withhe transport process in phase � should be inversely proportionalo the volume fraction of that phase.

At this point, we examine parameters that use line counting

31001-6 / Vol. 8, JUNE 2011


methods. These characterization parameters are defined using thenumber of contacts of a particular type of interface per unit linelength. A measurement grid is overlaid on a sample. All measure-ments that are provided were performed on the 2D RVE structuresusing grids with a spacing that was equal to that of the resolutionlimit of the Auger map used to obtain the RVE.

We begin with the contiguity, which provides a representativemeasure of a sample’s 3D properties from features observable in a2D micrograph �31�. More recently, this concept has been ex-tended from a two phase description to a three-phase descriptionand applied to the SOFC anode microstructure �32�. The contigu-ity is a dimensionless parameter that provides an effective mea-sure of the agglomeration of a particular phase �31�. It is consid-ered a measure of the self grain contact or the connectivity of thatphase. Only self-similar contacts are considered in this work;however, it can be extended to interphase contiguity �31�. Thecontiguity of phase � averaged over the RVE is defined as

C�� =2NL��

2NL��+ NL��

+ NL��

�19�

within a three-phase sample. The contiguity can formally be con-sidered as the contiguity between phase � and �. In Eq. �19�, NL��is the number of grain boundaries crossed corresponding to phases� and � per unit line length with a given measurement grid �32�.Because the contiguity is a measure of the connectivity of a phaseof interest, the ohmic losses should increase as the contiguitydecreases due to decreased interparticle contact in the conductingphase.

At this point, an assumption is required. The RVE domain doesnot contain grain boundary information within an individualphase. Similar grain boundaries are considered in the definition ofcontiguity. This requires us to assume an average grain size withinthe domain. To better understand the impact of the average grainsize, three sets of average grain sizes are considered on an indi-vidual basis for the Ni and YSZ. For the Ni phase, 1 m, 3 m,and 5 m grains are considered. The 1 m and 5 m grainsizes represent reasonable limits of Ni grains in an anode; the3 m is considered to be representative for actual systems�21,22�. Grain sizes of 0.5 m, 5.4 m, and 25 m are consid-ered for the YSZ phase. The YSZ phase of common cermet an-odes is comprised of �80% volume 0.5 m grains and �20%volume 25 m grains to provide both percolation and mechanicalsupport �21,22�. The constituent grain sizes are used as boundingvalues for grain sizes considered within the microstructure and5.4 m is the volume averaged YSZ grain size. Use of the vol-ume averaged YSZ grain size is considered a reasonable approxi-mation because for an even distribution of the 0.5 m and25 m grains, portions of both grains may be present within anyRVE considered given the 10�10 m2 size of the RVEs. This isa generalization that leaves room for improving on the character-ization reported here. Also, any RVE maintaining a 25 m grainshould be readily apparent as it would dominate the 10�10 m2 structure and approach the condition representative ofan ideal conductor. However, with this assumption, the 3 m Nigrain and 5.4 m YSZ grain sizes are considered as representa-tive of the structure. It is also assumed that the grains are spheri-cally shaped, so that the mean linear transverse of the grain is2 /3Dg, where Dg is the grain diameter. The mean linear traverse isused to represent the expected size of an individual grain withinthe microstructure.

A mean free path can also be defined with respect to an indi-vidual phase in the microstructure. The mean free path is a mea-sure of the mean distance of the secondary phases that must becrossed to travel from one grain to a second grain of the samephase �31�. It does not consider adjacent grains of a similar phase.As the name would suggest, the mean free path is a measure of theconstriction of a phase. This could also be interpreted as the cross-
section of the phase path that follows. Using a similar definition to


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revious studies �31�, the extension of the two phase mean freeath to a three phase mean free path requires the consideration ofhe additional dissimilar grain contacts. The mean free path forhase � is considered in the form

�� =2V�

NL��+ NL��

·1

LRVE�20�

here the length of the RVE LRVE is used to make the mean freeath dimensionless. From a conceptual standpoint, the mean freeath is a measure of phase constriction. Therefore, it is anticipatedhat ohmic losses will increase in magnitude as the mean free pathor the primary conducting phase decreases.

