Characterization and Quantification of Electronic and Ionic Ohmic Overpotential and Heat Generation in a Solid Oxide Fuel Cell Anode 1

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1 Kyle N. Grew John R. Izzo, Jr. Wilson K. S. Chiu 2 wchiu@engr.uconn.edu Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road, Storrs, CT Characterization and Quantification of Electronic and Ionic Ohmic Overpotential and Heat Generation in a Solid Oxide Fuel Cell Anode 1 The development of a solid oxide fuel cell (SOFC) with a higher efficiency and power density requires an improved understanding and treatment of the irreversibilities. Losses due to the electronic and ionic resistances, which are also known as ohmic losses in the form of Joule heating, can hinder the SOFC s performance. Ohmic losses can result from the bulk material resistivities as well as the complexities introduced by the cell s microstructure. In this work, two-dimensional (2D), electronic and ionic transport models are used to develop a method of quantification of the ohmic losses within the SOFC anode microstructure. This quantification is completed as a function of properties determined from a detailed microstructure characterization, namely, the tortuosity of the electronic and ionic phases, phase volume fraction, contiguity, and mean free path. A direct modeling approach at the level of the pore-scale microstructure is achieved through the use of a representative volume element (RVE) method. The correlation of these ohmic losses with the quantification of the SOFC anode microstructure are examined. It is found with this analysis that the contributions of the SOFC anode microstructure on ohmic losses can be correlated with the volume fraction, contiguity, and mean free path. DOI: / Keywords: SOFC, ohmic losses, Joule, characterization, representative volume element, pore-scale 1 Presented at the 5th ASME International Fuel Cell Science, Engineering and Technology Conference, Paper No. FuelCell , Brooklyn, NY, June Corresponding author. Manuscript received July 27, 2007; final manuscript received June 26, 2010; published online February 15, Assoc. Editor: Abel Hernandez. 1 Background Significant losses can arise in the solid oxide fuel cell SOFC, arising from the cell s electronic and ionic resistances. Depending on the cell design and operating conditions, resistive losses in the form of Joule heating can be among the more substantial losses within the cell 1,2. These losses need to be understood and minimized to develop a more efficient SOFC. In this work, a focus is placed on the contributions of the SOFC anode microstructure to these losses. An analysis of the impact of the pore-scale microstructure s features on mass transfer process is being examined in a complementary effort 3 6. The reduction of resistive losses in a state of the art SOFC anode, comprised of a porous nickel-yttria-stabilized zirconia Ni- YSZ cermet is not just a mater of shortening the respective transport path lengths, enhancing transport properties, or changing the volumetric composition to improve percolation. The optimization of the SOFC anode requires the balance of the ohmic, activation, and concentration losses with additional consideration of the mechanical, thermal, and chemical stability requirements 7. This balance necessitates sufficient ionic and electronic conduction pathways, pore-space for the transport of fuel and product species, electrocatalytically active three-phase boundary TPB regions, and a robust and stable structure 7. The complexity of the anode microstructure makes it difficult to optimize a particular aspect of the SOFC anode design without impacting another. Numerous micrographs of the Ni-YSZ cermet anode microstructures show this complexity 8,9. These micrographs show a complex heterogeneous microstructure to nanostructure, which has complex Ni and YSZ phase-networks, irregular pores, and unique features. To examine the charge transfer processes, these complex features must be considered at the length scales of the features i.e., pore-scale microstructure to understand their effects. In a recent review, Reifsnider et al. suggest that the losses originate at nanometer length scales in these complex structure 8. These length scales are characteristic of the grains structure and pore-scale features. The details of these effects are difficult to capture in aggregated cell or electrode models, which use effective transport and averaged microstructural properties. Due to a lack of understanding regarding the impact that the features at these length scales have on SOFC performance, we explore the effect that the anode microstructure has on ohmic losses in this work. The goal of this microstructural or pore-scale approach is to provide a tool for the quantitative characterization of the SOFC microstructure with respect to the ohmic losses. This is completed using several characterization parameters that provide descriptions of the cell s microstructure i.e., tortuosity of the conducting phase, volume fraction, contiguity, and mean free path. A focus is placed on the two-dimensional 2D electronic and ionic charge transfer processes within the anode. This effort includes the analysis of ohmic losses resulting from ionic conduction in the electrochemically active regions of the anode. Phase-specific micrographs of the SOFC anode are collected using Auger electron spectroscopy AES on several samples. Conceptual microstructures are used to supplement these AES samples. Both are used to develop a better understanding of the microstructure s role in ohmic losses. Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 / Copyright 2011 by ASME

2 Fig. 1 Schematic of the 1D SOFC button cell model used to model experimental work in literature 10. The electronic RVE is at an arbitrary location within the anode support while the ionic RVE is localized to the anode/electrolyte interface. 2 Method of Approach This paper begins by presenting a one-dimensional 1D cell model that is used to specify boundary conditions for a RVE. The RVE method is used to consider detailed electrode structure. Once the RVE method is defined, AES is used to provide detailed RVEs of actual SOFC samples. A discussion on the structural characterization and performance quantification method is then presented. 