2.5 Performance Quantification Methods. To interpret theVE structure’s impact on the ohmic losses, it is important to

orm a consistent metric for quantifying the microstructure’s per-ormance. We will quantify these ohmic losses in terms of anhmic performance parameter. The ohmic performance refers tohe ability of the structure to transport charge without losses at-ributed to these processes �i.e., Joule heating�. The optimal ohmicerformance would that be of a continuous bulk phase devoid ofhe other phases considered. Increased losses that result from theransport processes occurring in the structure are consistent with aecrease in performance due to an increase in Joule heating. Theoule heating is produced from this irreversibility of the chargeransport process

Q�joule =

1

A��

A�

i2�� · dA� �21�

nd scales with the square of the current density. The currentensity can be electronic or ionic in the respective Ni and YSZhases.

The ohmic losses within the RVE must also be able to be quan-ified on a consistent basis for the unique structures considered. Aseen in Eq. �21�, Joule heating is defined in terms of the localurrent density. Therefore, the Joule heating must be corrected forhe area of the conducting phase � with respect to the total area ofhe RVE

Q�joule,corrected = Q�

joule ·ARVE

A�

�22�

here the area corrected Joule heating in phase �, Q�joule,corrected, is

escribed in terms of the originally calculated Joule heating Q�joule

rom Eq. �21�. With the area corrected Joule heating, a nondimen-ional heat release, Q�

� , can be defined. This is calculated by form-ng a ratio of the area corrected heat release Qjoule,corrected to ainimum loss RVE. The minimum loss RVE is a RVE of the same

omain size and boundary conditions but with a pure conductinghase �. The nondimensional heat release takes the form

Q�� =

Q�joule,corrected

Q�joule,min =

�A�

i2��dA

�ARVE

i2��dA

·ARVE

A�

�23�

here the minimum loss condition for phase �, Q�joule,min, is used

o normalize Eq. �22�. This provides a nondimensional and effec-ive measure of ohmic performance or heat release due to Jouleeating in the RVE.

An additional correction in the form of the tortuosity of theonductive phase or, more formally, pseudotortuosity as defined inhe discussion on the methods used, must be considered to ac-ount for additional conduction length within a RVE. This correc-ion accounts for conduction through the tortuous paths within the

aterial. This is an effect that cannot be captured by the lineounting based characterization methods and has a direct impact
n interpretation of the ohmic performance of the RVE. Conse-


quently, results will be presented in the form of �Q�� /��

A�, where ��A

is the conducting phase’s tortuosity defined in Eq. �17�. This defi-nition provides a consistent basis for reporting a loss due to ohmicheating between unique RVEs.

To define the total effective heating, it is necessary to sum theelectronic and ionic contributions

Qnet� = QNi,el

� + QYSZ,ion� �24�

because both can contribute to the net Joule heating. As defined,the effective Joule heating or heat release Q�

� is a nondimensionalnumber that can range from one to infinity. It is a figure of meritwith respect to the ohmic performance of any RVE case consid-ered and its respective current density. It could also be consideredas an effective resistivity correction based on a given structure ofthe form ��

eff=Q��.

3 Results and DiscussionWith the characterization parameters defined, correlations to the

effective ohmic performance of the RVE can be identified. Corre-lations between the tortuosity corrected effective ohmic perfor-mance �Q�

� /��A�, and the microstructure parameters are examined

for actual 2D AES and micrographs as well as some conceptualmicrostructures. In all, eight AES samples and 42 conceptualstructures are examined. The conceptual microstructures are usedto supplement the actual samples and to improve the size of thestatistical set. Due to the finite sizes of the RVEs, the RVE prop-erties may extend above or below what is typically anticipated inthe SOFC anode.