2.1 One-Dimensional Cell Level Model. A cell model is needed to prescribe boundary conditions to a detailed RVE, which permits the microstructural details contained in the RVE to be used for more detailed analysis. Experimental data published by Jiang and Virkar 10 is used to parameterize and validate a 1D model that is developed here. The work by Jiang and Virkar considers a laboratory SOFC button cell operating on hydrogen H 2 fuel diluted with water vapor H 2 O and oxygen O 2 taken from atmospheric air as the oxidant. A schematic of the 1D model is provided in Fig. 1. A Ni-YSZ cermet anode, bilayer YSZ-SDC thin film electrolyte, and LSC-SDC cathode are used in this study. Table 1 provides a summary of the conditions and cell properties that were used for the cell-level model that will be considered within this section 10. The cell model is used to prescribe boundary conditions to a localized and finite sized RVE. As a part of this effort, this model must consider electrochemical reaction process within the electrode structures so that the local electronic and ionic fluxes and potentials can be identified. Specifically, the Faradaic portion of the electrochemical oxidation processes must be considered, Table 1 Parameters used in 1D cell-level model. Bulk conductivities were taken from the provided references. Anode effective conductivities are corrected based on percolation theory 16. Parameter Value Reference T 1073 K 10 L an m 10 L lyte m 10 L cat m 10 D H2 H 2 O m 2 /sec 10 k D H m 2 /sec 10 k D H2 O m 2 /s 10 r pore m 10 an = / eff,an el S/m 11 lyte el S/m 7 eff,cat el 25.5 S/m eff,an ion 1.05 S/m 15 lyte ion 5.31 S/m 15 eff,cat ion 1.86 S/m mol/m 3 ref C t eff A TPB m 2 /m 3 16 G H2 O 1073 K kj/mol 7 G H K kj/mol 7 G O K kj/mol 7 G O K kj/mol 17 which represent the electronic neutralization of the ionic flux with the consumption of hydrogen and generation of water. This allows the ohmic loss contributions to be examined in the electrochemically active region of the anode. Similar processes may be considered for the electrochemical oxygen reduction in the cathode. Experimental literature suggests this exchange region is of comparable length to that of the electrolyte in the anode of an anodesupported cell 18. The ionic current will be depleted in these regions due to its electrical neutralization through the Faradaic portion of the electrochemical reactions; however, even with an ionic current that is being neutralized as it extends into the anode region, the losses can be substantial and contribute to the cell s polarization resistance. Structural complexities may magnify these losses. To consider these processes, a volumetric source term is used to treat the electrochemical conversion of species that results from the Faradaic process and/or due to Joule heating 16,19. The source terms in the respective transport equations effectively couples the physics within the ionic, electronic, heat, and mass transport equations. We begin to describe the approach that is used, we begin with the governing transport equations. These transport equations take the form of Poisson s equation. eff eff el el = i n A TPB 1 eff eff ion ion = i n A TPB D eff i C i = A eff TPB zf i n k eff T = Q joule + Q reaction + Q activation 4 where the active area, per unit volume, in the vicinity of the TPB eff A TPB is prescribed using percolation theory 16,19. Similarly, the local Faradaic current density resulting from the electrochemical oxidation i n, electronic potential el, ionic potential ion, concentration C i, temperature T, effective electronic or ionic conductivity eff, effective molecular diffusivity D eff, and effective thermal conductivity k eff are used. The signs on the source terms for the electronic and ionic conduction in Eqs. 1 and 2 are written with respect to the anodic electrochemical oxidation process; they are flipped for reduction in the cathode. The effective diffusivities for gas-phase species i is expressed as D i eff = D ij D i k D ij + D i k to treat the parallel continuum diffusion and Knudsen diffusion effects, where the superscript k indicates the Knudsen diffusion coefficient. In Eq. 5, the porous electrode structure is scaled by an empirical parameter that represents the porosity-tortuosity factor of the structure. The local Faradaic current densities i n due to the respective anodic and cathodic i.e., oxidation and reduction reactions in the electrodes are treated using Butler Volmer equations i an n = i o an P an H 2 zf ref exp P H 2O ref exp 1 zf RT P H2 P H2 O i cat n = i o cat P cat O 2 zf ref exp exp 1 zf RT RT P O2 RT an cat 7 for the respective electrodes. The signs for the overpotentials in the respective electrodes have been modified for consistency. The exchange current densities i o i o an = P H 2 P H2 ref P H 2 O ref P H2 O exp E an a RT / Vol. 8, JUNE 2011 Transactions of the ASME

3 Table 2 1D cell-level model boundary conditions Interface Physics Boundary condition Anode/channel Diffusion C ref H2, C H2 O ref ref Electronic conduction el = G H2 O G H2 G O 2 / 2F RT/ 2Fln P H2 / P H2 O Ionic conduction ion = el el ion eq + an = an Overpotential n an an,eff =i cell / el Anode/electrolyte Overpotential n an an,eff = i cell / ion Cathode/electrolyte Overpotential n cat cat,eff = i cell / ion ref Cathode/channel Diffusion C O2 Electronic conduction i el =i cell Ionic conduction ion = el 1 / 2G O2 G O 2 / 2F + RT/ 4Fln P O2 Overpotential n cat cat,eff =i cell / el ref + cat i cat o = P 1/4 8 O 2 P O2 ref exp E cat a 9 RT are experimental fits 20. Consistent with previously reported methods 16,19, the electrode and total overpotentials are reported in terms of i cell or the operational current density. 2 an = ln RT P H 2 2F 2 eff,an + 1 eff eff,an A TPB i cell 2 cat = RT P H2 O + 1 el 4F 2 ln P O2 + 1 cell = an + cat + el ion eff,cat + 1 eff eff,cat A TPB i cell ion Llyte icell lyte ion The overpotentials shown in Eqs are used to calculate the 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. The boundary conditions are provided in Table 2. The top and bottom domain boundaries are considered periodic. In these transport equations, the heat equation is provided; however, it is not explicitly considered in this study. Preliminary analyses produced small temperature gradients across the detailed RVE i.e., on the order of K/ m due to its finite length. Should it be considered, it is rather straight-forward to implement due to the consistency in the form of the equation to the charge and mass transfer processes considered. Results and validation for the cell-level model are provided in Fig. 2. In Fig. 2 a, comparisons between the experimental data and the model results are provided. There are some deviations between the model and experimental data near the open circuit voltage OCV. There are several possible explanations for this deviation, which can include experimental fuel leakage and/or pinholes in the thin film electrolyte. Jiang and Virkar 10 provided a similar explanation and note that the discrepancy from the theoretical Nernst potential is on the order of 50 mv to 100 mv, which we incorporate into our model. Figure 2 b provides a result from the cell-level model that shows the local distribution of cell voltage and current density at i total =2.5 A/cm 2 and a 50%:50%, H 2 :H 2 O fuel ratio. The cell voltage is set so that it provides the Nernst open circuit voltage at the anode current collector. Therefore, its value at the cathode current collector is representative of the operational cell voltage. The total current density is conservative. As expected, the Faradiac neutralization of the ionic current in the form of an electronic current occurs over small but discrete distances into the electrodes. 2.2 Representative Volume Element Method. The problem can now be defined at the microstructural level. Representative volume elements or RVEs are used in this effort. This concept is demonstrated in Fig. 1 with the labeled squares of the anode. The first RVE is associated with the electronic conduction processes. Because the electronic charge transfer process occurs through the bulk of the electrode, this RVE can be considered at any position in the anode. The second RVE corresponds to ionic charge transfer process. The ionic current exchange occurs over a finite region. Therefore, the ionic RVE should be located near the anode/ electrolyte interface. In this study, the ionic charge transfer RVE will be fixed with one face on the theoretical anode/electrolyte interface from the cell model. The RVE boundary conditions are set using the results obtained from the cell model. A local ionic potential, electronic potential, and current density value are extracted from the cell model. To properly constrain the system, a potential and current density boundary condition must be applied to each RVE. Assuming unit depth for each 2D RVE is considered, the current density boundary condition must also be modified to account for the area fraction of the phase/region considered relative to the total area of the interface for which it is prescribed. This modification is necessary because the experimentally measured current density is typically defined per unit area irrespective of phase or void fraction. The respective current densities i.e., electronic in the Ni and ionic in the YSZ are therefore modified to maintain consistency with the values prescribed from the cell model results. A potential boundary 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 the system. Boundaries to phases being examined are insulated due to Fig. 2 Validation of 1D SOFC button cell model lines against experimental data symbols 10. a Voltage versus current density curves for different fuel feed streams. b Local cell potential, normalized at anode current collector to Nernst OCV and local electronic, and ionic current densities. It can be noted that total current density is conservative and current exchange occurs a finite distance into the respective electrodes. Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 /

4 the large discrepancies in the electronic conductivity for the Ni and YSZ phases; the ionic charge transfer process is treated in a similar manner 7. The details of the transport processes within the RVE require further consideration for examining the effects that the microstructure has on the ohmic losses. An assumption that the processes fall within the continuum transport regime is made for this analysis. This assumption is merited by the length scales of the grains and features that are observed within the anode microstructure. Furthermore, this assumption provides the use of the general charge transport equations that take on the form of a Poisson s or Laplace equation. Experimental work with ionic conduction in epitaxial YSZ films has been shown to exhibit continuum behavior down to lengths of 60 nm 15. This length is nearly an order of magnitude smaller than the minimum YSZ grain size considered in this work, 0.5 m 21,22. A continuum assumption also maintains merit for electron conduction in Ni. Nickel is a metallic conductor with a large number of electrons in the conduction bands. This permits its approximation as an electron gas. Furthermore, Drude s free electron theory of metals would suggest a mean free path on the order of several nanometers 23. This is several orders of magnitude smaller than the minimum Ni grain size considered in this work, 1 m 21,22. Although the length scales are appropriate for the conduction processes to be assumed continuum, the present model also neglects grain boundary diffusion processes. This could also be stated as the neglecting of space charge effects 24. These types of effects have been recognized as being important for ionic conduction as length scales approach nanometer length scales 9. A mean field approximation is used to address this issue. The mean field approximation considers that grain boundary diffusion effects are contained in the bulk charge transport coefficients and processes. The model additionally does not consider long range effects from the secondary conducting or gas-phase potentials or the nature of the capacitive double layer. The charge transfer processes are inherently linked to the Faradaic electrochemical oxidation and activation processes in the SOFC. Only the ohmic contributions are considered at this time. A discrete electrochemical reaction mechanism is not considered. A volumetric source term is used in these regions, where the Faradaic processes are occurring. The magnitude of this source term is determined using the observed change in ionic current density over the RVE region that was identified in the 1D cell model. This change in current represents the electrical neutralization of the ionic current density due to the electrochemical oxidation kinetics. This source term is scaled by the area of the ionic phase being considered i.e., YSZ of the RVE interface. Within the RVE microstructure, electronic conduction is constrained to the form of Laplace s equation. The current density is prescribed by Ohm s law. el el =0 13 i el = el el 14 The ionic conduction processes are governed by Poisson s equation, with a source term representing the Faradaic electrochemical oxidation processes, as well as Ohm s law ion ion zf = i ion,rve ARVE L RVE A YSZ 15 i ion = ion zf ion 16 where the change in ionic current density within the RVE i ion,rve is that due to Faradaic processes in the cell model. The electronic conduction processes shown in Eqs. 13 and 14 can also be considered in terms of the electrochemical potential ; however, the chemical potential associated with the Fermi-level energy is assumed spatially invariant in the metallic Ni phase. This means a standard electrical potential gradient can be used to represent the force driving transport. Equations are discretized using a finite element analysis FEA method and solved using COMSOL multiphysics modeling suite 25. The detailed FEA solutions are checked to ensure charge conservation is satisfied, the solutions are independent of the grid size, and individual solutions are independently verified with the 1D cell model. These detailed FEA solutions were verified by comparing the average current density on the second boundary to that predicted by the 1D model to ensure consistency. 2.3 Auger Electron Spectroscopy. Using the RVE concept, 2D phase-specific structures are collected from anode samples using AES. The samples examined with AES are used for the detailed RVE studies. AES is an electron based method that collects Auger electrons that are emitted from the sample when excited using a focused electron beam. A detector analyzes the energies associated with the emitted Auger electrons and provides an elemental map of a prepared sample surface. Discrete phases are inferred from this elemental map. The samples used in this study are taken from a SOFC anode sample. The region that is examined is taken from a plane that lies parallel to the anode/electrolyte interface and nearly midway through the electrode. The electrode that the samples are prepared from is composed of comparable volume fractions of Ni, YSZ, and pores. Preparation of these samples for AES analysis is completed by using consecutive surface polishing steps. This process begins with 1200 grit silicon carbide sandpaper. Four intermediate polishing steps are completed with diamond paste slurry down to a 0.5 m particle size. The sample surfaces are prepared in this manner to remove hills and valleys that can obstruct the electron beam. These types of surface topographic effects can conceal parts of the microstructure 26,27. A PHI 595 Auger electron spectroscopy unit with a 50 nm probe diameter using a 10 kv Argon ion sputter gun in a Torr ultrahigh vacuum chamber is used. The AES maps are completed using an 8 kev