We begin this effort by introducing the conceptual or phenom-enological cases. These conceptual cases begin with three casesthat consist of pure Ni and pore phases as seen on the left handside of Fig. 5. The tortuosity of the Ni phase and Joule heating can

Fig. 5 Phenomenological electronic conduction geometriesconsidered. Red: Ni, white: YSZ, and black: pore. All geom-etries have a symbol associated with them that will be usedduring analysis of results. For each, the left denotes the basecase containing only Ni and pores while the right case denotesa variation on the case with inclusions of secondary phases.Complementary cases are considered for ionic conduction inYSZ. „a… Straight Ni phase with pore constriction, „b… straight Niphase with a blind branch, and „c… tortuous pore cuttingthrough Ni substrate. All geometries are 10Ã10 �m2 in size.

be examined for these three geometries. The right hand column of

JUNE 2011, Vol. 8 / 031001-7


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ig. 5 denotes an intermediate case with YSZ and inclusions. Inll of the cases, the YSZ inclusions are white, pores are black, andemainder �red� is Ni. These additional intermediates contain dif-erent size, shapes, and placements of YSZ and pore inclusionsithin the bulk Ni phase. Random distributions and sizes of round

nclusions were considered with inclusion diameters ranging fromractions that of the smallest single YSZ grains inclusion to sev-ral times larger. Similar sizes have been considered for pore in-lusions. Inclusions have been placed at random through the struc-ure. The overlapping of inclusions provided unique and arbitrarynclusion shapes and configurations. This can be seen in the rightand column of Fig. 5. For the conceptual cases that consideronic conduction, the Ni and YSZ phases are interchanged andithin the basic structures in consideration.Using the conceptual structures shown in Fig. 5 and their inter-ediates as well as the actual anode structures from the Augeraps such as the one seen in Fig. 3, the tortuosity corrected ef-

ective heat release can be examined with respect to the conduct-ng phase volume fraction V�, contiguity C��, and mean free path�. Prior to the discussing these individual parameters, a set oficrographs with detailed normal current density and constant

Fig. 6 Normalized distributions of current denrent density: „a… Current density magnitude forarea corrected left Ni phase boundary current dcurrent density flux lines, „c… current density mmalized to area corrected left YSZ phase bounconstant ionic current density flux lines

urrent density flux lines are provided in Fig. 6. In Figs. 6�a� and

31001-8 / Vol. 8, JUNE 2011


6�c�, the magnitude of the electronic and ionic current densities inthe Ni and YSZ phases are, respectively, shown after to the cur-rent density boundary condition is normalized. These figures rep-resent local variations in current density with respect to the ap-plied boundary condition. Figures 6�b� and 6�d� show the constantcurrent density flux lines for the respective electronic and ionicconduction processes. The variation in current density is noted forboth electronic and ionic conduction. With respect to the elec-tronic conduction found in Fig. 6�a�, there is approximately8.5� variation in current density from the specified current den-sity boundary condition of 7.65�104 A /m2 �i.e., area correctedcurrent density� to the constriction, where the maximum currentdensity occurs. However, as seen in Fig. 6�c�, the ionic conductionRVE does not see a dramatic increase in current density relative tothe current density boundary condition due to the current ex-change process �i.e., boundary condition of an area corrected ioniccurrent density of 4.85�105 A /m2�. Constricted regions containcurrent densities near that of the inlet current density by approach-ing 0.95� of the inlet current density. Despite the current ex-change process, these constricted regions still play a prominent

in anode samples for 2.0Ã104 A/m2 net cur-ctronic conduction in Ni phase normalized to

sity of 7.65Ã104 A/m2, „b… constant electronicnitude for ionic conduction in YSZ phase nor-ry current density of 4.85Ã105 A/m2, and „d…

sityeleenagda

role in the RVE ohmic performance.