focused electron beam. A spatial resolution of better than 0.26 m is obtained. The AES map resolution is limited by the resolution of the raster used during the elemental mapping process. This resolution is set prior to the elemental mapping of the sample s surface with the AES. Greater spatial resolutions can be obtained for future efforts; however, the increase in resolution scales with the time that is required to acquire the map. AES is a suitable characterization technique because it only excites Auger electrons from the first few monolayers of the surface sample Furthermore, it has a fine electron probe diameter on the order of nm for practical incident beam energies and can provide an elemental map. A secondary electron detector SED is simultaneously used to form an electron micrograph from secondary electron sources during the Auger electron excitation process. This SED micrograph is used to verify pore locations. The Auger map is further verified by checking the consistency of the locations, where only trace amounts of the considered elements are found with the pores identified in the SED image. Additional Environmental Scanning Electron Microscope ESEM micrographs are used for verification. A sample AES map from the SOFC anode samples is provided in Fig. 3. A 2D section of the structure is obtained using Auger maps to represent the SOFC anode microstructure. The AES method is used because the full 3D characterization of the microstructure at spatial resolutions suitable for capturing the pore-scale features is not trivial 5,6. By using 2D slices of the heterogeneous 3D structure, the full connectivity of the three-phase structure cannot be captured. This is a limitation of the methods used and requires additional examination in future work. To combat the limitation of using 2D representations of the structure, RVEs of finite size are used. It is assumed that a single phase with a continuous path across the RVE approaches the connectivity of the 3D structure. This is accomplished by using finite sized RVEs and signifies a limiting assumption of this analysis. However, it may not be unreasonable to interpret this as an averaged description of the pore / Vol. 8, JUNE 2011 Transactions of the ASME

5 Fig. 3 A sample Auger electron spectroscopy map of polished SOFC anode sample. Note distinct phases: red is Ni, white is YSZ, and black is pore. scale structure. The 3D connectivity of the phase s examined in the 2D plane can be both enhanced and/or exacerbated through the out-of-plane connectivity for the broader structure. Likewise, similar features such as constricted regions of transport, blind or dead end paths, and isolated regions may be observed. The tortuosity of a given phase must be larger in the full 3D structure to support the connectivity of all three phases; however, this can be appropriately handled for these systems. The use of 2D structure to represent the full structure can be thought of as a first approach to the problem. This approach finds validity in the selfconsistency with both the characterization and transport analysis in the proceeding sections. The extension to the full 3D structure has been considered in 5,6, and the methods used can be extended to a 3D network to support these efforts. 2.4 Microstructure Characterization Methods. Microstructural characterization of the samples considered in this study is needed to understand the details and effects of geometry of the pore-scale anode microstructure. In this study, the microstructure is quantitatively characterized using the tortuosity, volume fraction, contiguity, and mean free path of the phase being examined These parameters provide a unique approach to characterize the microstructure compared with more traditional means such as percolation theory. Percolation theory treats the microstructure as an effective medium, based on factors such as mean grain size and phase volume fractions, with bulk contributions for the structure s impact on ohmic losses 11,16,19,20. The percolation limit for a three-phase media is the vicinity of a 30% volume fraction for any of the incorporated phases, barring substantial difference in grain size. When a sample contains a phase with a volume that falls below the percolation limit, the effective conductivity asymptotes toward zero and transport losses dominate. However, percolation theory lumps the microstructural description and cannot necessarily account for the localized microstructural effects. Structures that fall below the traditional percolation limit may be observed in this study when examining the problem at the pore-scale; however, these structures still maintain phase connectivity. Because there is connectivity within the structures, an effective conductivity asymptote toward zero is not observed. This is possible because the structures that are considered are of a finite size. However, the sample RVEs also may not be representative of the larger anode structure. Therefore, several samples are considered. This is done so that a statistical basis can be formed, which provides a range of results due to localized features that may be observed within the boarder structure. A variety of features are examined by taking a statistical approach, but this does not mean that the analysis is representative of the full anode properties. Fig. 4 Definition of the phase tortuosity, defined as the primary type-a conductor, which represents a continuous path of the conducting phase across the structure. a 1D domain considering type-a cluster continuity with remainder of domain. The gray region is the remainder of the domain that is considered continuous with the light blue type-a conductor in the first Laplace solution, the dotted line is a continuity boundary, and the solid black lines at top/bottom of domain are insulation boundaries. b Local Laplace solutions, where the crosshatched region is not considered, and the solid black lines represent insulation boundaries between bulk type-a cluster and the rest of the RVE. Instead, these analyses provide snapshots of localized regions and their effects. The individual results are self-consistent and provide a range of features within the structure. For this study, m 2 RVEs are used. This cross-section is considerably larger than that, which has previously been used in 3D to show volumetric independence for porosity and tortuosity of the pore regions 5. Furthermore, the correlations shown in the results and discussion section of this effort demonstrate an underlying trend indicating that RVE sizing is acceptable. Regardless, the deviations observed in these pore-scale structures from that of the bulk structure are of interest because they can provide substantial contributions to the ohmic losses. Literature has suggested that these types of pore-scale effects, in particular, those associated with the YSZ phase, may be a primary contributor to the overall cell resistivity 33. This also motivates the pore-scale approaches to the problem. To characterize the pore-scale structure, we begin with the definition of the tortuosity of a primary conducting phase, for the regions of the microstructure which continuously traverse the micrograph. We refer to these regions as type-a clusters. The tortuosity is required for domains that exhibit irregular structures that result in a conduction path length greater than that of the nominal domain length. The ratio of the two defines the tortuosity. The tortuosity is considered because the line counting methods that will be considered cannot account for the additional path length. The tortuosity of this region traversing the sample can be defined by considering the two complementary Laplace solutions as illustrated in Fig. 4. An area weighted integration of the streamlines is performed. To complete this analysis, a primary conducting phase devoid of any inclusions of the secondary solid phase that are Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 /

6 enclosed by the continuous conduction path is examined. In other words, a secondary solid phase surrounded by the primary conducting phase has temporarily been redefined as the bulk conducting material. It is important to recognize that this redefinition of the secondary phases is completed only during the analysis of the tortuosity. This is done to simplify the analysis such that even a visual estimation could be used as a rough approximation by someone attempting to repeat or use this work. Among the goals of this work is to be able to correlate the microstructural effects on the ohmic losses within the structure without needing to perform detailed meshing and analysis of the micrographs obtained. More specifically, that line counting methods can be used to repeat the analysis. By removing the inclusions of secondary phases from the continuous path s being studied, the tortuosity identified is not a true tortuosity. The removal of these secondary regions reduces the tortuosity and must be considered as a pseudo- or simplified-tortuosity. The effects of the portions that are redefined are captured and treated by the additional characterization parameters. As described, the tortuosity can be defined as A = = A da Continuity A da Insulation 2 x A L RVE + 2 y 17 A 1/2 da Insulation where A is the tortuosity of the type-a conducting path i.e., continuous, traversing the sample of solid phase, is an arbitrary scalar field in Laplace s equation i.e., 2 =0, continuity refers to a continuous solution through the RVE for the first Laplace solution Fig. 4 a, and insulation refers to the boundary condition of the primary regions being examined with the rest of the domain for the second Laplace solution Fig. 4 b. By applying scalar boundary conditions of =1 and =0 on the left and right boundaries, continuity between phases effectively leads to a 1D solution. Using the same scalar boundary conditions, the insulated boundaries of that particular phase lead to the development of discrete streamlines within the region studied in the form of following lines with a constant gradient of the scalar field. By integrating the magnitude of the vector field defined by the gradient of the scalar over the area of the region examined and then taking the ratio of the two definitions, the area weighted streamline integration is arrived at as defined in Eq. 17. This is performed analytically for the 1D solution. In examination of the tortuosity, even seemingly simple variations within a RVE can lead to a tortuosity substantial enough to warrant consideration. For example, even a phase path moving at an angle to the primary direction of transport can result in an increased tortuosity. The actual samples, which support a threephase structure and are typically made from sintered and reduced powders, are quite complex. The method is readily extendable to 3D structures. The second quantification parameter that will be defined is the phase volume fraction. Assuming unit depth for the 2D geometry, the volume fraction of phase is defined as V = A A RVE 18 within a RVE, where the area of phase, A is divided by the total area of the 2D RVE A RVE. The ohmic losses associated with the transport process in phase should be inversely proportional to the volume fraction of that phase. At this point, we examine parameters that use line counting methods. These characterization parameters are defined using the number of contacts of a particular type of interface per unit line length. A measurement grid is overlaid on a sample. All measurements that are provided were performed on the 2D RVE structures using grids with a spacing that was equal to that of the resolution limit of the Auger map used to obtain the RVE. We begin with the contiguity, which provides a representative measure of a sample s 3D properties from features observable in a 2D micrograph 31. More recently, this concept has been extended from a two phase description to a three-phase description and applied to the SOFC anode microstructure 32. The contiguity is a dimensionless parameter that provides an effective measure of the agglomeration of a particular phase 31. It is considered a measure of the self grain contact or the connectivity of that phase. Only self-similar contacts are considered in this work; however, it can be extended to interphase contiguity 31. The contiguity of phase averaged over the RVE is defined as 2N L C = 19 2N L + N L + N L within a three-phase sample. The contiguity can formally be considered as the contiguity between phase and. InEq. 19, N L 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 phase of interest, the ohmic losses should increase as the contiguity decreases due to decreased interparticle contact in the conducting phase. At this point, an assumption is required. The RVE domain does not contain grain boundary information within an individual phase. Similar grain boundaries are considered in the definition of contiguity. This requires us to assume an average grain size within the domain. To better understand the impact of the average grain size, three sets of average grain sizes are considered on an individual 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 grain sizes represent reasonable limits of Ni grains in an anode; the 3 m is considered to be representative for actual systems 21,22. Grain sizes of 0.5 m, 5.4 m, and 25 m are considered for the YSZ phase. The YSZ phase of common cermet anodes is comprised of 80% volume 0.5 m grains and 20% volume 25 m grains to provide both percolation and mechanical support 21,22. The constituent grain sizes are used as bounding values for grain sizes considered within the microstructure and 5.4 m is the volume averaged YSZ grain size. Use of the volume averaged YSZ grain size is considered a reasonable approximation because for an even distribution of the 0.5 m and 25 m grains, portions of both grains may be present within any RVE considered given the m 2 size of the RVEs. This is a generalization that leaves room for improving on the characterization reported here. Also, any RVE maintaining a 25 m grain should be readily apparent as it would dominate the m 2 structure and approach the condition representative of an ideal conductor. However, with this assumption, the 3 m Ni grain and 5.4 m YSZ grain sizes are considered as representative of the structure. It is also assumed that the grains are spherically shaped, so that the mean linear transverse of the grain is 2/3D g, where D g is the grain diameter. The mean linear traverse is used to represent the expected size of an individual grain within the microstructure. A mean free path can also be defined with respect to an individual phase in the microstructure. The mean free path is a measure of the mean distance of the secondary phases that must be crossed to travel from one grain to a second grain of the same phase 31. It does not consider adjacent grains of a similar phase. As the name would suggest, the mean free path is a measure of the constriction of a phase. This could also be interpreted as the crosssection of the phase path that follows. Using a similar definition to / Vol. 8, JUNE 2011 Transactions of the ASME