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With this detailed look at the local magnitudes of the currentensities, the quantification and characterization can be examined.e begin by examining the behavior of the volume fraction, con-

iguity, and mean free path of the participating phase on an indi-idual basis as related to the tortuosity corrected effective heatelease. Figure 7 provides a look at the nature of each of thendividual quantification parameters with respect to electronic andonic conduction. Figures 7�a�–7�c� show the tortuosity correctedffective heat release versus the volume fraction, contiguity, andean free path of the Ni phase for the electronic conductor. Fig-

res 7�d�–7�f� provide the tortuosity corrected effective heat re-ease versus the volume fraction, contiguity, and mean free path ofhe YSZ phase for the ion conductor. It is recognized that thectual SOFC anode structures obtained using AES and phenom-nological cases demonstrate the same qualitative trends. The vol-me fraction, contiguity, and mean free path of the YSZ phaselso consistently show an inverse relationship with increasingosses. The mean grain size that is used to define the contiguitylays a substantial role in the relationship between the correctedffective ohmic performance parameter and the respective conti-uity. For both the electronic and ionic conduction processes, Fig.�b� and 7�e� show that a change in average grain size shifts thentire contiguity curve and has a moderate impact on the disper-ion of the contiguity. However, given the sizes of RVEs consid-red, the differences in average grain sizes studied are quite sub-tantial in comparison to the deviations from the average grainize that would likely be found in the RVE or for a completeicrostructure �21,22�. These cases effectively represent the spe-

ial and limiting conditions for the system described, where a

Fig. 7 Electronic and ionic tortuosity correctedization parameters: „a… electronic corrected effelectronic corrected effective heat release versu3 �m, and 5 �m, „c… electronic corrected effeionic corrected effective heat release versus Yheat release versus YSZ-YSZ contiguity for mea„f… ionic corrected effective heat release versus

VE with only that particular grain size is found. This result only



bounds the problem. If an even distribution of the large and meangrains that comprise the YSZ phase is assumed, it is reasonable toassume the RVE as being comprised of grains near the averagegrain size. To check the importance of variations in mean grainsize, a secondary sensitivity analysis is also completed with regardto the contiguity. In this study, it is found that a + /−10% deviationin average grain size for the cases of 3 m Ni grains and 5.4 mYSZ grains have less than + /−5% variation on contiguity.

To correlate the characterization parameters to the effectiveohmic performance, a Levenberg–Marquardt �LM� nonlinearleast-squares method is used to fit the effective ohmic perfor-mance to a power law in terms of the quantification parameters.

QNi,el�

�NiA = A1 · VNi

A2 · Clow,Ni–NiA3 · Cavg,Ni–Ni

A4 · Chigh,Ni–NiA5 · �Ni–Ni

A6

�25a�

QYSZ,ion�

�YSZA = B1 · VYSZ

B2 · Clow,YSZ-YSZB3 · Cavg,YSZ-YSZ

B4

· Chigh,YSZ-YSZB5 · �YSZ

B6 �25b�

where A1,A2, . . . ,A6 are fit constants for electronic conductionand B1,B2 , . . . ,B6 are fit constants for ionic conduction. Linearregression analysis is also used whenever possible and to validatethe LM nonlinear least-squares method. In Eq. �25�, the “low,”“avg,” and “high” contiguity definitions refer to the minimum,average, and maximum constituent particle size considered in

fective heat release as a function of character-ve heat release versus Ni volume fraction, „b…i–Ni contiguity for mean particle size of 1 �m,

ve heat release versus Ni mean free path, „d…volume fraction, „e… ionic corrected effective

article size of 0.5 �m, 5.4 �m, and 25 �m, andZ mean free path

efectis NctiSZ

n pYS

their definition, respectively.

JUNE 2011, Vol. 8 / 031001-9


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Prior to completing a comprehensive fit, the dependence of thehmic performance on the individual parameters is performed sohat their individual effects can be better understood. This regres-ion analysis is completed using Eqs. �25a� and �25b� and theethod providing the best fit data are presented in Tables 3 and 4.ll results are presented in the form of Eq. �25�, where the powersf parameters not considered for a particular fit are set to zero.ables 3 and 4 contain these fits to the tortuosity corrected ohmicerformance. The correlation of the individual characterizationarameters correspond to the first five data sets in Tables 3 and 4.n these first five data sets, as also can be observed in Fig. 7, all ofhe characterization parameters show an inverse relation to in-reases in Joule heating, denoted by an increased �Q�

� /��A�.