7 previous studies 31, the extension of the two phase mean free path to a three phase mean free path requires the consideration of the additional dissimilar grain contacts. The mean free path for phase is considered in the form 2V 1 = 20 N L + N L L RVE where the length of the RVE L RVE is used to make the mean free path dimensionless. From a conceptual standpoint, the mean free path is a measure of phase constriction. Therefore, it is anticipated that ohmic losses will increase in magnitude as the mean free path for the primary conducting phase decreases. 2.5 Performance Quantification Methods. To interpret the RVE structure s impact on the ohmic losses, it is important to form a consistent metric for quantifying the microstructure s performance. We will quantify these ohmic losses in terms of an ohmic performance parameter. The ohmic performance refers to the ability of the structure to transport charge without losses attributed to these processes i.e., Joule heating. The optimal ohmic performance would that be of a continuous bulk phase devoid of the other phases considered. Increased losses that result from the transport processes occurring in the structure are consistent with a decrease in performance due to an increase in Joule heating. The Joule heating is produced from this irreversibility of the charge transport process Q joule = 1 A A i 2 da 21 and scales with the square of the current density. The current density can be electronic or ionic in the respective Ni and YSZ phases. The ohmic losses within the RVE must also be able to be quantified on a consistent basis for the unique structures considered. As seen in Eq. 21, Joule heating is defined in terms of the local current density. Therefore, the Joule heating must be corrected for the area of the conducting phase with respect to the total area of the RVE Q joule,corrected joule = Q ARVE 22 A where the area corrected Joule heating in phase, Q joule,corrected,is joule described in terms of the originally calculated Joule heating Q from Eq. 21. With the area corrected Joule heating, a nondimensional heat release, Q, can be defined. This is calculated by forming a ratio of the area corrected heat release Q joule,corrected to a minimum loss RVE. The minimum loss RVE is a RVE of the same domain size and boundary conditions but with a pure conducting phase. The nondimensional heat release takes the form Q = Q joule,corrected Q joule,min = A i 2 da A RVE i 2 da ARVE A 23 Fig. 5 Phenomenological electronic conduction geometries considered. Red: Ni, white: YSZ, and black: pore. All geometries have a symbol associated with them that will be used during analysis of results. For each, the left denotes the base case containing only Ni and pores while the right case denotes a variation on the case with inclusions of secondary phases. Complementary cases are considered for ionic conduction in YSZ. a Straight Ni phase with pore constriction, b straight Ni phase with a blind branch, and c tortuous pore cutting through Ni substrate. All geometries are 10Ã10 m 2 in size. where the minimum loss condition for phase, Q joule,min, is used to normalize Eq. 22. This provides a nondimensional and effective measure of ohmic performance or heat release due to Joule heating in the RVE. An additional correction in the form of the tortuosity of the conductive phase or, more formally, pseudotortuosity as defined in the discussion on the methods used, must be considered to account for additional conduction length within a RVE. This correction accounts for conduction through the tortuous paths within the material. This is an effect that cannot be captured by the line counting based characterization methods and has a direct impact on interpretation of the ohmic performance of the RVE. Consequently, results will be presented in the form of Q / A, where A is the conducting phase s tortuosity defined in Eq. 17. This definition provides a consistent basis for reporting a loss due to ohmic heating between unique RVEs. To define the total effective heating, it is necessary to sum the electronic and ionic contributions Q net = Q Ni,el + Q YSZ,ion 24 because both can contribute to the net Joule heating. As defined, the effective Joule heating or heat release Q is a nondimensional number that can range from one to infinity. It is a figure of merit with respect to the ohmic performance of any RVE case considered and its respective current density. It could also be considered as an effective resistivity correction based on a given structure of the form eff =Q. 3 Results and Discussion With the characterization parameters defined, correlations to the effective ohmic performance of the RVE can be identified. Correlations between the tortuosity corrected effective ohmic performance Q / A, and the microstructure parameters are examined for actual 2D AES and micrographs as well as some conceptual microstructures. In all, eight AES samples and 42 conceptual structures are examined. The conceptual microstructures are used to supplement the actual samples and to improve the size of the statistical set. Due to the finite sizes of the RVEs, the RVE properties may extend above or below what is typically anticipated in the SOFC anode. We begin this effort by introducing the conceptual or phenomenological cases. These conceptual cases begin with three cases that consist of pure Ni and pore phases as seen on the left hand side of Fig. 5. The tortuosity of the Ni phase and Joule heating can be examined for these three geometries. The right hand column of Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 /

8 Fig. 6 Normalized distributions of current density in anode samples for 2.0Ã10 4 A/m 2 net current density: a Current density magnitude for electronic conduction in Ni phase normalized to area corrected left Ni phase boundary current density of 7.65Ã10 4 A/m 2, b constant electronic current density flux lines, c current density magnitude for ionic conduction in YSZ phase normalized to area corrected left YSZ phase boundary current density of 4.85Ã10 5 A/m 2, and d constant ionic current density flux lines Fig. 5 denotes an intermediate case with YSZ and inclusions. In all of the cases, the YSZ inclusions are white, pores are black, and remainder red is Ni. These additional intermediates contain different size, shapes, and placements of YSZ and pore inclusions within the bulk Ni phase. Random distributions and sizes of round inclusions were considered with inclusion diameters ranging from fractions that of the smallest single YSZ grains inclusion to several times larger. Similar sizes have been considered for pore inclusions. Inclusions have been placed at random through the structure. The overlapping of inclusions provided unique and arbitrary inclusion shapes and configurations. This can be seen in the right hand column of Fig. 5. For the conceptual cases that consider ionic conduction, the Ni and YSZ phases are interchanged and within the basic structures in consideration. Using the conceptual structures shown in Fig. 5 and their intermediates as well as the actual anode structures from the Auger maps such as the one seen in Fig. 3, the tortuosity corrected effective heat release can be examined with respect to the conducting phase volume fraction V, contiguity C, and mean free path. Prior to the discussing these individual parameters, a set of micrographs with detailed normal current density and constant current density flux lines are provided in Fig. 6. In Figs. 6 a and 6 c, the magnitude of the electronic and ionic current densities in the Ni and YSZ phases are, respectively, shown after to the current density boundary condition is normalized. These figures represent local variations in current density with respect to the applied boundary condition. Figures 6 b and 6 d show the constant current density flux lines for the respective electronic and ionic conduction processes. The variation in current density is noted for both electronic and ionic conduction. With respect to the electronic conduction found in Fig. 6 a, there is approximately 8.5 variation in current density from the specified current density boundary condition of A/m 2 i.e., area corrected current density to the constriction, where the maximum current density occurs. However, as seen in Fig. 6 c, the ionic conduction RVE does not see a dramatic increase in current density relative to the current density boundary condition due to the current exchange process i.e., boundary condition of an area corrected ionic current density of A/m 2. Constricted regions contain current densities near that of the inlet current density by approaching 0.95 of the inlet current density. Despite the current exchange process, these constricted regions still play a prominent role in the RVE ohmic performance / Vol. 8, JUNE 2011 Transactions of the ASME

9 Fig. 7 Electronic and ionic tortuosity corrected effective heat release as a function of characterization parameters: a electronic corrected effective heat release versus Ni volume fraction, b electronic corrected effective heat release versus Ni Ni contiguity for mean particle size of 1 m, 3 m, and 5 m, c electronic corrected effective heat release versus Ni mean free path, d ionic corrected effective heat release versus YSZ volume fraction, e ionic corrected effective heat release versus YSZ-YSZ contiguity for mean particle size of 0.5 m, 5.4 m, and 25 m, and f ionic corrected effective heat release versus YSZ mean free path With this detailed look at the local magnitudes of the current densities, the quantification and characterization can be examined. We begin by examining the behavior of the volume fraction, contiguity, and mean free path of the participating phase on an individual basis as related to the tortuosity corrected effective heat release. Figure 7 provides a look at the nature of each of the individual quantification parameters with respect to electronic and ionic conduction. Figures 7 a 7 c show the tortuosity corrected effective heat release versus the volume fraction, contiguity, and mean free path of the Ni phase for the electronic conductor. Figures 7 d 7 f provide the tortuosity corrected effective heat release versus the volume fraction, contiguity, and mean free path of the YSZ phase for the ion conductor. It is recognized that the actual SOFC anode structures obtained using AES and phenomenological cases demonstrate the same qualitative trends. The volume fraction, contiguity, and mean free path of the YSZ phase also consistently show an inverse relationship with increasing losses. The mean grain size that is used to define the contiguity plays a substantial role in the relationship between the corrected effective ohmic performance parameter and the respective contiguity. For both the electronic and ionic conduction processes, Fig. 7 b and 7 e show that a change in average grain size shifts the entire contiguity curve and has a moderate impact on the dispersion of the contiguity. However, given the sizes of RVEs considered, the differences in average grain sizes studied are quite substantial in comparison to the deviations from the average grain size that would likely be found in the RVE or for a complete microstructure 21,22. These cases effectively represent the special and limiting conditions for the system described, where a RVE with only that particular grain size is found. This result only bounds the problem. If an even distribution of the large and mean grains that comprise the YSZ phase is assumed, it is reasonable to assume the RVE as being comprised of grains near the average grain size. To check the importance of variations in mean grain size, a secondary sensitivity analysis is also completed with regard to the contiguity. In this study, it is found that a +/ 10% deviation in average grain size for the cases of 3 m Ni grains and 5.4 m YSZ grains have less than +/ 5% variation on contiguity. To correlate the characterization parameters to the effective ohmic performance, a Levenberg Marquardt LM nonlinear least-squares method is used to fit the effective ohmic performance to a power law in terms of the quantification parameters. Q Ni,el A Ni A 3 =A A2 1 V Ni C low,ni Ni Q YSZ,ion A YSZ B =B 1 V 2 YSZ A C 4 avg,ni Ni A C 5 A high,ni Ni 6 Ni Ni B C 3 B low,ysz-ysz C 4 avg,ysz-ysz 25a B C 5 B high,ysz-ysz 6 YSZ 25b where A 1,A 2,...,A 6 are fit constants for electronic conduction and B 1,B 2,...,B 6 are fit constants for ionic conduction. Linear regression analysis is also used whenever possible and to validate the 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 their definition, respectively. Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 /

10 Table 3 Tortuosity corrected effective electronic ohmic loss correlation table for Eq. 25a performed on detailed RVEs of both actual anode samples and generated phenomenological structures. x denotes that the parameter was not considered for a particular case. This was completed in the first eight correlations to investigate the dependencies of both individual and coupled parameter effects. The bold case represents the final fit that was determined for the data sets. The complete data ranges that were observed within the RVEs considered are provided in the last row. A 1 A 2 V Ni A 3 C low,ni Ni A 4 C avg,ni Ni A 5 C high,ni Ni A 6 Ni Ni R 2 RMSE x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Parameter data range x x Prior to completing a comprehensive fit, the dependence of the ohmic performance on the individual parameters is performed so that their individual effects can be better understood. This regression analysis is completed using Eqs. 25a and 25b and the method providing the best fit data are presented in Tables 3 and 4. All results are presented in the form of Eq. 25, where the powers of parameters not considered for a particular fit are set to zero. Tables 3 and 4 contain these fits to the tortuosity corrected ohmic performance. The correlation of the individual characterization parameters correspond to the first five data sets in Tables 3 and 4. In these first five data sets, as also can be observed in Fig. 7, all of the characterization parameters show an inverse relation to increases in Joule heating, denoted by an increased Q / A. In Tables 3 and 4, the following four data sets demonstrate how the system performs when combinations of these parameters are combined in the fit, reported