In Tables 3 and 4, the following four data sets demonstrate howhe system performs when combinations of these parameters areombined in the fit, reported in a power law form of Eq. �25�. Theystem is reported in the power law form because each of thearameters follows a power law form on an individual basis.hile examining the four combined correlation data sets in Tablesand 4, the conducting phase volume fraction is recognized as

eing of great importance. When it is removed from the correla-ion, poor statistical agreement is observed. The combinations ofhe phase volume fraction, mean free path, and average grainized contiguity for the respective conducting phase are examined.his trend is shown in bold in Tables 3 and 4. It represents thenal overall correlation for connection between the characteriza-

ion parameters and the tortuosity corrected ohmic performance.

Table 3 Tortuosity corrected effective electrperformed on detailed RVEs of both actual anstructures. x denotes that the parameter wascompleted in the first eight correlations to invcoupled parameter effects. The bold case repdata sets. The complete data ranges that wervided in the last row.

A1 A2�VNi� A3�Clow,Ni–Ni� A

1.476 2.175 x3.227 x 4.0911.934 x x1.350 x x1.678 x x0.741 1.622 x0.847 1.729 x1.631 x x0.748 �1.626 x

Parameter data range 0.63–0.14 0.90–0.50

Table 4 Tortuosity corrected effective ionicformed on detailed RVEs of both actual anstructures. x denotes parameter was not consin the first eight correlations to investigate thparameter effects. The bold case represents tThe complete data ranges that were observedrow.

B1 B2�VYSZ� B3�Clow,YSZ-YSZ� B4�Ca

1.466 2.137 x3.112 x 4.5201.094 x x 0.289 x x1.867 x x1.162 1.333 x0.870 1.332 x 1.365 x x 1.124 1.331 x

Parameter datarange 0.66–0.14 0.90–0.50 0.4

31001-10 / Vol. 8, JUNE 2011


For these final fits, an R2 of 0.88 and 0.91 and a root mean squareerror �RMSE� of 6.02 and 5.46 were found for Tables 3 and 4,respectively. This represents a reasonable statistical agreementand indicates a clear trend.

Using the correlation dependencies that have been bolded inTables 3 and 4, the tortuosity corrected ohmic performance fromthe individual model RVE models �i.e., both the Auger and phe-nomenological models� can be examined. Figure 8 plots the tor-tuosity corrected ohmic performance from the individual RVEmodels versus the predicted value using the correlation found inEq. �25� using the bolded fitting parameters in Tables 3 and 4.Because the power-series fit performed in the preceding discus-sion was a linearization, the fit value for this correlation can alsobe represented. This is shown in Fig. 8 along with the 95% con-fidence intervals. The reasonable agreement between both the con-ceptual and physical anode RVE cases is noted. Only moderatescatter exists and this primarily falls within the 95% confidenceintervals. Most importantly, these figures effectively show that thecharacterization parameters have been used to correlate a tortuos-ity corrected ohmic performance. Such a correlation indicates thatnot only the conducting phase volume fraction but also the phasetortuosity and measures of constriction and agglomeration are im-portant to understanding the structure’s effective ohmic perfor-mance. These parameters aspects are quantified with the phasemean free path and contiguity, respectively. This improvement incombined form is substantiated by the improved R2 and RMSE inTables 3 and 4 when compared with the individual parameters.

ic ohmic loss correlation table for Eq. „25a…e samples and generated phenomenologicalt considered for a particular case. This wasgate the dependencies of both individual andents the final fit that was determined for thebserved within the RVEs considered are pro-

avg,Ni–Ni� A5�Chigh,Ni–Ni� A6��Ni–Ni� R2 RMSE

x x x 0.82 7.45x x x 0.47 12.90.395 x x 0.48 12.85x 2.047 x 0.48 12.85x x 1.214 0.59 11.42x x 0.646 0.88 6.05.121 x x 0.88 6.26