in a power law form of Eq. 25. The system is reported in the power law form because each of the parameters follows a power law form on an individual basis. While examining the four combined correlation data sets in Tables 3 and 4, the conducting phase volume fraction is recognized as being of great importance. When it is removed from the correlation, poor statistical agreement is observed. The combinations of the phase volume fraction, mean free path, and average grain sized contiguity for the respective conducting phase are examined. This trend is shown in bold in Tables 3 and 4. It represents the final overall correlation for connection between the characterization parameters and the tortuosity corrected ohmic performance. For these final fits, an R 2 of 0.88 and 0.91 and a root mean square error RMSE of 6.02 and 5.46 were found for Tables 3 and 4, respectively. This represents a reasonable statistical agreement and indicates a clear trend. Using the correlation dependencies that have been bolded in Tables 3 and 4, the tortuosity corrected ohmic performance from the individual model RVE models i.e., both the Auger and phenomenological models can be examined. Figure 8 plots the tortuosity corrected ohmic performance from the individual RVE models versus the predicted value using the correlation found in Eq. 25 using the bolded fitting parameters in Tables 3 and 4. Because the power-series fit performed in the preceding discussion was a linearization, the fit value for this correlation can also be represented. This is shown in Fig. 8 along with the 95% confidence intervals. The reasonable agreement between both the conceptual and physical anode RVE cases is noted. Only moderate scatter exists and this primarily falls within the 95% confidence intervals. Most importantly, these figures effectively show that the characterization parameters have been used to correlate a tortuosity corrected ohmic performance. Such a correlation indicates that not only the conducting phase volume fraction but also the phase tortuosity and measures of constriction and agglomeration are important to understanding the structure s effective ohmic performance. These parameters aspects are quantified with the phase mean free path and contiguity, respectively. This improvement in combined form is substantiated by the improved R 2 and RMSE in Tables 3 and 4 when compared with the individual parameters. Table 4 Tortuosity corrected effective ionic ohmic loss correlation table for Eq. 25b performed on detailed RVEs of both actual anode samples and generated phenomenological structures. x denotes parameter was not considered for a particular case. This was completed in the first eight correlations to investigate the dependencies of both individual and coupled parameter effects. The bold case represents the final fit that was determined for the data sets. The complete data ranges that were observed within the RVEs considered are provided in last row. B 1 B 2 V YSZ B 3 C low,ysz-ysz B 4 C avg,ysz-ysz B 5 C high,ysz-ysz B 6 YSZ R 2 RMSE x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Parameter data range x x / Vol. 8, JUNE 2011 Transactions of the ASME

11 Fig. 8 Fit of parameterized results based on quantification parameters shown with 95% confidence interval bands: a tortuosity corrected effective electronic heat release versus correlated characterization parameter fit and b tortuosity corrected effective ionic heat release versus correlated characterization parameter fit 4 Conclusions A method of quantifying electronic and ionic ohmic performance in a SOFC anode has been developed. This method relies on line counting based characterization methods from detailed phase-specific AES micrographs. It has been shown that the characterization parameters in a power law form can be correlated with the continuum-level model of detailed transport within conceptual and AES-based microstructures with reasonable statistical agreement. Such a method provides a useful tool to an engineer who wishes to make a statistical measure of a microstructure s anticipated ohmic performance without completing detailed experimental work and without having to complete detailed and computationally intensive studies. Acknowledgment The authors gratefully acknowledge financial support from the Army Research Office Young Investigator Program Award CH-YIP, the National Science Foundation Award CBET , an Energy Frontier Research Center on Science Based Nano-Structure Design and Synthesis of Heterogeneous Functional Materials for Energy Systems funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences Award DE-SC and the ASEE National Defense Science and Engineering Graduate Fellowship program. The authors would like to thank Adaptive Materials Inc. AMI of Ann Arbor, MI for the SOFC samples. Nomenclature A area, m 2 AES Auger electron spectroscopy C concentration, mol/m 3 C contiguity between phases and D diffusivity, m 2 /s D g mean grain diameter, m E a activation energy, J/mol F Faraday s constant, 96,485 A s/mol H height, m G Gibbs free energy, J/mol i current density, A/m 2 i o exchange current density, A/m 2 k thermal conductivity, W/m/K L length, m LSM lanthanum strontium manganite N L number of - grains crossed per unit line length, m P pressure, Pa Q heat source, W/m 3 R universal gas constant, J/mol/K T 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 path electrochemical potential, J/mol resistivity, ohm m conductivity, S/m tortuosity potential, V porosity/tortuosity diffusion correction factor Subscripts,, arbitrary phases char characteristic el electronic i chemical species ion ionic net net Ni Nickel phase RVE representative volume element TPB three phase boundary YSZ YSZ phase Superscripts A type-a cluster designation an anode cat cathode cell cell-level corrected corrected eff effective joule Joule heating k Knudsen lyte electrolyte min minimum ref 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 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 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-Component Mass Diffusion in a Solid Oxide Fuel Cell Anode, ASME J. Fuel Cell Sci. Technol., 7, p Joshi, A. S., Grew, K. N., Peracchio, A. A., and Chiu, W. K. S., 2007, Lattice Boltzmann Modeling of 2D Gas Transport in a Solid Oxide Fuel Cell Anode, J. Power Sources, 164 2, pp 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 Fuel Cell Anodes Using X-Ray Computed Tomography at Sub-50 nm Resolution, J. Electrochem. Soc., 155 5, pp. B504 B 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 Elemental Mapping and Analysis of a Solid Oxide Fuel Cell Anode, J. Electrochem. Soc., 157, pp. B783 B Singhal, S. C., and Kendall, K., 2003, High Temperature Solid Oxide Fuel Cells Fundamentals, Design and Applications, Elsevier, New York. 8 Marinšek, M., Pejovnik, S., and Macek, J., 2007, Modeling of Electrical Properties of Ni-YSZ Composites, J. Eur. Ceram. Soc., 27, pp Reifsnider, K., Huang, X., Ju, G., and Solasi, R., 2006, Multi-Scale Modeling Approaches for Functional Nano-Composite Materials, J. Mater. Sci., 41, pp Jiang, Y., and Virkar, A. V., 2003, Fuel Composition and Dilutent Effect on Gas Transport and Performance of Anode-Supported SOFCs, J. Electrochem. Soc., 150, pp. A942 A Nam, J. H., and Jeon, D. H., 2006, A Comprehensive Micro-Scale Model for Journal of Fuel Cell Science and Technology JUNE 2011, Vol. 8 /