.388 x 1.406 0.59 11.41.389 x 0.459 0.88 6.02–0.25 0.65–0.17 0.63–0.06 x x

mic loss correlation table for Eq. „25b… per-samples and generated phenomenological

red for a particular case. This was completedependencies of both individual and couplednal fit that was determined for the data sets.

thin the RVEs considered are provided in last

SZ-YSZ� B5�Chigh,YSZ-YSZ� B6��YSZ� R2 RMSE

x x 0.81 8.07x x 0.56 12.10

98 x x 0.60 11.621.324 x 0.60 11.61

x 1.161 0.61 11.46x 0.631 0.91 5.45

68 x x 0.91 5.4834 x 0.484 0.60 11.5505 x 0.410 0.91 5.46

.09 0.16–0.02 0.63–0.06 x x

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ConclusionsA method of quantifying electronic and ionic ohmic perfor-ance in a SOFC anode has been developed. This method relies

n line counting based characterization methods from detailedhase-specific AES micrographs. It has been shown that the char-cterization parameters in a power law form can be correlatedith the continuum-level model of detailed transport within con-

eptual and AES-based microstructures with reasonable statisticalgreement. Such a method provides a useful tool to an engineerho wishes to make a statistical measure of a microstructure’s

nticipated ohmic performance without completing detailed ex-erimental work and without having to complete detailed andomputationally intensive studies.

cknowledgmentThe authors gratefully acknowledge financial support from the

rmy Research Office Young Investigator Program �Award6964-CH-YIP�, the National Science Foundation �Award CBET-828612�, an Energy Frontier Research Center on Science Basedano-Structure Design and Synthesis of Heterogeneous Func-

ional Materials for Energy Systems funded by the U.S. Depart-ent of Energy, Office of Science, Office of Basic Energy Sci-

nces �Award DE-SC0001061� and the ASEE National Defensecience and Engineering Graduate Fellowship program. The au-

hors would like to thank Adaptive Materials Inc. �AMI� of Annrbor, MI for the SOFC samples.

omenclatureA � area, m2

AES � Auger electron spectroscopyC � concentration, mol /m3

C�� contiguity between phases � and �D � diffusivity, m2 /s

Dg � mean grain diameter, mEa � activation energy, J/molF � Faraday’s constant, 96,485 A s /molH � height, mG � Gibbs free energy, J/moli � current density, A /m2

io � exchange current density, A /m2

k � thermal conductivity, W/m/KL � length, m

LSM � lanthanum strontium manganiteNL�� number of �-� grains crossed per unit line

length, mP � pressure, Pa

3

ig. 8 Fit of parameterized results based on quantification pa-ameters shown with 95% confidence interval bands: „a… tortu-sity corrected effective electronic heat release versus corre-

ated characterization parameter fit and „b… tortuosity correctedffective ionic heat release versus correlated characterizationarameter fit

Q � heat source, W /m



R � universal gas constant, 8.314 J/mol/KT � temperature, K

V� � volume fraction of phase �YSZ � yttria-stabilized zirconia

z � moles of electrons per mole of reactant

Greek� � charge transfer symmetry constant� � porosity� � overpotential, V� � dimensionless mean free path̄ � electrochemical potential, J/mol� � resistivity, ohm m� � conductivity, S/m� � tortuosity� � potential, V� � porosity/tortuosity diffusion correction factor

Subscripts� ,� ,� � arbitrary phases

char � characteristicel � electronici � chemical species

ion � ionicnet � netNi � Nickel phase

RVE � representative volume elementTPB � three phase boundaryYSZ � YSZ phase

SuperscriptsA � type-A cluster designationan � anodecat � cathode

cell � cell-levelcorrected � corrected

eff � effectivejoule � Joule heating

k � Knudsenlyte � electrolytemin � minimumref � reference

� � nondimensional of effective

References�1� Hussain, M. M., Li, X., and Dincer, I., 2006, “Mathematical Modeling of

Planar Solid Oxide Fuel Cells,” J. Power Sources, 161, pp. 1012–1022.�2� Greene, E. S., Chiu, W. K. S., and Medeiros, M. G., 2006, “Mass Transfer In

Graded Microstructure Solid Oxide Fuel Cell Electrodes,” J. Power Sources,161, pp. 225–231.

�3� Joshi, A. S., Grew, K. N., Izzo, J. R., Jr., Peracchio, A. A., and Chiu, W. K. S.,2010, “Lattice Boltzmann Modeling of Three-Dimensional, Multi-ComponentMass Diffusion in a Solid Oxide Fuel Cell Anode,” ASME J. Fuel Cell Sci.Technol., 7, p. 011006.

�4� Joshi, A. S., Grew, K. N., Peracchio, A. A., and Chiu, W. K. S., 2007, “LatticeBoltzmann Modeling of 2D Gas Transport in a Solid Oxide Fuel Cell Anode,”J. Power Sources, 164�2�, pp. 631–638.

�5� Izzo, J. R., Jr., Joshi, A. S., Grew, K. N., Chiu, W. K. S., Tkachuk, A., Wang,S. H., and Yun, W., 2008, “Structural Characterization of Solid Oxide FuelCell Anodes Using X-Ray Computed Tomography at Sub-50 nm Resolution,”J. Electrochem. Soc., 155�5�, pp. B504–B508.

�6� Grew, K. N., Chu, Y. S., Yi, J., Peracchio, A. A., Izzo, J. R, Jr., De Carlo, F.,Hwu, Y., and Chiu, W. K. S., 2010, “Nondestructive Nanoscale 3D ElementalMapping and Analysis of a Solid Oxide Fuel Cell Anode,” J. Electrochem.Soc., 157, pp. B783–B792.

�7� Singhal, S. C., and Kendall, K., 2003, High Temperature Solid Oxide FuelCells Fundamentals, Design and Applications, Elsevier, New York.

�8� Marinšek, M., Pejovnik, S., and Macek, J., 2007, “Modeling of ElectricalProperties of Ni-YSZ Composites,” J. Eur. Ceram. Soc., 27, pp. 959–964.

�9� Reifsnider, K., Huang, X., Ju, G., and Solasi, R., 2006, “Multi-Scale ModelingApproaches for Functional Nano-Composite Materials,” J. Mater. Sci., 41, pp.6751–6759.

�10� Jiang, Y., and Virkar, A. V., 2003, “Fuel Composition and Dilutent Effect onGas Transport and Performance of Anode-Supported SOFCs,” J. Electrochem.Soc., 150, pp. A942–A951.

�11� Nam, J. H., and Jeon, D. H., 2006, “A Comprehensive Micro-Scale Model for

JUNE 2011, Vol. 8 / 031001-11


0

Downlo

Transport and Reaction in Intermediate Temperature Solid Oxide Fuel Cells,”Electrochim. Acta, 51, pp. 3446–3460.

�12� Jiang, Y., and Virkar, A. V., 2001, “La1−xSrxCoO3−� �LSC�-Ce0.8Sm0.2O2−�

�SCD� Composite Cathodes for Anode Supported, YSZ-SDC Bi-Layer Elec-trolyte, Thin Film, Solid Oxide Fuel Cells,” Electrochemical Society Proceed-ings on Ionic and Mixed Conducting Ceramics IV, T. A. Ramanarayanan, ed.,The Electrochemical Society, Inc., Pennington, NJ, Vol. 2001-28, pp. 374–383.

�13� Bucher, E., Sitte, W., Rom, I., Papst, I., Grogger, W., and Hofer, F., 2002,“Microstructure and Ionic Conductivity of Strontium-Substituted LanthanumCobaltites,” Solid State Ionics, 152–153, pp. 417–421.

�14� Koep, E., Jin, C., Haluska, M., Das, R., Narayan, R., Sandhage, K., Snyder, R.,and Liu, M., 2006, “Microstructure and Electrochemical Properties of CathodeMaterials for SOFCs Prepared via Pulsed Laser Deposition,” J. Power Sources,161, pp. 250–255.

�15� Kosacki, I., Rouleau, C. M., Becher, P. F., Bentley, J., and Lowndes, D. H.,2005, “Nanoscale Effects on the Ionic Conductivity of Highly Textured YSZThin Films,” Solid State Ionics, 176, pp. 1319–1326.

�16� Chan, S. H., and Xia, Z. T., 2001, “Anode Micro Model of Solid Oxide FuelCell,” J. Electrochem. Soc., 148�4�, pp. A388–A394.

�17� Bessler, W. G., Warnatz, J., and Goodwin, D. G., 2007, “The Influence ofEquilibrium Potential on the Hydrogen Oxidation Kinetics of SOFC Anodes,”Solid State Ionics, 177, pp. 3371–3383.

�18� Brown, M., Primdahl, S., and Mogensen, M., 2000, “Structure/PerformanceRelations for Ni/Yttria-Stabilized Zirconia Anodes for Solid Oxide FuelCells,” J. Electrochem. Soc., 147�2�, pp. 475–485.

�19� Cannarozzo, M., Grosso, S., Agnew, G., Borghi, A. D., and Costamagna, P.,2007, “Effects of Mass Transport on the Performance of Solid Oxide FuelCells Composite Electrodes,” ASME J. Fuel Cell Sci. Technol., 4, pp. 99–106.

�20� Costamagna, P., Slimovic, A., Borghi, M. D., and Agnew, G., 2004, “Electro-chemical Model of the Integrated Planar Solid Oxide Fuel Cell �IP-SOFC�,”Chem. Eng. J., 102, pp. 61–69.

31001-12 / Vol. 8, JUNE 2011


�21� 2007, Private Communication With Prof. Nigel Sammes, University of Con-necticut, Mar. 20.

�22� Brown, M. S., 1998, “Fabrication and Characterization of Nickel/Yttria Stabi-lized Zirconia Cermet Anodes for Solid Oxide Fuel Cells,” Ph.D. thesis, Uni-versity of Waikato, New Zealand.

�23� Livingston, J. D., 1999, Electronic Properties of Engineering Materials, Wiley,New York.

�24� Maier, J., 1995, “Ionic Conduction in Space Charge Regions,” Prog. SolidState Chem., 23, pp. 171–263.

�25� COMSOL, 2006, COMSOL Multiphysics User’s Guide, Burlington, MA.�26� Brandon, D., and Kaplan, W. D., 1999, Microstructural Characterization of

Materials, Wiley, New York, NY.�27� Brundle, C. R., Evans, C. A., Jr., Wilson, S., and Fitzpatrick, L. E., 1992,

Encyclopedia of Materials Characterization: Surfaces, Interfaces, Thin Films,Manning, Greenwich, CT.

�28� Reniers, F., and Tewell, C., 2005, “New Improvements in Energy and Spatial�x,y,z� Resolution in AES and XPS Applications,” J. Electron Spectrosc. Relat.Phenom., 142, pp. 1–25.

�29� Smith, C. S., and Guttman, L., 1953, “Measurement of Internal Boundaries inThree-Dimensional Structures by Random Sectioning,” J. Met., 197, pp. 81–87.

�30� Gurland, J., 1966, “An Estimation of Contact and Continuity of Dispersions inOpaque Samples,” Trans. Metall. Soc. AIME, 236, pp. 642–646.

�31� Gurland, J., 1958, “The Measurement of Grain Contiguity in Two-Phase Al-loys,” Trans. Metall. Soc. AIME, 212, pp. 452–455.

�32� Lee, J.-H., Moon, H., Lee, H.-W., Kim, J., Kim, J.-D., and Yoon, K.-H., 2002,“Quantitative Analysis of Microstructure and Its Related Electrical Property ofSOFC Anode, Ni-YSZ Cermet,” Solid State Ionics, 148, pp. 15–26.

�33� Zhao, F., and Virkar, A. V., 2005, “Dependence of Polarizations in Anode-Supported Solid Oxide Fuel Cells on Various Cell Parameters,” J. PowerSources, 141�1�, pp. 79–95.



characterization and quantiﬁcation of electronic and ionic

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